Nondestructive selective probing of phononic excitations in a cold Bose gas using impurities

We introduce a detector that selectively probes the phononic excitations of a cold Bose gas. The detector is composed of a single impurity atom confined by a double-well potential, where the two lowest eigenstates of the impurity form an effective probe qubit that is coupled to the phonons via density-density interactions with the bosons. The system is analogous to a two-level atom coupled to photons of the radiation field. We demonstrate that tracking the evolution of the qubit populations allows probing both thermal and coherent excitations in targeted phonon modes. The targeted modes are selected in both energy and momentum by adjusting the impurity's potential. We show how to use the detector to observe coherent density waves and to measure temperatures of the Bose gas down to the nanokelvin regime. We analyze how our scheme could be realized experimentally, including the possibility of using an array of multiple impurities to achieve greater precision from a single experimental run.

Figure 1

(a) The phonon detector consists of a single impurity atom in a double-well potential $V_a$ with well width $\sigma$ and separation $2L$, and barrier height $\Delta V$. The impurity is restricted to the two lowest states with symmetric and antisymmetric wave functions ${\varphi }_0$ and ${\varphi }_1$, separated by the energy splitting $2J$. This detector is immersed in a weakly interacting Bose gas and acts as a probe. (b) In two and three dimensions, the impurity is trapped by a potential that has the same double-well shape in one direction, but is tightly confined in the remaining orthogonal directions.

Figure 2

(a) Dependence of the energy splitting $2J/{\omega }_a$ according to Eq. (4) as a function of (a) the relative well separation $\ell$, and (b) $L$ and $\sigma$ in terms of the natural Bose gas length and energy scales, the healing length $\xi =1/\sqrt{m_{b}g{n}_0}$ and chemical potential $g{n}_0$. The shaded region in the bottom right corner is inaccessible since here $L<\sigma$. The other parameters for both plots are $D=1,m_a=0.5m_b$, and $\Delta V/{\omega }_a=D$.

Figure 3

Spectral densities ${\mathcal{J}}_D\left(\omega \right)$ in the linear regime [Eq. (12)] for $D=1$ (green solid line), 2 (red dotted line), 3 (blue dashed line). The detector parameters used are $\sigma =\xi ,L=5\xi$, and $\Delta V={\omega }_a$, the detector-Bose gas coupling is $\kappa =5g$, and the Bose gas density $n_0{\xi }^D=1$.

Figure 4

The detector in a two-dimensional setup with $m_a=0.5m_b,\Delta V={\omega }_a,\sigma =0.1/\left|\mathbf{p}\right|$, and $\mathbf{L}=0.27\mathbf{p}/{\left|\mathbf{p}\right|}^2$ is tuned close to resonance with a mode with the wave vector $\mathbf{p}=3\widehat{\mathbf{y}}/\xi$ along the $y$ axis. (a) The amplitude $A={\Omega }_{\mathbf{p}}^2/{\tilde{\Omega }}_{\mathbf{p}}^2$ of coherent oscillations exhibits a clear dependence on both magnitude and direction of $\mathbf{p}=(p_x,p_y)$, i.e., the direction and energy of the density waves. (b) Angular dependence of the Rabi frequency ${\Omega }_{\mathbf{p}}$ on the direction of $\mathbf{p}$. The other parameters (for both plots) are $\kappa =10g,|{\beta }_{\mathbf{p}}|=5,n_0{\xi }^2=0.5$, and $\mathcal{V}={\left(100\xi \right)}^2$.

Figure 5

Simulation of the temperature measurement in two dimensions under ideal conditions using the values presented in Table 1. The detector is initialized in its excited state $|1\rangle$ and evolves according to Eq. (11). The fictional measurement points (green) are distributed around the expectation value ${\rho }_{11}\left(t\right)$ according to a Bernoulli distribution obtained from 5000 measurements at each of the 50 points. The corresponding plots show the decay of the population ${\rho }_{11}\left(t\right)$ for temperatures (a) $T_0=0.5g{n}_0/k_B$ and (b) $T_1=10g{n}_0/k_B$, resulting in the markedly different decay times.

Figure 6

Detection of a coherently occupied phonon mode with $|{\beta }_{\mathbf{p}}|=5$ and frequency ${\omega }_{\mathbf{p}}=10g{n}_0$ in one dimension (left) and two dimensions (right). The detector is tuned close to resonance such that $2J\approx {\omega }_{\mathbf{p}}$. The resulting oscillations of ${\rho }_{00}$ (solid blue line) and ${\rho }_{11}$ (dashed green line) (a) in one dimension with $Q$ factor $Q=7.93$ and (b) in two dimensions with $Q=2.32$ are clearly visible. Panels (c) and (d) show the sensitivity of the oscillation amplitude (solid blue line) on changes of $2J$ away from ${\omega }_{\mathbf{p}}$, realized as $L$ is varied away from its near-resonant value, and $\sigma$ fixed. Panels (e) and (f) show the dependence of the $Q$ factor upon the same changes in $L$. The values of $2J$ corresponding to those chosen for the examples (a) and (b) are indicated by the vertical dashed green lines. The specific parameters of the Bose gas in 1D are $T=0.01g{n}_0/k_B$ and $n_0\xi =1$; in 2D we set $T=0.1g{n}_0/k_B,n_0{\xi }^2=10$. The system volume is set to $\mathcal{V}={\left(100\xi \right)}^D$. The parameters of the detector in 1D are $\kappa =0.5g,\sigma =0.3/\left|\mathbf{p}\right|,L=0.62/\left|\mathbf{p}\right|$ and in 2D we set $\kappa =2g,\sigma =0.05/\left|\mathbf{p}\right|,L=0.14415/\left|\mathbf{p}\right|$. The impurity mass is $m_a=0.5m_b$, and the barrier height $\Delta V=D{\omega }_a$ in both cases.

Figure 7

If an array of impurities with lattice vector $\mathbf{a}$ is arranged such that $\mathbf{a}·\mathbf{L}=0$ and $\left|\mathbf{a}\right|\gg L$, no Bose gas-mediated correlations between the impurities can arise.