Experimental Determination of Dynamical Lee-Yang Zeros

Statistical physics provides the concepts and methods to explain the phase behavior of interacting many-body systems. Investigations of Lee-Yang zeros—complex singularities of the free energy in systems of finite size—have led to a unified understanding of equilibrium phase transitions. The ideas of Lee and Yang, however, are not restricted to equilibrium phenomena. Recently, Lee-Yang zeros have been used to characterize nonequilibrium processes such as dynamical phase transitions in quantum systems after a quench or dynamic order-disorder transitions in glasses. Here, we experimentally realize a scheme for determining Lee-Yang zeros in such nonequilibrium settings. We extract the dynamical Lee-Yang zeros of a stochastic process involving Andreev tunneling between a normal-state island and two superconducting leads from measurements of the dynamical activity along a trajectory. From the short-time behavior of the Lee-Yang zeros, we predict the large-deviation statistics of the activity which is typically difficult to measure. Our method paves the way for further experiments on the statistical mechanics of many-body systems out of equilibrium.

Figure 1

Phase transitions and Lee-Yang zeros. (a) Density of a gas with $⟨N⟩$ particles as a function of the inverse temperature $\beta =1/(k_{B}T)$. With increasing volume $V$, the density develops a discontinuity at the critical inverse temperature ${\beta }_c$ corresponding to a first-order phase transition. (b) As $V$ increases, the zeros of the partition function for complex values of $\beta$ (blue dots) approach the critical value ${\beta }_{\mathrm{c}}$ (red dot) on the real axis. (c) Average activity of an open quantum system as a function of the biasing field $s$ that couples to the number of random events $K$ along a trajectory of length $t$. As $t$ increases, the system may exhibit a phase transition between a dynamical phase with a large activity (left box) and one with a low activity (right box). (d) With increasing time, the zeros of the dynamical partition function (blue dots) approach the value $s_c$ (red dot) where the phase transition occurs.

Figure 2

Experimental scheme. (a) The setup consists of a normal-state metallic island ($N$) coupled to superconducting leads ($S$) via insulating tunnel barriers ($I$) [29]. Single-electron events that bring the system from the ground state (0) to one of the excited states ($\pm$) occur with the rate ${\mathrm{\Gamma }}_u=12\mathrm{Hz}$. The reverse process happens with the rate ${\mathrm{\Gamma }}_d=252\mathrm{Hz}$. Andreev events between the excited states happen with the rate ${\mathrm{\Gamma }}_a=615\mathrm{Hz}$. (b) The current in a nearby single-electron transistor switches between three levels corresponding to the three charge states of the island. Colored regions indicate Andreev events, where the current switches between a high level (above dashed blue line) and a low level (below dashed red line). (c) Absolute value of four cumulants $⟪K^n⟫(t)$ of the number of Andreev events along trajectories of length $t$.

Figure 3

Determination of dynamical Lee-Yang zeros. (a) The leading pair of dynamical Lee-Yang zeros (blue) in the complex plane of the biasing field $s$. The Lee-Yang zeros are extracted from the cumulants of the activity in Fig. 2 for $t=1,\dots ,35\mathrm{ms}$. The extracted Lee-Yang zeros are unchanged if the cumulant order is increased, ensuring us that subleading zeros can safely be neglected. At longer times, subleading zeros start to interfere and the extraction method is no longer accurate. The Lee-Yang zeros converge towards the points $s_c$ and $s_c^{*}$ indicated in red. (b) The convergence points are obtained by linearly extrapolating the dependence of the real part and the imaginary part on the inverse observation time for $t=10–35\mathrm{ms}$ [27]. By taking the inverse time to zero, we find the convergence points $s_{\mathrm{c}}^{(*)}=-0.23\pm 0.15i$ shown with red in both panels.

Figure 4

Large-deviation statistics of the dynamical activity. The solid line is the upper part of the tilted ellipse in Eq. (11) using the convergence points extracted in Fig. 3. The dashed line is the lower part of the ellipse. The ellipse is delimited by the average activity in the two phases $J_1\simeq 0$ and $J_2=615\mathrm{Hz}$ (the rate at which Andreev events occur between the excited states of the island). To fit the data, the ellipse has been shifted slightly downwards to compensate for an offset which is not included in Eq. (11). The distribution measured over $t=3\mathrm{s}$ is indicated in blue. The distribution takes on the large-deviation form at around 3 s which is much longer than the 35 ms used to extract the convergence points in Fig. 3.