aa r X i v : . [ qu a n t - ph ] J un Berry’s phase for coherent states of Landau levels
Wen-Long Yang and Jing-Ling Chen ∗ Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, P.R.China
The Berry’s phase for coherent states and squeezed coherent states of Landau levels are calculated.Coherent states of Landau levels are interpreted as a result of a magnetic flux moved adiabaticlyfrom infinity to a finite place on the plane. The Abelian Berry’s phase for coherent states of Landaulevels is an analogue of the Aharonov-Bohm effect. Moreover, the non-Abelian Berry’s phase iscalculated for the adiabatic evolution of the magnetic field B . PACS numbers: 03.65.Vf, 03.65.-w, 47.27.De, 71.70.Di
I. INTRODUCTION
Since the famous work of Berry [1], the geometric phasehas been widely investigated and its generalizations havebeen made in many ways [2][3][4][5]. Recently much con-cern has been concentrated on the geometric phase ofentangled states as well as mixed states [6][7]. Coherentstate is an important physical concept both theoreticallyand experimentally [8][9]. It can be generated from anarbitrary reference state, and in this Brief Report Lan-dau levels are chosen to be such reference states. Thoughcoherent states of Landau level have been studied in Refs.[10][11] and Berry’s phase for coherent states as well assqueezed coherent states of a one-dimensional harmonicoscillator has been illustrated in Ref. [12], Berry’s phasefor coherent states of Landau levels which is highly de-generate and with an additional parameter, i.e., the mag-netic field B , is still worthy of further investigation.This paper is organized as follows. In Sec. II, we showhow to get the coherent states of Landau levels, andthese states can be regarded as a result of a magneticflux moved adiabaticly from infinity to a finite place onthe plane. In Sec. III, we calculate the Abelian andnon-Abelian Berry’s phase for coherent states of Landaulevels. The Abelian Berry’s phase is just like an alter-native version of the Aharonov-Bohm (AB) effect; thedifference between them is that in our case the cyclicmotion of the magnetic flux results in the phase shift. InSec. IV, we provide the explicit form of the Hamiltonianand Berry’s connections for squeezed coherent states ofLandau levels. Conclusion and discussion are made inthe last section. II. COHERENT STATES OF LANDAU LEVELS
The motion of a free electron in a two-dimensional xy -plane in a static magnetic field along the z -direction isdescribed by the following Hamiltonian H = ¯ h µ [( p x + ec A x ) + ( p y + ec A y ) ] , (1) ∗ Electronic address: [email protected] where µ is the mass of the electron, ¯ h is the Planck con-stant, − e is the electron charge, c is the speed of lightin the vacuum, p x and p y are the linear momentums, A x and A y are vector potentials of the magnetic field sat-isfying ∂ x A y − ∂ y A x = B . For simplicity the Zeeman’sterm is not included.We introduce the following operators π x = p x + ec A x , π y = p y + ec A y , π ± = π x ± iπ y . (2)from which one can form a pair of operators b + and b which satisfy the commutation relation [ b, b + ] = 1: b = r c heB π − , b + = r c heB π + . (3)In this case we can rewrite the Hamiltonian in a moresimpler form H = ¯ hω ( b + b + 1 / , (4)where ω = eB/cµ , and b + , b are raising and low-ering operators between Landau levels with b + | n i = √ n + 1 | n + 1 i , b | n i = √ n | n − i . The energy of Lan-dau levels is E n = ¯ hω ( n + 1 / a = r c heB (cid:16) − p x − ip y + ec A x + i ec A y (cid:17) ,a + = r c heB (cid:16) − p x + ip y + ec A x − i ec A y (cid:17) . We choose the symmetric gauge A x = − y B and A y = x B , and then a , a + are commutative with b , b + and[ a, a + ] = 1. Therefore the Hamiltonian commutes with a and a + . The ground state is defined as | , i = p Be/ (2 πc ¯ h ) exp[ − Be ( x + y ) / (4 c ¯ h )], and all othereigenstates of this system can be generated from | , i state with raising operators | n, m i = √ n ! m ! b + n a + m | , i .The states | n, m i are also orthogonal and normalizedbases for this system. States with the same n are inthe same energy level, and states with the same n butdifferent m stand for the different degenerate states onthe same energy level.The coherent states of Landau levels are generated inthe following way as in [12]: | n ( α ) , m i = exp( αb + − α ∗ b ) | n, m i , where α = X + iX . The Hamiltonian for the coherentstates is: H = D ( α ) H D + ( α ) = ¯ hω (cid:20) ( b + − α ∗ )( b − α ) + 12 (cid:21) , (5)where D ( α ) = exp( αb + − α ∗ b ), and the eigenstates of thisHamiltonian are always combinations of the degeneratestates with the same energy: | n ( α ) i = X m f m | n ( α ) , m i , (6)where f m are arbitrary complex numbers which make | n ( α ) i normalized. We put Eq. (3) back to Eq. (5),this would make the Hamiltonian easier to understand,we get H = ¯ h µ [( π x − r heBc X ) + ( π y + r heBc X ) ] . (7)We found that the magnetic vector potential is addedby a constant vector potential. This can be regarded asa result of a magnetic flux perpendicular to the planemoving adiabaticly from infinitely far to a finitely farposition on the plane. In the following we would like toshow how we get the result.We can assume that the added magnetic flux is aGaussian form magnetic field centered ( x , y ), B ′ = Φ π ∆ exp h − ( x − x ) +( y − y ) ∆ i , where ∆ is refered to as thespread or standard deviation for the Gaussian function.And we may choose the symmetric gauge with respect to( x , y ), i.e., ∇ · A | x = x ,y = y = 0, the nonsingular vectorpotential for this added magnetic field is A ′ x = Φ { exp[ − ( x − x ) +( y − y ) ∆ ] − } ( y − y )2 π [( x − x ) + ( y − y ) ] ,A ′ y = − Φ { exp[ − ( x − x ) +( y − y ) ∆ ] − } ( x − x )2 π [( x − x ) + ( y − y ) ] . (8)One may observe that B ′ has nothing to do with theHamiltonian (1) when x , y → ∞ . Now we assume thatthe electron in the plane is in a certain eigenstate, for ex-ample, | , i , in this case the electron is localized near theorigin because of h x i = h , | x | , i = h y i = 0, h x + y i =2 c ¯ h/Be . Let ∆ > h p x + y i , so when the flux movesadiabaticly to a place ( x , y ) which is finitely far fromthe electron (i.e. x , y ≫ h p x + y i ≃ p c ¯ h/Be ),we assume that the electron is still distributed aroundthe origin and A ′ x , A ′ y near the origin of the plane canbe regarded as constants. Then the Hamiltonian for the electron will be of the form (7) with X = r e hBc Φ y [exp (cid:16) − x + y ∆ (cid:17) − π ( x + y ) ,X = r e hBc Φ x [exp (cid:16) − x + y ∆ (cid:17) − π ( x + y ) . (9)This modification of the Hamiltonian also correspondsto the following transformation ( x, y ) → ( x + δx, y + δy ),where δx = Φ x [exp( − x + y ∆ ) − πB ( x + y ) ,δy = Φ y [exp( − x + y ∆ ) − πB ( x + y ) . (10)So the state ψ ( x, y ) = | , i will become ψ ( x + δx, y + δy ). Since the distribution of the electron is near theorigin, one also makes sure that δx ≪ x , δy ≪ y .The coherent states of Landau levels is nothing butthe shifted eigenstates of Landau levels in the phasespace, here we assume such a shift happens in real space ψ ( x, y ) → ψ ( x + δx, y + δy ). When the conditionsabove are satisfied, this assumption is reasonable. It is B ′ who causes this small shift. We may see from Eq.(10) that the direction of B ′ , i.e., the sign of Φ is relatedto the direction of the shift. When Φ > III. BERRY’S PHASE FOR COHERENTSTATES OF LANDAU LEVELS
We know that the Landau levels are highly degenerate,and so is the coherent states of Landau levels. Berry’sphase for degenerate states was presented in [3] and mayhave a non-Abelian nature. We calculated the Berry’sconnections as follows: h n ( α ) , m | ∂ X | n ′ ( α ) , m ′ i = ( − iX ) δ n,n ′ δ m,m ′ +( √ n ′ + 1 δ n,n ′ +1 − √ n ′ δ n,n ′ − ) δ m,m ′ , h n ( α ) , m | ∂ X | n ′ ( α ) , m ′ i = iX δ n,n ′ δ m,m ′ +( i √ n ′ + 1 δ n,n ′ + i √ n ′ δ n,n ′ − ) δ m,m ′ , h n ( α ) , m | ∂ B | n ′ ( α ) , m ′ i = 12 B ( α √ m ′ δ m,m ′ − − α ∗ √ m ′ + 1 δ m,m ′ +1 ) δ n,n ′ + 12 B √ n ′ m ′ δ n,n ′ − δ m,m ′ − − B p ( n ′ + 1)( m ′ + 1) δ n,n ′ +1 δ m,m ′ +1 . (11)In the degenerate space, i.e., between states with thesame n , the Berry’s connections become A m,m ′ X = − iX δ m,m ′ , A m,m ′ X = iX δ m,m ′ A m,m ′ B = 12 B ( α √ m ′ δ m,m ′ − − α ∗ √ m ′ + 1 δ m,m ′ +1 ) . (12)We found that A X and A X are Abelian. With the adi-abatic theorem for degenerate states proved in [13] andto a higher order in [14], we know that the | f m | in Eq.(6) will not change during the arbitrary slow evolution of X and X . Also because the Berry’s connections of X and X are Abelian, f m ’s will gain a Berry’s phase factor,which is the same of all m , after an adiabatic evolutionin the X - X plane. The Berry’s phase is γ n = i I C h n ( α ) | ∂ X | n ( α ) i dX + h n ( α ) | ∂ X | n ( α ) i dX = I C X dX − X dX = − S, (13)where C is the path of the adiabatic evolution of ( X , X )in X - X plane and S is the area of C . This result ap-peared in [12] for non-degenerate coherent states. Theresult is also the same as the phase in the paper [15].However, in our case it is the moving magnetic flux thatmoves the electron instead of a moving potential well.With the interpretation in the above section, we cansee from Eq. (9) that when the magnetic flux circles theelectron for one loop, the ( X , X ) will also enclose anarea, and this gives the Berry’s phase. For example, welet ( x , y ) moves around the origin in a circle with theradius R for one loop. The Berry’s Phase will be γ ′ = − S ′ = − e Φ (1 − e − R / ∆ ) π ¯ hcBR . (14)This can be viewed as an alternative version of the ABeffect. The different between them is that we move themagnetic flux instead of the electron.One may see from Eq. (11) that A m,m ′ B is non-Abelian,so the change of B will give non-Abelian Berry’s phase.As the non-Abelian Berry’s phase in such a system hasnot been shown in the literature before, in the followingwe would like to give a simple examples to illustrate it.We assume X = 0 during the evolution and the othertwo parameters undergo the loop in Fig. 1. We can getthe eigenvalues of matrix A B as ε ξ / (2 B ) and the corre-sponding eigenstates | n ( α ) , ξ i . The states before evolu-tion Eq. (6) can be rewritten in the new base as | n ( iX , t = 0) i = X ξ h n ( iX ) , ξ | n ( iX ) i | n ( iX ) , ξ i . (15)After the system undergoes an evolution as shown in Fig. FIG. 1: This is the loop of the adiabatic evolution in X -ln( B )plane. The magnetic field B appears in Berry’s phase in thelogarithm form.
1, the state may become | n ( iX , t = τ ) i = X ξ e − iε ξ S ′ h n ( iX ) , ξ | n ( iX ) i | n ( iX ) , ξ i , (16)where S ′ is the area enclosed by X and ln( B ) as in Fig.1. However, if X = 0, the calculation will involve pathordered integral and beome very complicated. IV. BERRY’S PHASE FOR SQUEEZEDCOHERENT STATES OF LANDAU LEVELS
The Hamiltonian for squeezed coherent state is H = D ( α ) S ( β ) H S + ( β ) D + ( α ) , (17)where H is defined in Eq. (4), β = re iθ and S ( β ) = exp( 12 βb +2 − β ∗ b ) . (18)The eigenstates for this Hamiltonian, i.e., the squeezedcoherent states are | n ( α, β ) , m i = D ( α ) S ( β ) | n, m i . (19)In the same way,we put Eq. (3) to Eq. (17), and we canget H = 12 µ n e − r (cid:2) cos( θ/ π ′ x + sin( θ/ π ′ y (cid:3) + e r (cid:2) − sin( θ/ π ′ x + cos( θ/ π ′ y (cid:3) o , (20)where π ′ x = π x − p heB/cX , π ′ y = π y + p heB/cX .For r = 0, Eq. (20) reduces to Eq. (7). For r = 0,one can see from this Hamiltonian that the squeez-ing operation S ( β ) caused an anisotropy in the plane.More clearly if we set θ = 0 Eq. (20) will become H = (cid:0) e − r π ′ x + e r π ′ y (cid:1) / µ , in other words, the kineticenergies π ′ x / µ , π ′ y / µ are squeezed by the factors e − r and e r respectively.Now we consider the Berry’s connections of squeezedcoherent states. h n ( α, β ) , m | ∂∂r | n ′ ( α, β ) , m ′ i = (cid:20) − e − iθ p n ′ ( n ′ − δ n,n ′ − + 12 e iθ p ( n ′ + 1)( n ′ + 2) δ n,n ′ +2 (cid:21) δ m,m ′ , h n ( α, β ) , m | ∂∂θ | n ′ ( α, β ) , m ′ i = i sinh(2 r )4 h e − iθ p n ′ ( n ′ − δ n,n ′ − + e iθ p ( n ′ + 1)( n ′ + 2) δ n,n ′ +2 i δ m,m ′ + i sinh r n ′ + 1) δ n,n ′ δ m,m ′ , h n ( α, β ) , m | ∂∂B | n ′ ( α, β ) , m ′ i = 12 B h r √ n ′ m ′ δ n,n ′ − δ m,m ′ − − r p ( n ′ + 1)( m ′ + 1) δ m,m ′ +1 δ n,n ′ +1 + e iθ sinh r p m ′ ( n ′ + 1) δ n,n ′ +1 δ m,m ′ − − e − iθ sinh r p ( m ′ + 1) n ′ δ m,m ′ +1 δ n,n ′ − i + 12 B ( α √ m ′ δ m,m ′ − − α ∗ √ m ′ + 1 δ m,m ′ +1 ) δ n,n ′ . (21)To our knowledge, the Berry’s connection with respectto B has not been appeared in the literature before.With these, the Berry’s phase is not hard to obtain. Ifanisotropy exist in 2-dimensional electron gas systems,its Hamiltonian would be of the form of Eq. (20). V. CONCLUSION AND DISCUSSION
In this Brief Report, we have calculated the Berry’sphase for coherent states as well as squeezed coherentstates of Landau levels. The Hamiltonian of the coher-ent states of Landau levels is interpreted as a result of amagnetic flux perpendicular to the plane, and it is movedadiabaticly from infinity to a distance away from the electron so that some approximations are satisfied. Thecyclic adiabtic motion of this magnetic flux caused theBerry’s phase of coherent states of Landau levels, thisis an analogue of the AB effect. And the non-Abelianphase is also of interest, the magnetic field B appears inthe Berry’s phase in the form of ln( B ). So when B → B and becomesindefinite, and the reversion of the magnet field is pro-hibited if we want to get this phase. Ref. [16] also statesthis phenomenon that near the level crossing point theBerry’s phase sometimes vanishes.We thank Prof. M. L. Ge and Prof. Jie Liu for theirvaluable discussions. J. L. Chen acknowledges financialsupport from NSF of China (Grant No.10605013). [1] M. V. Berry, Proc. Roy. Soc. London, Ser. A , 45(1984).[2] B. Simon, Phys. Rev. Lett. , 2167 (1983).[3] F. Wilczek, and A. Zee, Phys. Rev. Lett. , 2111 (1984).[4] Y. Aharonov, and J. Anandan, Phys. Rev. Lett. , 1593(1987).[5] J. Samuel, and R. Bhandari, Phys. Rev. Lett. , 2339(1988).[6] E. Sj¨oqvist, Phys. Rev. A , 022109 (2000).[7] E. Sj¨oqvist, A. K. Pati, A. Ekert, J. S. Anandan, M. Er-icsson, D. K. L. Oi, and V. Vedral, Phys. Rev. Lett. ,2845 (2000).[8] R. J. Glauber, Phys. Rev. Lett. , 84 (1963).[9] W. M. Zhang, D. H. Feng, and R. Gilmore, Rev. Mod. Phys. , 867 (1990).[10] M. K. Fung, and Y. F. Wang, Chin. J. Phys. , 435(1999)[11] H. Fakhri, J. Phys. A: Math. Gen. , 5203 (2004)[12] S. Chaturvedi, M. S. Sriram, and V. Srinivasan, J. Phys.A , L1091 (1987).[13] T. Kato, J. Phys. Soc. J. Jpn. , 435 (1950).[14] J. E. Avron, R. Seiler, and L. G. Yaffe, Commum. Math.Phys. , 33 (1987).[15] P. Exner and V. A. Geyler, Phys. Lett. A , 16 (2000).[16] S. Deguchi, and K. Fujikawa Phys. Rev. A72