Comparison between microscopic methods for finite-temperature Bose gases

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.

Figure 1

Condensate density $N_c|{\varphi }_c(x)|{}^2$ (dashed, black) and a typical realization of the noncondensate density $|\psi {}^{\prime }(x)|{}^2\equiv |{\psi }_{\perp }(x)|{}^2$ (noisy red curve; data are multiplied by 5 to be visible on this scale). The depletion correction to the condensate mode, ${\varphi }_2(z)$, is illustrated by plotting the “interference term” $2\hspace{0.5em}\mathrm{Re}\hspace{0.5em}[{\varphi }_0^{*}(z){\varphi }_2(z)]$ (dot-dashed, green; also multiplied by 5). Note that the condensate atom number $N_c\approx 19\hspace{0.5em}500$ depends on the realization. The zero-temperature Thomas-Fermi shape for the same parameters (inverted parabola; thin, solid brown line) is also plotted. Here (and in most of the paper) we fix $\mu =22.41\hspace{0.5em}\hslash \omega$ (corresponding to $N=20\hspace{0.5em}000$ for the GPE at $T=0$), choosing here $k_{B}T=46\hspace{0.5em}\hslash \omega$. Densities are plotted in units of $g/\mu$.

Figure 2

Growth to equilibrium obtained by numerical solution to the SGPE. (a) Snapshots of the atomic density profile $\langle |\psi (z,t)|{}^2\rangle$, with time increasing from bottom to top. (b) Growth in the average central density vs. time—here the times highlighted by colored squares label the corresponding colored density profiles in (a). (c) Growth in the particle number as the system approaches equilibrium. The inset shows trajectories from single numerical realizations, which illustrates the fluctuating particle number between these, compared to the result of the main plot in (c), which was obtained by averaging over $1000$ such trajectories. (Parameters as stated in Sec. 2f, but with $k_{B}T=860\hspace{0.5em}\hslash \omega$.)

Figure 3

Illustration of the procedure for solving the modified Popov theory. (Left) Graphical determination of the self-consistent quasicondensate density: crossing point between the dashed and solid lines [left- and right-hand sides of Eq. (20), $n_{\mathrm{qc}}$ is the quasicondensate density and $n$ the total density].The thin (red) line shows the classical (high-temperature) approximation to Eq. (19). If we take $\mu$ as in the rest of the paper, the two temperatures correspond to $N\approx 23\hspace{0.5em}800$, $T\approx 1.3T_{\varphi }\approx 0.63T_c$ and $N=20\hspace{0.5em}000$, $T\approx 0.16T_{\varphi }\approx 0.074T_c$, respectively [see Eq. (1)]. (Right) Temperature-dependent Thomas-Fermi radius $R(T)$ determined by numerically solving for the (local) chemical potential, where $\mu (z)=2g{n}_{\mathrm{th}}^{\prime }(z)$. The trap is harmonic, chemical potential at the center $\mu =22.41\hslash \omega$ as in the rest of the paper, and $g=0.01(\hslash {}^3\omega /m){}^{1/2}$. The arrows mark the temperatures chosen for the simulations (the threeleftmost arrows correspond to those shown in Table ).

Figure 4

Characteristic temperatures and atom numbers of the SGPE and NCB simulations (hollow black circles). The filled black circle shows the higher temperature regime at which only SGPE simulations were undertaken. The ($1D$) characteristic temperatures in a trap are $T_c$ [Eq. (1); Bose-Einstein condensation in an ideal gas] and $T_{\varphi }<T_c$ [Eq. (1); phase coherence], shown by the dotted blue and solid black lines, respectively. The red squares mark the parameters chosen for the condensate statistics comparison of Sec. 3d1 at a fixed ratio $T/T_c=0.23$, as indicated by the dashed brown line.

Figure 5

Data points representing the ensemble of stochastic field values $\psi (z)$ at three positions $z$ in the trap (from left to right, the average density decreases). Temperature increases from top to bottom. Black: SGPE simulation, red: NCB simulation. The dashed green circle indicates, for $\left|z\right|<R$, the modulus of the Thomas-Fermi wave function, and for $z=1.5\hspace{0.5em}R$ the square root of the Bose-Einstein density (26). In the trap center, the Thomas-Fermi approximation yields for these parameters $|{\psi }_{\mathrm{TF}}(0)|\approx 47/\sqrt{\ell }$. (Atom number $N\approx 20\hspace{0.5em}000$.)

Figure 6

Average density profiles $n(z)$, normalized as $gn(z)/\mu$, returned by the stochastic Gross-Pitaevskii equation (solid black) after an equilibration time $t_{\mathrm{eq}}$ and by the number conserving Bogoliubov expansion (dashed red). Plots (a)–(c) are for temperatures $k_{B}T=46,140,430\hspace{0.5em}\hslash \omega$ listed in Table , $R$ is the Thomas-Fermi radius at $T=0$ [Eq. (24)]. The dot-dashed green curves give the prediction of the modified Popov theory, as developed by Andersen et al. [6]. Plot (d) analyzes, on a logarithmic scale, the density in the “thermal wings” of plot (c) for $R<z\lesssim z_T$ where $z_T$ (grey vertical line) is the point where the trap energy becomes comparable to temperature, $V(z_T)=k_{B}T+\mu$. The Rayleigh-Jeans density (dashed cyan) and the Bose-Einstein density (dot-dashed green) are calculated for an ideal gas. The brown solid line corresponds to the NCB density plus the quantum density level which asymptotes to the latter, $n_q$ [Eq. (4)], indicated by the blue dotted line.

Figure 7

Condensate (top row) and thermal cloud (bottom row) densities for temperatures (a),(d) $k_{B}T=46\hslash \omega$; (b),(e) $k_{B}T=140\hslash \omega$; and (c),(f) $k_{B}T=430\hslash \omega$. Condensate density $n_c(z)=\langle N_c\rangle |{\varphi }_c(z)|{}^2$ obtained from Penrose-Onsager analysis of the one-body density matrix [solid (black), SGPE; dashed (red), TWNCB method]. Also shown for reference is the $T=0$ stationary solution to the GP equation with eigenvalue $\mu$ (thin solid, turquoise), which coincides with the zeroth order condensate mode within the NCB approach. In (c) the condensate density $\langle N_c\rangle |{\varphi }_c(z)|{}^2$ following the second-order correction within the NCB expansion is also shown (dot-dashed, blue), as well as the modified Popov condensate density (dot-double dashed, green) [see Sec. 4 Eq. (56) for details]. Bottom row: thermal density $n_{\mathrm{th}}(z)=n(z)-n_c(z)$ [solid (black) SGPE; dashed (red), TWNCB method]. As in Fig. 6, the Wigner correction was made for the TWNCB method case. Shown in (d)–(f) is the $T=0$ Bogoliubov prediction, Eq. (A4), with $N$ (solid brown) and, in (f) only, also $N=N_{\mathit{PO}}$ of the SGPE (dotted maroon), alongside the modified Popov result (dot-double dashed, green).

Figure 8

Analysis of the back-action of the noncondensate atoms on the shape of the condensate mode ${\varphi }_c$. We plot the effective potential of Eq. (34) with (lower curves, HFB theory) and without the last term (upper curves, HF theory), normalized to $\mu$. The condensate mode function ${\varphi }_c(z)$ is taken from a Penrose-Onsager analysis of the one-body density matrix. Solid black/brown lines: SGPE data, (dot-)dashed red/green lines: TWNCB method data. The HFB theory that includes the back action from the anomalous average is closer to the actual chemical potential.

Figure 9

First-order correlation function $g^{(1)}(z)=g^{(1)}(0,z)$, as defined in Eq. (37), from the SGPE data (solid black) and the NCB data (dashed red). Temperatures in (a)–(c) as given in Table , in (d), we have $T\approx 1.3T_{\varphi }$ (only SGPE data shown). We plot the real part of $g^{(1)}(z)$, the imaginary part being negligibly small. Dot-dashed green lines—Eq. (39), based on modified Popov theory. Brown, thin solid lines—Eq. (40). In (d), we plot Eq. (40) using $R(T)<R$ (solid thin brown) and $R(T)=R$ (dotted maroon). The contributions from the SGPE Penrose-Onsager condensate mode (dashed blue) and from noncondensate modes (dot-dashed red) are shown separately, from Eq. (43). In (a)–(d) the vertical dashed thin lines indicate $R(T)$. (e) Estimated temperature based on fitting $g^{(1)}(z)$ near $z=0$ to Eq. (42) (dashed line) for SGPE (black circles) and NCB approach (red squares).

Figure 10

Density correlation function $g^{(2)}(z)$ [Eq. (44)] from the SGPE simulations (solid black) and the NCB simulations (dashed red). For the NCB method case, the corrections of Eq. (45) are applied, so that in the quantum field theory, $g^{(2)}(z)$ has the meaning of a second-order coherence function. The vertical dashed thin lines indicate $R(T)$ at the temperatures in (a)–(c), which are as in Table . (d) Comparison of results for $g^{(2)}(0)$ (black circles, SGPE; red squares, TWNCB method) with Lieb-Liniger theory, Eq. (46), taken from Ref. [75] (dashed brown).

Figure 11

Real part of the squeezing correlation (anomalous average) $m(z)$, as defined in Eq. (47), for the temperatures of Table . Solid black, SGPE result; dashed red, NCB approach result. The Bogoliubov result [solid brown, Eq. (32)] is calculated with the mode functions of the $T=0$ BdG operator (A3), and also, in (c) only, using the Bogoliubov spectrum calculated for a condensate number equal to the SGPE PO condensate number (blue, dot-dashed).

Figure 12

Representation of the ensemble of stochastic fields $\psi (z)$, locked to the phase of the condensate mode [we multiply with $a_c^{*}/\left|a_c\right|$ where $a_c$ is the amplitude (27) of the (Penrose-Onsager) condensate mode]. Black, SGPE simulation; red, NCB approach simulation. Trap position and temperatures as in Fig. 5. Note the elliptical shape of the point cloud, illustrating the relative enhancement of phase fluctuations (imaginary part). The SGPE data deform into a “crescent” shaped cloud at intermediate and high temperatures, while the NCB distribution remains aligned to orthogonal quadratures. The dashed green circles represent the Thomas-Fermi ($\left|z\right|<R$) and Bose-Einstein ($z=1.5\hspace{0.5em}R$) predictions for the square root of the density at each trap position and temperature, as in Fig. 5.

Figure 13

Counting statistics $P(N_c)$ for the condensate number, with $N_c=\left|a_c\right|{}^2$, and the condensate mode in Eq. (27) obtained from the Penrose-Onsager criterion. Solid black, SGPE; dashed red, NCB approach; dot-dashed green, theory of Ref. [71]. Temperatures in (a)–(c) increase from left to right, (as in Table ), ($N\approx 20\hspace{0.5em}000$, $\mu =22.41\hslash \omega$). Panel (d) has $N\approx 1000$ ($\mu =3.11\hslash \omega$) and a temperature $T\approx 0.23T_c$, the same ratio as in (c). Dotted blue line, NCB approach with Rayleigh-Jeans instead of Bose-Einstein occupation numbers, i.e., in Eq. (7), $\sigma {}_k^2\mapsto k_{B}T/E_k$.

Figure 14

First three moments of $P(N_c)$ vs. total particle number $\langle N\rangle$ of SGPE (open circles) vs. theory of Ref. [71] (filled green squares): (a) mean value $\langle N_c\rangle$ scaled to $\langle N\rangle$ (condensate fraction), (b) relative standard deviation $\sigma (N_c)/\langle N_c\rangle$, (c) skewness or third centered moment, $\mathrm{skew}(N_c)=\langle (N_c-\langle N_c\rangle ){}^3\rangle /\sigma {}^3(N_c)$. The total particle number $N$ is calculated over the region $\left|z\right|<2R$.

Figure 15

Schmidt number versus scaled temperature for the SGPE simulations. Inset: at the temperature $T\approx 1.3T_{\varphi }$, the fractional occupation $N_k/\langle N\rangle$ for the $30$ lowest modes with the largest occupation; condensate fraction (in $k=0$ mode above) $\langle N_c\rangle /\langle N\rangle \approx 0.2$, and $\langle N\rangle \approx 23\hspace{0.5em}800$.

Figure 16

Normalized density profiles showing: Total density (SGPE, solid black, noisy; modified Popov, dashed, brown), quasicondensate density (SGPE, solid orange, noisy; modified Popov, dashed, blue) and (phase coherent) condensate densities (SGPE PO, noisy, turquoise; modified Popov, dot-dashed/dotted, maroon). The modified Popov condensate density is shown dotted at small distances from the trap center, where the relation $n_c^{\prime }(z)=n_c(z)$ [see Eq. (56)] breaks down. The grey shaded region shows the Penrose-Onsager condensate density. The dashed red line shows $n(z)-n_{\mathrm{qc}}(z)$ of the modified Popov theory. Here $T=1.3T_{\varphi }=0.63T_c$ and $\langle N\rangle \approx 23\hspace{0.5em}800$. There are no NCB data because the NCB expansion no longer works at this temperature.

Figure 17

Comparison between Eq. (56) (dot-dashed, maroon) and the PO mode due to diagonalizing the density matrix (solid turquoise and shaded) at three temperatures. Also shown is the quasicondensate density from Eq. (54) (noisy orange curve), used to generate the dot-dashed maroon densities. The data here is extracted from the SGPE simulations.

Figure 18

(1D) Quasicondensate and PO condensate numbers from the SGPE (black circles) and modified Popov theory (hollow green triangles). The SGPE results are based on the condensate mode given by PO analysis of the one-body density matrix (Sec. 3b) and on Eq. (53) for the quasicondensate. For the integration over $n_{\mathrm{qc}}(z)$, only the region $\left|z\right|⩽R(T)$ is taken into account. The modified Popov data are based on the quasicondensate density described in Sec. 2e and on the condensate density $n_c^{\prime }(z)$ given in Eq. (56), where $g^{(1)}$(0,$z$) is calculated from Eq. (40).

Figure 19

Top row (a),(b): total atomic density for the SGPE (solid, black) at equilibrium and the TWNCB method (a) before (dashed red) and (b) after (dashed blue) evolution via the GPE. The TWNCB data contain the correction to normal order; the evolution period is $5000\omega {}^{-1}$. Bottom row (c),(d): condensate (thermal) density [SGPE, solid green (brown); TWNCB method, dashed], as obtained by Penrose-Onsager analysis (see Fig. 7), (c) before (dashed red) and (d) after (dashed blue) evolution of the NCB state over $5000\omega {}^{-1}$. Upper curves: condensate density, lower curves with maxima at $\left|z\right|\lesssim R$: thermal density.

Figure 20

Left: condensate statistics at different evolution times under the GPE, at initial nominal temperature $T\approx 0.48T_{\varphi }$. Solid blue, TWNCB data; dashed green, equilibrium Scully and co-workers theory [71]. The SGPE data at equilibrium are not shown, as the condensate statistics essentially stays the same as in Fig. 13. Right, mean condensate number $\langle N_c(t)\rangle$ for SGPE (upper, black line); TWNCB method (lower, red curve) at the temperatures of Table , with temperature increasing from top (g) to bottom (i).

Figure 21

Top row: $g^{(1)}(z)$ for the SGPE (solid black), (a) TWNCB method initially at $(t=0)$ (dashed red) and (b) TWNCB method after GPE evolution up to $t=5000\omega {}^{-1}$ (dashed blue); bottom row: as per top row for $g^{(2)}(z)$. In each plot, the vertical dashed line indicates $R(T)$ at $T=430\hslash \omega$.

Figure 22

Temperature measurement using correlation functions as a function of GPE evolution time: (upper data) $g^{(1)}(z)$, for $0<z<R/2$ and Eq. (42) (blue triangles); (lower data) $g^{(2)}(0)$ and Eq. (46) (black squares). Initial temperature (dashed red); best horizontal fit through $g^{(1)}(z)$ data (dotted brown) and through $g^{(2)}(0)$ data (dot-dashed green); temperatures corresponding to the horizontal lines are indicated.