Projected Gross-Pitaevskii equation theory of finite-temperature collective modes for a trapped Bose gas

We develop a formalism based on the projected Gross-Pitaevskii equation to simulate the finite-temperature collective mode experiments of Jin et al. [Phys. Rev. Lett. 78, 764 (1997)]. We examine the $m=0$ and $m=2$ quadrupolar modes on the temperature range $0.51T_c–0.83T_c$ and calculate the frequencies of, and phase between, the condensate and noncondensate modes, and the condensate mode damping rate. This study is the first quantitative comparison of the projected Gross-Pitaevskii equation to experimental results in a dynamical regime.

Figure 1

(Color online) Schematic view of the quadrupolar modes of an oblate condensate. (a) The $m=0$ excitation where the axial and radial widths of the condensate oscillate out of phase and (b) the $m=2$ excitations and the widths of the condensate in the $x$ and $y$ directions oscillate out of phase.

Figure 2

(Color online) Schematic view of the $c$ field and the incoherent regions for a Bose gas in a harmonic trap potential, and the approximations we employ in our treatment of the collective mode dynamics.

Figure 3

Experimental time sequence and form of trap perturbations used to excite the Bose gas. (a) Overview of time sequence used in the experiment to excite and observe collective excitations in the system. (b)–(d) Schematic representations of the various trap perturbations used (see text). Contours of equipotential in the $xy$ plane are shown for the unperturbed (solid lines) and for the perturbed (dashed lines) traps. (b) Symmetric perturbation used to drive the $m=0$ mode. (c) The perturbation used to drive the $m=2$ mode corresponds to a rotating ellipse. (d) Trap center displacement used to drive the dipole mode.

Figure 4

Equilibrium state properties. (a) Temperature, (b) total atom number, and (c) condensate number as a function of $T∕T_c$. PGPE results (diamonds) and lines are guides to the eye.

Figure 5

(Color online) Line densities evolution after perturbation. (a) A single trajectory position space line density $n_l^{\left(j\right)}(x,t)$. (b) Trajectory averaged position space line density $n_l(x,t)$. (c) Condensate position space line density $n_l^{\mathrm{cond}}(x,t)$. (d) A single trajectory momentum space line density $n_l^{\left(j\right)}(p_x,t)$. (e) Trajectory averaged momentum space line density $n_l(p_x,t)$. Results for a system with $T=154\mathrm{nK}$, $N=2.0\times {10}^4$, $N_{\mathrm{cond}}=5.7\times {10}^3$.

Figure 6

(Color online) Surface plots of trajectory averaged line density plots. (a) Trajectory averaged position space line density $n_l(x,t)$. (b) Trajectory averaged position space line density $n_l(p_x,t)$. Same data as displayed in Figs. 5b, 5e.

Figure 7

(a) Single trajectory and (b) trajectory averaged momentum line density at $t=32\mathrm{ms}$ for the condensate domain (dark shaded region) and the noncondensate domain (light shaded region) are shown (see text). Results for the same parameters given in Fig. 5.

Figure 8

(a) Condensate domain observable, $P_c\left(t\right)$, and (b) noncondensate domain observable, $P_n\left(t\right)$, as a function of time. For a system excited by the $m=0$ symmetry perturbation: Trajectory averaged result (black line), and single trajectory result (gray line). For unperturbed equilibrium system: Trajectory averaged result (black diamond line), and single trajectory result (gray diamond line). System parameters: $T=159\mathrm{nK}\approx 0.75T_c$, $N=2.1\times {10}^4$, $N_{\mathrm{cond}}=4.9\times {10}^3$.

Figure 9

Fits to observables: (a) Condensate domain observable, $P_c\left(t\right)$, (line) and (b) noncondensate domain observable, $P_c\left(t\right)$, (line) as a function of time for a system excited by the $m=0$ symmetry perturbation. Fits to observables using Eq. (40) are shown (dotted lines). System parameters: $T=159\mathrm{nK}\approx 0.75T_c$, $N=2.1\times {10}^4$, $N_{\mathrm{cond}}=4.9\times {10}^3$.

Figure 10

Results for the frequency dependence on temperature for PGPE and experimental results of [2]. (a) $m=0$ mode frequencies. (b) $m=2$ mode frequencies. Experimental results (gray symbols) and PGPE results (black symbols). Frequency for condensate (solid symbols) and noncondensate (open symbols). Error bars on some PGPE results indicate the spread in values from different calculations at the same temperature.

Figure 11

Results for the damping rates variation with temperature for PGPE and experimental results. Results are given for the condensate $m=0$ modes (triangles) and $m=2$ modes (circles). Experimental results (gray symbols) and PGPE results (black symbols). Error bars on some PGPE results indicate the spread in values from different calculations at the same temperature.

Figure 12

(Color online) Dependence of equilibrium variables of down-projected state on the energy cutoff ${ϵ}_{\mathrm{cut}}^{\prime }$. (a) Average occupation of highest energy $\mathbf{C}$ region mode $n_{\min }$ (diamonds), (b) temperature $T$ (diamonds), (c) condensate number $N_{\mathrm{cond}}$ (diamonds), and (d) total atom number $N$ (diamonds) and $\mathbf{C}$ region atom number $N_{\mathbf{C}}$ (squares). Dashed line in (a) is a linear fit to $n_{\min }$ in the variable $1∕{ϵ}_{\mathrm{cut}}^{\prime }$, solid lines in (b)–(d) are linear fits to the data. In (d) the shaded regions indicate the relative number of atoms in the $\mathbf{C}$ and $\mathbf{I}$ regions for each value of ${ϵ}_{\mathrm{cut}}$.

Figure 13

Frequency dependence of mode excitation on energy cutoff, for (a) $m=0$ mode and (b) $m=2$. Condensate frequency (solid symbols) and noncondensate frequency (open symbols). Solid lines are linear fits to the condensate data, and dashed line in (a) is fit to the noncondensate data.

Figure 14

Frequency of dipole mode as a function of temperature. Condensate (solid squares) and noncondensate (open squares). Error bars on some results indicate the spread in values from different calculations at the same temperature. Linear fit to the condensate (line) and noncondensate (dashed) results.

Figure 15

Relative phase between the condensate and noncondensate collective modes for the different symmetries studied: $m=0$ mode (triangles) and $m=2$ (circles). Error bars on some PGPE results indicate the spread in values from different calculations at the same temperature. Solid line is a linear fit to the $m=0$ data and dashed line is a linear fit to the $m=2$ data. Same parameters as in Fig. 13.

Figure 16

Relative phase between the condensate and noncondensate collective modes for the $m=0$ (triangles) and $m=2$ modes (circles) as a function of the down-projected energy cutoff ${ϵ}_{\mathrm{cut}}^{\prime }$. Solid line is a linear fit to the $m=0$ data and dashed line is a linear fit to the $m=2$ data. Same parameters as in Fig. 13.