OPTIMAL CONTROL OF MIXED-STATE QUANTUM

SYSTEMS BASED ON LYAPUNOV METHOD

Shuang Cong, Yuanyuan Zhang

Department of Automation, University of Science and Technology of China, Hefei, Anhui, 230027, Republic of China

Kezhi Li

Dept.of Electrical and Electronic Engineering, Imperial College London, South Kensington Campus

London SW7 2AZ, U.K.

Keywords: Optimal control, Lyapunov-based control, Quantum systems, Mixed state, Oscillator.

Abstract: An optimal control strategy of mixed state steering in finite-dimensional closed quantum systems is

proposed in this paper. Two different situations are considered: one is the target state is in statistical

incoherent mixtures of energy eigenstates in which the target states are diagonal. Another is not all of the

off-diagonal elements in the target states are zeros. We change the trajectory tracking problem into the state

steering one by introducing the unitary transformation with all energy eigenstates in the inner Hamiltonian

of system controlled. Based on Lyapunov stability theorem the stable parameters of controller designed is

selected and the optimality of the control law proposed is proven. Moreover, two numerical system control

simulations are performed on the diatomic molecule described by the Morse oscillator model under the

control law proposed. The system control simulation experimental results demonstrate that the control

strategies proposed are efficient even when the controlled system is not completely controllable

.

1 INTRODUCTION

As one of the greatest achievements in the 20th

century, quantum mechanics has urged the human

view of the matter to the microcosm. An enormous

amount of revolution in theory and engineering

science have been undertaken due to the

development and applications of quantum physics,

quantum chemistry, quantum computation and

quantum information. (Nielsen and Chuang, 2000).

In these new interdisciplinary fields, how to control

the quantum systems has become a challenging

subject. One part of the quantum control theories is

about the applications of classical and modern

control theory to quantum systems. (

Wang and

Schirmer, 2008) Now there have been various control

schemes applied to the quantum systems, such as the

Lyapunov-based method (Grivopoulos and Bamieh,

2003; Mirrahimi, Rouchon, and Turinici, 2005;

Beauchard et al. 2007; Cong and Kuang, 2007;

Kuang and Cong, 2008), optimal control method

(Peirce, et al. 1988; D’Alessandro and Dahleh,

2001; Girardeau, et al. 1998; Schirmer, et al. 2000),

learning control method (

Judson and Rabitz, 1992;

Phan and Rabitz, 1999), state estimation method

(

Doherty and Jacobs, 1999; Zhang, Li, and Guo, 2000),

and stochastic control method (Belavkin, 1992;

Bouten, et al. 2004), etc. Generally, the control aim

of a quantum system is to search for a control field

by means of minimizing an energy-type cost

function of system that usually requires a maximal

transition probability from an initial state to a

particular target state. Among all of the quantum

control strategies, optimal control methods are the

most popular approaches that have been widely used

specially in quantum chemistry fields. Since the mid

1980s, the quantum optimal control theory has

attracted attentions from many researchers.

However, many proposed optimal control methods

are generally obtained by means of complex numeral

iterative algorithms, which are off-line control

methods and quite inconvenient to operate and

realize. Thus, how to obtain an optimal method

without iterative solutions is of great significance.

22

Zhang Y., Cong S. and Li K..

OPTIMAL CONTROL OF MIXED-STATE QUANTUM SYSTEMS BASED ON LYAPUNOV METHOD.

DOI: 10.5220/0003126000220030

In Proceedings of the International Conference on Bio-inspired Systems and Signal Processing (BIOSIGNALS-2011), pages 22-30

ISBN: 978-989-8425-35-5

Copyright

c

2011 SCITEPRESS (Science and Technology Publications, Lda.)

There is another method called local optimal

control which defines a general performance index

()yt as a function of the expectation values of

physical observables. Then it designs a control field

that drives the quantum system satisfying the

monotonous increasing condition of

()yt (Ohtsuki,

1998; Sugawara, 2003). Local optimal law is

explicitly derived without iteration and can satisfy

the necessary condition for a solution to the optimal

control problem. In the applications of optimal

control theory, we can select different performance

index and get different control laws. Here we will

select the error between the states as a performance

index of the control law. The difference between

Ohtsuki (1998), Sugawara (2003) and this paper is

that the derived control law in this paper satisfies the

sufficient condition for optimal. We have applied

this method to a pure state quantum system (

Zhang

and Cong, 2008). As a further step, in this paper we

would like to consider the mixed-state control

problem based on the formulas of statistical

mechanics, and apply the idea in Ref. Zhang and

Cong (2008) to the Liouville equation. In quantum

system, two reasons lead to mixed-state: one is

quantum dissipation due to quantum system entangle

with environment. In such a situation, the system

will be open. Quantum state will become a mixed-

state even though it is a pure state at the beginning.

Here, evolution of density matrix in this open system

will not be unitary. Second, a mount of same

particles in different pure states are incoherent

mixed, which would be a quantum ensemble.

Particles in different pure states are in this ensemble

with some probability, viz. average statistically. In

this paper, we only consider closed system without

action with environment, so mixed-state here refers

to mixed-state in ensemble.

The rest of the paper is organized as follows. In

Sec. 2 we introduce the system models in Hilbert

space and in Liouville space. Section 3 gives the

control law theorem and its proof based on the

Lyapunov stability theorem and principle of

optimality under the condition that the target state is

diagonal and non-diagonal, respectively. The

numerical simulation on the diatomic molecule

described by the Morse oscillator model is presented

in Sec. 4. Finally, Sec. 5 concludes the study of the

paper.

2 MODEL OF THE SYSTEM

CONTROLLED

The state of a quantum mechanical system can be

described in various ways. When a system is in a

pure state un-entangled with its environment, the

state of the system can be described by a wave

function that evolves according to a control-

dependent Schrödinger equation. One can also

describe the state of the system by a density operator

ˆ

()t

ρ

, which can not only represent a pure state but

also a mixed state. The density operator

ˆ

()t

ρ

acting on the system’s Hilbert space H evolves

with time according to the quantum Liouville

equation:

ˆ

ˆˆ

() [ (), ()]itHtt

t

ρρ

∂

=

∂

= ,

0

1

ˆ

ˆ

ˆ

() ()

M

mm

m

H

tH ftH

=

=+

∑

(1)

where

0

ˆ

H

is the system’s internal (or free)

Hamiltonian, and

ˆ

m

H

is the interaction (or control)

Hamiltonian, respectively, all of them will be

assumed to be time-independent.

()

m

f

t is the

admissible real-valued external control field. We set

the Planck constant 1

=

= for convenience.

Because

ˆ

()t

ρ

is a NN× density matrix in

Hilbert space, it’s difficult to solve the differential

Eq. (1). One may introduce the Liouville operator in

the Liouville space according to the concept of Dirac

operator to simplify this problem. There is a natural

connection between the density matrix and Liouville

space (Barnett and Dalton, 1987; Ohtsuki, et al.

1989). In the Liouville space, Eq. (1) can be

represented in the same form as the Schrödinger

equation

() () ()it tt

t

ρρ

∂

=

∂

L ,

0

1

() ()

M

mm

m

tft

=

=+

∑

LL L

(2)

where

()t

ρ

is defined as a Liouville ket, and L

is the Liouville operator defined by the dual

correspondence

ˆ

ˆ

() () [ , ()]tt Ht

ρρ

↔L

(3)

The basis vectors respectively belonging to

Liouville space and Hilbert space are defined by the

OPTIMAL CONTROL OF MIXED-STATE QUANTUM SYSTEMS BASED ON LYAPUNOV METHOD

23

bijective correspondence

mn m n↔

(Schirmer, 2000). Then one has

,

*

ˆ

[, ]

ˆ

([, ])

ˆˆ

()

ˆ

ˆ

jk mn

i

nk jm jm nk kn jm

jk mn j H m n k

tr k j H m n

ik jHmni ik jmnHi

jHm nHk H H

δδδδ

==

=

=−

=−=−

∑

LL

(4)

For an NN× density matrix

ˆ

()t

ρ

in Hilbert

space, its replacement form

()t

ρ

is an

2

N

column vector in Liouville space, and L is an

22

NN× matrix. In such a way it is much easier to

solve Eq. (2) than Eq. (1) expressed in terms of

some commutators. Hence, Eq. (2) will be adopted

as the investigated model in following sections of

the paper.

3 CONTROL LAW DESIGN

The quantum control problems can be formulated in

state steering (or transfer) problem, that is to say

steer the system from a given initial state to a

desired target state. In this section we’ll develop an

optimal control method based on Lyapunov theorem

for the Liouville equation.

First we will introduce principle of optimality

and the sufficient condition for optimality. Suppose

the controlled system is in the form

of () [ (), (),]tttt=

xfxu

, where

12

[, ]tTT∈ ,

12

() [ , ]

n

tTT⊂×x \ ,

12

() [ , ]

m

tTT⊂×u \ . Let X be

a given region in

12

[, ]

n

TT×\ and contain the

target set

S . For each

00

(,)tx in X , one need

determine the control

u which transfers

00

(,)tx

to

S and minimizes the performance index

1

0

(, ,) [(), (),]d

t

t

J

tLtttt=

∫

xu x u . Define

*

(,)

J

tx is the

minimum of ( , , )

J

txu . The Hamiltonian

(, , ,)

H

txpu is given by

(, , ,) (, ,) ,(, ,)

H

tL t t=+〈 〉xpu xu pf xu

Principle of Optimality: If

*

()tu is an optimal

control and if

*

()tx , for

01

[,]ttt∈ , is the optimal

trajectory corresponding to

*

()tu , then the

restriction of

*

()tu to a subinterval

1

[, ]tt of

01

[,]tt is an optimal control for the initial pair

*

((),)ttx .

Sufficient Condition for Optimality (Athans and

Falb, 1966): Suppose that

12

(, )

n

XTT=×\ ,

H

is

normal relative to

12

(, )

n

TT×\ , and ( , , )tuxp is

the

H

-minimal control relative to

12

(, )

n

TT×\ .

Let

*

()tu be an admissible control such that:

a.

*

()tu transfers

00

(,)tx to S .

b. There is a solution

*

(,)

J

tx of the

Hamilton-Jacobi equation

(,) [, (,), (, (,),),] 0

JJJ

t H t ttt

t

∂∂∂

+

=

∂∂∂

xxxuxx

xx

satisfying the boundary condition ( , ) 0Jt=

x for

(,)tS

∈

x , such that

*

** *

( ) ( ( ), ( ( ), ), )

J

tt ttt

∂

=

∂

ux x

x

for t in

01

(,)tt .

Then

*

()tu is an optimal control.

3.1 Stationary Target States

Assume the target state is the statistical incoherent

mixtures of energy eigenstates:

1

ˆ

N

fn

n

wnn

ρ

=

=

∑

,

ˆ

f

ρ

is a stationary target state, e.g.

10

131

ˆ

0011

03

444

f

ρ

⎛⎞

=+=

⎜⎟

⎝⎠

. In this case, all of

the off-diagonal elements in the target state are

zeros. If so, the optimal control law is given by the

following theorem 1.

Theorem 1.

For the system defined in the Liouville

space by Eq. (2), given the performance index

2

0

1

11

{ [Im( )] ( ) ( )}d

2

M

fm

m

m

J

PtRtt

r

ρρ ρ

∞

=

=−+

∑

∫

T

ffL

(5)

where

12

() [ () () ()]

T

M

tftftft=f " ,

R

is a

diagonal matrix with positive elements, 0

m

r > ,

(1,2,,)mM=

" , and

P

is a positive definite

symmetric matrix that satisfies the equation

†

00

0PP

−

=LL

(6)

Then there exists an optimal control law

1

Im( )

mfm

m

fP

r

ρρ ρ

∗

=− − L ,

(1,2,,)mM=

"

(7)

BIOSIGNALS 2011 - International Conference on Bio-inspired Systems and Signal Processing

24

such that the system (2) is stable and the

performance index (5) is minimum.

In fact, according to the Lyapunov indirect stability

theorem,

P

is a positive definite symmetric matrix

that should satisfy Lyapunov equation

†

00

()()Pi i P Q+=−LL . Because

0

L is a linear

Hermitian operator, whose eigenvalues are real. So

0

iL is a skew Hermitian operator, whose

eigenvalues are pure imaginary. Accordingly,

0Q =

, which results in the condition (6).

Proof: (1) Proof of stability

Select the following Lyapunov function

1

()

2

ff

VP

ρρρρρ

=− −

(8)

where

P

is a positive definite symmetric matrix

satisfying Eq. (6). The first-order time derivative of

()V

ρ

is

()Re( )

f

VP

ρρρρ

=−

(9)

Substituting Eq. (2) into Eq. (9) yields

0

1

()Im( )

()Im( )

f

M

mfm

m

VP

ft P

ρρρρ

ρρ ρ

=

=− +

−

∑

L

L

(10)

Since

†

00

0PP−=LL

and

0

0

f

ρ

=

L

,

0

Im( ) 0

f

PL

ρρ ρ

−=

holds. Hence, Eq. (10) can

be re-written as

1

() ()Im( )

M

mfm

m

Vft P

ρρρρ

=

=−

∑

L

(11)

Substituting control law (7) into Eq. (11) yields

2

1

1

() [Im( )]0

M

fm

m

m

VP

r

ρρρρ

=

=− − ≤

∑

L

(12)

Thus, the system (2) is stable under the control law

(7). Next we will prove this control law is optimal.

(2) Proof of optimality

a) The Sufficient Condition for Optimality says

that if a system can be transferred from some initial

state to a target set by applying admissible control,

then an optimal control exists and may be found by

determining the admissible control

m

f

∗

that causes

the system to reach the target set S. A description of

the target set S is assumed to be known. So for the

system (2) it now only remains that one needs to

construct a proper target set S. Here we use the

similar way we have proven in reference 8 to

construct the target set S. In fact, in the Lyapunov-

based control design, the Lyapunov function V can

be seen as a target set S. So one can define the target

set S is the Lyapunov function V by constructing an

appropriate matrix P.

P

is selected a positive definite symmetric

matrix that satisfies the Eq. (6). At the same time,

the eigenvectors with the largest eigenvalue are the

maxima of V, the eigenvectors with the smallest

eigenvalue are the minima and all others are saddle

points. Then select the smallest eigenvalue of P is

f

P

with the corresponding target state

f

ρ

. In such

a way, a target set S with a monotonic function and

the target state as the minima value are constructed,

in which the initial state can be transferred to the

target state by the control law

m

f

∗

.

b) From Eq. (7) and Eq. (12), we can get

*

(,)

J

t

ρ

as following

*

2

1

2

1

(,)

11

{[Im( )]()()}d

2

1

{[Im( )]}d

()t()

M

fm

t

m

m

M

fm

t

m

m

t

Jt

PtRtt

r

Pt

r

VdV

ρ

ρρ ρ

ρρ ρ

ρρ

∞

=

∞

=

∞

=−+

=−

=− =

∑

∫

∑

∫

∫

*T *

ff

L

L

(13)

Thus, the Hamiltonian function of the system can be

†

0

1

(,)

()

(,)Im[( )( ())]

M

mm

m

H

V

Lft

ρ

ρ

ρ

ρ

ρ

=

∂

=+ +

∂

∑

f

f LL

(14)

where

2

1

11

[Im( )] ( ) ( )

2

M

fm

m

m

L

PtRt

r

ρρ ρ

=

=−+

∑

T

ffL

Because

*

(,)0

J

t

t

ρ

∂

=

∂

, a part of the sufficient

condition for optimality is

min[ ( , )] 0

M

R

H

ρ

∈

=

f

f

(15)

From Eq. (14), one can obtain

2

1

0

1

2

1

2

11

2

1

11

((),) [Im( )] () ()

2

Im( ( ( ) ) )

11

[Im( )]

2

() ()Im( )

11

[Im( ) ( )] 0

2

M

fm

m

m

M

fmm

m

M

fm

m

m

MM

mm m f m

mm

M

fm mm

m

m

H

tPtRt

r

Pft

P

r

rft ft P

Prft

r

ρρρρ

ρρ ρ

ρρ ρ

ρρ ρ

ρρ ρ

=

=

=

==

=

=

−++

−+

=−+

+−

=−+≥

∑

∑

∑

∑∑

∑

T

fffL

LL

L

L

L

(16)

Substituting Eq. (7) into Eq. (16) yields

OPTIMAL CONTROL OF MIXED-STATE QUANTUM SYSTEMS BASED ON LYAPUNOV METHOD

25

((),)0Ht

ρ

=

*

f

Thus, the control law (7) is optimal and minimizes

the performance index (5). The proof of theorem 1 is

completed.

The design steps of the optimal control law proposed

based on Theorem 1 are as follows:

(1) Select the weighting on the control vector

diag( )

i

R

r= , 0

i

r > , 1, 2, ,im= "

(2) Solve Eq. (6) for obtaining the positive define

matrix

P

.

(3) Calculate the optimal stabilizing control law

from (7).

3.2 Non-stationary Target States

If not all of the off-diagonal elements in the target

state are zeros, which is also a case of a mixed-state,

e.g.

122122

ˆ

11( 0 1)( 0 1)

22 222 2

11

1

13

4

f

ρ

=+ + +

⎛⎞

=

⎜⎟

⎝⎠

In this case, the target state

ˆ

()

f

t

ρ

is in fact not

stationary which evolves under

0

ˆ

H

according to

the Liouville-von Neumann equation

0

ˆ

ˆˆ

() [ , ()]

ff

itHt

t

ρρ

∂

=

∂

(17)

Now the target state is a time-dependent function,

and the control problem becomes a trajectory

tracking problem. From the system control point of

view, a trajectory tracking problem can be easily

solved by translating it into the state steering

problem. To do so, we first carry out the following

unitary transformations

†

ˆ

() () ()tUtUt

ρρ

=

(18)

And

†

ˆ

() () ()

ff

tUtUt

ρρ

=

(19)

in which

f

ρ

is a stationary target state which

equals

12

() ( , , , )

N

iE tiE t iE t

U t diag e e e

−−−

= " and

, 1,...,

i

E

iN= satisfy

012

ˆ

(, , , )

N

H

diag E E E= " in

Eq. (1).

Substituting Eq. (18) into Eq. (1), one can obtain

1

() [ () (), ()]

M

mm

m

it ftHtt

t

ρρ

=

∂

=

∂

∑

(20)

where

ˆ

() () ()

mm

H

tUtHUt

+

=

.

Owing to the unitary transformation,

ˆ

()t

ρ

and

()t

ρ

have the same populations, which means that

Eq. (1) and Eq. (20) describe the same physical

system. In such a way, the problem of system (1)

tracking a time-dependent target state

ˆ

()

f

t

ρ

in Eq.

(17) is equivalent to a problem of steering the state

in system (20) to the stationary target state

f

ρ

.

In the Liouville space, Eq. (20) can be

represented as

1

() () () ()

M

mm

m

it fttt

t

ρρ

=

∂

=

∂

∑

L

(21)

The optimal control law of Eq. (21) is given by

the following theorem 2.

Theorem 2.

For the system defined by Eq. (21), give

the performance index

2

0

1

11

{ [Im( ( ) )] ( ) ( )}d

2

M

fm

m

m

J

Pt tRtt

r

ρρ ρ

∞

=

=−+

∑

∫

T

ff

L

(22)

where

12

() [ () () ()]

T

M

tftftft=f " ,

R

is a

diagonal matrix with positive elements, 0

m

r > ,

(1,2,,)mM

= " , and

P

is a positive definite

symmetric matrix. Then there exists an optimal

control law

1

Im( ( ) )

mfm

m

fPt

r

ρρ ρ

∗

=− −

L ,

(1,2,,)mM

= "

(23)

such that the system (21) is stable and the

performance index (22) is minimum.

The proof method of theorem 2 is the same as that of

theorem 1, thus it will not be repeated here. In

computer simulation, we need to choose an

appropriate discrete propagation method to solve the

differential equation (2) or (21). A simple approach

would be adopting the first-order Euler method. But

to obtain more efficient result, we employ four-order

Runge-Kutta method, which has higher precision

and faster convergence rate.

4 NUMERICAL SIMULATIONS

AND RESULTS ANALYSIS

As an explicit example we consider a typical

diatomic molecule model with N discrete

BIOSIGNALS 2011 - International Conference on Bio-inspired Systems and Signal Processing

26

vibrational energy levels

n

E

corresponding to

independent states

n of the system. The internal

Hamiltonian is given by

0

1

ˆ

N

n

n

H

Enn

=

=

∑

(24)

Assume that the diatomic molecular system is

controlled by a single control field ( )

f

t . Then the

total Hamiltonian of the system can be represented

as

01

ˆˆ ˆ

() ()

H

tHftH=+ , and the corresponding

Liouville operator is

01

ˆ

ˆ

ˆ

() ()tft

=+LL L. The

interaction Hamiltonian can be chosen as the dipole

form

1

1

1

ˆ

(11)

N

n

n

H

dnn n n

−

=

=+++

∑

(25)

Next we will separately study the diatomic

molecules described by the Morse oscillator model

and the Harmonic oscillator model.

4.1 Morse Oscillator Model

To simplify the calculation, we consider a hydrogen

fluoride (HF) molecule described by a four-level

Morse oscillator model. The vibrational energy

levels are as follows (Schirmer, etc. 2001)

0

111

1

222

n

E

nnB

ω

⎡⎤

⎛⎞ ⎛⎞

=−−−

⎜⎟ ⎜⎟

⎢⎥

⎝⎠ ⎝⎠

⎣⎦

=

(26)

where

14 1

0

7.8 10 s

ω

−−

=× and 0.0419B = . Thus

the corresponding energy levels are

1

0.4948E = ,

2

1.4529E = ,

3

2.3691E = and

4

3.2434E = in

units of

0

ω

= . In the following calculations, all the

parameters are expressed in atomic units (a.u.). Here

the dipole moments in Eq. (25) are

n

dn= ,

(1,2,3)n = . This system is completely controllable

verified in Ref. Schirmer, etc. 2001.

Assume that the system is initially in the thermal

equilibrium, i.e.,

4

0

1

ˆ

n

n

wnn

ρ

=

=

∑

with weights

41

exp[ /( )]

nn

wC EEE=− −. This is a Boltzmann

distribution, and the normalization constant

3

12 4

/// /

1

()

EkTEkT EkT E kT

Ce e e e

−−− −

−

=+++ with

41

kT E E=− . Concretely,

1

0.3877w = ,

2

0.2736w = ,

3

0.1961w = , and

4

0.1426w = . The

control task is to determine the control field ( )

f

t

so as to steer the system from the initial state

0

ˆ

ρ

to

the target state

4

5

1

ˆ

fn

n

wnn

ρ

−

=

=

∑

. The state

control problem and the observable control problem

are inter-convertible. Thus the problem in this paper

is equivalent to that in Refs. 13 and 14 with the goal

to maximize the expectation value of the observable

0

ˆ

ˆ

AH= .

According to theorem 1, the optimal control law

can be obtained as

1

1

1

() Im( )

f

ft P

r

ρ

ρρ

=− − L

(27)

The initial state of the system lies within the set of

states resulting in

010

Im( ) 0

f

P

ρρ ρ

−=L , at

the moment the control field

0

0f = . This problem

can be solved by applying an initial small magnitude

disturbance to excite the system out of its initial

equilibrium state (

Beauchard, et al. 2007). In our

numerical system simulations, the initial control

field

0

0.05 a.u.f = , the target time 200 . .

f

tau= ,

and the sampling time 0.1 a.u.dt = . The suitable

choice of the parameters

1

r and

P

is crucial to

get good results. In order to obtain a higher

probability of the target state,

P

can be chosen to

make the Lyapunov function described by Eq. (8)

larger at the initial time, and the diagonal elements

of the initial state are ordered in a non-increasing

sequence, the corresponding elements of

P

are

also arrayed in non-increasing sequence (

Kuang and

Cong, 2008

). After several times of tuning, we select

1

1r

=

and

(18, 1,1, 1,1,1.5, 1,1,1,1,1, 1,1, 1,1, 0.01)Pdiag=

The numerical simulation results are shown in

Figures 1 to 4, in which Figure 1 shows the control

field. The corresponding evolution populations of

energy levels 1 through 4 are shown in Figure 2,

from which one can see that the populations are

inverted, i.e., the most energetic state

4 has the

highest population, and the second one has the

second highest population, etc. The final populations

of energy levels are 0.1547, 0.1927, 0.2680, and

0.3845, respectively. Figure 3 shows the

performance index, and Figure 4 shows the distance

from the target state. At the target time, the distance

is

2

ˆˆ

0.0034

f

ρρ

−= , so that the mixed-state

control is completed. In Ref. Schirmer, et al. (2000)

OPTIMAL CONTROL OF MIXED-STATE QUANTUM SYSTEMS BASED ON LYAPUNOV METHOD

27

0

ˆ

ˆ

AH= , and at the target time 200 a.u.

f

t = the

expectation value

ˆ

()

f

A

t is 99% of the

theoretical maximum. While in this paper, this ratio

is also 99% . Under the condition that the

simulation result is the same, the design process of

the control law in this paper is easier than that in

Ref. Schirmer, et al. (

2000) which needs iteration.

Also, by comparing the results, we can find that the

inverted rate of the levels is faster here, that is

because the initial control value is larger. In the real

applications, the control value can be tuned

according to the requirement.

0 50 100 150 200

-0.45

-0.3

-0.15

0

0.15

0.3

0.45

Time (a.u.)

Field (a.u.)

Figure 1: Optimal control field for a four-level Morse

oscillator model.

0 50 100 150 200

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (a.u.)

Populations

|4>

|3>

|2>

|1>

Figure 2: Evolution of populations for a four-level Morse

oscillator model.

0 50 100 150 200

0

0.1

0.2

0.3

0.4

0.5

0.6

Time (a.u.)

Index

Figure 3: Performance index for a four-level Morse

oscillator model.

0 50 100 150 200

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Time (a.u.)

Distance

Figure 4: Distance from target state for a four-level Morse

oscillator model.

4.2 Harmonic Oscillator Model

Comparing with the above mentioned completely

controllable Morse oscillator model, here we

consider the diatomic molecule described by a four-

level Harmonic oscillator model. The vibrational

energy levels are determined by

1

2

n

En

=

−

(28)

Thus the energy levels are

1

0.5E = ,

2

1.5E = ,

3

2.5E = and

4

3.5E = . The dipole moments in

this model are 1

n

d

=

, ( 1, 2,3)n = . The system is

not completely controllable because the dimension

of the Lie algebra generated by

0

ˆ

H

and

1

ˆ

H

is less

than 16 (

Barnett and Dalton, 1987). We still suppose

BIOSIGNALS 2011 - International Conference on Bio-inspired Systems and Signal Processing

28

that the initial density is

4

0

1

ˆ

n

n

wnn

ρ

=

=

∑

, in

which

1

0.3850w = ,

2

0.2758w = ,

3

0.1976w =

and

4

0.1416w = . The target state and the control

law are the same as that in the situation of the Morse

oscillator model. Starting with

0

0.15a.u.f = ,

0.1a.u.dt = ,

1

1r = , and

(4,1,1,1,1,3,1,1,1,1,2,1,1,1,1,1)Pdiag= , the

simulation curves are shown in Figures 5-8. At the

target time, the populations of energy levels are

0.1482, 0.2003, 0.2732, and 0.3783, respectively,

and the distance from the target state is

2

ˆˆ

0.0036

f

ρρ

−= . Despite the system is not

completely controllable, the method in this paper is

still efficient.

0 50 100 150 200

-0.12

-0.08

-0.04

0

0.04

0.08

0.12

Time (a.u.)

Field (a.u.)

Figure 5: Optimal control field for a four-level Harmonic

oscillator model.

0 50 100 150 200

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (a.u.)

Populations

|4>

|3>

|2>

|1>

Figure 6: Evolution of populations for a four-level

Harmonic oscillator model.

0 50 100 150 200

0

0.03

0.06

0.09

0.12

0.15

0.18

Time (a.u.)

Index

Figure 7: Distance from target state for a four-level

Harmonic oscillator model.

5 CONCLUSIONS

In this paper we have developed an optimal control

method based on Lyapunov theorem for the

Liouville equation to realize the quantum control of

the mixed states. The detailed design processes of

the control laws have been given both in the cases of

the target density operator of the system of interest

being a diagonal form and a general one,

respectively. Moreover, the numerical simulations

were performed for the diatomic molecule described

by the Morse oscillator model. The simulation

results show that the method proposed is as efficient,

even in the case that the system is not completely

controllable.

ACKNOWLEDGEMENTS

This work was supported in part by the National Key

Basic Research Program under Grants No.

2006CB922004 and No. 2009CB929601, the

National Science Foundation of China under Grant

No. 61074050.

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