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.