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Displaced squeezed number states: Position space representation, inner product, and some applications
Møller, Klaus Braagaard; Jørgensen, Thomas Godsk; Dahl, Jens Peder
Published in: Physical Review A
Link to article, DOI: 10.1103/PhysRevA.54.5378
Publication date: 1996
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Citation (APA): Møller, K. B., Jørgensen, T. G., & Dahl, J. P. (1996). Displaced squeezed number states: Position space representation, inner product, and some applications. Physical Review A, 54(6), 5378-5385. https://doi.org/10.1103/PhysRevA.54.5378
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PHYSICAL REVIEW A VOLUME 54, NUMBER 6 DECEMBER 1996
Displaced squeezed number states: Position space representation, inner product, and some applications
K. B. Mo”ller, T. G. Jo”rgensen, and J. P. Dahl Department of Chemistry, Chemical Physics, DTU-207, Technical University of Denmark, DK-2800 Lyngby, Denmark ͑Received 25 March 1996͒ For some applications the overall phase of a quantum state is crucial. For the so-called displaced squeezed number state ͑DSN͒, which is a generalization of the well-known squeezed coherent state, we obtain the position space representation with the correct overall phase, from the dynamics in a harmonic potential. The importance of the overall phase is demonstrated in the context of characteristic or moment generating func- tions. For two special cases the characteristic function is shown to be computable from the inner product of two different DSNs. ͓S1050-2947͑96͒06212-9͔
PACS number͑s͒: 42.50.Dv, 03.65.Ϫw
I. INTRODUCTION two applications of the CF are presented in Sec. V, and con- cluding remarks are found in Sec. VI. For many purposes the overall phase of a quantum state is unimportant—for instance, when calculating transition prob- II. DEFINITION OF A DISPLACED SQUEEZED abilities or expectation values. However, in some applica- NUMBER STATE tions the inclusion of the correct overall phase is crucial. A Since the displacement and squeezing operators may be special class of such applications is the computation of ex- defined in a variety of ways we find it useful to present our pectation values using characteristic or generating functions definitions along with notational aspects. ͓1–3͔ where the overall phase may depend on the parameter With a and a† as the canonical annihilator ͑see, e.g., ͓16͔͒ with respect to which differentiation is performed, as dem- and creator, respectively, onstrated in this paper for two specific cases. In the present work we are concerned with various prop- 1 1 † erties of a generalization of the squeezed coherent state aϭ ͑qϩip͒, a ϭ ͑qϪip͒, ͑1͒ ͱ2ប ͱ2ប ͑SCS͒ of the harmonic oscillator ͓4–8͔͑for reviews on squeezed states see, e.g., ͓9,10͔͒ which we call the displaced obeying the canonical commutator squeezed number state ͑DSN͒. This state is equivalent to the † generalized harmonic-oscillator state ͑GHO͓͒11,12͔, the ͓a,a ͔ϭ1, ͑2͒ only difference being that the DSN is defined using we define the reference harmonic-oscillator Hamiltonian, ‘‘squeezed state terminology.’’ with unity mass and frequency, as In particular, we are interested in the characteristic func- tion ͑CF͒ for Cahill-Glauber ordered products ͓13͔ of the † 1 2 2 HH.O.ϭប͑a a ϩ 1/2͒ϭ 2 ͑p ϩq ͒. ͑3͒ canonical annihilator and creator a and a†, respectively, and in the CF for powers of the number operator a†a, and it is As defined in Eq. ͑1͒ a and a† have the standard properties shown that the CFs in these cases may be computed as inner † products of two DSNs. a͉n͘ϭͱn͉nϪ1͘, a ͉n͘ϭͱnϩ1͉nϩ1͘, ͑4͒ The inner product is most conveniently evaluated in posi- with ͉n͘ as the nth eigenstate of the Hamiltonian Eq. ͑3͒. tion space and for this purpose we present an alternative way In accordance with Hollenhorst ͓7͔ and Caves ͓8͔ we de- of deriving the correct position space representation of the fine the displacement and squeezing operators as DSN—correct in the sense that we obtain the correct overall phase. Our approach differs from previously reported meth- D͑z͒ϭexp͑za†Ϫz*a͒ ods for the SCS ͓6,14,15͔ in exploiting the relation between 1 †2 2 rotations in phase space and time evolution in a harmonic S͑͒ϭexp͕ 2 ͑a Ϫ*a ͖͒, ͑5͒ potential, thereby rendering our procedure physically more i i intuitive. respectively. Here, zϭ͉z͉e and ϭre are complex num- bers and z is related to the real numbers q ,p by The paper is organized as follows: In Sec. II the basic z z framework is put forward along with the definition of a DSN. 1 Section III derives the correct position space representation zϭ ͑qzϩipz͒ ͑6͒ of the DSN and the general expression for the inner product ͱ2ប is presented. In Sec. IV the above mentioned CFs are ob- or tained as inner products from group-theoretical arguments and for two special cases they are given explicitly. Finally, qzϭͱ2ប͉z͉cos, pzϭͱ2ប͉z͉sin. ͑7͒
1050-2947/96/54͑6͒/5378͑8͒/$10.0054 5378 © 1996 The American Physical Society 54 DISPLACED SQUEEZED NUMBER STATES: POSITION... 5379
One may also introduce a unitary rotation operator as ͗q͉z,,n͘ϭ͗q͉D͑z͒S͉͑͒n͘. ͑15͒ R()ϭexp͓i(a†aϩ1/2)͔, where is real valued. Then, from the relation R()aR†()ϭa exp(Ϫi) and by use of To our knowledge such an expression does not exist—not the unitary property of R we have that ͑see also the appendix even for the ground state. It should be noted, though, that of ͓7͔͒ due to the relation in Eq. ͑11͒ an expression for ͗q͉D(z)S()͉n͘ can be obtained from an expression for † i R͑͒D͑z͒R ͑͒ϭD͑ze ͒, ͗q͉S()D(z)͉n͘. An expression for the latter ͑although only for nϭ0) has already been given by Yuen 6 and it has been † 2i ͓ ͔ R͑͒S͑͒R ͑͒ϭS͑e ͒. ͑8͒ rederived later in different ways by several authors ͓14,15͔. However, it is common to all these derivations that they These relations serve as a basis for the following. involve evaluation of complicated integrals which is not the With the displacement and squeezing operators ͑5͒ we case in the approach we present below. Furthermore, our now define a DSN by approach treats squeezing and displacement equally for all ͉z,,n͘ϭD͑z͒S͉͑͒n͘. ͑9͒ n. It has previously been argued that the coordinate repre- We immediately recover the squeezed state of Caves ͓8͔ as sentation of a DSN can be written on the form of a general- ͉z,͘ϵ͉z,,0͘. At this point it should be noted that Yuen ͓6͔ ized harmonic-oscillator state ͑GHO͓͒11,12͔ which takes the introduced the squeezed state as a squeezed coherent state form ͑SCS͒, i.e., as S()D(z)͉0͘. However, it follows from the definitions in Eq. ͑5͒ that ͑see, e.g., ͓9͔͒ 1 1 1/4 ͗q͉ ͘ϭ H ͓ͱ͑qϪqЈ͔͒G͑q͒eϪin, n n ͩ បͪ n D͑z͒S͑͒ϭS͑͒D͑˜z ͒, ͑10͒ ͱ2 n! ͑16͒ where where ˜zϭz coshrϪz*eisinhr, ͑11͒ i G͑q͒ϭexp ͓␣͑qϪqЈ͒2ϩp ͑qϪqЈ͒ϩ␥͔ , ͑17͒ and one may therefore easily get from one definition to the ͭ ប z ͮ other. For this reason we also refer to ͉z,͘ as a SCS. As for the reference oscillator we may introduce an anni- and ϭ2Im␣/ប. In this expression ␣ and ␥ may be complex hilator and a creator for the DSNs. The DSN annihilator A is while the other parameters are real. With the parameter val- defined through the requirement ues ␣ϭi/2, ϭ␥ϭ0, and qЈϭpЈϭ0, the nth GHO equals the coordinate representation of harmonic-oscillator eigen- A͉z,,n͘ϭAD͑z͒S͉͑͒n͘ϭD͑z͒S͑͒a͉n͘. ͑12͒ state ͉n͘. It was shown in ͓12͔ that a GHO with given pa- From this we immediately obtain rameters is proportional to a DSN and the z and parameters were determined in terms of the parameters in the GHO. AϭD͑z͒S͑͒aS†͑͒D†͑z͒ However, the proportionality constant was left undeter- mined. In the present work we determine uniquely the pa- † i ϭ͑aϪz͒coshrϪ͑a Ϫz*͒e sinhr, rameters in Eq. ͑16͒ so that
† † † † A ϭD͑z͒S͑͒a S ͑͒D ͑z͒ ͗q͉z,,n͘ϭ͗q͉n͘. ͑18͒ † Ϫi ϭ͑a Ϫz*͒coshrϪ͑aϪz͒e sinhr, ͑13͒ To evaluate the effect of the squeezing operator with the complex argument we decompose S(), as in ͓15͔, into a resulting in A and A† satisfying A,A† ϭ1 and having the ͓ ͔ product of rotations and a squeezing with a real parameter as following properties, analogous to Eq. ͑4͒ for a and a†: given by Eq. ͑8͒, A z, ,n ϭ n z, ,nϪ1 , ͉ ͘ ͱ ͉ ͘ S͑͒ϭS͑rei͒ϭR͑/2͒S͑r͒R†͑/2͒. ͑19͒ † A ͉z,,n͘ϭͱnϩ1͉z,,nϩ1͘. ͑14͒ Then the squeezing part is easy to evaluate since the squeez- ing operator with a real argument is just a scale transforma- In essence, for a fixed pair (zЈ,Ј) the set of DSNs with tion with the scale factor exp(Ϫr) ͓5,7͔, that is, nϭ0,1,... arethe eigenstates of another oscillator with a ͑complex-valued͒ frequency different from unity carrying ͗q͉S͑r͉͒͘ϭeϪr/2͗eϪrq͉͘. ͑20͒ the same properties as the eigenstates of the reference oscil- lator ͓17͔. To evaluate the effect of the rotation operators we make use of the relation between rotations in phase space and time III. POSITION SPACE REPRESENTATION evolution in a harmonic potential. The classical time evolu- AND INNER PRODUCT tion in a harmonic potential is a rigid rotation in phase space about the origin and since the dynamics in a harmonic po- A. DSN in position space tential is essentially the same in classical and quantum me- In this section we derive an explicit expression for a DSN chanics the same is to be expected in quantum mechanics. In in position space, i.e., fact, it follows from the previous section that 5380 K. B. MO” LLER, T. G. JO” RGENSEN, AND J. P. DAHL 54
R͑/2͒ϭUH.O.͑Ϫ/2͒, ͑21͒ As a special case we have the coordinate representation of a SCS where 1 1/4 i Ϫ1/2 ϪiHH.O.t/ប ͗q͉z,͘ϭ ͑coshrϩe sinhr͒ UH.O.͑t͒ϭe ͑22͒ ͩ បͪ is the time evolution operator for the harmonic oscilla- 1 coshrϪeisinhr tor. Thus ϫexp Ϫ ͑qϪq ͒2 ͭ 2ប ͩ coshrϩeisinhrͪ z S͑͒ϭUH.O.͑Ϫ/2͒S͑r͒UH.O.͑/2͒. ͑23͒ i ϩ pz͑qϪqz/2͒ , ͑29͒ The form of a GHO is well known to be preserved under ប ͮ time evolution in a harmonic potential ͓12,18͔, the param- eters ␣,,␥ satisfying where qzϭͱ2ប Rez and pzϭͱ2ប Imz. Invoking Eq. ͑11͒ we find that this expression coincides with Eq. ͑3.24͒ of Yuen 1 2␣0costϪsint ͓6͔. ␣ ϭ , t 2 ͩ 2␣ sintϩcostͪ We end this section by mentioning that the momentum 0 space and the Wigner phase-space representations of a GHO t were found in ͓12͔ and with Eq. ͑27͒ the expressions in ͓12͔ tϭ0ϩ2 ͵ dtЈIm␣tЈ , also give the momentum space and the Wigner phase-space 0 representations of a DSN. ␥ ϭ ␥ ϩ 1 ͑q p Ϫq p ͒ t 0 2 t t 0 0 B. Inner products of DSNs iប ϩ ln ͑2␣ sintϩ cost͒. ͑24͒ As mentioned previously, a set of DSNs with 2 0 nϭ0,1,... forafixed pair (zЈ,Ј) has the same properties as the reference oscillator. This means that for each pair of Here ␣0, 0,␥0 are initial values and qt ,pt are given by (zЈ,Ј) the set of DSNs constitutes a complete basis ͑the classical evolution of their initial values q0 ,p0. harmonic-oscillator eigenbasis being a special case͒. The in- The displacement operator D(z) displaces the expectation ner product of two DSNs characterized by (z1 ,1) and value of the position by q and the expectation values of the z (z2 ,2) therefore gives the transition amplitude between ba- momentum by pz according to sis states belonging to different bases. State expansion into a basis of DSNs is particularly convenient in the study of dy- iqpz /ប Ϫiqzpz /͑2ប͒ ͗q͉D͑z͉͒͘ϭ͗qϪqz͉͘e e . ͑25͒ namics in quadratic potentials ͓11,12,20,21͔. For the present purpose the inner product of two DSNs is desired in order to Putting it all together we therefore find that a DSN can be evaluate various characteristic functions and their derivatives written as shown in Sec. IV. ͗q͉z,,n͘ϭ͗q͉ ͘, ͑26͒ Since we have the coordinate representation of the DSNs, n the obvious way to calculate the inner product would be to where the right-hand side is a GHO defined in Eq. ͑16͒ with evaluate the integral the parameters given by dq͗z , ,n ͉q͗͘q͉z , ,n ͘. ͑30͒ i coshrϪeisinhr i coshrϩeϪisinhr ͵ 1 1 1 2 2 2 ␣ϭ , ϭ ln , 2 coshrϩeisinhr 2 coshrϩeisinhr ͩ ͪ ͩ ͪ This has been done for several special cases in the literature ͑in connection with the evaluation of Franck-Condon factors, iប qzpz ␥ϭ ln͑coshrϩeisinhr͒ϩ , ͑27͒ see, e.g., ͓22,23͔͒ but is rather cumbersome in the general 2 2 case. A different approach using annihilators and creators was used by Meyer ͓11͔ in the evaluation of the inner prod- and qЈϭqz , pЈϭpz . The uncertainties in position and mo- uct of a harmonic-oscillator eigenstate and a DSN. Meyer mentum of a DSN are ͓12͔ takes advantage of the linearity of the transformation be- tween A,A† and a,a† as given in Eq. ͑13͒. This approach can ប ប 2 easily be generalized to the evaluation of the inner product of ͑⌬q͒nϭ͑1ϩ2n͒ ϭ͑1ϩ2n͒ , 4Im␣ 2Res two DSNs since, in general, the transformation between A ,A† and A ,A† is a linear canonical transformation, given ប␣2 ប s 2 1 1 2 2 2 ͉ ͉ ͉ ͉ by ͑⌬p͒nϭ͑1ϩ2n͒ ϭ͑1ϩ2n͒ , ͑28͒ Im␣ 2Res A2 21 ␦21 Ϫ21 A1 where we have introduced the complex squeezing parameter † † A2 ϭ ␦21* 21* Ϫ21* A1 , ͑31͒ sϭϪ2i␣ of Schleich and Wheeler ͓19͔. We see that the uncertainty product equals the uncertainty product for a ͩ 1 ͪ ͩ 00 1ͪͩ 1 ͪ harmonic-oscillator eigenstate if s is real, that is, if ϭl with l integral. where 54 DISPLACED SQUEEZED NUMBER STATES: POSITION... 5381
i͑kϪl͒ klϭcoshrkcoshrlϪe sinhrksinhrl , For the cases of current interest we define the CFs, for future convenience, slightly differently by il ik ␦klϭe sinhrlcoshrkϪe sinhrkcoshrl , Cs͑,*͒ϭes*/2͗z,,n͉exp͑a†Ϫ*a͉͒zЈ,Ј,nЈ͘, ik klϭ͑zkϪzl͒coshrkϪ͑zk*Ϫzl*͒e sinhrk . ͑32͒ C͑͒ϭ͗z,,n͉exp͑ia†a͉͒zЈ,Ј,nЈ͘, ͑36͒ Having left the details for the Appendix, we find that the inner product of two DSNs can be written as where the ‘‘dummy’’ parameters and are complex val- ued and real valued, respectively. and * are considered ͗z1 ,1 ,n1͉z2 ,2 ,n2͘ independent during differentiation. From this the transition moments are given by n1 n2 1 2112* 1 21 12* ϭ exp ϩ ͑z2z1*Ϫz2*z1͒ †k l k l s ,͒*C ͑, ץ ץͱ ͭ 221 2 ͮͩ 21ͪ ͩ 21ͪ ͗z,,n͉͕a a ͖s͉zЈ,Ј,nЈ͘ϭ lim 21 Ϫ* 0 → [n1/2] [n2/2] min͑m1 ,m2͒ 1/2 ͑n1!n2!͒ ϫ ͚ ͚ ͚ † k k iC͑͒. ͑37͒ץ jϭ0 kϭ0 lϭ0 j!k!l!͑m1Ϫl͒!͑m2Ϫl͒! ͗z,,n͉͑a a͒ ͉zЈ,Ј,nЈ͘ϭ lim 0 → j k l Ϫ␦2121 ␦21* 21 21 ϫ 2 2 , ͑33͒ To evaluate the CFs Eq. ͑36͒ we invoke the properties of ͩ 2͑21͒ ͪ ͩ 2͑* ͒ ͪ ͩ 21* ͪ 12 12 the so-called two-photon Lie group H6 and its corresponding Lie algebra h6 ͓26͔. The generators of the algebra h6 are where m1ϭn1Ϫ2 j and m2ϭn2Ϫ2k. † 1 †2 2 † ͕a aϩ 2,a ,a ,a ,a,I͖ and the elements of the correspond- ing group, H , may be obtained by exponentiation of the IV. CHARACTERISTIC FUNCTIONS 6 h generators ͑see, e.g., ͓27͔͒. A general unitary element h of AS INNER PRODUCTS 6 the group H6 can be expressed as ͓26͔ In this section we consider how the general inner product obtained in Eq. ͑33͒ may be employed to facilitate the evalu- hϭDS R eiI, ͑38͒ ation of certain transition matrix elements using characteris- tic functions ͑CFs͒. Specifically, we are interested in the where D, S, and R are the operators introduced in Sec. II, transition matrix elements of the following two types of op- I is the identity, and is a real-valued parameter. Hence, the erators: most general state that can be formed by action of a unitary †k l iI ͑i͒ Cahill-Glauber ͓13͔ ordered products ͕a a ͖s , element of H6 on a number state ͉n͘ is DSRe ͉n͘. How- ͑ii͒ powers of the number operator (a†a)k. ever, since ͉n͘ is an eigenstate of the operator R the most Here, s is an ‘‘ordering’’ parameter where sϭ(Ϫ)1 means general state reduces to DS͉n͘eiЈ. In other words, acting on ͑anti-͒normal ordering and sϭ0 is symmetrical or Weyl or- a number state ͉n͘ with a unitary element of the Lie group dering ͑see, e.g., ͓24͔͒. H6 results in a DSN multiplied with a phase factor. In principle, we only need the transition matrix elements Keeping this in mind, we then return to the definition of of the first of the above types of operators since the powers the CFs in Eq. ͑36͒. The exponential operators appearing of the number operator can be expressed as a function of there are also seen to be unitary elements of H6. Hence, we Cahill-Glauber type products ͓25͔. However, because the have that number operator is ‘‘special’’ we find it appropriate to con- sider the powers of it separately. CFϭconstϫ͗z,,n͉h ͉zЈ,Ј,nЈ͘ In this section we generalize the notion of a ‘‘character- 1 Ϫ1 istic function,’’ in that both expectation values—as usual— ϭconstϫ͗n͉h h1hЈ͉nЈ͘ and transition matrix elements should be computable from it. ϭconstϫ͗z,,n͉z,Љ,ЉnЈ͘ , ͑39͒ Hence, in the general case the CF C⍀(⑀;⌿,⌿Ј) for the ‘‘transition moments’’ of an operator ⍀ between the states ͉⌿͘ and ͉⌿Ј͘ is chosen such that ͗⌿͉⍀k͉⌿Ј͘ is given by where ‘‘CF’’ means any of the two CFs of present interest ͓28͔. Thus, the CFs are nothing but the inner product of two k k different DSNs, multiplied with a constant. Below, the diag- C⍀͑⑀;⌿,⌿Ј͒, ͑34͒⑀ץ⌿͉⍀ ͉⌿Ј͘ϭ lim͗ ⑀ 0 onal CFs for the moments are given explicitly. → which may be accomplished by defining the CF as A. CF for Cahill-Glauber products ⑀⍀ C⍀͑⑀;⌿,⌿Ј͒ϭ͗⌿͉e ͉⌿Ј͘. ͑35͒ The diagonal CF for Weyl ordering, in general terms, is perhaps one of the most well-known CFs in quantum me- In the CF ⑀ is a ‘‘dummy’’ parameter used for the temporary chanics, since it is the Fourier transform of the celebrated differentiations, according to Eq. ͑34͒. Equation ͑35͒ is a Wigner distribution function ͑see, e.g., ͓1,24,29,30͔͒. Also straightforward generalization of the expectation value CF. the diagonal CFs for normal and antinormal ordering have In the following the expectation value CF is referred to as the been discussed extensively in the literature ͓1,30͔ and ac- diagonal CF. cordingly we shall not go into great detail with these. 5382 K. B. MO” LLER, T. G. JO” RGENSEN, AND J. P. DAHL 54
1. General case B. CF for powers of the number operator For a DSN the diagonal CF for Weyl ordering has been 1. General case obtained by Zhang, Feng, and Gilmore ͓26͔ in the special The CF C() for powers of the number operator, case ͉z,,0͘ ͓31͔. The extension to the general case with (a†a)k, is obtained in a manner similar to the previous one. DSNs is straightforward using the result above together with Hence, we immediately obtain the inner product Eq. ͑33͒. Using the Baker-Campbell-
Hausdorff relation for displacement operators ͑see, e.g., † Ϫi/2 ͓26,27͔͒ we immediately have C͑͒ϭ͗z,,n͉R͑͒D͑zЈ͒S͑Ј͒R ͑͒R͉͑͒nЈ͘e i 2i inЈ s ͑z Ϫ z ͒/2 s 2/2 ϭ͗z,,n͉zЈe ,Јe ,nЈ͘e , ͑43͒ C ͑,*͒ϭ͗z,,n͉ϩzЈ,Ј,nЈ͘e Ј* * Ј e ͉ ͉ . ͑40͒ where we have used that ͉n͘ is an eigenstate of R. 2. Diagonal case 2. Diagonal case In the diagonal case of the above CF the parameters Eq. ͑32͒ for the inner product are particularly simple, The parameters Eq. ͑32͒ for the diagonal CF are in this situation all nonvanishing, 21ϭ1, ␦21ϭ0, 2 2i 2 21ϭcosh rϪe sinh r i ˜ 21ϭϪ12ϭ coshrϪ*e sinhrϵ, ͑41͒ i 2i ␦21ϭe sinhr coshr͑1Ϫe ͒, and since ␦21ϭ0 the only contribution from the j and k sums in Eq. ͑33͒ is unity for jϭkϭ0. This, together with i/2 i i n1ϭn2ϭn resulting in m1ϭm2ϭn, gives us the final ex- 21ϭ͉z͉e ͑e Ϫ1͕͒cosЈ͑coshrϩe sinhr͒ pression ϩsinЈ͑coshrϪeisinhr͖͒, s ˜ 2 2 Cdiag͑,*͒ϭexp[ z*Ϫ*zϪ͉͉͑ Ϫs͉ ͉ ͒/2] Ϫi/2 Ϫi i n 12* ϭϪ͉z͉e ͑e Ϫ1͕͒cosЈ͑coshrϩe sinhr͒ n ͑Ϫ͉˜͉2͒k ϫ , ͑42͒ ϪsinЈ͑coshrϪeisinhr͒ , ͑44͒ k͚ϭ0 ͩ k ͪ k! ͖
where we have replaced the summation index nϪl in Eq. where ЈϭϪ/2. This results in the more complicated ͑33͒ by k. final expression
in 2 2i 2 Ϫ͑nϩ1/2͒ 2 Ϫ1 2 2 2 Cdiag͑͒ϭe ͑cosh rϪe sinh r͒ ϫexp͕Ϫ͉z͉ ͑1Ϫcos͒͑s cos ЈϪssin Ј͒ϩi͉z͉ sin͖ [n/2] [n/2] min͑m ,m ͒ 1 2 n! ͑e2iϪ1͒sinh rcosh r j ϫ ͚ ͚ ͚ 2 i 2 Ϫ1 2 2 jϭ0 kϭ0 lϭ0 j!k!l!͑m1Ϫl͒!͑m2Ϫl͒! ͩ 2͉z͉ ͑e Ϫ1͒ ͑s cos Јϩssin Јϩ2cosЈsinЈ͒ͪ 1ϪeϪ2i sinhr coshr k ͑ ͒ 2 Ϫ1 2 2 nϪl ϫ 2 Ϫi 2 Ϫ1 2 2 ͓͉z͉ ͑1Ϫcos͒͑s cos ЈϪssin Ј͔͒ , ͩ 2͉z͉ ͑1Ϫe ͒ ͑s cos Јϩssin ЈϪ2 cosЈsinЈ͒ͪ ͑45͒
where s is given by displacement and squeezing angles. Hence, any part of the DSN depending on these angles is to be differentiated—even the overall phase. coshrϪeisinhr sϭ i . ͑46͒ coshrϩe sinhr V. APPLICATIONS Different aspects of the results obtained in the previous In the case of a SCS ͉z,͘ the sums in Eq. ͑45͒ reduce to sections may be applied in different contexts, which in this unity, simplifying the CF considerably. section is demonstrated by two examples. From the expressions of the CFs, Eq. ͑45͒ and Eq. ͑42͒, and the prescriptions for the evaluation of the moments, Eq. A. Energy transfer in quadratic potentials ͑37͒, it is evident why the overall phase of a DSN is crucial in this context. The parameters with respect to which we A simple application of the moments of the number op- differentiate in Eq. ͑37͒ enters the DSN together with the erator is in the study of vibrational energy transfer in heavy 54 DISPLACED SQUEEZED NUMBER STATES: POSITION... 5383 particle scattering. A model for this is a forced harmonic oscillator with time-dependent frequency ͓11͔. In scaled co- ordinates the model Hamiltonian reads
1 2 2 H͑t͒ϭ 2 ͓p ϩ͑t͒q ͔ϪqF͑t͒, ͑47͒ where F(t Ϫϱ)ϭF(t ϱ)ϭ0 and (t Ϫϱ) ϭ(t ϱ)(→t Ϫϱ)ϭ(→t ϱ)ϭ1. If the initial→ state (t Ϫ→ϱ) is expanded→ in eigenstates→ of the harmonic oscil- lator→ Hϭ(p2ϩq2)/2, i.e.,
͉⌿i͘ϭ cn͉n͘, ͑48͒ ͚n the final state (t ϱ) is a sum of DSNs with the same ex- pansion coefficients→ ͓11,12͔ FIG. 1. Atomic inversion for an initial CS field as a function of the dimensionless time defined in the text. At low intensity the ͉⌿ f ͘ϭ͚ cn͉z,,n͘. ͑49͒ n familiar collapse is apparent whereas at high intensities the inver- sion approximates the classical Rabi oscillation. The energy transfer is given by
† † ⌬Eϭ͗a a͘ f Ϫ͗a a͘i . ͑50͒ where nϭ2gͱn/ប is the n-photon quantum Rabi fre- quency ͓34͔, g is the dipole coupling matrix element, and Similarly, the variance in the energy transfer can be defined Pn is the photon number distribution associated with the ini- as tial field state. ⌬2ϭ͗͑a†a͒2͘ Ϫ͗a†a͘2ϩ͗͑a†a͒2͘ Ϫ͗a†a͘2 Since Eq. ͑53͒ is a sum over n, the standard expression is f f i i inconvenient for high-intensity fields ͑see the example be- † † † † Ϫ2Re͗͑a a͒f͑a a͒i͘ϩ2͗a a͘f͗a a͘i. ͑51͒ low͒. However, by a Taylor expansion of the square of the sine in the above expression, and by employing the fact that Thus the energy transfer and the variance are expressed in ϱ k k ͚nϭ0Pnn ϭ͗n ͘, we may obtain the following alternative terms of the moments expression for the inversion:
† k 2 k ͗͑a a͒ ͘iϭ͚ ͉cn͉ n , n ϱ Ϫ4 k nk ͑ ͒ 2k ͗ ͘ Pba͑t͒ϭϪ͚ , ͑54͒ kϭ1 2͑2k͒! ¯nk † k † k ͗͑a a͒ ͘ f ϭ cm*cn͗z,,m͉͑a a͒ ͉z,,n͘, ͚m,n
where is a dimensionless time given by ϭgtͱ¯n/ប and † † † k ͗͑a a͒ f͑a a͒i͘ϭ͚ ncm*cn͗z,,m͉͑a a͒ ͉z,,n͘, ¯nϭ͗n͘. This expression is more suitable than Eq. ͑53͒ for m,n comparatively large field intensities, since the sum is inde- ͑52͒ pendent of n. A thorough discussion of the expression Eq. where kϭ1,2. One way to evaluate the moments in the sec- ͑54͒, along with an expression for the case of nonvanishing ond line is to rewrite (a†a)k in terms of A and A† and make detuning, is to appear elsewhere ͓35͔. use of Eq. ͑14͒ as done in ͓11͔ but an alternative way would For all practical purposes the series Eq. ͑54͒ must, of be to evaluate them using the CF method discussed in Sec. course, be truncated and in this case it is only valid for short IV ͓32͔. times. Hence, for the kth order truncation we need the mo- ments up to order k. However, for k much larger than two the direct evaluation of the moments becomes an enormous B. Atomic inversion in the Jaynes-Cummings model task. This is where the CF method comes into play ͓36͔. Again in this example we are concerned about moments To illustrate the usefulness of Eq. ͑54͒ we have calculated of the number operator. However, unlike the preceding ex- the inversion numerically for two different initial fields at ample, where one could argue that the CF method to a cer- four different intensities: tain degree is ‘‘overkill,’’ it is in this case essential for prac- ͑i͒ CS with ¯nϭ49,100,900,109 ͑Fig. 1͒, tical purposes. ͑ii͒ SCS with ¯nϭ49,100,900,109, ϭ0.01, and ϭ0 On exact one-photon resonance, the atomic inversion in ͑Fig. 2͒, the Jaynes-Cummings model ͑JCM͓͒33͔, with the atom ini- where is an alternative parameter for the squeezing mag- tially in its ground state, is customarily given by nitude r ͓37͔. In all cases we have truncated Eq. ͑54͒ at ϱ kϭ50. In comparison, using the standard expression for, e.g., 2 a CS one should at least include ͱ¯n terms which in the case Pba͑t͒ϭ͚ Pnsin ͑nt/2͒, ͑53͒ 9 nϭ0 of ¯nϭ10 would amount to approximately 30 000 terms. 5384 K. B. MO” LLER, T. G. JO” RGENSEN, AND J. P. DAHL 54
able to express the CFs for Cahill-Glauber ordered products and powers of the number operator as inner products of DSNs. Since the parameters, with respect to which we have to differentiate the CFs when evaluating moments, enter the wave function together with the squeezing and displacement angles and since the overall phase depends on these angles it is obvious that omitting the overall phase in this context would lead to erroneous results for the moments. The usefulness of the CFs in calculating moments was demonstrated by two examples: ͑a͒ calculation of mean value and variance of the energy transfer between states in a quadratic potential, and ͑b͒ calculation of the atomic inver- sion in the Jaynes-Cummings model for high-intensity quan- tized fields. The results of the first example was simplified by the CF method but could as well have been obtained by standard operator reordering, as done by Meyer for a special FIG. 2. Atomic inversion for an initial SCS field as a function of case. In the second example, however, the power of the CF the dimensionless time defined in the text. The collapse of the method emerged. In that case we obtained the well-known inversion is seen to persist at even high intensities, which manifests result that the atomic inversion approximates the classical squeezed light as being truly nonclassical. Rabi oscillations for high-intensity coherent fields. More in- terestingly, we were able to show that the atomic inversion Figure 1 shows the results for the CS and the well-known does not become a classical Rabi oscillation when a high fact that the inversion for a CS at high intensities approxi- intensity squeezed field is applied. mates the classical Rabi oscillation is recovered. In Fig. 2 the inversion for the SCS is shown. Unlike the CS case, the ACKNOWLEDGMENTS inversion does not approximate the classical Rabi oscillation at high intensities which, to the best of the authors’ knowl- The authors would like to thank Dr. Niels E. Henriksen edge, has not been demonstrated previously. and Professor Dr. Wolfgang P. Schleich for stimulating dis- cussions on the dynamics in harmonic potentials and the Jaynes-Cummings model. This research was supported by VI. CONCLUSION The Danish Natural Science Foundation. The purpose of the present work has been twofold: ͑a͒ We have obtained the position representation of a displaced APPENDIX squeezed number state with the correct overall phase using the relations between geometric operations in phase space In this appendix we briefly review the method used by and dynamics, and ͑b͒ we have shown by group-theoretical Meyer ͓11͔ to find the inner product of two generalized num- arguments that certain characteristic or moment generating ber states where the annihilator and creator defining one functions are obtained as inner products between different number state can be obtained by a linear canonical transfor- DSNs. mation of the annihilator and creator defining the other. The The position representation with the correct overall phase essence of Meyers approach is to use the annihilator and has been given previously by several authors in the special creator for the two DSNs to relate the inner product of these case of a squeezed state. However, these derivations all in- to the inner product of the corresponding SCSs. This can be volved the evaluation of complicated integrals and for the done invoking the relation general DSN one has not previously been concerned about the overall phase. The scheme put forward in this paper for n obtaining the position representation for the general DSN t † † ͚ ͉z,,n͘ϭetA ͉z,,0͘ϭetA ͉z,͘ ͑A1͒ avoids the evaluation of complicated integrals by exploiting n ͱn! that rotations in harmonic-oscillator phase space are nothing but time translations in the corresponding harmonic poten- tial. Hence, the decomposition of the general squeezing op- in order to write erator into a product of rotations in phase space and a squeezing along the q axis enables the construction of a DSN sn1tn2 to be regarded as two geometrical operations and two dy- ͚ ͗z1 ,1 ,n1͉z2 ,2 ,n2͘ namical operations: a forward time evolution, a squeeze n1,n2 ͱn1!n2! along the q axis, a backward time evolution, and finally a sA tA† ϭ z , ͉e 1e 2͉z , . ͑A2͒ displacement. In this way the construction of a DSN be- ͗ 1 1 2 2͘ comes physically very intuitive. † The reason for worrying about the overall phase, at all, Using Eq. ͑31͒ and ͓Ak ,Ak ͔ϭ1, the right-hand side of Eq. became very clear in connection with the considered CFs. ͑A2͒ can be expanded into a power series in s and t times the Because of the group relation between the displacement, inner product ͗z1 ,1͉z2 ,2͘ ͑see ͓11͔ for details͒. Comparing squeezing, and rotation ͑time-evolution͒ operators we were like powers of s and t then gives 54 DISPLACED SQUEEZED NUMBER STATES: POSITION... 5385
n1 n2 ͓n1/2͔ ͓n2/2͔ min͑m1 ,m2͒ 21 12* ͗z1 ,1 ,n1͉z2 ,2 ,n2͘ϭ͗z1 ,1͉z2 ,2͘ ͚ ͚ ͚ ͩ 21ͪ ͩ 21ͪ jϭ0 kϭ0 lϭ0
1/2 j k l ͑n1!n2!͒ Ϫ␦2121 ␦21* 21 21 ϫ 2 2 , ͑A3͒ j!k!l!͑m1Ϫl͒!͑m2Ϫl͒! ͩ 2͑21͒ ͪ ͩ 2͑* ͒ ͪ ͩ 21* ͪ 12 12 where m1ϭn1Ϫ2 j and m2ϭn2Ϫ2k. Since the SCSs are Gaussians in position space one easily gets
i͑2Ϫ1͒ Ϫ1/2 Ϫi͑qz pz Ϫqz pz ͒/͑2ប͒ ͗z1 ,1͉z2 ,2͘ϭ ͵dq͗z1 ,1͉q͗͘q͉z2 ,2͘ϭ͑coshr2coshr1Ϫe sinhr2sinhr1͒ e 2 2 1 1 ␣ ␣*͑q Ϫq ͒2ϩ͑p Ϫp ͒2/4Ϫ͑p Ϫp ͒͑␣ q Ϫ␣*q ͒ 2 1 z2 z1 z2 z1 z2 z1 2 z2 1 z1 ϫexp , ͭ iប͑␣2Ϫ␣1*͒ ͮ ͑A4͒ where again qzϭͱ2ប Rez, pzϭͱ2ប Imz and ␣ is given in Eq. ͑27͒. Introducing the parameters in Eq. ͑32͒ the inner product ͗z1 ,1͉z2 ,2͘ can be written in a more compact form as
1 2112* 1 ͗z1 ,1͉z2 ,2͘ϭ exp ϩ ͑z2z1*Ϫz2*z1͒ . ͑A5͒ ͭ 221 2 ͮ ͱ21
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