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arXiv:cond-mat/0009258 · Journal of the Physical Society of Japan

Cluster Property and Robustness of Ground States of Interacting Many Bosons

Akira Shimizu, Takayuki Miyadera

Published Jan 1, 2002• 10 citations
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Abstract

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.

DOI
10.1143/jpsj.71.56

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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 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 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 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.
Already a few bosons with contact interparticle interactions in small optical lattices feature a variety of quantum phases: superfluid, Mott-insulator, and fermionized Tonks gases can be probed in such systems. To detect these phases---pivotal for both experiment and theory---as well as their many-body properties we analyze several distinct measures for the one-body and many-body Shannon information entropies. We exemplify the connection of these entropies with spatial correlations in the many-body state by contrasting them to the Glauber normalized correlation functions. To obtain the ground state for lattices with commensurate filling (i.e., an integer number of particles per site) for the full range of repulsive interparticle interactions we utilize the multiconfigurational time-dependent Hartree method for bosons (MCTDHB) in order to solve the many-boson Schr\"odinger equation. We demonstrate that all emergent phases---the superfluid, the Mott insulator, and the fermionized gas can be characterized equivalently by many-body entropy measures and by Glauber's normalized correlation functions. In sharp contrast, single-particle entropy cannot capture these phases.
The properties of cold Bose gases at unitarity have been extensively investigated in the last few years both theoretically and experimentally. In this Letter we use a family of interactions tuned to two-body unitarity and very weak three-body binding to demonstrate the universal properties of both clusters and matter. We determine the universal properties of finite clusters up to 60 particles and, for the first time, explicitly demonstrate the saturation of energy and density with particle number and compare with bulk properties. At saturation in the bulk we determine the energy, density, two- and three-body contacts, and the condensate fraction. We find that uniform matter is more bound than three-body clusters by nearly 2 orders of magnitude, the two-body contact is very large in absolute terms, and yet the condensate fraction is also very large, greater than 90%. Equilibrium properties of these systems may be experimentally accessible through rapid quenching of weakly interacting boson superfluids.
The properties of a Bose system of particles with repulsive interactions have previously been treated using perturbation theory in the formalism of second quantization. Others have also considered this problem by dealing with the wave function in configuration space, using the theory of cluster expansions. In these latter papers, variation with respect to a parameter in a trial function for the ground state has been shown to yield a ground-state energy close to the exact asymptotic expressions obtained from perturbation theory. The connection between the two methods is not immediately obvious from these cluster expansion treatments. It is shown here that, as one might expect, the cluster integral method can be handled so that it is completely equivalent to the pair approximation in perturbation theory.