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

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 $\ensuremath{\Psi}.$