Measurements of Tan’s Contact in an Atomic Bose-Einstein Condensate

A powerful set of universal relations, centered on a quantity called the contact, connects the strength of short-range two-body correlations to the thermodynamics of a many-body system with zero-range interactions. We report on measurements of the contact, using rf spectroscopy, for an ${}^{85}\mathrm{Rb}$ atomic Bose-Einstein condensate (BEC). For bosons, the fact that contact spectroscopy can be used to probe the gas on short time scales is useful given the decreasing stability of BECs with increasing interactions. A complication is the added possibility, for bosons, of three-body interactions. In investigating this issue, we have located an Efimov resonance for ${}^{85}\mathrm{Rb}$ atoms with loss measurements and thus determined the three-body interaction parameter. In our contact spectroscopy, in a region of observable beyond-mean-field effects, we find no measurable contribution from three-body physics.

Figure 1

Example of rf contact spectroscopy. (a) rf line shape, $S(\omega )$, normalized so that ${\int }_{-\infty }^{\infty }S(\omega )d\omega =1{\mathrm{s}}^{-1}$. The data at large detunings (circles) are multiplied by a factor of 300 to make the tail visible. Here the mean BEC density is $⟨n⟩=4.9\times {10}^{12}{\mathrm{cm}}^{-3}$. (b) Additional release energy of the outcoupled atom cloud. We calculate the energy from the width of the expanded cloud, $\sigma$, using $E=\frac{3}{2}m\frac{{\sigma }^2-{\sigma }_0^2}{\Delta t^2}$, where $\Delta t$ is the time between the middle of the rf pulse and the absorption image (4.5 ms) and ${\sigma }_0$ is the size of the expanded cloud measured at $\omega =0$. The solid line is $\frac{1}{2}\frac{|\omega |}{2\pi }$.

Figure 2

The contact vs $a$, measured at $|\omega |=2\pi \times 40\mathrm{kHz}$. (a) The contact per BEC atom $\frac{C_2}{N_0}$. (b) The raw signal before final-state corrections. The solid lines in (a) and (b) show the mean-field predictions. The dashed line includes the next-order LHY correction. For this data, $⟨n⟩$ is typically $5.8\times {10}^{12}{\mathrm{cm}}^{-3}$, with $(n{a}^3{)}^{1/2}$ reaching a maximum of 0.043. We linearly scale the points to account for $\sim 10\%$ variation in density. The final-state effects shift the solid line from a parabola centered about $a=0$ in (a) to one centered about $a^{\prime }=-565$ $a_0$ in (b), which enhances the raw signal at small $a$.

Figure 3

A three-body loss resonance for ${}^{85}\mathrm{Rb}$. We plot the three-body event constant $K_3$ vs $a$. From fitting Eq. (7) to the solid points, for which $a<1/k_{\mathrm{thermal}}$, we extract $a_{-}=-759(6)a_0$ and $\eta =0.057(2)$.

Figure 4

(a) The frequency dependence of $G_{\mathrm{RF}}(\omega )$, given our measured value for ${\kappa }_{*}$. (b) Frequency dependence of the tail of the rf spectrum for $a=982\pm 10$ $a_0$. The solid line is a fit of the data (●) to the expected frequency dependence of the two-body contact $C_2/N_0$ including final-state effects. The dotted line corresponds to the same value of $C_2/N_0$, but ignores final-state effects. For comparison, the fit plus a trial $C_3/N_0$ term of $0.1\mu {\mathrm{m}}^{-2}$ is shown with the dashed line. Our measurements are consistent instead with a $C_3/N_0$ of zero. Here $⟨n⟩=1.0\times {10}^{13}{\mathrm{cm}}^{-3}$.