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)
OUTLINE
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”
Note --- Chemistry operates near this classical regime
Introduce Coherent Control
Traditional Photoexcitation in Photochemistry/Photophysics
Excited
states
A+BC, AB+C
E.g. degenerate state in
The continuum to different
final arrangements
Laser excitation
Ground state
ABC
That is, one route to the final state of interest
No active control over product ratio
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
w1
w2
w3
3w1
f2
f1
f1
Bichromatic Control
1 vs 3
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)
Lots of successful experimental an theoretical implementations of
this basic interference-bassed quantum approach.
(pardon occasional “english-greek”)
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.
The 1 vs. 2 scenario and symmetry breaking
Energy
can produce a current:
2nd order
‘2-photon’
absorption
1-photon
absorption
couples states with
opposite parity
couples states with
the same parity
Final State:
Not a parity eigenstate:
Broken symmetry
Laser controllable
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:
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 theoretically
of 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 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
(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) 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).
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.
That is, omega + 2 omega excitation --- which, e.g.
e-
Laser-induced
symmetry breaking
left/right
symmetry
no bias voltage
metal
trans-polyacetylene oligomer
This is a type of rectification:
AC source
DC response!
Control current direction by varying relative laser phase.
metal
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
Classical nuclei
Quantum electrons
5. Results :
Sketch only; extensive details in: JCP X 2
1 vs. 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 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
Usual rectification mechanism: multiphoton absorption
Tune the laser frequencies at or near resonance
The resulting multiphoton
absorption processes
create laser-induced rectification
transfer population to transporting states


F
But -- decoherence time-scale in isolated chains
~10 fs!
Control wise, basically the worst case scenario
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
(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
Field frequency
= 0.13 eV


<<
F
energy gap ~ 1.3 eV
Far from resonance
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
Orbital
energies
Almost all excited electrons are deposited in the right contact only
Ensemble
averages
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)
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.
28
E.g. in Biology: consider electronic energy transfer in the PC645 antenna
Protein in Marine Algae:
Side view
From bottom facing up
Light-induced electronic excitation of DVB dimer flows to MBV’s
and then to PCB’s
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
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)
31
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,
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.
However
An w + 2w 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)
Or papers on classical ratchet transport, e.g. Gong and Brumer
How are the classical and quantum versions of
34
symmetry breaking related, if at all?
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?
35
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 w + 2w field in a quartic oscillator
Simplest model with welldefined 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.
36
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)
37
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)
38
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
interference between an i-th order and a j-th order optical route
39
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 evenorder 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
40
Final Result
Operator expression for the net dipole:
where
Some properties:
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.
41
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.
42
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 w +2w field
Quantum Case
Classical Case
43
The fine ~-dependent structure can change the magnitude and sign of the
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 ---
44
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)
1 vs. 2
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
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:
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?
47
2. Then: can we do an experiment that shows these features clearly?
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
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:
49
Gives Schrodinger equation with effective, controllable, “hbar”
Related to standard dipole driven form by defining:
50
Sample numerical results --- first the classical limit by hbar 0
51
Sample numerical results ---
Embellish:
Phase Variables: “Absolute phase
between envelope
oscillations
.
Relative Phase;
defines temporal shift
and the underlying
between the two driving frequencies.
52
P_theta
altitude
plot
53
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:
54
note dependence on Φabs arising entirely from the chaotic region,
which eventually disappears in the quantum limit. Would be enlightening
to see experimentally!
55
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.
56
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 emission—e.g. in delta kicked rotor model –
G. Ball, K. Vant, and N. Christensen, Phys. Rev. E 61, 1299 (2000).
57
In small hbar regime – classical mechanical control does emerge and
decoherence only serves to smooth out the dynamics. E.g.
58
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.
59
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 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
cases
60
where classical control exists.
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
Ignacio Franco –
Univ of Toronto graduate student. Now postdoc at Northwestern.
and Michael Spanner –
postdoc at University of Toronto, now at NRC, Ottawa