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The Berry Phase Johar M. Ashfaque In the standard approach to quantum mechanics, pure quantum states are represented by vectors in a complex Hilbert space H. Each vector ψ ∈ H describes a state by the collection of expectation values A→
hψ|A|ψi hψ|ψi
where A is a self-adjoint operator in H representing some physical quantity. For this reason, two vectors ψ and φ describe the same physical state if an only if they are linearly dependent that is to say ψ = λφ,
λ ∈ C.
If we normalize the state vector for example hψ|ψi = 1 there is still a freedom to choose an overall phase factor eiα . Two normalized state vectors ψ and φ are physically equivalent ψ ∼ φ ⇐⇒ ψ = eiα φ. Hence, one usually says that the above phase factor, eiα , has no physical meaning. For this reason we may equivalently represent pure quantum states as one-dimensional projectors in H ψ =⇒ Pψ = |ψihψ|. Clearly ψ ∼ φ ⇐⇒ Pψ = Pφ . However, we know that it is the phase that controls the key effect of quantum mechanics - quantum interference. This effect is governed by the relative phase. If we have two normalized state vectors ψ and φ such that ψ = eiα φ then one calls α a relative phase between ψ and φ. The relative phase or equivalently the phase difference does have a physical meaning and hence can be measured. Superposition of two states ψ ∼ φ differing in phase by α leads to the following interference formula I ∝ |1 + eiα |2 = 2(1 + cos α) = 4 cos2 (α/2) which enables one to measure α. In the interference experiment the overall phases of ψ and φ are still unknown and are not important. It is evident that eiλ ψ and eiλ φ will produce the same interference as ψ and φ. Only the relative phase counts. I.
THE STANDARD DERIVATION
Consider a curve C on a manifold of external parameters M t −→ xt ∈ M and the adiabatic evolution of the quantum system described by the parameter dependent Hamiltonian H = H(x) along the curve C. Then the Hamiltonian depends on time solely via the time-dependence of the external parameters: H(t) = H(xt ). Suppose that for any x ∈ M the Hamiltonian H(x) has a purely discrete spectrum that is H(x)|n(x)i = En (x)|n(x)i with hn(x)|m(x)i = δnm .
2 Moreover, let us assume that the eigenvectors |n(x)i are single-values (as functions of x ∈ M ) that is we assume the existence of the map x ∈ M −→ |n(x)i ∈ H with H being the system’s Hilbert space. Obviously, this map need not be defined globally on M . Therefore, we assume its existence only locally. Let us assume that the n-th eigenvalue En (x) is nondegenerate and let Pn (x) = |n(x)ihn(x)| be the corresponding one-dimensional projector onto the n-th eigenspace Hn (x) which we write as Hn (x) = {α|n(x)i|α ∈ C}. The eigenvectors |n(x)i are not uniquely defined and so one can arbitrarily change the phase of |n(x)i. Due to the adiabatic theorem ψ(t) stays in the n-th eigenspace of H(xt ) during the adiabatic evolution that is ψ(t) ∈ Hn (xt ). Therefore, if the evolution is cyclic that is a curve C is closed then ψ(0) and ψ(T ) both belong to Hn (x0 ) and hence they may differ only by a phase factor ψ(T ) = eiγ ψ(0). The obvious guess for the phase γ would be γ=−
1 ~
Z
T
En (t)dt 0
but Berry showed this guess to be wrong. There is an additional component that has a purely geometric origin. It depends upon the geometry of the manifold M and the circuit C itself. To find this, let us begin by noting that ψ(t) and |n(xt )i differ by a time-dependent phase factor Z i t ψ(t) = exp − En (τ )dτ eiφn (t) |n(xt )i ~ 0 where the Schrödinger equation implies the following equation for the function φn φ̇n = ihn|ṅi where for simplicity the argument for |ni has been omitted. This equation defines the following one-form on M A(n) = ihn|dni. Note that hn|dni is purely imaginary. We can solve φ̇n = ihn|ṅi by a simple integration t
Z
Z hn(τ )|ṅ(τ )idτ =
φn (t) = i 0
A(n)
C
(n)
where one integrates the one-form A along the curve C between x0 and xt . This shows that the guess for γ needs to be supplemented by the following geometric quantity γn (C) = φn (T ) that is the total phase shift γ splits into two parts namely the dynamical phase and the geometric phase. The geometric phase defines the much celebrated Berry phase corresponding to the cyclic adiabatic evolution along C. Using Stoke’s theorem one may rewrite Berry’s phase as Z γn (C) = F (n) Σ
3 where Σ is an arbitrary two-dimensional submanifold in M such that ∂Σ = C and F (n) = dA(n) . The quantity A(n) is usually called the Berry vector potential or Berry’s potential one-form. The quantity F (n) plays the role of a magnetic field for the potential A(n) and the equation Z γn (C) = F (n) Σ
shows that the Berry phase is an analog of the magnetic flux in the electromagnetic theory.