Steering a quantum harmonic oscillator state along cyclic trajectories leads to a path-dependent geometric phase. Here we describe its experimental observation in an electronic harmonic oscillator. We use a superconducting qubit as a nonlinear probe of the phase, which is otherwise unobservable due to the linearity of the oscillator. We show that the geometric phase is, for a variety of cyclic paths, proportional to the area enclosed in the quadrature plane. At the transition to the nonadiabatic regime, we study corrections to the phase and dephasing of the qubit caused by qubit-resonator entanglement. In particular, we identify parameters for which this dephasing mechanism is negligible even in the nonadiabatic regime. The demonstrated controllability makes our system a versatile tool to study geometric phases in open quantum systems and to investigate their potential for quantum information processing.
When the control field of a qubit, a polarization state, or a spin-
21 system is swept through a level degeneracy, its direction traces an open curve on the Bloch sphere whose endpoints are antipodal, and the geodesic rule for the open-path geometric phase becomes ambiguous: infinitely many geodesics close the path, and different closures enclose different solid angles. We resolve this ambiguity in closed form. A coordinate-free monopole connection defines the open-path solid angle
Ω[C] intrinsically, and displacing the degeneracy by $ε\uhat$ closes the path with enclosed solid angle $Ω(ε\uhat)=Ω[C]+2α+O(ε)$, where
α is the azimuth of the transverse part of $\uhat$ measured from the principal normal of the control curve at the crossing. The identity between geometric phase and enclosed solid angle therefore holds for exactly one closing geodesic---the great circle in the osculating plane (
α=0)---supplied by the curvature at the degeneracy. Berry's
π invariant under reversal of the displacement and the values
±π/2 under a reflection symmetry follow as corollaries, and the pure-state limit of the finite-temperature Uhlmann phase selects the osculating-plane closure automatically, turning the heuristic closing rules of the open-path literature into a computable prescription.
A wave function picks up, in addition to the dynamic phase, the geometric (Berry) phase when traversing adiabatically a closed cycle in parameter space. We develop a general multidimensional theory of the geometric phase for (double) cycles around exceptional degeneracies in non-Hermitian Hamiltonians. We show that the geometric phase is exactly $\ensuremath{\pi}$ for symmetric complex Hamiltonians of arbitrary dimension and for nonsymmetric non-Hermitian Hamiltonians of dimension 2. For nonsymmetric non-Hermitian Hamiltonians of higher dimension, the geometric phase tends to $\ensuremath{\pi}$ for small cycles and changes as the cycle size and shape are varied. We find explicitly the leading asymptotic term of this dependence, and describe it in terms of interaction of different energy levels.
We unveil that even an infinitesimal deviation of the initial state from the eigenstates of the initial Hamiltonian can yield a significant change in the geometric phase accompanying adiabatic evolution without degeneracies of the energy levels. This leads to the counterintuitive result that the values of two geometric phases computed from the general definition of the Berry phase [Samuel and Bhandari, Phys. Rev. Lett. 60, 2339 (1988)] can dramatically differ from each other in the adiabatic limit even if their evolution paths are sufficiently close in a pointwise manner. The result of this work indicates that the widely accepted theory of geometric phases is not very consistent in the context of adiabatic evolution involving superposed eigenstates.
The wave function of a quantum system may acquire an additional phase factor (the geometric phase), when its parameters undergo a cyclic change. An optical manifestation of the geometric phase is the change in the phase of a light wave produced when its state of polarization is taken around a circuit on the Poincare sphere. This phase change can be verified by interferometric measurements. We describe an interferometer in which the two beams traverse the same path in opposite directions, so that their optical paths are always equal, and the phase difference between them is varied by operating on the geometric phase. Measurements made with this interferometer, at a light level at which the probability of more than one photon being present simultaneously in the interferometer was negligibly small, confirmed that the interference effects observed due to the geometric phase persist down to the single photon level.