Geometric phase and its applications to fundamental

Geometric phase and its applications to fundamental physics Antonio Capolupo and Giuseppe Vitiello Dipartimento di Fisica E.R.Caianiello and INFN gruppo collegato di Salerno, Universit´ a di Salerno, Fisciano (SA) - 84084, Italy We report on recent results showing that the geometric phase can be used as a tool in the analysis of many different physical systems, as mixed boson systems, CPT and CP violations, Unruh effects, and thermal states. We show that the geometric phases appearing in the time evolution of mixed ¯s0 and the K 0 − K ¯ 0 are linked to the parameter z describing the CP T meson systems like Bs0 − B violation. A non zero phase difference between particle and antiparticle arises only in the presence of CP T symmetry breaking. Then the geometric phase can represent a completely new test for the CP T invariance. Moreover, we study the geometric phase of systems represented by mixed state and undergoing a nonunitary evolution and propose the realization of interferometers which can prove the existence of the Unruh effect and can allow very precise measurements of temperature. We are glad to dedicate this paper to Professor Gaetano Vilasi in the occasion of his 70th birthday. PACS numbers: 03.65.Vf, 11.10.-z, 04.62.+v

INTRODUCTION

In the recent years, many studies have been devoted to the analysis of the geometric phase [1]–[24]. It characterizes the evolution of many physical systems and has been detected in different ways [25]-[29]. On the other hand, it has been shown [30, 31] that in all the systems where the vacuum condensates are generated [32]-[40], the Aharonov–Anandan invariant [4] (AAI) and the geometric phase are produced in their evolution. This fact has suggested that phenomena like Unruh [36], Hawking [37] and Parker effects [38, 39], characterized by the presence of a vacuum condensate and very hard to be detected, togheter with some aspect of quantum field theory in curved space, could be analyzed by means of the geometric phase of atomic systems which simulate such phenomena [30, 31]. It has been shown also that geometric phases and invariants can be used to test CPT symmetry in meson ¯s0 system [41], to prove the existence of postulated particles like the axions [42], to test systems like kaons and Bs0 − B SUSY violation in thermal states [43], to reveal the Hawking [30] and Unruh [30, 31, 44, 45] effect and to build a quantum thermometer [30, 31, 46]. In the present paper, we report the results obtained by studying the geometric phase for mixed meson systems [41] and for quantum open systems represented by atoms accelerated in an electromagnetic field and interacting with thermal states [31]. We reveal the relation between the geometric phase appearing in the evolution of mixed mesons and the parameter denoting the CP T violation [41]. We show that a non-trivial phase difference between particle and antiparticle indicates unequivocally the CP T symmetry breaking in mixed bosons. We also show that the phases ¯ 0 mesons [41]. can be useful for the study of the CP violation and we do a numerical analysis for Bs0 − B s We then study the geometric phase for quantum open systems [16] and we show that a detectable difference of the geometric phases can be revealed between atoms which are accelerated in an arm of an interferometer and atoms which are inertial in the other branch. Such a phase difference is due only to the Unruh effect [30, 31]. Moreover, 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 other one is known [30, 31]. The use of the phase defined in [16] allows to consider time intervals arbitrary small and 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 quasi-stable. These facts permits to improve the results obtained in previous works studying different systems [44–46]. We consider the structure of the atomic levels of 85 Rb, 87 Rb and 133 Cs which improves the detection of Unruh effect and permits very precise temperature measurements. The structure of the paper is the following: in Section 2 we diagonalize the effective Hamiltonian of mixed meson systems by means of a complete biorthonormal set of states; we compute the geometric phase for mixed mesons and we show their links with CP T and CP violations. A numerical analysis for Bs mesons is also presented. In Section 3 we analyze the geometric phase for a two level atom undergoing a non-unitary evolution and study the possible applications to the detection of the Unruh effect and to build a very precise thermometer. Section 4 is devoted to the conclusions. We are glad to dedicate this paper to Professor Gaetano Vilasi in the occasion of his 70th birthday.

2 MESON MIXING, CP AND CPT VIOLATIONS AND GEOMETRIC PHASE

The particle mixing phenomenon has been analyzed thoroughly in the contexts of quantum mechanics (QM) [47]– [50] and of quantum field theory (QFT) [33]–[35], [51] –[55]. Although the QFT analysis discloses features which cannot be ignored (see for example Refs.[34] [53]–[56]), nevertheless a correct phenomenological description of systems ¯ 0 can be also dealt with in the context of QM. Therefore, in the following we analyze the mixed bosons in as B 0 − B the context of QM. The state vector of mixed boson systems as K 0 , Bd0 , Bs0 and D0 can be represented as |ψ(t)⟩ = ψM 0 (t)|M 0 ⟩ + ¯ 0 ⟩ + ∑ dn (t)|n⟩ , where M 0 is the meson state (K 0 , B 0 , Bs0 or D0 ); M ¯ 0 is the corresponding antiparticle, ψM¯ 0 (t)|M d n |n⟩ are the decay products, t is the proper time, ψM 0 (t), ψM¯ 0 (t) and dn (t) are time dependent functions. At initial ¯ 0 ⟩ . The time evolution of |ψ(t)⟩ can be described, in the space time t = 0 one has |ψ(0)⟩ = ψM 0 (0)|M 0 ⟩ + ψM¯ 0 (0)|M T d 0 0 ¯ , where Ψ = (ψM 0 (t) , ψM¯ 0 (t)) and formed by |M ⟩ and |M ⟩ by means of the Schr¨odinger equation i ( dt Ψ = H Ψ ) H11 H12 H is the effective non-Hermitian Hamiltonian of the system, H = . Here H = M − i Γ2 , with M and H21 H22 Γ Hermitian matrices. The matrix elements of H are constrained by the conservation of discrete symmetries [50]. Indeed, CP T conservation implies H11 = H22 , T conservation requires |H12 | = |H21 | and CP conservation imposes H11 = H22 and |H12 | = |H21 |. In particular, the CP violation, (i.e. |H12 | ̸= |H21 |), makes the Hamiltonian H non-Hermitian, H = ̸ H† and non† 1 normal, [H, H ] ̸= 0 . Thus, the left and right eigenstates of H are independent sets of vectors that are not connected by complex conjugation, then the diagonalization of H needs the use of non-Hermitian quantum mechanics. We will use the biorthonormal basis formalism [57] - [59]. We denote with λj = mj − iΓj /2, with j = L, H (L denotes the light mass state and H the heavy mass state) the eigenvalues of the Hamiltonian H and with |Mj ⟩, the corresponding eigenvectors, H|Mj ⟩ = λj |Mj ⟩ . Denoting fj ⟩, (j = L, H) the eigenvectors of H† , the eigenvalues of H† are the complex conjugate of those of H, with |M fj ⟩ . Notice that the conjugate states ⟨M fj |† ≡ |M fj ⟩ and |Mj ⟩† ≡ ⟨Mj | are not isomorphic to their fj ⟩ = λ∗ |M H † |M j fj ⟩ ̸= |Mj ⟩ and ⟨Mj | ̸= ⟨M fj |. In this case, a complete biorthonormal set for H is given by {|Mj ⟩, ⟨M fj |}, with duals: |M f f j = L, H. Indeed one has the following biorthogonality relation ⟨Mi |Mj ⟩ = ⟨Mj |Mi ⟩ = δij , and the completeness ∑ fj ⟩⟨Mj | = 1 . The existence of a complete biorthonormal set of eigenvector of H implies fj | = ∑ |M relations j |Mj ⟩⟨M j that H is diagonalizable. Moreover, since the time evolution operator associated with H, U (t) = e−iHt is not unitary, then we introduce the e (t) = e−iH† t , such that U U e† = U e † U = 1. The time evolved of the states |Mk ⟩ and time evolution operator of H† , U −iλ t fk ⟩, (k = L, H) are thus |Mk (t)⟩ = U (t)|Mk ⟩ = e k |Mk ⟩ and |M fk (t)⟩ = U e (t)|M f ⟩ = e−iλ∗k t |M fk ⟩. |M √ k |p/q|−|q/p| |H12 |−|H21 | By introducing the CP parameter ε = |p/q|+|q/p| = |H12 |+|H21 | , where q/p = H21 /H12 , and the CP T parameter, −H11 ) ¯ 0 (t)⟩ in terms of the eigenstates of H |ML ⟩ and , we derive the correct meson states |M 0 (t)⟩ and |M z = (Hλ22 L −λH |MH ⟩ [41], which have to be used in the computations, 1 [√ 1 − z |ML ⟩ e−iλL t 2p [√ ¯ 0 (t)⟩ = 1 |M 1 + z |ML ⟩ e−iλL t 2q [√ iλL t g0 (t)| = p g ⟨M 1 − z ⟨M + L| e [√ iλL t g ¯ 0 (t)| = q g ⟨M 1 + z ⟨M − L| e |M 0 (t)⟩ =

+

√

] 1 + z |MH ⟩ e−iλH t ,

√ ] 1 − z |MH ⟩ e−iλH t , ] √ iλH t g 1 + z ⟨M , H| e ] √ iλH t g 1 − z ⟨M . H| e

−

(1) (2) (3) (4)

We now study the geometric phase for mixed mesons. We consider the phase for pure states with a diagonalizable non-Hermitian Hamiltonian HN H (t) [5] and analyze its extension to the biorthonormal basis formalism. In this ∫t case, the geometric phase is defined as [41] ΦN H (t) = arg⟨ψeN H (0)|ψN H (t)⟩ − ℑ 0 ⟨ψeN H (t′ )|ψ˙ N H (t′ )⟩dt′ , which is reparametrization invariant and it is invariant under the complex gauge transformations [41]. Here |ψN H (t)⟩ and |ψeN H (t)⟩ are the solution to the Schr¨ odinger equation i(d/dt)|ψN H (t)⟩ = HN H (t)|ψN H (t)⟩ and to its adjoint equation † e e i(d/dt)|ψN H (t)⟩ = HN H (t)|ψN H (t)⟩, respectively.

1

the matrices M and Γ do not commute, [M, Γ] ̸= 0

3 ¯ 0 one has the following phases [41]: In the particular case of mixed meson systems M 0 − M ∫ t 0 g g0 (t′ )|M˙ 0 (t′ )⟩dt′ , 0 ΦM 0 M 0 (t) = arg⟨M (0)|M (t)⟩ − ℑ ⟨M 0 ∫ t 0 g g 0 ¯ ¯ ¯ 0 (t′ )|M ¯˙ 0 (t′ )⟩dt′ , ΦM¯ 0 M¯ 0 (t) = arg⟨M (0)|M (t)⟩ − ℑ ⟨M

(5) (6)

0

g0 (0)|M ¯ 0 (t)⟩ − ℑ ΦM 0 M¯ 0 (t) = arg⟨M

∫

t

g0 (t′ )|M ¯˙ 0 (t′ )⟩dt′ , ⟨M

(7)

g ¯ 0 (t′ )|M˙ 0 (t′ )⟩dt′ . ⟨M

(8)

0

g ¯ 0 (0)|M 0 (t)⟩ − ℑ ΦM¯ 0 M 0 (t) = arg⟨M

∫

t

0

¯ 0 , respectively. They are connected ΦM 0 M 0 (t) and ΦM¯ 0 M¯ 0 (t) are the phases of the particle M 0 and of the antiparticle M to CP T violation parameter. ΦM 0 M¯ 0 (t) and ΦM¯ 0 M 0 (t) are the phases due to particle-antiparticle oscillations which are linked to CP violation (see below). Denoting with m = mL + mH , ∆m = mH − mL and ∆Γ = ΓH − ΓL , and assuming ∆Γt 2 small, which is valid in the ¯ 0 system [60]–[62], Eqs.(5) and (6) become range |∆t| < 15ps used in the experimental analysis on B 0 − B ) [ ( ) ( )] ( ∆mt ∆mt t ∆Γ ΦM 0 M 0 (t) ≃ arg cos + (ℑz − iℜz) sin + ∆m ℜz + ℑz , 2 2 2 2 [ ( ) ( )] ( ) ∆mt ∆mt t ∆Γ ΦM¯ 0 M¯ 0 (t) ≃ arg cos − (ℑz − iℜz) sin − ∆m ℜz + ℑz , 2 2 2 2 respectively (the explicit form of ΦM 0 M 0 and ΦM¯ 0 M¯ 0 for the general case are reported in Ref.[41]). Notice that ΦM 0 M 0 and ΦM¯ 0 M¯ 0 depend on the real and imaginary part of the CP T parameter z and the difference phase, ∆Φ(t) = ΦM 0 M 0 (t) − ΦM¯ 0 M¯ 0 (t) is due only to terms related to z. Indeed it is non–zero only the)]presence of [ in ( ∆mt T CP T CP T violation. In the case of CP T invariance, z = 0, one has ΦCP , which is ¯ 0M ¯ 0 (t) = arg cos M 0 M 0 (t) = ΦM 2 trivially equal to 0 or π and ∆Φ(t) = 0. On the other hand, the phases ΦM 0 M¯ 0 (t) and ΦM¯ 0 M 0 (t), for ∆Γt 2 ≪ 1 and omitting second order terms in z, (see Ref.[41] for the general expressions) are [ ( )] ( ) ( ) π mt p ∆mt ∆mt p ∆Γt p ΦM 0 M¯ 0 (t) ≃ − + arg sin − ℜ − ℑ , 2 2 q 2 2 q 2 q [ ( )] ( ) ( ) π mt q ∆mt ∆mt q ∆Γt q ΦM¯ 0 M 0 (t) ≃ − + arg sin − ℜ − ℑ . 2 2 p 2 2 p 2 p The phase difference is ΦM 0 M¯ 0 (t) − ΦM¯ 0 M 0 (t) ̸= 0. On the contrary, in the case of CP conservation one has [ ( )] π t ∆mt CP CP ΦM 0 M¯ 0 (t) = ΦM¯ 0 M 0 (t) = − (m + ∆m) + arg sin , 2 2 2

(9)

and there is no phase difference. Numerical analysis: We present a numerical analysis of the phase related to z, ∆Φ = ΦM 0 M 0 − ΦM¯ 0 M¯ 0 for the Bs system. Such a system is particulary appropriate for the study of non-cyclic phases since many particle oscillations occur within its lifetime. We use the following experimental data: ms = 1.63007 × 1013 ps−1 , ∆ms = 17.77ps−1 , Γs = 0.678ps−1 , ∆Γs = −0.062ps−1 . Moreover, we assume, −0.1 ≤ ℜz ≤ 0.1 and −0.1 ≤ ℑz ≤ 0.1 [62]. Notice that, in such intervals for ℜz and ℑz, the phase ∆Φ is weakly depending on the value of ℑz, so that one can fix an arbitrary value of ℑz in the values interval [−0.1, 0.1] and study the non-cyclic phases as functions of time for different values of ℜz. In Figs. (1), the phase is drawn for ℑz = 0. In order to better show the behavior of the phases, the figures contain two plots A) and B) of the same phase for sample values of ℜz belonging to the intervals [−0.1, 0] and [0, 0.1], respectively. A non–zero phase difference ∆Φ appears in the case of CP T violation. UNRUH EFFECT, QUANTUM THERMOMETER AND GEOMETRIC PHASE

In refs.[30, 31] we have shown that the phenomena characterized by the presence of vacuum condensate [36]-[40] are also characterized by the presence of Aharonov-Anandan invariant and of geometric phase. Thus the geometric phase

4

AL DΦ HtL for values of Re z Î @-0.1, 0D

BL DΦ HtL for values of Re z Î @0, 0.1D

2.0

2.0 0.1 0.075

1.5

1.5

0.05

DΦ Hþ L

DΦ Hþ L

0.025

1.0

-0.1

0

1.0

-0.075 -0.05

0.5

0.5

-0.025 0 0.0

0.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0

0.2

tH psL

0.4

0.6

0.8

1.0

1.2

1.4

tH psL

Figure 1: Plots of ∆Φ = ΦBs0 Bs0 − ΦB¯s0 B¯s0 as a function of time t for ℑz = 0 and different values of ℜz. In picture A) ∆Φ(t) is reported for sample values of Rez ∈ [−0.1, 0] as indicated in the inset. In picture B) ∆Φ(t) is plotted for sample values of Rez ∈ [0, 0.1] as shown in the inset.

can used to study the properties of such systems and to reveal phenomena like Unruh, Casimir effects, or quantum field theory (QFT) in curved background, which are elusive to the observations. Here we focus our attention on the Unruh effect and on the possibility of the realization of a quantum thermometer. To do that we consider quantum open systems and use the Wang and Liu approach [16] to define the geometric phase for mixed states in nonunitary, noncyclic evolution. - Geometric phase for mixed states in nonunitary, noncyclic evolution: The geometric phase is defined as Φg =

N ∑

[√ ] λk (t0 )λk (t) ⟨φk (t0 )|φk (t)⟩

arg

(10)

k=1

− ℑ

N ∫ ∑

k=1

t

t0

λk (t′ )⟨φk (t′ )|

∂ |φk (t′ )⟩dt′ , ∂t′

where |φk (t)⟩ and λk (t) are eigenstates and eigenvalues of the matrix density ρ(t) describing the quantum open system for N-level mixed states. The first term of Eq.(10) represents the total phase, the second √ term the dynamic one. In the case of a two level open system, N = 2, the radius of Block sphere is r(t) = n21 + n22 + n23 , with n1 = ρ12 + ρ21 , n2 = i(ρ12 − ρ21 ) , n3 = ρ11 − ρ22 . Defining the angles θ = cos−1 (n3 /r) and ϕ = tan (n2 /n1 ) , the eigenvectors of ρ, |φ1 (t)⟩ and |φ2 (t)⟩ are (apart overall phase factors) ( ) ( ) θ(t) cos θ(t) sin 2 2 |φ1 (t)⟩ = , |φ2 (t)⟩ = , eiϕ(t) sin θ(t) −eiϕ(t) cos θ(t) 2 2 and the eigenvalues are λ1 (t) = 12 [1 + r(t)] and λ2 = 21 [1 − r(t)]. ∑ - Interaction: Let us now consider the Hamiltonian [63], H = ~2 ω0 σ3 + HF − mn µmn · E(x(t))σmn , which describes a two level atom as an open system with a non-unitary evolution in the reservoir of the electromagnetic field. In H, 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′ ([63]), σmn = σm σn , and E is the strength of the electric field. We consider a weak interaction between atom and field and study the evolution of the total density matrix ρtot = ρ(0) ⊗ |0⟩⟨0|, in the frame of the atom. Here |0⟩ and

5 ρ(0) are the vacuum and the initial reduced density matrix of the atom. The evolution is given by [64, 65] 3 ∂ρ(τ ) i 1 ∑ = − [Hef f , ρ(τ )] + aij (2σj ρ σi − σi σj ρ − ρ σi σj ) , ∂τ ~ 2 i,j=1

(11)

with τ proper time, Hef f effective hamiltonian, Hef f ≃ ~2 ω0 σ3 , (ω0 is the atomic transition frequency, we neglect the Lamb shift terms), aij coefficients of the Kossakowski matrix, aij = Σ δij − iΥ ϵijk δk3 − Σ δi3 δj3 , with Σ = ∫∞ 1 1 iωτ + [G(ω ) + G(−ω )] , Υ = [G(ω ) − G(−ω )] , and G(ω) = dτ e G (x(τ )) Fourier transform of 0 0 0 0 4 4 −∞ 2 ∑3 e + G (x − y) = ~2 i,j=1 ⟨+|ri |−⟩⟨−|rj |+⟩⟨0|Ei (x)Ej (x)|0⟩. ( ) ( ) For an initial state of the atom given by |ψ(0)⟩ = cos θ2 |+⟩ + sin θ2 |−⟩ , with θ ≡ θ(0), one has the reduced density matrix ρ(τ ) [30, 31, 45] √ ( ) 1 χ+1 e−iΩτ ξ 2 − χ2 √ (12) ρ(τ ) = , 2 eiΩτ ξ 2 − χ2 1−χ √ −4Στ where ξ(τ ) = χ2 + e−4Στ sin2 θ , and χ(τ ) = e−4Στ cos θ + Υ − 1) . Σ (e The eigenvalues and the corresponding eigenvectors of ρ(τ ) are: λ± = 12 (1 ± ξ) , and ( ) ( ) θ(τ ) θ(τ ) iΩτ |ϕ+ (τ )⟩ = cos |+⟩ + sin e |−⟩ , 2 2 (13) ) ( ) ( θ(τ ) iΩτ θ(τ ) |+⟩ − cos e |−⟩ . |ϕ− (τ )⟩ = sin 2 2 Considering the initial time t0 = 0, the geometric phase (10) becomes [ ] ∫ θ θ(t) θ θ(t) iΩt Ω t Φg (t) = arg cos cos + sin sin e − [1 − ξ(τ ) cos θ(τ )] dτ , 2 2 2 2 2 0

(14)

with θ ≡ θ(0). Such a phase can be used in the detection of Unruh effect and in the building of a quantum thermometer. - Unruh effect: In the case of Unruh effect, we study the difference between two geometric phase (14) computed for the two level system in the presence of an acceleration and in the inertial case. Such a phase difference 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. Considering the atom acceleration through Minkowski spacetime the Rindler coordinates are √ in(the x direction, ) θ(t) χ(t) c2 aτ c aτ 1 x(τ ) = a cosh c , t(τ ) = a sinh c . Then the function sin 2 = ± 2 1 − ξ(t) in Eq.(14) becomes v u1 θa (t) Ra − Ra e4Σa t + cos θ u sin = ±t − √ , 2 2 2 2 4Σ t 4Σ t a a 2 e sin θ + (Ra − Ra e + cos θ)

(15)

( ) 2πcω /a ( ) 2 2 0 +1 and similar for cos θa2(t) , where Ra = Υa /Σa , with Σa = γ40 1 + c2aω2 ee2πcω0 /a −1 and Υa = γ40 1 + c2aω2 , and γ0 0 0 spontaneous emission rate. Eq. (14) holds also for an inertial atom, a = 0. In this case, sin θa2(t) , cos θa2(t) and cos θa (t) are replaced by (t) (t) , cos θa=0 and cos θa=0 (t) in which Σa , Υa , Ra are replaced by Σa=0 = Υa=0 = γ0 /4, Ra=0 = 1. sin θa=0 2 2 The phase difference ∆ΦU (t) = Φa (t) − Φa=0 (t), gives the geometric phase in terms of the acceleration a. Notice that the atomic element which has to be used in the interferometer plays an important role, since a non trivial value of ∆ΦU (t) can be achieved when γ0 /ω0 > 10−5 . In Fig.2 we plot ∆ΦU as function of the acceleration a for different hyperfine level structures of 85 Rb, 87 Rb and 133 Cs, as reported in the caption of the picture. Values of ∆ΦU ∼ 10−4 π, which are accessible with the current technology, can be obtained for accelerations of order of 1016 m/s2 and speeds of order of (0.2 − 0.3)c. The time intervals considered are of order of 1/ω0 in order to have negligible spontaneous emission (N (t ∼ 1/ω0 ) ∼ 0.98N (0)). The geometric phase above described can be

6 DΦU H10-4 Π L for 85 Rb, 87 Rb and

133

Cs

DΦU H10-4 Π L

1.5

1 1017 0.5 0.5

3 1017

6 1017

0.3

0.1 0 4 1016

6 1016

8 1016

1017

aHms2 L

Figure 2: Plots of ∆ΦU 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: 87 Rb, 52 P1/2 line, splitting for F = 1 → F = 2 transition (ω0 = 814.5M Hz, γ0 = 36.129M Hz [66]); - (green) solid line: 133 Cs, 62 P1/2 line, splitting of the between the F = 3 and F = 4 levels (ω0 = 1167.68M Hz, γ0 = 28.743M Hz [67]). Pictures in the inset: - (blue) dashed line: 85 Rb, 52 S1/2 line, energy splitting between the levels F = 1 and F = 2 (ω0 = 3.035GHz, γ0 = 36.129M Hz for D1 transition, γ0 = 38.117M Hz for D2 transition [68]); - (red) dot dashed line: 87 Rb, 52 S1/2 line, energy splitting between the levels F = 1 and F = 2 (ω0 = 6.843GHz, γ0 = 36.129M Hz for D1 transition, γ0 = 38.117M Hz for D2 transition [66]).

revealed with a Mach-Zehnder interferometer with branches of length of 4 cm. In particular, a difference in the arm lengths of the interferometer of about 0.1µm permits to remove almost completely the dynamical phase differences δ and to reveal only the geometric phase [31]. - Quantum thermometer: The geometric phase of Eq.(14) appears also when an atom interacts with thermal states. Therefore, its analysis could allow very precise measurement of the temperature. In the case of thermal states, the above coefficients Υa and Σa are replaced by ΣT and ΥT which 2 2 = (γ0 /4) (1 + depend on the temperature [30, 31], ΥT = (γ0 /4) (1 + 4π 2 kB T /~2 ω02 ) , and ΣT E0 /kB T E0 /kB T 2 2 2 2 2 + 1)/(e − 1) , E0 = ~ω0 . 4π kB T /~ ω0 ) (e Then a quantum thermometer can be built by means of an interferometer in which an atom follows two different paths interacting with two thermal states at different temperatures. Assuming known the temperature of one thermal state, the temperature of the other one can be obtained by measuring the difference between the geometric phases generated in the two paths. In the following we assume known the temperature Th of the hotter source and we derive the temperature Tc of the colder source by measuring ∆ΦT . Also in this case we consider the hyperfine structure of 133 Cs, and 87 Rb and we plot in Fig. 3, ∆ΦT as function of the temperatures of cold sources Tc , for different lines and different values of Th . We consider time intervals t ≃ 4ω1 0 s in order that the particle decay can be neglected and we obtain temperatures of the cold source of ∼ 2 orders of magnitude below the reference temperature of the hot source. Paths of slightly different lengths can be chosen in order to let the geometric phase be dominating over the relative dynamical phase.

CONCLUSIONS

We have studied the geometric phase for different phenomena and analyzed its possible applications. We have shown that the geometric phases generated by the evolution of mixed meson systems depend on the CP T violating parameter z [41]. In particular, we have shown that a non zero difference of phase ∆Φ between particle and antiparticle appears only in the case of CP T symmetry breaking. Therefore, in the next future, accurate analysis of the geometric phases for mixed mesons like the neutral Bs system and the kaons, might represent a completely alternative method to test one of the most important symmetries of the nature. The geometric phase can allow also the study of the CP symmetry breaking in mixed mesons. Moreover, we have analyzed the geometric phase for mixed state with a non-unitary evolution and we have shown that the geometric phase of atoms accelerated in an interferometer could permit the laboratory detection of the Unruh effect. Similar atoms, interacting with two different thermal states can be utilized in an interferometer to have very precise temperature measurements [31].

7 DFT HΠ L for 87 Rb and

133

Cs

DFT HΠ L

0.03

0.02

0.01

0.00 10-5

10-4

0.001

0.01

0.1

1

TemperatureHKL

Figure 3: Plots of ∆ΦT as function of the temperatures of cold sources Tc , for the splitting between the hyperfine energy levels and Th values: - (blue) dashed line: 133 Cs, 62 P3/2 line, splitting for F = 4 → F = 5 transition (ω0 = 251.09M Hz, γ0 = 32.889M Hz [67]) and Th = 10−2 K; - (red)dot-dashed line: 87 Rb, 52 P1/2 line, splitting for F = 1 → F = 2 transition (ω0 = 814.5M Hz, γ0 = 36.129M Hz [66]), and Th = 3 × 10−2 K; - (gray) solid line: 133 Cs, 62 P1/2 line, splitting for F = 3 → F = 4 transition (ω0 = 1167.68M Hz, γ0 = 28.743M Hz [67]) and Th = 6 × 10−2 K; - (black) dotted line: 133 Cs, 62 S1/2 line, splitting for F = 3 → F = 4 transition (ω0 = 9.192GHz, γ0 = 28.743M Hz [67]) and Th = 1K. The time considered are t ≃ 4ω1 0 s.

Partial financial support from MIUR and INFN is acknowledged.

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]

Berry M. V., Proc. Roy. Soc. Lond. A 392, 45 (1984). Aharonov Y., Anandan J., Phys. Rev. Lett. 58, 1593 (1987). Samuel J., Bhandari R., Phys. Rev. Lett. 60, 2339 (1988). Anandan J., Aharonov Y., Phys. Rev. Lett. 65, 1697 (1990). Mukunda N., Simon R., Ann.Phys.(N.Y) 228, 205 (1993). Garcia de Polavieja G., Phys. Rev. Lett. 81, 1 (1998) . Pancharatnam S., Proc. Indian Acad. Sci. A 44, 1225 (1956). Shapere A., Wilczek F., Geometric Phases in Physics, World Scientific, Singapore, (1989). Simon B., Phys. Rev. Lett. 51, 2167 (1983). Garrison J. C., Wright E. M., Phys. Lett. A 128, 177 (1988). Garrison J. C., Chiao R. Y., Phys. Rev. Lett. 60, 165 (1988). Pati A. K., J. Phys. A 28, 2087 (1995). Pati A. K., Phys. Rev. A 52, 2576 (1995). Anandan J., Phys. Lett. A 133, 171 (1988). Mostafazadeh A., J. Phys. A 32, 8157 (1999). Wang Z.S., Liu Q., Phys. Lett A 377, 3272 (2013). Uhlmann A., Rep. Math. Phys. 24, 229 (1986). Sjqvist E., et al., Phys. Rev. Lett. 85, 2845 (2000). Carollo A., Fuentes-Guridi I., Santos M.F., Vedral V., Phys. Rev. Lett. 90, 160402 (2003). Wang Z.S., Kwek L.C., Lai C.H., Oh C.H., Europhys. Lett. 74, 958 (2006). Chaturvedi S., Ercolessi E., Marmo G., Morandi G., Mukunda N., Simon R., Eur. Phys. J. C 35, 413 (2004). Ericsson M., Pati A.K., Sjqvist E., Brnnlund J., Oi D.K.L., Phys. Rev. Lett. 91, 090405 (2003). Tong D.M., Sjoqvist E., Kwek L.C., Oh C.H., Phys. Rev. Lett. 93, 080405 (2004). Bruno A.,Capolupo A.,Kak S.,Raimondo G., Vitiello G., Mod. Phys. Lett. B 25, 1661 (2011). Tomita A., Chiao R. Y., Phys. Rev. Lett. 57, 937 (1986). Jones J. A., Vedral V., Ekert A., Castagnoli G., Nature 403, 869 (2000). Leek P. J., et al., Science 318, 1889 (2007); Neeley M., et al.,Science 325, 722 (2009). Pechal M., et al., Phys. Rev. Lett. 108, 170401 (2012). Capolupo A., Vitiello G., Phys. Rev. D 88, no. 2, 024027 (2013). Capolupo A., Vitiello G., Adv. High Energy Phys. Volume 2015, 878043, (2015). Umezawa H., Advanced field theory: Micro, macro, and thermal physics, New York, USA: AIP (1993). Blasone M., Capolupo A., Romei O., Vitiello G., Phys. Rev. D 63, 125015 (2001). Capolupo A., Ji C. R., Mishchenko Y., Vitiello G., Phys. Lett. B 594, 135 (2004).

8 [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53]

[54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68]

Blasone M., Capolupo A., Vitiello G., Phys. Rev. D 66, 025033 (2002) and references therein. Unruh W. G., Phys. Rev. D 14, 870 (1976). Hawking S. W., Commun. Math. Phys. 43, 199 (1975) [Erratum-ibid. 46, 206 (1976)]. Parker L., Phys. Rev. Lett. 21, 562 (1968). Schr¨ odinger E., Physica 6, 899 (1939). Birrell N. D., Davies P. C. W., Quantum Fields in Curved Space (Cambridge Monographs on Mathematical Physics) (1984). Capolupo A., Phys. Rev. D 84, 116002 (2011), Capolupo A., Lambiase G., Vitiello G., Adv. High Energy Phys. 2015, 826051, (2015). Capolupo A., Vitiello G., Adv. High Energy Phys. 2013, 850395 (2013). Martin-Martinez E., Fuentes I., Mann R. B., Phys. Rev. Lett. 107, 131301 (2011). Hu J., Yu H., Phys. Rev. A 85, 032105 (2012). Martin-Martinez E., Dragan E. A., Mann R. B., Fuentes I., New J. Phys. 69, 053036 (2013). Kabir P. K., The CP Puzzle, Academic Press, London (1968); Nachtmann O., Elementary Particle Physics: Concepts and Phenomena, Springer, Berlin (1990). Bilenky S.M., Pontecorvo B., Phys. Rep. 41, 225 (1978). Lenz A., Nierste U., J. High Energy Phys. 06, 072 (2007). Fidecaro M., Gerber H. J., Rept. Prog. Phys. 69, 1713, (2006), [(2006) Erratum-ibid. 69, 2841] and references therein. Blasone M., Capolupo A., Terranova F., Vitiello G., Phys. Rev. D 72, 013003 (2005). Blasone M., Capolupo A., Ji C. -R., , Vitiello G. Int. J. Mod. Phys. A 25, 4179 (2010). Capolupo A., Capozziello S., Vitiello G., Phys. Lett. A 363, 53 (2007); Int. J. Mod. Phys. A 23, 4979 (2008), Blasone M., Capolupo A., Capozziello S., Vitiello G., Nucl. Instrum. Meth. A 588, 272 (2008), Blasone M., Capolupo A., Vitiello G., Prog. Part. Nucl. Phys. 64, 451 (2010). Blasone M., Capolupo A., Capozziello S., Carloni S., Vitiello G., Phys. Lett. A 323, 182 (2004). Capolupo A., Capozziello S., Vitiello G., Phys. Lett. A 373, 601 (2009). Capolupo A., Di Mauro M., Iorio A., Phys. Lett. A 375, 3415 (2011). Capolupo A., Di Mauro M., Phys. Lett. A 376, 2830 (2012); Acta Phys. Polon. B 44, 81 (2013); Vacuum condensates as a mechanism of spontaneous supersymmetry breaking, Adv. High Energy Phys. in press. Bender C. M., Boettcher S., Phys. Rev. Lett. 80, 5243 (1998). Tanaka T., J. Phys. A: Math. Gen. 39, 7757 (2006). Baker H. C., Singleton Jr. R. L., Phys. Rev. A 42, 10 (1990). Dattoli G., Torre A., Mignani R., Phys. Rev. A 42, 1467 (1990). Aubert B. et al. , [BABAR Collaboration], Phys. Rev. D 70, 012007 (2004) Aubert B. et al., [BABAR Collaboration], Phys. Rev. Lett. 96, 251802 (2006) Aubert B. et al., [BABAR Collaboration], Phys. Rev. Lett. 100, 131802 (2008) Compagno G., Passante R., Persico F., Atom-field interactions and dressed atoms (Cambridge University Press, Cabridge, 1995). G. Lindblad, Commun. Math. Phys. 48, 119 (1976). Gorini V., Kossakowski A., Sudarshan E. C., J. Math. Phys. 17, 821 (1976). Steck D. A., Rubidium 87 D Line Data, available online at http://steck.us/alkalidata (revision 2.1.4, 23 December 2010). Steck D. A., Cesium D Line Data, available online at http://steck.us/alkalidata (revision 2.1.4, 23 December 2010). Steck D. A., Rubidium 85 D Line Data, available online at http://steck.us/alkalidata (revision 2.1.6, 20 September 2013).