**STUDIES IN NONCLASSICAL STATES ** **OF OPTICAL FIELDS **

**by **

**ANIL KUMAR ROY **

**Department of Physics **

**THESIS SUBMITTED **

**IN FULFILMENT OF THE REQUIREMENTS ****FOR THE DEGREE OF **

**DOCTOR OF PHILOSOPHY **

**to the **

**INDIAN INSTITUTE OF TECHNOLOGY, DELHI **

**NEW DELHI-110016. INDIA **

**SEPTEMBER 1992 **

### Q- -11-1: qc)--1

*Let noble thoughts come to us from every Side, *

*laGVEDA, 1-89-i . *

*TO MY BHABHI *

### DECLARATION

hereby, declare that the work being presented in this thesis entitled, "Studies in Nonclassical States of Optical Fields", in the partial fulfillment of the requirements towards the award of the degree of Doctor of Philosophy, submitted in the department of Physics, Indian Institute of Technology, Delhi, is an authentic record of research work carried out by me under the supervision of Prof. C.L. Mehta of this department. The matter embodied in this thesis has not been submitted by me or anybody else for the award of any other degree.

(Anil Kumar Roy) 88RP11004.

### CERTIFICATION BY THE SUPERVISOR

This is to certify that the above statements made by the candidate Mr. Anil Kumar Roy are correct to the best of my knowledge. I feel satisfied with his work and allow him to submit his thesis.

(C.L. Mehta) Professor

Department of Physics

Indian Institute of Technology, Delhi New Delhi, 110 016. INDIA.

### ACKNOWLEDGEMENTS

*I (11'1 privil(ged to exprd.vs my sincere sense of gratitude to my supervisor *
*Mehta fOr his valuable guidance ,for canying out my research -vork towards my *
*Ph I). clegrec. During this period, I have benefitted greatly from his deep physical insight *
*and unitizing mathematical skills through numerous long sessions of discussion on *
*research problems. His critical reading of the manuscript and thd changes suggested by *
*him have helped me substantially in finalizing the form and content of the thesis. All *
*through the course of my research, it was his support, advice and etwouragetnent, which *
*kept my enthusiasm alive, I am also indebted to him for passing an' those nuggets *
*of ~wisdom form port of his philosophy of life, *

*I am greatful to Dr. I), Ranganathun Delhi), Dr, G.M. Saxena (National *
*Physical Laboratory, Delhi), Or. Rapt (;bosh (Jawaharlal Nehru (Iniversity, Delhi), Prof. *

*S. Chopra (I Delhi) and ,Dr. Ajit Kumar (LIT. Delhi) for their highly informal and *
*fruitful discussions, I hurl with them On various topics. I. avail this vportunity to eApress *
*my thanks to them. *

*No words can :tidily cApres,s ttt,y uppreciatiort of the co,ttribution of my friends. It *
*is a pleasure to convey my leelings to Anurag Kumar, A. Sudarvhan, DX. Singh, flentant *
*Singh., Kornai Pant, KUMIY111 Manish (Daly), Pratik, Raj Kumar, Rakesh Jain, *
*Sanjay 94„Satyavir Singh, S.K. Singh, V.A. Raghaw, Vipul (Poucla) and Yashveer Singh *
*for their help and inspiration. Special thanks are due to my oldest friend Mithilesh *
*(Baba), whose computational skill, literary knowledge *and *philosophical attitude helped *
*me immensely, Time lighter moments shared with Meenakshi, too, is acknowledged with *
*gratitude, *

*There are some people who are not directly involved in this research work, but *
*without whose emotional support and encouragement this work would not have been *
*possible. The loving care and affection which my parents have always lavished upon time *
*and the concern of my Bhaiya-Bhabhi for my well-being, have contributed a lot in *
*helping m . to stay focussed on acheiving my goals. It will be impossible for me to find *
*appropriate *^{/WW1'S to }*fidfil the eternal debt which I owe them. *

*The list of my well wishers would be incomplete without including Meeta and Dr. *

*KK Thakur, whose warmth I can only tty to reciprocate by extending a hearty thank *
*you for all you have been to me *

*(Anil Kumar * ^{)y) }
*flatter **Khas, *

*Septetilb *

**ABSTRACT **

This thesis presents studies of some nonclassical states of optical fields. The usual squeezed states generated by exp[1/2 (aat1-a`a2)] form the basis of several recent publications. We consider a generalization by including a term proportional to (ata aat) in the exponent. The states generated by such a generalized operator is denoted by I a,a,B>. We obtain various representations of this state and study some of its nonclassical properties.

Following the normal ordering technique we show that the state I a,a,13> may
he generated by boson creation operator, i.e., the state I a,a,B > may be shown to be
equivalent to exp(V2 at2 + ^{ n(t) } > where and n are complex parameters related
to a, *a *and B, This form of the state is found very useful in obtaining expectation
values of the operators which are the functions of a and at.

Though a and at are singular operators, it is shown that we can introduce their inverses in the generalized sense. We define these inverse operators by their action on the number states and denote them by 8.-1. and â. We show that a-1 (at.i ) is the right (left) inverse of ti (at). Also, a"'

### (el

behaves as a creation (an annihilation) operator.### We,

then, construct three combinations of operators which have normalizable right eigenstates with non zero eigenvalues. These three operatorsift, dam and az are the two-photon annihilation operators (TAO). We solve their eigenvalue equations and discuss

### their eigenstates

in detail. We study some of their nonclassical properties as well. We### show that a family of the eigenstates of the TAO (1"A is essentially the customary squeezed vacuum and that of the eigenstates

of Cifft"' is the squeezed first number state. We

### obtain a new eigenvalue

equation for these states. A novel method of summing some series by using the eigenvalue equation is also considered.We show that the photon added coherent states Atm' a> [Agarwal and Tara, 1991] are essentially the simultaneous eigenstates of the operators (a-me-1) and at-w-

### with eigenvalue a. We introduce another family of such photon added coherent states I con> = CC"' a > . Analogously, we introduce 'photon depleted coherent states' I a,-m > = a>. , We obtain the normalization constants and discuss the completeness

of these states. We study nonclassical properties of I### >

### states and show that while these states show squeezing for some range

of the eigenvalue, they### never exhibit antibunching of photons.

### CONTENTS

*Acknowledgement *

*Ali.stract * ^{iil }

1. PREF'AC'E
*1./ haroduction *

*1.2 * *Review of the Harmonic Oscillator States *
*1.3 Outline of t1w work *

2. EXPONENTIAL OF A GENERAL QUADRATIC IN BOSON OPEATORS AND ASSOCIATED SQUEEZED STATES

*2.1 Introduction * *20 *

*2.2 Transformation relations. * *21 *

*2.3 Quadroture uncertainties * *24 *

*2.4 71w quadratic operator 0(0,p) *

*2. 5 ** The states I a.,(7,f3., * *40 *

*Squeezing and antibunc ling properties * *46 *

SQUEEZED STA' PS GENERAT 'ON CREATION C)PRRATOR

*3_1. Introduction * *49 *

*3,2 Another form of squeezed state * *50 *

*, rypectation value of a general "Unction anti a * *55 *
*1/11cf ,rtaintics **and nonclassical properties *

*3,5 Conclusion * *60 *

4. BOSON INVERSE OPERATORS, TWO-PHOTON ANtiIITILATION OPERATORS AND THEIR REPRESENTATIONS

*4.1 Introduction * *61 *

*4.2 Buyout Int'erve Op ,rti tors * *62 *

*4.1 Thvphoron Anin'hilation Oper**a**tors * *(hi; *

5. EIGENSTATES OF TWO-PHOTON ANNIHILATION OPERATORS AND THEIR NONCLASSICAL PROPERTIES

*5.1 Introduction * ^{71 }

*5.2 Eigenstate.s of a * *r*

*ir *

^{72 }*5,3 17.1genstates of (at * ^{74 }

*5.4 Eigievistates of (1 * *70 *

*Relation ',chive,' 12, I , I A, * , *and *IA, *states * *77 *
*5.6 * *Nonclassical properties of **the eigen.s.tates of the *

*two-photon annihilation operators * *81 *

*5.7 Conclusion * *105 *

6. TWO-PHOTON ANNIHILATION OPERATOR AND SQUEEZED VACUUM

*6, 1 Introduction * *107 *

6.*2 Eigenralue **equation * *110 *

7. BOSON INVERSE OPERATORS AND ASSOCIATED COHERENT STATES

*7.1 Introduction * *11.1 *

*7.2 Eigentalue equation of Photon ftcitlerl C'oheret,St**ates * *110 *
*7.3 Completeness of the eigenstates Of tlw **operator *d *118 *
7.-/ *Another *fouilly of *Photon Added Coherent **States * *119 *

*7.5 * *Photon Depleted Coherent States * *121 *

*7,6 CoMpit'frn *OA'S *of Ire,**-**nt> **states * *120 *
7 *7 * *Nonclassical propertie,s• of I rt, ,,n 0141eS * *127 *

*7.i Conclusion * *133 *

8. APPENDICES

rl.l *Trails ormution **Relation * *1. 5 *

*4.2 Proof of Eqs. (4.2.9) and (4.2,10) **of the malt text * _{137 }*A **3 Summation of series using the eigenvalue **equation * *139 *

9, REFI :R N ' _{148 }