Quantum trajectories: applied quantum measurement

1
Quantum trajectories applied quantum measurement
Andrew Doherty
APS March Meeting Tutorial Session, New Orleans,
March 2008
2
Outline
In this talk I hope to address the question of
how a model such as Todd discussed could be a
good model of real open quantum systems.
In quantum optics this kind of picture is
associated with the Markov approximation, master
equations, input-output theory and quantum
trajectories. Some standard references are below
and my own notes are (will be) available (my web
page).
Ive tried to use only elementary methods and use
an approach that may be more familiar to a broad
audience. Gardiner, Parkins, Zoller,
Wave-function quantum stochastic differential
equations and quantum-jump simulation methods,
Phys. Rev. A 46 4363 (1992). Gardiner, Zoller,
Quantum Noise, 2nd ed, Springer H. M. Wiseman
Ph.D thesis.
3
Step One
Toy model of physical system
4
Quantum system coupled to 1-D field
Data processor
A bosonic field A(x), with canonical momentum
density p(x) is coupled to a localised quantum
system (X is a system observable). This kind of
impurity model is much studied of course, we are
interested in very weak coupling with k(x)
localised near x0 and methods that can be
applied to essentially any impurity system. If
the system has characteristic frequency W the
field will result in damping at some rate g and
we are interested in
5
Quantum system coupled to 1-D field
Data processor
An atom spontaneously emitting into free
space. An optical Fabry-Perot cavity coupled to
its transmitted field. A superconducting
microwave resonator coupled to a transmission
line. The same approach could be used for Fermi
fields Quantum point contacts measuring quantum
dots A nanomechanical resonator measured by a
single electron transistor Basically the field
here is the detector from Aashs talk. Well see
that once again all we need to know is the noise
spectrum of the field.
6
Goals of calculation
Data processor
We would like to predict the evolution of the
system. We aim to calculate the properties of
the observed signal, so calculate correlation
functions of the field in some approximation.
Given the measurement results we would like to
be able to say something more about the state of
the system. We will see how to obtain the
picture of continuous measurement that Todd
described, where individual systems (here modes
of the field) interact sequentially with the
system and propagate away to be measured.
7
Physical Picture
Data processor
In sufficiently weak coupling only energy
conserving processes are important. Narrow
temporal modes of the field approach the
reflecting boundary, interact very briefly with
system and then propagate freely back towards the
detector. We can hope that this kind of
description will provide an accurate description
for sufficiently high Q resonances of the system,
for frequencies of the field that are
sufficiently close to the resonance. (Lorentzian
lineshape) The methods will be familiar from the
Wigner-Weiskopf theory of spontaneous emission.
8
Step Two
Markov Approximation
9
Model in terms of field modes
In the absence of coupling solve the model. Waves
propagate from the right, reflect off the
boundary at x0 and propagate back towards the
detector.
10
Moving to Interaction Picture
The Markov approximation relies on a difference
of timescales. To see what is going on we should
move to the interaction picture. We will restrict
to the case where And Heisenberg picture
evolution under is such that We can
then move to the interaction picture with respect
to
The first line is the energy conserving terms
that will dominate. The second line is energy
non-conserving terms that we will drop (rotating
wave approximation) and counter-terms to zero out
shifts of W
11
Properties of the input field operators
We have written the interaction picture evolution
in terms of certain mode operators of the field,
called input fields in the quantum optics
literature. The input field operator at time t is
a linear combination of Schroedinger picture bath
operators. It corresponds to the mode of the bath
that interacts with the system at t. Dont
confuse it with a Heisenberg picture operator.
The commutator of the input field is
This bath correlation function g(t) has some
non-zero value at early times, but it is
anticipated that for many problems of interest
g(t) will be very small for times larger than
some correlation time tC. The Markov
approximation depends on being able to choose a
time Dt that is larger than tC and much less
than the timescales of the interaction picture
evolution.
12
Time-averaged fields
When we come to do perturbation theory time
averages of the field will arise. Lets define
They have commutators
If the average is over times longer than the bath
correlation time we get
The extension to the integrals requires
13
Time-averaged fields
When we come to do perturbation theory time
averages of the field will arise. Lets define
They have commutators
Time averaged fields separated by more than Dt
approximately commute. In perturbation theory we
will perform the manipulations that led to this
result in all integrals and consistently work up
to order Dt. Notice that this formula implies
that we should regard DB as suggesting that we
will need second order perturbation theory.
14
Perturbation Expansion
Recall interaction picture Hamiltonian
Perturbation expansion for time evolution operator
Notice that the time-averaged fields we defined
earlier occur here. I have suppressed the
non-energy conserving terms and the counter terms.
15
Retain Only Dominant Contributions
Not all of the second order contributions are in
fact large, its possible to show the
following (We can assume that the bath at the
initial time is in the thermal state.)
This relies on the following identities, that can
be proved by taking matrix elements of the
operators on coherent states of the field and
allowing only frequencies close to W
16
Resum These Terms
To get the evolution over longer times we just
iterate. So if and
then
Physically the propagator corresponds to a series
of independent bath modes interacting with the
system, and then propagating away. (Exactly as in
Todds talk.)
The limit is for fixed t and t.
(This requires rescaling parameters so that the
correlation time remains smaller than Dt). This
is sometimes called the van Hove or weak coupling
limit. The differential operators dB describe a
quantum stochastic differential equation. These
were defined rigorously by Hudson and Parasarathy
and have found much use in quantum optics.
17
Physical Requirements
Interaction picture time evolution described by g
is slow compared to the system time-evolution
described by W.
Both the coupling and
are slowly varying functions of
frequency for frequencies close to W. In
practice this will require slow variation over a
frequency range of order g which will become the
width of the resonance.
This will mean that the coupling is
non-zero over a range such that
and so the field propagates across the impurity
very rapidly compared to the interaction picture
dynamics.
In practice this will mean that the following
correlation functions are short-lived.
18
Physical Requirements
Interaction picture time evolution described by g
is slow compared to the system time-evolution
described by W.
Both the coupling and
are slowly varying functions of
frequency for frequencies close to W. In
practice this will require slow variation over a
frequency range of order g which will become the
width of the resonance.
This will mean that the coupling is
non-zero over a range such that
and so the field propagates across the impurity
very rapidly compared to the interaction picture
dynamics.
In the description of the field modes we are
interested in frequencies close to the system
resonance.
19
Specific Bath Models
We can consider a specific model density of states
Where we have a assumed that the bath density of
states is a simple power law described by s (s3
for spontaneous emission) and we have introduced
a cut-off frequency
This results in a zero temperature correlation
time that depends on
However at sufficiently high frequencies
And so the thermal correlation function h(t) has
a correlation time that increases with
temperature. (Low frequencies and issue for
slt1)
20
Step Three
System Evolution
21
Master Equation
If we are only interested in the dynamics of the
system we may average over the field states
(partial trace). If the state of the system and
field at time t is c(t) then
The first line refers to processes where an
excitation leaks out of the system into the bath.
The third line is the reverse process.
Clearly we get a differential equation, called
the master equation, when we consider the
limit (If we organised the perturbation
according to some closed time path method, the
first contribution on each line would have
contributions from interaction vertices on either
side of the contour, with the remaining terms on
one side or the other.)
22
Master Equation
The two rates that appear in the master equation
are what we would have guessed based on the Fermi
Golden rule.
The ratio of these rates is a Boltzmann factor
for the (effective) temperature of the bath.
They depend on the correlation functions of the
bath
Recall Aashs talk, the non-commutativity of the
fields results is important.
23
Expectation Values and Correlation Functions
The master equation easily generates differential
equations for system expectation values and
correlations operators Y,Z
For expectation values
For correlation functions
Where the notation
represents the solution of the master equation
with initial condition
24
Damped Oscillator
This should describe a damped oscillator for
which
As expected the model describes exponential
decay of the oscillator energy with time constant
1/g, towards the thermal equilibrium value.
Alternatively we can consider an initial state
that is a mixture of number states For which we
find from the master equation
This rate equation is exactly what one would
expect for an oscillator coupled to a bath and
the steady state distribution is precisely the
thermal distribution of number states. The first
two terms reflect the oscillator jumping down one
number state as an excitation enters the bath
while the second pair of terms reflects processes
where the oscillator absorbs an excitation from
the bath.
25
Output Field
We imagine a detector measuring the reflected
field at some much later time. This requires
knowing how the averaged fields evolve
Recall that the evolution operator is
So we have
Physically this is a result of the Markov
approximation after interacting with the system
at time t the subsequent dynamics of the mode
bin(t) are unaffected by the interaction and
simply correspond to free propagation.
26
Correlation Functions of Output Field
The input output relation and the time evolution
operator allow us to find correlation functions
for the reflected field.
These things are pretty easy to calculate
The input field at t doesnt affect the system
until t, the output field at t doesnt affect the
system after t.
Where T is time ordering and is reverse time
ordering. And I have used the notation
27
Quantum Trajectories Diffusive/Homodyne
We can now model measurements on the output field.
Say we measure
to get value
The mean value is The variance is
Firstly we should check that in the Markov
approximation these observables commute for times
separated by Dt.
The physical picture of the previous slide is
that this observable has essentially Gaussian
statistics, with a mean shifted by the decay out
of the system.
We should regard this as a model of a measurement
of in a white noise background
(infer measurement sensitivity).
28
Quantum Trajectories II
We can find the state of the system given this
measurement by projecting on this value of the
field and renormalising.
The measurement outcome is
Where is a Gaussian random variable with
mean zero and variance
Then we can show that
This equation keeps up with the back-action due
to coupling the system to the detecting field AND
what we learn about the system when we get the
measurement result DQ
29
Quantum Trajectories Particle Counting
In photon counting we detect particles in the
output field. Specialise to zero temperature.
Intensity in input field
Roughly we have We get either zero or one
photon at any time. DN is a projection operator.
The detector measures the reflected version Find
that Evolution with detection Evolution with no
detection
If we see a photon in the field, we extract it
from the system. If we dont see a photon it is
less likely there was one in the system in the
first place.
30
Goals of calculation
Data processor
We would like to predict the evolution of the
system. We aim to calculate the properties of
the observed signal, so calculate correlation
functions of the field in some approximation.
Given the measurement results we would like to
be able to say something more about the state of
the system. We will see how to obtain the
picture of continuous measurement that Todd
described, where individual systems (here modes
of the field) interact sequentially with the
system and propagate away to be measured.
31
Physical Requirements
Interaction picture time evolution described by g
is slow compared to the system time-evolution
described by W.
Both the coupling and
are slowly varying functions of
frequency for frequencies close to W. In
practice this will require slow variation over a
frequency range of order g which will become the
width of the resonance.
This will mean that the coupling is
non-zero over a range such that
and so the field propagates across the impurity
very rapidly compared to the interaction picture
dynamics.
In the description of the field modes we are
interested in frequencies close to the system
resonance.
32
(No Transcript)