Simulating Dynamic Quantum Phase Transitions in Photonic Quantum Walks

Signaled by nonanalyticities in the time evolution of physical observables, dynamic quantum phase transitions (DQPTs) emerge in quench dynamics of topological systems and possess an interesting geometric origin captured by dynamic topological order parameters (DTOPs). In this Letter, we report the experimental study of DQPTs using discrete-time quantum walks of single photons. We simulate quench dynamics between distinct Floquet topological phases using quantum-walk dynamics and experimentally characterize DQPTs and the underlying DTOPs through interference-based measurements. The versatile photonic quantum-walk platform further allows us to experimentally investigate DQPTs for mixed states and in parity-time-symmetric nonunitary dynamics for the first time. Our experiment directly confirms the relation between DQPTs and DTOPs in quench dynamics of topological systems and opens up the avenue of simulating emergent topological phenomena using discrete-time quantum-walk dynamics.

Figure 1

(a) Experimental setup for the simulation of DQPTs using QWs. Pairs of single photons are generated via type-I spontaneous parametric down-conversion using a nonlinear $\beta$-barium-borate (BBO) crystal. One photon serves as a trigger and the other signal photon is prepared in an arbitrary linear polarization state using polarizing beam splitters (PBSs), half-wave plates (HWPs) and quarter-wave plates (QWPs) with certain setting angles, and a nonpolarizing beam splitter (NPBS). Coin rotations and conditional translations are realized by two HWPs and a beam displacer (BD), respectively. For nonunitary QWs, a sandwich-type HWP-PPBS-HWP setup is inserted to introduce the partial measurement, where PPBS is an acronym for a partially polarizing beam splitter. Avalanche photodiodes (APDs) detect the signal and heralding photons. (b) Phase diagram for QWs governed by Floquet operators $U$ and $\tilde{U}$, labeled by the winding number $\nu$ as a function of coin parameters $({\theta }_1,{\theta }_2)$. Note that phase boundaries and winding numbers for unitary QW dynamics governed by $U$ and nonunitary dynamics governed by $\tilde{U}$ are the same. Dashed red lines represent boundaries between $\mathcal{P}\mathcal{T}$-symmetry-unbroken and broken regimes for $\tilde{U}$, with $\mathcal{P}\mathcal{T}$-symmetry-broken regimes lying in between the red lines near topological phase boundaries, which are represented by solid black lines. The black star represents coin parameters of the initial Floquet operator $U^i$ or ${\tilde{U}}^i$; other symbols indicate coin parameters of final Floquet operators in different cases.

Figure 2

Rate function (upper) and ${\nu }^m(t)$ (lower) of seven-step unitary QWs as functions of time steps. The initial state of the walker-coin system is $|x=0⟩\otimes |{\psi }_{-}^i⟩$. QWs are governed by $U^f$ with (a) $({\theta }_1^f=-\pi /2,{\theta }_2^f=3\pi /8)$ and (b) $U^f$ with $({\theta }_1^f=-\pi /2,{\theta }_2^f=\pi /4)$, respectively. Error bars are derived from simulations where we consider all the systematic inaccuracies of the experiment. Asymmetry in error bars is due to decoherence.

Figure 3

Rate function (upper) and ${\nu }^m(t)$ (lower) of a seven-step unitary QW. The walker starts at $x=0$ and the QW is governed by the final Floquet operator $U^f$ with $({\theta }_1^f=-\pi /2,{\theta }_2^f=3\pi /8)$. The initial coin state is a mixed state with (a) $p=0.9$ and (b) $p=0.7$, respectively. Error bars are derived from simulations, where we consider all the systematic inaccuracies of the experiment.

Figure 4

(a) Rate function and (b) ${\nu }^m(t)$ of a seven-step unitary QW. The QW is governed by the final Floquet operator $U^f$ with $({\theta }_1^f=-\pi /16,{\theta }_2^f=-3\pi /16)$, which has the same winding number as that of the initial state. Error bars are derived from simulations, where we consider all the systematic inaccuracies of the experiment.

Figure 5

(a) Rate function and ${\tilde{\nu }}^m(t)$ of a seven-step nonunitary QW with a loss parameter $l=0.36$. The initial state of the walker-coin system is $|0⟩\otimes |{\tilde{\psi }}_{-}^i⟩$. The QW is governed by the nonunitary Floquet operator ${\tilde{U}}^f$ with $({\theta }_1^f=-\pi /3,{\theta }_2^f=\pi /5)$. (b) Rate function of the QW governed by ${\tilde{U}}^f$ with $[{\theta }_1^f=-\pi /2,{\theta }_2^f=(\pi -\xi )/2]$, where $\xi =\arccos (1/\alpha )$ and $\alpha =(1+\sqrt{1-l})/(2\sqrt[4]{1-l})$. The two critical timescales are $t_0=\{1.7183,2.1482\}$, which give rise to nonanalyticities in the rate function as indicated by vertical dashed lines in (a). Theoretically calculated fixed points are located at $k_{1,2,3,4}=\{-1.0094\pi ,-0.4470\pi ,-0.0094\pi ,0.5530\pi \}$ and the critical momenta $k_c=\{-0.7888\pi ,-0.1534\pi ,0.2112\pi ,0.8466\pi \}$. Experimental errors are due to photon-counting statistics.