Quantum measurement spintronic engine powered by quantum coherence enhanced by bosonic catalysis

Quantum information engines can advantageously convert information into work, usually through a feedback mechanism operated by a Maxwell demon. In principle, this opens prospects of autonomous machines that challenge the classical second law of thermodynamics. Yet prior research, which typically tests a given concept in isolation, usually makes strong assumptions on turning on/off internal interactions or the connection to the baths, which has fueled skepticism. Meanwhile, several recent experiments describe an autonomous spintronic engine that purportedly combines several concepts using inherent electronic interactions to harvest spin fluctuations and quantum correlations. So far, no full-fledged theory has described it. In this article, we address these several interdisciplinary shortcomings by presenting a theory for a solid-state, fully electronic spintronic quantum information engine that converts the energy of quantum coherence into useful electrical work thanks to quantum measurements. Our simple two-stroke engine operates on two correlated atomic, single-spin quantum dots (QDs) that are connected in series with two ferromagnetic electrodes. The ultrafast measurement stroke destroys the quantum coherence stored within the QDs and projects the system into a local high-energy separated state. The resulting energy is then released into the leads as electrical current when the partial thermalizing stroke restores quantum coherence. Using a master equation approach, we show that a robust steady-state current can flow between the electrodes despite the absence of a tunneling path and at thermal equilibrium. Our model proves the feasibility of harvesting quantum fluctuations by the measurement back-action through the on-site Coulomb interaction, and can reproduce experimental output power levels if bosons mediate this transport in the presence of an additional nonequilibrium resource. This can for instance be accomplished by the spin bias that is inherently generated by the ferromagnetic electrodes. Our work raises the prospect of measuring the “spintronic temperature” of the leads using magnetization dynamics experiments, while the model's underpinnings of built-in, materials-inherent measurements of quantum information can in principle be adapted to describe fundamental quantum processes in chemistry and biology.

Figure 1

Illustration of the model quantum spintronic engine, featuring two quantum dots trapped in series between two ferromagnetic leads in an antiparallel configuration with fully spin-polarized interactions. Blue/red levels represent spin $\downarrow /\uparrow$ energy levels. Green double arrows represent the magnetic couplings; yellow double arrows, capacitive couplings; and black arrows, tunnel couplings.

Figure 2

Illustration of the engine cycle. (a) Energy-entropy illustration of the engine cycle. The unselective measurement stroke (straight blue arrows) instantly projects the thermalized steady state (red) onto several possible separated states (green) with higher entropy and potentially higher energy. The thermalizing strokes (curved black arrows) reset the system to the steady state while allowing for work extraction on average. (b) Illustration of the population of the energy levels during the two phases of the cycle. In the thermalized state, coherence energy is stored in the system; then the measurement stroke projects the right QD, destroying the superposition. The excessive energy of this localized ground state is then dissipated into the baths during the thermalizing stroke which restores coherence. In a solid-state implementation [19, 20], these strokes reflect the inherent electronic interactions between a pair of exchange-coupled paramagnetic atoms, and with fully spin-polarized leads such as the ferromagnet/molecule interface [38].

Figure 3

Simulation results of $\mathrm{\Delta }E$ for (a) ${\rho }_0=|\downarrow \downarrow \rangle \langle \downarrow \downarrow |$ and (b) ${\rho }_0=\frac{1}{2}|\uparrow \downarrow \rangle \langle \uparrow \downarrow |+\frac{1}{2}|\downarrow \downarrow \rangle \langle \downarrow \downarrow |$. The corrected perturbative results (orange) derived in Sec. 2 of the Supplemental Material [36], and the numerically calculated solution at 4 ps (blue) are shown. Here ${\epsilon }_{\downarrow }=8,{\epsilon }_{\uparrow }=-3,{\epsilon }_R=1,J=8,U=1000,\gamma =0.1,{\mathcal{T}}_L^{+}={\mathcal{T}}_L^{-}=0.1,{\mathcal{T}}_R^{-}={\mathcal{T}}_R^{+}=0.01$; i.e., the two leads are both at infinite temperature (all units may be taken in meV and justification for their values may be found in Sec. 4 of the Supplemental Material [36]).

Figure 4

Parameter pair dependence of the coherence energy $-\langle C\rangle$, calculated at 4 ps starting from the pure state ${\rho }_0=|\uparrow \downarrow \rangle \langle \uparrow \downarrow |$, with ${\epsilon }_{\downarrow }=8,{\epsilon }_{\uparrow }=-3,{\epsilon }_R=1,J=8,U=1000,\gamma =1000,{\mathcal{T}}_L^{+}={\mathcal{T}}_L^{-}={\mathcal{T}}_R^{+}={\mathcal{T}}_R^{-}=0.1$. (a) ${\mathcal{T}}_L^{+}={\mathcal{T}}_L^{-}$ versus $U$, (b) ${\mathcal{T}}_L^{+}={\mathcal{T}}_L^{-}$ versus ${\mathcal{T}}_R^{+}={\mathcal{T}}_R^{-}$, (c) ${\mathcal{T}}_R^{-}$ versus ${\mathcal{T}}_R^{+}$, (d) $U$ versus $\gamma$.

Figure 5

Simulation results of $\mathrm{\Delta }E$ for ${10}^6$ cycles when measuring the charge $Q$ of the left QD using (a) unselective and (b) selective measurements. The numerical calculation (orange) and the perturbative solution (blue) are shown. ${\epsilon }_{\downarrow }=8,{\epsilon }_{\uparrow }=-3,{\epsilon }_R=1,J=8,U=1000,\gamma =1000,{\mathcal{T}}_L^{+}={\mathcal{T}}_R^{+}={\mathcal{T}}_L^{-}={\mathcal{T}}_R^{-}=0.1$ (all units may be taken in meV), and with the initial condition ${\rho }_0=|\uparrow \downarrow \rangle \langle \uparrow \downarrow |$.

Figure 6

(a) Simulation results of $\mathrm{\Delta }E$ for ${10}^6$ cycles powered by a nonthermal bosonic bath when measuring the population $n_R$ in the case of two identical electrodes at thermal equilibrium with infinite temperature. Parameters are ${\epsilon }_{\downarrow }=8,{\epsilon }_{\uparrow }=-3,{\epsilon }_R=1,J=8,U=1000,\gamma =1000,{\mathcal{T}}_L^{+}={\mathcal{T}}_R^{+}={\mathcal{T}}_L^{-}={\mathcal{T}}_R^{-}=0.1$, and ${\mathrm{\Lambda }}^{+}=2{\mathrm{\Lambda }}^{-}=0.01$. (b) Color plot of $-\langle C\rangle$ calculated at 4 fs as a function of the bosonic coupling parameters ${\mathrm{\Lambda }}^{+}$ and ${\mathrm{\Lambda }}^{-}$, starting from the pure state ${\rho }_0=|\uparrow \downarrow \rangle \langle \uparrow \downarrow |$. All other parameters are the same as in (a).

Figure 7

Simulation of the engine with bosonic catalysis powered by a spin bias. (a) and (b) Simulation results of $\mathrm{\Delta }E$ for ${10}^6$ cycles powered by a spin bias bath when measuring the population $n_R$. Parameters are ${\epsilon }_{\downarrow }=8,{\epsilon }_{\uparrow }=-3,{\epsilon }_R=1,J=8,U=1000,\gamma =1000,{\mathcal{T}}_L^{+}=0.2,{\mathcal{T}}_R^{+}={\mathcal{T}}_L^{-}={\mathcal{T}}_R^{-}=0.1, {\mathrm{\Lambda }}^{+}=0.01$, and ${\mathrm{\Lambda }}^{-}=0.0101$. (c) Color plot of $-\langle C\rangle$ calculated for the steady state as a function of the bosonic coupling parameters ${\mathcal{T}}_L^{-}$ and ${\mathcal{T}}_L^{+}$, starting from the pure state ${\rho }_0=|\uparrow \downarrow \rangle \langle \uparrow \downarrow |$. All other parameters are the same as in (a). (d) Color plot of $-\langle C\rangle$ as a function of ${\mathcal{T}}_R^{-}$ and ${\mathcal{T}}_R^{+}$ with the same set of parameters.