Cooling of a One-Dimensional Bose Gas

We experimentally study the dynamics of a degenerate one-dimensional Bose gas that is subject to a continuous outcoupling of atoms. Although standard evaporative cooling is rendered ineffective by the absence of thermalizing collisions in this system, we observe substantial cooling. This cooling proceeds through homogeneous particle dissipation and many-body dephasing, enabling the preparation of otherwise unexpectedly low temperatures. Our observations establish a scaling relation between temperature and particle number, and provide insights into equilibration in the quantum world.

Figure 1

Analysis of the outcoupling mechanism. (a) A typical density speckle pattern formed by a 1D quasicondensate expanding for 10.5 ms in time of flight. A small fraction was outcoupled by a single 2 ms rf pulse 2.5 ms before the gas was released. The outcoupled cloud is visible below the source cloud, showing the same speckle pattern. (b) Comparing the longitudinal density profiles of the source cloud (blue line) and the outcoupled cloud (red line), averaged over many realizations, reveals the same form of the profile. (c) Sketch of the outcoupling mechanism. Transversal to the 1D axis the atoms occupy the ground state of the harmonic potential (blue). Outcoupling is achieved by coupling these atoms to untrapped states, e.g., $|2,0⟩$ (red), which only feel the linear gravitational potential. The rate of this transition, following from a simple calculation taking only the transversal degrees of freedom into account, is plotted in (d) over the detuning of the driving field, given in units of the longitudinal trap frequency. The large width of this rate compared to the energy scale of the longitudinal dynamics leads to a nearly homogeneous outcoupling of atoms.

Figure 2

Three different examples for measurements of temperature $T$ over atom number $N$. The green triangles are obtained through a continuous rf-induced outcoupling from a harmonic trap with trap frequencies ${\omega }_{\perp }=2\pi \times (2.1\pm 0.1)\mathrm{kHz}$ and ${\omega }_{\parallel }=2\pi \times (11\pm 1)\mathrm{Hz}$ over a time scale of 25 ms. The blue circles differ in that they start from a much colder initial temperature and were measured over a time scale of 90 ms in a transversally slightly anharmonic trap created through a dressed state potential [23]. There the trap frequencies are ${\omega }_{\perp }=2\pi \times (1.6\pm 0.1)\mathrm{kHz}$ and ${\omega }_{\parallel }=2\pi \times (8\pm 1)\mathrm{Hz}$. Finally, the red diamonds were acquired through a very different measurement procedure, where microwave radiation was used to couple atoms from the dressed trap directly to the antitrapped $|1,1⟩$ state. Furthermore, in contrast to the other data points, the temperature was measured after a constant period of outcoupling with a different outcoupling rate for each data point. The temperature error bars give the 68% confidence interval obtained from a bootstrap. The inset depicts the speckle correlations of the coldest red and blue data points together with their respective thermal fits, still showing good agreement.

Figure 3

The data from Fig. 2 are plotted rescaled to their respective initial temperature $T_0$ and atom number $N_0$ values revealing a clear linear scaling.

Figure 4

Dissipative cooling mechanism visualized through the Wigner function $W_k$ (a) of the phase and density quadrature ${\varphi }_k$ and ${\rho }_k$ of a single mode $k$, which rotates with the eigenfrequency ${\omega }_k$. The homogeneous outcoupling of atoms squeezes the Wigner function in the density quadrature, indicated by the black arrows, thereby reducing the energy of the mode. The rotation with ${\omega }_k$ together with a slow continuous squeezing leads to a (nearly) symmetric narrowing of the Wigner function (b) reflecting a reduced occupation of the mode. In our multimode system this happens in all modes at the same time, leading to a reduced temperature.