Decay of a quantum vortex: Test of nonequilibrium theories for warm Bose-Einstein condensates

The decay of a vortex from a nonrotating high temperature Bose-Einstein condensate is modeled using the stochastic projected Gross-Pitaevskii equation (SPGPE). In order to exploit the tunability of temperature in SPGPE theory while maintaining the total atom number constant, we develop a simple and accurate Hartree-Fock method to estimate the SPGPE parameters for systems close to thermal equilibrium. We then calculate the lifetime of a vortex using three classical field theories that describe vortex decay in different levels of approximation. The SPGPE theory is shown to give the most complete description of the decay process, predicting significantly shorter vortex lifetimes than the alternative theories. Using the SPGPE theory to simulate vortex decay for a trapped gas of $5\times {10}^5$ ${}^{87}\mathrm{Rb}$ atoms, we calculate a vortex lifetime $\mathrm{t̄}$ that decreases linearly with temperature, falling in the range $20\mathrm{s}>\mathrm{t̄}>1.5\mathrm{s}$ corresponding to the temperature range $0.78T_c⩽T⩽0.93T_c$. The vortex lifetimes calculated provide a lower bound for the lifetime of a persistent current with unit winding number in our chosen trap geometry in the limit of vanishing vortex pinning potential.

Figure 1

Schematic of different vortex decay processes in two dimensions. (a) Under Hamiltonian evolution, an eigenstate of the GPE containing a displaced vortex will evolve so that the vortex follows trajectories of constant radius about the center of the condensate. At finite temperatures, the presence of a nonrotating thermal cloud creates a damping mechanism for vortex motion. (b) Dissipative decay: in the absence of noise, a damped, off-center vortex will decay to the edge of the condensate along a regular spiral trajectory. (c) Diffusive decay: thermal fluctuations lead to stochastic vortex dynamics resembling Brownian motion [24].

Figure 2

(a) Separation of the high-temperature Bose gas using the single particle energy cutoff ${\epsilon }_{\mathrm{cut}}$ into coherent ($C$, with energies ${\Omega }_C⩽{\epsilon }_{\mathrm{cut}}$) and incoherent ($I$, with energies ${\Omega }_I>{\epsilon }_{\mathrm{cut}}$) regions. (b) The $I$-$C$ collisions that lead to Bose-enhanced growth or loss: during a collision of two $I$ atoms, one atom absorbs most of the collision energy, allowing a stimulated transfer of the other into the coherent region, $C$. The time-reversed process also occurs.

Figure 3

Estimated and SPGPE parameters as a function of input temperature using an input value of $N_T=1\times {10}^6$ for a spherical system with trap frequency $\omega =2\pi \times 10$ ${\mathrm{s}}^{-1}$. (a) Hartree-Fock estimates of the cutoff energy ${\epsilon }_{\mathrm{cut}}$ for a range of cutoff mode occupations $n_{\mathrm{cut}}$. The dashed lines give the curves for ${\epsilon }_{\mathrm{cut}}$ and $\mu$ used in (b). Total atom number from SPGPE simulations evolved to equilibrium using HF estimated parameters (circles) and fixed $\mu$ (squares). $\mu$ is held at the value giving $N_T$ at $T=34$ nK. The solid line is the target value of $N_T$.

Figure 4

Vortex dynamics using the DPGPE and PGPE in a system with a 4:1 geometry. The vortex is most easily observed from the phase singularity (circle), and the center of the condensate is marked by a cross ($+$). (a) DPGPE vortex evolution for $T/T_c=0.78$, where the vortex has an initial offset from the origin of the condensate. The initial noise is damped out within $~$1.2 s, after which the vortex decay is governed by $\gamma$. (b) PGPE vortex evolution for $T/T_c=0.78$, with an initially centered vortex. Despite initial thermal fluctuations kicking the vortex off the center, the size of the thermal component is too small at this temperature to see a finite lifetime. (c) PGPE vortex evolution for a system at $T/T_c=0.93$ where the initially centered vortex has a finite lifetime due to increased classical fluctuations at this temperature.

Figure 5

Column densities and phase slices showing SPGPE vortex evolution in a system with aspect ratio of 4:1 at temperatures (a) $T/T_c=0.78$ and (b) $T/T_c=0.93$. The vortex is most easily identified from the phase singularity (circle), and the center of the condensate is marked by a cross ($+$). Other parameters are for a gas of $5\times {10}^5$ ${}^{87}\mathrm{Rb}$ atoms in a harmonic trap with $({\omega }_r,{\omega }_z)=2\pi (12.4,49.7)$ Hz.

Figure 6

Lifetime of a single vortex, $\mathrm{t̄}$, as a function of temperature for a system containing $5\times {10}^5$ ${}^{87}\mathrm{Rb}$ atoms. The trap has fixed $\mathrm{\omega ̄}=2\pi \times 19.7\hspace{0.5em}{\mathrm{s}}^{-1}$, so that $T_c=69.7$ nK is the same for both geometries.

Figure 7

The $z$ component of the averaged angular momentum per particle as a function of $t/\mathrm{t̄}$, where $\mathrm{t̄}$ is the vortex lifetime, for a system with an aspect ratio of 4:1 as found from an ensemble average of SPGPE trajectories. The case of overdamped purely dissipative decay is given by the dotted line. Note the wide contrast in the form of the vortex decay, where, for $T/T_{\mathrm{c}}=0.78$, we see dissipation dominating the decay, while for higher temperatures, diffusive decay becomes more prevalent.