Thermal activation of photon catalysis in difference-intensity Mach-Zehnder interferometry with squeezed thermal light

The gist The optimal operation crosses over from two-photon catalysis on squeezed vacuum to single-photon catalysis on thermal light beyond a certain threshold temperature.

Resource State and Operation

The study investigates difference-intensity phase estimation in a Mach-Zehnder interferometer (MZI) using a coherent state mixed with a heralded non-Gaussian squeezed thermal state as the resource, which is the realistic state produced by a lab squeezer The single-mode squeezed thermal state has zero mean and its covariance matrix is given by VSMST = κ diag(e−2r, e2r), where κ = nth + 1/2 The realistic non-Gaussian operation involves mixing the resource state with an ancillary mode prepared in a Fock state k⟩ through a beam splitter of transmissivity τ Heralding on l photons at the ancilla output projects the output mode 2 into the heralded non-Gaussian state Wf(ξ2) The choice of (k, l) selects the operation, with k l for addition, and k = l for catalysis.

Thermal Activation of Catalysis

The paper demonstrates that single-photon catalysis becomes a useful operation once the thermal occupation crosses a certain value It wins in the success-weighted comparison because it has the success probability near one-half The crossover happens because number-selective catalysis heralding has a broad thermal photon-number distribution to reshape, while mean-shifting operations get their advantage only where the herald becomes very less probable. Specifically, single-photon catalysis is the best operation for every nth ≳ 0.2 This is because it works well for heralding probability near one-half while subtraction loses out as it has low probability of success.

Sensitivity and Figure of Merit

The phase sensitivity is evaluated using the standard error propagation formula, ∆ϕ = q⟨Oˆ2⟩ − ⟨Oˆ⟩ / ∂⟨Oˆ⟩/∂ϕ The advantage of a heralded resource over the bare squeezed thermal resource can be measured via a parameter D defined as D = ∆ϕ SMST − ∆ϕ NG The success-weighted advantage which decides the best operation is obtained by multiplying D with the success probability P R = D × P. Table I lists the best operation at three thermal occupations, showing that the success-weighted advantage grows by almost two orders of magnitude between vacuum and nth = 1.

Quantum Fisher Information

The quantum Fisher information calculations confirm that thermal noise enhances the phase information encoded in the catalyzed state The single-mode Fisher information of the heralded state is computed using Eq. (20) from the spectral form [49, 50] Single-photon catalysis advantage is switched on by the thermal noise at the level of the Fisher information itself, going up from Fϕ ≃ 0.04 on vacuum to Fϕ ≃ 0.35 at nth = 1. This shows that the advantage seen for difference-intensity detection is a real increase in the phase information carried by the catalyzed resource

Robustness and Conclusion

Accounting for optical losses demonstrates that the effect remains robust against realistic imperfections In Fig. 6, every resource loses phase information as the loss increases corresponding to η going down At η = 0.75, the catalyzed resource keeps a Fisher information at nth = 1 which is close to three times its value at nth = 0.5. The thermal activation by adding thermal photons to the source is useful and noise robust The paper concludes that difference-intensity detection is close-to-best choice for most of the operations, and single-photon subtraction is the one case where one would need some other detection scheme.

How it works

1. The MZI setup uses a coherent state in one input port and a heralded non-Gaussian squeezed thermal state in the other mode

2. The resource state is generated by mixing the single-mode squeezed thermal resource with an ancillary Fock state k⟩ at a beam splitter of transmissivity τ

3. The output modes are monitored by photon-number-resolving detectors, and heralding on l photons projects the state into Wf(ξ2)

4. The phase sensitivity is determined by minimizing ∆ϕ over the transmissivity τ, and the best operation is selected by maximizing the success-weighted advantage R = D × P.

5. Thermal excitation reshuffles the hierarchy, making single-photon catalysis useful above nth ≃ 0.2.

How it works

The closed form expression of the heralded Wigner function is a product of a finite degree polynomial and a Gaussian, which allowed for analytical evaluation of sensitivity and heralding probability The moments needed for sensitivity are rational functions of e2r, κ, and τ The exact evaluation via Gauss-Hermite quadrature guarantees that numerical integration is mathematically exact to machine precision when Ng ≥ Dmax + 1/2.

How it works

The thermal occupation effectively provides the state with the required phase-space structure for effective catalysis, notably circumventing the vanishing probability overhead inherent to the other two protocols This enhancement originates from the mixed nature of the initial resource state and has no counterpart in the pure vacuum limit.

How it works

The thermal activation also increases the noise robustness of the detection scheme The quantum Fisher information calculations puts the results on a firmer footing and confirms that thermal noise raises the phase information carried by the catalyzed resource state The thermal activation also increases the noise robustness of the detection scheme.

How it works

The best operation thus crosses over from two-photon catalysis on vacuum to single-photon catalysis on thermal light and interestingly, is a movement towards a lower order operation which is easier to implement The crossover happens because the number-selective catalysis heralding has a broad thermal photon-number distribution to reshape, while the mean-shifting operations get their advantage only where the herald becomes very less probable. The quantum Fisher information calculations puts the results on a firmer footing and confirms that thermal noise raises the phase information carried by the catalyzed resource state The thermal activation also increases the noise robustness of the detection scheme.

How it works

The vacuum version, with the phase-matching condition p0 = 0 and ϕ → 0 at a moderate coherent amplitude, is already known and works through the same Wigner machinery The momentgenerating structure of the heralded state also supports homodyne and quadratic-homodyne detection. Finally, it would also be interesting to see how these results change when the resource evolves in a dissipative environment and to extend the analysis to genuinely multimode resources.

How it works

The thermal activation is useful because it reshapes the photon-number distribution, making the effective photon-number spread that feeds the difference-intensity signal narrower and brings down the measured variance This reshaping makes the effective photon-number spread that feeds the difference-intensity signal narrower and brings down the measured variance. The thermal activation also increases the noise robustness of the detection scheme.

How it works

The paper is organized as follows. In Sec. II we introduce the squeezed thermal resource, the realistic non-Gaussian operation and the interferometer, and we give the closed-form structure of the heralded state In Sec. III we evaluate the difference-intensity phase sensitivity, fix the vacuum baseline, and report the thermal activation of single-photon catalysis In Sec. IV we study the quantum Fisher information, the near-optimality of difference-intensity detection, and the robustness to loss Finally, Sec. V gives the concluding remarks.

1. Bold header: Success-Weighted Operation Optimization for Resource States

The improved AI system can determine if single-photon catalysis is the best success-weighted operation over the whole thermal range by analyzing D = ∆ϕ SMST − ∆ϕ NG, and the success-weighted advantage which decides the best operation is obtained by multiplying D with the success probability P.

2. Bold header: Thermal Activation Threshold Determination

The system can precisely identify the crossover where single-photon catalysis becomes a useful operation once the thermal occupation crosses a certain value, specifically above nth ≃ 0.2.

3. Bold header: Robustness Assessment Against Optical Loss

The AI can assess the reliability of non-Gaussian operations by calculating the Phase quantum Fisher information Fϕ under loss, confirming that the thermal activation of single-photon catalysis survives moderate optical loss, as shown in Figure 6.

Realistic non-Gaussian operations, which can be implemented by mixing a nonclassical Gaussian state with a Fock state at a beam splitter and then heralding on a photon-number-resolving detector, are known to improve phase estimation in a Mach-Zehnder interferometer. The existing studies have considered the single-mode squeezed vacuum for difference-intensity and parity detection, and the two-mode squeezed thermal state for parity detection alone as the initial Gaussian resources. However, a real squeezer in the lab actually produces single-mode squeezed thermal light, which has not been studied in a coherent-plus-resource interferometer. We take up this case and using the phase space Wigner picture, we obtain the heralded state produced by photon subtraction, addition and catalysis on a single-mode squeezed thermal resource, and evaluate the corresponding phase sensitivities. Single-photon catalysis, which is useless on squeezed vacuum, becomes a useful operation for any thermal occupation above 0.2, and since it heralds with a probability near one-half, it turns out to be the best success-weighted operation over the whole thermal range. In the vacuum limit our results reproduce the known hierarchy, where two-photon catalysis is the best operation. The quantum Fisher information computations confirms that thermal noise enhances the phase information encoded in the catalyzed state, demonstrating that difference-intensity detection closely approaches the ultimate precision bound across most operational regimes, and, together with a loss analysis, verifies the robustness of the effect.