Quantum metrology with boundary time crystals

The work addresses whether and how decoherence—typically detrimental to quantum metrology—can be harnessed to achieve quantum-enhanced parameter estimation. Prior approaches to criticality-assisted metrology have focused on ground states of critical Hamiltonians and dissipative phase transitions. Here, the context is boundary time crystals (BTCs), exotic non-equilibrium phases of open quantum systems where time-translational symmetry is spontaneously broken at the thermodynamic limit, leading to persistent oscillations. The central hypothesis is that the second-order phase transition into the BTC phase can be exploited as a metrological resource, providing super-linear scaling of sensitivity (beyond the standard quantum limit) with system size, and that this enhancement can be accessed with simple measurements without stringent state preparation.

- Quantum metrology promises precision advantages but is challenged by noise and probe preparation. Strategies include measurement-based and control-based schemes, as well as leveraging criticality. - Critical ground states: Near quantum critical points, ground states show enhanced susceptibility to parameters driving the transition; numerous studies explored critical metrology in closed systems. - Dissipative phase transitions: In open systems, Liouvillian gap closing leads to critical steady-state properties and enhanced susceptibilities; metrological applications have been proposed including symmetry-breaking scenarios, Kerr interactions, and continuous monitoring. - Time crystals and BTCs: Following Wilczek’s proposal of time-translation symmetry breaking, extensive theoretical and experimental works have explored discrete and continuous time crystals in closed and driven systems. BTCs are open-system analogs featuring persistent oscillations in stationary dynamics, with Liouvillian spectra showing bands in the imaginary parts and vanishing real parts at large sizes. Prior works characterized BTC formation and properties in collective spin models, mean-field analyses, continuous monitoring signatures, and the build-up of correlations. However, open problems remained regarding full critical characterization, metrological utility of BTC transitions, and whether simple measurements can reveal enhanced sensitivity.

- Model: N spin-1/2 particles forming a collective spin S = N/2 with collective operators Sx, Sy, Sz. Hamiltonian H = ω Sz (ω the single-particle coherent splitting). Open-system dynamics via a Lindblad master equation with collective dissipation rate κ: dρ/dt = −i [ω Sz, ρ] + κ L[S−](ρ), where L denotes the standard Lindblad superoperator with jump S−. - BTC phases: By varying ω/κ, the steady state exhibits a transition from a static phase (ω < κ) to a BTC phase (ω > κ) with long-lived oscillations at large N. The critical point in the thermodynamic limit is ωc = κ. - Numerical analysis: The Lindblad equation is solved numerically for finite N across phases. Observables include magnetization dynamics ⟨Sz(t)⟩/N, steady-state magnetization ⟨Sz⟩ss, and Liouvillian spectrum (real and imaginary parts of eigenvalues) to diagnose oscillatory bands and slowing dynamics. - Finite-size scaling for order parameter: The order parameter is |⟨Sz⟩ss|/N. Curves vs ω/κ for various N are collapsed using finite-size scaling to extract the critical point and exponents. Scaling ansatz of the form |⟨Sz⟩ss|/N ∼ N^{-β/ν} f[N^{1/ν}(ω−ωc)/κ]. The Python package physas is used to optimize collapse and estimate ω* and exponents with uncertainties. - Quantum Fisher information (QFI): The QFI FQ(ω) for estimating ω/κ is computed from steady states across sizes N. The peak value FQ^max = FQ(ωmax) is analyzed vs N to determine scaling (standard quantum limit vs super-linear). The shift of ωmax with N is fitted to assess convergence to the thermodynamic critical point. - QFI finite-size scaling and ansatz: Near criticality, scaling invariance suggests FQ(ω) collapses under FQ ∼ N^{-νQ} G[N^{1/νQ}(ω−ωc)/κ], and an empirical ansatz FQ(ω) = a / (N^b + (ω/κ)^η) is considered to connect finite-N and thermodynamic scaling. Independent determinations of exponents (b, η, ν) are compared for consistency. - Classical Fisher information (CFI) with simple measurement: Consider measuring spin projection S·n for a unit vector n(θ, φ). Projective measurements in the eigenbasis of S·n yield classical probabilities, from which classical Fisher information FC(θ, φ) is computed. Angles are optimized per N at ω = ωmax to assess achievable precision with feasible measurements and its scaling with N. The efficiency ratio FC^max/FQ^max is evaluated vs N. - Time-constrained sensing: When total evolution time T is a resource, the relevant figure of merit is FQ/T. Bounds for Markovian dynamics are derived using operator norm techniques, yielding an upper bound proportional to S = N/2 (hence linear in N), FQ/T ≤ N/(2κ). The slowest Liouvillian decay rate (smallest nonzero |Re E2|) sets the relaxation time T ≈ 1/|Re E2|; its scaling with N is fitted from spectra to assess practical time-constrained performance.

- BTC phase transition and Liouvillian spectrum: • Static phase (ω < κ): Dynamics relax to a unique steady state without oscillations; relevant Liouvillian eigenvalues are real and non-positive with a unique zero mode. • BTC phase (ω > κ): Imaginary parts of Liouvillian eigenvalues form approximately equally spaced bands; real parts approach zero with increasing N, indicating persistent oscillations in the thermodynamic limit. - Second-order transition and critical point: • Steady-state magnetization |⟨Sz⟩ss|/N shows a sharpening transition with N and vanishes in the BTC phase at large N. • Finite-size scaling collapse yields ω*/κ = 0.995 ± 0.002, consistent with ωc = κ in the thermodynamic limit. • Critical exponents from magnetization scaling: ν = 1.453 ± 0.064, β = 0.434 ± 0.055. - Quantum Fisher information (QFI) and enhanced sensitivity: • FQ(ω) exhibits a pronounced peak near ωc; the peak position ωmax approaches ω* as N increases (fit: ωmax = ω* [1 − N^{−0.776}]). • The peak QFI scales super-linearly with probe size: FQ^max ∝ N^b with b ≈ 1.345 (fit indicates crossover exponent b > 1), surpassing the standard quantum limit and evidencing quantum-enhanced sensitivity enabled by the dissipative transition. • Finite-size scaling and ansatz comparisons for QFI identify consistent critical behavior with exponents b ≈ 1.345 and η ≈ 1.511; independent analyses support relations among exponents. - Classical Fisher information (CFI) with simple measurements: • Projective measurements of S·n with optimized angles (θ, φ) at ω = ωmax achieve FC^max ∼ N^{1.38}, i.e., super-linear scaling close to the QFI. • The efficiency ratio FC^max/FQ^max maintains a finite fraction over N, showing that simple collective spin measurements can extract a sizable part of the ultimate precision. - Time-constrained metrology: • For fixed total time T, a rigorous bound for Markovian dynamics yields FQ/T ≤ N/(2κ), implying that when time is a resource, the ultimate scaling becomes linear in N. • The relaxation time is set by the slowest Liouvillian mode E2; fits indicate 1 − |E2| ∼ N^{-0.9}, consistent with slowing dynamics near criticality.

The results confirm that the dissipative second-order transition into a boundary time crystal can be harnessed for metrology: proximity to criticality amplifies sensitivity to the Hamiltonian parameter ω/κ, as quantified by a super-linear scaling of the QFI peak with system size. Importantly, the enhancement arises from decoherence-driven criticality—turning a typically detrimental factor into a resource. The Liouvillian spectral structure explains persistent oscillations and critical slowing down, underpinning scaling behaviors. Finite-size scaling of both the order parameter and the QFI provides a consistent critical point near ωc = κ and compatible critical exponents, reinforcing the second-order nature of the BTC transition. From a practical standpoint, simple projections of the collective spin already achieve super-linear CFI scaling close to the QFI, reducing the need for complex optimal measurements. When accounting for total time as a resource, the achievable scaling becomes linear in N, in line with fundamental bounds for Markovian dynamics; nonetheless, the BTC-enhanced prefactors and practical accessibility (initialization independence, simple readout) make the scheme attractive for realistic quantum sensors.

This work demonstrates that boundary time crystal transitions, driven by collective dissipation, can be used to achieve quantum-enhanced sensitivity in estimating ω/κ. Through numerical solutions and finite-size scaling, we establish the second-order nature of the transition, determine the critical point and exponents, and show that the QFI peak scales super-linearly with system size. We further show that a simple collective spin measurement can capture a substantial fraction of the ultimate precision and that, under time constraints, performance adheres to a linear-in-N upper bound. The protocol is robust in that it does not require special initialization and relies on accessible measurements. Potential future directions include experimental validation in cold atom platforms (e.g., collectively driven Rb ensembles), exploring other BTC-capable models and noise mechanisms, optimizing measurement schemes for higher FC/FQ ratios, and extending analysis to non-Markovian regimes and adaptive control to approach bounds under realistic constraints.

- The analysis is largely numerical and relies on finite-size scaling; while care is taken to extract critical exponents, finite-N effects and fitting choices can lead to uncertainties and apparent inconsistencies between different scaling analyses. - The reported critical exponents from different observables and ansatz fits (e.g., ν, η, b) carry uncertainties and may vary across methods; establishing universal relations conclusively could require larger system sizes and analytical treatments. - The metrological advantage in the steady-state regime relies on approaching criticality, which may be sensitive to experimental imperfections and parameter drifts. - Under time constraints, fundamental bounds reduce scaling to linear in N, limiting asymptotic gains; reaching steady state near criticality can require long times due to critical slowing down. - The simple measurement strategy, while practical, recovers only a fraction of the QFI and requires angle optimization that may depend on N and ω.