Photon-Recycled DSU(1,1) Interferometer

• The paper proposes a novel interferometric design that recycles photons via a feedback loop to enhance phase sensitivity, nearing the quantum Cramér–Rao bound.
• It employs a multi-pass DSU(1,1) structure with active displacement and phase adjustments to boost the effective photon number and improve SID and HD performance.
• The study demonstrates that optimal displacement and recycling parameters yield superior metrological gain compared to conventional setups, emphasizing practical improvements in phase estimation.

The photon-recycled displacement-assisted SU(1,1) interferometer, often abbreviated PR-DSU(1,1), is an active nonlinear interferometric architecture in which a conventional displacement-assisted SU(1,1) interferometer is augmented by a feedback loop that returns the output field of one mode to the corresponding input mode after an additional phase shift and loss channel. In the formulation proposed in "Enhancement in phase sensitivity in displacement-assisted SU(1,1) interferometer via photon recycling" (Kumar et al., 4 Oct 2025), the recycled field is fed from output mode $a$ back into input mode $a$, while mode $b$ remains the reference/input mode. The stated purpose of the scheme is phase estimation enhancement under both single-intensity detection (SID) and homodyne detection (HD), with the paper reporting better performance than the conventional DSU(1,1) interferometer for some conditions and practical sensitivities that approach the quantum Cramér–Rao bound.

1. Baseline DSU(1,1) structure

The underlying DSU(1,1) interferometer consists of two optical parametric amplifiers, local displacement operations on both modes, a phase shift in one arm, and a final nonlinear recombination stage (Kumar et al., 4 Oct 2025). The OPA unitary is written as

$$\hat{U}_{\mathrm{OPA}} = e^{-s \hat{a}^\dagger \hat{b}^\dagger + s^* \hat{a}\hat{b}}, \qquad s = g e^{i\eta},$$

where $g$ is the gain and $\eta$ is the pump phase. The local displacement operators are

$$\hat{D}_a(\gamma)=e^{\gamma \hat{a}^\dagger-\gamma^*\hat{a}}, \qquad \hat{D}_b(\gamma)=e^{\gamma \hat{b}^\dagger-\gamma^*\hat{b}},$$

with $\gamma=|\gamma|e^{i\delta_\gamma}$. The phase shift in arm $b$ is

$\hat{U}_\phi=e^{i\phi \hat{b}^\dagger \hat{b}}.$

For the conventional DSU(1,1), the output modes are given by

$a$0

$a$1

with

$a$2

$a$3

$a$4

$a$5

The balanced interferometer specialization used in the paper is

$a$6

The probe state is vacuum in mode $a$7 and squeezed vacuum in mode $a$8,

$a$9

with

$b$0

and the parameter choices

$b$1

Within this construction, the adjective “displacement-assisted” refers to the local displacements $b$2 and $b$3, which are retained in the photon-recycled extension and play a nontrivial role in the eventual metrological gain.

2. Recycling loop and multi-pass dynamics

The defining modification of the PR-DSU(1,1) is that the output field in mode $b$4 is re-injected into the next pass through the interferometer after acquiring an additional phase $b$5 and experiencing loss characterized by transmission $b$6 (Kumar et al., 4 Oct 2025). The feedback rule is

$b$7

for $b$8, where $b$9 is a vacuum noise operator. Thus only mode $$\hat{U}_{\mathrm{OPA}} = e^{-s \hat{a}^\dagger \hat{b}^\dagger + s^* \hat{a}\hat{b}}, \qquad s = g e^{i\eta},$$0 is recycled, while mode $$\hat{U}_{\mathrm{OPA}} = e^{-s \hat{a}^\dagger \hat{b}^\dagger + s^* \hat{a}\hat{b}}, \qquad s = g e^{i\eta},$$1 is held fixed across passes.

This creates a photon recycling loop, so photons that would normally be discarded after one pass are reused, increasing the effective number of interacting photons and enhancing the phase signal (Kumar et al., 4 Oct 2025). The iteration is represented as a sequence of DSU(1,1) stages,

$$\hat{U}_{\mathrm{OPA}} = e^{-s \hat{a}^\dagger \hat{b}^\dagger + s^* \hat{a}\hat{b}}, \qquad s = g e^{i\eta},$$2

and yields, after $$\hat{U}_{\mathrm{OPA}} = e^{-s \hat{a}^\dagger \hat{b}^\dagger + s^* \hat{a}\hat{b}}, \qquad s = g e^{i\eta},$$3 passes,

$$\hat{U}_{\mathrm{OPA}} = e^{-s \hat{a}^\dagger \hat{b}^\dagger + s^* \hat{a}\hat{b}}, \qquad s = g e^{i\eta},$$4

$$\hat{U}_{\mathrm{OPA}} = e^{-s \hat{a}^\dagger \hat{b}^\dagger + s^* \hat{a}\hat{b}}, \qquad s = g e^{i\eta},$$5

where

$$\hat{U}_{\mathrm{OPA}} = e^{-s \hat{a}^\dagger \hat{b}^\dagger + s^* \hat{a}\hat{b}}, \qquad s = g e^{i\eta},$$6

$$\hat{U}_{\mathrm{OPA}} = e^{-s \hat{a}^\dagger \hat{b}^\dagger + s^* \hat{a}\hat{b}}, \qquad s = g e^{i\eta},$$7

In the limit $$\hat{U}_{\mathrm{OPA}} = e^{-s \hat{a}^\dagger \hat{b}^\dagger + s^* \hat{a}\hat{b}}, \qquad s = g e^{i\eta},$$8, the model reaches a steady recycled form,

$$\hat{U}_{\mathrm{OPA}} = e^{-s \hat{a}^\dagger \hat{b}^\dagger + s^* \hat{a}\hat{b}}, \qquad s = g e^{i\eta},$$9

$g$0

with

$g$1

$g$2

$g$3

The conventional DSU(1,1) is recovered by setting $g$4.

3. Detection models and operational phase sensitivity

The paper evaluates the recycled interferometer under two readout models, each treated by linear error propagation (Kumar et al., 4 Oct 2025). In SID, the observable is the output photon number in mode $g$5,

$g$6

and the phase sensitivity is

$g$7

The corresponding mean is

$g$8

and the variance is

$g$9

In HD, the measured quadrature is

$\eta$0

with sensitivity

$\eta$1

The moments entering this expression are

$\eta$2

and

$\eta$3

The reported comparison is that both detection strategies benefit from photon recycling, but HD performs better than SID and HD shows a broader region of parameter values where sensitivity beats the SNL (Kumar et al., 4 Oct 2025). The same source also notes that with $\eta$4, the HD scheme becomes problematic because $\eta$5, making the sensitivity ill-defined. Nonzero displacement therefore has an operational status: it is not merely an auxiliary Gaussian preprocessing step, but a condition for a useful homodyne signal in the recycled architecture.

4. Shot-noise reference, QFI, and quantum Cramér–Rao bound

The paper defines the total mean photon number inside the interferometer as (Kumar et al., 4 Oct 2025)

$\eta$6

with

$\eta$7

The shot-noise limit is then

$\eta$8

This establishes the classical reference against which the recycled configuration is assessed.

Because the state remains pure and Gaussian under the assumed operations, the paper uses the quantum Fisher information to characterize the ultimate phase limit,

$\eta$9

where

$$\hat{D}_a(\gamma)=e^{\gamma \hat{a}^\dagger-\gamma^*\hat{a}}, \qquad \hat{D}_b(\gamma)=e^{\gamma \hat{b}^\dagger-\gamma^*\hat{b}},$$0

This simplifies to

$$\hat{D}_a(\gamma)=e^{\gamma \hat{a}^\dagger-\gamma^*\hat{a}}, \qquad \hat{D}_b(\gamma)=e^{\gamma \hat{b}^\dagger-\gamma^*\hat{b}},$$1

The quantum Cramér–Rao bound is

$$\hat{D}_a(\gamma)=e^{\gamma \hat{a}^\dagger-\gamma^*\hat{a}}, \qquad \hat{D}_b(\gamma)=e^{\gamma \hat{b}^\dagger-\gamma^*\hat{b}},$$2

with $$\hat{D}_a(\gamma)=e^{\gamma \hat{a}^\dagger-\gamma^*\hat{a}}, \qquad \hat{D}_b(\gamma)=e^{\gamma \hat{b}^\dagger-\gamma^*\hat{b}},$$3 in the paper.

To compare the recycled and non-recycled devices, the work introduces the improvement factors

$$\hat{D}_a(\gamma)=e^{\gamma \hat{a}^\dagger-\gamma^*\hat{a}}, \qquad \hat{D}_b(\gamma)=e^{\gamma \hat{b}^\dagger-\gamma^*\hat{b}},$$4

Values larger than unity indicate an advantage for photon recycling. The reported conclusions are that the PR-DSU(1,1) outperforms the conventional DSU(1,1) for suitable parameter choices, with $$\hat{D}_a(\gamma)=e^{\gamma \hat{a}^\dagger-\gamma^*\hat{a}}, \qquad \hat{D}_b(\gamma)=e^{\gamma \hat{b}^\dagger-\gamma^*\hat{b}},$$5 in significant regions of parameter space, and that the QCRB is also improved, with $$\hat{D}_a(\gamma)=e^{\gamma \hat{a}^\dagger-\gamma^*\hat{a}}, \qquad \hat{D}_b(\gamma)=e^{\gamma \hat{b}^\dagger-\gamma^*\hat{b}},$$6 exceeding unity in part of the same landscape (Kumar et al., 4 Oct 2025). The paper further states that the practical phase sensitivity under SID or HD approaches the QCRB.

5. Physical mechanism and parameter dependence

The paper attributes the metrological improvement to three effects (Kumar et al., 4 Oct 2025). First, recycling causes photons to make multiple passes through the interferometer, so the field acquires the phase information more than once. This increases the effective phase imprint and can boost the signal. Second, the recycled field interferes with fresh input in a phase-sensitive way; the recycled amplitude depends on $$\hat{D}_a(\gamma)=e^{\gamma \hat{a}^\dagger-\gamma^*\hat{a}}, \qquad \hat{D}_b(\gamma)=e^{\gamma \hat{b}^\dagger-\gamma^*\hat{b}},$$7, $$\hat{D}_a(\gamma)=e^{\gamma \hat{a}^\dagger-\gamma^*\hat{a}}, \qquad \hat{D}_b(\gamma)=e^{\gamma \hat{b}^\dagger-\gamma^*\hat{b}},$$8, gain $$\hat{D}_a(\gamma)=e^{\gamma \hat{a}^\dagger-\gamma^*\hat{a}}, \qquad \hat{D}_b(\gamma)=e^{\gamma \hat{b}^\dagger-\gamma^*\hat{b}},$$9, displacement $\gamma=|\gamma|e^{i\delta_\gamma}$0, and transmission $\gamma=|\gamma|e^{i\delta_\gamma}$1, and favorable matching reduces output fluctuations relative to the signal slope. Third, local displacement operations are crucial: without displacement, the advantage from recycling is weaker and more restricted, whereas nonzero $\gamma=|\gamma|e^{i\delta_\gamma}$2 makes the recycled light contribute much more strongly to the measured signal, especially for homodyne detection.

The stated parameter trends are monotonic in the explored regime: the improvement increases with higher recycling transmission $\gamma=|\gamma|e^{i\delta_\gamma}$3, larger OPA gain $\gamma=|\gamma|e^{i\delta_\gamma}$4, larger displacement amplitude $\gamma=|\gamma|e^{i\delta_\gamma}$5, and larger squeezing $\gamma=|\gamma|e^{i\delta_\gamma}$6, while photon loss in the recycling arm limits the effect (Kumar et al., 4 Oct 2025). The enhancement also depends strongly on the recycled-arm phase $\gamma=|\gamma|e^{i\delta_\gamma}$7 and the sensing phase $\gamma=|\gamma|e^{i\delta_\gamma}$8, with certain phase regions producing strong constructive interference. In addition, the paper notes a threshold-like behavior in $\gamma=|\gamma|e^{i\delta_\gamma}$9: there exists a regime $b$0 where adding displacement in the recycling loop becomes especially advantageous, and increasing $b$1 while decreasing $b$2 shifts this favorable threshold toward smaller $b$3.

Within the broader SU(1,1) literature, these observations fit a more general resource-based interpretation. A theoretical analysis of SU(1,1) metrology identifies large intramode correlations, indicated by the Mandel $b$4-parameter, as the crucial resource for quantum enhancement (Gong et al., 2016). This suggests that the PR-DSU(1,1) gain is not reducible to repeated traversal alone: a plausible implication is that photon recycling is useful insofar as it increases the effective internal excitation and the relevant photon-number fluctuations. Related work on internal non-Gaussian operations and Kerr phase encoding also finds that internal processing can enhance phase sensitivity and improve robustness against internal losses (Kang et al., 2024, Chang et al., 2021). This suggests that the recycling loop should be viewed as a metrologically critical internal channel rather than a passive add-on.

6. Relation to the wider SU(1,1) landscape and practical scope

The PR-DSU(1,1) belongs to the general class of two-stage active interferometers in which the first nonlinear stage generates or amplifies the probe and the second stage reprocesses it. In integrated and broadband realizations, this two-stage picture is naturally interpreted as a form of coherent reuse of internally generated photons. A spectrally multimode integrated SU(1,1) interferometer based on two PDC sections and an internal phase element makes this point explicit, while also showing that perfect destructive interference is impossible in the full multimode device; only the central spectral region can be nearly canceled, and residual side-lobes remain (Ferreri et al., 2020). For photon-recycled DSU(1,1), this is a direct caution against reading the single-mode formulas as mode-independent truths.

A second relevant development is the integrated two-colour broadband SU(1,1) interferometer used to retrieve ultrashort bi-photon correlation times by means of a double-pass nonlinear geometry (Roeder et al., 2023). There, the first parametric down-conversion stage generates a biphoton state and the same waveguide is used again as the recombining stage. The experiment reports correlation-time reconstruction around $b$5 and emphasizes that spectral phase and dispersion are inseparable from the interferometric observable. This indicates that realistic photon recycling in active interferometers is conditioned by bandwidth, dispersion engineering, and the joint spectral structure, not only by nominal pass number.

The phrase “recycled photons” also has a distinct meaning outside phase metrology. In a fiber optical loop mirror integrated with a ferri-magnetic sphere resonator, the same photon is intentionally reused to interact twice with the matter system so that entanglement created on the first pass can be removed on the second (Buks et al., 2021). In that setting the relevant observable is the transmission probability $b$6, and the purpose is to distinguish unitary evolution from collapse-like nonunitary evolution. The common thread is repeated use of the same optical resource, but the operational objective differs from DSU(1,1) phase sensing.

Taken together, these adjacent results delimit the scope of the PR-DSU(1,1) proposal. The architecture is a concrete photon-recycling extension of a displacement-assisted SU(1,1) interferometer, with explicit multi-pass input–output relations, SID and HD sensitivities, a shot-noise benchmark, and a QFI-based QCRB analysis (Kumar et al., 4 Oct 2025). At the same time, the broader literature implies that realistic performance is inseparable from internal loss, multimode structure, dispersion, and the preservation of the correlations that make active interferometry useful.