Observation of Critical Phenomena in Parity-Time-Symmetric Quantum Dynamics

We experimentally simulate nonunitary quantum dynamics using a single-photon interferometric network and study the information flow between a parity-time- ($\mathcal{P}\mathcal{T}$-)symmetric non-Hermitian system and its environment. We observe oscillations of quantum-state distinguishability and complete information retrieval in the $\mathcal{P}\mathcal{T}$-symmetry-unbroken regime. We then characterize in detail critical phenomena of the information flow near the exceptional point separating the $\mathcal{P}\mathcal{T}$-unbroken and $\mathcal{P}\mathcal{T}$-broken regimes, and demonstrate power-law behavior in key quantities such as the distinguishability and the recurrence time. We also reveal how the critical phenomena are affected by symmetry and initial conditions. Finally, introducing an ancilla as an environment and probing quantum entanglement between the system and the environment, we confirm that the observed information retrieval is induced by a finite-dimensional entanglement partner in the environment. Our work constitutes the first experimental characterization of critical phenomena in $\mathcal{P}\mathcal{T}$-symmetric nonunitary quantum dynamics.

Figure 1

Experimental setup. A photon pair is created via spontaneous parametric down-conversion (SPDC). One of the photons serves as a trigger, the other is projected into the polarization state $|H⟩$ or $|V⟩$ with a polarizing beam splitter (PBS) and a half-wave plate (HWP), and then goes through various optical elements. Upper layer: the experimental setup for the nonunitary dynamics of a single-qubit $\mathcal{P}\mathcal{T}$-symmetric system. Lower layer: experimental setup for the unitary dynamics of a two-qubit system, where the two qubits are encoded in polarization and spatial degrees of freedom. To prepare the initial state of a two-qubit system, heralded single photons pass through a PBS and a HWP with certain setting angles and are split by a birefringent calcite beam displacer (BD) into two parallel spatial modes: upper and lower modes. After passing through wave plates inserted into different spatial modes, the photons are prepared in the state $|\overline{0}⟩$ or $|\overline{1}⟩$ (see the main text for the definition). We construct the time-dependent density matrices by performing quantum-state tomography for both of these states.

Figure 2

Information retrieval in the $\mathcal{P}\mathcal{T}$-symmetry-unbroken regime. (a)–(c) Oscillations of the distinguishability $D(t)$ for $a<1$, between the two time-evolved states starting from $|H⟩$ and $|V⟩$. Dots with error bars represent the experimental results, while the curves show the theoretical predictions. (d) Recurrence time $T$ of the distinguishability as a function of $ϵ=1-a$.

Figure 3

Irreversible information flow to the environment in the $\mathcal{P}\mathcal{T}$-symmetry-broken regime. (a) Decay of the distinguishability in the $\mathcal{P}\mathcal{T}$-broken regime with different coefficients $a>1$. (b) Relaxation time $\tau$ of the distinguishability as a function of $ϵ=1-a$. The blue solid curve shows the theoretical result $\tau =1/2\sqrt{a^2-1}$. The relaxation time is determined by fitting the experimental results (dots with error bars) to an exponential function.

Figure 4

(a) Power-law behavior of the distinguishability $D(t)\sim t^{-2}$ of the $\mathcal{P}\mathcal{T}$-symmetric system for initial states $\{|H⟩,|V⟩\}$. (b),(c) Power-law behavior of the distinguishability of the time-reversal-symmetric system at the exceptional point $a=1$ for initial states (b) $(|H⟩\pm |V⟩)/\sqrt{2}$ and (c) $\{|H⟩,|V⟩\}$. The power-law behaviors $D(t)\sim t^{-2}$ in (b) and $D(t)\sim t^{-1}$ in (c) demonstrate the dependence of the critical phenomena on the initial state. In the long-time limit, the experimentally measured distinguishability agrees well with theoretical asymptotes. The blue solid curves show the theoretical prediction, and the red dashed lines indicate their asymptotes.

Figure 5

$\mathcal{P}\mathcal{T}$ dynamics embedded in a two-qubit system with $a=0.5$. (a) Quantum oscillations of the distinguishability. (b) Entanglement entropy between the $\mathcal{P}\mathcal{T}$-symmetric system and its ancilla, which oscillates with half the period of that of the distinguishability.