STUDIES IN NONCLASSICAL STATES OF OPTICAL FIELDS
ANIL KUMAR ROY
Department of Physics
IN FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
INDIAN INSTITUTE OF TECHNOLOGY, DELHI
NEW DELHI-110016. INDIA
Q- -11-1: qc)--1
Let noble thoughts come to us from every Side,
laGVEDA, 1-89-i .
TO MY BHABHI
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.
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,
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"'
(elbehaves as a creation (an annihilation) operator.
We,then, construct three combinations of operators which have normalizable right eigenstates with non zero eigenvalues. These three operators
ift, dam and az are the two-photon annihilation operators (TAO). We solve their eigenvalue equations and discuss
their eigenstatesin 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 eigenstatesof Cifft"
' is the squeezed first number state. We
obtain a new eigenvalueequation 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 completenessof these states. We study nonclassical properties of I
states and show that while these states show squeezing for some rangeof the eigenvalue, they
never exhibit antibunching of photons.
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 Operators (hi;
5. EIGENSTATES OF TWO-PHOTON ANNIHILATION OPERATORS AND THEIR NONCLASSICAL PROPERTIES
5.1 Introduction 71
5.2 Eigenstate.s of a r
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,States 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
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