Analogue many-body gravitating quantum systems with a network of dipolar Bose-Einstein condensates

Abstract

Operational probes of the interface between quantum mechanics and general relativity in the Newtonian regime -- via mass-energy equivalence in clocks or spatial superpositions in interferometers -- share a common description in terms of an effective qubit-qubit Ising coupling. Here we generalize both paradigms to interacting $(N+1)$-level effective qudits made of atomic ensembles with particle number, $N$. The many-body enhancement boosts the signal-to-noise and increases the effective interaction rate, facilitating the observation of gravitationally induced entanglement and decoherence, certified by metrological witnesses based on local and collective measurements. Furthermore, we show that quantum effects induced by gravitational interaction can be simulated by trapped bimodal Bose-Einstein condensates with long-range (e.g. dipolar) coupling, providing a programmable analogue platform to explore gravitating quantum dynamics at accessible time and energy scales. Finally, extending the protocol to a sensor network broadens the entanglement-detection window.

Figures (5)

• Figure 1: Generalization of the CGB and BMV proposals to trapped bimodal BECs. (a) Schematic of two quantum clocks, labeled as A and B, each made of $N$-particle bimodal BEC placed at a distance $d$. The two modes are two internal levels, with energy gap $\Delta E$. (b) Schematic of two $N$-particle path atom interferometers (top). The two nearby arms are separated by a distance $d$ while the inter-arm separation is $d'$. Beam-splitters are shown in red and mirrors in blue. This configuration can be realized with BECs in two double-well potentials (bottom), each well acting as an external mode.
• Figure 2: GIE dynamics. We start with all the atoms in the same mode for each BEC. At $t=0$ a $\pi/2$ pulse distributes the particles among the two modes. The subsequent dynamics is a free evolution according to the Hamiltonian $J_A^z\otimes J_B^z$. (a) Local ($\xi^{2}_{\rm loc}$, solid black) and collective ($\xi^{2}_{\rm col}$, dashed red) spin squeezing and inverse Fisher information ($2N/F_{\rm col}$ solid red line) as a function of the effective time $\tau =\chi_{\rm nloc}^G t$. From these quantities we compute the coefficients ${\cal C}_{1}$ and ${\cal C}_{2}$, shown as red solid line and red dashed line in panel (b), respectively. Accessing the gray region (${\cal C}_{1}, {\cal C}_{2} < 1$) witnesses entanglement between the two ensembles. The inset shows ${\cal C}_{1}$ (red solid) and ${\cal C}_{2}$ (red dashed) as well as a tighter version of these criteria, $\tilde{{\cal C}}_{1}$ (green solid) and $\tilde{{\cal C}}_{2}$ (green dashed), based on the local Fisher information matrix instead of the local covariance matrix, see Methods for more details. For both figures, we have taken $N = 1000$ for the main panels, and $N=10$ for the inset of figure (b). Results shown in the main panels of figures (a-b) are fully analytical and obtained following Ref. KurkjianPRA2013, see details in Appendix F.
• Figure 3: GID dynamics. (a) Schematic of the experimental sequence. The local interaction is turned on at time $t=0$, to generate spin squeezing in each ensemble. It is turned off at $t=t_{\rm rot}$, when an additional local rotation of the state by an angle $\theta$ is considered. The subsequent dynamics is a free evolution coupling the two ensembles. (b) Local spin squeezing as a function of the effective time $\tau$. In particular, $\tau = (\chi_{\rm loc}^G + \chi_{\rm cont}) t$ for $0 \leq t \leq t_{\rm rot}$ and $\tau = (\chi_{\rm loc}^G + \chi_{\rm cont}) t_{\rm rot} + \chi_{\rm nloc}^G (t-t_{\rm rot})$ for $t \geq t_{\rm rot}$. The black solid line corresponds to the spin squeezing obtained before $\tau_{\rm rot}$. The different colored lines, starting from $\tau_{\rm rot}$, correspond to the different orientations of the local spin squeezing: $\beta = 0$ (purple), $\beta = \pi/24$ (blue), and $\beta = \pi/2$ (red). The inset shows the corresponding state on the Bloch sphere after the rotation at time $\tau_{\rm rot}$. (c) Parameters ${\cal C}_{1}$ (dashed lines) and ${\cal C}_{2}$ (solid). Different colors correspond to the different orientations of the state at time $t_{\rm rot}$, as in panel (b), while the black lines correspond to the local case. The gray-shaded area is only accessible by bipartite entangled states. For both figures (b) and (c), we have taken $N = 1000$. The numerical methods used here are detailed in Appendix G.
• Figure 4: GIE dynamics in a network of 3 and 4 nodes. Quantities ${\cal C}_{1}$ (red solid) and ${\cal C}_{2}$ (red dashed) as a function of $\tau = \chi_{\rm nloc} t$. The protocol generalizes the one described in Fig \ref{['Fig1']} (a), with the initial state at time $\tau=0$ being a product of coherent spin states $\bigotimes_{j=1}^M \ket{CSS_x}_j$, for which ${\cal C}_{1}={\cal C}_{2}=1$. Thin lines are for $M=3$ BECs, while thick lines are for $M=4$ BECs. The gray region witnesses entanglement according to ${\cal C}_{1}, {\cal C}_{2}<1$. Insert: schematic of the network with 3 and 4 nodes. Each BEC is represented by a circle with two modes. In the numerics we have used $N= 1000$ in each ensemble. All the analytical formulae used to obtain these results are detailed in Appendix H.
• Figure 5: Extension to a network of 3 and 4 nodes of the GID experiment. (a) Evolution of the local (solid) and collective (dashed) squeezing, for $M=3$, $N= 1000$, following the protocol described in Fig \ref{['Fig2']}. Different colors correspond to different orientations of the local spin squeezing. Insets show the local squeezing angle for the different colors, and the geometric configuration for the $M=3$ network. (b) Evolution of the entanglement criteria ${\cal C}_{1}$ (solid) and ${\cal C}_{2}$ (dashed) following the same protocol. (c-d) Same figures as (a-b), but with $M=4$ BECs.