Universal cooling of quantum systems via randomized measurements

Abstract

Designing cooling protocols is believed to require knowledge of the system spectrum. In contrast, cooling in nature occurs whenever the system is coupled to a cold bath. How does nature know how to cool? A natural cold bath can be mimicked with a reservoir of "meter" qubits that are initialized in their ground state. We show that a quantum system can be cooled without knowledge of system details when system-meter interactions and meter splittings are chosen randomly. For sufficiently small interaction strengths and long interaction times, the protocol ensures that resonant energy-exchange processes, leading to cooling, dominate over heating. Effectively, the dynamics is then captured by the rotating-wave approximation, which we identify as the basic mechanism for robust and scalable cooling of complex quantum systems through generic, structure-independent protocols. This offers a versatile universal framework for controlling quantum matter far from equilibrium, in particular, for quantum computing and simulation.

Figures (8)

• Figure 1: Universal cooling via repeated, stochastically chosen interactions with meter qubits: Each constituent of the system is coupled to a meter qubit with coupling strength $\gamma$. The meter qubits are initialized in their ground states and interact sequentially with the system for time $t_M$ after which they are discarded.
• Figure 2: Cooling a two-level system: Steady-state energy $E_\text{S}(\rho_\infty)$ as a function of system-meter coupling strength $\gamma$ and interaction time $t_M$ with cuts along $t_M$ and $\gamma$ presented, respectively, in (a) and (b) by the corresponding (dotted, dashed, and solid) magenta and black curves. The steady-state energy follows the estimate \ref{['eq:es_estimate']} for $\omega_\text{S}\gg 1/t_M$ and $\omega_\text{S}\gg \gamma$, as indicated by the black curves.
• Figure 3: Dynamics of cooling a two-level system via repeated interactions: (a) System energy as a function of time for three different types of system-meter interaction corresponding to different levels of knowledge about the system. Stochastically averaged interactions (green) are used in the case of no knowledge, whereas $\sigma_x\tau_x$ (black) and $\sigma_+\tau_-$ (blue) couplings imply, respectively, knowledge of the quantization axis and knowledge of both the quantization axis and the gap sign. Note that in general the system attains a stroboscopic steady state, i.e, at $t=n t_M$ when the meter is reset (indicated by faint vertical lines). In between measurements, the system energy can still show oscillations. (b) Steady-state energy (black) and the ratio of contributions to the steady-state energy associated with the counter- and co-rotating interactions (red), displaying revivals of the co-rotating terms at integer multiples of $\nu t_M/2\pi$, where $\nu$ is an eigenfrequency of $H_\text{tot}$, Eq. \ref{['eq:tls_nu']}. Except for the close vicinity of these values, the RWA is valid since the co-rotating contributions to the system energy dominate over the counter-rotating ones. Hence, except for singular unfortunate choices of $t_M$, the system energy can be reduced, implying efficient cooling. Parameters for both panels are $\omega_\text{S}>0$, $\gamma=0.1\omega_\text{S}$, and $\omega_\text{M}=\omega_\text{S}$.
• Figure 4: Cooling a Heisenberg chain: (a-c) Steady-state energy as a function of system-meter interaction strength $\gamma$ and interaction time $t_M$ for $N=3$. As in the case of a two-level system (cf. Fig. \ref{['fig:tls_3panels']}), cooling works best for weak interactions and long interaction times, where it follows the estimate \ref{['eq:es_estimate']}. (d) Steady-state energy of the system as a function of interaction time $t_M$ for chain sizes $N=3,4,5,6$, all showing a scaling of the steady-state energy with $1/t_M$. The dashed lines indicate the lower bound on the energy obtainable by steering. Parameters are $\omega_\text{M}\in[0,1.1\left\lVert H_\text{S}\right\rVert]$ and $\gamma=4 \times10^{-3} J$ in (d). The averaging of the CPTP map was done using Monte Carlo integration. The error due to the finite number of samples is of the order of $10^{-6}$ and thus not visible in this plot. All panels are for translation-invariant chains with open boundary conditions.
• Figure 5: Implementation of the cooling protocol: The normalized energy expectation value of an $N=4$ anisotropic Heisenberg chain with $J_a= \{J, 1.2J, 0.7J\}$, cf. Eq. \ref{['eq:Anisotropic_Heisenberg_model']}, as a function of the time. (a) Dependence on the coupling strength $\gamma$: Weak coupling (with respect to the many-body gap $\Delta=2.134J$) guarantees cooling towards the ground state of the system (yellow line). The shaded regions depict the standard deviations around the mean values for the yellow and black curves; for the rest of the curves the standard deviations are omitted for clarity. The meter splittings are sampled from a box distribution with range from $0$ to $\omega_{\text{M},\max}=\left \Vert H_\text{S} \right \Vert$. (b) Sampling of the meter splittings with $\gamma=0.04\Delta$ where box distributions (blue) cover $0$ to $\omega_{\text{M},\max}$ and Half-Gaussian distributions are centered at $0$ with variance $\sigma$ and sampling only $\omega_\text{M}\geq0$ (black) : Cooling is possible irrespective of the precise width of the box except for very narrow boxes (lightest blue, with the horizontal dashed line indicating the steady state energy). For sufficiently broad box distributions, the width only affects the cooling rate. Full agnosticity can be ensured by Gaussian distributions as their tails cover the entire range of possible system transitions.