# Vacuum Condensate, Geometric Phase, Unruh Effect, and Temperature Measurement

Authors: Antonio Capolupo; Giuseppe Vitiello
Journal: Advances in High Energy Physics
 (2015)

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## Abstract

In our previous work the possibility to use the Aharonov-Anandan invariant as
a tool in the analysis of disparate systems has been shown, 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.


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## Body

## 1. Introduction

It is hard to observe phenomena like Unruh [1], Hawking [2], and Parker effects [3, 4]. However it has been shown [5] that in such phenomena as well as in all the systems where the vacuum condensates are generated [6–12], the Aharonov-Anandan invariant (AAI) [13] is produced in their evolution. Moreover, it has been shown [14, 15] that such an invariant is related to the geometric phase [16–37]. This fact suggests that in all the above phenomena, in which the presence of AAIs has been revealed [5], an associated geometric phase also appears. Then, the study of such a phase could open a new way to the detection of effects elusive to the detection.Here, instead of using AAI, which is experimentally difficult to be observed, we use geometric phases to study the Unruh effect and the possibility to perform very precise measurement of temperature. Geometric phases have been detected in many physical systems [38–42] and have been also related to CPT symmetry [43] and SUSY violation in thermal states [44].In the present paper, we focus our attention on systems of atoms accelerated in an electromagnetic field and atoms interacting with thermal states, so that we study geometric phases for mixed states in nonunitary, noncyclic evolution. Different approaches exist in dealing with geometric phases and mixed states [31–37]; in this paper we mostly follow the Wang and Liu approach [30].In our treatment, besides the relation between boson condensation and geometric phase, we obtain two novel results, one concerning atoms accelerated in an electromagnetic field and the Unruh effect and the other one exhibiting the possibility to perform very precise measurement of temperature. In the first case, 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, 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 other one (assumed as reference temperature) is known.The idea of using geometric phases and invariants to probe the Unruh effect [5, 45, 46] and to build a thermometer [5, 47] has been already presented in previous works (the study of dynamical phase to have precise estimation of the temperature has been proposed in [48]). In this paper we consider a realistic scheme for experimental implementations and we study the geometric phase defined in [30], which generalize the Berry phase (used in [46, 47]), to the case of quantum open systems. The use of the phase presented in [30] allows to consider time intervals arbitrary small (we do not need to consider cyclic evolutions and their related period). In a similar way the discussion reported in [30] allows to consider the cases of very low transition frequencies, as well as spontaneous emission rates characterizing fine and hyperfine atomic structures. Indeed, in the short time intervals which we consider, the number of spontaneously emitted particle is negligible and the systems are quasistable.The Berry phase for mixed states cannot be used in the study of the systems which we consider. One reason is that our systems do not undergo cyclic evolutions because of their interaction with the environment [30]. Even in the approximation of a quasicyclicity of mixed states, the number of spontaneously emitted particles becomes not negligible in time intervals of the order of T=2π/ω0 (with ω0 atomic transition frequency). Thus the analysis of the geometric phase loses meaning.Our treatment offers other improvements, which cannot be obtained by using the Berry phase. For example, in the present treatment, by considering the fine and hyperfine atomic structure, we have access to lower acceleration regimes, which in turn improves the detection of Unruh effect and permits very precise temperature measurements.We consider the structure of the atomic levels of85Rb, 87Rb, and 133Cs. We suggest that a Mach-Zehnder interferometer of 4 cm, where the hyperfine level of these atoms is considered, may reveal the Unruh effect with accelerations of order of 1016 m/s2 and that a similar device could be used also for very precise temperature measurements.In Section2 we link together AAI, geometric phase, and vacuum condensates and introduce the geometric phase for mixed states in nonunitary evolution. In Section 3 we analyze the geometric phase for a two-level atom undergoing a nonunitary evolution and in Sections 4 and 5 we study the realization of a Mach-Zehnder interferometer to reveal the Unruh effect and to build a very precise thermometer. Section 6 is devoted to the conclusions.


## 2. Geometric Phase

The geometric phase for pure states can be expressed as [15] (1)ΦΓ=∫ΓdD2-dS2=∫Γϕϕiddt-ψ˙tϕϕdt,which also establishes the relation with the AAI. Here dD2 denotes the infinitesimal “reference distance” in the projective Hilbert space P and is given by (2)
d
D
2
=
d
d
t
ϕ
ϕ
∣
d
d
t
ϕ
ϕ
+
ψ
˙
2
-
2
i
ψ
˙
ϕ
ϕ
∣
d
d
t
ϕ
ϕ
d
t
2
,with ψ(t)=i/2ln⁡〈ϕ(0)∣ϕ(t)〉/〈ϕ(t)∣ϕ(0)〉. In (1), dS2 is the Fubini-Study metric given by(3)
d
S
2
=
d
d
t
ϕ
ϕ
∣
d
d
t
ϕ
ϕ
-
i
ϕ
ϕ
∣
d
d
t
ϕ
ϕ
2
d
t
2
.The AAI is the total length of the path measured using the Fubini-Study metric S expressed as S=(2/ħ)∫0t‍ΔE(t′)dt′ [13]. In the above formulas |ϕ(t)〉 is a nonstationary state with energy uncertainty ΔE(t) given by ΔE2(t)=〈ϕ(t)|H2|ϕ(t)〉-(〈ϕ(t)|H|ϕ(t)〉)2. Since the length of the path “S” is the minimum length measured by the “reference distance” function “D” [14, 15], 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 (1). As already mentioned in the Introduction, the AAI is produced in the evolution of systems with vacuum condensates [5] (see the Appendix); then (1) suggests that vacuum condensation is also related with geometric phases.Vacuum fluctuations may produce particle creation from vacuum or vacuum condensate in disparate ways [1–12]. Then the geometric phase may provide the possibility to study the properties of many systems and could be used 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.As said, in the present paper we focus on the Unruh effect and on thermal states, studying the possible realization of a quantum thermometer. In our analysis we consider quantum open systems and analyze the geometric phase for mixed states in nonunitary, noncyclic evolution. In particular, we use the Wang and Liu approach [30] and the geometric phase defined as(4)Φg=1N·∑k=1Narg⁡1+brt0φk1+brtφkφkt0∣φkt-1NI∑k=1N∫t0t1+brt′φkφkt′∂∂t′φkt′dt′,where the first term of the right side represents the total phase, while the second term is the dynamic one. In (4), r(t) is the Block radius, satisfying the condition 0≤r(t)≤1, b=N(N-1)/2, and φk are the eigenvalues of the operator ∑i=1N2-1‍ni(t)λi (with λi traceless and hermitian N×N operators and ni(t)=(N/2b)Trλiρ(t)) which permits to write ρ(t) as ρ(t)=(1/N)(1+b∑i=1N2-1‍ni(t)λi). Moreover, |φk(t)〉 and λk(t)≡1+br(t)φk/N are eigenstates and eigenvalues of the density matrix ρ of N-level mixed states, respectively:(5)ρφkt=1+brtφkNφkt.Then the geometric phase (4) can be expressed also as(6)Φg=∑k=1Narg⁡λkt0λktφkt0∣φkt-I∑k=1N∫t0tλkt′φkt′∂∂t′φkt′dt′.We consider a two-level open system. In this case,N=2, b=1, and the density matrix ρ is expressed in terms of Pauli matrices σi as (7)ρt=121+∑i=13nitσi,with ni=Tr(ρσi) and radius of Block sphere given by r(t)=n12+n22+n32. Explicitly, one has(8)n1=ρ12+ρ21,n2=iρ12-ρ21,n3=ρ11-ρ22.Introducing the angles such as (9)θ=cos-1⁡n3r,ϕ=tan⁡n2n1,the eigenvectors |φ1(t)〉 and |φ2(t)〉 (apart from overall phase factors which do not contribute to Φg) are(10)
φ
1
t
=
cos
⁡
θ
t
2
e
i
ϕ
t
sin
⁡
θ
t
2
,
φ
2
t
=
sin
⁡
θ
t
2
-
e
i
ϕ
t
cos
⁡
θ
t
2
,and the eigenvalues of the operator ∑i=13‍ni(t)σi are φ1=1 and φ2=-1. Therefore λ1(t)=(1/2)[1+r(t)] and λ2=(1/2)[1-r(t)].


## 3. Geometric Phase and Two-Level Atoms

We consider the interaction of an atom with vacuum modes of the electromagnetic field in the multipolar scheme [49] and treat the atom as an open system with a nonunitary evolution in the reservoir of the electromagnetic field. The Hamiltonian of the atom and the reservoir is (11)H=ħ2ω0σ3+HF-∑mnμmn·Extσmn,where ω0 is the energy level spacing of the atom, σ3 is the Pauli matrix, HF is the electromagnetic field Hamiltonian, μmn is the matrix element of the dipole momentum operator connecting single-particle states un and un′ (see [49]), σmn=σmσn, 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 ρtot=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 [50, 51] (12)∂ρτ∂τ=-iħHeff,ρτ+12∑i,j=13aij2σjρσi-σiσjρ-ρσiσj.Here τ is the proper time and aij are the coefficients of the Kossakowski matrix (13)aij=Σδij-iΥϵijkδk3-Σδi3δj3,with (14)Σ=14Gω0+G-ω0,Υ=14Gω0-G-ω0,(15)Gω=∫-∞∞dτeiωτG+xτ.Gω is the Fourier transform of G+(x-y):(16)G+x-y=e2ħ2∑i,j=13+ri--rj+0EixEjx0.Heff is the effective hamiltonian, Heff=(ħ/2)Ωσ3, where 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 ω0; that is Ω~ω0.By writingρ(τ) in terms of Pauli matrices, as in (7), and considering the initial state of the atom, (17)ψ0=cos⁡θ2++sin⁡θ2-,with θ≡θ(0), one can derive the reduced density matrix ρ(τ) [5, 45](18)
ρ
τ
=
e
-
4
Σ
τ
cos
2
⁡
θ
2
+
Υ
-
Σ
2
Σ
e
-
4
Σ
τ
-
1
1
2
e
-
2
Σ
τ
-
i
Ω
τ
sin
⁡
θ
1
2
e
-
2
Σ
τ
+
i
Ω
τ
sin
⁡
θ
1
-
e
-
4
Σ
τ
cos
2
⁡
θ
2
-
Υ
-
Σ
2
Σ
e
-
4
Σ
τ
-
1
.Denoting byξ(τ) and χ(τ) the following quantities(19)ξτ=χ2+e-4Στsin2⁡θ,χτ=e-4Στcos⁡θ+ΥΣe-4Στ-1,the matrix (18) can be written as(20)
ρ
τ
=
1
2
χ
+
1
e
-
i
Ω
τ
ξ
2
-
χ
2
e
i
Ω
τ
ξ
2
-
χ
2
1
-
χ
.Then, (8), the Block radius and the angles defined in (9) are given by (21)n1τ=ξ2-χ2cos⁡Ωτ,(22)n2τ=ξ2-χ2sin⁡Ωτ,(23)n3τ=χ,(24)rτ=ξ,(25)θτ=cos-1⁡χξ,ϕτ=Ωτ,respectively. The eigenvalues of ρ(τ) are (26)λ±=121±r=121±ξ,and the corresponding eigenvectors are (27)ϕ+τ=cos⁡θτ2++sin⁡θτ2eiΩτ-,ϕ-τ=sin⁡θτ2+-cos⁡θτ2eiΩτ-.We consider the initial time t0=0. Being λ-(0)=0, the phase (6) becomes (28)Φgt=arg⁡λ+0λ+tϕ+0∣ϕ+t-I∫t0tλ-t′ϕ-t′∂∂t′ϕ-t′dt′-I∫t0tλ+t′ϕ+t′∂∂t′ϕ+t′dt′,which reduces to (29)Φgt=arg⁡cos⁡θ2cos⁡θt2+sin⁡θ2sin⁡θt2eiΩt-Ω2∫0t1-ξτcos⁡θτdτ,with θ≡θ(0).In the next sections we consider the use of (29) in the detection of Unruh effect and in the building of a quantum thermometer.


## 4. Unruh Effect

For an accelerated observer the ground state of an inertial system appears at a nonzero temperature depending on the acceleration of the observer. Such a phenomenon is called the Unruh effect. It has not yet been detected. Recently it has been shown that geometric phases and invariants could allow its detection in table top experiments [5, 45, 46].Here we show that the realization of an interferometer is possible, 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 geometric phase (29) 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 thex direction with acceleration a, through Minkowski spacetime, is conveniently described with Rindler coordinates x(τ)=c2/acosh⁡(aτ/c), t(τ)=c/asinh⁡(aτ/c). For this system, the field correlation function is given by [5, 45] (30)G+x,x′=e2-r+216π2ε0ħc7a4sinh4⁡a/2cτ-τ′-iε,and its Fourier transform is(31)Gω=ω3e2-r+26πε0ħc31+a2c2ω21+coth⁡πcωa.Then, the coefficients Σ and Υ in (14) become [5, 45](32)Σa=γ041+a2c2ω02e2πcω0/a+1e2πcω0/a-1,Υa=γ041+a2c2ω02,where γ0 is the spontaneous emission rate and ω0 is the atomic transition frequency. The function sin⁡(θt/2)=±(1/2)1-χ(t)/ξ(t) in (29) becomes (33)sin⁡θat2=±12-Ra-Rae4Σat+cos⁡θ2e4Σatsin2⁡θ+Ra-Rae4Σat+cos⁡θ2,and similar for cos⁡(θat/2). Here Ra=Υa/Σa.For an inertial atom,a=0, the geometric phase Φa=0 assumes the identical expression of (29), with sin⁡(θat/2), cos⁡(θat/2), and cos⁡θa(t) replaced by sin⁡(θa=0t/2), cos⁡(θa=0t/2), and cos⁡θa=0(t) in which the coefficients Σa, Υa, Ra are replaced by Σa=0=Υa=0=γ0/4, with γ0 spontaneous emission rate, and Ra=0=1, respectively.The phase difference between the accelerated and inertial atoms,ΔΦU(t)=Φa(t)-Φa=0(t), gives the geometric phase in terms of the acceleration of the atom. The value of ΔΦU(t) depends on the ratios a/(cω0) and γ0/ω0 and on the time interval t. Indeed ΔΦU(t) increases for values of the acceleration a which approach cω0; that is a~cω0. Moreover, ΔΦU(t) is detectable when γ0/ω0>10-5. Therefore a crucial role is played by the choice of the atomic systems used in the interferometer. Moreover, ΔΦU(t) depends on θ(0); it reaches the maximum value for θ(0)=π/2. We consider an initial state with angle θ(0)~π/2 and, in order to decrease the value of the acceleration, we consider the hyperfine level structure of different atoms.In Figure1 we plot the difference of geometric phase ΔΦU as function of the acceleration a for different systems.Figure 1
Plots ofΔΦU as a function of the atom acceleration a, for time intervals t≃1/ω0 and the splitting between the hyperfine energy levels: main pictures: (gray) dot dashed line: 87Rb, 52P1/2 line, splitting for F=1→F=2 transition (ω0=814.5MHz, γ0=36.129 MHz [53]); (green) solid line: 133Cs, 62P1/2 line, splitting of the between the F=3 and F=4 levels (ω0=1167.68 MHz, γ0=28.743MHz [54]). Pictures in the inset: (blue) dashed line: 85Rb, 52S1/2 line, energy splitting between the levels F=1 and F=2 (ω0=3.035GHz, γ0=36.129MHz for D1 transition, γ0=38.117 MHz for D2 transition [52]); (red) dot dashed line: 87Rb, 52S1/2 line, energy splitting between the levels F=1 and F=2 (ω0=6.843GHz, γ0=36.129MHz for D1 transition, γ0=38.117 MHz for D2 transition [53]).The (blue) dashed line in the inset of the figure is obtained considering the energy splitting between the levelsF=1 and F=2 of the ground state 52S1/2 of 85Rb (F=J+I is the total atomic angular momentum, with J total electron angular momentum and I total nuclear angular momentum). For this system one has ω0=3.035 GHz, and we considered the D1 transition for which γ0=36.129 MHz [52]. A similar plot is obtained if we consider the D2 transition for which γ0=38.117 MHz [52]. We derive the (red) dot dashed line in the inset by studying the energy splitting between the levels F=1 and F=2 of the 52S1/2 line of 87Rb. In this case, ω0=6.843 GHz and we consider the D1 transition with γ0=36.129 MHz [53]. Similar plot can be derived in the case of the D2 transition with γ0=38.117 MHz [53]. Such plots show that a phase ΔΦU~10-4π is obtained for accelerations of order of 1017 m/s2 and the times t of order of t~1/ω0, (inset of Figure 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/ω0)~0.99N(0). The values of the phases obtained are accessible with the current technology.Better results can be also obtained by considering other splitting between the levels of the87Rb, and the 62P1/2 energy splitting between the levels of the 133Cs, as shown in the main pictures of Figure 1. Here, the (gray) dot dashed line represents ΔΦU for the 52P1/2 energy splitting between the F=1 and F=2 levels of the 87Rb. In this case, ω0=814.5 MHz and γ0=36.129 MHz [53]. The (green) solid line is achieved by considering the 62P1/2 line splitting between the F=3 and F=4 levels of 133Cs. For this system, ω0=1167.68 MHz and γ0=28.743 MHz [54]. In these cases ΔΦU~10-4π can be achieved for accelerations of order of 1016 m/s2 and speeds of order of (0.2-0.3)c, as shown in the main plots of Figure 1 and one has N(t~1/ω0)~0.98N(0).We now analyze the characteristic of a Mach-Zehnder interferometer able to reveal the geometric phase (28) related to the Unruh effect. The geometric 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 Φg by the formula (34)Φtott=Φgt+Ω2∫0t1-ξτcos⁡θτdτ,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(35)
δ
=
Ω
2
∫
0
t
′
1
-
ξ
a
τ
cos
⁡
θ
a
τ
d
τ
-
∫
0
t
1
-
ξ
a
=
0
τ
cos
⁡
θ
a
=
0
τ
d
τ
≪
Δ
Φ
U
.In this case the difference of total phases ΔΦtot detected in the cross point of the interferometer corresponds almost completely to the difference of geometric phases ΔΦ; that is ΔΦtot≃ΔΦ. For example, by considering as two-level system the 52P1/2 energy splitting between the F=1 and F=2 levels of 87Rb, and an acceleration of order of 5 × 1016 m/s2, one has that, in an interferometer with branches of length of 4 cm, the dynamical phase differences δ can be completely neglected when in such interferometer there is a difference in the arm lengths of about 0.1 μm.


## 5. Quantum Thermometer

In this section we consider the interaction of an atom with thermal states. A geometric phase identical to the one in (29) appears also in this case. The analysis of the geometric phase (29) in a Mach-Zehnder interferometer could allow very precise measurement of the temperature.For thermal states, the coefficientsΣa and Υa are replaced by the coefficients ΣT and ΥT depending on the temperature [5], ΣT=(γ0/4)(1+4π2kB2T2/ħ2ω02)(eE0/kBT+1)/(eE0/kBT-1),E0=ħω0, and ΥT=(γ0/4)(1+4π2kB2T2/ħ2ω02).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 the reference temperature of one thermal state is known, 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 measuringΔΦT, one can derive precise estimations of the temperature Tc of the colder source, if the temperature Th of the hotter source is known.We consider the hyperfine structure of atoms and we plot in Figure2  ΔΦT as function of the temperatures of cold sources Tc, for different 133Cs, and 87Rb lines and Th values. In such a figure, the (blue) dashed line is obtained for the energy splitting between the levels F=4 and F=5 of the 62P3/2 line of 133Cs. In this case ω0=251.09 MHz and γ0=32.889 MHz [54] and we considered a reference temperature Th=10-2 K. The (red) dot-dashed line represents ΔΦT for the F=1→F=2 transition of the 52P1/2 line of 87Rb, for which ω0=814.5 MHz and γ0=36.129 MHz [53]. A value of Th=3×10-2 K has been taken into account. We derive the (gray) solid line by considering Th=6×10-2 K and the splitting between F=3 and F=4 levels of the 62P1/2 line of 133Cs, for which ω0=1167.68 MHz and γ0=28.743 MHz [54]. Moreover, we obtain the (black) dotted line by analyzing the F=3→F=4 transition of the 62S1/2 line of 133Cs. In this case ω0=9.192 GHz, γ0=28.743 MHz [54], and Th=1 K.Figure 2
Plots ofΔΦT as function of the temperatures of cold sources Tc, for the splitting between the hyperfine energy levels and Th values: (blue) dashed line: 133Cs, 62P3/2 line, splitting for F=4→F=5 transition (ω0=251.09 MHz, γ0=32.889 MHz [54]) and Th=10-2 K; (red) dot-dashed line: 87Rb, 52P1/2 line, splitting for F=1→F=2 transition (ω0=814.5 MHz, γ0=36.129 MHz [53]), and Th=3×10-2 K; (gray) solid line: 133Cs, 62P1/2 line, splitting for F=3→F=4 transition (ω0=1167.68 MHz, γ0=28.743 MHz [54]) and Th=6×10-2 K; (black) dotted line: 133Cs, 62S1/2 line, splitting for F=3→F=4 transition (ω0=9.192 GHz, γ0=28.743 MHz [54]) and Th=1 K. The time considered is t≃1/4ω0 s.The time considered ist≃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 [47] the Berry phase generated by an atom coupled just to a single mode of a quantum field within a cavity is studied. Here we have studied the role of the geometric phase in the realistic case of the nonunitary evolution of specific atoms interacting with thermal states. Notice that any quantum system interacting with an external field is an open system. Therefore, what is really needed is the analysis of a geometric phase, such as the one in (29), which is defined for mixed states with nonunitary evolution. Moreover, the phase equation (29), contrarily to the Berry phase and its generalization to the mixed state [37], covers the case of noncyclic and nonadiabatic 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 negligible spontaneous decay in such intervals for the energy level splitting analyzed. 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.Our results are thus realistic and new since they refer to specific atoms and their effective nonunitary evolution.


## 6. Conclusions

We have shown that all the phenomena where vacuum condensates appear generate geometric phases in their time evolution. In particular, we have analyzed the geometric phase for mixed state with a nonunitary evolution for a two-level atom system. We have shown that atoms with hyperfine structure of the energy levels, as for example87Rb, accelerated in an interferometer with branches of length of 4 cm and a difference in the branch lengths of 0.1 μm 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.



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*Source: 878043-2015-07-07.xml*