Physics Reports 375 (2003) 327 – 410
www.elsevier.com/locate/physrep
Method of multiple scales in quantum optics
Maciej Janowicz∗
Institute of Physics, Polish Academy of Sciences, Al. Lotnikow 32/46, 02-668 Warsaw, Poland
Accepted 1 October 2002
editor: J: Eichler
Abstract
Applications of the method of multiple scales to the quantum-optical problems are reviewed. After a preliminary study of applications to the classical and quantum anharmonic oscillator, several examples in the
spontaneous emission, resonance /uorescence, and cavity quantum electrodynamics are analyzed. A preliminary account of application of the method to a model of Bose–Einstein condensates is given. A common
thread throughout the main body of the review is the stabilization of the wave function and population of a
level due to the phase modulation by an external agent.
c 2002 Published by Elsevier Science B.V.
PACS: 42.50−p; 02.30Mv; 42.50Hz; 42.65Sf
Keywords: Quantum optics; Method of multiple scales; Anharmonic oscillator; Spontaneous emission; Resonance
/uorescence; Cavity quantum electrodynamics
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Anharmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1. Heisenberg equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Time evolution operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. Comparison with numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1. Classical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2. Quantum-mechanical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Spontaneous emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. Resonance /uorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
∗
Corresponding author. Carl-von-Ossietzky University, Fachbereich 8-Physik, 26111
Tel.: +49-441-798-3996; fax: +49-441-798-3201.
E-mail addresses: mjanow@ifpan.edu.pl, janowicz@marvin.uni-oldenburg.de (M. Janowicz).
c 2002 Published by Elsevier Science B.V.
0370-1573/03/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 2 ) 0 0 5 5 1 - 3
Oldenburg,
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332
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Germany.
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4.1. General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Modulation of atomic frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Population trapping in cascade systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4. Bichromatic excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. Atom–Deld interactions in cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1. Two-level externally excited atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2. Two-mode cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3. The four-level “Lambda” system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1. Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2. Description of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3. Application of multiple scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.4. SchrIodinger-cat states and population dynamics in a restricted two-mode two-photon interaction . . .
6. Fields and atoms in cavities with oscillating mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1. Electromagnetic Deld in the oscillating cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2. Atom–Deld interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7. Multiple scales and cold boson gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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407
1. Introduction
This introductory section serves the following purposes: Drstly, it explains the motivation to study
the method of multiple scales (MMS) in Quantum Theory. Then, a most elementary example is
shown to give an idea why the method can be successful. Thirdly, very brief historical and bibliographical remarks are given to place the topic in a suitable context in the development of non-linear
physics.
Undoubtedly, the computer methods play an ever-growing role in the recent development of scientiDc research, especially in physics. This is true in quantum optics and electrodynamics as well.
Even very complicated master equations, typical for this Deld of research, can be handled by modern high-speed computers. In the 1990s, an additional very powerful tool for computational analysis
of quantum optical systems was developed, namely the Monte-Carlo simulation of the stochastic
SchrIodinger equations. In this connection, one might rightly ask whether the approximate methods
of applied mathematics can still be of any use, and, in particular, whether they are still useful in such
a subject as quantum optics. This work deals with these methods, especially with perturbation theory,
and it tries to justify their usefulness. The most important fact about them is that they can strengthen
and stimulate our intuition, leading to better understanding of the problem considered. Sometimes,
the role of the approximate methods is still greater: they give us the language to speak about the
behavior of physical systems. Probably the most striking example are the Feynman diagrams which
give graphical representation of perturbation expansions.
In this work we concentrate on applications of a speciDc method, called the method of multiple
scales, developed in classical mechanics to deal with non-linear equations containing a small parameter. It will be shown that this method is by no means restricted to the classical domain but
can also be applied to the Heisenberg equations of motion as well as to the SchrIodinger equation.
Whenever possible and meaningful, we shall try to apply this method directly to the SchrIodinger
M. Janowicz / Physics Reports 375 (2003) 327 – 410
329
equation for the time-evolution operator. This is because the time-evolution operator—together with
an initial density matrix—provides the most general description of a physical system.
There is a simple reason why the method of multiple scales can be eLcient in the quantum-optical
context. In quantum optics, we very often have to do with systems of few degrees of freedom
described by some Hamiltonian operators. These operators have the dimension of energy and usually
consist of several terms some of which contain numerical factors of the dimension of energy or
frequency much smaller than analogous factors standing at other terms. But a Hamiltonian is a
generator of the time evolution of the quantum system. Therefore, in the time evolution of the
system one can often very naturally distinguish several diMerent time scales deDned by diMerent
terms in the Hamiltonian.
Let us now brie/y explain why an ordinary—or regular—perturbation expansion is very often
insuLcient to obtain reasonable approximation in both classical and quantum dynamics. Indeed, as
is well known in non-linear mechanics, the regular perturbation expansion applied to systems with
the so-called internal resonance brings about secular terms in the approximate solutions. The term
“secular” has been ascribed in astronomy to expressions which grow with time t as powers of t. Such
expressions cannot be usually accepted even in solutions to problems in classical mechanics because
they like to appear even in the systems which are known to be periodic. But the situation in quantum
case is much worse. Indeed, if an approximate expression for the time evolution operator was to
contain terms growing (say—linearly) in time, the canonical commutation relations would be broken
and the probability would not be conserved. This is in striking contradiction to the most essential
postulates of Quantum Theory. Hence the need for a “good” quantum perturbation expansion method,
which does not allow such pathological situations. In the following chapters we will show such a
method—the method of multiple scales—in action.
To illustrate the diLculties encountered when using regular perturbation expansions let us consider
a trivial example. Let a(t) be a quantity depending on time and let it satisfy the following Drst-order
diMerential equation:
ȧ = −i!a − a ;
(1)
with the initial condition a(0) = b where “dot” denotes diMerentiation over time, and !. Let
us pretend for a moment that we are not able to solve this equation exactly and must resort to a
perturbation expansion; that is, we can solve it if either ! or is present in the right-hand side,
but not for both of them multiplying a. To proceed, let us introduce a dimensionless time variable
= !t so that d=dt = !d=d. Let, in addition, j = =!1. Then we have
da
= −ia − ja :
(2)
d
According to the prescription of the regular perturbation theory we write a = a0 + ja1 + j2 a2 + · · · ;
substitute this expansion into Eq. (2), and compare terms on both sides standing at the same power
of j assuming the initial conditions to be a0 (0) = b, ai (0) = 0; i = 1; 2; : : : : Then we obtain in the
zeroth order
d
a0 = −ia0 ;
d
and in the Drst order
d
a1 = −ia1 − a0 :
d
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
And it is in the Drst order that the diLculty appears. When we solve for a0 , a0 (t) = be−i , substitute
to the equation for a1 , and try to solve the resulting inhomogeneous equation, we Dnd
a1 (t) = −be−i
which means that our perturbation expansion can be valid only for extremely short times. In the
higher orders the situation becomes even worse, the powers of t which enter the approximate solution
grow with the order of perturbation expansion.
To Dnd a remedy, let us make use of the fact that in the above example ! and deDne two
diMerent time scales. Thus, we can try to improve the situation by assuming that all ai (i = 0; 1; : : :)
depend separately on two “times”, say, T0 and T1 , such that T0 = and T1 = j. Then we have
d=d = 9=9T0 + j9=9T1 . The perturbation expansion is now slightly diMerent:
9a0
= −ia0 ;
9T0
9a1
9a0
+
= −ia1 − a0 :
9T0 9T1
The zeroth-order solution is now a0 = b(T1 )e−iT0 . On substituting this solution into the equation for
a1 we can avoid the troubles connected with the inhomogeneous term on the right-hand side if we
“absorb” it to the equation satisDed by 9a0 =9T1 . This means that we should write
9b(T1 ) −iT0
9a0
=
e
= −a0 = −b(T1 )e−iT0
9T1
9T1
so that b(T1 ) = be−T1 while a1 becomes zero in view of the initial conditions. This way we have
actually obtained the exact solution (after expressing T1 and T0 in terms of t, ! and ). Needless to
say, in less trivial examples getting exact solutions by the method of multiple scales happens very
rarely. The rest of our work is devoted to an analysis of some of those less trivial problems which
appear in quantum optics.
The great eLciency of MMS has Drst been recognized in the Deld of non-linear diMerential
equations and classical non-linear mechanics. The major developments have included non-linear
oscillations, boundary layer problems, /uid dynamics and aerodynamics, and the theory of spacecraft
motion. In this connection we should mention that there already exist several excellent books dealing
with MMS. Among the monographs written from the point of view of applied mathematics one should
mention the books by Kevorkian and Cole [1,2], by Nayfeh [3,4] and by Nayfeh and Mook [5].
An outstanding exposition of approximate and asymptotic methods of applied mathematics including
MMS as their crown is contained in the book by Bender and Orszag [6]. A classical exposition of
perturbation methods of /uid mechanics including MMS is given in [7]. In quantum mechanics, one
of the early attempts to apply the methods of non-linear oscillations to the Heisenberg equations
of motion appeared in the work by Ackerhalt and RzTażewski [8], which also contained famous
Ackerhalt’s exact solutions to the Heisenberg equations in the Jaynes–Cummings model [9,10] (we
shall use the abbreviation “JCM” for the Jaynes–Cummings model). The approach used in [8] is
similar to the Lindstedt–Poincare method of non-linear mechanics which may be considered to Dt in
the general framework of the multiple scales techniques. But still before [8] there had been several
ideas to use multiple scales to describe the spontaneous emission in the SchrIodinger picture [11,12].
Those ideas were later applied to the case of the radiation (including superradiance eMects) from thin
M. Janowicz / Physics Reports 375 (2003) 327 – 410
331
Dlms [13–15] as well as to the spontaneous decay processes in the presence of radiationless transitions
[16]. Another quantum-optical Deld of principal interest in which MMS found its application in 1970s
was the interaction of a single- and many-atom systems with external laser radiation: problems of this
type were considered in [17–19]. The years 1980 –1995 brought about the interest in the asymptotic
methods in laser physics. Both the method of multiple scales [20,21,24] and other methods, Drst of all
the Krylov–Bogolyubov–Mitropolskii techniques [22,23,95], were employed, mostly, however, on the
level of semi-classical equations. Those interesting applications in the laser theory are already covered
in the review papers by Erneux [25,26] and will not be discussed here. In the 1990s, MMS was
also applied to provide a general link between the internal (quantum) and external (semi-classical)
motion of atomic and molecular systems within the framework of the Wigner-function approach
[27]. The coupling between external and internal degrees of freedom of an atom in the laser Deld
was considered—using the method of multiple scales—in [28]. Further applications of MMS, mostly
in the cavity quantum electrodynamics, are described in [29–31].
The remaining part of our review is organized as follows. In Section 2 we discuss the problem of
the dynamics of an anharmonic oscillator from the point of view of MMS. Using the results of [36,37]
as well as our own, we show that the anharmonic oscillator admits high-quality perturbative solutions
for both the Heisenberg and the SchrIodinger equations. In Section 3 some applications to the problem
of spontaneous emission are studied. Section 4 is devoted to the interaction of few-level atoms
with external laser Delds. The excitations of model atoms with non-monochromatic radiation, e.g.,
having modulated amplitude, modulated phase, or coming from two independent lasers with diMerent
frequencies, are especially interesting for us. Section 5 deals with the interaction of atoms with one
or two modes of quantized electromagnetic Deld; such model interactions are very often employed in
the cavity quantum electrodynamics. In Section 6 some aspects of the Deld quantization in cavities
with moving mirrors and atomic interaction with quantized Delds in such cavities are discussed in
adiabatic approximations. Section 7 contains attempts to apply the multiple-scales methodology to
the problems of tunneling of the cold gases in the double-well potential. In Section 8 we provide
several remarks, both optimistic and skeptical, about the possible further applications of the method
of multiple scales in the domain of quantum optics and, more generally, quantum mechanics of
simple systems.
It is to be noted that the present work has been written on the so-called “physical” level of rigor.
No attempt has been made for mathematical justiDcation in terms of proofs; thus, all the calculations
are purely “formal” from the mathematical point of view. In fact, the series which we will have to
deal with, are, in generic cases, asymptotic and divergent. A heuristic rule has been even oMered by
Van Dyke [7] which states that an asymptotic solution to a problem is usually divergent if it depends
on two independent scales (length scales in his case or time scales in ours). This statement does
not, of course, depreciate the usefulness of asymptotic multiple-scales expansions as will be seen in
the following sections on explicit examples. On the other hand, except for the following section, we
shall try to concentrate as much as possible on the physical contents of solutions provided by the
method we discuss. Our strategy in Section 2 which deals with anharmonic oscillator is to provide
numerical evidence for eLciency (as well as limitations) of MMS; in the following sections we
shall rather try to obtain qualitative and semi-quantitative predictions from MMS.
We Dnish this introduction with two remarks about notation. All the operators in our
formulae will be denoted by letters with carets. Dots over symbols denote diMerentiation over the
time t.
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
2. Anharmonic oscillator
This section is devoted to the anharmonic oscillator, one of the best investigated model systems in
quantum mechanics. It has been studied in much detail and from many points of view, in particular,
by Bender and Wu [33], Hioe et al. [34], and by Stevenson [35]. It has acquired such a popularity
because of the following important characteristics:
• Its classical counterpart, associated with the so-called DuLng equation, is a very useful model to
study non-linear oscillations (in particular, it forms an approximation to the equation of pendulum
and shares some of its features), and to test methods of non-linear mechanics.
• It is still relatively simple and allows for detailed study with almost arbitrary precision, but at the
same time it is amazingly non-trivial, cf. [33].
• It is considered to be the simplest model of the quantum Deld theory corresponding to selfinteracting boson theory with local quartic self-interaction; therefore, the theory of quantum anharmonic oscillations is sometimes called a “zero-dimensional” Deld theory .
2.1. Heisenberg equations of motion
Quite recently, Bender and Bettencourt [36,37] have tested the method of multiple scales for
possible usefulness in the quantum Deld theory by applying it to the Heisenberg equations of motion
in the anharmonic oscillator case, thus oMering the Drst “pure” example of using this tool of non-linear
mechanics in quantum dynamics.
They have started with a brief analysis of the classical DuLng equation with a small parameter
multiplying the non-linear term and have shown how the ordinary (regular) perturbative expansion
leads to secular terms proportional to time due to resonant coupling in successive order. The DuLng
equation is, in fact, the most standard example used in all books about perturbation theory (cf.
[1–3,6]), but the authors of [36,37] have discussed it as classical counterpart of their quantum results.
To make our review reasonably self-contained, we shall brie/y summarize the application of MMS
to the DuLng equation (in this we shall follow [36]), referring the reader to the above-mentioned
books for details. The DuLng equation reads
d2
x + x + 4jx3 = 0 ;
(3)
dt 2
where x is a classical variable—the position of a non-linearly oscillating particle, and j is a small
parameter. We shall solve it perturbatively with the initial conditions x(0) = 1 and x (0) = 0. Let us
Drst try to use just an ordinary expansion into a series of j,
∞
x(t) =
jn x n (t) ;
n=0
with the initial conditions x n (0) = n0 and xn (0) = 0. In the zeroth and Drst order in j we get
x0 + x0 = 0
(4)
x1 + x1 = −4x03 :
(5)
and
M. Janowicz / Physics Reports 375 (2003) 327 – 410
333
The solution to Eq. (4) that satisDes the initial conditions is x0 (t)=cos t, so that the right-hand side
of Eq. (5) becomes equal to −cos(3t) − 3 cos t. But then Eq. (5) becomes an equation of the forced
harmonic oscillator with the force being in resonance with the free oscillations. This inevitably leads
to a solution with the envelope growing linearly in time. In our case it is
x1 = 18 cos(3t) − 18 cos t − 32 t sin t :
(6)
This solution contains the secular term ∼ t sin t, which has, of course, nothing to do with the correct
solution which remains bounded for all times. This is perhaps even more self-evident physically than
mathematically; on one hand, the potential grows very fast to inDnity with x → ±∞ so that the
motion must be bounded; on the other hand, the energy of any closed physical system cannot grow
indeDnitely without being added from the environment.
One can improve the ordinary expansion by reordering and resummation of the most secular terms
[36,37] provided that one is able to recognize them. But the less secular terms would still remain,
and, in order to get rid of them, one would rather like to employ a less cumbersome approach from
the very beginning. MMS enables one to do this. Indeed, let us assume that the variable x depends
separately on two time variables, t and = jt:
x(t) = X0 (t; ) + jX1 (t; ) + O(j2 )
(7)
Then, in the zeroth and the Drst order with respect to j we Dnd
92
X0 + X 0 = 0
9t 2
(8)
92
92
3
X
+
X
=
−4X
−
2
X0 :
1
1
0
9t 2
9t9
(9)
and
The general solution to Eq. (8): X (t) = A()cos t + B()sin t is then substituted into (9), and the
amplitudes A() and B() are found from the requirement of absence of the term proportional to
time. This can be achieved by the requirement that the coeLcients of sin t and cos t on the right-hand
side vanish. As a result we get
dB
3
3
= − A3 − AB2 ;
d
2
2
(10)
dA 3 3 3 2
= B + AB :
d
2
2
(11)
By multiplication of (10) by B() and (11) by A() one obtains a “conservation law”:
d
C() = 0 ;
d
where
C() =
1
([A()]2 + [B()]2 ) ;
2
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
so that C() plays the role of an adiabatic invariant: it changes only in the higher orders of perturbation expansion. Using the above two formulae, we rewrite (10) and(11) in terms of C:
dB
= −3C(0)A ;
(12)
d
dA
= 3C(0)B ;
(13)
d
which, on using the initial conditions, provides us with the secularity-free O(1) approximation to x:
3
X0 (t; ) = cos 1 + j t :
(14)
2
which contains the Drst-order corrections to the frequency. In the following, Bender and Bettencourt
have proceeded with the quantum version of the DuLng equation:
d2
q̂ + q̂ + 4jq̂3 = 0
(15)
dt 2
(where now q̂ is the position operator), which can be derived from the following Hamiltonian:
Ĥ = 12 p̂2 + 12 q̂2 + jq̂4
(16)
by writing down the Heisenberg equations of motion and eliminating the momentum operator. The
initial conditions q̂(0) = q̂0 and p̂(0) = p̂0 are such that the canonical commutation relations
[q̂0 ; p̂0 ] = i˝
are fulDlled. To apply MMS, Bender and Bettencourt write
q̂(t) = Q̂(t; ) = Q̂0 (t; ) + jQ̂1 (t; ) + O(j2 )
(17)
so that the operator of position of the oscillator now depends on two independent time variables.
Substitution of Eq. (17) into (15) and collecting coeLcients of consecutive powers of j gives:
92
Q̂0 + Q̂0 = 0 ;
9t 2
(18)
and
92
92
3
+
Q̂
=
−4
Q̂
−
2
Q̂
Q̂0 :
1
1
0
9t 2
9t9
The general solution to (18) is given by
ˆ
t + B̂()sin t ;
Q̂0 (t) = A()cos
(19)
(20)
from which it follows that in the zeroth order
p̂(t) = B̂ cos t − Aˆ sin t + O(j) ;
where Aˆ and B̂ are operators which depend on but not on t. To fulDll the canonical commutation
relations these operators have to satisfy
ˆ
[A();
B̂()] = i˝ ;
ˆ = q̂0 and B̂(0) = p̂0 .
while the initial conditions require that A(0)
M. Janowicz / Physics Reports 375 (2003) 327 – 410
335
Now the right-hand side of Eq. (19) is evaluated. In order to prevent secularity, the coeLcients
of cos t and sin t are set to zero to obtain
and
d B̂ 1
= (−3Aˆ3 − B̂AˆB̂ − B̂B̂Aˆ − AˆB̂B̂)
d
2
d Aˆ 1
ˆ :
= (3B̂3 + AˆB̂Aˆ + AˆAˆB̂ + B̂AˆA)
d
2
The above equations imply the “conservation” law for the operator Ĥ 1 = 1=2(Aˆ2 + B̂2 ):
(21)
(22)
d Ĥ 1
=0 ;
d
from which it follows that
Ĥ 1 = 12 (p̂20 + q̂20 ) :
Canonical commutation relations allow to rewrite (22) and (21) as
and
3
d B̂
= − (Ĥ 1 Aˆ + AˆĤ 1 )
d
2
(23)
d Aˆ 3
= (Ĥ 1 B̂ + B̂Ĥ 1 ) :
(24)
d
2
These (still apparently complicated) operator diMerential equations admit an elegant solution in terms
of a generalized Weyl ordering of operators
ˆ = W [q̂0 cos(3Ĥ 1 ) + p̂0 sin(3Ĥ 1 )] ;
A()
(25)
B̂() = W [p̂0 cos(3Ĥ 1 ) − q̂0 sin(3Ĥ 1 )]
(26)
and
(cf. [36,37] for the deDnition of the generalized Weyl product), which can be rewritten in a compact
form to give the complete O(1) solution
Q̂0 (t; ) =
q̂0 cos(t + 3Ĥ 1 jt) + cos(t + 3Ĥ 1 jt)q̂0
2 cos(3jt˝=2)
p̂0 sin(t + 3Ĥ 1 jt) + sin(t + 3Ĥ 1 jt)p̂0
:
(27)
2 cos(3jt˝=2)
Taking into account that the coeLcient which multiplies t in the arguments of trigonometric function
is an operator, Bender and Bettencourt have inferred that MMS allows to perform a kind of an
operator mass renormalization. Taking the matrix elements of Q̂0 between the states n − 1| and |n
of the unperturbed, linear oscillator they have concluded that the energy level diMerences are, to the
Drst order in j, equal to 1 + 3n˝j in full accord with the standard Drst-order Rayleigh–SchrIodinger
perturbation calculations.
In the following development Bender and Bettencourt have solved an old problem of discrepancy
between the Rayleigh–SchrIodinger perturbation theory and the WKB method regarding the form
+
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
of asymptotic wave functions. By suitable reorganization and summation of perturbation series
they have shown that both approaches actually give the same results. This is of interest from the
point of view of MMS since the WKB approach can be viewed as a variant of the multiple scales
technique [6].
The asymptotic solutions obtained in [36,37] can be used to obtain other characteristics of the
anharmonic oscillator. A simple example will be discussed in Section 2.3.
2.2. Time evolution operator
The following considerations are parallel to those of [36,37] but use the method of multiple scales
applied directly to the SchrIodinger equation for the time-evolution operator. Our model of a (scalar,
quartic) anharmonic oscillator is deDned by the following Hamiltonian operator:
1 2 1
p̂ + m!2 x̂2 + gx̂4 ;
(28)
2m
2
where x̂ and p̂ are position and momentum operators, m is the mass of the oscillator, ! is the
frequency of linear (unperturbed) oscillations, and g is the “coupling constant” measuring anharmonicity. For a moment we shall keep all the constants (like the Planck constant, frequency, and the
mass of the oscillator) in our formulae, because we want to deDne a dimensionless small parameter
in terms of them. It is convenient—and very much in the spirit of quantum optics—to introduce the
creation and annihilation operators a and a† deDned as
i
m!
x̂ +
p̂ ;
â =
2˝
m!
i
m!
†
x̂ −
p̂ :
â =
2˝
m!
Ĥ =
The inverse formulae are
˝
(â + ↠)
x̂ =
2m!
and
p̂ = −i
˝m!
(â − ↠) :
2
The harmonic terms in Hamiltonian (28) obviously give ˝!(↠â+1=2). Expanding the operator x̂4 in
terms of the operators â and ↠gives, after a little of algebra, the following form of the anharmonic
term:
˝2 g
1
4
2
†4
†3
†2
+ [â + 4â â + 6â + h:c:] ;
6 N̂ + N̂ +
(29)
gx̂ =
4m2 !2
2
where N̂ denotes the excitation number operator, N = ↠â, and “h.c.” means “Hermitian-conjugate
terms”. On the right-hand side we can now recognize the Rayleigh–SchrIodinger Drst-order correction
to the energy of the nth level, except that we still have the operator N̂ instead of the number n.
M. Janowicz / Physics Reports 375 (2003) 327 – 410
337
Let us now rewrite the Hamiltonian as
Ĥ = HC + Ĥ 0 + Ĥ 1 ;
where the c-number quantity HC is given by
3 ˝2 g
1
HC = ˝! +
;
2
4 m2 ! 2
the new zeroth-order Hamiltonian Ĥ 0 reads
Ĥ 0 = ˝!1 ↠â ;
where
3 ˝g
2 m2 ! 2
and, Dnally, the perturbing term is
!1 = ! +
Ĥ 1 = ˝(6N̂ 2 + [â†4 + 4â†3 â + 6â†2 + h:c]) ;
where
g˝
:
4m2 !2
The SchrIodinger equation for the complete time-evolution operator reads
=
d Û
= (HC + Ĥ 0 + Ĥ 1 )Û ;
dt
but we immediately factor out the c-number phase and write
−i
Û = exp
HC t Û
˝
i˝
(30)
so that Û satisDes the equation
d Û
1
i
= (Ĥ 0 + Ĥ 1 )Û :
dt
˝
Let us now introduce a dimensionless time variable
(31)
= !1 t ;
so that
d
d
= !1
:
dt
d
The SchrIodinger equation for Û now reads
where
d Û
= (ĥ0 + jĥ1 )Û ;
d
(32)
ĥ0 = ↠â = N̂ ;
ĥ1 = 6N̂ 2 + [â†4 + 4â†3 â + 6â†2 + h:c:] ;
(33)
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
and we have Dnally deDned the parameter which in the following will be considered small, i.e.,
much smaller than 1:
˝g
=
j=
:
(34)
!1 4m2 !3 + 6˝g
The above derivation of a simple basic equation may be thought too long, but we believe the expression of a small dimensionless expansion parameter in terms of physical constants and parameters
is a matter of principal importance.
We are now ready to apply MMS and obtain approximate solutions. Under the assumption that j
is small, we may look for the perturbative solution in the terms of the expansion:
Û () = Û 0 (T0 ; T1 ; : : :) + jÛ 1 (T0 ; T1 ; : : :) + j2 Û 2 (T0 ; T1 ; : : :) + · · · ;
(35)
where all the operators Û i are supposed to depend separately on all the “times” T0 ; T1 ; T2 ; : : : ; and
are subject to the initial conditions:
U0 (0; 0; : : :) = 1;
U1 (0; 0; : : :) = U2 (0; 0; : : :) = · · · = 0 :
(36)
The dimensionless times Ti correspond to various scales of dynamics of the system and are deDned
as
Tm = jm :
In the following sections we shall encounter situations in which such simple deDnitions of time
scales turn out to be insuLcient.
On substituting expansion (35) into (32) and collecting terms with equal powers of j, we obtain
in the zeroth order
i
9Û 0
= ĥ0 Û 0 ;
9T0
(37)
in the Drst order
i
9Û 0
9Û 1
+i
= ĥ0 Û 1 + ĥ1 Û 0 ;
9T0
9T1
(38)
in the second order
i
9Û 1
9Û 0
9Û 2
+i
+i
= ĥ0 Û 2 + ĥ1 Û 1 ;
9T0
9T1
9T2
(39)
and so on. We can try to solve them systematically order by order. In the zeroth order we obtain
Û 0 = e−iĥ0 T0 V̂ 0 (T1 ; T2 ; : : :) = exp(−iN̂ T0 )V̂ 0 (T1 ; T2 ; : : :)
(40)
while in the Drst order, on writing
Û 1 (T0 ; T1 ; T2 ; : : :) = e−iN̂ T0 UŴ 1 (T0 ; T1 ; T2 ; : : :) ;
(41)
we obtain
i
9V̂ 0
9UŴ 1
ˆ 0 )V̂ 0 ;
+i
= (eiN̂ T0 ĥ1 e−iN̂ T0 )V̂ 0 = 6N̂ 2 V̂ 0 + A(T
9T0
9T1
(42)
M. Janowicz / Physics Reports 375 (2003) 327 – 410
339
where
ˆ 0 ) = â†4 e4iT0 + 4â†3 âe2iT0 + 6â†2 e2iT0 + h:c :
A(T
It is clear that on the right-hand side of Eq. (42) there exists one term—6N̂ 2 —which, after integration
over T0 , would give a result linear in T0 , that is, a secular term of the kind which we attempt to
avoid. Therefore, we absorb this dangerous term in the derivative of V0 over T1 by writing
i
9V̂ 0
= 6N̂ 2 V̂ 0 ;
9T1
(43)
with the result
V̂ 0 (T1 ; T2 ; : : :) = e−6iN̂
2
T1
Ŵ 0 (T2 ; : : :) :
(44)
All other terms on the right-hand side of Eq. (42) can be integrated to give
UŴ 1 (T0 ; T1 ; T2 ) = X̂ 1 (T1 ; T2 ; : : :) − B̂(T0 )V̂ 0 ;
(45)
with
B̂(T0 ) = 14 â†4 e4iT0 + 2â†3 âe2iT0 + 3â†2 e2iT0 − h:c :
The above equations Dnish our calculations up to the Drst order.
In the second order the formalism becomes somewhat algebraically involved, but computations
are still fairly straightforward. We have
i
9Û 2
9X̂ 1 (T1 ; : : :)
9Ŵ 0
2
+ ie−iN̂ T0
+ ie−iN̂ T0 e−6iN̂ T1
9T0
9T1
9T2
=N̂ Û 2 + ĥ1 e−iN̂ T0 [X̂ 1 (T1 ; : : :) − B̂(T0 )e−6iN̂
2
T0
Ŵ 0 ] :
(46)
As before, we factor out the operator exp(−iN̂ T0 )
Û 2 = e−iN̂ T0 UŴ 2 ;
and obtain
i
9UŴ 2
9Ŵ 0
9X̂ 1 (T1 ; : : :)
2
+i
+ ie−6iN̂ T1
9T0
9T1
9T2
=(eiN̂ T0 ĥ1 e−iN̂ T0 ) [X̂ 1 (T1 ; : : :) − B̂(T0 )e−6iN̂
2
ˆ 0 )] [X̂ 1 (T1 ; : : :) − B̂(T0 )e−6iN̂
=[6N̂ 2 + A(T
T1
2
Ŵ 0 (T2 ; : : :)]
T1
Ŵ 0 (T2 ; : : :)] :
(47)
ˆ 0 )X̂ 1 (T1 ; : : :)
Now, we have to identify possible sources of secular terms. It is clear that both A(T
2
and 6N̂ B̂(T0 )V̂ 0 (T1 ; T2 ; : : :) which appear on the right-hand side of Eq. (47) can be integrated over
T0 without any problem. On the other hand, integration over T0 of the term 6N̂ 2 X̂ 1 (T1 ; : : :) would
340
M. Janowicz / Physics Reports 375 (2003) 327 – 410
give a term linear in T0 . We still cannot require, however, that
i
9X̂ 1 (T1 ; : : :)
= 6N̂ 2 X̂ 1 (T1 ; : : :)
9T1
with the obvious solution:
X̂ 1 (T1 ; : : :) = e−6iN̂
2
T1
Ŷ 1 (T2 ; : : :) ;
(48)
because the terms
−e6iN̂
2
T0
ˆ 0 )B̂(T0 )e−6iN̂ 2 T0 = Ĉ
A(T
Drst need to be checked for the presence and character of operators which would possibly cause
secular terms to appear. A somewhat boring algebra, characterized by an extensive application of
the commutation relations for the creation and destruction operators leads to the following result:
Ĉ = −2(2N̂ + 1)[17N̂ (N̂ + 1) + 21]Ŵ 0 (T2 ; : : :) + Ĉ 1 (T0 ; T1 )Ŵ 0 (T2 ; : : :) ;
where Ĉ 1 (T0 ; T1 ) can already be integrated over T0 without giving any dangerous terms. Thus,
we must require
Ŵ 0 (T2 ) = exp(2i(2N̂ + 1)[17N̂ (N̂ + 1) + 21]T2 )Ẑ ;
(49)
where the operator Ẑ can depend on higher-order times Tm , m ¿ 3. We can now set X̂ 1 as in Eq. (48)
for Ĉ does not contain any operators which would have to be absorbed in 9X1 =9T1 . Integration over
T0 in Eq. (47) (under the condition that we have already eliminated “bad” terms by “absorbing” them
in X̂ 1 and Ŵ 0 as shown just above) completes the second-order solution. We shall not provide the
whole complicated formula. The important second-order contributions are contained in X̂ 1 and in W0 .
One can, in principle, continue the computation to obtain (more and more complicated) higher-order
ˆ 0 ) and B̂(T0 ) contains four-linear terms in
terms. It is interesting that, although the product of A(T
the creation and annihilation operators, the exponential multiplying Ẑ contains N̂ in a polynomial
of the third, and not the fourth order. On the other hand, if we stop computation at the present
stage, the only remaining exercise is to impose the initial conditions. In particular, Ẑ must be the
unit operator and Ŷ 1 (0; : : :) has to be equal to B̂(0).
The above procedure and results for asymptotic forms of the time-evolution operator well demonstrate the power of MMS. Although the fact that in the lowest order the eMective Hamiltonian should
be proportional to the diagonal part of the “intensity” x̂2 could be guessed just by looking at Hamiltonian (28), MMS not only provides a precise numerical coeLcient of such eMective energy operator,
but also gives corrections not contained in the exponential terms as well as allows for eLcient
(if long) computations of higher-order approximations containing numerical coeLcients which are
rather diLcult to guess, cf. Eq. (49).
As our Hamiltonian is time-independent, the approximate evolution operator can be used to obtain
both the approximate time dependence of the operators in the Heisenberg picture and the evolution
of the wave function. The second part of the following section is devoted to an analysis of the
dynamics of the expectation value of coordinate x̂ of the anharmonic oscillator, computed using our
asymptotic results as well as “exact” (numerical) results.
M. Janowicz / Physics Reports 375 (2003) 327 – 410
341
Comparison of exact and approximate solutions to the Duffing equation
3
exact solution
purely harmonic
ordinary perturbative solution
multiple-scales solution
2
x
1
0
-1
ε = 0.1, x0 = 1
-2
-3
0
2
4
6
8
10
12
14
t
Fig. 1. Comparison of exact and approximate solutions to the DuLng equation for the time-dependence of the dimensionless
coordinate x of the anharmonic oscillator. The initial conditions are: p(0) = 0, x(0) = 1:0, and the parameter j = 0:1.
2.3. Comparison with numerical results
2.3.1. Classical case
To begin with, we will show Drst how good (or bad) the method of multiple scales can be when
applied to the anharmonic oscillator in the classical case. Extensive comparisons of this type already
exist, cf., e.g., [6], so we shall be very concise. In Figs. 1–6 we have compared four types of
solutions: (i) exact solutions (the simple anharmonic oscillator admits, of course, an exact analytical solution in terms of elliptic integrals; we have not, however, plotted these analytic solution, but
simply integrated numerically the DuLng equation and displayed obtained numbers); (ii) solutions
for purely harmonic oscillations without any non-linearity; (iii) ordinary (regular) perturbative solutions containing secular terms; and Dnally, (iv) classical solutions obtained in [36,37] with the help
of MMS. In all cases, the initial value of momentum has been taken equal to zero. Three families
of initial conditions of position x0 and parameters j have been considered: (i) x0 = 1:0, j = 0:1;
(ii) x0 = 5:0, j = 0:005; (iii) x0 = 0:5, j = 1. Figs. 1–3 display the position of the oscillator x as a
function of time t, while Figs. 4–6 show the corresponding pattern in the phase plane (x; p).
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
Comparison of exact and approximate solutions to the Duffing equation
10
exact solution
purely harmonic
multiple-scales solution
x
5
0
-5
ε = 0.005, x0 = 5
-10
0
2
4
6
8
10
12
14
t
Fig. 2. The same as in Fig. 1, but for diMerent initial conditions: p(0) = 0, x(0) = 5:0, and for j = 0:005.
It is clear that in all three cases (i) – (iii) the MMS solutions are much superior to the simple harmonic approximation and regular perturbative solution. For the parameters of the family (i)
(Figs. 1 and 4) we observe that the approximate solution becomes out of phase with the exact
one at time of the order of 10, but the qualitative picture in the phase space is excellent for all
times.
It is quite interesting that MMS works well even for large amplitudes (Fig. 2), provided that j
is small enough—there is only a little “dephasing” even for t = 15, the picture in the phase space
(Fig. 5) is quite satisfactory as well (the regular perturbative approximation is not shown since it
blows up very fast). It is also interesting that even though the “small” parameter j is actually equal
to 1, MMS again provides reliable results for relatively large times on condition that the initial amplitude is O(1). Let us notice, however, that the time for which one might still expect MMS to be a
good approximation evidently depends on the initial conditions, and for very large initial amplitudes
one should expect large deviations from the true solution. Let us repeat the last statement in a somewhat diMerent context. Let us imagine we try to describe a non-linear classical system with a small
parameter in the “SchrIodinger picture”, i.e., using the classical distribution function in the phase
space (classical density matrix), #, which satisDes the Liouville equation. We can write down the
M. Janowicz / Physics Reports 375 (2003) 327 – 410
343
Comparison of exact and approximate solutions to the Duffing equation
1.5
ε = 1, x0 = 0.5
1
0.5
x
0
-0.5
-1
exact solution
purely harmonic
ordinary perturbative solution
multiple-scales solution
-1.5
-2
0
2
4
6
8
10
t
Fig. 3. The same as in Fig. 1, but for the initial conditions: p(0) = 0, x(0) = 0:5, and for j = 1:0.
solution to the Liouville equation if we know the solutions to the Hamilton equations for q(t); p(t).
Now, if # is initially spread out in a large region of the phase space, there will be some trajectories
for which MMS will work well, but there is a non-zero probability that the system will evolve from
such a point in the phase space that our approximation will become completely unrealistic. This
means that we have to be very careful when large /uctuations are possible in the system; they
impose additional constraints on the usefulness of MMS, beside the necessity of the presence
of a small parameter. We have indicated this problem—somewhat trivial, perhaps, in the classical
setting—anticipating more subtle and diLcult situation in quantum mechanics. On the other hand, let
us stress that in our examples we have used only the O(1) MMS solution which has only contained
corrections to the phase, but no amplitude corrections. This shows that even the minimum eMort to
obtain such absolutely lowest-order approximation can bring impressive success.
2.3.2. Quantum-mechanical case
Here we shall compare exact numerical solutions and their MMS approximations assuming that
the system is initially in a coherent state. We shall discuss the dynamics of expectation values x̂
of the position operator x̂ of the oscillator as functions of time. Coherent states have been chosen
here as initial states since, on the one hand, they are usually assumed to be, in a sense, very
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
Comparison of exact and approximate solutions to the Duffing equation - phase space picture
ε = 0.1, x0 = 1
1
p
0.5
0
-0.5
-1
exact solution
purely harmonic
ordinary perturbative solution
multiple-scales solution
-1
-0.5
0
0.5
1
x
Fig. 4. Comparison of exact and approximate solutions to the DuLng equation in the phase space (x; p). The initial
conditions are: p(0) = 0, x(0) = 1:0, and the parameter j = 0:1.
close to classical states, while, on the other hand, quantum-mechanical /uctuations are already very
pronounced for coherent states. They are conveniently expressed in terms of the number states of
harmonic oscillator, which are eigenstates |n of the operator N̂ :
N̂ |n = n|n ;
(50)
where n is a non-negative integer. In the coordinate representation number states are products of
Hermite polynomials and a Gaussian function. Coherent states are eigenstates of the annihilation
operators
â|$ = $|$ ;
(51)
where $ is a complex number. Any coherent state can be written as the following superposition of
the number states:
|$ =
∞
$n
2
√ |n e−(1=2)|$| :
n!
n=0
(52)
M. Janowicz / Physics Reports 375 (2003) 327 – 410
345
Comparison of exact and approximate solutions to the Duffing equation - phase space picture
8
exact solution
purely harmonic
multiple-scales solution
6
4
p
2
0
-2
-4
-6
ε = 0.005, x0 = 5
-8
-8
-6
-4
-2
0
2
4
6
8
x
Fig. 5. The same as in Fig. 4, but for diMerent initial conditions: p(0) = 0, x(0) = 5:0, and for j = 0:005.
To compute asymptotic dynamics of the system, we have used the O(1) approximation to the
time-evolution operator, which already contains Drst-order “phase” corrections, and is given by
Û 0 = e−iN̂ e−6ijN̂
2
:
(53)
Let us consider
the dynamics of expectation values of x̂, which are displayed in Figs. 7–12 (in
units of ˝=(2m!)) and form a counterpart of the previous classical comparison. Let us Drst notice
that, as is clear from Eqs. (53) (as well as from intuitive representations), for our asymptotic
approximation to be meaningful, it is not enough that j itself is small. We should, in fact, have at
least jN̂ (0) = jn1.
W
Figs. 7–9 deal with the case where this product is equal to 3 × 10−3 , while
in Figs. 10–12 we have jnW = 3 × 10−2 .
In Fig. 7 we show the short-time behavior of x̂ for j = 10−4 and nW = 30:0 as given by exact
numerical solution to the SchrIodinger equation as well as by application of U0 . To plot this Dgure
we have deliberately chosen quite large ’s in order to be able to see the diMerence between the
exact and approximate results. For shorter times the agreement is, naturally, still better, for, in fact,
the exact and the MMS solutions are almost identical. Obviously, for jnW still smaller, we obtain
even better agreement.
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
Comparison of exact and approximate solutions to the Duffing equation - phase space picture
1
ε = 1, x0 = 0.5
p
0.5
0
-0.5
-1
exact solution
purely harmonic
ordinary perturbative solution
multiple-scales solution
-1
-0.5
0
x
0.5
1
Fig. 6. The same as in Fig. 4 but for the initial conditions p(0) = 0, x(0) = 0:5, and for j = 1.
In Fig. 8, keeping jnW = 3 × 10−3 we have changed j to 10−3 and nW to 3, and displayed our
solutions for longer time. Now, the anharmonic oscillator is an example of systems in which the
“grain-like” nature of the Quantum Theory manifest itself by the presence of the so-called collapses
and revivals of oscillations. Fig. 8 shows the dynamics approaching the Drst collapse; it is clear that
MMS solutions give good qualitative approximation for the location of collapse on time axis (the
same is true about the Drst revival, not shown in the Dgure). The quantitative agreement of exact
and approximate solution is satisfactory, to say the least, for times smaller than the time of the Drst
collapse.
The situation changes for worse when j becomes equal to 10−2 . In Fig. 9 we can see that for
dimensionless times of about 40 the approximate solution, while qualitatively still sound, becomes
strongly out of phase from the exact solution. This may be somewhat surprising in view of our
expectations associated with the classical theory when MMS usually gives satisfactory results for the
dimensionless time ∼ 1=j. But in the quantum case with j ∼ 10−2 the /uctuations start to spoil the
results even for nW as small as ∼ 0:1. With growing nW this eMect becomes more and more serious,
and Dnally devastating. Nevertheless, for the parameters of Fig. 9, MMS provides reasonably good
qualitative picture of oscillations and good approximation of their amplitudes.
M. Janowicz / Physics Reports 375 (2003) 327 – 410
347
Quantum anharmonic oscillator: comparison of exact and approximate results
6
exact solution
O(1) first-order MMS
ε = 0.0001, <N(0)> = 30.0
4
<x(t)>
2
0
-2
-4
-6
250
255
260
265
τ
270
275
280
Fig. 7. Comparison of exact and approximate
solutions to the problem of an anharmonic oscillator: time-dependence of the
expectation value x(t) in units of ˝=(2m!). Initially the system is in the coherent state, and the important parameters
are j = 10−4 ; nW = 30:0.
In Figs. 10–12 the product jnW is one order of magnitude larger than in Figs. 7–9 which leads to
large inaccuracies of MMS for not very short times. In Figs. 10 and 11 we display the long-time
dynamics of x̂ for j = 0:001 and nW = 30. It is clear that the MMS completely misses the true
revivals (both small and large ones) of the oscillations, and gives a large-scale false revivals. In
addition, when we have attempted to plot the O(j) MMS approximation (i.e. that obtained with
the help U0 + jU1 ) we have realized that, for jnW ∼ 10−2 , the norm of the wave function is rather
poorly conserved. It oscillates with the amplitude of up to 0:2. This is because the time-evolution
operator is unitary only up to the order j, and the factor multiplying j2 can become fairly large as
it contains fourth-order polynomials of the creation and annihilation operators. Thus, in the case of
still quite small jnW we would have to use the second-order correction to keep unitarity. Let us be
quick to mention, however, that even with this imperfect unitarity, MMS is much superior to the
ordinary regular perturbation expansion of the time-evolution operator (the results of the latter are
not even shown since they blow up very fast).
The short-time agreement (up to ∼ 10) between MMS and exact results is also displayed in
Fig. 12, where j = 0:01 and nW = 3:0. The MMS approximation predicts collapses and revivals of the
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
Quantum anharmonic oscillator: comparison of exact and approximate results
exact solution
O(1) first-order MMS
0.3
ε = 0.001, <N(0)> = 3.0
0.2
<x(t)>
0.1
0
-0.1
-0.2
-0.3
120
130
140
150
160
τ
170
180
190
200
Fig. 8. The same as in Fig. 7, but for diMerent parameters j and n:
W j = 10−3 , nW = 3.
oscillations, but, starting with the Drst revival, its predictions about times of collapses and revivals are
unreliable. Again, the Drst-order approximation gave here rather large (up to 20%) non-conservation
of probability.
As a general conclusion from these quantum-mechanical comparison, we notice that MMS usually
gives good results for shorter times and smaller j than might be expected in view of the spectacular
success of its classical version. But this only means that “quantum” MMS—like every method
of applied mathematics—has its own region of validity which depends on a particular problem.
Another conclusion which we would like to draw in the end of this section is that the Drst-order
approximation may become insuLcient, and that both phase- and amplitude corrections of higher
order may be necessary if the results are to be reliable.
3. Spontaneous emission
By “spontaneous emission” we mean the emission of light by an excited atom in the absence of
initial excitations of electromagnetic modes. As such, this process belongs to the most important
M. Janowicz / Physics Reports 375 (2003) 327 – 410
349
Quantum anharmonic oscillator: comparison of exact and approximate results
1.5
exact solution
O(1) first-order MMS
ε = 0.01, <N(0)> = 0.3
1
<x(t)>
0.5
0
-0.5
-1
-1.5
0
10
20
30
40
50
60
τ
Fig. 9. The same as in Fig. 7, but for diMerent parameters j and n:
W j = 10−2 , nW = 0:3.
ones in nature. In addition, correct theory of spontaneous emission, Drst developed by Dirac within
his time-dependent quantum perturbation formalism, opened, together with fundamental works of
Heisenberg, Pauli and Jordan, the era of quantum Deld theory. Hence the problem of spontaneous
emission (including the problem of widths of spectral lines of spontaneously emitted light) under
various conditions remains one of the most interesting ones in quantum theory and is still intensively
studied.
As attempts of Fermi [38] as well as Crisp and Jaynes [39] (cf. [40] as well) to build a
“neoclassical” theory of spontaneous emission seem to have shown, the details of the process of
spontaneous emission of light require the Deld quantization for their description. Hence our discussion of spontaneous emission from the point of view of MMS must start with the concept of
quantized electromagnetic Deld. Usually, quantum opticians Dnd it reasonable and convenient to
quantize the electromagnetic Deld in a large-volume Dctitious box or cavity by imposing the periodic boundary conditions on the walls of the cavity (cf., e.g., [41]). In all practical calculations we
Dnally take the limit of inDnite size of the cavity and replace summation over all cavity modes by
integration over the wave vector.
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
Quantum anharmonic oscillator: exact results
15
exact solution
ε = 0.001, <N(0)> = 30.0
10
<x(t)>
5
0
-5
-10
-15
0
200
400
600
800
1000
τ
Fig. 10. Long-time behavior of x(t) for j = 10−3 , nW = 30—exact numerical results.
The electromagnetic vector potential in the cavity is written in the form
2
˝
{el; âl; eikl ·r + el;? â†l; e−ikl ·r } ;
Â(r; t) =
2!l j0 V
l
(54)
=1
where V is the cavity volume, the subscript l is the index of cavity modes (it in fact consists of
three indices, l1 ; l2 ; l3 , is the polarization index, el; , el;? are the polarization vectors, kl is the
wave vector, !l = kl c is the frequency of the (l; ) mode, while âl and â†l are the annihilation and
creation operators which satisfy the canonical commutation relations:
[âl; ; â†l ; ] = ll ; ;
(55)
whereas all other commutators made of these operators vanish. In the following we shall
restrict ourselves to the description in terms of linear polarizations in free space, so that the polarization vectors el; will be assumed to be real. The wave vector k can be expressed by the
M. Janowicz / Physics Reports 375 (2003) 327 – 410
351
Quantum anharmonic oscillator: approximate results
15
O(1) first-order MMS
ε = 0.001, <N(0)> = 30.0
10
<x(t)>
5
0
-5
-10
-15
0
200
400
600
800
1000
τ
Fig. 11. Long-time behavior of x(t) for j = 10−3 , nW = 30—MMS results.
multi-index l as
k=
2(
(l1 ; l2 ; l3 ) ;
L
and the summation over l in (55) is the summation over all three indices l1 ; l2 ; l3 . It is to be noted
that the sums over l are very often written as a sum over k, and then the creation and annihilation
operators bear the index k as well.
In order to Dnd the decomposition of the electric Deld in terms of the annihilation and creation operators, it is necessary to choose the gauge. The most natural though not explicitly Lorentz-invariant
choice is the Coulomb (radiation) gauge in which we have
∇ · Â(r; t) = 0 ;
and the total electric Deld is given by
Ê(r; t) = −
9Â(r; t)
ˆ t)
− ∇*(r;
9t
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
Quantum anharmonic oscillator: comparison of exact and approximate results
4
exact solution
O(1) first-order MMS
ε = 0.01, <N(0)> = 3.0
3
2
<x(t)>
1
0
-1
-2
-3
-4
0
20
40
60
80
100
τ
Fig. 12. The same as in Fig. 7, but for diMerent parameters j and n:
W j = 10−2 , nW = 3:0.
where *ˆ is the static potential to be expressed by the operator variables characterizing matter Delds.
In the regions of space free of charges we can put *ˆ = 0, and, by noticing that for non-interacting
Delds
âl; t = âl e−i!l t ;
? i !l t
â?
;
l; t = âl e
(56)
we obtain immediately:
Ê(r; t) = i
2
l
=1
˝!l
el; {âl; exp(ikl · r) − â†l; exp(−ikl · r)} ;
2j0 V
while the magnetic induction decomposition reads
2
˝
(el; × kl ){âl; exp(ikl · r) − â†l; exp(−ikl · r)} :
B̂(r; t) = −i
2!l j0 V
l
=1
(57)
(58)
M. Janowicz / Physics Reports 375 (2003) 327 – 410
353
Relation (57) remains true in the regions where the matter Delds are non-zero provided, however,
that Ê is replaced by ÊT which is the transverse part of the electric Deld operator. Now we are
ready to express the radiation Deld Hamiltonian in terms of modes
1
1 2
3
2
d r j0 ÊT (r) +
Ĥ =
˝!l â†l âl ;
(59)
B̂ (r) =
2
,0
l;
where we have omitted the inDnite vacuum energy term (1=2) (˝!l ).
The advantage of the box quantization is that one has to do with operators â; ↠enumerated by
discrete indices, and not with the operator-valued distributions, and one can easily make the limit
of atomic coupling with only one mode, as is standard in Cavity Quantum Electrodynamics. The
passage from the discrete mode—decomposition of quantized Delds given above, to the decomposition
in terms of operator-valued
distributions in k space is quite simple; one should only write 1=(2()3=2
√
instead of 1= V and replace summation by integration over k.
The major and very widespread assumption about the atomic degrees of freedom which is to be
employed here is that the atom has only Dnite number of discrete levels. In fact, for the purposes
of this section it is enough to consider the model consisting of just two or three levels. There is a
huge amount of experiments in which such a two-level model was perfectly suLcient and justiDed,
cf., e.g., [42–44].
If the energy of the Drst (|1) atomic level is E1 and of the second (|2) E2 , then we may write
the atomic energy operator as
Ĥ A = E1 |11| + E2 |22| ;
(60)
or, using the convenient “.” notation
n
Ei .̂ii ;
Ĥ A =
(61)
i=1
where .̂ij = |ij| and n = 2 for two-level atoms. Many authors have found it convenient to use
symbols .̂+ to denote .̂21 , and .̂− to denote .̂12 . The operators R̂± = (1=2).̂± have also been
extensively used. As always in non-relativistic physics, we can choose the zero of energy according to our convenience. If we choose, for instance, E1 = 0, the two-level-atom Hamiltonian can be
written as
Ĥ A = ˝!.̂22 ;
(62)
where ! = E2 =˝; or, we can choose such zero of energy that the upper level has the energy
+(1=2)(E2 − E1 ), and the lower one the energy −(1=2)(E2 − E1 ). Then the atomic Hamiltonian
reads
1
(63)
Ĥ A = ˝!(.̂22 − .̂11 ) :
2
The diMerence .̂22 − .̂11 is often denoted by .̂z , since in the SchrIodinger picture it acts in the Hilbert
space C2 as one of the ordinary Pauli matrices, the latter being deDned as
0 1
0 −i
1 0
.̂x =
; .̂y =
; .̂z =
:
1 0
i 0
0 −1
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
The problem of the quantum-optical interaction Hamiltonian in the non-relativistic setting is surprisingly non-trivial. There exists an elegant and authoritative derivation of the most correct nonrelativistic Hamiltonian from the coupled operator Maxwell–Dirac equations, provided in [50].
However, further approximations are usually made to render it in a maximally simpliDed form.
There are two basic forms of such simpliDed interaction Hamiltonian Ĥ AF which are extensively
used in quantum optics (which, after all, may be considered as a part of non-relativistic quantum
electrodynamics). The Drst one stems from the “minimal-coupling” Hamiltonian and can be written
as
e
e2 2
Ĥ AF =
|ii| − p̂ · Â(r) +
(64)
 (r) |jj| ;
m
2m
ij
where m is the electron mass, e is its charge, and  is the electromagnetic potential. In the context
of spontaneous emission one usually makes two further approximations. Firstly, one assumes that,
due to the smallness of atomic size, the electromagnetic potential changes only very little within the
atom. Thus, it should be a very good approximation to calculate the potential not at the position
of the electron, but at the center of mass of the atom. In addition, to obtain the overall picture of
spontaneous emission one may neglect the A2 term present in (64). Let us notice, however, that the
latter approximation could be very dangerous if one is interested in the details of atomic dynamics,
(cf. [45,46]), or the exact non-relativistic values of radiative energy-level shifts. If we disregard this
objection, we obtain from Eq. (64)
e
Ĥ AF = − Â(r0 )
i|p̂|j.ij ;
(65)
m
ij
where r0 is the position of the atomic center of mass.
The second form of the interaction Hamiltonian: the “−d · E” coupling stems from the coupling
of the dipole to the electric Deld already known from the elementary electrostatics. The interaction
energy of a dipole in an external electric Deld is equal to minus scalar product of the dipole moment
and the Deld itself. Hence we write
Ĥ AF = −d̂ · ÊT (r0 ) ;
where d̂ is the dipole moment operator of the atom, and ÊT is the operator of the transverse electric
Deld at the center of mass of the atom, the mode decomposition of which is given by (57). This form
of the interaction Hamiltonian already assumes the electric dipole approximation. It can be obtained
from the minimal coupling form by a unitary transformation called the Power–Zienau transformation
provided, however, that one neglects an additional term containing the square of the (transverse)
atomic polarization, [47–49]. If we now take into account that atoms do not have permanent dipole
moments (unless they are in an external static Deld), so that the diagonal matrix elements of d
vanish, we obtain
Ĥ AF = −
i|d̂|j.̂ij · Ê(r0 ) :
(66)
i=j
In the two-level model of an atom the above summation only involves two terms.
With the energy operators given above, we are ready to consider the spontaneous emission from
the point of view of MMS. The method of multiple scales has been Drst applied to the problem of
M. Janowicz / Physics Reports 375 (2003) 327 – 410
355
spontaneous emission in the papers by Yung-Chang Lee and coworkers [11–13] in the context of
superradiance [51,52]. Superradiance is a term given to the spontaneous emission of a collection of
N atoms characterized by the intensity proportional to N 2 under the absence of any external agent
(excitation); such an eMect clearly involves high degree of coherence of emission in spite of the fact
that the spontaneous emission is usually (and rightly) considered as a powerful coherence-destroyer.
For the superradiance to be present, all atoms of the sample must “feel” the presence of all remaining
atoms (hence the sample should not have large spatial size), and each atom should experience a
signiDcant energy loss, cf. [51].
In their Drst paper on the subject of spontaneous emission, Lee et al. studied one- and two-atom
system coupled to the electromagnetic Deld. They showed that the usual exponential ansatz used
rather ad hoc by Weisskopf and Wigner [53] in their theory of natural line shape follows “naturally”
from the multiple-scales analysis. They started from the following Hamiltonian governing a system
containing a collection of atoms (in the two-level approximation) and the radiation Deld, written in
the interaction picture:
Ĥ =
i
j[d?
k exp(ik · xi ) exp(i!0k t)
k
·Ĉ k R̂+ (i) + dk exp(−ik · xi ) exp(−i!0k t)Ĉ †k R̂− (i)] ;
(67)
where Ĉ k and Ĉ †k denote the photon creation and annihilation operators of the mode labeled by k;
R̂+ and R̂− are the atomic raising and lowering operators for the ith atom, j is a parameter explained
√
below, while dk and d?
k are the atom-Deld coupling constants proportional to ek ·d12 = !k with ek are
the polarization vectors. Finally, !0 denotes the energy diMerence (divided by ˝) between the atomic
excited and ground states (assumed identical for all atoms), !k is the frequency associated with the
wave vector k, and !0k = !0 − !k . In [11] the authors have clearly assumed that summation over
k also involves the summation over polarizations, and the anti-resonant terms leading to emission
together with excitation as well as spontaneous absorption have been omitted.
The possibility of application of MMS in the present context relies on the fact that, physically,
there are at least two natural and very diMerent time scales in the system. The Drst one is associated
with the atomic transition frequency !0 (relevant electromagnetic mode frequencies are close to
it), the second one is the inverse of the spontaneous decay rate (typically ∼ 10−7 !0 ), which,
however, does not appear explicitly in the interaction Hamiltonian. For this reason Lee et al. have
introduced an artiDcial “small” parameter j which in the end will be set equal to 1. The true, natural
small parameter appears at a later stage of the MMS procedure.
Following the Wigner–Weisskopf SchrIodinger-picture approach [53,54], Lee et al. have written
the wave function of the system in the form
|6(t) = a(t)|2; 0k +
bk (t)|1; 1k ;
(68)
k
where |2; 0k denotes the state with the atom excited and zero photons in the Deld, while |1; 1k describes the atom in the ground state and one photon present in the mode k. Then, amplitudes a(t)
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
and bk (t) satisfy the following system of linear diMerential equations:
d
i!0k t
jd?
bk ;
i a(t) =
ke
dt
k
d
bk = jdk e−i!0k a(t) :
dt
The above system of equations, to be solved with the initial conditions
i
(69)
bk (0) = 0 ;
a(0) = 1;
although linear, is still fairly complicated, Drst of all because it describes a physical system with
inDnitely many degrees of freedom. The authors of [11] proceed by deDning the new time variables Tn = jn t and using MMS with respect to the perturbation parameter j. Thus, the SchrIodinger
amplitudes are looked for in the form
∞
a(t) =
jn a(n) (T0 ; T1 ; T2 ; : : :) ;
n=0
bk (t) =
∞
jn bk(n) (T0 ; T1 ; T2 ; : : :) ;
n=0
with the initial conditions
a(T0 ; T1 ; T2 ; : : :) = 1 ;
bk (T0 ; T1 ; T2 ; : : :) = 0 :
Substituting the above expansions into (69) and collecting terms standing at equal powers of j one
obtains
i
9a(0)
=0 ;
9T0
9b(0)
i k =0 ;
9T0
(1)
9a(0)
9a
i!0k T0 (0)
=
+
d?
bk ;
i
ke
9T0
9T1
k
9b(0)
9b(1)
k
= dk e−i!0k T0 a(0) ;
+ k
i
9T0
9T1
(2)
9a(1) 9a(0)
9a
i!0k T0 (1)
=
+
+
d?
bk ;
i
ke
9T0
9T1
9T2
k
(1)
(0)
9b
9b
9b(2)
k
= dk e−i!0k T0 a(1) :
+ k + k
i
9T0
9T1
9T2
(70)
M. Janowicz / Physics Reports 375 (2003) 327 – 410
357
The requirement of absence of secular terms leads to the conclusion that both a(0) and b(0)
are
k
independent not only of T0 , as is clear from Eqs. (70), but also of T1 . The same statement is true
about a(1) and b(1)
k . But in the second order in j one gets
s
T0 (0)
9a
a(2) = −T0
−
ds
|dk |2
ds1 ei!0k s1 a(0) :
(71)
9T2
0
0
k
In addition to MMS, Lee et al. had to employ the assumption that T0 is large, that is, much larger
than 1=!0 . From this assumption it follows that
T0
P
i!0k s
ds e
= i8(!0k ) = i
− i((!0k ) ;
(72)
!0k
0
where P denotes the Cauchy principal value, and
T0 s
T0
1 − ei!0k T0
i!0k s1
∼ iT0 8(!0k ) :
ds
ds1 e
=i
−i
2
!0k
!0k
0
0
The latter equation is only valid for large T0 , as well. Using (73) in (72) we obtain Dnally
9a(0)
+i
|dk |2 8(!0k )a(0) = 0 ;
9T2
(73)
(74)
k
which can also be written as
a(0) (T2 ) = exp(−i!s T2 − s T2 ) ;
(75)
where we have introduced the symbols
1
!s =
|dk |2 P
;
!0k
k
s = (
|dk |2 (!0k ) :
k
If Eq. (75) is fulDlled, the right-hand side in (71) is equal to zero for large T0 , and no secular terms
arise in the second order on time scales much larger that 1=!0 .
The authors of [11] have also shown that a(2n+1) = 0 for n ¿ 1, and that, for large T0 , a(2n) = 0
for n ¿ 2. Thus, when t becomes large, we Dnally obtain
a(t) ≈ e−i!s t −s t ;
(76)
where j has been replaced by 1. The amplitudes bk can be obtained from (76) by simple integration
which gives
b k = dk
e−i!0k t e−i(!s −is )t − 1
:
!0k + !s − i
(77)
The last formula shows that the spectrum of emitted light is Lorentzian.
It is to be noted that the results contained in Eqs. (76) and (77) are obviously well known. Also,
MMS alone cannot actually provide the correct spectral line width . The additional relation (72),
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
which is valid, strictly speaking, only for inDnite T0 , has had to be used; and one can comment that
it is also used in standard approaches to the problem of spontaneous emission. In fact, the need to
use Eq. (72) (or an equivalent relation) is related to time-reversal invariance of the Hamiltonian and
SchrIodinger equation: to get any non-zero one has to break this invariance “by hand” regardless
of which method one uses (this is not less true about the von Neumann equation for the density
matrix). Nevertheless, it is to be stressed that no additional exponential or other ansatz has had to
be employed in [11] to obtain the solution.
In [11], the problem of spontaneous emission from two atoms located at the points x1 and x2
has been addressed as well, and the method of multiple scales has been applied to solve it. The
SchrIodinger-picture wave function has been assumed to have the form
|6(t) = b1 (t)|1; 0; 0k + b2 (t)|0; 1; 0k +
bk (t)|1; 1; 1k ;
(78)
k
which means that the subspace of the Hilbert space allowed by the model consists of (a) the state
with the Drst atom in the excited state, and the second one in the ground state with no photon in the
Deld; (b) the state with the Drst atom in the ground state, the second atom excited, and no excitation
of the electromagnetic Deld; (c) the state with both atoms in their ground states and one quantum
of Deld in the mode k. The initial conditions are such that only the Drst atom is excited, b1 (0) = 1,
b2 (0) = bk (0) = 0. Now, MMS works almost algorithmically (up to the necessity of using (72) once
again) and gives in the second order
9b(0)
(0)
1
= −f11 b(0)
1 − f12 b2 ;
9T2
9b(0)
(0)
2
= −f21 b(0)
1 − f22 b2 ;
9T2
(79)
where
fij = i!ij = ij ;
!ij =
ij = (
|dk |2 P
k
1 ikxij
e
;
!0k
|dk |2 eikxij (!0k ) ;
and xij = xi − xj . After solving the above pair of equations, Lee et al. could obtain results for all
three SchrIodinger amplitudes. In the next paper, Lee and Lee [12] generalized the procedure to the
case of N atoms. Instead of system (79), using MMS they obtained the following equations for the
amplitudes of having precisely one, jth atom in the excited state:
N
dbi
=−
fij bj (t) ;
dt
j=1
where
fij = i
k
|dk |2 eikxij 8(!0 − !k ) :
(80)
M. Janowicz / Physics Reports 375 (2003) 327 – 410
359
Using these formulae, Lee and Lee discussed several geometrical arrangements of atoms in lattices
and obtained intensities and dispersion relations by numerically solving system (80). It is clear
that MMS served there only as a starting point for the analysis while the main physical results of
excitation energy trapping and single-peaked intensity spectrum [13] were not related to the method.
MMS provided here a system of equations to be solved numerically, which is much more convenient
than the initial system of diMerential equations for the SchrIodinger amplitudes. Investigations of
many-atom spontaneous emission found their continuation in the paper [14] and, in the context of
exciton superradiance, in [15].
System (69) is an inDnite system of, in fact, uncountably many (the sum is to be understood as
an integral if we get rid of the quantization box) ordinary diMerential equations. It is clear that, if
we know the amplitude a(t), all the amplitudes bk can be obtained just by integration. Therefore,
an interesting question arises: can we replace all the modes bk by just one or a few time-dependent
variables, so that the dynamics of a(t) is unchanged, and all bk can be computed on the Dnal stage
of calculations by integrating the second equation of system (69) with already known a(t). In some
cases, this question has been answered aLrmatively (cf. e.g., [55] as an important example) leading
to the concept of “faked vacuum” [56] or “pseudomodes”, [57–59]. The concept of pseudomodes
turns out to be especially convenient in the theoretical analyses of radiating atoms in cavities, in
photonic crystals, waveguides, or any other systems in which the coupling constant is to a good
approximation not a /at, but rather a well-localized function of frequency. As an example, let us
consider such a coupling constant in a Lorentzian form:
1 √
dk = g! = √
;
(81)
g
(! − !c ) − i
(
where !c is a central frequency of “reservoir” which may consist, for instance, of the electromagnetic
modes of a cavity. The parameter is the rate of damping of the excitations of reservoir (the leakage
rate of, say, the cavity excitations), while g measures the eMective strength of the interactions of the
atom and the reservoir. The total Hamiltonian of the system reads
Ĥ = Ĥ A + Ĥ F + Ĥ AF ;
(82)
where the Hamiltonian of a two-level atom Ĥ A is given by
Ĥ A = ˝!0 .̂22
(83)
and ˝!0 is the energy gap between two atomic levels. The Deld Hamiltonian is given by
∞
d! â†! â! ;
Ĥ F = ˝
(84)
while the interaction Hamiltonian is written in the form
∞
Ĥ FA = ˝
d!(g! â†! .̂12 + g!? .̂21 â! ) ;
(85)
0
0
where .̂ij = |ij| are the atomic raising and lowering operators, while â! and â†! are annihilation
and creation operators of the Deld. We have clearly ignored the dependence of the above operators
on the k vector; this is justiDed by the mode structure of the Delds in cavity, as well as by the fact
that we are not interested here in any spatial relations.
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
Dividing the total Hamiltonian into two parts, Ĥ = Ĥ 0 + Ĥ 1 with
∞
†
d! â! â! ;
Ĥ 0 = ˝!0 .̂22 +
0
(86)
we write the total time evolution operator as the product:
Û = exp((−i=˝)Ĥ 0 t)Û I ;
(87)
where the interaction-picture time-development operator satisDes the equation
i˝
d Û I
= Ĥ I Û I :
dt
(88)
The interaction-picture Hamiltonian HI is given by
∞
∞
†
†
?
d! <! â! â! +
d![g! â! .̂12 + g! .̂21 â! ] ;
Ĥ I = ˝
0
0
(89)
where <! = ! − !0 . As before, the subspace of the Hilbert space allowed by the model is very
restricted, the wave function is assumed to have the form
∞
d! =! (t)|1; 1! ;
(90)
|6(t) = $(t)|2; 0 +
0
where |2; 0 is the state with the excited atom and no excitation of the Deld, and |1; 1! is a state
with the atom deexcited and one quantum present in the Deld. Writing the SchrIodinger equation and
eliminating the amplitudes =! from the resulting system, we obtain the simple integro-diMerential
equation for $:
t
$̇ = −
G(t; t )$(t ) dt (91)
0
with the kernel G(t; t ) given by
∞
d!|g! |2 exp[ − i<! (t − t )] :
G(t; t ) =
0
(92)
The integral can obviously be performed exactly, but to simplify its form one can observe that the
lower limit of integration can be set approximately equal to −∞ because the two frequencies !0
and !c are assumed to be approximately equal and very large, that is, much larger than both and
g! . From now on we shall assume the perfect resonance between the atom and the cavity Deld,
!0 = !c . Then the result of integration in (92) is
G(t; t ) = g exp(−|t − t |) :
Now, it is a very simple matter to Dnd out the appropriate pseudomode. We immediately verify that
the system
√
$̇ = −i gB ;
√
Ḃ = −B − i g$
(93)
M. Janowicz / Physics Reports 375 (2003) 327 – 410
361
gives, upon the elimination of B, the same integro-diMerential equation for $ as the system containing
all amplitudes b! . The variable B is thus the required pseudomode. The solution of Eq. (93) is
evidently trivial. But let us now suppose that the atomic energy is not constant. That is, the atom is
in/uenced by some external low-frequency laser Deld which serves to modulate !0 . A practical setup
to perform such modulation is described in [60]. We can take into account the eMect of modulation
by the replacement of !0 by !0 (1 − j sin ?t) in the atomic Hamiltonian. In the following we shall
use the parameter f = j!0 (this is the modulation depth which is much smaller than !0 but may or
may not be much smaller—or much larger—than ?). In this case of a modulated atomic frequency,
we can work again in the (slightly modiDed) interaction picture to obtain the following diMerential
equations for the SchrIodinger amplitude $ and pseudomode B
√
f
$̇ = −i g exp i cos(?t) B
(94)
?
and
f
Ḃ = −B − i g exp −i cos(?t) $ :
?
√
(95)
This system is somewhat more complicated due to the time-dependent coeLcients and has been
analyzed numerically in [61]. Quite naturally, a Floquet-type analysis could be performed, but we
can apply here a simple version of MMS to strengthen intuitive reasoning. In the following we shall
need the simple equality:
exp[ ± ia cos(?t)] = J0 (a) + 2
∞
(±i)n Jn (a)cos(n?t) ;
(96)
n=1
where a = f=? and Jn (x) are the Bessel functions of an integer order n and argument x.
√
We shall assume that g is much smaller than either ? or d. We deDne new dimensionless time
√
variable , = ?t, and the small parameter = g=?. Then Eqs. (94) and (95) take the form
d$
f
= −i exp i cos() B ;
d
?
f
dB
= −
B − i exp −i cos() $ :
(97)
d
g
?
There is a family of various non-trivial relations between parameters which can easily be explored.
We shall restrict ourselves, however, to two cases only. Let us Drst take the limit of the weak
coupling, g. More speciDcally, let us assume that =(1=2 )g21 and the (dimensionless) parameter
1 is of the order of 1. Then we have
d$
f
= −i exp i cos() B ;
dt
?
f
dB
= −1 B − i exp −i cos() $ :
d
?
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
Although
the example is almost too simple to apply MMS, let us still do this. We write Tn = n ,
$ = n=0 $n (T0 ; T1 ; : : :), and B = n=0 Bn (T0 ; T1 ; : : :). From the zeroth-order equations in we obtain
$0 = $0 (T1 ; T2 ; : : :) ;
B0 = e−1 T0 B00 (T1 ; T2 ; : : :) :
In the Drst order the secular terms cannot appear, so it turns out that $00 and B00 do not depend on
T1 , while $1 and B1 are given by the following expressions:
inT0
∞
J0 (a) n+1
e
e−inT0
e−1 T0 B00 (T2 ; : : :) ;
−
i Jn (a)
+
$1 = $10 (T1 ; T2 ; : : :) + i
1
in
−
−in
−
1
1
n=1
and
J0 (a)
$00 (T2 ; : : :)
1
inT0
∞
e
−inT0
n+1
+
$00 (T2 ; : : :) :
(−i) Jn (a)
+
in + 1 −in + 1
n=1
B1 = B10 (T1 ; T2 ; : : :)e−1 T0 − i
In the second order, however, we obtain
9$2
9$1
9$00
J0 (a)
+
+
= −ieia cos T0 B10 (T1 ; T2 ; : : :)e−1 T0 − i
$00 (T2 ; : : :) ;
9T0 9T1
9T2
1
and this means that the secular terms will appear unless we require that
∞
−J02 (a) Jn2 (a)
9$00
$00 ;
=
−
9T2
1
n
−
i
1
n=1
which means that we have got an eMective decay constant
∞
J 2 (a) nJn2 (a)
A2 = 0
+
;
2 + 2
1
n
1
n=1
so the probability that the excited state is still populated at time t is approximately equal to
Pe (t) = |$00 |2 ≈ e−2A2 T2 = e−2(gA2 =?)t :
By choosing an appropriate a = f=? we can to some extent manipulate the rate of the decay of the
excited state. However, we cannot “turn oM” the spontaneous emission this way since ? and are
of the same order and driving Deld cannot detune the atom from the reservoir.
As the second case we shall consider the medium-strength-coupling with =g21 , where, as before,
1 is of the order of 1, so that g and are of the same order. Then, we have the following diMerential
equations for $ and B:
d
$ = −ieia cos B ;
d
dB
= −1 B − ie−ia cos $ :
d
M. Janowicz / Physics Reports 375 (2003) 327 – 410
363
Now, using MMS again, we Dnd that in the zeroth order $0 and B0 do not depend on T0 , while in
the Drst order the secular terms vanish provided that $0 and B0 satisfy
9$0
= −iJ0 (a)B0 ;
9T1
9B0
= −1 B0 − iJ0 (a)$0 :
9T1
From the above formulae it is clear that the quantity J0 (a) plays the role of an eMective coupling
constant between the atom and the pseudomode, and that it is possible to change the dynamics
of the spontaneous emission in quite a spectacular way by changing the modulation depth f. In
particular, we may choose f and ? in such a way that J0 (a) = 0. This way, by driving the atom
√
in and out of resonance with the reservoir (large ?, ? g) as well as imposing the resonance
condition J0 (a) = 0 we can suppress the spontaneous emission. Obviously, such a suppression of
spontaneous decay may happen only in the case of a reservoir with a well-localized spectrum,
Suppression of spontaneous emission by modulation
|α|, for a = 2.40483
|α|, for a = 0.0
|B|, for a = 2.40483
|B|, for a = 0.0
1.2
1
|α|, |B|
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
τ
30
35
40
45
50
Fig. 13. Suppression of spontaneous emission of an atom coupled to a Lorentzian reservoir. Time dependence of the
moduli of amplitudes |$()| and |B()| is shown for a = 2:40483 and 0. The other parameters are 1 = 2:0, = 0:1.
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
Suppression of population exchange by modulation
|α1|, for a = 2.40483
|α2|, for a = 2.40483
|α1|, for a = 0.0
|α2|, for a = 0.0
1.2
1
|α1|, |α2|
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
τ
30
35
40
45
50
Fig. 14. Suppression of energy exchange between two atoms coupled to a Lorentzian reservoir. Time dependence of |$1 ()|
and |$2 ()| is shown for a = 2:40483 and 0. The other parameters are 1 = 2:0, = 0:1.
√
otherwise the condition g? is not fulDlled. The above conclusions remain true in the case of
the strong-coupling limit g.
The suppression of spontaneous emission in a system with modulation is illustrated in Fig. 13,
where we have plotted the dependence of the square root of population of the excited state on time
as well as the time dependence of the pseudomode amplitude B for diMerent values of a and for
= 0:1, 1 = 2: From Fig. 13 it is clear that the suppression of the decay of |$| is very strong even
for quite large values of the small parameter .
Let us also address the problem of interaction of two spontaneously decaying atoms coupled to the
Lorentzian reservoir and such that they are driven by an external Deld in the same way as above.
(2)
This means that the interaction Hamiltonian contains the term .̂(1)
ij + .̂ij instead of .̂ij with the
obvious meaning of the superscripts (we still omit any spatial dependencies), and the wave function
is given by
∞
|6(t) = $1 (t)|2; 1; 0 + $2 (t)|1; 2; 0 +
=! (t)|1; 1; 1! 0
M. Janowicz / Physics Reports 375 (2003) 327 – 410
365
which means that our Hilbert space consists of the states with either the Drst or the second atom
excited and no Deld excitation, and the states with both atoms in the ground state and one photon
in the mode ! present. The equations of motion for the amplitudes $1 and $2 read
√
$̇1 = −i geia cos ?t B ;
√
$̇2 = −i geia cos ?t B ;
√
Ḃ = −B − i ge−ia cos ?t ($1 + $2 ) ;
a rather simple generalization of Eqs. (94) and (95). In the case of medium-strength and strong
coupling we immediately infer from the Drst-order of MMS that not only the emission can be
√
suppressed if ? g and J0 (a) = 0, but also the energy exchange between the two atoms will be
very slow, because they are coupled via the pseudomode only.
Indeed, simple numerical calculations conDrm this result, as is seen in Fig. 14, where |$1 | and |$2 |
are plotted versus for a = 0 and for a = 2:40483. The initial values are $1 (0) = 1, $2 (0) = 0:001,
B = 0. This interesting example of resonance suppression of interaction Dnishes our section devoted
to the spontaneous emission.
4. Resonance uorescence
In this section we consider our Drst examples from the circle of problems associated with coherent
excitation and dynamics. In fact we shall only touch several important topics connected with the
interaction of atomic matter with the electromagnetic Deld under the assumption that atoms can
be approximated by just few levels while the electromagnetic Deld is classical. What is more, the
in/uence of atoms on the Deld itself is assumed to be negligible; the latter assumption allows the
Deld to be identical with “applied” or “external” one. The only dynamical variables of the system are
atomic variables. On the other hand, the variation of Delds in time will be considered as prescribed.
In addition, in all but one example it is assumed that the frequency or frequencies of the applied
Deld are close to resonance with one or more of the characteristic frequencies of atomic transitions.
At Drst the damping will be ignored, but it will appear in latter stages of computations to make the
models more realistic.
The major source of references containing a huge amount of information about methods and results
in coherent atomic excitations is the monumental treatise by Shore [62]. Among other valuable
references one should mention [51,63].
After some very brief description of Drst important papers on the application of MMS to the dynamics of coherently excited atoms, we present basic ideas associated with the correlation-function
approach. Then we derive an eMective Hamiltonian describing a two-level atom under the in/uence
of a strongly non-resonant and weak external Deld. It is shown that such a Deld produces eMective modulation of atomic frequency. This way we provide a justiDcation for one of the eMective
Hamiltonians used in the previous section. Then we consider the problem of the atomic population
trapping in a three-level cascade system. The last subsection is devoted to the application of MMS
in the case of bichromatic excitation.
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
4.1. General considerations
The method of multiple scales has been Drst applied to the problem of coherent atomic excitations
by Wong et al. [17–19]. In the Drst paper of the series, Wong et al. have developed their general
theory and applied it to two test cases: the Bloch–Siegert shift and the two-photon resonance excitation of a molecule. An interesting and important feature of the approach of these authors is that
they have developed a technique to avoid the problem of small denominators (cf., e.g., [64]) in all
required orders, and not just to elimination of secular terms. They have worked in the SchrIodinger
picture, but a realization of their technique in terms of the Heisenberg equations is possible as
well. We shall not attempt to show somewhat involved details of their computations, but to apply
techniques similar to theirs in some speciDc examples.
In their second work, Garrison et al. have presented their multiple scales analysis of the Doppler-free
population inversion induced by a two-photon resonance. Both these works contain comparisons of
MMS predictions with numerical results. The paper [19] is devoted to the problem of a two-photon
resonance coupled to a one photon resonance through a common energy level.
In one of the remaining parts of this section we shall use MMS to study multichromatic excitations
of two- and three-level atoms using MMS in the setting close to that oMered in [17]. We shall,
however, use the Heisenberg equations of motion in order to include losses. The Heisenberg equations
of motion lead naturally to study the evolution of correlation functions in time. Our exposition of
the basic equations for the correlation-function dynamics will be patterned after that of [65].
To derive the Heisenberg equation of motion, we write the two-level atomic Hamiltonian in one
of the forms shown in the previous section:
Ĥ A = 12 ˝!0 .̂z :
(98)
The Hamiltonian describing interactions with the external electromagnetic Deld is chosen to be
Ĥ AF = −d̂ · E(t) ;
(99)
where, as in the previous section, E(t) denotes the time-dependent (classical) electric Deld, and
d̂ is the atomic dipole moment operator. For two-level atoms the latter operator has the following
representation:
d̂ =
2
dij .̂ij ;
(100)
i; j=1
where dij = i|d̂|j, and for the atoms not exposed to static electric or magnetic Delds the diagonal
elements are equal to zero. One has to notice, however, that in order to account for scattering and
damping phenomena, the atom has to be coupled to a reservoir. This can be modeled in many
diMerent ways. For instance, one can start with the full Hamiltonian of the form
Ĥ T = Ĥ A + Ĥ AF + Ĥ R + Ĥ AR ;
(101)
where Ĥ R denotes the Hamiltonian of the reservoir and Ĥ AR the coupling between the atom and the
reservoir. The Heisenberg equations of motion are then solved partially by “adiabatic elimination”
of all the degrees of freedom of the reservoir (in principle this can be done with the help of
MMS as well), as shown, e.g., in [41]. The remaining equations of motion contain non-Hamiltonian
M. Janowicz / Physics Reports 375 (2003) 327 – 410
367
terms—damping constants T and L , and quantum-mechanical noise operators B̂+ (t), B̂− (t), and
B̂z (t), about which we shall assume that they are correlated to the delta function and that their mean
values vanish. Taking into account the form of the Hamiltonian and the above simple considerations
concerning damping, we obtain
C1 (t)
.̂z + B̂− (t) ;
.̂˙− = −(T + i!).̂− − i
2
C1 (t)
.̂z + B̂+ (t) ;
.̂˙+ = −(T − i!).̂+ + i
2
.̂˙z = −L .̂z + iC1 (t)(.̂+ − .̂− ) − L + B̂z (t) ;
(102)
where C1 (t) = 2d12 · E(t)=˝, and we have written .̂− instead of .̂12 as well as .̂+ instead of .̂21 .
We shall now perform a transformation to a rotating frame, rotating with a frequency !L close to
one of the characteristic frequencies of the Deld E(t). The transformation is deDned by the unitary
operator
−i
Û (t) = exp
(!L t − DL ).̂z ;
2
so that the Hamiltonian of the system atom plus Deld reads
Ĥ A + Ĥ F = 12 \.̂z − d̂ · E(t)[ei(!L t −DL ) .̂+ + e−i(!L t −DL ) .̂− ] ;
(103)
where < = !0 − !L . As a result, the raising and lowering operators transform as
.̂− → .̂− e−i(!L t −DL ) ;
.̂+ → .̂+ ei(!L t −DL ) ;
and Eq. (102) takes the form
C(t)
.̂z + B̂− (t) ;
.̂˙− = −(T + i<).̂− − i
2
C? (t)
.̂z + B̂+ (t) ;
.̂˙+ = −(T − i<).̂+ + i
2
.̂˙z = −L .̂z + i(C(t).̂+ − C? (t).̂− ) − L + B̂z (t) ;
(104)
where C(t) = C1 (t) exp[i(!L t − DL )]. From this system of equations we obtain two other systems for
the expectation values and for the correlation functions which will play the key role in the following
subsections. By taking the expectation values of Eqs. (104) we get
.̂˙− = −(T + i<).̂− − i
C(t)
.̂z ;
2
.̂˙+ = −(T − i<).̂+ + i
C(t)
.̂z ;
2
.̂˙z = −L .̂z + iC(t)(.̂+ − .̂− ) − L :
(105)
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
On the other hand, deDning /uctuations of the operators .̂± and .̂z by the formulae
.̂± = .̂± − .̂± ;
.̂z = .̂z − .̂z ;
and three correlation functions c+ , c− , and cz :
c+ (t; t ) = .̂+ (t).̂− (t ) ;
c− (t; t ) = .̂− (t).̂− (t ) ;
cz (t; t ) = .̂z (t).̂− (t ) ;
(106)
we obtain the following system for the latter:
d
C(t)
c− (t; t ) = −(T + i<)c− (t; t ) − i
cz (t; t );
dt
2
C? (t)
d
c+ (t; t ) = −(T − i<)c+ (t; t ) + i
cz (t; t );
dt
2
d
cz (t; t ) = −L cz (t; t ) − iC? (t)c− (t; t ) + iC(t)c+ (t; t ) ;
dt
(107)
where we have used the assumption about B̂ operators speciDed above, as well as the assumption
that .̂±; z (t ) and B̂±; z (t) are uncorrelated for t 6 t. This assumption leads to results equivalent to
those obtained using the so-called Quantum Regression Hypothesis.
Eqs. (107) are subject to the following initial conditions which follow from the Pauli matrix
algebra:
c+ (t ; t ) = 12 [1 + .z (t )] − |.+ (t )|2 ;
c− (t ; t ) = −.− (t )2 ;
cz (t ; t ) = −[1 + .z (t )].− (t ) :
(108)
In the case of excitation by just one external resonant Deld, the above equations (as well as
their density-matrix-elements counterparts) have led to the theoretical predictions [66–69] of the
three-peaked spectra of resonance /uorescence (called Mollow triplets), the triumphant experimental
vindication [43] of which has been one of the major achievements of the heroic period of the physics
of laser-atom interactions.
4.2. Modulation of atomic frequency
We begin with an MMS description of the dynamics of a two-level atom coupled to a Deld with
the frequency much smaller than the characteristic frequency of atomic transitions. The losses
will not be taken into account and we shall concentrate on the eMective time evolution operator. We
will show that a simple eMective Hamiltonian emerges in a natural way in this system.
M. Janowicz / Physics Reports 375 (2003) 327 – 410
369
We start with the energy operator
Ĥ = 12 ˝!0 .̂z − 2d̂ · E0 cos(!t) ;
(109)
and assume that both ! and ?, the latter deDned by
?=
2d12 · E0
;
˝
(110)
are much smaller than !0 . We shall deDne our small parameter as j = ?=!0 and write ! = jE!0 ,
where E is a dimensionless parameter and E=O(1). Introduction of a new dimensionless time variable
= !0 t leads to the following SchrIodinger equation for the time-evolution operator:
1
d Û
=
.̂z − j cos(jE).̂x Û ;
(111)
i
d
2
where .̂x = .̂+ + .̂− . We now introduce new time variables deDned as
T0 = ;
T1 = j;
T2 = j2 (j; ) ;
for it will shortly become evident that the simplest choice of Tn =jn does not work in this example.
From the above equation it follows that
9
9 (j; ) 9
9
9
=
+j
+ j2
:
9 9T0
9T1
9
9T2
The expansion of the time-evolution operator is standard:
U = U0 + jU1 + j2 U2 + · · · :
In the zeroth-order approximation we Dnd immediately
−i
.z T0 V̂ (T1 ; T2 ; : : :) ;
Û 0 = exp
2
(112)
which naturally can be written as a linear combination of .̂z and .̂0 = 1. In the Drst order we get
i
where
9V̂
9UŴ 1
+i
= −cos(ET1 )M̂ (T0 )V̂ (T1 ; : : :) ;
9T0
9T1
i
.̂z T0 U1
UŴ 1 = exp
2
and the matrix M̂ (T0 ), equal to
0
eiT0
;
M̂ (T0 ) =
0
e−iT0
(113)
(114)
(115)
can be written as a linear combination of the matrices .̂x and .̂y . We can see from Eq. (113) that
no secular terms can appear in the Drst order, hence (9V̂ )=(9T1 ) = 0, and we may simply integrate
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
over T0 to obtain
UŴ 1 = cos(ET1 )M̂ 1 (T0 ) ;
where
M̂ 1 (T0 ) =
0
ei T 0
−e−iT0
0
(116)
V̂ (T2 ; : : :) + X̂ (T1 ; T2 ; : : :) :
(117)
Writing
Û 2 = e−(i=2).̂z T0 UŴ 2 ;
we get in the second order
i
9UŴ 1
9 9V̂
9UŴ 2
+i
+i
= −cos2 (ET1 )M̂ (T0 )M̂ 1 (T0 )V̂ − cos(ET1 )M̂ (T0 )X̂ :
9T0
9T1
9 9T2
(118)
Since, however,
M̂ (T0 ) · M̂ 1 (T0 ) = −.̂z ;
the Drst term on the right-hand side of (118) is equal to
cos2 (ET1 ).̂z V̂ = 12 (1 + cos(2ET1 ))V̂ ;
and, in order to avoid secular terms, we must require that V̂ fulDlls the condition
i
1
1
9 9V̂
= [1 + cos(2ET1 )].̂z V̂ = [1 + cos(2jE)].̂z V̂ :
9 9T2 2
2
(119)
On the other hand, X̂ is independent of T1 . We would be unable to integrate (119) without the
presence of on the left-hand side. But as it is present, we simply write
9
1
= [1 + cos(2jE)] ;
9
2
so that
1
1
+
sin(jE) ;
(j; ) =
2
2jE
(120)
while
V̂ (T2 ; : : :) = e−i.̂z T2 :
It follows that, up to the second order, Û 0 can be written as
1
j
2
[1 + j ] +
sin(2jE) :
U0 = exp −i.̂z
2
2E
(121)
(122)
By recalling the deDnition of , we can recast Eq. (122) into the form
Û eM ≡ Û 0 = exp[ − iĤ eM (t)=˝] ;
(123)
M. Janowicz / Physics Reports 375 (2003) 327 – 410
where the eMective Hamiltonian HeM reads
1
?2
?2
!0 +
+
sin(2!t) .̂z :
HeM = ˝
2
!0 2!0
371
(124)
This means that slowly varying and very strongly detuned external Delds mainly give rise to the
modulation of the atomic frequency, which is a very intuitive result, for such a Deld should not
induce any transitions—that is, no changes in the population inversion .z should take place.
4.3. Population trapping in cascade systems
In this section we shall use MMS and results of the previous subsection to obtain population
trapping in the two-photon system analogous to that considered by Agarwal and Harshawardhan
[77]. It consists of a three-level atom with the levels |1, |2, |3, coupled to two external Delds.
The Drst one couples only the levels |1 and |2, and, having very small frequency, leads just to
the modulation of the frequency of the atomic transition |1 → |2. On the other hand, the second
Deld is in a two-photon resonance with the levels |1 and |3, that is, the Deld frequency is equal
to the energy separation of the two latter levels (divided by ˝). However, it is assumed that the
matrix elements of the dipole moment of the form 1|d̂|3 vanish, while 1|d̂|2 as well as 2|d̂|3
are diMerent from zero. Such a ladder system with the energies E3 ¿ E2 ¿ E1 is sometimes called a
I system. In the following we shall choose zero of the energy such that E1 = 0 and write E2 = ˝!21 ,
E3 = ˝!31 . The Hamiltonian which we begin with is thus
Ĥ = ˝!21 .̂22 + ˝j!ˆ 21 (.̂22 − .̂11 ) · sin(!m t) + ˝!31 .̂33
−2d12 · E0 cos(!L t)(.̂12 + .̂21 ) − 2d12 · E0 cos(!L t)(.̂23 + .̂32 ) ;
(125)
where !m is the modulation frequency, corresponding to ! from the previous subsection. To get rid
of unwanted terms in the above Hamiltonian, we perform the unitary transformation generated by
Ĥ 0 = ˝(!L .̂22 + 2!L .̂33 + j(.̂22 − .̂11 )sin !m t) :
The transformed Hamiltonian reads
Ĥ = ˝\.̂22 − 2d12 · E0 cos !L t (.̂12 e−i!L t+2ir(t) + .̂21 ei!L t −2ir(t) )
−2d23 · E0 cos !L t (.̂23 ei!L t −ir(t) + .̂32 e−i!L t+2ir(t) ) ;
(126)
where < = !21 − !L and
r(t) =
j!21
(cos !m t − 1) :
!m
We shall now perform the rotating wave approximation, disregarding all terms in (126) which
oscillate with the frequency ∼ 2!L . These terms provide small corrections of the Bloch–Siegert-shift
type, analyzed from the point of view of MMS in [17]. On neglecting these fast oscillating terms,
we obtain the following simpler Hamiltonian which is already suitable for a simple multiple scales
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
analysis:
Ĥ = ˝\.̂22 − 12 ˝?12 (.̂12 e2ir(t) + .̂21 e−2ir(t) ) − 12 ˝?23 (.̂23 e−ir(t) + .̂32 eir(t) ) ;
(127)
where ?ij = 2dij · E0 =˝. We assume that ?ij <, !m <. On using the new symbols Jij and Jm
such that ?ij = j\Jij and !m = j\Jm we may write the following SchrIodinger equation for the
time-evolution operator (the dynamics of which is generated by Ĥ ) in time = \t:
d Û
= ĥ0 + jĥ1 ;
d
(128)
where ĥ0 = .̂22 and
h1 = − 12 (J12 [e2ir() .̂12 + h:c:] + J23 [e−ir() .̂23 + h:c:]) :
We deDne the diMerent “times” in the standard way, Tn = jn , n = 0; 1; 2; : : :, and obtain in the zeroth
order
Û 0 = exp[ − i.̂22 T0 ]V̂ (T1 ; T2 ; : : :) :
(129)
In the Drst order, upon using a deDnition which has already become standard
Û 1 = exp(−i.̂22 T0 )UŴ 1 (T0 ; T1 ; : : :) ;
we Dnd that V̂ does not depend on T1 and that UŴ 1 is given by
UŴ 1 = B̂(T0 ; T1 ; : : :) + X̂ (T1 ; T2 ; : : :) ;
where
1
B̂(T0 ; T1 ; : : :) = [J12 (.̂21 e−2ir(T1 ) eiT0 − h:c:) + J23 (.̂23 e−ir(T1 ) e−iT0 − h:c:)]V̂ (T2 ) :
2
In the derivation of the last equation we have used the equalities
ei.̂22 T0 .̂12 e−i.̂22 T0 = .̂12 e−iT0 ;
ei.̂22 T0 .̂23 e−i.̂22 T0 = .̂23 eiT0 ;
as well as analogous formulae for .̂21 and .̂32 . The variable r() has been written as r(T1 ) since
we have !m t = j\Jm =< = jJm = Jm T1 .
ˆ 0 ; T1 ; : : :) such that
To proceed, let us also introduce a new operator A(T
ˆ 0 ; T1 ; : : :) = ei.̂22 T0 ĥ1 e−i.̂22 T0 :
A(T
Then in the second order of the perturbation expansion we Dnd
i
9UŴ 2
9UŴ 1
9V̂
ˆ 0 ; T1 ; : : :)(X̂ (T1 ; : : :) + B̂(T0 ; T1 ; : : :)V̂ ) :
+i
+i
= A(T
9T0
9T1
9T2
(130)
M. Janowicz / Physics Reports 375 (2003) 327 – 410
373
Using simple operator algebra of the .̂ matrices we realize that the terms on the right-hand side of
(130) that can exhibit secular behavior after integration over T0 are
−1 2
(131)
{J12 (.̂11 − .̂22 ) + J223 (.̂33 − .̂22 ) + J12 J23 (.̂13 eir(T1 ) + .̂31 e−ir(T1 ) )}V̂ (T2 ; : : :) :
4
The physical meaning of these terms is quite clear. The diagonal ones in .̂ij represent the energy
shift of the three involved levels proportional to the intensity of the applied Deld. More importantly,
there are terms representing the eMective two-photon coupling between the levels |1 and |3. This
coupling is, however, modulated. Using Eq. (96) we may extract the terms in Eq. (131) which
do not contain T1 explicitly, and write
9V̂
!21
−1 2
2
{J (.̂11 − .̂22 ) + J23 (.̂33 − .̂22 )} + J0
i
J12 J23 (.̂13 + .̂31 )V̂ ;
=
(132)
9T2
4 12
\Jm
while the part of (131) which does depend explicitly on T1 can be absorbed into (9X̂ )=(9T1 ). The
expression multiplying V̂ on the right-hand side of Eq. (132) can be considered as an eMective
Hamiltonian generating dynamics of a two-photon transition in the cascade system we consider. If
it happens, however, that
!21
=0 ;
J0
\Jm
the coupling between levels |1 and |3 will eMectively be equal to zero and the initial population
of these levels will be stabilized. Of course, one cannot turn the coupling completely oM, since the
transition operators will be present in X̂ (T1 ). Nevertheless, such a resonant dynamical suppression
of the two-photon oscillations should be observable—both the absorption and resonance /uorescence
lines should be very weak. Suppression of the Rabi oscillations caused by this type of modulation
has been Drst predicted for another system in [77,78]. Several other interesting interference eMects
which can be found in the phase-modulated systems have been analyzed in [79–81].
4.4. Bichromatic excitation
In the previous example we have considered, in fact, excitations of a three-level system by two
applied Delds with very diMerent frequencies. Now, we want to consider the bichromatic excitations
in a stricter sense: the two applied laser Delds will be assumed to have almost equal frequencies
close to the one-photon resonance. Thus, the two-level atom model will suLce for our purposes.
Such a model of bichromatic excitations was studied extensively in the 1980s and 1990s, see, e.g.,
[70–76]. Here we shall concentrate on the case of one Deld being much stronger that the other,
while the frequency diMerence between them will be of the same order as the Rabi frequency of the
weak (probe) Deld. Such a problem has been analyzed in [75]. Our starting point are systems (105)
and (107). The external Deld C(t) is given by
C(t) = 2(?1 cos(!1 t) + ?2 cos(!2 t))ei!L t :
(133)
We can still choose an appropriate !L . For our purposes it is convenient to use !L = !1 , so that the
strong Deld labeled “1” is in the exact “resonance” with the frequency of the rotating system. Let
us denote the diMerence of frequencies of the two excitation Delds by B, B = !1 − !2 . On neglecting
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
the oscillations with very large frequencies ∼ 2!L , we obtain
C(t) = ?1 + ?2 e−iBt :
(134)
We shall assume that ?1 is much larger than all the other parameters of the dimension of frequency.
Naturally, many diMerent relations between the other parameters are possible (and it is possible to
analyze the other cases using MMS). We shall, however, consider only the case
?1 ?2 ∼ B ∼ < ∼ ;
where we have assumed for simplicity that L = T .
To begin with, we shall address the problem of the dynamics of the expectation values. As in all
considerations before, we introduce the dimensionless time variable such that = ?1 t. Let us also
denote the vector made of the expectation values of the atomic operators .̂± , .̂z by S, that is
  

s−
.̂−   

  
(135)
S=
 .̂+  =  s+  :
sz
.̂z We shall look for the solution for S in the form of the perturbation series:
S = S0 + jS1 + j2 S2 ;
where, say, j = max(<=?1 ; B=?1 ; ?2 =?1 ; =?1 ). System (105) takes the form
d
S = (M0 + jM1 )S + jA ;
d
where the matrices M̂ 0 , M̂ 1 are given by


0 0 − 2i


i ;
M̂ 0 = 
 0 0
2 
−i i
0


−(E + i)
0
− 2i J2 e−ij8


ij8 
i
M̂ 1 = 
0
−(E
−
i)
J
e
2
2

;
−iJ2 eij8
iJ2 eij8
−E
(136)
(137)
and new parameters are deDned by the scalings
< = j?1 ;
?2 = j?1 J2 ;
B = j?1 8 ;
= j?1 E :
(138)
The vector A is equal to (0; 0; −E)T . Clearly, our system is such that MMS can be useful, so we
deDne the new time scales Tn = jn , n = 0; 1; 2; : : :, and obtain in the zeroth order
9S0
= M̂ 0 S0 :
(139)
9T0
M. Janowicz / Physics Reports 375 (2003) 327 – 410
375
The matrix M̂ 0 has the eigenvalues i, −i and 0. Let us denote the system of right eigenvectors
of this matrix by vj , while uj will form the system of its left eigenvectors. The index j = 1 will
correspond to the eigenvalue i, j = 2—to eigenvalue −i, and j = 3—to eigenvalue 0. We have the
biorthogonality relation
uj · vk = jk :
(140)
The solution to (139) is given by
S0 =
3
Aj (T1 ; T2 ; : : :)ej T0 vj :
(141)
j=1
The explicit form of the normalized eigenvectors is given by
v1 = 12 (−1; 1; 2)T ;
v2 = 12 (1; −1; 2)T ;
v3 =
(1; 1; 0)T ;
√1
2
and
u1 = 12 (−1; 1; 1) ;
u2 = 12 (1; −1; 1) ;
u3 =
√1
2
(1; 1; 0) :
In the Drst order we have
9S1 9S0
+
= M̂ 0 S1 + M̂ 1 S0 − A ;
9T0
9T1
(142)
and it seems quite natural to represent S1 in terms of the eigenvectors of M0 :
S1 =
3
Bj (T0 ; T1 ; T2 ; : : :)ej T0 vj :
(143)
j=1
Substitution of (143) into (142) gives
3
9Bj
9Aj
+
= e− j T0
Ak (uj M̂ 1 vk )ek T0 − uj · A ;
9T0 9T1
(144)
k=1
where we have used the fact that vk are eigenvectors of M0 . Computation of all products uj M̂ 1 vk is
straightforward and gives the following equations for Bj :
A3
E
9B1 9A1
−iT0
i T0
;
(145)
A1 (iJ2 cos(8T1 ) − E)e + √ (i + J2 sin(8T1 )) −
+
=e
9T0
9T1
2
2
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
A3
E
9B2 9A2
i T0
i T0
;
A2 (−iJ2 cos(8T1 ) − E)e + √ (−i + J2 sin(8T1 )) −
+
=e
9T0
9T1
2
2
(146)
9B3 9A3
A1
A2
+
= √ (i − J2 sin(8T1 ))eiT0 + √ (−i − J2 sin(8T1 ))e−iT0 − EA3 :
9T0
9T1
2
2
(147)
We infer that the following equations for Aj should hold:
9A1
= (iJ2 cos(8T1 ) − E)A1 ;
9T1
9A2
= (−iJ2 cos(8T1 ) − E)A2 ;
9T1
9A3
= −EA3 ;
9T1
(148)
from which we obtain the Drst-order solutions for Aj :
A1 (T1 ; : : :) = $1 (T2 ; : : :) exp[i(J2 =8) sin(8T1 ) − ET1 ] ;
A2 (T1 ; : : :) = $2 (T2 ; : : :) exp[ − i(J2 =8) sin(8T1 ) − ET1 ] ;
A3 (T1 ; : : :) = $3 (T3 ; : : :) exp[ − ET1 ] :
(149)
From Eqs. (145)–(147) and (149) it follows that Bj are given by
iE
−iT0 A3
√ (− + iJ2 sin(8T1 )) −
;
B1 = C1 (T1 ; : : :) + e
2
2
iE
iT0 A3
√ (− − iJ2 sin(8T1 )) +
B2 = C2 (T1 ; : : :) + e
;
2
2
1
B3 = C3 (T1 ; : : :) + √ [eiT0 A1 ( + iJ2 sin(8T1 )) + e−iT0 A2 ( − iJ2 sin(8T1 ))] :
2
(150)
(151)
These results Dnish the Drst-order computations. In the second order we have
9S2 9S1 9S0
+
+
= M̂ 0 S2 + M̂ 1 S1 :
9T0
9T1
9T2
(152)
Again, we represent the solution for S2 as an expansion in terms of the right eigenvectors of the
matrix M̂ 0 :
S2 =
3
Dk (T0 ; : : :)vk ek T0 ;
k=1
to obtain
3
9Dj (T0 ; : : :) 9Bj (T0 ; : : :) 9Aj (T1 ; : : :) +
+
=
Bk (T0 ; : : :)uj M̂ 1 · vk e(k −j )T0 :
9T1
9T2
9T0
k=1
(153)
M. Janowicz / Physics Reports 375 (2003) 327 – 410
377
Performing the multiplication of the matrix N̂ , N̂ jk = uj M̂ 1 vk by the vector with entries Bk we
convince ourselves that in order to avoid secular terms in the second order the following equations
must hold:
9C1 9A1
1
+
= (iJ2 cos(8T1 ) − E)C1 + ( + iJ2 sin(8T1 ))(i + J2 sin(8T1 ))A1 ;
9T1
9T2
2
9C2 9A2
1
+
= (−iJ2 cos(8T1 ) − E)C1 + ( − iJ2 sin(8T1 ))(−i + J2 sin(8T1 ))A2 ;
9T1
9T2
2
9C3 9A3
E
+
= −EC3 + √ :
9T1
9T2
2
(154)
(155)
(156)
Writing
C1; 2 = exp[ ± i(J2 =8) sin(8T1 ) − ET1 ]E1; 2 ;
where the plus sign refers to the index 1, and minus sign to the index 2, we Dnally get
2
J22
+
T2 ;
$1 = $10 exp i
2
4
2
J22
+
T2 ;
$2 = $20 exp −i
2
4
$3 = $30 ;
(157)
and
E1 = F1 (T2 ; : : :) −
i
$1 sin(28T1 ) ;
8
E2 = F2 (T2 ; : : :) +
i
$2 sin(28T1 ) ;
8
E3 = F3 (T2 ; : : :)e−ET1 + √ ;
2
(158)
where $i0 , i = 1; 2; 3 can depend on T3 and higher times. The formal MMS analysis has thus been
accomplished up to the second order except that we do not provide long and not very impressive
solutions for Dj . There is only one qualitatively important feature contained in the Dj , namely, D1
and D2 have terms which guarantee that sz will eventually reach the value −1, and not 0; this is
necessary to make the model self-consistent.
Before addressing the problem of the time-dependent /uorescence spectra let us just observe that,
if the atom is initially in, say, the ground state, the approximate expression for sz is given by
J2
1 2
−ET1
2
(159)
sz = −e
cos T0 + sin(8T1 ) + (2 + J2 )T2 ;
8
4
which can also be rewritten by recalling the deDnition of Tn and as
2
<
?2
?22
1
−t
2
t :
+
sz (t) = −e cos ?1 t +
sin(Bt) +
B
4
? 1 ?1
(160)
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
Besides a small correction to the frequency of the oscillations of inversion sz , we also have the
modulation of frequency due to the sinusoidal term under the cosine function. This means that the
atomic inversion actually oscillates with all the frequencies ?1 ± nB with natural n, where B is the
diMerence between the carrier frequencies of the excitation Delds, and ?1 = ?1 + (1=4)(2<2 =?1 +
?22 =?1 ). It is interesting that the roles of B and < are completely diMerent, the latter providing just a
shift of the frequencies. It might seem at Drst sight that we do not observe any particular resonance
condition involving B and ?2 . But the resonance is in fact present: if we use the identities
∞
cos(z sin s) = J0 (z) + 2
J2n (z) cos(2ns)
n=1
and
sin(z sin s) = 2
∞
J2n+1 (z) sin((2n + 1)s) ;
n=0
we realize that once again there is a kind of resonance if the Rabi frequency of the weak Deld and
the diMerence of the carrier frequencies are such that J0 (?2 =B) = 0. In this case the oscillations with
the frequency ?1 are turned out and the inversion oscillates just with the frequencies ?1 ± nB with
non-zero n.
Under the above initial conditions, we have the following simple zeroth-order solutions for s± :
?2
−t
s± = ∓ie sin ?t +
sin(Bt)t ;
B
where ? = ?1 + <2 =(2?1 ) + ?12 =(4?1 ).
From the expression for s+ we can immediately obtain the coherent part of the time-dependent
spectrum, as it is given by [82]
2
1 T
Scoh (!; T ) = s+ (t) exp[ − i(! − !L )t]
T 0
(cf. [83] for extensive discussion of the time-dependent spectrum). Using the above deDnition
we obtain
2
∞
1 f1 (!; T ) − f2 (!; T )
Jm (a) 2
(161)
Scoh (!; T ) =
;
2T m=−∞
+ (? + mB − (! − !L ))2 where
f1 (!; T ) = g(1 − e−T cos(gT )) ;
f2 (?; T ) = sin(gT ) ;
and
g = ? + mB − (! − !L ) :
It can be seen that, quite naturally, the spectra contain lines centered at ! − !L = ±(? + mB), m
being an integer (positive or negative), and the factor which is responsible for the intensity of each
line contains the Bessel functions of the argument a = ?2 =B. Thus, by choosing a particular value of
M. Janowicz / Physics Reports 375 (2003) 327 – 410
379
a we may get rid of one or more lines, or at least strongly diminish its intensity. This is also true
about the principal sidebands of the Mollow spectrum which can be “turned oM” by choosing such
a that J0 (a) = 0. This oMers an interesting possibility of an experiment with slow (adiabatic) change
of either the Rabi frequency of the weak component, or the detuning B, or both, and monitoring the
spectrum—we should observe appearance and disappearance of certain lines when a crosses zeros
and local maxima of consecutive Bessel functions. Let us Dnally notice that the spectrum itself
oscillates in time, being modulated by harmonics of the detuning.
The expression for incoherent part of the spectrum is somewhat complicated (and will not be
provided here), because of the long asymptotic formula for the correlation function necessary to
compute it, i.e. c+ (t; t ). This correlation function is obviously obtained in the same manner as s+ ,
except that the initial conditions are not that simple. But recalling the fact that c+ (t; t ) must satisfy
almost the same equation as s+ , the initial conditions, and the deDnition of the incoherent part of
the spectrum:
T t
2
Sinc (!; T ) = Re
c+ (t; t ) exp[ − i(! − !L )(t − t )] dt dt ;
T
0
0
we can easily realize that the incoherent part of the spectrum has lines centered at ! − !L at the
distance equal to either zero, or ±(? + mB), or ±(2? + mB) for integer m.
These results, the analogs of which can also be found, e.g., in the case of amplitude- or phasemodulated excitation Delds, Dnish our discussion of applications of MMS in the Deld of coherent
atomic excitations.
5. Atom–#eld interactions in cavities
In this section we shall discuss some examples of application of MMS to the problem of interaction
of few-level atoms with the quantized electromagnetic Deld in microwave cavities. In such systems
the mean number of photons is usually not large enough to neglect the in/uence of the atom on
the Deld. Therefore, one should treat the whole atom–Deld system self-consistently. Nevertheless,
it turns out that striking similarities exist between the atom–Deld systems in cavities, and atomic
system driven by external laser Delds.
Let us provide some numbers related to the considered interactions. In the microwave cavities, the
characteristic frequencies of electromagnetic modes are of the order of 109 –1010 Hz. The interaction
strengths, on the other hand, are of the order of 104 –105 Hz. The cavity losses (due to the interactions
of electromagnetic Deld with anything other than the atom under consideration) can be as small as
100 Hz, but obviously larger losses are also possible.
It is to be noted that careful analysis of systems, examples of which are discussed here, led, in
the 1980s, to the discovery of collapses and revivals of the oscillations in the atomic population
inversion, one of the most important achievements of the quantum theory in that decade, since it
provided one of the most direct and convincing proofs of the quantum, “grain-like” nature of light
[84–88,10,44].
We shall discuss three examples in this section. The Drst one is a two-level atom interacting
with both one mode of the cavity Deld and with a strong microwave external Deld. This system
will be used to show how MMS leads to eMective multi-photon coupling. Then we shall address
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
the problem how the interactions of a two-level atom with one cavity mode are in/uenced by
the presence of a second detuned mode of electromagnetic Deld. The third example involves a
four-level atom interacting with two cavity modes in the “M” conDguration. Again, building up
eMective two-photon Hamiltonians is shown, but in a very diMerent context: one of the possible
realizations of the SchrIodinger-cat states and a system with fully predictable collapses and revivals
are discussed.
5.1. Two-level externally excited atom
Let us consider a two-level atom with the energy gap between its levels equal to ˝!0 , which is
coupled to both the quantized cavity Deld and strong external Deld which can be considered classical.
It will be assumed that the coupling of the atom to all cavity modes except one is inessential (though
it can be studied using MMS).
The Hamiltonian of the system is given by
Ĥ = ˝!0 .̂22 + ˝!↠a − 2d12 · E0 cos(!L t)(.̂12 + .̂21 ) − ˝g(.̂12 + .̂21 )(â + ↠) ;
(162)
where â; ↠are the cavity Deld destruction and creation operators, g represents the coupling of the
Deld with the atom, E0 is the amplitude of the external Deld, and d12 is the dipole moment matrix
element assumed to be real. We perform a unitary transformation of this Hamiltonian using the
following unitary operator:
Û = exp(−i[!L (.̂22 + ↠â)])
and neglect all the terms that oscillate with frequencies ±2!L in the resulting expression. This way
we obtain the Hamiltonian
1
Ĥ = ˝\.̂22 + ˝<0 ↠â − ˝?(.̂12 + .̂21 ) + ˝g(.̂21 â + ↠.̂12 ) ;
2
where < = ! − !L , <0 = !0 − !L , and ? = 2d12 · E0 =˝. The problem is exactly solvable if < = <0 .
Our goal here is to obtain a family of eMective Hamiltonians if the coupling between the atom and
the cavity Deld is much smaller than the Rabi frequency of the external Deld, that is, if g?. As
usual, we start with changing the time variable form t to = ?t, and deDning the small parameter
j = g=?. The SchrIodinger equation for the time evolution operator becomes
1
d Û
†
†
= − (.̂12 + .̂21 ) + j[.̂22 + 0 â â + (.̂21 â + â .̂12 )] Û ;
(163)
d
2
where we have assumed for a moment that < and <0 are of the same order of smallness as g and
we substituted = <=(j?) as well as 0 = <0 =(j?). In the simplest case, the external Deld is just
in exact resonance with the atom, = 0. For simplicity we shall concentrate on this case, but the
case of = 0 presents actually no diLculties. Eq. (163) can be rewritten as
i
d Û
= (ĥ0 + jĥ1 )Û ;
d
(164)
with the obvious meanings of the operators ĥ0 and ĥ1 . We expand the time-evolution operator Û
as Û = Û 0 + jÛ 1 + j2 Û 2 + · · ·, and the time scales are deDned as Tn = jn , n = 0; 1; 2; : : : . In the
M. Janowicz / Physics Reports 375 (2003) 327 – 410
381
zeroth order we get
i
Û 0 = exp (.̂12 + .̂21 )T0 V̂ (T1 ; T2 ; : : :) ;
2
while in the Drst order we must consider
9UŴ 1
9V̂
i
+i
= [0 ↠a + exp(iĥ0 T0 )g(.̂21 â + ↠.̂12 ) exp(−iĥ0 T0 )]V̂ ;
9T0
9T1
with already standard meaning of UŴ 1 . The right-hand side can be computed to obtain
g
[0 a† a + (.12 + .21 )(a + a† )
2
ig
g
cos T0 (.̂21 − .̂21 )(â − ↠) + sin T0 (.̂22 − .̂11 )(â − ↠)]V̂ :
2
2
To avoid any secular terms we must require that
9V̂
g
i
= 0 ↠â + (.̂12 + .̂21 )(â + ↠) V̂ :
9T1
2
+
The solution for V̂ (T1 ) is clearly given by
g
V̂ (T1 ; T2 ; : : :) = exp[ − i(0 ↠â + (.̂12 + .̂21 )(â + ↠))T1 ]Ŵ (T2 ; : : :) ;
2
which we shall also write as
(165)
V̂ = exp[ − iK̂T1 ]Ŵ
(this equality deDnes the operator K̂), while for UŴ 1 we get
ig
g
†
†
sin T0 (.̂21 − .̂12 )(â − â ) − cos T0 (.̂22 − .̂11 )(â − â ) V̂ :
UŴ 1 = X̂ (T1 ; : : :) −
(166)
2
2
It is interesting to note that the Drst-order eMective Hamiltonian which is, up to a constant, equal
to K̂, contains anti-resonant terms of the form of .̂12 â as well as of .̂21 ↠, as if the rotating wave
approximation was not performed. This is absolutely correct because all the time the atom is excited
by the external Deld, and emitting a photon from the ground state means essentially that the atom
transfers photons from the external Deld to the cavity mode.
Let us check how the Deld operators evolve in slow time T1 if we neglect Û 1 . From Eq. (165)
we Dnd immediately
g
g
â(T1 ) = â(0) +
(.̂12 + .̂21 ) e−i0 T1 −
(.̂12 + .̂21 ) ;
20
20
g
g
†
†
â (T1 ) = â (0) +
(.̂12 + .̂21 ) ei0 T1 −
(.̂12 + .̂21 ) ;
(167)
20
20
from which it follows that the photon number operator behaves according to the formula
g2
N̂ (T1 ) = ↠(0)â(0) + 2 (1 − cos(0 T1 ))
20
+
g
(.̂12 + .̂21 )[â(0)(1 − e−i0 T1 ) + ↠(0)(1 − ei0 T1 )] ;
20
(168)
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
which means that, in accordance with our expectations, that the photon number operator oscillates
even if initially there were no photons in the cavity and the atom enters it in the ground state; the
frequency of oscillations is g<0 =?.
In the second order of our perturbation expansion we Dnd that, to avoid secular terms, W (T2 )
must satisfy the following equation:
9Ŵ
g2
g2
g
†
†
(.̂12 + .̂21 ) 2â â + 2 + 1 − (â + â ) Ŵ :
i
=−
9T2
4
0
0
We can see that in the second order the cavity Deld starts to provide additional corrections to the
Rabi frequency “felt” by the atom, which are proportional to the cavity photon number, but there
are also constant factors multiplying .̂12 + .̂21 .
This subsection has provided a short introduction to the techniques used to extract eMective Hamiltonians in Cavity QED. A similar approach will be used in the following considerations where there
are no external Delds, but both the atomic and cavity subsystems are more complicated.
5.2. Two-mode cavity
In this subsection we are interested in the in/uence of the second highly detuned cavity mode on
the standard interaction of the Drst cavity mode with the two-level atom. It is quite evident that in
every cavity there are plenty of out-of-resonance modes and we shall try to provide some insight
into the problem of their signiDcance.
We start with the Hamiltonian of the entire system:
Ĥ = ˝[!0 .̂22 + !1 â†1 â1 + !2 â†2 â2 + 1 (.̂12 + .̂21 )(â1 + â†1 ) + 2 (.̂12 + .̂21 )(â2 + â†2 )] ;
(169)
where the subscripts at â and ↠are introduced to distinguish the two modes. For reasons which
will become clear later we shall assume that the expectation value of the operator N̂ = .̂22 + â†1 a1
does not vanish. We write the time-evolution operator Û = exp[ − (i=˝)Ĥ t] as Û = Ûp Û , where
Ûp = exp{−i[!0 (.̂22 + a†1 a1 + a†2 a2 )]} ;
and assume that the atom and the Drst mode are in the exact resonance, !0 = !1 . The dynamics of
the time-evolution operator Û are generated by the Hamiltonian
Ĥ = ˝[\â†2 â + 1 (â†1 .̂12 + .̂21 â) + 2 (â†2 .̂12 + .̂21 â)] ;
(170)
where the terms oscillating like exp(±2!0 t) have been neglected, and < = !2 − !0 . As always,
we introduce a new dimensionless time equal to the product of the largest frequency—< in our
case—and the physical time t, = \t. Also, we rescale the coupling constants i : i = jgi <,
i = 1; 2. This way we obtain the SchrIodinger equation:
i
d Û
= (ĥ0 + jĥ1 )Û ;
d
where ĥ0 = â†2 â2 and
ĥ1 = g1 (â†1 .̂12 + .̂21 â1 ) + g2 (â†2 .̂12 + .̂21 â) :
M. Janowicz / Physics Reports 375 (2003) 327 – 410
383
Again, we introduce the time variables: Tn =jn ; n=0; 1; 2; : : :, and in the zeroth-order approximation
we obtain
Û 0 = exp(−iâ†2 â2 T0 )V (T1 ; T2 ; : : :) :
As in all our examples with the time-evolution operator, we shall simplify the Drst-order equations
by writing Û 1 = exp(−iĥ0 T0 )UŴ 1 so that the Drst-order equation is given by
i
9UŴ 1
9V̂
+i
= k̂ 1 V̂ ;
9T0
9T1
(171)
where
k̂ 1 = eiĥ0 T0 ĥ1 e−iĥ0 T0 = g1 (a†1 .̂12 + .̂21 â) + g2 (â†2 .̂12 eiT0 + .̂21 âe−iT0 ) :
It is therefore clear that we have to solve the following two equations:
i
9V̂
= k̂ 2 V̂ = g1 (â†1 .̂12 + .̂21 â)V̂
9T1
(172)
i
9UŴ 1
= g2 (.̂21 â2 e−iT0 + â†2 .̂12 eiT0 ) :
9T0
(173)
and
The solution for V̂ is easy to obtain since we essentially just have to write down the time-evolution
operator for the Jaynes–Cummings model with exact resonance. An elegant way to obtain suchan
operator has been shownin [89]. We deDne the operators Ŝ ij such that Ŝ ii = .̂ii and Ŝ 12 = â†1 .̂12 = N̂
as well as Ŝ 21 = .̂21 â= N̂ , and easily check that Ŝ ij have the same algebraic properties as .̂ij .
Therefore, our operator V̂ can be represented as
V̂ (T1 ; T2 ; : : :) = exp[ − ig1 N̂ (S12 + S21 )T1 ]Ŵ (T2 )
= [cos(g1 N̂ T1 ) + i sin(g1 N̂ T1 )(Ŝ 12 + Ŝ 21 )]Ŵ (T2 ) ;
where we have taken advantage of the fact that N̂ commutes with all Ŝ ij . For UŴ 1 we get
UŴ 1 = g2 (.̂21 â2 e−iT0 − â†2 .̂12 eiT0 )V̂ + X̂ (T1 ; : : :) :
In the second order we have
i
9UŴ 2
9UŴ 1
9V̂
+i
+i
= k1 UŴ 1 ;
9T0
9T1
9T2
where UŴ 2 is deDned by Û 2 = exp[ − iĥ0 T0 ]UŴ 2 . A simple algebra leads to the following form of the
right-hand side of the above equation:
g1 (â†1 .̂12 + .̂21 â)X̂ + g22 (.̂11 â†2 â2 − .̂22 â2 â†2 )V̂ + g1 g2 (.̂11 â†1 â2 e−iT0 − .̂22 â1 â†2 eiT0 )V̂ :
Taking into account that the only term in UŴ 1 depending on T1 is X̂ , we obtain that the following
conditions must be fulDlled if we require absence of the secular terms:
i
9Ŷ
9Ŵ
ˆ
ˆ
+i
= eik 2 T1 g22 (.̂11 â†2 â2 − .̂22 â2 â†2 )e−ik 2 T1 Ŵ ;
9T1
9T2
(174)
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
where Ŷ (T1 ; T2 ; : : :) is deDned by the relation
X̂ = exp(−ik̂ 2 T1 )Ŷ :
Simple computation leads to the following expression for the right-hand side of Eq. (174):
g2
R:H:S:(174) = 2 [1 − (2â†2 â2 + 1){cos(2g1 N̂ T1 )(Ŝ 22 − Ŝ 11 )
2
− i sin(2g1 N̂ T1 )(Ŝ 21 − Ŝ 12 )}]Ŵ :
(175)
It follows that Ŵ must contain just a trivial c-number term:
g2
Ŵ = exp −i 2 T2 ;
2
while the expression for Ŷ is somewhat more complicated:
g22
1
†
sin(2g1 N̂ T1 )(Ŝ 22 − Ŝ 11 )
Ŷ (T1 ; : : :) = i (2â2 â2 + 1)
2
2g2 N̂
i
cos(2g1 N̂ T1 )(Ŝ 12 − Ŝ 21 ) Ŵ + Ŷ 1 (T2 ; : : :) :
+
(176)
2g1 N̂
We shall stop our perturbative analysis at this point. Let us summarize what we know about the
time evolution operator. We have Û () = Û 0 () + jÛ 1 () + j2 Û 2 (), and
ˆ
2
Û 0 = e−iĥ0 T0 e−ik 2 T1 e−i=2g2 T2 ;
2
Û 1 = e−iĥ0 T0 e−i=2g2 T2
+e
+
−ikˆ2 T1
2g1
i
N̂
(177)
ˆ
g2 (.̂21 â2 e−iT0 − â†2 .̂12 )e−ik 2 T1
g2
i 2 (2â†2 â2 + 1)
2
cos(2g1
1
2g1
sin(2g1
N̂
N̂ T1 )(S21 − S12 )
N̂ T1 )(Ŝ 22 − Ŝ 11 )
;
(178)
and
ˆ
Û 2 = g1 g2 e−iĥ0 T0 e−i=2g2 T2 (.̂11 â†1 â2 e−iT0 + .̂22 â1 â†2 eiT0 e−ik 2 T1 ) :
2
(179)
Although the above expressions are a bit long, we can draw from them several qualitative conclusions. First of all, it is interesting to note that Û 0 does not contain any coupling between the atom
and the second mode. Such a coupling we would obtain in Û 0 only in the third-order analysis, and
it would express itself by the purely dispersive terms, proportional to the photon number in the
second mode and to the inversion, but it would not contain raising and lowering atomic operators.
This means that the in/uence of the second mode on the atomic dynamics is contained only in
Û 1 and higher-order Û n . And, as we can see from (178), this in/uence is either just through the
intensity of radiation in the second mode, with no change in the number of quanta in that mode, or,
M. Janowicz / Physics Reports 375 (2003) 327 – 410
385
although the second mode does couple to the atomic dipole moment, this coupling it accompanied
by the fast-varying factors exp(±iT0 ). In Û 2 we Dnd the term responsible for the coupling of the
Drst and the second mode of radiation Deld without any change of the population of atomic levels.
However, in this low order the coupling is again accompanied by phase factors which vary fast
in time. We might thus reasonably expect that, for times of the order of ∼ 1=j2 (which means
t ∼ <=g2 where g ∼ min(g1 ; g2 )) the in/uence of the detuned mode can be visible through very
fast and very low-amplitude oscillations of inversion or the resonant-mode photon number, without
causing transitions. Let us note that the amplitude of the oscillations depends linearly on the number
of photons in the detuned mode, so that certain caution seems to be necessary (taking into account
conclusions of the section about an anharmonic oscillator) in the case of quasi-mesoscopic cavity
Delds when the mean photon number in detuned modes could be of the order of 102 . But even in
this case the product of j and the second-mode photon number is of the order of 10−3 for realistic
detunings and coupling constants.
5.3. The four-level “Lambda” system
5.3.1. Introductory remarks
In this section, devoted to an analysis of the dynamics of a four-level atom with two cavity modes,
we want to provide an insight into how eMective Hamiltonians of quite a complicated quantum
system can be obtained using MMS. Our considerations turn out to be much simpliDed by the fact
that the system under consideration is actually exactly solvable, that is, the time-evolution operator can be written down exactly. This is done using an algebraic technique developed for the
Jaynes–Cummings-like models by Yu et al. [89]. Application of MMS is justiDed, nevertheless,
because the exact solution is too diLcult to be of extensive use; simpler eMective Hamiltonians and
eMective time-evolution operators are to be obtained. The following considerations are based on
the paper [29].
Our system is a four-level generalization of the three-level non-resonant “M-system” considered,
e.g., in [91,92]. Let us note that there has been some discrepancy between the authors of [91] and
[92] regarding the structure of the eMective Hamiltonian which describes the coupling between two
lower levels of their model. We believe that the method of multiple scales, since it is universal
and eLcient, brings resolution to this discrepancy. It also provides several other interesting eMective
Hamiltonians not considered in [91] or [92].
We shall also provide a two-mode system which can exhibit predictable collapses and
revivals—this is a special type of the four-level “M-system” with two degenerate upper levels.
Unlike in the similar one-mode case, however, the dynamics of inversion, though described by
a closed-form expression, can also be only quasi-periodic and thus quite erratic. In addition, we
propose a Stern–Gerlach type of experiment to produce a SchrIodinger-cat-type state in the cavity.
Again, the best candidate for such an experiment is the degenerate (in the two upper levels) system
if the coupling constants are such that the Stark shifts are identical.
The rest of this section is organized as follows. First, we present the model and attempt to solve
it exactly. In the following subsection we use MMS to extract a family of eMective Hamiltonians
for our generalized “M-system”. Then we analyze the production of the SchrIodinger-cat state of the
cavity modes and the two-mode regular collapses-and-revivals pattern.
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
5.3.2. Description of the model
We consider a model system consisting of a four-level atom interacting with two modes of the
cavity electromagnetic Deld. We shall assume a very speciDc coupling between the atom and the
modes which is a direct generalization of the “M-system” considered, e.g., in [91]. More precisely,
the system is deDned by the following Hamiltonian:
4
Ĥ =
Ei .̂ii + ˝
!i â†i âi + ˝
(g1j â1 .̂j1 + g2j â2 .̂j2 + h:c:) :
(180)
i=1
i=1;2
j=3;4
Thus, the Drst (lowest) level is coupled to the third and fourth only via the Drst mode of the cavity
Deld, while the second level is coupled to the upper ones only via the second mode. So far, we
have not assumed any relationship between the atomic energies and the mode frequencies except
for the rotating wave approximation; we will try to solve the problem exactly. By “exact solution”
we mean an explicit expression for the time-evolution operator. We start by writing the Hamiltonian
as a linear combination of operators with simple algebraic properties. This task is simpliDed by the
observation that for some given m photons in the Drst mode and n photons in the second one, the
following states of the system can be populated: |4; m; n, |3; m; n, |1; m + 1; n and |2; m; n + 1. This
suggests that the problem can be reduced to the four-level system in an external time-independent
(within the rotating wave approximation) Deld if we Dnd appropriate constants of motion. Let us
further observe that, beside the Hamiltonian itself, the system indeed possesses at least two further
constants of motion (the excitation number operators) which turn out to be very useful:
N̂ 1 = â†1 â1 + .̂22 + .̂33 + .̂44
(181)
N̂ 2 = â†2 â2 + .̂11 + .̂33 + .̂44 :
(182)
and
The operators N̂ 1 and N̂ 2 commute not only with the Hamiltonian, but also with the free part of the
Hamiltonian and with all products of the type âi .̂ji ; i = 1; 2; j = 3; 4, as well as â†i âj .̂ij ; i; j = 1; 2.
Let us deDne the new operators as
1
1
Ŝ 31 = â1 .̂31 ; Ŝ 41 = â1 .̂41 ;
N̂ 1
N̂ 1
1
Ŝ 32 = â2 .̂32 ;
N̂ 2
1
Ŝ 42 = â2 .̂42 ;
N̂ 2
Ŝ ii = .̂ii ; i = 1; 2; 3; 4;
Ŝ 34 = .̂34 :
(183)
These operators, together with their Hermitian conjugates and the operator Ŝ 12 = Ŝ 13 Ŝ 32 = Ŝ 14 Ŝ 42 (and
its conjugate) span the Lie algebra su(4) with the usual commutator as the Lie product. Actually, it
is of even more importance that they form an associative matrix algebra, isomorphic to that spanned
by the operators .̂ij . Let us for instance calculate the product
1
1
1
Ŝ 12 Ŝ 24 = Ŝ 13 Ŝ 32 Ŝ 24 = â†1 .̂13 â2 .̂32 â†2 .̂24
N̂ 1
N̂ 2
N̂ 2
1
1
= â†1 .̂13 (â†2 â2 + 1).̂34 :
N̂ 2
N̂ 1
M. Janowicz / Physics Reports 375 (2003) 327 – 410
387
But
1 †
1
1 †
(â2 â2 + 1).̂34 −
(â2 â2 + 1 − N̂ 2 ).̂34
N̂ 2 .̂34 =
N̂ 2
N̂ 2
N̂ 2
=
1 †
(â2 â2 + 1 − â†2 â2 − .̂11 − .̂33 − .̂44 ).̂34 = 0 ;
N̂ 2
hence Ŝ 12 Ŝ 24 = (1= N̂ 1 )a†1 .̂14 = Ŝ 14 .
One can now write the Hamiltonian in terms of the operators Ŝ ij :
Ĥ =
4
Ei Ŝ ii + ˝
i=1
!i â†i âi + ˝
i=1;2
(gij Ŝ ji + h:c:) :
(184)
i=1;2 j=3;4
By applying the unitary transformation:
Ŵ = exp[(−i=˝)(!1 N̂ 1 + !2 N̂ 2 )t] ;
we get the following “interaction” Hamiltonian which only contains the operators Ŝ ij and N̂ i without
the explicit presence of the free Deld terms:
Ĥ = (E1 − ˝!2 )Ŝ 11 + (E2 − ˝!1 )Ŝ 22 + (E3 − ˝!1 − ˝!2 )Ŝ 33
+ (E4 − ˝!1 − ˝!2 )Ŝ 44 + ˝
N̂ i (gij Ŝ ji + gij? Ŝ ij ) :
i=1;2 j=3;4
If we choose the zero of energy in such a way that E4 = ˝(!1 + !2 ), we obtain
3
Ĥ = −˝
<i Ŝ ii + ˝
N̂ i (gij? Ŝ ij + gij Ŝ ji ) ;
i=1
(185)
i=1;2 j=3;4
where <1 = (E4 − E1 )=˝ − !1 , <2 = (E4 − E2 )=˝ − !2 , <3 = (E4 − E3 )=˝.
The matrix representation of the Hamiltonian is given by


−<1
0
ˆ†13 ˆ†14


 0
−<2 ˆ†23 ˆ†24 


(186)

;
 ˆ

ˆ
−<
0
23
3
 13

ˆ24
0
0
ˆ14
?
where ˆij = gij N̂ i and ˆ?
=
g
N̂ i . All the ˆ operators commute.
ij
ij
Thus, the problem of the exact solution for our model has been reduced to the problem of
the diagonalization of a quadratic matrix of the fourth degree. This can always be performed, but
the necessary intermediate step—the computation of the eigenvalues and eigenvectors—introduces
a terrible algebraic mess which obscures both the formal structure of the Hamiltonian and the
underlying physics. Therefore, despite the fact that the system admits an exact solution, in the next
section we will apply an eLcient perturbation scheme—the method of multiple scales—to extract
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
the physical information contained in the Hamiltonian Ĥ , provided that a natural small parameter
exists in it.
5.3.3. Application of multiple scales
To apply MMS, it will be√ assumed that the absolute values of at least two of <i , i = 1; 2; 3,
are much larger than all |gij | N i , where Ni = N̂ i . In spite of this simplifying assumption, several
internal resonances are possible in the system and hence we have to consider several cases; in some
of them we will obtain interesting eMective Hamiltonians to be compared with those in [91,92].
First we will outline the general procedure
√ assuming that all the three <i are of the same order
and that they are much larger than all gij N i . The eMective coupling constants
are also assumed
√
to be of the same order. Let , = min(|<1 |; |<2 |; |<3 |) and J = max(|gij | N i ). We deDne a small
dimensionless parameter j as j = J=,.
Let i = <i =, and m̂ij = gij N̂ i =J. Please note that the operator character of ˆij and m̂ij is rather
trivial: they are products of complex c-numbers and operator constants of motion.
In addition, let û denote the time-evolution operator (with exp[−i(!1 N̂ 1 +!2 N̂ 2 )t] already factored
out) as a function of time .
We introduce once again a new dimensionless time = ,t. The dynamics in are governed by
the Hamiltonian
3
ĥ = −
i Ŝ ii + j
mij Ŝ ij = ĥ0 + jĥ1 ;
i=1
(187)
i=1;2 j=3;4
that is, we have
i
d û
= ĥû :
d
There are two natural time scales of the evolution: one connected with the part of the Hamiltonian
containing detunings, the other connected with the coupling constants.
Let us therefore introduce the new standard time variables:
T0 = ;
T1 = j;
T2 = j2 ;
so that we have again
9
d
9
9
=
+j
+ j2
+ ··· :
d 9T0
9T1
9T2
(188)
Before proceeding any further, let us introduce the following notation that will be used throughout
this section. Namely, in all the formulae below, the symbols that start with F̂, like F̂(T0 ), denote
operators which contain exponential functions of T0 such that the factors multiplying T0 in the
exponents are O(1). Such operators can be safely integrated without any danger of obtaining small
(or zero) denominators. This notation has been introduced here because the system considered in
this section is the most complicated of those analyzed in this paper.
To obtain an asymptotic expression for û we expand it as
û = û 0 + jû 1 + j2 û 2 + · · · :
M. Janowicz / Physics Reports 375 (2003) 327 – 410
389
Taking into account Eqs. (187) and (188) we get
i
9û 0
= ĥ0 û 0 ;
9T0
(189)
i
9û 1 9û 0
+
= ĥ0 û 1 + ĥ1 û 0 ;
9T0
9T1
(190)
i
9û 2 9û 1 9û 0
+
+
= ĥ0 û 2 + ĥ1 û 1 + ĥ2 û 0 :
9T0
9T1
9T2
(191)
The Hamiltonian in (187) does not contain terms of the second-order in j so that Eqs. (189)–(191)
become a bit simpler in this case since one has ĥ2 = 0. However, in the following considerations
we shall meet the case of ĥ2 = 0 as well.
Even in the case of purely dispersive interactions with no transitions between levels we can have
internal resonances, the appearance of which completely changes the physical situation. In order to
analyze them, we must consider several sub-cases.
A. Case 1: i − j = O(1)
Here we have to consider the case in which the diMerences between the ’s are of the same order
as the ’s themselves. Eq. (189) can easily be solved:
û 0 = exp[ − i ĥ0 T0 ]v̂(T1 ; T2 ) = exp(i(1 Ŝ 11 + i2 Ŝ 22 + i3 Ŝ 33 )T0 )v̂(T1 ; T2 ) :
(192)
To solve Eq. (190) we write
û 1 = exp[ − iĥ0 T0 ]uŴ 1 (T0 ; T1 ; T2 ) ;
and Eq. (190) takes the form
i
9uW 1
9v
+i
= eiĥ0 T0 ĥ1 e−iĥ0 T0 v̂(T1 ; T2 ) :
9T0
9T1
(193)
The Hamiltonian ĥ1 in the “interaction representation” with respect to ĥ0 is
eiĥ0 T0 ĥ1 e−iĥ0 T0 = (m̂†13 Ŝ 13 e−i13 T0 + m̂13 Ŝ 31 ei13 T0 ) + (m̂†14 Ŝ 14 e−i1 T0 + m̂14 Ŝ 41 ei1 T0 )
+ (m̂†23 Ŝ 23 e−i23 T0 + m̂23 Ŝ 32 ei23 T0 ) + (m̂†24 Ŝ 24 e−i2 T0 + m̂24 Ŝ 42 ei2 T0 ) ;
(194)
where ij = i − j .
We can see that, in the Drst order, all the terms on the right-hand side of Eq. (193) are of the
type F̂(T0 ) and hence they can be integrated without the appearance of secular terms. Therefore, we
put 9v̂=9T1 = 0, and integrate Eq. (193). Let x̂(T1 ; T2 ) denote a solution to the homogeneous equation
i 9uŴ 1 =9T0 = 0.
To solve Eq. (191) we write
û 2 = e−iĥ0 T0 uŴ 2 (T0 ; T1 ; T2 ) :
(195)
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
Then Eq. (191) becomes
i
9uŴ 1
9v̂
9uŴ 2
+i
+i
= eiĥ0 T0 ĥ1 e−iĥ0 T0 uŴ 1 :
9T0
9T1
9T2
(196)
Here 9uŴ 1 =9T1 = 9x̂=9T1 since v̂ does not depend on T1 . Taking into account (194) we Dnd the
right-hand side of (196):
e
iĥ0 T0
ĥ1 e
−iĥ0 T0
uŴ 1 =
|m̂13 |2
|m̂14 |2
(Ŝ 33 − Ŝ 11 ) +
(Ŝ 44 − Ŝ 11 )
13
1
|m̂23 |2
|m̂24 |2
+
(Ŝ 33 − Ŝ 22 ) +
(Ŝ 44 − Ŝ 22 ) v + F̂(T0 ) :
23
2
(197)
By |m̂ij |2 we mean |gij |2 N̂ i =J2 . Now, the Drst four terms on the right-hand side of Eq. (197) do not
depend on T0 . Thus, if we try to integrate over T0 , this will immediately lead to unphysical terms
proportional to T0 . We can avoid their appearance very easily by equating i 9v̂=9T2 to the Drst four
terms in Eq. (197). The solution to the equation produced in this way is obvious:
|m̂13 |2
|m̂14 |2
(Ŝ 33 − Ŝ 11 ) +
(Ŝ 44 − Ŝ 11 )
v̂(T2 ) = exp −i
13
1
|m̂23 |2
|m̂24 |2
+
(Ŝ 33 − Ŝ 22 ) +
(Ŝ 44 − Ŝ 22 ) T2 :
23
2
(198)
If we combine this result with Eq. (192), and then go back to time t and deDne the eMective
Hamiltonian as Ĥ eM = i˝Û †0 d Û 0 =dt, Û 0 = Ŵ û 0 , we obtain
3
|g14 |2 N̂ 1 |g24 |2 N̂ 2
Ĥ eM
= !1 N̂ 1 + !1 N̂ 2 −
Ŝ 44
<i Ŝ ii +
+
˝
<1
<2
i=1
|g13 |2 N̂ 1 |g23 |2 N̂ 2
|g13 |2 N̂ 1 |g14 |2 N̂ 1
Ŝ 33 −
Ŝ 11
+
+
+
<13
<23
<13
<1
|g23 |2 N̂ 2 |g24 |2 N̂ 2
Ŝ 22 :
−
+
<23
<2
(199)
Our eMective Hamiltonian in Eq. (199) is thus diagonal which is a characteristic feature of purely
dispersive interactions. It is to be observed that the shifts of all atomic levels depend on the mean
excitation numbers. We shall see later that very diMerent forms of eMective Hamiltonians arise if an
internal resonance is present in the system.
M. Janowicz / Physics Reports 375 (2003) 327 – 410
391
B. Case 2: 1 − 2 = O(j)
Let us now assume that, although all the <’s are much larger than the eMective coupling constants,
one of their diMerences, <1 − <2 is a small quantity, comparable with the eMective couplings. In
√
this case we write <2 = <1 − (<1 − <2 ) and deDne , as min(|<1 |; |<3 |), and J as max(|<12 |; gij ni )
where <ij = <i − <j . The dimensionless small parameter is deDned as before, j = J=, and = ,t.
The Hamiltonian which generates the dynamics in time , is now given by


2 4
(m̂†ij Ŝ ij + m̂ij Ŝ ji )
ĥ = −1 (Ŝ 11 + Ŝ 22 ) − 3 Ŝ 33 + j 12 Ŝ 22 +
i=1 j=3
= ĥ0 + jĥ1 ;
(200)
where lower case and m̂ij ’s are deDned as before and 12 = <12 =J. However, this is not enough to
substantially change the eMective Hamiltonian as given by Eq. (199). Indeed, proceeding as before,
we obtain almost exactly the same eMective Hamiltonian, with the only diMerence being that all the
<2 ’s are replaced by <1 ’s and all the <23 ’s by <13 ’s in all those (and only those) terms which
contain |gij |2 . Substantial changes are introduced to the @rst-order correction only; the operator x̂ is
no longer T1 -independent: x(T1 ; T2 ) = exp[ − i12 Ŝ 22 T1 ]ŷ(T2 ), where ŷ(T2 ) can be expressed in terms
of the new v̂.
C. Case 3: 1 − 2 = O(j2 )
Now we assume that we have almost exact internal resonance:E1 + ˝!1 is almost equal to
E2 + ˝!2 , so their diMerence is small even when compared with |gij | N̂ i . While , and J are deDned
as before, we have 12 = <12 ,=(J2 ). Again, j = J=,.
The Hamiltonian (for the evolution in time ) is now given by
†
ĥ = −1 (Ŝ 11 + Ŝ 22 ) − 3 Ŝ 33 + j
(m̂ij Ŝ ij + m̂ij Ŝ ji ) + j2 12 Ŝ 22 = ĥ0 + jĥ1 + j2 ĥ2 :
(201)
i; j
In the Hamiltonian above, the index i takes the values of 1; 2 while the index j takes the values
3; 4. The zeroth-order approximation is given by
û 0 = exp[i(1 (Ŝ 11 + Ŝ 22 ) + 3 Ŝ 33 )T0 ]v̂(T1 ; T2 ) :
(202)
To obtain the Drst-order approximation we compute the Hamiltonian ĥ1 in the “interaction representation” with respect to ĥ0 :
eiĥ0 T0 ĥ1 e−iĥ0 T0 = (m̂†13 Ŝ 13 e−i13 T0 + m̂13 Ŝ 31 ei13 T0 )
+ (m̂†14 Ŝ 14 e−i1 T0 + m̂14 Ŝ 41 ei1 T0 ) + (m̂†23 Ŝ 23 e−i13 T0 + m̂23 Ŝ 32 ei13 T0 )
+ (m̂†24 Ŝ 24 e−i1 T0 + m̂24 Ŝ 42 ei1 T0 ) :
(203)
Because the right-hand-side of Eq. (203) is of the type F̂(T0 ), we have 9v̂=9T1 = 0 and hence
9uŴ 1 =9T1 = 9x̂=9T1 , where uŴ 1 , x̂ and uW 2 are deDned as for Case 1.
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
We have to solve to the second order an expression similar in Eq. (197):
i
9uŴ 2
9x̂
9v̂
+i
+i
= eiĥ0 T0 ĥ1 e−ih0 T0 uŴ 1 + eiĥ0 T0 ĥ2 e−iĥ0 T0 v̂
9T0
9T1
9T2
|m̂14 |2
|m̂23 |2
|m̂24 |2
|m̂13 |2
(Ŝ 33 − Ŝ 11 ) +
(Ŝ 44 − Ŝ 11 ) +
(Ŝ 33 − Ŝ 22 ) +
(Ŝ 44 − Ŝ 22 ) v̂
=
13
1
13
1
m̂†13 m̂23 m̂†14 m̂24
−
Ŝ 12 + h:c: v + 12 Ŝ 22 + F̂ 1 (T0 ) :
+
(204)
13
1
Note the presence of non-diagonal terms in the above expression. We may put i 9x̂=9T1 = 0 and
i 9uŴ 2 =9T0 = F̂ 1 (T0 ). Then the solution for v̂ is obvious and we immediately Dnd the eMective Hamiltonian:
3
Ĥ eM
|g13 |2 N̂ 1
= !1 N̂ 1 + !2 N̂ 2 −
<i Sii +
(Ŝ 33 − Ŝ 11 )
˝
<13
i=1
|g14 |2 N̂ 1
|g23 |2 N̂ 2
(Ŝ 44 − Ŝ 11 ) +
(Ŝ 33 − Ŝ 22 )
<1
<13
?
?
g23 N̂ 1 N̂ 2 g14
g24 N̂ 1 N̂ 2
g13
Ŝ 12 + h:c: :
+
−
<13
<1
+
(205)
The above eMective Hamiltonian again contains intensity-dependent Stark shifts of all levels as well
as the coupling between the Drst and the second level. Its form may justify the assumption of Gerry
and Eberly that to study an eMective interaction between the states |1 and |2 it is enough to restrict
oneself to near resonant cases, i.e. two detunings, <1 and <2 , which are equal or almost equal. If this
is not the case, one can use a diagonal eMective Hamiltonian and investigate the eMective coupling
between the two lower levels only as a small correction. Needless to say, having linearized both the
total Hamiltonian and all eMective Hamiltonians, we could very easily obtain an explicit expression
for all eMective time-evolution operators.
D. Case 4: 1 − 2 = O(j2 ); 3 = O(j2 )
Let us now consider the case of a very
√ small energy gap between the levels |3 and |4. We
deDne: , = min(|<1 |; |<2 |), J = max(|gij | Ni ), 12 = <12 ,=(J2 ), 3 = <3 ,=(J2 ), j = J=,. This case is
quite interesting because we obtain an eMective coupling not only between the two lower levels, but
also between the two upper ones. In fact, calculations of exactly the same type as before lead to the
eMective Hamiltonian
1
Ĥ eM = −
<i Sii + [N̂ 1 (|g13 |2 (Ŝ 33 − Ŝ 11 ) + |g14 |2 (Ŝ 44 − Ŝ 11 ))
<1
i
?
?
+ N̂ 2 (|g23 |2 (Ŝ 33 − Ŝ 22 ) + |g24 |2 (Ŝ 44 − Ŝ 22 )) + ((g13 g14
N̂ 1 + g23 g24
N̂ 2 )Ŝ 34 + h:c:)
?
?
− ((g13
g23 + g14
g24 ) N̂ 1 N̂ 2 Ŝ 12 + h:c:)] :
(206)
M. Janowicz / Physics Reports 375 (2003) 327 – 410
393
As it has been pointed out in [91], the eMective interaction between the two lower levels is
zero-photon: there is neither gain nor loss of photons but any transition from one level to another is connected with the exchange of photons between modes. The Hamiltonian above shows
that between the two upper levels we have a zero-photon coupling in an even stricter sense: there
are Rabi oscillations in the sub-system consisting of states |3 and |4, but there is no exchange of
photons between the two modes.
E. Case 5: 1 − 2 = O(1); 3 = O(j2 )
Now , = min(|<1 |; |<2 |), and the other variables are deDned as before. In this case we obtain
an eMective Hamiltonian which contains just the eMective interaction between the two upper levels,
without any coupling between the lower ones:
3
Ĥ eM
1
=−
<i Ŝ ii + (|g13 |2 N̂ 1 (Ŝ 33 − Ŝ 11 ) + |g14 |2 N̂ 1 (Ŝ 44 − Ŝ 11 ))
˝
<1
i=1
1
2
N̂ 2 (Ŝ 44 − Ŝ 22 ))
(|g23 |2 N̂ 2 (Ŝ 33 − Ŝ 22 ) + g24
<2
?
?
N̂ 1 g23 g24
N̂ 2
g13 g14
Ŝ 34 + h:c: :
+
+
<1
<2
+
(207)
This Hamiltonian provides a two-mode generalization of that proposed by [96]. Unlike in the Case C,
this time only the upper levels are eMectively coupled, and the operator coupling constant signiDcantly
diMers from the one present in Eq. (205). We will use the Hamiltonian given by (207) in the next
subsection to “produce” a kind of SchrIodinger-cat state in the cavity.
In this section we apply a variant of the method of multiple scales to extract eMective interactions
in a “M-type” four-level system with weak coupling. It is to be stressed that the notation has been
greatly simpliDed by the previous linearization of our system, but MMS would work equally well
even if we had no idea about how to linearize the initial Hamiltonian.
5.3.4. SchrAodinger-cat states and population dynamics in a restricted two-mode two-photon
interaction
We shall now consider the possible appearance of some special states of the cavity Delds from
a superposition of the products of simple coherent states with large mean number of photons. By
a turn of phrase, such states are called “SchrIodinger-cat” states. They are quite important from a
fundamental point of view as their existence is a reply to Einstein’s and SchrIodinger’s objections
against quantum theory, based on the fact that it is diLcult to Dnd any coherent superposition of
quantum states on a macroscopic level. In fact, on the one hand, we can now produce such coherent
superpositions in high-Q cavities (see, e.g., [90]) and in optical traps, but on the other, there is a
satisfactory explanation of the diLculty in observing such superpositions in terms of the decoherence
introduced by the coupling with the reservoir which is always present.
One of the most important non-classical features of models of the Jaynes–Cummings type is the
presence of the so-called collapses and revivals in the dynamics of the populations of atomic states.
Usually, however, one cannot Dnd a closed formula to express these dynamics. Asymptotic and
numerical methods (e.g. see [84,86]) show that when time increases, collapses and revivals usually
appear less and less regularly and then Dnally in an erratic way, due to the strong overlap between
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
neighboring revivals. Phoenix and Knight [96] have found a system which exhibits perfectly regular
dynamics of collapses and revivals. The system described by our eMective Hamiltonian in Case 5 is
another example which may have this feature. In this section we will assume—without the loss of
generality—that the gij ’s are real. Let us begin with SchrIodinger-cat type states. We will assume—
just in order to obtain a closed-form expression—that g13 =g14 ; g23 =g24 and <3 =0. This means that
the upper levels are perfectly degenerate: not only are their energies (without interactions) equal;
the Stark shifts are also the same.
Let us rewrite the Hamiltonian containing the eMective coupling between the levels |3 and |4 as
Ĥ eM = E11 Ŝ 11 + E22 Ŝ 22 + ˝(!1 + !2 )(Ŝ 33 + Ŝ 44 ) + ˝!1 n̂1 + ˝!2 n̂2 + Ĥ I ;
(208)
where n̂i = â†i âi , i = 1; 2, and Ĥ I is given by
2
2
g13
2g13
2g2
g2
N̂ 1 Ŝ 11 + 23 N̂ 2 Ŝ 22 + ˝
N̂ 1 + 23 N̂ 2 (Ŝ 33 + Ŝ 44 + Ŝ 34 + Ŝ 43 ) : (209)
Ĥ I = −˝
<1
<2
<1
<2
Products of the excitation number operators and Ŝ jk ’s can be written in terms of the photon number
2
=<i .
operators n̂i : N̂ i Ŝ jk = (n̂i + 1)Ŝ jk , i = 1; 2, j; k = 3; 4. We will write i instead of gi3
Let us suppose that initially the cavity Delds are in coherent states and the atom is in a coherent
superposition of the upper states:
|6(0) = (c3 |3 + c4 |4)|$|= :
(210)
Let us denote the time of /ight of the atom through the cavity by t0 (we will identify this time with
the time of the atom–cavity interaction). After a time t0 + t1 , we perform a measurement to project
the wave functions onto one of the states 3| or 4|; because the atomic energies in these two states
are the same, we propose to use a Stern–Gerlach type of experiment if the upper levels diMer in
their mJ value—atoms in state |3 /y to a diMerent spatial region than those in state |4 and we
can then measure the atomic energy in an ionization chamber to project on states |3 or |4. Let us
suppose that we have found the atom in state |4. Then the Deld in the cavity will be in the state
|6Deld = 12 exp(−i(!1 + !2 )(t0 + t1 )) exp(−i(!1 n̂1 + !2 n̂2 )(t0 + t1 ))
×[(c4 − c3 )|$|= + (c4 + c3 )e−2i(1 +2 )t0 |$e−2i1 t0 |=e−2i2 t0 ] :
(211)
If we choose i and t0 appropriately, the expression in the square bracket above can be equal, e.g.,
to (c4 − c3 )|$|= + (c4 + c3 )| − $| − =, taking a shape characteristic to cavity SchrIodinger-cats.
Let us note that, unlike in the usual dispersive interaction of the Jaynes–Cummings atom with a
cavity mode, it is not actually necessary to prepare the atom in a superposition of two states. It
can be prepared in state |3 or in state |4, or in any superposition of these states (provided that
c3 = c4 and c3 = −c4 ), and we can perform projection onto either of states 3| or 4|—in any
case, a SchrIodinger-cat state will arise, without any additional (=2 pulses applied to the atom before
measurement. On the other hand, if there is no degeneracy in the two upper levels and the interactions
are purely dispersive (the eMective Hamiltonian is as in Case 1 of the previous section), we can
prepare the atom in a superposition of, say, the two upper levels before it enters the cavity and then
apply an additional pulse after it leaves the cavity. The subsequent measurement of atomic energy
will produce two-mode SchrIodinger cats.
M. Janowicz / Physics Reports 375 (2003) 327 – 410
395
Let us now consider the population dynamics for the case of the exact degeneracy of levels |3
and |4; that is, the assumptions about gij are the same as above in this section. Let the system be
initially prepared in the state |6(0) = |4|$|=.
Then by straightforward calculation we Dnd that
Ŝ 44 (t) = 12 [1 + Re(exp[2i(1 + 2 )t + |$|2 (e2i1 t − 1) + |=|2 (e2i2 t − 1)])] :
(212)
Thus the population dynamics in our system can be expressed by a closed-form formula. The overlap between neighboring revivals does not appear as a consequence of the fact that in our eMective
Hamiltonian the coupling contains the actual Deld intensities rather than their square roots. Nevertheless the population dynamics are, in general, only quasi-periodic in time, since in the time
dependence of Ŝ 44 we get all frequencies of the type k1 + l2 , where k; l are arbitrary integers.
Contributions of various frequencies are weighted by the products of Bessel and modiDed Bessel
functions. Thus, the behavior of Ŝ 44 (t) can be both quite regular and fairly bizarre, depending on
the relation between the ’s and mean number of photons, as well as on the scale of time of the
observation.
The above results Dnish our considerations about the four-level two-mode “M” system. Analogous
approach can obviously be developed for other interesting conDgurations of levels and modes in
Cavity QED.
6. Fields and atoms in cavities with oscillating mirror
There has been considerable interest in studies of the behavior of the electromagnetic Deld in a
cavity with oscillating mirror due to interesting eMects which may be found in such system; these
eMect include: quantum vacuum radiation [98–103], non-classical statistics of these photons
[104–108], as well as dynamical modiDcation of the Casimir forces cf., e.g., [109–111].
The problem of the interaction dynamics of an atom and radiation in a cavity with oscillating
mirrors has certain peculiarities. First, it is to be noted that the problem is analytically very diLcult,
almost intractable, if the cavity walls are not assumed to be perfectly conducting, that is, perfectly
re/ecting. But then it turns out that very fundamental troubles appear since, in general, the quantum
Deld theory of systems with boundary conditions depending on time is by no means well understood
[112]. More speciDcally, in perfectly conducting cavities with oscillating mirrors the probability is
not conserved, in other words, the SchrIodinger picture does not exist. This has been proved by
showing the non-existence of the time-evolution operator [98], the very object in which we are most
interested in here. The situation can be improved in several ways. On the one hand, one can still try
to get rid of the assumption of inDnitely conducting walls and try to use a relatively simple but less
idealized model of the wall [109,110]. Another solution consists in including the oscillating wall as
a part of the dynamical system [113]; a Hamiltonian which includes the position and momentum
of the wall is well deDned. Finally, motivated by the present-day experimental possibilities, one
can consider only adiabatic (slow) motion of the walls [114,100,30]; for such motion one can
Dnd an approximate eMective Hamiltonian of the Deld in the cavity, and the atom–Deld interaction
Hamiltonian is always well-deDned. It is precisely the third approach we shall examine in this section.
In the following subsection we shall consider the electromagnetic Deld in a one-dimensional cavity
with one oscillating mirror to get an eMective free-Deld Hamiltonian. Then we shall couple the Deld
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
with a two-level atom to obtain an interesting eMect of stabilization of the excited-state population.
In the whole discussion of Section 6 we shall consider any cavity losses to be negligibly small.
6.1. Electromagnetic @eld in the oscillating cavity
We shall consider here idealized one-dimensional cavities only. In such cavities, the electromagnetic Deld is speciDed completely if the electromagnetic vector potential A is known. Due to the
one-dimensional character of the problem, the polarizations of the Deld completely decouple and the
vectorial character of A is irrelevant. For the Deld of TE polarization, one can prove [98] that the
electric Deld should vanish on the walls (even if they are moving), and the same is true for the
component of A perpendicular to the walls. Let this component be called u. It depends on time and
one spatial coordinate, say, z. Let us also assume that one mirror is Dxed at z = 0, while the other
one oscillates periodically so that it is located at z = q(t) = L + Af(?t) = Lg(?t) where L is the size
of the cavity at rest. The variable A is the amplitude of the oscillations, f is a periodic function of
time, and g(?t) is deDned as 1+(A=L)f(?t). Let us introduce the dimensionless variables =(c=L)t,
x = z=q(t), and J = (L=c)?. Then, from the wave equation satisDed by u:
1 92 u 92 u
= 2
(213)
c2 9t 2
9x
we obtain
2 2
ġ 2 9u
1 92 u
92 u
gI 9u
ġ 92 u
2 ġ 9 u
+
2x
=
;
(214)
−
x
−
2x
+
x
92
g2 9x
g 9x
g 9x9
g2 9x2
g2 9x2
where ġ = dg=d. The boundary conditions to be fulDlled by u in terms of the new variables are
very simple: u has to vanish for x = 0 and 1. We look for the solution for u in terms of the series
∞
2
u=
An () sin(n(x)
(215)
L n=1
so that the boundary conditions are fulDlled automatically. Substitution of (215) into (214) gives
ġ 2 gI ġ AI m + 2 2
wmn An −
wmn An − 2
wmn Ȧn
g n
g n
g n
−
ġ 2 1
vmn An + 2 (m()2 Am = 0 ;
g2 n
g
where
wmn = 2n(
and
vmn = 2(n()
1
x sin(m(x) cos(n(x) d x
0
2
(216)
0
1
x2 sin(m(x) sin(n(x) d x :
We shall make the following calculations somewhat easier by assuming that g is a purely sinusoidal
function:
A
(217)
g() = 1 + sin(J) :
L
M. Janowicz / Physics Reports 375 (2003) 327 – 410
397
We shall now introduce the following adiabatic assumptions: (i) the amplitude of the oscillations of
the mirror is much smaller than the size of the cavity, AL; (ii) the frequency of the oscillations
of the moving mirror is much smaller than the fundamental frequency of the cavity, !1 = (c=L, that
is, ?!1 ; (ii) the two ratios just deDned are of the same order, that is, if we deDne the small
parameter j as j = A=L, then J can be written as J = j, with , = O(1).
Eq. (216) can be conveniently rewritten including only terms up to the order j2 :
wmn Ȧn + (m()2 [1 − 2j sin(j,) + 3j2 sin2 (j,)]Am = 0 :
(218)
AI m − 2j2 , cos(j,)
n
We now apply the method of multiple scales to obtain the approximate solution to (218). The
amplitudes Am are written as
(1)
2 (2)
Am = A(0)
m + jAm + j Am + · · · ;
and we introduce the following “time” variables:
T0 = ;
T1 = j;
T2 = jD(T1 ) ;
so that
9
9
9
d
=
+j
+ j2 D (T1 )
+ ···
d 9T0
9T1
9T2
and
92
92
92
=
+
2j
+ j2
92 9T02
9T0 9T1
92
92
+
D
(T
)
1
9T0 9T2
9T12
;
where the superscript “prime” standing at D denotes the derivative of the function D over its
argument T1 .
In the zeroth-order of the perturbation expansion we obtain
2
9
2
+ (m() A(0)
(219)
m =0
9T02
with the solution
−im(T0
im(T0
+ a?
:
A(0)
m = am (T1 ; T2 ; : : :)e
m (T1 ; T2 ; : : :)e
(220)
In the Drst order we get
2
9
9am −im(T0
9a?
m im(T0
2
(1)
A
+
(m()
=
2
im(
e
−
im(
e
m
9T1
9T1
9T02
im(T0
):
+ 2(m()2 sin (,T1 )(am e−im(T0 + a?
me
(221)
To avoid secular terms we must require that the following conditions are fulDlled:
9am
= im( sin(,T1 )am ;
9T1
9a?
m
= −im( sin(,T1 )a?
m :
9T1
(222)
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M. Janowicz / Physics Reports 375 (2003) 327 – 410
Their solutions are given by
m(
[cos(,T1 ) − 1] ;
am = $(T2 ) exp −i
,
m(
?
[cos(,T
a?
=
$(T
)
exp
i
)
−
1]
:
2
1
m
,
(223)
We may stop at this point, considering $’s as constants independent of T2 and writing the solution
to Eq. (221) A(1)
m = 0, which can be done since the inhomogeneities have been absorbed in the
amplitudes am and a?
m . In this case we realize that the combined solutions (220) and (223) can
be obtained if we write equations for the amplitudes am , a?
m as Hamilton’s equations, where the
Hamiltonian is given by
Ĥ eM =
!m (1 − j sin(?t))a?
where !m = m(c=L ;
(224)
m am
m
with the canonical Poisson brackets:
{am ; ia?
n } = mn :
Then we can try to perform a quite naive quantization just by replacing the Poisson brackets by the
corresponding commutators and raise the complex amplitudes am , a?
m to the status of destruction and
creation operators while the form of the Hamiltonian remains the same (Eq. (224)).
We can also continue the multiple-scales procedure to obtain Am in the second order. We write
−im(T0
im(T0
A(1)
+ b?
m = bm (T1 ; T2 )e
m (T1 ; T2 )e
and obtain the following equation for the second-order correction A(2)
m :
2
9
92
2
(2)
A
+
(m()
=
−2
+ 2(m()2 sin(,T1 ) A(1)
m
m
2
9T
9T
9T0
0
1
2
9A(0)
9
92
n
(0)
A
−
+
2D
(T
)
+
2,
cos(,T
)
w
− 3(m()2 sin2 (,T1 )A(0)
1
1
mn
m
m :
9T0 9T2
9T
9T12
0
n
(225)
(226)
To avoid any terms proportional to powers of T0 in the solution, the following relations must hold:
9bm
= im( sin(,T1 )bm ;
9T1
9b?
m
= −im( sin(,T1 )b?
m ;
9T1
?
so that the dependence of bm , b?
m on T1 is the same as that of am , am . In addition, we must require
that the relation
iD (T1 )
9$m 1
= [i, cos(,T1 )$m + 2i, cos(,T1 )wmm $m + 2m( sin2 (,T1 )$m ]
9T2
2
M. Janowicz / Physics Reports 375 (2003) 327 – 410
399
holds (together with its complex conjugate). This leads to the solution for D:
i
m(
m(
T1 −
sin(2,T1 ) ;
D(T1 ) = sin(,T1 )(1 + 2wmm ) +
2
2
4,
and for $m , $m = $Wm e−iT2 = $Wm e−ijD(T1 ) . It is to be noted that the solution obtained in the second
order does not allow us to write down an eMective Hamiltonian of the form
Ĥ eM =
!m (t)$m? $m :
m
If, however, j is suLciently small, we can restrict ourselves to the Drst-order approximation and
use the quantum version of Hamiltonian (224). This is what we shall do in our description of the
atom–Deld interaction. Realistic values of j will be discussed at the end of this subsection together
with conditions of experimental veriDcation of our results.
6.2. Atom–@eld interactions
We shall use here the two-level model of an atom and the approximate Deld Hamiltonian given by
Eq. (224), except that we shall disregard coupling of the atom to any modes except of the “lower”
ones (coupling with the other modes produces very fast-varying terms). Thus, we start with the
energy operator
Ĥ = ˝!1 (1 − j sin?t)↠â + ˝!22 .̂22 + ˝G(.̂21 â + ↠.̂12 ) ;
(227)
↠,
â and neglected anti-resonant terms. In the above equation
where we have omitted the index 1 at
G denotes the atom–Deld coupling constant. To proceed, let us Drst notice that the excitation-number
operator
N̂ = .̂22 + ↠â
is an operator constant of motion despite the modulation. We may simplify the form of the Hamiltonian by writing the total time-evolution operator Û as the product Û = Ŵ Û 1 , where
j!1
Ŵ = exp −i!1 N̂ t − i
(228)
[cos(?t) − 1]↠â :
?
The operator Û 1 satisDes the SchrIodinger equation
d
i Û 1 = Ĥ 1 (t)Û 1 ;
dt
where
Ĥ 1 (t) = \Ŝ 22 + G N̂ (Ŝ 21 e−ir(t) + Ŝ 12 eir(t) ) ;
and we have introduced the following notation:
< = ! − !1 ;
Ŝ ii = .̂ii ;
for i = 1; 2 ;
Ŝ 12 = N̂ −1=2 ↠.̂12 ;
Ŝ 21 = N̂ −1=2 .̂21 â ;
(229)
(230)
400
M. Janowicz / Physics Reports 375 (2003) 327 – 410
and
r(t) =
j!1
[cos(?t) − 1] :
?
The excitation-number operator commutes with all operators Ŝ ij , and the latter form a realization of
the same algebra (both in the sense of Lie algebra and associative algebra) as the atomic operators
.̂ij . Using once more expansion (96) we get
√
j!1
N̂
d Û 1
= G N √ J0
Ŝ 21 eij!1 =? + Ŝ 12 e−ij!1 =?
i
dt
?
N
∞
√
j!1
N̂ i[j!1 =?−k((=2)]
−i[j!1 =?−k((=2)]
cos k?t Û 1 ;
+ 2G N √
(Ŝ 21 e
+ Ŝ 12 e
) Jk
?
N
k=1
(231)
where N = N̂ and we have assumed that < = 0, that is, under the absence of the oscillations of
the wall the cavity Deld is in exact resonance with the atom. It is now important to realize that the
arguments of the Bessel functions are actually O(1) if A=L and ?=!1 are of the same order, which
is what we assume. But this means that we can quite easily Dnd such ? and A that J0 (j!1 =?) will
actually be equal to zero. Using again the method of multiple scales with the new small parameter
= GN=? we Dnd the approximate solution for Û 1 in the Drst order:
√ ∞
G N 1
Û 1 = 1 − 2i
(Ŝ 21 )ei[j!1 =?−k((=2)]
?
k
k=1
j!1
−i[j!1 =?−k((=2)]
sin(k?t) V̂ 1 ;
+ Ŝ 12 e
Jk
(232)
?
where
j!1
j!1
i(j!1 =?)
−i(j!1 =?)
G N̂ t − i(e
G N̂ t :
Ŝ 21 + e
Ŝ 12 ) sin J0
V̂ 1 = cos J0
?
?
(233)
From Eqs. (232) and (233) it follows that the Rabi oscillations in the cavity will be strongly inhibited
if the zeroth Bessel function vanishes since in this case V̂ 1 ≈ 1 and the population inversion Ŝ 22 would oscillate only due to very small-amplitude and high-frequency terms containing higher Bessel
functions in (232). Let us notice that for microwave cavities we have !1 ∼ 1010 Hz, L ∼ 1 cm,
and G ∼ 104 –105 Hz. This means that if we make one wall of the cavity from a piezoelement, all
assumptions made during our MMS analysis will be fulDlled if the piezoelement oscillates with the
amplitude A ∼ 1 m and frequency ? ∼ 106 Hz, and the initial number of photons in the cavity
is of the order of 1. To the author’s knowledge, preparation of such cavities and piezoelements
is entirely within the possibilities of the current experimental technique. Very similar suppression
of the Rabi oscillations have been predicted for a cavity partially Dlled with the dielectric with a
time-varying dielectric constants. The quantitative aspect of this prediction also heavily relies on the
M. Janowicz / Physics Reports 375 (2003) 327 – 410
401
application of MMS [31]. The above remarks Dnish our examples of the application of MMS in the
Cavity QED.
7. Multiple scales and cold boson gases
In this section we shall only touch the extremely fashionable topics of the physics of Bose–
Einstein condensates (cf., e.g., [115–118]). The scope of this review as well as (lack of) the author’s
competence do not allow for even a very short presentation of this somewhat cross-disciplinary
branch of quantum optics and statistical mechanics. We shall only consider the interesting behavior
of a system of cold bosons conDned to a double-well potential in the two-mode approximation
without and with external modulation of the well. Such a system has been considered in the context
of Bose–Einstein condensates e.g. in [119–123]. Our strategy in this section is as follows: after the
formulation of the problem in terms of the many-body Hamiltonian, we shall make the two-mode
approximation and consider the semi-classical problem. We shall then Dnd approximate solutions
of the latter problem using MMS. In our semi-classical treatment we shall take into account losses
due to the coupling with the remaining modes as well as due to the in/uence of the environment.
We utilize here the simplest way to introduce losses, just by adding the damping constants to the
semi-classical and mean-Deld equations. That such an approach is suLcient in the context of BEC
is by no means evident, and we employ it for the sake of simplicity. A more reDned approach has
been outlined in [124] and applied in [125].
Let us start with writing down the Hamiltonian for the system of many bosons in the conDning
potential U (r; t):
2
˝
V
0
3
†
†
†
†
ˆ ˆ ˆˆ ;
Ĥ = d r
∇ ˆ · ∇ ˆ + ˆ U (r; t) ˆ +
(234)
2m
2
where ˆ and ˆ † are the boson Deld operators, m is the mass of a single atom, V0 =4(˝2 a=m measures
the strength of the two-body interactions (assumed to be modeled by the Dirac-delta-function potential
in the Drst-quantization picture), and a is the s-wave scattering length. For instance, the trap potential
can be taken to be a symmetric double well in the x direction and harmonic in the y, z directions
[119,123]:
U (r) = b(x2 − q02 )2 + 12 m!t2 (y2 + z 2 ) :
(235)
In the above equation !t is the trap frequency in the y–z plane and b measures the strength of the
conDnement in the x direction. The linearized motion in each well is harmonic with the frequency
!0 = q0 (8b=m)1=2 , and we shall take !0 = !t following [123]. The energy eigenstates of the double
well may be approximated as symmetric and antisymmetric combinations of the localized states with
energy eigenvalues given in the Drst-order perturbation theory by E± = E0 ± ˝?=2, where E0 is
the ground-state energy of the almost harmonic potential near the bottom of each well, while the
energy-level splitting ? is given by
3
q2
!0 20 exp[ − q02 (2r02 )] ;
8
r0
where r0 = ˝=2m!0 .
?=
(236)
402
M. Janowicz / Physics Reports 375 (2003) 327 – 410
The two-mode approximation consists in the following almost trivial mode decomposition of the
Deld operators:
ˆ (r) =
2
ĉn un (r) ;
n=1
ˆ † (r) =
2
ĉ†n un? (r) ;
(237)
n=1
where un (r) = u0 (r − rn ), n = 1; 2, and u0 (r) is the normalized single-particle ground state of the
harmonic potential near each of the two stable minima of the double-well potential, located at rn .
The operators ĉn , ĉ†n are the annihilation and creation operators in the local modes in the nth well.
In this two-mode approximation the Hamiltonian takes the form
Ĥ 1 = ˝!0 (ĉ†1 ĉ1 + ĉ†2 ĉ2 ) +
where
2(a˝
R=
m
˝? †
(ĉ ĉ1 + ĉ†1 ĉ2 ) + ˝R[ĉ†12 ĉ21 + ĉ†22 ĉ22 ] ;
2 2
(238)
d 3 r|u0 (r)|4
is a measure of strength of the interparticle interaction. The creation and annihilation operators satisfy
the obvious commutation relations
[ĉ1 ; ĉ†1 ] = [ĉ2 ; ĉ†2 ] = 1 ;
and all other commutators vanish.
From Hamiltonian (238) we derive the following semi-classical equations of motion supported by
the damping:
i
db1 ?
= b2 + 2Rn1 b1 − 1 b1 ;
dt
2
i
db2 ?
= b1 + 2Rn2 b2 − 2 b2 ;
dt
2
(239)
plus corresponding equations for b?
i , where we have replaced the operators ĉi with complex variables
bi exp(−i!0 t), and where ni = b?
b
i i . The parameters 1 , 2 are damping constants which in principle
need not be equal. These semi-classical counterparts of the Heisenberg equations contain the damping
terms due to the coupling to other modes as well as small coupling to the environment.
In addition, let us now rescale the amplitudes:
√
= → N= ;
√
=? → N =?
M. Janowicz / Physics Reports 375 (2003) 327 – 410
403
(where N is the total number of particles) to obtain the equations for which the name “mean-Deld”
is more appropriate than “semi-classical”:
d=1 ?
i
= =2 + 2RN |=1 |2 =1 − 1 =1 ;
dt
2
d=2 ?
= =1 + 2RN |=2 |2 =2 − 2 =2 :
dt
2
Let us now deDne the following new variables:
1
Sx = (|=2 |2 − |=1 |2 ) ;
2
i
(240)
−i ?
(b b2 − c:c:) ;
2 1
1
b2 + c:c:) :
(241)
Sz = (b?
2 1
The closed system of equations for Sx , Sy , Sz can be obtained only in the case of 1 = 2 , since
we have
dSx
1
1
= −?Sy − (1 + 2 )Sx + (2 − 1 )(|=1 |2 + |=2 |2 ) ;
dt
2
2
Sy =
dSy
= ?Sx − 4RNSx Sy − (1 + 2 )Sy ;
dt
dSz
= 4RNSx Sy − (1 + 2 )Sz :
(242)
dt
Let us concentrate on this case, and let = 1 = 2 . The sign of ? is irrelevant and we shall change
it, ? → −?. Let us also introduce a new time variable = 2t. Finally, since we have to do with
the semi-classical theory, we may perform an arbitrary scaling of the dependent variables. Let us
choose it in the following way:
Sx =
Rx ;
2RN
Sy =
2
Ry ;
RN?
2
Rz :
RN?
With these substitutions, the mean-Deld equations become simply
dRx
= Ry − Rx ;
d
Sz =
dRy
= #Rx − Ry − Rx Rz ;
d
dRz
= R x Ry − R z ;
d
(243)
404
M. Janowicz / Physics Reports 375 (2003) 327 – 410
where # = −?2 =(42 ). This is evidently the system of the Lorenz equations (confer e.g. with
[126,127]) with the “Prandtl number” . = 1, the “aspect ratio” equal to 1, but with the
negative “Rayleigh’s number”. It is somewhat surprising that the only parameter in the above equations that is not simply a constant does not contain the particle number N , and depends exclusively
on the ratio ?=. Naturally, this is true only if = 0 and 1 =2 . Unfortunately, despite the interesting
relation to the Lorenz model which is so rich in spectacular properties, our system with negative #
is in fact very trivial. Indeed, the origin (Rx = Ry = Rz = 0) is the only Dxed point of the system, it is
asymptotically stable, and it actually is a strong attractor. The system usually reaches the origin quite
fast for not very large values of −#. Physically, it means that the system performs rather trivial,
though non-linear, oscillations, moving from one well to the other with the exponentially decaying
amplitude, and no bifurcations, let alone more complicated eMects, can ever appear. And if we allow
for 1 = 2 , the situation does not improve: the dynamics of the variable S0 = (1=2)(|=1 |2 + |=2 |2 )
do not depend on the other variables of the system, and, as a result, the origin is again the only
Dxed point.
In the paper [121] it has been shown that system (240) without damping has two types of stationary
solutions (except the zero solution): one with populations equally distributed in both wells, and the
other with an unbalanced distribution of atoms over the wells. By repeating the procedure of [121]
we realize that even the smallest damping present in the system completely destroys both types of
non-zero stationary solutions, radically trivializing the mean Deld dynamics; this corresponds to the
above Dndings about the negative “Rayleigh’s number” in the associated Lorenz system.
So far our considerations have rather had negative character. Let us, however, following Holthaus
and Stenholm [121,122], introduce the modulation of the double well with the frequency !, so that
instead of (240) we have
d=1 ?
i
= =2 + 2RN |=1 |2 =1 + (, sin(!t) − )=1 ;
dt
2
d=2 ?
= =1 + 2RN |=2 |2 =2 − (, sin(!t) + )=2 ;
(244)
dt
2
where we have again assumed for simplicity that 1 = 2 = . As a matter of fact, the two-mode
approximation is meaningful only for small condensates with N ∼ 103 at most (cf. [119,121] for
more precise estimations). We shall now assume that the number of condensed atoms is just about
102 and we will try to perform perturbation expansion assuming that 2RN ? and i ? for i = 1; 2.
Our goal is to check to what extent modulation and losses can aMect the tunneling.
To make our computations easier, we introduce the following complex variables:
i
S+ = 12 (Sz + iSy );
S− = 12 (Sz − iSy ) ;
(245)
to obtain the following system of equations for the dynamics of S− , S+ , Sx in the “time” = ?t:
2
i
4iRN
,
Sx S− + 2i sin(jJ) −
S− ;
Ṡ − = − Sx + i
2
?
?
?
2
i
4iRN
,
Sx − i
Sx S+ − 2i sin(jJ) −
S+ ;
2
?
?
?
2
Sx ;
Ṡ x = i(S+ − S− ) −
?
Ṡ + =
(246)
M. Janowicz / Physics Reports 375 (2003) 327 – 410
405
where the dot over the symbol denotes diMerentiation over and jJ = !=?—we have thus assumed
that the modulation frequency is much smaller than ?. One might ask what happens if ! and ? are
of the same order, in particular, what happens when they are equal. Unfortunately, in this case MMS
leads to only very slight simpliDcation, and equations we have to solve in the Drst order are only a
little bit easier than the system we have started with. Below, we obtain the Drst-order approximate
solutions under the above assumption about !. In addition, we require that 4RN=?=j1, 2,=?=jp,
p = O(1), and 2=? = jE, E = O(1).
We form the vector S having the components (S− ; S+ ; Sx ), and write
S = S0 + jS1 + · · ·
(247)
n
as well as Tn = j , and we observe that the zeroth-order problem has already been solved in the
section about the bichromatic excitation of the two-level atom, since in the lowest order we have
9S0
= M̂ 0 S0 ;
(248)
9T0
where

i 
0 0 −

2



i 
(249)
M̂ 0 = 
:
0
0

2 
i
−i
0
The solution is given by
S0 =
3
Aj (T1 ; : : :)vj ej T0 ;
(250)
j=1
where vj , j = 1; 2; 3, are the eigenvectors of the matrix M̂ 0 to the eigenvalues j , and 1 = i,
2 = −i, 3 = 0. Thus, our problem is again singular, since the zeroth-order matrix is non-invertible.
Nevertheless, MMS allows to obtain an approximate solution in this non-linear problem as well.
Since all the Si , i = x; y; z must be real, we have A2 = A?
1.
In the Drst order we have
9S1 9S0
+
= M̂ 0 S1 + N(S) ;
(251)
9T0
9T1
where the vector N contains non-linear terms, modulation and losses:


iSx0 S−0 + ip sin(JT1 )S+0 − ES−0



(252)
N=
 −iSx0 S+0 − ip sin(JT1 )S−0 − ES+0  :
−ESx0
As in the case of the analysis of bichromatic excitation of an atom, we look for the Drst-order
solution in the form
S1 =
3
j=1
Bj (T0 ; T1 ; : : :)vj ej T0 ;
(253)
406
M. Janowicz / Physics Reports 375 (2003) 327 – 410
to obtain, after somewhat boring though simple calculations:
−i
9B1 9A1
+
= √ (A1 A3 + A2 A3 e−2iT0 + pA3 sin(JT1 )e−iT0 ) − EA1 :
9T0
9T1
2
(254)
The equation for B2 and A2 is obtained by taking complex conjugate of (253), while for B3 and A3
we have
i
9B3 9A3
+
= √ (−A21 e2iT0 + A22 e−2iT0 + p sin(JT1 )(A1 eiT0 − A2 e−iT0 )) − EA3 :
9T0
9T1
2
(255)
From the above equations it follows that, to avoid singular terms in the Drst order, we must require
that the following conditions determining Aj be fulDlled:
i
9A1
= − √ A1 A3 − EA1 ;
9T1
2
9A2
i
= √ A2 A3 − EA2 ;
9T1
2
9A3
= −EA3 :
9T1
(256)
Let us observe that the modulation does not change the phases of the zeroth-order solutions, the
latter given by
i
−ET1
;
A1 = $1 exp −ET1 + √ $3 e
E 2
i
−ET1
A2 = $2 exp −ET1 − √ $3 e
;
E 2
A3 = $3 e−ET1 :
(257)
Since we restrict ourselves to the Drst-order approximation, the variables $i , i = 1; 2; 3, which
in principle depend on all higher times Tn , n ¿ 2, have been set constant.
Integrating over T0 we immediately Dnd three Bj :
1
1
B1 (T0 ; T1 ) = C1 + √ A2 (T1 )A3 (T1 )e−2iT0 + √ pA3 (T1 ) sin(JT1 )e−iT0 ;
2 2
2
1
1
B2 (T0 ; T1 ) = C2 + √ A1 (T1 )A3 (T1 )e2iT0 + √ pA3 (T1 ) sin(JT1 )eiT0 ;
2 2
2
1
p
B3 (T0 ; T1 ) = C3 − √ (A21 e2iT0 + A22 e−2iT0 ) + √ sin(JT1 )(A1 eiT0 + A2 e−iT0 ) ;
2 2
2
(258)
(259)
(260)
where Cj , j = 1; 2; 3 are constants of integration which no more depend on T1 ; T2 ; : : : and are to be
found from the initial conditions.
We can see
√ that in the zeroth order the dynamics of Sz = S+ + S− is almost trivial, for in this
order Sz ∼ 2A3 so the dominant feature of the evolution of Sz in time are damped oscillations
with the frequency 1. On the other hand, the zeroth-order dynamics of Sy and Sx contain damped
M. Janowicz / Physics Reports 375 (2003) 327 – 410
407
modulation of the frequency, with the initial modulation depth depending on the initial conditions
for Sz . The latter fact is obviously a consequence of the non-linearity of the system.
8. Conclusions
In this paper we have reviewed several examples of quantum-optical applications of a peculiar
method of changing the variables called the method of multiple scales, with the restriction that only
time scales are multiple. Evidently, there exist systems described by partial diMerential equations in
which we have to introduce many spatial variables as well; the foremost example is the system of
Maxwell–Bloch equations (cf., e.g., [128]) for which the application of MMS leads to the famous
reduced system integrable by the Inverse Scattering Transform. The multiple-scales analyses oMered
in this review have usually been dealing with only few degrees of freedom. For non-linear systems
with many degrees of freedom, MMS on its own does not, in general, oMer very much help. For
instance, in the case of spontaneous emission, the application of the method could be made sound
only via the introduction of the concept of pseudomodes, thus eMectively reducing the number of
degrees of freedom to three. Usually, we cannot expect that MMS provides us with substantial
simpliDcation of a problem with many degrees of freedom without additional insight speciDc to this
problem. This is particularly true about systems with uncountably many resonant and quasi-resonant
modes. It is to be noted, however, that the analysis of partial diMerential equations can also involve
MMS if we may expect, on physical grounds, that one or very few fundamental frequencies are
distinguished. This is illustrated, for instance, in a book by Nayfeh [3] where the non-linear relativistic
SchrIodinger equation serves as an example.
On the other hand, one can also apply MMS in some systems where there no natural small
parameter can be found. For instance, in the case of parametric ampliDers and lasers one can study
the dynamics of the systems close to the Hopf bifurcations, where the small parameter appears
due to the expansion of a physical control parameter near its critical value at which the bifurcation
takes place [22,23].
It is the author’s hope that the examples provided above constitute a convincing argument for further application of MMS in the domain of quantum optics, non-relativistic quantum electrodynamics,
and quantum statistical mechanics, sometimes just conDrming our intuition in a semi-quantitative
way, and sometimes developing it.
Acknowledgements
On various stages of development of the manuscript, the author has discussed the method of
multiple scales with Professors Luiz Davidovich and Stig Stenholm. It is a great pleasure and honor
to thank them for their time and advice.
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