Active Quantum Reservoir Engineering: Using a Qubit to Manipulate its Environment

TL;DR

This work introduces active reservoir engineering as a framework where a controllable qubit repeatedly initializes and interacts with its environment to sculpt the environment’s state, rather than passively hosting decay channels. Central to the approach is the master equation for active reservoir engineering (MARE), which tracks both the system and a bath observable (magnetization m) and reveals a conserved quantity M = m + 1/2(↑-↓) that enables analytic, block-structured solutions and clear thermodynamic interpretation via observational entropy. The authors apply the framework to two platforms—superconducting qubits with TLS baths and quantum-dot spin qubits with nuclear baths—demonstrating bath cooling, narrowing of P_m, and enhanced coherence times, especially when exploiting initial-system correlations such as Θ and Ramsey correlations. Correlations emerge as a powerful resource, enabling sharper bath state control and formation of satellite peaks in P_m, with practical strategies to suppress unwanted features. Collectively, the work provides a versatile theoretical toolkit for designing and understanding active reservoir engineering in open quantum systems and highlights pathways to map bath properties and extend to quantum-Fokker-Planck-type descriptions.

Abstract

Quantum reservoir engineering leverages dissipative processes to achieve desired behavior, with applications ranging from entanglement generation to quantum error correction. Therein, a structured environment acts as an entropy sink for the system and no time-dependent control over the system is required. We develop a theoretical framework for active reservoir engineering, where time-dependent control over a quantum system is used to manipulate its environment. In this case, the system may act as an entropy sink for the environment. Our framwork captures the dynamical interplay between system and environment, and provides an intuitive picture of how finite-size effects and system-environment correlations allow for manipulating the environment by repeated initialization of the quantum system. We illustrate our results with two examples: a superconducting qubit coupled to an environment of two-level systems and a semiconducting quantum dot coupled to nuclear spins. In both scenarios, we find qualitative agreement with previous experimental results, illustrating how active control can unlock new functionalities in open quantum systems.

Figures (6)

• Figure 1: Actively engineering the environment of a qubit. (a) Cyclic operation implementing active reservoir engineering. After initializing the qubit (step 1), it may be correlated with the environment (step 1') before exchanging energy (step 2). These steps are repeated multiple times to manipulate the magnetization of the environment. (b) Active reservoir engineering using a superconducting qubit. Probability of the reservoir's magnetization before (red) and after (blue) active reservoir engineering are shown for two different protocols, together with the Bloch vectors for the state after initialization (arrows with open heads) and the direction of the magnetic field $\vec{B}_m$ (arrows with closed heads). Whenever the Bloch vector points in the same (opposite) direction as $\vec{B}_m$, the magnetization increases (decreases), as illustrated by the curved arrows. The middle panel shows how repeated preparation of the state $\ket{\downarrow_z}$ results in cooling of the environment. The bottom illustrates how a correlated state can be used for cooling the environment if it has positive $m$ and heating it for negative $m$, resulting in a narrowing of the magnetization. (c) Active reservoir engineering using a spin qubit. The middle illustrates how repeatedly preparing the state $\ket{\uparrow_z}$ results in narrowing of the magnetization, because the field $\vec{B}_m$ rotates. The bottom shows how a correlated state can be used to create periodic narrowing, resulting in a distribution with multiple peaks. Dotted red boxes denote attractors of the magnetization, where the state's Bloch vector is perpendicular to the field. Here $\Delta=0$. For the superconducting qubit, we consider $N_r = 10^3$ iterations, and for the spin qubit $N_r=10^6$.
• Figure 2: TLSs bath cooling. We consider a bath with $N=1000$ TLSs and, $A/\omega_S = 0.1/N$, $\beta\omega_S = 0.001, \kappa/\omega_S = 10^{-5}$. (a) Change in bath (blue) and system (yellow) magnetization in the first iteration of the cooling protocol. (b) Decrease in the bath variance for the first iteration. (c) $P_m$ at the end of $N_r$ repetitions initializing the $\ket{\downarrow_z}$ state in the system. Dotted lines indicate population inversion protocol, in which we initialize $\ket{\uparrow_z}$ at each cycle. (d) Von Neumann entropy of the system during different iterations of the cooling protocol (c). We observe that the maximum entropy achieved decreases, witnessing the cooling of the reservoir through the qubit. (e) Average magnetization by the end of each repetition. The yellow dashed curve represents the linear term in $N_r$, from Eq. \ref{['sc-avg-cycles']}. (f) Decrease in the variance, the yellow dashed is the linear term contribution in $N_r$ from Eq. \ref{['sc-var-cycles']}. (g) As the bath cools, $P_m$ peaks at less degenerate values of $m$, decreasing the observational entropy. The change in observational entropy contains relevant contributions from the change in the average Boltzmann entropy. (h) Decrease in the Shannon entropy of the bath due to narrowing of the distribution.
• Figure 3: TLSs reservoir engineering using correlated states. We consider $N=1000,~A/\omega_S = 0.1/N, ~\beta=0, \kappa/\omega_S = 10^{-5},$ and $\alpha = 2\pi/20,~ \varphi = \pi/2$ for the Ramsey state. (a) $P_m$ at after $N_r =100$ repetitions using each correlated state. We observe optimal narrowing for $\rho^\Theta$ and formation of satellite peaks in integer multiples of $20$ for $\rho^R$. (b) Angle between the initial Ramsey state's field and $\vec{B}_m$. The stars denote the points at which $\theta_m = \pi/2$, where peaks in $P_m$ are expected. (c) Shannon entropy of the bath by the end of each repetition. For the $\rho^\Theta$ we observe a huge decrease, and for $\rho^R$ the presence of peaks limits the decrease of entropy. (d) The change of observational entropy is dominated by the Shannon entropy since the distribution is always centered at $m=0$.
• Figure 4: Narrowing of nuclear-bath with uncorrelated states. We use nominal values for the electron spin of a GaAs quantum-dot from Nguyen2023, listed in Table \ref{['t:sq-exp-values']}. In (a--d) we simulate a single cycle of driving Rabi oscillations at Hartmann-Hahn resonance initializing state $\ket{\uparrow_z}$ in the qubit. (a) Slight decrease of the standard deviation during a single interaction cycle. (b) The von Neumann entropy of the system tends to the maximum value, but exhibits oscillations at short time scales (see inset) due to the development of sizeable system-bath correlations. In (c--f) we simulate the repeated process of narrowing the bath up to $N_r=10^7$ repetitions. We consider reseting the spin after the system-bath compound reach the steady state, $\Gamma_\text{avg} t_c \gg 1$ and at finite time, $\Gamma_\text{avg} t_c =3.42\times 10^{-5}$ [vertical-dash in (a/b)], corresponding to $t_c=1000$ ns. (c) $P_m$ after narrowing. (d) Standard deviation at the end of each cycle. (e) Sizeable decrease in the Shannon entropy of $P_m$. (f) The change in observational entropy is dominated by the change in Shannon entropy.
• Figure 5: Active reservoir engineering with Ramsey-correlated states. Fixed parameters from Table \ref{['t:sq-exp-values']} and sensing time, $|A_{\rm c}| \tau = 1/100$. (a) Spiked distribution after $10^6$ repetitions and initial distribution. (b) $\theta_m$ is the angle between $\vec{B}_m$ and the Bloch vector of $\ket{\Psi^R_m}$. The peaks in (a) are formed every time the angle is $\pi/2$. (c) Decrease in observational entropy. (d) The decrease in observational entropy is dominated by the decrease in Shannon entropy of the bath. Although many peaks remain, the uncertainty on the value of $m$ reduces.