What is the minimum time required to take a temperature? In this paper, we solve this question for a large class of processes where temperature is inferred by measuring a probe (the thermometer) weakly coupled to the sample of interest, so that the probe's evolution is well described by a quantum Markovian master equation. Considering the most general control strategy on the probe (adaptive measurements, arbitrary control on the probe's state and Hamiltonian), we provide bounds on the achievable measurement precision in a finite amount of time, and show that in many scenarios these fundamental limits can be saturated with a relatively simple experiment. We find that for a general class of sample-probe interactions the scaling of the measurement uncertainty is inversely proportional to the time of the process, a shot-noise like behaviour that arises due to the dissipative nature of thermometry. As a side result, we show that the Lamb shift induced by the probe-sample interaction can play a relevant role in thermometry, allowing for finite measurement resolution in the low-temperature regime. More precisely, the measurement uncertainty decays polynomially with the temperature as $T\rightarrow 0$, in contrast to the usual exponential decay with $T^{-1}$. We illustrate these general results for (i) a qubit probe interacting with a bosonic sample, where the role of the Lamb shift is highlighted, and (ii) a collective superradiant coupling between a $N$-qubit probe and a sample, which enables a quadratic decay with $N$ of the measurement uncertainty.
The rapidly developing quantum technologies and thermodynamics have put forward a requirement to precisely control and measure the temperature of microscopic matter at the quantum level. Many quantum thermometry schemes have been proposed. However, precisely measuring low temperature is still challenging because the obtained sensing errors generally tend to diverge with decreasing temperature. Using a continuous-variable system as a thermometer, we propose non-Markovian quantum thermometry to measure the temperature of a quantum reservoir. A mechanism to make the sensing error $δT$ scale with …
We investigate the sensing performance of a single-qubit quantum thermometer within a non-Markovian dynamical framework. By employing an exactly numerical hierarchical equations of the motion method, we go beyond traditional paradigms of the Born-Markov theory, the pure dephasing mechanism, and the weak-coupling approximation, which were commonly used in many previous studies of quantum thermometry. We find (i) the non-Markovian characteristics may boost the estimation efficiency, (ii) the sensitivity of quantum thermometry can be effectively optimized by engineering the proportions of differe…
Relativistic Quantum Thermometry in AdS Spacetime via Non-Markovian Temperature Sensing
Faculty of Sciences, Mohammed V University in Rabat（拉巴特穆罕默德五世大学理学院）；Centre of Physics and Mathematics, CPM, Faculty of Sciences, Mohammed V University in Rabat（拉巴特穆罕默德五世大学理学院物理与数学中心）
Nonequilibrium thermometry via an ensemble of initially correlated qubits
Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche (CNR-INO)（意大利国家光学研究所）；Department of Physics and Astronomy, Università di Firenze（佛罗伦萨大学物理与天文学系）；European Laboratory for Non-linear Spectroscopy, Università di Firenze（佛罗伦萨大学欧洲非线性光谱学实验室）
The dynamics of two spins coupled to bosonic baths at different temperatures is studied. The analytical solution for the reduced density matrix of the system in the Markovian and Post-Markovian case with exponential memory kernel is found. The dynamics and temperature dependence of spin-spin entanglement is analyzed.
The thermometry precision of a sample is a question of both fundamental and technological importance. In this paper, we consider a ring-structure system as our probe to estimate the temperature of a bath. Based on the Markovian master equation of the probe, we calculate the quantum Fisher information (QFI) of the probe at any time. We find that for the thermal equilibrium thermometry, the ferromagnetic structure can measure a lower temperature of the bath with a higher precision compared with the non-structure probe. While for the dynamical thermometry, the antiferromagnetic structure can make…
In this work, we investigate the non-Markovianity, quantum Fisher information (QFI) and quantum coherence of a qubit in a nonequilibrium environment and have obtained the expressions of QFI and quantum coherence as well as their relationship. We have also discussed in detail the influences of the different noise parameters on these quantum sffects. The results show that the suitable parameters of the nonequilibrium environment can retard the QFI and quantum coherence in both Markovian and non-Markovian regions. In addition, the smaller memory effects and the larger the jumping rate, the greate…
Precise thermometry for quantum systems is important to the development of new technology, and understanding the ultimate limits to precision presents a fundamental challenge. It is well known that optimal thermometry requires projective measurements of the total energy of the sample. However, this is infeasible in even moderately-sized systems, where realistic energy measurements will necessarily involve some coarse graining. Here, we explore the precision limits for temperature estimation when only coarse-grained measurements are available. Utilizing tools from signal processing, we derive t…
One of the fundamental issues in the field of open quantum systems is the classification and quantification of non-Markovianity. In the contest of quantity-based measures of non-Markovianity, the intuition of non-Markovianity in terms of information backflow is widely discussed. However, it is not easy to characterize the information flux for a given system state and show its connection to non-Markovianity. Here, by using the concepts from thermodynamics and information theory, we discuss a potential definition of information flux of an open quantum system, valid for static environments. We pr…
Decoherence often happens in the quantum world. We try to utilize quantum dephasing to build an optimal thermometry. By calculating the Cram$\acute{e}$r-Rao bound, we prove that the Ramsey measurement is the optimal way to measure the temperature for uncorrelated particles. Using the optimal measurement, the metrological equivalence of product and maximally entangled states of initial quantum probes that always holds. However, using Ramsey measurement, the metrological equivalence only holds in special situation. Contrary to frequency estimation, the quantum limit can be surpassed under the ca…
We theoretically study the non-Markovian dynamics of qubit systems coupled to nonequilibrium environments with nonstationary and non-Markovian statistical properties. The reduced density matrix of the single qubit system satisfies a closed third-order differential equation with all the higher-order environmental correlations taken into account and the reduced density matrix of the two qubit system can be expressed as the Kraus representation in terms of the tensor products of the single qubit Kraus operators. We derive the relation between the entanglement and nonlocality of the two qubit syst…
Understanding heat transfer between a quantum system and its environment is of undisputed importance if reliable quantum devices are to be constructed. Here, we investigate the heat transfer between system and bath in non-Markovian open systems in the process of adiabatic speedup. Using the quantum state diffusion equation method, the heat current, energy current, and power are calculated during free evolution and under external control of the system. While the heat current increases with increasing system-bath coupling strength and bath temperature, it can be restricted by the non-Markovian n…
The time reversal of a completely-positive, nonequilibrium discrete-time quantum Markov evolution is derived via a suitable adjointness relation. Space-time harmonic processes are introduced for the forward and reverse-time transition mechanisms, and their role for relative entropy dynamics is discussed.
We consider a quantum harmonic oscillator coupled to a general nonequilibrium environment. We show that the decoherence factor can be expressed in terms of a measurable effective temperature, defined via a generalized fluctuation-dissipation relation. We further propose a simple experimental scheme to determine the time-dependent effective temperature in a linear Paul trap with engineered reservoirs. Our formalism allows quantitative description of nonequilibrium decoherence in the presence of an arbitrary number of non-Markovian noise sources in a unified manner.
We investigate dynamics of an optomechanical system under the Non-Markovian environment. In the weak optomechanical single-photon coupling regime, we provide an analytical approach fully taking into account the non-Markovian memory effects. When the cavity-bath coupling strength crosses a certain threshold, an oscillating memory state for the classical cavity field (called bound state) is formed. Due to the existence of the non-decay optical bound state, a nonequilibrium optomechanical thermal entanglement is preserved even without external driving laser. Our results provide a potential usage …
We theoretically study the geometric effect of quantum dynamical evolution in the presence of a nonequilibrium noisy environment. We derive the expression of the time dependent geometric phase in terms of the dynamical evolution and the overlap between the time evolved state and initial state. It is shown that the frequency shift induced by the environmental nonequilibrium feature plays a crucial role in the geometric phase and evolution path of the quantum dynamics. The nonequilibrium feature of the environment makes the length of evolution path becomes longer and reduces the dynamical decohe…
A systematic approach to the non-Markovian quantum dynamics of open systems is given by the projection operator techniques of nonequilibrium statistical mechanics. Combining these methods with concepts from quantum information theory and from the theory of positive maps, we derive a class of correlated projection superoperators that take into account in an efficient way statistical correlations between the open system and its environment. The result is used to develop a generalization of the Lindblad theory to the regime of highly non-Markovian quantum processes in structured environments.
We propose a method to study the thermodynamic behaviour of small systems beyond the weak coupling and Markovian approximation, which is different in spirit from conventional approaches. The idea is to redefine the system and environment such that the effective, redefined system is again coupled weakly to Markovian residual baths and thus, allows to derive a consistent thermodynamic framework for this new system-environment partition. To achieve this goal we make use of the reaction coordinate mapping, which is a general method in the sense that it can be applied to an arbitrary (quantum or cl…
We investigate the transient dynamics of photon statistics through two-time correlation functions for optical fields. We find that the transient correlations at different time t yield a smooth transition from antibunching to bunching photon statistics in the weak system-environment coupling regime. In the strong-coupling regime, the two-time correlations exhibit bunching-antibunching oscillations that persists both in the transient process and in the steady-state limit. The photon bunching-antibunching oscillations is a manifestation of strong non-Markovian dynamics, where the system remains i…
Specific nonequilibrium states of the quantum harmonic oscillator described by the Lindblad equation have been hereby suggested. This equation makes it possible to determine time-varying effects produced by statistical operator or statistical matrix. Thus, respective representation-varied equilibrium statistical matrixes have been found. Specific mean value equations have been found and their equilibrium solutions have been obtained.
We develop a general theory describing the thermodynamical behavior of open quantum systems coupled to thermal baths beyond perturbation theory. Our approach is based on the exact time-local quantum master equation for the reduced open system states, and on a principle of minimal dissipation. This principle leads to a unique prescription for the decomposition of the master equation into a Hamiltonian part representing coherent time evolution and a dissipator part describing dissipation and decoherence. Employing this decomposition we demonstrate how to define work, heat, and entropy production…
Quantum probes are atomic-sized devices mapping information of their environment to quantum mechanical states. By improving measurements and at the same time minimizing perturbation of the environment, they form a central asset for quantum technologies. We realize spin-based quantum probes by immersing individual Cs atoms into an ultracold Rb bath. Controlling inelastic spin-exchange processes between probe and bath allows mapping motional and thermal information onto quantum-spin states. We show that the steady-state spin-population is well suited for absolute thermometry, reducing temperatur…
The nonequilibrium dynamics of a small quantum system coupled to a dissipative environment is studied. We show that (1) the oscillatory dynamics close to a coherent-to-incoherent transition is surprisingly different from the one of the classical damped harmonic oscillator and that (2) non-markovian memory plays a prominent role in the time evolution after a quantum quench.
We present a general scheme to map correlated nonequilibrium quantum impurity problems onto an auxiliary open quantum system of small size. The infinite fermionic reservoirs of the original system are thereby replaced by a small number $N_B$ of noninteracting auxiliary bath sites whose dynamics is described by a Lindblad equation. Due to the presence of the intermediate bath sites, the overall dynamics acting on the impurity site is non-Markovian. With the help of an optimization scheme for the auxiliary Lindblad parameters, an accurate mapping is achieved, which becomes exponentially exact up…
We address the real-time dynamics of lattice quantum spin models coupled to single or multiple Markovian dissipative reservoirs using the method of closed hierarchies of correlation functions. This approach allows us to solve a number of quantum spin models exactly in arbitrary dimensions, which is illustrated explicitly with two examples of driven-dissipative systems. We investigate their respective nonequilibrium steady states as well as the full real-time evolution on unprecedented system sizes. Characteristic time scales are derived analytically, which allows us to understand the nontrivia…
This work provides a detailed derivation of a generalized quantum Fokker-Planck equation (GQFPE) appropriate for photo-induced quantum dynamical processes. The path integral method pioneered by Caldeira and Leggett (CL) [Caldeira and Leggett, Physica A 121, 587 (1983)] is extended for a nonequilibrium influence functional, which has been obtained for general cases where the ground and the excited electronic state baths can be different. Both nonequilibrium and non-Markovian effects are accounted for consistently by expanding the paths in the exponents of the influence functional with respect t…
Entropy production plays a fundamental role in nonequilibrium thermodynamics to quantify the irreversibility of open systems. Its positivity can be ensured for a wide class of setups, but the entropy production rate can become negative sometimes. This is often taken as an indicator of non-Markovian dynamics. We make this link precise by showing under which conditions a negative entropy production rate implies non-Markovian dynamics and when it does not. This is established within a unified language for two setups: (i) the dynamics resulting from a coarse-grained description of a system in cont…
The breaking of the continuous time-translation symmetry manifests, in Markovian open quantum systems, through the emergence of non-stationary dynamical phases. Systems that display nonequilibrium transitions into these phases are referred to as time-crystals, and they can be realized, for example, in many-body systems governed by collective dissipation and long-ranged interactions. Here, we provide a complete analytical characterization of a boundary time-crystal phase transition. This involves exact expressions for the order parameter and for the dynamics of quantum fluctuations, which, in t…
The possibility of maintaining entanglement in a quantum system at finite, even high, temperatures -- the so-called `hot entanglement' -- has obvious practical interest, but also requires closer theoretical scrutiny. Since quantum entanglement in a system evolves in time and is continuously subjected to environmental degradation, a nonequilibrium description by way of open quantum systems is called for. To identify the key issues and the contributing factors that may permit `hot entanglement' to exist, or the lack thereof, we carry out a model study of two spatially-separated, coupled oscillat…
