Open and Isolated System Quantum Control: Interferences in the Classical

Limit

Paul Brumer

Chemical Physics Theory Group, Chemistry Dept.and

Center for Quantum Information and Quantum Control

University of Toronto

ITAMP, August 2010

A change in topic from the original title/topic due to a remark of Tommaso.

Other recent work relevant to the meeting, and can discuss:

a. General approach to building Kraus operators (with A. Biswas)

b. Theorem on specific control scenario showing deleterious effects of the environment (with A. Bharioke and L-A. Wu)

c. Theorem on a specific control scenario showing the benefits of the environment ( with M. Spanner and C. Arango)

d. Overlapping resonances approach to analyzing stability of states against decoherence (with A. Biswas)

e. Entanglement issues in EET (with Y. Khan, G Scholes)

f. Control in semiconductor quantum dots (with D. Gerbasi)

g. (Electric field) decoherence issues and coherent control (with J. Gong)

h. Coherent vs coherent light as an enviroment (with K. Hoki)

But here, something that we “may sort-of know” from various other areas,But which has profound influence on way of thinking about interference.

If time permits:

c. Theorem on a specific control scenario showing the benefits of the environment ( with M. Spanner and C. Arango)

1. Introduction to the nature of coherent control of molecular processes, with emphasis on interfering pathways (“intuitive”, not optimal control). 2. Comments on the successes of this approach to the control of processes in isolated systems. (e.g. one photon vs. two photon current control)

3. But realistic systems are not isolated --- the open system problem/i.e. decoherence ---clarification on decoherence vs. dephasing.

4. Sample open system c.c. challenge (1 vs. 2 currents in polyacetylene)

5.Decoherence --- clarification. a. expectations b. surprises when expectations surpassed c. decoherence - the classical limit

6. Expect: Decoherence classical loss of control. But ratchets??

7. Hence, can control survive in the classical limit?

OUTLINE

8. Propose experiment to observe this limiting process/theory to understand

I.e. inroads into control competing with decoherence --- but long way to go to identify “negative decoherence” vs. “benign decoherence”

OUTLINE

Note --- Chemistry operates near this classical regime

Introduce Coherent Control

Traditional Photoexcitation in Photochemistry/Photophysics

Ground state

Excitedstates

Laser excitation

That is, one route to the final state of interest

E.g. degenerate state inThe continuum to differentfinal arrangements

ABC

A+BC, AB+C

No active control over product ratio

Associated Coherent Control Scenario

Ground state

Excitedstates

Coherent Control and "Double Slits" inPhotochemistry/Photophysics

Two (or more) indistinguishable interfering routes to the desired products. Control laser characteristics Control Interferences Control relative cross sections

Active control overproduct ratio via quantum interferences

Hence typical successful coherent control scenarios rely upon multiple pathway interferences such as those below -- the essence ofquantum control.

1 vs 3

Bichromatic Control

Common to rely upon analogy of double slit experiment.

Obvious reminder – double slit interference pattern disappears as hbar 0.

More sophisticated analyses (entanglement, double slit analogy explored):

Gong and Brumer, J. Chem. Phys. 132, 054306 (2010)Franco, Spanner and Brumer, Chem. Phys. 370, 143 (2010)

Lots of successful experimental an theoretical implementations ofthis basic interference-bassed quantum approach.

One example of interest: omega vs. 2 omega current generation(without bias voltage) --- model for this talk.

Specifically, omega + 2 omega fields are incident on a molecule:

Control current direction by varying relative laser phase.

(pardon occasional “english-greek”)

The 1 vs. 2 scenario and symmetry breaking

Laser controllable

En

erg

y

couples states with the same parity

couples states with opposite parity

1-photon absorption

2nd order‘2-photon’ absorption

Not a parity eigenstate:Broken symmetry

Final State:

can produce a current:

The 1 vs. 2 scenario: role of interference

I.e, :

After the + 2 field, the excitation left on the system:

from the 2-photon absorptionfrom the 1-photon absorption

Net photoinduced momentum:

Direct terms Interference contribution

Only the interference contribution survives:

Laser control:Changing the relative phase of the lasers changes the magnitude and sign of the current.

E.g. done exptly in quantum wells by Corkum’s group, PRL 74, 3596 (1995)

Numerous successes in isolated systems, both experimentally and theoreticallyof interference-based control scenarios. See, e.g. Rice book, our book

But real systems are Open systems --- i.e coupled to an environment

Issue of Decoherence.

Clarification:

“Bath”

System

Bath = Part being traced over = Not measured

System dynamics:

(A) Measure of continuous system decoherence:

Pure state:

Mixed state:

Termed “purity”; Related, Renyi entropy (a rather amazing man)

Clarification

Clarify the statement defining decoherence

Adopted definition (e.g., E. Joos, or Schlosshauer)

DECOHERENCE (OR TRUE DECOHERENCE): Unitary dynamical evolution of the system + bath, without any dynamical change in the system states:

|1> |Phi> |1> |Phi_1> |i> is system, |Phi_i> is bath

|2> |Phi> |2> |Phi_2>

So that (|1> + |2>) |Phi> |1> |Phi_1> + |2> |Phi_2>

And off diagonal density matrix element of system, resulting from trace over (ignoring) the bath, has term |1><2| <Phi_1 | Phi_2>.

Hence loss of coherence due to system entangling with different bath components that are dissimilar.

Important Clarification

Hence, in pure decoherence, the system and the bath entangle in unitarydynamics of the pair. Ignoring the bath causes loss of quantum information, and hence decoherence of the system.

Tr(rho_s^2) is good measure

“FAKE DECOHERENCE” (E. Joos, Schlosshauer, etc.) or “DEPHASING”:

Loss of coherence arising from some averaging mechanism – e.g.

(i) similar Hamiltonian evolution to members of an ensemble but differentinitial conditions (e.g. thermal effects), or

(ii) Collection of identically prepared systems subjected to different Hamiltonians.

Joos: “Here there is no decoherence at all from a microscopic viewpoint”.

Are distinctions in effects on control between Decoherence and Dephasing,but not discussed here. Use term D&D = Decoherence and Dephasing

From experimental analyses in Chemistry--

The challenge to overcome D&D is considerable. For example, we require no coherence to explain two optimal control-in liquid examples:

Control of “vibrational populations” in methanol in liquid – (Bucksbaum expt)Analyzed in Spanner and Brumer, Phys. Rev. A 73, 023809 (2006) andPhys. Rev. A 73, 023810 (2006).

Control of isomerization in NK88 – (Gerber expt) --- analyzed in Hoki and Brumer, Phys Rev Lett. 85, 168305 (2005).

(I know of none other – in open systems in Chmistry -- than have been theoretically analyzed.)

Note. we focus on natural D&D; no attempt to engineersystem against D&D (as in quantum computing).

What does, from our viewpoint, decoherence do? --

Decoherence causes loss of quantum features - classical limit argument

(E.g. Joos et al, “Decoherence and the Appearance of the Classical WorldIn Quantum Theory”, Springer, 2004)

E.g. consider laser-induced current generation in molecule like polyacetylene

System = electrons

Bath causing decoherence = nuclei/nuclear motion.

That is, omega + 2 omega excitation --- which, e.g.

trans-polyacetylene oligomer metalmetal

left/right symmetry

Laser-induced symmetry breaking

AC source DC response!

no bias voltage

This is a type of rectification:

e-

Control current direction by varying relative laser phase.

4. Mixed quantum-classical photoinduced dynamics

5. Results :

Outline III : The Practical Issue

1. Laser-induced symmetry breaking and the 1 vs. 2 coherent control scenario

2. Applications to molecular wires: challenges and motivations

3. The model (SSH Hamiltonian):

1 vs. 2

Classical nuclei

Quantum electrons

Sketch only; extensive details in: JCP X 2

Sketch of computation:

1. Nuclei move classically in the average of electronic potential energies (Ehrenfest Approx).

2. Electrons evolve quantum mechanically and respond instantaneously to changing nuclear positions.

3. Electron density matrix elements are expanded in time evolving electron orbitals.

4. Leads to which molecules are attached are quantitatively incorporated as sinks for electrons over the lead Fermi level (which is computed as a function of time). This one hard part.

5. Gives set of N(N+2) coupled first order ODE, where N is number of Carbons in the chain. (typically N= 20)

6. Average the dynamics over the initial nuclear configuration---e.g.40,000 initial conditions (since decoherence must converge), or 1000for Stark case. Another hard part.

7. Variety of pulses, with “weak” being 10^9 W/cm^2 for 2 omega “strong” being 2 X 10^10 W/cm^2

Usual rectification mechanism: multiphoton absorptionTune the laser frequencies at or near resonance

εF

π

π∗

The resulting multiphoton absorption processes

create laser-induced rectification

transfer population to transporting states

But -- decoherence time-scale in isolated chains

~10 fs!Control wise, basically the worst case scenario

20 site wire

weak, 300 fs pulse

strong, 300 fs pulse

weak, 10 fs pulse strong, 10 fs pulse

flexible wirerigid wire

This regime is fragile to electronic decoherence processes induced by the vibronic couplings

Currents through multiphoton absorption: efficiency

The vibronic couplings make the rectification inefficient

= 0.13 eVField frequency

100-site chainField intensity ~109 W/cm2

energy gap ~ 1.3 eV

εF

π

π∗

Far from resonance

(Nice) aside: Can “get around it” : Robust ultrafast currents in molecular wires through the dynamic

Stark effect

Currents through the dynamic Stark effect

Almost all excited electrons are deposited in the right contact only

Stark shifts ~ E(t) 2 (symmetric systems)

The laser closes the energy gap causing crossings between the valence and conduction band in individual trajectories

Case 1:

Field

Current

Orbital energies

Ensemble averages

Efficiency and phase control

The mechanism is robust to electron-vibrational couplings and is able to induce large currents with efficiencies as high as 90%!

rigid wire

Net rectification

Efficiency

flexible wire

Almost complete laser control in the presence of strong decoherence

Note that for certain range of phases the currents are phonon-assistedI. Franco, M. Shapiro, P. Brumer, Phys. Rev. Lett. 99, 126802 (2007) I. Franco, M. Shapiro, P. Brumer, J. Chem. Phys. 128, 244905 and 244906

(2008)

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Hence, one may be able to bypass decoherence effects but, in general,this is difficult. Indeed, when decoherence is not as effective as expected,leads to surprise/incredulity/enthusiasm:

E.g.

E.g. in Biology: consider electronic energy transfer in the PC645 antennaProtein in Marine Algae:

Light-induced electronic excitation of DVB dimer flows to MBV’s and then to PCB’s

Side view From bottom facing up

2DPE experiment – at room temperature! Does the energy transfer display coherence despite the vast vibrational background?

E. Collino, K.E. Wilk, P.M.G. Curmi, P. Brumer and G.D. Scholes, Nature, 463, 644 (2010)

Indeed, coherence (unexpectedly) evident for over 400 fs

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Typically, decoherence will be highly disturbing, with classical mechanicsemerging. E.g.:

Han and Brumer, J. Chem. Phys. 122,144316 (2005)

Reaction H + HD HH + D

Here is the current understanding:

1. Coherent Control of a system requires maintenance of system matter coherence. Control is based on quantum interference

2. System coherence can be destroyed by decoherence associated with system-bath interactions.

3. Such decoherence can result in quantum mechanics going over to classical mechanics.

4. But classical mechanics does not show interference based control – hence decoherence can be expected to cause loss of control.

Thus understanding of the control of systems in a bath relates to an understanding of the classical limit, and what happens in the classical limit.In particular, we ask: does control vanish in the classical limit? Does decoherence always cause loss of control? Are there benign forms of decoherence from the control perspective?

Consider these issues via our one control example,

Symmetry breaking via 1 vs 2 photon excitation

1. Examine classical limit – control still manifest examine “why”

2. Propose an optical lattice experiment to examine the quantum to classical transition,

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The classical correspondence issue--

Quantum interpretation of laser-induced symmetry breaking

Quantum interference

Parity

However

These concepts do not have a classical analogue and the effect seems completely quantum mechanical.

An + 2 field generates phase-controllable symmetry breaking in completely classical systems as well!

See, for example, S. Flach, O. Yevtushenko, and Y. Zolotaryuk, Phys. Rev. Lett. 84, 2358 (2000)

How are the classical and quantum versions of symmetry breaking related, if at all?

Or papers on classical ratchet transport, e.g. Gong and Brumer

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The classical correspondence issue--

Quantum interpretation of laser-induced symmetry breaking

Quantum interference

Parity

These concepts do not have a classical analogue and the effect seems completely quantum mechanical.

How are the classical and quantum versions of symmetry breaking related, if at all? Are they the same physical phenomenon?

If yes, what happened to the double slit analog where, no doubt, the interference terms vanish in the classical limit?

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The Earlier Strategy (Franco and Brumer, PRL 97,040402, 2006)

Quantum Control

Classical Limit

Analytically consider the quantum-to-classical transition of the net dipole induced by an + 2 field in a quartic oscillator

(a) time-dependent perturbation theory in the Heisenberg picture that admits an analytic classical (~ ! 0) limit in the response of the oscillator to the field.

(b) Anharmonicities included to minimal order in a multiple-scale approximation; interaction with the radiation field is taken to third order.

Simplest model with well-defined classical analog wherein induced symmetry breaking is manifest

?

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Perturbation Theory in the Heisenberg Picture

Advantages

1. The result of the perturbation is independent of the initial state

2. The classical limit of the solution coincides with true classical result.Osborn and Molzahn, Ann. Phys. 241, 79-127 (1995)

Main drawbacks

1. Operators and their algebraic manipulation (not always easy)

2. One needs to begin with a system for which an exact solution in Heisenberg picture exists (e.g. harmonic oscillator)

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Calculation

Symmetry breaking is characterized through the long-time average of the position operator in Heisenberg representation

We employ the Interaction picture where

Evolution operator in the absence of the field

Captures the effects induced by the field

This splits the problem into two steps

Perturbative analysis to include the oscillator anharmonicities; C. M. Bender and L. M. A Bettencourt, Phys. Rev. Lett. 77, 4114 (1996)

Subsequent perturbation to incorporate the effect of the field (to third order in the field)

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The perturbative expansion for is given by

zeroth order term

nth order correction

The result up to third order in the field (34370 Oscillatory operator terms)

Note that the terms :

describe the contribution to the dipole coming from the interference between an i-th order and a j-th order optical route

Calculation-II

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Which terms contribute to symmetry breaking?

Using reflection symmetry and not parity:

1. Only those terms allowed by the symmetry of the initial state

Reflection symmetryParity?

have a non-zero contribution to the trace

Symmetry breaking comes from the interference between an even-order and an odd-order response to the field

2. Only those terms that have a zero-frequency (DC) component

The remaining terms, with a residual frequency dependence, average out to zero in time

Calculation-III

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1. The sign and magnitude of the dipole can be manipulated by varying the relative phase between the frequency components of the laser --irrespective of the initial state.

2. In the zero-anharmonicity limit all symmetry breaking effects are lost

It is precisely because of the anharmonicities that the system can exhibit a nonlinear response to the laser, mix the frequencies of the field and generate a zero harmonic component in the response.

Final Result

Operator expression for the net dipole:

Some properties:

where

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The Classical Limit

The ~! 0 limit is analytic and nonzero, despite the fact that individualperturbative terms can exhibit singular behavior as ~! 0

The field induced interferences responsible for symmetry breaking survive in the classical limit and are the source of classical control.

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Quantum Corrections

In the quantum case, the net dipole can be written as

~-independent classical-like contribution

Quantum Corrections

The nature of the quantum corrections can be associated with the ~ dependence of the resonance structure of the oscillator

Resonances sampled by the +2 field

Quantum Case Classical Case

The fine ~-dependent structure can change the magnitude and sign of the effect

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Different initial states emphasize the classical part of the solution or the quantum corrections depending on the nature of the state

Classical limit reached as quantum state level increased.

Note then --- the 1 vs 2 scenario can persist classically ---

Quantum Corrections-II

Classical Quantum solution with increasing energy

Related experimental and theoretical references.

Theoretically:

Note the general phenomenon can be accounted for from:

1) The coherent control perspective of interfering optical pathways G. Kurizki, M. Shapiro and P. Brumer Phys. Rev. B 39, 3435 (1989); M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Processes (Wiley, 2003)

2) Nonlinear response theory arguments I. Franco and P. Brumer Phys. Rev. Lett. 97, 040402 (2006); Goychuk and P. Hänggi, Europhys. Lett. 43, 503 (1998)

3) Space-time symmetry analyses of the equations of motion I. Franco and P. Brumer J. Phys. B 41, 074003 (2008) S. Flach, O. Yevtushenko, and Y. Zolotaryuk, Phys. Rev. Lett. 84, 2358 (2000)

1 vs. 2

Spatial symmetry breaking using + 2 fields

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Hence,

Ratchet effect is qualitatively the same effect in classical and quantum mechanics.

One has directional transport = classical part + quantum part.

Magnitude may differ, but control survives in the classical limit.

Motivates examination of other scenarios from perspective of Nonlinear response (not at all the prior general direction in c.c.

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1. hence if you see (experimentally) dependence of features -on relative

laser phase, this does not necessarily imply that it is quantum effect.

2. It connects to classical language “here and there”. E.g. early work and

language of Bucksbaum/Corkum

3. On (often major) quantitative difference in classical vs. quantum

response functions --- see several papers by Maksym:

M. Kryvohuz and J.Cao, Phys. Rev. Lett. 95, 180405, (2005)M. Kryvohuz and J. Cao, Phys. Rev. Lett. 96, 030403 (2006)And by Loring:

S.M. Gruenbaum and R. Loring, J. Chem. Phys. 128, 124106 (2008)

Some conceptual issues:

1. So the question becomes --- is the quantum interference, and if it is,

how/why does it survive in the classical limit? Is the standard analogy

with the double slit in need of supplementing?

2. Then: can we do an experiment that shows these features clearly?

Asides:

Return to origin of the symmetry breaking

I.e, :

After the + 2 field, the excitation left on the system:

from the 2-photon absorptionfrom the 1-photon absorption

c1 proportional to єω2

c2 proportional to є2ω

Crucial difference from the double slit analog is that the interferenceterm is driven by external fields.

Significantly --- driven interference terms need not vanish in the classicalLimit.

Analysis substantiated by recent Heisenberg representation analysis ofinterference processes (Franco, Spanner & Brumer, Chem. Phys. 370, 143, 2010) – not a competition between terms

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Consider an atom interacting with a longitudinally shaken 1D opticallattice. Hamiltonian is:

Proposed experimental examination of the quantum – classical transition(M. Spanner, I. Franco and P. Brumer, Phys. Rev. A 80, 053402, 2009)

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Gives Schrodinger equation with effective, controllable, “hbar”

Related to standard dipole driven form by defining:

51Sample numerical results --- first the classical limit by hbar 0

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Phase Variables: “Absolute phase defines temporal shift between envelope and the underlying oscillations .

Relative Phase; between the two driving frequencies.

Sample numerical results ---

Embellish:

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P_thetaaltitudeplot

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Focus on full calculation:

Evidently:

a. Quantum goes over to classical as he goes to zero --- i.e. the classicallimit is indeed classical mechanics, which does show nice control.

b. The fully quantum shows no dependence on the absolute phase, unlikethe small he and classical cases --- origin is in the chaotic region that issensitive to the detailed initial conditions:

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note dependence on Φabs arising entirely from the chaotic region,which eventually disappears in the quantum limit. Would be enlighteningto see experimentally!

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And range of control?

Solid is quantum;Dashed is classical’Essentially sameorder of magnitude.

Although the situationCan be quite different inthe strong quantum regimedue to highly resonant contributions.

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Consider an atom interacting with a longitudinally shaken 1D opticallattice. Hamiltonian is, as above:

Proposed experimental examination of the quantum – classical transitionThe effect of decoherence

Source of decoherence is photon emission—e.g. in delta kicked rotor model – G. Ball, K. Vant, and N. Christensen, Phys. Rev. E 61, 1299 (2000).

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In small hbar regime – classical mechanical control does emerge anddecoherence only serves to smooth out the dynamics. E.g.

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So clear that for small hbar, reasonable decoherence does little to the control --- i.e. classical control is the result. But for larger hbar? WorkIn progress that depends upon the nature of the decoherence. E.g.

Large hbar --- requires (as in Han and Brumer, JCP 112,114316 (2005) on reactive scattering) modifying classical as well. Not problem if control still survives.

Now looking at higherDecoherence to see ifQuantum + decoh =Classical + decoh resultsin loss of control.

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Summary:

Control can survive into the classical limit. It is qualitatively the samephenomenon, but can differ greatly quantitatively.

Control IS due to interference effects, but they can differ from thedouble slit paradigm insofar as they can be field driven. Such fielddriven interference terms may survive to the classical limit. (Some control cases, e.g. collisional control scenarios based on entanglement will lose control in the classical limit – not driven)

Optical lattice experiment proposed to examine the quantum to classical transition.==================Decoherence shows little effect on control for small hbar systems. ThatIs, this decoherence was “benign” with respect to control, both pure classical as well as classical+decoherence showed control.

Decoherence that would lead to classical mechanics is also “benign”in cases where classical limit control exists.

Work ongoing to determine what types of decoherence are “benign”And what types destroy control. Work ongoing to further extend caseswhere classical control exists.

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Issue of decoherence and control is just beginning to be examined, with lots of enlightenment to come.

Ad: Postdoc in this and quantum effects in biology, entangled photon generation in quantum dots, etc.

Thanks to NSERC as well as to:

Ignacio Franco –Univ of Toronto graduate student. Now postdoc at

Northwestern.

and Michael Spanner – postdoc at University of Toronto, now at NRC, Ottawa