Classical Approach to Computing Quantum Decoherence Dynamics

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Classical Approach to Computing Quantum
Decoherence Dynamics
Paul Brumer Dept. of Chemistry, and Center for
Quantum Information and Quantum
Control University of Toronto
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Original Motivation

• For Decoherence Possibility of controlling
  atomic and molecular processes via quantum
  interference (coherent control)
• Ability of decoherence to destroy quantum
  effects, and hence destroy quantum control
• Formal --Attempts to understand decoherence and
  entanglement (will not get to entanglement
  today) --- since these are quantum properties

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Specifically --- applications

• Main Determine decoherence rates in realistic
  systems, e.g., molecules in solution
• Hence
• develop useful methods to evaluate
  decoherence in realistic systems these, as seen
  below, essentially classical (the classical
  analog approach)
• assess the utility of model
  master equation methods to quantitatively provide
  decoherence rates
• if valid, determine the correct
  Lindblad operator to describe decoherence in
  these systems
• Then Develop scenarios to counter decoherence in
  realistic systems

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Essence of Coherent Control
Original application
AB C ABC
Control the ratio AC B

Basic principle (1) Construct two or more
indistinguishable routes to the final state (2)
Manipulate resultant interference via laboratory
knobs
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• Lots of applications done
• But preservation of quantum mechanics required.
  Hence concern about loss of quantum effects via
  decoherence and concern about developing methods
  to counter decoherence effects.

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Here sketch of ongoing program

• Decoherence computation via semiclassical
• Perturbation and proof of utility of classical
  analog
• at short times and at all times for strong
  decoherence.
• Numerical demo of validity over all time (small
  systems).
• Application to I2 in Liquid (Lennard-Jones) Xe.
• Interesting observation on temperature
  dependence/bath chaos of decoherence (Wilkies
  conjecture).

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General problem
Bath
System
Bath Part being traced over Not measured
System dynamics
(A) Measure of decoherence
Pure state
Mixed state
Termed purity or Renyi entropy advantage ---
basis indpt.
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• E.g. for two levels

Both two level as well as multilevel examined
below

Includes two effects, but here Interested in
short time where population changes are
small. Also time dependence of , in (basis
dependent) energy eigenstates of the system.
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First look Semiclassical IVR(Elran and Brumer,
JCP 121,2673, 2004)

• Sample system I2 linearly coupled to an
  harmonic oscillator bath (Bill Millers group---
  Wang et al, JCP 114, 2562, 2001)
• Parameters qualitative.

v
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Semiclassical IVR Approach

• Consider time correlation function
• To obtain for system in thermal bath,
  choose

Propagate using semiclassical forward-backward
Initial Value Representation
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Recurrences
Fig. 1 Decoherence dynamics Purity as a
function of time for the multilevel coherent
state. T 300 K, ? 0.25. Iodine in Harmonic
bath. Note three regimes And vast
dependence on initial state.
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6-level coherent state
2-level superposition
CAT
60-level coherent state
Note times
Figure 5 Purity as a function of times for
different initial states at T 300 K, ? 0.25.
cat state (solid line), multilevel coherent state
(dotted line), six-level coherent state (dashed
line), superposition state (dotted-dashed line).
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Decay Rates depend on the nature of the
distribution
Consistent with earlier work Indicating that the
greater the phase space structure of the state,
the faster the decoherence Pattanayak Brumer,
PRL. 79, 4131 (1997)
Figure 6 Relative population of the initial
superposition state (dot-dashed line), the
initial multilevel coherent state (dotted line),
the initial six-level coherent state (dashed
line) and the initial cat state (solid line).
The initial cat state wavefunction appears in the
inset.
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• Computations successful but very intensive.
  Possible
• approximations?
• Here look at two directions to a classical
  (analog) approach
• Perturbative argument
• Classical analog (linearized IVR)

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Classical Analog

• Recall correlation function structure for
• In general, we have some correlation function of
  the form
• C(t) Tr B(t) A(0) Tr BW(t) AW(0)
• Classical Analog Propagate BW(t) classically,
  even if
• distribution is negative
• Application here Start with Quantum system
  bath,
• propagate dynamics classically and trace over
  bath

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Long History

• Considerable work now using this type of
  approximation. Historically
• Classical analog
• Brumer and Jaffe, J. Chem. Phys. 82, 2330 (1985)
  Jaffe, Kanfer and Brumer, Phys. Rev. Lett. 54, 8,
  (1985) Wilkie Brumer, Phys. Rev. A 55, 27
  (1997) Wilkie Brumer, Phys. Rev. A 55, 43
  (1997)
• Linearized semiclassical IVR
• l
• Wang, Sun and Miller, J. Chem. Phys. 108, 9726
  (1998) Sun and Miller, J Chem. Phys. 110, 6635
  (1999)
• Shao, Liao and Pollak, J. Chem. Phys. 108, 9711
  (1998)

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Classical Analog vs. Full Semic. FB-IVRSample
Test on I2 in Harmonic bath
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Similarly sample matrix elements at t64 fs
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In support of this approximation--- conceptual
and practical for decoherence (and entanglement)
computations
Quantum Mechanics --- (drop s subscript
throughout)
Density Phase-space repn

Dynamics
One of several complete phase space repns of
quantum dynamics Classical mechanics
Density
Dynamics
Poisson Bracket
Conceptual Note beautiful classical/quantum
analog
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E.G. Can define Eigenfunctions, Eigenvalues,
etc. of time evolution OP (Liouville OP) etc.
Hilbert space, etc. e.g., Koopmans,
Prigogine (Our) Prior applications Quantum
classical correspondence
Jaffe Brumer, J. Chem. Phys. 82, 2330
(1985) Wilkie Brumer, Phys. Rev. A 55, 27
(1997) Wilkie Brumer, Phys. Rev. A 55, 43
(1997)
Classical analog of superposition state Jaffe,
Kanfer Brumer,
Phys. Rev.
Lett. 54, 8 (1985)
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Support --- formal Consider, for simplicity,
one-D system coupled to harmonic bath
(1)
Coupling
System coordinate
linear
N.B. f(Q) can be or nonlinear
common
(2) Define reduced system density, both class.
quant.

• Measures of decoherence (sample)
• Linear entropy
• Quantal
• Classical

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(b) Off-diagonal Matrix Element
Definition Quantum
Classical !

• Possible treatments
• (A) Exact dynamics
• (B) Perturbative for short time
• (C) Strong decoherence for all time
• Perturbation theory

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where
Recall
(time zero)
Hence The sole difference between quantum and
classical (perturbative short
time) is
i.e.,
Note result applicable to any coupling f(Q)
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Qualitative consequences
E.g., Zurek / Caldeira-Leggett

• If then
  i.e., classical is exact
  for linear quadratic system-bath coupling!
  
  Of course, but . . .
• (2) For any wherever
  decays fast enough with ?Q so that
  then class ? quant.
• (3) For any nonlinear/nonquadratic ,
  then class ?
  quant.

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For longer time? Can do strong decoherence case
(i.e., Hs 0) and obtain both
And again all expressions, including phases are
See J. Gong P. Brumer, Phys. Rev. Lett. 90,
050402 (2003) J. Gong P. Brumer, Phys.
Rev. A 68, 022101 (2003)
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Hence, if you set up an initial superposition
state, the subsequent decoherence dynamics
is Short time
(1) Classical if coupling (2) Classical for
any coupling if (3) Nonclassical if NOT (1)
or (2)
over
All time Strong decoherence
Even if the state is nonclassical
As above
Can we use to compute, etc ? Sample intrinsic
decoherence case
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But what about dynamics over longer times? Try
sample simple systems E.g.,
Two types of
Quartic oscillator
Integrable
Chaotic
Highly nonlinear
Note Zeroth order is not harmonic oscillators
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Figure 3.1 Comparison between ?q(t) (solid
line) and ?c(t) (discrete filled circular points)
for the quartic oscillator model in the case of
integrable dynamics (? 0.03). ?Q ?P
, Q0 0.4, P0 0.5, q0 0.6, with H (Q0,
P0, q0, p0) 0.24. All variables are in
dimensionless units.
Note excellent classical / quant
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t 0
t 5
t 10
t 15
Figure 3.37 Time evolution of
in energy
representation for the integrable case considered
in Fig. 3.1. The left (right) panels correspond
to the quantum (classical) system.
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Chaotic much faster decay
Figure 3.4 Same as Fig. 3.1 except for strongly
chaotic dynamics (?1.0).
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t 0
t 5
t 10
t 15
Figure 3.38 Time evolution of
in energy
representation for the chaotic case considered in
Fig. 3.4. The left (right) panels correspond to
the quantum (classical) system.
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Figure 3.13 A comparison between
(solid line) and (discrete filled
circular points) for the highly nonlinear
coupling potential case
with
All variables are in
dimensionless units.
But for strongly nonlinear coupling (as predicted)
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Realistic System Application to Breathing Sphere
I2 in Lennard-Jones Bath

• Model due to Egorov and Skinner, JCP 105, 7047
  (1996)
• Compute both and in energy
  basis times far shorter than T1

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Computational approach

• Thermalize bath (from 23 to 824 particles)
• Set up initial wavefunction for I2, compute
  associated Wigner function, propagate using
  classical mechanics
• Produce system by ignoring other
  variables ( averaging).

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Decoherence of initial coherent states F(x_i)
Time units ? 1 unit 3.316 ps. Decoherence
time scale here is 0.8 ps
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Correlates well with size of coherent state.
Also predicts harmonic oscillator slower decay
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Typical decay of system matrix elements in energy
representation
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Sample decay of superpositions of vibrational
states
V3 v4

• Decoherence times
• On order of 0.3 ps
• 4 times slower for
• Harmonic oscillaior
• I2.

V3 v6
V3 v8
Again---increasing structure ? increasing
decoherence rate//note harmonic much slower due
to collisional selection rules.
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Does a simple Caldeira-Leggett model work?

• Still computationally intensive, Can we replace
  by simple master equation. Try standard
  Caldeira-Leggett model
• System linearly coupled to an harmonic bath
• In high temperature, low coupling limit.
  Gives, for the
• Wigner function

Where D is the coupling term
How does Tr(rho2) behave?
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Can show directly

• Consider

Then can show directly that for this model that
Hence, (1) dS/dt increases with structure of The
system (2) We can test Caldeira-Leggett utility
by Computing terms and extracting D. Is it
Constant, and of what size?
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Is D constant? Sample results for I2 in
Lennard-Jones bath various cases-- Indeed very
far from constant --- fall off much faster at
short time, Strongly dependent on initial state.
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Indeed, did not even work well for I2 coupled
linearly to harmonic bath
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Wilkie conjecture

• Decoherence of system interacting with a chaotic
  bath is slower than that of a system in an
  uncoupled system --- at least at low temperature
• Possible changeover in behavior at higher
• temperature to be consistent with others
• Behavior confirmed for spin bath. But for
  collisional bath?
• Can test by decoupling LJ bath.

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Test on I2 in coupled and uncoupled LJ
bathFirst, is the bath chaotic?
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yes
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Does the coupling slow down decoherence at low
T?Sample case
YES
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How does this reconcile with higher T
predictions? Is there an inversion?
YES
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Summary

• Decoherence can be efficiently computed using
  the classical analog approach
• Relative decoherence rates are in accord with
  predictions based upon phase space structure.
• The simple Caldeira-Leggett model unfortunately
  not useful in Iodine in liquid Xe.
• Interesting behavior to explore, such as the
  reduction of decoherence at low temperatures upon
  strongly coupling up the bath.

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THANKS TO Dr. Yossi Elran (semiclassical
decoherence and classical analog) Dr. Jiangbin
Gong (theory and analytics ) Prof. Raymond
Kapral (classical analog) Dr. Angel Sanz (Wilkie
conjecture) Ms. Heekung Han (further studies)
ONR, Photonics Research Ontario NSERC