Using nanokelvin quantum thermometry to detect timelike Unruh effect in a Bose–Einstein condensate

It is found that the Unruh effect can not only arise out of the entanglement between two sets of modes spanning the left and right Rindler wedges, but also between modes spanning the future and past light cones. Furthermore, an inertial Unruh–DeWitt detector along a spacetime trajectory in one of these cones may exhibit the same thermal response to the vacuum as that of an accelerated detector confined in the Rindler wedge. This feature thus could be an alternative candidate to verify the “Unruh effect”, termed as the timelike Unruh effect correspondingly. In this paper we propose to detect the timelike Unruh effect by using an impurity immersed in a Bose–Einstein condensate (BEC). The impurity acts as the detector which interacts with the density fluctuations in the condensate, working as an effective quantum field. Following the paradigm of the emerging field of quantum thermometry, we combine quantum parameter estimation theory with the theory of open quantum systems to realize a nondemolition Unruh temperature measurement in the nanokelvin (nK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{nK}$$\end{document}) regime. Our results demonstrate that the timelike Unruh effect can be probed using a stationary two-level impurity with time-dependent energy gap immersed in a BEC within current technologies.


I. INTRODUCTION
The Unruh effect [1] is a conceptually subtle quantum field theory result in relativistic framework.It plays a crucial role in our understanding that vacuum fluctuations and the particle content of a field theory are observer-dependent.It predicts that uniformly accelerating observers perceive the quantum field vacuum defined by inertial observers as a thermal state, rather than a zero-particle state.Since, to produce an experimentally appreciable Unruh temperature, prohibitively large accelerations have to be required (e.g., about 1 Kelvin even for accelerations as high as 10 20 m/s 2 ), the direct experimental confirmation of the Unruh effect until now still remains elusive.
A uniformly accelerating observer is conveniently described as a stationary observer in Rindler coordinates [2,3].From the perspective of the accelerating observer, the Minkowski vacuum state of a quantum field, e.g., a massless scalar field, defined by the inertial observers can be written as a spacelike entangled state between two sets of modes, respectively, spanning the left and right Rindler wedges [4].Since the accelerating observer is confined to just one of these wedges, the thermalized vacuum is obtained (i.e., Unruh effect arises as the re- * tianzh@ustc.edu.cnsult of the spacelike entanglement) when tracing out the unobserved modes.To detect the Unruh effect, many experimental scenarios have been designed in the analogue gravity [5] regime, including various physical systems ranging from ultracold atoms systems [6][7][8][9][10][11][12][13] to a graphene nanosheet system [14].In this regard, let us note that quantum simulation of Unruh effect has been reported through the BEC system [11] and NMR system [15] experimentally.These simulations rely on functional equivalence (i.e., simulating two-mode squeezed mechanics), while one significant step forward is to embody the essential of Unruh effect-acceleration-induced particle creation-in the simulation.Furthermore, other scenarios to enhance the detection of the acceleration-induced emission has been successfully theoretically put forward [16][17][18][19][20].However, the intractable, but required, relativistic motion is still the main obstacle to verifying the Unruh effect in practice.
Recently, it has been shown that the Minkowski vacuum state of a quantum field could also be written down similarly as entangled states between modes spanning in the future and past light cones [21][22][23].The Unruh effect may also arise as a result of this timelike entanglement if an observer (or detector interacting with the field) is confined only in one of these cones.In this case, a detector in the future/past light cone with world line (τ, 0) (see Fig. 1) corresponds to a detector with energy gap scaled with 1/at in the laboratory frame (see more details be-low).Therefore, one can in principle detect this timelike Unruh effect by considering a stationary detector with a special form of time-dependent energy gap [21], while without involving any real relativistic motion.Note that extraction of timelike entanglement from the quantum vacuum [24] has been investigated with this model.Besides, Berry phase from the entanglement of future and past light cones has been proposed to detect this timelike Unruh effect [25].However, a concrete experimentally feasible scenario to estimate the timelike Unruh effect is still lacking.
In this paper, we aim at resolving this issue by proposing to detect the timelike Unruh effect with an experimentally accessible platform consisting of a BEC and an immersed impurity [26,27].In our scenario, the density fluctuations in the BEC are modeled as the quantum field.The impurity, analogously dipole coupled to the density fluctuations, acts as an Unruh-DeWitt detector, and its energy gap can be designed as a time-dependent one by an external electromagnetic field (see more details below).We treat the impurity as an open system and derive its dynamical evolution by tracing over the degree of freedom of the analogous quantum field.Adopting tools from the theory of quantum parameter estimation, we show that our proposed scheme achieves experimentally accessible precision for the Unruh temperature detection in the BEC system within current technologies.
The outline of this paper is as follows.In Sec.II we review some basic concepts and physics of the Unruh effect from the perspective of quantum field theory and Unruh-DeWitt detector approach.In Sec.III we propose a concrete scenario to realize the detection of the timelike Unruh effect using the BEC platform.Experimental feasibility of the relevant analysis within the current technologies of BEC is discussed in Sec.IV.Finally, discussion and summary of the main results are present in Sec.V.

II. TIMELIKE UNRUH EFFECT
In this section we will simply review the timelike Unruh effect from the perspective of quantum field vacuum state and Unruh-DeWitt detector response.For more details, we refer the reader to Refs.[4,21].

A. Quantum field vacuum state
Let us begin with the spacetime broken into quadrants (F, P, R, L) shown in Fig. 1.The corresponding four coordinate systems can be used to define a set of field modes, complete in each region.To derive the Unruh

𝑥 𝑡
A spacetime diagram divided into four quadrants: the future (F) and past (P) light cones, and the left (L) and right (R) Rindler wedges.The transform between the usual Minkowski coordinate (t, x) and that of the four regimes are shown.From the perspective of an accelerating observer, the Minkowski vacuum of massless quantum field can be expanded in terms of modes confined in R and L or in terms of modes restricted to F and P. Therefore, if the observer in one of these quadrants (e.g., the F light cone), tracing over the unobserved modes (e.g., in the P light cone) leads to the (timelike) Unruh effect.
effect, we here, for simplification, consider a two dimensional massless scalar field case for the example.Because of the conformal invariance of the massless wave equation in two dimensions, all the Klein-Gordon equations in the four coordinate systems are the same as that of two-dimensional Minkowski case.They are given by [4], where the subscripts (F, P, R, L) denote the regions.Correspondingly, the set of field modes defined in the respective quadrants are given by where χ = η + ϵ, χ = −η − ε, ν = τ + ζ, and ν = −τ − ζ denote the "light-cone" in the four coordinate systems above.
One can quantize the field and define the field vacuum and field creation/annihilation operators in the respective quadrants.For example, one can define the Rindler vacuum as âR ω |0 R ⟩ = 0 with âR ω being the annihilation operator for a right Rindler particle, corresponding to the solution ϕ R ω (χ).Here ω here is the particle frequency.The particle number state is given by From the perspective of an accelerated observer, the vacuum of quantum massless scalar field, |0 M ⟩, defined by an inertial observer in the Minkowski spacetime can be rewritten as an entangled state between two sets of modes, respectively, spanning the right and left Rindler wedges [1,4]: where C i = √ 1 − e −2πωi/a , and n i and ω i respectively denote the Rindler particle number and energy in corresponding regions, R and L. However, since the R region and L region are causally disconnected (spacelike), the uniformly accelerated observer can only access one set of Rindler modes.The tracing out of the unobserved modes (e.g., the left Rindler modes) leads to the prediction that such an accelerated observer sees a thermalized vacuum, i.e., Note that this is density matrix for the system of free bosons with temperature T = a/2π.It means that the Minkowski vacuum state |0 M ⟩ of quantum field is viewed as a thermal state from the perspective of the accelerated observer, known as the Unruh effect.Modes in R is independent from modes in L, and modes in F is independent from modes in P , but modes in F/P are not independent of the modes in R/L [21].Actually, ϕ F ω (ν) is the same solution as ϕ R ω (χ), extended from R into F, which is pointed out in Refs.[4,21].This is also hold for ϕ P ω (ν) and ϕ L ω (χ).Therefore, the demonstration of F-P entanglement of the Minkowski vacuum is exactly the same as the standard demonstration of R-L entanglement shown in Eq. ( 5), with a change of labels R → F and L → P. That is to say, the Minkowski vacuum restricted to F − P takes a symmetrical form when expressed in terms of the "conformal modes" ϕ F ω and ϕ P ω : Analogously, the state of the field in the future (or the past) alone is a "thermal" state of the ϕ F ω (ϕ P ω )-modes, given by Note that this thermal phenomenon in (8) arises out of the timelike entanglement between modes of F and P light cones, so it is also called as the timelike Unruh effect [21,24].

B. Unruh-DeWitt detector response function
To detect the timelike Unruh effect, one can consider an Unruh-DeWitt detector evolving in the F light cone with world line x = y = z = 0, t = a −1 e aτ .Along this world line the Schrödinger equation in the conformal time τ reads i∂ψ/∂τ = Hψ, and the eigenvalues of H is assumed to be a constant gap ω 0 .In Minkowski coordinates or laboratory framework, this Schrödinger equation can be rewritten as The 1/at factor is due to the change of variables to Minkowski time.This means that in the laboratory a detector with energy gap (denoted by the Hamiltonian H) scaled with 1/at corresponds to a detector with the fixed Hamiltonian H on the x = y = z = 0, t = a −1 e aτ world line [21].
We can assume the interaction between the detector and the field to be the standard Unruh-DeWitt term.Specifically, for a two-level detector it takes the form H I = λ φ(x(t))σ x .Therefore, consider the full Hamiltonian in the conformal time framework, we have i∂ψ/∂τ = (H + e aτ H I )ψ.The interaction term acquires the exponential factor due to the change of variables to conformal time.For this case, the detector response function is found to be (10) where D + (τ, τ ′ ) = ⟨0 M |ϕ(τ )ϕ(τ ′ )|0 M ⟩ denotes the Wightman function of the field.The Wightman function along the inertial trajectory x = y = z = 0, t = a −1 e aτ can be calculated to take the form [21,24] while it along a uniformly accelerated trajectory t = a −1 sinh(aη), x = a −1 cosh(aη), y = z = 0 in the R region takes the form With the Wightman functions in Eqs. ( 11) and ( 12), we can find the response function integral for the inertial detector in F region is formally identical to that for the accelerated detector in the R region.Furthermore, through the standard evaluation of the response function integral [3], this leads us to a thermal response function at temperature T = a 2π , called as the Unruh temperature.Note that the accelerated detector in R region can demonstrate the Unruh effect in terms of its response function.However, the inertial detector in F region may also demonstrate the Unruh effect, while without involving any acceleration motion.Instead of that, it requires its energy gap scaled with 1/at in the laboratory [21,24].

III. DETECT THE TIMELIKE UNRUH EFFECT IN A BEC
We will in the following observe the timelike Unruh effect with an impurity immersed into a BEC, and adopt tools from the theory of quantum parameter estimation to estimate the Unruh temperature.

A. Dynamics of the impurity detector in a BEC
To demonstrate the timelike Unruh effect with the Unruh-DeWitt detector model in a BEC, let us begin with the standard one-dimensional Gross-Pitaevskii equation [28][29][30] where Ψ is the condensate wave function, V ext is the externally imposed trapping potential, and g denotes the two-body contact interaction coupling.In the secondquantized formalism, one can decompose the field operator for the dilute Bose gas as Ψ = Ψ 0 (1 + φ) with Ψ 0 = √ ρ 0 e iθ0 , where |Ψ 0 (x)| 2 = ρ 0 ≃ const represents the condensate density, and φ describes the perturbations (excitations) on the top of the condensate.Within the Bogoliubov theory [31,32], density fluctuations in Heisenberg representation can be written in the form of φ as which closely resembles the quantized scalar field in terms of bosonic operators bk In the laboratory frame where the condensate is at rest, the frequency of quasiparticle reads ω k = c 0 k 1 + (ξ 0 k/2) 2 with c 0 and ξ 0 being the speed of sound and the healing length, respectively.Note that u k and v k are Bogoliubov parameters satisfying ( We consider an impurity as the simulator of the Unruh-DeWitt detector, which is immersed into the condensate and collisionally coupled to the Bose gas.The impurity is assumed to be illuminated by a monochromatic external electromagnetic field at the frequency close to resonance with the impurity's internal level transition, with a timedependent Rabi frequency.The effective Hamiltonian of the whole system, the impurity plus the density fluctuations, is given by (see Supplemental Material [33] for details) where ω 0 (t) is the time-dependent Rabi frequency which can be experimentally controlled, and g − is the coupling parameter.Therefore, the dynamics of this system yields , then in this time frame one can find where ω 0 (t) = c 0 ω 0 /at has been taken.This choice means that a two-level detector with energy gap scaled with c 0 /at corresponds to a two-level detector with fixed energy gap ω 0 in the time τ frame.Note that this scenario also corresponds to a two-level detector with the fixed energy gap ω 0 on the (τ, 0) world line, which has been proposed to detect the timelike Unruh effect [21,24,25].
The correlation function of the density fluctuations with respect to the laboratory time t along the world line (τ, 0) reads (see Supplemental Material [33] for details) where ∆τ = τ − τ ′ and ⟨•⟩ = ⟨0| • |0⟩.With this, one can calculate the detector's response function as where Γ(ω 0 ) = (ρ 0 ω 0 g 2 − )/(2mc 3 0 ) is the spontaneous emission rate.Note that this response function is similar to that of the uniformly accelerating Unruh-DeWitt detector in R region [3].As such, this thermal response implies that the detector views the vacuum fluctuations as a thermal bath with temperature T = |a|/2π.The timelike Unruh effect is thus demonstrated.To further explore this effect, we will explore the quantum dynamics of the impurity, and estimate the Unruh temperature with the quantum metrology approach below.
Since we are interested in the dynamics of the impurity, we trace over the degree of freedom of the analogue quantum field.Besides, we perform the Born approximation and the Markov approximation as a results of the weak coupling between the impurity and field.We thus can find from Eq. ( 16), the dynamics evolution of the impurity is given by the Lindblad master equation ρA where is the time-dependent state of the impurity, and Ω = ω 0 + ω L with ω L being the Lamb shift [34] as a result of the interaction with vacuum fluctuations.Specifically, it reads with ω−ω0 .Note that P denotes principle value.Because of Γ(ω 0 )/ω 0 ≪ 1, the Lamb shift is quite small and thus usually is assumed to be negligible.
Assuming the initial state of the impurity is prepared at |ψ(0)⟩ = sin(θ/2)|g⟩ + cos(θ/2)e −iϕ |e⟩, one can find the solution to Eq. ( 19) where , and σ = (σ 1 , σ 2 , σ 3 ) are Pauli matrixes.In the long evolution time limit, δ + τ ≫ 1, the impurity eventually evolves to the state regardless of the initial state.Here H 0 = ω0 2 σ z , and β = 1/T denotes the inverse Unruh temperature.It means that the Unruh effect can be understood as a manifestation of thermalization phenomena that involves decoherence and dissipation in open quantum systems [35].
With the evolving state of the detector in Eq. ( 21), in the following we will estimate the Unruh temperature through quantum metrology approach, similar to our previous studies [36,37].

B. Optimal estimation of the Unruh effect
As shown above we have proved that the impurity with a specially scaled energy gap may be exploited to observe the timelike Unruh effect in BEC.The Unruh temperature of the density fluctuations of BEC, viewed by the impurity probe, parametrizes the probe state ρ A (τ ) shown in Eq. (21).Since the dependence of ω(τ ) on T is well understood, one can infer this temperature from the statistics of measurements which are made on a large ensemble of identically prepared probes.However, as a result of the random character of quantum measurement and the finite size of the ensemble, uncertainty is unavoidable in any such temperature estimate.In this regard, the theory of quantum parameter estimation plays a crucial role in finding the optimal measurement that minimizes this uncertainty [38,39].
In general, the solution of parameter estimation problem is to find an estimator which denotes a mapping T = T (x 1 , x 2 , . . . ) from the set of measurement outcomes into the space of parameters.Optimal estimators in classical estimation theory means that they can saturate the Cramér-Rao inequality [40,41] where N is the number of measurements, F (T ) is the socalled Fisher Information, and is the mean square error.This inequality establishes a lower bound on the mean square error of any estimator of the parameter T .Specifically, the Fisher Information Here p(x|T ) denotes the conditional probability of obtaining the value x when the parameter has the value T .Furthermore, note that actually the mean square error is equal to the variance Var = E T [ T 2 ]−E T [ T ] 2 for unbiased estimators.
In quantum version, the conditional probability according to the Born rule reads p(x|T ) = Tr[Π x ρ T ], where {Π x } denotes a positive operator-valued measure satisfying dxΠ x = I.Moreover, ρ T is the density operator parametrized by the temperature T we want to estimate.By introducing the Symmetric Logarithmic Derivative (SLD) LT as the self-adjoint operator satisfying the Fisher information (24) then can be rewritten as Therefore, for a given quantum measurement, Eqs. ( 23) and ( 26) establish the classical bound on precision, which may be achieved by a proper data processing, e.g., by maximum likelihood, which is known to provide an asymptotically efficient estimator.Since each measurement may have a corresponding Fisher information of its own, to find the ultimate bounds to precision one has to optimize the Fisher information over the quantum measurements.In this regard, the Fisher information F (T ) of any quantum measurements is bounded by the so-called Quantum Fisher Information (QFI), which leads the Cramér-Rao bound of quantum version, This inequality is valid for the variance of any estimator.We also define the quantum signal-to-noise ratio (QSNR) , which bounds the signal-to-noise ratio as T /∆ T ≤ √ N Q.Hence, Q here can be used to quantify the ultimate sensitivity limit of our impurity thermometer.
For a qubit probe, the QFI has a simple expression in terms of the Bloch vector [42].Applying that to our Unruh-DeWitt detector case shown in (21), we can obtain Note that the QFI for the mixed states cases (the first line of Eq. ( 28)) can be rewritten as where λ ± = (1±|ω|)/2 are two eigenvalues of the density operator of the qubit state.The corresponding eigenstates are with Eq. ( 29) means that the QFI in (28) consists of two terms, respectively corresponding to the Fisher information for measurements of |p + (τ )⟩⟨ p + (τ )| and |p − (τ )⟩⟨ p − (τ )|.The SLD is given by with Since the SLD is directly related to the QFI and optimizes the quantum Cramér-Rao bound, measuring LT means to minimize the uncertainty in the Unruh temperature estimate due to the finite number of samples.Furthermore, it is needed to point out that the SLD depends on the temperature, and thus some prior information on T is assumed when constructing its corresponding measurement.Seen from the expression of the SLD in Eq. (33), if one wants to measure LT in practice, then one is required to be able to efficiently evaluate the Bloch vector ω(τ ) and its temperature derivatives from an accurate theoretical model for the detector's state ρ A (τ ).In addition, the ability to measure an arbitrary combination of Π + = 1+n•σ 2 and Π − = 1−n•σ 2 is also needed.

IV. EXPERIMENTAL IMPLEMENTATION
Recent experimental advances have allowed for groundbreaking observations of BEC, its excitation spectrum, and dynamics of impurity immersed in the BEC [43][44][45].These relevant technologies in principle hold promise to realize our experimental scenario about timelike Unruh effect proposed above.We here make an estimate of the experimental parameters that are required to observe single-detector thermalization resulting from the timelike Unruh effect.

A. Unruh temperature and the conformal evolution time
As shown above, a "conformal observer" (i.e., in the τ framework with constant conformal frequency gap) actually will see a constant temperature By taking typical value for the speed of sound in the condensate c 0 ∼ 10 −3 m/s, the Unruh temperature via the effective acceleration a is shown in Fig. 2 (a).It is found that 1 K timelike Unruh temperature in the BEC approximately requires the effective acceleration as high as 10 8 m/s 2 , which is quite smaller than that for the massless scalar field case, on the order of 10 20 m/s 2 .This is because that the massless scalar field (or electromagnetic field) usually considered to observe the Unruh effect is replaced with the phononic field here, and the corresponding sound speed to which is much smaller than the speed of light in the vacuum.However, for the perspective of scaling the detector energy gap, we find that this scaling a/c 0 plays the key role in the temperature (35), and 1 K Unruh temperature requires a scaling on the order of 100 GHz.We now need to clarify the correspondence between these conformal parameters and the laboratory frame parameters (i.e., Minkowski time intervals and frequencies).In particular, the relation between the time interval of observation in the laboratory frame and that of the conformal time ∆τ reads where ω Mi denotes the initial frequency of the two-level Unruh-DeWitt detector in the laboratory frame.Usually, since in practice the change of the frequency is not arbitrary, the ratio of the initial, ω Mi and final, ω Mf , frequencies in the laboratory frame needs to be confined and clarified.This ratio is given by It means the conformal time interval ∆τ is confined.However, it is pointed out in Ref. [46] that unlike the electromagnetic case, the Rabi frequency in our cold atom setup could be tuned to zero, which gives ω 0 = 0. Therefore, here we won't limit the ratio of the initial and final frequencies in the laboratory frame.We plot this ratio as a function of the effective acceleration and the conformal evolution time of the detector in Fig. 2 (b).To observe the timelike Unruh effect, the high enough effective acceleration is required to result in high enough temperature, and the long enough conformal evolution time ∆τ of the detector is also needed to accumulate more effects on the detector (i.e., to thermalize the detector or to achieve the thermal equilibrium between the detector and the Unruh thermal bath).However, considering the practical limited time interval of observation in the laboratory frame, we should choose proper effective acceleration and the conformal evolution time because of Eq. ( 36).We will in the following analyze the thermometric performance with considering these conditions.Furthermore, in Fig. 2 (c) we plot the detector's response function (or spontaneous excitation rate) as a function of the dimensionless acceleration parameter.With the increase of the acceleration, the spontaneous excitation rate increases monotonously.This function may be explored to verify the thermal character of the timelike Unruh effect.

B. Thermometric performance
Note that the thermometer which measures the temperature of a BEC in the sub-nK regime has been investigated recently [47][48][49][50][51].We here consider the similar thermometer to estimate the Unruh temperature.
Fig. 3 shows the QSNR as a function of the Unruh temperature and the conformal evolution time of the detector for θ = π (i.e., the initial ground state).At a given temperature, the optimal measurement time corresponds to the maximum sensitivity, i.e., Q max = max τ Q(τ ) = Q(τ max ).It would be a certain moment before the thermal equilibrium between the detector and the Unruh thermal bath, or the long evolution time limit, δ + τ ≫ 1, (i.e., the thermal equilibrium case), which depends on the Unruh temperature.It is found that the 4. QSNR at T = 20 nK as a function of the conformal evolution time of the detector for various Γ = Γ(ω0)/ω0 (which can be considered as the parameterized coupling strength) by taking θ = π (i.e., the initial ground state).Here we take a typical value for the speed of sound in the condensate c0 ∼ 10 −3 m/s, and the fixed energy gap of the detector in conformal time framework ω0 = 2π × 500Hz.
QSNR may achieve its maximum, shown as the large red region in Fig. 3, in the relevant temperature range which is valid within current experiments [52][53][54].For example, if T = 20 nK we find Q max ≈ 0.59, meaning that an error of ∆ T /T ≈ 10% can be achieved with N ≈ 280 measurements after a time ∆τ max ≈ 0.265 s.According to Eq. ( 36), this conformal evolution time corresponds to an approximately infinite long laboratory time, i.e., ∆ t → ∞.Therefore, the maximum sensitivity Q max seems not to be achieved.However, one can still achieve the same error by properly choosing the measurement time and the measurement number.For example, if ∆τ = 0.55 ms, and the measurement number N = 50000, one can achieve the error ∆ T /T ≈ 10% with the laboratory time ∆ t ≈ 0.55 s, which is eminently feasible since a single gas sample may have a lifetime of several seconds [55][56][57].Furthermore, instead of looking at a single impurity, we can consider, say, 1000 to a few thousand of independent impurities [58,59].In this case, we can effectively reduce the measurement (e.g., to N = 50) while to achieve the same expected error ∆ T /T ≈ 10%.The coupling strength between the detector and the field, which can be experimental controlled, plays an important role in the thermometric performance.In our model, the coupling strength can be embodied by the spontaneous emission rate Γ(ω 0 ) shown under Eq.(18).Therefore, different spontaneous emission rates could represent different coupling strength.In Fig. 4, we fix the Unruh temperature T = 20 nK and show the dynamical QSNR for various coupling strength.We find that the coupling strength does not affect the maximum sensitivity, while influences the optimal measurement time at which the maximum sensitivity can be achieved.The maximum sensitivity shifts to progressively later times as the coupling strength decreases.As discussed above, it seems that the maximum sensitivity can not be obtained experimentally since the optimal measurement time in conformal frame usually might correspond to infinite time in the laboratory according to Eq. (36).Note that the correspondence between the conformal time and the laboratory time does not depend on the coupling strength.Therefore, although the experimentally feasible laboratory time ( or the conformal evolution time) is fixed and usually smaller than the optimal measurement time, by controlling the coupling strength (i.e., increasing the coupling strength) one can still in principle obtain a considerable QSNR which approaches to the maximum sensitivity.For example, if we choose the laboratory time ∆ t = 2 s which is smaller than the lifetime of a single gas sample and thus is eminently experimentally feasible, then the corresponding conformal evolution time ∆τ ≈ 0.63 ms, thus the corresponding QSNR Q ≈ 0.05, 0.07, 0.1 for Γ(ω 0 )/ω 0 = 10 −3 , 2×10 −3 , 4×10 −3 cases, respectively.
To further understand how the coupling strength affects the estimation of the Unruh temperature, in Fig. 5 (a) we fix an eminently experimentally feasible time ∆τ ≈ 0.63 ms for the Unruh temperature T = 20 nK and show the relative error ∆ T /T as a function of the effective coupling strength for different measurement number N .Remarkably, increasing the coupling strength can effectively reduce the error, meaning that the estimation of the Unruh temperature is more precise.For example, after 40000 measurements, the relative error for the Γ(ω 0 )/ω 0 = 0.2 × 10 −3 case is about 22.3%, while for the Γ(ω 0 )/ω 0 = 4 × 10 −3 case, this achieved relative error is around 5% and thus is effectively reduced.Furthermore, as shown in Fig. 5 (b) we can improve the estimation precision (or reduce the error) by increasing the measurements since ∆ T /T ∝ 1/ √ N .However, on the other hand, the huge number of measurement may take a lot of time and lower the experimental efficiency.To achieve the same measurement error, the stronger coupling strength can effectively reduce the number of measurement under the same condition.For example, if our target error is around 10%, for the Γ(ω 0 )/ω 0 = 1 × 10 −3 case, the required measurements are about 39788, while for the Γ(ω 0 )/ω 0 = 4 × 10 −3 case, the required measurements are around 10028.In Fig. 5 (c) we fix an eminently experimentally feasible time ∆ t = 4 s and show the relative error ∆ T /T as a function of the experimentally feasible Unruh temperature T in nK and even sub-nK regime.We find that in this case the relative error decreases with the increase of the Unruh temperature, meaning that we can obtain more precision when estimating the Unruh temperature in the relatively higher temperature regime.

V. DISCUSSIONS AND CONCLUSIONS
The parameters were chosen above just as an example that our proposed setup is feasible, which should not be considered as the only available configuration.Actually, we can also choose a higher effective acceleration a to create a higher temperature than nK, Repeat the same analysis, we can get the thermometric performance, going beyond the nK regime.Note that the main limited factor in our proposal is the time interval of observation in the laboratory frame since even the finite product of the acceleration and the conformal time, i.e., a∆τ , may corresponds to a very huge laboratory time (see Eq. ( 36)), which can not be accessible experimentally.However, for a fixed a∆τ , it is possible to choose the different combinations of the acceleration and the conformal evolution time to improve and optimize the performance of our proposal.
Usually the Unruh effect is notoriously difficult to observe, since the temperature is so tiny for accessible values of the acceleration, a, namely T = ℏ a 2π ck B .In other wards, to achieve an experimentally accessible Unruh temperature, quite high acceleration, which actually has been far beyond our ability, has to be required (e.g., 1K temperature requires about 10 20 m/s 2 acceleration).Here we propose to detect the so-called timelike Unruh effect [21,24,25] in a BEC system.In our scenario, an inertial two-level detector, whose energy gap is continuously scaled as 1/at responds to the Minkowski vacuum in a manner identical to an accelerated detector with a fixed proper-energy gap.Instead of doing the real relativistic motion, e.g., quite high linear acceleration, we here just need to control the time-dependent Rabi frequency of the detector, and thus it seems to be more accessible for experiment.
Thermalization is a key feature of Unruh effect.Thus, an important question arises: is the energy gap of the detector scaled over a long enough period to allow thermalization?Let us consider a detector in conformal time is scaled between times τ 1 and τ 2 , and assume that within the interaction time period many oscillations happens at the constant frequency (in the conformal time τ frame) ω 0 of the detector.This requirement means ∆τ = τ 2 − τ 1 ≫ ω −1 0 .In the laboratory frame, it corresponds to t2 t1 = e a∆τ ≫ e a/ω0 = e 1/(t1 ω1) = e 1/(t2 ω2) , where t i and ωi with i = {1, 2} are respectively the laboratory time and the corresponding detector's energy gap at which.If t 1 ≈ 1/ω 1 , then thermalization requires t 2 ≫ 2.71t 1 .In this regard, these thermalization conditions could lead us to choose certain appropriate experimental parameters to access to the optimal measurement time and thus realize the maximum sensitivity.
In our scheme, we estimate the Unruh temperature by monitoring the impurity atoms only, while without measuring the BEC itself destructively.Moreover, our scheme is inherently nonequilibrium and all the underlying analysis does not assume thermalization of the impurity at the temperature of the expected Unruh bath, thus alleviating the need for thermalization of the probe before accurate temperature estimation is feasible.
In summary, we present a concrete experimental proposal to detect the timelike Unruh effect that arises out of the entanglement between future and past light cones.Specifically, our model is based on an impurity with a time-dependent energy gap immersed in a BEC.We choose some typical experimentally accessible parameters to investigate the thermometric performance and find very low relative error of the estimated Unruh temperature can be obtained.Therefore, the preliminary estimates indicate that the proposed experimental implementation of the timelike Unruh effect is within reach of current state-of-the-art ultracold-atom experiments.
Our proposed quantum fluid platform may also allow us in the experimentally accessible regime to explore interesting questions concerning extraction of timelike entanglement from the quantum field vacuum [24], Unruh effect induced geometric phase [17,25], Lorentz-invariance-violation-induced nonthermal Unruh effect [9], and so on.

FIG. 2 .
FIG. 2. (a)The temperature of the timelike Unruh effect via the effective acceleration; (b) the ratio of the initial, ωMi and final, ω Mf , frequencies in the laboratory frame as a function of the effective acceleration and the conformal evolution time of the detector; (c) the detector's response function (in the units of detector's spontaneous emission rate Γ(ω0)) as a function of the dimensionless acceleration parameter.Here we take a typical value for the speed of sound in the condensate c0 ∼ 10 −3 m/s.

FIG. 3 .
FIG.3.QSNR as a function of the Unruh temperature and the conformal evolution time of the detector for θ = π (i.e., the initial ground state).Here we take a typical value for the speed of sound in the condensate c0 ∼ 10 −3 m/s, the spontaneous emission rate Γ(ω0)/ω0 ∼ 10 −3 , and the fixed energy gap of the detector in conformal time framework ω0 = 2π × 500Hz.

3 Γ = 2 × 10 - 3 Γ = 4 × 10 1 FIG. 5 .
FIG. 5. (a) The relative error ∆ T /T at T = 20 nK as a function of the spontaneous emission rate Γ = Γ(ω0)/ω0 (which can be considered as the parameterized coupling strength) for measurement number N by taking θ = π (i.e., the initial ground state); (b) the relative error ∆ T /T at T = 20 nK as a function of the measurement number N for different spontaneous emission rates Γ = Γ(ω0)/ω0 by taking θ = π (i.e., the initial ground state); (c) the relative error ∆ T /T at Γ = 4 × 10 −3 and N = 4 × 10 4as a function of the experimentally feasible Unruh temperature T by taking θ = π (i.e., the initial ground state).Here we take a typical value for the speed of sound in the condensate c0 ∼ 10 −3 m/s, and the fixed energy gap of the detector in conformal time framework ω0 = 2π × 500Hz.