Temperature sensing lies at the heart of the thermodynamics of quantum devices, which has many potential applications in microelectronics,^[1] biochemistry,^[2] and material science.^[3,4] How to achieve an accurate measurement of unknown temperature of a heat reservoir has attracted much attention over the last few years.^[5–10] The most simple and straightforward strategy is to contact the heat reservoir with an individual quantum probe, which acts as a thermometry. Due to the probe–reservoir coupling, the message of reservoir’s temperature is labeled in the quantum state of the probe and can be extracted out directly. Commonly, the steady state of the probe is utilized to draw out the information of temperature. The above scheme has been extensively studied in literature,^[5–15] for example, quantum dot is used as the temperature probe to accurately estimate the temperature of both fermionic^[12,13] and bosonic^[14,15] reservoirs.

Physically speaking, the probe and the heat reservoir comprise a so-called quantum dissipative system,^[16] which is a prominent physical model to simulate relaxation and decoherence in a noisy environment. In this sense, one can gain additional insight into the problem of quantum thermometry from a quantum dissipative system perspective. Unfortunately, the exact expression of the final steady-state of the system (probe), namely , is generally difficult to obtain. To derive a tractable result, a series of assumptions, for example, the weak system–reservoir coupling condition and the Markovian approximation, are employed in many previous studies.^[5,7,8] Under these restrictions, can be approximated as a canonical equilibrium state with respect solely to the system’s Hamiltonian, if the system–reservoir interaction time is sufficiently long. The above process, i.e., thermalization from a given initial state to a canonical equilibrium state, can be alternatively achieved by using the hypothesis of ergodicity in statistical mechanics, suggesting ensemble and time averages are equivalent. Consequently, many of previous temperature estimating schemes neglect the influence of system–reservoir correlations, which shall modify the canonical equilibrium state. Several studies^[17–21] have pointed out that the system–reservoir correlations play a significant role in strong system-reservoir coupling regimes, resulting in the occurrence of noncanonical distribution properties. Thus, a more precise temperature measuring scheme should take system–reservoir correlations into account.

On the other hand, in quantum metrology theory, the fundamental limitation of an arbitrary parameter estimation is bounded by the Cramér–Rao theorem: for a family of quantum state , carrying a unknown parameters θ, the estimating precision of θ from the quantum state is theoretically determined by^[22]

In many previous temperature sensing schemes,^[5–10] the temperature is read out from the Gibbs state of the system (probe), namely , where denotes the partition function and β ≡ T⁻¹ denotes inverse temperature (we set the Boltzmann constant k_B = 1 through this paper). The QFI of temperature with respect to the above Gibbs state can be exactly evaluated, and the result is given by^[5]

In this paper, we investigate the effect of system–reservoir correlations on the precision of temperature estimation in a spin-boson (SB) model with arbitrary spin sizes, in which a spin acts as the probe to measure the temperature of a bosonic reservoir, by making use of the reaction coordinate mapping method.^{[20,21,25–27]} The reaction coordinate mapping, which was originally proposed by Garg et al. in Ref. [25], can exactly convert the conventional SB model into an equivalent model without any approximations and has been widely applied in studies of electron transfer in biomolecules^[25] as well as heat transport in a nonequilibrium environment.^[28] As shown in Fig. 1, in this equivalent model, instead of interacting with the bosonic reservoir directly, the spin couples to an intermediate harmonic oscillator (IHO), which is embedded into a residual bosonic reservoir in turn. Compared with the conventional SB model, the “system’s part” of the mapped spin-IHO-boson (SIB) model includes system–environment correlations, because parts of environmental information (reflected by the spin-IHO coupling) are incorporated into the mapped system’s Hamiltonian. In Ref. [21], the authors have demonstrated that the robust system–reservoir correlations can be even persisted into equilibrium state of the mapped system. Therefore, the long-lived system–reservoir correlations lead to a clear department from the canonical equilibration dynamics of a quantum dissipative system.^[20,21] By tracing out the degrees of freedom of the IHO, a corrected steady state of the spin can be obtained. We use this corrected state to compute the QFI and explore the influence of system-reservoir correlations on the estimation precision.

Fig. 1.

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	Fig. 1. (a) Original scheme: a spin directly coupled to a bosonic reservoir. (b) Our scheme: a spin interacting with an IHO, which is in turn embedded in a residual bosonic reservoir.

The full SB model with arbitrary spin sizes considered in our temperature estimation scheme is given by (we adopt the unit ħ = 1 throughout the paper)

To take system–reservoir correlations into account, a transformation is applied to the original Hamiltonian to formulate an exactly equivalent SIB model. Following the procedure outlined by Garg et al. in Ref. [25], one can obtain the equivalent system, in which, the spin interacts with a collective coordinate of reservoir, known as an IHO^[26,27] or a reaction coordinate,^[20,21] which is in turn coupled to the residual bosonic reservoir. The Hamiltonian of this equivalent SIB model is given by^{[20,21,25–27]}

As pointed out in Refs. [20, 21, 26], the map from Eq. (4) to Eq. (5) does not involve the spin, i.e., the spin is unaffected by the transformation. Thus, one can replace the spin with a classical coordinate, which moves in a potential. By comparing the Fourier transformed equations of motion for the classical coordinate before and after the reaction coordinate mapping, one can connect the spectral density function to the original spectrum , see Appendix A for more details. If has an Ohmic form, i.e., , the spectral density function of its equivalent SB model is described by a Lorentzian structured spectrum, it reads^[20,21]

In Refs. [20, 21], the authors have studied the reduced dynamics (tracing out the degrees of freedom of the residual bosonic reservoir) of the SIB model by a numerically exact hierarchical equations of motion^[31,32] as well as a perturbative quantum master equation approach. Both methods showed that the final steady state can be faithfully described by the Gibbs state of the mapped spin-IHO Hamiltonian . Therefore, eliminating the degrees of freedom of the IHO, a corrected steady state of the spin can be extracted from the canonical equilibrium state of , namely,

The corrected steady state in Eq. (10) can be numerically evaluated. However, to obtain a more clear physical picture, we prefer an analytical approach. Motivated by Refs. [33–35], we find can be cast in a form of canonical equilibrium state with respect to an effective Hamiltonian, which preserves the same size with the original bare spin.

To this aim, we first separate into two parts , where and . The next key step is the employment of the following identity^[33–35]

The above effective Hamiltonian can be perturbatively computed as follows:^[34,35]

The QFI from the corrected steady state given in Eq. (13) can be analytically computed, and we express the result as

In Figs. 2(a) and 2(b), we plot versus/temperature T for spin-1 and spin-3/2 systems. In both cases, we find the QFI vanishes as T → 0 and decays exponentially at high temperatures. This result suggests there exists an optimal temperature that maximizes the value of QFI. In Figs. 2(c) and 2(d), we display as the function of Ω with different spin sizes. We find when Ω is small, which suggests the system–reservoir correlations destroy the estimation precision; as Ω increases, becomes positive, indicating the system-reservoir correlations may improve the value of QFI. Our result implies that the system–reservoir correlations play a complicated role in the estimation scheme, it effect (improving or decreasing the estimation precision) depends strongly on the details of the reservoir spectral density function. Due to the fact that , our result is consistent with the result reported in Ref. [38], in which the authors found the ratio of C_cor/C_bare, C_cor is the heat capacity of a harmonic oscillator strongly correlated with its surrounding environment, while C_bare is the heat capacity of a bare harmonic oscillator without system-reservoir coupling, is sensitive to parameters in the spectral density: by adjusting the coupling strength, the ratio of C_cor/C_bare can be larger or smaller than 1 at different temperatures. Moreover, as demonstrated in Ref. [39], parameters in the spectral density function are controllable by tuning experimental conditions, when simulating the dephasing process of trapped ultracold atoms. Thus, our result suggests that one can improve the estimation precision by employing the spectrum engineering technique.

Fig. 2.

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Fig. 2. (a) The total QFI with J = 1 is plotted as the function of temperature T: Ω = 50 (red dashed line), Ω = 75 (purple dotdashed line), and Ω = 100 (blue solid line). Other parameters are chosen as λ = 0.1Ω and ε = Ω. (b) The total QFI with J = 3/2 is plotted as the function of temperature T: Ω = 10 (red dashed line), Ω = 12 (purple dotdashed line), and Ω = 15 (blue solid line). Other parameters are chosen as λ = 0.1Ω and ε = Ω. (c) The corrected QFI at T = 50 is plotted as the function of Ω: J = 1 (blue solid line) and J = 3/2 (red dashed line). Other parameters are chosen as λ = 100 and ε = Ω. (d) The corrected QFI at T = 5 is plotted as the function of Ω: J = 1 (blue solid line) and J = 3/2 (red dashed line). Other parameters are chosen as λ = 30 and ε = Ω. In our calculation, temperature, the frequency of IHO, and QFI are expressed in arbitrary units.

At the end of this section, we would like to briefly discuss the experimental feasibility of measuring temperature of a reservoir by making use of spins. Many previous studies^[40] have showed that a superconducting circuit based on the Cooper-pair box behaves as a spin-1/2 system. Nuclear spins and molecular nano-magnets,^[41] which are small clusters of a few atoms embedded into a crystal, can be used to simulate spins greater than one half in experiments. As pointed out in Ref. [13], the level populations of a spin can be measured by recording resonance fluorescence signals, then via fitting experimental data into a Fermi–Dirac distribution or a Bose–Einstein distribution, which are the thermal distributions of spins in steady state, one can extract the temperature of the back contact (reservoir).

In summary, we investigate the reduced equilibration dynamics of an SB model, which consists of an arbitrary size spin interacting with a finite temperature bosonic reservoir. By making use of the reaction coordinate mapping approach, an exact equivalent SIB model is obtained, in which an IHO (a collective coordinate) is partitioned into the mapped system’s Hamiltonian, reflecting the influence of system–reservoir correlations on the reduced equilibration dynamics. We find the steady state of the spin can be expressed in a form of Gibbs state with respect to an effective Hamilton, which is size consistent with the original bare spin. Employing this corrected steady state, we explore how the system–reservoir correlations affect the estimation precision of reservoir’s temperature. It is revealed that the effect of the system–reservoir correlations on the estimation precision is highly sensitive to the details of the spectral density function of the reservoir. Varying the parameters of the spectrum, the system–reservoir correlations play opposite roles in different parameter spaces. Technically speaking, our treatment is acceptable only in high temperature regimes, due to the the omission of the higher order cumulant averaging terms. It would be very interesting to further extend this study to the entire range of temperatures. Recently, in Ref. [42], the authors have generalized the reaction coordinate mapping to impurity systems coupled to fermionic reservoirs, which suggests our method can be also exploited to measure the temperature of a reservoir of free fermions. Finally, due to the generality of the SB model, we expect our result to be of interest for a wide range in quantum metrology tasks.