Optimal Quantum Control in Dissipative Environments: General

Chapter 9

Downloaded by MONASH UNIV on November 29, 2016 | http://pubs.acs.org Publication Date: June 21, 2002 | doi: 10.1021/bk-2002-0821.ch009

Optimal Quantum Control in Dissipative Environments: General Formalism and Perturbative Limits Seogjoo Jang and Jianshu Cao Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139

Optimal quantum control in dissipative environments is formu -lated combining the optimal control theory from the Wilson group with the projection operator technique. In the weak response limit, a set of formally exact equations defines the optimal weak field for arbitrary target states and arbitrary system-bath inter -actions. Perturbative approximations are then made to derive general equations for weakly dissipative non-Markovian systems, which are amenable to analytical and/or numerical solutions. Po -tential applications of these results are discussed.

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Bandrauk et al.; Laser Control and Manipulation of Molecules ACS Symposium Series; American Chemical Society: Washington, DC, 2002.

133


Introduction Quantum processes are counter-intuitive in many respects. The notable features are uncertainty, interference, and tunneling, which seem to make the quantum system more difficult to control. W i t h the advance of laser techniques, however, active utilization of such novel features for control has become possible through the manipulation of matter-radiation interactions on the relevant microscopic time scale. Impressive advances have been made for isolated gas phase systems, and recent reviews [1-4] provide a general perspective. Most natural or practically important processes occur i n condensed phases. The underlying microscopic events are governed by quantum princi ples, although the complexity of the system in general disguises such details. Therefore, the lessons learned from the studies of gas phase systems can in principle be extrapolated to condensed phase systems. Such possibilities have been demonstrated by recent experiments [4-9] and new theoretical works [10-14] suggest various novel ideas. Two fundamental issues arise in condensed phase quantum control. One is the complexity of the Hamiltonian, and the other is how to deal with dis sipation and decoherence [11,15]. The former aspect causes difficulties i n finding robust schemes insensitive to small details of the underlying Hamil tonian. Recent advances in learning algorithms [4] provide ways to bypass these difficulties and to obtain information of the underlying Hamiltonian. The latter issue of decoherence and dissipation is under active investigation and some successful schemes have been devised i n various limiting situa tions [11-14]. These studies indicate that a key factor is for the variation of the optical field to be i n time scales comparable with those of the bath relaxation dynamics, thus invoking subtle interplay between the two pro cesses. Therefore, the non-Markovian nature of the bath dynamics should be considered i n devising a successful control scheme in condensed phase. Quantum control i n dissipative environments can be considered as an extension of time dependent dissipative quantum dynamics [11,16], where general formalisms have been introduced. However, extension of these for malisms to controlling molecular systems has not been well-established yet. Semi-group approaches are available [17,18], but these implicitly assume Markovian dynamics i n a phenomenological way. Direct wavepacket dy namics [7] is possible, but may be limited in providing a general qualitative understanding. Cao, Messina, and Wilson [19] addressed the issue based on the system-bath Hamiltonian, by combining the weak field limit of the W i l son group formalism [20,21] with the Feynman-Vernon influence functional approach [22-24]. This approach provides a framework for investigating the



134 role of the non-Markovian bath. However, practically, the use of the influ ence functional approach is limited to the harmonic bath, and the number of system degrees of freedom is small. A complementary approach is to apply the projection operator formalism, which can be applied to more general types of bath as long as certain criteria are met. This is the main subject of the present work. We first review the general formalism based on the Wilson group approach [20,21]. The resulting expression includes the gas phase system as a special case and also serves as a starting point for the explicit treatment of dissipative dynamics. Then, in the weak weak response limit, formally exact equations for the material response function and their second order perturbation approximations are derived. The ar ticle concludes with several comments on the implications of the present result.

Optimal Equation Consider an ensemble of molecular systems with two electronic states within the optical range of interest. The effective system-bath Hamiltonian is written as E

M

= H \g)(g\ + (H + e )\e)(e\ g

e

eg

where H and H are nuclear Hamiltonians i n the ground and excited states and e is the energy difference between the potential minima of the excited and ground states. B o t h nuclear Hamiltonians respectively consist of three terms, H = H° + H\ + H and H = 4- H] + H , where the terms with superscript 0 represent the system Hamiltonian and the terms with superscript 1 are those of the system-bath interaction. These expressions represent general situations where the system and the system-bath interac tion Hamiltonians on the excited state surface are different from those on the ground state. In these definitions, the system Hamiltonians are renormalized i n the sense that Trb{Hgpb} — Trt{Hlpb} = 0, where p& is the canonical equilibrium density operator of the bath. In the presence of the optical field, the time dependent Hamiltonian is given by g

e

eg

9

g

b

H(t) = H ~ M

where the rotating h = 1 through out Given a target mize the following

e

fc

Ai(£*(t)e "*|p) 0 2

which is not a true density operator unless t\ — 0. Taking the trace over the bath degrees of freedom leads to a reduced T G D O (rTGDO), G (t ,ti) = Trb{G(t i h)}. Then, the material response function in Eq. (7) is expressed as , t > t' (9) s

2


2

where G\ is the Hermitian conjugate of G . The use of two different expres sions in Eq. (9), depending on the time ordering of t and t', is necessary because the trace over the bath and the ensuing approximations break the time reversal symmetry of the full Hamiltonian dynamics. The eigenvalue equation and the corresponding expression for the ma terial response function can be transformed into the frequency domain. Taking the Fourier expansion of the electric field, s

E(t) =

Inn

—iQ t

£ n

n

Eq. (6) can be transformed into

m where T M nm

xTr

1 ;

s

iQmt

dt' (e 'G (t,t') s

1

+ e-* "*'Gl(t,f))


138

Master Equation for Reduced Two-time Generalized Density Operator (rTGDO) We now use the projection operator technique [25,26] to derive a for mally exact equation for G {t , h), which is then reduced to a set of approx imate equations valid to the second order of the system-bath interaction. First, we define the following interaction picture T G D O : s

i

+ H

G{t ,ti) = e ^ °

2

i

^G(t ,t )e- ^

2

2

+ H

1

£

^

+£

9

= e'< 2 *>* G(t ,*i) 2


which satisfies the quantum Liouville equation: £-G(t ,h) Ot

= -i{Hl{t ),G{t ,h)]

2

2

=

2

-iC\{t )G{t ,h) 2

2

2

£

+ £ b

2

where Hl(t ) = e ^ « ^ i f g . The projection operator is defined as V(-) = pbTr (') and its complement is denoted as Q = 1 -V. Under the conditions of Trt,{Hg( jpb} = 0, one can show that 2

b

e

JL G(t ,h) ot

=

-iVC\(t )QG(t ,h)

^-QG{t M) ot

=

-iQC\{t )QG{t M)-iQC\{t2)VG{t ,h)

V

2

2

(10)

2

2

2

2

2

(11)

2

2

The formal solution of Eq. (11) is QG(t ,n)

= ex

2

-

P ( + )

j-i^

dt' exp 2

{+)

2

dt' QCl(t' ) J 2

| - z £

2

2

QG(0,h)

Optimal Quantum Control in Dissipative Environments: General

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