An investigation on the nonclassical and quantum phase properties of a family
of engineered quantum states
A Thesis submitted by
Priya Malpani
in partial fulfillment of the requirements for the award of the degree of
Doctor of Philosophy

Indian Institute of Technology Jodhpur
Department of Physics
September, 2020

## Certificate

This is to certify that the thesis titled An investigation on the nonclassical
and quantum phase properties of a family of engineered quantum states,
submitted by Priya Malpani (P14EN001) to the Indian Institute of Technology
Jodhpur for the award of the degree of Doctor of Philosophy, is a bonafide
record of the research work done by her under my supervision. To the best of
my knowledge, the contents of this report, in full or in parts, have not been
submitted to any other Institute or University for the award of any degree or
diploma.

## Acronyms

BS | Binomial state
---|---
DFS | Displaced Fock state
ECS | Even coherent state
HOA | Higher-order antibunching
HOSPS | Higher-order sub-Poissionian photon statistics
HOS | Higher-order squeezing
KS | Kerr state
LE | Linear entropy
PADFS | Photon added displaced Fock state
PSDFS | Photon subtracted displaced Fock state
PABS | Photon added binomial state
PAECS | Photon added even coherent state
PAKS | Photon added Kerr state
PASDFS | Photon added then subtracted displaced Fock state
VFBS | Vacuum filtered binomial state
VFECS | Vacuum filtered even coherent state
VFKS | Vacuum filtered Kerr state

## Declaration

I hereby declare that the work presented in this thesis entitled An
investigation on the nonclassical and quantum phase properties of a family of
engineered quantum states submitted to the Indian Institute of Technology
Jodhpur in partial fulfillment of the requirements for the award of the degree
of Doctor of Philosophy, is a bonafide record of the research work carried out
under the supervision of Dr. V. Narayanan. The contents of this thesis in full
or in parts, have not been submitted to, and will not be submitted by me to,
any other Institute or University in India or abroad for the award of any
degree or diploma.

## Acknowledgment

It is my great pleasure to express my gratitude to the person (Prof. Anirban
Pathak) without whom my Ph.D. journey was impossible. He has not only guided
me in my thesis but also inspired my life with his valuable suggestions and
magical words “Tension mat lo..sab ho jayega..”., I was in the dark with no
hope then Pathak Sir was candle of hope. Thank you for returning to me faith
in myself. Nothing better than these words can express my feelings “koi mujhko
yun mila hai..jaise banjaare ko ghar..”.

Special Thanks to my supervisor Dr. V. Narayanan. Thank you sir for all the
support and being one of the best human being in research field. I am grateful
to Dr. Subhashish Banerjee for his helpful discussion and motivation. I am
thankful to ample number of people involved in my research activity. I am
deeply indebtied to Dr. Kishore Thapliyal (Google Scholar- apko jitna bhi
thanku bolu bahut kam hi hoga!! Thankyou Sir for ignoring my silly mistakes,
your continuous support, motivation and everlasting inspiration) and Dr. Nasir
Alam (Thank you Sir for all valuable help), who had always supported me and I
feel blessed having seniors like you both. I acknowledge my colleagues, Sanjoy
Chatterjee (Dada, the best lab-mate), Javid Naikoo (Best Advisor …without whom
the journey of Ph.D. was next to impossible), Khushboo Dixit, Vandana Dahiya
(Cutest friend-Jatni…yaara teri yari ko maine to khuda mana.. thanku for all
the support, without your help survival in IITJ was not possible), Shilpa
Pandey (Best roommate), Mitali Sisodia (thank you for listening my endless
drama), Swarn Rajpoot (naam hi kafi hai), Satish Sangwan (Tau- the best
buddy), Vishwadeepak Kumar (Rockstar), Ashwin Saxena, and other research
scholars, teaching and non-teaching staff in IITJ department, who had become
part of my life during this period of time. Now, I would like to thank
everyone who had played a prominent role in my life. Special thanks to my
parents for their unconditional love and support, without your support it was
not possible for me to reach this level. I would like to thank my grandpa
(Baba). I will thank my bua (Sheelam Bua) for her continuous moral support. A
big thanks to my lovely Sadi, Vaibhav (no words can express my thank you to
you bro. Thank you for handling all my moods with extreme patience), Meghai
(Friendster), Doll (my soul sister), Labbu (blessed to have you in my life
bro), Yashivangi (strong pillers of my life) and a big thanks to someone who
has completely changed my way of thinking. I would also like to acknowledge
the financial helps I received in different phases of my research work from
IITJ, JIIT and MHRD without which this work would not have been possible.

## Abstract

The main focus of this thesis is to study the nonclassical and phase
properties of a family of engineered quantum states, most of which show
various nonclassical features. The beauty of these states is that these states
can be used to establish quantum supremacy. Earlier, a considerable amount of
works has been reported on various types of quantum states and their
nonclassical properties. Here, complementing the earlier works, the effect of
non-Gaussianity inducing operators on the nonclassical and phase properties of
displaced Fock states have been studied. This thesis includes 6 chapters. In
Chapter 1, motivation behind performing the present work is stated explicitly,
also the basic concepts of quantum optics are discussed with a specific
attention on the witnesses and measures of nonclassicality. In Chapter 2,
nonclassical properties of photon added and subtracted displaced Fock states
have been studied using various witnesses of lower- and higher-order
nonclassicality which are introduced in Chapter 1. In Chapter 3, we have
continued our investigation on photon added and subtracted displaced Fock
states (and their limiting cases). In this chapter, quantum phase properties
of these states are investigated from a number of perspectives, and it is
shown that the quantum phase properties are dependent on the quantum state
engineering operations performed. In Chapter 4, we have continued our
investigation on the impact of non-Gaussianity inducing operators on the
nonclassical and phase properties of the displaced Fock states. In Chapter 5,
we have performed a comparison between to process that are used in quantum
state engineering to induce nonclassical features. Finally, this thesis is
concluded in Chapter 6, where we have summarized the findings of this thesis
and have also described scope of the future works.

Dedicated to my parents for their unconditional love and support.

###### Contents

  1. Acknowledgment
  2. Abstract
  3. 1 Introduction
    1. 1.1 Introduction
    2. 1.2 Quantum theory of radiation field
      1. 1.2.1 Creation and annihilation operator
      2. 1.2.2 Some more quantum operators of relevance
        1. 1.2.2.1 Displacement operator
        2. 1.2.2.2 Squeezing operator
      3. 1.2.3 Eigen states of the field operators
        1. 1.2.3.1 Fock state: Eigen state of the number operator
        2. 1.2.3.2 Coherent state: Eigen state of the annihilation operator
    3. 1.3 Quantum states of our interest
      1. 1.3.1 Displaced Fock state
      2. 1.3.2 Photon added and photon subtracted displaced Fock state
      3. 1.3.3 Photon added then subtracted displaced Fock state
      4. 1.3.4 Even coherent state and states generated by holeburning on it
        1. 1.3.4.1 Vacuum filtered even coherent state
        2. 1.3.4.2 Photon added even coherent state
      5. 1.3.5 Binomial state and the states generated by holeburning on it
        1. 1.3.5.1 Vacuum filtered binomial state
        2. 1.3.5.2 Photon added binomial state
      6. 1.3.6 Kerr state and the states generated by holeburning on it
        1. 1.3.6.1 Vacuum filtered Kerr state
        2. 1.3.6.2 Photon added Kerr state
    4. 1.4 The notion of nonclassical states
    5. 1.5 Nonclassical states: witnesses and measures
      1. 1.5.1 Witnesses of nonclassicality
        1. 1.5.1.1 Lower- and higher-order antibunching
        2. 1.5.1.2 Lower- and higher-order sub-Poissionian photon statistics
        3. 1.5.1.3 Higher-order squeezing
        4. 1.5.1.4 Klyshko’s criterion
        5. 1.5.1.5 Vogel’s criterion
        6. 1.5.1.6 Agarwal-Tara’s criterion
        7. 1.5.1.7 Mandel $Q_{M}$ parameter
      2. 1.5.2 Other quasiprobability distributions
        1. 1.5.2.1 $Q$ function
        2. 1.5.2.2 Wigner function
      3. 1.5.3 Measures of nonclassicality
        1. 1.5.3.1 Linear entropy
    6. 1.6 Analytic tools for the study of phase properties of nonclassical states
      1. 1.6.1 Phase distribution function
      2. 1.6.2 Phase dispersion
      3. 1.6.3 Angular Q function
      4. 1.6.4 Phase fluctuation
      5. 1.6.5 Quantum phase estimation parameter
    7. 1.7 Structure of the rest of the thesis
  4. 2 Lower-and higher-order nonclassical properties of photon added and subtracted displaced Fock state
    1. 2.1 Introduction
    2. 2.2 Higher-order moment for PADFS and PSDFS
    3. 2.3 Nonclassical features of PADFS and PSDFS
      1. 2.3.1 Mandel $Q_{M}$ Parameter
      2. 2.3.2 Lower- and higher-order antibunching
      3. 2.3.3 Higher-order sub-Poissonian photon statistics
      4. 2.3.4 Higher-order squeezing
      5. 2.3.5 $Q$ function
      6. 2.3.6 Agarwal-Tara’s criterion
      7. 2.3.7 Klyshko’s Criterion
    4. 2.4 Conclusions
  5. 3 Quantum phase properties of photon added and subtracted displaced Fock states
    1. 3.1 Introduction
    2. 3.2 Quantum phase distribution and other phase properties
    3. 3.3 Phase properties of PADFS and PSDFS
      1. 3.3.1 Phase distribution function
      2. 3.3.2 Angular $Q$ function of PADFS and PSDFS
      3. 3.3.3 Quantum phase fluctuation of PADFS and PSDFS
      4. 3.3.4 Phase Dispersion
      5. 3.3.5 Phase sensing uncertainity for PADFS and PSDFS
    4. 3.4 Conclusions
  6. 4 Impact of photon addition and subtraction on nonclassical and phase properties of a displaced Fock state
    1. 4.1 Introduction
    2. 4.2 Moments of the field operators for the quantum states of our interest
    3. 4.3 Nonclassicality witnesses and the nonclassical features of PASDFS witnessed through those criteria
      1. 4.3.1 Lower- and higher-order antibunching
      2. 4.3.2 Higher-order sub-Poissonian photon statistics
      3. 4.3.3 Lower- and higher-order squeezing
      4. 4.3.4 Klyshko’s Criterion
      5. 4.3.5 Agarwal-Tara’s criterion
      6. 4.3.6 Vogel’s criterion
    4. 4.4 Phase properties of PASDFS
      1. 4.4.1 Phase distribution function
      2. 4.4.2 Phase Fluctuation
    5. 4.5 Quasidistribution function: $Q$ function
    6. 4.6 Conclusions
  7. 5 Manipulating nonclassicality via quantum state engineering processes: Vacuum filtration and single photon addition
    1. 5.1 Introduction
    2. 5.2 Quantum states of interest
      1. 5.2.1 Expressions for moments of annihilation and creation operators
        1. 5.2.1.1 Expectation values for even coherent states and the corresponding engineered states
        2. 5.2.1.2 Expectation values for binomial state and the corresponding engineered states
        3. 5.2.1.3 Expectation values for Kerr state and the corresponding engineered states
    3. 5.3 Nonclassicality witnesses
      1. 5.3.1 Lower- and higher-order antibunching
      2. 5.3.2 Lower- and higher-order squeezing
      3. 5.3.3 Higher-order sub-Poissonian photon statistics
    4. 5.4 Nonclassicality measure
    5. 5.5 Conclusion
  8. 6 Conclusions And Scope For Future Work
    1. 6.1 Conclusion
    2. 6.2 Scope for future work

###### List of Figures

  1. 1.1 Phase picture for coherent state and squeezed state.
  2. 1.2 Hanbury Brown and Twiss setup. Here, APD is avalanche photo diode.
  3. 2.1 A schematic diagram for the generation of PSDFS (in (a) and (b)) and PADFS (in (c) and (d)). In (a) and (c) ((b) and (d)), single-mode (two-mode) squeezed vacuum is used for generation of the desired state. Here, NLC corresponds to nonlinear crystal and D1 and D2 are photon number resolving detectors.
  4. 2.2 Variation of Mandel $Q_{M}$ parameter for PADFS (in (a) and (b)) and PSDFS (in (c) and (d)) is shown with displacement parameter $\alpha$. In (a) and (c), the value of number of photons added/subtracted (i.e., $u$ or $v$) is changed for the same initial Fock state $|1\rangle$. Different initial Fock states $|n\rangle$ are chosen to be displaced in (b) and (d) for the single photon addition/subtraction. The blue curve corresponds to vacuum for $\alpha=0$ and thus starts from $0$ unlike rest of the states which are Fock state ($n\neq 0)$ in the limiting case. Therefore, nonclassicality increases first with increasing $\alpha$ before decreasing as in the rest of the cases.
  5. 2.3 The presence of higher-order antibunching is shown as a function of $\alpha$ for PADFS (in (a)-(c)) and PADFS ((d)-(f)). Specifically, (a) and (d) illustrate comparison between lower- and higher-order antibunching. It should be noted that some of the curves are multiplied by a scaling factor in order to present them in one figure. Figures (b) and (e) show the effect of photon addition/subtraction, and (c) and (f) establish the effect of Fock state chosen to displace in PADFS and PSDFS, respectively. Here, without the loss of generality, we have used the notation $\psi_{+}(u,n,l-1)$ ($\psi_{-}(u,v,l-1)$) for nonclassicality in photon added (subtracted) scenarios, and will follow this notation in subsequent figures of this chapter.
  6. 2.4 Dependence of higher-order sub-Poissonian photon statistics on $\alpha$ for PADFS (in (a)-(c)) and PSDFS ((d)-(f)) is illustrated here. Specifically, (a) and (d) show the increase in depth of nonclassicality witness with order, (b) and (e) depict the effect of photon addition and subtraction, respectively, and (c) and (f) establish the effect of choice of Fock state to be displaced in PADFS and PSDFS, respectively.
  7. 2.5 Illustration of the higher-order squeezing using Hong-Mandel criterion as the function of displacement parameter. In (a) and (d), dependence of the observed nonclassicality on different orders ($l$) is shown for PADFS and PSDFS, respectively; while in (b) and (e), the effect of variation in the number of photon added/subtracted is shown in case of PADFS and PSDFS, respectively. In (c) and (f), the variation due to change in the initial Fock state chosen to be displaced is shown for PADFS and PSDFS, respectively.
  8. 2.6 Hong-Mandel type higher-order squeezing for PADFS and PSDFS is shown dependent on the phase of the displacement parameter $\alpha=|\alpha|\exp\left(i\theta\right)$ in (a) and (b), respectively.
  9. 2.7 Contour plots of the $Q$ function for (a) single photon added displaced Fock $|1\rangle$ state, (b) two photon added displaced Fock $|1\rangle$ state, (c) single photon added displaced Fock $|2\rangle$ state, (d) single photon subtracted displaced Fock $|1\rangle$ state, (e) two photon subtracted subtracted displaced Fock $|1\rangle$ state, (f) single photon subtracted displaced Fock $|2\rangle$ state. In all cases, $\alpha=\sqrt{2}\exp\left(\frac{i\pi}{4}\right)$ is chosen.
  10. 2.8 Variation of Agarwal-Tara’s parameter with $\alpha$ for PADFS and PSDFS is shown in (a)-(b) and (c)-(d), respectively. Specifically, the effect of photon addition/subtraction (in (a) and (c)) and the choice of Fock state ((b) and (d)) on the presence of nonclassicality in PADFS and PSDFS is illustrated.
  11. 2.9 Illustration of the Klyshko’s criterion. Variation of $B(m)$ with respect to $m$ (a) and (c) for different values of the number of the photon additon/subtraction for PADFS and PSDFS, respectively; (b) and (d) for different values of the number of the Fock state parameter for PADFS and PSDFS, respectively. Here, we have chosen $\alpha=1$ in all cases because almost all of the other criteria detected nonclassicality for this choice of $\alpha$ so does the Klyshko’s criterion.
  12. 3.1 Variation of phase distribution function with phase parameter for PADFS with displacement parameter $\left|\alpha\right|=1$ for different values of photon addition ((a) and (c)) and Fock parameters ((b) and (d)). The phase distribution is shown using both two-dimensional ((a) and (b) with $\theta_{2}=0$) and polar ((c) and (d)) plots. In (c) and (d), $\theta_{2}=\frac{n\pi}{2}$ with integer $n\in\left[0,3\right]$, and the legends are same as in (a) and (b), respectively.
  13. 3.2 Variation of phase distribution function with phase parameter for PSDFS with displacement parameter $\left|\alpha\right|=1$ for different values of photon subtraction ((a) and (c)) and Fock parameters ((b) and (d)). The phase distribution is shown using both two-dimensional ((a) and (b) with $\theta_{2}=0$) and polar ((c) and (d)) plots. In (c) and (d), $\theta_{2}=\frac{n\pi}{2}$ with integer $n\in\left[0,3\right]$, and the legends are same as in (a) and (b), respectively.
  14. 3.3 The polar plots for angular $Q$ function for PADFS (in (a) and (b)) and PSDFS (in (c) and (d)) for displacement parameter $\left|\alpha\right|{\rm=}1$ and $\theta_{2}=\frac{n\pi}{2}$ with integer $n\in\left[0,3\right]$ for different values of photon addition/subtraction and Fock parameters. In (a) and (c), for $n=1$, the smooth (blue), dashed (red), dot-dashed (magenta), and dotted (brown) lines correspond to photon addition/subtraction 0, 1, 2, and 3, respectively. In (b) and (d), for the single photon added/subtracted displaced Fock state, the smooth (blue), dashed (red), dot-dashed (magenta), and dotted (brown) lines correspond to Fock parameter 1, 2, 3, and 4, respectively.
  15. 3.4 Variation of three phase fluctuation parameters introduced by Carruthers and Nieto with the displacement parameter with $\theta_{2}=0$. The values of photon addition ($u$), subtraction ($v$), and Fock parameter $n=1$ are given in the legends. Parameter $U\left(i,n\right)\,\forall i\in\left\\{u,v\right\\}$ also illustrates antibunching in the states for values less than $\frac{1}{2}$.
  16. 3.5 Variation of phase dispersion for PADFS (in (a) and (b)) and PSDFS (in (c) and (d)) with displacement parameter for an arbitrary $\theta_{2}$. Dependence on different values of photon added/subtracted and the initial Fock state $\left|1\right\rangle$ (in (a) and (c)), while on different values of Fock parameter for single photon added/subtracted state (in (b) and (d)).
  17. 3.6 Phase sensing uncertainty for (a) PADFS and (b) PSDFS as a function of phase to be estimated $\phi$ for different number of photon addition/subtraction with $n=1$. The dependence for (c) PADFS and (d) PSDFS is also shown for different values of Fock parameters with $u=1$ and $v=1$, respectively. In all cases, we have chosen $\alpha=0.1$.
  18. 4.1 For PASDFS the lower-and higher-order antibunching is given as a function of displacement parameter $\alpha$. (a) Lower-order antibunching for different values of parameters of the state. (b) Higher-order antibunching for particular state. HOSPS for PASDFS for different values of (c) state parameters and (d) order of nonclassicality.
  19. 4.2 Dependence of the Hong-Mandel-type higher-order squeezing witness on displacement parameter for (a) different state parameters and (b) order of squeezing. (c) Lower-order squeezing as a function of phase parameter of the state, i.e., phase $\theta$ of displacement parameter $\alpha=1.\exp[\iota\theta].$
  20. 4.3 Illustration of Klyshko’s parameter $B\left(z\right)$ with respect to the photon number $z$ for different values of state parameters with (a) $\alpha=0.5$ and (b) $\alpha=1$.
  21. 4.4 Nonclassicality reflected through the negative values of (a) Agarwal-Tara’s and (b) Vogel’s criteria as a function of $\alpha$ or different state parameters.
  22. 4.5 Polar plot of phase distribution function for PASDFS $|\psi\left(k,q,n,\alpha\right)\rangle$ with respect to variation in displacement parameter for (a) $n=1,\,k=2$ and $q=1,$ 2, and 3 represented by the smooth (cyan), dashed (magenta), and dot-dashed (purple) lines, respectively; (b) $n=2,\,q=2$ and $k=1,$ 2, and 3 illustrated by the smooth (cyan), dashed (magenta), and dot-dashed (purple) lines, respectively; and (c) $n=1$ with $k=q=1,$ 2, and 3 shown by the smooth (cyan), dashed (magenta), and dot-dashed (purple) lines, respectively.
  23. 4.6 Variation of phase fluctuation parameter with displacement parameter for various state parameters in PASDFS.
  24. 4.7 $Q$ function for PASDFS $|\psi\left(k,q,n,\alpha\right)\rangle$ with (a) $k=q=n=1,$ (b) $k=2,\,q=n=1,$ and (c) $q=2,\,k=n=1$ with $\alpha=\frac{1}{5\sqrt{2}}\exp\left(\iota\pi/4\right)$. (d) Similarly, $Q$ function of PASDFS with $q=2,\,k=n=1$ and $\alpha=\sqrt{2}\exp\left(\iota\pi/4\right)$. $Q$ function for $|\psi\left(k,q,n,\alpha\right)\rangle$ with (e) $k=q=1,\,n=2$ and (f) $q=1,\,k=n=2$ for $\alpha=\frac{1}{5\sqrt{2}}\exp\left(\iota\pi/4\right)$.
  25. 5.1 Lower- and higher-order antibunching witnesses as functions of displacement parameter $\alpha$ (for ECS and KS) and probability $p$ (for BS with parameter $M=10$) for (a) ECS, PAECS and VFECS, (b) BS, PABS and VFBS, and (c) KS, PAKS and VFKS. The quantities shown in all the plots are dimensionless.
  26. 5.2 Illustration of lower- and higher-order squeezing for (a) BS, PABS and VFBS; (b) KS, PAKS and VFKS at the fixed value of $\chi=0.02$; (c) KS, VFKS and PAKS as a function of $\chi$ with $\alpha=1$. The negative regions of the curves illustrate the presence of squeezing.
  27. 5.3 The dependence of HOS witness ($l=4$) on Kerr parameter $\chi$ and displacement parameters $\left|\alpha\right|$ and $\theta$ for PAKS with (a) $\left|\alpha\right|=3$, (b) $\chi=0.02$, (c) $\theta=0$.
  28. 5.4 Illustration of HOSPS as a function of displacement parameter $\alpha$ (for ECS and KS) and probability $p$ (for BS) for (a) ECS, (b) BS, and (c) KS and corresponding engineered states. HOSPS is not observed in KS.
  29. 5.5 Illustration of linear entropy for (a) ECS, PAECS and VFECS, (b) BS, PABS and VFBS, (c) KS, PAKS and VFKS with $\alpha$ or $p$ for $\chi=0.02$. (d) Dependence of nonclassicality in KS, PAKS and VFKS on $\chi$ for $\alpha=1$.
  30. 5.6 Illustration of linear entropy for (a) KS (b) VFKS (c) PAKS.

###### List of Tables

  1. 3.1 Various states that can be obtained as the limiting cases of the PADFS and PSDFS.
  2. 4.1 Summary of the nonclassical properties of PASDFS.

##  Chapter 1 Introduction

###  1.1 Introduction

As the title of the thesis suggests, in this thesis, we aim to study the
nonclassical and phase properties of a family of engineered quantum states.
Before, we introduce such states and properties, it would be apt to lucidly
introduce the notion of nonclassical and engineered quantum states. By
nonclassical state we refer to a quantum state having no classical analogue.
Such states are characterized by the negative values of Glauber-Sudarshan
$P$-function or $P$-function more singular than Dirac delta function Sudarshan
[1963]; Glauber [1963a] and witnessed by various operational criteria (to be
described in Section 1.5.1). To visualize the relevance of nonclassical states
we may first note that quantum supremacy refers to the ability of performing a
task using quantum resources in such a manner that either the task itself
cannot be performed using classical resources or the speed/efficiency achieved
using quantum resources cannot be achieved in the classical world Grover
[1997]. A recent experiment performed by Google aimed at estabilishing quantum
supremacy has drawn much of public attention Courtland [2017]. The relevance
of the present study lies in the fact that to establish quantum supremacy or
to perform a fundamental test of quantum mechanics, we would require a state
having some features that would not be present in any classical state. As we
have already mentioned, such a state having no classical analogue is referred
to as the nonclassical state. Frequently used examples of nonclassical states
include squeezed, antibunched, entangled, steered, and Bell nonlocal states.
The relevance of the states having nonclassical features has already been
established in the various domains of physics. For example, we may mention,
teleportation of coherent states Furusawa et al. [1998], continuous variable
quantum cryptography Hillery [2000], quantum radar Lanzagorta [2011], and many
more. Further, we may note that the art of generating and manipulating quantum
states as per need is referred to as the “quantum state engineering” Dakna et
al. [1998]; Sperling et al. [2014]; Vogel et al. [1993]; Miranowicz and
Leonski [2004]; DellAnno et al. [2006]. Particularly interesting examples of
such engineered quantum states are Fock state, photon added/subtracted
coherent state Agarwal and Tara [1991], displaced Fock state (DFS) which is
also referred to as generalized coherent state and displaced number state
Satyanarayana [1985]; Wunsche [1991]; Ziesel et al. [2013]; Zavatta et al.
[2004]; De Oliveira et al. [1990]; Malpani et al. [2019a], photon added DFS
(PADFS) Malpani et al. [2019a], and photon subtracted DFS (PSDFS) Malpani et
al. [2019a]. In what follows, we will state the relevance of such engineered
quantum states in the implementation of different tasks exploiting their
nonclassical and phase properties. The relatively new area of research on
quantum state engineering has drawn much attention of the scientific community
because of its success in experimentally producing various quantum states
Zavatta et al. [2004]; Torres et al. [2003]; Rauschenbeutel et al. [2000]; Gao
et al. [2010]; Lu et al. [2007] having nonclassical properties and
applications in realizing quantum information processing tasks, like quantum
key distribution Bennett and Brassard [1984] and quantum teleportation
Brassard et al. [1998]; Chen [2015]. Engineered quantum states, such as cat
states, Fock state and superposition of Fock states, are known to play a
crucial role in performing fundamental tests of quantum mechanics and in
establishing quantum supremacy in the context of quantum computation and
communication (Kues et al. [2017] and references therein).

As mentioned in the previous paragraph, with the advent of quantum state
engineering Vogel et al. [1993]; Sperling et al. [2014]; Miranowicz and
Leonski [2004]; Marchiolli and José [2004] and quantum information processing
(Pathak [2013] and references therein), the study of nonclassical properties
of engineered quantum states have become a very important field. This is so
because only the presence of nonclassical features in a quantum state can
provide quantum supremacy Grover [1997]. In the recent past, various
techniques for quantum state engineering have been developed Vogel et al.
[1993]; Sperling et al. [2014]; Miranowicz and Leonski [2004]; Agarwal and
Tara [1991]; Lee and Nha [2010]; Marchiolli and José [2004]. If we restrict
ourselves to optics, these techniques are primarily based on the clever use of
beam splitters, detectors, and measurements with post selection, etc. Such
techniques are useful in creating holes in the photon number distribution
Escher et al. [2004] and in generating finite dimensional quantum states
Miranowicz and Leonski [2004], both of which are nonclassical Pathak and
Ghatak [2018]. The above said techniques are also useful in realizing non-
Gaussianity inducing operations, like photon addition and subtraction Zavatta
et al. [2004]; Podoshvedov [2014]. Motivated by the above, in this thesis, we
aim to study the nonclassical properties of a set of engineered quantum states
such as photon added, photon subtracted, and photon added then subtracted
displaced Fock states which can be produced by using the above mentioned
techniques. In the present thesis, we also wish to investigate the phase
properties of the above mentioned engineered quantum states for the reasons
explained below.

The impossibility of writing a Hermitian operator for quantum phase is a
longstanding problem (see Peřinová et al. [1998]; Carruthers and Nieto [1968];
Lynch [1987] for review). Early efforts of Dirac Dirac [1927] to introduce a
Hermitian quantum phase operator were not successful, but led to many
interesting proposals Susskind and Glogower [1964]; Pegg and Barnett [1989];
Barnett and Pegg [1986]. Specifically, Susskind-Glogower Susskind and Glogower
[1964], Pegg-Barnett Pegg and Barnett [1988, 1989]; Barnett and Pegg [1990],
and Barnett-Pegg Barnett and Pegg [1986] formalisms played very important role
in the studies of phase properties and the phase fluctuation Imry [1971].
Thereafter, phase properties of various quantum states have been reported
using these formalisms Sanders et al. [1986]; Gerry [1987]; Yao [1987];
Carruthers and Nieto [1968]; Vaccaro and Pegg [1989]; Pathak and Mandal
[2000]; Alam and Mandal [2016b]; Alam et al. [2017a]; Verma and Pathak [2009].
Other approaches have also been used for the study of the phase properties.
For example, quantum phase distribution is defined using phase states Agarwal
et al. [1992], while Wigner Garraway and Knight [1992] and $Q$ Leonhardt and
Paul [1993]; Leonhardt et al. [1995] phase distributions are obtained by
integrating over radial parameter of the corresponding quasidistribution
function. In experiments, the phase measurement is performed by averaging the
field amplitudes of the $Q$ function Noh et al. [1991, 1992]; Pegg-Barnett and
Wigner phase distributions are also reported with the help of reconstructed
density matrix Smithey et al. [1993]. Further, quantum phase distribution
under the effect of the environment was also studied in the past leading to
phase diffusion Banerjee and Srikanth [2007]; Banerjee et al. [2007]; Abdel-
Aty et al. [2010]; Banerjee [2018]. A measure of phase fluctuation named phase
dispersion using quantum phase distribution has also been proposed in the past
Peřinová et al. [1998]; Banerjee and Srikanth [2007]. Recently, quantum phase
fluctuation Zheng-Feng [1992] and Pancharatnam phase Mendas and Popovic [1993]
have been studied for DFS. The quantum phase fluctuation in parametric down-
conversion Gantsog et al. [1991] and its revival Gantsog [1992] are also
reported. Experiments on phase super-resolution without using entanglement
Resch et al. [2007] and role of photon subtraction in concentration of phase
information Usuga et al. [2010] are also performed. Optimal phase estimation
Sanders and Milburn [1995] using different quantum states Higgins et al.
[2007] (including NOON and other entangled states and unentangled single-
photon states) has long been the focus of quantum metrology Giovannetti et al.
[2006]; Giovannetti et al. [2011]. Nonclassicality measure based on the
shortening of the regular distribution defined on phase difference interval
broadbands due to nonclassicality is also proposed in the recent past Peřina
and Křepelka [2019]; Thapliyal and Perina [2019]. In brief, quantum phase
properties are of intense interest of the community since long (see Pathak
[2002]; Peřinová et al. [1998] and references therein), and the interest in it
has been further enlightened in the recent past as many new applications of
quantum phase distribution and quantum phase fluctuation have been realized.

To be specific, this work is also motivated by the fact that recently several
applications of nonclassical states and quantum phase properties have been
reported. Specifically, squeezed states have played an important role in the
studies related to phase diffusion Banerjee and Srikanth [2007]; Banerjee et
al. [2007], the detection of gravitational waves in LIGO experiments Abbasi
and Golshan [2013]; Abbott et al. [2016b, a]. The rising demand for a single
photon source can be fulfilled by an antibunched light source Yuan et al.
[2002]. The study of quantum correlations is important both from the
perspective of pure and mixed states Chakrabarty et al. [2011]; Dhar et al.
[2013]; Banerjee et al. [2010a, b]. Entangled states are found to be useful in
both secure Ekert [1991] and insecure Bennett and Wiesner [1992]; Bennett et
al. [1993] quantum communication schemes. Stronger quantum correlation present
in the steerable states are used to ensure the security against all the side-
channel attacks on devices used in one-side (i.e., either preparation or
detector side) for quantum cryptography Branciard et al. [2012]. Quantum
supremacy in computation is established due to quantum algorithms for unsorted
database search Grover [1997], factorization and discrete logarithm problems
Shor [1999], and machine learning Biamonte et al. [2017] using essentially
nonclassical states. We may further stress on the recently reported
applications of quantum phase distribution and quantum phase fluctuation by
noting that these have applications in quantum random number generation Xu et
al. [2012]; Raffaelli et al. [2018], cryptanalysis of squeezed state based
continuous variable quantum cryptography Horak [2004], generation of solitons
in a Bose-Einstein condensate Denschlag et al. [2000], storage and retrieval
of information from Rydberg atom Ahn et al. [2000], in phase encoding quantum
cryptography Gisin et al. [2002], phase imaging of cells and tissues for
biomedical application Park et al. [2018]; as well as have importance in
determining the value of transition temperature for superconductors Emery and
Kivelson [1995].

Now to achieve the above advantages of the nonclassical states, we need to
produce these states via the schemes of quantum state engineering. For the
same, there are some distinct theoretical tools, like quantum scissoring
Miranowicz et al. [2001], hole-burning Escher et al. [2004]; Gerry and
Benmoussa [2002]; Malpani et al. [2020a] or filtering out a particular Fock
state from the photon number distribution Meher and Sivakumar [2018], applying
non-Gaussianity inducing operations Agarwal [2013]. However, these distinct
mechanisms are experimentally realized primarily by appropriately using beam
splitters, mirrors, and single photon detectors or single photon counting
module. Without going into finer details of the optical realization of quantum
state engineering tools, we may note that these tools can be used to generate
various nonclassical states, e.g., DFS De Oliveira et al. [1990], PADFS
Malpani et al. [2019a], PSDFS Malpani et al. [2019a], photon added squeezed
coherent state Thapliyal et al. [2017b], photon subtracted squeezed coherent
state Thapliyal et al. [2017b], number state filtered coherent state Meher and
Sivakumar [2018]. Some of these states, like photon added coherent state, have
already been realized experimentally Zavatta et al. [2004].

Many of the above mentioned engineered quantum states have already been
studied in detail. Primarily, three types of investigations have been
performed on the engineered quantum states- (i) study of various nonclassical
features of these states (and their variation with the state parameters) as
reflected through different witnesses of nonclassicality. Initially, such
studies were restricted to the lower-order nonclassical features. In the
recent past, various higher-order nonclassical features have been predicted
theoretically Alam et al. [2018a, b]; Pathak and Garcia [2006]; Pathak and
Verma [2010]; Verma et al. [2008]; Thapliyal et al. [2017b] and confirmed
experimentally (Hamar et al. [2014]; Peřina Jr et al. [2017] and references
therein) in quantum states generated in nonlinear optical processes. (ii)
Phase properties of the nonclassical states have been studied El-Orany et al.
[2000] by computing quantum phase fluctuations, phase dispersion, phase
distribution functions, etc., under various formalisms, like Susskind and
Glogower Susskind and Glogower [1964], Pegg-Barnett Pegg and Barnett [1989]
and Barnett-Pegg Barnett and Pegg [1986] formalisms. (iii) Various
applications of the engineered quantum states have been designed. Some of them
have already been mentioned.

Motivated by the above observations, in this thesis, we would like to perform
an investigation on nonclassical and phase properties of a particularly
interesting set of engineered quantum states which would have the flavor of
the first two facets of the studies mentioned above. Applications of the
engineered quantum states will also be discussed briefly, but will not be
investigated in detail. To begin with we would like to briefly describe the
physical and mathematical concepts used in this thesis, and that will be the
focus of the rest of this chapter.

The rest of this chapter is organized as follows. In Section 1.2, we will
briefly discuss quantum theory of radiation and introduce annihilation,
creation, and number operators as well as Fock and coherent states. In Section
1.3 this will be followed by an introduction to a set of quantum states which
will be studied in this thesis. After this, the notion of nonclassicality will
be introduced mathematically in Section 1.4, and a set of operational criteria
for observing nonclassical properties will be introduced in Section 1.5.
Subsequently, the parameters used for the study of quantum phase properties
will be introduced in Section 1.6. These witnesses of nonclassicality and the
parameters for the study of phase properties will be used in the subsequent
chapters, to investigate the nonclassical and phase properties of the quantum
states discussed in Section 1.3. Finally, the structure of the rest of the
thesis will be provided in Section 1.7.

###  1.2 Quantum theory of radiation field

Historically, quantum physics started with the ideas related to quanta of
radiation. To be precise, Planck’s work on black body radiation Planck [1901]
and Einstein’s explanation of photoelectric effect Einstein [1905] involved a
notion of quantized radiation field. In Planck’s work, light was considered to
be emitted from and absorbed by a black body in quanta; and in Einstein’s
work, it was also considered that the radiation field propagates from one
point to another as quanta. These initial works contributed a lot in the
development of quantum mechanics, but after the introduction of quantum
mechanics, in 1920s, in the initial days, most of the attention was given to
the quantization of matter. A quantum theory of radiation was introduced by
Dirac in 1927 Dirac [1927]. In what follows, we will describe it briefly as
this would form the backbone of the present thesis.

####  1.2.1 Creation and annihilation operator

Maxwell gave a classical description of electromagnetic field. But here the
objective is to study light and its properties apart from Maxwell’s equations.
So, to begin with, it would be reasonable to write Maxwell’s equations in free
space:

$\nabla\,.\,E=0,$ (1.1)

$\nabla\,.\,B=0,$ (1.2)

$\nabla\times E=-\frac{\partial B}{\partial t},$ (1.3)

and

$\nabla\times B=\frac{1}{c^{2}}\frac{\partial E}{\partial t},$ (1.4)

where $c=\frac{1}{\sqrt{\mu_{0}\epsilon_{0}}}$ is the speed of light without
any medium (i.e. vacuum). Using the set of above equations one can express
magnetic and electric fields in the form of the solution of wave equations,
like

$\nabla^{2}E-\frac{1}{c^{2}}\frac{\partial^{2}E}{\partial t^{2}}=0.$ (1.5)

The quantization of radiation field can be done by assuming a cavity of length
$L$ having a linearly polarized electric field whose direction of propagation
is in $z$ direction. Because of the linearity of the wave equation (1.5), we
are allowed to write the electric field in the form of linear combination of
all the normal modes as

$E_{x}\left(z,t\right)=\sum_{n}A_{n}q_{n}\left(t\right){\rm
sin}\left(k_{n}z\right),$ (1.6)

where $q_{n}$ being the amplitude of the $n$th normal mode with
$k_{n}=\frac{n\pi}{L},$$V$ is the volume of the resonator,
$A_{n}=\frac{2m_{n}\nu_{n}^{2}}{\epsilon_{0}V}$ with $\nu_{n}=ck_{n}$, and
$m_{n}$ is a constant (in the units of mass). With the help of this, we try to
form an analogy of radiation with mechanical oscillator. In analogy to Eq.
(1.6) we are able to write the corresponding magnetic field equation in cavity
as

$B_{y}\left(z,t\right)=\sum_{n}A_{n}\left(\frac{\dot{q_{n}}}{c^{2}k_{n}}\right){\rm
cos}\left(k_{n}z\right).$ (1.7)

So, the total energy of the field can be written as a classical Hamiltonian

$H=\frac{1}{2}\int_{V}d\tau\left(\epsilon_{0}E_{x}^{2}+\frac{1}{\mu_{0}}B_{y}^{2}\right),$
(1.8)

$H=\frac{1}{2}\sum_{n}\left(m_{n}\nu_{n}^{2}q_{n}^{2}+m_{n}\dot{q}_{n}^{2}\right)=\frac{1}{2}\sum_{n}\left(m_{n}\nu_{n}^{2}q_{n}^{2}+\frac{p_{n}^{2}}{m_{n}}\right),$
(1.9)

Substituting position and momentum variables by corresponding operators to
obtain the quantum mechanical Hamiltonian, where $p_{n}=m_{n}\dot{q_{n}}.$ The
position ($q_{n}$) and momentum ($p_{n}$) operators follow the commutation
relations

$\left[q_{n},p_{m}\right]=\iota\hbar\delta_{nm},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[q_{n},q_{m}\right]=\left[p_{n},p_{m}\right]=0,$

where $\hbar$ is the reduced Planck’s constant. Using these one may define a
new set of operators which can be analytically written as

$\hat{a_{n}}{\rm
exp\left[-\iota\nu_{n}t\right]=\frac{1}{\sqrt{2\hbar\text{$m_{n}$}\nu_{n}}}}\left(\text{$m_{n}$}\nu_{n}q_{n}+\iota
p_{n}\right)$ (1.10)

and

$\hat{a}_{n}^{\dagger}{\rm
exp\left[\iota\nu_{n}t\right]=\frac{1}{\sqrt{2\hbar\text{$m_{n}$}\nu_{n}}}}\left(\text{$m_{n}$}\nu_{n}q_{n}-\iota
p_{n}\right).$ (1.11)

Thus, the Hamiltonian can be written as

$H=\sum_{n}\hbar\nu_{n}\left(\hat{a_{n}}^{\dagger}\hat{a_{n}}+\frac{1}{2}\right),$
(1.12)

and the commutation relations

$\left[\hat{a}_{n},\hat{a}_{m}^{\dagger}\right]=\delta_{nm},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[\hat{a}_{n},\hat{a}_{m}\right]=\left[\hat{a}_{n}^{\dagger},\hat{a}_{m}^{\dagger}\right]=0,$

with corresponding electric and magnetic fields, as given by Eq. 1.1.27 in
Scully and Zubairy [1997]

$E\left(\overrightarrow{r},t\right)=\sum_{k}\hat{\epsilon_{k}}\xi_{k}\hat{a}_{k}{\rm
exp}\left[-\iota\nu_{k}t+\iota k.\overrightarrow{r}\right]+{\rm H.c}.$

and

$B\left(\overrightarrow{r},t\right)=\sum_{k}\frac{k\times\hat{\epsilon_{k}}}{\nu_{k}}\xi_{k}\hat{a}_{k}{\rm
exp}\left[-\iota\nu_{k}t+\iota k.\overrightarrow{r}\right]+{\rm H.c}.,$

where $\xi_{k}=\left(\frac{\hbar\nu_{k}}{\epsilon_{0}V}\right)^{1/2}$ is a
constant, and $\hat{\epsilon_{k}}$ is a unit polarization vector with the wave
vector $k$.

The above analysis shows that a single-mode field is identical to harmonic
oscillator. So, in the domain of quantum optics, harmonic oscillator system
plays an important role.

Notice that the quantum treatment of electromagnetic radiation hinges on
annihilation $\hat{a}$ (depletes photon) and creation $\hat{a}^{\dagger}$
(creates photon) operators. The annihilation operator $\hat{a}$ depletes one
quantum of energy and thus lowers down the system from harmonic oscillator
level $\left|n\right\rangle$ to $\left|n-1\right\rangle$, given by

$\hat{a}\left|n\right\rangle=\sqrt{n}\left|n-1\right\rangle.$ (1.13)

Here, $\left|n\right\rangle$ is called Fock or number state. Further, an
application of annihilation operator on vacuum leads to $0$, i.e.,
$\hat{a}\left|0\right\rangle=0$. The creation operator $\hat{a}^{\dagger}$
creates one quantum of energy by raising the state from $\left|n\right\rangle$
to $\left|n+1\right\rangle$. Therefore, the creation operator in the number
state can be represented as

$\hat{a}^{\dagger}\left|n\right\rangle=\sqrt{n+1}\left|n+1\right\rangle.$
(1.14)

If creation operator is applied to vacuum it creates a photon so these
operators enables one to write a Fock state ($\left|n\right\rangle$) in terms
of the vacuum state as

$\left|n\right\rangle=\frac{\left(\hat{a}^{\dagger}\right)^{n}}{\sqrt{n!}}\left|0\right\rangle.$

In the above, we have seen that annihilation and creation operators are the
important field operators and are required for the quantum description of
radiation. These operators can induce nonclassicality and non-Gaussianity when
applied on classical states Zavatta et al. [2004]; Agarwal [2013]. In the
present thesis, we study the role of these non-Gaussianity inducing operations
in controlling the nonclassicality of the quantum states which are often
already nonclassical. For instance, enhacement in squeezing in a nonclassical
state does not ensure advantage with respect to use as a single photon source
and vice-versa. In the following subsection, we will introduce a set of other
operators which can be expressed in terms of annihilation and creation
operators and which play a crucial role in our understanding of the quantum
states of radiation field.

####  1.2.2 Some more quantum operators of relevance

So far we have introduced some non-unitary operations (operations $\hat{O}$
which are not norm preserving and do not satisfy
$\hat{O}^{\dagger}=\hat{O}^{-1}$, where $\hat{O}^{\dagger}$ and $\hat{O}^{-1}$
are the Hermitian conjugate and inverse of $\hat{O}$, respectively), namely
photon addition and subtraction. We now aim to introduce some more unitary
operations important in the domain of quantum state engineering in general,
and in this thesis in particular. To begin with let us describe displacement
operator.

#####  1.2.2.1 Displacement operator

Displacement operator is a unitary operator. The mathematical from of
displacement operator is given as

$\hat{D}(\alpha)=\exp\left(\alpha\hat{a}^{\dagger}-\alpha^{\star}\hat{a}\right).$
(1.15)

This operator can be used as a tool to generate coherent state from vacuum.
Specifically, a coherent state $\left|\alpha\right\rangle$ is defined as
$\left|\alpha\right\rangle=\hat{D}(\alpha)\left|0\right\rangle.$

#####  1.2.2.2 Squeezing operator

The squeezing operator for a single mode of electromagnetic field is

$\hat{S}(z)=\exp\left(\frac{1}{2}\left(z^{\star}\hat{a}^{2}-z\hat{a}^{\dagger
2}\right)\right).$ (1.16)

The description of light is given by two quadratures namely phase $(X_{1})$
and amplitude $(X_{2})$ in the domain of quantum optics, mathematically
defined as

$\hat{X_{\theta}}=\frac{1}{\sqrt{2}}\left(i\hat{a}^{\dagger}\exp\left[\iota\theta\right]-\iota\hat{a}\exp\left[-\iota\theta\right]\right),$

The corresponding uncertainty of these two quadratures is observed by relation
$\Delta X_{1}\Delta X_{2}\geq\hbar\,/2$, where $\Delta X_{1}$ $\left(\Delta
X_{2}\right)$ is variance in the measured values of quadrature
$\hat{X_{1}}=\hat{X_{1}}\left(\theta=0\right)$
$\left(\hat{X_{2}}=\hat{X_{2}}\left(\theta=\frac{\pi}{2}\right)\right)$.
Specifically, $\Delta X_{i}=\sqrt{\left\langle
X_{i}^{2}\right\rangle-\left\langle X_{i}\right\rangle^{2}},$where
$\left\langle\hat{A}\right\rangle=\left\langle\psi\left|\hat{A}\right|\psi\right\rangle$
is the expectation value of the operator $A$ with respect to the quantum state
$\left|\psi\right\rangle$. Coherent state has an equal uncertainty in both
quadratures so they form a circle in the phase picture (shown in Fig. 1.1).
Least value of the variance for suitable $\theta$ is studied as principle
squeezing. With the advent of nonlinear optics, a very special branch of
optics, the uncertainty in one of the quadratures can be reduced at the cost
of increment in other quadrature’s uncertainty, which means that the circle
can be squeezed.

####  1.2.3 Eigen states of the field operators

Here, we will discuss eigen states of some of the operators we have
introduced. The eigenvalue equation can be defined as
$\hat{A}\lambda=a\lambda$ with eigen operator $\hat{A}$, eigenvalue $a$, and
eigen function $\lambda$. For example, Schrodinger equation
$H\psi_{i}=E_{i}\psi_{i}$ has Hamiltonian $H$ as eigen operator with eigen
functions $\psi_{i}$ and eigenvalues as allowed energy levels.

#####  1.2.3.1 Fock state: Eigen state of the number operator

In case of quantum optics or quantized light picture, photon number state is
known as number state. The single-mode photon number states are known Fock
states, and its ground state is defined as vacuum state. As the set of number
states are a full set of orthonormal basis so any quantum state can be written
in terms of these basis. The method of representing a quantum state as
superposition of number states is known as number state representation. Now
using Eq. (1.13) and (1.14), we can introduce an operator

$\hat{N}=\hat{a}^{\dagger}\hat{a}.$

which would satisfy the following eigen value equation

$\hat{N}\left|n\right\rangle=n\left|n\right\rangle.$

Clearly Fock states are the eigen states of the number operators and in
consistency with what has already been told, a Fock state
$\left|n\right\rangle$ represent a $n$ photon state.

#####  1.2.3.2 Coherent state: Eigen state of the annihilation operator

Coherent state Fox [2006] is considered as a state of the quantized
electromagnetic field which shows classical behaviour (specifically, beavior
closest to classical states). According to Erwin Schrodinger it is a minimum
uncertainty state, having same uncertainity in position and momentum
Schrödinger [1926]. According to Glauber, any of three mathematical
definitions described below can define coherent state:

(i) Eigen vectors of annihilation operator
$\hat{a}\left|\alpha\right\rangle=\alpha\left|\alpha\right\rangle$, $\alpha$
being a complex number.

(ii) Quantum states having minimum uncertainty $\Delta X_{1}=\Delta
X_{2}=1/\sqrt{2}$, with $X_{2}$ and $X_{1}$ as momentum and position
operators.

(iii) States realized by the application of the displacement operator
$D(\alpha)$ on the vacuum state. Thus, is also known as displaced vacuum state
and can be expressed as

$\left|\alpha\right\rangle=D\left(\alpha\right)\left|0\right\rangle.$

In Fock basis, it is expressed as infinite superposition of Fock state as

$\left|\alpha\right\rangle=\exp\left[-\frac{\mid\alpha\mid^{2}}{2}\right]\sum\limits_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle,$
(1.17)

where $\alpha$ is a complex number. Experimentally established state very
close to this coherent state was possible only after the successful
development of laser. Finally, one can easily see that
$\hat{a}\left|\alpha\right\rangle=\alpha\left|\alpha\right\rangle$ implies
$\left\langle\alpha\right|\hat{a}^{\dagger}=\left\langle\alpha\right|\alpha^{\star}$
and consequently
$\left\langle\alpha\right|\hat{a}^{\dagger}\hat{a}\left|\alpha\right\rangle=\left\langle\alpha\right|\hat{N}\left|\alpha\right\rangle=N=\left|\alpha\right|^{2}$
or average photon number in a coherent state is $\left|\alpha\right|^{2}$.

---
Figure 1.1:  Phase picture for coherent state and squeezed state.

###  1.3 Quantum states of our interest

In this section, we provide basic mathematical details of the set of
engineered quantum states studied in the present thesis.

####  1.3.1 Displaced Fock state

Displaced Fock state Satyanarayana [1985] are formed by applying displacement
operator on Fock state and thus a DFS is defined as

$\left|\phi\right\rangle=D\left(\alpha\right)\left|n\right\rangle.$

Analytically it is given as

$|\phi(n,\alpha)\rangle=\frac{1}{\sqrt{n!}}\sum_{p=0}^{n}{n\choose
p}(-\alpha^{\star})^{(n-p)}\exp\left(-\frac{\mid\alpha\mid^{2}}{2}\right)\sum_{m=0}^{\infty}\frac{\alpha^{m}}{m!}\sqrt{(m+p)!}|m+p\rangle.$
(1.18)

Various nonclassical properties of DFS are reported in literature De Oliveira
et al. [1990]; El-Orany et al. [2000]; Lvovsky and Babichev [2002]; Mendas and
Popovic [1993].

####  1.3.2 Photon added and photon subtracted displaced Fock state

Using DFS, we can define a $u$ photon added DFS (i.e., a PADFS) as

$\displaystyle|\psi_{+}(u,n,\alpha)\rangle$ $\displaystyle=$ $\displaystyle
N_{+}\hat{a}^{\dagger
u}|\phi(n,\alpha)\rangle=\frac{N_{+}}{\sqrt{n!}}\sum_{p=0}^{n}{n\choose
p}(-\alpha^{\star})^{(n-p)}\exp\left(-\frac{\mid\alpha\mid^{2}}{2}\right)\sum_{m=0}^{\infty}\frac{\alpha^{m}}{m!}$
(1.19) $\displaystyle\times$ $\displaystyle\sqrt{(m+p+u)!}|m+p+u\rangle.$

Similarly, a $v$ photon subtracted DFS (i.e., a PSDFS) can be expressed as

$\displaystyle|\psi_{-}(v,n,\alpha)\rangle$ $\displaystyle=$ $\displaystyle
N_{-}\hat{a}^{v}|\phi(n,\alpha)\rangle=\frac{N_{-}}{\sqrt{n!}}\sum_{p=0}^{n}{n\choose
p}(-\alpha^{\star})^{(n-p)}\exp\left(-\frac{\mid\alpha\mid^{2}}{2}\right)\sum_{m=0}^{\infty}\frac{\alpha^{m}}{m!}$
(1.20) $\displaystyle\times$
$\displaystyle\dfrac{(m+p)!}{\sqrt{(m+p-v)!}}|m+p-v\rangle,$

where $m$ and $p$ are the real integers. Here,

$\displaystyle N_{+}=\left[\frac{1}{n!}\sum_{p,p^{\prime}=0}^{n}{n\choose
p}{n\choose
p^{\prime}}(-\alpha^{\star})^{(n-p)}(-\alpha)^{(n-p^{\prime})}\exp\left[-\mid\alpha\mid^{2}\right]\sum_{m=0}^{\infty}\frac{\alpha^{m}(\alpha^{\star})^{m+p-p^{\prime}}(m+p+u)!}{m!(m+p-p^{\prime})!}\right]^{-\frac{1}{2}}$
(1.21)

and

$N_{-}=\left[\frac{1}{n!}\sum_{p,p^{\prime}=0}^{n}{n\choose p}{n\choose
p^{\prime}}\left(\alpha^{\star}\right)^{(n-p)}\left(-\alpha\right){}^{(n-p^{\prime})}\exp\left[-\mid\alpha\mid^{2}\right]\sum_{m=0}^{\infty}\frac{\alpha^{m}\left(\alpha^{\star}\right){}^{m+p-p^{\prime}}(m+p)!}{m!(m+p-p^{\prime})!(m+p-v)!}\right]^{-\frac{1}{2}}.$
(1.22)

are the normalization constants, and subscripts $+$ and $-$ correspond to
photon addition and subtraction. Thus, $|\psi_{+}(u,n,\alpha)\rangle$ and
$|\psi_{-}(v,n,\alpha)\rangle$ represent $u$ photon added DFS and $v$ photon
subtracted DFS, respectively, for the DFS which has been produced by
displacing the Fock state $|n\rangle$ by a displacement operator $D(\alpha)$
characterized by the complex parameter $\alpha.$ Clearly, the addition and the
subtraction of photons on the DFS can be mathematically viewed as application
of the creation and annihilation operators from the left on the Eq. (1.18).
Here, it may be noted that different well-known states can be obtained as
special cases of these two states. For example, using the notation introduced
above to define PADFS and PSDFS, we can describe a coherent state
$|\alpha\rangle$ as
$|\alpha\rangle=|\psi_{+}(0,\alpha,0)\rangle=|\psi_{-}(0,\alpha,0)\rangle$,
naturally, coherent state can be viewed as a special case of both PADFS and
PSDFS. Similarly, we can describe a single photon added coherent state as
$|\psi\rangle_{+1}=|\psi_{+}(1,\alpha,0)\rangle$, a Fock state as
$|n\rangle=|\psi_{+}(0,0,n)\rangle=|\psi_{-}(0,0,n)\rangle$ and a DFS as
$|\psi\rangle_{{\rm
DFS}}=|\psi_{+}(0,\alpha,n)\rangle=|\psi_{-}(0,\alpha,n)\rangle.$

####  1.3.3 Photon added then subtracted displaced Fock state

A PASDFS can be obtained by sequentially applying appropriate number of
annihilation (photon subtraction) and creation (photon addition) operators on
a DFS. Analytical expression for PASDFS (specifically, a $k$ photon added and
then $q$ photon subtracted DFS) in Fock basis can be shown to be

$\displaystyle|\psi(k,q,n,\alpha)\rangle$ $\displaystyle=$ $\displaystyle
N\hat{a}^{q}\hat{a}^{\dagger
k}|\psi(n,\alpha)\rangle=\frac{N}{\sqrt{n!}}\sum_{p=0}^{n}{n\choose
p}(-\alpha^{\star})^{(n-p)}\exp\left(-\frac{\mid\alpha\mid^{2}}{2}\right)$
(1.23) $\displaystyle\times$
$\displaystyle\sum_{m=0}^{\infty}\frac{\alpha^{m}\left(m+p+k\right)!}{m!\sqrt{(m+p+k-q)!}}|m+p+k-q\rangle,$

where

$N=\left[\frac{1}{n!}\sum_{p,p^{\prime}=0}^{n}{n\choose p}{n\choose
p^{\prime}}(-\alpha^{\star})^{(n-p)}(-\alpha)^{(n-p^{\prime})}\exp\left[-\mid\alpha\mid^{2}\right]\right]^{-\frac{1}{2}}$

is the normalization factor.

####  1.3.4 Even coherent state and states generated by holeburning on it

Even coherent state can be defined as the superposition of two coherent states
having opposite phase
($|\phi(\alpha)\rangle\propto|\alpha\rangle+|-\alpha\rangle$). The analytical
expression for ECS in number basis can be written as

$\begin{array}[]{lcl}|\phi(\alpha)\rangle&=&\frac{\,\exp\left[-\frac{\mid\alpha\mid^{2}}{2}\right]}{\sqrt{2\left(1+\exp\left[-2\mid\alpha\mid^{2}\right]\right)}}\sum\limits_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}\left(1+\left(-1\right)^{n}\right)|n\rangle.\end{array}$
(1.24)

The parameter $\alpha=|\alpha|\exp(i\theta)$, in Eq. (1.24), is complex in
general and $\theta$ is phase angle in the complex plane. Various schemes to
generate ECS are reported in Brune et al. [1992]; Ourjoumtsev et al. [2007];
Gerry [1993]. The nonclassical properties (witnessed through the antibunching
and squeezing criteria, $Q$ function, Wigner function, and photon number
distribution, etc.) of ECS have been studied in the recent past Gerry [1993].

#####  1.3.4.1 Vacuum filtered even coherent state

As mentioned above, experimentally, an ECS or a cat state can be generated in
various ways, and the same can be further engineered to produce a hole at
vacuum in its photon number distribution. Specifically, filtration of vacuum
will burn a hole at $n=0$ and produce VFECS, which can be described in Fock
basis as

$\begin{array}[]{lcl}|\phi_{1}(\alpha)\rangle&=&N_{{\rm
VFECS}}\sum\limits_{n=0,\,n\neq
0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}\left(1+\left(-1\right)^{n}\right)|n\rangle,\end{array}$
(1.25)

where

$\begin{array}[]{lcl}N_{{\rm VFECS}}&=&\\{4{\rm
cosh}\left(\mid\alpha\mid^{2}\right)-1\\}^{-1/2}\end{array}$ (1.26)

is the normalization constant. For simplicity, we may expand Eq. (1.25) as a
superposition of a standard ECS and a vacuum state as follows

$\begin{array}[]{lcl}|\phi_{1}(\alpha)\rangle&=&N_{{\rm
VFECS}}\left(\sum\limits_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}\left(1+\left(-1\right)^{n}\right)|n\rangle-2|0\rangle\right).\end{array}$
(1.27)

In what follows, Eq. (1.27) will be used to explore various nonclassical
features that may exist in VFECS.

#####  1.3.4.2 Photon added even coherent state

One can define a single photon added ECS as

$|\phi_{2}(\alpha)\rangle=N_{{\rm
PAECS}}\hat{a}^{\dagger}|\phi(\alpha)\rangle=N_{{\rm
PAECS}}\sum\limits_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}\left(1+\left(-1\right)^{n}\right)\sqrt{n+1}|n+1\rangle,$
(1.28)

where

$\begin{array}[]{lcl}N_{{\rm PAECS}}&=&\\{{\rm
cosh}\left(\mid\alpha\mid^{2}\right)+\mid\alpha\mid^{2}{\rm
sinh}\left(\mid\alpha\mid^{2}\right)\\}^{-1/2}/2\end{array}$ (1.29)

is the normalization constant for PAECS.

####  1.3.5 Binomial state and the states generated by holeburning on it

Binomial state is a finite superposition of Fock states having binomial photon
number distribution. It is quite similar to the coherent state which is the
linear combination of Fock states having the Poissonian photon number
distribution Stoler et al. [1985]. BS can be defined as

$\begin{array}[]{lcl}|p,M\rangle&=&\sum\limits_{n=0}^{M}\left[\frac{M!}{n!(M-n)!}p^{n}\left(1-p\right)^{M-n}\right]^{1/2}|n\rangle.\end{array}$
(1.30)

The binomial coefficient describes the presence of $n$ photons with
probability $p$ in $M$ number of ways. Recently, the study of nonclassical
properties of BS, specifically, antibunching, squeezing, HOSPS Verma et al.
[2008]; Verma and Pathak [2010]; Bazrafkan and Man’ko [2004], etc., have been
studied very extensively. However, no effort has yet been made to study the
nonclassical properties of VFBS and PABS.

#####  1.3.5.1 Vacuum filtered binomial state

The vacuum filtration of BS can be obtained by simply eliminating vacuum state
from the BS as

$\begin{array}[]{lcl}|p,M\rangle_{1}&=&N_{{\rm
VFBS}}\sum\limits_{n=0}^{M}\left[\frac{M!}{n!(M-n)!}p^{n}\left(1-p\right)^{M-n}\right]^{1/2}|n\rangle-
N_{VFBS}\left[\left(1-p\right)^{M}\right]^{1/2}|0\rangle,\end{array}$ (1.31)

where

$\begin{array}[]{lcl}N_{{\rm
VFBS}}&=&\\{1-\left(1-p\right)^{M}\\}^{-1/2}\end{array}$ (1.32)

is the normalization constant for the VFBS.

#####  1.3.5.2 Photon added binomial state

A hole at $n=0$ at a BS can also be introduced by the addition of a single
photon on the BS. A few steps of computation yield the desired expression for
PABS as

$\begin{array}[]{lcl}|p,M\rangle_{2}&=N_{{\rm
PABS}}&\sum\limits_{n=0}^{M}\left[\frac{M!(n+1)!}{\left(n!\right)^{2}(M-n)!}p^{n}\left(1-p\right)^{M-n}\right]^{1/2}|n+1\rangle,\end{array}$
(1.33)

where

$\begin{array}[]{lcl}N_{{\rm PABS}}&=&\left(1+Mp\right)^{-1/2}\end{array}$
(1.34)

is the normalization constant for single photon added BS.

####  1.3.6 Kerr state and the states generated by holeburning on it

A KS can be obtained when electromagnetic field in a coherent state interacts
with nonlinear medium with Kerr type nonlinearity Gerry and Grobe [1994]. This
interaction generates phase shifts which depend on the intensity. The
Hamiltonian involved in this process is given as

$H=\hbar\omega\hat{a}^{\dagger}\hat{a}+\hbar\chi\left(\hat{a}^{\dagger}\right)^{2}\left(\hat{a}\right)^{2},$
(1.35)

where $\chi$ depends on the third-order susceptibility of Kerr medium.
Explicit contribution of $H$ is ${\rm exp}\left[-i\chi n(n-1)\right]$. Thus,
the compact analytic form of the KS in the Fock basis can be given as

$\begin{array}[]{lcl}|\psi_{K}\left(n\right)\rangle&=&\sum\limits_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}\exp\left(-\frac{\mid\alpha\mid^{2}}{2}\right)\exp\left(-\iota\chi
n\left(n-1\right)\right)|n\rangle.\end{array}$ (1.36)

#####  1.3.6.1 Vacuum filtered Kerr state

Similarly, a VFKS, which can be obtained using the same quantum state
engineering process that leads to VFECS and VFBS, is given by

$\begin{array}[]{lcl}|\psi_{K}\left(n\right)\rangle_{1}&=&N_{{\rm
VFKS}}\left[\sum\limits_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}\exp\left(-\iota\chi
n\left(n-1\right)\right)|n\rangle-|0\rangle\right]\end{array},$ (1.37)

where

$\begin{array}[]{lcl}N_{{\rm
VFKS}}&=&{{\left(\exp\left[\mid\alpha\mid^{2}\right]-1\right)}^{-1/2}}\end{array}$
(1.38)

is the normalization constant for the VFKS.

#####  1.3.6.2 Photon added Kerr state

An addition of a photon to Kerr state would yield PAKS which can be expanded
in Fock basis as

$\begin{array}[]{lcl}|\psi_{K}\left(n\right)\rangle_{2}&=N_{{\rm
PAKS}}&\sum\limits_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}\exp\left(-\iota\chi
n\left(n-1\right)\right)\sqrt{\left(n+1\right)}|n+1\rangle,\end{array}$ (1.39)

where

$\begin{array}[]{lcl}N_{{\rm
PAKS}}&=&{\left(\exp\left[\mid\alpha\mid^{2}\right]\left(1+\mid\alpha\mid^{2}\right)\right)^{-1/2}}\end{array}$
(1.40)

is the normalization constant for the PAKS.

###  1.4 The notion of nonclassical states

Quantum states which do not have any classical analogue have been referred to
as nonclassical states Agarwal [2013]. In other words, states having their
$P$-distribution more singular than delta function or having negative values
are referred to as nonclassical states Dodonov and Renó [2003]. This idea was
possible only when Glauber and Sudarshan published papers in 1963 Sudarshan
[1963]; Glauber [1963b, a]. Sudarshan found a mathematical form to represent
any state in the coherent basis, mathematically given as

$\rho=\int
P\left(\alpha\right)\left|\alpha\right\rangle\left\langle\alpha\right|d^{2}\alpha,$

where $P\,(\alpha)$ is known as Glauber-Sudarshan $P$-function, which follows
normalization condition as $\int dP(\alpha)=1$, but it may have negative
values. Thus, it is defined as quasidistribution function or quasiprobability
distribution. When $P\,(\alpha)$ attains a positive probability density
function, immediately it indicates that the state is classical. This leads to
the definition of nonclassicality. If an arbitrary quantum state is failed to
represent as mixture of coherent states, that is known as nonclassical state.
To establish quantum supremacy, these nonclassical states play very essential
role, for instance in theses states are useful in establishing quantum
supremacy of quantum information processing, quantum communication, etc.
Although $P$-function is not reconstructable for any arbitrary state yet it
has been of major interest as it provides an important signature of
nonclassicality. The negativity (positivity or non-negativity) of the
$P$-function essentially provides the nonclassical (classical) behavior of the
state under consideration. The experimental difficulty associated with the
easurement of $P$-function in its reconstruction led to various feasible
substitutes as nonclassicality witnesses. Here, we list some of those
nonclassicalty witnesses. These witnesses can be viewed as operational
criterion of nonclassicality.

###  1.5 Nonclassical states: witnesses and measures

Using nonclassical states, the essence of quantum theory of light can be
understood. There are various tools for characterization of nonclassical
states. In this section, some tools are described which are used to
characterize such states. If historically seen, the first such approach was
aimed to check the deviation from Poissonian photon statistics, the second is
to evaluate the volume of the negative part of the quasiprobability
distribution in the phase space, etc. An infinite set of moments based
criteria is available in literature which is used as witness of
nonclassicality equivalent to $P$-function Shchukin and Vogel [2005b]. Any
subset of this infinite set may detect nonclassicality or fail to do so.
Example of these witnesses are lower- and higher-order antibunching, sub-
Poissonian photon statistics, squeezing as well as Mandel $Q_{M}$ parameter,
Klyshko’s, Vogel’s, and Agarwal-Tara’s criteria, $Q$ function, etc. To
quantify the amount of nonclassicality, a number of measures have been
proposed, like linear entropy, Wigner volume, concurrence and many more. A
small description of these criteria is given here.

####  1.5.1 Witnesses of nonclassicality

#####  1.5.1.1 Lower- and higher-order antibunching

In this section, we study lower- and higher-order antibunching. To do so, we
use the following criterion of $(l-1)$th order antibunching (Pathak and Garcia
[2006] and references therein) in terms of nonclassicality witness ($d(l-1)$)
as

$d(l-1)=\langle\hat{a}^{\dagger
l}\hat{a}^{l}\rangle-\langle\hat{a}^{\dagger}\hat{a}\rangle^{l}<0.$ (1.41)

This nonclassical feature characterizes suitability of the quantum state to be
used as single photon source as the negative values of $d(l-1)$ parameter show
that the probability of photons coming bunched is less compared to that of
coming independently. The signature of lower-order antibunching can be
obtained as a special case of Eq. (1.41) for $l=2$, and that for $l\geq 3$,
the negative values of $d(l-1)$ correspond to higher-order antibunching of
$(l-1)$th order. Figure 1.2 illustrates the scheme for studying antibunching
experimentally (corresponds to $l=2$). For higher values of $l$, we require
more beamsplitters and APDs. On these cascaded beamplitters signal is mixed
with vacuum and measured higher-order correlation Avenhaus et al. [2010].

---
Figure 1.2: Hanbury Brown and Twiss setup. Here, APD is avalanche photo diode.

#####  1.5.1.2 Lower- and higher-order sub-Poissionian photon statistics

The lower-order counterparts of antibunching and sub-Poissonian photon
statistics are closely associated as the presence of latter ensures the
possibility of observing former (see Thapliyal et al. [2014a, 2017b] for a
detailed discussion). However, these two nonclassical features were shown to
be independent phenomena in the past (Thapliyal et al. [2014a, 2017b] and
references therein). Higher-order counterpart of sub-Poissonian photon
statistics can be introduced as

$\begin{array}[]{lcccc}\mathcal{D}_{h}(l-1)&=&\sum\limits_{e=0}^{l}\sum\limits_{f=1}^{e}S_{2}(e,\,f)\,^{l}C_{e}\,\left(-1\right)^{e}d(f-1)\langle
N\rangle^{l-e}&<&0,\end{array}$ (1.42)

where $S_{2}(e,\,f)$ is the Stirling number of second kind, and
$\,{}^{l}C_{e}$ is the usual binomial coefficient.

#####  1.5.1.3 Higher-order squeezing

As mentioned beforehand, the squeezing of quadrature is defined in terms of
variance in the measured values of the quadrature (say, position or momentum)
below the corresponding value for the coherent state, i.e., minimum
uncertainty state. The higher-order counterpart of squeezing is studied in two
ways, namely Hong-Mandel and Hillery-type squeezing Hong and Mandel [1985b,
a]; Hillery [1987a]. Specifically, the idea of the higher-order squeezing
originated from the pioneering work of Hong and Mandel Hong and Mandel [1985b,
a], who generalized the lower-order squeezing using the higher-order moments
of field quadrature. According to the Hong-Mandel criterion, the $l$th order
squeezing can be observed if the $l$th moment (for even values of $l>2$) of a
field quadrature operator is less than the corresponding coherent state value.
The condition of Hong-Mandel type higher-order squeezing is given as follows
Hong and Mandel [1985b, a]

$S(l)=\frac{\langle(\Delta
X)^{l}\rangle-\left(\frac{1}{2}\right)_{\frac{l}{2}}}{\left(\frac{1}{2}\right)_{\frac{l}{2}}}<0,$
(1.43)

Here, $S_{(l)}$ is higher-order squeezing, $\Delta X$ is the quadrature (
$\Delta X_{1}$) as defined in Section 1.2.2. Further, $(x)_{l}$ is
conventional Pochhammer symbol. The inequality in Eq. (1.43) can also be
rewritten as

$\begin{array}[]{lcl}\langle(\Delta
X)^{l}\rangle&<&\left(\frac{1}{2}\right)_{\frac{l}{2}}=\frac{1}{2^{\frac{l}{2}}}(l-1)!!\end{array}$
(1.44)

with

$\langle\left(\text{$\Delta$}\text{X}\right)^{l}\rangle=\sum\limits_{r=0}^{l}\sum\limits_{i=0}^{\frac{r}{2}}\sum\limits_{k=0}^{r-2i}\left(-1\right)^{r}\frac{1}{2^{\frac{l}{2}}}\left(2i-1\right)!^{2i}C_{k}{}^{l}C_{r}{}^{r}C_{2i}\langle\hat{a}^{\dagger}+\hat{a}\rangle^{l-r}\langle\hat{a}^{\dagger
k}\hat{a}^{r-2i-k}\rangle.$ (1.45)

#####  1.5.1.4 Klyshko’s criterion

This criterion is relatively simpler as to calculate this witness of
nonclssicality, only three consecutive probability terms are required rather
than all the terms. Negative values of $B(m)$ are symbol of nonclassicality
present in the state. Klyshko introduced this criterion Klyshko [1996] to
investigate the nonclassical property using only three successive photon-
number probabilities. In terms of the photon-number probability $p_{m}=\langle
m|\rho|m\rangle$ of the state with density matrix $\rho$, the Klyshko’s
criterion in the form of an inequality can be written as

$B(m)=(m+2)p_{m}p_{m+2}-(m+1)\left(p_{m+1}\right)^{2}<0.$ (1.46)

#####  1.5.1.5 Vogel’s criterion

The moments-based nonclassicality criterion of the previous subsection was
later extended to Vogel’s nonclassicality criterion Shchukin and Vogel [2005b]
in terms of matrix of moments as

$v=\left[\begin{array}[]{ccc}1&\langle\hat{a}\rangle&\langle\hat{a}^{\dagger}\rangle\\\
\langle\hat{a}^{\dagger}\rangle&\langle\hat{a}^{\dagger}\hat{a}\rangle&\langle\hat{a}^{\dagger
2}\rangle\\\
\langle\hat{a}\rangle&\langle\hat{a}^{2}\rangle&\langle\hat{a}^{\dagger}\hat{a}\rangle\end{array}\right].$
(1.47)

The negative value of the determinant $dv$ of matrix $v$ in Eq. (1.47) is
signature of nonclassicality.

#####  1.5.1.6 Agarwal-Tara’s criterion

There were certain quantum states having negative $P$-function yet showing no
squeezing and sub-Poissonian behavior, to witness the nonclassicality residing
in those particular types of states Agarwal and Tara Agarwal and Tara [1992]
introduced this criterion which is again a moments based criterion. This can
be written in a matrix form and expressed as

$A_{3}=\dfrac{\det m^{(3)}}{\det\mu^{(3)}-\det m^{(3)}}<0,$ (1.48)

where

$m^{(3)}=\begin{bmatrix}1&m_{1}&m_{2}\\\ m_{1}&m_{2}&m_{3}\\\
m_{2}&m_{3}&m_{4}\end{bmatrix}$

and

$\mu^{(3)}=\begin{bmatrix}1&\mu_{1}&\mu_{2}\\\ \mu_{1}&\mu_{2}&\mu_{3}\\\
\mu_{2}&\mu_{3}&\mu_{4}\end{bmatrix}.$

The matrix elements are defined as $m_{i}=\langle\hat{a}^{\dagger
i}\hat{a}^{i}\rangle$ and
$\mu_{j}=(\langle\hat{a}^{\dagger}\hat{a}\rangle)^{j}=(m_{1})^{j}$.

#####  1.5.1.7 Mandel $Q_{M}$ parameter

The Mandel $Q_{M}$ parameter Mandel [1979] illustrates the nonclassicality
through photon number distribution in a quantum state. The Mandel $Q_{M}$
parameter is defined as

$Q_{M}=\frac{\langle(\hat{a}^{\dagger}\hat{a})^{2}\rangle-\langle\hat{a}^{\dagger}\hat{a}\rangle^{2}-\langle\hat{a}^{\dagger}\hat{a}\rangle}{\langle\hat{a}^{\dagger}\hat{a}\rangle}.$
(1.49)

The negative values of $Q_{M}$ parameter essentially indicate the negativity
for $P$-function and so it gives a witness for nonclassicality. For the
Poissonian statistics it becomes 0, while for the sub-Poissonian (super-
Poissonian) photon statistics it has negative (positive) values.

####  1.5.2 Other quasiprobability distributions

Inability to give a phase space description of quantum mechanics is exploited
in terms of quasiprobability distributions (Thapliyal et al. [2015] and
references therein). Later, it was found that they are useful as witnesses of
nonclassicality. These real and normalized quasiprobability distributions
allow to calculate the expectation value of an operator as any classical
probability distribution. One such quasiprobability distributions is $Q$
function Husimi [1940], and zeros of this function are signature of
nonclassicality. Another example is Wigner Wigner [1932] function whose
negative values corresponds to the nonclassicality.

#####  1.5.2.1 $Q$ function

$Q$ function Husimi [1940] is defined as

$Q=\dfrac{1}{\pi}\langle\beta|\rho|\beta\rangle,$ (1.50)

where $|\beta\rangle$ is the coherent state (1.17).

#####  1.5.2.2 Wigner function

Another quasiprobability distribution is Wigner function formulated by Wigner
in 1932 Wigner [1932] in the early stage of quantum mechanics, the motive was
to connect the wavefunction approach to a probability distribution in phase
space. Negativity of Wigner function represents the nonclassicality present in
an arbitrary quantum state. Also the ability to reconstruct the Wigner
function experimentally makes this approach more impactful than any other
approach. Specifically, Wigner function obtained through optical tomography
can be used to obtain other quasidistributions, however, Wigner function is
stronger witness of nonclassicality than $Q$ function while is not singular
like $P$-function. Mathematically, it is expressed as

$W\left(\gamma,\gamma^{\star}\right)=A\exp\left[-2\left|\gamma\right|^{2}\right]\int
d^{2}\lambda\langle-\lambda|\rho|\lambda\rangle\exp\left[-2\left(\gamma^{\star}\lambda-\gamma\lambda^{\star}\right)\right].$
(1.51)

The zeros of $Q$ function while the negativity of $P$-function and Wigner
function correspond to the nonclassical behavior of any arbitrary quantum
state. It is worth stressing here that only $P$-function is both necessary and
sufficient criterion of nonclassicality, while rest of the quasidistribution
functions are only sufficient.

####  1.5.3 Measures of nonclassicality

In the above section, we have seen that there exist numerous criteria of
nonclassicality. However, most of these criteria only witness the
nonclassicality. They do not provide any quantification of the
nonclassicality. Except $P$-function and infinite set of vogel’s criteria, all
other criteria are sufficient but not necessary. However, many efforts have
been made for the quantification of nonclassicality, e.g., in 1987, a
distance-based measure of nonclassicality was introduced by Hillery Hillery
[1987b]. A trace norm based measure Mari et al. [2011] was introduced by Mari
et al., for the set of all states having the positive Wigner function. In
1991, Lee gave a measure of nonclassicality known as nonclassical depth Lee
[1991]. However, in this work, we will not study these measures. There are
certain measures those can be exploited in terms of entanglement, like linear
entropy Wei et al. [2003], which we will use for our calculations and the same
is described below.

#####  1.5.3.1 Linear entropy

In 2005, a measure of nonclassicality was proposed as entanglement potential,
which is the amount of entanglement in two output ports of a beam splitter
with the quantum state $\rho_{in}$ and vacuum $|0\rangle\langle 0|$ sent
through two input ports Asbóth et al. [2005]. The amount of entanglement
quantifies the amount of nonclassicality in the input quantum state as
classical state can not generate entanglement in the output. The post beam
splitter state can be obtained as
$\rho_{out}=U\left(\rho_{in}\otimes|0\rangle\langle 0|\right)U^{\dagger}$ with
$U=\exp[-iH\theta]$, where
$H=(\hat{a}^{\dagger}\hat{b}+\hat{a}\hat{b}^{\dagger})/2$, and
$\hat{a}^{\dagger}\,(\hat{a})$, $\hat{b}^{\dagger}\,(\hat{b})$ are the
creation (annihilation) operators of the input modes. For example, considering
quantum state ($|\psi\rangle=\sum\limits_{n=0}^{\infty}c_{n}|n\rangle$) and a
vacuum state $|0\rangle$ as input states, we can write the analytic expression
of the two-mode output state as

$|\phi\rangle=U\left(|\psi\rangle\otimes|0\rangle\right)\equiv
U|\psi,0\rangle=\sum_{n=0}^{\infty}\,\frac{c_{n}}{2^{n/2}}\sum_{j=0}^{n}\sqrt{{}^{n}C_{j}}\,\,|j,\,n-j\rangle.$
(1.52)

We can measure the amount of entanglement in the output state to quantify the
amount of input nonclassicality in $|\psi\rangle$. Here, we use linear entropy
of single mode subsystem (obtained after tracing over the other subsystem) as
entanglement potential. The linear entropy for an arbitrary bipartite state
$\rho_{AB}$ is defined as Wei et al. [2003]

$\mathcal{L}=1-{\rm Tr}\left(\rho_{B}^{2}\right),$ (1.53)

where $\rho_{B}$ is obtained by tracing over subsystem $A$.

###  1.6 Analytic tools for the study of phase properties of nonclassical
states

In this section, we aim to introduce the parameters that are used to study
phase properties of a given quantum state under consideration in this section.

####  1.6.1 Phase distribution function

A distribution function allows us to calculate expectation values of an
operator analogous to that from the corresponding density matrix. Phase
distribution function for a given density operator Banerjee and Srikanth
[2007]; Agarwal et al. [1992] can be defined as

$P_{\theta}=\frac{1}{2\pi}\langle\theta|\varrho|\theta\rangle,$ (1.54)

where the phase state $|\theta\rangle$, complementary to the number state
$|n\rangle$, is defined Agarwal et al. [1992] as

$|\theta\rangle=\sum_{n=0}^{\infty}e^{\iota n\theta}|n\rangle.$ (1.55)

####  1.6.2 Phase dispersion

A known application of phase distribution function (1.54) is that it can be
used to quantify the quantum phase fluctuation. Although the variance is also
used occasionally as a measure of phase fluctuation, it has a drawback that it
depends on the origin of phase integration Banerjee and Srikanth [2007]. A
measure of phase fluctuation, free from this problem, is phase dispersion
Peřinová et al. [1998] defined as

$D=1-\left|\intop_{-\pi}^{\pi}d\theta\exp\left[-\iota\theta\right]P_{\theta}\right|^{2}.$
(1.56)

####  1.6.3 Angular Q function

Analogous to the phase distribution $P_{\theta}$, phase distributions are also
defined as radius integrated quasidistribution functions which are used as the
witnesses for quantumness Thapliyal et al. [2015]. One such phase distribution
function based on the angular part of the $Q$ function is studied in Leonhardt
and Paul [1993]; Leonhardt et al. [1995]. Specifically, the angular $Q$
function is defined as

$Q_{\theta_{1}}=\intop_{0}^{\infty}Q\left(\beta,\beta^{\star}\right)\left|\beta\right|d\left|\beta\right|,$
(1.57)

where the $Q$ function Husimi [1940] is defined in Eq. (1.50).

####  1.6.4 Phase fluctuation

In attempts to get rid of the limitations of the Hermitian phase operator of
Dirac Dirac [1927], Louisell Louisell [1963] first mentioned that bare phase
operator should be replaced by periodic functions. As a consequence, sine
$(\mathcal{\hat{S}})$ and cosine $(\hat{\mathcal{C}})$ operators appeared,
explicit forms of these operators were given by Susskind and Glogower Susskind
and Glogower [1964], and further modified by Barnett and Pegg Barnett and Pegg
[1986] as

$\mathcal{\hat{S}}=\frac{\hat{a}-\hat{a}^{\dagger}}{2\iota\left(\bar{N}+\frac{1}{2}\right)^{\frac{1}{2}}}$
(1.58)

and

$\hat{\mathcal{C}}=\frac{\hat{a}+\hat{a}^{\dagger}}{2\left(\bar{N}+\frac{1}{2}\right)^{\frac{1}{2}}}.$
(1.59)

Here, $\bar{N}$ is the average number of photons in the measured field, and
here we refrain our discussion to Barnett and Pegg sine and cosine operators
Barnett and Pegg [1986]. Carruthers and Nieto Carruthers and Nieto [1968] have
introduced three quantum phase fluctuation parameters in terms of sine and
cosine operators

$U=\left(\Delta
N\right)^{2}\left[\left(\Delta\mathcal{S}\right)^{2}+\left(\Delta\mathcal{C}\right)^{2}\right]/\left[\langle\mathcal{\hat{S}}\rangle^{2}+\langle\hat{\mathcal{C}}\rangle^{2}\right],$
(1.60) $S=\left(\Delta N\right)^{2}\left(\Delta\mathcal{S}\right)^{2},$ (1.61)

and

$Q=S/\langle\hat{\mathcal{C}}\rangle^{2}.$ (1.62)

These three phase fluctuation parameters $U$, $S$ and $Q$ show phase
properties of PADFS and PSDFS, while $U$ parameter is shown relevant as a
witness of nonclassicality (antibunching).

####  1.6.5 Quantum phase estimation parameter

Quantum phase estimation is performed by sending the input state through a
Mach-Zehnder interferometer and applying the phase to be determined ($\phi$)
on one of the arms of the interferometer. To study the phase estimation using
Mach-Zehnder interferometer, angular momentum operators Sanders and Milburn
[1995]; Demkowicz-Dobrzański et al. [2015], defined as

$\hat{J_{x}}=\frac{1}{2}\left(\hat{a}^{\dagger}\hat{b}+\hat{b}^{\dagger}\hat{a}\right),$
(1.63)
$\hat{J_{y}}=\frac{\iota}{2}\left(\hat{b}^{\dagger}\hat{a}-\hat{a}^{\dagger}\hat{b}\right),$
(1.64)

and

$\hat{J_{z}}=\frac{1}{2}\left(\hat{a}^{\dagger}\hat{a}-\hat{b}^{\dagger}\hat{b}\right),$
(1.65)

are used. Here, $\hat{a}$ and $\hat{b}$ are the annihilation operators for the
modes corresponding to two input ports of the Mach-Zehnder interferometer. The
average value of $\hat{J_{z}}$ operator in the output of the Mach-Zehnder
interferometer, which is one-half of the difference of photon numbers in the
two output ports (1.65), can be written as

$\langle\hat{J_{z}}\rangle=\cos\text{$\phi$}\langle\hat{J_{z}}\rangle_{in}-\sin\text{$\phi$}\langle\hat{J_{x}}\rangle_{in}.$
(1.66)

Therefore, variance in the measured value of operator $\hat{J_{z}}$ can be
computed as

$\left(\Delta
J_{z}\right)^{2}=\cos^{2}\phi\left(\Delta{J_{z}}\right)_{in}^{2}+\sin^{2}\phi\left(\Delta{J_{x}}\right)_{in}^{2}-2\sin\text{$\phi\,\cos\phi\,$cov}\left(\hat{J_{x}},\hat{J_{z}}\right)_{in},$
(1.67)

where covariance of the two observables is defined as

$\text{{\rm
cov}}\left(\hat{J_{x}},\hat{J_{z}}\right)=\frac{1}{2}\langle\hat{J_{x}}\hat{J_{z}}+\hat{J_{z}}\hat{J_{x}}\rangle-\langle\hat{J_{x}}\rangle\langle\hat{J_{z}}\rangle.$
(1.68)

This allows us to quantify precision in phase estimation Demkowicz-Dobrzański
et al. [2015] as

$\Delta\phi=\frac{\Delta{J_{z}}}{\left|\frac{d\langle\hat{J_{z}}\rangle}{d\phi}\right|}.$
(1.69)

Before we proceed further and conclude this chapter by noting the structure of
the rest of the thesis, it would be apt to to note that there exist various
methods of quantum state engineering (some of which have already been
mentioned) and photon addition, subtraction, filteration, punching etc., which
can be viewed as examples of quantum state engineering processes. In rest of
the thesis these processes will be studied with detail.

###  1.7 Structure of the rest of the thesis

This thesis has 6 chapters. The next 4 chapters are focused on the study of
nonclassical and phase properties of the engineered quantum states and the
last chapter is dedicated to conclusion. These chapters and thus the rest of
this thesis is organized as follows.

In Chapter 1, in Section 1.3, quantum states of our interest (i.e., PADFS and
PSDFS) have been introduced in detail. In Chapter 2, in Section 2.3, the
analytical expressions of various witnesses of nonclassicality are reported.
Further, the existence of various lower- and higher-order nonclassical
features in PADFS and PSDFS are shown through a set of plots. Finally, we
conclude in Section 2.4.

In Chapter 3, in Section 3.3, we investigate the phase properties of PADFS and
PSDFS from a number of perspectives. Finally, the chapter is concluded in
Section 3.4.

In Chapter 4, we describe the quantum state of interest (i.e., PASDFS) in Fock
basis and calculate the analytic expressions for the higher-order moments of
the relevant field operators for this state. In Section 4.3, we investigate
the possibilities of witnessing various nonclassical features in PASDFS and
its limiting cases by using a set of moments-based criteria for
nonclassicality. Variations of nonclassical features (witnessed through
different criteria) with various physical parameters are also discussed here.
In Section 4.4, phase properties of PASDFS are studied. $Q$ function for
PASDFS is obtained in Section 4.5. Finally, we conclude in Section 4.6.

In Chapter 5, in Section 5.2, we have introduced the quantum states of our
interest which include ECS, BS, KS, VFECS, VFBS, VFKS, PAECS, PABS, and PAKS.
In Section 5.3, we have investigated the nonclassical properties of these
states using various witnesses of lower- and higher-order noncassicality as
well as a measure of nonclassicality. Specifically, in this section, we have
compared nonclassicality features found in vacuum filtered and single photon
added versions of the states of our interest using the witnesses of Higher-
order antibunching (HOA), Higher-order squeezing (HOS) and Higher-order sub-
Poissonian photon statistics (HOSPS). Finally, in Section 5.5, the results are
analyzed, and the chapter is concluded.

Finally, the thesis is concluded in Chapter 6, where we have summarized the
findings reported in Chapter 2-5 and have emphasized on the main conclusion of
the present thesis. We have also discussed the scopes of future work.

##  Chapter 2 Lower-and higher-order nonclassical properties of photon added
and subtracted displaced Fock state

In this chapter, which is based on Malpani et al. [2019a], we aim to study the
nonclassical properties of the PADFS and PSDFS (which are already introduced
in Section 1.3.2) using the witnesses of nonclassicality introduced in Section
1.5.1.

###  2.1 Introduction

As we have mentioned in Chapter 1, with the advent of quantum state
engineering Vogel et al. [1993]; Sperling et al. [2014]; Miranowicz and
Leonski [2004]; Marchiolli and José [2004] and quantum information processing
(Pathak [2013] and references therein), the study of nonclassical properties
of engineered quantum states have become a very important field. Quantum state
engineering is helpful in realizing non-Gaussianity inducing operations, like
photon addition and subtraction Zavatta et al. [2004]; Podoshvedov [2014].
Keeping this in mind, in what follows, in this chapter, we aim to study the
nonclassical properties of a set of engineered quantum states (both photon
added and subtracted) which can be produced by using the above mentioned
techniques.

---
Figure 2.1:  A schematic diagram for the generation of PSDFS (in (a) and (b))
and PADFS (in (c) and (d)). In (a) and (c) ((b) and (d)), single-mode (two-
mode) squeezed vacuum is used for generation of the desired state. Here, NLC
corresponds to nonlinear crystal and D1 and D2 are photon number resolving
detectors.

It is already mentioned in Chapter 1 that a state having a negative
$P$-function is referred to as a nonclassical state. Such a state cannot be
expressed as a mixture of coherent states and does not possess a classical
analogue. In contrast to these states, coherent states are classical, but
neither their finite dimensional versions Miranowicz and Leonski [2004]; Alam
et al. [2017c] nor their generalized versions are classical Satyanarayana
[1985]; Thapliyal et al. [2016]; Thapliyal et al. [2015]; Banerjee and
Srikanth [2007]. Here, we would like to focus on photon added and subtracted
versions of a particular type of generalized coherent state, which is also
referred to as the displaced Fock state (DFS). To be precise, state of the
form $|\psi\rangle=D(\alpha)|n\rangle$, where $D(\alpha)$ is the displacement
operator with Fock state $|n\rangle$, is referred to as generalized coherent
state (see Section 1.3 of Chapter 1), as this state is reduced to a coherent
state in the limit $n=0$. However, from the structure of the state it seems
more appropriate to call this state as the DFS, and this seems to be the
nomenclature usually adopted in the literature Keil et al. [2011]; Wunsche
[1991]; Blažek [1994]; Moya-Cessa [1995]. In some other works, it is referred
to as displaced number state Ziesel et al. [2013]; De Oliveira et al. [2006];
Dodonov and De Souza [2005], but all these names are equivalent; and in what
follows, we will refer to it as DFS. This is an extremely interesting quantum
state for various reasons. Specifically, its relevance in various areas of
quantum optics is known. For example, in the context of cavity QED, it
constitutes the eigenstates of the Jaynes-Cummings systems with coherently
driven atoms Alsing et al. [1992]. Naturally, various lower-order nonclassical
properties and a set of other quantum features of DFS have already been
studied. Specifically, quasiprobability distributions of DFS were studied in
Wunsche [1991], phase fluctuations of DFS was investigated in Zheng-Feng
[1992], decoherence of superposition of DFS was discussed in Dodonov and De
Souza [2005], $Q$ function, Wigner function and probability distribution of
DFS were studied in De Oliveira et al. [1990], Pancharatnam phase of DFS has
been studied in Mendas and Popovic [1993]. Further, in the context of non-
optical DFS, various possibilities of generating DFS from Fock states by using
a general driven time-dependent oscillator has been discussed in Lo [1991];
and in the trapped-ion system, quantum interference effects have been studied
for the superposition of DFS Marchiolli and José [2004]. Thus, DFS seems to be
a well studied quantum state, but it may be noted that a little effort has yet
been made to study higher-order nonclassical properties of DFS. It is a
relevant observation as in the recent past, it has been observed that higher-
order nonclassicality has various applications Hillery et al. [1999]; Banerjee
et al. [2018]; Sharma et al. [2017]; Thapliyal et al. [2017c], and it can be
used to detect the existence of weak nonclassical characters Thapliyal et al.
[2014a]; Thapliyal et al. [2014b]; Verma and Pathak [2010]; Thapliyal et al.
[2017b]; Alam and Mandal [2016a]; Alam et al. [2015]; Thapliyal et al.
[2017a]; Prakash and Mishra [2006]; Prakash et al. [2010]; Das et al. [2018].
Further, various higher-order nonclassical features have been experimentally
detected Allevi et al. [2012b, a]; Avenhaus et al. [2010]; Peřina Jr et al.
[2017]. However, we do not want to restrict to DFS, rather we wish to focus on
lower- and higher-order nonclassical properties of a set of even more general
states, namely photon added DFS (PADFS) and photon subtracted DFS (PSDFS). The
general nature of these states can be visualized easily, as in the special
case that no photon is added (subtracted) PADFS (PSDFS) would reduce to DFS.
Further, for $n=0$, PADFS would reduce to photon added coherent state which
has been widely studied (Agarwal and Tara [1991]; Verma et al. [2008];
Thapliyal et al. [2017b] and references therein) and experimentally realized
Zavatta et al. [2004, 2005]. Here it is worth noting that DFS has also been
generated experimentally by superposing a Fock state with a coherent state on
a beam splitter Lvovsky and Babichev [2002]. Further, an alternative method
for the generation of DFS has been proposed by Oliveira et al., de Oliveira et
al. [2005]. From the fact that photon added coherent state and DFS have
already been generated experimentally, and the fact that the photon added
states can be prepared via conditional measurement on a beam splitter, it
appears that PADFS and PSDFS can also be built in the lab. In fact, inspired
by these experiments, we have proposed a schematic diagram for generating the
PADFS and PSDFS in Figure 2.1 using single-mode and two-mode squeezed vacuum
states. Specifically, using three (two) highly transmitting beam splitters, a
conditional measurement of single photons at both detectors D1 and D2 in
Figure 2.1 (a) (Figure 2.1 (b)) would result in single photon subtracted DFS
as output from a single-mode (two-mode) squeezed vacuum state. Similarly, to
generate PADFS conditional subtraction of photon is replaced by photon
addition, using a nonlinear crystal and heralding one output mode for a
successful measurement of a single photon to ensure generation of single
photon added DFS. This fact, their general nature, and the fact that
nonclassical properties of PADFS and PSDFS have not yet been accorded
sufficient attention, has motivated us to perform this study.

Motivated by the above facts, in what follows, we investigate the
possibilities of observing lower- and higher-order sub-Poissonian photon
statistics, antibunching and squeezing in PADFS and PSDFS. We have studied
nonclassical properties of these states through a set of other witnesses of
nonclassicality, e.g., zeros of $Q$ function, Mandel $Q_{M}$ parameter,
Klyshko’s criterion, and Agarwal-Tara’s criterion. These witnesses of
nonclassicality successfully establish that both PADFS and PSDFS (along with
most of the states to which these two states reduce at different limits) are
highly nonclassical. Thus, making use of the analytical expressions of moments
of creation and annihilation operators, discussed below, facilitates an
analytical understanding for most of the nonclassical witnesses.

###  2.2 Higher-order moment for PADFS and PSDFS

We have already mentioned that we are interested in PADFS and PSDFS. In what
follows, we will see that various experimentally measurable nonclassicality
witnesses can be expressed as the moments of annihilation and creation
operators Allevi et al. [2012b, a]; Avenhaus et al. [2010]; Peřina Jr et al.
[2017]; Miranowicz et al. [2010]. To utilize those witnesses to identify the
signatures of nonclassicality, we will compute an analytic expression for the
most general moment, $\langle\hat{a}^{\dagger q}\hat{a}^{r}\rangle$, with $q$
and $r$ being non-negative integers. This is the most general moment in the
sense that any other moment can be obtained as a special case of it. For
example, if we need $\langle\hat{a}^{2}\rangle,$ we would just require to
consider $q=2$ and $r=0$. Thus, an analytic expression for
$\langle\hat{a}^{\dagger q}\hat{a}^{r}\rangle$ would essentially help us to
obtain analytic expression for any moment-based witness of nonclassicality.
Further, the analytic expressions of moment obtained for PADFS and PASDFS
would also help us to obtain nonclassical features in the set of states
obtained in the limiting cases, like Fock state, DFS, photon added coherent
state. Keeping this in mind, we have computed
$\langle\psi_{+}(u,n,\alpha)|\hat{a}^{\dagger
q}\hat{a}^{r}|\psi_{+}(u,n,\alpha)\rangle$ and
$\langle\psi_{-}(v,n,\alpha)|\hat{a}^{\dagger
q}\hat{a}^{r}|\psi_{-}(v,n,\alpha)\rangle$ using Eq. (1.19) and (1.20), and
provide the final analytic expressions of these moments without going into the
mathematical details to maintain the flow of the chapter. The obtained
expressions for the above mentioned moments for PADFS and PSDFS are

$\displaystyle\langle\hat{a}^{\dagger q}\hat{a}^{r}\rangle_{{\rm PADFS}}$
$\displaystyle=$ $\displaystyle\langle\psi_{+}(u,n,\alpha)|\hat{a}^{\dagger
q}\hat{a}^{r}|\psi_{+}(u,n,\alpha)\rangle$ $\displaystyle=$
$\displaystyle\frac{N_{+}^{2}}{n!}\sum\limits_{p,p^{\prime}=0}^{n}{n\choose
p}{n\choose p^{\prime}}(-\alpha^{\star})^{(n-p)}(-\alpha)^{(n-p^{\prime})}$
$\displaystyle\times$
$\displaystyle\exp\left[-\mid\alpha\mid^{2}\right]\sum\limits_{m=0}^{\infty}\frac{\alpha^{m}(\alpha^{\star})^{m+p-p^{\prime}-r+q}(m+p+u)!(m+p+u-r+q)!}{m!(m+p-p^{\prime}-r+q)!(m+p+u-r)!},$

and

$\displaystyle\langle\hat{a}^{\dagger q}\hat{a}^{r}\rangle_{{\rm PSDFS}}$
$\displaystyle=$ $\displaystyle\langle\psi_{-}(v,n,\alpha)|\hat{a}^{\dagger
q}\hat{a}^{r}|\psi_{-}(v,n,\alpha)\rangle$ (2.2) $\displaystyle=$
$\displaystyle\frac{N_{-}^{2}}{n!}\sum\limits_{p,p^{\prime}=0}^{n}{n\choose
p}{n\choose p^{\prime}}(-\alpha^{\star})^{(n-p)}(-\alpha)^{(n-p^{\prime})}$
$\displaystyle\times$
$\displaystyle\exp\left[-\mid\alpha\mid^{2}\right]\sum\limits_{m=0}^{\infty}\frac{\alpha^{m}(\alpha^{*})^{m+p-p^{\prime}-r+q}(m+p)!(m+p-r+q)!}{m!(m+p-p^{\prime}-r+q)!(m+p-v-r)!},$

respectively. The values of normalization constants for PADFS and PSDFS are
already given in Eqs. (1.21) and (1.22), respectively. In the following
section, we shall investigate the possibilities of observing various types
lower- and higher-order nonclassical features in PADFS and PSDFS by using Eqs.
(2.2) and (2.2).

###  2.3 Nonclassical features of PADFS and PSDFS

The moments of number operators for PADFS and PSDFS states obtained in the
previous section enable us to study nonclassical properties of these states
using a set of moments-based criteria of nonclassicality Miranowicz et al.
[2010]Naikoo et al. [2018]. In the recent past, an infinite set of these
moments-based criteria is shown to be equivalent to the $P$-function-based
criterion, i.e., it becomes both necessary and sufficient Richter and Vogel
[2002]; Shchukin and Vogel [2005b]. However, in this section, we will use a
subset of this infinite set as witnesses of nonclassicality to investigate
various nonclassical properties of the PADFS and PSDFS. Specifically,
nonclassicality will be witnessed through Mandel $Q_{M}$ parameter, criteria
of lower- and higher-order antibunching, Agarwal-Tara’s criterion, Klyshko’s
criterion, criteria of higher-order sub-Poissonian photon statistics, zeros of
$Q$ function, etc. As all these criteria are already introduced in Section
2.3, here we may discuss the plots and results.

####  2.3.1 Mandel $Q_{M}$ Parameter

Negativity of this parameter indicates nonclassicality which can be calculated
using Eqs. (2.2) and (2.2). In Figure 2.2, the dependence of $Q_{M}$ on the
state parameter $\alpha$ and non-Gaussianity inducing parameters (i.e., photon
addition, subtraction, and Fock parameters as they can induce non-Gaussianity
in a quantum state) is shown. Specifically, variation of $Q_{M}$ parameter for
PADFS and PSDFS is shown with state parameter $\alpha$, where the effect of
the number of photons added/subtracted and the initial Fock state is also
established. For $\alpha=0$, the PADFS with an arbitrary number of photon
addition has $Q_{M}$ parameter -1, which can be attributed to the fact that
final state, which reduces to the Fock state ($|1\rangle$ chosen to be
displaced in this case) is the most nonclassical state (cf. Figure 2.2 (a)).
With increase in the number of photons added to the DFS, the depth of
nonclassicality witness $Q_{M}$ increases. However, the witness of
nonclassicality becomes less negative for higher values of the displacement
parameter. In contrast to the photon addition, with the subtraction of photons
from the DFS the $Q_{M}$ parameter becomes almost zero for the smaller values
of displacement parameter $\alpha$ in DFS as shown in Figure 2.2 (c). This
behavior can be attributed to the fact that photon subtraction from
$D\left(\alpha\right)|1\rangle$ for small values of $\alpha$ will most likely
yield vacuum state. Also, with the increase in the displacement parameter the
witness of nonclassicality becomes more negative as with a higher average
number of photons in DFS photon subtraction becomes more effective. However,
for the larger values of displacement parameter the nonclassicality disappears
analogous to the PADFS. For large values of $\alpha$, this parameter dominates
in the behavior of the state and thus it behaves analogous to coherent state.

As Fock states are known to be nonclassical, and photon addition and
subtraction are established as nonclassicality inducing operations, it would
be worth comparing the effect of these two independent factors responsible for
the observed nonclassical features in the present case. To perform this study,
we have shown the variation of the single photon added (subtracted) DFS with
different initial Fock states in Figure 2.2 (b) (Figure 2.2 (d)).
Specifically, the nonclassicality present in PADFS decays faster for the
higher values of the Fock states with increasing displacement parameter (cf.
Figure 2.2 (b)). However, such nature was not present in PSDFS shown in Figure
2.2 (d). Note that variation of $Q_{M}$ parameter with $\alpha$ starts from 0
(-1) iff $u\leq n$ $\left(u>n\right)$. For instance, if $u=n=1$, i.e.,
corresponding to state $\hat{a}D\left(\alpha\right)|1\rangle$, nonclassicality
witness is zero for $\alpha=0$ as it corresponds to vacuum state. Therefore,
the present study reveals that photon addition is a stronger factor for the
nonclassicality present in the state when compared to the initial Fock state
chosen to be displaced. Whereas photon subtraction is a preferred choice for
large values of displacement parameter in contrast to the higher values of
Fock states to displace with small $\alpha$. Among photon addition and
subtraction, addition is a preferred choice for the smaller values of
displacement parameter, while the choice between addition and subtraction
becomes immaterial for large $\alpha$.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 2.2:  Variation of Mandel $Q_{M}$ parameter for PADFS (in (a) and (b))
and PSDFS (in (c) and (d)) is shown with displacement parameter $\alpha$. In
(a) and (c), the value of number of photons added/subtracted (i.e., $u$ or
$v$) is changed for the same initial Fock state $|1\rangle$. Different initial
Fock states $|n\rangle$ are chosen to be displaced in (b) and (d) for the
single photon addition/subtraction. The blue curve corresponds to vacuum for
$\alpha=0$ and thus starts from $0$ unlike rest of the states which are Fock
state ($n\neq 0)$ in the limiting case. Therefore, nonclassicality increases
first with increasing $\alpha$ before decreasing as in the rest of the cases.

####  2.3.2 Lower- and higher-order antibunching

The nonclassicality reflected by the lower-order antibunching criterion
obtained here is the same as Mandel $Q_{M}$ parameter
$\left(Q_{M}=\frac{d\left(1\right)}{\left\langle\hat{a}^{\dagger}\hat{a}\right\rangle}\right)$
illustrated in Figure 2.2. Therefore we will rather discuss here the
possibility of observing higher-order antibunching in the quantum state of our
interest using Eqs. (2.2) and (2.2) in Eq. (1.41). Specifically, the depth of
nonclassicality witness can be observed to increase with order for both PADFS
and PSDFS as depicted in Figure 2.3 (a) and (d). This fact is consistent with
the earlier observations (Thapliyal et al. [2014a]; Thapliyal et al. [2014b];
Thapliyal et al. [2017b]; Thapliyal et al. [2017a]; Alam et al. [2017b] and
references therein) that higher-order nonclassicality criteria are useful in
detecting weaker nonclassicality. On top of that the higher-order antibunching
can be observed for larger values of displacement parameter $\alpha$, when
lower-order antibunching is not present. The presence of higher-order
nonclassicality in the absence of its lower-order counterpart establishes the
relevance of the present study.

The depth of nonclassicality parameter ($d\left(l-1\right)$) was observed to
decrease with an increase in the number of photons subtracted from DFS for
small values of $\alpha$ in Figure 2.2 (c). A similar nature is observed in
Figure 2.3 (e), which shows that for the higher values of displacement
parameter, the depth of higher-order nonclassicality witness increases with
the number of photon subtraction. Therefore, not only the depth of
nonclassicality but the range of displacement parameter for the presence of
higher-order antibunching also increases with photon addition/subtraction (cf.
Figure 2.3 (b) and (e)). With the increase in the value of Fock state
parameter $n$, the depth of higher-order nonclassicality witness increases
(decreases) for smaller (larger) values of displacement parameter in both
PADFS and PSDFS as shown in Figure 2.3 (c) and (f), respectively. Thus, we
have observed that the range of $\alpha$ with the presence of nonclassicality
increases (decreases) with photon addition/subtraction (Fock state) in DFS.

|  |
---|---|---
(a) | (b) |
|  |
(c) | (d) |
|  |
(e) | (f) |
Figure 2.3:  The presence of higher-order antibunching is shown as a function
of $\alpha$ for PADFS (in (a)-(c)) and PADFS ((d)-(f)). Specifically, (a) and
(d) illustrate comparison between lower- and higher-order antibunching. It
should be noted that some of the curves are multiplied by a scaling factor in
order to present them in one figure. Figures (b) and (e) show the effect of
photon addition/subtraction, and (c) and (f) establish the effect of Fock
state chosen to displace in PADFS and PSDFS, respectively. Here, without the
loss of generality, we have used the notation $\psi_{+}(u,n,l-1)$
($\psi_{-}(u,v,l-1)$) for nonclassicality in photon added (subtracted)
scenarios, and will follow this notation in subsequent figures of this
chapter.

####  2.3.3 Higher-order sub-Poissonian photon statistics

This allows us to study higher-order sub-Poissonian photon statistics using
Eqs. (2.2) and (2.2) in Eq. (1.42). The presence of higher-order sub-
Poissonian photon statistics (as can be seen in Figure 2.4 (a) and (d) for
PADFS and PSDFS, respectively) is dependent on the order of nonclassicality
unlike higher-order antibunching, which is observed for all orders.
Specifically, the nonclassical feature was observed only for odd orders, which
is consistent with some of the earlier observations Thapliyal et al. [2017b],
where nonclassicality in those cases could be induced due to squeezing. Along
the same line, we expect to observe the nonclassicality in such cases with
appropriate use of squeezing as a useful quantum resource, which will be
discussed elsewhere. In case of photon addition/subtraction in DFS, a behavior
analogous to that observed for higher-order antibunching is observed, i.e.,
the depth of nonclassciality increases with the photon addition while it
decreases (increases) for small (large) values of $\alpha$ (cf. Figure 2.4 (b)
and (e)). Similar to the previous case, nonclassicality can be observed to be
present for larger values of displacement parameter with photon
addition/subtraction, while increase in the value of Fock parameter has an
opposite effect.

|  |
---|---|---
(a) | (b) |
|  |
(c) | (d) |
|  |
(e) | (f) |
Figure 2.4:  Dependence of higher-order sub-Poissonian photon statistics on
$\alpha$ for PADFS (in (a)-(c)) and PSDFS ((d)-(f)) is illustrated here.
Specifically, (a) and (d) show the increase in depth of nonclassicality
witness with order, (b) and (e) depict the effect of photon addition and
subtraction, respectively, and (c) and (f) establish the effect of choice of
Fock state to be displaced in PADFS and PSDFS, respectively.

####  2.3.4 Higher-order squeezing

The analytical expressions of the nonclassicality witness of the Hong-Mandel
type higher-order squeezing criterion for PADFS and PSDFS can be obtained with
the help of Eqs. (2.2), (2.2), and (1.43)-(1.45). We have investigated the
higher-order squeezing and depict the result in Figure 2.5 assuming $\alpha$
to be real. Incidentally, we could not establish the presence of higher-order
squeezing phenomena in PADFS (Figure 2.5 (a)-(c)). However, the depth of the
higher-order squeezing witness increases with order for small values of
$\alpha$ as shown in Figure 2.5 (a), while for the higher values of the
displacement parameter, higher-order squeezing disappear much quicker (cf.
Figure 2.5 (d)). With increase in the number of photons subtracted, the
presence of this nonclassicality feature can be maintained for the higher
values of displacement parameter as well (cf. Figure 2.5 (e)). In general,
photon subtraction is a preferred mode for nonclassicality enhancement as far
as this nonclassciality feature is concerned. The choice of the initial Fock
state is also observed to be relevant as the depth of squeezing parameter can
be seen increasing with value of the Fock parameter for PSDFS in the small
displacement parameter region (shown in Figure 2.5 (f)), where this
nonclassical behavior is also shown to succumb to the higher values of Fock
and displacement parameters. Unlike the other nonclassicalities discussed so
far, the observed squeezing also depends on phase $\theta$ of the displacement
parameter $\alpha=|\alpha|\exp\left(i\theta\right)$ due to the second last
term in Eq. (1.45). We failed to observe this nonclassicality behavior in
PADFS even by controlling the value of the phase parameter (also shown in
Figure 2.6 (a)). For PSDFS, the squeezing disappears for some particular
values of the phase parameter, while the observed squeezing is maximum for
$\theta=n\pi$ (see Figure 2.6 (b)). It thus establishes the phase parameter of
the displacement operator as one more controlling factor for nonclassicality
in these engineered quantum states.

|  |
---|---|---
(a) | (b) | (c)
|  |
(d) | (e) | (f)
Figure 2.5:  Illustration of the higher-order squeezing using Hong-Mandel criterion as the function of displacement parameter. In (a) and (d), dependence of the observed nonclassicality on different orders ($l$) is shown for PADFS and PSDFS, respectively; while in (b) and (e), the effect of variation in the number of photon added/subtracted is shown in case of PADFS and PSDFS, respectively. In (c) and (f), the variation due to change in the initial Fock state chosen to be displaced is shown for PADFS and PSDFS, respectively.  |
---|---
(a) | (b)
Figure 2.6:  Hong-Mandel type higher-order squeezing for PADFS and PSDFS is
shown dependent on the phase of the displacement parameter
$\alpha=|\alpha|\exp\left(i\theta\right)$ in (a) and (b), respectively.

####  2.3.5 $Q$ function

Using Eqs. (1.19) and (1.20) in (1.50), we obtain the analytic expressions for
the Husimi $Q$ function for PADFS and PSDFS as

$\begin{array}[]{lcl}Q_{+}&=&\frac{N_{+}^{2}}{\pi}\frac{\exp\left[-\mid\beta\mid^{2}\right]}{n!}\sum\limits_{p,p^{\prime}=0}^{n}{n\choose
p}{n\choose
p^{\prime}}(-\alpha^{\star})^{(n-p)}(-\alpha)^{(n-p^{\prime})}\exp\left[-\mid\alpha\mid^{2}\right]\\\
&\times&\sum\limits_{m,m^{\prime}=0}^{\infty}\frac{\alpha^{m}(\alpha^{\star})^{m^{\prime}}\beta^{(m^{\prime}+p^{\prime}+u)}(\beta^{\star})^{(m+p+u)}}{m!m^{\prime}!}\end{array}$
(2.3)

and

$\displaystyle\begin{array}[]{lcl}Q_{-}&=&\frac{N_{-}^{2}}{\pi}\frac{\exp\left[-\mid\beta\mid^{2}\right]}{n!}\sum\limits_{p,p^{\prime}=0}^{n}{n\choose
p}{n\choose
p^{\prime}}(-\alpha^{\star})^{(n-p)}(-\alpha)^{(n-p^{\prime})}\exp\left[-\mid\alpha\mid^{2}\right]\\\
&\times&\sum\limits_{m,m^{\prime}=0}^{\infty}\frac{\alpha^{m}(\alpha^{\star})^{m^{\prime}}\beta^{(m^{\prime}+p^{\prime}-v)}(\beta^{\star})^{(m+p-v)}(m+p)!(m^{\prime}+p^{\prime})!}{m!m^{\prime}!(m+p-v)!(m^{\prime}+p^{\prime}-v)!},\end{array}$
(2.6)

respectively. We failed to observe the nonclassical features reflected beyond
moments based nonclassicality criteria through a quasiprobability
distribution, i.e., zeros of the $Q$ function. We have shown the $Q$ function
in Figure 2.7, where the effect of photon addition/subtraction and the value
of Fock parameter on the phase space distribution is shown. Specifically, it
is observed that the value of Fock parameter affects the quasidistribution
function more compared to the photon addition/subtraction.

---
Figure 2.7:  Contour plots of the $Q$ function for (a) single photon added
displaced Fock $|1\rangle$ state, (b) two photon added displaced Fock
$|1\rangle$ state, (c) single photon added displaced Fock $|2\rangle$ state,
(d) single photon subtracted displaced Fock $|1\rangle$ state, (e) two photon
subtracted subtracted displaced Fock $|1\rangle$ state, (f) single photon
subtracted displaced Fock $|2\rangle$ state. In all cases,
$\alpha=\sqrt{2}\exp\left(\frac{i\pi}{4}\right)$ is chosen.

####  2.3.6 Agarwal-Tara’s criterion

The analytic expression of $A_{3}$ parameter defined in Eq. (1.48) can be
obtained for PADFS and PSDFS using Eqs. (2.2) and (2.2). The nonclassical
properties of the PADFS and PSDFS using Agarwal-Tara’s criterion are
investigated, and the corresponding results are depicted in Figure 2.8, which
shows highly nonclassical behavior of the states generated by engineering.
Specifically, the negative part of the curves, which is bounded by -1, ensures
the existence of the nonclassicality. From Figure 2.8, it is clear that
$A_{3}$ is 0 (-1) for the displacement parameter $\alpha=0$ because then DFS,
PADFS, and PSDFS reduce to Fock state, and $A_{3}=0$ (-1) for the Fock state
parameter $n=0,\,1$ $\left(n>1\right)$. Nonclassicality reflected through
$A_{3}$ parameter increases (decreases) with photon addition (subtraction)
(shown in Figure 2.8 (a) and (c)). In contrast, the Fock parameter has a
completely opposite effect that it leads to decrease (increase) in the
observed nonclassicality for PADFS (PSDFS), which can be seen in Figure 2.8
(b) and (d). However, for larger values of displacement parameter, the depth
of nonclassicality illustrated through this parameter can again be seen to
increase (cf. Figure 2.8 (b)).

|
---|---
(a) | (b)
|
(c) | (d)
Figure 2.8:  Variation of Agarwal-Tara’s parameter with $\alpha$ for PADFS and
PSDFS is shown in (a)-(b) and (c)-(d), respectively. Specifically, the effect
of photon addition/subtraction (in (a) and (c)) and the choice of Fock state
((b) and (d)) on the presence of nonclassicality in PADFS and PSDFS is
illustrated.

####  2.3.7 Klyshko’s Criterion

The analytic expression for the $m$th photon-number distribution for PADFS and
PSDFS can be calculated (using $q=r=1$) from Eqs. (2.2) and (2.2),
respectively.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 2.9:  Illustration of the Klyshko’s criterion. Variation of $B(m)$ with
respect to $m$ (a) and (c) for different values of the number of the photon
additon/subtraction for PADFS and PSDFS, respectively; (b) and (d) for
different values of the number of the Fock state parameter for PADFS and
PSDFS, respectively. Here, we have chosen $\alpha=1$ in all cases because
almost all of the other criteria detected nonclassicality for this choice of
$\alpha$ so does the Klyshko’s criterion.

The advantage of the Klyshko’s criterion over any other existing moments based
criteria is that a very small amount of information is required. Specifically,
probability of only three successive photon numbers is sufficient to
investigate the nonclassical property. The negative values of $B(m)$ serve as
the witness of nonclassicality. Klyshko’s criterion in Eq. (1.46) is derived
analytically and the corresponding nonclassical properties for both PADFS and
PSDFS are investigated (cf. Figure 2.9). Specifically, the negative values of
$B(m)$ are observed for different values of $m$ in case of photon addition and
subtraction (cf. Figure 2.9 (a) and (c)) being the signature of
nonclassicality induced via independent operations. Additionally, one can also
visualize that due to the photon addition (subtraction) the negative peaks in
the values of $B(m)$ shift to higher (lower) photon number regime. A similar
observation is obtained for different values of the Fock state parameter for
the PADFS and PSDFS, where the negative values of the witness of
nonclassicality get amplified towards higher photon number regime and becomes
more negative, and the corresponding results are shown in Figure 2.9 (b) and
(d), respectively. This further establishes the relevance of operations, like
photon addition, subtraction, and starting with Fock states in inducing
nonclassicality in the engineered quantum states.

###  2.4 Conclusions

The only Fock state that does not show any nonclassical feature is the vacuum
state Miranowicz et al. [2015], and displacement operator preserves its
classicality. All the rest of the Fock states are maximally nonclassical and
they are shown to remain nonclassical even after application of a displacement
operator. Here, we set ourselves a task: What happens when the displacement
operator applied on a Fock state is followed by addition or subtraction of
photon(s)? Independently, photon addition and subtraction are already
established as nonclassicality inducing operations in case of displaced vacuum
state (i.e., coherent state). In this chapter, we have established that photon
addition/subtraction is not only nonclassicality inducing operation, it can
also enhance the nonclassicality present in the DFS. It’s expected that these
operations would increase the amount of nonclassicality (more precisely, the
depth of the nonclassicality witnessing parameter) present in other
nonclassical states, too. There is one more advantage of studying the
nonclassical features of PADFS and PSDFS. These states can be reduced to a
class of quantum states, most of which are already experimentally realized and
found useful in several applications. Inspired by the available experimental
results, we have also proposed optical designs for generation of PADFS and
PSDFS from the squeezed vacuum state.

To analyze the nonclassical features of the engineered final states, i.e.,
PADFS and PSDFS, we have used a set of moments based criteria, namely Mandel
$Q_{M}$ parameter, Agarwal-Tara’s $A_{3}$ parameter, criteria for higher-order
antibunching, sub-Poissonian photon statistics, and squeezing. In addition,
the nonclassical features have been investigated through Klyshko’s criterion
and a quasiprobability distribution–$Q$ function. The states of interest are
found to show a large variety of nonclassical features as all the
nonclassicality witnesses used here (except $Q$ function) are found to detect
the nonclassicality. They show that state is useful as squeezed, antibunched,
as well as generation of entangled state.

This study has revealed that the amount of nonclassicality in PADFS and PSDFS
can be controlled by the Fock state parameter, displacement parameter, the
number of photons added or subtracted. In general, the amount of
nonclassicality with respect to the witness used here is found to increase
with the number of photons added/subtracted, while smaller values of Fock
state and displacement parameters are observed to be preferable for the
presence of various nonclassical features. On some occasions, nonclassicality
has also been observed to increase with the Fock state parameter, while larger
values of displacement parameter always affect the nonclassicality adversely.
Most of the nonclassicality criteria used here, being moments-based criteria,
could not demonstrate the effect of phase parameter of the displacement
parameter. Here, higher-order squeezing witness and $Q$ function are found to
be dependent on the phase of the displacement parameter. However, only higher-
order squeezing criterion was able to detect nonclassicality, and thus
established that this phase parameter can also be used to control the amount
of nonclassicality.

Further, in the past, it has been established that higher-order
nonclassicality criteria have an advantage in detecting weaker
nonclassicality. We have also shown that the depth of nonclassicality witness
increases with order of nonclassicality thus providing an advantage in the
experimental characterization of the observed nonclassical behavior.

##  Chapter 3 Quantum phase properties of photon added and subtracted
displaced Fock states

In this chapter, the motive is to observe the phase properties of PADFS and
PSDFS. The main findings of this chapter are published in Malpani et al.
[2019b]

###  3.1 Introduction

In the previous chapter, the nonclassical properties of PADFS and PSDFS were
studied. Here, our specific interest is to study the phase properties of PADFS
and PSDFS and their limiting cases. In the recent past, the nonclassical
properties of this set of engineered quantum states, many of which have been
experimentally generated Lvovsky et al. [2001]; Lvovsky and Babichev [2002];
Zavatta et al. [2004, 2005, 2008], were focus of various studies (see Malpani
et al. [2019a] and references therein). In Section 1.3.2, we have already
expressed PADFS and PSDFS as superposition of Fock states. Further, in Section
1.6 we have described the parameters used for the study of phase properties of
a quantum state. In that context we have already mentioned several
applications of quantum phase distribution and quantum phase fluctuation.

To stress on the recently reported applications of quantum phase distribution
and quantum phase fluctuation, we note that these have applications in quantum
random number generation Xu et al. [2012]; Raffaelli et al. [2018],
cryptanalysis of squeezed state based continuous variable quantum cryptography
Horak [2004], generation of solitons in a Bose-Einstein condensate Denschlag
et al. [2000], in phase encoding quantum cryptography Gisin et al. [2002],
phase imaging of cells and tissues for biomedical application Park et al.
[2018]; as well as have importance in determining the value of transition
temperature for superconductors Emery and Kivelson [1995]. Keeping these
applications and the general nature of engineered quantum states PADFS and
PSDFS in mind, in what follows, we aim to study phase distribution, $Q$ phase,
phase fluctuation measures, phase dispersion, and quantum phase estimation
using the concerned states and the states obtained in the limiting cases. As
PADFS and PSDFS are already described, we may begin this study by describing
limiting cases of these states as our states of interest.

We have already mentioned that our focus would be on PADFS and PSDFS. Due to
the general form of PADFS and PSDFS, a large number of states can be obtained
in the limiting cases. Some of the important limiting cases of PADFS and PSDFS
in the present notation are summarized in Table 3.1. This table clearly
establishes that the applicability of the results obtained in the present
study is not restricted to PADFS and PSDFS; rather an investigation of the
phase properties of PADFS and PSDFS would also reveal phase properties of many
other quantum states of particular interest.

Reduction of state |  Name of the state | Reduction of state |  Name of the state
---|---|---|---
$|\psi_{+}(u,n,\alpha)\rangle$ |  $u$-PADFS | $|\psi_{-}(v,n,\alpha)\rangle$ |  $v$-PSDFS
|$\psi_{+}(0,n,\alpha)\rangle$ |  DFS |  |$\psi_{-}(0,n,\alpha)\rangle$ |  DFS
$|\psi_{+}(0,0,\alpha)\rangle$ |  Coherent state | $|\psi_{-}(0,0,\alpha)\rangle$ |  Coherent state
|$\psi_{+}(0,n,0)\rangle$ |  Fock state |  |$\psi_{-}(0,n,0)\rangle$ |  Fock state
|$\psi_{+}(u,0,\alpha)\rangle$ |  $u$-Photon added coherent state |  |$\psi_{-}(v,0,\alpha)\rangle$ |  $v$-Photon subtracted coherent state
Table 3.1: Various states that can be obtained as the limiting cases of the
PADFS and PSDFS.

###  3.2 Quantum phase distribution and other phase properties

Quantum phase operator $\hat{\phi}$ was introduced by Dirac based on his
assumption that the annihilation operator $\hat{a}$ can be factored out into a
Hermitian function $f(\hat{N})$ of the number operator
$\hat{N}=\hat{a}^{\dagger}\hat{a}$ and a unitary operator $\hat{U}$ Dirac
[1927] as

$\hat{a}=\hat{U}\,f\left(\hat{N}\right),$ (3.1)

where

$\hat{U}=e^{\iota\hat{\phi}}.$ (3.2)

However, there was a problem with the Dirac formalism of phase operator as it
failed to provide a meaning to the corresponding uncertainty relation.
Specifically, in the Dirac formalism, the creation ($\hat{a}^{\dagger}$) and
annihilation ($\hat{a}$) operators satisfy the bosonic commutation relation,
$\left[\hat{a},\,\hat{a}^{\dagger}\right]=1$, iff
$\left[\hat{N},\,\hat{\phi}\right]=\iota$, which leads to the number phase
uncertainty relation $\Delta N\,\Delta\phi\geq 1$. Therefore, in order to
satisfy the bosonic commutation relation under Dirac formalism, the phase
uncertainty should be greater than 2$\pi$ for $\Delta N$ < $\frac{1}{2\pi}$
which lacks a physical description. Subsequently, Louisell Louisell [1963]
proposed some periodic phase based method, which was followed by Susskind and
Glogower formalism based on Sine and Cosine operators Susskind and Glogower
[1964]. An important contribution to this problem is the Barnett-Pegg
formalism Barnett and Pegg [1986] which is used in this thesis. In what
follows, we will also briefly introduce notions, such as quantum phase
distribution, angular $Q$ phase function, phase fluctuation parameters, phase
dispersion, quantum phase estimation to study the phase properties of the
quantum states of our interest.

###  3.3 Phase properties of PADFS and PSDFS

The description of the states of our interest given in the previous section
can be used to study different phase properties and quantify phase fluctuation
in the set of quantum states listed in Table 3.1. Specifically, with the help
of the quantum states defined in Eqs. (1.19)-(1.20), we have obtained the
analytic expressions of phase distribution and other phase parameters defined
in Section 3.2.

####  3.3.1 Phase distribution function

From the definition of the phase distribution (1.54), it can be observed that
for a Fock state, $P_{\theta}=\frac{1}{2\pi}$, implying it has a uniform
distribution of phase. Interestingly, the states of our interest, PADFS and
PSDFS, are obtained by displacing the Fock state followed by photon
addition/subtraction. Therefore, we will study here what is the effect of
application of displacement operator on a uniformly phase distributed (Fock)
state and how subsequent photon addition/subtraction further alters the phase
distribution. Using phase distribution function, the information regarding
uncertainty in phase and phase fluctuation can also be obtained. To begin
with, we compute the analytic expressions of $P_{\theta}$ for the PADFS and
PSDFS, using Eq. (1.54) as

$\displaystyle\begin{array}[]{lcl}P_{\theta}\left(u,n\right)&=&\frac{1}{2\pi}\dfrac{\left|N_{+}\right|^{2}}{n!}\sum\limits_{p,p^{\prime}=0}^{n}{n\choose
p}{n\choose
p^{\prime}}\exp\left[-\mid\alpha\mid^{2}\right]\left|\alpha\right|^{2n-p-p^{\prime}}\\\
&\times&\sum\limits_{m,m^{\prime}=0}^{\infty}\frac{(-\left|\alpha\right|)^{m+m^{\prime}}\sqrt{(m+p+u)!(m^{\prime}+p^{\prime}+u)!}}{m!m^{\prime}!}\exp[\iota\left(\theta-\theta_{2}\right)(m^{\prime}+p^{\prime}-m-p)],\end{array}$
(3.5)

and

$\displaystyle\begin{array}[]{ccc}P_{\theta}\left(v,n\right)&=&\frac{1}{2\pi}\dfrac{\left|N_{-}\right|^{2}}{n!}\sum\limits_{p,p^{\prime}=0}^{n}{n\choose
p}{n\choose
p^{\prime}}\exp\left[-\mid\alpha\mid^{2}\right]\left|\alpha\right|^{2n-p-p^{\prime}}\\\
&\times&\sum\limits_{m,m^{\prime}=0}^{\infty}\frac{(-\left|\alpha\right|)^{m+m^{\prime}}(m+p)!(m^{\prime}+p^{\prime})!}{m!m^{\prime}!\sqrt{(m+p-v)!(m^{\prime}+p^{\prime}-v)!}}\exp[\iota\left(\theta-\theta_{2}\right)(m^{\prime}+p^{\prime}-m-p)],\end{array}$
(3.8)

respectively. Here, $\theta_{2}$ is the phase associated with the displacement
parameter $\alpha$ ($\alpha=|\alpha|e^{\iota\theta_{2}}$). Since the obtained
expressions in Eqs. (3.5) and (3.8) are complex in nature, we depict numerical
(graphical) analysis of the obtained results in Figs. 3.1 and 3.2 for PADFS
and PSDFS, respectively. Specifically, in Figure 3.1 (a), we have shown the
variation of phase distribution with phase parameter $\theta$ for different
number of photon added in the displaced single photon Fock state
($D\left(\alpha\right)\left|1\right\rangle$) for $\theta_{2}=0$. A uniform
phase distribution for Fock state (with a constant value of $\frac{1}{2\pi}$)
is found to transform to one that decreases for higher values of phase and
possess a dip in the phase distribution for $\theta=0$, which can be thought
of as an approach to the Fock state. In fact, in case of classical states,
$P_{\theta}$ has a peak at zero phase difference $\theta-\theta_{2}$, and
therefore, this contrasting behavior can be viewed as signature of quantumness