Open and Isolated System Quantum Control: Interferences in the Classical Limit

1. 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

2. 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: • General approach to building Kraus operators (with A. Biswas) • 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) • Overlapping resonances approach to analyzing stability of • states against decoherence (with A. Biswas) • Entanglement issues in EET (with Y. Khan, G Scholes) • Control in semiconductor quantum dots (with D. Gerbasi) • (Electric field) decoherence issues and coherent control (with J. Gong) • Coherent vs coherent light as an enviroment (with K. Hoki)

3. 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)

4. OUTLINE • 1. Introduction to the nature of coherent control of molecular processes, with emphasis on interfering pathways (“intuitive”, not optimal control). • Comments on the successes of this approach to the control of processes in isolated systems. (e.g. one photon vs. two photon current control) • But realistic systems are not isolated --- the open system problem/i.e. decoherence ---clarification on decoherence vs. dephasing. • Sample open system c.c. challenge (1 vs. 2 currents in polyacetylene) • Decoherence --- clarification. • a. expectations • b. surprises when expectations surpassed • c. decoherence - the classical limit • Expect: Decoherence  classical  loss of control. But ratchets?? • Hence, can control survive in the classical limit?

5. 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” Note --- Chemistry operates near this classical regime

6. Introduce Coherent Control Traditional Photoexcitation in Photochemistry/Photophysics A+BC, AB+C E.g. degenerate state in The continuum to different final arrangements Excited states Laser excitation Ground state ABC That is, one route to the final state of interest No active control over product ratio

7. Associated Coherent Control Scenario Coherent Control and "Double Slits" in Photochemistry/Photophysics Excited states Active control over product ratio via quantum interferences Ground state Two (or more) indistinguishable interfering routes to the desired products. Control laser characteristics  Control Interferences  Control relative cross sections

8. w1 w2 w3 3w1 f2 f1 f1 1 vs 3 Bichromatic Control Hence typical successful coherent control scenarios rely upon multiple pathway interferences such as those below -- the essence of quantum 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)

9. Lots of successful experimental an theoretical implementations of this 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: (pardon occasional “english-greek”) Control current direction by varying relative laser phase.

10. The 1 vs. 2 scenario and symmetry breaking can produce a current: 2nd order ‘2-photon’ absorption 1-photon absorption Energy couples states with opposite parity couples states with the same parity Final State: Not a parity eigenstate: Broken symmetry Laser controllable

11. The 1 vs. 2 scenario: role of interference I.e, : After the w + 2w field, the excitation left on the system: from the 1-photon absorption from the 2-photon absorption Net photoinduced momentum: Interference contribution Direct terms Only the interference contribution survives: Changing the relative phase of the lasers changes the magnitude and sign of the current. Laser control: E.g. done exptly in quantum wells by Corkum’s group, PRL 74, 3596 (1995)

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

13. But real systems are Open systems --- i.e coupled to an environment  Issue of Decoherence.

14. 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)

15. 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.

16. Important Clarification • Hence, in pure decoherence, the system and the bath entangle in unitary • dynamics 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 different • initial conditions (e.g. thermal effects), or • 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

17. 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) and Phys. 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 engineer system against D&D (as in quantum computing).

18. 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 World In Quantum Theory”, Springer, 2004) E.g. consider laser-induced current generation in molecule like polyacetylene System = electrons Bath causing decoherence = nuclei/nuclear motion.

19. e- That is, omega + 2 omega excitation --- which, e.g. Laser-induced symmetry breaking left/right symmetry no bias voltage metal metal trans-polyacetylene oligomer This is a type of rectification: AC source DC response! Control current direction by varying relative laser phase.

20. 1 vs. 2 Outline III : The Practical Issue • 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): • 4. Mixed quantum-classical photoinduced dynamics • Results : Classical nuclei Quantum electrons Sketch only; extensive details in: JCP X 2

21. Sketch of computation: • Nuclei move classically in the average of electronic potential • energies (Ehrenfest Approx). • 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 1000 • for 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

22. create laser-induced rectification The resulting multiphoton absorption processes π∗ transfer population to transporting states εF π Usual rectification mechanism: multiphoton absorption Tune the laser frequencies at or near resonance

23. But -- decoherence time-scale in isolated chains ~10 fs! Control wise, basically the worst case scenario

24. Currents through multiphoton absorption: efficiency This regime is fragile to electronic decoherence processes induced by the vibronic couplings 20 site wire weak, 300 fs pulse strong, 300 fs pulse flexible wire rigid wire The vibronic couplings make the rectification inefficient weak, 10 fs pulse strong, 10 fs pulse

25. π∗ εF π (Nice) aside: Can “get around it” : Robust ultrafast currents in molecular wires through the dynamic Stark effect 100-site chain Field intensity ~109 W/cm2 = 0.13 eV energy gap ~ 1.3 eV Field frequency << Far from resonance

26. Currents through the dynamic Stark effect 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 Ensemble averages Orbital energies Almost all excited electrons are deposited in the right contact only

27. Efficiency and phase control Almost complete laser control in the presence of strong decoherence Net rectification Efficiency flexible wire rigid wire The mechanism is robust to electron-vibrational couplings and is able to induce large currents with efficiencies as high as 90%! Note that for certain range of phases the currents are phonon-assisted I. 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)

28. 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.

29. E.g. in Biology: consider electronic energy transfer in the PC645 antenna Protein in Marine Algae: From bottom facing up Side view Light-induced electronic excitation of DVB dimer flows to MBV’s and then to PCB’s

30. 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

31. Typically, decoherence will be highly disturbing, with classical mechanics emerging. E.g.: Reaction H + HD  HH + D Han and Brumer, J. Chem. Phys. 122, 144316 (2005)

32. Here is the current understanding: • Coherent Control of a system requires maintenance of system matter • coherence. Control is based on quantum interference • System coherence can be destroyed by decoherence associated with • system-bath interactions. • Such decoherence can result in quantum mechanics going over to • classical mechanics. • 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?

33. Consider these issues via our one control example, • Symmetry breaking via 1 vs 2 photon excitation • Examine classical limit – control still manifest • examine “why” • 2. Propose an optical lattice experiment to examine the quantum to • classical transition,

34. The classical correspondence issue-- Quantum interference Quantum interpretation of laser-induced symmetry breaking Parity 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! However See, for example, S. Flach, O. Yevtushenko, and Y. Zolotaryuk, Phys. Rev. Lett. 84, 2358 (2000) Or papers on classical ratchet transport, e.g. Gong and Brumer How are the classical and quantum versions of symmetry breaking related, if at all?

35. The classical correspondence issue-- Quantum interference Quantum interpretation of laser-induced symmetry breaking 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? 35

36. The Earlier Strategy(Franco and Brumer, PRL 97,040402, 2006) Classical Limit Quantum Control ? Analytically consider the quantum-to-classical transition of the net dipole induced by an  + 2 field in a quartic oscillator Simplest model with well-defined classical analog wherein induced symmetry breaking is manifest (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.

37. 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)

38. 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)

39. Calculation-II 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 interferencebetween an i-th order and a j-th order optical route

40. Calculation-III Which terms contribute to symmetry breaking? 1. Only those terms allowed by the symmetry of the initial state ? Parity Reflection symmetry Using reflection symmetry and not parity: 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

41. Final Result Operator expression for the net dipole: where Some properties: • 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. • 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.

42. The Classical Limit The ~! 0 limit is analytic and nonzero, despite the fact that individual perturbative 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.

43. Quantum Corrections In the quantum case, the net dipole can be written as Quantum Corrections ~-independent classical-like contribution 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

44. Quantum Corrections-II Different initial states emphasize the classical part of the solution or the quantum corrections depending on the nature of the state Classical Quantum solution with increasing energy Classical limit reached as quantum state level increased. Note then --- the 1 vs 2 scenario can persist classically ---

45. Spatial symmetry breaking using w + 2w fields 1 vs. 2 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)

46. 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. 46

47. Asides: • 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: • 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? 47

48. Return to origin of the symmetry breaking I.e, : After the w + 2w field, the excitation left on the system: from the 1-photon absorption from the 2-photon absorption c1 proportional to єω2 c2 proportional to є2ω Crucial difference from the double slit analog is that the interference term is driven by external fields. Significantly --- driven interference terms need not vanish in the classical Limit. Analysis substantiated by recent Heisenberg representation analysis of interference processes (Franco, Spanner & Brumer, Chem. Phys. 370, 143, 2010) – not a competition between terms

49. Proposed experimental examination of the quantum – classical transition (M. Spanner, I. Franco and P. Brumer, Phys. Rev. A 80, 053402, 2009) Consider an atom interacting with a longitudinally shaken 1D optical lattice. Hamiltonian is:

50. Gives Schrodinger equation with effective, controllable, “hbar” Related to standard dipole driven form by defining: