Off-Diagonal Geometric Phases

A collaboration project within the joint ESRF (Nicola Manini) and ILL (Fabio Pistolesi) theory group.

Summary

We investigate the phase relation among different eigenstates of a parameterized Hamiltonian evolving adiabatically along an open path. This phase relation can be described in terms of gauge-invariant, measurable quantities, extending the concept of Berry phase. We analyze several practical occurrences of these quantities, including an experiment of deformed microwave cavities where they can be determined from the published data.

Background

The Berry phase of the adiabatic, parallel-transported eigenstates of a deformed "quantum billiard" has been studied experimentally[Lauber94]. This animation illustrates the evolution of the lowest of the three states of the quantum billiard when its corner is driven along a loop around the degenerate point:

animation White/black indicate +/- signs of the wavefunction extrema.

A cyclic Berry phase of corresponds to a sign change of the corresponding eigenstate: the "jump" of the plotted wavefunction when the loop restarts reflects this cyclic phase.

Consider in general the parallel adiabatic evolution of the nondegenerate normalized eigenstates | psi _i(r)> of a parameterized Hamiltonian H(r). The idea that, with a suitable definition, the phase of the scalar product <_j(r₁)|_j(r₂)> can be measurable dates back to the pioneering work of Pancharatnam[Pancharatnam]. When r₁=r₂ and the state |_j(r)> is transported adiabatically along a closed loop, we have the usual cyclic Berry phase. Since its formalization, considerable work has been devoted to interpretation generalization, and experimental determination of these geometric phase factors. Surprisingly, for r₁r₂, the phase of <_j(r₁)|_k(r₂)> between two different eigenstates has not been equally well investigated so far[simon:note].

New ideas

This is even more surprising if one considers that for some pairs of points it may occur that |_k(r₂)>=eⁱ|_j(r₁)> (with j and k different). Thus, both scalar products <_j(r₁)|_j(r ₂)> and <_k(r₁)|_k(r₂)> vanish: the evolved state gets orthogonal to the initial one. Here the usual Pancharatnam-Berry phase on any path connecting r₁ to r₂ gets undefined for the states k and j. The only phase information left is contained in the off-diagonal scalar product <_j(r₁)|_k(r₂)>.

In our figure above, this happens exactly at mid path: the evolution starts from a wave of type {2,4} (i.e. with 2 semi-oscillations in the horizontal direction and 4 in the vertical one). It then undergoes some mixing, to become a {7,1} wave at mid way. Finally it evolves to become again {2,4}, but with the changed sign (its Berry phase discussed above). It is known by elementary Fourier analysis that the waves {2,4} and {7,1} are indeed orthogonal.

As the figure below indicates, it turns out that, following the same path, the state {7,1} goes at mid path to {2,4}: the two states exchange.

The observed initial (=0), intermediate (=) and final (=2) eigenstates of the microwave cavities deformed following adiabatically the path of Ref. [Lauber94]. These three eigenstates of the rectangular resonator get degenerate for the undistorted rectangular geometry.

Clearly, at mid path, each state has no meaningful phase relation with respect to the state at departure. However there should be some phase relation hidden somewhere in the adiabatic evolution. The purpose of this work is to determine the measurable quantities associated to the phases of the off-diagonal matrix elements <_j(r₁)|_k(r₂)> for a general open path in the parameter space connecting r₁ to r₂.

Results

We find[NM & FP] the minimal set of independent off-diagonal phase factors that exhaust the geometrical phase information carried by the basis of eigenstates along the path. The formalism is then applied [FP & NM] to the experiment on quantum billiards[Lauber94], where the off-diagonal phase factors can be extracted directly from the published experimental data.

An illustration of these ideas is available as a pdf presentation. More details are available in our publications (published paper in PRL / html preprint / latex/ps/pdf in the arXiv).

Also, recently in collaboration with a group of experts in neutron interferometry, we measured at the ILL the off-diagonal phase of the neutron spin wavefunction, confirming the theoretical predictions.

References

S. Pancharatnam, Proc. Ind. Acad. Sci. A 44, 247 (1956).
M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984).
H.-M. Lauber, P. Weidenhammer, and D. Dubbers, Phys. Rev. Lett. 72, 1004 (1994).
N. Manini and F. Pistolesi, Phys. Rev. Lett. 85, 3067 (2000).
F. Pistolesi and N. Manini, Phys. Rev. Lett. 85, 1585 (2000).

created: 23 November 1999

last modified: 2 Dec. 2004 by Nicola Manini