Algebraic approach and Berry phase of a Hamiltonian with a general SU(1,1) symmetry
aa r X i v : . [ m a t h - ph ] A ug Algebraic approach and Berry phase of a Hamiltonian with a general SU (1 ,
1) symmetry
E. Chore˜no a , R. Valencia a , D. Ojeda-Guill´en b ∗ a Escuela Superior de F´ısica y Matem´aticas, Instituto Polit´ecnico Nacional, Ed. 9, Unidad Profesional AdolfoL´opez Mateos, Delegaci´on Gustavo A. Madero, C.P. 07738, Ciudad de M´exico, Mexico. b Escuela Superior de C´omputo, Instituto Polit´ecnico Nacional, Av. Juan de Dios B´atiz esq. Av. Miguel Oth´onde Mendiz´abal, Col. Lindavista, Delegaci´on Gustavo A. Madero, C.P. 07738, Ciudad de M´exico, Mexico.
Abstract
In this paper we study a general Hamiltonian with a linear structure given in terms of two different real-izations of the SU (1 ,
1) group. We diagonalize this Hamiltonian by using the similarity transformations of the SU (1 ,
1) and SU (2) displacement operators performed to the su (1 ,
1) Lie algebra generators. Then, we computethe Berry phase of a general time-dependent Hamiltonian with this general SU (1 ,
1) linear structure.
PACS: 02.20.Sv, 03.65.Fd, 42.65.Yj, 42.50.-pKeywords: Berry phase, Lie algebra, SU (1 ,
1) bosonic Hamiltonian, tilting transformation
Group theory has become a very valuable tool when studying and solving various problems in theoretical physics.This theory has been applied in high-energy physics, condensed matter, atomic, molecular, and nuclear physics.Two of the main groups that are frequently used to describe these physical phenomena are the SU (1 ,
1) and SU (2)groups. In particular, the SU (1 ,
1) and SU (2) groups and their simple generalizations have been used to study manyproperties of relevant Hamiltonians in Quantum Optics like the Jaynes-Cummings model [1], the Tavis-Cummingsmodel [2, 3] and the optical parametric amplifiers [4, 5].The Jaynes-Cummings model describes the interaction between radiation and matter, and is the simplest andcompletely soluble quantum-mechanical model. The exact solution of this theoretical model has been found inthe rotating wave approximation [6]. Despite the simplicity of the Jaynes-Cummings model, it presents interestingquantum phenomena [7–14], all of them being experimentally corroborated, as can be seen in references [15–17].The Tavis-Cummings model emerged from the study of N identical two-level molecules interacting through adipole coupling with a single-mode quantized radiation field at resonance. This model has been studied throughdifferent methods, among which are the Holstein-Primakoff transformation [18], quantum inverse methods [19, 20],and polynomially deformed su (2) algebras [21]. Both the Jaynes-Cummings model and the Tavis-Cummings modelare still widely studied nowadays [22–27].In the optical parametric amplifier, one photon of a pump field transforms, via the nonlinear medium, into twophotons called signal and idler. These output beams have the same frequency and polarization in the degeneratecase and different ones in the non-degenerate case [28]. In Ref. [29], Gerry used an su (1 ,
1) Lie algebra realizationto study the Berry phase in the degenerate parametric amplifier.The Berry phase [30] is a phase factor gained by the wavefunction after the system is transported througha closed path via adiabatic variation of parameters. Since its introduction, it has been extensively studied inseveral quantum systems [31–33]. We recently have applied the theory of the SU (1 ,
1) and SU (2) groups to obtainthe energy spectrum, eigenfunctions and the Berry phase of some of these Quantum Optics models [34–38]. In ∗ E-mail address: [email protected] Sp (4 , R ) group (and contains the SU (1 ,
1) and SU (2) groups)was introduced to solve exactly the interaction part of the most general Hamiltonian of a two-level system intwo-dimensional geometry.The aim of the present work is to introduce an algebraic method to solve exactly and compute the Berry phaseof a general Hamiltonian with an SU (1 ,
1) symmetry.This work is organized as follows. In Section 2, we construct a general Hamiltonian with a linear SU (1 , su (1 ,
1) Lie algebra. Then, we introduce a methodto diagonalize this Hamiltonian based on the similarity transformations of the SU (1 ,
1) and SU (2) displacementoperators performed to the SU (1 ,
1) generators. These transformations allow us to obtain the energy spectrum andeigenfunctions of this general Hamiltonian. In Section 3, we calculate the transformations of the operator i ∂∂t interms of the SU (1 ,
1) and SU (2) displacement operators introduced in Section 2. With this result, we compute theBerry phase for a general time-dependent Hamiltonian with this SU (1 ,
1) linear structure. Section 4 is dedicatedto study a particular case of the general SU (1 ,
1) Hamiltonian introduced in Section 2. Finally, we give someconcluding remarks. SU (1 , bosonic Hamiltonian In many Quantum Optics problems related to parametric amplifiers we have the one-mode (also known as the“squeezed oscillator Hamiltonian”) and two-mode Hamiltonians H a = ω ˆ a † ˆ a + g ˆ a † + g ∗ ˆ a , (1) H ab = ω ˆ a † ˆ a + ω ˆ b † ˆ b + λ ˆ a ˆ b + λ ∗ ˆ a † ˆ b † , (2)which are expressed in terms of the bosonic annihilation ˆ a , ˆ b and creation ˆ a † , ˆ b † operators. These operators obeythe commutation relations [ˆ a, ˆ a † ] = [ˆ b, ˆ b † ] = 1 , (3)[ˆ a, ˆ b ] = [ˆ a † , ˆ b † ] = [ˆ a † , ˆ b ] = [ˆ a, ˆ b † ] = 0 . (4)The Hamiltonians of equations (1) and (2) can be studied separately in terms of an appropriate SU (1 ,
1) realization.As it is well known, the su (1 ,
1) Lie algebra is defined in terms of the commutation relations [40][ K , K ± ] = ± K ± , [ K − , K + ] = 2 K . (5)With the operators ˆ a † ˆ a , ˆ b † ˆ b , ˆ a † ˆ b † , ˆ b ˆ a , ˆ a † and ˆ a we can construct the following two realizations of the su (1 ,
1) Liealgebra K ( ab )+ = ˆ a † ˆ b † , K ( ab ) − = ˆ b ˆ a, K ( ab )0 = 12 (ˆ a † ˆ a + ˆ b † ˆ b + 1) N ( ab ) d = ˆ a † ˆ a − ˆ b † ˆ b, (6)and K ( a )+ = 12 ˆ a † , K ( a ) − = 12 ˆ a , K ( a )0 = 12 (cid:18) ˆ a † ˆ a + 12 (cid:19) . (7)Here, the operator N ( ab ) d is the difference of the number operators of the two oscillators and commutes with allthe generators of the algebra. Therefore, the SU (1 ,
1) Hamiltonians H a and H ab of equations (1) and (2) can bewritten as H a = ω (cid:18) K ( a )0 − (cid:19) + 2 gK ( a )+ + 2 g ∗ K ( a ) − , (8) H ab = ( ω + ω ) (cid:18) K ( ab )0 − (cid:19) + 12 ( ω − ω ) N ( ab ) d + λK ( ab ) − + λ ∗ K ( ab )+ . (9)Similarly, the su (2) Lie algebra is spanned by the generators J + , J − and J , which satisfy the commutationrelations [40] [ J , J ± ] = ± J ± , [ J + , J − ] = 2 J . (10)2ith the bilinear products ˆ a † ˆ a , ˆ b † ˆ b , ˆ a † ˆ b and ˆ b † ˆ a we can construct an su (2) Lie algebra realization by introducingthe operators J + = ˆ a † ˆ b, J − = ˆ b † ˆ a, J = 12 (ˆ a † ˆ a − ˆ b † ˆ b ) . (11)Therefore, the operator N ( ab ) d of the su (1 ,
1) Lie algebra is related to the su (2) Lie algebra, since N ( ab ) d = J .Based on all these results, we can introduce a more general Hamiltonian with an SU (1 ,
1) linear structure H = ω (cid:16) ˆ a † ˆ a + ˆ b † ˆ b (cid:17) + λ ˆ a ˆ b + λ ∗ ˆ a † ˆ b † + g ˆ a † + g ∗ ˆ a + c ˆ b † + c ∗ ˆ b . (12)We can write this general Hamiltonian in terms of the su (1 ,
1) Lie algebra realizations of equations (6) and (7) asfollows H = α ( a ) − K ( a ) − + α ( a )+ K ( a )+ + α ( b ) − K ( b ) − + α ( b )+ K ( b )+ + α ( ab ) − K ( ab ) − + α ( ab )+ K ( ab )+ + α ( ab )0 K ( ab )0 , (13)where the α ’s are complex constants such that α − = α ∗ + . This Hamiltonian can be diagonalized by using of the SU (2) displacement operator D ( χ ) and the SU (1 ,
1) displacement operator D ( ξ ) ab = D ( ξ a ) D ( ξ b ) (see AppendixA). Here, the complex constants χ , ξ a and ξ b are explicitly given by χ = − θ e − iφ ; ξ a = − θ a e − iφ a ; ξ b = − θ b e − iφ b . (14)We use the SU (2) and SU (1 ,
1) displacement operators to transform each of the operators of the different su (1 , K ( a ) ± , K ( b ) ± , K ( ab ) ± and K ( ab )0 , as it is shown in the Appendix A. Thus, in order to remove theladder operators K ( ab )+ and K ( ab ) − of the Hamiltonian (13), we first apply the similarity transformation in terms of D ( χ ) as H ′ = D † ( χ ) HD ( χ ) . (15)By using the equations (66)-(69) we can write the new Hamiltonian H ′ as H ′ = β ( a ) − K ( a ) − + β ( a )+ K ( a )+ + β ( b ) − K ( b ) − + β ( b )+ K ( b )+ + β ( ab ) − K ( ab ) − + β ( ab )+ K ( ab )+ + β ( ab )0 K ( ab )0 , (16)where ( β ( i ) ± ) † = β ( i ) ∓ and the new complex constants β ’s are given as β ( a ) − = 12 α ( a ) − (cos (2 | χ | ) + 1) − ( χ ∗ ) | χ | α ( b ) − (cos (2 | χ | ) − − α ( ab ) − χ ∗ | χ | sin (2 | χ | ) , (17) β ( b ) − = − χ | χ | α ( a ) − (cos (2 | χ | ) −
1) + 12 α ( b ) − (cos (2 | χ | ) + 1) + α ( ab ) − χ | χ | sin (2 | χ | ) , (18) β ( ab ) − = χ | χ | α ( a ) − sin (2 | χ | ) − χ ∗ | χ | α ( b ) − sin (2 | χ | ) + α ( ab ) − cos (2 | χ | ) , (19) β ( ab )0 = α ( ab )0 = α . (20)Therefore, if we choose the parameters of the complex number χ = − θ e − iφ astan θ q ( α ( ab )+ α ( a ) − + α ( ab ) − α ( b )+ )( α ( ab )+ α ( b ) − + α ( ab ) − α ( a )+ ) α ( a ) − α ( a )+ − α ( b ) − α ( b )+ , e iφ = vuut α ( ab )+ α ( a ) − + α ( ab ) − α ( b )+ α ( ab )+ α ( b ) − + α ( ab ) − α ( a )+ , (21)we can eliminate the coefficients β ( ab ) ± , and the Hamiltonian of the equation (16) is reduced to H ′ = β ( ab )0 K ( ab )0 + β ( a ) − K ( a ) − + β ( a )+ K ( a )+ + β ( b ) − K ( b ) − + β ( b )+ K ( b )+ . (22)Following the above procedure, we now apply the similarity transformation in terms of the displacement operator D ( ξ ) ab to this Hamiltonian H ′ as follows H ′′ = D † ( ξ ) ab H ′ D ( ξ ) ab . (23)3iven that the boson operators ˆ a and ˆ b commute and by using the equations (71) and (72) of the Appendix A, wecan show that the Hamiltonian H ′′ is transformed to H ′′ = q α − β ( a )+ β ( a ) − K ( a )0 + q α − β ( b )+ β ( b ) − K ( b )0 . (24)Here, the parameters of the complex numbers ξ a = − θ a e − iφ a and ξ b = − θ b e − iφ b were chosen astanh ( θ i ) = 2 α q β ( i ) − β ( i )+ ; e iφ i = vuut β ( i ) − β ( i )+ , (25)with i = a, b .From expression (24) we can see that the eigenstates | ϕ ′′ i of the Hamiltonian H ′′ are the direct product of thenumber states of the modes { a, b } , that is | ϕ ′′ i = | n a i ⊗ | n b i . Hence, these states explicitly are the eigenfunctionsof the two-dimensional harmonic oscillator ϕ ′′ n l ,m n ( ρ, φ ) = 1 √ π e im n φ ( − n l s n l )!( n l + m n )! ρ m n L m n n l ( ρ ) e − / ρ , (26)where n l is the left chiral quantum number. From this result we obtain that the energy spectrum of the Hamiltonian H ′′ , and therefore of the general SU (1 ,
1) Hamiltonian H of equation (13), is given by E n l ,m n = (cid:18)q α − β ( a )+ β ( a ) − + q α − β ( b )+ β ( b ) − (cid:19) ( n l + 1)4 + (cid:18)q α − β ( a )+ β ( a ) − − q α − β ( b )+ β ( b ) − (cid:19) m n . (27)Moreover, the eigenfunctions of this general SU (1 ,
1) Hamiltonian H are obtained from the relationship | ϕ i = D ( χ ) D ( ξ ) ab | ϕ ′′ i , (28)where the term D ( ξ ) ab | ϕ ′′ i can be identified as the SU (1 ,
1) Perelomov number coherent states for the two-dimensional harmonic oscillator [38]. SU (1 , time-dependent Hamiltonian In this Section we shall now consider the Hamiltonian (13) as an explicit function of time H ( t ), that is H ( t ) = α ( t ) K ( ab )0 + α ( ab )+ ( t ) K ( ab )+ + α ( ab ) − ( t ) K ( ab ) − + α ( a )+ ( t ) K ( a )+ + α ( a ) − ( t ) K ( a ) − + α ( b )+ ( t ) K ( b )+ + α ( b ) − ( t ) K ( b ) − , (29)where the α ’s are complex constants such that α ( j ) ± = (cid:16) α ( j ) ∓ (cid:17) ∗ and can be written as α ( j )+ ( t ) = λ j ( t ) e iγ j ( t ) . (30)Here λ (1) j ( t ) and φ (1) j ( t ) with j = a, b, ab, are arbitrary real functions of time. Since this Hamiltonian is time-dependent, to describe quantum dynamics we shall use the Schr¨odinger picture i ~ ddt | ψ ( t ) i = H ( t ) | ψ ( t ) i . (31)Thus, in order to study the time evolution of the states of Hamiltonian (29), we will use the time-dependentnontrivial invariant Hermitian operator I ( t ) [41, 42], which satisfies the conditions i ∂∂t I ( t ) + [ I ( t ) , H ( t )] = 0 . (32)Now, we shall use the time-dependent versions of the SU (1 ,
1) and SU (2) displacement operators of equations (70)and (65), where the complex parameters θ ’s and φ ’s of the expressions (14) are arbitrary real functions of time.Then, with these considerations we can define the invariant operator I ( t ) as I ( t ) = D ( χ ( t )) D ( ξ ( t )) ab K ( ab )0 D † ( ξ ( t )) ab D † ( χ ( t )) , (33)4r explicitly as I ( t ) = β J + β J + + β ∗ J − + β K ( ab )0 + β K ( ab )+ + β ∗ K ( ab ) − + β K ( a )+ + β ∗ K ( a ) − + β K ( b )+ + β ∗ K ( b )+ , (34)where the β ’s coefficients are given by β = cosh( θ a ) − cosh( θ )2 cos( θ ) , (35) β = cosh( θ a ) − cosh( θ )4 sin( θ ) e iφ , (36) β = cosh( θ a ) + cosh( θ )2 , (37) β = sin( θ ) (cid:16) sinh( θ a ) e i ( φ − φ a ) − sinh( θ b ) e − i ( φ + φ b ) (cid:17) , (38) β = sinh( θ a ) (cos( θ ) + 1) e − iφ a − sinh( θ b ) (cos( θ ) − e − iφ , (39) β = sinh( θ b ) (cos( θ ) + 1) e − iφ b − sinh( θ a ) (cos( θ ) − e iφ . (40)From the condition of equation (32) and the form of the invariant operator I ( t ), we obtain that the time-dependentphysical parameters of the complex constants α j ± are related to coefficients β ’s as follows˙ β + α ( a )+ β ∗ − α ( a ) − β − α ( b )+ β ∗ + α ( b ) − β = 0 , ˙ β + α ( ab )+ β ∗ + α ( a )+ β ∗ − α ( ab ) − β − α ( b ) − β = 0 , ˙ β + 2 α ( ab )+ β ∗ − α ( ab ) − β + α ( a )+ β ∗ − α ( a ) − β + α ( b )+ β ∗ − α ( b ) − β = 0 , ˙ β + α ( b )+ β + α ( a )+ β ∗ + α ( ab )+ β − α β = 0 , ˙ β + α ( a )+ β + 2 α ( ab )+ β + α ( a )+ β − α β = 0 , ˙ β + α ( b )+ β + 2 α ( ab )+ β ∗ − α ( b )+ β − α β = 0 , (41)together with their corresponding conjugate equations.On the other hand, the transformations of the algebra generators K ’s and J ’s under time-dependent displacementoperators D ( ξ ( t )) ab and D ( χ ( t )) remain unchanged and are given by the expressions of the Appendix A. In addition,we can transform the operator i ∂∂t under the time-dependent displacement operators D ( ξ ( t )) ab and D ( χ ( t )) as D † ( ξ ( t )) ab D † ( χ ( t )) (cid:18) i ∂∂t (cid:19) D ( χ ( t )) D ( ξ ( t )) ab = (cid:18) i ∂∂t (cid:19) ′′ . (42)to obtain (cid:18) i ∂∂t (cid:19) ′′ = i ∂∂t + ( b + c ) K ( ab )0 + ( b + c ) J + c J + + c ∗ J − + c K ( ab )+ + c ∗ K ( ab ) − + ( b + c ) K ( a )+ + ( b ∗ + c ∗ ) K ( a ) − + ( b + c ) K ( b )+ + ( b ∗ + c ∗ ) K ( b ) − . (43)The explicit form of the constants b ’s and c ’s are given in the Appendix B. Also, as it is shown in Ref. [42], if theeigenstates of the invariant operator satisfy the Schr¨odinger equation its eigenvalues are real. Therefore, given that K | k, n i = ( k + n ) | k, n i we have D ( χ ) D ( ξ ) ab K ( ab )0 | k, n i = ( k + n ) D ( χ ) D ( ξ ) ab | k, n i , which implies that I ( t ) D ( χ ) D ( ξ ) ab | k, n i = ( k + n ) D ( χ ) D ( ξ ) ab | k, n i . Thus, the states of the invariant operator I ( t ) are D ( χ ) D ( ξ ) ab | k, n i = D ( χ ) | ζ ( t ) , k, n i where | ζ ( t ) , k, n i are the SU (1 ,
1) Perelomov number coherent states. 5oreover, if the states | ψ ( t ) i satisfy the relationship (31) for the Hamiltonians H ( t ), these states can be expandedthrough the sates D ( χ ) | ζ ( t ) , k, n i in the form | ψ ( t ) i su (1 , = X n a n e iα n D ( χ ) | ζ ( t ) , k, n i , (44)where according to Lewis [42] the phase α is given as α = Z t dt ′ h λ, κ | i ∂∂t ′ − H ( t ′ ) | λ, κ i . (45)Here, | λ, κ i are the eigenstates and λ are the eigenvalues of the invariant operator I ( t ). Therefore, the phase of theeigenstates D ( χ ) | ζ ( t ) , k, n i in a non-adiabatic process is given by α n,µ = ( n + k ) Z t (cid:20) b ( t ) + c ( t ) − A ( t ) + B ( t )2 (cid:21) dt ′ − µ Z t (cid:20) A ( t ) + B ( t )2 (cid:21) dt ′ , (46)where as it is shown in the Ref. [30], the terms A ( t ) and B ( t ) are given as A ( t ) = cosh( θ a ( t )) α − λ ′ a sinh( θ a ( t )) cos( φ a ( t ) + γ ′ a ( t )) , (47) B ( t ) = cosh( θ b ( t )) α − λ ′ b sinh( θ b ( t )) cos( φ b ( t ) + γ ′ b ( t )) . It is worth mentioning that for simplicity we have taken the β ’s coefficients of the Hamiltonian (24) as β ( j )+ ( t ) = λ ′ j ( t ) e iγ ′ j ( t ) , (48)where λ ′ j ( t ) and φ ′ j ( t ) with j = a, b, are arbitrary real functions of time.Unlike in a non-adiabatic process, in an adiabatic process we have that ˙ θ = ˙ φ = 0. Therefore, from the relations(41) we can obtain the time-dependent versions of the expressions (21) and (25). Therefore, in an adiabatic processthe phase of the states D ( χ ) | ζ ( t ) , k, n i are reduced to α n l ,m n = − ( n l + 1)4 Z t (cid:18)q α ( t ′ ) − β ( a )+ ( t ′ ) β ( a ) − ( t ′ ) + q α ( t ′ ) − β ( b )+ ( t ′ ) β ( b ) − ( t ′ ) (cid:19) dt ′ − m n Z t (cid:18)q α ( t ′ ) − β ( a )+ ( t ′ ) β ( a ) − ( t ′ ) − q α ( t ′ ) − β ( b )+ ( t ′ ) β ( b ) − ( t ′ ) (cid:19) dt ′ . (49)These are known as the dynamical phases and are defined as˙ ǫ n = h λ, κ | H ( t ′ ) | λ, κ i . (50)while the Berry phase is defined as ˙ γ κ = i h λ, κ | ∂∂t | λ, κ i . (51)Thus, the Berry phase of the states D ( χ ) | ζ ( t ) , k, n i is obtained in the adiabatic limit as follows γ n,µ ( T ) = ( n + k ) Z T ( b + c ) dt + µ Z T ( b + c ) dt, (52)where T denotes the period. According to the values of the constants b and c shown in the Appendix B, the Berryphase is γ k,n,µ ( T ) = ( n + k )2 "Z T ˙ φ a (cosh( θ a ) − dt + Z T ˙ φ b (cosh( θ b ) − dt + Z T ˙ φ (cos( θ ) −
1) (cosh( θ a ) − cosh( θ b )) dt + µ "Z T ˙ φ a (cosh( θ a ) − dt − Z T ˙ φ b (cosh( θ b ) − dt + Z T ˙ φ (cos( θ ) −
1) (cosh( θ a ) + cosh( θ b )) dt . (53)6o see the topological aspect of the Berry phase explicitly for our problem, let us suppose that the γ ab , γ a and γ b phases are not independent of each other but they are related as follows 2 γ ab − γ a − γ b = nπ . If n = 0, thecoherent parameters of the ξ b , ξ a and χ complex constants are related to λ ′ j and γ ′ j physical constants of the α j ± coefficients as φ = γ b − γ ab = γ ab − γ a , φ a = − γ a , φ b = − γ b , (54)and θ = tan − (cid:20) λ ab λ a − λ b (cid:21) , θ a = f a ( λ ab , λ a , λ b ) , θ b = f b ( λ ab , λ a , λ b ) . (55)Hence, the Berry phase is reduced to the closed integral γ k,n,µ ( C ) = ( n + k )2 (cid:20) (cosh( θ a ) − I dγ a + (cosh( θ b ) − I dγ b + (cos( θ ) −
1) (cosh( θ a ) − cosh( θ b )) I ( dγ ab − dγ a ) (cid:21) + µ (cid:20) (cosh( θ a ) − I dγ a − (cosh( θ b ) − I dγ b + (cos( θ ) −
1) (cosh( θ a ) + cosh( θ b )) I ( dγ ab − dγ a ) (cid:21) . (56)Therefore, the Berry phase of the states D ( χ ) | ζ ( t ) , k, n i is finally given by γ k,n,µ ( C ) = 2 π µ θ b ) − cosh( θ a )] − π ( n + k )2 [cosh( θ a ) + cosh( θ b ) − . (57)It is obvious that in the cases where the condition 2 γ ab ( t ) − γ a ( t ) − γ b ( t ) = nπ is satisfied, the Berry phases do notdepend on an explicit form of the functions γ ab ( t ), γ a ( t ) and γ b ( t ), which are part of the complex coefficients of theHamiltonian (29). The time-independent Hamiltonian that considers all possible linear interactions in the phase space of momentsand positions is given by the expression [31, 43] H = H + X i =1 , h ω i u i x i p i + p i x i ) i + s p p m + √ ω ω ( ux p + u ′ x p ) + ω ω mvx x , (58)where H is the Hamiltonian of the two-dimensional harmonic oscillator, u i , u , u ′ , s , v are real constants, and ω , ω are the oscillation frequencies. By introducing the bosonic operators a i = ( mω i x i + ip i ) / √ mω i and using therealizations (6), (7) and (11), the above Hamiltonian can be rewritten as [39] H = β J + β + J + + β − J − + α K (12)0 + α (12)+ K (12)+ + α (12) − K (12) − + α (1)+ K (1)+ + α (1) − K (1) − + α (2)+ K (2)+ + α (2) − K (2) − , (59)where the α ′ s and β ′ s are complex constants such that α + = α ∗− and β + = β ∗− . Here, these complex constants aregiven by the following expressions α = ω + ω , β = ω − ω , (60) α (1)+ = − iω u , α (2)+ = − iω u , (61) α (12)+ = √ ω ω v − s ) − i ω u + ω u ′ ) , β + = √ ω ω v + s ) + i ω u − ω u ′ ) . (62)The general harmonic oscillator Hamiltonian of equation (59) can be mapped onto a Hamiltonian of the type(13). Choosing the real constants u , u ′ , v , s so that u = u ′ , v = − s , and taking the isotropic case ( ω = ω ), wehave that the Hamiltonian (59) can be written as H = α K (12)0 + α (12)+ K (12)+ + α (12) − K (12) − + α (1)+ K (1)+ + α (1) − K (1) − + α (2)+ K (2)+ + α (2) − K (2) − , (63)where now the complex constants α ′ s are given by α = 2 ω, α (1)+ = − i u ω , α (2)+ = − i u ω , α (12)+ = ω v − i ω u. (64)7herefore, the Hamiltonian (63) could be considered as a particular case of the Hamiltonian of the two-dimensionalisotropic harmonic oscillator with linear interactions in the 4-dimension x − p phase. On the other hand, if theconstants v , u , u and u are arbitrary real functions of time which vary smoothly with time, its respective dynamicaland Berry phases in the adiabatic limit are given by the expressions (49) and (57), respectively. In this paper we introduced a general Hamiltonian with a general linear structure given in terms of two differentrealizations of the SU (1 ,
1) group. We developed a method to diagonalize this Hamiltonian based on the similaritytransformations of the SU (1 ,
1) and SU (2) displacement operators performed to the SU (1 ,
1) generators. Withthese transformations, we were able to obtain the energy spectrum and eigenfunctions of our general SU (1 , i ∂∂t , we computed the Berry phase of ageneral time-dependent Hamiltonian with this SU (1 ,
1) linear structure.It is important to note that, even though our general Hamiltonian was written only in terms of the SU (1 , SU (2) group theory to be able to diagonalize it. This fact can be explainedby remembering that the SU (1 ,
1) and SU (2) groups, together with the so-called potential group SU p (1 ,
1) canbe imbedded into a larger group, Sp (4 , R ). Also, since the Hamiltonian studied in this paper is very general, ourresults can be adequately transferred to more specific problems with these symmetries, such as the degenerate andnon-degenerate parametric amplifier, among others. Acknowledgments
This work was partially supported by SNI-M´exico, EDI-IPN, SIP-IPN Project Number 20200225.
Appendix A. The similarity transformations of the su (1 , Lie algebrarealizations
In this Appendix we shall compute some similarity transformations of the operators K ( a ) ± , K ( b ) ± , K ( ab ) ± and K ( ab )0 of the different su (1 ,
1) Lie algebra realizations introduced in Section 2. Therefore, by introducing the SU (2)displacement operator D ( χ ) = exp[ χJ + − χ ∗ J − ] , (65)and considering the commutation relations of the Sp (4 , R ) Lie algebra [39], we can find the following results D † ( χ ) K ( a ) − D ( χ ) = χ | χ | K ( ab ) − sin(2 | χ | ) + 12 K ( a ) − (cos(2 | χ | ) + 1) − χ | χ | K ( b ) − (cos(2 | χ | ) − , (66) D † ( χ ) K ( b ) − D ( χ ) = − χ ∗ | χ | K ( ab ) − sin(2 | χ | ) − ( χ ∗ ) | χ | K ( a ) − (cos(2 | χ | ) −
1) + 12 K ( b ) − (cos(2 | χ | ) + 1) , (67) D † ( χ ) K ( ab ) − D ( χ ) = K ( ab ) − cos(2 | χ | ) − χ ∗ | χ | K ( a ) − sin(2 | χ | ) + χ | χ | K ( b ) − sin(2 | χ | ) , (68) D † ( χ ) K ( ab )0 D ( χ ) = K ( ab )0 . (69)Similarly, we can introduce the SU (1 , × SU (1 ,
1) displacement operator as follows D ( ξ ) ab = D ( ξ a ) D ( ξ b ) = exp[ ξ a K ( a )+ − ξ ∗ a K ( a ) − + ξ b K ( b )+ − ξ ∗ b K ( b ) − ] . (70)Thus, the similarity transformations of the operators K ( a ) ± , K ( b ) ± , K ( ab ) ± and K ( ab )0 in terms of this displacementoperator are presented below D † ( ξ ) ab K ( i )+ D ( ξ ) ab = sinh(2 | ξ i | ) ξ ∗ i | ξ i | K ( i )0 + (cosh(2 | ξ i | ) + 1) K ( i )+ | ξ i | ) − ξ ∗ i ξ i K ( i ) − , (71)8 † ( ξ ) ab K ( i )0 D ( ξ ) ab = cosh(2 | ξ i | ) K ( i )0 + sinh(2 | ξ i | ) (cid:18) ξ i | ξ i | K ( i )+ + ξ ∗ i | ξ i | K ( i ) − (cid:19) , (72)where i = a, b and (cid:0) D † K ± D (cid:1) † = D † K ∓ D . Moreover, the bosonic operators ˆ a and ˆ b are transformed in terms ofthis displacement operator as D † ( ξ a )ˆ aD ( ξ a ) = ˆ a cosh( | ξ a | ) + ˆ a † ξ a | ξ a | sinh( | ξ a | ) , D † ( ξ b )ˆ bD ( ξ b ) = ˆ b cosh( | ξ b | ) + ˆ b † ξ b | ξ b | sinh( | ξ b | ) . (73) Appendix B. The similarity transformations of the operator i ∂∂t Now, we shall compute the similarity transformation of the operator i ∂∂t in terms of the displacement operators D ( χ ) and D ( ξ ) ab of equations (65) and (70), respectively. Thus, we proceed to apply these transformations in thefollowing order D † ( ξ ( t )) ab D † ( χ ( t )) (cid:18) i ∂∂t (cid:19) D ( χ ( t )) D ( ξ ( t )) ab = D † ( ξ ( t )) ab (cid:18) i ∂∂t (cid:19) ′ D ( ξ ( t )) ab = (cid:18) i ∂∂t (cid:19) ′′ . (74)As it is shown in Ref. [39], from the first transformation (cid:0) i ∂∂t (cid:1) ′ we obtain (cid:18) i ∂∂t (cid:19) ′ = i ∂∂t + a J + a J + + a ∗ J − , (75)where the complex constants a ’s are explicitly given as a = ˙ φ (cos( θ ) − , a = − e − iφ (cid:16) ˙ φ sin( θ ) + i ˙ θ (cid:17) . (76)With this result we obtain that the transformation (cid:0) i ∂∂t (cid:1) ′′ can be written as D † ( ξ ( t )) ab (cid:18) i ∂∂t (cid:19) ′ D ( ξ ( t )) ab = D † ( ξ ( t )) ab (cid:18) i ∂∂t (cid:19) D ( ξ ( t )) ab + D † ( ξ ( t )) ab { a J + a J + + a ∗ J − } D ( ξ ( t )) ab . (77)The first term of this expression results to be [39] D † ( ξ ( t )) ab (cid:18) i ∂∂t (cid:19) D ( ξ ( t )) ab = i ∂∂t + b K ( ab )0 + b J + b K ( a )+ + b ∗ K ( a ) − + b K ( b )+ + b ∗ K ( b ) − , (78)with the values of the complex constants b ’s given by b = 12 (cid:16) ˙ φ a (cosh( θ a ) −