Decoherence Control by Tracking a Hamiltonian Reference Molecule

Gil Katz,1 Mark A. Ratner,1 and Ronnie Kosloff2

1Department of Chemistry, Northwestern University, Evanston, Illinois 60208-3113, USA2Fritz Haber Research Center for Molecular Dynamics, Hebrew University of Jerusalem, Jerusalem, 91904, Israel

(Received 10 February 2006; revised manuscript received 15 February 2007; published 17 May 2007)

A molecular system in contact with a bath undergoes strong decoherence processes. We examine acontrol scheme to minimize dissipation, while maximally retaining coherent evolution, by relating theevolution of the molecule to that of an identical freely propagating system. We seek a driving field thatmaximizes the projection of the open molecular system onto the freely propagated one. The evolution intime of a molecular system consisting of two nonadiabatically coupled electronic states interacting with abath is followed. The driving control field that overcomes the decoherence is calculated. A proposition toimplement the scheme in the laboratory using feedback control is suggested.

DOI: 10.1103/PhysRevLett.98.203006 PACS numbers: 33.80.Ps, 02.30.Yy, 82.50.Nd, 82.53.Kp

Coherent control originated as a method to steer achemical reaction to a desired outcome [1–3]. The usualagent of control is quantum interference [4]. The purposeof the endeavor is to find means, usually through shapingan electromagnetic field (EM), to direct the process to itstarget. In condensed phases coherent evolution is degradedby dissipative forces in the environment, termed decoher-ences. Since generation of interference pathways is a glo-bal process in time, an intervention at one instant influen-ces the subsequent evolution and can interfere at later timeswith another intervention. To address this issue optimalcontrol theory (OCT) has been developed as a mathemati-cal tool common in various engineering fields [5] thatsearches forward and backwards iteratively for the optimalpulse [6,7].

Local control theory (LCT) is a unidirectional timepropagation scheme to determine the instantaneous phaseof the EM field that monotonically increases the expecta-tion of a target operator [8–10]. Although OCT can lead tosuperior results, LCT has the advantage that the controlmechanism can be interpreted, and the iterative time propa-gation procedure is absent. Tracking control originatedfrom the idea of changing the local field in order to followa reference state [11–14]. Tracking can be considered aslocal control with a time dependent target. This trackingmechanism differs from OCT since it follows the reactionpath rather than optimizing backward from a chosen out-come [15].

Our basic idea here is to use a freely propagating mole-cule as a tracking target for an identical molecule in a dis-sipative environment. The perfect tracking field will thennull the dissipative forces and protect the molecule fromdecoherence. The question to be addressed is how can anexternal field that generates a unitary transformation pro-tect the system against nonunitary dissipative forces? Acomputational model of an online dynamic decoherencecontrol method is explored. The method is established bysimultaneously following the evolution in time of a systemwith and without the bath, thus using the overlap betweenthe two systems to construct a correcting electric field

which is directly applied back on the system. The methodis explored for the fastest dissipative mechanism,environment-induced electronic quenching from an ex-cited state to a lower state (the approximate decay timein solid state systems is less than 50 fsec). The coherencetime of the system is extended by an order of magnitudeusing the simplest tracking method.

Measuring the overlap between the controlled and thetarget system also underlies a proposed experimental ap-proach. This observable can then be employed as a basisfor a learning loop used to optimize a control field.

The molecular system is described by the density opera-tor �, a function of nuclear and electronic degrees offreedom. The reference or target system �tar evolves freelyunder its Hamiltonian H0:

_� tar � �i�H0; �tar�; (1)

where @ � 1 is chosen. The subjected system �S is coupledto a bath: �S � TrBf�SBg. This evolves according to:

_� SB � �i�HT; �SB�; (2)

where �SB is the combined system-bath operator. Thedynamics is generated by the combined system-bathHamiltonian HT � H0 � HB � HSB. The first approachemployed to solve the open system dynamics is based onrenormalized bath operators HB � HSB represented ex-plicitly. The equations of motion are solved in a largerHilbert space (the surrogate Hamiltonain method [16] ).The system density operator is obtained from �S �TrBf�SBg.

In the second approach, based on the Markovian limit,the reduced dynamics of the open quantum system �S [17]can be described by the generator _�S � �i�H0; �S� �LD��S�, where LD describes the bath-induced decoher-ence dynamics, conveniently cast into the Lindblad semi-group form:

L D��S� �Xj

�Fj�SFyj �12fFyj Fj; �Sg�; (3)

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where fA; Bg � A B�B A is the anticommutator and Fare operators from the Hilbert space of the system. Thenature of the bath interaction determines the form of theLindblad operators F. The choice of F determines thedissipative model considered.

Our purpose is to control the system coupled to the bathand force it to follow as closely as possible the dynamics ofthe freely evolving reference system, �tar. The externalcontrol field ��t� is coupled to the system through thetransition dipole of the molecule. The control Hamilton-ian becomes Hc � ���t� leading to the a combined dy-namics of the controlled system �C generated by: HT �

H0 � HB � HSB � Hc.The challenge of decoherence control is to find the field,

��t�, that induces the dynamics of the system �C�t� to be asclose as possible to �tar. This is attacked by maximizing theoverlap functional J�t� between the target and controlledstates:

J�t� � ��C �tar� � Tr f�C�targ � h�tari: (4)

J�t� can be interpreted as the expectation of the targetdensity operator in the controlled state. To achieve thistarget the control field should increase the expectation ofthe target operator. The Heisenberg equation of motion forthe observable h�tari generated by the control Hamiltonianbecomes:

ddth�tari � �ih�Hc; �tar�i � (5)

� i��t�Tr f��tar�C � �tar��Cg: (6)

The control field is constructed by requiring maximal J�t�,so that d

dt h�tari 0 at any instant, leading to:

��t� � �iKTr f�C��tar � �tar��Cg�; (7)

where K � K�t� is a positive envelope function with di-mension energy

�dipole�2. From Eq. (7) it is clear that the correcting

field ��t� � 0 if �C approaches �tar.We adopt a molecular model system with two electronic

states described by the density operator:

� ��e �eg�ge �g

� �; (8)

where the indices g and e designate the ground and excitedelectronic states and the submatrices are functions of thenuclear coordinates. The Hamiltonian of the system has theform:

H 0 �He Veg

Vge Hg

!(9)

with Hg=e � T� Vg=e. T is the kinetic energy operator,Vg and Ve are the potential energy operators on the groundand excited electronic states, and Veg is the nonadiabaticcoupling potential. The control Hamiltonian is chosen as:

H c �0 ��y��t�

����t� 0

� �; (10)

where � is the coordinate dependent electronic transitiondipole element. The dissipation phenomena included in themodel were fast electronic quenching, vibrational, andelectronic dephasing. Fq �

�������qp�jeihgj� is chosen to model

fast electronic quenching. The choice Fvd ��������vdp

�@

Hvd

dictates pure vibrational dephasing of the system. Fed ����������edp

�jeihej � jgihgj� represents electronic dephasing [16].These choices of system/bath interaction terms will lead toprocesses with a characteristic dissipation rates �y. Thedominant term would be the fast electronic quenching rate.

The control field form Eq. (7) becomes:

��t� � KIm�TrQf�gec ��

gtarg � TrQf�

ec��

getarg

� TrQf�getar��

gcg � TrQf�

etar��

gec g

� TrQf�gtar��

egc g � TrQf�

gc��

egtarg

� TrQf�egtar��

gec g � TrQf�

egc ��etarg�; (11)

where TrQ is a partial trace over the coordinates.The computational model is constructed from two elec-

tronic states of a diatomic molecule represented by twodiabatic Morse potential energy surfaces. Using dimen-sionless normal coordinates:

Vg�Q� � ���Dg�1� exp��ag�Q�Qg�2��;

Ve�Q0� � ��De�1� exp��ae�Q�Qe�2��;

(12)

where Dg and De are the dissociation energies of theground and excited electronic states, 2� is the adiabatic ex-citation energy, and a is the width of the state. Qe and Qg

are the equilibrium bond lengths of the two electronicstates. (Dg � De � 1:25 eV, � � 0:3 eV, ae � 0:8 �A�2,ag � 1:5 �A�2, Vge=eg � 0:05 eV,Qg � 0 �A,Qe � 0:1 �A,and � � 1:0.)

The time evolution was obtained by solving either thenon-Markovian Surrogate Hamiltonian method or thesemigroup time dependent Liouville-von Neumann equa-tion. A grid was used for the spatial coordinates and timepropagation used the Chebychev scheme [18]. The initialground vibronic state of the system was calculated via arelaxation scheme [19]. A pump pulse transfers a signifi-cant fraction of the population from the ground to theexcited electronic state. A Gaussian form is chosen forthis pulse:

�pump�t� � �0e��t�tmax�

2=2�2Lei!L

t: (13)

The carrier frequency!L is chosen to match the differencebetween the ground and excited electronic potentials at theminimum of the ground state Q � 0. The width of thepulse �L was adjusted to a FWHM duration of 12 fsecand the amplitude �0� � 0:228 eV.

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At this point three parallel propagations of the densityoperator were performed. The first for �tar was carried outonly with the Hamiltonian term Eq. (1). The system densityoperator, �S, was propagated with the dissipative LD in-cluded, Eq. (2). The controlled density operator, �C, waspropagated with both the dissipative LD and the correctingfield applied terms. The correction field was calculatedaccording to Eq. (11) at each time step and fed back inthe next time step. Both Markovian and non-Markovianmethods gave similar results.

The systems free dynamics represents a complex popu-lation oscillation between the two electronic states. Theexcited state population is shown in Fig. 1 for the con-trolled �c, uncontrolled �S, and target �tar for differentquenching parameters � and energy gap �. In all cases afast decay of the excited state is seen for the uncontrolledstate. The controlled state is able to track the target stateand maintain the population. When the quenching in-creases the controlled system is still able to follow theoverall dynamics of the target but not the finer details.As can be seen in Fig. 1, the control ability increaseswith the energy gap �. The control field � is shown inFig. 2 in time and frequency. In general the field is com-posed of a central frequency corresponding to the energygap modulated by the vibrational frequency. The actualvibronic lines are missing from the spectrum [20]. Wignerplots (not shown) show phase locking between the fre-quency components.

The changes in purity Tr f�2g and in the scalar productwith the target state (� �tar) show a different viewpoint onthe decoherence control cf. Fig. 3. The uncontrolled state�S undergoes fast exponential decay of both measures intime. The controlled purity Tr f�2

Cg maintains a high valueas does the scalar product (�C �tar) The general trend is a

linear decay in time. The oscillations around this decay andthe local increase in purity suggest that a cooling mecha-nism is taking place [20]. To gain insight on this possibilitythe correcting field was stopped for 100 fsec and resumedagain, Fig. 4. As expected, once the control field is turnedoff the purity and the scalar product of the controlled state�C decease sharply. When the controlling field is resumedan almost linear increase in both parameters can be ob-served. Such an effect can only be the result of a coolingmechanism which represents an interplay between thenonunitary dissipation and the unitary control field [20].It is clear [21] that simple unitary transformation cannotinduce cooling.

The present results show that the decoherence can besuppressed, extending good overlap with the target by anorder of magnitude in time. The model calculation estab-lishes the principle that a correcting external field can befound which protects the system from decoherence. Here

0

0.2

0.4

0

0.2

0.4

Popu

latio

n of

exc

ited

stat

e

0 100 200 300 4000

0.2

0.4

0 100 200 300 400

Time(fsec)0 100 200 300 400

γ

2

3

∆=

γ

γ

γ

γ

γ γ

γ

γ∆= ∆=

∆=

∆=∆=

∆=∆=

∆=

2 2

3 3

0.0eV

0.0eV

0.0eV

0.2eV

0.2eV

0.2eV

0.5eV

0.5eV

0.5eV

FIG. 1 (color). The population of the excited state as a functionof time for the target state �tar (red), the controlled densityoperator �C (black), and the uncontrolled system density opera-tor �S (green) for different magnitudes of quenching (columns)and �’s (rows). � � 0:003 fsec�1.

0 5000 10000Frequency(cm-1)

0

0 100 200 300 400 500Time(fsec)

0

FIG. 2 (color). Typical control field in frequency (top left) andtime domain (bottom) for � � 0:003 (blue) and � �0:006 fsec�1 (red).

0.6

0.8

1

0 100 200 300 400 500Time(fsec)

0

0.1

0.2

0.3

0.4

popu

latio

n of

exc

ited

stat

eT

r(

)

Tr(

)

ρ∗ρ

xy

ρ2

FIG. 3 (color). Top: the purity Tr f�2g (solid line) and thescalar product with the target state (� �tar) (dashed line) forthe target state �tar (red) the controlled density operator �C(black) and the reference uncontrolled density operator �S(green). Bottom: the population of the excited state, same colorcodes.

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we discuss control to overcome fast electronic dephasing;while the rapid modulations in Fig. 2 are not currentlyattainable, the essential behavior is correct.

The nearly complete distinction between coherent (gas)and decoherent (condensed) phase molecular dynamics hasdecreased with the evolution of nanoscale science. It iseasy to imagine applications that require preservation ofthe decoherence-free properties of a system upon becom-ing a subsystem of a larger apparatus. This ability mayhave significant importance in many future applications,for example, extending lifetime of the read or write processin quantum information processing, or memory chips andswitches in molecular electronics. One might even imagineimplementations for localizing intramolecular gas phaseproperties to a specific part or chemical bond of a macro-molecule. Thus retaining a level of decoherence-free be-havior in a given subspace is a significant goal. The schemesuggested allows a driving process to control such deco-herence and to retain coherent evolution in the subspace.

The proof of principle then shifts attention to findingsuch a field in the laboratory. The idea is to use an actualfreely evolving molecule as the target for an identicalmolecule subject to a dissipative environment such as insolution or on a surface. The correcting feedback should beset such that the difference in transient absorption of thetwo molecules is minimized.

To implement the present scheme in the laboratory thekey element is to apply a scalar product between the statesof the reference and controlled systems: (�C �tar) (dis-played in Fig. 3). To make this task possible the twoisolated states should become part of the same quantumsystem. This can be carried out by placing the controlledsystem �C and the reference system �tar in two branches of

an interferometer [22]. A pair of twin ultrashort photonswhich can be created by down conversion are split spa-tially. Then they are directed to the two branches of theinterferometer interrogating the two systems. The trans-mitted photons are then redirected to interfere with eachother. Any difference between the target and the controlledsystem will degrade this interference. The interferencesignal can then be used as a feedback to generate thecorrecting field [23].

We are grateful to the BSF and the NSF chemistrydivision for support of this work.

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0 100 200 300 400 5000.4

0.6

0.8

1

0 100 200 300 400 500Time(fsec)

0

0.2

0.4

(ρ ∗

ρ )

Tr

xy

Popu

latio

n of

exc

ited

stat

e

Tr(

)ρ2

C

R

T

T

C

R

FIG. 4 (color). Turning off the control field. Top: the purityTr f�2g (solid line) and the scalar product with the target state(� �tar) (dashed line) for the target state �tar (red), the con-trolled density operator �C (black), and the reference uncon-trolled density operator �S (green). The vertical lines indicatethe time interval where the control field is turned off. Bottom: thepopulation of the excited state as a function of time, same colorcodes.

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