Measuring the adiabatic non-Hermitian Berry phase in feedback-coupled oscillators

The geometrical Berry phase is key to understanding the behavior of quantum states under cyclic adiabatic evolution. When generalized 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 $\mathcal{PT}$-symmetric Hamiltonian that is inspired by the Hatano-Nelson model. We realize 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 noncyclical 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 $\mathcal{PT}$ 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 further studies of non-Hermitian and topological physics in synthetic metamaterials.

Figure 1

Exploration of the non-Hermitian Berry phase in the two-site Hatano-Nelson model. (a) In non-Hermitian systems with $\mathcal{PT}$ symmetry, adiabatic paths in parameter space generically results in amplification or attenuation of the time-dependent population $N\left(t\right)$. This results directly from the imaginary portion of the adiabatic non-Hermitian Berry phase $\varphi$ acquired by a state along its trajectory. (b) (Top) The Hatano-Nelson (HN) dimer, a minimal non-Hermitian lattice model with nonreciprocal left/right hopping rates $J\pm \delta J$ and an intersite frequency imbalance $2\mathrm{\Delta }$. (Bottom) Implementation of the HN dimer in a mechanical system via measurement-and-feedback. Nonreciprocal coupling between mechanical oscillators, as well as shifts to their resonance frequencies, are realized through applied forces that are responsive to real-time measurements. (c) $\mathcal{PT}$-symmetry breaking phase diagram of the HN dimer. A conical surface of exceptional points in the $J\text{−}\delta J\text{−}\mathrm{\Delta }$ parameter space separates regions of broken and preserved $\mathcal{PT}$ symmetry, respectively lying inside and outside of the conical surface.

Figure 2

$\mathcal{PT}$-symmetry breaking phase diagram of the unbiased ($\mathrm{\Delta }=0$) Hatano-Nelson dimer. (a) White points mark the experimentally measured exceptional points (EPs). Critical $\delta J$ values for these points are determined by detecting the breakdown of adiabaticity as the EP is crossed. (b) Experimental energy dynamics of prepared eigenstates along the ramp of $\delta J$ [dashed red line in (a)] for $J=6.0$ mHz. The measured energy at site 1 decays (while the site 2 and total energy grow) until the EP is reached at $\delta J\sim J$. Here, we plot the site 1 ($E_1$), site 2 ($E_2$), and total energy ($E_T$) normalized to their respective initial values ($E_i$). (c) Crossing of the EP is marked by the onset of growth of the otherwise decaying site 1. The red dashed line is an empirical fit to the data, with the fit minimum defining $\delta J_{\text{crit}}$. Here, we plot the energy in oscillator 1 normalized to its initial value ($E_1/E_{1,i}$).

Figure 3

Geometric energy amplification and attenuation in the Hatano-Nelson dimer. (a) (Left) Dynamics of the total energy $E_T=E_1+E_2$ for adiabatic transformations (normalized to its initial value $E_{T,i}$). (Inset) Path of the adiabatic transformation, for fixed $J=6.5$ mHz, ramping from $\delta J=0$ to/from different maximum values of $\delta J_{\text{max}}=3.4$ (yellow), 4.3 (red), and 5.2 mHz (blue). (Right) Plot of the total energy vs the instantaneous $\delta J$ value, for the yellow and blue paths. The black dotted line shows the expected parametric dependence of $E_T/E_{T,i}$ on $\delta J$ for fully adiabatic evolution under $H$. (b) Dynamics of $E_T$ for CW (purple) and CCW (gold) paths as specified by the inset, centered at $(J,\delta J)=(6.5,2.2)$ mHz and having a radius of 2.2 mHz. The colored dashed lines in the main panels relate to the time-dependent solutions of the Schrödinger-type equation based on evolution under the ideal HN model, including nonadiabatic effects caused from finite ramp durations of $T_{\text{ramp}}=500$ for (a) and $1000s$ for (b).

Figure 4

Cyclic amplification and attenuation along adiabatic paths enclosing $PT$-broken sources of non-Hermitian Berry phase. (a) Illustration of the paths traversed in parameter space. For trajectories in the $J-\mathrm{\Delta }$ plane at fixed and finite $\delta J$, the paths enclose a conical $\mathcal{PT}$-broken region that acts as a source of non-Hermitian Berry flux. (b) Dynamics of the total energy $E_T=E_1+E_2$ for CCW (purple) and CW (gold) paths in the $J-\mathrm{\Delta }$ plane as specified in panel (a), shown for three values of the tunneling asymmetry $\delta J$. The solid purple (gold) lines are the measured trajectories from experiment for CCW (CW) paths. The long-dashed and short-dashed lines are theory comparisons, described below. (c) Ratio of the final total energy $E_{T,f}$ to the initial total energy $E_{T,i}$ as a function of the tunneling asymmetry $\delta J$. The upper and lower panels show the energy ratios for one half cycle and one full cycle, respectively. The purple (gold) points are the experimentally measured ratios for the CCW (CW) paths (with error bars smaller than the data points) and the long-dashed and short-dashed lines are theory comparisons. For (b) and (c), the black long-dashed lines are the analytical predictions (detailed in Ref. [38]) for the amplification/attenuation under fully adiabatic evolution according to Eq. (9). The dotted lines are the trajectories determined by numerical simulation of the experimental ramping procedure, also incorporating weak nonlinear contributions that serve to capture the saturation observed for large amplification (detailed in Ref. [38]).