Transcript Decoherence in the Quantum Control of Molecular Processes

Classical Approach to Computing
Quantum Decoherence Dynamics
Paul Brumer
Dept. of Chemistry, and
Center for Quantum Information and Quantum Control
University of Toronto
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”
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
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”
• 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.
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 (“Wilkie’s conjecture”).
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.
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.
First look; Semiclassical IVR
(Elran and Brumer, JCP 121,2673, 2004)
Sample system: I2 linearly
coupled to an harmonic
oscillator bath (Bill Miller’s
group--- Wang et al, JCP
114, 2562, 2001):
Parameters qualitative.
v
Semiclassical IVR Approach
• Consider time
correlation function:
• To obtain
for
system in thermal bath,
choose
Propagate using semiclassical forward-backward Initial Value Representation
“Zeno effect”
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.
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).
Decay Rates depend on the nature of the distribution
Slowest
Fastest
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.
Computations successful but very intensive. Possible
approximations?
Here look at two directions to a classical (analog) approach
1. Perturbative argument
2. Classical analog (linearized IVR)
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
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)
Classical Analog vs. Full Semic. FB-IVR:
Sample Test on I2 in Harmonic bath
Similarly: sample matrix elements at
t=64 fs
In support of this approximation--- conceptual and
practical for decoherence (and entanglement) computations
Quantum Mechanics --- (drop s subscript throughout)
Density
; Phase-space rep’n
Dynamics
One of several complete phase space repn’s of quantum dynamics
Classical mechanics:
Density
Dynamics
Poisson Bracket
Conceptual: Note beautiful classical/quantum analog:
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)
Support --- formal
Consider, for simplicity, one-D system coupled to harmonic bath:
(1)
Coupling:
Gen’l possible
System coordinate
N.B. f(Q) can be
linear
or nonlinear
common
(2) Define reduced system density, both class. + quant.
(3) Measures of decoherence (sample)
Linear entropy
Quantal
Classical
(b) Off-diagonal Matrix Element
Definition: Quantum
Classical !
Possible treatments
(A) Exact dynamics
(B) Perturbative for short time
Importance for decoherence control
(C) Strong decoherence for all time
Quantum computing
Perturbation theory
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)
Qualitative consequences
(1) If
then
i.e., classical is exact for
linear + quadratic system-bath coupling!
Of course, but . . .
(2) For any
that
then class  quant.
(3) For any nonlinear/nonquadratic
 quant.
decays fast enough with ΔQ so
; wherever
,
then class
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)
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
over
(3) Nonclassical if NOT (1) or (2)
All time: Strong decoherence
As above
Even if the state
is nonclassical
Can we use to compute, etc ?
Sample intrinsic decoherence case
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
Entangles & decoheres
Figure 3.1: Comparison between ζq(t) (solid line) and ζc(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.
Quartic oscillator
Regular regime
Note excellent classical / quant
Linear coupling
Gaussians
Quantum
Classical
t=0
t=5
t = 10
Same in
energy basis
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.
Figure 3.4: Same as Fig. 3.1 except for strongly chaotic dynamics (=1.0).
Quantum
Classical
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.
Figure 3.13: A comparison between
(solid line) and
the highly nonlinear coupling potential case
All variables are in dimensionless units.
(discrete filled circular points) for
with
But for strongly nonlinear
coupling (as predicted)
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
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).
Decoherence of initial coherent states: F(x_i)
Time units === 1 unit = 3.316 ps.
Decoherence time scale here is ~ 0.8 ps
Correlates well with “size” of coherent state. Also predicts
harmonic oscillator slower decay
Typical decay of system matrix elements in energy representation
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.
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(rho^2) behave?
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?
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.
Indeed, did not even work well for I2 coupled linearly to harmonic
bath
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.
Test on I2 in coupled and uncoupled LJ bath
First, is the bath chaotic?
yes
Does the coupling slow down decoherence at low T?
Sample case
YES
How does this reconcile with higher T predictions? Is
there an inversion?
YES
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.
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)
$$ ONR, Photonics Research Ontario
NSERC$$