Berry s Phase. Erik Lötstedt November 16, 2004

Transcription

1 Berry s Phase Erik Lötstedt lotstedt@kth.se November 16, 2004 Abstract The purpose of this report is to introduce and discuss Berry s phase in quantum mechanics. The quantum adiabatic theorem is discussed and Berry s phase is introduced and examined in some detail. Also the geometrical meaning of Berry s phase, such as Berry s connection and curvature are included. Finally experimental verifications of Berry s phase are pointed out. Contents 1 Introduction 2 2 The quantum adiabatic theorem Adiabatic processes The adiabatic theorem Berry s phase Anholonomic processes Derivation of Berry s phase Geometrical aspects of Berry s phase Berry s connection and curvature Holonomy Example of Berry s phase The monopole analogy Experimental measurement 8 6 Further reading 8 1

2 1 Introduction Berry s phase, discovered by Berry in 1984 [3], points out that the phase of a quantum system is in some cases a nonarbitrary quantity that can be measured. This phase is geometrical in its character and introduces a geometrical approach to quantum theory. The ideas behind are simple, so you may find it remarkable that the discovery was made so many years after the invention of quantum mechanics. 2 The quantum adiabatic theorem In this section we follow [8]. 2.1 Adiabatic processes An adiabatic process is one where there are two time scales, one external timescale T e, characteristic for the change rate of the environment of the system, and one internal timescale T i for the system itself. A process is called adiabatic if T i T e. A familiar classical example is a pendulum. Here T i would be the period of one oscillation and T e the time for the environment of the pendulum to change. We could have, for example, a pendulum with a slowly changing length or a pendulum in a slowly changing gravitational field. In the classical case, we expect the system to behave as if it were solved with the external time held fixed. The pendulum has period T = 2π L/g, so with a gradually changing length L(t), the period of the pendulum would be T = 2π L(t)/g. 2.2 The adiabatic theorem When speaking about quantum adiabatic processes, we mean that the Hamiltonian H changes gradually with time. The time period T e under which the Hamiltonian changes must be much longer than the internal time scale T i of the wave function. The quantum adiabatic theorem now says that if the Hamiltonian is changed slowly from one initial form H i to a final form H f, the nth eigenstate ψ i n of H i will be time translated via the Schrödinger equation into the nth eigenstate ψ f n of H f. Using formulas we have H(t i ) ψ n (t i ) = E n (t i ) ψ n (t i ), H(t f ) ψ n (t f ) = E n (t f ) ψ n (t f ) (1) if the adiabatic approximation holds. Here H(t i,f ) = H i,f, ψ n (t f,i ) = ψ f,i and E n (t f,i ) = En f,i. We assume that the spectrum of the Hamiltonian H(t) is discrete and nondegenerate for t i < t < t f, not to have any problems with ordering of the states. Although not treated here, the nondegeneracy requirement can be relaxed, leading to natural appearance of non-abelian gauge fields [14]. The adiabatic theorem tells us that there will be no transitions from an eigenstate ψ i n of the initial Hamiltonian H i to a different eigenstate ψ f m of the final Hamiltonian H f. To prove the adiabatic theorem is not that easy, even if it 2

3 sounds reasonable. One proof can be found in [9]. In [8], a simpler proof is given, with the assumption that the Hamiltonian is of the form H(t) = f(t)v, (2) where f rises gradually from f(t i ) = 0 to f(t f ) = 1. It is now possible to use perturbation theory to show that for φ(t i ) = ψ i n no transitions occur: 3 Berry s phase φ(t f ) ψ f m 2 = { 1, if m = n, 0, if m n. In this section we follow [3], [7] and [8]. 3.1 Anholonomic processes An anholonomic process is a process where a system does not return to its original state when the external parameters of the system are changed so that they eventually come back to where they started. Again, a familiar example is the Foucault pendulum, where the change is also adiabatic. The direction of swing does not return to its starting value after travelling once around the Earth at a circle of latitude, see [4]. In geometry, anholonomy is expressed by parallel transport round a circuit. This geometrical aspect will be treated in section Derivation of Berry s phase We now ask what will happen with a quantum system, transported adiabatically round some circuit in parameter space. We let the Hamiltonian H depend on a collection of N parameters R = (R 1, R 2,..., R N ) and let R depend adiabatically on time, that is R = R(t) changes slowly with t. Furthermore, we assume that the Hamiltonian does not commute with itself at different times, [H(R(t)), H(R(t ))] 0. Now, if the system at time t = 0 is in the nth eigenstate (assumed normalized to one), H(R(0)) n, R(0) = E n (R(0)) n, R(0), (4) where ψ(0) = n, R(0), what about the solution at later times, ψ(t)? The solution ψ(t) satisfies the Schrödinger equation (3) H(R(t)) ψ(t) = i d ψ(t), (5) dt and under the adiabatic assumption, the adiabatic theorem tells us that the eigenstate n, R(0) continues to be an eigenstate at later times: H(R(t)) n, R(t) = E n (R(t)) n, R(t). (6) The obvious guess for the solution to the Schrödinger equation is that the state picks up a dynamical phase factor [ ψ(t) = exp i t ] E n (R(t ))dt n, R(t). (7) 0 3

4 This is, however, not correct for a Hamiltonian satisfying [H(t), H(t )] 0. Inserting (7) into the Schrödinger equation (5), it is easily seen that the guess (7) is not a solution. It turns out that we have to add an extra phase to the state: [ ψ(t) = exp i t ] E n (R(t ))dt exp [iγ n (t)] n, R(t). (8) 0 The function γ n is found by requiring that ψ(t) satisfies the Schrödinger equation. Careful calculations lead to (see [8], page ) γ n (t) = i t 0 n, R(t ) d dt n, R(t ) dt = i R(t) R(0) n, R n, R dr µ, (9) R µ where summation over µ is implied. This extra phase factor γ n is called Berry s phase, first published by Berry in 1984 [3]. In order to obtain an adiabatic cyclic evolution, the Hamitonian is now brought back to its initial form, so that R(0) = R(T ) for some T large enough for the adiabatic approximation to hold. The phase change of the state vector, besides the dynamical phase i T 0 dt E n (R(t )), is γ n (C) = i n, R n, R dr µ, (10) R µ C where C is a closed curve in parameter space. For the state vector we then have [ ψ(t ) = exp iγ n (C) i ] T E n (R(t ))dt ψ(0). (11) Note that γ n only depends on the path taken and not on the elapsed time T. Neither does it depend on the choice of phase for the eigenstates n, R, a statement that will be further clarified in section 4. In fact, Berry s phase is a geometrical quantity, depending on the geometry of the parameter space. Even though the state returns to the same physical state, the state remembers its evolution through Berry s phase. If the dynamic phase contains information of how long time the evolution took, the geometric phase provides the best answer to the question: where did the system go? We also want to check that γ n is real, not to violate the normalization of ψ. That this is really the case is seen by 0 = ( ) n, R n, R = n, R n, R + n, R n, R, (12) R µ R µ R µ so n, R R µ n, R is imaginary. This also shows that Berry s phase vanishes when the eigenfunctions n, R are real. Since Berry s phase is just a phase factor, you would think that it is of no physical significance. However, it is possible to measure Berry s phase, for example by dividing a beam of particles into two, and let one of the beams pass through an adiabatically changing potential. When the two beams are combined, Berry s phase can be measured by looking at constructive and destructive interference. Another example of a measurement of Berry s phase is found in [6], discussed in section

5 4 Geometrical aspects of Berry s phase In this section we follow [10] and [12]. We will see that equation (6) can be used to describe a line bundle over the parameter space and that Berry s phase can be described as holonomy associated with the connection induced by the adiabatic limit. Explained in the language of bundles, the geometric nature of Berry s phase becomes evident. 4.1 Berry s connection and curvature Let the parameter R = (R 1,..., R N ) be the local coordinate of the manifold M (a space looking locally like R N ) describing the parameter space. To each point R of M we associate the normalized eigenstate n, R of the Hamiltonian H(R). Since the state n, R cannot be distinguished from u n, R, where u U(1), the group of unit length complex numbers, the physical state is an equivalence class: [ n, R ] = { u n, R u U(1) }. (13) Thus, at each point R we attach all states u n, R, and this is the fibre of our U(1) bundle. A fibre bundle is a space that locally looks as a direct product between two manifolds, the base space and the fibre, but not globally. In our case, the bundle locally looks like the direct product M U(1), but may globally be twisted in some way. This topological information is contained in the connection, or gauge field, of the bundle. If the bundle is twisted, it will not be possible to have a non-singular gauge field defined on all of M. We then locally define a connection one form (a gauge field), called Berry s connection, as A = A µ dr µ = n, R d n, R, (14) where d = R dr µ is the exterior derivative in parameter space. This A can µ be checked to satisfy all demands on a connection. In particular, A transforms under a gauge transformation as A A + g 1 dg, (15) where g belongs to the gauge group U(1). Note that the gauge group is Abelian. The corresponding field strength, called Berry s curvature, is given by F = da = n, R R µ R ν n, R (drµ dr ν dr ν dr µ ). (16) Using the connection and the curvature, we can write Berry s phase (10) as γ n (C) = i A = i F, (17) where S is the surface bounded by the loop C. 4.2 Holonomy C What a connection does is to provide a rule for parallel transport. Parallel transport in a fibre bundle along a curve C in M is provided by the so-called horizontal lift of the curve C. A horizontal lift is a curve in the bundle, which S 5

6 projected down onto M gives C and at each point is horizontal to the fibre. If a vector φ belonging to the fibre of R is parallel transported around a circuit back to the fibre of R, then, in general, it does not return to the same point on the fibre. The difference is called a holonomy (it should be anholonomy, really, but in mathematics this is the term used) and we will show that this can be identified with Berry s phase. First, we redefine the Hamiltonian to get rid of the dynamic phase: so that H n(r) = H(R) E n (R), (18) H n(r) n, R = 0. (19) The solution to the new Schrödinger equation H n(r) ψ n (t) = i d dt ψ n(t) now becomes ψ n (t) = exp [iγ n (t)] n, R(t). (20) In order to parallel transport a vector around a curve, we let C = R(t), t [0, T ] be a curve in M. Our rule for parallel transport of a vector φ(t) is given by If the parallel transported vector φ(t) is written as where u(r(0)) = 1, then u U(1) satisfies φ(t) d φ(t) = 0. (21) dt φ(t) = u(r(t)) n, R(t), (22) du(r(t)) u(r(t)) 1 = n, R(t) d n, R(t) = A(d/dt). (23) dt dt If we let u = exp[iϕ], then the solution of equation (23) is ϕ(t) = i n, R d n, R. C Thus, we can identify γ n with ϕ and the parallel transported vector is given by (22): φ(t) = exp[iγ n (t)] n, R(t), (24) precisely what we obtained for the solution (20) to the Schrödinger equation. 4.3 Example of Berry s phase Here we will show an example of Berry s phase. The adiabatically changing parameters of the Hamiltonian will be a magnetic field. Consider a spin 1 2 particle in a magnetic field given by B = (B 1, B 2, B 3 ). The Hamiltonian is given by ( ) B H(B) = B i σ i = 3 B 1 ib 2, (25) B 1 + ib 2 B 3 where σ i are the Pauli sigma matrices. Here we have absorbed the constant involving the gyromagnetic ratio and into B. The Hamiltonian H has two eigenvalues, E ± = ± B. As in (18), we redefine the Hamiltonian to be H ±(B) = H(B) E ±. (26) 6

7 Now, consider the Hamiltonian H +(B). One normalized eigenstate with eigenvalue zero is ( ) 1 B + B3 1+, B =. (27) 2 B ( B + B3 ) B 1 + ib 2 In order to observe Berry s phase, it is convenient to introduce spherical coordinates for the magnetc field, B 1 = B sin θ cos φ, B 2 = B sin θ sin φ and B 3 = B cos θ with 0 θ π, 0 φ 2π. The eigenstate becomes ( 1+, B = cos(θ/2) exp(iφ) sin(θ/2) ). (28) We now let θ and φ depend adiabatically on time, fix the length of B = B 0 and let the magnetic field vector B sweep out a closed curve on the sphere with radius B 0. In three dimensions, formula (10) can be reformulated with Stokes s theorem to read γ + = i ε ijk 1+, R 1+, R da j, (29) S B k B i where S is the area on the sphere bounded by the circuit, and da the area element. Calculating the curl in spherical coordinates, we obtain γ + = 1 B j 2 B 3 da j = 1 Ω, (30) 2 S where Ω is the solid angle of the path, as seen from the origin. (Remember da j = B B j sin θdθdφ = B B j dω.) This is very much like the classical pendulum, which, if carried around a circuit on a sphere, will have its swinging direction altered with the solid angle of the circuit. 4.4 The monopole analogy We note that in the above example, the point R = 0 is a singularity. All the eigenvalues of the Hamiltonian become degenerate (H is just the zero matrix) and the adiabatic assumption breaks down. We will note that this singularity behaves like a magnetic monopole. ( ) cos(θ/2) Looking at the state 1+, B =, we observe that it is exp(iφ) sin(θ/2) not defined for θ = π (where B 3 = B ). In order to obtain a well-defined state on the whole sphere, we use the U(1) freedom and take the eigenvector ( ) exp[ iφ] cos(θ/2) 2+, B = exp[ iφ] 1+, B =, (31) sin(θ/2) defined everywhere on the sphere except at θ = 0. Berry s gauge fields corresponding to the above states are computed to be A 1 = 1+, B d 1+, B = i B 2dB 1 B 1 db 2 = i (1 cos θ) dφ, 2B(B + B 3 ) 2 A 2 = 2+, B d 2+, B = i (32) 2 (1 + cos θ) dφ. 7

8 The gauge fields cover the whole sphere and are related by the gauge transformation A 2 = A 1 + exp[iφ]d exp[ iφ] = A 1 idφ. (33) These gauge fields A 1 and A 2 describe a Wu-Yang magnetic monopole of strength 1 2. A magnetic monopole is a magnetic point charge, or an Abelian gauge theory defined on R 3 with the origin removed and is a simple example of a twisted U(1) bundle, see [10]. If the field strength is calculated, we indeed obtain F 1 = da 1 = i 2 B 3 (B 1dB 2 db 3 +B 2 db 3 db 1 +B 3 db 1 db 2 ) = i B 2 B 3, (34) the magnetic field of a magnetic point 1 2 charge. Berry s phase can now be interpreted as the magnetic flux through the area enclosed by the circuit. 5 Experimental measurement When trying to measure Berry s phase, the fact that the dynamical phase i t 0 dt E n (R(t )) will be much larger than the geometrical phase γ n suggests that the measurement will be difficult. However, measurements of Berry s phase that have been performed show that it is possible to separate out the two contributions. For an experiment realizing the example in 4.3, neutrons moving through a rotating magnetic field, see [6]. Here a superposition of spin eigenstates is used, so that ψ i = c n n, (35) n which after cycling the magnetic field becomes ψ f = n exp [iθ n + iγ n ] c n n, (36) where iθ n is the dynamic phase. If now an observable A which do not commute with the Hamiltonian is measured, then Berry s phase can be read from the oscillating part of the expectation value of A: ψ f A ψ f = n, m c nc m exp [i(θ m θ n + γ m γ n )] n A m. (37) For an experiment with bosons, see [13]. Here a beam of photons are sent into a coiled fibre. After travelling through the coil the direction of linear polarization will be rotated by plus or minus the solid angle of the coil. This is Berry s phase for light. In this case, Berry s phase can be understood on a classical level, in fact the parallel transport law for the electric field can be derived from the classical Maxwell equations [5]. 6 Further reading In this report we have, like Berry in his original paper [3], only considered adiabatic transport of vectors, depending very much on the Hamiltonian of the system. It is possible to generalize this and associate a geometric phase with 8

9 the physical motion of the quantum state itself instead of the motion of the Hamiltonian. By disregarding which Hamiltonian that caused the motion, it is possible to relax the conditions that the state should be an eigenstate of the Hamiltonian and that the motion is adiabatic. Berry s case then appears as a limit. It also becomes possible to consider cyclic motions in isolated systems, see [1] and [2]. So far we have always had a time-dependent Hamiltonian, which corresponds to a system interacting with the environment. In fact, it turns out to be possible [11] to associate a geometric phase with every path, closed or not, using the natural metric associated with the Hilbert space to close the path with a geodesic curve. References [1] Y. Aharonov and J. Anandan, Physical Review Letters 58, 1593 (1987). [2] J. Anandan, Physics Letters A 129, 201 (1988). [3] M. V. Berry, Proceedings of the Royal Society of London A 392, 45 (1984). [4] M. V. Berry, Physics Today December Issue, 34 (1990). [5] M. V. Berry, Nature 326, 277 (1987). [6] T. Bitter and D. Dubbers, Physical Review Letters 59, 251 (1987). [7] A. Bohm, B. Kendrick and M. E. Loewe, The Berry Phase in Molecular Physics, Center for Particle Theory, University of Texas, DOE-ER (1990). [8] D. J. Griffiths, Introduction to Quantum Mechanics, chapter 10, Prentice Hall (1995). [9] A. Messiah, Quantum Mechanics, volume II, chapter XVII, section 12, John Wiley and Sons, New York (1962). [10] M. Nakahara, Geometry, Topology and Physics, second edition, chapter 10, Institute of Physics Publishing (2003). [11] J. Samuel and R. Bhandari, Physical Review Letters 60, 2339 (1988). [12] B. Simon, Physical Review Letters 51, 2167 (1983). [13] A. Tomita and R. Y. Chiao, Physical Review Letters 57, 937 (1986). [14] F. Wilczek and A. Zee, Physical Review Letters 52, 2111 (1984). 9