PHYSICAL REVIEW A 75, 033407 共2007兲
Monotonically convergent algorithms for solving quantum optimal control problems described by an integrodifferential equation of motion Yukiyoshi Ohtsuki* and Yoshiaki Teranishi Department of Chemistry, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan and CREST, Japan Science and Technology Agency, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan
Peter Saalfrank Institut für Chemie, Universität Potsdam, Karl-Liebknecht-Strasse 24-25, D-14476 Potsdam-Golm, Germany
Gabriel Turinici CEREMADE, Universite Paris Dauphine, Place du Marechal De Lattre De Tassigny, 75775 Paris Cedex 16, France
Herschel Rabitz Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA 共Received 6 January 2007; published 19 March 2007兲 A family of monotonically convergent algorithms is presented for solving a wide class of quantum optimal control problems satisfying an inhomogeneous integrodifferential equation of motion. The convergence behavior is examined using a four-level model system under the influence of non-Markovian relaxation. The results show that high quality solutions can be obtained over a wide range of parameters that characterize the algorithms, independent of the presence or absence of relaxation. DOI: 10.1103/PhysRevA.75.033407
PACS number共s兲: 32.80.Qk
I. INTRODUCTION
Quantum optimal control theory provides a flexible way of designing a control to steer a dynamical system from the initial state to a desired target state 关1兴. The control design equations are derived by maximizing or minimizing a cost functional that measures control performance. The control design equations are nonlinear coupled equations, and the development of efficient solution algorithms is essential to numerically obtain the optimal control design 关2–8兴. The purpose of the present study is to extend the previous algorithms 关8兴 for solving optimal control problems to now operate with a general inhomogeneous integrodifferential equation of motion. We adopt the non-Markovian master equation, which also connects the limiting case of no relaxation with that of Markovian relaxation. Relaxation effects on the convergence behavior are systematically examined. Through a case study, we will show the present algorithm can solve quantum optimal control problems with high quality, independent of the presence or absence of relaxation. The systems of concern here are assumed to obey the following inhomogeneous integro-differential equation of motion:
兩u共t兲典 = 关␣ − E共t兲兴兩u共t兲典 − t
冕
t
d␥共t, 兲兩u共兲典 + 兩i共t兲典.
0
共1兲
time-dependent control, E共t兲, interacts with the system through the operator, . The operators ␣, , and ␥, which can be non-Hermitian, do not depend on the control. The inhomogeneous term 兩i共t兲典 appears, for example, in the master equation to capture an initial correlation between the relevant system and its environment. The following two cost functionals describe a wide range of applications for designing controls 关5,8兴. For convenience, they are referred to as type I and type II, which are linear or bilinear in the system state, respectively. The type-I functional is expressed as JI = 2 Re具X兩u共t f 兲典 + 2 Re −
冕
dt具Y共t兲兩u共t兲典
tf
dt
1 关E共t兲兴2 , A共t兲
共2兲
where Re refers to the real part. The state vector, 兩X典, specifies the target state at the final time t f and 兩Y共t兲典 denotes the time-dependent target over the control period. The last term is a penalty over the integral of 关E共t兲兴2, in which A共t兲 ⬎ 0 weighs the physical significance of the penalty. When E共t兲 is an electric field, the penalty term is the pulse fluence. In the case of the type-II functional, the physical targets are specified by two positive, semidefinite Hermitian operators, X and Y共t兲, such that JII = 具u共t f 兲兩X兩u共t f 兲典 + −
冕
冕
tf
dt具u共t兲兩Y共t兲兩u共t兲典
0
tf
0
1050-2947/2007/75共3兲/033407共5兲
tf
0
0
With suitable interpretation, the vector 兩u共t兲典 specifying the system state can be a wave function, a quantum density operator, or a product state of a density operator 关5,9兴. The
*Electronic address:
[email protected]
冕
dt
1 关E共t兲兴2 . A共t兲
共3兲
Note that the type-II functional can be reduced to the type-I 033407-1
©2007 The American Physical Society
PHYSICAL REVIEW A 75, 033407 共2007兲
OHTSUKI et al.
functional if we introduce a product space, in which a direct product u 丢 共t兲 = 兩u共t兲典具u共t兲兩 specifies the system state. Introduction of the direct product state enables us to remove the positive constraints of the target operators; however, we have to deal with higher-dimensional dynamics that require timeconsuming computation. The choice of either the type-I or type-II functional depends on target objectives and available computational resources. An optimal control E共t兲 is defined as one that maximizes a cost functional. The coupled control design equations are derived by applying the calculus of variations to the basic functionals, Eqs. 共2兲 and 共3兲, under the constraint of satisfying the equation of motion, Eq. 共1兲. Then the optimal control is expressed as E共t兲 = − A共t兲Re具共t兲兩兩u共t兲典,
共4兲
where 兩共t兲典 is the Lagrange multiplier state that assures satisfaction of Eq. 共1兲. In the type-I and type-II cases, the time evolution of the Lagrange multiplier is, respectively, given by
兩共t兲典 = − 关␣† − †E共t兲兴兩共t兲典 + t
冕
tf
¯E共k兲共t兲 = 共1 − 兲E共k−1兲共t兲 − A共t兲Re具共k兲共t兲兩兩u共k−1兲共t兲典 k k 共9兲 and ¯ 共k兲共t兲 − A Re具共k兲共t兲兩兩u共k兲共t兲典, 共10兲 E共k兲共t兲 = 共1 − k兲E k where the convergence parameters, k and k, characterize the iteration algorithm 关6,8兴. As shown below, the parameters 兵k , k其 苸 关0 , 2兴 guarantee monotonic convergence behavior of the present algorithm. In the present algorithm, both the state vector and the Lagrange multiplier are improved at each iteration step. If we set one of the convergence parameters to zero, one of the possible means of improving the control is neglected. That is, we can set either ¯E共k兲共t兲 = E共k−1兲共t兲 or E共k兲共t兲 = ¯E共k兲共t兲; however, this does not lead to any significant reduction in the number of computational steps. To reveal the convergence behavior, we consider the difference in the cost functionals between the kth and 共k − 1兲th adjacent iteration steps:
␦JI共k,k−1兲 = JI共k兲 − JI共k−1兲 = 2 Re具X兩␦u共k,k−1兲共t f 兲典
d␥†共,t兲兩共兲典 − 兩Y共t兲典
+ 2 Re
t
共5兲 with the final condition 兩共t f 兲典 = 兩X典, and
兩共t兲典 = − 关␣† − †E共t兲兴兩共t兲典 + t
冕
tf
− d␥†共,t兲兩共兲典 共6兲
− Y共t兲兩u共t兲典
with the final condition 兩共t f 兲典 = X兩u共t f 兲典. The optimal control design equations, Eqs. 共1兲, 共4兲, and 共5兲 as well as Eqs. 共1兲, 共4兲, and 共6兲, are derived by a local maximum condition of the cost functional. In this sense, the present algorithms are local optimization schemes.
¯ 共k兲共t兲 − E共k−1兲共t兲兴 − 2 Re具共k兲共t兲兩兩u共k−1兲共t兲典关E + 2 Re
d␥†共,t兲兩共k兲共兲典
t
共7兲
with the final condition 兩共k兲共t f 兲典 = 兩X典, and
+ 兩i共t兲典
冕
冕 冕
tf
d具共k兲共兲兩␥共,t兲兩␦u共k,k−1兲共t兲典
t
− 2 Re
t
d具共k兲共t兲兩␥共t, 兲兩␦u共k,k−1兲共兲典,
共12兲
0
− 兩Y共t兲典
共k兲 兩u 共t兲典 = 关␣ − E共k兲共t兲兴兩u共k兲共t兲典 − t
共11兲
= − 2 Re具共k兲共t兲兩兩u共k兲共t兲典关E共k兲共t兲 − ¯E共k兲共t兲兴
The solution algorithm starts the iteration with 兩u 共t兲典 using an appropriate initial trial control E共0兲共t兲, and the type-I control design equations at the kth iteration step 共k 艌 1兲 are
冕
1 兵关E共k兲共t兲兴2 − 关E共k−1兲共t兲兴2其, A共t兲
d 共k,k−1兲 P 共t兲 + 2 Re具Y共t兲兩␦u共k,k−1兲共t兲典 dt
共0兲
t
dt
where 兩␦u共k,k−1兲共t兲典 = 兩u共k兲共t兲典 − 兩u共k−1兲共t兲典. It is convenient to introduce the function, P共k,k−1兲共t兲 = 2 Re具共k兲共t兲 兩 ␦u共k,k−1兲共t兲典, which satisfies P共k,k−1兲共0兲 = 0 at t = 0 and P共k,k−1兲共t f 兲 = 2 Re具X 兩 ␦u共k,k−1兲共t f 兲典 at t = t f . Differentiating with respect to time produces
II. SOLUTION ALGORITHM AND ITS CONVERGENCE BEHAVIOR
tf
dt具Y共t兲兩␦u共k,k−1兲共t兲典
0
tf
0
t
共k兲 兩 共t兲典 = − 关␣† − †¯E共k兲共t兲兴兩共k兲共t兲典 + t
冕
冕
tf
where we have used the control design equations 共7兲 and 共8兲 at the kth iteration step. Substituting Eqs. 共9兲 and 共10兲 into Eq. 共12兲 and integrating over t 苸 关0 , t f 兴 permits expressing Eq. 共11兲 in terms of the controls
␦JI共k,k−1兲 =
d␥共t, 兲兩u共k兲共兲典
冕
tf
0
0
1 A共t兲
再冉 冊
冉 冊
2 2 − 1 关E共k兲共t兲 − ¯E共k兲共t兲兴2 + −1 k k
冎
¯ 共k兲共t兲 − E共k−1兲共t兲兴2 . ⫻关E
共8兲
with the initial condition 兩u共k兲共0兲典 = 兩u0典. The controls at the kth step are expressed as
dt
共13兲
Summing both sides of Eq. 共13兲 from k = 1 to N, yields
␦JI共N,0兲 = JI共N兲 − JI共0兲 艌 0, which establishes the monotonic con-
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vergence of the cost functional for 兵k , k其 苸 关0 , 2兴 except for the two special cases of 兵k = k = 0其 and 兵k = k = 2其. A similar algorithm can be developed for the typeII case, which again identifies monotonic convergence as long as the operators X and Y共t兲 are positive, semidefinite. The latter restriction arises from the final conditions in the type-II case. Finally, we note some possible extensions of the algorithms. The algorithms allow the equation of motion to include higher-order integral terms 兰t0d1兰01d2 ¯ 兰0n−1dn␥共t , 1 , . . . , n兲兩u共n兲典, while retaining their monotonic convergence behavior. The algorithms are also applicable to classical systems modeled as linear in the system state. III. NUMERICAL RESULTS
Numerical tests are implemented in a four-level model system having a time-dependent electric field, E共t兲, coupled through a dipole interaction. In the double-space 共Liouvillespace兲 notation, the equation of motion for the reduced density operator, 兩共t兲典典, is given by
iប 兩共t兲典典 = 关L0 − ME共t兲兴兩共t兲典典 − iប t
冕
t
d⌫共t, 兲兩共兲典典.
FIG. 1. Convergence behavior for several values of the convergence parameters, and , in the non-Markovian case.
0
共14兲 The Liouvillians L0 and M correspond to the commutators 关H0 , . . . 兴 and 关 , . . . 兴, respectively, with H0 and being the field-free system Hamiltonian and the electric dipole moment operator. The memory kernel is ⌫共t , 兲. In Eq. 共14兲, we have neglected an inhomogeneous term assuming that the initial state of the system is uncoupled from the environment 关10兴. Consider the cost functional that aims at achieving the largest transition probability from the initial state 兩1典 to an objective state 兩典 at a target time, t f . The objective is specified by a Hermitian target operator, W = 兩典具兩, and the cost functional in type-I form is J = 具具W兩共t f 兲典典 −
冕
tf
0
dt
1 关E共t兲兴2 , A共t兲
共15兲
where 具具W 兩 共t f 兲典典 = tr兵W†共t f 兲其. The four-level system, 兩n典 共n = 1 , 2 , 3 , 4兲, is characterized by the energy eigenvalues of H0 as 1 = 0.0 cm−1, 2 = 69.6 cm−1, 3 = 139.2 cm−1, and 4 = 1811.0 cm−1, and by transition moment elements 具4兩兩m典 = 具m兩兩4典 = 1.0D 共m = 1 , 2 , 3兲. We assume phase relaxation processes, with nonzero matrix elements given by 具具m4兩⌫共t , 兲兩m4典典 = 具具4m兩⌫共t , 兲兩4m典典 = ⌳T exp共−兩t − 兩 / T兲共m = 1 , 2 , 3兲. Here ⌳ and T are the coupling strength and coherence time, respectively. The following simulations consider three cases with differing relaxation conditions: 共a兲 no relaxation 共⌳ = 0兲, 共b兲 non-Markovian 共⌳ = 15.0 cm−1 and T = 20.0 fs兲, and 共c兲 Markovian 共⌳ = 15.0 cm−1 and T = 0兲 dynamics. The control objective is to transfer the initial state population in 兩1典 to the superposition state 兩典 = 冑12 关兩2典 + 兩3典兴. The amplitude weight is A共t兲 = 5 ⫻ 105, and the final time is t f = 1000 fs. For simplicity, the convergence parameters are taken as constant over the iteration, i.e., k = and k = .
Figure 1 shows the behavior of log10共␦J共k,k−1兲兲 with respect to log10k for several sets of convergence parameters 兵 , 其 in the non-Markovian case. Here, the convergence limit is defined by the value of the cost functional at the final indicated iterative step, beyond which further monotonic convergence breaks down due to numerical limitations. More detailed discussions about the convergence limit will be given later. The results in Fig. 1 confirm the monotonic convergence behavior up to high numerical accuracy, except when employing a value of = 0, i.e., the Krotov algorithm 关12,13兴. We have checked that an increase in the number of time grid points did not significantly change the convergence behavior 关11兴. When ⫽ 0, the converged values of the cost functionals 共0.74518兲 are virtually independent of the choice of the convergence parameters, the difference among values being less than 5.3⫻ 10−4%. In these examples, smaller values of accelerate the convergence. As the present cost functional has a value of ⬃0.75, the difference in the finally attained cost functional, ␦J共k,k−1兲 ⬍ 10−16, cannot be evaluated due to the limited precision of the computer’s floating-point capabilities. When ⫽ 0, the final values of ␦J共k,k−1兲 are 10−14 – 10−12. When = 0, the final values of ␦J共k,k−1兲 are 10−7 – 10−5, which is far from the computer’s precision limit. Employment of the value of = 0 neglects either the improvement of the state vector or that of the Lagrange multiplier at each iteration step, which appears to explain the poor behavior in this case. Since the = 0 case leads to a larger final value of ␦J共k,k−1兲, it needed the smallest number of iteration steps to reach its own low quality convergence limit shown in Fig. 1. Figure 2 shows the effects of relaxation when the convergence parameters are chosen to be 共a兲 共 = 1.0, = 1.0兲 and 共b兲 共 = 1.0, = 0.5兲. The solutions are characterized by small
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FIG. 2. Relaxation effects on the convergence behavior. The solid, dashed, and dotted line show the results for the cases of no relaxation, non-Markovian, and Markovian dynamics, respectively.
final converged intervals ␦J共k,k−1兲 艋 10−12, indicating that high numerical accuracy is achieved independent of the presence or absence of relaxation. The presence of relaxation leads to rapid convergence without loss of numerical accuracy. Figure 3 shows the convergence behavior for several values of with a fixed value of = 1.0 to clarify the parameter range that yields high numerical accuracy. In the examples, the value of = 0.1 produces high numerical accuracy even in the non-Markovian case. These results together with those in Fig. 1 suggest that the present iteration scheme can achieve high numerical accuracy over a wide range of convergence parameters. To further analyze this numerical observation, we rewrite Eq. 共13兲 using Eqs. 共9兲 and 共10兲:
␦JI共k,k−1兲 =
冕
tf
0
dt
1 ¯ 共k兲 兵关E 共t兲 − E共k−1兲共t兲兴2 + 共2 − 兲 A共t兲 共k兲
⫻关A共t兲Re具 共t兲兩兩␦u
共k,k−1兲
共t兲典兴 其,
FIG. 3. Convergence behavior for small values of 共fixed value of = 1.0兲 in the 共a兲 Markovian and 共b兲 non-Markovian cases.
with through the prefactor 共2 − 兲 ⬃ 2, which can explain the rapid recovery of numerical accuracy with an increase in the value of . IV. SUMMARY
We have presented a family of monotonically convergent algorithms for solving a wide class of quantum optimal control problems under dynamics described by an inhomogeneous integrodifferential equation of motion. The algorithm family is characterized by two convergence parameters, , 苸 关0 , 2兴. Numerical tests showed that high numerical accuracy can be realized over a wide range of parameter values, independent of the presence or absence of relaxation. Extremely precise solutions often require long computational time; however, the convergence behavior can be monitored by means of extrapolation 关8兴. If slow convergence is detected, the calculation can be restarted using more suitable convergence parameters. ACKNOWLEDGMENTS
共16兲
where k = and k = 1.0. As → 0, the first term in the integrand reduces to 关E共k兲共t兲 − E共k−1兲共t兲兴2, which is the integrand in the case of = 0, and the second term goes to zero. Thus the second term gives a measure of how close the parameter, , is to the range in which high numerical accuracy is realized. For a small value of , the second term varies linearly
One of the authors 共Y.O.兲 acknowledges stimulating discussions with Professor Takamasa Momose and Professor Kenji Ohmori. This work was supported in part by a Grantin-Aid from MEXT of Japan 共Grant No. 17550005兲. P.S. acknowledges support by the Deutsche Forschungsgemeinschaft through Grant No. SFB 450. G.T. acknowledges support from INRIA Rocquencourt and GIP-ANR Grant No. C-QUID. H.R. acknowledges support form the NSF.
关1兴 R. J. Levis, G. M. Menkir, and H. Rabitz, Science 292, 709 共2001兲. 关2兴 W. Zhu, J. Botina, and H. Rabitz, J. Chem. Phys. 108, 1953 共1998兲.
关3兴 W. Zhu and H. Rabitz, J. Chem. Phys. 109, 385 共1998兲. 关4兴 Y. Ohtsuki, W. Zhu, and H. Rabitz, J. Chem. Phys. 110, 9825 共1999兲. 关5兴 Y. Ohtsuki and H. Rabitz, CRM Proc. Lecture Notes 33, 163
2
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MONOTONICALLY CONVERGENT ALGORITHMS FOR… 共2003兲. 关6兴 Y. Maday and G. Turinici, J. Chem. Phys. 118, 8191 共2003兲. 关7兴 Y. Ohtsuki, J. Chem. Phys. 119, 661 共2003兲. 关8兴 Y. Ohtsuki, G. Turinici, and H. Rabitz, J. Chem. Phys. 120, 5509 共2004兲. 关9兴 Y. Ohtsuki, K. Nakagami, Y. Fujimura, W. Zhu, and H. Rabitz, J. Chem. Phys. 114, 8867 共2001兲. 关10兴 K. Blum, Density Matrix Theory and Applications 共Plenum
Press, New York, 1981兲. 关11兴 Y. Maday, J. Salomon, and G. Turinici, Numer. Math. 103, 323 共2006兲. 关12兴 J. Somlói, V. A. Kazakov, and D. J. Tannor, Chem. Phys. 172, 85 共1993兲. 关13兴 A. Bartana, R. Kosloff, and D. J. Tannor, J. Chem. Phys. 106, 1435 共1997兲.
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