Reservoir interactions during Bose-Einstein condensation: Modified critical scaling in the Kibble-Zurek mechanism of defect formation

As a test of the Kibble–Zurek mechanism (KZM) of defect formation, we simulate the Bose–Einstein condensation transition in a toroidally confined Bose gas by using the stochastic projected Gross–Pitaevskii equation, with and without the energy-damping reservoir interaction. Energy-damping alters the scaling of the winding-number distribution with the quench time—a departure from the universal KZM theory that relies on equilibrium critical exponents. Numerical values are obtained for the correlation-length critical exponent $\nu$ and the dynamical critical exponent $z$ for each variant of reservoir interaction theory. The energy-damping reservoir interactions cause significant modification of the dynamical critical exponent of the phase transition, while preserving the essential KZM critical scaling behavior. Comparison of numerical and analytical two-point correlation functions further illustrates the effect of energy damping on the correlation length during freeze-out.

Figure 1

(a) Adiabatic-impulse model of dynamical symmetry breaking in KZM. The boundary between regions $\widehat{t}$ is defined as the instant when the time remaining until the critical point is equal to the system relaxation time, i.e., when $\widehat{t}=\tau (-\widehat{t})$. (b) Schematic of defect formation during Bose–Einstein condensation in a ring trap. The scale of regions acquiring a $U\left(1\right)$ symmetry-broken phase is set by the coherence length at the freeze-out time, $\widehat{\xi }=\xi (-\widehat{t})$. Phase differences between adjacent domains, $\mathrm{\Delta }{\theta }_i$, are resolved through the formation of defects. (c) Reservoir interactions in the stochastic projected Gross–Pitaevskii theory of the dilute Bose gas. The number-damping $\left(\gamma \right)$ interaction drives condensate growth. The number-conserving energy-damping process $\left(\varepsilon \right)$ has a significant role far from equilibrium.

Figure 2

Single trajectories of the $\mathbf{C}$ field during a quench $\left({\tau }_Q=e^5{\omega }_{\perp }^{-1}\right)$, showing (a) number density and (c) phase for the number-damping SPGPE, and (b) number density and (d) phase for the full SPGPE. During initial density growth there are numerous solitons, evident as low-density notches with an associated phase jump, that quickly decay leaving a persistent current. The decay of solitons appears to be more rapid for the full SPGPE simulations.

Figure 3

The condensate number dynamics for several quench times, using (a) the number-damping SPGPE and (b) the full SPGPE. The condensate number was extracted from the numerical data by using the Penrose–Onsager criterion [see (43)].

Figure 4

The results of our self-similarity algorithm with respect to the condensate number for the number-damping SPGPE and the full SPGPE. The numerical data for the number-damping (full) SPGPE is represented by blue (red) points, while the green (cyan) line is a least-squares fit of a power law to the numerical data. The power-law exponents are $\alpha =0.5119\pm 0.0178$ and $\alpha =0.7145\pm 0.0358$ for the number-damping SPGPE and full SPGPE, respectively.

Figure 5

The condensate number over time for several quench times using (a) the number-damping SPGPE and (b) the full SPGPE. The time axis has been scaled by the freeze-out time as predicted by our self-similarity algorithm for each theory. The yellow shaded region indicates the impulse regime where KZM approximates the system dynamics as frozen. For comparison, the total $\mathbf{C}$-field population is also shown for one of the quenches (black dashed line).

Figure 6

Standard deviation of the final winding for various quench times. The blue and red data points are the result of simulations of the number-damping SPGPE and the full SPGPE, respectively. The green line is a a best fit for the number-damping SPGPE data in the regime that obeys a power law (14), giving an exponent of $\beta =0.1236\pm 0.0098$. The cyan line is the equivalent for the full SPGPE data, giving an exponent of $\beta =0.0966\pm 0.0128$.

Figure 7

The two-point correlation function $\rho (r,t)=\langle {\psi }^{*}(x,t)\psi (x^{\prime },t)\rangle$, where $r\equiv x-x^{\prime }$, at various times for three different quench times simulated using both the number-damping SPGPE and the full SPGPE. The quench times are (a), (d) ${\tau }_Q{\omega }_{\perp }=e^6$, (b), (e) ${\tau }_Q{\omega }_{\perp }=e^7$, and (c), (f) ${\tau }_Q{\omega }_{\perp }=e^8$. The top row (a)–(c) shows the normalized numeric correlation function at $t=-\widehat{t}$ (blue), $t=0$ (red), and $t=\widehat{t}$ (magenta) resulting from simulations of the number-damping SPGPE, as well as the normalized analytic correlation function (36) at $t=-\widehat{t}$ (green) and $t=0$ (cyan). The bottom row (d)–(f) shows the normalized numeric correlation function at $t=0$ (red) resulting from simulations of the full SPGPE, as well as the normalized correlation function (36) at $t=0$ (cyan).