Conserved quantities in parity-time symmetric systems

Conserved quantities such as energy or the electric charge of a closed system, or the Runge-Lenz vector in Kepler dynamics, are determined by its global, local, or accidental symmetries. They were instrumental in advances such as the prediction of neutrinos in the (inverse) beta decay process and the development of self-consistent approximate methods for isolated or thermal many-body systems. In contrast, little is known about conservation laws and their consequences in open systems. Recently, a special class of these systems, called parity-time ($\mathcal{PT}$) symmetric systems, has been intensely explored for their remarkable properties that are absent in their closed counterparts. A complete characterization and observation of conserved quantities in these systems and their consequences is still lacking. Here, we present a complete set of conserved observables for a broad class of $\mathcal{PT}$-symmetric Hamiltonians and experimentally demonstrate their properties using a single-photon linear optical circuit. By simulating the dynamics of a four-site system across a fourth-order exceptional point, we measure its four conserved quantities and demonstrate their consequences. Our results spell out nonlocal conservation laws in nonunitary dynamics and provide key elements that will underpin the self-consistent analyses of non-Hermitian quantum many-body systems that are forthcoming.

Figure 1

Nonunitary dynamics of a four-site $\mathcal{PT}$-symmetric system. (a) Schematic of a four-site system with nearest-neighbor tunneling, gain for sites $|1\rangle$ and $|2\rangle$, and corresponding loss for sites $|4\rangle$ and $|3\rangle$. Spectrum of $H_{\mathcal{PT}}$. (b) Schematic of the optical circuit used to measure the state norm and the inner product between the initial and final states. BS: beam splitter; HWP: half-wave plate; QWP: quarter-wave plate; APD: avalanche photodiode; PBS: polarizing beam splitter; CC: compensated crystal; $G_{\mathcal{L}}\left(t\right)$: lossy time evolution circuit [35]. (c) Schematic of quantum state tomography used to reconstruct the time-evolved state $\left|\psi \right(t)\rangle$ [35]. (d) In the Hermitian limit with $\gamma =0$, the measured state norm $N\left(t\right)$ is conserved and remains unity (open symbols). In the $\mathcal{PT}$-symmetric region with $\gamma =0.2J, N\left(t\right)$ oscillates with a period $T\left(\gamma \right)$ (solid symbols). (e) At the $\mathcal{PT}$ transition point $\gamma =J$, the state norm grows algebraically, $N\left(t\right)\propto t^6$. (f) In the $\mathcal{PT}$-symmetry broken region with $\gamma =1.2J$, the state norm grows exponentially with time. Experimental errors are due to photon-counting statistics; when not shown, the error bars are smaller than the symbol size.

Figure 2

Nonlocal, conserved observables across the $\mathcal{PT}$ transition. (a)–(d) For a symmetric initial state $|\psi \left(0\right)\rangle =|{\psi }_2\rangle$, the measured expectation values ${\eta }_i\left(t\right)$ of the four dimensionless observables depend on $\gamma$, but remain time invariant. The symmetry of the initial state is responsible for the $\gamma$-independent behavior of the positive ${\eta }_1\left(t\right)$ and negative ${\eta }_2\left(t\right)$. Error bars are smaller than the symbol size and not shown. Conserved quantities ${\eta }_i\left(t\right)$ for $t<1.2$ and $t>4$ in the unitary and $\mathcal{PT}$-symmetric cases, and that for $t<1.2$ in the $\mathcal{PT}$-broken case (and at the exceptional point) are obtained by direct measurement [35]. By contrast, ${\eta }_i\left(t\right)$ for $1.2<t<4$ are measured through state tomography. The experimental results of conserved quantities obtained from both methods are consistent with each other, and match well with theoretical predictions.

Figure 3

Consequences of nonlocal, conserved observables. (a), (b) At the EP, the adjacent-site phase differences ${\theta }_k\left(t\right)$ reach a steady-steady value $\pi /2$ irrespective of the initial state. (c), (d) The same phase locking occurs in the $\mathcal{PT}$-symmetry broken region. The diverging state norm and the constancy of ${\eta }_i\left(t\right)$ are responsible for this phenomenon. (e) Conserved quantities ${\eta }_i$ measured in the $\mathcal{PT}$-symmetric eigenstates $|E_j\rangle$ for $\gamma =0.2J$. (f) At the fourth-order EP, $\gamma =J$, all conserved quantities ${\eta }_i$ are zero for the sole eigenstate of $H_{\mathcal{PT}}$. (g) In the $\mathcal{PT}$-symmetry broken region, $\gamma =1.2J$, all ${\eta }_i$ vanish for each eigenstate $|E_j\left(\gamma \right)\rangle$. (h) Measured time evolution of the angle ${\mathrm{\Phi }}_j\left(t\right)$ between $|v_j\rangle$ and $\left|\psi \right(t)\rangle$.