Berry Phase Quantum Thermometer

We show how Berry phase can be used to construct an ultra-high precision quantum thermometer. An important advantage of our scheme is that there is no need for the thermometer to acquire thermal equilibrium with the sample. This reduces measurement times and avoids precision limitations.

Introduction.-The state of a point-like discrete-energy leveled quantum system (e.g. an atom) interacting with a quantum field acquires a geometric phase [1] that is dependent on the state of the field. If the field is in a thermal state, this geometric phase encodes information about its temperature [2], and so it was used in a proposal to measure the Unruh effect at low accelerations [2].
In this letter we use Berry phase to construct an ultrahigh precision thermometer. The thermometer consisting of an atomic interferometer measures the temperature of a cold medium by comparison with a hotter thermal source of approximately known temperature. Since our scheme does not require the thermometer to reach thermal equilibrium with the sample, measurement times are shorter and precision limitations due to thermalization are avoided [3].
In principle the thermometer that we introduce can be physically implemented in different ways. Here, as a simple example, we consider the point-like system to be an atom whose internal level structure is described by a quantum harmonic oscillator. The sample will be a thermal source modeled as a quantum scalar field contained in a cavity. The sampled field could correspond to many possible different physical situations: a phonon field, an infinite number of harmonic oscillators coupled to the atom (a way to model thermal reservoirs), the electromagnetic field in one dimension or a fixed polarization component of a full 3D EM field.
In the considered scenario optimal precision is obtained for temperatures of cold sources 3 orders of magnitude below the reference temperature of the hot source. Moreover, our thermometer is very sensitive to temperature variations of the cold sample but almost insensitive to variations of the hot source, making this setting a very precise and reliable method of measuring low temperatures with high precision. The temperature range in which the thermometer operates optimally is determined by the natural frequency of the harmonic oscillator (in this case the atom). Therefore, the thermometer will be tuned to a particular temperature range by choosing an atom with a given energy gap. It is possible to make measurements within different temperature ranges by using atoms with various energy gaps. If one employs atoms with energy gaps going from 1 Mhz to 1 Ghz, it is possible to measure with high precision temperatures over a range of 10 −1 K to 10 −6 K. Our proposal is readily scalable using atoms with energy gaps of radio-frequency width to measure nanokelvin temperatures.
To construct the Berry phase based thermometer consider an atomic interferometer in which a single atom follows two different paths. In one branch the atom goes through a cavity containing the cold sample while in the second branch the atom moves through a cavity where the field is in a thermal state of a known temperature. In both arms the atom will acquire a Berry phase due to the interaction with the field which will depend on the field's temperature. The geometric phase difference measured at the output of the interferometer will determine the temperature of the cold sample.
Geometrical phase.-Let the atom to follow a trajectory given by x(t) and its internal harmonic degrees of freedom be described by creation and annihilation operators d † and d. We consider the interaction Hamiltonian between the atom and the field is given by: The λ m 's are coupling constants, Ω and t are the atom's frequency and proper time, respectively. The field operators f † m are associated with the field mode solutions u m (x(t)) which are labeled by the frequency index m. These functions are evaluated at the atom's position implying that the interaction between the field and the atom takes place locally. For simplicity we are considering a 1 + 1-dimensional field.
This model is a type of Unruh-DeWitt detector [4][5][6][7][8] which has been previously studied in [2,9,10]. It generally describes a multi-level quantum emitter (typically an atom) interacting with an infinite number of harmonic oscillators that may correspond to a scalar field. This model describes any harmonic detector coupled to the field (as, e.g. a many-levels atom in a very good approximation) and it is strongly related to the standard twolevel approaches typically used to model atoms coupled to a scalar or EM field. In fact, there is a well-known mapping between this model and the standard Jaynes-Cummings model via the Holstein-Primakoff transformations [11]; the equivalence of the two models for the relevant regimes can be seen in [12,13].
Since we consider the field to be inside a cavity, the mode frequencies m are discrete. For small cavities the mode separation is large and in this case it is possible to assume that the atom interacts effectively only with a single near-resonant cavity mode. In this case the analysis is considerably simplified and the Hamiltonian becomes and has been empirically verified [14,15]. For a cavity of length L with walls placed at x = ±(L/2), the normalized mode function is given by where ω is the field frequency. We assume the atom's trajectory is orthogonal to x such that x(t) =const. The effective coupling λ seen by the atom would depend on the configuration of the cavity. We will work in the standard coupling regime λ ≈ 1 − 1.5 Khz [14]. The most convenient procedure for computing the Berry phase is to operate in a mixed picture [2], where the atom's free evolution is absorbed in the atoms' operators. This situation is mathematically more convenient for Berry phase calculations and the Hamiltonian (2) can be diagonalized analytically; its eigenstates are the dressed states U † |m, n , where |m, n are eigenstates of the noninteracting and are the two-mode displacement, single-mode squeezing and phase rotation operators [14], respectively.
The diagonalization procedure determines the parameters characterizing the squeezing, displacement and rotation, giving rise to U as functions of the effective coupling strength λ and the atom and field frequencies Ω and ω. In fact, it turns out that four parameters -v, ϕ, ν d and ν f -are independent. With these results it is now possible to calculate the Berry phase acquired by the atom due to its interaction with the field.
Berry showed that an eigenstate of a quantum system acquires a phase, in addition to the usual dynamical phase, when the parameters of the Hamiltonian are adiabatically and cyclicly varied with time [1,16,17]. In the case of a point-like atom interacting with a quantum field, the motion of the atom in spacetime gives rise to Berry's phase provided that the changes in the hamiltonian are adiabatic and cyclic. This phenomenon was recently exploited to propose an experiment to detect the Unruh effect with much smaller accelerations than previously considered [2]. Here we use it to build a quantum thermometer: when an atom interacts with a thermal state of a bosonic field, the geometric phase acquired is a function of the temperature of the probed state. The Berry phase acquired by the eigenstate |ψ(t) of a system, whose Hamiltonian depends on k cyclicly, is obtained by adiabatically varying the vector of parameters: where R is a closed trajectory in the parameter space and As a first step, we calculate the Berry phase acquired by an eigenstate of the Hamiltonian under cyclic and adiabatic evolution of the parameters (v, ϕ, ν f , ν d ). The only non-zero component of A is , n , and so The Berry phase acquired by an eigenstate U † |0, n is is the density matrix of the thermal state, where Before turning on the interaction between the field and atom the system is in the mixed state |0 0|⊗ρ T . Turning the interaction on adiabatically leaves the system in the state ρ = U † (|0 0| ⊗ ρ f ) U .
For a system in a mixed state ρ = i ω i |i i| where |i are eigenstates of the Hamiltonian, after a cyclic and adiabatic evolution [18] the state acquires a geometric phase γ = Re η where where γ i is the Berry phase acquired by the eigenstate |i . Under one cycle of evolution for the state ρ T we have that e iη = 1 cosh 2 r T n tanh 2n r T e iγI n . where Therefore, the Berry phase accrued by the state of the atom interacting with a thermal state is Let us compare the Berry phase acquired by the same multi-level system interacting with two thermal sources at different temperatures. After a complete cycle in the parameter space (near-resonance this means waiting a time 2πΩ −1 ) the phase difference between these two atoms is equal to: For realistic coupling values for atoms in cavities the phase difference δ is of considerable magnitude as shown in Fig. 1; it is also very sensitive to a particular range of temperatures, depending on the atomic gap. Adjusting this gap, we tune δ to a particular temperature range, where we assume a hot source to have the temperature 3 orders of magnitude greater than the temperature of the target source. While δ is quite sensitive to variations of the cold source, it is rather insensitive to changes in the hot source. Consequently we can obtain ultra-high precision measurements of the temperature of the cold source with almost no need to control the temperature of the hot one. In Fig. 2 we illustrate the effects of underestimating the temperature of the hot source on the precision of the measurement of δ. Large variations of the hot source temperature translate into very small variations of the measured phase, providing us with an ultra-high precision thermometer, able to measure temperatures 3 orders of magnitude smaller than the hot source, over which no high precision temperature control is required.
The Berry phase is always a global phase. In order to detect it, it is necessary to prepare an interferometric  experiment. For example, a multi-level system in a superposition of trajectories that go through different thermal sources would allow for detection of the phase. Any experimental set-up in which such a superposition can be implemented would serve our purposes. A possible scenario can be found in the context of atomic interferometry. This technology has already been successfully employed to measure with great precision general relativistic effects such as time dilation due to Earth's gravitational field [19]. Paths (of slightly different length) can be chosen such that the dynamical relative phase cancels. It is sufficient that the dynamical phase difference throughout both trajectories be equal or smaller than the geometric phase to allow for its detection. Although such cancellation depends upon the specific experimental setup, we find that even for a simple setting with current length metrology technology [20], we can control the relative dynamical phase with a precision ∆φ ≈ 10 −4 , several orders of magnitude smaller than the Berry phase acquired in one cycle. Possible limitations of the results.-One might expect the atoms to lose coherence upon interacting with thermal sources, thus rendering the interferometry experiment useless. However for energy gaps and temperatures discussed here, to have decoherence times comparable with the time for one cycle the thermal sources must be at temperatures several orders of magnitude above the ones we are considering. Furthermore, for this formalism to be valid we require that the probability of finding  FIG. 2: Relative error in the Berry phase (and therefore, the determination of the temperature for the cold source) as a function of the relative error in determining the temperature of the hot source. As we see, the setting is very robust: huge changes of temperature of the hot source translate into small changes in the phase δ.
the atom in an excited state after one cycle of evolution should be much smaller than 1 (weak adiabadicity), in turn implying coherence is not lost. In general we expect this condition to fail in three regimes: small atomic gap, high temperatures or high coupling. When it holds, the interaction time of the multi-level atom with the thermal state should be short enough so that the only change in the atom's state is that it acquires a global phase (dynamical + geometrical).
To investigate under what regimes the time evolution of the coupled field-atom system fulfills this weak adiabaticity condition, we check that the ground state of the atom for the Hamiltonian H(t 0 ) evolve after a time t − t 0 (equal to one cycle of evolution) to the ground state of the Hamiltonian H(t) when the field is in a thermal state [2] [21], a fact easily demonstrated for realistic values of the coupling λ.
Solving the exact Schwinger equation (in the interaction picture): numerically [22], we find that indeed, after a short period of time (approximately 10 4 · 2πΩ −1 for realistic couplings), the probability of finding the atom in the excited state cannot be distinguished from thermal noise, as expected. However, for our purposes, we will only have the atoms interacting with the thermal bath for very short times (1 cycle of evolution t ≈ 2πΩ −1 ), and for these times we can see that for realistic values, the hypothesis holds perfectly. In Fig. 3 we can see that for the cases considered, even in the worst case scenario (1 Mhz gap and 1 mK temperature) the probability of excitation is P ≈ 10 −3 ≪ 1. In the best case scenario considered (1 Ghz gap and 1 K temperature) the probability is P ≈ 10 −9 ≪ 1 and the approximation holds very well. In other words, the atoms interact with the thermal bath a time short enough to avoid being excited; the only effect of the interaction is that they acquire a phase dependent on the bath's temperature.
If we consider the atom interacting with a continuum of modes, we can see how the resonant component is dominant in this case too: a computation in first-order perturbation theory with the contimuum gives similar bounds to the maximum temperature of the sources so that the adiabaticity condition holds after considering more than 100 cycles of evolution.
Furthermore, the probability remains negligible for many evolution cycles. Even in the worst-case-scenario considered, with realistic couplings and for an atomic gap of 1 Ghz, the approximation remains valid for more than 100 cycles until thermal fluctuations render the hypothesis invalid. Considering one cycle of evolution, we need to demand that the transition probability P ≪ 1. For the scenarios considered we find for the best case scenario P ≈ 10 −9 (Ghz gap), while for the worst case scenario P ≈ 10 −3 (Mhz gap).
As a final remark, the thermometer has a target temperature range for which it is highly sensitive. As seen in Fig. 1, this range is typically of 3 orders of magnitude. To see an interference pattern in an atomic interference experiment that allows measurement of the target temperature, we need also a reference source at a fixed temperature for which we need to have some control but whose temperature may experience variations. To satisfy the necessary condition for reducing the influence of fluctuations in the reference source, in a manner shown in Fig. 2, we need the temperature of the source to be 3 orders of magnitude different from the target temperature. In principle, the source could be hotter or colder than the target , but of course, it is much easier to control warmer thermal sources than colder ones. As a final note, geometric phases can be generalized to non-adiabatic and non-cyclic cases. We are currently considering generalizations of the thermometer in these directions.