Temporal Interference from Topological Transitions in Monitored Quantum Dynamics

The gist: Temporal interference patterns can be detected with stroboscopic monitoring that treats the back action of measurements and the unitary dynamics, focusing on how this interference emerges near topological transitions in monitored quantum dynamics.

General Formalism

The recurrence problem gives rise to long-lived interference patterns in time, implying that the amplitude A˜(n) is decaying very slowly For the recurrence problem, the process starts and ends at the target state xT ⟩, so a detected recurrence effectively closes a loop in Hilbert space The amplitude for first detection at step n is given by [7, 10] ϕ(n, xT) = ⟨xT U(τ)S n−1xT ⟩ The eigenvalues of the survival operator S provide the key to the mechanism behind temporal interference

Spectral Decomposition and Interference

The first-detection amplitude admits an interference structure, specifically ϕ(n, xT) = A˜1(n)e iΦ˜ 1(n) + A˜2(n)e iΦ˜ 2(n) Interference is strongest when the condition A˜1(n) = A˜2(n) is realized, leading to Fn = 4A˜2 (n) cos squared ∆Φ (˜ n) / 2 The first-detection amplitude admits the spectral decomposition ϕ(n, xT) = wX−1 k=0 rke iβk ξ n k, where the coefficients rke iβk are the corresponding spectral weights For large n, the behavior of ϕ(n, xT) is controlled by a pair of eigenvalues, ξ1 and ξ2, which approach the unit circle simultaneously This coexistence gives rise to temporal interference

Topological Transitions

The key questions are what sets the oscillation frequency, what controls the envelope and slow relaxation, and how these features are connected to the emergence of dark states near topological transitions Topological transitions occur when phase factors of the unitary operator U(τ) become degenerate, which happens when e−iE1τ = e−iE2τ Long-lived temporal interference appears close to transitions in which the winding number changes by two, w → w − 2 This transition is identified by plotting the mean recurrence time ⟨n⟩ as a function of sampling interval τ For example, in the monitored quantum walk on a four-dimensional hypercube, topological transitions are directly visible as jumps in ⟨n⟩, for example from 5 to 3

Hamiltonian Origin of Temporal Interference

The long-lived interference observed in Fn occurs close to, but not exactly at, the condition where two independent pairs of unitary phase factors approach each other while remaining well separated from the rest This near-merging corresponds to two dark states being weakly coupled to the detector, and their corresponding eigenvalues ξ1 and ξ2 move slightly inside the unit circle The oscillation frequency is set by the phase difference ∆θ = θ1 − θ2 ≃ (E34 − E12)τ This phase-matching condition arises naturally in systems with bipartite symmetry, which provides a natural route to the w → w−2 transition and to the resulting long-lived temporal interference

Scenarios of Interference Regimes

The paper describes two distinct regimes for realizing temporal interference based on how phase factors merge 1) Merging of pairs of phase factors: This scenario is governed by the leading-order small corrections to Eq. (7), where the frequency is ∆θ = (E34 − E12)τ + δ, with δ = 2η ∆]Eτ 2) Merging of three phase factors: This scenario involves a zero-energy stationary state, leading to an oscillation pattern qualitatively distinct from the finite-frequency fringes found above In both regimes, Fn decays slowly, like e−λ(∆]Eτ) 2n

Discussion and Outlook

The mechanism is structurally protected by the bipartite symmetry of the underlying graph, whose mirror-symmetric spectrum guarantees the balanced weights and conjugate phases of the two dominant modes The fringe frequency distinguishes between scenarios: in bipartite systems with a stationary E = 0 we find slow oscillations, as in Fig. 3(c), whereas the merging of two pairs of phase factors yields faster oscillations, see Fig. 3(a) Looking forward, observing these long-lived interference fringes offers a timely experimental opportunity A critical next step is to investigate how realistic environmental noise and finite coherence times modify or smear the interference patterns identified here

Appendix A

The eigenvalues of the survival operator S = (1 − xT ⟩⟨xT)U(τ) introduced in Eq. (2) can be analytically determined through an electrostatic mapping [7] Applying the matrix determinant lemma to the characteristic polynomial of S yields 0 = det [ξ1 − S] = det[ξ1 − U(τ)]⟨xT [ξ1 − U(τ)]−1xT ⟩ξ Evaluating the matrix element via spectral decomposition in the energy basis, we find that the non-trivial eigenvalues ξ strictly inside the unit disk, which enter the spectral decomposition of the first-detection amplitude in Eq. (3), are given by the roots of Xw k=1 Xgk m=1 ⟨xT Ek,m⟩2 1 e−iEkτ − ξ = 0

Appendix B

The global phase shift β appearing in the interference pattern [Eq. (5)] is explicitly given by β = 1/2 Arg Qw−1 j=1 (1 − ξ∗j)ξk squared Q j̸=1(ξj − ξk) - 1/2 Arg Qw−1 j=2 (1 − ξ∗j)ξk squared Q j!=2(ξj − ξk)

Appendix C

Without a partner mode of comparable magnitude to generate a relative phase, Fn carries no oscillations and decays as a single exponential This single-mode asymptotic behavior analytically explains the slow monotonic decay observed in Fig. 1(a) (blue triangles) and aligns with the isolated quasidark state configuration shown in Fig. 2(c)

Appendix D

The vertices of the graph are the computational basis states z⟩ = Nd j=1 zj ⟩ with zj ∈ [0, 1], and nearest-neighbor hopping corresponds to single spin flips The energy levels are Ek = −(d − 2k) for k = 0,..., d, with degeneracy gk = d/k For the recurrence problem initialized at a single vertex, xT ⟩ = 0⟩ ⊗d j=1 z j ⟩, the spectral weights are binomial, Pk = d/k squared − d The spectrum and weights satisfy Ek = −Ed−k and Pk = Pd−k, realizing the bipartite symmetry that protects the interference visibility discussed in the main text

Appendix E

The system possesses a discrete spectrum Ek = −2γ cos kπ L+1 for k = 1,...

1. Noisy quantum walk dynamics modeling: The improved AI system can simulate long-lived temporal interference patterns by accurately predicting Fn ∼ 4(1 − ξ2)2ξ2/2n ε squared cos2(n/2∆θ) + β as a function of sampling time τ and Hamiltonian parameters.

2. Topological transition detection: The system can identify critical points where the mean recurrence time w = 5 reflects five distinct energy levels and undergoes discrete transitions at resonances, specifically detecting when a pair of eigenvalues approaches the unit circle signaling a w → w − 2 transition.

3. Spectral signature analysis: The AI can analyze the spectrum of the survival operator S to determine if the system is exhibiting slow decay, fast relaxation, or oscillatory behavior by monitoring whether a pair of eigenvalues, denoted ξ1 and ξ2, approaches the unit circle simultaneously or if a single eigenvalue dominates.

4. Hamiltonian-driven interference prediction: The AI can predict oscillation frequencies by calculating ∆θ = θ1 − θ2 ≃ (E34 − E12)τ, directly linking the observed fringe frequency to the energy difference between merged pairs of Hamiltonian states.

5. Bipartite symmetry exploitation: For systems with bipartite symmetry, the AI can leverage this property to guarantee the balanced weights and conjugate phases of the two dominant modes, ensuring a generic route to stable temporal interference.

Quantum interference is conventionally associated with the coherent superposition of alternative paths. We uncover a temporal form of interference in repeatedly monitored quantum systems, appearing directly as fringes in the stochastic first-detection-time distribution. This temporal interference becomes remarkably long lived near a class of topological transitions at which a pair of dark states emerges in Hilbert space. Close to such transitions, the corresponding quasi-dark modes dominate the late-time dynamics and interfere coherently, producing pronounced temporal fringes whose decay becomes arbitrarily slow upon approaching the transition. We identify the symmetry conditions underlying this mechanism and relate the fringe frequency to the energy spectrum of the Hamiltonian. Our results reveal how the interplay of quantum superposition, topological transitions, and dark-state formation generates long-lived temporal interference in monitored quantum dynamics.