Journal of Molecular Structure, 223 (1990) 425-456
Elsevier Science Publishers RV., Amsterdam - Printed in The Netherlands
425
OPTIMAL DESIGN OF EXTERNAL FIELDS FOR CONTROLLING
MOLECULAR MOTION: APPLICATION TO ROTATION·
R.S.JUDSON
Department of Chemistry and Department of Physics, University of Houston, Houston,
TX 77204-5641 (U.S.A.)
K.K. LEHMANN, H. RABITZand W.S. WARREN
Department of Chemistry, Princeton University, Princeton, NJ 08544-1009 (U.S.A.)
(Received 19 October 1989)
ABSTRACT
A general discussion of quantum controllability leads to the specific focus of this work, namely
the use of tailored radiation to excite rotational states, either specific IJM) states or superposition states which correspond to a high degree of molecular orientation. It is shown that starting
from the 100> state it is in principle possible to produce any eigenstate or superposition state
given a long enough pulse and specific examples are presented. Highly ordered states, which are
useful in a variety of spectroscopic applications, can be prepared by realistic tailored microwave
fields.
INTRODUCTION
It is a great pleasure to be able to contribute to this volume honoring E. Bright
Wilson and his extraordinary contributions to science in general. One of us
(W. S. W.) worked with Wilson as his last undergraduate; two of us (K. K. L.
and H. R.) were graduate students at Harvard, and K. K. L. was also a Harvard
Junior Fellow. We have vivid and fond recollections of Wilson as a teacher and
mentor. His work at the forefront of developments in vibrational and microwave spectroscopy, detailed elsewhere in this book, has set a standard for excellence that can only be admired.
This paper will concentrate on extensions into the microwave regime of a
recent development in other types of molecular spectroscopy; the availability
of precisely controlled radiation fields, with arbitrary phase shifts and arbitrary amplitude or phase modulation. The uses of highly controlled, complex
pulse sequences to fundamentally alter the quantum mechanical dynamics of
the system was pioneered in NMR, where it has been possible for twenty years
to turn off intermolecular dipole-dipole couplings (a homogeneous broadening
mechanism) with repetitions of a sequence of four rectangular envelope pulses
*Dedicated to Professor E. Bright Wilson, Jr.
0022-2860/901$03.50
© 1990 Elsevier Science Publishers RV.
426
with phase shifts [1]. As the mathematics and technology progressed, ever
more sophisticated NMR sequences permitted vast improvements in sensitivity, resolution, and applicability ofthe technique (see, for example, ref. 2). In
the last few years the restriction to constant amplitude, constant frequency
pulses has disappeared (for a review see ref. 3). The effects of pulses with
envelopes as sophisticated as (sech(aT) )l+5i have been analytically derived
[4,5] and experimentally explored [5] in NMR and its medical cousin, magnetic resonance imaging [3]. In addition, computerized optimization [3,6] has
produced quite nonintuitive waveforms which are extremely successful in selective excitation of a small region of a spectrum [6,7], suppression of solvent
peaks [8], or effective time reversal of the dynamical evolution [9].
In a very different frequency regime, theoretical investigations of the effects
of highly controlled optical or IR fields [10-12] have been spurred by technological breakthroughs permitting arbitrarily shaped, phase and amplitude
modulated laser pulses with 100 fs resolution [13,14] and 10 MW peak powers;
neither of these is a fundamental limitation. One important potential application of tailored pulses is atomic site-specific modification of complex molecules (e.g. selective bond breaking). This task has been an elusive goal ever
since the invention of the first practical laser. In essentially every large molecule, energy can only be deposited in a specific bond for a short period of time
due to nonlinear resonances which transfer energy into other modes. In quantum language, the optically bright state, which carries the oscillator strength,
"dephases" because it is not an eigenstate. This intramolecular vibrational
"relaxation" (IVR) is blamed for the failed attempts to do "laser selective
chemistry" except in a few simple cases, though it is important to recognize
there is precious little experimental evidence that indicates that the dephased
state is like a microcanonical distribution as the term "relaxation" implies.
Intense radiation fields can fundamentally change the molecular eigenstates, and can thus affect intramolecular dynamics. Under these conditions
the optical field can no longer be treated as a perturbation, and knowledge of
the weak field atomic spectra may be of little use in designing the pulse field.
Mathematically, designing a field for selective bond breaking poses an inverse
problem for which we seek a stable solution; the complexity of this task, recalling the IVR problem mentioned above, precludes obtaining a field design
by simple intuition alone. The overall approach proposed by Rabitz is based
on techniques of optimal control theory developed primarily for other applications in engineering (see, for example, ref. 15). In essence, a functional J is
defined which contains the physical objective (for example, the desired nature
of the ultimate molecular state), costs or penalties (for example, the production of other excited states and wasteful pump field fluence) and the system
equations of motion. The functional J contains the unknown optical field f (t ),
which is optimized by minimizing J with respect to the field. One specific success of this approach has been the design of pulses which quantitatively trans-
427
fer vibrational energy from one end of a linear chain molecule to the other
. [12,16]. The optimum waveforms can be extremely sensitive to uncertainties
in the Hamiltonian, and this question of robustness is one of the focuses of
current research efforts; explicit inclusion of minimal sensitivity criteria and
statistical averaging into J can improve robustness [17].
After a general discussion of the concept of quantum controllability, we will
specifically concentrate in this manuscript on the use of tailored radiation fields
to excite interesting rotational states-either specific IJM) states or superposition states which correspond to producing a high degree of molecular orientation. In fact, this case is much more attractive for design and laboratory
investigation than the vibrational problem mentioned above, because like the
NMR case the exact nature of the Hamiltonian and its eigenstates (the spherical harmonics) is known. The Hamiltonian is taken here to be a rigid rotor,
which will be valid for virtually any molecule at short times; known vibrationrotation coupling effects or any other spectral perturbation could be included
as well. We show that, in principle, starting from the 100) state it is possible
to produce any eigenstate or superposition state given a long enough pulse, and
present specific examples of our numerical results for experimentally realizable laboratory fields and pulse lengths. In the simplest cases (for example,
exciting the 111) state with a small enough Rabi frequency to permit neglect
of all but the 0-+ 1 transition) the answers are obvious, but the optimal waveforms become less intuitive as intensities increase or the target state gets more
complex. We also briefly summarize current technology, explain how these
waveforms could be generated in the laboratory, and discuss the general consequences of control over rotational motion.
GENERAL FORMULATION OF THE CONTROL PROBLEM FOR THE MATRIX FORMULATION OF QUANTUM MECHANICS
Initial conditions and equation of motion
In this section we briefly review the equations of motion for a general system
in the density matrix framework to fix our notation. In the following section
we discuss what can be learned about quantum controllability in general and
then proceed to specialize to a rigid-rotor diatom.
We first define the Hamiltonian as
Yt'(t) =Ho + V[f(r,t)]
(1)
where H o is the nominal time independent Hamiltonian with eigenstates In)
and eigenvalues Em and the time dependence ofthe external potential V[ to (r,t) ]
is determined by a set of control fields f(r,t) whose temporal and possibly
spatial distribution is to be determined. In most cases of practical importance,
the external potential is V[f(r,t)] =f(r,t)*,uwhere,u is a time independent
428
quantum operator that depends only on atomic coordinates. For interaction
with electromagnetic fields in the dipole approximation, f(r,t) is just a component of the applied electromagnetic field and J1 the corresponding component of the dipole moment operator. We treat the control fields as classical
variables that we can manipulate at will, which implicitly implies that we are
neglecting the back reaction of fields generated by the molecules, Le. propagation effects. At sufficiently low density this will always be an adequate approximation. Since realistic laboratory fields have negligible gradients over the
size of a molecule, we will treat the fields as functions of time alone, though it
is recognized that in practice this time dependence may come about at least in
part by a molecule traveling through spatially modulated static control fields.
The initial density matrix for the system in the basis that diagonalizes Ho is
assumed to have matrix elements of the form
(2)
where P n is the fractional population in the nth eigenstate. At equilibrium the
values of P n satisfy a Boltzmann distribution. This is the way we find systems
in the laboratory before we begin interaction with them. Of course, this matrix
is not generally diagonal if a basis other than the eigenbasis is chosen. The
time evolution of the system is given by the propagator U
p(t) =U (t,O )p(O )ut (t,O)
(3)
which sometimes does not have to be completely evaluated. For example, if
only N levels are initially populated, only N columns of U are needed to find
p (t ). The equation of motion for the propagator is
ilz(dUjdt) =H(t)U
(4)
Note that the columns of U can be propagated independently of one another.
The expectation value of any operator 0 is given by Tr (pO).
THE QUANTUM CONTROL PROBLEM
Theoretical considerations: weak and strong controllability
Our objective is to find a set of optimal controlling fields f (t) which drive
the initial state to a final state that produces desired values for a set of observabIes. The feasibility of the goal is embodied in the controllability of the system. Thus, before we outline the general optimal control formalism we have
used, we want to discuss what we expect is and is not achievable given the
Hamiltonian structure of quantum mechanics. Within quantum mechanics
some formal general results are known [18]. For example, within the density
matrix framework as we have presented it (i.e. without relaxation mechanisms), eqn. (3) implies that the final density matrix is related to p (0) by a
429
unitary transformation, so the eigenvalues of the density matrix are conserved
quantities; equivalently, we can note that Tr(pN) is conserved, where N is
arbitrary. This implies that a pure state (which corresponds to a density matrix with one eigenvalue of 1 and the rest 0 and Tr (p2) = 1) and a mixed state
(which has more than one nonzero eigenvalue and Tr(p2) < 1) cannot be interconverted. The entropy of a quantum system cannot be changed by control
fields in the absence of relaxation processes.
Given the above limitation we define a quantum problem as strongly controllable if a set of time dependent control fields exist such that any unitary
transformation can be produced as the time evolution operator. If we can produce this transformation at any time we wish we call the problem strongly
controllable in any time. If there is a minimum time necessary to produce this
transformation we say the problem is strongly controllable in finite time, and if
we can only generate some transformations as a limit as time goes to infinity
then we say the problem is strongly controllable in infinite time.
If we can generate any unitary transformation, we can map any pure or mixed
state into any other pure or mixed state, consistent with conservation of the
eigenvalues of the density matrix as described above. It may happen that we
cannot generate any unitary transformation, but that we can at least map any
pure state into any other pure state. This is equivalent to being able to generate
a unitary transformation with any desired column, but not being able to specify
the other columns at the same time. We call this a weakly controllable system.
As before we can distinguish between the three time classes of weak
controllability.
It turns out that a straightforward procedure exists to decide, at least in
principle, if a system is strongly controllable in either any or infinite time. Any
control field can be treated as the limit of a chain of infinitesimal, delta function kicks by the control fields with free propagation for an infinitesimal time
between kicks. Each kick generates a small rotation in the Hilbert space. Each
control field, such as EX' Ey , E z , will have a generator /1k associated with it and
will produce a rotation given by exp(iE/1k)' By applying four infinitesimal rotations of E/1j, E/1k' - E/1j, and - E/1k we generate a second order rotation of
iE 2[/1j,/1k]' Thus the set of all possible rotations, without time propagation is
just a Lie group associated with the Lie algebra given by the linear vector space
that includes the generators and is closed under commutation. For finite n
dimensional quantum spaces, this Lie group will include all possible unitary
transformations of the n states if and only if the Lie algebra covers the n X n
dimensional real vector space of n X n Hermetian operators. If the space is
closed except for the identity matrix, we also consider that controllable since
the transformed density matrix is invariant to an addition rotation by a constant times the identity. Such a system will be strongly controllable in any
time. If adding the Hamiltonian operator as a generator produces a Lie algebra
that covers the space of Hermetian operators, then the system will be control-
430
lable in infinite time. If the eigenvalues of the Hamiltonian are rationally related, then free time evolution by H will be periodic, and as a consequence if
the system is controllable in infinite time it will also be controllable in a finite
time as well.
As an example, consider a two level system, such as an isolated spin -1/2.
To be fully controllable we must span a four dimensional space. In the rotating
frame we have two control fields, Ex and E y , which are related by a phase shift
and on resonance have generators (Jx and (Jy. The commutator of these two
generates (Jz, and then the algebra is closed. Adding in the identity we directly
have a complete four dimensional basis and thus this system is strongly controllable in any time; if the initial density matrix is
p(O) = 1 +a(Jx + p(Jy +Y(Jz
we can produce any other density matrix
p(T) = 1 +a' (Jx + p' (Jy +Y' (Jz
as long as
a 2+ {f+y2=a,2+ p'2+y'2
We can do this with a series of pulses with any desired total length, although
the pulse intensity obviously goes up as the length goes down.
If we only have one control field, the two-level problem will clearly not be
controllable in any time. However, time evolution generates rotations around
Do =flJw(Jz
One fundamental difference from the case above is that, in principle, large
enough (Jx and (Jy fields can generate any rotation we want in an arbitrarily
short time, but the Hamiltonian can only help to generate an arbitrary rotation
if the length T of the sequence is such that IIHo TII ~ 1. Thus the problem will
be strongly controllable in finite time since the Lie algebra produced by the
generator and Do is complete (except for the identity), as long as the irradiation is not at exact resonance (Jw = 0 ).
A more interesting example is provided by a nondegenerate quantum spectrum that is connected, by which we mean any two states have a path (which
may require several intermediate states) of allowed transitions connecting
them. The system will not be connected if there is a symmetry operator that
commutes with both the generators and H o; this could be an obvious spatial
symmetry, but it could also be due to some "hidden" symmetry or to some
accidental combination of coupling matrix elements and thus be missed. If we
consider pumping times as long compared to the separation in frequencies of
allowed transitions, then off resonance pumping can be neglected and we get
effectively two independent generators for each resonance frequency that corresponds to the in-phase and out-of-phase components of the control field at
431
that resonance frequency. These are proportional to (Jx(ij) and (Jy(ij), connecting the two states i,j. Commutation of these generators with (Jx Uk) and
(JyUk) produced on a transitionj,k will produce (JAik) and (J)ik) [19]. Each
(Jx and (Jy generates the corresponding (Jz. This set will clearly be complete
except for the identity. Thus we see that a general connected quantum system
with a nondegenerate spectrum .will be strongly controllable in finite time as
long as we can neglect off resonance pumping. If one uses a smooth pulse, the
errors introduced by off resonance pumping will decrease exponentially as the
pulse becomes longer. This case includes many important problems including
the bound part of an anharmonic oscillator.
Strong controllability is important if we are dealing with a system that starts
out in a mixed state. For most molecules even at 1 K a number of rotational
states are populated and thus we start in a mixed state, unless we state select
using electrostatic fields or a laser pump. Controllability in a finite time is
important in that collisions and time of flight will limit the coherent interaction time of a molecule; the practical constraint of reasonable field intensities
implies that some minimum time will be needed to effect a transformation as
well. More subtly, longer time interaction means we must consider weaker
intramolecular interactions which complicate the Hamiltonian structure, such
as centrifugal distortion and hyperfine interactions. If our pulse can be made
short compared to the frequency splittings produced, we can ignore these effects.
It is important to note that at least some quantum problems are not even
weakly controllable in infinite time. If we consider a harmonic oscillator with
linear dipole moment, then the Lie algebra closes with x (the generator), H
and p. Thus only propagators of the form exp (i (aH + fix + yp» can be produced and the system is far from controllable. If one starts in the ground state
then only coherent states can be produced, a well known result from the quantum theory of light. A pure number state can never be generated.
Another illustrative example is provided by a spin 1 or greater nucleus in a
static magnetic field with no quadrupolar coupling. Thus H is proportional to
1z , the generators of the control fields are just I XJ 1y , and 1z , and the Lie algebra
closes with only these three operators. Notice that all the allowed transitions
are degenerate. As an example of what this implies consider the case of spin 1,
with all ofthe initial population starting in m[= -1. This initial density matrix
can be written as
0 0 0]
0 0 = (Iz 2-1z )/2
[ 001
p(t=O) = 0
(5 )
Since the set of possible transformations is equivalent to the set of rotations
of the coordinate axis, the propagator must take 1z to a linear combination
a1x + fi1y +y1z , where a 2+ ~+y2= 1
432
P (T) = «alx+ply+ylz )2- (alx +Ply + ylz ) )/2
(-cdx-Ply-ylz+a2Ix 2+p 2Iy2+fT;+ap(lx1y+lylx)
=
(6)
+ yP(lzIy + Iylz) +)la (lzIx+ lJz) ) /2
Explicit calculation of the 3 X3 matrices corresponding to each of these operators shows that the only ones with P22 (the population in m/=O) #0 are l x2
and I y 2; both ofthese have a numerical value of 1 for that matrix element. Thus
(7)
and it is impossible to transfer more than 50% of the population to the M = 0
state. In NMR parlance, the quadrupolar order of the system is conserved.
Note, however, that if the degeneracy of the two transitions is broken, it is
possible to populate the middle state completely; the total set of operators in
the Hamiltonian will then include a quadrupolar term proportional to I z 2 , and
more complex rotations are possible. The system is then strongly controllable
in finite time because the spectrum is nondegenerate.
Computational considerations
In this section we outline the general optimal control formalism appropriate
for the matrix formulation of quantum mechanics. Given some initial density
matrix Po and some target time T with corresponding target values of a set of
observables 0;;, we wish to find values for EO (t) that minimize the cost functional
T
C< = ~wa[Oa(T) _O;;]2+~ f dt[EO(t} PW(t)f(t)
o
(8)
T
+ ~Ta~f dt[Oa(t) -O'!'x]2+ other terms
o
The factors W ao T a and W(t) are freely adjustable parameters that allow different components of the cost functional to be given different weights. Our
immediate goal of having the system observables approach the target values
O'!'x at t =T will be effected by minimizing the first term in Ce The second term
in C< allows us to restrict the magnitude of the control fields. Minimizing the
third term attempts to ensure that the trajectory in the space of observables
achieves the final state as rapidly as possible. It is a flexible option to keep such
terms and the cost functional can include a host of other possible terms to
balance experimental, theoretical, and numerical issues.
In order to find the time behavior of f (t) that minimizes en we use the
standard Lagrange multiplier formalism [20] that allows us to directly incor-
433
porate the equations of motion as a constraint. We first define a new cost
functional
(9)
4
4
where Ano (t) are (complex) Lagrange multipliers corresponding to Uno (t).
Notice that CE=CE(the term in square brackets vanishes) so that minimizing
CE will likewise minimize CE'
The next step is to find the functional gradient of CE with respect to f (t)
that will allow us to define the optimal direction to change f in order to minimize the cost functional. To simplify the following discussion, we will only
treat the first two terms in eqn. (5). The procedure is identical when other
terms are included. The functional derivative, denoted by ~CJM(t') is given
by
_
T
~CE _"
0
OT~Oa(T) ~f 4) t
)~(
f
~t(tf)-~2Wa[ a(T)- a] ~t(tf) +T dt[f(t ] W(t u t-t)
o
(10)
We now need an expression for ~Oa(T)/M(tf) which is given by
(11)
Equation (10) can be simplified by integrating by parts the term containing
(12)
T
-
~
t
fd tAno
4
()t ~Uno
•~t(tf )
o
The second term
on the right-hand-side is identically zero because the initial
4
conditions [Uno (t =0 ) ] are independent of the field. If we now combine eqns.
(10)-(12) we arrive at
434
...
If we choose final conditions and time evolution for Ano (t) as
(14)
(15)
then this reduces the expression for the functional derivative of the cost functional to
(16)
The following algorithm is used to find the minimizing values for the fields:
(1) choose t(t);
...
(2) integrate the equatio~s for the propagator Uno forward in time;
(3) calculate Oa(T) and A no (T);
...
(4) integrate the equations of motion for Ano (t) backward in time;
(5) calculate E/ O€ (t' );
(6) determine an improved set of 1(t );
(7) repeat steps (2)-(6) until convergence is reached.
This procedure will typically converge after a relatively small number of iterations. Note, however, that it is possible that it will converge to a point in
observable space that is not near the desired target point. This may indicate
that there is nO time dependence of the set of fields being used that will drive
the system to a state that produces that set of values for the observables. Al-
be
435
ternatively, it may mean that the weights used in the cost functional prohibit
satisfactory optimizing fields from being found.
The improved form of f (t) (step (6) above) is optimally calculated using a
conjugate gradient algorithm which we outline here. First, we make the
definition
-+
<5C,
gi(t) = - <5f;(t)
(17)
where the subscript i specifies which iteration is being considered. We then
define the conjugate gradient function [21] as
-+
-+
fo(t) =go(t)
-+
(18)
-+
-+
fi+l (t) =gi+l (t) +Yi(t)fi(t)
where
(19)
The new fields are then given by
-+
-+
...
fi+l (t) =fi(t) + Pfi(t)
(20)
The value of P used minimizes C, and is found from a one-dimensional line
search.
As a special case of this formalism, we might wish to start with all the population in the ground state of the system, and find a control field that will
produce a desired superposition state, with all relative phases specified, i.e.
where each <m I U( t,O) Ino) is specified up to some overall phase. The control
formalism in this case is very similar to that just derived and here we briefly
indicate the differences. In this case, we wish to specify the final wavefunction
rather than the values of some set of observables so we use the cost functional
(21)
As before, additional terms could be included. We form C, (eqn. 9) except the
sum over no is dropped because only one initial state is included. The Lagrange
multipliers satisfy the same equations as before, except that their initial conditions are now
-»
A(T)
-»
=Uno(T) -
-+r
Uno
(22)
The gradient of the cost functional (eqn. 16) is unchanged as is the algorithm
for finding the optimal control field.
436
APPLICATION TO ROTATIONAL CONTROL OF POLAR DIATOMS
Initial conditions and equations of motion
In this section, we give the equations of motion for a polar diatom, treated
as a rigid rotor, in a time varying electric field given by
...
E(t) = [Ex(t),E)t),EAt)].
The equations of motion are
:t<jml U(t,O) liomo) =
-;/:1, {Boj,j' Om,m' i' (j' +
1)
(23)
TtJ ,m
+ <j,ml V(t) Ii' ,m')} <j' ,m'! U(t,O) liomo)
where B=1i 2/ (211'2), J1 is the diatom reduced mass and r is the internuclear
separation. The matrix elements of the potential are given by
21<
<j,mIV(t)lj',m')=
=
+1
fd¢>
f
o
-1
d(cos())Yjm«(),¢»E(t)·ilYj'm'«()'¢»
'ill [EAt) <j,mlxli' ,m')
(24)
(25)
+ Ey(t) <j,m Iyli' ,m') + Ez(t) <j,m lili' ,m') ]
and
1
<j,m!xli' ,m') = (-1)m+1j2J
(2j+ 1) (2j'
x{(j
1
1
m
j' )_(j
m'
m
1
+ 1) ("~
-1
1
0 J")
o
j')}
m'
<j,mlYIi',m')=(-1)mfiJ(2j+l)(2j'+1)~~ ~)
X
{( ' 1"')+(" 1 ~,",)}
J_
m
1 J ,
m
J_
m
(26)
(27)
-1
(28)
(" 1"')
The terms such as J_ m 0 ~, are 3- j symbols.
437
Controllability of the rigid rotor problem
The issue of controllability of the diatomic rotor problem is nontrivial, since
transitions involving different m levels are degenerate in the absence of some
additional perturbing field. If we must rely on breaking this degeneracy by
electric and magnetic fields to get controllability, then the sequence will surely
have to be very long compared to the induced splittings, and probably will be
extremely difficult to implement. Fortunately, we can prove that this system
is weakly controllable, at least in infinite time. The proof is by induction; it
works as long as we consider only initial and final states that have only levels
with finite J excited, and actually construct a field which would create the
desired transformation. We will prove this general result in pieces.
(1) Starting from the ground state I0 0> we can create any other Ii
state. The selection rules are I1j = + 1 for absorption, 11m = 0 for + z-polarized
fields, 11m = 1 for x + iy-circularly polarized fields, and 11m = -1 for x - iy-circularly polarized fields. Furthermore, since absorption for different values of
j' are at vastly different frequencies, the Rabi frequency (w to) /h can be made
sufficiently small (and of course then the pulse sufficiently long) to make this
a collection of perfect two level systems. Thus, as an example, the state 152 >
could be populated by the pathway 10 0> ~ 11 0 > ~ 12 0 > ~ 13 0 > ~ 14 1> ~ 15
2>, where each arrow represents a n pulse (constant field, length T such that
(w to) T/ h = n) at the exact resonance frequency, the first four arrows represent
z fields, and the last two arrows represent x + iy fields.
(2) Starting from I0 0>, we can create any pure state density matrix we
desire consisting solely of population and coherences for j = 0 and the three
j = 1 states. This is done with three consecutive electric field pulses with different polarizations at the o~ 1 resonance frequency, transferring population
from 10 0> exclusively to 11 1>, 11 0> or /1 -1> respectively. The phases of
the coherences are completely controlled by the phases ofthe irradiating fields,
which are adjusted to produce the exact desired state.
(3) Given the ability to produce any arbitrary pure state density matrix
involving states up to some value j, we can produce any desired distribution of
population in the j + 1 states, with the rest of the population in the I0 0> state
(which can then, by this same assumption, be redistributed to fill all of the
states up to j as desired). This is done by the following:
(a) First populating Ii + 1 j + 1> by j + 1 consecutive pulses with x + iy circularly polarized light. The first pulse (O~1 resonance frequency) transfers
the exact amount of population to 11 1> which will ultimately be wanted in
Ii + 1 j + 1 >; the last j pulses are n pulses. The phase of anyone of these pulses
can be adjusted to give the proper phase to the coherence.
(b) Next populating Ii + 1- (j + 1) >in exactly the same way, using x - iy
circularly polarized light.
m>
438
(c) Creating a superposition state with each of the Ij m> levels having the
population that we will ultimately want in the Ii + 1 m> level, and coherence
with the 100> level which is related to the ultimately desired coherence as
specified below.
(d) Performing an adiabatic sweep from far below the j -+j + 1 resonance to
far above resonance with Z polarized light. In practice, the (sech(aT»1+5i
pulses measured in the introduction do such a sweep, with essentially no perturbation of states outside of the nominal bandwidth. The phase shift induced
by the adiabatic passage is solely a function of the value of the Rabi frequency,
which is of course different for the different Ii m> components, but since the
dipole moments of these transitions are calculable the initial phases of the
coherences are simply chosen to counteract this shift.
Step (3) corresponds to a proof by induction that any pure state density
matrix can be created starting from I00>. In each case the highest j levels are
filled first.
(4) This entire process can be inverted to give a prescription for starting
from an arbitrary superposition of lim> eigenstates and putting all of the population in 100>. From there it could be redistributed into any other superposition state, as in step (3).
Formally this proof by induction only works for infinite time, because it uses
n pulses and adiabatic sweeps with infinitely low power to avoid populating
any but the desired transition. In practice, however, the separation between
transitions is very large, and extremely good approximations to the desired
states can be made in quite short times.
Furthermore, we have numerically created the Lie algebra generated from
the 3 finite dimensional generators for x, y and Z fields. The system is clearly
not controllable in any time, as can be seen by examining only the four states
corresponding to J = 0 and J = 1. One can choose basis functions for J = 1 that
are x, y and z polarized such that each is connected to the J = 0 state only by
the electric field of the same direction. Without time evolution, if we start in
the 10,0> state we can create any state a 10,0> + b \l,x >+ c 11,y >+ d Il,z > but
with only real values for a, b, c and d. With time evolution, the phase of J =1
states will change relative to J = 0 and complex expansion coefficients can be
generated. Looked at from the density matrix perspective, any magnitude of
coherence can be generated but with fixed phase unless time evolution occurs.
Thus a time period on the order of 1/ (2B) is needed to fully control the problem.
Considering the nine dimensional state space for J =0-2, we find that the
Lie algebra closes with only 36 dimensions out of a total of 81. Including H,
the algebra closes only after all possible 9 X9 Hermetian operators are spanned.
If we restrict ourselves to only two polarizations of the control fields, then the
problem is definitely not controllable even in infinite time. For J =0 through
J = 2 only a 44 dimensional subspace (including the identity) ofthe 81 dimensional total space is generated from the two field generators and H. Adding the
439
Hamiltonian does not change the size ofthe space. Based upon these numerical
calculations, we believe that with three fields the rotor problem is strongly
controllable in finite time, at least for any finite limit on J. At present we know
of no general proof of this claim but are working on one.
Formal controllability and practical controllability are two different issues:
the "prescription" provided above for preparing a superposition is certainly
highly inefficient and probably impractical because of relaxation effects. A
proof of formal controllability only shows that some set of control fields, with
no limits on either their amplitudes or bandwidth, can achieve the desired
transformation. In the laboratory, there are constraints on the fields and bandwidths that we can apply because of available sources. It is important to note
that there is a strong anticorrelation of power and bandwidth. The role of the
optimization is to find practical waveforms that take into account realistic
laboratory constraints (for example on field, bandwidth, and interaction time).
These kinds of practical constraints are extremely difficult to include in attempting to calculate waveforms from first principles, but easy to include in
computerized optimization. Further, we may wish to look for solutions that are
robust to small errors in the laboratory fields or our knowledge of the precise
molecular Hamiltonian. An important part of this procedure is to not ask of
the system any more than we require since overspecifying the final state will
restrict the size of the solution space.
Specific applications: pure initial state-+ Ijm) eigenstate target
In this section we present results for a series ofcalculations where the diatom
was in a pure eigenstate Uomo) of the system initially (in all cases the state
100) ) was chosen) and the objective was to drive all molecules to a specific
eigenstate UTm T). To simplify the notation we will make the definition
aj,m(t) = <jml U(t,O) Uomo)
(29)
Thus the final wavefunction is
(30)
In all cases we have minimized the cost functional
C, = L lajm(t) -ajmT l2
j,m
(31)
where ajm T = t5j,jT t5m,m T. This is not the only cost function that could be used.
This functional demands that the total overall phase of the final wavefunction
be specified, while in practice such a control is never needed unless this is just
a subset of some larger, coherent problem. It is also somewhat artificial, because it requires knowledge of the absolute phase of the initial wavefunction
as well. However, the additional constraint imposed by dictating the absolute
440
phase is trivially satisfied by shifting the time origin. Another cost function
would be to maximize the square overlap of the final state with the target; this
would treat all solutions that differ only in final phase as equivalent. All calculations are done in a basis including states up to j = 4.
The molecule we have chosen is KCI both because of its large dipole moment
(u=10.27 D) and its small rotational constant (v o=2B/h=6.97 GHz). An
even more favorable case with respect to laboratory considerations is CsI which
has a rotational constant of 708 MHz and a dipole moment of 11.6 D. The
results are a function only ofreduced time=B*t and reduced field=E*,u and
are thus applicable to any spinless diatom by appropriate scaling. For molecules with rotational constants less than about 20 GHz, it is not possible to
create a pure state merely by cooling in a strong expansion. A more realistic
goal would be a final density matrix, but for our initial calculations we have
taken a much more modest goal.
The first target state we will consider is 110). If the target time is very long,
the algorithm should only irradiate at the 0-+ 1 transition frequency (6.97 GHz)
with z-polarized light, and should give a n pulse. However, within these constraints there are many different possibilities. For example, any purely amplitude modulated waveform (w EAt) =Qh (t) cos(2n Vo t» which satisfies the
constraint
+00
f
Qh (t )dt= (2N + 1)n
(N arbitrary)
(32)
t= -en
will perform equivalently at the level of this calculation; however, cost functions which minimize the peak power for a given pulse duration will make the
program tend toward a rectangular pulse with N = O. Frequency swept waveforms which approximate a full adiabatic passage will work as well and would
have been favored if the cost function tried to minimize sensitivity to field
amplitudes.
Figure 1 (a) shows the results of this optimization for a pulse duration of 1
ns. The approximation that the peak power is low enough to make only two
levels nearly resonant is quite good, but not perfect (Wl =2nXO.5 GHz, so the
1-+2 transition at 13.94 GHz is about 14wl from resonance). Nonetheless, a
small amount of population would be transferred into 120) by a simple rectangular pulse. The optimized waveform prevents this with a small frequency
component at the 1-+2 transition frequency, as can be seen by the Fourier
transform in Fig. 2(a). A solution that tried to minimize bandwidth instead of
amplitude would have solved this problem with a smooth (e.g. Gaussian) n
pulse at 2B. Such a pulse would have higher peak amplitude but would have
exponentially reduced the level of off resonance pumping.
As the pulse duration is decreased the necessary peak power to satisfy eqn.
441
Target state (1.0) T=1000
(a)
Te.rget state (1.0) T=500
(bl
10
10
a
(x 10)
a
(x 10)
u
u
'-
'OJ
5
~
=:0
~
5
~
...
.d
:i
0
llll
.....
~
.....
d
0
III
III
rn
rn
'";;
'"
.:
;; -5
-5
..
1..
I
-10
-10
1000
0
t (psee)
Te.rget state (1,0) T=200
(e)
500
0
t (psee)
Te.rget state (1,0) T=100
(d)
10
10
e
e
u
"
'OJ
'-
5
!..
~0
5
~
.d
..
.d
0
~
III
......
;;
'"
.:
0
~
b
rn
;g -5
III
III
rn
-5
1..
.
I
-10
-10
200
0
t (psee)
Te.rget state (1,0) T=50
(e)
a
100
t (p,-ee)
10
e
u
'OJ
...
5
l..
.d
0
~
III
b
rn
'";;
r..
-5
-10
L/
a
50
t (psee)
Fig. 1. Waveforms produced by computerized optimization to transfer population from 100) to
110> without populating higher states. given a specific total pulse duration. If the duration is very
long, a simple n pulse can be resonant with the 0-> 1 transition with a low enough Rabi frequency
to avoid pumping higher j states. As the duration decreases perturbations to other levels increase,
the waveform becomes more complicated. and ultimately the objective of 100% transfer becomes
impossible as noted in Table 1.
442
la)
Target state (1,0) T",lOOO
Target state (1,0) T"'500
(b)
..,...
...'a'"
~
,...,
'a::>
::>
....
t
.....
p,
III
III
~
:0
..0
!!p,
!!
'"
"§
lJ
t-
::::
l:l
.....s
.....s'"
III
0
Ie)
10
20
30
40
Frequency (GHz)
0
50
Target state (1,0) T=200
ld)
..,::::
,~
10
20
30
40
Frequency (GHz)
50
Target state (1,0) T=100
...'a'"
~
:§
....
::::
::>
p,
p,
!!p,
!!p,
....
III
III
~
..0
..0
::::
~
.....s'"
.....s'"
l:l
l:l
III
III
o
(e)
o
10
20
30
40
Frequency (GHz)
50
0
10
Target state (1,0) T=50
10
20
30
40
Frequency (GHz)
50
Fig. 2. Magnitude spectra of the waveforms in Fig. 1.
20
30
40
Frequency (GHz)
50
443
(32) increases, and thus the two-level approximation weakens. Thus the waveform must become more complicated (as seen in Fig. 1) with new frequency
components introduced (Fig. 2). For T ~ 200 ps substantial asymmetry appears in the waveform, and for T ~ 100 ps the computer program could not find
a solution which completely populated 110) (see Table 1) despite the fact that
we are assured such a solution exists by the arguments given above.
In Figs. 3 and 4 we show corresponding results when the target state is j = 1,
m = 1. In this case, the x andy fields drive the excitation by producing circularly
polarized light. Initially one might suspect that, except for the difference of
producing x + iy polarization or z polarization, the results would be identical to
the case of trying to pump the 110) state, but comparison with Figs. 1 and 2
shows substantial differences when the field intensity is large. The reason is
straightforward. The standard derivation of the effects of a 7C pulse relies on
the rotating wave approximation (RWA) [22], which resolves the linearly polarized field into its two circularly polarized components. The effects of the
cosine wave z-polarized field are approximated by a rotating wave, with the
assumption that the counter-rotating component is so far from resonance that
it is negligible. In this calculation the relatively large Rabi frequency implies
that this is not a perfect assumption, so particularly at high peak powers the
TABLE 1
Final population in target state and maximum required field strength
Run
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16d
Target state8
T (ps)
Populationb
(1,0)
(1,0)
(1,0)
(1,0)
(1,0)
(1,1 )
(1,1)
(1,1)
(1,1)
(1,1)
(2,0)
(2,0)
(2,0)
(2,0)
(2,0)
k=5
1000
500
200
100
50
1000
500
200
100
50
1000
500
200
100
50
1000
1.00
1.00
1.00
0.88
0.71
1.00
1.00
1.00
0.97
0.67
1.00
1.00
0.98
0.97
0.56
1.00
0.17
0.36
0.90
9.88
4.49
0.12
0.25
0.78
3.09
5.95
0.52
1.06
5.97
13.37
21.27
0.89
8Numbers in parentheses are (j,m) of the target state. bThis number is the final fraction of the
wavefunction in the target state. cEmax is the maximum value ofthe driving field during the time
the field is on. Values are in kV cm-l. dThis run is for an oriented target with k=5 as described
in the text.
444
Target state (1.1) T=1000
(a)
Target state (1.1) T=500
(b)
10
10
(x 10)
a
(x 10)
a
Cl
Cl
'OJ
'III
5
:=0
5
:=0
~
~
...
.....
Gl
::l
:f
Gl
fIl
fIl
'tl
:g -5
Gl
;;::
..d
0
tllI
0
....
d
Ol -5
.
f.
I
-10
-10
1000
0
t (psec)
Target state (1.1) T=200
(e)
0
(d)
10
500
t (psec)
Target state (1.1) T=100
10
(x 10)
a
a
Cl
Cl
'III
'III
5
:=0
:=0
5
~
~
...
..d
...
..d
0
tllI
tllI
0_
)/-
.....
d
d
Gl
Gl
b
fIl
--,
/'
'\
fIl
'tl
'tl
-5
Ol
r..
]
I
-5
.
i
-10
I
-10
0
200
t (psec)
100
0
t (psec)
Target state (1.1) T=50
(e)
10
a
Cl
'OJ
!...
..d
:f
Gl
.....
5
0
fIl
:g -5
Gl
7.
-10
0
50
t (psec)
Fig. 3. Waveforms produced by computerized optimization to transfer population from 100> to
111 >without populating higher states, given a specific total pulse duration. At low powers the
only real difference between Figs. 1 and 3 is the choice of x+ iy polarization instead of z polarization, but at higher powers the waveforms are significantly different because the rotating wave
approximation has to be made for the linearly polarized field of Fig. 1.
445
Target state (1,1) T=1000
(a)
Target state (1,1) T=500
(b)
...
'OJ
-a:;J
.
I>.
.....
:a
CIS
~
I>.
::::
III
....!3l:l
Ql
\ ~.........
.....
I
0
(e)
10
20
30
40
Frequency (GHz)
Target state (1,1) T=200
50
0
10
20
30
40
Frequency (GHz)
Target state (1,1) T=100
50
0
10
50
(d)
.,...
'E
..
.-.
III
:a
:;J
:;J
I>.
I>.
~
...
:a
CIS
.t:
:a
~
~
I>.
::::
l:l
....!3l:l
t'til
III
....!3
Ql
Ql
0
10
20
30
Frequency (GHz)
40
50
Target state (1,1) T=50
(e)
.-.
...
III
'E
:;J
»
.
::::
~
~
~
I>.
::::
III
....!3l:l
Ql
o
10
20
30
40
Frequency (GHz)
50
Fig, 4. Magnitude spectra of the waveforms in Fig. 3.
20
30
40
Frequency (GHz)
446
corrections needed to prevent population migration into higher states should
be different (as they clearly are, comparing Figs. 3 and 4).
The situation becomes more complicated when we attempt to design fields
which drive the ground state to a state having j = 2 or higher because only
L1j = ± 1 transitions are allowed by the Hamiltonian. For the target state 120)
this means that any population must be excited through the j = 1 levels. This
qualitatively explains the optimum field (Fig. 5) which starts off with a large
amplitude at the frequency Wo corresponding to the 0-+ 1 transition, then gradually gets a component at the 2wo frequency of the 1-+ 2 transition. This structure is best seen at the bottom of Fig. 5, which was obtained by Fourier transforming the optimum waveform, windowing out either a region around Wo or
2wo, then back Fourier transforming. As the pulse length decreases the optimum field to pump the 120) state also becomes more complicated (see Fig. 6).
As Table 1 shows, for a given pulse length the algorithm is less successful in
pumping 120) than 110) or 111). This is because population must first be
transferred out of 100) to 110), leaving effectively less time for the transfer
to 120).
Calculations were performed for target states up to 133) and in all cases a
field was found which produced the desired population inversion. In each case,
the fields had components corresponding to transitions between each of the j
levels between the ground and target states.
Target state (2.0) T=1000
(al
10
(b)
(x~-l
e
='"
5
='o" ,
1 ~I
I
~
..cl
tlO
-----1
"--I
"-.. I
0
...
...en...
---.--~--
i
_1
~
tl
"-
Target state (2.0) T=1000
1 -- - - - -..
0
~I
I:l
...en...
Q)
.~
,
-\-;f-'d
I
i
i
i
" I
I
'<:I
:g -5
b..--...,e>""""cA:cf"-I'-"d·-'T+·T+~c+T+
"F~~~
i'
"
I
N
-10
1000
0
t (psec)
1000
0
t (psec)
Fig. 5. Left: Waveform produced by computerized optimization to transfer population from 100 >
to 120> without ultimately populating either higher states or the j = 1 levels, given a total pulse
duration of 1 ns. Right: components of this pulse near the 0... 1,1 ... 2,2 ... 3 and 3... 4 resonance
frequencies (bottom to top), obtained by Fourier transforming, windowing in a region of width
± (/)0 around those frequencies, and than back transforming. The relative timing of the two fundamental frequencies is apparent.
447
(al
Target state (2.0) T=200
10 . . . . . . . - - - - - - - - - - - - - - - - - - ,
(b)
Target state (2.0) T=500
10 , - - - - - - - - - - . . . , , - - - - - ,
8'o
...
~ 5
~
~
...
0 I--\---f------'\------,I----+-fi
d
..,...
Q)
CIl
:g -5
.2
iN
-10
'--------------~
o
-10
WO
L-
t (psec)
t (psec)
Target state (2.0) T=100
(d)
____'
200
0
Target state (2.0) T=50
10 . r - - - - - - - - - - - - - - - - - - , I
(x .5)
(x.5)
~
~
I
~
0 I----+-~-----
Q)
.b
CIl
:g -5
.2
iN
-10
100
L-_---'~.L_
t (psec)
____'
50
0
t (psec)
Fig. 6. Waveforms produced by computerized optimization to transfer population from 100> to
120> without ultimately populating either higher states or the j = 1 levels, for different total pulse
durations.
Pure initial state-oriented target distribution
In the case of a pure initial state, we are allowed quite a 'bit of flexibility in
the final target wavefunction. One particularly interesting goal is to design
fields which drive diatoms in their ground state to a superposition state which
is oriented in the laboratory. Alternatively, we could attempt to simply achieve
alignment (i.e. no distinction is made between the two ends ofthe molecule).
In either case we specify a target probability distribution p ((),¢) (where () is
the angle from the laboratory z axis to the diatom internuclear vector and ¢ is
the azimuthal angle measured from the laboratory x axis). We can then specify
the target wavefunction as
448
rpT «(),</» = IaIm }l,m «(),</» =JpT «(),</»
j,m
(33)
which can be inverted to give
2"
+1
f f
aIm = d</> d(cos () }l,m «(),</» JpT«(),</»
o
(34)
-1
It is important to note that the target probability distribution depends on the
relative phases between the eigenstates and that during subsequent free propagation (t > T), the probability distribution will spread out due to dephasing
unless only a single j value is included. However, since any j state has either
even or odd inversion symmetry, in this case only alignment is possible [23].
An oriented molecule will require a mixture of even and odd j states, and will
dephase (although in the rigid rotor approximation there are recurrences, as
discussed below).
The free time evolution of the probability distribution produced by the a
coefficients of eqn. (33) is given by
p«(),</>,t-T)=I I (aIm)*a!.m' Yj,m((),</»Yj',m' «(),</»
j,mj' ,m'
exp[ -i(Ej-Ej')(t-T)/Ii]
(35 )
where Ej =Bj (j + 1). The function p is periodic in time with period nfllB. Around
each recurrence (i.e., when p(t- T) =p(T) =pT) there will be a window in
time during which p(t- T) _pT. In order for the aligned or oriented distribution to be useful or detectable, this window of agreement must be long enough
for some chemical process to occur. For instance, one might wish to orient
molecules along the laboratory z axis and then probe the degree of orientation
by using a picosecond laser pulse polarized in the z direction to photodissociate
the molecules. In this situation, it is necessary that the window of agreement
be long compared to the duration of the laser pulse. Typically, as one narrows
the target distribution in space, the window of agreement also narrows. We will
illustrate this point using a simple oriented distribution given by
pT(()
= 2k+ 1(cos ()2k
O~ ()~ n/2
=0
~2~()~n
2n
(36)
where k= 1,2,3,... There is no </> dependence. The distribution sharpens up as k
increases. Figure 7(a) showspT(() for k=l, 5 and 10; Fig. 7(b) shows laTo 12
for the same distributions as a function of j. (Note that aJm = 0 for m'l: 0.) By
the uncertainty relation, to localize the rotor to within <5() requires a rotational
angular momentum spread of ~ iii <5(), or equivalently an energy spread of hB I
«<5()2). Qualitatively, we expect the width of the window where all of the
449
(a)
Probability as a function of x=cos(8)
Probability In state j
(b)
2.0
0.6
..-----,----,---r-----,
0.5
k=10
1.5
..;;;:
~
0.4
.~
1.0
0.3
0.2
0.5
0.1
0.0 l..-_-===......L-_........l.-=::::::""-':L-_-'
1.0
0.2
0.0
0.6
0.6
0.4
2
4
6
8
x=cos(8)
Fig. 7. Target oriented angular distributions pT(l1), defined by eqn. (26), as a function of the
parameter k in that equation (a) and as a function of rotational quantum number j (b).
states are in phase to be of order (J()/2n) X (hI (2BJmax ) ~ (J()2/2B. In order
to quantitatively measure the width of the window of agreement, we compute
the error function
2"
Ep(t) =
+1
fd~ f
°
d(cos ()[p(O,~,t) _pT«()
F
(37)
-1
Figure 8 shows Ep (t) versus (t - T) I (nh.1B) for the three distributions shown
in Fig. 7. Clearly, as the degree of orientation increases, the window in time
during which the freely evolving distribution agrees with the target function
narrows. For KCI, nh.1B is equal to 143 ps so that the window of agreement for
k = 5 is a few picoseconds.
The preceding discussion has shown how to construct a target state which
will be oriented in the laboratory. Figure 9 shows the z component of the driving field; alternatively, the bottom ofthe figure shows driving fields at Wo, 2wo,
and so forth, obtained by windowing in the frequency domain as in Fig. 5. The
modulation functions on the individual transition frequencies are quite simple.
The high frequency components at the beginning of the pulse are an artefact
of the nonzero first derivative of the waveform at t=O. Figure 10 shows its
spectrum. The x and y fields were not needed because this distribution, being
independent of ~ was made up only of m = 0 components. Figure 11 shows the
resulting probability distribution produced by the control algorithm (dotted
line) and the target distribution (solid line) for k=5 (eqn. 36). The two are
virtually indistinguishable. As we have seen previously, the field initially os-
450
DephMing Error
2.0 r - - - , - - - - , - - - , - - - - - , - - - ,
1.5
~
<:)
1.0
0.5
0.2
0.4
0.11
0.8
1.0
A=(t-T)/(1Th/B)
Fig. 8. Error function E p (t) for the three distributions given in Fig. 7. As the sharpness of the
distribution increases, the window in time during which the freely evolving distribution agrees
with the target function narrows.
Oriented Target (k=5)
(a)
Oriented Target (k=5)
(b)
1000.
S
.
S
.
0
'-.
;::
0
500.
'-.
;::
0
0
~
.......
~
.<:l
d
.<:l
....
...
O•
d
.......
rn
Q)
.......
rn
Q)
""
:s
N
N
Gl -500.
r
Q)
r
-1000.
0
500
t (psec)
1000
0
500
1000
t (psec)
Fig. 9. Left: Optimized field to force a diatomic molecule initially in the completely unoriented
100) state into the strongly oriented state characterized by k=5 in Fig. 7. Right: Decomposition
of this field into the frequency components near Wo, 2wo, ... showing how the different frequencies
grow in.
cillates at frequency Wo and higher harmonics add in later to boost the probability into successively higher j states. The frequency spectrum is very clean,
having sharp peaks at nwo for n = 1, 2, 3, 4 and a small peak at 5.
451
Oriented Target (k=5)
o
10
20
30
40
50
60
Frequency (GHz)
Fig. 10. Magnitude frequency spectrum of the field in Fig. 9.
Target distribution (k=5)
10 ,---,------.-------,------,
B
6
M
p;
)
4
2
0
-1.0
-0.5
0.0
0.5
1.0
x=cos(O)
Fig. 11. Probability distribution produced by the control algorithm (dotted line) and the target
distribution (solid line). These distributions are nearly superimpo!!able.
EXPERIMENTAL IMPLEMENTATION
The experimental realization of general phase and amplitude modulated microwave pulses for coherent control of molecular rotation is a substantial but
manageable challenge. It is useful to separate the problem into several parts.
Creation of an arbitrary microwave waveform at low peak powers with the
kinds of bandwidths required here is, in fact, quite feasible by many different
452
approaches. For example, Haner and Warren [24] have built GaAs FET- or
HEMT-based circuits which are programmable and can generate any waveform with approximately 20 GHz of bandwidth. One example of a quite complicated waveform (the real or imaginary component of sech (aT) 1 +5i) is shown
in Fig. 12. In fact, the waveforms we are using here are quite simple; for the
most part, the require multiple phase coherent frequency sources (for example,
at the 0-.1,1-.2 and 2-.3 transition frequencies) which are each only slowly
modulated and then combined. This is particularly easy in the rigid rotor diatom problem because the frequency sources are harmonically related.
A more substantial difficulty, as mentioned earlier, is that bandwidth and
power are anticorrelated. For example, as of this writing, Varian sells pulsed
Klystrons with at most few percent tuning range and peak power of several
megawatts, yet pulsed traveling wave tubes with an octave of bandwidth are
available (8-16 GHz) at peak powers ofthe order of 1 kW, and CW at 200 W.
To coherently excite molecules from 0 through J requires a bandwidth from
2B through 2BJ at a minimum. If we start from some intermediate J, (for
example, by laser pumping), our fractional bandwidth requirements are less.
..
0
~
w
a
:::l
0.50
(al Micn:lwave sech
Real Component
0.25
0.0
~
:::;
...:IE
-0.25
«
-0.50
0
100
200
300
400
500
600
PICOSECONDS
0.50
~
"0
~
w
a
:::l
(bl Microwave sech
Imaginal')' Component
0.25
0.0
~
...:::;::Iii
-0.25
«
-0.50
a
100
200
300
400
500
600
PICOSECONDS
Fig. 12. Illustration of the complexity of microwave pulse shapes which is possible with currently
available technology. The two parts of this waveform correspond quite closely to the real and
imaginary parts of the function (sech (aT) ) 1+5;, which is used to generate localized and uniform
population inversions in laser spectroscopy and nuclear magnetic resonance (see refs. 3-5). This
figure is reprinted from ref. 24, where the waveform generator based on GaAs HEMTs is characterized and used with an electro-optic modulator to make shaped laser pulses.
453
Another consideration is that throughout the microwave region, commercially
available power also falls with increasing frequency, while the price increases.
Again as of this writing, covering the band from 2-40 GHz would require four
10 W amplifiers from Hughes, at a cost of about $100,000.
A significant factor in experimental feasibility will be the pulse time. The
peak power requirement falls approximately as the inverse square of the pulse
duration, as long as the pulse time is long compared to the rotational period.
As these calculations show, in that case only radiation in narrow bands near
multiples of 2B is needed. This will allow use of multiple, narrow band sources
that are phase locked. One can greatly increase the power as well by using an
open space resonance cavity as is used in Fourier Transform microwave spectroscopy [25]. These devices have resonances that are multiples of the cavity
fundamental and thus are well suited to simultaneously enhance frequencies
near the rotor transitions. A possible difficulty will be simultaneous broadband
coupling, but this would probably be solvable as long as only modest Qs are
required. The Q that can be used, and thus the power enhancement, will scale
as the pulse time. Another advantage of longer pulses will be in modulators;
PIN diode microwave switches with high power handling capabilities have nanosecond switching times. Higher speeds can be achieved with double balanced
mixers, but at increased loss and with reduced isolation.
In order to make the power requirements more tangible, we note that in a
cavity or waveguide, the microwaves will have to occupy cross sections of ~ 1
cm2 , which implies that 1 kW is required to produce the fields on the order of
1000 V cm -1 in Fig. 9. Such fields could be reached with cavity Qs of 100~1000,
but then the photon lifetime in the cavity would be ~ 10 ns. A more typical
dipole moment of 1 D instead of 10 D would require 100 times the power and
clearly be out of the question. But extending the pulse time from 1 to 10 ns
would reduce the power requirements 100 fold such that KCl would be feasible
with a few watts, which could be generated directly, and molecules with aID
moment would be feasible with a low Q cavity.
The chief disadvantage of longer pumping times is that our calculation has
explicitly neglected small terms in the Hamiltonian (such as D constants),
inhomogeneous broadening, and hyperfine effects, and relaxation. This approximation is only valid for times short compared to the reciprocal of the
appropriate splitting or broadening. In addition, as noted earlier, the orientation only persists for a short period of time, although there are recurrences for
the rigid rotor (a complete recurrence occurs every hi (2B) seconds, independent of the number of J states involved). Finally, the required field accuracy
grows as the molecules undergo more rotations during the preparation cycle;
as long as the amplitudes are stable, it should be possible to at least partly
compensate for errors by shifting the observation time.
An experimental realization of the distribution is given in Fig. 13. A cold
beam of KCl is created by supersonic expansion from a heated nozzle source.
454
Side view:
(Microwave Fabry-Perot)
=-::-=1-'-:-:0:..
)
cBeai,OCk»)
~
~
:III~~
~~
~(Probe
uadrupole
Rods
lase()
Skimmers
End view of rods:
End view of cavity:
(Microwave Fabry-Perot)
~
Fig. 13. Experimental apparatus to generate oriented molecules in the laboratory.
A terminal temperature of perhaps 10 K could realistically be expected. This
will produce a Boltzmann distribution which peaks near J = 5. Thus in order
to selectively populate single J, M, we could velocity select the beam if need be
and then pass it through a state selecting quadrupole lens. J = 1, M =0 molecules will focus lower in field than any other and thus bend behind the beam
stop, so essentially a pure state can be generated [26]. The beam will then pass
through a Fabry-Perot microwave cavity to generate a good Q at frequencies
corresponding to the 0---+ 1, 1---+ 2, 2---+ 3, ... transitions; or two crossed microwave
cavities so that x, y and z polarizations can be selected. In this region, pulsed
microwave fields would strike the molecules. A picosecond laser pumping to a
direct dissociating state, followed by an ionization laser would provide a means
of sampling the angular probability distribution of the molecules to compare
with predictions.
A possible collisional experiment could be to selectively pump an M =J state
which rotates in the x,y plane. One could use d.c. fields to realign the z axis to
be anywhere in space. Thus one could compare the cross-section for collisions
in the x,y plane with collisions in the x,z plane. Intuitively, we expect that
collision partners approaching along the z axis would find a bigger target than
those along the x axis.
CONCLUSIONS
We have shown that controlled radiation fields can produce any possible
superposition of lim> states in principle, starting from all population in the
ground level, and we have given explicit prescriptions for creating interesting
455
rotational states (eigenstates or oriented molecular states) with reasonable
microwave peak powers. In particular, the prospect of enforcing rotational orientation has many potential applications in reaction chemistry. To name only
two examples, control over orientation would dramatically simplify measurement of bimolecular reaction dynamics, and scattering of oriented molecules
off a surface would provide a sensitive and valuable probe of surface structure
and reactivity.
In general, implementing rotational control in the laboratory will always
involve a critical evaluation of experimental constraints (maximum microwave power, time of flight in a beam, spectral perturbations due to hyperfine
interactions, inhomogeneous broadening, and the like). The power of the optimal control method lies in its ability to incorporate such constraints to produce usable fields. Microwave spectroscopy has long been the premier technique for the determination of molecular structure; what we have shown is that
it may become the premier technique for influencing molecular reactivity and
characterizing reaction dynamics as well.
ACKNOWLEDGEMENTS
We acknowledge support by the National Science Foundation under grants
CHE-8719545 (W.S.W.) and CHE-8552757 (K.K.L.). H.R. would like to thank
the Army Research Office and the Office of Naval Research for support. R.H.J.
would like to acknowledge support from the Texas Advanced Research Projects, Texas Higher Education Coordinating Board.
REFERENCES
1 J. S. Waugh, L. Huber and U. Haeberlen, Phys. Rev. Lett., 20 (1968) 180.
2 R. R. Ernst, G. Bodenhausen and A. Wokaun, Principles of Magnetic Resonance in One and
Two Dimensions, Oxford University Press, London, 1987.
3 W. S. Warren and M. Silver, Adv. Magn. Reson., 12 (1988) 248.
4 L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Wiley, New York, 1975;
F. T. Hioe, Phys. Rev. A, 30 (1984) 2100.
5 M. S. Silver, R. I. Joseph, D. I. Hoult, Phys. Rev. A, 31 (1985) 2753.
6 J. Murdoch, A. Lent and M. Kritzer, J. Magn. Reson., 74 (1987) 226; S. Connolly, D. Nishimura, A. Macovski, IEEE Trans. Med. Imaging, MI5 (1986) 106; D. J. Lurie, Magn. Reson.
Imag., 3 (1985) 235; F. Loaiza et aI., Health Care Instrum., 1 (1986) 188; R. R. Edelman, D.
Atkinson, M. Silver, F. Loaiza and W. S. Warren, J. Radiol., 166 (1988) 231.
7 F. Loaiza, M. A. McCoy, S. L. Hammes and W. S. Warren, J. Mag. Reson., 77 (1988) 175.
8 M. A. McCoy, F. Loaiza, K. Valentine and W. S. Warren, J. Mag. Reson., 80 (1988) 155;
J. Gutow, M. McCoy, F. Spano and W. S. Warren, Phys. Rev. Lett., 55 (1985) 1090.
9 C. J. Lee, N. Murali and W. S. Warren, J. Magn. Reson., 82 (1989) 185. Adv. Magn. Reson.,
14 (1990) 241.
10 D. Tannor and S. A. Rice, J. Chern. Phys., 83 (1985) 5013.
11 P. Brumer and M. Shapiro, Chern. Phys. Lett., 126 (1986) 541.
456
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
H. Rabitz, in A. Bandrauk (Ed.), Atomic and Molecular Processes with Short Intense Laser
Pulses, Plenum, New York, 1988, p. 389.
A. M. Weiner, J. P. Heritage and R. N. Thurston, Opt. Lett., 10 (1985) 609; 11 (1986) 153.
M. Haner and W. S. Warren, Appl. Phys. Lett., 52 (1988) 1458; Ultrafast Phenomena VI,
Springer, Berlin, 1988, p. 139.
D. Luenberger, Introduction to Dynamic System Theory, Models and Applications, Wiley,
New York, 1989.
S. Shi, A. Woody and H. Rabitz, J. Chem. Phys., 88 (1988) 6870; A. Pierce and H. Rabitz,
Phys. Rev. A, 37 (1988) 4950.
C. Schweiters, H. Beumee and H. Rabitz, J. Opt. Sci. Am., in press.
A. G. Butkovski and Yu. I. Samorlenko, Automatikai Telemeckhanika, 4 (1979) 5; C. Huang,
T. Tam and J. Clark, J. Math. Phys., 24 (1983) 2608.
A. Wokaun and R. R. Ernst, J. Chem. Phys., 67 (1977) 1752.
A. E. Bryson and Y. Ho, Applied Optical Control, Blaisdell-Ginn, Waltham, MA, 1969.
W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes, Cambridge University Press, New York, 1986.
L. Allen andJ. H. Eberly, Optical Resonance and Two-Level Atoms, Wiley, New York, 1975.
S. Kais and R. D. Levine, J. Phys. Chem., 91 (1987) 5462.
M. Haner and W. S. Warren, Appl. Opt., 26 (1987) 3687.
T. J. Balle, E. J. Campbell, M. R. Keenan and W. H. Flygare, J. Chem. Phys., 71 (1979) 2723.
H. G. Bennewitz, W. Paul and Ch. Schlier, Z. Phys., 141 (1955) 6.