Full-range first-order spatial coherence of an interacting trapped Bose gas across the Bose–Einstein condensation transition: finite-size decay and quantitative benchmark against finite-temperature theory

A dilute gas of bosonic atoms confined in a magnetic trap undergoes Bose–Einstein condensation below a critical temperature $T_c$, and the onset of condensation is diagnosed by the first-order spatial correlation function $g^{(1)}(z_1,z_2)$, the normalized one-body density matrix. The landmark experiment of Bloch, Hänsch and Esslinger (Nature 403, 166 (2000)) measured $g^{(1)}$ of a trapped, weakly interacting rubidium gas using two radio-frequency fields that out-coupled atoms from two spatially separated 'slit' regions, observing a Gaussian decay above $T_c$ and a slow decay towards a plateau below $T_c$. The measurement, however, covered only a restricted range of slit separations, left the finite-size behavior of the coherence at separations approaching the cloud size uncharacterized, and achieved only qualitative agreement with quasiparticle-based finite-temperature theory. This problem asks for the completion of that characterization: an experimental determination of the equilibrium $g^{(1)}$ over the full range of separations up to the finite cloud size, across the transition, with systematics (mean-field distortion of the output-coupler geometry, back-action and phase dynamics of the output coupling) controlled, together with a quantitative comparison with finite-temperature many-body theory of the trapped interacting gas in the critical region. Characterization of the full three-dimensional correlation function and of second-order coherence in the same system is an explicitly linked extension of the same objective. The later literature (one-dimensional and crossover coherence measurements, nonequilibrium order-formation studies, second-order correlation measurements, and mature classical-field (stochastic classical treatment of the highly occupied low-energy modes) and finite-size theory) leaves this equilibrium three-dimensional core open.

Background

Bose–Einstein condensation (BEC) occurs when a dilute gas of bosonic atoms is cooled below a critical temperature $T_c$, so that a macroscopic occupation of a single-particle state builds up; it was realized in 1995 in trapped ultracold vapors of rubidium, sodium and lithium. The order parameter of the transition is captured by the first-order correlation function, the normalized one-body density matrix $g^{(1)}(z_1,z_2)=\frac{\langle\hat{\psi}^{\dagger}(z_1)\hat{\psi}(z_2)\rangle}{\sqrt{n(z_1)\,n(z_2)}},$ where $\hat{\psi}$ annihilates an atom at position $z$ and $n(z)$ is the density. Penrose and Onsager, and later Yang's concept of off-diagonal long-range order (non-vanishing correlations of the one-body density matrix at arbitrarily large separation), identify condensation with the long-range persistence of $g^{(1)}$: above $T_c$ a thermal cloud has a short coherence length of order the thermal de Broglie wavelength and $g^{(1)}$ decays approximately Gaussian to zero, while below $T_c$ the condensate contribution makes $g^{(1)}$ decay only towards a plateau whose height reflects the condensate fraction. In a finite trapped cloud, the density vanishes at the cloud edge, so the correlation function must eventually fall to zero at the largest separations; how interactions, thermal depletion and the trap's finite size shape this large-separation decay is the finite-size behavior of the coherence. The theoretical benchmark is provided by finite-temperature many-body theory of the weakly interacting trapped gas, in which excitations above the condensate are treated as quasiparticles (Bogoliubov theory, in which the gas is described as weakly interacting collective excitations; in traps usually combined with a local-density treatment (each region of the inhomogeneous cloud treated as a uniform gas at the local density) or a Hartree–Fock mean-field treatment of the thermal cloud, as in Naraschewski and Glauber's computation of first- and second-order correlations of trapped Bose gases, Phys. Rev. A 59, 4595 (1999), DOI: 10.1103/PhysRevA.59.4595).

Experimentally, $g^{(1)}$ of a trapped gas is hard to access because one cannot split and recombine parts of a single trapped cloud in place. Bloch, Hänsch and Esslinger (Nature 403, 166 (2000), DOI: 10.1038/35003132) solved this with an atom-laser technique: two frequency components of a radio-frequency field resonantly transferred atoms from the magnetically trapped state to an untrapped state at two spatially separated 'slit' positions inside the cloud (the resonance condition selects where out-coupling occurs), and the two falling matter-wave beams interfered; the fringe visibility as a function of slit separation gives the modulus of $g^{(1)}$. This extended the earlier MIT measurement of condensate coherence by Hagley et al. (Phys. Rev. Lett. 83, 3112 (1999), DOI: 10.1103/PhysRevLett.83.3112). Above $T_c$ the measured correlation function decayed rapidly (Gaussian, as expected for a thermal gas); below $T_c$ it showed a slow decay towards a plateau, the expected signature of the condensate's long-range phase coherence. The agreement with theoretical predictions based on the quasiparticle excitation spectrum was, however, only qualitative, and the measurement covered a restricted range of slit separations; the behavior at separations approaching the size of the finite trapped cloud was not resolved.

Problem Statement

Determine the equilibrium first-order spatial correlation function $g^{(1)}$ of a weakly interacting, magnetically trapped Bose gas across the Bose–Einstein condensation transition, over the complete range of separations up to the finite size of the trapped cloud, and establish quantitative agreement between that measurement and finite-temperature many-body theory of the trapped interacting gas.

Concretely, the required results are:

1. A measurement of $g^{(1)}$ resolving the correlation function at separations approaching the radial extent of the cloud, where the finite-size behavior — the decay of the correlation function to zero as the cloud density vanishes — becomes visible, rather than only the small-separation regime and the condensate plateau, for temperatures spanning above, near and below $T_c$.
2. Control of the systematic effects that limited the original measurement to qualitative comparisons: the mean-field distortion of the effective slit geometry (the quoted 3–50 nm uncertainty in slit separation), the perturbation of the trapped gas by the output coupling, and the phase evolution of the out-coupled matter waves, such that the measured fringe visibility maps to $g^{(1)}$ with quantified uncertainty.
3. A quantitative comparison of the measured $g^{(1)}$ — its thermal Gaussian component, its condensate plateau, the interaction-induced intermediate-distance behavior, and its finite-size decay — with finite-temperature quasiparticle predictions (Bogoliubov-type quasiparticles, local-density or Hartree–Fock treatments, or classical-field methods treating the highly occupied low-energy modes as classical stochastic fields) for the same trapped interacting gas, including in the critical region near $T_c$, with agreement or disagreement established within controlled uncertainties rather than qualitatively.

Beyond those required results (and additional rather than required for the core completion), the following are explicitly linked extensions of the same characterization objective, not yet realized for the three-dimensional trapped gas: the full three-dimensional correlation function beyond a single probing direction, and second-order (density–density) correlations $g^{(2)}$ in the same system at the transition. The common objective is a complete, quantitative experimental and theoretical characterization of how spatial coherence develops and saturates in a finite, interacting, trapped Bose gas at the phase transition.

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