Multi-mode cooling of a Bose-Einstein condensate with linear quantum feedback

TL;DR

This work develops a linear-quadratic-Gaussian (LQG) framework for measurement-based, multi-mode feedback cooling of a quasi-2D Bose-Einstein condensate under dispersive monitoring. It introduces a realistic derivative-current (cold-damping) control scheme actuated by a spatiotemporal optical potential, avoiding full quantum-state estimation while achieving ground-state cooling of many Bogoliubov modes. The analysis provides analytic steady-state cooling limits, optimal gains and filter bandwidths, and demonstrates via numerical simulations that ground-state cooling is achievable for dozens of modes with modest detection efficiency (η ≥ 1/9). The results offer practical guidance for experimental implementation and point to applications in quantum sensing and atomtronics where motional stabilization enhances coherence and performance.

Abstract

We theoretically investigate measurement-based feedback control over the motional degrees of freedom of an oblate quasi-2D atomic Bose-Einstein condensate (BEC) subject to continuous density monitoring. We develop a linear-quadratic-Gaussian (LQG) model that describes the multi-mode dynamics of the condensate's collective excitations under continuous measurement and control. Crucially, the multi-mode cold-damping feedback control we consider uses a realistic state-estimation scheme that does not rely upon a particular model of the atomic dynamics. We present analytical results showing that collective excitations can be cooled to below single-phonon average occupation (ground-state cooling) across a broad parameter regime and identify the conditions under which the lowest steady-state phonon occupation is asymptotically achieved. Further, we develop multi-objective optimization methods that explore the trade-off between cooling speed and the final energy of the cloud and provide numerical simulations demonstrating the ground-state cooling of the lowest ten motional modes above the condensate ground state. Our investigation provides concrete guidance on the feedback control design and parameters needed to experimentally realize a feedback-cooled BEC.

Figures (6)

• Figure 1: Schematic of the feedback control loop. (a) Non-destructive imaging of the atomic density is performed by continuous illumination of the atomic cloud with a coherent light field far detuned from resonance, i.e. $\Delta\gg \Gamma$. (b) Measured fluctuations of the atomic density are decomposed onto a finite basis of mode functions, $m_j(\mathbf{r})$ [see Eq. \ref{['eq:linearizedMeasurementOperator']}]. (c) The projected measurement current associated with the $j$th mode, $y_j(t)$, is constructed from the time series of many measurements (red). High-frequency noise is removed from the measurement current associated with each mode, using a low-pass filtered with bandwidth $\Omega_j$ (grey shading indicates a temporal averaging window of $1/\Omega_j$). (d) The derivative of low-pass-filtered measurement current, $I_j(t)$, is used to inform the control, which is actuated by a high-bandwidth spatiotemporal optical-dipole potential (green). Specifically, we consider a multi-mode cold-damping protocol realised by choosing the spatiotemporal control potential to be linear combinations of the Bogoliubov $f^+_j(\mathbf{x})$ functions (see Eq. \ref{['eq:SpatiotemporalPotential_ModeBasis']}) with gain proportional to the time derivative of the filtered measurement current, $\dot{I}_j(t)$.
• Figure 2: Collective mode structure for the first ten excitations of a quasi-2D cylindrically symmetric BEC, with $N_0=10^5$ condensate atoms and $g_{\rm 2D} = 0.035 E_0/l_0^2$. (a) Mode functions describing density fluctuations, $m_j(x)$, c.f. Eq. \ref{['eq:linearizedMeasurementOperator']}. (b) Measurement induced couplings, $\mathcal{M}_{jk}$, for each mode (diagonals) and between modes (off-diagonals). Note that while diagonal terms dominate, there are non-negligible cross-mode couplings between modes $8$ and $9$.
• Figure 3: Steady-state fluctuation spectra of a continuously-monitored and feedback-cooled mode with frequency $\omega_j$, in the case of perfect detection ($\eta=1$). (a) Absolute value of the mechanical susceptibility, Eq. \ref{['eq:MechanicalSusceptDef']}, for $\tilde{\Gamma}_j=c_j\sqrt{\omega_j}=\omega_j/\Omega_j = 0.1$ (b-c). (b) Power spectral densities (PSDs) of each quadrature, for the same parameters as (a). (c) Quadrature PSDs for a highly non-thermal steady-state, associated with the parameters $\tilde{\Gamma}_j=0.4$, $\omega_j/\Omega_j=0.35$, and $c_j\sqrt{\omega_j}=0.75$. For both sets of parameters, the shift in the effective resonance frequency due to the control is negligible, i.e. $\omega_j^{\rm eff}\approx \omega_j$.
• Figure 4: Steady-state phonon occupation of a single mode, $\bar{n}_j$, for a quantum efficiency of $\eta=0.5$ -- using Eq. \ref{['eq:MinPhononLimit']}, this sets the bound $\bar{n}_j\geq\bar{n}_{\rm min}\approx 0.08$. Empty spaces indicate parameter regimes of large steady-state phonon occupations, i.e. $\bar{n}_j>10$. In (a) the filter bandwidth is fixed as a function of $\Gamma_j$, by the analytical result, Eq. \ref{['eq:FBBandwidthOptimal']}. Similarly, in (b) the optimal control gain is chosen using Eq. \ref{['eq:ControlGainOptimal']} for each value of $\Gamma_j$. Dashed (orange) lines indicate the analytical results for (a) the optimal control gain, Eq. \ref{['eq:ControlGainOptimal']}, and (b) the optimal filter bandwidth, Eq. \ref{['eq:FBBandwidthOptimal']}.
• Figure 5: Pareto front characterizing the family of optimal solutions to the cost function, Eq. \ref{['eq:MultiobjectiveCostFunction_NoLoss']}, for a single mode of frequency $\omega_j$. Each point along the Pareto front represents the minimum of Eq. \ref{['eq:MultiobjectiveCostFunction_NoLoss']} for a particular weighting, $w$, for both perfect detection (black line) and imperfect detection (blue lines, decreasing opacity with $\eta$). For each value of $\eta<1$, the horizontal dashed lines give the theoretical minimum achievable phonon occupation $\bar{n}_{\rm min}$, given by Eq. \ref{['eq:MinPhononLimit']}. For $\eta=1$, $\bar{n}_{\rm min}=0$. The shaded region indicates $\bar{n}\leq 1$, i.e. the regime of ground-state cooling.