Nonlinear Interferometer with Feedback Loops

• Nonlinear interferometers with feedback loops are systems that use nonlinear media to repeatedly mix and recycle optical fields, enhancing phase sensitivity in quantum measurements.
• Sequential and partial feedback schemes recycle one or both optical modes to amplify phase-dependent signals, yielding a quadratic boost in quantum Fisher information for phase estimation.
• Feedback mechanisms also adjust photon statistics and induce bistable, hysteresis behaviors, improving resource efficiency and control in experimental quantum metrology setups.

A nonlinear interferometer with feedback loops is an interferometric system in which the propagation and mixing of optical fields occur via nonlinear media, and part or all of the output is recycled (“fed back”) into the device to interact further with the nonlinear element(s). Feedback modifies both the steady-state and dynamical response of the interferometer and can be engineered to optimize phase sensitivity, induce multistability, or realize enhanced nonclassical states in quantum metrology. These architectures—rooted in the SU(1,1) interferometer formalism and continuous-variable Gaussian state analysis—serve as a platform to demonstrate nonlinear quantum-limited estimation, optical bistability, and complex control over photon statistics.

1. Architectures and Feedback Schemes

Two canonical feedback schemes define the recent theoretical model of a nonlinear interferometer with feedback loops (Singh et al., 23 Jul 2025):

• Sequential Feedback Scheme: The output from both arms (modes) of the SU(1,1) interferometer is fed back, so both channels sequentially undergo repeated operations—two-mode squeezing, phase shift, and another squeezing—in each loop. After $N$ feedback loops, the total evolution operator is $$\mathcal{W}_\text{s} = [S(\Gamma_2) U(\phi) S(\Gamma_1)]^N$$, where $S(\Gamma)$ is a two-mode squeezing operator and $U(\phi)$ imparts the unknown phase. If $\Gamma_1 = \Gamma_2 = r$, the feedback effectively “reuses” and amplifies both modes simultaneously, leading to a highly phase-dependent final state.
• Partial (One-Mode) Feedback Scheme: Only a single mode is sent back into the feedback loop while the other is measured and reset (typically to vacuum) after each cycle. The corresponding evolution is

$$\mathcal{W}_\text{a} = S(\Gamma_2) U(\phi) S(\Gamma_1) [\Pi S(\Gamma_2) U(\phi) S(\Gamma_1)]^N,$$

where $\Pi$ projects the measured mode onto vacuum. This results in monotonic amplification and state nonclassicality primarily in the recycled channel.

The table below summarizes key structural distinctions:

Scheme	Feedback Routing	Output Dynamics
Sequential (both modes)	Both modes recycled	Oscillatory intensity and QFI; phase-dependent interference
Partial (one-mode)	One mode fed back	Monotonic intensity and QFI growth

Both configurations are implemented by tuning switching and detection logic that routes the appropriate modes back through the nonlinear region, allowing for explicit experimental realization via switches or passive optical circulators.

2. Quantum Fisher Information Scaling and Phase Estimation

The quantum Fisher information (QFI) for phase estimation quantifies the metrological advantage gained from feedback-enhanced nonlinear interferometry. For an $N$-loop setup with vacuum input, the QFI can be analytically related to the covariance matrix $\sigma$ of the resultant Gaussian state: $$H(\phi) = \frac{1}{4} \operatorname{Tr}\big[(\sigma^{-1} \partial_\phi \sigma)^2\big].$$ In a standard SU(1,1) configuration (two-mode squeezer, no feedback), this produces

$H = \sinh^2(2r) = \bar{n}(\bar{n}+2),$

with $\bar{n}=2\sinh^2(r)$.

With feedback:

• The QFI in the sequential feedback scheme contains a quadratic scaling term $N^2$:

$$H_\text{s}(\phi) = N^2\,\operatorname{Tr}[(A_s \sigma_s^{-1})^2] + \text{(lower-order terms)},$$

with $A_s$ reflecting the accumulated phase sensitivity from repeated phase interactions.

• For the partial feedback scheme, the QFI increases monotonically with $N$ without oscillations, and scaled QFI $H_\text{a}(\phi)/N^2$ robustly increases for higher $r$ and small $\phi$.

Both schemes exhibit enhanced phase estimation efficiency compared to standard single-pass interferometers, with the repeated feedback yielding a quadratic boost in the QFI for small phases.

3. Photon Loss Effects and Robustness

Photon loss degrades the metrological performance of both standard and feedback-based nonlinear interferometers, but this sensitivity is especially pronounced for vacuum-state input configurations (Singh et al., 23 Jul 2025):

• After each loop, the covariance matrix is updated to account for photon loss $\eta$ as

$$\sigma_{n}^{\text{(th)}} = t(\sigma_{n-1}^{\text{(th)}} - \mathbb{I}/2) + \mathbb{I}/2, \qquad t = 1-\eta.$$

• The QFI in the presence of loss decays exponentially as a function of total loss, reducing the advantage of feedback-enhanced operation. This mirrors the known loss sensitivity of SU(1,1) interferometry where $H \propto \eta$ for small $\eta$.

This suggests that feedback-enhanced schemes require stringent optical loss minimization throughout all passes, particularly in photon-starved (vacuum seeded) quantum-limited metrology settings.

4. Resource Efficiency and Scaling

Feedback enables resource-efficient phase estimation by effectively “recycling” photons through multiple phase interactions per input excitation:

• Low-Loss Regime: Recycling in feedback loops boosts the effective mean photon number interacting with the phase shifter, yielding a lower phase variance for a given input squeezing compared to the standard device. In sequential feedback, for small loop numbers and phases, resource efficiency (defined as QFI per total photon number) is optimal.
• High-Loop Numbers/Partial Feedback: For large $N$ or larger phase shifts, partial feedback can outperform sequential configurations, as the latter’s oscillatory interference can reduce sensitivity away from the phase-matched regime.

The feedback protocol thus allows for flexible optimization, matching resource deployment to the desired measurement scenario.

5. Bistability, Hysteresis, and Input–Output Relations in Atomic-Vapor Nonlinearities

In classic models, such as ring-cavity rubidium vapor systems with two feedback loops (1011.3617), the interplay of feedback and nonlinearity produces:

• Optical Bistability Domains: Multiple stable steady-states exist in the space of input field intensities, with the bistable region’s boundaries and area determined by detuning, feedback strength, and atomic parameters.
• Cross-Hysteresis: Feedback enables a channel-cross effect—hysteresis in the output of one field as a function of the input intensity in the other channel. This nontrivial interdependence offers a means to implement all-optical control or transistor-like behavior.
• Input–Output Equations:

$$I_j^{\text{in}} = I_{j0} + R_j I_j^{\text{out}},\quad I_j^{\text{out}} = I_j^{\text{in}} \exp\left[ -n_j(I_1^{\text{in}}, I_2^{\text{in}}) \right]$$

with $n_j$ the intensity-dependent absorption obtained from master-equation solutions.

• Linear Stability Analysis: Only branches with Jacobian matrix norm $|\mathcal{L}| < 1$ are stable and observable, imposing boundaries on experimentally feasible steady-state operation.

Feedback thus not only enhances metrological sensitivity but induces new steady-state optical phases.

6. Advanced Tuning: Squeezing/Anti-Squeezing Swapping and Phase Optimization

A special operational protocol involves alternating the sign of the squeezing operator (switching between $S(r)$ and $S(-r)$ after every $k$ loops, $k$ set to half the oscillation period):

$$\Psi_\text{ss}(\phi) = [S(r) U(\phi) S(r)]^k \cdot [S(-r) U(\phi) S(-r)]^k \cdot ...$$

This “swap” mechanism resets the phase evolution, maintaining optimal probe state properties even when the unknown phase $\phi$ is large. Numerical evaluation shows that QFI can be enhanced and phase variance minimized for arbitrary phase values by proper tuning of the swapping interval, overcoming destructive interference effects that otherwise limit standard sequential feedback at high phase.

7. Mathematical Formulation

The dynamics of a nonlinear interferometer with feedback loops are compactly captured by the following sequence of transformations:

• State evolution per loop:

$$\Psi(\phi) = S(\Gamma_2) U(\phi) S(\Gamma_1) \Psi_{\text{in}}$$

• Sequential feedback after $N$ loops:

$$\Psi_\text{s}(\phi) = [S(\Gamma_2) U(\phi) S(\Gamma_1)]^N |\mathbf{0}\rangle$$

• Quantum Fisher information for pure squeezed Gaussian states:

$$H(\phi) = \frac{1}{4} \operatorname{Tr}\left[\left(\sigma^{-1}\partial_\phi \sigma\right)^2\right]$$

All quantities—intensity, QFI, stability—are derived from the covariance matrix formalism and are directly computable for arbitrary loop number $N$, squeezing parameters, feedback routing, and loss factor.

Summary

Nonlinear interferometers augmented with feedback loops exhibit enhanced phase estimation sensitivity arising from the repeated nonlinear transformation and phase-shifting of optical fields. Resource efficiency is greatly improved in the low-loss, small-phase regime, with feedback-driven recycling leading to quadratic QFI scaling with the number of loops. The schemes are inherently sensitive to loss, and bistability or hysteresis can emerge in atomic vapor implementations due to the self-consistent interplay between nonlinearity and feedback. Advanced control protocols, such as swapping between squeezing and anti-squeezing, allow further tuning to maintain optimal estimation performance even for large phase shifts. This architecture thus unites quantum-limited metrology, nonlinear optics, and feedback control in a coherent formalism useful for both fundamental studies and practical quantum sensor design.