Dark-State Cooling of Atoms by Superfluid Immersion

We propose and analyze a scheme to cool atoms in an optical lattice to ultralow temperatures within a Bloch band and away from commensurate filling. The protocol is inspired by ideas from dark-state laser cooling but replaces electronic states with motional levels and spontaneous emission of photons by emission of phonons into a Bose-Einstein condensate, in which the lattice is immersed. In our model, achievable temperatures correspond to a small fraction of the Bloch bandwidth and are much lower than the reservoir temperature. This is also a novel realization of an open quantum optical system, where known tools are combined with new ideas involving cooling via a reservoir.

Figure 1

(a) Cooling setup: Atoms $a$ in an optical lattice are coupled to the first excited motional state via a Raman process and decay to the ground motional state due to collisional interactions with a BEC of species $b$ in which the lattice is immersed. Tunneling between neighboring sites with amplitude $J^{\alpha }$ gives rise to Bloch bands. (b) Momentum space picture: Atoms with higher quasimomentum $q$ are excited to the upper Bloch band and decay to random quasimomentum states. After several cycles, atoms $a$ collect in a dark-state region near $q=0$ with low excitation probability.

Figure 2

(a)–(c) Excitation probability $P_j(q)$ for a sequence of $N_p=5$ pulses: first [(a)–(c), dashed-dotted line], second [(b),(c), dotted line], and the remaining pulses (c). Parameters used: $\Omega =(27.9,13.7,13.7,2.37,2.37)J^0$, $\delta q=(0.16,-1.75,1.75,-2.63,2.63)/d$, and $(\delta -\omega )=-(28.4,25.8,25.8,24.7,24.7)J^0$ for the different pulses. (d) Successive narrowing of the momentum distribution in the lower Bloch band after 0, 1, 3, 5, and 10 cooling cycles from numerical simulations ($M=101$ lattice sites) based on the pulse sequence in (a)–(c). We choose parameters for ${}^{87}\mathrm{Rb}$ in the lattice and ${}^{23}\mathrm{Na}$ in the reservoir, with $\Gamma =53J^0$ from $m_a/m_b=3.73$, ${\rho }_b=5\times {10}^{14}{\mathrm{cm}}^{-3}$, and scattering length $a_{ab}=14\mathrm{nm}$. $V_0=10{\omega }_R$, and ${\omega }_R=2\pi \times 3.8\mathrm{kHz}$. (e) Temperature vs time for a single atom: Crosses and circles denote numerical results and solid lines analytical results based on Lévy statistics. Pulse sequences for circles: Same as in (d); crosses: $N_p=3$ pulses with $\Omega =(32.6,7.9,7.9)J^0$, $\delta qd=(0.31,2.12,-2.12)$, and $(\delta -\omega )=-(28.4,25.3,25.3)J^0$.

Figure 3

Numerical simulation of the QBME: (a) Temperature as a function of cooling time for bosons (crosses) and fermions (circles). (b) Momentum distribution in band $\alpha =0$ for fermions after 0 (dashed line), 1 (dotted-dashed line), 2 (dotted line), and 20 (solid line) cooling cycles, each with $N_p=4$ pulses. Parameters used: Bosons: As for Figs. 2a, 2b, 2c, but $N=51$ particles. Fermions: $N=71$, $M=101$, Blackman pulses with $\tau J^0=(1.78,1.78,14.2,14.2)$, $\delta qd=(0.19,-0.19,0.75,-0.75)$, $(\omega -\delta )=(28.4,28.4,27.9,27.9)J^0$; $V_0=10{\omega }_R$, ${\omega }_R=2\pi \times 6\mathrm{kHz}$, $m_a/m_b=1.74$, and $\Gamma =52.6J^0$.