Vacuum condensate, geometric phase, Unruh effect and temperature measurement

In our previous work it has been shown the possibility to use the Aharonov-Anandan invariant as a tool in the analysis of disparate systems, including Hawking and Unruh effects, as well as graphene physics and thermal states. We show that the vacuum condensation, characterizing such systems, is also related with geometric phases and we analyze the properties of the geometric phase of systems represented by mixed state and undergoing a nonunitary evolution. In particular, we consider two level atoms accelerated by an external potential and interacting with a thermal state. We propose the realization of Mach-Zehnder interferometers which can prove the existence of the Unruh effect and can allow very precise measurements of temperature.


INTRODUCTION
Phenomena like Unruh [1], Hawking [2] and Parker effects [3,4] are hardly revealed in the experiments. However it has been shown [5] that in such phenomena and in general in all the systems where vacuum condensates are generated [6]- [21], the Aharonov-Anandan invariant (AAI) [22] is produced in their evolution. Such an invariant represents the distance between quantum states along a given curve in projective Hilbert space as measured by the Fubini-Study metric. It has a geometric meaning analogous to the geometric phase [22]. Indeed, it has been demonstrate [23,24] that, by introducing a new distance function in the projective Hilbert space, the AAI is related to the geometric phase [25]- [39]. This fact implies that in all the above phenomena, in which the presence of AAIs has been revealed [5], an associated geometric phase also appears.
In contrast with the AAI, which is very hard to be experimentally observed, the geometric phase has been detected in many physical systems [40]- [44]. Therefore, we focus our attention on the analysis of the geometric phase and consider the Mukunda-Simon phase.
It has been shown that geometric phases and invariants can be used to test CPT symmetry in meson systems [45], to prove the existence of postulated particles like the axions [46], to test SUSY violation in thermal states [47], to reveal the Unruh effect [5,48,49] and to build a quantum thermometer [5,50,51]. The presence of geometric invariants has been shown also in other context [52,53].
The novelty resulting from the present paper is not solely in the exhibition of the intriguing relation between boson condensation, AAI and the geometric phase, rather, it is in the fact that it becomes possible to detect effects so far elusive to the observations, ranging from the Unruh effect to Casimir effect, including other phenomena of quantum field theory (QFT) in curved background. In addition our discussion provides us with the application of the construction of devices able to measure temperature accurately.
We reveal the relation between vacuum condensates and geometric phase, by using our previous results presented in Ref. [5], in which we have shown the relation between AAI and vacuum condensates, together with the results presented in [23,24], in which it is presented the relation between AAI and geometric phase. Moreover, we analyze the properties of the Mukunda-Simon (MS) [28] geometric phase generated by the non-unitary evolution of a two level atom and we study its implications in two particular cases: 1) atoms accelerated in electromagnetic field and 2) atoms interacting with thermal states. In the first case, we show that a detectable difference of the geometric phases appears between the accelerated and the inertial atoms. Such a phase difference is due only to the Unruh effect. In the second case we show that the difference between geometric phases produced by atoms interacting with two different thermal states allows to determine the temperature of a sample once the temperature of the another one (assumed as reference temperature) is known. This implies that the geometric phase can represent a useful instrument in the realization of a very precise quantum thermometer. The ideas of using geometric phases and invariants to probe the Unruh effect [5,48,49], and to build a thermometer [5,50] have been already presented in previous works, (the study of dynamical phase to have precise estimation of the temperature has been proposed in [51]). In this paper, besides the relation between boson condensation and geometric phase, we consider a realistic scheme for experimental implementations and we study the MS phase which generalize the Berry phase (used in [49,50]), to the case of non-cyclic, non-adiabatic and non unitary evolution. The use of the MS phase allows to consider time interval arbitrary small (we do not need to consider a cyclic system and a time interval equal to the period of the cycle), and to take into account transition frequencies very low and spontaneous emission rates characteristic of fine and hyperfine structure of the atoms. Indeed, in the time interval which we consider, the number of spontaneous emitted particle is negligible and the systems can be consider almost stable. The analysis of fine and hyperfine structure of atoms permits lower accelerations to detect Unruh effect and more precise temperature mea-surements with respect to the ones obtained by considering non-fine transition frequencies of the atoms, thus improving results limited to the use of Berry phase.
We consider in our analysis the structure of the atomic levels of 85 Rb, 87 Rb and 133 Cs. We show that a Mach Zendher interferometer of 4cm in which the hyperfine level of the above atoms is taken in consideration can be used to reveal Unruh effect with accelerations of order of 10 16 m/s 2 and a similar device could be used to have very precise measurement of the temperature.
In Section II we show the links relating AAI, geometric phase and vacuum condensates. In Section III we analyze the MS phase for an atom undergoing to a non-unitary evolution and in Sections IV and V we study the realization of a Mach Zehnder interferometer to reveal the Unruh effect and to build a very precise thermometer. Setion VI is devoted to the conclusions.

II. GEOMETRIC PHASE, AHARONOV-ANANDAN INVARIANT AND VACUUM CONDENSATE
The AAI is defined as S = (2/ ) t 0 ∆E(t ′ ) dt ′ [22]. It is generated in the time evolution of any system which has its state |φ(t) not in a stationary state, i.e. it has a nonzero value of the uncertainty ∆E(t) in energy, The AAI is the total length of the S-path measured using the Fubini-Study metric, which can be written as [24] Introducing the infinitesimal "reference distance" in the projective Hilbert space P, with , the geometric phase can be defined as [24], Such a formula matches the Aitchinson-Wanelik formula for the geometric phase and relates the AAI (which is two times the Fubini-Study metric) to the geometric phase. Moreover, we note that, since the length of the S-path is the minimum length of the path measured by the "reference distance" function D [23,24], the presence of the AAI in the evolution of a system implies the presence of the D invariant and consequentially the existence of the geometric phase (3).
The presence of the AAI has been shown to occur [5] in all the phenomena in which the vacuum condensate appears [1]- [21]. For these systems, the physically relevant states |Ψ(θ) , (θ ≡ θ(ξ, t), with ξ some physically relevant parameter) have indeed nonzero energy variance,

and AAI given by
with θ ′ ≡ θ(ξ, t ′ ). Here U k (θ) and V k (θ) are the Bogoliubov coefficients entering in the transformation |Ψ(θ) = J −1 (θ)|ψ(t) , with |ψ(t) original state and J −1 (θ) generator of the Bogoliubov transformation, U k and V k , depend on the system one consider and satisfy the relation |U k | 2 ± |V k | 2 = 1, with + for fermions and − for bosons. Notice that the vacuum state |0(θ) for such systems is related to the original one |0 by the relation, |0(θ) = J −1 (θ)|0 . Therefore, Eq. (3) shows that all the phenomena characterized by the presence of modifications of vacuum fluctuations, (which are all described by Bogoliubov transformations) are also characterized by the presence of the geometric phase in their evolution. The modifications of vacuum fluctuations are induced in disparate ways [1]- [21] and can produce particle creation from vacuum or vacuum condensate. Then the geometric phase may provide the possibility to study the properties of many systems and could be used to reveal phenomena very hard to be detected.
For example, in the case of the Unruh effect, the Bogoliubov coefficients that allow to express the Minkowski vacuum in terms of Rindler states are for bosons U k = e 2πω k /a e 2πω k /a +1 and V k = 1 e 2πω k /a −1 and similar for fermions. Here a is the acceleration of the observer. The relation between the Minkowski |0 M and Rindler |0 R vacua in the case of a single scalar field [54] is: where R and L refer to modes supported in the right and left Rindler wedges respectively.
Here we analyze the Mukunda-Simon phase generated by the non-unitary evolution of an open system. We consider a two level atom and we propose the use of a Mach-Zehnder interferometer to detect the Unruh effect and to build a quantum thermometer.

III. GEOMETRIC PHASE AND TWO LEVEL ATOMS
We consider the interaction of an atom with vacuum modes of the electromagnetic field in the multipolar scheme [55] and treat the atom as an open system with a non-unitary evolution in the reservoir of the electromagnetic field. The Hamiltonian of the atom and the reservoir is where ω 0 is the energy level spacing of the atom, σ 3 is the Pauli matrix, H F is the electromagnetic field Hamiltonian, µ mn is the matrix element of the dipole momentum operator connecting single-particle states u n and u n ′ (see [55]), σ mn = σ m σ n , and E is the strength of the electric field. Denoting by |0 and ρ(0) the vacuum and the initial reduced density matrix of the atom, respectively, we analyze the evolution of the total density matrix ρ tot = ρ(0) ⊗ |0 0|, in the frame of the atom. We assume a weak interaction between atom and field, then the evolution can be written as [56,57] ∂ρ with τ the proper time, L[ρ(τ )] the usual contribution containing dissipative terms (see [56]) and H ef f the effective hamiltonian, Here Ω is the renormalized energy level spacing containing the Lamb shift terms. Such terms can be neglected in the computation of the geometric phase, thus we approximate the effective level spacing of the atoms Ω with the atomic transition frequency ω 0 , i.e. Ω ∼ ω 0 . By writing ρ(τ ) in terms of Pauli matrices and considering the initial state of the atom, with θ ≡ θ(0), one can derive the reduced density matrix ρ(τ ) [5,48]. Such a matrix has non zero eigenvalue λ + only for the corresponding eigenvector (λ − (0) = 0) where and Σ and Υ are the Kossakowski coefficients [57].
The Mukunda-Simon phase for the state |φ + (t) is which reduces to with θ ≡ θ(0). If one consider a time interval equal to t = 2π/Ω, for which arg sin θ 2 sin θ(t) 2 + cos θ 2 cos θ(t) 2 e iΩt = 0, the result of Eq.(13) coincide with Eq.(15) of ref. [48], in which another definition of the geometric phase has been used [39]. Notice that the phase introduced in [39] represents the geometric phase defined for non-unitary evolution. On the other hand, the MS phase can be used for systems undergoing both a unitary evolution and a non-unitary evolution [28]. Moreover it generalize the Berry phase to the non-cyclic and non-adiabatic evolution case, therefore, the analysis of the MS phase is advantageous also in systems such as the one presented in [49].
In the next Sections we consider the use of Eq.(13) in the detection of Unruh effect and in the building of a quantum thermometer.

IV. UNRUH EFFECT
The Unruh effect consists in the fact that for an accelerated observer the ground state of an inertial system appears at a non-zero temperature depending on the acceleration of the observer. Such a phenomenon has not yet been detected. However, recently it has been shown that geometric phases and invariants could allow the detection of the Unruh effect in table top experiments [5,48,49].
Here, in particular, we show the possible realizability of an interferometer in which paths of slightly different lengths can be chosen in order to let the geometric phase be dominating over the relative dynamical phase. We compute the Mukunda-Simon phase for the two level system in the presence of an acceleration and in the inertial case. The atom interaction with the electromagnetic field itself produces a geometric phase; however, the difference between the two phases is due only to the atom acceleration and then to the Unruh effect, since the accelerated system sees the Minkowski vacuum as a thermal Rindler vacuum.
A two-level atom uniformly accelerated in the x direction with acceleration a, through Minkowski spacetime is conveniently described with Rindler coordinates x(τ ) = c 2 a cosh aτ c , t(τ ) = c a sinh aτ c . For this system, the coefficients Σ and Υ in Eqs.  [5,48], where γ 0 is the spontaneous emission rate and ω 0 is the atomic transition frequency. The function sin θ(t) and similar for cos θ(t) 2 . Here R a = Υ a /Σ a . For an inertial atom, a = 0, the geometric phase Φ a=0 assumes the identical expression of Eq. (13), with sin θa(t)  and cos θ0(t) 2 , in which the coefficients Σ a , Υ a , R a are replaced by Σ 0 = Υ 0 = γ 0 /4, with γ 0 spontaneous emission rate, and R 0 = 1, respectively.
The phase difference between the accelerated and inertial atoms, ∆Φ U (t) = Φ a (t) − Φ a=0 (t), gives the geometric phase in terms of the acceleration of the atom. The value of ∆Φ U (t) depends on the ratios a/(cω 0 ) and γ 0 /ω 0 and on the time interval t. Indeed ∆Φ U (t) increases for values of the acceleration a which approach to cω 0 , i.e. a ∼ cω 0 . Moreover, ∆Φ U (t) is detectable when γ 0 /ω 0 > 10 −5 . Therefore a crucial role is played by the choice of the atomic systems used in the interferometer. We consider an initial state with angle θ(0) = π/5 and, in order to decrease the value of the acceleration, we consider the hyperfine level structure of different atoms. If, for example, we consider the ground state 5 2 S 1/2 energy splitting of 85 Rb and 87 Rb, and we take into account the gap between the hyperfine energy levels F = 1 and F = 2 of 87 Rb, and the gap between the levels F = 2 and F = 3 of 85 Rb (F = J + I is the total atomic angular momentum, with J total electron angular momentum and I total nuclear angular momentum) we have a phase ∆Φ U ∼ 10 −4 π for accelerations of order of 10 17 m/s 2 and the times t of order of t ∼ 1/ω 0 , as shown in the inset of Fig.1. For such time interval, the speed of the atoms is of order of (0.2 − 0.3)c and the spontaneous emission can be neglected, since N (t ∼ 1/ω 0 ) ∼ 0.99N (0). The values of the phases obtained are accessible with the current technology. Better results can be also obtained by considering the 5 2 P 1/2 energy splitting between the F = 1 and F = 2 levels of the 87 Rb, and the 6 2 P 1/2 energy splitting between the F = 3 and F = 4 levels of the 133 Cs. In these cases ∆Φ U ∼ 10 −4 π can be achieved for accelerations of order of 10 16 m/s 2 and speeds of order of (0.2 − 0.3)c, as shown in the main plots of Fig.1. In these case, one has N (t ∼ 1/ω 0 ) ∼ 0.98N (0).
We now analyze the characteristic of a Mach-Zendher interferometer able to reveal the MS phase related to the Unruh effect. The MS phase can be detected when the dynamical phase is negligible compared with the geometric one. The total phase, for the accelerated and for the inertial atoms, is given in terms of the geometric phase Φ by the formula where the second term on the right side is the dynamical phase. We note that the dynamic phases can be made negligible compared to the geometric ones, if the branches of the interferometer are built in order that the two dynamical phases are almost equal, that is In this case the difference of total phases ∆Φ tot detected in the cross point of the interferometer corresponds almost completely to the difference of geometric phases ∆Φ, i.e. ∆Φ tot ≃ ∆Φ. For example, by considering as two level system the 5 2 P 3/2 energy splitting between the F = 0 and F = 1 levels of 87 Rb, and an acceleration of order of 5 × 10 16 m/s 2 , one has that the dynamical phase is less than the geometric one, δ ∼ 0.3∆Φ U in an interferometer with identical branches of length of 4 cm. In particular, the dynamical phase differences δ can be completely neglected when in a such interferometer there is a difference in the branch lengths of about 1µm.

V. QUANTUM THERMOMETER
In this Section we consider the interaction of an atom with thermal states. A geometric phase identical to the one in Eq.(13) appears also in this case. The analysis of the Mukunda-Simon phase in a Mach Zehnder interferometer could allow very precise measurement of the temperature.
For thermal states the coefficients Σ a and Υ a are replaced by the coefficients Σ T and Υ T depending on the temperature [5], . Thus an interferometer in which an atom follows two different paths and interacts with two thermal states at different temperatures can represent a very precise quantum thermometer. Indeed, if it is known the reference temperature of one thermal state, the temperature of the other one can be defined by measuring the difference between the geometric phases generated in the two paths.
For example, by measuring ∆Φ T , one can derive precise estimations of the temperature T c of the colder source, if it is known the temperature T h of the hotter source.
We consider the hyperfine structure of atoms and we plot ∆Φ T as function of the temperatures of cold sources T c , for different 133 Cs, and 87 Rb lines as reported in Fig. 2. The time considered are t ≃ 1 4ω0 s in order that the particle decay can be neglected. The result we obtain is that, considering the hyperfine structure of the atoms, one can measure temperatures of the cold source of ∼ 2 orders of magnitude below the reference temperature of the hot source.
In ref. [50] it has been considered the Berry phase generated by a generic atom coupled to only a single mode of a quantum field inside a cavity. Here we have studied the role of the geometric phase in the realistic case of the non unitary evolution of specific atoms interacting with thermal states. Notice that any quantum system interacting with an external field is an open system. Therefore, what it is really needed is the analysis of a geometric phase, such as the Mukunda-Simon phase, which can be used for non-unitary evolution. Moreover, the MS phase, contrarily to the Berry phase, covers the case of non-cyclic and non-adiabatic evolution, therefore we are not forced to consider time intervals equal to the period, but we can consider times arbitrarily small, in order to have negli- Also in this case, paths of slightly different lengths can be chosen in order to let the geometric phase be dominating over the relative dynamical phase.

VI. CONCLUSIONS
We have shown that all the phenomena where vacuum condensates appear generate geometric phases in their time evolution. This fact implies that the geometric phase could represents an useful tool to be used for the study and the understanding of phenomena in different fields of physics. In this paper, we have shown the advantages of using the MS phase for a two level atom system. We have shown that atoms with hyperfine structure of the energy levels, as for example 87 Rb, accelerated in an interferometer with branches of length of 4 cm and a difference in the branch lengths of 1µm could represent an efficient tool in the laboratory detection of the Unruh effect. On the other hand, we have shown that similar atoms, interacting with two different thermal states can be utilized in an interferometer to build a very precise quantum thermometer.