Momentum-Resolved SU(1,1) Amplifiers

• Momentum-resolved SU(1,1) parametric amplifiers are phase-sensitive quantum devices that generate correlated photon pairs across continuous momentum modes using an extended SU(1,1) framework.
• They employ Bogoliubov input-output relations and Schmidt-mode decomposition to quantify mode-specific gain and squeezing, crucial for optimizing multimode quantum amplification.
• Experimental designs leverage dispersion management and tailored local oscillator profiles to achieve high-dimensional, sub-shot noise performance in quantum metrology and communications.

Momentum-resolved SU(1,1) parametric amplifiers are a class of phase-sensitive quantum amplifiers in which parametric processes generate correlated photon pairs across multiple spatial (momentum) modes. In such devices, the full plane-wave (momentum-resolved) structure of the underlying nonlinear interaction is explicitly treated, enabling the amplification and quantum control of high-dimensional, multimode fields. Their description requires extending the standard SU(1,1) framework into the domain of continuous momentum bases, naturally leading to a Schmidt-mode formalism for both theory and experiment. This architecture is central to spatially multiplexed quantum information, quantum metrology, and the next generation of broadband, quantum-enhanced amplifiers (Scharwald et al., 27 Apr 2025, Reep, 2018, Frascella et al., 2019).

1. Theoretical Framework and Mode Structure

The quantum dynamics of momentum-resolved SU(1,1) parametric amplifiers are governed by Bogoliubov input–output relations in the continuous transverse momentum basis $k$. The transformation between input and output plane-wave annihilation operators, $\hat{a}^{(\rm in)}(k)$ and $\hat{a}^{(\rm out)}(k)$, is given by

$$\hat a^{(\rm out)}(k) = \int dk'\, U(k,k')\, \hat a^{(\rm in)}(k') + \int dk'\, V(k,k')\, \hat a^{(\rm in)\,\dagger}(k')$$

where $U(k,k')$ describes phase-insensitive mode mixing and $V(k,k')$ encapsulates the multimode squeezing. Bosonic commutation relations are preserved provided

$$\int dk'\,[U(k,k')U^*(k'',k') - V(k,k')V^*(k'',k')] = \delta(k-k'')$$

This integral-kernel description admits a joint Schmidt decomposition:

$$\begin{aligned} U(k,k') &= \sum_{n=0}^{\infty} \cosh g_n\, \psi_n(k)\, \phi_n^*(k') \ V(k,k') &= \sum_{n=0}^{\infty} \sinh g_n\, \psi_n(k)\, \phi_n(k') \end{aligned}$$

The mode functions $\psi_n(k)$ and $\phi_n(k)$ form complete orthonormal sets for input and output, respectively; $g_n$ are real, positive gain eigenvalues. Each Schmidt mode undergoes independent two-mode squeezing: for the $n$-th mode,

$$\hat B_n^{(\rm out)} = \cosh g_n\, \hat A_n^{(\rm in)} + \sinh g_n\, \hat A_n^{(\rm in)\,\dagger}$$

Design and performance are thus naturally characterized at the level of these orthogonal gain eigenmodes (Scharwald et al., 27 Apr 2025).

2. Hamiltonian Formulation and SU(1,1) Algebra

In superconducting Josephson travelling-wave lines and nonlinear optical platforms, the microscopic Hamiltonian for four-wave mixing and parametric amplification is expressible in the momentum basis as

$$H = \sum_k \hbar\omega_k (a_k^\dagger a_k + \tfrac{1}{2}) + H_\text{int}$$

For Josephson TWPAs, the interaction Hamiltonian resulting from the expansion of the Josephson nonlinearity is

$$H_\text{int} = \sum_{k_1,k_2,k_3,k_4} g(k_1, k_2; k_3, k_4) a_{k_1}^\dagger a_{k_2}^\dagger a_{k_3} a_{k_4} \,\delta_{k_1+k_2-k_3-k_4,0} + \mathrm{H.c.}$$

Under the undepleted pump and phase-matched (momentum-conserving) regime, the effective Hamiltonian for a signal–idler pair $(k,-k)$ reduces to an SU(1,1) form using the generators $K_0,K_+,K_-$:

$$H_k = \hbar\Omega(k) K_0(k) + \hbar \chi(k)[K_+(k) + K_-(k)]$$

where

$$K_0(k) = \frac12 [a_k^\dagger a_k + a_{-k} a_{-k}^\dagger], \quad K_+(k) = a_k^\dagger a_{-k}^\dagger, \quad K_-(k) = a_{-k} a_k$$

These follow the SU(1,1) algebra: $[K_0, K_\pm] = \pm K_\pm$, $[K_+, K_-] = -2K_0$. The Heisenberg evolution yields a Bogoliubov transformation parameterized by the squeeze parameter $r(k) = |\chi(k)| t$ and phase $\phi(k) = \arg \chi(k)$, with power gain

$G(k) = \cosh^2 r(k)$

The regime of validity is restricted to weak nonlinearity, undepleted pump, slowly varying amplitude, and effective phase matching (Reep, 2018).

3. Schmidt-Mode Analysis: Gain, Squeezing, and Overlaps

Experimentally and theoretically, gain and squeezing are distributed over Schmidt modes. For a laboratory trial mode $u_m(k)$, its overlap with the $n$-th Schmidt mode is

$c_{mn} = \int dk\, u_m^*(k) \psi_n(k)$

The output in such modes is

$$\hat a_m^{(\rm out)} = \sum_n c_{mn} [\cosh g_n\, \hat A_n^{(\rm in)} + \sinh g_n\, \hat A_n^{(\rm in)\,\dagger}]$$

Photon number and quadrature squeezing in channel $m$ are weighted sums over Schmidt modes:

$$\langle \hat a_m^\dagger \hat a_m \rangle = \sum_n |c_{mn}|^2 \sinh^2 g_n$$

Residual phase-mismatch or imperfect detection basis (non-diagonal $|c_{mn}|^2$) lead to mode mixing and reduced effective gain per detected channel. The independence of squeezing transformations for different Schmidt modes underpins parallel quantum amplification across distributed momentum modes (Scharwald et al., 27 Apr 2025).

4. Spatial Mode Content and Experimental Reconstruction

In wide-field optical SU(1,1) interferometers, spatial (momentum) Schmidt modes are defined and reconstructed by analysis of intensity covariance in the far field. Using the field operator $E^{(+)}(q)$ for transverse momentum $q$, the intensity covariance is

$$C(q, q') = \langle \Delta I(q) \Delta I(q') \rangle = |\langle E^{(-)}(q) E^{(+)}(q') \rangle|^2$$

Diagonalization of $C(q, q')$ yields orthonormal spatial coherent (Schmidt) modes $u_k(q)$ and weights $\lambda_k$. High-gain SU(1,1) operation modifies the weighting of these modes: the mean signal photon number in Schmidt mode $(\ell,p)$ is

$\Lambda_{\ell p} = |w^{\rm eff}_{2,\ell p}|^2$

with

$$w^{\rm eff}_{2,\ell p}(G_1, G_2, \Phi) = \sinh[\sqrt{\lambda_{\ell p}G_1}] \cosh[\sqrt{\lambda_{\ell p}G_2}] + e^{i\Phi} \cosh[\sqrt{\lambda_{\ell p}G_1}] \sinh[\sqrt{\lambda_{\ell p}G_2}]$$

The total number of spatial Schmidt modes—quantified as $K_{\rm total} = 1/\sum_{k} \lambda_k^2$—can exceed 30 in wide-field geometries, enabling parallel, momentum-resolved phase measurements below the shot-noise limit (Frascella et al., 2019).

Table: Selected Experimental Results for Wide-Field SU(1,1) Interferometer (Frascella et al., 2019)

Observable	Value/Description	Context
Number of azimuthal (OAM) modes	$K_\mathrm{azim}$ rises from $3.8\pm0.2$ to $18.2\pm0.6$	As function of $\theta_0=4\to14$ mrad in far field
Number of radial modes	$K_\mathrm{rad}=4.5\pm0.2$	Hermite–Gaussian-like radial profiles
Total spatial modes	$K_\mathrm{total}\approx34\pm2$	Factorizes as $K_\mathrm{rad} \times K_\mathrm{azim}$

5. Phase Profile, Dispersion, and Measurement Considerations

Each Schmidt mode $\psi_n(k)$ features a $k$-dependent phase $\Phi_n(k) = \arg \psi_n(k)$, determined by phase-matching and group-delay dispersion:

$\Phi_n(k) \approx \frac12 \Delta k(k) L$

where $\Delta k(k)$ incorporates dispersion via expansion around the central momentum $k_0$. The correspondence between local oscillator phase and Schmidt-mode chirp is essential for achieving the full, mode-resolved squeeze gain in homodyne detection. Imperfect phase compensation leads to degraded squeezing. To maximize amplification and squeezing visibility, local oscillators should be shaped in both amplitude and phase to match $\psi_n(k)$, necessitating accurate dispersion management and phase control (e.g., insertion of glass wedges) (Scharwald et al., 27 Apr 2025).

6. Design, Optimization, and Regimes of Applicability

The Schmidt-mode formalism provides precise guidance for device engineering:

• Computation of initial Schmidt pairs $\{\psi_n, \phi_n\}$ determines the spatial bandwidth and purity of the gain modes.
• Pump-waist and crystal length are adjustable parameters to target a desired momentum bandwidth ($\Delta k \approx 1/w_{\rm eff}$ with $w_{\rm eff} \sim w \exp(g)$ for high gain).
• Optimal SU(1,1) amplifier and interferometer performance is achieved by simultaneously matching pump and local oscillator profiles to the Schmidt modes with highest gain, tuning relative phases to minimize crosstalk, and compensating for quadratic phase chirps.
• Proper regime of validity must be maintained: weak nonlinearities (e.g., $I_p\lesssim0.8 I_c$ in Josephson devices), undepleted pumps, and efficient phase-matching.

Experimental protocols involve numerical calculation of kernels $U,V$, their joint singular value decomposition, and subsequent tailoring of detection modes by spatial light modulators or equivalent means (Scharwald et al., 27 Apr 2025, Reep, 2018).

7. Applications and Implications

Momentum-resolved SU(1,1) parametric amplifiers enable multiplexed, high-bandwidth quantum metrology, spatially distributed phase sensing, and the generation of broadband squeezed vacuum across tens of orthogonal spatial modes. The presence of multiple Schmidt modes, each offering near-optimal phase sensitivity scaling as $\delta\Phi_{n} = [\Lambda_n(\Lambda_n + 2)]^{-1/2}$ in the high-photon-number limit, facilitates spatially parallel sub-shot-noise measurements. These techniques generalize beyond optics, applying as well to superconducting circuit-based TWPAs and other traveling-wave, multimode quantum amplifiers (Scharwald et al., 27 Apr 2025, Frascella et al., 2019, Reep, 2018).

A plausible implication is that continued refinement of mode-selective amplification, dispersion control, and quantum detection, as guided by the Schmidt-mode analysis, will further expand the capabilities of multimode quantum-enhanced technologies in communications, computation, and sensing.