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Physical review. B, Condensed matter

Bose-Einstein condensation and long-range phase coherence in the many-particle Schrödinger wave function

J. Mayers

Published Nov 26, 2001• 5 citations
Publisher

Abstract

The properties of the many-particle Schr\"odinger wave function \ensuremath{\Psi} are examined in the presence of Bose-Einstein condensation (BEC). It is shown that it is possible to define, in terms of \ensuremath{\Psi}, a function $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s}),$ which can be regarded as the single-particle wave function of an arbitrary particle for a fixed configuration $\stackrel{\ensuremath{\rightarrow}}{s}$ of all other particles. It is shown that $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ plays an analogous role to the field operator of standard field-theoretical treatments of superfluidity. It is shown that in the presence of a Bose-Einstein condensate fraction f, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ must be nonzero and phase coherent within at least a fraction f of the total volume of the N-particle system for essentially all $\stackrel{\ensuremath{\rightarrow}}{s}.$ Examination of the form of variational many-particle wave functions shows that in liquid ${}^{4}\mathrm{He},$ $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ extends throughout the spaces left between the hard cores of the other atoms at $\stackrel{\ensuremath{\rightarrow}}{s}.$ By contrast, in the absence of BEC, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ in the ground state must be nonzero only over a localized region of space. It is shown that in order for long-range phase coherence in $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ to be maintained in the presence of velocity fields, any circulation must be quantized over macroscopic length scales. Some numerical calculations of the properties and fluctuations of liquid helium are presented. These suggest that the approach outlined in this paper may have significant advantages for the numerical calculations of the properties of Bose-Einstein condensed systems. The properties of $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ are used to show that there is no general connection between the static structure factor and the size of the Bose-Einstein condensate fraction in a Bose fluid. It is suggested that the observed connection in liquid ${}^{4}\mathrm{He}$ is due to the creation of vacancies in the liquid structure, which are required so that $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ can delocalize, in the presence of hard-core interactions. It is shown that this suggestion is quantitatively consistent with observations on liquid ${}^{4}\mathrm{He}.$ It is suggested that the superfluid transition in ${}^{4}\mathrm{He}$ occurs when the vacancies in the liquid structure form a connected network.

DOI
10.1103/physrevb.64.224521

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The properties of the many-particle Schr"odinger wave function \ensuremath{\Psi} are examined in the presence of Bose-Einstein condensation (BEC). It is shown that it is possible to define, in terms of \ensuremath{\Psi}, a function \ensuremathψ(r\ensuremath→∣s\ensuremath→),\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s}),\ensuremathψ(r\ensuremath→∣s\ensuremath→), which can be regarded as the single-particle wave function of an arbitrary particle for a fixed configuration s\ensuremath→\stackrel{\ensuremath{\rightarrow}}{s}s\ensuremath→ of all other particles. It is shown that \ensuremathψ(r\ensuremath→∣s\ensuremath→)\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})\ensuremathψ(r\ensuremath→∣s\ensuremath→) plays an analogous role to the field operator of standard field-theoretical treatments of superfluidity. It is shown that in the presence of a Bose-Einstein condensate fraction f, \ensuremathψ(r\ensuremath→∣s\ensuremath→)\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})\ensuremathψ(r\ensuremath→∣s\ensuremath→) must be nonzero and phase coherent within at least a fraction f of the total volume of the N-particle system for essentially all s\ensuremath→.\stackrel{\ensuremath{\rightarrow}}{s}.s\ensuremath→. Examination of the form of variational many-particle wave functions shows that in liquid 4He,{}^{4}\mathrm{He},4He, \ensuremathψ(r\ensuremath→∣s\ensuremath→)\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})\ensuremathψ(r\ensuremath→∣s\ensuremath→) extends throughout the spaces left between the hard cores of the other atoms at s\ensuremath→.\stackrel{\ensuremath{\rightarrow}}{s}.s\ensuremath→. By contrast, in the absence of BEC, \ensuremathψ(r\ensuremath→∣s\ensuremath→)\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})\ensuremathψ(r\ensuremath→∣s\ensuremath→) in the ground state must be nonzero only over a localized region of space. It is shown that in order for long-range phase coherence in \ensuremathψ(r\ensuremath→∣s\ensuremath→)\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})\ensuremathψ(r\ensuremath→∣s\ensuremath→) to be maintained in the presence of velocity fields, any circulation must be quantized over macroscopic length scales. Some numerical calculations of the properties and fluctuations of liquid helium are presented. These suggest that the approach outlined in this paper may have significant advantages for the numerical calculations of the properties of Bose-Einstein condensed systems. The properties of \ensuremathψ(r\ensuremath→∣s\ensuremath→)\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})\ensuremathψ(r\ensuremath→∣s\ensuremath→) are used to show that there is no general connection between the static structure factor and the size of the Bose-Einstein condensate fraction in a Bose fluid. It is suggested that the observed connection in liquid 4He{}^{4}\mathrm{He}4He is due to the creation of vacancies in the liquid structure, which are required so that \ensuremathψ(r\ensuremath→∣s\ensuremath→)\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})\ensuremathψ(r\ensuremath→∣s\ensuremath→) can delocalize, in the presence of hard-core interactions. It is shown that this suggestion is quantitatively consistent with observations on liquid 4He.{}^{4}\mathrm{He}.4He. It is suggested that the superfluid transition in 4He{}^{4}\mathrm{He}4He occurs when the vacancies in the liquid structure form a connected network.
The basic assumption of this paper is that in the presence of Bose-Einstein condensation (BEC) in liquid 4He^{4}\mathrm{He}4He, the wave functions of occupied many particle states are the superposition of two components; a phase coherent component, proportional to the ground state wave function and a phase incoherent component. It is shown that this assumption satisfies necessary conditions imposed by the presence of BEC and that the wave functions of the ideal Bose gas and wave functions of the Bijl-Feynman type are of this form. It is shown that this single assumption provides simple microscopic explanations of essentially all the exotic properties of helium II; why BEC implies two fluid behavior, how the superfluid and condensate fractions are linked, how Landau theory is linked to the presence of BEC, why the superfluid exhibits flow without viscosity and macroscopic quantum effects while the normal fluid does not, how the anomalous expansion and loss of spatial order, observed in helium II as it is cooled, is linked to BEC, and how the presence of sharp peaks observed in the dynamic structure factor are linked with BEC. The theory provides predictions that are in quantitative agreement with a wide range of presently unexplained experimental data on helium II.
It is shown that any Bose-Einstein-condensed fluid in its ground state will exhibit macroscopic single-particle quantum behavior (MSPQB). That is, (1) the many-particle wave function $\ensuremath{\Psi}({\mathbf{r}}_{1},\dots{},{\mathbf{r}}_{n})$ factors into a single-particle product ${\ensuremath{\prod}}_{n}\ensuremath{\eta}({\mathbf{r}}_{n})$; (2) the function $\ensuremath{\eta}(\mathbf{r})$ extends over macroscopic length scales and obeys the usual quantum equations for particle flux in a single-particle system; and (3) $\ensuremath{\eta}(\mathbf{r})$ obeys a nonlinear single-particle Schr\"odinger equation. The latter equation reduces to the Gross-Pitaevskii equation when interactions are weak and determines the density distribution of the fluid and the time development of this distribution. The arguments used rely only on elementary concepts of probability theory and many-particle wave mechanics and are valid even in strongly interacting fluids such as superfluid 4He^{4}\mathrm{He}4He. It is shown that Bose-Einstein condensation implies that the NNN-particle wave function $\ensuremath{\Psi}$ is delocalized. That is, if one considers a single-particle coordinate r\mathbf{r}r, then for all values that occur of the other $N\ensuremath{-}1$ coordinates, $\ensuremath{\Psi}$ is a nonzero function of r\mathbf{r}r over a region of space proportional to VVV, where VVV is the total volume within which the fluid is contained. MSPQB is a consequence of this delocalization and the absence of long-range correlations between particle positions in fluids. The results are accurate provided that only averages over regions of space containing many particles are considered. For averages over volumes of space containing ${N}_{\ensuremath{\Omega}}$ particles, inaccuracies due to quantum fluctuations are $\ensuremath{\sim}1∕\sqrt{{N}_{\ensuremath{\Omega}}}$.
It is shown that the assumption that the many particle wave functions describing superfluid 4He have some phase coherence properties of correlated basis functions provides a new explanation of the link between Bose-Einstein condensation and two fluid behavior. This assumption also implies that the condensate fraction is proportional to the superfluid fraction, in agreement with experiment, and provides a simple quantitative explanation of the anomalous reduction in spatial order and liquid density, observed as the temperature is lowered in superfluid 4He.
It is shown that in the presence of Bose–Einstein condensation (BEC) in any N particle system, the N particle Schrödinger wave functions of thermally occupied states are the sum of a 'localized' component and a 'delocalized' component, identical to the ground state wave function. It is shown that if N is sufficiently large, this implies that all physical properties of the system are the sum of two independent contributions from these two components. These results are used here to provide quantitative explanations of fundamental properties of BE condensed liquid 4 He, unexplained even qualitatively by existing theory; why BE condensed liquid 4 He is the only known physical system in which pair correlations between atomic positions reduce as it is cooled, why it is the only known liquid with sharp peaks in its dynamic structure factor, why the liquid expands with cooling and how the condensate fraction is related to the superfluid fraction. It is shown that these results also provide a relatively simple, physically transparent and quantitative explanation from first principles of macroscopic quantum effects. A new algorithm is given for the calculation of the time development of the macroscopic density of any BE condensed liquid or gas at any temperature. Unlike the Gross–Pitaevskii equation, this algorithm is valid for both strongly and weakly interacting systems. It is used here to show that macroscopic quantum interference fringes, observed between overlapping clouds of BE condensed atoms, are a necessary consequence of BEC and the N particle Schrödinger equation for the atoms in the clouds. It follows that the widely held view that these fringes are created by measurement is unnecessary. New, experimentally testable predictions are made of how the visibility of these fringes will vary with temperature.
The mathematical description of B.E. (Bose-Einstein) condensation is generalized so as to be applicable to a system of interacting particles. B.E. condensation is said to be present whenever the largest eigenvalue of the one-particle reduced density matrix is an extensive rather than an intensive quantity. Some transformations facilitating the practical use of this definition are given.An argument based on first principles is given, indicating that liquid belium II in equilibrium shows B.E. condensation. For absolute zero, the argument is based on properties of the ground-state wave function derived from the assumption that there is no "long-range configurational order." A crude estimate indicates that roughly 8% of the atoms are "condensed" (note that the fraction of condensed particles need not be identified with $\frac{{\ensuremath{\rho}}_{s}}{\ensuremath{\rho}}$). Conversely, it is shown why one would not expect B.E. condensation in a solid. For finite temperatures Feynman's theory of the lambda-transition is applied: Feynman's approximations are shown to imply that our criterion of B.E. condensation is satisfied below the lambda-transition but not above it.
By extending Einstein's separation of wave and particle parts of the second order thermal fluctuation to encompass "generalized fluctuations" in any Bose field, P. E. Gordon has proposed alternative definitions for nth order coherence and nth order coherent states. The main point of this paper is to explore some of the physical insights to be gained by extending dualism to higher orders. Recent experiments have examined aspects of the coherence of Bose-Einstein condensates. It has been argued that the condensate state is coherent to (at least) second or third order, but the coherence properties of Bose-Einstein condensates remain somewhat controversial. Using probability distributions developed by M. O. Scully and V. V. Kocharovsky et. al., we apply Gordon's dualistic expression of the coherence conditions to investigate coherence properties in Bose-Einstein condensation. Via numerical calculations, we present a graphical survey of wave-like and particle-like fluctuations in condensed and uncondensed fractions. Near the critical point, we find a very marked peak in the ratio of nth order wave to nth order particle fluctuations in the condensate. Not surprisingly, n-point correlations between the positions of condensate atoms also peak near the critical temperature, and this apparently mirrors, to higher orders, the well-known relation between the integral of the 2-point correlation function over a certain volume and the rms fluctuation in the number of particles in that volume.