Experimental Observation of the Yang-Lee Quantum Criticality in Open Quantum Systems

The Yang-Lee edge singularity was originally studied from the standpoint of mathematical foundations of phase transitions. However, direct observation of anomalous scaling with the negative scaling dimension has remained elusive due to an imaginary magnetic field required for the nonunitary criticality. We experimentally implement an imaginary magnetic field with an open quantum system of heralded single photons, directly measure the partition function, and demonstrate the Yang-Lee edge singularity via the quantum-classical correspondence. We also demonstrate unconventional scaling laws for finite-temperature quantum dynamics.

Figure 1

Experimental setup. Heralded single photons are created via type-II spontaneous parametric down-conversion in a PPKTP crystal. The polarizing beam splitter ${\mathrm{PBS}}_1$ and the half-wave plate (HWP) $H_0$ are used to generate initial polarization states $|H⟩$ and $|V⟩$. After photons pass through a $50:50$ beam splitter (BS), the transmitted photons enter the evolution path, and the reflected photons act as reference photons and interfere with the transmitted photons at ${\mathrm{PBS}}_3$. The evolution process involves nonunitary evolutions $U_1$ and $U_2$ realized by two beam displacers (BDs) and sets of wave plates and the projector $P$ is realized by two HWPs and ${\mathrm{PBS}}_2$. The measurement part involves ${\mathrm{PBS}}_3$, two quarter-wave plates (QWPs), and two HWPs, followed by another two PBSs. Finally, photons are detected by avalanche photodiodes (APDs), and recorded by the coincidence counts of $D_0$, $D_1$, and $D_0$, $D_2$, respectively.

Figure 2

Yang-Lee scaling laws of physical quantities for a finite-temperature quantum system in the $\mathcal{P}\mathcal{T}$-unbroken phase in the limit $\varphi \to \pi /4-0$ after ${\beta }^{-1}\to 0$. (a) Imaginary part of $m$ as a function of $\mathrm{\Delta }\varphi$. (b) Imaginary part of $\chi$ as a function of $\delta \varphi =(\mathrm{\Delta }\varphi +\mathrm{\Delta }{\varphi }^{\prime })/2$, where $\mathrm{\Delta }\varphi =(10,30,50,100,\dots ,350)\times {10}^{-6}$ and $\mathrm{\Delta }{\varphi }^{\prime }=(30,50,100,150,\dots ,400)\times {10}^{-6}$. Dependencies of real (c) and imaginary (d) parts of $G(t_2,t_1)$ on $\mathrm{\Delta }\varphi$. Experimental data are shown as open squares and theoretical predictions are represented by solid curves. We choose $R=0.05$, $\beta ={10}^5$, and $\mathrm{\Delta }t=3000$. Dashed curves in (a) and (b) correspond to the results fitted by different power laws. Error bars indicate the statistical deviation, obtained by Monte Carlo simulations under the assumption of Poissonian photon-counting statistics. Some error bars are smaller than the size of the symbols.

Figure 3

Temperature dependences of anomalous scaling laws and $\mathrm{\Delta }t$ dependence of $\mathrm{Re}(⟨{\sigma }_z(t_2){\sigma }_z(t_1)⟩Z)$ in the limit of ${\beta }^{-1}\to 0$ after $\varphi \to \pi /4-0$. Imaginary part of $m$ (a) and that of $\chi$ (b) as functions of ${\beta }^{-1}$. (c) Real part of $G(t_2,t_1)$ as a function of ${\beta }^{-1}$. (d) Real part of $⟨{\sigma }_z(t_2){\sigma }_z(t_1)⟩Z$ as a function of $\mathrm{\Delta }t$. We choose $R=0.05$, $\varphi =\pi /4–{10}^{-6}$ (${\varphi }^{\prime }=\pi /4–{10}^{-2}$), $\mathrm{\Delta }t=0.1$ in (a), (b), and (c), and $R=0.05$, $\varphi =\pi /4–{10}^{-6}$, $\beta ={10}^4$ in (d).

Figure 4

(a) Phase diagram of the Yang-Lee quantum critical system. Experimental values of $Z$ on the ${\beta }^{-1}-\varphi$ plane. The critical point is located at $\varphi =\pi /4$ and ${\beta }^{-1}=0$. The Yang-Lee zeros appear in the $\mathcal{P}\mathcal{T}$-broken ($\varphi >\pi /4$) phase (gray region) given by Eq. (13). In the $\mathcal{P}\mathcal{T}$-unbroken phase, a crossover between Yang-Lee scaling laws and unconventional scaling laws occurs around the region indicated by the blue-dashed curve. The experimental data are obtained with $n=0$, $n=1$, $n=10$ from top to bottom, where $n$ is given in Eq. (11), and the color indicates the value of $Z$. (b) $Z$ as a function of $\varphi$ for $n=10$, $n=1$, and $n=0$. We choose $R=0.01\pi$ for (a)–(b). For (c)–(d) we choose $R=0.05$ and $\beta ={10}^3$. (c) $Z$ versus $-\mathrm{\Delta }\varphi$. The region colored in light blue shows the $\mathcal{P}\mathcal{T}$-broken regime, and the inset is an enlarged view of the regime. (d) Density of zeros $1/({\varphi }_{n+1}-{\varphi }_n)$ for $n=0,1,2,\dots ,9$ from left to right. The horizontal axis is taken as $({\varphi }_n+{\varphi }_{n+1})/2$, which is measured from the critical point $\varphi =\pi /4$.