Quantum dynamics of the parametric oscillator
P. Kinsler,
PhD Thesis, University of Queensland, (1994).
Email: Dr.Paul.Kinsler@physics.org
(a) Abstract
(b) Table of contents
(c) Conclusions
Abstract
========
The purpose of this thesis is to study a number of different types of
quantum mechanical effects. The parametric oscillator is a device
which causes a non linear interaction between a light beam and its
second harmonic in an optical cavity, and it is sed in this thesis
because it exhibits a wide variety of behaviours. I investigate two
general topic by studying particular phenomena in both conservative and
driven non w#equilibrium systems. The first topic is the difference
between quantum mechanical predictions and those from a stochastic
electrodynamics theory. The second is how the qantum effects scale
with the number of photons in the system.
The time taken for the degenerate parametric oscillator to tunnel
between its two stable states is calcuklated. This tunneling is a
quantum stchasitic process, and is not due to the coherent oscillation
of a quantum system. A comparison between the quantum results and
those from a stochastic electrodynamics model is instructive in that
the limitations of such approximations to quantum mechanics are clearly
demonstrated. For these tunnelling calculations, the semiclassical
theory only worked close to threshold, where the critical fluctuations
become large.
I compare numerical calculations of the transient squeezing in the
lossless models to two previous analytic calculations. These both used
asymptotic expansions, but gave conflicting results. Suggested
applications for squeezing include high resolution spectroscopy and
gravity wave detection -- it is useful because it enables the standard
qantum limit to be beaten. Here I find that the semiclassical theory
agreed with the quantum for large photon numbers. The scaling law
obtained for the amount of squeezing suggests the presence of a
"conservation of phase information" law. Thisis a significant factor in
deciding how useful squeezed light measurement techniques are.
The size of the critical fluctuations at threshold are calculated using
an adiabatic approach. The depleteion of the pump mode and the amount
of squeezing in the orthogonal quadrature are also worked out. This
comprehensively describes thisnon equilibrium quantum system at
threshold. In addition this theory provides a more realisti version of
the lossless squeezing calculations. Their scaling laws are found to
be equivalent.
A systematic approach is needed to properly investigate the limitations,
as well as the uses, of treating the quantum vacuum as a source of
classical noise. I consider particular third order correlations that
show a dramatic distinction between quantum mechanics and stochastic
electrodynamics. This sugests that new "all or nothing" tests of
hidden variable theories may be possible without having to construct N
particle correlated states.
Table of contents
=================
Chapter 0 - Front matter
Acknowledgements i
Abstract iii
List of Publications v
Table of Contents vi
List of Figures viii
Chapter 1 - Introduction 1
1.1 Summary of thesis 1
1.2 Stochastic methods 3
1.3 The Positive-P representation 10
1.4 The Wigner representation 16
1.5 Numerical stochastic calculations 20
1.6 Numerical number state techniques 27
Chapter 2 - The degenrate parametric oscillator 35
2.2 The Hamiltonian and the master equation 37
2.3 The number state basis 43
2.4 The Positive-P representation 46
2.5 The Wigner representation 52
Chapter 3 - The non degenrate parametric oscillator 59
3.2 The Hamiltonian and the master equation 60
3.3 The number state basis 62
3.4 The Positive-P representation 64
3.5 The Wigner representation 66
Chapter 4 - Tunnelling 67
4.2 Introduction 67
4.3 Number state results 72
4.4 The Positive-P representation 74
4.5 The Wigner representation 81
4.6 Near threashold comparisons 86
4.7 Conclusion 90
Chapter 5 - Transient squeezing 93
5.2 Introduction 94
5.3 Asymptotic methods 98
5.4 The degenrate parametric oscillator 102
5.5 The non degenrate parametric oscillator 107
5.6 Discussion 113
Chapter 6 - Critical fluctuations and squeezing 117
6.2 Linearized theory 120
6.3 An exactly soluble model 121
6.4 A non linear approach 123
6.5 Critical slowing down 126
6.6 Near threashold 131
6.7 Mean field theory: spectrum 135
6.8 The non degenrate parametric oscillator 142
6.9 Conclusion 146
Chapter 7 - Limits to squeezing 149
7.2 A phase "signal to noise" ratio 150
7.3 Making squeezed coherent light 155
7.4 Discussion 163
7.5 Information theory 165
Chapter 8 - Vacuum noise 167
8.2 Stochastic electrodynamics 168
8.3 Operator calculation 172
8.4 Positive-P calculation 176
8.5 A simple Raman model 180
8.6 Discussion 188
8.7 Conclusion 194
Chapter 9 - Conclusion 197
Appendix A - Computers: equipment and software 201
Chapter R - Bibliography 203
Conclusion
==========
The purpose of this thesis was to study a number of different types
of quantum mechanical effects,
with numerical confirmation of the theoretical results.
Quantum mechanical models of the parametric oscillator
were studied because despite their simplicity,
they exhibit a wide variety of behaviours.
Both conservative and driven non equilibrium models were treated.
Franken et al [1961] first observed the second order non linear effect
that occurs in parametric amplifiers by focissing a ruby laser beam
onto a quartz crystal.
At about the same time,
Louisell et al [1961] and Gordon et al [1963],
were pioneering theoretical work on the quantum parametric amplifier.
Since that time,
this system has been widely sudied,
as has the parametric oscillator,
which is simply a parametric amplifier in an optical cavity.
In 1973,
Graham described both the non degenerate and degenerate systems
using the Wigner representation [Wigner 1932],
and in 1980 Drummond etal used the coherent state basis
positive-P represntation to describe the degenerate case.
One of the themes in this thesis is the difference between QM predictions
and those from a stochastic electrodynamics theory.
The differences are highlighted with examples from the behaviour
of the degenerate parametric oscillator.
A stochastic electrodynamics theory [Marshall 1963; Boyer 1980]
is one in which the classical field equations are supplemented
by the addition of some classical noise
that represents ``vacuum fluctuations''.
This added noise replaces a quantum picture
with its instrinsic quantum uncertainty.
Equations equivalent to the stochastic electrodynamics theory
can be obtained by truncating the equation of motion
for the Wigner representation for this system.
In the linear regime,
there is usually no difference between the predictions
of these two theories.
However,
in non linear systems,
the differences are often large.
The tunnelling calculation in chapter 4 showed
that the average quantum stochastic tunnelling time
is much shorter than that obtained in a stochastic electrodynamics model.
For high driving strengths,
the quantum tunnelling time is exponentially proportional to the driving,
but the stochastic electrodynamics tunnelling time
varies exponentially with the square of the driving.
This is analagous to the differences between quantum tunneling
through a potential barrier and thermal activation over it.
With a sufficient reduction in thermal noise,
this quantum tunnelling rate could be observed at microwave frequencies
in a Josephson junction parametric oscillator.
However,
stochastic electrodynamics approaches can sometimes work well.
The squeezing calculations done in chapter 5 are an example of this.
For large photon numbers both the quantum
and semiclassical theories give the same results for the best squeezing
in a lossless parametric oscillator,
assuming a coherent state initial condition.
It would be interesting to check whether this equivalence remains
in a driven non equilibrium system at threshold
(see chapter 6),
but it is the expected result since the dynamics at threshold
are dominated by large ``classical-like'' critical fluctuations.
Although quantum theory and stochastic electrodynamics agree
for this squeezing calculation,
they are still fundamentally inequivalent.
Not only are the tunnelling results different,
but examination of the third order field moments is also instructive.
Chapter 8 included results which show that
in a lossledd parametric oscillator, for a particular field moment,
the quantum prediction and the stochastic electrodynamics prediction
have a dramatic disparity.
This occurs even at short times,
where quantum and classical dynamics agree.
The difference is significant
because it is not just a difference of scale
(as for the tunnelling time calculations),
but is a difference in character of the initial dynamics.
These moment results need to be generalized,
and two directions seem promising.
Firstly,
the results in chapter 8 are for a lossless system --
so they need to be made more realistic by adding damping and a driving field.
Then the differences should show up in the output spectra of the system.
Alternatively,
a travelling wave of pulsed system could be investigated.
Such an improved calculation would give predictions
that were more experimentally accessible.
Secondly,
since stochastic electrodynamics is a ``hidden variable'' theory,
it may be possible to construct a ``Bell inequality''
in a similar manner as Greenberger, Horne, and Zeilinger [1989]
did for e three particle spin system.
This would lead to a more elegant characterisation
of the nature of the system.
The other theme is that of photon number scaling laws.
As an example,
conjsider the behaviour of ferromagnets at their critical temperature,
or substances at their liquid/gas phase boundary.
These are known as critical phenomena,
and there is usually some scaling law
describing the fluctuations in magnetism or density near threshold.
These fluctuations scale with the size of the system.
In addition,
semiclassical or stochastic electrodynamics theories
are often assumed to agree with quantum mechanics in the limit
of large photon number N --
with the leading order quantum correction being proportional
to some inverse power of N.
In the tunneling calculations in chapter 4,
the tunnelling time was found to scale with the photon number N,
for both the quantum and stochastic electrodynamics calculations.
In particular,
the two models were shown to give equal results
in the near threshold limit.
The times still scaled differently with the driving field,
however.
In chapter 5 it was found the best squeezing also had a scaling law,
with the squeezing variance varying as $N^{-1/2}$.
Scaling laws for the behaviour of the degenerate parametric oscillator
at its critical point were calculated in chapter 6.
However,
not only was the size of the large critical fluctuations calculated
(the scaled as $N^{1/2}$)
but so was the size of the squeezed fluctuations in the orthogonal quadrature.
This sqeezing scaled as $N^{-1/2}$ above its minimium of 1/2
caused by quantum fluctuations from the outside leaking
into the optical cavity through the output mirrors.
Consideration of the experimentally accessible output spectra
gave somewhat similar results,
the zero frequency part of the critical fluctuations scaled as $N$
becaue of the effect of the critical slowing downm,
and the squeezing spectra scaled as $N^{-1/2}$.
The results from chapter 6 are an analytic calculation
of the threshold behaviour in a non equilibrium quantum system
with an external driving field.
Such examples are not common,
since the usual linear approaches fail at threshold
and a non linear theory like this cannot always be found.
The same $N^{-1/2}$ squeezing scaling is found in both
the spectra of this non equilibrium model,
and the quadrature moments of the lossless model.
This prediction then led to chapter 7,
where I considered the "phase resolution" of the initial coherent state
or driving field,
and how that limits the pump to signal mode conversion efficiency
and the amount of squeezing produced.
It was also shown that the use of a linear device such as a beamsplitter
interferometer will not increase the "phase information" once all
the energy is accounted for.
These results in turn suggested the possibility of using information theory
to provide rigorous constraints on physical systems by using a
"conservation of information" law.
Further work could be done on the threshol behaviour of this system
to provide a more detailed picture.
The non degenerate case could be analysed further,
an the entire output spectrum need to be calculated --
not just the zero frequency part.
The entire spectrum is important in the above threshold regime,
where squeezing improves for small detunings.
To conclude,
this thesis is an investigation of various quantum features
of the parametric oscillator.
In it I draw a distinction between quantum and stochastic electrodynamics
theories,
and back this up with several examples.
Also,
I have investigated how the various properties
of the parametric oscillator scale with the number of photons in the system.