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arXiv:2112.13765 · Physical Review A

Designing refrigerators in higher dimensions using quantum spin models

Tanoy Kanti Konar, Srijon Ghosh, Amit Kumar Pal, Aditi Sen

Published Mar 2, 2023• 11 citations
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Abstract

We design quantum refrigerators based on spin-$j$ quantum $XYZ$ and bilinear-biquadratic models with individual spins attached to bosonic thermal baths. By considering both local and global master equations, we illustrate an enhancement in the performance of the refrigerators with an increase in the spin dimension irrespective of the choice of the spin models. To assess the performance of the refrigerators, we introduce a distance-based measure to quantify the local temperature of a particle with arbitrary spin quantum number $j$. Interestingly, we find that the local temperature quantifier, defined via minimizing the distance between a spin-$j$ thermal state and the evolved state of the spin-$j$ particle in the steady state, coincides with the population-based definition of local temperature known in the literature for spin-$\frac{1}{2}$ particles. Moreover, we demonstrate that the qualitative behavior of the distance-based local temperature is independent of the choice of the distance measure by comparing the trace distance, Uhlmann's fidelity, and relative entropy distance. We further observe by computing a local master equation that the quantum refrigerator consisting of a spin-$1/2$ and a spin-$j$ particle can lead to a lower local temperature compared to a refrigerator with two identical spin-$j$ particles following the $XYZ$ interactions.

DOI
10.1103/physreva.107.032602

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We design quantum refrigerators based on spin-jjj quantum XYZXYZXYZ and bilinear-biquadratic models with individual spins attached to bosonic thermal baths. By considering both local and global master equations, we illustrate an enhancement in the performance of the refrigerators with an increase in the spin dimension irrespective of the choice of the spin models. To assess the performance of the refrigerators, we introduce a distance-based measure to quantify the local temperature of a particle with arbitrary spin quantum number jjj. Interestingly, we find that the local temperature quantifier, defined via minimizing the distance between a spin-jjj thermal state and the evolved state of the spin-jjj particle in the steady state, coincides with the population-based definition of local temperature known in the literature for spin-12\frac{1}{2}21​ particles. Moreover, we demonstrate that the qualitative behavior of the distance-based local temperature is independent of the choice of the distance measure by comparing the trace distance, Uhlmann's fidelity, and relative entropy distance. We further observe by computing a local master equation that the quantum refrigerator consisting of a spin-1/21/21/2 and a spin-jjj particle can lead to a lower local temperature compared to a refrigerator with two identical spin-jjj particles following the XYZXYZXYZ interactions.
We explore a small quantum refrigerator in which the working substance is made of paradigmatic nearest-neighbor quantum spin models, the XYZXYZXYZ and the XYXYXY model with Dzyaloshinskii-Moriya interactions, consisting of two and three spins, each of which is in contact with a bosonic bath. We identify a specific range of interaction strengths which can be tuned appropriately to ensure a cooling of the selected spin in terms of its local temperature in the weak-coupling limit. Moreover, we report that in this domain, when one of the interaction strengths is disordered, the performance of the thermal machine operating as a refrigerator remains almost unchanged instead of degrading, thereby establishing the flexibility of this device. However, to obtain a significant amount of cooling via ordered as well as disordered spin models, we observe that one has to go beyond the weak-coupling limit and compute the figures of merit by using global master equations.
Abstract We propose a model of two identical autonomous quantum refrigerators that interact through their qubits coupled to cold thermal baths. We consider the same specific disposition for each refrigerator related to a three-level system coupled to different thermal baths and, in turn, to a qubit as the object to be cooled. By coupling the two systems, we show that under certain conditions on the model parameters, the devices can operate in the regime of an independent autonomous quantum refrigerator or be subject to an adjustment of the thermodynamic performances. We further investigate the correlations induced by the interaction between the two refrigerators, and illustrate their behavior with respect to their thermodynamic performances. We have found that adjusting the different interaction constants of the model can lead to an evolution of the quantum correlations of the two machines that is inversely related to the behavior of the thermodynamic quantities of both refrigerators. However, this is not necessarily the case when the temperatures of the model are considered as the reference parameters.
We examine a quantum absorption refrigerator that comprises three qubits, each of which is connected with a separate spin-star environment, with the three qubit-bath units coupled through an effective six-body interaction. The refrigerator exhibits the feature of transient cooling, i.e., lowering of the temperature of the first qubit in sufficiently small timescales, rather than steady-state refrigeration. A key advantage of our model is that the symmetries of the Hamiltonian enable a semi-analytic solution of the reduced density matrices of the refrigerator qubits, even in the presence of a large number of environmental spins. We derive the condition for autonomous refrigeration and analyze how the optimal cold-qubit temperature scales with the number of bath spins. We find a power-law scaling towards a constant asymptotic value. We also find the scaling of the minimum time required for optimal cooling as a function of the number of bath spins. Furthermore, we quantify the non-Markovianity of the cold-qubit dynamics using a restricted Breuer-Laine-Piilo information-backflow measure and observe that stronger backflow correlates with lower transient minimum temperatures across the sampled parameter regime. The transient-cooling performance is found to be robust under broad parameter variations. Compared to a conventional Markovian three-qubit refrigerator, the CSQAR achieves lower cold-qubit temperatures on shorter timescales. We further analyze the heat currents associated with the three qubits and their respective baths.
We show that quantum absorption refrigerators, which have traditionally been studied as of three qubits, each of which is connected to a thermal reservoir, can also be constructed by using three qubits and two thermal baths, where two of the qubits, including the qubit to be locally cooled, are connected to a common bath. With a careful choice of the system, bath, and qubit-bath interaction parameters within the Born-Markov and rotating-wave approximations, one of the qubits attached to the common bath achieves a cooling in the steady state. We observe that the proposed refrigerator may also operate in a parameter regime where no or negligible steady-state cooling is achieved, but there is considerable transient cooling. The steady-state temperature can be lowered significantly by an increase in the strength of the few-body interaction terms existing due to the use of the common bath in the refrigerator setup. The proposed refrigerator built with three qubits and two baths is shown to provide steady-state cooling for both Markovian qubit-bath interactions between the qubits and canonical bosonic thermal reservoirs, and a simpler reset model for the qubit-bath interactions.
We develop a method to transform a collection of higher-dimensional spin systems from the thermal state with a very high temperature of a local spin-s Hamiltonian to a low-lying energy eigenstate of the same. The procedure utilizes an auxiliary system, interactions between all systems, and appropriate projective measurements of arbitrary rank performed on the auxiliary system. We refer to this process as subspace cooling. The performance of the protocol is assessed by determining the fidelity of the target state with the output one and the success probability of achieving the resulting state. For this analysis, spin-s XXZ and bilinear biquadratic models are employed as the evolving Hamiltonian. We demonstrate that in both scenarios, unit fidelity can be attained after a reasonable number of repeated measurements and a finite amount of evolution time when all the systems are aligned in an open chain, but it fails when the interactions between the spin follow the star configuration. We report that the success probability increases with the rank of the projectors in the measurement for a fixed dimension and that for each dimension, there exists a range of interaction strength and evolution period for which the fidelity gets maximized. Even when some subsystems are in contact with the thermal bath, the method proves to be resistant to decoherence.
We design a measurement-based quantum refrigerator with an arbitrary number of qubits situated in a one-dimensional array that interact through variable-range XYXYXY interactions. The method proposed is based on repeated evolution followed by a measurement on the single accessible qubit, which has the potential to reduce the temperature in the rest of the subsystems, thereby demonstrating cooling in the device. The performance of the refrigerator is quantified by the fidelity of each local subsystem with the ground state of the local Hamiltonian and the corresponding probability of success. We identify system parameters, which include the interaction strength, range of interactions, initial temperature of each qubit, and the position of the measured qubit, so that the fidelities of all the unmeasured qubits approach unity with a nonvanishing probability. We observe that although strong interactions during evolution are required to achieve cooling, the long-range interactions typically deteriorate the performance of the refrigerator, which indicates that interactions are not ubiquitous. We report the scalability and the saturation property of the success probability with respect to the system size, which turns out to be independent of the involved system parameters and the number of repeated measurements. Furthermore, we show that the number of subsystems which can be cooled changes depending on the odd or even number of sites in the refrigerator. We argue that the distribution of entanglement between unmeasured qubits can give a possible explanation of the dependence of cooling process on the measured and unmeasured sites.