Experimental Observation of Dynamical Phase Transitions in a Dephased Photonic Quantum Walk

Abstract: Dynamical phase transitions in open quantum systems govern how non-equilibrium states relax toward a stationary state. We study these transitions experimentally using a discrete-time photonic quantum walk on a three-node graph. A tunable synthetic gauge flux and calibrated dephasing allow us to control time-reversal symmetry and the detailed balance properties of the effective Markovian dynamics. With detailed balance, we observe a first-order dynamical phase transition marked by a crossing of real Liouvillian eigenvalues. When detailed balance is broken, we observe a second-order dynamical phase transition at an exceptional point where eigenvalues and eigenvectors coalesce. By progressively reducing the dephasing strength, we track the crossover toward the quantum-coherent regime and determine that the transitions persist down to a finite threshold. Our results link Liouvillian spectral topology to relaxation criticality and demonstrate a controllable platform for engineered dissipative dynamics.

• The paper demonstrates experimental observation of both first- and second-order dynamical phase transitions in a dephased photonic quantum walk by precisely controlling dephasing and synthetic gauge flux.
• It employs quantum process tomography and spectral analysis to identify diabolic crossings and exceptional points, directly linking spectral topology with relaxation dynamics.
• The findings pave the way for engineered dissipation and enhanced sensing in scalable non-Hermitian open quantum systems.

Experimental Observation of Dynamical Phase Transitions in Dephased Photonic Quantum Walks

Overview

The paper "Experimental Observation of Dynamical Phase Transitions in a Dephased Photonic Quantum Walk" (2606.15935) presents a comprehensive experimental investigation of dynamical phase transitions (DPTs) in open quantum systems, specifically leveraging a discrete-time photonic quantum walk (OQW) on a minimal three-node graph. Through precise control of dephasing and a tunable synthetic gauge flux, the authors realize both first- and second-order DPTs mediated by the spectral properties of the associated Floquet–Liouvillian superoperator. This work characterizes the phase transitions by their spectral signatures—either eigenvalue crossings (diabolic points, DPs) or eigenvalue/eigenvector coalescences (exceptional points, EPs)—and quantitatively benchmarks the transition behavior via state tomography and detailed Floquet eigenmode analysis.

Theoretical Framework

The studied system is a three-site OQW described by a discrete-time evolution: coherent unitary propagation controlled by a parameter $\beta$ and a phase $\phi$ (synthetic gauge flux), interleaved with local dephasing of calibrated strength $q$. The interplay between these components determines whether the effective Markovian dynamics obeys or breaks detailed balance. Under time-reversal symmetry (TRS, $\phi = 0$), $Q$ is symmetric, enforcing real spectra and DPTs by eigenvalue crossings (FOPTs). Breaking TRS ($\phi \neq 0$) introduces complex spectra, thus enabling EPs and second-order DPTs (SOPTs).

The open-system Floquet map,

$$\rho^{(s+1)} = (1 - q) \mathcal{U} \rho^{(s)} \mathcal{U}^{\dagger} + q \sum_n \mathcal{K}_n \mathcal{U} \rho^{(s)} \mathcal{U}^{\dagger} \mathcal{K}_n^{\dagger},$$

interpolates between fully coherent evolution ($q=0$) and classical population dynamics ($q=1$), allowing spectral signatures of relaxation to be probed across regimes. The paper emphasizes that in the $q=1$ Markovian limit, analytical identification of DPTs is possible via the eigenvalue structure of the transition matrix $\phi$0; for $\phi$1, the full superoperator is reconstructed and analyzed.

Experimental Implementation

The experiment employs a reconfigurable photonic network, with three modes encoded in combinations of polarization and spatial state. The network synthesizes the desired unitary step operator $\phi$2 using waveplate and beam displacer arrangements, followed by controlled dephasing. Heralded single photons are injected, and after each step, mode-resolved detection yields stepwise population data. Full quantum process tomography is performed to reconstruct $\phi$3 or the Liouvillian superoperator as required.

This setup enables precise control of dephasing and phase flux, facilitating direct observation of the change in relaxation dynamics. The protocol distinguishes between monotonic and oscillatory relaxation, directly correlating relaxation patterns with underlying spectral physics.

(Figure 1)

Figure 1: Schematic of the interferometric OQW apparatus, highlighting the implementation of coherent evolution and stepwise dephasing via single-photon optics.

Observation of First-Order Dynamical Phase Transitions

Under TRS, the spectrum of $\phi$4 is strictly real and diagonalizable, so DPTs manifest as abrupt crossings of the relaxation rates of non-stationary modes (i.e., FOPTs). Experimental population measurements reveal a discontinuous switch in the slow relaxation mode as $\phi$5 traverses a critical point $\phi$6. This is quantified by constructing a relaxation order parameter from the relevant eigenmode; a sharp jump in this order parameter at $\phi$7 characterizes the first-order nature of the transition.

The experimentally extracted Floquet exponents show a clear diabolic crossing, with the order parameter shift matching theoretical expectations and confirming the abrupt change in relaxation behavior.

Observation of Second-Order Dynamical Phase Transitions and Exceptional Points

With broken TRS ($\phi$8), the Markov generator $\phi$9 becomes non-symmetric, and its non-stationary eigenvalues can coalesce at an EP. Near the EP, the spectrum exhibits the characteristic square-root branch point: the splitting between non-stationary eigenvalues,

$q$0

vanishes as $q$1, scaling as $q$2.

Figure 2: Square-root scaling of eigenvalue (a) and eigenmode (b) splitting near the EP, establishing the critical behavior associated with SOPTs.

Experimental relaxation data exhibit a crossover from non-oscillatory to oscillatory population dynamics, marking the emergence of a conjugate pair of complex exponents above the EP. Simultaneous coalescence of eigenvectors (quantified by a normalized overlap parameter $q$3 approaching unity) further validates EP formation. This non-analytic behavior in the spectral and eigenmode structure rigorously identifies the SOPT as an EP-driven critical phenomenon.

Influence of Quantum Coherence: Transition Persistence and Suppression

Beyond the fully classical ($q$4) regime, the authors systematically reduce dephasing, reconstructing the full evolution superoperator for intermediate $q$5. The DPTs—the FOPT under TRS and the SOPT under broken TRS—persist for moderate $q$6, with the critical point $q$7 shifting as $q$8 varies. As quantum coherence dominates ($q$9), signatures of DPTs, such as spectral crossings and EP features, become indistinguishable due to spectral smearing and non-analyticity suppression.

These observations demonstrate that DPTs, as defined here, are not restricted to the strict classical (Markovian) limit, but persist deep into the quantum regime, up to a finite coherence threshold.

Implications and Future Directions

The demonstration of controlled DPTs and EP phenomena in a minimal-size open quantum system has multiple ramifications:

• Spectral Topology in Open Systems: The experiment links the topology of the Floquet–Liouvillian spectrum (diabolic points and EPs) to macroscopic relaxation dynamics, positioning OQWs as an accessible testbed for non-Hermitian criticality and spectral topology.
• Engineered Dissipation and Relaxation Engineering: The ability to switch between FOPT and SOPT by tuning synthetic gauge flux allows for programmable control of system relaxation, relevant for quantum simulators and future dissipation-based state-preparation protocols.
• Sensitivity Enhancement via EPs: Operation near EPs is known to enhance parametric sensitivity; optical quantum walk systems may thus offer robust platforms for sensing applications leveraging non-Hermitian degeneracies.
• Scalability and Topological Phenomena: Extending the architecture to larger graphs could enable exploration of phenomena such as the Liouvillian non-Hermitian skin effect and dissipative bulk-boundary correspondence, with implications for topological classification in open quantum systems.
• Non-Markovian Effects and Quantum Enhancements: The flexibility of the setup enables probing routes to non-Markovian DPTs, quantum Mpemba effects, and non-Hermitian speedup in relaxation—a direction of increasing current interest.

Conclusion

This work provides the first experimental observation of both first- and second-order dynamical phase transitions in a dephased OQW controlled by a tunable gauge flux. The results rigorously connect the topology of the Liouvillian spectrum to observable critical behavior in relaxation, validate the persistence of these transitions into the quantum-coherent regime, and position discrete-time photonic OQWs as versatile platforms for investigating fundamental and applied aspects of non-Hermitian physics and engineered dissipation (2606.15935).