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.
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
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〉, which is obtained by operating a unitary operator
eiG(y) on the number state of free bosons. Using
eiG(y), we identify a ``natural coordinate''
b0 of the interacting bosons, by which many physical properties can be simply described. The
∣N,y〉 can be represented simply as a number state of
b0; 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
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 $|\ensuremath{\xi},N,\mathbf{y}〉$ 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 $\ensuremath{\Psi}.$
As is well-known, in the conventional formulation of Bogoliubov's theory of an interacting Bose gas, the Hamiltonian is written as a decoupled sum of contributions from different momenta of the form . Then, each of the single-mode Hamiltonians is diagonalized separately, and the resulting ground state wavefunction of the total Hamiltonian is written as a simple product of the ground state wavefunctions of each of the single-mode Hamiltonians . While this way of diagonalizing the total Hamiltonian may seem to be valid from the perspective of the standard, number non-conserving Bogoliubov's method, where the k = 0 state is removed from the Hilbert space and hence the individual Hilbert spaces where the Hamiltonians {} are diagonalized are disjoint from one another, we argue that from a number-conserving perspective this diagonalization method may not be adequate since the true Hilbert spaces where the Hamiltonians {} should be diagonalized all have the k = 0 state in common, and hence the ground state wavefunction of the total Hamiltonian may not be written as a simple product of the ground state wavefunctions of the 's. In this paper, we give a thorough review of Bogoliubov's method, and discuss a variational and number-conserving formulation of this theory in which the k = 0 state is restored to the Hilbert space of the interacting gas, and where, instead of diagonalizing the Hamiltonians separately, we diagonalize the total Hamiltonian . When this is done, we find that the ground state energy is lowered below the Bogoliubov result, and the depletion of bosons is significantly reduced with respect to the one obtained in the number non-conserving treatment. We also find that the spectrum of the usual αk excitations of Bogoliubov's method changes from a gapless one, as predicted by the standard, number non-conserving formulation of this theory, to one which exhibits a finite gap in the k → 0 limit. We discuss the presence of a gap in the spectrum of the αk's in light of Goldstone's theorem, and show that there is no contradiction with the latter.
We analyze the condensation of interacting bosons confined in a finite volume V , by evaluating the time evolution of the density operator
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 of interacting bosons, using which many physical properties can be simply described. In particular, the time evolution of
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. 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.
A simple model of a boson system is considered for which the exact Schr\"odinger equation reduces to a difference equation which can be solved numerically. Application of the Bogoliubov method and comparison of its results with the numerical results then yield some insight into the validity of the Bogoliubov approximations. The simple model considered has, in the absence of interactions, only a zero energy state and two states of unit energy available to each boson. Initially there is assumed to be no interaction between bosons in the degenerate excited states and all existing interactions are assumed to be repulsive. The ground-state energy is calculated in the Bogoliubov approximation, with and without the depletion effect considered, and in another numerical approximation. These results and the results of the numerical solution of the exact Schr\"odinger equation converge to the Bogoliubov result with depletion ignored when the number of particles in the system approaches infinity. It is surprising to note that, for the interaction strengths considered, the Bogoliubov result is within 3% of the numerical result for as few as 32 particles in the system, and within 10% for only 4 particles. A modification of the above system is considered in which there exists an additional two-body interaction between particles in the degenerate excited states which may be attractive or repulsive. It is shown that the ground-state energy, with this additional interaction present, deviates from the Bogoliubov value linearly with the strength of the added interaction, i.e., $W={W}_{\mathrm{Bog}}+\ensuremath{\alpha}F$ (with
F the strength parameter of the interaction). The factor $\ensuremath{\alpha}$ is found to be small and constant over a wide range of
F and in this range the deviation of
W from the Bogoliubov value is small. There is, however, a transition point
FT, beyond which $\ensuremath{\alpha}$ is constant and large and the usual Bogoliubov approximation is invalid for interactions more attractive than that characterized by
FT. A Bogoliubov-like approximation is shown to be quite accurate in this region. It is then shown that $F_{T}^{\ensuremath{-}1}$ is a linear function of the number of particles in the system.