Condensation and quasicondensation in an elongated three-dimensional Bose gas

We study the equilibrium correlations of a Bose gas in an elongated three-dimensional harmonic trap using a grand-canonical classical-field method. We focus in particular on the progressive transformation of the gas from the normal phase, through a phase-fluctuating quasicondensate regime to the so-called true-condensate regime, with decreasing temperature. Choosing realistic experimental parameters, we quantify the density fluctuations and phase coherence of the atomic field as functions of the system temperature. We identify the onset of Bose condensation through analysis of both the generalized Binder cumulant appropriate to the inhomogeneous system, and the suppression of the effective many-body $T$ matrix that characterizes interactions between condensate atoms in the finite-temperature field. We find that the system undergoes a second-order transition to condensation near the critical temperature for an ideal Bose gas in the strongly anisotropic three-dimensional geometry but remains in a strongly phase-fluctuating quasicondensate regime until significantly lower temperatures. We characterize the crossover from a quasicondensate to a true condensate by a qualitative change in the form of the nonlocal first-order coherence function of the field and compare our results to those of previous works employing a density-phase Bogoliubov–de Gennes analysis.

Figure 1

Variation of system densities on the long ($z$) axis with temperature: condensate density $n_0\left(z\right)$ (dotted blue line), quasicondensate density $n_{\mathrm{QC}}\left(z\right)$ (dashed blue line), anomalous density $\kappa \left(z\right)$ (dot-dashed red line), and total ($\mathbf{C}$-region plus semiclassical $\mathbf{I}$-region) density $n\left(z\right)$ (solid red line).

Figure 2

(a) Dependence of condensate (pluses) and quasicondensate (crosses) fractions on system temperature. Also shown are the ideal-gas BEC transition temperature $T_c^0$ (vertical dotted line) and condensate fraction (dot-dashed line). (b) Ratio of condensate and quasicondensate populations $N_0/N_{\mathrm{QC}}$ (asterisks) and generalized Binder cumulant $C_B$ (squares). The horizontal dashed line indicates the critical value ${\left(C_B\right)}_{\mathrm{crit}}$, and the vertical dashed line indicates the corresponding critical temperature. The solid line is a smooth line of best fit used to identify the intersection with ${\left(C_B\right)}_{\mathrm{crit}}$. (c) (Negative of the) integrated anomalous density (circles), and ratio of expectation values of the effective many-body $T$ matrix and two-body $T$ matrix in the condensate (diamonds).

Figure 3

First-order correlation functions $g^{\left(1\right)}(-z,z)$ (solid green) and $g^{\left(1\right)}(0,z)$ (dot-dashed green), and densities of the condensate $n_0\left(z\right)$ (dotted blue) and quasicondensate $n_{\mathrm{QC}}\left(z\right)$ (dashed blue).

Figure 4

(a) Fitting parameter $\xi$, which characterizes the functional form of $g^{\left(1\right)}(-z,z)$ (see text), in SPGPE (open stars) and BdG [36] (filled stars) calculations. The horizontal dot-dashed line indicates the condition $\xi =0.5$, by which we define the crossover temperature. The solid lines are smooth lines of best fit used to identify the intersection with $\xi =0.5$. (b) Phase coherence length $L_{\varphi }$ obtained from SPGPE (open circles) and BdG [36] (filled circles) calculations, and fitted Thomas-Fermi lengths $L_z^{\left(0\right)}$ (pluses) and $L_z^{\left(\mathrm{QC}\right)}$ (crosses) of the condensate and quasicondensate, respectively. Vertical lines through both panels indicate the crossover temperature $T_{\varphi }$ obtained from SPGPE (vertical dot-dashed line) and BdG [36] (vertical dotted line) calculations, and the critical point identified using $C_B$ (vertical dashed line).

Figure 5

Energy cutoff ${\epsilon }_{\mathrm{cut}}$ (circles) and ratio $({\epsilon }_{\mathrm{cut}}-{\epsilon }_0)/U_0{n}_{\mathbf{C}}\left(0\right)$ of cutoff and interaction energies (triangles) for simulations with (a) varying $T$ at constant $N$, (b) varying $T$ at constant $\mu$, and (c) varying $\mu$ at constant $T$.

Figure 6

$\mathbf{C}$-region population $N_{\mathbf{C}}$ (pluses) and semiclassical Hartree-Fock calculations of total ($\mathbf{C}$-region plus $\mathbf{I}$-region) atom number $N(\mu ,T)$ (crosses) and chemical potential $\mu (N,T)$ (triangles). Quantities are shown for (a) varying $T$ at constant $N$, (b) varying $T$ at constant $\mu$, and (c) varying $\mu$ at constant $T$.

Figure 7

Properties of the system evaluated at varying $T$ and fixed $N$ (a), (d), (g), (j), (m), and (p), varying $T$ and fixed $\mu$ (b), (e), (h), (k), (n), and (q), and varying $\mu$ and fixed $T$ (c), (f), (i), (l), (o), and (r): (a)–(c) Temperature $T$ (squares) and chemical potential $\mu$ (triangles). (d)–(f) Condensate fraction $N_0/N$ (pluses) and quasicondensate fraction $N_{\mathrm{QC}}/N$ (crosses). For reference the condensate fraction of the ideal Bose gas and the ideal-gas critical temperature (chemical potential) are also indicated (dot-dashed line and vertical dotted line, respectively) [121]. (g)–(i) Ratio $N_0/N_{\mathrm{QC}}$ of condensate and quasicondensate populations (asterisks) and generalized Binder cumulant $C_B$ (squares). The horizontal dashed line represents the critical value ${\left(C_B\right)}_{\mathrm{crit}}$ for the Binder cumulant, and the vertical dashed line indicates the inferred estimate of the critical temperature. The solid line is a smooth line of best fit used to identify the intersection with ${\left(C_B\right)}_{\mathrm{crit}}$. (j)–(l) (Negative of the) integrated anomalous thermal density (circles) and effective suppression of scattering between condensate atoms due to associated processes (diamonds). (m)–(o) Fitting parameter $\xi$, which characterizes the functional form of $g^{\left(1\right)}(-z,z)$ (see text), for SPGPE (open stars) and BdG [36] (filled stars) calculations. The horizontal dot-dashed line indicates the condition $\xi =0.5$, by which the crossover is characterized for SPGPE (vertical dot-dashed line) and BdG [36] (vertical dotted line) calculations. The solid lines are smooth lines of best fit used to identify the intersection with $\xi =0.5$. (p)–(r) Phase coherence length $L_{\varphi }$ for SPGPE (open circles) and BdG [36] (filled circles) calculations, alongside fitted Thomas-Fermi lengths $L_z^{\left(0\right)}$ (pluses) and $L_z^{\left(\mathrm{QC}\right)}$ (crosses) of the condensate and quasicondensate, respectively.