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)

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
Traditional Photoexcitation in Photochemistry/Phot
ophysics
ABC, ABC
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
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
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 the same parity
couples states with opposite 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 2-photon absorption
from the 1-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 Corkums
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 Adopte
d 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 1gt Phigt ? 1gt Phi_1gt
igt is system, Phi_igt is
bath 2gt Phigt ? 2gt Phi_2gt So that (1gt
2gt) Phigt ? 1gt Phi_1gt 2gt Phi_2gt And off
diagonal density matrix element of system,
resulting from trace over (ignoring) the bath,
has term 1gtlt2 ltPhi_1 Phi_2gt. 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_s2) 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 DD
  Decoherence and Dephasing

17
From experimental analyses in Chemistry--
The challenge to overcome DD 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 DD no attempt to
engineer system against DD (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
That is, omega 2 omega excitation --- which,
e.g.
Laser-induced symmetry breaking
left/right symmetry
no bias voltage
trans-polyacetylene oligomer
metal
metal
This is a type of rectification
AC source
DC response!
Control current direction by varying relative
laser phase.
20
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)
  

• 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(N2) 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.

22
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
(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
Field frequency
energy gap 1.3 eV
ltlt
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/enthusias
m 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 MBVs
and then to PCBs
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

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
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
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
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
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
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
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
40
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
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 ! 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
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
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
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,

47
48
Return to origin of the symmetry breaking
I.e,
After the w 2w field, the excitation left on
the system
from the 2-photon absorption
from the 1-photon absorption
c1 proportional to ??2 c2 proportional to
?2? 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
51
Sample numerical results --- first the classical
limit by hbar ?0
52
Sample numerical results ---
Embellish
Phase Variables Absolute phase
defines temporal shift
between envelope and the
underlying
oscillations . Relative Phase
between the two driving frequencies.
53
P_theta altitude plot
54
Focus on full calculation
Evidently a. Quantum goes over to classical as
he goes to zero --- i.e. the classical limit is
indeed classical mechanics, which does show nice
control. b. The fully quantum shows no
dependence on the absolute phase, unlike the
small he and classical cases --- origin is in the
chaotic region that is sensitive to the detailed
initial conditions
55
note dependence on Fabs arising entirely from
the chaotic region, which eventually disappears
in the quantum limit. Would be enlightening to
see experimentally!
56
And range of control?
Solid is quantum Dashed is classical Essentially
same order of magnitude.
Although the situation Can be quite different
in the strong quantum regime due to highly
resonant contributions.
57
Proposed experimental examination of the quantum
classical transition The effect of decoherence
Consider an atom interacting with a
longitudinally shaken 1D optical lattice.
Hamiltonian is, as above
Source of decoherence is photon emissione.g. in
delta kicked rotor model G. Ball, K. Vant, and
N. Christensen, Phys. Rev. E 61, 1299 (2000).
57
58
In small hbar regime classical mechanical
control does emerge and decoherence only serves
to smooth out the dynamics. E.g.
59
So clear that for small hbar, reasonable
decoherence does little to the control --- i.e.
classical control is the result. But for larger
hbar? Work In 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 higher Decoherence to see
if Quantum decoh Classical decoh results in
loss of control.
60
Summary Control can survive into the classical
limit. It is qualitatively the same phenomenon,
but can differ greatly quantitatively. Control
IS due to interference effects, but they can
differ from the double slit paradigm insofar as
they can be field driven. Such field driven
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.
That Is, this decoherence was benign with
respect to control, both pure classical as well
as classicaldecoherence showed
control. Decoherence that would lead to
classical mechanics is also benignin 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 cases where classical
control exists.
60
61
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
61
62
Ignacio Franco Univ of Toronto graduate
student. Now postdoc at Northwestern.
63
and Michael Spanner postdoc at University of
Toronto, now at NRC, Ottawa