Precision measurements of temperature and chemical potential of quantum gases

We investigate the sensitivity with which the temperature and the chemical potential characterizing quantum gases can be measured. We calculate the corresponding quantum Fisher information matrices for both fermionic and bosonic gases. For the latter, particular attention is devoted to the situation close to the Bose-Einstein condensation transition, which we examine not only for the standard scenario in three dimensions, but also for generalized condensation in lower dimensions, where the bosons condense in a subspace of Hilbert space instead of a unique ground state, as well as condensation at fixed volume or fixed pressure. We show that Bose-Einstein condensation can lead to sub-shot-noise sensitivity for the measurement of the chemical potential. We also examine the influence of interactions on the sensitivity in three different models and show that mean-field and contact interactions deteriorate the sensitivity but only slightly for experimentally accessible weak interactions.

Figure 1

Log-log-plot of the optimal relative error $1/[\beta \sqrt{{\left(F_d^{\mathrm{hom}}\right)}_{\beta ,\beta }]}$ for a homogenous gas of ${}^6$Li atoms in three dimensions as function of temperature: $L=20\mu$m (left) and $L=500\mu$m (right); and ${\langle N\rangle }_3^{\mathrm{hom}}=10$ (circles), ${\langle N\rangle }_3^{\mathrm{hom}}=1000$ (squares), and ${\langle N\rangle }_3^{\mathrm{hom}}=10000$ (diamonds). Dotted [dashed] lines are the corresponding small temperature limits (59) [classical limits (57) and (58)]. For a given temperature, we computed the chemical potential inverting the equation of state (42) with the Newton's method, then we used this result in (45) and plotted the corresponding values of $1/\left[\beta \sqrt{{\left(F_d^{\mathrm{hom}}\right)}_{\beta ,\beta }}\right]$.

Figure 2

Log-log plot of the optimal relative error $1/\left[\beta \sqrt{{\left(F_d^{\mathrm{hom}}\right)}_{\beta ,\beta }}\right]$ for a homogenous gas of ${}^6$Li atoms in three dimensions: $T=1$ pK (top left), $T=1$ nK (top right), $T=1\mu$K (bottom left), and $T=1$ mK (bottom right); and ${\langle N\rangle }_3^{\mathrm{hom}}=10$ (circles), ${\langle N\rangle }_3^{\mathrm{hom}}=1000$ (squares), and ${\langle N\rangle }_3^{\mathrm{hom}}=10000$ (diamonds). Dotted [dashed] lines are the corresponding small temperature limits (59) [classical limits (57) and (58)]. For a given $L$, we computed the chemical potential inverting the equation of state (42) with Newton's method, then we used this result in (45) and plotted the corresponding values of $1/\left[\beta \sqrt{{\left(F_d^{\mathrm{hom}}\right)}_{\beta ,\beta }}\right]$.

Figure 3

Log-log plot of the upper bounds of the rescaled Fisher information ${\left(F_d^{\mathrm{hom}}\right)}_{\mu ,\mu }/{\beta }^2$ for homogenous Bose gas above the condensation temperature against the average number of particles within the continuum approximation in three dimensions (dot-dashed line) (62), two dimensions (dashed line) (64), and one dimension (dotted line) (67). The continuous line is the shot noise reproduced by the classical gas [Eq. (57)]. The thick line is the Fisher information if all particles are in the ground state, namely the zero-temperature case [Eq. (81)].

Figure 4

Upper bounds of the rescaled Fisher information ${\left(F_d^{\mathrm{harm}}\right)}_{\mu ,\mu }/{\beta }^2$ for harmonically trapped bosons above the condensation temperature against the average number of particles in double logarithmic scale within the continuum approximation in three dimensions (dot-dashed line) (75), two dimensions (dashed line) (76), and one dimension (dotted line) (78). The continuous line is the shot noise reproduced by the classical gas. The thick line is the Fisher information if all particles are in the ground state, i.e., the zero-temperature case.

Figure 5

Double logarithmic plot of the rescaled Fisher information $F_{\mu ,\mu }/{\beta }^2$ of an ideal Bose gas versus the average number of particles of a gas of ${}^{87}$Rb atoms in a slab with exponential anisotropy $L_{x,y}\sim \gamma e^{\alpha L_z}$. The curves refer to different temperatures: classical limit (continuous line, linear in $\langle N\rangle$), first critical temperature $T_c^{3\mathrm{D}}=100$ nK (dotted line), $T_c^{3\mathrm{D}}/2$ (dashed line), second critical temperature $T_c^{2\mathrm{D}}=20$ nK (dot-dashed line), zero temperature (thick continuous line, quadratic in $\langle N\rangle$ [see Eq. (81)]). The two critical temperatures are uniquely determined by $\rho =13\times {10}^{12}$ cm${}^{-3}$ and $\alpha =10\mu$m${}^{-1}$. Furthermore, $\ell ={\lambda }_T$.

Figure 6

Log-log plot of the rescaled Fisher information $F_{\mu ,\mu }/{\beta }^2$ in the perturbative regime (97) against the average number of particles. The results are for a gas of ${}^{87}$Rb atoms confined in $L_x=4.5\mu$m at $T=510$nK and interaction strength $c=2.93\times {10}^{-40}$ J m (dotted line), $c=2.93\times {10}^{-38}$ J m (dashed line), and $c=0$ (dot-dashed line). The thin continuous line is the shot noise resulting from the classical limit, the thick dashed line is the best scaling achievable within the continuum approximation (67), and the thick line is the Fisher information if all particles are in the ground state, i.e., the zero-temperature case.