Advances in chemical physics vol 119 modern nonlinear optics part i

In early theoretical descriptions of nonlinear optical phenom-ena, the quantum nature of optical fields has been ignored on the grounds thatlaser fields are so strong, that is, the numbe

MODERN NONLINEAR OPTICS

Part 1 Second Edition

ADVANCES IN CHEMICAL PHYSICS

VOLUME 119

Edited by Myron W Evans Series Editors: I Prigogine and Stuart A Rice.

Copyright # 2001 John Wiley & Sons, Inc ISBNs: 0-471-38930-7 (Hardback); 0-471-23147-9 (Electronic)

Depart-RUDOLPHA MARCUS, Department of Chemistry, California Institute of Technology,Pasadena, California, U.S.A.

G NICOLIS, Center for Nonlinear Phenomena and Complex Systems, Universite´Libre de Bruxelles, Brussels, Belgium

THOMASP RUSSELL, Department of Polymer Science, University of Massachusetts,Amherst, Massachusetts

DONALD G TRUHLAR, Department of Chemistry, University of Minnesota,Minneapolis, Minnesota, U.S.A

JOHND WEEKS, Institute for Physical Science and Technology and Department ofChemistry, University of Maryland, College Park, Maryland, U.S.A

PETERG WOLYNES, Department of Chemistry, University of California, San Diego,California, U.S.A

MODERN NONLINEAR

OPTICS

Part 1 Second Edition

ADVANCES IN CHEMICAL PHYSICS

Center for Studies in Statistical Mechanics and Complex Systems

The University of Texas Austin, Texas and International Solvay Institutes Universite´ Libre de Bruxelles Brussels, Belgiumand

STUART A RICE

Department of Chemistry

and The James Franck Institute The University of Chicago Chicago, Illinois

AN INTERSCIENCE1PUBLICATION

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Carlos, Sa˜o Carlos, SP, Brazil and Moscow Institute of Physics andTechnology, Lebedev Physics Institute of the Russian Academy ofSciences, Moscow, Russia

MILOSLAV DUSˇEK, Department of Optics, Palacky´ University, Olomouc,Czech Republic

ZBIGNIEW FICEK, Department of Physics and Centre for Laser Science, TheUniversity of Queensland, Brisbane, Australia

JAROMI´R FIURA ´ SˇEK, Department of Optics, Palacky´ University, Olomouc,Czech Republic

Applications, University d’Angers, Faculte´ des Sciences, Angers, France

ONDR ˇ EJHADERKA, Joint Laboratory of Optics of Palacky´ University and theAcademy of Sciences of the Czech Republic, Olomouc, Czech Republic

MARTINHENDRYCH, Joint Laboratory of Optics of Palacky´ University and theAcademy of Sciences of the Czech Republic, Olomouc, Czech Republic

ZDENE ˇ KHRADIL, Department of Optics, Palacky´ University, Olomouc, CzechRepublic

NOBUYUKIIMOTO, CREST Research Team for Interacting Carrier Electronics,School of Advanced Sciences, The Graduate University of AdvancedStudies (SOKEN), Hayama, Kanagawa, Japan

v

MASATOKOASHI, CREST Research Team for Interacting Carrier Electronics,School of Advanced Sciences, The Graduate University for AdvancedStudies (SOKEN), Hayama, Kanagawa, Japan

Applications, Universite´ d’Angers, Faculte´ des Sciences, Angers,France

WIESLAWLEON ´ SKI, Nonlinear Optics Division, Adam Mickiewicz University,Poznan´, Poland

ANTONI´NLUKSˇ, Department of Optics, Palacky´ University, Olomouc, CzechRepublic

ADAM MIRANOWICZ, CREST Research Team for Interacting CarrierElectronics, School of Advanced Sciences, The Graduate Universityfor Advanced Studies (SOKEN), Hayama, Kanagawa, Japan andNonlinear Optics Division, Institute of Physics, Adam MickiewiczUniversity, Poznan, Poland

JAN PER ˇ INA, Joint Laboratory of Optics of Palacky´ University and theAcademy of Sciences of the Czech Republic, Olomouc, Czech Republic

JAN PER ˇ INA, JR., Joint Laboratory of Optics of Palacky´ University and theAcademy of Sciences of the Czech Republic, Olomouc, Czech Republic

VLASTA PER ˇ INOVA´, Department of Optics, Palacky´ University, Olomouc,Czech Republic

JAROSLAV RˇEHA ´ CˇEK, Department of Optics, Palacky´ University, Olomouc,Czech Republic

MENDEL SACHS, Department of Physics, State University of New York atBuffalo, Buffalo, NY

ALEXANDERS SHUMOVSKY, Physics Department, Bilkent University, Bilkent,Ankara, Turkey

RYSZARD TANAS´, Nonlinear Optics Division, Institute of Physics, AdamMickiewicz University, Poznan´, Poland

Few of us can any longer keep up with the flood of scientific literature, even

in specialized subfields Any attempt to do more and be broadly educatedwith respect to a large domain of science has the appearance of tilting atwindmills Yet the synthesis of ideas drawn from different subjects into new,powerful, general concepts is as valuable as ever, and the desire to remaineducated persists in all scientists This series, Advances in ChemicalPhysics, is devoted to helping the reader obtain general information about awide variety of topics in chemical physics, a field that we interpret verybroadly Our intent is to have experts present comprehensive analyses ofsubjects of interest and to encourage the expression of individual points ofview We hope that this approach to the presentation of an overview of asubject will both stimulate new research and serve as a personalized learningtext for beginners in a field

I PRIGOGINE

STUARTA RICE

vii

of quantum mechanics opposite that proposed by the Copenhagen School Theformal structure of quantum mechanics is derived as a linear approximation for

a generally covariant field theory of inertia by Sachs, as reviewed in his article.This also opposes the Copenhagen interpretation Another review providesreproducible and repeatable empirical evidence to show that the Heisenberguncertainty principle can be violated Several of the reviews in Part 1 containdevelopments in conventional, or Abelian, quantum optics, with applications

In Part 2, the articles are concerned largely with electrodynamical theoriesdistinct from the Maxwell–Heaviside theory, the predominant paradigm at thisstage in the development of science Other review articles develop electro-dynamics from a topological basis, and other articles develop conventional orU(1) electrodynamics in the fields of antenna theory and holography There arealso articles on the possibility of extracting electromagnetic energy fromRiemannian spacetime, on superluminal effects in electrodynamics, and onunified field theory based on an SU(2) sector for electrodynamics rather than aU(1) sector, which is based on the Maxwell–Heaviside theory Several effectsthat cannot be explained by the Maxwell–Heaviside theory are developed usingvarious proposals for a higher-symmetry electrodynamical theory The volume

is therefore typical of the second stage of a paradigm shift, where the prevailingparadigm has been challenged and various new theories are being proposed Inthis case the prevailing paradigm is the great Maxwell–Heaviside theory and itsquantization Both schools of thought are represented approximately to the sameextent in the three parts of Volume 119

As usual in the Advances in Chemical Physics series, a wide spectrum ofopinion is represented so that a consensus will eventually emerge Theprevailing paradigm (Maxwell–Heaviside theory) is ably developed by severalgroups in the field of quantum optics, antenna theory, holography, and so on, butthe paradigm is also challenged in several ways: for example, using generalrelativity, using O(3) electrodynamics, using superluminal effects, using an

ix

extended electrodynamics based on a vacuum current, using the fact thatlongitudinal waves may appear in vacuo on the U(1) level, using a reproducibleand repeatable device, known as the motionless electromagnetic generator,which extracts electromagnetic energy from Riemannian spacetime, and inseveral other ways There is also a review on new energy sources UnlikeVolume 85, Volume 119 is almost exclusively dedicated to electrodynamics, andmany thousands of papers are reviewed by both schools of thought Much of theevidence for challenging the prevailing paradigm is based on empirical data,data that are reproducible and repeatable and cannot be explained by the Max-well–Heaviside theory Perhaps the simplest, and therefore the most powerful,challenge to the prevailing paradigm is that it cannot explain interferometric andsimple optical effects A non-Abelian theory with a Yang–Mills structure isproposed in Part 2 to explain these effects This theory is known as O(3)electrodynamics and stems from proposals made in the first edition, Volume 85.

As Editor I am particularly indebted to Alain Beaulieu for meticulouslogistical support and to the Fellows and Emeriti of the Alpha Foundation’sInstitute for Advanced Studies for extensive discussion Dr David Hamilton atthe U.S Department of Energy is thanked for a Website reserved for some ofthis material in preprint form

Finally, I would like to dedicate the volume to my wife, Dr Laura J Evans

MYRONW EVANS

Ithaca, New York

By Adam Miranowicz, Wieslaw Leon´ski, and Nobuyuki Imoto

QUANTUM-OPTICALSTATES INFINITE-DIMENSIONALHILBERT SPACE 195

II STATEGENERATION

By Wieslaw Leon´ski and Adam Miranowicz

CORRELATED SUPERPOSITION STATES IN TWO-ATOM SYSTEMS 215

By Zbigniew Ficek and Ryszard Tanas´

MULTIPOLARPOLARIZABILITIES FROMINTERACTION-INDUCED 267

RAMAN SCATTERING

By Tadeusz Bancewicz, Yves Le Duff, and Jean-Luc Godet

NONSTATIONARYCASIMIREFFECT ANDANALYTICAL SOLUTIONS 309

FORQUANTUMFIELDS IN CAVITIES WITH MOVING BOUNDARIES

By V V Dodonov

QUANTUMMULTIPOLERADIATION 395

By Alexander S Shumovsky

NONLINEARPHENOMENA IN QUANTUMOPTICS 491

By Jirˇı´ Bajer, Miloslav Dusˇek, Jaromı´r Fiura´sˇek, Zdeneˇk Hradil,Antonı´n Luksˇ, Vlasta Perˇinova´, Jaroslav Rˇ eha´cˇek, Jan Perˇina,

Ondrˇej Haderka, Martin Hendrych, Jan Perˇina, Jr.,

Nobuyuki Imoto, Masato Koashi, and Adam Miranowicz

A QUANTUMELECTRODYNAMICALFOUNDATION FOR 603

MOLECULARPHOTONICS

By David L Andrews and Philip Allcock

xi

SYMMETRY INELECTRODYNAMICS: FROMSPECIAL TO GENERAL 677

RELATIVITY, MACRO TOQUANTUMDOMAINS

By Mendel Sachs

QUANTUM NOISE IN NONLINEAR

II Basic Definitions

III Second-Harmonic Generation

Modern Nonlinear Optics, Part 1, Second Edition: Advances in Chemical Physics, Volume 119.

Edited by Myron W Evans Series Editors: I Prigogine and Stuart A Rice.

Copyright # 2001 John Wiley & Sons, Inc ISBNs: 0-471-38930-7 (Hardback); 0-471-23147-9 (Electronic)

1

experimentally observed and described theoretically and now are the subject oftextbooks [1,2] In early theoretical descriptions of nonlinear optical phenom-ena, the quantum nature of optical fields has been ignored on the grounds thatlaser fields are so strong, that is, the number of photons associated with them are

so huge, that the quantum properties assigned to individual photons have nochances to manifest themselves However, it turned out pretty soon thatquantum noise associated with the vacuum fluctuations can have importantconsequences for the course of nonlinear phenomena Moreover, it appearedthat the quantum noise itself can change essentially when the quantum field issubject to the nonlinear transformation that is the essence of any nonlinearprocess The quantum states with reduced quantum noise for a particularphysical quantity can be prepared in various nonlinear processes Such stateshave no classical counterparts; that is, the results of some physical measure-ments cannot be explained without explicit recall to the quantum character ofthe field The methods of theoretical description of quantum noise are thesubject of Gardiner’s book [3] This chapter is not intended as a presentation ofgeneral methods that can be found in the book; rather, we want to compare theresults obtained with a few chosen methods for the two, probably mostimportant, nonlinear processes: second-harmonic generation and downconver-sion with quantum pump

Why have we chosen the second-harmonic generation and the sion to illustrate consequences of field quantization, or a role of quantum noise,

downconver-in nonldownconver-inear optical processes? The two processes are at the same time similarand different Both of them are described by the same interaction Hamiltonian,

so in a sense they are similar and one can say that they show different faces ofthe same process However, they are also different, and the difference betweenthem consists in the different initial conditions This difference appears to bevery important, at least at early stages of the evolution, and the properties of thefields produced in the two processes are quite different With these two best-known and practically very important examples of nonlinear optical processes,

we would like to discuss several nonclassical effects and present the mostcommon theoretical approaches used to describe quantum effects The chapter

is not intended to be a complete review of the results concerning the twoprocesses that have been collected for years We rather want to introduce thereader who is not an expert in quantum optics into this fascinating field bypresenting not only the results but also how they can be obtained with presentlyavailable computer software The results are largely illustrated graphically foreasier comparisons In Section II we introduce basic definitions and the mostimportant formulas required for later discussion Section III is devoted topresentation of results for second-harmonic generation, and Section IV resultsfor downconversion In the Appendixes A and B we have added examples ofcomputer programs that illustrate usage of really existing software and were

actually used in our calculations We draw special attention to symboliccalculations and numerical methods, which can now be implemented even onsmall computers.

In classical optics, a one mode electromagnetic field of frequency o, with thepropagation vector k and linear polarization, can be represented as a plane wave

Eđr; tỡ Ử 2E0cosđk  r  ot ợ jỡ đ1ỡwhere E0 is the amplitude and j is the phase of the field Assuming the linearpolarization of the field, we have omitted the unit polarization vector to simplifythe notation Classically, both the amplitude E0 and the phase j can be well-defined quantities, with zero noise Of course, the two quantities can beconsidered as classical random variables with nonzero variances; thus, theycan be noisy in a classical sense, but there is no relation between the twovariances and, in principle, either of them can be rendered zero giving thenoiseless classical field Apart from a constant factor, the squared real ampli-tude, E2, is the intensity of the field In classical electrodynamics there is no realneed to use complex numbers to describe the field However, it is convenient towork with exponentials rather than cosine and sine functions and the field (1) isusually written in the form

Eđr; tỡ Ử Eđợỡeiđk  rotỡợ Eđỡeiđk  rotỡ đ2ỡwith the complex amplitudes EỬ E0eij The modulus squared of such anamplitude is the intensity of the field, and the argument is the phase Bothintensity and the phase can be measured simultaneously with arbitrary accuracy

In quantum optics the situation is dramatically different The electromagneticfield E becomes a quantum quantity; that is, it becomes an operator acting in aHilbert space of field states, the complex amplitudes Ebecome the annihilationand creation operators of the electromagnetic field mode, and we have

^

EỬ

ffiffiffiffiffiffiffiffiffiffi

o2e0V

r

ơ^aeiđk  rotỡợ ^aợeiđk  rotỡ đ3ỡ

with the bosonic commutation rules

of laws of quantum mechanics, optical fields exhibit an inherent quantumindeterminacy that cannot be removed for principal reasons no matter howsmart we are The quantity

E0¼

ffiffiffiffiffiffiffiffiffiffi

o2e0V

r

ð5Þ

appearing in (3) is a measure of the quantum optical noise for a single mode ofthe field This noise is present even if the field is in the vacuum state, and for thisreason it is usually referred to as the vacuum fluctuations of the field [4].Quantum noise associated with the vacuum fluctuations, which appears because

of noncommuting character of the annihilation and creation operators expressed

by (4), is ubiquitous and cannot be eliminated, but we can to some extentcontrol this noise by ‘squeezing’ it in one quantum variable at the expense of

‘‘expanding’’ it in another variable This noise, no matter how small it is incomparison to macroscopic fields, can have very important macroscopicconsequences changing the character of the evolution of the macroscopic fields

We are going to address such questions in this chapter

The electric field operator (3) can be rewritten in the form

^

E¼ E0Q cosðk  r  otÞ þ ^^ P sinðk  r  otÞ

ð6Þwhere we have introduced two Hermitian quadrature operators, ^Q and ^P, definedas

^

Q¼ ^aþ ^aþ; P^ ¼ ið^a ^aþÞ ð7Þwhich satisfy the commutation relation

The two quadrature operators thus obey the Heisenberg uncertainty relation

hð ^QÞ2ihð^PÞ2i 1 ð9Þwhere we have introduced the quadrature noise operators

 ^Q¼ ^Q h ^Qi ; ^P¼ ^P h^Pi ð10ÞFor the vacuum state or a coherent state, which are the minimum uncertaintystates, the inequality (9) becomes equality and, moreover, the two variances areequal

hð ^QÞ2i ¼ hð^PÞ2 ¼ 1 ð11Þ

The Heisenberg uncertainty relation (9) imposes basic restrictions on theaccuracy of the simultaneous measurement of the two quadrature components

of the optical field In the vacuum state the noise is isotropic and the twocomponents have the same level of quantum noise However, quantum statescan be produced in which the isotropy of quantum fluctuations is broken—theuncertainty of one quadrature component, say, ^Q, can be reduced at the expense

of expanding the uncertainty of the conjugate component, ^P Such states arecalled squeezed states [5,6] They may or may not be the minimum uncertaintystates Thus, for squeezed states

hð ^QÞ2i < 1 or hð^PÞ2i < 1 ð12Þ

Squeezing is a unique quantum property that cannot be explained when the field

is treated as a classical quantity —field quantization is crucial for explaining thiseffect

Another nonclassical effect is referred to as sub-Poissonian photon statistics(see, e.g., Refs 7 and 8 and papers cited therein) It is well known that in acoherent state defined as an infinite superposition of the number states

jai ¼ exp jaj

2

2

!

X1 n¼0

an

ffiffiffiffin!

the photon number distribution is Poissonian

pðnÞ ¼ jhnjaij2¼ exp ðjaj2Þjaj

If the variance of the number of photons is smaller than its mean value, the field

is said to exhibit the sub-Poissonian photon statistics This effect is related to thesecond-order intensity correlation function

Gð2ÞðtÞ ¼ h: ^nðtÞ^nðt þ tÞ :i ¼ h^aþðtÞ^aþðt þ tÞ^aðt þ tÞ^aðtÞi ð16Þwhere : : indicate the normal order of the operators This function describes theprobability of counting a photon at t and another one at tþ t For stationaryfields, this function does not depend on t but solely on t The normalized

second-order correlation function, or second-order degree of coherence, isdefined as

anti-gð2Þð0Þ ¼h^a

þ^þ^a ^aih^aþ^i2 ¼

h^nð^n 1Þih^ni2 ¼ 1 þ

hð^nÞ2i  h^nih^ni2 ð18Þwhich gives the relation between the photon statistics and the second-ordercorrelation function Another convenient parameter describing the deviation ofthe photon statistics from the Poissonian photon number distribution is theMandel q parameter defined as [9]

q¼hð^nÞ

2

ih^ni  1 ¼ h^niðg

ð2Þð0Þ  1Þ ð19Þ

Negative values of this parameter indicate sub-Poissonian photon statistics,namely, nonclassical character of the field One obvious example of thenonclassical field is a field in a number statejni for which the photon numbervariance is zero, and we have gð2Þð0Þ ¼ 1  1=n and q ¼ 1 For coherentstates, gð2Þð0Þ ¼ 1 and q ¼ 0 In this context, coherent states draw a somewhatarbitrary line between the quantum states that have ‘‘classical analogs’’ and thestates that do not have them The coherent states belong to the former category,while the states for which gð2Þð0Þ < 1 or q < 0 belong to the latter category.This distinction is better understood when the Glauber–Sudarshan quasidistri-bution function PðaÞ is used to describe the field

The coherent states (13) can be used as a basis to describe states of the field

In such a basis for a state of the field described by the density matrix r, we canintroduce the quasidistribution function PðaÞ in the following way:

operators to the right) has the form

From the definition (13) of coherent state it is easy to derive the ness relation

complete-1p

ð

and find that the coherent states do not form an orthonormal set

jhajbij2¼ exp ðja  bj2Þ ð23Þand only forja  bj2 1 they are approximately orthogonal In fact, coherentstates form an overcomplete set of states

To see the nonclassical character of squeezed states better, let us express thevariancehð ^QÞ2i in terms of the P function

Similarly, for the photon number variance, we get

Again, hð^nÞ2i < h^ni only if PðaÞ is not positive definite, and thus Poissonian photon statistics is a nonclassical feature.

sub-In view of (24), one can write

hð ^QÞ2i ¼ 1 þ h: ð ^QÞ2:i ; hð^PÞ2i ¼ 1 þ h: ð^PÞ2:i ð26Þwhere : : indicate the normal form of the operator Using the normal form of thequadrature component variances squeezing can be conveniently defined by thecondition

h: ð ^QÞ2:i < 0 or h: ð^PÞ2:i < 0 ð27ÞTherefore, whenever the normal form of the quadrature variance is negative, thiscomponent of the field is squeezed or, in other words, the quantum noise in thiscomponent is reduced below the vacuum level For classical fields, there is nounity coming from the boson commutation relation, and the normal form of thequadrature component represents true variance of the classical stochasticvariable, which must be positive

The Glauber–Sudarshan P representation of the field state is associated withthe normal order of the field operators and is not the only c-number represen-tation of the quantum state Another quasidistribution that is associated withantinormal order of the operators is the Q representation, or the Husimi function,defined as

Generally, according to Cahill and Glauber [10], one can introduce the parametrized quasidistribution functionWðsÞðaÞ defined as

s-WðsÞðaÞ ¼1

pTrfr ^TðsÞðaÞg ð30Þ

where the operator ^TðsÞðaÞ is given by

^DðaÞjni sþ 1

The phase y is the quantity representing the field phase

With the quasiprobability distributionsWðsÞðaÞ, the expectation values of thes-ordered products of the creation and annihilation operators can be obtained byproper integrations in the complex a plane In particular, for s¼ 1; 0; 1, the s-ordered products are normal, symmetric, and antinormal ordered products of thecreation and annihilation operators, and the corresponding distributions are theGlauber–Sudarshan P function, Wigner function, and Husimi Q function By

virtue of the relation inverse to (34), the field density matrix can be retrievedfrom the quasiprobability function

From Eqs (38) and (40) we see that a system in a number state is equally likely

to be found in any statejymi, and a system in a phase state is equally likely to befound in any number state jni

The Pegg–Barnett Hermitian phase operator is defined as

^

y¼Xs

Of course, the phase states (38) are eigenstates of the phase operator (40) withthe eigenvalues ym restricted to lie within a phase window between y0 and

y0þ 2ps=ðs þ 1Þ The Pegg–Barnett prescription is to evaluate any observable

of interest in the finite basis (38), and only after that to take the limit s! 1.Since the phase states (38) are orthonormal,hymjym 0i ¼ dmm 0, the kth power

of the Pegg–Barnett phase operator (41) can be written as

^

ky¼Xs m¼0

The Pegg–Barnett Hermitian phase formalism allows for direct calculations

of quantum phase properties of optical fields As the Hermitian phase operator isdefined, one can calculate the expectation value and variance of this operator for

a given statej f i Moreover, the Pegg–Barnett phase formalism allows for theintroduction of the continuous phase probability distribution, which is a re-presentation of the quantum state of the field and describes the phase properties

of the field in a very spectacular fashion For so-called physical states, that is,states of finite energy, the Pegg–Barnett formalism simplifies considerably Inthe limit as s! 1 one can introduce the continuous phase distribution

PðyÞ ¼ lim

s!1

sþ 12p jhymj f ij2 ð45Þwhere ðs þ 1Þ=2p is the density of states and the discrete variable ym isreplaced by a continuous phase variable y In the number-state basis the

Pegg–Barnett phase distribution takes the form [15]

PðsÞðyÞ ¼

ð1

djaj WðsÞðaÞjaj ð51Þ

which, after performing of the integrations, gives the formula similar to thePegg–Barnett phase distribution

l l

mþ n

2  l þ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðm  lÞ!ðn  lÞ!

The phase distributions obtained by integration of the quasidistribution tions are different for different s, and all of them are different from the Pegg–Barnett phase distribution The Pegg–Barnett phase distribution is alwayspositive while the distribution associated with the Wigner distribution (s¼ 0)may take negative values The distribution associated with the Husimi Qfunction is much broader than the Pegg–Barnett distribution, indicating thatsome phase information on the particular quantum state has been lost Quantumphase fluctuations as fluctuations associated with the operator conjugate to thephoton-number operator are important for complete picture of the quantumnoise of the optical fields (for more details, see, e.g., Refs 16 and 17)

Second-harmonic generation, which was observed in the early days of lasers [18]

is probably the best known nonlinear optical process Because of its simplicityand variety of practical applications, it is a starting point for presentation ofnonlinear optical processes in the textbooks on nonlinear optics [1,2] Classi-cally, the second-harmonic generation means the appearance of the field atfrequency 2o (second harmonic) when the optical field of frequency o(fundamental mode) propagates through a nonlinear crystal In the quantumpicture of the process, we deal with a nonlinear process in which two photons ofthe fundamental mode are annihilated and one photon of the second harmonic iscreated The classical treatment of the problem allows for closed-form solutionswith the possibility of energy being transferred completely into the second-harmonic mode For quantum fields, the closed-form analytical solution of the

problem has not been found unless some approximations are made The earlynumerical solutions [19,20] showed that quantum fluctuations of the fieldprevent the complete transfer of energy into the second harmonic and thesolutions become oscillatory Later studies showed that the quantum states ofthe field generated in the process have a number of unique quantum featuressuch as photon antibunching [21] and squeezing [9,22] for both fundamentaland second harmonic modes (for a review and literature, see Ref 23) Nikitinand Masalov [24] discussed the properties of the quantum state of thefundamental mode by calculating numerically the quasiprobability distributionfunction QðaÞ They suggested that the quantum state of the fundamental modeevolves, in the course of the second-harmonic generation, into a superposition

of two macroscopically distinguishable states, similar to the superpositionsobtained for the anharmonic oscillator model [25–28] or a Kerr medium [29,30].Bajer and Lisoneˇk [31] and Bajer and Perˇina [32] have applied a symboliccomputation approach to calculate Taylor series expansion terms to findevolution of nonlinear quantum systems A quasiclassical analysis of the secondharmonic generation has been done by Alvarez-Estrada et al [33] Phaseproperties of fields in harmonics generation have been studied by Gantsog et

al [34] and Drobny´ and Jex [35] Bajer et al [36] and Bajer et al [37] havediscussed the sub-Poissonian behavior in the second- and third-harmonicgeneration More recently, Olsen et al [38,38] have investigated quantum-noise-induced macroscopic revivals in second-harmonic generation and criteriafor the quantum nondemolition measurement in this process

Quantum description of the second harmonic generation, in the absence ofdissipation, can start with the following model Hamiltonian

is conserved by the interaction ^HI The free evolution associated with theHamiltonian ^H0 leads to ^aðtÞ ¼ ^að0Þexp ðiotÞ and ^bðtÞ ¼ ^bð0Þexp ði2otÞ.This trivial exponential evolution can always be factored out and the importantpart of the evolution described by the interaction Hamiltonian ^H, for the slowly

varying operators in the Heisenberg picture, is given by a set of equations

d

dt^ðtÞ ¼ 1

i½^a; ^HI ¼ 2ik ^aþðtÞ^bðtÞd

dt

^ðtÞ ¼ 1i½^b; ^HI ¼ ik ^a2ðtÞ ð56Þwhere for notational convenience we use the same notation for the slowlyvarying operators as for the original operators — it is always clear from thecontext which operators are considered In deriving the equations of motion (56),

it is assumed that the operators associated with different modes commute, whilefor the same mode they obey the bosonic commutation rules (4)

Usually, the second-harmonic generation is considered as a propagationproblem, not a cavity field problem, and the evolution variable is rather the path

z the two beams traveled in the nonlinear medium In the simplest, discrete twomode description of the process the transition from the cavity to the propagationproblem is done by the replacement t¼ z=v, where v denotes the velocity ofthe beams in the medium (we assume perfect matching conditions) We will usehere time as the evolution variable, but it is understood that it can be equallywell the propagation time in the propagation problem So, we basically consider

an idealized, one-pass problem In fact, in the cavity situation the classical fieldpumping the cavity as well as the cavity damping must be added into the simplemodel to make it more realistic Quantum theory of such a model has beendeveloped by Drummond et al [39,40] Another interesting possibility is tostudy the second harmonic generation from the point of view of the chaoticbehavior [41] Such effects,however, will not be the subject of our concern here

A Classical FieldsBefore we start with quantum description, let us recollect the classical solutionswhich will be used later in the method of classical trajectories to study somequantum properties of the fields Equations (56) are valid also for classical fieldsafter replacing the field operators ^a and ^b by the c-number field amplitudes aand b, which are generally complex numbers They can be derived from theMaxwell equations in the slowly varying amplitude approximation [1] and havethe form

d

dtaðtÞ ¼ 2ikaðtÞbðtÞd

and the fundamental field amplitude is real and equal to að0Þ ¼ a0the solutionsfor the classical amplitudes of the second harmonic and fundamental modes aregiven by [1]

aðtÞ ¼ a0sechð ffiffiffi

2

p

a0ktÞbðtÞ ¼ a0ffiffiffi

The solutions (58) are monotonic and eventually all the energy present initially

in the fundamental mode is transferred to the second-harmonic mode

In a general case, when both modes initially have nonzero amplitudes, a06¼ 0and b06¼ 0, introducing a ¼ jajeifaand b¼ jbjeifb, we obtain the following set

dt#¼ k jaj

2

jbj  4jbj

!cos #d

dtfa¼ 2kjbj cos #d

dtfb¼ kjaj

2

where #¼ 2fa fb The system (59) has two integrals of motion

C0¼ jaj2þ 2jbj2; CI ¼ jaj2jbjcos # ð60Þwhich are classical equivalents of the quantum constants of motion ^H0 and ^HI(C0¼ h ^H0i, CI ¼ h ^HIi) Depending on the values of the constants of motion C0

and CI, the dynamics of the system (59) can be classified into several gories [42,43]:

cate-1 Phase-stable motion, CI ¼ 0, in which the phases of each mode arepreserved and the modes move radially in the phase space The phasedifference # is also preserved, which appears for cos #¼ 0 and

#¼ p=2 The solutions (58) belong to this category

2 Phase-changing motion, CI 6¼ 0, in which the dynamics of each modeinvolves both radial and phase motion In this case both modes must beinitially excited and their phase difference cannot be equal top=2

3 Phase-difference-stable motion, which is a special case of the changing motion that preserves the phase difference # between the modeseven though the phases of individual modes change This corresponds tothe no-energy-exchange regime when sin #¼ 0 and the initial amplitudes

phase-of the modes are preserved

Introducing new (scaled) variables

dtub¼ u2

asin #d

dtfa¼ ubcos #d

where nb¼ 1  na Since the normalized variable namust be less, than or equal

to unity, the maximum value that can be obtained by E2 is equal to 4

27 (forcos #¼ 1) From (66) we immediately obtain

27

In case of three different real roots nb1< nb2 < nb3ð < 0or E2 < 4

27Þ, wecan effect a substitution

n ¼ n þ ðn  n Þ sin2f ð72Þ

which leads to the elliptical integral

k2 This means that even very small E makes the solution periodic The values of

nbðtÞ are restricted to the region between the two smallest roots of the thirdorder polynomial nb1 nbðtÞ  nb2 To illustrate the behavior of the classicalsolutions, we plot in Fig 1 the time evolution of the intensities of the twomodes, naðtÞ and nbðtÞ, for the case when the second-harmonic mode is initiallyweak with respect to the fundamental mode (nbð0Þ ¼ 0:001) and the initialphases are both zeros (fað0Þ ¼ fbð0Þ ¼ 0) In this case the constant of motion

E¼ 0:0316 We see the regular periodic oscillations of the two intensities

In the limiting case, for which k¼ 1, we have nb1¼ 0, nb2 ¼ nb3¼ 1, andsnðx j 1Þ ¼ tanhðxÞ which is the phase-stable motion case and reproduces theclassical result (70) The other limiting case appears when k¼ 0, whichcorresponds to the situation with E2¼ 4

motions in the phase space and they are special cases of the general case of thephase changing motion of the system.

The solution (77) for radial variables uaðtÞ ¼ ffiffiffiffiffiffiffiffiffiffiffi

naðtÞ

p

and ubðtÞ ¼ ffiffiffiffiffiffiffiffiffiffiffi

nbðtÞpmust be supplemented with the corresponding solution for the phase variables

faðtÞ and fbðtÞ in order to find the trajectory in the phase space The equationsgoverning the evolution of the individual phases of the two modes can berewritten in the form

by p=2 whenever the intensity naðtÞ reaches its minimum and a jump by p ofthe phase fbðtÞ when nbðtÞ reaches its minimum The phase difference

#ðtÞ ¼ 2faðtÞ  fbðtÞ jumps between the values p=2 To plot these figures,

we have solved numerically the set of equations (63)

Solutions of equations (66) and (78), or equivalently the set (63), for giveninitial values describe the deterministic trajectories in the phase space for both

second-modes, the mode at frequency o and the mode at frequency 2o, in a general case

of the system that describes coupling of the two modes via the wð2Þnonlinearity

It is a matter of initial conditions whether we have a purely second-harmonicgeneration case [nbð0Þ ¼ 0, nað0Þ ¼ 1] or a purely downconversion case[nað0Þ ¼ 0, nbð0Þ ¼ 1] It is clear from (63) that for the purely downconversionregime [uað0Þ ¼ 0] the classical description does not allow for generating signal

at the fundamental frequency from zero initial value The quantum fluctuationsare necessary to obtain such a signal In a general case both processes take placesimultaneously and compete with each other If the initial amplitudes are welldefined, that is, there is no classical noise, the amplitudes at time t are also welldefined For quantum fields, however, the situation is different because of theinherent quantum noise associated with the vacuum fluctuations Some quantumfeatures, however, can be simulated with classical trajectories when the initialfields are chosen as random Gaussian variables with appropriately adjustedvariances, and examples of such simulations will be shown later

B Linearized Quantum EquationsAssuming that the quantum noise is small in comparison to the mean values ofthe field amplitudes, one can introduce the operators

which describe the quantum fluctuations On inserting the fluctuationoperators (79) into the original evolution equations (56) and keeping only thelinear terms in the quantum fluctuations, we get the equations

d

dt^a¼ 2ikð^aþh^bi þ h^aþi^bÞd

whereh^ai and h^bi are the solutions for the mean fields and can be identified withthe classical solutions With the scaled variables (61) and (62) we can rewriteequations (80) in the form

 ^QaðtÞ ¼ ^aðtÞeifaðtÞþ ^aþðtÞeifaðtÞ

^PaðtÞ ¼ i½^aðtÞeifa ðtÞ ^aþðtÞeifa ðtÞ

^PbðtÞ ¼ ^bðtÞeifb ðtÞþ ^bþðtÞeifb ðtÞ

 ^QbðtÞ ¼ i½^bðtÞeifb ðtÞ ^bþðtÞeif b ðtÞ ð82Þfor which we get from (81) the following set of equations:

d

dt ^Qa¼  ^Qaubsin # 2^Paubcos #

 ^Pb

ffiffiffi2

p

uasin #  ^Qb

ffiffiffi2

p

uacos #d

dt^Pa¼ ^Paubsin # ^Pb

ffiffiffi2

p

uacos #

þ  ^Qb

ffiffiffi2p

dt^Pb¼  ^Qa

ffiffiffi2

p

uasin #þ ^Pa

ffiffiffi2

dt ^Qb¼  ^Qa

ffiffiffi2

p

uacos # ^Pa

ffiffiffi2

In the case of pure second-harmonic generation, that is, for ubð0Þ ¼ 0 and

uað0Þ ¼ 1, we have from (59) that cos # ¼ 0 or # ¼ p=2, which implies that,according to (77) for k¼ 1, the scaled intensities obey the equations

psech td

dt^Pb¼  ^Qa

ffiffiffi2

psech td

dt^Pa¼ ^Patanh tþ  ^Qb

ffiffiffi2

psech td

 ^QaðtÞ ¼  ^Qað0Þð1  t tanh tÞ sech t  ^Pbð0Þ ffiffiffi

2

ptanh t sech t

Now, assuming that the two modes are not correlated at time t¼ 0, it isstraightforward to calculate the variances of the quadrature field operators andcheck, according to the definition (12), whether the field is in a squeezed state Ifthe initial state of the field is a coherent state of the fundamental mode and avacuum for the second-harmonic mode,jc0i ¼ juað0Þij0i, for which we have

h½ ^Qað0Þ 2i ¼ h½ ^Qbð0Þ 2i ¼ h½^Pað0Þ 2i ¼ h½^Pbð0Þ 2i ¼ 1 ð87Þthe variances of the two quadrature noise operators are described by thefollowing analytical formulas [44,45]:

h½ ^QaðtÞ 2i ¼ ð1  t tanh tÞ2sech2tþ 2 tanh2

t sech2th½^PaðtÞ 2i ¼ sech2tþ1

2ðsinh t þ t sech tÞ2h½ ^QbðtÞ 2i ¼ 2 tanh2tþ ð1  t tanh tÞ2

2 According tothe definition of squeezing (12), we find that the quadratures ^Qaand ^Pb becomesqueezed as t increases while the other two quadratures, ^Pa and ^Qb, arestretched For very long times (lengths of the nonlinear medium) the noise inthe amplitude quadrature of the fundamental mode is reduced to zero (perfectsqueezing), while for the second-harmonic mode it approaches the value1

2(50%squeezing) Quantum fluctuations in the other quadratures of both modesexplode to infinity as t goes to infinity Of course, we have to keep in mindthat the results have been obtained from the linearized equations that requirequantum fluctuations to be small In Fig 3a we have shown the evolution of thequadrature variances h½ ^QaðtÞ 2i and h½^PbðtÞ 2i exhibiting squeezing ofquantum fluctuations in both fundamental and second harmonic-modes Withdotted lines the classical amplitudes of the two modes are marked for reference.The value of unity for the quadrature variances sets the level of vacuumfluctuations (coherent states experience the same fluctuations), and we findthat indeed the quantum noise can be suppressed below the vacuum level in theamplitude quadrature h½ ^QaðtÞ 2i of the fundamental mode and the phasequadrature h½^PðtÞ 2i of the harmonic mode It becomes possible at theexpense of increased fluctuations in the other quadratures as to preserve the

validity of the Heisenberg uncertainty relation (9) We have

and as t! 1 both uncertainty products are divergent as t2=2 The evolution ofthe uncertainty products is illustrated in Fig 3b Since, except for the initialvalue, the value of the uncertainty product is larger than unity, the quantumstates produced in the second-harmonic generation process are not the minimumuncertainty states.

The linear approximation to the quantum noise equations presented in thissection shows that even in linear approximation the inherent property ofquantum fields — the vacuum fluctuations which are ubiquitous and alwayspresent — undergo essential changes when transformed nonlinearly The lineari-zed solutions suggest that perfect squeezing (zero fluctuations) is possible in thefundamental mode for long evolution times (long interaction lengths) Thismeans that one can produce highly nonclassical states of light in such a process.Later we will see to what extent we can trust in the linear approximation

C Symbolic CalculationsThe linear approximation with respect to quantum noise operators, whichassumes that the mean values of the fields evolve according to the classicalequations and the quantum noise represents only small fluctuations around theclassical solutions is a way to solve the operator equations (56) Anotheralternative is to use Taylor series expansion of the operator solution and makethe short time (or short length of the medium) approximation to find theevolution of the quantum (operator) fields This approach has been proposed byTanas´ [46] for approximate calculations of the higher-order field correlationfunctions in the process of nonlinear optical activity and later used byKozierowski and Tanas´ [21] for calculations of second order correlationfunction for the second-harmonic generation Mandel [9] has used this approach

to discuss squeezing and photon antibunching in harmonic generation Whendoing calculations with operators it is crucial to keep track of the operatorordering and use the commutation relations to rearrange the ordering Thismakes the calculations cumbersome and error-prone The first calculations wereperformed by hand, but now we have computers that can do the job for us Thecomputer symbolic calculations of the subsequent terms in a series expansionhave been performed by Bajer and Lisoneˇk [31] and Bajer and Perˇina [32].Bajer and Lisoneˇk [31] have written their own computer program for thispurpose (about 3000 lines of code in Turbo Pascal) We want to show here how

to do the same calculations with the freely available version of the computerprogram FORM [47] with only few lines of coding (see Appendix A).The main idea of the approximate symbolic computations is based on theseries expansion of any operator ^OðtÞ into a power series

^OðtÞ ¼ ^Oð0Þ þX1 tk

k!

dk

dtkOðtÞ^ 

t¼0 ð90Þ

where the subsequent derivatives are obtained from the Heisenberg equations ofmotion

ddt

k¼1

ti

k Dk

where

Dk¼ ½Dk1; ^H ¼ ½   ½½ ^Oð0Þ; ^H ; ^H ; ; ^H ð93Þ

is the kth-order commutator with D0 ¼ ^Oð0Þ

Implementing the algorithm sketched above in the computer symbolicmanipulation program FORM, as exemplified in Appendix A, and applyingthe method to the second-harmonic-generation (SHG) process, which is de-scribed by the interaction Hamiltonian ^HI given by (55), one can easilycalculate subsequent terms of the series (92) Restricting the calculations tothe fourth-order terms, we get

k 3; The latter products appeared as a result of application the bosoniccommutation relations (4) for the operators of the two modes, and these termsrepresent purely quantum contributions that would not appear if the fields wereclassical For classical fields, only the highest-order products survive Thequantum noise contributions appear in terms t3 and higher in the expansion(94) for the fundamental mode operators and in terms  t2 and higher in theexpansion (95) for the second harmonic mode operators However, for the initialconditions representing the purely second-harmonic generation process, speci-fically, under the assumption that the harmonic mode is initially in the vacuumstate such that ^bj0i ¼ 0, we can drop all the terms containing operators ^b or ^bþbecause they give zero due to the normal ordering of the operators Assuming,moreover, that the pump beam is in a coherent stateja0i we find the followingexpansions for the mean values of the operators ^aðtÞ and ^bðtÞ [7]

!

þ   

ð97Þ

On neglecting the quantum noise terms,  1=ja0j2, one can easily recognize

in (97) the first terms of the power series expansions of sech t and tanh t, whichare the classical solutions Whenja0j2 1, the quantum noise introduces onlysmall corrections to the classical evolution of the field amplitudes It is also seenthat the phase of the second harmonic field is phase-locked so as to satisfy

#¼ 2fa fb¼ p=2

We can thus expect from the short-time approximation that quantum noisedoes not significantly affect the classical solutions when the initial pump field isstrong We will return to this point later on, but now let us try to find the short-time solutions for the evolution of the quantum noise itself—let us take a look

at the quadrature noise variances and the photon statistics Using the operatorsolutions (94) and (95), one can find the solutions for the quadrature operators ^Qand ^P as well as for ^Q2 and ^P2 It is, however, more convenient to use thecomputer program to calculate the evolution of these quantities directly Let usconsider the purely SHG process, we drop the terms containing ^b and ^bþ afterperforming the normal ordering and take the expectation value in the coherent

stateja0i of the fundamental frequency mode, and in effect we arrive at

þ   h^P2aðtÞi ¼ 1 þ 2ja0j2 ða2

0Þi

þ   h^P2bðtÞi ¼ 1 þ ðktÞ2 2ja0j4þ a4

0þ a4 0

is probably the best known nonlinear optical process Because of its simplicityand variety of practical applications, it is a starting point for presentation ofnonlinear optical processes in. .. modesexplode to infinity as t goes to infinity Of course, we have to keep in mindthat the results have been obtained from the linearized equations that requirequantum fluctuations to be small In Fig 3a... distribution associated with the Wigner distribution (s¼ 0)may take negative values The distribution associated with the Husimi Qfunction is much broader than the Pegg–Barnett distribution, indicating