Abstract The precise estimation of small parameters is a challenging problem in quantum metrology. Here, we introduce a protocol for accurately measuring weak magnetic fields using a two-level magnetometer, which is coupled to two (hot and cold) thermal baths and operated as a two-stroke quantum thermal machine. Its working substance consists of a two-level system (TLS), generated by an unknown weak magnetic field acting on a qubit, and a second TLS arising due to the application of a known strong and tunable field on another qubit. Depending on this field, the machine may either act as an engine or a refrigerator. Under feasible conditions, determining this transition point allows to reduce the relative error of the measurement of the weak unknown magnetic field by the ratio of the temperatures of the colder bath to the hotter bath.
Abstract The precise estimation of small parameters is a challenging problem in quantum metrology. Here, we introduce a protocol for accurately measuring weak magnetic fields using a two-level magnetometer, which is coupled to two (hot and cold) thermal baths and operated as a two-stroke quantum thermal machine. Its working substance consists of a two-level system (TLS), generated by an unknown weak magnetic field acting on a qubit, and a second TLS arising due to the application of a known strong and tunable field on another qubit. Depending on this field, the machine may either act as an engine or a refrigerator. Under feasible conditions, determining this transition point allows to reduce the relative error of the measurement of the weak unknown magnetic field by the ratio of the temperatures of the colder bath to the hotter bath.
We identify and interpret the possible quantum thermal machine regimes with a transverse-field Ising model as the working substance. In general, understanding the emergence of such regimes in a many-body quantum system is challenging due to the dependence on the many energy levels in the system. By considering infinitesimal work strokes, we can understand the operation from equilibrium properties of the system. We find that infinitesimal work strokes enable both heat engine and accelerator operation, with the output and boundaries of operation described by macroscopic properties of the system, in particular the net transverse magnetization. At low temperatures, the regimes of operation and performance can be understood from quasiparticles in the system, while at high temperatures an expansion of the free energy in powers of inverse temperature describes the operation. The understanding generalises to larger work strokes when the temperature difference between the hot and cold reservoirs is large. For hot and cold reservoirs close in temperature, a sufficiently large work stroke can enable refrigerator and heater regimes. Our results and method of analysis will prove useful in understanding the possible regimes of operation of quantum many-body thermal machines more generally.
We show how one can implement a quantum heat machine by using two interacting trapped ions, in presence of a thermal bath. The electronic states of the ions act like a working substance, while the vibrational mode is modelled as the cold bath. The heat exchange with the cold bath is mimicked by the projective measurement of the electronic states. We show how such measurement in a suitable basis can lead to either a quantum heat engine or a refrigerator, which undergoes a quantum Otto cycle. The local magnetic field is adiabatically changed during the heat cycle. The performance of the heat machine depends upon the interaction strength between the ions, the magnetic fields, and the measurement cost. In our model, the coupling to the hot and the cold baths is never switched off in an alternative fashion during the heat cycle, unlike other existing proposals of quantum heat engines. This makes our proposal experimentally realizable using current tapped-ion technology.
We present a measurement-based quantum thermal machine that extracts work from the back-action of generalized quantum measurements whose working medium is a coupled two-level quantum system. Specifically, we derive universal optimization criteria for a three-stroke measurement-based engine cycle with coupled two-level system of Ising-like interaction as a working medium. Furthermore, we present two numerical algorithms to optimize the engine work extraction and enhance its performance. Our numerical results demonstrate: (i) efficiency peaks in the projective-measurement limit; (ii) symmetry breaking (detuning or weak coupling) enlarges the exploitable energy gap; and (iii) performance remains robust (>50% of optimum) under ∼10∘ feedback-pulse errors. The framework is platform-agnostic and directly implementable with current superconducting, trapped-ion, or NMR technologies, providing a concrete route to scalable, measurement-powered quantum thermal machines.
We study the performance of quantum sensors composed of four qubits arranged in different geometries for magnetometry and thermometry. The qubits interact via the transverse-field Ising model with both ferromagnetic and antiferromagnetic couplings, maintained in thermal equilibrium with a heat bath under an external magnetic field. Using quantum Fisher information, we evaluate the metrological precision of these sensors. For ferromagnetic couplings, weakly connected graphs (e.g., the chain graph, P_4) perform optimally in estimating weak magnetic fields, whereas highly connected graphs (e.g., the complete graph, K_4) excel at strong fields. Conversely, K_4 achieves the highest sensitivity for temperature estimation in the weak-field regime. In the antiferromagnetic case, we uncover a fundamental trade-off dictated by spectral degeneracy: configurations with non-degenerate energy spectra - such as the pan-like graph (three qubits in a triangle with the fourth attached) - exhibit strong magnetic field sensitivity due to their pronounced response to perturbations. In contrast, symmetric structures like the square graph, featuring degenerate energy levels (particularly ground-state degeneracy), are better suited for precise thermometry. Notably, our four-qubit sensors achieve peak precision in the low-temperature, weak-field regime. Finally, we introduce a spectral sensitivity measure that quantifies energy spectrum deformations under small perturbations, providing a simple heuristic indicator of metrological sensitivity.
ABSTRACT Quantum sensors are devices that utilize quantum systems' properties to achieve high‐precision measurements of physical quantities, such as magnetic fields and temperature. The metrological precision of quantum sensors refers to their ability to provide accurate and reliable measurements with minimal uncertainty. In this paper, we propose a quantum sensor consisting of multiple atoms arranged in a spin chain, exposed to a thermal bath at a temperature and an external weak magnetic field with a magnetic‐field gradient for quantum thermometry and magnetometry. Employing the Hilbert‐Schmidt speed (HSS), a popular measure of quantum statistical speed to quantify precision limits in parameter estimation theory, we have investigated in detail how the magnetic‐field gradient and spin chain position influence the quantum sensor's metrological capabilities. We show that the performance and precision of our quantum sensor depend on the spin chain position and the magnetic field and its gradient; increasing these factors' values reduces estimation precision and weakens quantum sensor performance. Finally, increasing the number of atoms in a quantum sensing structure improves the accuracy of temperature and field estimation.
We propose a thermodynamic cycle similar to the Otto cycle with a two-qubit working substance such that one of its isochoric strokes is provided by a controllable quantum measurement. We show that the measurement device can affect the working substance in four possible ways, as well as the energy exchanges between the working substance and an external work source during the isentropic strokes take place in two possible ways. By these considerations, we obtain eight thermal machines, each of which works individually in engine, accelerator, or refrigerator modes. We show that, for half of the obtained thermal machines, the applied quantum measurement causes their respective performances to go higher than the corresponding regular Otto limits, while the remainder go lower than the mentioned limits. Interestingly, it is shown that the performance of each thermal machine explicitly depends on the strength parameters of applied measurement, which in turn are related to coupling constants of the working substance and measurement device. These findings highlight that the role of generalized measurements as nonequilibrium quantum resources is not always constructive in powering the thermal machines.