Quantum Depletion of a Homogeneous Bose-Einstein Condensate

We measure the quantum depletion of an interacting homogeneous Bose-Einstein condensate and confirm the 70-year-old theory of Bogoliubov. The observed condensate depletion is reversibly tunable by changing the strength of the interparticle interactions. Our atomic homogeneous condensate is produced in an optical-box trap, the interactions are tuned via a magnetic Feshbach resonance, and the condensed fraction is determined by momentum-selective two-photon Bragg scattering.

Figure 1

Momentum distribution of a zero-temperature homogeneous Bose gas. We consider a gas of density $n$ and size $L$ and two different values of the scattering length $a$. We show the expected 1D momentum distribution $\tilde{n}(k)$ (see the text), normalized so that $\tilde{n}(0)=1$ would correspond to no quantum depletion [setting $\gamma$ in Eq. (1) to 0]. The total $\tilde{n}(k)$ consists of the BEC peak (blue), with a Heisenberg-limited width $\propto 1/L$, and a broad quantum-depletion pedestal (orange) of characteristic width $1/\xi$, where $\xi$ is the healing length. To a good approximation, the low-$k$ distribution is the same as for a pure BEC, just scaled by a factor of $1-\gamma \sqrt{n{a}^3}$, indicated by the dashed lines. For this illustration, we use experimentally relevant values of $L/\xi$, but exaggerated values of $\sqrt{n{a}^3}$, to make the orange shading visible in the main panels. Also note that we assume that the very broad ${\tilde{n}}_{\mathrm{QD}}(k)$ is not affected by finite-size effects. The cartoons on the left depict the coherent excitations out of the (blue) condensate, which occur as pairs of atoms with opposite momenta. The right insets highlight the fact that ${\tilde{n}}_{\mathrm{QD}}(k)\gg {\tilde{n}}_{\mathrm{BEC}}(k)$ at large $k$.

Figure 2

Bragg filtering and reversible interaction tuning of the condensed fraction. (a) Diffracted fraction (DF) as a function of the Bragg pulse duration $\tau$ for $\mathrm{\Omega }=2\pi \times 1.8\mathrm{kHz}$ and $a\approx 3000a_0$. Absorption images in the background show the stationary (bottom) and diffracted (top) clouds, for the data points indicated by the red diamonds. (b) Diffracted fraction for $\tau$ close to $\pi /\mathrm{\Omega }$, for three different preparations of the cloud (see the inset): at $700a_0$ (solid blue circles), after raising $a$ from $700a_0$ to $3000a_0$ in 80 ms (orange diamonds), and after reducing it back to $700a_0$ in another 80 ms (open green circles). We see that increasing $a$ reversibly reduces the maximal diffracted fraction. All error bars show standard statistical errors in the mean.

Figure 3

Measurement of the quantum depletion. We plot the maximal diffracted fraction $\eta$ versus the interaction parameter $\sqrt{n{a}^3}$. A linear fit (solid line) gives $\eta (0)=0.954(5)$ and $\gamma =1.5(2)$. Vertical error bars show fitting errors, while horizontal ones reflect the uncertainty in the position of the Feshbach resonance and a 10% uncertainty in $n$. Inset: Analysis of systematic effects. We show numerical simulations for $T=0$ (dashed line) and for initial temperatures (at $a=200a_0$) between 3.5 and 5 nK (orange shading, from top to bottom); see the text and [31] for more details.