Abstract We theoretically show that multiport photonic lattices emulating driven quantum harmonic oscillators give rise to Aharonov–Anandan geometric phase, which is proportional to the associated dynamic phase irrespective of the initial state. Consequently, the non-adiabatic geometric phase can be directly measured without the need of complex setups to remove the dynamic phase. Further, we propose the observation of such geometric phases in multiport photonic lattices via integrated interferometry. Our work provides insights into the potential development of integrated photonic platforms for modulation-based topological photonics.
Periodically time-dependent quantum systems have Floquet solutions. They consist of a phase factor and a Floquet mode which shares the periodicity of the driving. Consequently, any such solution possesses a non adiabatic geometric phase known as the Aharonov-Anandan (AA) phase. It is gauge-invariant and, therefore, is experimentally accessible. We propose a protocol for the measurement of AA phases based on the adiabatic change of the phase of the non-adiabatic driving. We argue that the experimental requirements of the protocol are those of previous measurements of adiabatic phases.
In a recent paper we have studied the peculiar features of the Berry and Aharonov-Anandan geometric phases for isolated spins $S\ensuremath{\geqslant}1.$ We have assumed that they are submitted to a dipole and quadrupole coupling to external
E and
B fields with the mild restriction $\mathbf{E}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{B}=0$. This implies discrete symmetries leading to remarkable simplifications of the geometry and algebra involved. The aim of the present work is to describe realistic proposals, within the realm of atomic physics, for the verification of some of our most significant theoretical predictions. There are several challenges to be overcome. For alkali-metal atoms, most commonly used in atomic interferometers, the only practical way to generate quadrupole coupling, with a strength comparable to the dipole one, is the ac Stark effect induced by a nearly resonant light beam. One then has to face the instability of the``dressed'' atom hyperfine (hf) level, a candidate for our isolated spin. One deleterious effect is the apparition of an imaginary part in the quadrupole to dipole coupling strength ratio, $\ensuremath{\lambda}$. Fortunately we have found a simple way to get rid of $\mathfrak{I}(\ensuremath{\lambda})$ by an appropriate detuning. We are left with an unstable isolated spin. This implies an upper bound to the quantum cycle duration
Tc. In the case of the Berry's phase,
Tc has a lower bound due to the necessity of keeping the nonadiabatic corrections below a predefined level. We have found a compromise in the case of the
F=2,m=0 87Rb ground-state hf level. This is our candidate for the measurement of the somewhat ``exotic'' Berry's phase acquired by the
S=2,m=0 state at the end of a quantum cycle involving a rotation of $\ensuremath{\pi}$ of the
E field---in practice, the linear polarization of the dressing beam ---about the
B field direction. We have found a way to implement, in a Ramsey-type interferometric measurement, the procedures aimed at controlling the nonadiabatic corrections, as described in detail in our previous theoretical paper. A numerical simulation of our experimental proposal shows that a
0.1 accurate determination of Berry's phase, free of nonadiabatic corrections, can be achieved. Measurements could also be considered for cold
52Cr chromium atoms with
S=3, where values of $\ensuremath{\lambda}\ensuremath{\simeq}1$ can be obtained with an instability smaller than in the
87Rb case, due to a more favorable spectroscopic structure. The
F=1,m=1 hf level of the
87Rb ground state offers the opportunity to extend the measurement of Aharonov-Anandan's phases beyond the case
S=21. We construct, using ``light shift,'' the Hamiltonian ${H}_{\ensuremath{\parallel}}(t)$ generating a closed circuit in the density-matrix space which satisfies at any time the ``parallel transport'' condition, thus making the quantum cycle free from the adiabaticity condition. We also consider the case of half-integer spins (e.g.,
201Hg,
135Ba, and
137Ba), with their own specific features. We show how the difference of Berry's phases for states
S=23 and
S=21, with
m=21, can be exploited to achieve a holonomic maximum entanglement of three qubits.
The geometrical Berry phase is key to understanding the behaviour of quantum states under cyclic adiabatic evolution. When generalised to non-Hermitian systems with gain and loss, the Berry phase can become complex, and should modify not only the phase but also the amplitude of the state. Here, we perform the first experimental measurements of the adiabatic non-Hermitian Berry phase, exploring a minimal two-site
PT-symmetric Hamiltonian that is inspired by the Hatano-Nelson model. We realise this non-Hermitian model experimentally by mapping its dynamics to that of a pair of classical oscillators coupled by real-time measurement-based feedback. As we verify experimentally, the adiabatic non-Hermitian Berry phase is a purely geometrical effect that leads to significant amplification and damping of the amplitude also for non-cyclical paths within the parameter space even when all eigenenergies are real. We further observe a non-Hermitian analog of the Aharonov--Bohm solenoid effect, observing amplification and attenuation when encircling a region of broken
PT symmetry that serves as a source of imaginary flux. This experiment demonstrates the importance of geometrical effects that are unique to non-Hermitian systems and paves the way towards the further studies of non-Hermitian and topological physics in synthetic metamaterials.