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arXiv:quant-ph/9703005 · Physical Review A

Particle-number-conserving Bogoliubov method which demonstrates the validity of the time-dependent Gross-Pitaevskii equation for a highly condensed Bose gas

Crispin W. Gardiner

Published Aug 1, 1997• 256 citations
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Abstract

The Bogoliubov method for the excitation spectrum of a Bose-condensed gas is generalized to apply to a gas with an exact large number $N$ of particles. This generalization yields a description of the Schr\"odinger picture field operators as the product of an annihilation operator $A$ for the total number of particles and the sum of a ``condensate wave function'' $\ensuremath{\xi}(\mathbf{x})$ and a phonon field operator $\ensuremath{\chi}(\mathbf{x})$ in the form $\ensuremath{\psi}(\mathbf{x})\ensuremath{\approx}A{\ensuremath{\xi}(\mathbf{x})+\ensuremath{\chi}(\mathbf{x})/\sqrt{N}}$ when the field operator acts on the $N$ particle subspace. It is then possible to expand the Hamiltonian in decreasing powers of $\sqrt{N}$, and thus obtain solutions for eigenvalues and eigenstates as an asymptotic expansion of the same kind. It is also possible to compute all matrix elements of field operators between states of different $N$. The excitation spectrum can be obtained by essentially the same method as Bogoliubov only if $\ensuremath{\xi}(\mathbf{x})$ is a solution of the time-independent Gross-Pitaevskii equation for $N$ particles and any chemical potential $\ensuremath{\mu}$, which yields a valid and stable solution of the Gross-Pitaevskii equation. The treatment within a subspace of fixed $N$ is identical in form to that usually used, but the interpretation of the operators is slightly different. A time-dependent generalization is then made, yielding an asymptotic expansion in decreasing powers of $\sqrt{N}$ for the equations of motion. In this expansion the condensate wave function has the time-dependent form $\ensuremath{\xi}(\mathbf{x},t)$, and the condition for the validity of the expansion is that $\ensuremath{\xi}(\mathbf{x},t)$ satisfies the time-dependent Gross-Pitaevskii equation $\ensuremath{\partial}\ensuremath{\xi}/\ensuremath{\partial}t=\ensuremath{-}({\ensuremath{\Elzxh}}^{2}/2m){\ensuremath{\nabla}}^{2}\ensuremath{\xi}+V\ensuremath{\xi}+Nu|\ensuremath{\xi}{|}^{2}\ensuremath{\xi}$. The physics is then described in a kind of interaction picture, called the condensate picture, in which the phonon operator can be expressed as $\ensuremath{\chi}(\mathbf{x},t)={\ensuremath{\sum}}_{k}{\ensuremath{\xi}}_{k}(\mathbf{x},t){\ensuremath{\alpha}}_{k},$ where the operators ${\ensuremath{\alpha}}_{k}$ are time-independent annihilation operators, and the state vector has a time evolution described by a Schr\"odinger equation in which the Hamiltonian is a time-dependent quadratic form in the phonon creation and annihilation operators, whose coefficients are explicitly determined in terms of the time-dependent condensate wave function $\ensuremath{\xi}(\mathbf{x},t)$.

DOI
10.1103/physreva.56.1414

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The Bogoliubov method for the excitation spectrum of a Bose-condensed gas is generalized to apply to a gas with an exact large number NNN of particles. This generalization yields a description of the Schr"odinger picture field operators as the product of an annihilation operator AAA for the total number of particles and the sum of a ``condensate wave function'' \ensuremathξ(x)\ensuremath{\xi}(\mathbf{x})\ensuremathξ(x) and a phonon field operator \ensuremathχ(x)\ensuremath{\chi}(\mathbf{x})\ensuremathχ(x) in the form \ensuremathψ(x)\ensuremath≈A\ensuremathξ(x)+\ensuremathχ(x)/N\ensuremath{\psi}(\mathbf{x})\ensuremath{\approx}A{\ensuremath{\xi}(\mathbf{x})+\ensuremath{\chi}(\mathbf{x})/\sqrt{N}}\ensuremathψ(x)\ensuremath≈A\ensuremathξ(x)+\ensuremathχ(x)/N​ when the field operator acts on the NNN particle subspace. It is then possible to expand the Hamiltonian in decreasing powers of N\sqrt{N}N​, and thus obtain solutions for eigenvalues and eigenstates as an asymptotic expansion of the same kind. It is also possible to compute all matrix elements of field operators between states of different NNN. The excitation spectrum can be obtained by essentially the same method as Bogoliubov only if \ensuremathξ(x)\ensuremath{\xi}(\mathbf{x})\ensuremathξ(x) is a solution of the time-independent Gross-Pitaevskii equation for NNN particles and any chemical potential \ensuremathμ\ensuremath{\mu}\ensuremathμ, which yields a valid and stable solution of the Gross-Pitaevskii equation. The treatment within a subspace of fixed NNN is identical in form to that usually used, but the interpretation of the operators is slightly different. A time-dependent generalization is then made, yielding an asymptotic expansion in decreasing powers of N\sqrt{N}N​ for the equations of motion. In this expansion the condensate wave function has the time-dependent form \ensuremathξ(x,t)\ensuremath{\xi}(\mathbf{x},t)\ensuremathξ(x,t), and the condition for the validity of the expansion is that \ensuremathξ(x,t)\ensuremath{\xi}(\mathbf{x},t)\ensuremathξ(x,t) satisfies the time-dependent Gross-Pitaevskii equation \ensuremath∂\ensuremathξ/\ensuremath∂t=\ensuremath−(\ensuremath\Elzxh2/2m)\ensuremath∇2\ensuremathξ+V\ensuremathξ+Nu∣\ensuremathξ∣2\ensuremathξ\ensuremath{\partial}\ensuremath{\xi}/\ensuremath{\partial}t=\ensuremath{-}({\ensuremath{\Elzxh}}^{2}/2m){\ensuremath{\nabla}}^{2}\ensuremath{\xi}+V\ensuremath{\xi}+Nu|\ensuremath{\xi}{|}^{2}\ensuremath{\xi}\ensuremath∂\ensuremathξ/\ensuremath∂t=\ensuremath−(\ensuremath\Elzxh2/2m)\ensuremath∇2\ensuremathξ+V\ensuremathξ+Nu∣\ensuremathξ∣2\ensuremathξ. The physics is then described in a kind of interaction picture, called the condensate picture, in which the phonon operator can be expressed as \ensuremathχ(x,t)=\ensuremath∑k\ensuremathξk(x,t)\ensuremathαk,\ensuremath{\chi}(\mathbf{x},t)={\ensuremath{\sum}}_{k}{\ensuremath{\xi}}_{k}(\mathbf{x},t){\ensuremath{\alpha}}_{k},\ensuremathχ(x,t)=\ensuremath∑k​\ensuremathξk​(x,t)\ensuremathαk​, where the operators \ensuremathαk{\ensuremath{\alpha}}_{k}\ensuremathαk​ are time-independent annihilation operators, and the state vector has a time evolution described by a Schr"odinger equation in which the Hamiltonian is a time-dependent quadratic form in the phonon creation and annihilation operators, whose coefficients are explicitly determined in terms of the time-dependent condensate wave function \ensuremathξ(x,t)\ensuremath{\xi}(\mathbf{x},t)\ensuremathξ(x,t).
A recent paper of Gardiner [Phys. Rev. A 56, 1414 (1997)] introduces a particle-number-conserving Bogoliubov method for the excitation spectrum of a Bose-condensed gas, for use in theories of recently experimentally produced trapped atomic Bose condensates. Gardiner's approach is compared and contrasted with the 1959 Girardeau-Arnowitt theory [Phys. Rev. 113, 755 (1959)], to which it is closely related and which is also fully number conserving. The number-conserving Bogoliubov quasiparticle operators of the Girardeau-Arnowitt theory satisfy Bose commutation relations exactly so long as states with the condensate totally depleted are neglected, whereas those of Gardiner satisfy Bose commutation relations only in an approximation that deteriorates progressively as the condensate is depleted.
The impact of interparticle correlations on the behavior of Bose-Einstein Condensates (BECs) is discussed using two approaches. In the first approach, the wavefunction of a BEC is encoded in the NNN-particle sector of an extended "catalytic state". Going to a time-dependent interaction picture, we can organize the effective Hamiltonian by powers of N−1/2{N}^{-1/2}N−1/2. Requiring the terms of order N1/2{N}^{1/2}N1/2 to vanish, we get the Gross-Pitaevskii Equation. Going to the next order, N0N^0N0, we obtain the number-conserving Bogoliubov approximation. Our approach allows one to stay in the Schrödinger picture and to apply many techniques from quantum optics. Moreover, it is easier to track different orders in the Hamiltonian and to generalize to the multi-component case. In the second approach, I consider a state of N=l×nN =l\times nN=l×n bosons that is derived by symmetrizing the nnn-fold tensor product of an arbitrary lll-boson state. Particularly, we are interested in the pure state case for l=2l=2l=2, which we call the Pair-Correlated State (PCS). I show that PCS reproduces the number-conserving Bogoliubov approximation; moreover, it also works in the strong interaction regime where the Bogoliubov approximation fails. For the two-site Bose-Hubbard model, I find numerically that the error (measured by the trace distance of the two-particle reduced density matrices) of PCS is less than two percent over the entire parameter space, thus making PCS a bridge between the superfluid and Mott insulating phases. Amazingly, the error of PCS does not increase, in the time-dependent case, as the system evolves for longer times. I derive both time-dependent and -independent equations for the ground state and the time evolution of the PCS ansatz. The time complexity of simulating PCS does not depend on NNN and is linear in the number of orbitals in use.
We encode the many-body wave function of a Bose-Einstein condensate (BEC) in the NNN-particle sector of an extended catalytic state. This catalytic state is a coherent state for the condensate mode and an arbitrary state for the modes orthogonal to the condensate mode. Going to a time-dependent interaction picture where the state of the condensate mode is displaced to the vacuum, we can organize the effective Hamiltonian by powers of ${N}^{\ensuremath{-}1/2}$. Requiring the terms of order N1/2{N}^{1/2}N1/2 to vanish gives the Gross-Pitaevskii equation. Going to the next order, N0{N}^{0}N0, we derive equations for the number-conserving Bogoliubov approximation, first given by Castin and Dum [Phys. Rev. A 57, 3008 (1998)]. In contrast to other approaches, ours is well suited to calculating the state evolution in the Schr\"odinger picture; moreover, it is straightforward to generalize our method to multicomponent BECs and to higher-order corrections.
We study the truncated Wigner method applied to a weakly interacting spinless Bose-condensed gas which is perturbed away from thermal equilibrium by a time-dependent external potential. The principle of the method is to generate an ensemble of classical fields ψ( r ) which samples the Wigner quasi-distribution function of the initial thermal equilibrium density operator of the gas, and then to evolve each classical field with the Gross–Pitaevskii equation. In the first part of the paper we improve the sampling technique over our previous work (Sinatra et al 2000 J. Mod. Opt. 47 2629–44) and we test its accuracy against the exactly solvable model of the ideal Bose gas. In the second part of the paper we investigate the conditions of validity of the truncated Wigner method. For short evolution times it is known that the time-dependent Bogoliubov approximation is valid for almost pure condensates. The requirement that the truncated Wigner method reproduces the Bogoliubov prediction leads to the constraint that the number of field modes in the Wigner simulation must be smaller than the number of particles in the gas. For longer evolution times the nonlinear dynamics of the noncondensed modes of the field plays an important role. To demonstrate this we analyse the case of a three-dimensional spatially homogeneous Bose-condensed gas and we test the ability of the truncated Wigner method to correctly reproduce the Beliaev–Landau damping of an excitation of the condensate. We have identified the mechanism which limits the validity of the truncated Wigner method: the initial ensemble of classical fields, driven by the time-dependent Gross–Pitaevskii equation, thermalizes to a classical field distribution at a temperature T class which is larger than the initial temperature T of the quantum gas. When T class significantly exceeds T a spurious damping is observed in the Wigner simulation. This leads to the second validity condition for the truncated Wigner method, T class − T ≪ T , which requires that the maximum energy max of the Bogoliubov modes in the simulation does not exceed a few k B T .
The evolution of Bose-Einstein condensates is amply described by the time-dependent Gross-Pitaevskii mean-field theory which assumes all bosons to reside in a single time-dependent one-particle state throughout the propagation process. In this work, we go beyond mean field and develop an essentially exact many-body theory for the propagation of the time-dependent Schr\"odinger equation of NNN interacting identical bosons. In our theory, the time-dependent many-boson wave function is written as a sum of permanents assembled from orthogonal one-particle functions, or orbitals, where both the expansion coefficients and the permanents (orbitals) themselves are time-dependent and fully determined according to a standard time-dependent variational principle. By employing either the usual Lagrangian formulation or the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the orbitals and one for the expansion coefficients. The first set comprises of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with time-dependent coefficients. We call our theory multiconfigurational time-dependent Hartree for bosons, or MCTDHB(M)\text{MCTDHB}(M)MCTDHB(M), where MMM specifies the number of time-dependent orbitals used to construct the permanents. Numerical implementation of the theory is reported and illustrative numerical examples of many-body dynamics of trapped Bose-Einstein condensates are provided and discussed. The convergence of the method with a growing number MMM of orbitals is demonstrated in a specific example of four interacting bosons in a double well.
We propose an algorithm for the numerical solution of the time-dependent Schrödinger equation for N >> 1bosons interacting via a two-body interaction potential in the mean-field scaling regime. Unlike existing numerical algorithms for quantum dynamics, the proposed algorithm has a computational complexity that is independent of the number N of particles. It is based on an ansatz for the wave function inspired by the Bogoliubov theory, which is an asymptotic theory for large N. In a first part, we follow Bogoliubov theory in evolving the condensate function by the Hartree equation, but we use the Dirac--Frenkel time-dependent variational principle to derive nonlinear equations of motion for the excitation functions in the Bogoliubov ansatz. By carefully exploiting the bosonic symmetry, the so obtained approximate wave functions are shown to have derivatives bounded independently of N apart from a time-dependent global phase factor which rotates with a frequency proportional to N. In the second part, we numerically approximate the wave function (modulo the phase) by a composite algorithm, which employs an appropriate splitting of the equations of motion and uses a dynamical low-rank approximation of the fundamental two-particle excitation function by linear combinations of products of single-particle functions. The low-rank excitation function is evolved in time by a basis update and Galerkin (BUG) integrator that is robust to small singular values of the coefficient matrix and allows for rank adaptivity. Numerical results are provided to assess the performance of the proposed algorithm.