Optimal Work Extraction from Finite-Time Closed Quantum Dynamics

TL;DR

The paper addresses how to maximize work extraction from finite-time closed quantum dynamics under a bounded control Hamiltonian. It introduces a Lie-algebraic control framework that reduces the optimization to a self-consistent, time-independent generator problem, yielding an exactly solvable protocol: a quick quench, time-invariant interaction-picture driving, and a final switch-off. A fundamental power–work trade-off is established, showing that maximum power occurs before the time needed to attain maximum work, highlighting the benefit of rapid protocols. The approach extends to many-body systems through Lie-algebra representation reduction, enabling efficient computation for SU(n)-Hubbard and Heisenberg-type models, and it generalizes to optimization of general observables and fidelity-related tasks. Overall, the work provides a saturable quantum speed limit for finite-time work extraction and a practical, scalable route to design optimal control under realistic constraints.

Abstract

Extracting useful work from quantum systems is a fundamental problem in quantum thermodynamics. In scenarios where rapid protocols are desired -- whether due to practical constraints or deliberate design choices -- a fundamental trade-off between power and efficiency is yet to be established. Here, we investigate the problem of finite-time optimal work extraction from closed quantum systems, subject to a constraint on the magnitude of the control Hamiltonian. We first reveal the trade-off relation between power and work under a general setup, stating that these fundamental performance metrics cannot be maximized simultaneously. We then introduce a general framework based on Lie algebras, which involves a wide range of control problems such as many-body control of the Heisenberg model and the SU(n)-Hubbard model. This framework enables us to reduce the optimal work extraction problem to an analytically or numerically efficiently tractable form. The resulting optimal protocol turns out to be remarkably simple: it suffices to use a time-independent control Hamiltonian in the interaction picture, determined by a nonlinear self-consistent equation. By exploiting the Lie-algebraic structure of the controllable terms, our approach is applicable to quantum many-body systems with an efficient numerical computation. Our results highlight the necessity of rapid protocols to achieve the maximum power and establish a theoretical framework for designing optimal work extraction protocols under realistic time constraints.

Figures (5)

• Figure 1: Schematic of the finite-time work extraction. The dynamics of a closed quantum system are governed by a time-dependent Hamiltonian of the form $H(t) = H_{\mathrm{c}}(t) + H_{\mathrm{u}}(t)$, where $H_{\mathrm{c}}(t)$ is the controllable part subject to optimization (green arrows), and $H_{\mathrm{u}}(t)$ denotes the uncontrollable part not subject to optimization (gray curves). Under the norm constraint $\|{ H_{\mathrm{c}}(t) }\|_{\mathrm{f}} \leq \omega$, the power attains its maximum strictly before the time $\mathcal{T}_{\ast}$ required for the maximum work extraction, indicating the trade-off relation between work and power. This relation highlights the importance of considering fast control processes with $T < \mathcal{T}_{\ast}$ to enhance the power (Proposition \ref{['thmM_TradeOff']}). We derive the optimal work extraction protocol under Lie-algebraic control, where $H_{\mathrm{c}}(t)$ is optimized over a subspace $\mathcal{V}$ closed under the commutation relation, and $H_{\mathrm{u}}(t)$ leaves $\mathcal{V}$ invariant, i.e., $i [{ H_{\mathrm{u}}(t) },{ \mathcal{V} }] \subseteq \mathcal{V}$ (Theorem \ref{['thmM_MainResult']}). The optimal protocol is remarkably simple and consists of three steps: (i) turn on the control via a quench to $H_{\mathrm{c}}(t=+0) = \omega \mathsf{H}$; (ii) steer the system under the control Hamiltonian $H_{\mathrm{c}}^{I}(t) = \omega \mathsf{H}$, which is time-independent in the interaction picture; and (iii) switch off the control at the final time $t=T$. Here, $\mathsf{H}$ is a time-independent, rescaled Hamiltonian that satisfies the closed-form self-consistent equation \ref{['eqM_SelfConsistentEq']}.
• Figure 2: Optimal work-extraction protocol within time $T$ under $\mathfrak{su}(2)$ control. We analytically find that the solution \ref{['eq_SU2OptimalHamiltonian']} is to rotate the projected density operator from $\rho^{\mathrm{i}}_\mathcal{V}$ to $\rho^{\mathrm{f}}_\mathrm{c} \coloneqq U_{\mathrm{c}}(T) \rho^{\mathrm{i}}_\mathcal{V} U_{\mathrm{c}}(T)^{\dagger}$ along the shortest possible path in the Bloch sphere so that $\rho^{\mathrm{f}}_\mathrm{c}$ becomes maximally aligned with $-H_{\mathcal{V}}^{\mathrm{f},I}\!(T)$. Here, we describe operators as vectors in the three-dimensional space of controllable operators.
• Figure 3: Optimal work extraction from a random three-level system under the full control ($\mathcal{V} \cong \mathfrak{su}(3)$). (a) Optimal amount of extracted work as a function of $T$ (the green curve). The blue curve shows the constant $C$ in Eq. \ref{['eqM_SelfConsistentEq']}, which is proportional to the derivative of the optimal work. It remains finite for $T \in (0, \ell/\omega)$ and vanishes as $T \to \ell/\omega$. (b) Minimum time $\mathcal{T}_{\mathrm{c}}(W_{\mathrm{c}})$ required to extract a given amount of work $W_{\mathrm{c}}$, which is obtained by inverting $W_{\mathrm{c}} = \mathcal{W}_{\mathrm{c}}(T)$. (c) Optimal power $\mathcal{P}_{\mathrm{c}}(T) \coloneqq \mathcal{W}_{\mathrm{c}}(T) / T$ of work extraction within time $T$, normalized by the reference power $\mathcal{P}_{\mathrm{c}\ast} \coloneqq \mathcal{W}_{\mathrm{c}\ast} / (\ell/\omega)$. Its maximum is attained strictly before the minimum time for the maximum work extraction. (d) Optimal power $\mathcal{P}_{\mathrm{c}}$ as a function of the extracted work $W_{\mathrm{c}}$. It decreases rapidly to $\mathcal{P}_{\mathrm{c}\ast}$ as $W_{\mathrm{c}} \to \mathcal{W}_{\mathrm{c}\ast}$.
• Figure 4: Optimal work extraction from an $\mathrm{SU}(6)$ Hubbard model with static flavor-dependent potentials $V_{\alpha}(t)\equiv V_{\alpha}$. The values of $V_{\alpha}$ and the flavor fractions $N_{\alpha}/N$ in Eq. \ref{['eq_SUnRhoProjection']} are randomly sampled as $V_{\alpha} \simeq \{0, 0.11, 0.34, 0.54, 0.84, 1.17\}$ and $N_{\alpha}/N \simeq \{0.022, 0.078, 0.15, 0.24, 0.25, 0.26\}$. The initial state is chosen as the one maximizing the energy expectation value. According to Eqs. \ref{['eq_SUnRhoProjection']} and \ref{['eqM_reducedRep_Work']}, the optimal work scales extensively with particle number for fixed flavor fractions and is independent of the lattice size $V$ and the parameters $J(t)$ and $U(t)$ in Eq. \ref{['hubbard']}. Moreover, $C$ has several cusps as a function of operational time $T$, where a discontinuous transition of the optimal operator $\mathsf{H}$ occurs.
• Figure 5: Weyl chambers for the $\mathfrak{su}(3)$ algebra, labeled $W_{1}, \dots, W_{6}$. The algebra has three roots up to sign, $\alpha_{1}, \alpha_{2}, \alpha_{3}$, with the relation $\alpha_{3} = \alpha_{1} + \alpha_{2}$, and their dual elements denoted by $T_{\alpha_{j}}$. The Weyl group is generated by reflections about the hyperplanes $\alpha_{j}(X) = 0$$(j=1,2,3)$ defined in Eq. \ref{['eqM_WeylGenerators']}. (One can confirm Proposition \ref{['prop_SimpleTransitivityOfWeylGroup']} by inspection.) Black dots show the Weyl group orbit of an element $X_{0} \in \mathfrak{h}$, which coincides with the intersection of the adjoint orbit of $X_{0}$ with $\mathfrak{h}$ (Proposition \ref{['prop_WeylChamberIntersection']}).

Theorems & Definitions (17)

• Proposition 1: Trade-off between power and work
• Theorem 1
• Proposition 2: Gradient of the cost function
• proof
• Lemma 1
• Definition 1: Simple Lie algebra
• Definition 2: Semisimple Lie algebra
• Definition 3: Cartan subalgebra
• Definition 4: Compact Lie algebra
• Definition 5: Root space decomposition for compact Lie algebras