Laser Physics, Vol. 14, No. 10, 2004, pp. 1342–1365. PHYSICS OF COLD Original Text Copyright © 2004 by Astro, Ltd. Copyright © 2004 by MAIK “Nauka /Interperiodica” (Russia). TRAPPED ATOMS Nonlinear Response Functions of Strongly Correlated Boson Fields: Bose–Einstein 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 Bose–Einstein condensates (BECs) in a one-dimensional harmonic trap driven by an external field coupled to particle density are calculated by solving the time-dependent Hartree–Fock–Bogoliubov equations. These pro- vide 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 Chern–Simons 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- Bose–Einstein condensation and associated phenom- ture BEC that take into account higher order collision ena, such as superfluidity and superconductivity [1], processes. They include the Hartree–Fock–Bogoliubov and the quantum Hall effect in two-dimensional elec- (HFB) theory [8–11], the time-dependent Bogoliubov– tron systems [2]. A useful tool for the study of such de Gennes equations [12], quantum kinetic theory [13– complex, correlated many-body systems is provided by 18], and stochastic methods [19–23]. 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 Gross– 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 Hartree–Fock (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 Gross–Pitaevskii 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 Hartree–Fock– 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 1342 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 fre- quency, and B is the magnetic field; e is the electron where charge (we set c = 1 and ប = 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 Chern–Simons 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 [26–28]. 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 Chern–Simons 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 rr– ' k r' l r , composite fermion system [31–33]. In this paper, we describe the collision between the atoms, with V(r – 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 Hartree–Fock–Bogoliubov 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 – 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 – 〈〉〈〉 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 iប------------------- = ᏸ 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), LASER PHYSICS Vol.