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arXiv:quant-ph/9811029 · Physical Review A

Time evolution of the condensed state of interacting bosons with reduced number fluctuation in a leaky box

Akira Shimizu, Jun‐ichi Inoue

Published Oct 1, 1999• 2 citations
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Abstract

We study the nonequilibrium time evolution of the Bose-Einstein condensate of interacting bosons confined in a leaky box, when its number fluctuation is initially $(t=0)$ suppressed. We take account of quantum fluctuations of all modes, including $\mathbf{k}=0,$ of the bosons. As the wave function of the ground state that has a definite number N of interacting bosons, we use a variational form $|N,\mathbf{y}〉,$ which is obtained by operating a unitary operator ${e}^{iG(\mathbf{y})}$ on the number state of free bosons. Using ${e}^{iG(\mathbf{y})},$ we identify a ``natural coordinate'' ${b}_{0}$ of the interacting bosons, by which many physical properties can be simply described. The $|N,\mathbf{y}〉$ can be represented simply as a number state of ${b}_{0};$ we thus call it the ``number state of interacting bosons'' (NSIB). To simulate real systems, for which if one fixes N at $t=0$ N will fluctuate at later times because of a finite probability of exchanging bosons between the box and the environment, we evaluate the time evolution of the reduced density operator $\stackrel{^}{\ensuremath{\rho}}(t)$ of the bosons in the box as a function of the leakage flux J. We concentrate on the most interesting and nontrivial time stage, i.e., the early time stage for which $\mathrm{Jt}\ensuremath{\ll}N,$ much earlier than the time when the system approaches the equilibrium state. It is shown that the time evolution can be described very simply as the evolution from a single NSIB at $t<0,$ into a classical mixture, with a time-dependent distribution, of NSIBs of various values of N at $t>0.$ Using ${b}_{0},$ we successfully define the cosine and sine operators for interacting many bosons, by which we can analyze the phase fluctuation in a fully quantum-mechanical manner. We define a new state $|\ensuremath{\xi},N,\mathbf{y}〉$ called the ``number-phase-squeezed state of interacting bosons'' (NPIB), which is characterized by a complex parameter $\ensuremath{\xi}.$ It is shown that $\stackrel{^}{\ensuremath{\rho}}(t)$ for $t>0$ can be rewritten as the phase-randomized mixture (PRM) of NPIBs. Among many possible representations of $\stackrel{^}{\ensuremath{\rho}}(t),$ this representation is particularly convenient for analyzing the phase fluctuations and the order parameter. We study the order parameter according to a few typical definitions, as well as their time evolution. It is shown that the off-diagonal long-range order (ODLRO) does not distinguish the NSIB and NPIB. Hence, the order parameter $\ensuremath{\Xi}$ defined from ODLRO does not distinguish them, either. On the other hand, the other order parameter $\ensuremath{\Psi},$ defined as the expectation value of the boson operator $\stackrel{^}{\ensuremath{\psi}}$, has different values among these states. In particular, for each element of the PRM of NPIBs, we show that $\ensuremath{\Psi}$ evolves from zero to a finite value very quickly. Namely, after the leakage of only two or three bosons, each element acquires a full, stable, and definite (nonfluctuating) value of $\ensuremath{\Psi}.$

DOI
10.1103/physreva.60.3204

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We study the nonequilibrium time evolution of the Bose-Einstein condensate of interacting bosons confined in a leaky box, when its number fluctuation is initially (t=0)(t=0)(t=0) suppressed. We take account of quantum fluctuations of all modes, including k=0,\mathbf{k}=0,k=0, of the bosons. As the wave function of the ground state that has a definite number N of interacting bosons, we use a variational form ∣N,y〉,|N,\mathbf{y}〉,∣N,y〉, which is obtained by operating a unitary operator eiG(y){e}^{iG(\mathbf{y})}eiG(y) on the number state of free bosons. Using eiG(y),{e}^{iG(\mathbf{y})},eiG(y), we identify a natural coordinate'' ${b}_{0}$ of the interacting bosons, by which many physical properties can be simply described. The $|N,\mathbf{y}〉$ can be represented simply as a number state of ${b}_{0};$ we thus call it the number state of interacting bosons'' (NSIB). To simulate real systems, for which if one fixes N at t=0t=0t=0 N will fluctuate at later times because of a finite probability of exchanging bosons between the box and the environment, we evaluate the time evolution of the reduced density operator \stackrel{^}{\ensuremath{\rho}}(t) of the bosons in the box as a function of the leakage flux J. We concentrate on the most interesting and nontrivial time stage, i.e., the early time stage for which Jt\ensuremath≪N,\mathrm{Jt}\ensuremath{\ll}N,Jt\ensuremath≪N, much earlier than the time when the system approaches the equilibrium state. It is shown that the time evolution can be described very simply as the evolution from a single NSIB at t<0,t<0,t<0, into a classical mixture, with a time-dependent distribution, of NSIBs of various values of N at t>0.t>0.t>0. Using b0,{b}_{0},b0​, we successfully define the cosine and sine operators for interacting many bosons, by which we can analyze the phase fluctuation in a fully quantum-mechanical manner. We define a new state ∣\ensuremathξ,N,y〉|\ensuremath{\xi},N,\mathbf{y}〉∣\ensuremathξ,N,y〉 called the ``number-phase-squeezed state of interacting bosons'' (NPIB), which is characterized by a complex parameter \ensuremathξ.\ensuremath{\xi}.\ensuremathξ. It is shown that \stackrel{^}{\ensuremath{\rho}}(t) for t>0t>0t>0 can be rewritten as the phase-randomized mixture (PRM) of NPIBs. Among many possible representations of \stackrel{^}{\ensuremath{\rho}}(t), this representation is particularly convenient for analyzing the phase fluctuations and the order parameter. We study the order parameter according to a few typical definitions, as well as their time evolution. It is shown that the off-diagonal long-range order (ODLRO) does not distinguish the NSIB and NPIB. Hence, the order parameter \ensuremathΞ\ensuremath{\Xi}\ensuremathΞ defined from ODLRO does not distinguish them, either. On the other hand, the other order parameter \ensuremathΨ,\ensuremath{\Psi},\ensuremathΨ, defined as the expectation value of the boson operator \stackrel{^}{\ensuremath{\psi}}, has different values among these states. In particular, for each element of the PRM of NPIBs, we show that \ensuremathΨ\ensuremath{\Psi}\ensuremathΨ evolves from zero to a finite value very quickly. Namely, after the leakage of only two or three bosons, each element acquires a full, stable, and definite (nonfluctuating) value of \ensuremathΨ.\ensuremath{\Psi}.\ensuremathΨ.
We analyze the condensation of interacting bosons confined in a finite volume V , by evaluating the time evolution of the density operator ρ^(t)\hat{\rho}(t)ρ^​(t) in the case where the number of bosons is exactly fixed for t < 0 and a small leakage occurs for t ≥0. We identify a “natural coordinate” b^0\hat{b}_{0}b^0​ of interacting bosons, using which many physical properties can be simply described. In particular, the time evolution of ρ^(t)\hat{\rho}(t)ρ^​(t) can be simply described as the evolution from the number state, to a phase-randomized mixture (PRM) of “number-phase squeezed states”, and finally to the PRM of coherent states, of b^0\hat{b}_{0}b^0​. The order parameter, defined as the expectation value of the boson field, of each element of the PRM grows rapidly from zero to the full value.
The kinetics of the formation of coherent correlation properties associated with Bose condensation is studied in detail. The evolution of a nonequilibrium state with no “condensate-seed” is related to a hierarchy of relaxation times. At the first stage, a particle flux in energy space toward low energies sets in. The evolution in this case is described by a nonlinear Boltzmann equation, with a characteristic time given by interparticle collisions. When the particles which will later form the condensate have a kinetic energy which is less than the potential energy, a quasicondensate starts to form. In this stage, fluctuations of the density (but not of the phase) are suppressed and short-range coherent correlation properties are governed by the equation of motion for a quasiclassical complex field. The next stage is connected with the formation of the long-range order. The time for forming topological order and therefore genuine superfluidity proves to be dependent on the system size. The off-diagonal long-range order, arising after the attenuation of long-wave phase fluctuations, has a size-dependent relaxation time as well. Introduction The problem of Bose–Einstein Condensation (BEC) kinetics, being interesting in itself, has acquired a special significance in connection with experimental efforts to observe this condensation in a number of systems with particles with a finite lifetime. Such systems include spin-polarized atomic hydrogen [1], excitons [2] and biexcitons [3] in semiconductors and, more recently, laser-cooled atomic systems [4].
We study the time evolution of an initially trapped weakly interacting Bose gas at positive temperature, after the trapping potential has been switched off. It has been recently shown in Deuchert–Seiringer (2021) that the one-particle density matrix of Gibbs states of the interacting trapped gas is given, to leading order in N , as N \rightarrow \infty , by that of the ideal gas, with the condensate wave function replaced by the minimizer of the Hartree energy functional. We show that this structure is stable with respect to the many-body evolution in the following sense: the dynamics can be approximated in terms of the time-dependent Hartree equation for the condensate wave function and in terms of the free evolution for the thermally excited particles. The main technical novelty of our work is the use of the Hartree–Fock–Bogoliubov equations to define a fluctuation dynamics.
We study the robustness, against the leakage of bosons, of wave functions of interacting many bosons confined in a finite box by deriving and analyzing a general equation of motion for the reduced density operator. We identify a robust wave function that remains a pure state, whereas other wave functions, such as the Bogoliubov's ground state and the ground state with a fixed number of bosons, evolve into mixed states. Although these states all have the off-diagonal long-range order, and the same energy, we argue that only the robust state is realized as a macroscopic quantum state.
We study spatial correlation functions of local operators of interacting many bosons confined in a box of a large, but volume V, for various `ground states' whose energy densities are almost degenerate. The ground states include the coherent state of interacting bosons (CSIB), the number state of interacting bosons (NSIB), and the number-phase squeezed state of interacting bosons, which interpolates between the CSIB and NSIB. It was shown previously that only the CSIB is robust (i.e., does not decohere for a macroscopically long time) against the leakage of bosons into an environment. We show that for the CSIB the spatial correlation of any local operators A(r) and B(r') (which are localized around r and r', respectively) vanishes as |r - r' | \sim V^{1/3} \to \infty, i.e., the CSIB has the `cluster property.' In contrast, the other ground states do not possess the cluster property. Therefore, we have successfully shown that the robust state has the cluster property. This ensures the consistency of the field theory of bosons with macroscopic theories.
The time-dependent non-equilibrium dynamics of a Bose–Einstein condensate (BEC) typically generates incoherent excitations out of the condensate due to the finite frequencies present in the time evolution. We present a detailed derivation of a general non-equilibrium Greenʼs function technique that describes the coupled time evolution of an interacting BEC and its single-particle excitations in a trap, based on an expansion in terms of the exact eigenstates of the trap potential. We analyze the dynamics of a Bose system in a small double-well potential with initially all particles in the condensate. When the trap frequency is larger than the Josephson frequency, , the dynamics changes at a characteristic time, , abruptly from the slow Josephson oscillations of the BEC to fast Rabi oscillations driven by quasiparticle excitations in the trap. For times , the Josephson oscillations are undamped, in agreement with the experiments. We analyze the physical origin of the finite scale as well as its dependence on the trap parameter Δ .