The elusive phase operator for oscillators has puzzled scientists since quantum inception. This talk explores the latest breakthroughs, tracing back to Dirac’s phase operator dilemma. Discover how phase and angle operators intertwine and the cutting-edge formulations shaping our understanding today.

- This presentation will take you through the fascinating journey of a phase operator for oscillators. Describing the phase of an electromagnetic field mode or a harmonic oscillator has been an obstacle since since the early days of modern quantum theory. I hope you find this talk both informative and engaging.
- The phase operator is a crucial in describing the phase of the oscillating system. Despite their importance, defining a proper phase operator has been a challenge since the inception of quantum theory. It has profound implications for the field, such as quantum optics and quantum information and computation.
- In 1927, Paul Dirac made the first significant attempt to define a phase operator for a quantum harmonic oscillator. William Louise in 1963 proposed an alternative approach to tackle the issues left unresolved by Dirac's definition. However, his approach still did not yield a hermitian phase operator.
- A perfect phase operator satisfying all the desired criteria is still eluding us. To determine the effectiveness of any proposed phase operator, it is important to consider certain desired properties. Meeting these criteria is crucial for the phase operator to be physically meaningful and mathematically consistent.
- The journey to define a phase operator for quantum harmonic oscillators has been both challenging and enlightening. Many open questions remain, particularly regarding the extension of these concepts to more complex quantum systems. Future research will likely focus on resolving these issues and exploring new applications in emerging quantum technologies.

Good day, everyone. I am delighted to present to you today on
the topic of the elusive phase unravelling
quantum mysteries. This presentation will take you through
the fascinating journey of a phase operator for oscillators,
a concept that has puzzled scientists since the inception of
quantum theory. Describing the phase
of an electromagnetic field mode or a harmonic
oscillator has been an obstacle since since the early days of
modern quantum theory. We will explore the historical
context, starting from Dirac's phase operator
dilemma, and trace the breakthroughs and advancements leading
up to the modern formulations that are shaping our understanding
today. I hope you find this talk both informative and engaging.
Let's dive in. To begin with, let's set
the stage by introducing the concept of phase operators in
the context of quantum harmonic oscillators,
or what we call as Qho, which is a fundamental
model in quantum mechanics, representing a particle in a
potential well oscillating around an equilibrium
position. The phase operator is a crucial
in describing the phase of the oscillating system,
analogous to how phase is represented
in in a classical wave theory.
Despite their importance, defining a proper phase
operator has been a challenge since the inception of
quantum theory. Since Dirac's first attempt
in 1927, which is almost a
century ago, scientists have struggled to find a
suitable definition that resolves the various theoretical
issues. Understanding the phase operator is not just an
academic exercise. It has profound implications
for the field, such as quantum optics and quantum information
and computation as well. The journey to understand
phase operators is as old as quantum mechanics itself,
with significant contributions and obstacles along the way.
Let's delve into the details of this elusive operator, shall we?
In 1927, Paul Dirac made
the first significant attempt to define a phase
operator for a quantum harmonic oscillator.
Simple enough. He proposed a definition where the annihilation operator,
a hat, is expressed as the following expression that is
given right over here. So a hat is
equal to e to the power of I, phi hat, n hat to the power
of half. Here we have n
hat being the number operator and phi hat
being our very own phase operator.
An annihilation operator, just to give you a brief
introduction, is nothing but an operator which
acts on a state such that the number state
is reduced by one. This is a
common notation in the quantum harmonic oscillator.
I do not want to bore you with more details, so I will move on
from this point. So the problem with
this approach is that it encountered
some major issues. The term here, e to
the power of I phi hat, was found to be non unitary.
That just means that it does not preserve the inner
product of the Hilbert space itself, which is a fundamental
requirement in quantum mechanics.
Additionally, there were ambiguities in the expectation
values of the phase operator phi hat given here,
which means that it is very difficult to interpret this physically.
These problems highlighted the need for an
alternative approach to defining this quantum phase operator.
Enter Louise contribution.
Almost 50 years later,
William Louise in 1963 proposed
an alternative approach to tackle the issues left unresolved by
Dirac's definition. To tackle the original problems
encountered by Dirac, let us transform the
operator such that the inner product of the operator
with the number states end up as unitary. Let the
operator become unitary is what he proposed.
He introduced the commutation relations for the cosine and
sine phase operators. Here we have the
commutator relations cos phi hat comma. The number
operator n hat is equal to I sine phi hat and
sine phi hat comma number operator n hat is equal to minus
I cosmic. Using this operation,
the phi hat has finally become a unitary operator.
While this is a step forward, Louisa's approach
still did not yield a hermitian phase operator.
Hermitian operators are essentially in quantum mechanics
because they ensure that observable quantities
like phase have real eigenvalues,
meaning that it has real physical
correlation to the real world.
The inability to define a hermitian phase operator continues
to be a significant hurdle, necessitating further exploration
and new ideas. We move to a very
popular and a similar kind of idea given by
Suskind and Glorgover in 1964.
Just a year later, Leonard Suskind
and John Glogoer proposed a novel approach
by decomposing the annihilation operator in a different manner
from dynac. They defined cosine and sine
operators, the c hat and the n hat s hat.
So, based on this following relation,
where e hat is our electric field operator,
e hat is a one sided unitary shift operator.
They derive the commutator relations for the following operations.
C hat comma n hat is equal to my I side is
hat, and s hat comma, n hat is equal to minus ic
hat. Do you see the relations and the ideas building
from previous models being incorporated
in their model as well? This approach was
really innovative, but unfortunately it was still
imperfect. The cosine and sine operators did not commute
and thus they did not form a complete set
of phase variables, highlighting the
ongoing complexity of defining a proper
phase of operator in quantum mechanics.
Now we come to Carter's and Nieto,
who further explored the properties of cosine and the sine
operators. Now what we call as the sg operators
standing for of course such kind and block over. They discovered
that their eigenvalues spectra are continuous in a particular
interval which is minus one to one. However,
they also found that these operators do not commute as shown
by the relation c hat s hat is equal to
one by two. I zero zero outer product
state. What does this zero zero actually mean?
So these are called as the number states and the number inside
represents the number of photons in the particular field at that time.
Here, zero photons basically mean vacuum and
what we are seeing is that the commutator relation leads to a relation
which directly computes to the outer product of the vacuum
states. This non commutative indicates
that a perfect phase operator satisfying all the desired
criteria are still eluding us.
Go another 20 years ahead to 1989,
in an attempt to overcome the issues faced by the earlier definitions,
Peg and Barnett introduced a novel approach.
They defined the phase operator as a finite dimensional
subspace of the Hilbert space as a whole, which allowed
for a more consistent mathematical framework.
Specifically, they used the definition in that the subspace
for the annihilation operator and expressed the exponential
of that phase operator as the following cyclic
summation. So here we have our phi hat
of the subspace being expressed as a
cyclic summation of all the number states up to
the state s, which is present in their particular chosen
subspace. Even this definition
looks very similar to Dirac's original definition,
and just see how the world is
very cyclic in nature. So specifically,
this approach, while effective in a finite
dimensional setting, becomes problematic as the
dimension s approaches infinity, that is,
the whole Hilbert space. Nonetheless, it provides a
significant step forward in our understanding
of a quantum phase operator.
So it is clear to us at this point that
phase measuring experiments respond to various
constructed operators differently.
Let us build our own phase operator.
To determine the effectiveness of any proposed
phase operator, it is important to consider certain
desired properties. There are three properties
that every phase operator must have and which
we would consider as a valid phase operator.
The phase operator should be hermitian, that is,
phi dagger is equal to five.
Now this means that the eigenvalues which
corresponds directly to physical measurements.
Secondly, the trigonometric identity should hold
analogous to the classical case cos square phi hat
plus sine square phi hat should be equal to identity.
This, if violated, might not get commutative
operators. As seen before, the phase operator should
exhibit proper time dependence described here,
where omega is nothing but our frequency
of the oscillator and phi hat,
or at time t can be
directly represented with phi hat, with the initial phi
hat and omega times the time
elapsed. Meeting these criteria is crucial for
the phase operator to be physically meaningful and mathematically
consistent. Now I just want
to talk about the recent advancements and applications.
In the current recent years,
where there have been significant advancements
in theoretical formulations of phase operators,
researchers have proposed new models that address some of
the historical challenges, such as maintaining hermesity,
ensuring consistent time dependency. These modern
formulations not only enhance our theoretical understanding,
but also have a practical application. For instance,
in quantum optics, accurate phase operators are
essential for precise control of the light fields
in various experiments. Additionally, in the realm
of quantum information, they play a critical role
in protocols for quantum communication and computation,
where phase coherence is crucial.
The journey to define a phase operator for
quantum harmonic oscillators has been both challenging and
enlightening. Starting from Dirac's initial attempts,
through the contributions of Loessel Saska and Logover
Carathis, we have made substantial
progress. However, the quest is far from over.
Many open questions remain,
particularly regarding the extension of these concepts to
more complex quantum systems. Future research
will likely focus on resolving these issues and exploring new
applications in emerging quantum technologies.
I want to take this opportunity to thank you all for your attention,
and here I would like to conclude my presentation, and I
thank the hosts for giving me the opportunity for this talk. So thank
you.

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