We study the robustness of wavefunctions of interacting many bosons againstthe leakage of bosons, by deriving and analyzing a general equation of motionfor the reduced density operator. We identify a robust wavefunction thatremains a pure state, whereas other wavefunctions, that are proposed as theground state, such as the Bogoliubov's ground state and the ground state with afixed number of bosons, evolve into mixed states. Although these states allhave the off-diagonal long-range order, and the same energy, we argue that onlythe 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 $\ensuremath{\Psi}.$
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.
I show that the dynamics of the weakly interacting bose gas can be described by a modified time dependent Bogoliubov theory. The novelty of the approach is to include decoherence steps that gradually transform the entanglement entropy of the pure state into the von Neumann entropy of a statistical mixture. This approximation drastically reduces the entanglement that is needed in order to represent the system's state while becoming exponentially accurate in the mean field limit. I argue that this scheme can be extended to all quantum systems whose ground state can be well approximated by a variational wave function. The upshot is that the dynamics of almost all quantum systems can be reduced to stochastic classical motion supplemented with small quantum fluctuations.