Thermometry Precision in Strongly Correlated Ultracold Lattice Gases

The precise knowledge of the temperature of an ultracold lattice gas simulating a strongly correlated system is a question of both, fundamental and technological importance. Here, we address such question by combining tools from quantum metrology together with the study of the quantum correlations embedded in the system at finite temperatures. Within this frame we examine the spin-$1/2$ XY chain, first estimating, by means of the quantum Fisher information, the lowest attainable bound on the temperature precision. We then address the estimation of the temperature of the sample from the analysis of correlations using a quantum non demolishing Faraday spectroscopy method. Finally, we demonstrate that for sufficiently low temperatures the proposed measurements are optimal to estimate accurately the temperature of the sample.


Introduction
Ultracold atomic samples are considered to be, nowadays, one of the most promising setups for implementing quantum simulators of condensed matter [1][2][3]. Such promise has been reinforced by several breakthroughs which include, among others, the celebrated Mott insulator to superfluid quantum phase transition for bosons [4], as well as recent simulations of antiferromagnetic spin chains with both, bosonic [5] and fermionic [6] ultracold atomic gases.
At zero temperature, the emergence of a new order in a strongly correlated system is signalled by the presence of quantum correlations at all length scales. At finite temperature, however, such emergence fades gradually away due to the presence of thermal fluctuations. As a result, for low dimensional systems, critical points signalling quantum phase transitions often broaden into "critical" regions. Those regions still separate different phases which keep track of their ground state correlations. Hence, the transition between those phases might appear as smooth crossovers [7,8], nonetheless carrying a footprint of the quantum phase transition occurring at zero temperature. In view of these facts, the behavior of quantum correlations at finite temperatures could be used as a way to estimate it. A natural question to pose, both for fundamental as well as for technological reasons, is how well can the temperature be determined in these systems.
As it is well known in quantum metrology, the quantum Cramér-Rao bound [9-11] settles a limit in the precision of the estimation of a given parameter. If the parameter to be estimated is temperature and the system is in thermal equilibrium, the Cramér-Rao bound yields a relation of the form ∆T ∆H ≥ T 2 , being H the Hamiltonian governing the system [12][13][14]. This relation indicates that the minimal error in temperature estimation of a thermal sample is realized by a projective measurement on its energy eigenbasis. In general, such type of measurements in ultracold lattice gases is not accessible. Instead, information about quantum phases and temperature is usually obtained from momentum and density distributions or from density-density (or spinspin) correlations. These quantities can be extracted by using destructive methods such as time of flight imaging (the latter via the study of noise correlations [15]) or in-situ imaging, for instance using single site addressability [16,17]. Despite their huge relevance, these methods might suffer limitations in certain occasions, due to their destructive character. For instance, in order to study spin-spin correlations in currently available setups for single site imaging, one needs to remove all particles from one of the two spin components. In this sense, quantum non demolition (QND) methods can provide clear advantages [18]. The quantum Faraday spectroscopy is a minimally disturbing matter-light interface that maps collective atomic quantum correlations into light quadrature fluctuations, the latter perceptible to be measured by homodyne detection. Here, we adapt this method to estimate the temperature of a strongly correlated system simulated by an atomic lattice gas. Furthermore, to asses the reliability of our method for precision thermometry, we compare the signal-to-noise ratio obtained from the measurement of collective atomic correlations with the minimal possible error provided by the quantum Cramér-Rao bound. Our results show that the measurement of collective quantum correlations can become optimal for temperature estimation in some integrable models.
The paper is organized as follows. In Section 2, we briefly review the basic properties of the spin-1/2 XY chain in a transverse field, both at zero and finite temperatures. Unlike the majority of quantum spin models, the XY model can be exactly solved by means of a Jordan-Wigner transformation mapping it onto a system of non-interacting fermions and giving access to the full energy spectrum [19]. In Section 3, we focus on the quantum metrology aspects of the problem. To this aim, we derive first a closed form of the quantum Fisher information (QFI) as a function of the temperature for the whole phase diagram. This, in turn, provides the minimal error on the temperature estimation when performing an optimal measurement. Section 4 reviews the basic concepts describing the QND Faraday spectroscopy, while Section 5 is devoted to the analysis of quantum correlations at finite temperatures with this method. We evaluate, for the whole phase diagram of the model, the signal-to-noise ratio, T /∆T , obtained with a Faraday interface. As we will show later, the thermal sensitivity of a given quantum phase strongly depends on the temperature of the sample. Remarkably, our results support the suitability of collective quantum correlations as optimal observables for quantum thermometry of strongly correlated systems. In Section 6 we conclude and present some open questions.

The XY model
The spin-1/2 XY chain in a transverse field (including the Ising and isotropic XX models as particular cases) is an exactly solvable model, and as such, it can be used as a prototype to understand the interplay between quantum and thermal fluctuations. The Hamiltonian governing the system can be written as: where σ α i are the usual Pauli matrices at site i, −1 ≤ γ ≤ 1 is the parameter that sets the XY anisotropy (γ = ±1 and γ = 0 for Ising and XX models respectively), h is the transverse magnetic field and N is the number of sites of the chain. The coupling constant J can be positive (ferromagnet) or negative (antiferromagnet). Throughout this paper, we will consider only the ferromagnetic case J > 0. However, equivalent results can be straightforwardly derived for the antiferromagnetic case J < 0. For simplicity, we consider here periodic boundary conditions with an even number of sites, but the results can be easily extended to an odd number of sites or an open chain. However, for large enough chains, one expects such variations not to influence the results [19].
The Hamiltonian (1) can be easily diagonalized by mapping it onto a non-interacting fermionic model that provides the full energy spectrum. As it is well known [19,20] the non-interacting fermionic representation of the XY model is obtained by means of the Jordan-Wigner transformation, followed by a unitary Bogoliubov transformation in the quasi-momentum space, yielding the separable Hamiltonian (up to a constant): and the energy dispersion relation being k the quasi-momentum, k = π N (2j + 1), and j = −N/2, . . . , N/2 − 1. The sign of this energy is arbitrary. Choosing a positive value corresponds to the particle-hole picture for the fermionic quasiparticles, which are defined for k ∈ (0, π) by the following Bogoliubov transformation: Here, cos(2θ k ) = γ sin k/(cos k − h/J) for θ k ∈ (0, π/2), and c † k are the Fourier transform of the on-site fermionic operators that directly relate to the spin operators via the Jordan-Wigner transformation The ground state of the system corresponds to the vacuum of the Bogoliubov quasiparticles, and excitations are obtained with creation operators acting on the vacuum. The energy gap between the ground state and the continuum of excited states is thus given by ∆E = min k ( k ). Note that the Hamiltonian is symmetric under the exchange h ↔ −h (by k ↔ π/2 − k) and under γ ↔ −γ (by σ x ↔ σ y ). A sketch of the phase diagram at zero temperature, together with the energy gap ∆E and the energy dispersion relation are displayed in Fig. 1. The system is always gapped, i.e. ∆E > 0, except for the quantum critical lines occurring at h/J = ±1 (Ising transitions), which separate the paramagnetic phases (PM) from the ferromagnetic (FM) ones (or antiferromagnetic if J < 0) and for γ = 0 and |h/J| ≤ 1, corresponding to the critical phase in the XX model (anisotropic transition). Moreover, Heisenberg systems with general anisotropies exhibit, for particular values of the couplings, a ground state which is doubly degenerated and which is factorizable as a product of on-site localized wave-functions [21,22]. In the XY model, for each value of γ, this product ground state corresponds to an external transverse field h/J = ± 1 − γ 2 , which is depicted by a dashed line in the phase diagram of Fig. 1(a).
In the thermodynamic limit (large N ), the system in thermal equilibrium at a given temperature T can be described by the density matrix in the macrocanonical ensemble (we set k B = 1): where Z denotes the partition function of the system. For compactness of notation we write from now on (k) (γ, h/J, T /J) simply as (k) (T ). Since the Hamiltonian (2) is separable, the density matrix can be directly written as a tensor product of the density matrices associated to each quasiparticle mode k. These quasiparticles obey fermionic commutation relations, and thus where |0 k (|1 k ) denotes an empty (occupied) quasiparticle state k. We take the above expression as the starting point to study correlations at finite temperatures. Finally, let us remark that the XY model can be realistically implemented in experiments. In particular, the isotropic XX model directly maps onto a system of hard-core bosons and it has been experimentally realized with cold atoms in optical lattices [23], while the Ising model has been also engineered with a similar system [5]. Moreover, other models that can be implemented with cold atoms, as the bond-charge Hubbard model, directly map onto the XY model [24].

Optimal strategy: lowest bound on the temperature error
Consider the state of our strongly correlated system given by (T ). This state depends on the value of the temperature T , which is unknown and that we want to estimate. In general, if a quantum state depends on an unknown parameter θ that we want to estimate, the typical strategy is to choose an unbiased estimatorθ for which θ = θ.
The standard deviation of this estimator, i.e. ∆θ = Var(θ), quantifies the error on estimation of θ. The quantum Cramér-Rao bound sets a lower bound on this error as follows [9,10]: with ν being the number of repetitions, and F(θ) the quantum Fisher information (QFI) associated to the parameter θ, which is given by: where the symmetric logarithmic derivative, Λ θ , is defined as For temperature estimation on a Gibbs state (T ), the QFI is explicitly given by [13,25]: where ∆H 2 ≡ Tr(H 2 (T )) − [Tr(H (T ))] 2 and we have used the identity H = T 2 ∂ T ln Z. For Gibbs states this can be connected to the specific heat [13]. Maximizing the quantum Fisher information is thus equivalent to maximize the variance of the Hamiltonian. Introducing the thermal energy as k B T , it is possible to express the quantum Cramér-Rao bound in the form of an uncertainty relation [13,14] ∆T T (equivalently, ∆H∆β ≥ 1/ √ ν), which provides a very useful insight to understand how the thermal energy, the energy spectrum of the Hamiltonian and the error on the temperature determination come into play. Indeed, according to (8), quantum states having a larger QFI can be estimated with a smaller error. To have a figure of merit, we define the thermal sensitity as the value of the bound obtained for ν = 1, which is given only by the QFI. In this way we withdraw the statistical dependence on the number of times the sample is probed.
In general, finding the QFI is a very difficult task, and different bounds on the QFI that are easier to evaluate have been suggested [26][27][28]. In the temperature estimation of a strongly correlated thermal state, the difficulty arises in the calculation of its intricate energy spectrum, and, in general, it is not possible to derive a closed expression for the QFI. However, such calculation becomes straightforward for the XY model due to the simple structure of a thermal state which corresponds to a product state in the fermionic representation (6). This allows us to express F(T, (T )) as the sum of the QFI F(T, k (T )) of each individual mode k, i.e.
being n k = (1 + e k /T ) −1 the Fermi-Dirac distribution of the quasiparticles.
Using the above expressions, the upper bound on the signal-to-noise-ratio is given by In the top panels of Fig. 2, we display this upper bound, normalized by the total number of sites N , for the whole phase diagram at different temperatures. For finite T , this quantity scales linearly with N . For very small temperatures, e.g. T /J = 0.05, the QFI becomes noticeable only close to the critical lines. This is not surprising, since for a gapless system, excitations to the lowest part of the energy spectrum will be created no matter how small the temperature is. Thus, as the uncertainty in energy of the state grows, so does the QFI, and accordingly the state becomes very sensitive to thermal fluctuations. In contrast, for a gapped phase, if k B T ∆E, the probability of creating excitations remains low. In such cases, the energy remains well defined, yielding a vanishing value of the QFI and correspondingly a large error in temperature estimation. On the other hand, for large enough values of the temperature, i.e. k B T ≥ ∆E, different modes become excited, and other regions of the phase diagram become more sensitive and optimal for thermometry. In fact, for a given value of k B T , the accurate estimation of the sample temperature depends on the energy spectrum but also on the density of states (DOS), as they play a crucial role in the QFI expression (13). This can be clearly seen in Fig. 2, where the value of the optimal signal-to-noise ratio (T /∆T ) 2 CRB is also displayed for T /J = 0.2 and T /J = 0.8. The more sensitive regions of the phase diagram are now clearly different than the "zero temperature transition points", i.e. h = ±1 and γ = 0 for |h/J| < 1. In the same figure, in the middle and bottom panels, we display the signal-to-noise ratio obtained from measuring collective correlations that we will analyze in Section 5.
Finally, the behavior of the QFI or thermal sensitivity with temperature is explicitly shown for some particular cases in Fig. 5 (solid lines). After displaying a maximum at certain value of T /J, this quantity decreases again as the state tends to be maximally disordered. Indeed, at very large temperatures (β → 0), and despite the variance ∆H is maximum and the error ∆β is minimum, the signal-to-noise ratio T /∆T = β/∆β will tend to zero.

Quantum Faraday Spectroscopy
Here, we briefly review a quantum non-demolition scheme for measuring quantum correlations in ultracold atomic lattices. The method is based on a light-matter interface [29] employing the quantum Faraday effect. It was adapted to determine quantum phases of strongly correlated systems in optical lattice systems in [18,30]. The scheme is extremely versatile and can detect superfluidity, superlattice ordering and itinerant magnetism for fermionic and bosonic lattice gases [31,32]. It also allows to reconstruct the phase diagram of non-trivial spin chain models [33,34] and to engineer quantum correlations by suitable post-selection [35]. In the following we review the basics of the scheme but we point the reader to the previous references for more details.
The basics of a QND Faraday spectroscopy assume a strongly linearly polarized light beam along e.g. the x-axis propagating on the z-axis and interacting off resonantly with the internal spin degree of freedom of an atomic sample. Due to the atom-photon interaction, the light polarisation is rotated by an amount that depends on the magnetic state of the sample. The light can be described by time-integrated canonical operators X = S 2 / N ph and P = S 3 / N ph , where S 2(3) denote the Stokes operators in the perpendicular directions of the incoming beam while N ph is the total number of photons of the beam. If the atomic sample is confined in an optical lattice, the light can be modulated in a standing wave configuration as schematically depicted in Fig. 3. After the Faraday interaction has taken place, the integrated equations of motion result into [29] where X in and X out represent, in the input-output formalism, the light quadratures before and after the Faraday interaction. The observable J z corresponds to the modulated collective angular momentum along z-direction and is defined as: The above sum extends on all lattice sites l, k p is the wave vector of the probing beam and d is the inter-site distance. Finally, the light-matter coupling constant κ = √ d o η depends on the optical depth of the atomic sample d o as well as on the spontaneous emission probability induced by the probe. Typical values of κ are in the range 1-10.
As the light and atom states are initially uncorrelated, it follows that where we assume the incoming light beam to be in a coherent state with zero mean and variance 1/2. For the ferromagnetic case (J > 0), the output signal is maximum when the wave vector of the probe beam is set to k p = π/d, i.e. the light is not modulated. For the antiferromagnetic case (J < 0), since the total magnetization of the sample is zero, it is necessary to modulate the incoming beam with half of the frequency k p = π/2d. After the outcoming light quadrature X out has been homodyne measured, the atomic sample is projected onto a subspace of fixed J z . Owing to the fact that the off resonant interaction with the light does not destroy the sample, we further assume that after the measurement thermalization will take place on such given subspace. Since typical thermalization times for ultracold lattice gases are on the order of ms and the many-body sample is stable on the time scale of seconds, the Faraday interface taking place in the µs regime can be considered as instantaneous. Thus, the Faraday interface could be repeated several times on the same sample preserving its QND character. Finally, we remark that in order to measure the other collective operators J x and J y using the same experimental setup, one should apply an appropriate spin rotation to the atomic sample so to map σ x → σ z or σ y → σ z [36].  After the second pass, the laser beam is outcoupled to a homodyne detector, where the light quadrature is measured and recorded.

Quantum thermometry for the XY model using a Faraday interface
The quantum polarization spectroscopy technique described in the previous section grants access, a priori, to any order of the statistical moments of the collective atomic angular momentum. For certain phases, as for instance the paramagnetic phase, the mean value of the transverse magnetization J z is sufficient to infer the temperature of the sample. However, the mean value might vanish for other observables in an unbroken symmetry phase (e.g. the longitudinal magnetization for a the thermal state in the Ising model). Instead, the ordering is clearly revealed when looking at the quantum fluctuations or variance of the observable. Here, for reasons that will become clearer later, we focus our study on the mean value of J z and the variance of J x . The latter can be written as: and corresponds to the sum over any two-site correlation function or, equivalently, to the magnetic structure factor at zero quasi-momentum. We start by analyzing the strength of the output signal when measuring the variance of the observable associated to the order parameter, i.e. VarJ x for γ > 0. Note that the results for J y and γ < 0 are equivalent to those for J x and γ > 0. We recall that for a coherent input beam, the shot noise is VarX in = 1/2. As expected, the variance of the operator associated to the order parameter always exceeds the variance of the angular momentum along the other two directions. Moreover, this is maximal for the Ising model (γ = 1) and continuously decreases when approaching the XX model (γ = 0).
A comparison between these two limiting cases (γ = 1 and γ = 0) is depicted in Fig. 4, where in the top panels, we display the output signal VarJ x /N normalized by the input shot-noise (VarX in ) as a function of T /J, for different values of h/J and two different system sizes N = 100 and N = 200. At zero temperature, and in the gapped FM phase (red line), the signal scales as κ 2 N , whereas in the PM phase (blue line), it scales as κ 2 . Strictly speaking, and since we are dealing with a 1D system, there exists no phase transition at finite temperature. This is reflected in the fact that, at any finite value of T , the signal in the gapped FM phase does not scale anymore as κ 2 N but rather as κ 2 , and the signal for the two system sizes overlap. Therefore, the plateau depicted in the top panel of in Fig. 4 is only a finite size effect and it disappears as the system size increases. In this region, the signal is constant with T and thus, it will not be useful for thermometry. The results for any γ = 0 are qualitatively similar to those for the Ising model. Moreover, for any value of the parameters γ and h it is always satisfied that VarJ x ≥ 1. If the optical depth d o is tuned so that κ ≥ 1, the signal of the output beam will be greater than the input beam shot-noise. This is, however, not the case for the other two observables VarJ y and VarJ z , which go well below the shot noise limit when approaching the XX model.
The output signal, when measuring the mean value of the J z observable ( J z / √ N ), is depicted in the bottom panels of Fig. 4. In contrast to the former observable, this is maximum (in absolute value) in the PM phase and it increases when approaching the γ = 0 limit.
In order to asses the optimality of measuring collective quantum correlations for precision thermometry, we focus on the signal-to-noise ratio (T /∆T ) 2 F achievable by using the Faraday interface, and compare it with the minimal possible error in temperature estimation, provided by the Cramér-Rao bound (T /∆T ) 2 CRB (14). To this aim, the error performed in measuring temperature using the observable A can be estimated as [11] ∆T ≈ ∂ A ∂T where ∆A is the standard deviation of the observable or (VarA) 1/2 . Therefore, For the analyzed model, the variance of the two observables of interest can be evaluated by making extensive use of Wick's theorem. A comparison between the optimal signal-to-noise ratio (T /∆T ) 2 CRB (top panels), and the one obtained measuring the two observables A = J 2 x(y) − J x(y) 2 for γ > 0 (γ < 0) (middle panels) and A = J z (bottom panels) is presented in Fig. 2, for the whole phase diagram and different temperatures. By fixing the value of h/J, a quantitative comparison between both signals can be performed as a function of temperature for different phases. In Fig. 5 we fix the anisotropy parameter to γ = 1, γ = 0.3 and γ = 0, These two figures show that, in general, A (A ) performs better in the FM (PM) regions. Also, in the FM regions the signal-to-noise ratio of A follows the same qualitative behavior as (T /∆T ) 2 CRB , shifting with temperature its maximum value from the multicritical points (γ = 0, |h/J| = 1) to the Ising model at h = 0. However, it decays faster with T /J than (T /∆T ) 2 CRB . Moreover, our results clearly show that, for T /J ≤ 0.35, the Faraday spectroscopy, when reading out the observable VarJ x , provides an almost optimal measurement for thermometry precision in the FM phase in the Ising model, and its optimality decreases when approaching the critical XX model (γ = 0). Instead, in the PM phases, J z approaches the ideal bound in the XX model for a temperature range 0 < T /J ≤ 0.45. It is worth to recall that this is indeed the range of temperatures of interest for present experiments with ultracold atomic gases simulating strongly correlated systems [6,23]. . VarJ x is optimal in the FM phase and γ = 1, J z is optimal in the PM phase and γ = 0. At very large T /J the signal of the Faraday interface becomes less optimal.

Summary
In summary, we have analyzed the suitability of QND Faraday interfaces to provide a precise estimate of the temperature of a sample of ultracold gases simulating the XY model. The Faraday interface, giving access, a priori, to any statistical moment of the collective angular momentum operators, might become optimal for this task. Their suitability depend upon the order displayed in the strongly correlated system and the temperature range. By borrowing concepts from quantum metrology, we have analytically derived the optimal signal-to-noise ratio for a thermal state governed by the XY Hamiltonian given by the quantum Cramér-Rao bound, and we have compared it with the one obtained from the Faraday interface. Remarkably enough, collective atomic correlations can be considered as optimal observables for precision thermometry in the temperature range of interest in present experiments of ultracold lattice gases simulating strongly correlated systems. Our results hold for the XY model, but it remains to be analyzed if the method can also be optimal for other quantum spin models, either integrable or not.