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Physical Review A

Excitations in a Bose Gas at Finite Temperatures. II. Relation between Single-Particle and Density Fluctuations

Allan Griffin, T. H. Cheung

Published Jun 1, 1973• 36 citations
Publisher

Abstract

By making a systematic analysis of correlation functions in terms of irreducible and reducible parts, Ma and others have developed a dielectric formulation of the dynamics of a condensed Bose system which fully includes the background (or excited) atoms. We show that this formulation quite generally leads to the density and single-particle-correlation functions both having two resonances ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ although with different weights. If the condensate ${n}_{0}=0$, the ${\ensuremath{\omega}}_{2}$ mode only appears in the single-particle spectrum while ${\ensuremath{\omega}}_{1}$ only appears in the density-fluctuation spectrum. For finite values of ${n}_{0}$, these modes are coupled and renormalized to ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$. As a specific illustration, we use the shielded-potential approximation (SPA) for the reducible self-energies. In this model, we find the free-particle excitations (${\ensuremath{\omega}}_{2}$) are coupled to the zero-sound density fluctuations (${\ensuremath{\omega}}_{1}$) through the action of the condensate. In the SPA, the reducible self-energies have a pole at ${\ensuremath{\omega}}_{1}$. The usual Bogoliubov, Hartree-Fock, and ordinary $t$-matrix approximations involve nonsingular self-energies and hence do not exhibit the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ mode. Experimentally, it appears that the high-frequency ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ mode has never been detected, but this is probably owing to the fact that it is strongly damped as a result of its coupling to the zero-sound mode. Using the two-fluid hydrodynamic equations, one can argue that at low frequencies the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ excitation corresponds to first sound and the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ excitation corresponds to second sound. Finally, we briefly discuss the possibility of singularities in the irreducible self-energies at high frequencies and the resulting two-roton bound states.

DOI
10.1103/physreva.7.2086

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By making a systematic analysis of correlation functions in terms of irreducible and reducible parts, Ma and others have developed a dielectric formulation of the dynamics of a condensed Bose system which fully includes the background (or excited) atoms. We show that this formulation quite generally leads to the density and single-particle-correlation functions both having two resonances \ensuremathω\ifmmode~\else˜\fi1{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}\ensuremathω\ifmmode~\else˜\fi​1​ and \ensuremathω\ifmmode~\else˜\fi2{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}\ensuremathω\ifmmode~\else˜\fi​2​ although with different weights. If the condensate n0=0{n}_{0}=0n0​=0, the \ensuremathω2{\ensuremath{\omega}}_{2}\ensuremathω2​ mode only appears in the single-particle spectrum while \ensuremathω1{\ensuremath{\omega}}_{1}\ensuremathω1​ only appears in the density-fluctuation spectrum. For finite values of n0{n}_{0}n0​, these modes are coupled and renormalized to \ensuremathω\ifmmode~\else˜\fi1{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}\ensuremathω\ifmmode~\else˜\fi​1​ and \ensuremathω\ifmmode~\else˜\fi2{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}\ensuremathω\ifmmode~\else˜\fi​2​. As a specific illustration, we use the shielded-potential approximation (SPA) for the reducible self-energies. In this model, we find the free-particle excitations (\ensuremathω2{\ensuremath{\omega}}_{2}\ensuremathω2​) are coupled to the zero-sound density fluctuations (\ensuremathω1{\ensuremath{\omega}}_{1}\ensuremathω1​) through the action of the condensate. In the SPA, the reducible self-energies have a pole at \ensuremathω1{\ensuremath{\omega}}_{1}\ensuremathω1​. The usual Bogoliubov, Hartree-Fock, and ordinary ttt-matrix approximations involve nonsingular self-energies and hence do not exhibit the \ensuremathω\ifmmode~\else˜\fi1{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}\ensuremathω\ifmmode~\else˜\fi​1​ mode. Experimentally, it appears that the high-frequency \ensuremathω\ifmmode~\else˜\fi2{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}\ensuremathω\ifmmode~\else˜\fi​2​ mode has never been detected, but this is probably owing to the fact that it is strongly damped as a result of its coupling to the zero-sound mode. Using the two-fluid hydrodynamic equations, one can argue that at low frequencies the \ensuremathω\ifmmode~\else˜\fi1{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}\ensuremathω\ifmmode~\else˜\fi​1​ excitation corresponds to first sound and the \ensuremathω\ifmmode~\else˜\fi2{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}\ensuremathω\ifmmode~\else˜\fi​2​ excitation corresponds to second sound. Finally, we briefly discuss the possibility of singularities in the irreducible self-energies at high frequencies and the resulting two-roton bound states.
In Chapter 3, we have shown how a Bose condensate leads to the single-particle Green's function G αβ ( Q , ω) appearing as a distinct component of the the dynamic structure factor S ( Q , ω). This feature is a direct consequence of (3.13) and is explicitly shown by the expression for S ( Q , ω) in (3.18). In the present chapter, we review the many-body diagrammatic analysis which specifies the precise relation between G αβ ( Q , ω) and S ( Q , ω) in a Bose-condensed fluid ( n 0 ≠ 0). The power of this “dielectric formalism” lies in showing that both functions mirror each other, as a result of the condensate-induced intermixing of single-particle and density excitations. At the same time, the formalism emphasizes the continuity of these excitations as the fluid passes through T λ, although these excitations have very temperature-dependent weights in G αβ ( Q , ω) and S ( Q , ω). While it is the poles of the single-particle Green's function which describe the elementary excitations, experimentally it is difficult to probe directly this field-fluctuation spectrum. In particular, it is only in a Bose-condensed fluid that one can obtain information about G αβ ( Q , ω) from inelastic neutron-scattering data on S ( Q , ω). In the absence of a condensate, there is little relation between the density-fluctuation modes and the single-particle excitations in a Bose fluid. After setting up the dielectric formalism in Section 5.1, the next two sections consider some model approximations at finite temperatures. These models are most appropriate to a dilute weakly interacting Bose gas (WIDBG), but we believe they also give crucial insight into various aspects of superfluid 4 He.
Making use of the finite-temperature version of Beliaev's field-theoretical description of an interacting Bose gas, we sum the self-energy diagrams which correspond to the collisionless shielded potential approximation (SPA). This generalizes Bogoliubov's first-order results by replacing the bare repulsive interaction by a dynamically shielded interaction and includes the effect of the excited atoms. This theory is completely equivalent to that of Tserkovnikov if we use the ideal Bose gas approximation for the polarization function which screens the two-particle interaction. The excitation spectrum is found to have a single resonance. We do not find any convincing evidence for the additional high frequency second sound mode obtained by Tserkovnikov.
This is the first of two papers in which the high-density charged Bose gas at zero temperature is treated by two independent and self-contained methods. In this paper, we are concerned with the excitation spectrum of the system. An analysis of the formal and physical aspects of the theory is followed by a simple numerical calculation. We apply the usual field-theoretic formulation for a Bose system. The Green's functions and other response functions, or propagators, are analyzed, emphasizing the role of the dielectric constant. The general features of the excitation spectrum are described. To apply and to further illustrate the formalism, the low-lying levels are numerically determined to the order of rs34{{r}_{s}}^{\frac{3}{4}}rs​43​ in units of ${\ensuremath{\omega}}_{\mathrm{pl}}$ (the next order beyond the Bogoliubov approximation). rs{r}_{s}rs​ is the ratio of the average interparticle distance to the Bohr radius and ${\ensuremath{\omega}}_{\mathrm{pl}}$ is the plasma frequency.
In contrast with Chapter 2, in this chapter we include the dynamics of the thermal cloud. As noted in Chapter 1, we treat the noncondensate atoms using the simplest microscopic model approximation that captures the important physics. In particular, we consider only temperatures high enough ( T ≥ 0.4 T BEC ) that the noncondensate atoms can be described by a particle-like Hartree–Fock (HF) spectrum. To extend the analysis to very low temperatures is in principle straightforward (see Chapter 7). However, the details are more complicated since the excitations of the thermal cloud take on a collective aspect (i.e. a Bogoliubov-type quasiparticle spectrum must be used). In trapped Bose gases, the HF single-particle spectrum gives a good approximation down to much lower temperatures than in the case of uniform Bose gases, as first emphasized by Giorgini et al . (1997). In Section 3.1, we derive a generalized form of the Gross–Pitaevskii equation for the Bose order parameter Φ( r , t ) that is valid at finite temperatures. It involves terms that are coupled to the noncondensate component (the thermal cloud) and thus its solution in general requires one to know the equations of motion for the dynamics of the noncondensate atoms. In Section 3.2 we restrict ourselves to finite temperatures high enough that the noncondensate atoms can be described by a quantum kinetic equation for the single-particle distribution function f ( p, r , t ). A detailed microscopic derivation of this kind of kinetic equation is given in Chapters 6 and 7 using the Kadanoff–Baym Green's function formalism. A characteristic feature of a Bose-condensed gas is that the kinetic equation governing f ( p, r , t ) involves a collision integral C 12 [ f , Φ] describing collisions between condensate and noncondensate atoms.
Single-particle Green's functions, density response functions and other correlation functions are calculated in many different ways in the literature on Bose-condensed gases. An in-depth comparison and classification of different approaches was first given in the classic paper by Hohenberg and Martin (1965), with an emphasis on the various exact identities (conservation laws) that are satisfied. A key feature to be included in any theory is that a Bose broken symmetry leads to a hybridization of the single-particle excitations and the collective density fluctuations in such a way that the two excitation spectra become identical. This key feature is demonstrated in Chapter 5 of Griffin (1993). How to relate and assess various approximations for correlation functions in a Bose superfluid has been a topic of continual interest (and some controversy) since the late 1950s. These questions were largely resolved by the early 1960s at a conceptual level but the detailed applications of the theory were limited to dilute Bose-condensed gases at T = 0. Since it was difficult to relate the theory to the properties of superfluid. He at a quantitative level, this formalism based on a Bose broken symmetry was of little interest to experimentalists. The creation of superfluid Bose condensed gases in 1995 changed all this and has given new life to the many body theory of dilute weakly interacting Bose condensed gases. Various approximations for the Beliaev single-particle self-energies Σ αβ were derived in Sections 4.3 and 4.4. The discussion in Chapter 4 was somewhat abstract.
In Chapter 6, we derived a generalized Gross–Pitaevskii condensate equation which is coupled to a kinetic equation for the distribution function for the thermal atoms. However, the kinetic equation in Chapter 6 is only valid in the semiclassical limit. It involves the assumption that the thermal energy k B T is much greater than the spacing between the harmonic trap energy levels ( k B T ≫ω 0 where ω 0 is the trap frequency) and also much greater than the average interaction energy ( k B T ≫ gn ). The ZNG model, based on HF excitations, is still expected to be adequate down to quite low temperatures in trapped Bose gases, as will be shown by the results in Chapter 12. However, the ZNG model will break down at very low temperatures, where the Hartree–Fock excitations must be replaced by the Bogoliubov spectrum. To deal with this, one has to derive a kinetic equation for the Bogoliubov quasiparticle excitations. This is the goal of the present chapter. In this chapter, we use the second-order Beliaev approximation to discuss the nonequilibrium dynamics of a trapped Bose-condensed gas at finite temperatures. In doing to, we combine the second-order Beliaev self-energies with the lower-order Bogoliubov excitation spectrum, including off-diagonal single-particle propagators but still omitting the anomalous correlation functions. This last condition defines the Bogoliubov–Popov approximation. In this chapter, we consider only the damping which arises from collisions. We will not explicitly calculate corrections that are second-order in g to the quasiparticle energy spectrum or to the condensate chemical potential, both of which are associated with the real parts of the second-order Beliaev self-energies. The present chapter is a natural generalization of work presented in Chapter 6 for the simpler HF excitation spectrum assumed in the ZNG analysis.