IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003
2107
Transfer Function Approach to Quantum Control–
Part I: Dynamics of Quantum Feedback Systems
Masahiro Yanagisawa and Hidenori Kimura, Fellow, IEEE
Abstract—This paper gives a unified approach to feedback
control theory of quantum mechanical systems of bosonic modes
described by noncommutative operators. A quantum optical
closed-loop, including a plant and controller, is developed and its
fundamental structural properties are analyzed extensively from a
purely quantum mechanical point of view, in order to facilitate the
use of control theory in microscopic world described by quantum
theory. In particular, an input–output description of quantum
mechanical systems which is essential in describing the behavior
of the feedback systems is fully formulated and developed. This
would then provide a powerful tool for quantum control and pave
an avenue that connects control theory to quantum dynamics.
This paper is divided into two parts. The first part is devoted to
the basic formulation of quantum feedback control via quantum
communication and local operations on an optical device, cavity,
that can be regarded as a unit of quantum dynamics of bosonic
modes. The formulation introduced in this paper presents the
feature intrinsic in quantum feedback systems based on quantum
stochastic differential equations. The input-output description
provides a basis for developing quantum feedback control through
the transfer function representation of quantum feedback systems.
In the follow-up paper, the quantum mechanical representation
of feedback is further elaborated to yield the control theoretical
representation of fundamental notions of quantum theory, uncertainty principle, e.g., and some applications are presented.
Index Terms—Bosonic mode, quantum mechanical feedback
and cascade connection, quantum stochastic differential equation,
quantum theory.
NOMENCLATURE
Throughout this paper, we use fairly standard notations listed
as follows.
Annihilation operator for a cavity mode.
Annihilation operator for a traveling field.
Hilbert space.
Set of linear operators on Hilbert space.
Trace of .
Partial trace of
over .
Unitary operator.
Density operator.
State vector.
Manuscript received September 6, 2001; revised December 9, 2002. Recommended by Associate Editor Y. Yamamoto.
M. Yanagisawa is with the Department of Complexity Science, School of
Frontier Science The University of Tokyo, 113-8654 Tokyo, Japan, and also
with the Control and Dynamical Systems, California Institute of Technology,
Pasadena, CA 91125 USA (e-mail: yanagi@cds.caltech.edu).
H. Kimura is with the Department of Complexity Science, School of Frontier
Science, The University of Tokyo, 113-8654 Tokyo, Japan, and also with the
Laboratory for Biological Control Systems, Bio-Mimetic Control Center, The
Institute of Physical and Chemical Research (RIKEN), 463-0003 Nagoya, Japan
(e-mail: kimura@crux.t.u-tokyo.ac.jp).
Digital Object Identifier 10.1109/TAC.2003.820063
Inner product of and .
Adjoint of an operator .
Mean value of an observable .
commutation relation.
anticommutation relation.
Symbol of tensor product.
.
.
I. INTRODUCTION
E
XTENSION of control theory to the quantum domain has
been a target of some researchers since the mid-1970s
[1]–[3]. The main motivation there was tied to the fact that measurements of any physical quantity inevitably disturbs the state
of the quantum system. The formulation of feedback control
under this circumstance seemed to be a great challenge for control theorists. On the other hand, variational principle used in
the optimal control manifests itself more explicitly in quantum
mechanics, because its fundamental governing equation is energy preserving. This is perhaps another reason why quantum
theory attracts control theorists [4].
In the early 1980s, more realistic pictures were brought
forward in the field by a group of chemists who tried to control chemical reactions by properly arranging electromagnetic
fields [5]–[8]. Their purpose was to increase the probability of
favorable chemical reaction by means of adjusting the phase
difference between two electromagnetic fields created by laser
beams. Theoretical, as well as experimental, verifications of
the possibility of materializing these attempts have been reported extensively in the literature of photochemistry [9]–[13].
In these papers by chemists, control is ascribed to the selection
of Hamiltonian due to the method of “inverse problem,” and
is therefore essentially a feedforward control, as Gordon and
Rice properly described [14].
The chemical experiments on the reaction between the electromagnetic field and two or three level atomic systems led to
one possible generalization of control theoretical notions, such
as controllability [15]–[18]. Since the evolution of a quantum
system is given by the unitary operators with continuous parameters, the generalization is based on the unitary representation
of Lie groups. This technique has resolved quantum feedforward control into the unitary operator construction problem
[19]–[22].
The first theoretical work on feedback for quantum systems
appeared in quantum optics, which treated the fluctuations of
the photocurrent in a quantum mechanical way [24], [23], [25].
The stochastic Schrödinger equation was first introduced in the
0018-9286/03$17.00 © 2003 IEEE
Authorized licensed use limited to: Univ of Calif Santa Cruz. Downloaded on September 11,2023 at 05:22:24 UTC from IEEE Xplore. Restrictions apply.
2108
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003
early 1990s [26]–[28]. This formulation enables one to control quantum systems via measurements, in which the quantum
system is driven by interactions conditioned by the measurement outcomes [29]–[32]. A definite class of states, referred to
as Gaussian, is of particular interest in not only classical but also
in quantum case. As a result, feedback control for the state via
measurement was studied [33].
Recent progress in quantum electronics has opened up the
possibility of quantum information technologies, which are expected to eliminate the bottlenecks of modern communication
and computation. They are based on the notion of entanglement which is thought of as a quantum information resource.
Entanglement is a quantum mechanical correlation which is
produced only by nonlocal quantum mechanical interactions.
In theoretical works, it is assumed that we can specify the
quantum state at our disposal whenever we need it, no matter
how the environment of the system would be. In other words,
it is presumed that the quantum state can be controlled for the
use of communication and computation. This presumption is
far from trivial taking into account the fact that the quantum
systems sometimes entangle with undesirable systems, which
results in a noisy information resource, and consequently, it has
been necessary to consider the production of entanglement in
the light of quantum control accordingly. Feedback is a method
whereby the performance and robustness of the system can
be improved considerably, even if the system includes some
uncertainty in its environment to which the system is highly
structured. This paper is devoted to the formulation of quantum
mechanical feedback, in order to introduce the concepts and
tools of control theory to quantum theory for understanding
quantum systems and developing quantum control.
For a system placed among a large number of degrees
of freedom interacting with one another, one may ignore
the detailed dynamics of the external degrees of freedom by
treating them statistically. If the system is weakly coupled to
the external field, then this treatment becomes effective as an
approximation. In the classical case, the statistical nature of the
external field is characterized by the singular correlation of the
field. This singularity constitutes the description of the system
through the stochastic differential equation or the forward
Fokker–Plank equation. In the quantum case, in order to deal
with quantum systems properly, physical variables should be
quantized through the canonical commutation relation, which is
essentially singular. An analogy between the singularities of the
classical correlation function and the quantized commutation
relation leads to a generalization of the stochastic differential
equation subject to the quantum mechanical law.
There is a dual relationship in the description of quantum dynamics analogous to a one-to-one correspondence between the
Fokker–Plank equation and the stochastic differential equation.
The former describes the evolution of probability distribution of
the system which interacts with the external field. The influence
of the external field is not explicitly represented in this description because the information of the external field is averaged
out. The latter is a dual description in the sense that it represents the evolution of physical variables, and the single path of
the system along with the external field is explicitly presented.
Both are basically equivalent, however the latter provides the
input–output relation of the system by which we can consider
various connections of systems. If quantum systems are connected in a complex way, it is sometimes hard to derive the
Hamiltonian which describes the behavior of the entire system
because the connected systems are entangled with each other
through the inputs/outputs, and consequently, the total Hamiltonian is not given by the sum of local Hamiltonians describing
each component system. Furthermore, the noncommutativity of
quantum variables complicates the difficulty of the description
if the field, after having interacted with the system at some time,
then interacts with it again at some later time through a closed
loop. This is why there has been little work on using nonclassical field to construct large quantum systems including closed
loops. This paper proposes a systematic procedure to obtain the
Hamiltonian and the quantum stochastic differential equation
that lead to a natural extension of control theory and some applications of quantum control.
This paper is divided into two parts. In the first part, we derive general dynamics of quantum feedback systems. Based on
the framework of quantum feedback systems established in the
first part, the applications of quantum feedback to some of the
most important problems in quantum theory are developed in
the second part.
The first part of this paper starts with Section II which is a
brief review of fundamental notions of quantum theory for introducing control theoretical viewpoints to quantum systems. In
particular, it focuses on introducing the interaction between a
system and environments in a quantum mechanical manner, because system control is essentially based on the plant-controller
interaction. Section III introduces a quantum stochastic process
as a noncommutative analog of Wiener process, in which the
quantized electromagnetic field traveling in free space is the
noncommutative input source. An optical system is treated in
terms of an idealized class of Hamiltonians describing a linear
coupling of a localized system to the noncommutative input.
The system then obeys the quantum stochastic differential equation which arises due to the stochastic nature of the noncommutative input operators. Section IV deals with the quantum
mechanical feedback in the proper context of quantum theory,
and derives Hamiltonian which describes the quantum feedback
system. The feedback connection of quantum systems has a
wide range of applications that enables us to derive Hamiltonians for various types of systems. Section V lists the great utilization of the applications for deriving the evolution of quantum
systems connected in a complex way.
II. BRIEF REVIEW OF QUANTUM THEORY
A. Quantum States and Observables
To each quantum mechanical system, there corresponds a
to describe its behaviors. A quantum system
Hilbert space
and the corresponding Hilbert space are sometimes identified.
Each state of the system is represented by an element of
and is usually written by
, which is called a ket, following
is denoted by
which is called a bra. In
[34]. A dual of
corresponds to a column vector
the context of linear algebra,
a row vector. Thus, the inner product of
and
and
is denoted by
. In quantum theory,
and
denote
Authorized licensed use limited to: Univ of Calif Santa Cruz. Downloaded on September 11,2023 at 05:22:24 UTC from IEEE Xplore. Restrictions apply.
YANAGISAWA AND KIMURA: TRANSFER FUNCTION APPROACH TO QUANTUM CONTROL—PART I
the same state for each scalar . Hence, we use normalized eleto denote physical states.
ments
the set of all linear operators on .
Denote by
An observable physical quantity, or simply an observable,
on
is represented by a self-adjoint operator
[34]. Assume that
is bounded for simplicity. Then,
has
a spectral decomposition
2109
the quantum case. The time evolution of the quantum state is
described by the Schrödinger equation [34]
(10)
This describes an evolution of the state in the entire Hilbert
space . We can rewrite (10) in the equivalent form
(11)
(1)
where
denotes a unitary operator given by
denotes the projection operwhere is an eigenvalue, and
ator on the associated eigenspace satisfying
(2)
(12)
Corresponding to (11), time evolution of the density operator
is governed by
(13)
If we measure the observable for the system which is in the
just prior to the measurement, then we obtain one of
state
as the measurement outcomes and the
the eigenvalues
is projected on the eigenspace. The probability of an
state
is given by
outcome
(3)
The mean value of the measured data is conventionally thought
of as the associated classical parameter. It is easy to see that the
mean value is given by
The relationship (13) is referred to as the Schrödinger picture of
is a density operator
the evolution. It should be noted that if
. Since
is unitary, the transformasatisfying (7), so is
tion (13) is rank preserving. This reflects the fact that quantum
evolution of an isolated system is always energy conserving.
From the evolution (13) of the operator-valued probability
distribution, the mean value of an arbitrary observable at time
is given by
(4)
Introducing the trace operator as an extension of the trace of a
matrix, the relations (3) and (4) are rewritten as
(14)
This form gives the alternative expression of the evolution in
which the state of the system is fixed at the initial time and
the observable evolves in time as
(5)
(15)
(6)
where the operator
is defined by
(7)
It is easy to see that
(8)
(9)
The operator satisfying the conditions (8) and (9) is important in
quantum theory and is usually referred to as a density operator.
is regarded as representing the state of a system, instead of the
, that is an operator-valued probability distribution [35].
ket
The condition (8) corresponds to the positivity of probability
distribution and (9) to its normalization. Probability distribution
functions are then described by commutative density operators.
A significant difference between classical and quantum systems
is noncommutativity.
B. Time Evolution
The Hamiltonian of a system is an observable of energy.
on
in
Thus, it is represented by a self-adjoint operator
This is referred to as the Heisenberg picture of the evolution
[34], which is the dual form of the Schrödinger picture (13) in
the sense that the density operator corresponds to a function
and the observable to . A significant difference between classical and quantum observables is noncommutativity.
C. Composite System
and
are given. To
Assume that two quantum systems
and
, we
describe a quantum system incorporating both
and
.
need a new Hilbert space which is composed of
The space is usually referred to as the tensor space of
and
, constructed in the following way.
and
. Then, we define a map from
Let
to which is denoted by
(16)
This map must be bilinear and satisfies the following conditions.
constitutes an orthonormal basis
a) If
and
of
, then so does
of
of .
and
. Then,
b) Let
.
Authorized licensed use limited to: Univ of Calif Santa Cruz. Downloaded on September 11,2023 at 05:22:24 UTC from IEEE Xplore. Restrictions apply.
2110
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003
In this case, the Hilbert space
is denoted by
, and denote
mean value of is given by
and
. Then, the
(17)
(26)
The tensor product
is defined by
of operators
and
is given by
where
(18)
(27)
(19)
A comparison of (5) and (26) shows that the original observable
on the tensor space
induces a new observable
on
through the partial trace.
From condition b), we have
E. Generalized Evolution
The following identities are easily proven:
(20)
(21)
. For example,
Note that
and
such
we cannot find operators
. The interaction between
that
quantum systems is represented by Hamiltonians in this class
of operators. This will be seen more precisely in the next two
subsections.
on the
Now, we consider a time evolution of the state
. We are interested in deriving a dynamtensor space
. From (11), we have
ical law that governs
(28)
Assume that at time
is represented as
(29)
for some
. Then, it is not difficult to see
D. Partial Trace
Sometimes, we must ignore what is happening in the system
while observing the system
. In such cases, it is necessary
, and concentrate
to average out the state of the system
. This is particularly important
on the behavior of system
when we deal with an open system, i.e., a system subject to
continuous interaction with the environment. In this case,
represents the environment whose information is not available.
We must consider the averaged effect of the environment to
deal with the system of interests.
Notice that the information of a quantum state is eliminated
of an operator
by the trace, as in (9). The partial trace
with respect to
is a map from
to
such that
(22)
for each
the partial trace
. To get a more concrete representation of
, note that, using a orthonormal basis of
is represented as a “matrix” form
(23)
(30)
where
(31)
are fundamental vectors, i.e.,
is the ket with 1 at
and
the th component and 0 at all other components. Since
is unitary, we see the relation [36]
(32)
The resulting operator (30) is also a density operator satisfying
(8) and (9). It should be noted that the time evolution (30) is no
longer invertible nor rank preserving, reflecting the fact that the
is interacting with the environment
and losing
system
is also a quantum
its information. For a quantum state , if
is, the map is said
state no matter what the environment
to be completely positive [36]. From the duality between the
Schrödinger picture (13) and the Heisenberg picture (15), one
is
can see that the generalized evolution of an observable
given by
Then, it is not difficult to see that
(33)
(24)
If
happens to be given by a tensor product
(34)
Clearly, we have
(25)
Let
teracting systems
be an observable on the mutually in. Assume that the state is
of the unitary operators
, then it is easy to see that
and
(35)
Authorized licensed use limited to: Univ of Calif Santa Cruz. Downloaded on September 11,2023 at 05:22:24 UTC from IEEE Xplore. Restrictions apply.
YANAGISAWA AND KIMURA: TRANSFER FUNCTION APPROACH TO QUANTUM CONTROL—PART I
2111
This implies that the evolution of the system
is not affected
. In other words,
and
do not interact with each
by
other essentially. However, such a case is not relevant. We are
specially interested in the case where the representation (34)
and
are
does not hold. Then, we say that the systems
entangled.
F. Quantizing Electromagnetic Field
One of the most important operators in quantum mechanics
which satisfies the
is the annihilation (creation) operator
commutation relation
(36)
Fig. 1. Stationary waves in a cavity. The frequency of each wave is determined
by mirrors placed to face in the opposite directions. Operators a and a are
responsible for the creation and annihilation of the modes inside the cavity.
denotes the commutator defined by
[34]. Although is not a self-adjoint operator, we
can define several observables with . For example, the operator
is self-adjoint so that it has real eigenvalues
where
(37)
Fig. 2.
is the associated eigenket of
where is an eigenvalue and
is referred to as the number
the operator . The operator
operator. From the commutation relation (36), we have
(38)
(39)
and
are also eigenkets
These relations imply that
of with an eigenvalue that is increased or decreased by one.
The eigenvalue is obviously positive from the definition of .
Furthermore, the commutation relation (36) indicates that the
with
eigenvalues are positive integers. If this is not the case,
is an eigenket of
having a negative eigenvalue
, which contradicts the positivity of . As a result, has
zero eigenvalue, and the associated eigenket, denoted by , is
and
called a vacuum state. From (38), the eigenkets
are identical up to a multiplicative constant. This expression for
defines the sense in which the vacuum state
the eigenket
represents a field in which there is nothing and the eigenket
is a state which consists of “particles” generated by
creation operators. Because of this nature, the term, creation
is deemed appropriate. In fact,
(annihilation) operator, for
the operators satisfying the commutation relation (36) describe
the creation and annihilation of the bosonic mode in physics
[34]. A typical example of this is the electromagnetic field in
a cavity.
in the Coulomb gauge is a fundaThe vector potential
mental quantity to describe the electromagnetic field. It obeys
the wave equation
(40)
in free space. The transverse electric field is then defined by
. We consider the case of a polarized field, i.e., the
vector potential is specified only by one component of ,
and its spatial variation is confined to one direction, say , and
the transverse spatial range of the beam is confined to a narrow
area.
Schematic representation of the cavity, input, and output fields.
A cavity is an optical device with two mirrors, positioned to
face in opposite directions, that produce stationary waves of discrete frequencies determined by the boundary condition inside
the cavity; see Figs. 1 and 2. A solution of (40) for the cavity
can be constructed using the complex function for the modes
as
of the discrete frequency
(41)
is an adequate constant, and
is a complete
where
set of functions. To treat the electromagnetic field properly, it is
necessary to introduce quantization to the cavity modes. This is
implemented by the commutation relation for the cavity modes
(42)
Because of this commutation relation for and , the vector
becomes an operator. The role of the operators
potential
and
is now obvious. The operator
creates (annihilates) the mode of the frequency inside the cavity.
The Hamiltonian of the cavity is written by
(43)
The energy of the cavity is determined by the number of the
is
modes inside the cavity, and the number operator
responsible for counting the modes. From the Hamiltonian (43)
and the commutation relation (42), one can see that the evolution
of the vector potential (41) is consistent with the Heisenberg
picture (15).
Next, we consider the electromagnetic field traveling in free
space. The vector potential obeys the same evolution (40) as
a cavity does, and the solution is constructed by the complex
Authorized licensed use limited to: Univ of Calif Santa Cruz. Downloaded on September 11,2023 at 05:22:24 UTC from IEEE Xplore. Restrictions apply.
2112
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003
function
. In this case, however, the frequency
uous, and we have
is contin-
where
is a weighting function which restricts the domain
of integration to a narrow frequency region around . Then,
is
the commutation relation of
(53)
(44)
and the vector potential and the electric field are given by
where is an adequate constant. The set of functions
is complete, and satisfying
(54)
(55)
(45)
Here,
and
, and
Since the photons which make up the beam of light propagating
direction are localized within the narrow frequency region
is a slowly varying operator compared to .
around ,
This fact allows us to introduce a considerable approximation
in the next section.
III. CAVITY AS A UNIT OF QUANTUM SYSTEM
Note that the function
becomes a conventional delta function for forward propagating solution of (40) satisfying
(46)
The quantization of the electromagnetic field traveling in free
space is implemented by the commutation relation for the function
(47)
This is the generalization of the commutation relation (36) for
the traveling field.
as
Now, we define a new operator
(48)
This satisfies the commutation relation
A. Hamiltonian of a Cavity System
In the preceding section, we have considered two kinds of
quantum systems, photons confined in the cavity and traveling
in free space. In this section, we consider the interaction
between them, i.e., an optical medium is placed in a cavity
composed of two mirrors, one of which is partially transparent
through which the traveling wave can come into the cavity and
interact with the cavity mode. Assume that the external field
is
is of the form (54) such that the dominant frequency
identical with that of one possible cavity mode . If the size of
the cavity is much smaller than the wave length of the external
field, the cavity is approximated as a localized system, say,
. Then, the traveling wave is incoming from
,
at
coupling to the cavity mode , and outgoing from the cavity to
.
the direction
and
be Hilbert spaces of the cavity and the exLet
ternal field, respectively. The entire system is expressed by the
. The total Hamiltonian is of the form
tensor space
(56)
(49)
If we are only concerned with the forward propagating field, the
commutation relation can be written in an alternative form
(50)
This operator represents the creation of all possible modes at the
spatial point because of a property
(51)
which implies that the total number of photons over all modes
is equivalent to that over space.
is photons localized
A simple but practical example of
is of the
at a frequency, say . In this case, the function
form
(52)
ized at
parts
describes the motion of the cavity mode local. This Hamiltonian may be decomposed into two
(57)
is given by (43) whereby the evolution of the cavity mode
, as in (41).
is the residual
is of the form
Hamiltonian determined by the optical medium placed in the
is the
cavity, referred to as a free Hamiltonian.
Hamiltonian of the external field described as in (54) and (55).
is assumed
The interaction Hamiltonian
, and given by
to be a weak dipole coupling at
(58)
denotes an operator for the dipole moment of the
where
cavity, defined by the real part of the cavity mode, and
Authorized licensed use limited to: Univ of Calif Santa Cruz. Downloaded on September 11,2023 at 05:22:24 UTC from IEEE Xplore. Restrictions apply.
YANAGISAWA AND KIMURA: TRANSFER FUNCTION APPROACH TO QUANTUM CONTROL—PART I
is the external field given by (55) at
. This weak interaction results in a slight deviation of the cavity mode from the
is proportional to
frequency . Thus, the dipole moment
2113
system is in a pseudoclassical state at temperature
ample, we can define the density operator as
(67)
(59)
is slowly varying compared to . From (55) and
where
(59), the interaction Hamiltonian can be written as
, for ex-
Then,
ments vanish [37].
and all other first and second mo-
C. Quantum Stochastic Differential Equation
(60)
(61)
The evolution of an arbitrary operator is given by (15) in
which the unitary operator is generated by the Hamiltonians (56)
and
drive the cavity and external field at ,
and (57).
generate small changes compared
respectively, and and
to . Since the slowly varying components, which results from
the interaction between the cavity and external field, is of inand . The
terest, we focus on an evolution generated by
unitary operator of the system is then given by
It is necessary to treat this Hamiltonian carefully because of the
singularity of the condition (53), which leads us to use stochastic
calculus to deal with the evolution of the quantum system.
(68)
where is a coupling constant. Since the terms of frequency
is fast oscillating compared to
, we can approximate
them to its cycle averaged value, i.e., zero. This is known as
the rotating wave approximation, which is extensively used in
quantum optics [37]. As a result of this approximation, we have
where the time argument has been omitted, and
B. Quantum Stochastic Process
Let us define an operator
is defined by
(69)
(62)
and
are commutative to order
Note that
. The increment of an arbitrary operator of the system driven
is given by
by the stochastic input
(63)
(70)
This represents the field immediately before it interacts with the
system, and we can regard it as an input to the system. From the
commutation relation (53), we have
Using (65) in expansion of the exponential of the unitary operator (68) and retaining the term of order , we obtain the
quantum stochastic differential equation
where
(64)
is of order
. The
This relation implies that the operator
analogy between the commutation relation (64) and Ito’s rule
for Wiener process leads to a natural definition of a quantum
stochastic process as
(65)
are
and all other products higher than the second order in
are real and complex numbers, respecequal to zero. and
tively, and satisfying [37]
(71)
Here, we omit the symbol of tensor product to simplify the
exposition. Derivation of the aforementioned equation is found
in Appendix A.
As a result of the same interaction (61), the incoming field
also evolves in time, and the cavity produces the outgoing field.
Let us define an operator as
(72)
(66)
It will be shown that (66) is concerned with the uncertainty
relation.
Remark 1: A stochastic process is uniquely determined by
specifying all the moments. In the quantum case, it is implemented by defining the density operator of a system. If the input
where
(73)
This represents the field immediately after the interaction with
can be regarded as the output from the
the system. Thus,
Authorized licensed use limited to: Univ of Calif Santa Cruz. Downloaded on September 11,2023 at 05:22:24 UTC from IEEE Xplore. Restrictions apply.
2114
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003
cavity at time . Expanding the unitary operator to the lowest
order in , we have
(74)
Fig. 3. Block diagram of the cavity.
is no longer inThis relation indicates that the output
and the
dependent of the cavity, because the cavity system
are entangled with each other through the insignal system
.
teraction
From (74), one can see that the motion of the operator
is essentially responsible for describing the output
. As. Taking
in (71) yields
sume here that
(75)
Fortunately, the output is linear in the operator . From (74) and
(75), we have
Fig. 4. Schematic representation of fully quantum mechanical closed loop.
The annihilation operators for the system A and B are denoted by a and a ,
respectively, while b ; b ; b ; b , and b represent traveling waves.
(76)
This is the transfer function of the quantum system coupling
to the traveling field through the Hamiltonian (61). It is worth
noting that this transfer function is unitary and, hence, power
preserving. The relation (76) is represented in Fig. 3, where the
“black box” denotes the cavity, and the input and output signals are carried by the bosonic modes. This is identical with
the input/output representation in system theory. The essential
difference between classical and quantum systems is that the
input, output, and state variables of the quantum system are represented by operators.
The input signals
and
the outputs and by
to the beam splitter are related to
(77)
where and are real and satisfy
input–output relation of each system, we have
. From the
(78)
These relations determine each signal in the feedback loop as
IV. QUANTUM MECHANICAL FEEDBACK
A cavity is thought of as a unit of quantum system with a
single operator-valued state variable, an operator-valued input
and output on the associated Hilbert space. A quantum mechanical feedback is constructed through the input–output which
store the information of the cavity. Here, we consider two cavities, and , that are positioned to interact with each other
and
be the
through the external field, shown in Fig. 4. Let
annihilation operators for the modes of and , respectively.
Usually, spatially separated fields are treated as statistically independent. The distinctive feature of the quantum mechanical
feedback is that because of the closed loop of the traveling field,
spatially separated quantum systems are entangled, i.e., is influenced by and vice versa. Then, the entanglement supplies
control resources.
Unlike the classical case, there are physical limitations in manipulating quantum signals, such as sum and split of signals. A
beam splitter is a useful optical device which allows us to perform the manipulations of quantum signals. In Fig. 4, the input
is sent to one port of a beam splitter, which is chosen
field
is from
to have reflectivity and transmissivity , and
to the other input port. Meanwhile, one of the transmitted fields
from the beam splitter is sent back to . Assume that the time of
propagation between the systems is negligible as the feedback
system does not include time delay in the closed loop. This is
an ideal physical situation, but it will be the case when the intracavity separation is of the same order as a length of the system.
(79)
The evolution of the feedback system is then given by the following theorem.
Theorem 1: An arbitrary operator on the feedback system
shown in Fig. 4 obeys the evolution law
(80)
where
is a unitary operator given by
(81)
where
(82)
Authorized licensed use limited to: Univ of Calif Santa Cruz. Downloaded on September 11,2023 at 05:22:24 UTC from IEEE Xplore. Restrictions apply.
YANAGISAWA AND KIMURA: TRANSFER FUNCTION APPROACH TO QUANTUM CONTROL—PART I
2115
This is equivalent to
(83)
where
(84)
Proof: See Appendix II
In (81), it seems that the unitary operator, after having acted
on , then acts on . However, the closed loop apparently provides an undirected control resource between and . The result of the evolution should be the same as if the unitary operator
for the coupling at acts first and at second. This leads to
the symmetry of operator ordering in the evolution of the feedback system. From the expression of the external field (79), the
unitary operators (82) are formally governed by the following.
, acting on first, is generated
I) The unitary operator
to
in the input to .
by the coupling of
, acting on second, is generII) The unitary operator
to
and
in the input to .
ated by the coupling of
In the alternative expression (83) where evolves first and
second, the unitary operators (84) obtained by the following.
, acting on first, is generIII) The unitary operator
to
in the input to .
ated by the coupling of
, acting on second, is genIV) The unitary operator
to
and
in the input to
erated by the coupling of
.
Note that these two schemes to obtain (83) and (81) are formally identical. This coincidence reflects the intrinsic nature of
quantum control resource supported by the feedback connection. We call I-IV) symmetric operator-ordering Scheme (SOS)
for convenience.
According to the unitary operators derived here, we can
obtain the stochastic differential equation for the quantum
feedback system.
Corollary 1: An arbitrary operator on the feedback system
shown in Fig. 4 obeys the stochastic differential equation
(85)
Proof: The proof is straightforward, expanding the unitary
operators (82) or (84) to order . Note that the term of the free
Hamiltonian is of order .
Finally, the Hamiltonian of the feedback system can be derived from Theorem 1.
Corollary 2: The Hamiltonian describing the feedback
system shown in Fig. 4 is given by
(86)
Proof: From the Hausdorff’s formula, for any two operand commuting with
, the following relation
ators
holds:
(87)
Applying this formula to the result of Theorem 1 and ignoring
the terms of order greater than , the total unitary operator is
written as
(88)
is given by (86). This is, therefore, the total Hamilwhere
tonian of the feedback system.
Authorized licensed use limited to: Univ of Calif Santa Cruz. Downloaded on September 11,2023 at 05:22:24 UTC from IEEE Xplore. Restrictions apply.
2116
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003
The first and second terms of the feedback Hamiltonian
describe the coupling of each system to the external
field. The third term describes the direct coupling between
the two systems, which is generated by the feedback through
the input/output of the systems. This direct coupling provides
control resources in the feedback system.
Laplace transform on the operators yields the transfer function
for each system
(95)
(96)
V. SYSTEM CONNECTIONS AND THEIR REPRESENTATION
In the preceding section, we derived SOS to obtain the unitary
evolution operator of the feedback system. We shall now show
that SOS works for deriving the evolution of various types of
systems.
A. Cascade Connection
The simplest configuration is the cascade connection of two
systems, say and , in which the output of is fed into .
The external field, after the interaction with the mode , then
.
interacts with the mode
The cascade connection is a special case of feedback discussed in the preceding section with zero reflectivity of the beam
. Theorem 1 verifies the physical meansplitter:
ings of the operator ordering for the cascade connection. Using
and
, one can see that an arbitrary
Theorem 1 with
operator of this system obeys the evolution
The relation between the input
written as
and the output
is, therefore,
(97)
As far as we are concerned with the input/output relation, the
cascade quantum system is surely represented by the cascade
of the transfer functions in the same way as is done in control
theory.
B. Feedback Connection of Three Systems
We shall consider a system with a closed loop in which three
, and , are positioned to form a cascade consystems,
, and
nection, as shown in Figs. 5 and 6. Denote by
the annihilation operators for the modes of the three systems,
respectively.
Each traveling wave is given by
(89)
where the total evolution operator is given by
(90)
with
(91)
(92)
where is the output from the system , given by (74). A typical feature of the cascade system is that the effect of interaction
with the external field only goes unilaterally. interacts with
the output field of and, consequently, is influenced by the
dynamics of along with the external field, but not vice versa.
Because of this unilaterality, the unitary operator at is identical with (68), regardless of the existence of . On the other
hand, is driven by , which is the output from , and the
unitary operator at is generated by the interaction between
and , as in (92). The total evolution is given by (90), in which
the operator ordering is logical as it indicates that the unitary
evolution at occurs before that at .
and
are responsible for the
As in the preceding section,
.
input/output relation of the cascade system. Assume that
and
in (85) and using the condition
Taking
and
, we obtain
(98)
According to SOS, the total unitary evolution can be obtained
by allowing the evolution of each system to occur successively
as
(99)
Here, each operator is given by
(93)
(94)
(100)
Authorized licensed use limited to: Univ of Calif Santa Cruz. Downloaded on September 11,2023 at 05:22:24 UTC from IEEE Xplore. Restrictions apply.
YANAGISAWA AND KIMURA: TRANSFER FUNCTION APPROACH TO QUANTUM CONTROL—PART I
Fig. 5.
2117
Cascade connection of two cavities.
Fig. 7. Schematic representation of a combination of three cavities with
feedback connection. The beam splitter is denoted by the symbol .
Fig. 6. Schematic representation of the closed loop including three cavities.
The annihilation operator is denoted by c, while the traveling wave is denoted
by b.
the cascade connection and, hence,
is the same as (88). The
is responsible for the cascade part of the
unitary operator
system. From the unilaterality of the cascade connection, one
is generated by the interaction between and
can see that
, i.e.,
SOS also gives the alternative expression of the total evolution
in the reverse order as
(104)
(101)
where is given by (79). The stochastic differential equation
for an arbitrary operator is
where
(105)
The total increment of
is calculated to be
(102)
Taking the noncommutativity of the external field into account,
one can see that the two expressions of evolution are precisely
equivalent.
C. Cascade Connection With Feedback
A certain class of systems, such as cavities coupling in time
symmetric/asymmetric ways, is reduced to simple combinations of feedback and cascade connections. Here, we consider
a system of three cavities as shown in Fig. 7.
The total Hamiltonian of this system is not trivial, even if the
connection is simple, because the three cavities interact with
each other through the same external field and a complicated
entanglement is generated between them accordingly. This
system, however, can be thought of as the cascade of a single
system and a closed loop. Thus, we can obtain the unitary operator in a similar fashion to the previous arguments. The total
evolution is decomposed into two parts—one is the evolution
of the closed loop and the other is of the cascade connection.
According to SOS, the total unitary operator is given by
(103)
describes the evolution of the closed loop. This part
Here,
is not influenced by the system , because of the property of
(106)
where the input is assumed to be in the vacuum state for
simplicity.
As stated previously, the interaction of the external field goes
unilaterally from the closed loop of and to . Actually, it
is not difficult to see that the stochastic differential equation for
is identical with (85), and that and
operator
Authorized licensed use limited to: Univ of Calif Santa Cruz. Downloaded on September 11,2023 at 05:22:24 UTC from IEEE Xplore. Restrictions apply.
2118
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003
(107)
the effect of quantum noise in the output of the system. This
paper provides a solid foundation for the design of the quantum
feedback system for noise reduction to be considered in the
follow-up paper.
(108)
APPENDIX I
are not affected by . For example, the differential equations
and
in the closed loop are rewritten as
for
where and are given by (79). These evolutions imply that
and are actually independent of in accordance with the
property of the cascade connection.
in (106) yields
On the other hand, letting
(109)
where is given by (79). This result indicates that takes
as an input, which is also consistent with the property of the
cascade connection.
In the treatment of Ito-type stochastic differential equations,
it is important that the noise and the system are independent at
prior to the interaction
the same time. The input operator
, while the
with the system is defined on the Hilbert space
. It is obvious that
system operators is on the Hilbert space
is independent of the system and comthe input operator
mutative with an arbitrary system operator. Then, the quantum
stochastic differential equation is derived by the standard calculus, expanding the Taylor series to order . Using (68), the
unitary operator (68) is expanded as
VI. CONCLUSION
The importance of the results obtained in this paper lies in
illustrating the structure of the quantum feedback system and
the systematic scheme for defining its Hamiltonian, no matter
how complex the system is. For optical systems, signals are carried by the continuum of the field modes, to which the canonical commutation relation is introduced in order to deal with
the signals in a quantum mechanical way. The infinite number
of independent field modes allow each infinitesimal time interval to constitute the interaction with one possible mode of
the system. Then, the input to the system is defined by the field
mode immediately before the interaction, and the output from
the system is the field mode immediately after the interaction.
Using the input–output description, the system is expressed by
a transfer function, which enables us to regard the system as a
map from the input operator to the output operator. The main
structural role of the transfer function is a path to relating a
quantum theory to control theory. It will provide a method to
evaluate how well systems deal with noise and uncertainties. To
this end, considerable emphasis was placed on the examination
of feedback connection of quantum systems. If the output signal
of the quantum system, after having interacted with a controller,
then interact with the system again, the information stored in
the output can be fed back through the closed loop. The field is
modulated by the controller that one can design to achieve desirable performances of the feedback system. In this case, the
closed loop generates the entanglement between each component of the system, and increases the degree of complexity of
the system dynamics accordingly. The systematic scheme for
unitary evolution operators obtained in this paper treats the essential elements of the closed loop in a quantum mechanical
way, and clarifies the structure of the system which is highly
complicated in dynamics.
The quantum feedback system considered in Section IV
leads to a natural extension of classical feedback. For linearized
quantum systems, what is known as state in control theory
is an operator-valued vector. Replacing each element of the
vector to its expectation value, this vector formally represents
the corresponding classical system. A potential application of
the quantum feedback treated by this formalism is to reduce
(110)
again and noting that the input
Using the same definition of
operator is independent of the system operator prior to the interaction and hence commutative with it, we obtain the quantum
stochastic differential equation (71).
APPENDIX II
The interaction arising at each cavity on the closed loop is
defined by the coupling (61), superposition of tensor product
of operators on the Hilbert space corresponding to the cavity
and the external field, respectively, that produces the entanglement over the entire system for a control resource. However,
the closed loop leads to the self-interaction of the cavity, and
results in an ambiguity of operator ordering, which constitutes
a difficulty to define the interaction Hamiltonian properly. For
consists of the term
instance, the coupling of to the input
(111)
where each of these quantities is an operator. Implicit within this
is ’s interaction with itself through the closed loop.
Consider a general form of a Hamiltonian describing the
and
. Let
and
be of
linear coupling of the modes
the form
(112)
(113)
, and are adequate constants. The Hamiltonians
where
should be symmetric. We, then, introduce a symmetric ordering
for the coupling of each mode to the external field
(114)
(115)
where
.
Authorized licensed use limited to: Univ of Calif Santa Cruz. Downloaded on September 11,2023 at 05:22:24 UTC from IEEE Xplore. Restrictions apply.
YANAGISAWA AND KIMURA: TRANSFER FUNCTION APPROACH TO QUANTUM CONTROL—PART I
Remark 2: Although the time argument of all the operators
and
in the above expression is supposed to be , the operators
in (114) and (115) may actually be of a slightly later time,
after the inputs have interacted with the system. As a result, we
have to consider that the system operators include the terms of
and . However, manipulations based on this
order
observation lead to the same form of the Hamiltonian, and hence
the time arguments in (114) and (115) really are all .
Now, suppose that the unitary operator at acts first, and that
at acts second. In this case, an arbitrary operator obeys the
evolution
(116)
where
(117)
(118)
(119)
Expanding these unitary operators to order
and
, we have operator is
and letting
(120)
(121)
where the time argument has been omitted. From the fact that
the diffusion term of the quantum stochastic differential equation for the system operator is constant as stated in Section III,
the system and should be driven by and in (79), re.
spectively. Therefore, we have
Having considered all of these, the unitary operators of the
feedback system are written as
where an indeterminable scalar function
has been included. We note that the only total unitary evolution leads to
the sensible result. The evolution of the feedback system is
thus independent of the function , and we suppose that this
. After all, the unitary
function is constantly zero,
operators are given by (82).
REFERENCES
[1] T. J. Tarn, G. Huang, and J. W. Clark, “Modeling of quantum mechanical
control system,” Math. Modeling, vol. 1, pp. 109–121, 1980.
[2] G. Huang, T. J. Tarn, and J. W. Clark, “On the controllability of quantummechanical systems,” J. Math. Phys., vol. 24, no. 11, pp. 2608–2618,
1983.
2119
[3] C. K. Ong, G. Huang, T. J. Tarn, and J. W. Clark, “Invertibility of
quantum-mechanical Control Systems,” Math. Syst. Theory, vol. 17,
pp. 335–350, 1984.
[4] H. H. Rosenblock, “A stochastic variational treatment of quantum mechanics,” in Proc. R. Soc. Lond. A, vol. 45, 1995, pp. 417–437.
[5] A. P. Peirce, M. A. Dahleh, and H. Rabitz, “Optimal control of quantummechanical systems: Existence, numerical approximation, and applications,” Phys. Rev. A, vol. 37, pp. 4950–4964, 1988.
[6] S. Shi, A. Woody, and H. Rabitz, “Optimal control of selective vibrational excitation in harmonic chain molecules,” J. Chem. Phys., vol. 88,
pp. 6870–6883, 1988.
[7] S. H. Tersigni, P. Gaspard, and S. A. Rice, “On using shaped light pulses
to control the selectivity of product formation in a chemical reaction:
An application to a multiple level system,” J. Chem. Phys., vol. 93, pp.
1670–1680, 1990.
[8] M. A. Dahleh, A. P. Peirce, and H. Rabitz, “Optimal control of uncertain
quantum systems,” Phys. Rev. A, vol. 42, pp. 1065–1079, 1990.
[9] R. S. Judson and H. Rabitz, “Teaching laser to control molecules,” Phys.
Rev. Lett., vol. 68, pp. 1500–1503, 1992.
[10] P. Gross et al., “Optimally designed potentials for control of electron-wave scattering in semiconductor nanodevices,” Phys. Rev. B, vol.
49, pp. 100–110, 1995.
[11] H. Tang, R. Kosloff, and S. A. Rice, “A generalized approach to the
control of the evolution of a molecular system,” J. Chem. Phys., vol.
104, pp. 5457–5471, 1996.
[12] J. M. Shapiro and P. Brumer, “Quantum control of chemical reactions,”
J. Chem. Soc., Faraday Trans., vol. 93, pp. 1263–1277, 1997.
[13] G. J. Toth, A. Lorincz, and H. Rabitz, “The effect of quantum dispersion on laboratory feedback optimal control,” J. Mod. Opt., vol. 44, pp.
2049–2052, 1997.
[14] R. J. Gordon and S. A. Rice, “Active control of the dynamics of atoms
and molecules,” Annu. Rev. Phys. Chem., vol. 48, pp. 601–641, 1997.
[15] V. Ramakrishna, M. V. Salapaka, M. A. Dahleh, H. Rabitz, and A. P.
Peirce, “Controllability of molecular systems,” Phys. Rev. A, vol. 51,
pp. 960–966, 1995.
[16] D. D’Alessandro, “On the controllability of of systems on compact Lie
groups and quantum mechanical systems,” J. Math. Phys., vol. 42, pp.
4488–4496, 2001.
[17] G. Turinici and H. Rabitz, “Quantum wave function controllability,”
Chem. Phys., vol. 267, pp. 1–9, 2001.
[18] S. Schirmer, J. V. Leahy, and A. I. Solomon, “Degrees of controllability
for quantum systems and applications to atomic systems,” J. Phys. A,
vol. 35, pp. 4125–4141, 2002.
[19] V. Ramakrishna, R. J. Ober, K. L. Flores, and H. Rabitz, “Quantum control by decompositions of SU(2),” Phys. Rev. A, vol. 62, p. 053 409,
2000.
[20] D. D’Alessandro and M. Dahleh, “Optimal control of two-level quantum
systems,” IEEE Trans. Automat. Contr., vol. 46, pp. 866–876, June 2001.
[21] N. Khaneja, R. Brockett, and S. J. Glaser, “Time optimal control in spin
systems,” Phys. Rev. A, vol. 63, p. 032 308, 2001.
[22] S. G. Schirner, A. D. Greentree, V. Ramakrishna, and H. Rabitz, “Constructive control of quantum systems using factorization of unitary operators,” J. Phys. A, vol. 35, pp. 8315–8339, 2002.
[23] Y. Yamamoto, N. Imoto, and S. Machida, “Amplitude squeezing in a
semiconductor laser using quantum nondemolition measurement and
negative feedback,” Phys. Rev. A, vol. 33, pp. 3243–3261, 1986.
[24] H. A. Haus and Y. Yamamoto, “Theory of feedback-generated squeezed
states,” Phys. Rev. A, vol. 34, pp. 270–292, 1986.
[25] J. M. Shapiro et al., “Theory of light detection in the presence of feedback,” J. Opt. Soc. Am. B, vol. 4, p. 1604, 1987.
[26] J. Dalibard, Y. Castin, and K. Molmer, “Wave-function approach to
dissipative processed in quantum optics,” Phys. Rev. Lett., vol. 68, pp.
580–583, 1992.
[27] C. W. Gardiner, A. S. Parkins, and P. Zoller, “Wave function quantum
stochastic differential equations and quantum jump simulation
methods,” Phys. Rev. A, vol. 46, pp. 4363–4381, 1992.
[28] H. J. Carmichael, “Quantum trajectory theory for cascaded open
system,” Phys. Rev. Lett., vol. 70, pp. 2273–2276, 1993.
[29] H. M. Wiseman, “Quantum theory of continuous feedback,” Phys. Rev.
A, vol. 47, pp. 5180–5192, 1993.
[30] H. M. Wiseman and G. J. Milburn, “Quantum theory of optical feedback
via homodyne detection,” Phys. Rev. Lett., vol. 70, pp. 548–551, 1993.
[31] A. C. Doherty, S. Habib, K. Jacobs, H. Mabuchi, and A. M. Tan,
“Quantum feedback control and classical control theory,” Phys. Rev. A,
vol. 62, p. 012 105, 2000.
[32] S. Lloyd, “Quantum feedback with weak measurements,” Phys. Rev. A,
vol. 62, p. 012 307, 2000.
Authorized licensed use limited to: Univ of Calif Santa Cruz. Downloaded on September 11,2023 at 05:22:24 UTC from IEEE Xplore. Restrictions apply.
2120
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003
[33] M. Yanagisawa and H. Kimura, “A control problem for gaussian states,”
in Learning, Control, and Hybrid Systems. ser. Lecture Notes in Control
and Information Sciences, Y. Yamamoto and S. Hara, Eds. New York:
Springer-Verlag, 1998, vol. 241, pp. 294–313.
[34] J. J. Sakurai, Modern Quantum Mechanics. Reading, MA: AddisonWesley, 1994.
[35] A. C. Holevo, Probabilistic and Statistical Aspects of Quantum
Theory. Amsterdam, The Netherlands: North-Holland, 1982.
[36] K. Kraus, States, Effects, and Operations: Fundamental Notions of
Quantum Theory. Berlin, Germany: Springer-Verlag, 1983.
[37] C. W. Gardiner, Quantum Noise. Berlin, Germany: Springer-Verlag,
1991.
[38] M. Yanagisawa and H. Kimura, “Transfer function approach to quantum
control systems,” presented at the 40th Conf. Decision Control, Orlando,
FL, 2001.
Masahiro Yanagisawa received the Ph.D. degree in engineering from the University of Tokyo, Tokyo, Japan, in 2001.
He was a Research Fellow of the Japan Society for the Promotion of Science from 2000 to 2002, and is currently a Postdoctoral Scholar in Control and
Dynamical Systems at the California Institute of Technology, Pasadena. His research interests are in quantum feedback control, quantum information, control
of stochastic processes, measurement theory, quantum computation, and entanglement control.
Hidenori Kimura (M’76–SM’87–F’90) graduated from the Department
of Mathematical Engineering and Instrumentation Physics, The University
of Tokyo, Tokyo, Japan, in 1965. He received the Dr.Eng. degree from the
University of Tokyo in 1970.
He joined the Faculty of Engineering Science, Osaka University, Osaka,
Japan, in 1970, where he studied nonlinear dynamics, multivariable control
systems, robust control and its applications, and signal processing. He stayed
at Warwick University and The Imperial College of Science and Technology,
Warwick, U.K., in 1974–1975, supported by British Council. He stayed at
Delft University of Science and Technology, Delft, The Netherlands, for three
months in 1994, and at the University of California, Berkeley, as a Springer
Professor in 1995.
Dr. Kimura has received various awards, including The Paper Awards from
the Society of Instrument and Control Engineers in 1972, 1978, 1983, and 1987;
The Automatica Paper Prize Awards from the Internation Federation of Automatic Control (IFAC) in 1984 and 1990; the George Axelby Award from The
IEEE Control Systems Society in 1985; The Distinguished Member Award from
The IEEE Control Systems Society. He was also awarded a Distinguished Technology Award from the Agency of Science and Technology in 1990. He served
as the General Chair of the Conference on Decision and Control 1996 in Kobe,
Japan. He served as a Member of the Editorial Board of numerous journals, including Automatica, the International Journal of Control, and the Asian Journal
of Control. He is currently the President of the Society of Instrument and Control Engineers (SICE).
Authorized licensed use limited to: Univ of Calif Santa Cruz. Downloaded on September 11,2023 at 05:22:24 UTC from IEEE Xplore. Restrictions apply.