Bose–Einstein Condensates and Fractional Quantum Hall System

Nonlinear Response Functions of Strongly Correlated Boson Fields: BoseÐEinstein Condensates and Fractional Quantum Hall Systems S. Choi1, O. Berman2, V. Chernyak3, and S. Mukamel1, 2 1 Department of Physics and Astronomy, University of Rochester, Rochester, New York, 14627 USA 2 Department of Chemistry, University of Rochester, Rochester, New York, 14627 USA 3 Corning Incorporated, Process Engineering and Modeling, Corning, New York, 14831 USA e-mail: [email protected] Received February 18, 2004

Abstract—Nonlinear response functions of strongly correlated boson fields subject to an external perturbation are calculated. In particular, the second-order response functions and susceptibilities of finite-temperature BoseÐEinstein condensates (BECs) in a one-dimensional harmonic trap driven by an external field coupled to particle density are calculated by solving the time-dependent HartreeÐFockÐBogoliubov equations. These provide additional insight into the BEC dynamics, beyond those of the linear response regime. We further demon- strate that the results directly apply to electron liquids in the fractional quantum Hall effect regime that can be mapped onto an effective boson system coupled to a ChernÐSimons gauge field.

1. INTRODUCTION ties were calculated [7]. In this paper, we extend the formalism of [7] to obtain the nonlinear response of The study of strongly correlated systems has been BEC and fractional quantum Hall system. one of the most vigorous areas of research in con- densed-matter physics. Some notable examples include Several dynamical theories exist for finite-tempera- BoseÐEinstein condensation and associated phenom- ture BEC that take into account higher order collision ena, such as superfluidity and superconductivity [1], processes. They include the HartreeÐFockÐBogoliubov and the quantum Hall effect in two-dimensional elec- (HFB) theory [8Ð11], the time-dependent BogoliubovÐ tron systems [2]. A useful tool for the study of such de Gennes equations [12], quantum kinetic theory [13Ð complex, correlated many-body systems is provided by 18], and stochastic methods [19Ð23]. The TDHFB the- linear and nonlinear response functions to external per- ory is a self-consistent theory of BEC in the collision- turbations. The linear response provides an adequate less regime that progresses logically from the GrossÐ description of systems subjected to weak external per- Pitaevskii equation by taking into account higher order turbations. For stronger perturbations, nonlinear effects correlations of noncondensate operators. Although contained in the higher order terms in the perturbation TDHFB neglects higher order correlations included in series in the driving field may not be ignored and pro- the various quantum kinetic theories, the TDHFB equa- vide useful insights. The nonlinear response formalism tions are valid at very low temperatures near zero, even has had remarkable successes in the field of nonlinear down to the zero-temperature limit, and are far simpler optics, where the coherent response of semiconductor than the kinetic equations, which can only be solved metals and molecular systems to multiple laser fields is under certain approximations [16]. Another attractive routinely analyzed in terms of multitime correlation feature of TDHFB from a purely pragmatic point of functions [3, 4]. view is that the fermionic version of the theory has already been well developed in nuclear physics [8]. We It is by now well known that strong similarities exist therefore work at the TDHFB level in this paper, and between nonlinear optics and the dynamics of Bose- our approach draws upon the analogy with the time- Einstein condensates (BECs) owing to the intrinsic dependent HartreeÐFock (TDHF) formalism developed interatomic interactions in Bose condensates. Previous for nonlinear optical response of many-electron sys- work done in this area includes the demonstration of tems [24]. four-wave mixing in zero-temperature BEC using the GrossÐPitaevskii equation (GPE) [5]. We have recently Another important, strongly correlated many-body studied an externally driven, finite-temperature BEC system that we consider in this paper is the two-dimen- described by the time-dependent HartreeÐFockÐ sional (2D) electron system in the fractional quantum Bogoliubov (TDHFB) equations [6]. A systematic pro- Hall effect (FQHE) regime [2]. The FQHE for a 2D cedure was outlined for solving these equations pertur- electron gas in a strong, perpendicular, external mag- batively in the applied external field, and position- netic field has been observed through the quantization σ ν dependent linear response functions and susceptibili- of the Hall dc conductivity H( ) as a function of the

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NONLINEAR RESPONSE FUNCTIONS 1343 ν ≡ π ω filling factor (2 n)/(m c), where n is the mean 2D Hˆ = Hˆ + Hˆ '()t , (1) ω 0 density of the electrons, c = eB/mc is the cyclotron frequency, and B is the magnetic field; e is the electron where charge (we set c = 1 and = 1). At ν = 1/(2k + 1), where σ ν k is an integer k = 1, 2, …, H( ) varies in discrete steps sp † 1 † † Hˆ 0 = ∑H aˆ aˆ + ---∑V aˆ aˆ aˆ aˆ , (2) and is given by σ (ν) = ν(e2/2π). ij i j 2 ijkl i j k l H ij ijkl Nonlinear response functions for FQHE systems have not been extensively studied so far. We demon- and strate in this paper that the FQHE is, in fact, formally related to a BEC driven by an external potential, allow- H'()t = η∑E ()t aˆ †aˆ , (3) ing the results obtained for BEC to be carried over ij i j directly to FQHE. The nonlinear response of a 2D elec- ij tron system in the FQHE regime at fixed temperature is with important, since it can be used to distinguish between ()≡ 3 φ () ()φ, () the Fermi liquid and Luttinger chiral liquid behavior Eij t ∫d r i* r V f r t j r . (4) [25]. ν † It has been shown that, for = 1/(2k + 1), the origi- The boson operators aˆ i (aˆ i ) create (annihilate) a parti- nal 2D fermion problem can be mapped into a boson cle from a basis state with wave functions φ (r). The system coupled to a ChernÐSimons gauge field added i single-particle Hamiltonian Hsp in Eq. (2) is diagonal if to the time-independent external magnetic field, where φ the original fermion system and the boson system have the basis state i(r) is chosen to be the eigenstates of the the same charge density and Hall conductivity [26Ð28]. trap, while the symmetrized interaction matrix ele- The static Hall conductivity (xy component of the con- ments, σ ductivity tensor) calculated for this boson system, H = 1 2 V = ---[]〈|ij Vkl|〉+ 〈|ji Vkl|〉, (5) ν(e /2π), coincides with the conductivity obtained by ijkl 2 using Laughlin’s ansatz for the many-electron wave function [29]. For an even inverse filling number ν = where 1/(2k), the original fermion problem is mapped onto a 〈|ij Vkl|〉 composite fermion system coupled to a ChernÐSimons ν (6) gauge field [30]. At = 1/2, this gauge field eliminates 3 3 φ*()φ*()()φ()φ() the external effective vector potential for the effective = ∫d rd r' i r j r' V rrÐ ' k r' l r , composite fermion system [31Ð33]. In this paper, we describe the collision between the atoms, with V(r Ð r') consider FQHE with the odd inverse denominator of ν being a general interatomic potential. H'(t) describes the filling number = 1/(2k + 1). Using this mapping, the effect of a general external force V (r, t) on the con- we can apply our results based on a generalized coher- f ent state (GCS) ansatz for the many-particle wave func- densate that mimics the mechanical force applied tion of a weakly interacting effective boson system [7, experimentally, such as shaking of the trap [34, 35]. 39, 40] to compute the second-order response of The dynamics of the system is calculated by solving FQHE. the time-dependent HartreeÐFockÐBogoliubov The paper is organized as follows. In Section 2, the (TDHFB) equations for the condensate mean field, zi = 〈〉 ρ 〈〉† 〈〉〈〉† second-order time and frequency domain response aˆ i , the noncondensate density ij = aˆ i aˆ j Ð aˆ i aˆ j , functions for an externally driven BEC are presented, κ 〈〉 including the discussion of numerical results for a con- and the noncondensate correlations ij = aˆ iaˆ j Ð 〈〉〈〉 densate of 2000 atoms in a one-dimensional harmonic aˆ i aˆ j . These are presented in Appendix A. These trap. We then show in Section 3 how the formalism may nonlinear coupled equations are solved by an order-by- ρ κ be applied for computing the second-order response for order expansion of the variables zi, ij, and ij; at each a fractional quantum Hall system. In Section 4, we con- order, the resulting equations to be solved become lin- clude. More detailed derivation of the second-order ear [7]. It is found that the sequence of linear equations response functions and susceptibilities are provided in to be solved has the general form Appendices C and D, respectively. ()n dψ ()t ()n ()n ()n i------= ψ ()t + λ ()t , (7) 2. NONLINEAR RESPONSE dt OF AN EXTERNALLY DRIVEN BEC where we have denoted the set of nth-order variables 2.1. Formalism ()n ρ ()n κ ()n × zi , ij , and ij as a 2N(2N + 1) 1 column vector We adopt the notation of [7] throughout the paper. in the Liouville space notation, where N is the number The Hamiltonian describing the system of an externally ()n ()n ()n ()n driven, trapped atomic BEC is given by of basis states used, ψ (t) = [z (t), z * (t), ρ (t),

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()n ()n ()n κ (t), ρ * (t), κ * (t)]T, and the 2N(2N + 1) × scale experimental effort. The position-dependent nth- (n) ()n ()n ()n ()n 2N(2N + 1) matrix is the nth-order Liouville oper- order solution ψ (r, t) ≡ [(z r, t), z *(r, t), ρ (r, t), ator obtained from the TDHFB equations [7]. The ()n ()n ()n matrices (n) for all orders n > 0 are identical, i.e., κ (r, t), ρ * (r, t), κ * (r, t)]T can be deﬁned in the (1) ≡ (2) ≡ … ≡ (n), so that only the matrices (0) Liouville space notation from relations (11)Ð(13) by and (1) are required to be calculated. (0) and (1) introducing a 2N(2N + 1) × 2N(2N + 1) square matrix are presented in Appendix B. The formal solution to ϒ˜ Eq. (7) is (r):

t ()n ()n ()n 1 i ()n ()n ψ ()ϒr, t ≡ ˜ ()ψr ()t , (14) ψ ()t = ----- exp Ð--- ()ttÐ ' λ ()t' dt'. (8) i∫ 0 where In the frequency domain, the solution to Eq. (7) takes the form ϒ˜ ()r (15) ()n 1 ()n ˜ ˜ ψ ()ω λ ()ω = diag[]φ()φr ,,,,*()Φr ρ()Φr κ()Φr ρ*()Φr ,κ*()r . = ------()n- , (9) ω Ð ()n Here, “diag[…]” denotes that ϒ˜ (r) is a block diagonal where ψ (ω) and λ(n)(ω) are the Fourier transforms of ()n square matrix made of N × N blocks φ˜ (r), φ˜ * (r) and ψ (t) and λ(n)(t), respectively. 2 × 2 Φ Φ Φ Φ φ˜ For the zeroth order (n = 0), we obtain the time-inde- N N blocks ρ(r), κ(r), ρ* (r), κ* (r). (r) is pendent HFB equations (TIHFB) itself a diagonal matrix with the ith diagonal element () () given by the basis states φ (r), and Φρ(r) and Φκ(r) are 0 ψ 0 () i t = 0, (10) also diagonal matrices whose ijth diagonal element is while, for the first order (n = 1), the equation solved is given by [Φρ(r)] = φ* (r)φ (r), and [Φκ(r)] = λ(1) × ζ ij, ij i j ij, ij Eq. (7) with (t) being a 2N(2N + 1) 1 vector (t) φ (r)φ (r), respectively. Individual real-space variables calculated in [7]; ζ(t) is also presented in Appendix B. i j (n) ρ(n) κ(n) Once the nth-order solution to TDHFB is found, we z (r, t), (r, t), and (r, t) are then obtained by can proceed to define the nth-order response functions. summing over the appropriate elements of the vector ()n The physical significance of the response functions ψ (r, t): becomes more transparent when the nth-order solutions (n) α , where α is one of the variables z, ρ, or κ, are n () ()n expressed in real space. We therefore introduce the z n ()r, t = ψ()r, t , position-dependent variables written in terms of the ∑ i trap eigenstate basis: i = 1 2 2nn+ () ()n () () ρ n (), ψ(), n (), n ()φ() r t = ∑ i r t , (16) z r t = ∑z j t j r , (11) i = 2n + 1 j 2 2n + 2n ()n ()n () () κ (), ψ(), ρ n (), ρn ()φ ()φ() r t = ∑ i r t . r t = ∑ ij t i* r j r , (12) 2 ij i = 2nn++1 The position-dependent second-order (n = 2) κ()n (), κ()n ()φ()φ() r t = ∑ ij t i r j r . (13) ()2 response function Kα (t, t1, t2, r, r1, r2) is defined as ij follows: Real-space noncondensate density and noncondensate correlations are, in general, nonlocal functions of () () α 2 (), 2 (),,,,, two spatial points ρ(r', r) and κ(r', r). We consider the r t = ∫Kα tt1 t2 rr1 r2 , (17) r = r' case in this paper, as this is most physically acces- (), (), sible. Measuring these quantities with r ≠ r' involves V f r1 t1 V f r2 t2 dt1dt2dr1dr2, observing atomic correlations, which is far more diffi- cult than for photonic counterparts, since, for a matter where α(2)(r, t) are second-order solutions, α = z, ρ, κ. wave, merely implementing individual components, The second-order response functions for the conden- such as beam splitters and mirrors, requires a large- sate, the noncondensate density, and the correlation that

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φ φ δ Ξ we use in our numerical calculations are given by the n* (r) j(r) im. The vector K of Eq. (22) may be writ- following expressions: ten () 2 (),,,,, Ξ [] ,,,,, * T Kz tt1 t2 rr1 r2 K = K *K K K K K* , (25) N (18) ()2 ()2 × 2 × ˜ with the N 1 matrix K and N 1 matrices K, K = ∑[]ϒ()r ()K I ()tt,,,t r ,r + K II ()tt,,,t r ,r , 1 2 1 2 1 2 1 2 i given in Appendix C (Eqs. (C.6)Ð(C.10)). i = 1 ()2 Having found the time domain response Kα (t, t1, ()2 (),,,,, Kρ tt1 t2 rr1 r2 t2, r, r1, r2), the second-order susceptibility is obtained 2 2NN+ by a Fourier transform to the frequency domain: ()2 [ϒ˜ ( ,,, , ∞ ∞ ∞ = ∑ ()r K I ()tt1 t2 r1 r2 (19) ()2 ()ΩΩ,,,,,Ω i = 2N + 1 Kα 1 2 rr1 r2 = ∫dtt∫d 1∫dt2 () (26) 2 ,,, , )] 0 0 0 + K II ()tt1 t2 r1 r2 i, × ()2 (),,,,, ()Ω Ω Ω Kα tt1 t2 rr1 r2 exp i ti++1t1 i 2t2 . ()2 (),,,,, Kκ tt1 t2 rr1 r2 The final expression for the susceptibility used in 2 2N + 2N our numerical calculations is given for the condensate, ()2 = [ϒ˜ ()r (K ()tt,,,t r ,r the noncondensate density, and the noncondensate cor- ∑ I 1 2 1 2 (20) relations in the following equations: 2 i = 2NN++1 ()2 ()2 K ()Ð Ω Ð Ω ; Ω ,,,,Ω rr r ,,, , )] z 1 2 1 2 1 2 + K II ()tt1 t2 r1 r2 i, N ˜ ()2 = [ϒ()r (K I ()Ω ,,,Ω r r (27) where 2N(2N + 1) × 2N(2N + 1) matrix ϒ˜ (r) was ∑ 1 2 1 2 i = 1 ()2 × () defined in Eq. (15) and the 2N(2N + 1) 1 vectors K I 2 Ω ,,,Ω )] ()2 + K II ()1 2 r1 r2 i, and K II are defined: ()2 ()Ω Ω Ω ,,,,Ω Kρ Ð 1 Ð 2; 1 2 rr1 r2 ()2 (),,, , 2 K I tt1 t2 r1 r2 2NN+ (21) ()2 () [ϒ˜ ( Ω ,,,Ω ()Φ˜ ()()Φ˜ ()ψ0 = ∑ ()r K I ()1 2 r1 r2 (28) = ttÐ 1 r1 t1 Ð t2 r2 , i = 2N + 1 () ()2 2 (),,, , + K II ()Ω ,,,Ω r r )] , K II tt1 t2 r1 r2 1 2 1 2 i t (22) ()2 ()Ω Ω Ω ,,,,Ω Kκ Ð 1 Ð 2; 1 2 rr1 r2 = dτ()Ξt Ð τ K()τ Ð t ,,,τ Ð t r r , ∫ 1 2 1 2 2 2N + 2N () 0 [ϒ˜ ( 2 Ω ,,,Ω = ∑ ()r K I ()1 2 r1 r2 (29) where 2 i = 2NN++1 ()2 () Ω ,,,Ω )] ()≡ i 2 () + K II ()1 2 r1 r2 i, ttÐ ' exp Ð,--- ttÐ ' (23) where () and 2 ()Ω ,,Ω , , K I 1 2 rr1 r2 () () Φ˜ () [Φ()Φ,,()ΦÐ ()Φ ,+ () 1 ()0 (30) r' = diag r' * r' r' r' , ------ϒ˜ ()()ΦΩ Ω ˜ ()()ΦΩ ˜ ()ψ (24) = Ð 2 r 1 + 2 r1 2 r2 , () () π Φ Ð *()Φr' ,]+ *()r' . 4 and ˜ Φ (r') is therefore a 2N(2N + 1) × 2N(2N + 1) block () 2 ()Ω ,,Ω , , diagonal square matrix with the blocks consisting of K II 1 2 rr1 r2 × Φ φ* φ 2 × 2 (31) N N square matrices [ (r)]ij = i (r) j(r) and N N 1 ˜ = Ð------ϒ()r ()ΞΩ + Ω K()Ω ,,,Ω r r . Φ(±) φ φ δ ± 3 1 2 1 2 1 2 square matrices [ (r)]ij, mn = i* (r) m(r) jn 8π i

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Here, ϒ˜ (r) and Φ˜ (r) are as defined in Eqs. (15) and equations is the most numerically involved step, as it (24) and requires solving nonlinear coupled equations. Griffin has provided a self-consistent prescription for solving ()ω ≡ 1 the TIHFB, in terms of the BogoliubovÐde Gennes ------()2 -. (32) ω Ð + i equations [10]. We have therefore followed the prescription of [10] to find the solution to TIHFB. Once the In addition, the vector Ξ (Ω , Ω , r , r ) of Eq. (31) zeroth-order solution is found, it is straightforward to K 1 2 1 2 calculate the first- and the second-order response func- may be written as tions. The calculation of the eigenvalues of the non- (2) Ξ ()Ω ,,,Ω r r Hermitian matrix required for computing the K 1 2 1 2 response functions was carried out using the Arnoldi T (33) algorithm [36]. ˜ ,,,,,˜ * ˜ ˜ ˜ * ˜ * = K K K K K K In order to provide an indication of the structure of the matrix (2), we first plot in Fig. 1 the linear suscep- × ˜ 2 × ˜ ˜ (1) Ω with the N 1 matrix K and N 1 matrices K , K tibility K ( , r = 0, r1 = 0) for zero and finite temper- given in Appendix D (Eqs. (D.8)Ð(D.12)). atures. Peak positions indicate the resonant frequencies. 2.2. Time Domain Response The second-order time domain response functions given in Eqs. (18)Ð(20) and Appendix C have been cal- So far, all our results were given in the trap basis and culated. We first obtained the numerical solution to hold for a general interatomic interaction potential. In ()0 the following numerical calculations, we approximate TIHFB; the 2N(2N + 1) × 1 vector ψ evaluated at the interatomic potential V(r Ð r') in Eq. (6) by a contact zero and finite temperatures; the 2N(2N + 1) × 2N(2N + 1) potential: matrices ϒ˜ , , and Φ˜ defined in Eqs. (15), (23), and π2 () δ() 4 a (24); and the 2N(2N + 1) × 1 vector Ξ defined in V rrÐ ' U0 rrÐ ' , U0 = ------, (34) K m Eqs. (25) and (C.8). Substituting these into Eqs. (21) where a is the s-wave scattering length and m is the and (22), the final calculation involves matrix multipli- ()0 atomic mass. This is valid because wave functions at cation of 2N(2N + 1) × 1 vectors ψ and Ξ with ultracold temperatures have very long wavelengths K compared to the range of interatomic potential imply- 2N(2N + 1) × 2N(2N + 1) matrices ϒ˜ , , and Φ˜ and ing that details of the interatomic potential become integration over the time variable τ. ϒ˜ and Φ˜ are con- unimportant. The tetradic matrices Vijkl defined in structed in terms of the harmonic oscillator basis states, Eq. (6) are then simply given by which are calculated numerically from the recursive π2 formula that involves the Gaussian function multiply- 4 a φ ()φ()φ()φ() V ijkl = ------∫ i* r *j r k r l r dr. (35) ing the Hermite polynomials [37]. The matrix is cal- m culated using a MATLAB function that uses the Padé We consider a 2000-atom one-dimensional (1D) approximation for matrix exponentiation [38]. condensate in a harmonic trap. As demonstrated by the We present the second-order response function in extensive body of literature so far, consideration of a the time domain K(2)(t, t , t , r, r , r ) as a function of r 1D BEC helps us to identify key features of a BEC 1 2 1 2 when full 3D simulation is computationally prohibitive. and r1 at various times t, t1, t2 and r2. This provides a The parameters used for our numerical calculation of way to depict graphically the correlation involving six 2 variables t, t1, t2, r, r1, and r2 on a two-dimensional plot ()0 4π a ψ are as follows: U = ------= 0.01, and tempera- and gives a “snapshot” of the position-dependent sec- 0 m ond-order correlations across the condensate. The times ω ω ω tures 0 trap/k and 10 trap/k, where trap is the trap t, t1, and t2 are, respectively, the time of detection and frequency, k is the Boltzmann constant, and the basis the time of the first and second applied short fields, set size of N = 5 is used, which was found to be suffi- while r, r1, and r2 denote the corresponding spatial vari- cient for the purposes of simulation; a higher basis ables. The position dependence is important since the number did not noticeably alter the final results. We experimentally produced condensates are mesoscopic keep the trap units throughout with 256 grid points for in size; in optical spectroscopy, however, the dipole position. The same parameters were used in the calcu- approximation usually applies and, consequently, the lations of the linear response in [7]. spatial dependence of the response is irrelevant. Figure 2 To solve for the second-order response, both the shows the absolute value of the second-order response zeroth- and the first-order solutions must be found. Cal- functions in the time domain with the time t2 fixed at culation of the zeroth-order solution from the TIHFB t2 = 0. Figure 2a is for the position of perturbation fixed

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(1) Ω Ω log [K (– 1; 1)]

5 Kz 0

5 Kρ = 0 T 0

5 Kκ

0 5 Kz 0

5 Kρ = 10 T 0

5 Kκ

0 12345678910 Ω 1 (1) Ω Fig. 1. Natural log of linear susceptibility K ( , r, r1) at r = r1 = 0 vs. frequency. Top three panels, zero temperature; bottom three ω Ω panels, ﬁnite temperature 10 /k. The frequency 1 is given in units of the trap frequency.

at r2 = 0, i.e., the center of the trapped atomic cloud, tures. The response functions attain spatial symmetry while Fig. 2b is for r2 = Ð5 at the edge of the cloud. All more rapidly at zero temperature owing to the weaker positions are referred to in harmonic oscillator length coupling between the variables z, ρ, and κ. units. The plots are for zero-temperature condensate at the short, intermediate, and long times ({t, t } = {5.89, 1 2.3. Frequency Domain Response 2.6}, {t, t1} = {15.7, 7.2}, {t, t1} = {31.4, 15.7}) as indicated at the top of each column of the figures. The times Using Eqs. (27)Ð(29), we have calculated the sec- ω ond-order susceptibility. It involves matrix multiplica- are given in units of the trap period, 1/ trap. The top, middle, and bottom rows give the response function for ()0 tion of 2N(2N + 1) × 1 vectors ψ and ΞK (ω) defined the condensate z, noncondensate density ρ, and non- × κ in Appendix D with the 2N(2N + 1) 2N(2N + 1) matri- condensate correlation , respectively. The dashed cir- ˜ ˜ cle represents the spatial extent of the trapped BEC. ces ϒ , (ω), and Φ . The Green’s function (ω) is The corresponding plot for a finite-temperature BEC at calculated as follows: ω temperature T = 10 trap/k is displayed in Fig. 3. † 1 ξνζν ()ω ------The spatially asymmetric function at r = Ð5 reverse ==------()2 - ∑ , (36) 2 ωωÐ ν + i ω Ð + i ν shape for r2 = 5, with similar changes also observed for the r = ±2.5 pair. With r = ±2.5, which is a point inside 2 2 ξ (2) the atomic cloud, the contours had a markedly less where ν is the right eigenvector of with eigenval- (2) ± ues ων such that ξν = ωνξν and ζν are the left eigen- symmetric shape than for r2 = 5 at the very edge of the (2) ξ ζ† atomic cloud. The response functions take a more sym- vectors of such that ∑ν ν ν = 1. The eigenval- metric shape as time increases. The r2 dependence of (2) ues ων of were calculated using the Arnoldi algo- the response function is found to be reduced at longer times. Comparing Figs. 2 and 3, the response functions rithm [36]. clearly show a strong temperature dependence, with the The absolute value of the second-order response func- | (2) Ω Ω Ω | functions giving distinct contours at different tempera- tion in the frequency domain K ( , 1, 2, r, r1, r2) is

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(a) |K(2)(t = 5.89, t' = 2.62)||K(2)(t = 15.7, t' = 7.2)||K(2)(t = 31.4, t' = 15.7)| 10 5 '

X 0 Kz –5 –10

10 5 '

X 0 Kρ –5 –10

10 5 '

X 0 Kκ –5 –10 –10 0 10 –10 0 10 –10 0 10 X X X (b)

|K(2)(t = 5.89, t' = 2.62)||K(2)(t = 15.7, t' = 7.2)||K(2)(t = 31.4, t' = 15.7)|

10 5 '

X 0 Kz –5 –10

10 5 '

X 0 Kρ –5 –10

10 5 '

X 0 Kκ –5 –10 –10 0 10 –10 0 10 –10 0 10 X X X

| (2) | Fig. 2. K (t, t1, t2, r, r1, r2) , i.e., the absolute value of the second-order response functions in the time domain with the time t2 ﬁxed at t2 = 0. (a) r2 = 0; (b) r2 = Ð5. The plots are for zero-temperature condensate at the short, intermediate, and long times t and t1 written at the top of each column of the ﬁgures. The top, middle, and bottom rows give the response function for the condensate, noncondensate density, and noncondensate correlation, respectively. The diameter of the dashed circle represents the spatial extent of the trapped BEC. The positions x and x' are given in harmonic oscillator length units.

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(a) |K(2)(t = 5.89, t' = 2.62)||K(2)(t = 15.7, t' = 7.2)||K(2)(t = 31.4, t' = 15.7)| 10 5 '

X 0 Kz –5 –10 10 5 '

X 0 Kρ –5 –10 10 5 '

X 0 Kκ –5 –10 –10 0 10 –10 0 10 –10 0 10 X X X (b) |K(2)(t = 5.89, t' = 2.62)||K(2)(t = 15.7, t' = 7.2)||K(2)(t = 31.4, t' = 15.7)| 10 5 '

X 0 Kz –5 –10 10 5 '

X 0 Kρ –5 –10 10 5 '

X 0 Kκ –5 –10 –10 0 10 –10 0 10 –10 0 10 X X X

Fig. 3. The same as in Fig. 2, but at a ﬁnite temperature of 10ω/k. The positions x and x' are given in harmonic oscillator length units.

Ω displayed in Fig. 4 with the position variable r2 set at and 2 indicated at the top of each column. We chose Ω Ω Ω Ω (a) r2 = 0 at the center of the atomic cloud and (b) r2 = the frequencies such that 1, 2, and 1 + 2 are off- Ð5 at the edge of the atomic cloud. The plots are for resonant with respect to the eigenvalues of (2) (ﬁrst Ω Ω Ω Ω Ω zero-temperature condensate at various frequencies 1 column, 1 = 2.23, 2 = 1.55); both 1 and 2 are on-

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(a) | (2) Ω Ω || (2) Ω Ω || (2) Ω Ω | K ( 1 = 2.23, 2 = 1.55)K ( 1 = 2.2, 2 = 1.5) K ( 1 = 0.7, 2 = 1.5) 10 5 '

X 0 Kz –5 –10 10 5 '

X 0 Kρ –5 –10 10 5 '

X 0 Kκ –5 –10 –10 0 10 –10 0 10 –10 0 10 X X X (b) | (2) Ω Ω || (2) Ω Ω || (2) Ω Ω | K ( 1 = 2.23, 2 = 1.55) K ( 1 = 2.2, 2 = 1.5) K ( 1 = 0.7, 2 = 1.5) 10 5 ' K X 0 z –5 –10 10 5 '

X 0 Kρ –5 –10

10 5 '

X 0 Kκ –5 –10 –10 0 10 –10 0 10 –10 0 10 X X X

| (2) Ω Ω Ω | Fig. 4. K ( , 1, 2, r, r1, r2) , i.e., the absolute value of the second-order response functions in the frequency domain with the Ω Ω variable r2 set at (a) r2 = 0 and (b) r2 = Ð5. The plots are for zero-temperature condensate at the frequencies 1 and 2 given at the top of each column. Denoting “off-resonant” when the frequency does not match an eigenvalue of (2) and “on-resonance” when Ω Ω Ω Ω Ω Ω a frequency matches an eigenvalue, these frequencies are chosen such that 1, 2, and 1 + 2 are off-resonant ( 1 = 2.23, 2 = Ω Ω Ω Ω Ω Ω 1.55); both 1 and 2 are on-resonance while 1 + 2 is off-resonant ( 1 = 2.2, 2 = 1.5); and, ﬁnally, frequencies chosen so that Ω Ω Ω Ω 1 + 2 is on-resonance ( 1 = 0.7, 2 = 1.5). The top, middle, and bottom rows give the response function for the condensate, noncondensate density, and noncondensate correlation, respectively. The diameter of the dashed circle represents the spatial extent of the trapped BEC. The positions x and x' are given in harmonic oscillator length units.

LASER PHYSICS Vol. 14 No. 10 2004 NONLINEAR RESPONSE FUNCTIONS 1351 (a) | (2) Ω Ω || (2) Ω Ω || (2) Ω Ω | K ( 1 = 2.45, 2 = 1.6) K ( 1 = 2.43, 2 = 1.5) K ( 1 = 0.92, 2 = 1.5) 10 5 '

X 0 Kz –5 –10 10 5 '

X 0 Kρ –5 –10 10 5 '

X 0 Kκ –5 –10 –10 0 10 –10 0 10 –10 0 10 X X X (b) | (2) Ω Ω || (2) Ω Ω || (2) Ω Ω | K ( 1 = 2.45, 2 = 1.6) K ( 1 = 2.43, 2 = 1.5) K ( 1 = 0.92, 2 = 1.5) 10 5 '

X 0 Kz –5 –10 10 5 '

X 0 Kρ –5 –10 10 5 '

X 0 Kκ –5 –10 –10 0 10 –10 0 10 –10 0 10 X X X Fig. 5. The same as in Fig. 4, but at a finite temperature of 10ω/k. At finite temperature, the resonant frequencies are shifted from Ω Ω Ω Ω the zero-temperature counterpart, so that the actual frequency combinations used are the following: 1, 2, and 1 + 2 are off- Ω Ω Ω Ω Ω Ω Ω Ω resonant ( 1 = 2.45, 2 = 1.6); both 1 and 2 are on-resonance with 1 + 2 off-resonant ( 1 = 2.43, 2 = 1.5); and, finally, Ω Ω Ω Ω 1 + 2 is on-resonance ( 1 = 0.92, 2 = 1.5). The positions x and x' are given in harmonic oscillator length units.

Ω Ω resonance, while 1 + 2 is off-resonant (second col- response function for the condensate, noncondensate Ω Ω Ω Ω Ω umn, 1 = 2.2, 2 = 1.5); and, finally, 1 + 2 and 2 density, and noncondensate correlation, respectively. Ω Ω are on-resonance ( 1 = 0.7, 2 = 1.5). As with the pre- The result for a finite-temperature BEC at temperature ω vious figures, the top, middle, and bottom rows give the T = 10 trap/k is displayed in Fig. 5. At finite tempera-

LASER PHYSICS Vol. 14 No. 10 2004 1352 CHOI et al.

| (2) Ω Ω ||(2) Ω Ω | | (2) Ω Ω | K ( 1, 2) K ( 1, 2) log K ( 1, 2)

2 1 Ω 1

0 1 2 0 1 2 0 1 2 Ω Ω Ω 2 2 2

| (2) Ω Ω | Fig. 6. (Left column) K ( 1, 2) , i.e., the absolute value of the second-order response functions in the frequency domain as a Ω Ω function of 1 and 2 with the position variables set at r = r1 = r2 = 0. There are three large peaks that completely dominate the | (2) Ω Ω | plot, so the remaining peaks are not represented in the plot. (Center column) Scaled K ( 1, 2) . The three largest peaks shown | (2) Ω Ω | in the left column were scaled down to the same magnitude as other peaks in the plot. (Right column) log K ( 1, 2) , i.e., loga- | (2) Ω Ω | rithm of the response functions presented in the left column to help visualize the large variation in the magnitude of K ( 1, 2) . The top, middle, and bottom rows give the response function for the condensate, noncondensate density, and noncondensate corre- Ω Ω lation, respectively. The frequencies 1 and 2 are given in units of the trap frequency. ture, the resonant frequencies are shifted from the zero- In Fig. 6, we plot the second-order susceptibility at temperature counterpart, so that the actual frequency zero temperature as a function of Ω and Ω with the Ω Ω Ω 1 2 combinations used are the following: 1, 2, and 1 + position variables set at r = r1 = r2 = 0, i.e., at the center Ω Ω Ω Ω (2) 2 are off-resonant ( 1 = 2.45, 2 = 1.6); both 1 and of the atomic cloud. The left column shows |K (Ω , Ω )|, Ω Ω Ω 1 2 2 are on-resonance, while 1 + 2 is off-resonant i.e., the absolute value of the second-order response Ω Ω Ω Ω ( 1 = 2.43, 2 = 1.5); and frequencies with 1 + 2 are function. The plot shows only few peaks near the fre- Ω Ω on-resonance ( 1 = 0.92, 2 = 1.5). quency of 1 because the difference between the magni- The susceptibilities become more spatially symmet- tude of these highest few peaks and the rest of the peaks ric as resonant frequencies are matched. The most sym- occurring at other frequencies is too large. This sug- gests that one should ideally tune into the combination metric function was generated when the sum of two fre- Ω Ω quencies Ω and Ω was on-resonance, while the least of frequencies 1 and 2 corresponding to these high- 1 2 est peaks to observe the maximum second-order symmetric function resulted when both frequencies Ω were off-resonant. When only of one of the frequencies response experimentally. The second frequency 2 was on-resonance, a result with an intermediate level of implies a completely different physics compared to the Ω symmetry was observed. The dependence of the linear response; 2 is the frequency of a new harmonic response function on r2 is strongest for the off-resonant being generated as a result of a strong external pertur- Ω Ω bation oscillating at frequency Ω . The middle column case, while the one where 1 + 2 is on-resonance 1 | (2) Ω Ω | remains more or less unaffected by the changes. As in of Fig. 6 gives K ( 1, 2) , where the largest peaks are the time domain response, the asymmetric function at scaled down to the magnitude of the smaller peaks r = Ð5 reverse shape for r = 5. Such change was also present. It clearly shows a number of peaks present at 2 2 Ω Ω observed for the r = ±2.5 pair; again, for r = ±2.5, frequencies 1 and 2 less than 1 and also shows that, 2 2 Ω Ω which is inside the atomic cloud, the contours were with 1 ( 2) ﬁxed at 1, there is a pronounced response Ω Ω found to be less symmetric than the corresponding ﬁg- for many values of 2 ( 1). The right column of Fig. 6 ± ures for r2 = 5. shows the logarithm of the left column. This enables the

LASER PHYSICS Vol. 14 No. 10 2004 NONLINEAR RESPONSE FUNCTIONS 1353

| (2) Ω Ω ||(2) Ω Ω | | (2) Ω Ω | K ( 1, 2) K ( 1, 2) log K ( 1, 2)

2 1 Ω 1

0 1 2 0 1 2 0 1 2 Ω Ω Ω 2 2 2

ω Ω Ω Fig. 7. The same as in Fig. 6, but at a ﬁnite temperature of 10 /k. The frequencies 1 and 2 are given in units of the trap frequency.

| (2) Ω Ω | large variations in the magnitude of K ( 1, 2) to be 3. NONLINEAR RESPONSE OF FRACTIONAL displayed. The top, middle, and bottom rows give the QUANTUM HALL SYSTEMS response function for the condensate, noncondensate In this section, we establish the correspondence density, and noncondensate correlation, respectively. between the FQHE effective Bose Hamiltonian and the Figure 7 shows the corresponding plot at the finite tem- Hamiltonian equations (1)Ð(6). With this correspon- ω perature T = 10 trap/k. The main difference is that dence, the nonlinear response functions derived above there are more peaks appearing around frequency may be applied directly. We note that nonlinear 0 < Ω < 1, i = 1, 2 than at zero temperature. response functions have not been extensively studied i for FQHE systems, and, hence, this mapping provides (2) Ω Ω Since the second-order susceptibility K ( 1, 2) is a useful starting point. We start with the many-electron by itself a product of several Green’s functions (ω) effective Hamiltonian Hˆ , where the electronÐelectron (1) ω ω Ω Ω and linear susceptibilities K ( ), = 1, 2, and interaction contains Coulomb repulsion V(r Ð r') = Ω Ω 1/|r Ð r'|, an external magnetic vector potential A(r), and 1 + 2, several features of Figs. 6 and 7 are closely related to the linear susceptibility. From Fig. 1, it is an external electrical scalar potential Vf(r, t) [7, 40]: clear that the linear susceptibility near the frequency of † Hˆ = dt drψˆ ()r, t 1 dominates the spectrum at both zero and finite tem- ∫ ∫ e peratures. It is therefore expected that any second-order 2 ()Ð i∇ Ð e()Ar() (2) Ω Ω Ω Ω × r µ ψ (), response K ( 1, 2) containing a 1 = 1 or 2 = 1 ------Ð ˆ e r t component in it will lend a much stronger contribution 2m Ω Ω 1 † † (37) to the spectrum than those not at 1 = 1 or 2 = 1. This ψˆ ()ψ, ˆ (), Ω Ω + ---∫dt∫drr∫d ' e r t e r' t explains the features around 1 = 1 and 2 = 1. In addi- 2 tion, the linear susceptibility K(1)(ω) at finite tempera- × ()ψ()ψ, (), V rrÐ ' ˆ e r' t ˆ e r t ture has more peaks around frequencies 0 < ω < 1 than ψ†(), ()ψ, (), at zero temperature, as there are more resonances below + ∫dt∫dr ˆ e r t V f r t ˆ e r t , the frequency of 1 at finite temperature. This leads to (2) Ω Ω ψ† ψ the second-order susceptibility K ( 1, 2) displaying where ˆ e (r, t) and ˆ e (r, t) are Fermi creation and Ω Ω more peaks around frequency 0 < 1 < 1 and 0 < 2 < 1, annihilation operators, m is the effective electron band as illustrated in Fig. 7. mass, and µ is the chemical potential.

LASER PHYSICS Vol. 14 No. 10 2004 1354 CHOI et al.

We next introduce a quasiparticle creation operator nˆ k with momentum k are ψ† ψ† ˆ (r, t), which is related to ˆ e (r, t) as [31] † nˆ k = ∑aˆ k aˆ k, (44) ψˆ †()r, t k † † Ð1 (38) ψ† ≡ψ (), []ν ()(), where aˆ k and aˆ k are Fourier components of ˆ (r) and ˆ e r t exp Ði ∫dr'arg rrÐ ' nˆ r' t , ψˆ (r). Using a basis set of single electron functions φ where arg(r Ð r') is the angle between the vector (r Ð r') i(r), the field operators may be expanded in the form and the direction of the Hall current (which is perpen- † ∇ × ψˆ †()r, t = ∑ φ*()r aˆ ()t , dicular to both the external magnetic field B = r A(r) i i ∇ i and the external electric field E = Ð rVf(r, t), which are (45) perpendicular to each other); νÐ1 is an odd integer; and ψˆ ()r, t = φ()r aˆ ()t , nˆ (r, t) is the charge density operator, which is the same ∑ i i for the actual Fermi and artificial Bose system, i where a† and a are Bose operators. nˆ ()ψr, t ≡ψˆ †()ψr, t ˆ ()r, t = ˆ †()ψr, t ˆ ()r, t . (39) i i e e We make the following GCS HartreeÐFockÐBogoli- The operators ψˆ † (r, t) and ψˆ (r, t) satisfy Bose com- ubov ansatz for the time-dependent many-boson wave mutation relations [26, 27] function [6, 9]: |〉ψ()t = exp( dt drα()ψr, t ˆ †()r ψˆ ()ψr ˆ †()r' Ð ψˆ †()ψr' ˆ ()r = δ()rrÐ ' ; ∫ ∫ ψˆ ()ψˆ () ψˆ ()ψˆ () (40) β()ψ,, †()ψ†())Ω|〉 r r' Ð0;r' r = + ∫dt∫drrd ' rr' t ˆ r ˆ r' 0 (46) ψˆ †()ψr ˆ †()r' Ð0. ψˆ †()ψr' ˆ †()r =  Using the Bose quasiparticle operators (Eq. (38)), the = exp∑α ()t aˆ † + ∑β ()t aˆ †aˆ † |〉Ω , i i ij i j 0 Hamiltonian equation (37) can be written in the form of i ij ˆ |Ω 〉 a many-boson effective Hamiltonian H by adding the where 0 is an arbitrary normalized reference state 〈Ω |Ω 〉 gauge ChernÐSimons vector potential a(r) [26, 28]: with 0 0 = 1 [6]. ˆ ψ†(), On the other hand, the operator set of GCS genera- H = ∫dt∫dr ˆ r t tors of the closed algebra [39, 40] in the exponent of the ansatz equation (46) creates an extended HeisenbergÐ 2 ()Ð i∇ Ð e()Ar()+ ar() Weyl algebra, which may be obtained by a repeated × ------r Ð µ ψˆ ()r, t 2m application of the standard boson commutators. The Hamiltonian that describes the system of many- 1 + --- dt drrd 'ψˆ †()ψr, t ˆ †()r', t (41) body interacting bosons is obtained and comes from the 2∫ ∫ ∫ basis set expansion of the Hamiltonian equation (41) and is given by [28] × V()ψrrÐ ' ˆ ()ψr', t ˆ ()r, t ˆ ˆ † † † H = ∑Hijaˆ i aˆ j + ∑V ijklaˆ i aˆ j aˆ kaˆ l † + dt drψˆ ()r, t V ()ψr, t ˆ ()r, t . ij ijkl ∫ ∫ f (47) In order for the static Hall conductivity of the boson + η∑E ()t aˆ †aˆ . Hamiltonian equation (41) in HartreeÐFockÐBogoli- ij i j ubov approximation [6] to coincide with the conductiv- ij Note that the Hamiltonian equation (47) coincides with ity obtained by using Laughlin’s ansatz for the many- sp electron wave function [29], the magnetic and gauge Eqs. (1)Ð(6) provided we replace in Eq. (2) Hij by Hij. potentials in k space should satisfy [28] In Eq. (47), the single-electron matrix element Hij is given by 2π αβikβ Aα()k + aα()k = ------ ------()nˆ Ð n . (42) ν 2 k Hij e k 2 αβ ()∇ ()() () (48) Here, is a unit antisymmetric tensor, φ ()Ð i r Ð e Ar + ar µ φ () = ∫dr i* r ------Ð j r ,  2m αβ 01 ≡ ; (43) Vijkl is a Coulomb repulsion between two electrons Ð01 (we put electron charge e = 1) νω π 1 n is the mean 2D electron density, n = (m c)/(2 ); and φ ()φ() φ ()φ() V ijkl = ∫dr1dr2 i* r1 *j r2 ------k r1 l r2 ,(49) the Fourier components of the charge density operator r1 Ð r2

LASER PHYSICS Vol. 14 No. 10 2004 NONLINEAR RESPONSE FUNCTIONS 1355 and the external electrical field E(t) field expanded in a higher order response functions and susceptibilities are basis set is expected to be indispensable in characterizing the BEC dynamics in the presence of strong perturbations and, () † φ () ()φ, ()† ∑Eij t aˆ i aˆ j = ∫dr i* r V f r t j r aˆ i aˆ j. (50) more importantly, in further development of matter- ij wave nonlinear optics [5]. We can therefore apply the results obtained for the We have also found that the second-order response Hamiltonian equations (1)Ð(6) to the FQHE by simply functions calculated for an externally driven BEC are sp directly applicable to the important physical phenome- replacing Hij in Eq. (2) by Hij (Eq. (48)). non of FQHE, and the GCS ansatz was applied for the Since, in a homogeneous system, the momentum is derivation of the second-order response of FQHE in a 2D conserved, we can use the plane wave basis, i.e., the electron liquid. Although the relationship between a rap- φ Ð1/2 eigenfunctions of momentum p ( p(r) = S exp(Ðipr), idly rotating BEC and QHE has been investigated previ- where S is the 2D volume of a system) by replacing the ously [41], our results correspond to a case in which indices i by the momentum p. BEC is not necessarily rotating, providing a potentially The second-order response is given by Eqs. (D.24)Ð useful conceptual connection between BEC and FQHE. (D.27) in the plane wave basis, provided that for the Hamiltonian we use Eqs. (1) and (2) in the Liouvillian ACKNOWLEDGMENTS equation (B.6) with Vtrap = 0 and replace the interparti- cle interaction Vijkl by The support of NSF (grant no. CHE-0132571) is gratefully acknowledged. 2πe2 4π2n V = ------+ ------, (51) ppÐ ' ppÐ ' ν2 2 m ppÐ ' APPENDIX A where the first term in the right-hand side represents the TDHFB EQUATIONS 2D Fourier transform component of the Coulomb repulsion and the second term comes from substituting The TDHFB equations of motion that couples z, ρ, Eq. (42) into Eq. (47). With these substitutions, the and κ are [7, 11] FQHE boson Hamiltonian equation (47) reduces to the dz classical form of Eq. (19) [28] and is equivalent to the i----- = [] + ηEt()z + z*, (A.1) classical boson Hamiltonian (Eq. (40) in [6]). The dou- dt z z* ble excitations are given by the eigenvalues of the Liou- dρ villian equation (B.6). The anisotropy in the gauge term i------= []κ∆h, ρ Ð ()* Ð ∆κ* + η[]Et()ρ, , (A.2) equation (42) in the Hamiltonian equation (47) gives a dt nonvanishing second-order response, shifting the dou- ble-excitation energies with respect to twice the single dκ i------= (hκκ+ h* )Ð () ρ∆Ð ∆ρ* excitations, as is the case in isotropic systems. dt (A.3) ∆η[]()κ, + + Et +, 4. CONCLUSIONS where […]+ denotes the anticommutator. Here, z, We have calculated the second-order response func- ∆ × tions and susceptibilities of both zero- and finite-tem- z∗, h, and are N N matrices, with n being the num- perature BoseÐEinstein condensates (BECs). The sys- ber of basis wave functions used: tematic, perturbative solution of the TDHFB equations [] µ []ρ for trapped, atomic BEC enables us to analyze the z ij, = Hij Ð + ∑V iklj zk*zl + 2 lk , (A.4) dynamics of finite-temperature BEC. The responses of kl both the noncondensate atoms and the condensate were calculated. The calculations apply for a general pertur- [] κ z* ij, = ∑V ijkl kl, (A.5) 3 φ* † kl bation of the form ∑ij ∫d r i (r)Vf(r, t)aˆ i aˆ j , where the shape of the external force V (r, t) is left arbitrary. µ []ρ f hij = Hij Ð2+ ∑V iklj zk*zl + lk , (A.6) The calculated second-order response functions were kl found to show a strong dependence on position and temperature. In the frequency domain, we have ∆ = ∑V []z z + κ . (A.7) observed a distinct nonlinear mixing effect, which dis- ij ijkl k l kl Ω Ω kl plays enhanced response at a number of 1Ð 2 combinations. The second-order response for BEC has not h is known as the “Hartree–Fock Hamiltonian,” and ∆ been reported yet, and most related recent work is lim- as the “pairing field” [9]. µ is the chemical potential ited to the linear response. As in optics, the second- and introduced in the Hamiltonian (Eq. (2)).

LASER PHYSICS Vol. 14 No. 10 2004 1356 CHOI et al.

(0) APPENDIX B where z and z∗ of are as defined in Eqs. (A.4) MATRIX AND VECTOR ζ and (A.5) and the remaining N2 × N2 submatrices are In this appendix, we define the 2N(2N + 1) × defined as 2N(2N + 1) Liouvillian matrix of Eq. (7). As discussed in the main text, it suffices to define the zeroth- and the ()Ð ()0 δ ()0 δ ij, mn = him jn Ð hnj im, (B.2) first-order matrices (0) and (1) only, since (n) for all ≥ (0) n 1 is identical. First of all, is defined as follows: ()+ ()0 δ ()0 δ ij, mn = him jn ++hnj im V ijmn, (B.3) ()0 ∆ ()0 δ ij, mn = im jn, (B.4) ()0 µ ()0 z Ð z 0000 * ∆ ()0 , = Ð∆* δ , (B.5) () () ij mn nj im  0 * 0 * µ z* z Ð000 0 (B.1) with h(0) and ∆(0) being the HartreeÐFock Hamiltonian ()Ð ∆ = 00 0 , and the pairing field defined in Eqs. (A.6) and (A.7). ∆ ()+ ≥ 00Ð * 0 For higher orders n 1 (in particular, n = 2), () ∆ Ð * * ()n ()0 000* ≡ + ', (B.6) ∆ ()+ 00*0Ð * where (0) is the zeroth-order matrix defined above and

zz1 zz2 z1 z2  00 zz2 zz1 z1 z2  * * 00 * * ρ ρ ρ κ∆ κ∆  z1 z2 h † ' = 0 . (B.7) κz1 κz2 κh ρ∆ κh†  0  ρz2 ρz1 κ∆† ρh κ∆  * * 0 () * () * () *  κz2 κz1 κh† κh ρ∆  * * () *0() * () *

The set of N × N submatrices zz1 and zz2; N × N2 sub- ρ () () () () z1 z2 2 × ρz1 ρz2 z2 0 ρ 0 0 ρ 0 matrices and ; N N submatrices , , ij, k = 2∑V iklrzl rj Ð V rkljzl ir κz1 κz2 2 2 , and ; and N × N component submatrices lr ρ ρ∆ κ κ∆ (B.11) h, , h, and of ' are given as follows: []κ*()0 *()0 ()0 Ð ∑ V rjklzl + V rjlkzl ir , zz1 *()0 ()0 zz2 ()0 ()0 lr il, = ∑V iklrzk zr , ik, = ∑V iklrzl zr , (B.8) κ () () () () kr lr z1 * 0 κ 0 * 0 κ 0 ij, k = 2∑V ilkrzl rj + V rkljzl ir lr z1 ()0 z2 ()0 ikl, = 2∑V z , ikl, = ∑V z , (B.9) () () () ilkr r iklr r + []ρV z 0 + V z 0 0 r r ∑ rjkl l rjlk l ir kr (B.12) ρz1 *()0 ρ ()0 *()0 ρ ()0 []ρ()0 ()0 *()0 ij, l = 2∑V iklrzk rj Ð V rkljzk ir + ∑ V irklzl + V irlkzl rj kr (B.10) lr []κ()0 ()0 *()0 []()0 ()0 + ∑ V irklzk + V irlkzk rj , + ∑ V ijklzl + V ijlkzl , kr l

LASER PHYSICS Vol. 14 No. 10 2004 NONLINEAR RESPONSE FUNCTIONS 1357

κz2 ()0 κ ()0 ()0 κ ()0 κh κ ()0 κh† κ ()0 ij, k = 2∑V iklrzl rj + V rlkjzl ir , (B.13) ij, kl ==∑V iklr rj , ij, kl ∑V rklj ir , (B.15) lr r r

ρh ()0 ()0 ρ ρ κ∆ () κ∆† () ij, kl = 2∑V iklr rj Ð V rklj ir , κ 0 * κ 0 ij, kl ==∑V irkl rj , ij, kl ∑V rjkl ir . r (B.14) r r (B.16) ρ∆ ρ ()0 * ρ ()0 ij, kl = ∑V irkl rj + V rjkl ir , In addition, the 2N(2N + 1) × 1 vector ζ(t) is deﬁned r as follows:

()0 Et() 000 0 0 z  0 E*()t 00 0 0 ()0 * z ()Ð  00 ()t 00 0 ρ()0 ζ()≡  t ()+ , (B.17) 00 0 ()t 00 ()0 κ ()Ð  00 0 0[] ()t *0 ()0 * ρ ()+  [] () ()0 00 0 0 0 t * κ *

where E(t) is as deﬁned in Eq. (4) and we further deﬁne ()2 Kρ ()tt,,,,,t rr r the N2 × N2 submatrices, 1 2 1 2 2 2NN+ () ()± ˜ 2 () ()δ ± ()δ = [ϒ()r (K I ()tt,,,t r ,r (C.2) t ij, kl = Eik t jl Elj t ik. (B.18) ∑ 1 2 1 2 i = 2N + 1 ()2 + K ()tt,,,t r ,r )] , APPENDIX C II 1 2 1 2 i

SECOND-ORDER RESPONSE FUNCTIONS ()2 (),,,,, Kκ tt1 t2 rr1 r2

2 In this appendix, we first summarize the final result 2N + 2N () for the response functions in Section C.1 and then pro- [ϒ˜ ( 2 ,,, , = ∑ ()r K I ()tt1 t2 r1 r2 (C.3) vide a more detailed derivation in Section C.2. In the 2 last subsection (Section C.3) of this appendix, we also i = 2NN++1 ()2 show an alternative form for the response functions + K II ()tt,,,t r ,r )] . written on the basis of the eigenvectors of the Liouvil- 1 2 1 2 i lian. Here, 2N(2N + 1) × 2N(2N + 1) matrix ϒ˜ (r) was ()2 defined in Eq. (15) and the 2N(2N + 1) × 1 vectors K I C.1. Final Expression ()2 and K II are defined as follows: The second-order response functions for the condensate, the noncondensate density, and the correlation ()2 K I ()tt,,,t r ,r that we use in our numerical calculations are given by 1 2 1 2 (C.4) Eqs. (18)Ð(20) [reproduced here as Eqs. (C.1)Ð(C.3)] as ()Φ˜ ()()Φ˜ ()ψ()0 follows: = ttÐ 1 r1 t1 Ð t2 r2 ,

()2 ()2 (),,,,, K II ()tt,,,t r ,r Kz tt1 t2 rr1 r2 1 2 1 2 N (C.1) t () () (C.5) []ϒ˜ ()2 ,,, , 2 ,,, , τ()Ξτ ()τ ,,,τ = ∑ ()r K I ()tt1 t2 r1 r2 + K II ()tt1 t2 r1 r2 i, = ∫d t Ð K Ð t1 Ð t2 r1 r2 , i = 1 0

LASER PHYSICS Vol. 14 No. 10 2004 1358 CHOI et al.

Φ˜ where (t Ð t') and (r') are given by Eqs. (23) and *()1 ()1 × []Kκ ()τ Ð t , r + ∑V []Kz ()τ Ð t , r Ξ 2 2 rj rjkl 1 1 k (24), respectively. The vector K of Eq. (22) rkl Ξ * * () [Eq. (C.5)] is written K = [ K, K , K, K, K , × []1 ()τ , κ ()0 Kz Ð t2 r2 k ir * T × 2 × K ] with the N 1 matrix K and N 1 matrices K, () () []1 ()τ , []1 ()τ , κ*()0 K given as follows: Ð V irkl Kz Ð t1 r1 k Kz Ð t2 r2 l rj , [] [] K i K ij

()1 () ()1 () () [[]*()τ , 0 []()τ , []1 ()τ , []1 ()τ , = ∑V iklr Kz Ð t1 r1 kzl Kz Ð t2 r2 r = 2∑V iklr Kκ Ð t1 r1 rj Kρ Ð t2 r2 lk klr rkl () 1 * ()1 ()0 ()1 ()1 + []Kz ()τ Ð t , r []Kz ()τ Ð t , r z + V []Kρ ()τ Ð t , r []Kκ ()τ Ð t , r 1 1 k 2 2 l r (C.6) rklj 1 1 kl 2 2 ir ()0 ()1 ()1 + z* []K ()τ Ð t , r []K ()τ Ð t , r () () k z 1 1 l z 2 2 r []1 ()τ , []1 *()τ , + ∑V irkl Kκ Ð t1 r1 kl Kρ Ð t2 r2 rj ()1 ()1 []()τ , []()τ , rkl +2 Kρ Ð t1 r1 lk Kz Ð t2 r2 r ()1 ()1 ()1 ()1 []()τ , []()τ , []()τ , []()τ , ] + V rjkl Kκ Ð t1 r1 kl Kρ Ð t2 r2 ir + Kκ Ð t1 r1 kl Kz Ð t2 r2 r ,

[] ()1 ()1 () K ij []* ()τ , []()τ , κ 0 +2∑V iklr Kz Ð t1 r1 k Kz Ð t2 r2 l rj

() () rkl []1 ()τ , []1 ()τ , = 2∑V Kρ Ð t r Kρ Ð t r () ()1 iklr 1 1 rj 2 2 kl []1 ()τ , []* ()τ , κ ()0 rkl + V rklj Kz Ð t1 r1 k Kz Ð t2 r2 l ir

()1 ()1 []()τ , []()τ , ()1 Ð V rklj Kρ Ð t1 r1 kl Kρ Ð t2 r2 ir [[]* ()τ , ()0 +2∑V iklr Kz Ð t1 r1 kzl () ()1 rkl []1 ()τ , []* ()τ , + ∑V iklr Kκ Ð t1 r1 kl Kκ Ð t2 r2 rj () () *()0 []1 ()τ , ][]2 ()τ , rkl + zk Kz Ð t1 r1 l Kκ Ð t1 r2 rj () (C.8) ()1 ()1 1 *()0 []* ()τ , []()τ , + V [[]Kz ()τ Ð t , r z + V rjkl Kκ Ð t1 r1 kl Kκ Ð t2 r2 ir rklj 1 1 k l () ()0 * 1 ()1 ()1 ()1 () + z []K ()τ Ð t , r ][]Kκ ()τ Ð t , r []* ()τ , []()τ , ρ 0 k z 1 1 l 2 2 ir +2∑V iklr Kz Ð t1 r1 k Kz Ð t2 r2 l rj rkl () [[]1 ()τ , ()0 ()1 () + ∑V rjkl Kz Ð t1 r1 kzl []* ()τ , []1 ()τ , ρ ()0 Ð V rklj Kz Ð t1 r1 k Kz Ð t2 r2 l ir rkl () () ()0 1 1 ()1 + z []K ()τ Ð t , r ][]Kρ ()τ Ð t , r [[]* ()τ , ()0 k z 1 1 l 2 2 ir +2∑V iklr Kz Ð t1 r1 kzl ()1 () rkl (C.7) [[]()τ , 0 + V irkl Kz Ð t1 r1 kzl *()0 ()1 ()1 + z []K ()τ Ð t , r ][]Kρ ()τ Ð t , r () ()1 k z 1 1 l 2 2 rj ()0 []1 ()τ , ][]* ()τ , + zk Kz Ð t1 r1 l Kρ Ð t2 r2 rj ()1 () [][]* τ , ()0 *()0 []1 τ , Ð V rklj Kz ()Ð t1 r1 kzl + zk Kz ()Ð t1 r1 l () () []1 ()τ , []1 ()τ , ρ ()0 + ∑V rjkl Kz Ð t1 r1 k Kz Ð t2 r2 l ir () ()1 1 ()0 × []()τ , [[]*()τ , * rkl Kρ Ð t2 r2 ir + ∑V rjkl Kz Ð t1 r1 kzl () () rkl []1 ()τ , []1 ()τ , ρ*()0 + V irkl Kz Ð t1 r1 k Kz Ð t2 r2 l rj () () *()0 []* 1 ()τ , ][]1 ()τ , + zk Kz Ð t1 r1 l Kκ Ð t2 r2 ir ()1 ()1 () () + ∑V []Kz ()τ Ð t , r []Kz ()τ Ð t , r , [][]1 ()τ , ()0 ()0 []1 ()τ , ijkl 1 1 k 2 2 l Ð V irkl Kz Ð t1 r1 kzl + zk Kz Ð t1 r1 l kl

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() () 1 τ 1 ρ ()1 ρ ()1 ρ ()1 ρ ()1 where the vectors Kz ( Ð ti, ri), Kρ (t Ð t1, ri), and ij = 2∑V iklr rj kl Ð V rklj kl ir () 1 τ rkl Kκ ( Ð ti, ri) found in Eqs. (C.6)–(C.8) are defined as ()1 ()1 ()1 ()1 ()1 ()α ()0 κ κ* κ* κ ()τ , ()Φτ ˜ ()ψ + ∑V irkl kl rj + V rjkl kl ir Kα Ð ti ri = Ð ti ri , (C.9) rkl α ==z,,ρκ, i 12., ()1 ()1 ()0 ()1 ()1 ()0 (α) τ +2∑V z* z ρ Ð V z* z ρ ( Ð ti) are the submatrices of (t Ð t1) such that iklr k l rj rklj k l ir rkl ()[()z ()τ , ()z *()τ ttÐ 1 = Ð ti Ð ti , ()1 ()0 ()0 ()1 ()1 ()ρ ()κ ()ρ +2∑V []ρz* z + z* z ()τ Ð t ,, ()τ Ð t *()τ Ð t , (C.10) iklr k l k l rj i i i rkl (C.15) ()κ T *()]τ []ρ()1 ()0 ()0 ()1 ()1 Ð ti , Ð V rklj zk* zl + zk* zl ir (z) τ × where ( Ð ti) is an N 2N(2N + 1) submatrix while ()1 ()0 ()0 ()1 ()1 (γ) τ γ ρ κ ρ κ 2 × + V []κz* z + z* z ( Ð ti), = , , *, * is an N 2N(2N + 1) sub- ∑ rjkl k l k l ir (z) τ rkl matrix; i.e., the submatrix ( Ð ti) is stacked on top of submatrix (z)*(τ Ð t ), which, in turn, is stacked on i []κ()1 ()0 ()0 ()1 ()1 (γ) τ Ð V irkl zk zl + zk zl rj* top of submatrices ( Ð ti). It is to be noted that ()1 τ × 2 × () () () () () () Kα ( Ð ti, ri) as defined here are N 1 and N 1 vec- + ∑V z 1 z 1 κ 0 Ð V z 1 z 1 κ 0 , tors, not scalar quantities obtained by integrating the rjkl k l ir irkl k l rj () rkl 1 τ scalar function Kα ( Ð ti, r, ri) over r. κ ()1 ρ ()1 ρ ()1 κ ()1 ij = 2∑V iklr rj lk + V rklj kl ir C.2. Derivation rkl () () () Writing the second-order solution to TDHFB + ∑V κ 1 ρ*()1 + V κ 1 ρ 1 explicitly, we have irkl kl rj rjkl kl ir rkl t () () ψ 2 () 1 i 2 ()Γ() ()1 ()1 ()0 ()1 ()1 ()0 t = ----- exp Ð--- ttÐ 1 t1 dt1 (C.11) +2∑V z* z κ + V z z* κ i∫ iklr k l rj rklj k l ir 0 rkl t []κ()1 ()0 ()0 ()1 ()1 ≡ 1 ()Γ() +2∑V iklr zk* zl + zk* zl rj -----∫ ttÐ 1 t1 dt1. (C.12) i rkl 0 Here, []κ()1 ()0 ()0 ()1 ()1 + V rklj zk zl* + zk zl* ir (C.16) ()1 Γ()t = ζ()ψt ()t + Ξ()t , (C.13) () () () () () 1 1 1 1 []ρ1 0 0 1 1 + ∑V rjkl zk zl + zk zl ir ()1 i.e., for the second-order response, λ(2)(t) ≡ ζ(t)(ψ t) + rkl Ξ (t) in Eq. (7), where ζ(t) is given in Appendix B and []ρ()1 ()0 ()0 ()1 ()1 + V irkl zk zl + zk zl rj* Ξ (t) is a 2N(2N + 1) × 1 vector originating from the ()1 ()1 ρ ()0 ()1 ()1 ρ*()1 terms in the expansion that are made up of products of + ∑V rjklzk zl ir + V irklzk zl rj two first-order variables, i.e., z(1), ρ(1), and κ(1). The vec- rkl tor Ξ (t) can be written as Ξ (t) = [, *, , , *, ()1 ()1 + ∑V z z . *]T with the N × 1 matrix , and N2 × 1 matrices , ijkl k l given as follows: kl Casting Eq. (C.12) in the form ()1 ()0 ()1 ()1 ()1 ()0 = V [z* z z + z* z z () ()2 i ∑ iklr k l r k l r ψ 2 (), (),,,,, (C.14) r t = ∫K tt1 t2 rr1 r2 klr (C.17) ()0 ()1 ()1 ρ ()001 ()1 κ ()1 ()1 ] × (), (), 3 3 + zk* zl zr ++2 lk zr kl zr , V f r1 t1 V f r2 t2 d r1dt1d r2dt2

LASER PHYSICS Vol. 14 No. 10 2004 1360 CHOI et al. involves rewriting Γ (t) of Eq. (C.13) in the position- The position-dependent second-order response dependent form: functions for the condensate, the noncondensate density, and the correlation are given by summing over ()2 t1 () appropriate indices of the vector K (Eq. (C.21)): Γ() 1 []Φ˜ () 1 (),, t1 = -----∫dt2∫dr1dr2 r1 K t1 t2 r2 i (C.18) ()2 (),,,,, 0 Kz tt1 t2 rr1 r2 × (), (), N (C.24) V f r1 t1 V f r2 t2 () () []ϒ˜ ()2 ,,, , 2 ,,, , = ∑ ()r K I ()tt1 t2 r1 r2 + K II ()tt1 t2 r1 r2 i, i = 1 t1 t1 []Ξ (),, ()2 (),,,,, + ∫∫dt2dt3∫dr2dr3 K t1 Ð t2 t1 Ð t3; r2 r3 Kρ tt1 t2 rr1 r2 (C.19) 2 0 0 2NN+ () × V ()r , t V ()r , t , [ϒ˜ ( 2 ,,, , f 2 2 f 3 3 = ∑ ()r K I ()tt1 t2 r1 r2 (C.25) i = 2N + 1 () ()1 2 + K II ()tt,,,t r ,r )] , where K (t1, t2, r2) is the linear response function for 1 2 1 2 i the combined variables z, ρ, and κ: ()2 (),,,,, Kκ tt1 t2 rr1 r2 () 1 ()0 2 (), ()Φ˜ ()ψ 2N + 2N K t1 Ð t2 r2 = t1 Ð t2 r2 , (C.20) () [ϒ˜ ( 2 ,,, , = ∑ ()r K I ()tt1 t2 r1 r2 (C.26) 2 i = 2NN++1 with the 2N(2N + 1) × 1 column vector ΞK (t Ð t , t Ð 1 2 1 ()2 + K II ()tt,,,t r ,r )] . t3; r2, r2' ) already been deﬁned above in Eqs. (25) and 1 2 1 2 i

() () (C.8). ΞK is derived from the vector Ξ (t) such that the 2 2 ()1 where K I and K II are defined in Eqs. (C.22) and components of the linear response vector K (t1, t2, r2), ϒ˜ Φ˜ Ξ ()1 ()1 ()1 (C.23) and , , , and K are defined in Eqs. (15), i.e., Kz (t1, t2, r2), Kρ (t1, t2, r2), and Kκ (t1, t2, r2) (23), (24), and (25) (or (C.8)), respectively. (1) defined in Eq. (C.9) replace, respectively, z (t1), ρ(1) κ(1) Ξ (t1), and (t1) in (t). C.3. Alternative Form for the Response Function Using these results, the position-dependent time To discuss the response functions in the frequency domain second-order response function for the com- ρ κ domain and to understand the physical processes bined variables z, , and may finally be written as involved, it is useful to expand the response functions

(2) () () () on the basis of the eigenvectors ξν of matrix such 2 ()ϒ,,,,, ˜ ()[]2 2 K tt1 t2 rr1 r2 = r K I + K II , (C.21) (2) that ξν = ων ξν , with ν = 1, 2, …, 2N(2N + 1). We τ deﬁne the Green’s function where, after the change of variables t1 , t2 t1, ()2 ()2 and t3 t2, K I and K II are given by () i ω () Gν ttÐ ' = exp Ð--- ν ttÐ ' (C.27) () 2 (),,, , K I tt1 t2 r1 r2 µ η δ (C.22) and the expansion coefﬁcients ν, ν(r), and ν(r) such ()Φ˜ ()()Φ˜ ()ψ()0 that = ttÐ 1 r1 t1 Ð t2 r2 , 2N()2N + 1 2N()2N + 1 ()0 () ψ µ ξ Φ˜ ()ξ η ()ξ 2 (),,, , ==∑ ν ν, r ν' ∑ ν r ν, K II tt1 t2 r1 r2 ν = 1 ν = 1 t (C.23) 2N()2N + 1 (C.28) τ()Ξτ ()τ ,,,τ ˜ = ∫d t Ð K Ð t1 Ð t2 r1 r2 . and ϒ()ξr ν = ∑ δν()ξr ν. 0 ν = 1

LASER PHYSICS Vol. 14 No. 10 2004 NONLINEAR RESPONSE FUNCTIONS 1361

()2 ()Ω Ω Ω ,,,,Ω On the basis of these eigenstates, ξν , the result is Kρ Ð 1 Ð 2; 1 2 rr1 r2 2 Eqs. (C.24)Ð(C.26) but with 2NN+ ()2 ()2 [ϒ˜ ( Ω ,,,Ω (D.2) (),,, , = ∑ ()r K I ()1 2 r1 r2 K I tt1 t2 r1 r2 i = 2N + 1 () 2N 2N + 1 (C.29) ()2 ()2 ()ξ,,, , + K II ()Ω ,,,Ω r r )] , = ∑ I, ν tt1 t2 r1 r2 ν, 1 2 1 2 i ν = 1 ()2 ()Ω Ω Ω ,,,,Ω Kκ Ð 1 Ð 2; 1 2 rr1 r2 ()2 (),,, , 2 K II tt1 t2 r1 r2 2N + 2N ()2 () [ϒ˜ ( Ω ,,,Ω 2N 2N + 1 (C.30) = ∑ ()r K I ()1 2 r1 r2 (D.3) ()2 ()ξ,,, , 2 = ∑ II, ν tt1 t2 r1 r2 ν, i = 2NN++1 ν = 1 ()2 + K II ()Ω ,,,Ω r r )] , where 1 2 1 2 i where ()2 (),,, , I, ν tt1 t2 r1 r2 ()2 () K I ()Ω ,,Ω rr, ,r 2N 2N + 1 (C.31) 1 2 1 2 (D.4) η ()η() ()µ() () = ∑ ν r1 ν' r2 Gν ttÐ 1 ν''Gν' t1 Ð t2 , 1 ϒ˜ ()()ΦΩ Ω ˜ ()()ΦΩ ˜ ()ψ0 = Ð------2 r 1 + 2 r1 2 r2 , ν', ν'' = 1 4π and and

2N()2N + 1 t ()2 ()2 ()Ω ,,Ω , , (),,, , τ()τ K II 1 2 rr1 r2 II, ν tt1 t2 r1 r2 = ∑ ∫d Gν t Ð (C.32) (D.5) ν', ν'' = 1 0 1 ϒ˜ ()()ΞΩ Ω ()Ω ,,,Ω = Ð------r 1 + 2 K 1 2 r1 r2 . × []η ()η()µµ ()τ ()τ π3 ν' r1 ν'' r2 ν' ν''Gν' Ð t1 Gν'' Ð t2 . 8 i Here, is the function given according to the expres- Here, ϒ˜ (r) and Φ˜ (r) are as deﬁned in Eqs. (15) and (24) and sion for ΞK but written in terms of µν, ην, and Gν with µ η Gν(t Ð t'), ν, and ν as deﬁned in Eqs. (C.27) and ()ω ≡ 1 ------()2 -. (D.6) (C.28). ω Ð + i

Ξ Ω Ω APPENDIX D In addition, the vector K ( 1, 2, r1, r2) of Eq. (31) may be written as SECOND-ORDER SUSCEPTIBILITIES Ξ ()Ω ,,,Ω In this appendix, we first summarize the final result K 1 2 r1 r2 for the second-order susceptibilities in Section D.1 and T (D.7) then provide a more detailed derivation in Section D.2. ˜ ,,,,,˜ * ˜ ˜ ˜ * ˜ * In the last subsection (Section D.3) of this appendix, we = K K K K K K show an alternative form for the susceptibilities written ˜ 2 ˜ ˜ on the basis of the eigenvectors of the Liouvillian. with the N × 1 matrix K and N × 1 matrices K , K given as follows: D.1. Final Expression ˜ []K i The final result that we use for our numerical calculation for the condensate, the noncondensate density, ()1 () [[]*()Ω , ()0 []1 ()Ω , and the noncondensate correlations are = ∑V iklr Kz 1 r1 kzl Kz 2 r2 r () klr K 2 ()Ð Ω Ð Ω ; Ω ,,,,Ω rr r z 1 2 1 2 1 2 ()1 () []*()Ω , []1 ()Ω , ()0 N + Kz 1 r1 k Kz 2 r2 lzr ()2 (D.8) [ϒ˜ ( Ω ,,,Ω () () = ∑ ()r K I ()1 2 r1 r2 (D.1) *()0 []1 ()Ω , []1 ()Ω , + zk Kz 1 r1 l Kz 2 r2 r i = 1 () () () 2 Ω ,,,Ω )] []1 ()Ω , []1 ()Ω , + K II ()1 2 r1 r2 i, +2 Kρ 1 r1 lk Kz 2 r2 r

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() () () () []1 ()Ω , []1 ()Ω , ] []1 ()Ω , []1 ()Ω , + Kκ 1 r1 kl Kz 2 r2 r , + V rjkl Kκ 1 r1 kl Kρ 2 r2 ir

[]˜ ()1 () K ij []* ()Ω , []1 ()Ω , κ ()0 +2∑V iklr Kz 1 r1 k Kz 2 r2 l rj ()1 ()1 rkl = 2 V []Kρ ()Ω , r []Kρ ()Ω , r ∑ iklr 1 1 rj 2 2 kl () ()1 []1 ()Ω , []* ()Ω , κ ()0 rkl + V rklj Kz 1 r1 k Kz 2 r2 l ir ()1 ()1 Ð V []Kρ ()Ω , r []Kρ ()Ω , r ()1 rklj 1 1 kl 2 2 ir [[]*()Ω , 00() +2∑V iklr Kz 1 r1 kzl ()1 *()1 rkl []κ ()Ω , []κ ()Ω , + ∑V iklr K 1 r1 kl K0 2 r2 rj () () *()0 []1 ()Ω , ][]2 ()Ω , rkl + zk Kz 1 r1 l Kκ 1 r2 rj (D.10) ()1 () ()1 () []* ()Ω , []1 ()Ω , [[]()Ω , * 0 + V rjkl Kκ 1 r1 kl Kκ 2 r2 ir + V rklj Kz 1 r1 kzl () ()0 * 1 ()1 ()1 ()1 []()Ω , ][]()Ω , []* ()Ω , []()Ω , ρ ()0 + zk Kz 1 r1 l Kκ 2 r2 ir +2∑V iklr Kz 1 r1 k Kz 2 r2 l rj rkl () [[]1 ()Ω , ()0 ()1 () + ∑V rjkl Kz 1 r1 kzl []* ()Ω , []1 ()Ω , ρ ()0 Ð V rklj Kz 1 r1 k Kz 2 r2 l ir rkl () () ()0 1 1 ()1 () + z []K ()Ω , r ][]Kρ ()Ω , r [[]* ()Ω , 0 k z 1 1 l 2 2 ir +2∑V iklr Kz 1 r1 kzl ()1 () rkl (D.9) [[]()Ω , 0 + V irkl Kz 1 r1 kzl *()0 ()1 ()1 + z []Kz ()Ω , r ][]Kρ ()Ω , r () ()1 k 1 1 l 2 2 rj ()0 []1 ()Ω , ][]* ()Ω , + zk Kz 1 r1 l Kρ 2 r2 rj ()1 () [][]* ()Ω , ()0 *()0 []1 ()Ω , Ð V rklj Kz 1 r1 kzl + zk Kz 1 r1 l ()1 ()1 ()0 + ∑V []Kz ()Ω , r []Kz ()Ω , r ρ () rjkl 1 1 k 2 2 l ir ()1 1 ()0 × []()Ω , [[]*()Ω , * rkl Kρ 2 r2 ir + ∑V rjkl Kz 1 r1 kzl () () rkl []1 ()Ω , []1 ()Ω , ρ*()0 + V irkl Kz 1 r1 k Kz 2 r2 l rj () () *()0 []* 1 ()Ω , ][]1 ()Ω , + zk Kz 1 r1 l Kκ 2 r2 ir ()1 ()1 () () + ∑V []Kz ()Ω , r []Kz ()Ω , r . [][]1 ()Ω , ()0 ()0 []1 ()Ω , ijkl 1 1 k 2 2 l Ð V irkl Kz 1 r1 kzl + zk Kz 1 r1 l kl

()1 ()1 () Ω × []* ()Ω , []1 ()Ω , Similar to Eq. (C.9) above, the quantities Kz ( i, ri), Kκ 2 r2 rj + ∑V rjkl Kz 1 r1 k () () 1 Ω 1 Ω rkl Kρ ( i, ri), and Kκ ( i, ri) used in Eqs. (D.8)Ð(D.10) () are deﬁned as × []1 ()Ω , κ ()0 Kz 2 r2 l ir ()1 ()α ()0 Kα ()Ω , r = ()ΦΩ ˜ ()ψr , (D.11) () () i i i i []1 ()Ω , []1 ()Ω , κ*()0 Ð V irkl Kz 1 r1 k Kz 2 r2 l rj , α ρ κ (α) Ω where = z, , , and i = 1, 2. ( i) are the subma- Ω ˜ trices of ( i) deﬁned in Eq. (32) such that []K ij ()z ()z ()Ω = [ ()Ω , *()Ω , () () i i i []1 ()Ω , []1 ()Ω , (D.12) = 2∑V iklr Kκ 1 r1 rj Kρ 2 r2 lk ()ρ ()Ω ,,()κ ()Ω ()ρ *()Ω ,()κ *()]Ω T i i i i , rkl (z) ()1 ()1 where (Ω ) is an N × 2N(2N + 1) submatrix while []()Ω , []()Ω , i + V rklj Kρ 1 r1 kl Kκ 2 r2 ir (γ) Ω γ ρ κ ρ κ 2 × ( i), = , , *, * is an N 2N(2N + 1) submatrix such that the submatrix (z)(Ω ) is stacked on top () () i []1 ()Ω , []1 *()Ω , (z) Ω + ∑V irkl Kκ 1 r1 kl Kρ 2 r2 rj of submatrix *( i), which, in turn, is stacked on top (γ) Ω rkl of submatrices ( i). It is to be noted that, as with

LASER PHYSICS Vol. 14 No. 10 2004 NONLINEAR RESPONSE FUNCTIONS 1363

() 1 Ω the time domain example discussed above, Kα ( i, ri) In addition, we have × 2 × as deﬁned here are N 1 and N 1 vectors, not scalar ∞ quantities obtained by integrating the scalar function ()2 1 () K ()ΩΩ,,,Ω rr, ,r = Ð------dωωd 'dω'' 1 Ω II 1 2 1 2 3 ∫ Kα ( i, r, ri) over r. 8π i Ð∞ (D.19) ×ϒ˜ ()()Ξω ()δωωω ,,,ω ()ω r K ' '' r1 r2 Ð ' Ð '' D.2. Derivation ×δ()δΩ Ω ω ()δΩω ()ω Ð 1 + ' 2 + '' . The second-order response function in frequency is given by the Fourier transform of the time domain We are able to write the Fourier transform for ΞK (t) in counterpart: Eq. (D.19) since the function ΞK (t) is made up of terms ∞ () () () which are simply products of two Green’s functions at 2 ()ΩΩ,,,,,Ω ϒ˜ ()[]2 2 different times. Equation (D.19) implies ω' = ÐΩ , ω'' = K 1 2 rr1 r2 = ∫dttd 1 r K I + K II 1 ÐΩ , ω = ω' + ω'' = ÐΩ Ð Ω , Ω = ω, so that 0 (D.13) 2 1 2 × exp()iΩti++Ω t iΩ t ()2 ()Ω Ω Ω ,,,,Ω 1 1 2 2 K II Ð 1 Ð 2; 1 2 rr1 r2 (D.20) ()2 1 ˜ ϒ˜ ()[ ()ΩΩ,,,,Ω = Ð------ϒ()r ()ΞΩ + Ω K()Ω ,,,Ω r r , = r K I 1 2 r1 r2 3 1 2 1 2 1 2 (D.14) 8π i ()2 + K ()]ΩΩ,,,,Ω r r , II 1 2 1 2 Ξ Ω Ω where K ( 1, 2, r1, r2) is as already given in ()2 ()2 Eq. (D.10). where K (Ω, Ω , Ω , r , r ) and K (Ω, Ω , Ω , r , r ) I 1 2 1 2 II 1 2 1 2 As for the time domain calculations, the susceptibil- are the Fourier transforms of the time domain expres- ities for the condensate, the noncondensate density, and sions (Eqs. (C.22), (C.23)). the noncondensate correlations are obtained by summing over the appropriate indices: Using the fact that the matrices (t Ð t1) are the Green’s functions with an implicit Heaviside step func- ()2 ()Ω Ω Ω ,,,,Ω ≡ θ Kz Ð 1 Ð 2; 1 2 rr1 r2 tion in time, i.e., (t Ð t1) (t Ð t1) (t Ð t1) such that N ()2 ∞ ˜ = ∑ [ϒ()r (K I ()Ω ,,,Ω r r (D.21) 1 1 1 2 1 2 θ()t ()t = Ð------dω------exp()Ðiωt (D.15) i = 1 π ∫ ()2 2 i ω Ð + i () Ð∞ 2 Ω ,,,Ω )] + K II ()1 2 r1 r2 i, ∞ ()2 Kρ ()Ð Ω Ð Ω ; Ω ,,,,Ω rr r = ∫ dω()ω exp()Ðiωt , (D.16) 1 2 1 2 1 2 2 2NN+ Ð∞ () [ϒ˜ ( 2 Ω ,,,Ω = ∑ ()r K I ()1 2 r1 r2 (D.22) we have i = 2N + 1 () ∞ 2 Ω ,,,Ω )] () + K II ()1 2 r1 r2 i, 2 ()ΩΩ,,,,Ω 1 ωω K I 1 2 r1 r2 = Ð------∫ d d ' π2 ()2 4 Kκ ()Ð Ω Ð Ω ; Ω ,,,,Ω rr r Ð∞ (D.17) 1 2 1 2 1 2 () 2 0 2N + 2N ×ϒ˜ ()r ()Φω ˜ ()r ()Φω' ˜ ()ψr δΩ()Ð ω () 1 2 ˜ 2 = ∑ [ϒ()r (K I ()Ω ,,,Ω r r (D.23) ×δ()δΩ Ω + ωωÐ ' ()+ ω' . 1 2 1 2 1 2 2 i = 2NN++1 ω Ω ω Ω Ω Ω ()2 This implies that ' = Ð 2, = Ð 1 Ð 2, and = + K ()Ω ,,,Ω r r )] . −Ω Ω II 1 2 1 2 i 1 Ð 2:

() K 2 ()Ð Ω Ð Ω ; Ω ,,,,Ω rr r D.3. Alternative Form for the Susceptibility I 1 2 1 2 1 2 (D.18) 1 ()0 ϒ˜ ()()ΦΩ Ω ˜ ()()ΦΩ ˜ ()ψ As before, expanding in the eigenstate basis ξν that = Ð------2 r 1 + 2 r1 2 r2 . 4π we introduced in Eq. (C.28), we may write the suscep-

LASER PHYSICS Vol. 14 No. 10 2004 1364 CHOI et al. tibility in a more useful form. The result is Eqs. (D.21)Ð where (D.23) but with ()2 ()2 ()Ω Ω Ω ,,,Ω ()Ω Ω Ω ,,,Ω I, ν Ð 1 Ð 2; 1 2 r1 r2 K I Ð 1 Ð 2; 1 2 r1 r2 2N()2N + 1 (D.26) () η ()η()µ 2N 2N + 1 (D.24) 1 ν r1 ν' r2 ν'' ()2 = Ð------. ()ξΩ Ω Ω ,,,Ω 2 ∑ ()ΩΩ Ω ω ()ω = ∑ I, ν Ð 1 Ð 2; 1 2 r1 r2 ν, π 1 ++2 Ð ν i 2 Ð ν' + i 4 ν', ν'' = 1 ν = 1 () 2 ()Ω Ω Ω ,,,Ω Since no additional information is gained by listing all K II Ð 1 Ð 2; 1 2 r1 r2 ()2 the terms in , ν (ÐΩ Ð Ω ; Ω , Ω , r , r ), we simply 2N()2N + 1 (D.25) II 1 2 1 2 1 2 ()2 note that, in the eigenstate basis, the typical term in = ∑ , ν ()ξÐ Ω Ð Ω ; Ω ,,,Ω r r ν, II 1 2 1 2 1 2 ()2 ν = 1 II, ν has the structure

2N()2N + 1 η ()η()µ ν r1 ν' r2 ν'' ∑ ------()ΩΩ Ω ω ()Ωω ()ω , (D.27) 1 ++2 Ð ν i 2 Ð ν + i 2 Ð ν' + i ν'ν''= 1

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