Degenerate Bose gases with uniform loss

We theoretically investigate a weakly interacting degenerate Bose gas coupled to an empty Markovian bath. We show that in the universal phononic limit the system evolves towards an asymptotic state where an emergent temperature is set by the quantum noise of the outcoupling process. For situations typically encountered in experiments, this mechanism leads to significant cooling. Such dissipative cooling supplements conventional evaporative cooling and dominates in settings where thermalization is highly suppressed, such as in a one-dimensional quasicondensate.

Figure 1

Schematic of the considered setup. Interacting degenerate bosonic atoms in the transversal ground state of a harmonic trap $|g\rangle$ are outcoupled to a continuum of free modes $|k\rangle$ using a microwave or an rf field with a Rabi frequency $\mathrm{\Omega }$ and detuning $\mathrm{\Delta }\omega$. The interactions between the atoms in $|g\rangle$ are manifested in the mean-field shift $\mu \equiv g\langle {\widehat{\mathrm{\Psi }}}^{†}\widehat{\mathrm{\Psi }}\rangle$, where $g=\mathrm{const}$ is the self-interaction strength. The mean-field shift is still considerably smaller than the energy of the first transversally excited level $|e\rangle$. The kinetic energy in the longitudinal direction (inside the plane of the paper), represented by the fine structure of the bands, is much smaller than any other relevant energy scale.

Figure 2

Approximate dispersion curves for a one-dimensional dissipative quasicondensate. The panels show real (a) and imaginary (b) parts of the quasiparticle energy ${\tilde{\epsilon }}_k$ (in units of the chemical potenial $\mu$) as a function of momentum (in units of the inverse healing length $1/\xi =mc$) and the dissipation strength. The quasiparticle decay rate is the negative imaginary part of the energy. Diffusive modes appear when the real part of the dispersion touches zero. The two diffusive modes, densitylike and phaselike, have different decay rates, which is represented by two branches in (b). Note that $\gamma$ is the dissipation rate of the order parameter, and not of the density, namely, $n\left(t\right)=n_0{e}^{-2\gamma t}$.

Figure 3

Log-log plot of the evolution of the Bogoliubov modes' occupation numbers $n_k$ as a function of momentum $k$ (in units of the inverse healing length $1/\xi =mc$) and time $t$ (solid lines, in units of the inverse dissipation rate $1/\gamma$) for initial thermal distribution at temperature $T_0={\mu }_0$. It is easy to see the two limits of the full Bose-Einstein distribution function: the phononic Rayleigh-Jeans limit $\left(k\lesssim mc\right)$ and the particlelike Boltzmann limit $\left(k\gtrsim mc\right)$. The dashed lines represent thermal distributions at temperatures, top to bottom, $T_{\mathrm{eff}}\left(t\right)/{\mu }_0\approx \{1.00,0.55,0.30,0.17,0.09\}$, fitted to agree with the calculated values in the phononic regime. Note that although the time-evolved distributions are clearly nonthermal in their high-energy tails, they however agree well with thermal predictions in the low-energy part of the spectrum, allowing the introduction of an effective temperature for phononic modes. Inset: time evolution of the chemical potential $\mu \left(t\right)={\mu }_0{e}^{-2\gamma t}$, solid line, in comparison with the fitted effective temperatures $T_{\mathrm{eff}}\left(t\right)$, dots. As explained in Sec. 4, in this special case $T\left(t\right)=\mu \left(t\right)$.

Figure 4

Temperature to chemical potential ratio as a function of dimensionless time $\gamma t$ for different initial values of $T_0/{\mu }_0$ (solid lines). Note the emergence of an asymptotic dissipative state $T\left(t\right)=\mu \left(t\right)$ for $t\to \infty$. For comparison, we present the classical mean-field prediction which neglects the quantum noise (dashed lines). The scale of the time axis is logarithmic.

Figure 5

Effective temperature scaling with the time-dependent chemical potential for different $T_0/{\mu }_0$ ratios. The lowermost curve represents the classical limit where the quantum noise is neglected. The chemical potential is proportional to the mean density in the uniform case, and to the central peak density in case of the harmonic potential.