Abstract In Ref. Gaudin (Nuclear Physics 15 , 89 1960), Gaudin established a generalized version of Wick’s theorem at finite temperature for systems described by a diagonal quadratic Hamiltonian. In this work, we extend Gaudin’s proof to statistical density operators generated by a general quadratic Hamiltonian, prior to diagonalization, as commonly encountered in interacting Bose systems. We show explicitly that the resulting thermal contractions depend on the Bogoliubov transformation parameters and on the quasiparticle spectrum, providing a transparent framework for evaluating thermal averages of arbitrary operator products. As an illustration of the usefulness of this generalized formulation, we evaluate particle-number fluctuations in a homogeneous, weakly interacting Bose gas below the Bose–Einstein condensation temperature. We show that particle-number fluctuations are generically non-Poissonian as a consequence of interaction-induced correlations and quantum depletion, with the Poissonian limit recovered only for an ideal Bose gas at zero temperature. Our results clarify the role of quadratic Hamiltonians and Bogoliubov quasiparticles in shaping fluctuation statistics and highlight the generalized Wick’s theorem as a controlled tool for computing correlation functions in interacting quantum gases.
Abstract In Ref. Gaudin (Nuclear Physics 15 , 89 1960), Gaudin established a generalized version of Wick’s theorem at finite temperature for systems described by a diagonal quadratic Hamiltonian. In this work, we extend Gaudin’s proof to statistical density operators generated by a general quadratic Hamiltonian, prior to diagonalization, as commonly encountered in interacting Bose systems. We show explicitly that the resulting thermal contractions depend on the Bogoliubov transformation parameters and on the quasiparticle spectrum, providing a transparent framework for evaluating thermal averages of arbitrary operator products. As an illustration of the usefulness of this generalized formulation, we evaluate particle-number fluctuations in a homogeneous, weakly interacting Bose gas below the Bose–Einstein condensation temperature. We show that particle-number fluctuations are generically non-Poissonian as a consequence of interaction-induced correlations and quantum depletion, with the Poissonian limit recovered only for an ideal Bose gas at zero temperature. Our results clarify the role of quadratic Hamiltonians and Bogoliubov quasiparticles in shaping fluctuation statistics and highlight the generalized Wick’s theorem as a controlled tool for computing correlation functions in interacting quantum gases.
In Gaudin (1960), Michel Gaudin showed the Wick's theorem at finite temperature using a diagonal Hamiltonian. We extend the Gaudin's prove for a statistical density operator which depend on a quadratic Hamiltonian. To illustrate the utility of the theorem, we evaluate the ratio [−2]/ of a homogeneous weakly interacting Bose gas at temperature below the Bose-Einstein condensation temperature. At this condition, the quadratic Hamiltonian approximation is valued and, in this evaluation, we show the sub-Poissonian behaviour of the fundamental state distribution at zero temperature.
Aiming at simplicity of explicit equations and, at the same time, controllable accuracy of the theory, we present our results for all the thermodynamic quantities and correlation functions for a weakly interacting Bose gas at short-to-intermediate distances obtained within an improved version of Beliaev's diagrammatic technique. With a controllably small (but essentially finite) Bogoliubov's symmetry-breaking term, Beliaev's diagrammatic technique becomes regular in the infrared limit. Up to higher-order terms (for which we present parametric order-of-magnitude estimates), the partition function and entropy of the system formally correspond to those of a non-interacting bosonic (pseudo-)Hamiltonian with a temperature-dependent Bogoliubov-type dispersion relation. Away from the fluctuation region, this approach provides the most accurate—in fact, the best possible within the Bogoliubov-type pseudo-Hamiltonian framework—description of the system with controlled accuracy. It produces accurate answers for the off-diagonal correlation functions up to distances where the behavior of correlators is controlled by generic hydrodynamic relations and, thus, can be accurately extrapolated to arbitrarily large scales. In the fluctuation region, the non-perturbative contributions are given by universal (for all weakly interacting U(1) systems) constants and scaling functions, which can be obtained separately—by simulating classical U(1) models—and then used to extend the description of the weakly interacting Bose gas to the fluctuation region. The technique works in all spatial dimensions, and we explicitly checked the validity of this technique against first-principle Monte Carlo simulations for various thermodynamic properties and the single-particle density matrix.
We analyze the equilibrium properties of a weakly interacting, trapped quasi-one-dimensional Bose gas at finite temperatures and compare different theoretical approaches. We focus in particular on two stochastic theories: a number-conserving Bogoliubov (NCB) approach and a stochastic Gross-Pitaevskii equation (SGPE) that have been extensively used in numerical simulations. Equilibrium properties like density profiles, correlation functions, and the condensate statistics are compared to predictions based upon a number of alternative theories. We find that due to thermal phase fluctuations, and the corresponding condensate depletion, the NCB approach loses its validity at relatively low temperatures. This can be attributed to the change in the Bogoliubov spectrum, as the condensate gets thermally depleted, and to large fluctuations beyond perturbation theory. Although the two stochastic theories are built on different thermodynamic ensembles (NCB, canonical; SGPE, grand-canonical), they yield the correct condensate statistics in a large Bose-Einstein condensate (BEC) (strong enough particle interactions). For smaller systems, the SGPE results are prone to anomalously large number fluctuations, well known for the grand-canonical, ideal Bose gas. Based on the comparison of the above theories to the modified Popov approach, we propose a simple procedure for approximately extracting the Penrose-Onsager condensate from first- and second-order correlation functions that is both computationally convenient and of potential use to experimentalists. This also clarifies the link between condensate and quasicondensate in the Popov theory of low-dimensional systems.
We derive an expression for the reduced density matrices of ideal Bose and Fermi gases in the canonical ensemble, which corresponds to the Bloch--De Dominicis (or Wick's) theorem in the grand canonical ensemble for normal-ordered products of operators. Using this expression, we study one- and two-body correlations of homogeneous ideal gases with N particles. The pair distribution function g(2)(r) of fermions clearly exhibits antibunching with g(2)(0)=0 due to the Pauli exclusion principle at all temperatures, whereas that of normal bosons shows bunching with g(2)(0)≈2, corresponding to the Hanbury Brown--Twiss effect. For bosons below the Bose--Einstein condensation temperature T0, an off-diagonal long-range order develops in the one-particle density matrix to reach g(1)(r)=1 at T=0, and the pair correlation starts to decrease towards g(2)(r)≈1 at T=0. The results for N→∞ are seen to converge to those of the grand canonical ensemble obtained by assuming the average ⟨ψ^(r)⟩ of the field operator ψ^(r) below T0. This fact justifies the introduction of the "anomalous" average ⟨ψ^(r)⟩=0 below T0 in the grand canonical ensemble as a mathematical means of removing unphysical particle-number fluctuations to reproduce the canonical results in the thermodynamic limit.
We consider equilibrium canonical-ensemble fluctuations of the Bose-Einstein condensate (BEC) in a trap using the particle-number conserving operator formalism of Girardeau and Arnowitt [Phys. Rev. 113, 755 (1959)] and the concept of the canonical-ensemble quasiparticles. We find explicit expressions for the characteristic function and all cumulants of the ground-state occupation statistics both for the dilute weakly interacting and ideal Bose gases. We present analytical formulas for the moments of the ground-state occupation fluctuations in the weakly interacting and ideal Bose gases in a box (``homogeneous gas'') as well as in the ideal Bose gas in an arbitrary power-law trap, in particular, in an arbitrary harmonic trap. We calculate the effect of Bogoliubov coupling between quasiparticles on suppression of the ground-state occupation fluctuations at moderate temperatures and their enhancement at very low temperatures. We clarify a crossover between ideal-gas and weakly interacting gas statistics, which is governed by a pair-correlation, squeezing mechanism. The important conclusion is that in most cases the ground-state occupation fluctuations are anomalously large and are not Gaussian even in the thermodynamic limit.
We re-examine the way in which Bogoliubov's theory of a dilute Bose gas at T=0 has been extended to describe the statistical mechanics of interacting bosons at finite temperature. We show explicitly that the field-theoretic calculation of the grand partition function in this formulation amounts to a canonical trace over the eigenfunctions of the Bogoliubov Hamiltonian at fixed total number of bosons N, and that the additional trace over N that is required in the grand-canonical formalism is never carried out. This implies that what usually passes as the grand-canonical treatment of the Bogoliubov Hamiltonian is not quite grand-canonical, and is in fact a canonical one. We also show that the discontinuity in the condensate density predicted by previous formulations of this theory as the temperature T goes past the critical transition temperature Tc is a direct consequence of an inappropriate generalization of the Bogoliubov prescription to finite temperatures, and that this discontinuity disappears when this prescription is either used as a zero temperature approximation or avoided altogether. Armed with the above findings, we reformulate the statistical mechanics of interacting bosons in the canonical ensemble and derive the thermodynamics of the system. We then show how the canonical treatment can be used to setup a truly grand-canonical description of the statistical mechanics of a weakly interacting Bose gas where the average number of bosons in the system varies with temperature, unlike existing formulations where the total number of bosons N is taken to be a constant that does not depend on T. Consequences on the physics of interacting bosons are briefly discussed.