Two-Mode Squeezing Operator in Quantum Optics

Updated 23 November 2025

Two-mode squeezing operator is a unitary transformation that entangles two bosonic modes via correlated photon-pair processes.
It applies Bogoliubov transformations to create two-mode squeezed vacuum states with controlled nonclassical correlations and entanglement.
Its implementation in circuit QED and optomechanics underpins practical advances in quantum-enhanced metrology and computational protocols.

The two-mode squeezing operator is a fundamental unitary transformation in continuous-variable quantum information theory and quantum optics. It generates entanglement between two independent bosonic modes via correlated photon-pair creation and annihilation processes, underpinning applications from quantum-enhanced metrology to measurement-based computation.

1. Mathematical Definition and Core Properties

The standard two-mode squeezing operator for bosonic modes—with annihilation operators $a$ and $b$ , satisfying $[a,a^\dagger]=[b,b^\dagger]=1$ —is defined as

$S_2(\zeta) = \exp\left[\zeta^* ab - \zeta a^\dagger b^\dagger\right], \qquad \zeta = r e^{i\theta}, \quad r \geq 0, \;\theta \in [0,2\pi)$

This unitary operator belongs to the noncompact group $SU(1,1)$ , with Hermitian generator $G_2(\zeta) = i(\zeta a^\dagger b^\dagger - \zeta^* ab)$ . Its action generates two-mode squeezed vacuum (TMSV) states: $|\text{TMSV}(\zeta)\rangle = S_2(\zeta)|0,0\rangle = \frac{1}{\cosh r} \sum_{n=0}^\infty [ - e^{i\theta} \tanh r ]^n |n,n\rangle$ The squeezing parameter $r$ controls the degree of entanglement and nonclassical correlations; $\theta$ sets the phase of the generated correlations (Gazeau et al., 2022).

Unitarity is manifest from the disentangled form: $S_2(\zeta) = \exp\left[e^{i\theta} \tanh r\, a^\dagger b^\dagger\right] \left(1/\cosh r\right)^{N_a+N_b+1} \exp\left[-e^{-i\theta} \tanh r\, ab\right]$ with $N_{a,b}$ the mode number operators.

2. Bogoliubov Transformations and Quadrature Structure

The two-mode squeezing operator implements a linear (Bogoliubov) transformation in the Heisenberg picture: $\begin{aligned} S_2^\dagger(\zeta)\, a\, S_2(\zeta) &= a\, \cosh r - e^{i\theta} b^\dagger\, \sinh r \ S_2^\dagger(\zeta)\, b\, S_2(\zeta) &= b\, \cosh r - e^{i\theta} a^\dagger\, \sinh r \end{aligned}$ The corresponding quadrature operators $X_j = (a_j + a_j^\dagger)/\sqrt{2}$ , $P_j = (a_j - a_j^\dagger)/(i\sqrt{2})$ transform such that linear combinations $X_1 - X_2$ and $P_1 + P_2$ display exponentially suppressed variances: $\text{Var}(X_1 - X_2) = \text{Var}(P_1 + P_2) = \frac{1}{2} e^{-2r}$ while antisqueezed quadratures grow as $e^{+2r}$ . In the large- $r$ limit, perfect EPR correlations are established (Gazeau et al., 2022).

3. Group-Theoretic and Geometric Perspectives

The operator $S_2(\zeta)$ realizes an element of $SU(1,1)$ , with generators

$K_+ = a^\dagger b^\dagger, \quad K_- = ab, \quad K_0 = \frac{1}{2}(a^\dagger a + b b^\dagger + 1)$

obeying $[K_0, K_{\pm}] = \pm K_{\pm}$ and $[K_+, K_-] = -2K_0$ (Dukalski et al., 2014). This structure generalizes further:

The $Sp(4;\mathbb{R})$ formulation realizes two-mode and four-mode squeezing as hyperbolic rotations on a four-dimensional Bloch hyperboloid (Hasebe, 2019).
The generalized operator $F_2(A,B,C,D)$ —implementing arbitrary real symplectic maps on $(Q_1\pm Q_2, P_1\pm P_2)$ —can be decomposed into two-mode squeezing ( $S_2$ ), single-mode squeezing, and phase-space transformations (free propagation/lens) (Gao et al., 2015).

4. Physical Realizations: Circuit QED and Optomechanics

Two-mode squeezing operators have been engineered in a range of bosonic platforms:

Circuit QED via Modulated Qubit Coupling: A superconducting qubit is dispersively coupled to two cavity modes (annihilation operators $a$ , $b$ ), and a sequence of sideband drives modulates the interaction, yielding the effective Hamiltonian

$H_{\rm TMS} = \pm r'\, (a^\dagger b^\dagger + ab), \qquad r' = \frac{|g|^2}{2\delta\Omega}$

Projecting onto a qubit eigenstate produces pure two-mode squeezing evolution $U(t) = \exp[\mp i r' t (a^\dagger b^\dagger + ab)] = S_2(\pm r't)$ , achieving up to $12\,{\rm dB}$ squeezing for realistic circuit QED parameters (Blumenthal et al., 30 Jul 2025, Diniz et al., 2018).

Cavity Optomechanics: Coupling two optical or microwave photon modes to a mechanical resonator and engineering optimal detuning and drives produces an effective Hamiltonian

$H_{\text{eff}} = g_{\rm eff}(e^{-i\theta} ab + e^{i\theta} a^\dagger b^\dagger)$

with $g_{\rm eff}$ set by the parametric drive and system couplings. This supports near-steadystate squeezing and allows surpassing the $3\,{\rm dB}$ limit typical of stable linear networks by optimizing cavity losses and driving (Qi, 4 Apr 2025).

5. Analytical Fock Representation and Multi-photon Interference

The action of $S_2(\zeta)$ on Fock states admits exact analytic expressions: $\langle k,l | S_2(\zeta) | m, n \rangle = \delta_{k-l, m-n} \sum_{j=0}^{\min(m,n)} (\cdots)$ capturing all multi-photon processes and phase-dependent interference pathways. This underpins the engineering of quantum interferometers and allows precise analysis of multi-crystal, high-gain nonlinear processes. Tuning the squeezing strength $r$ and phase $\theta$ can yield constructive or destructive interference in specific output photon-number channels, enabling applications in multi-photon quantum metrology and state engineering (Gu et al., 20 Nov 2025).

6. Open System Dynamics and Entanglement Under Dissipation

Subjected to loss, the two-mode squeezing scenario admits closed-form solutions for density matrix evolution using Wei-Norman factorization, with the key operators forming a representation of $SO(3,2)$ or $SO(4,2)$ in superoperator space. Entanglement persists in steady-state when the squeezing parameter exceeds a threshold set by thermal occupation and damping rates: $|r| > 4 K n_{\text{th}}$ and can be quantified using computable measures such as Horodecki's negativity. The structure of the covariance matrix and entanglement monotones provides necessary and sufficient criteria for inseparability of the state under arbitrary Gaussian (and some non-Gaussian) processes (Dukalski et al., 2014).

7. Generalizations and Quantum Information Applications

Two-mode squeezing extends well beyond bosonic Gaussian states. Replacement of bosonic creation/annihilation operators with arbitrary lowering operators $O_1, O_2$ on finite-dimensional subsystems yields a generalized squeezed state: $|\psi_G(r)\rangle = \mathcal{N} \sum_{m=0}^{m_{\max}} [-\tanh r]^m |m\rangle_1 \otimes |m\rangle_2$ with covariance and quantum Fisher information matrix structures mirroring the infinite-dimensional case. Associated Lindbladian dissipative processes can stabilize these states as unique dark states, enabling simultaneous Heisenberg-limited multi-parameter estimation in finite-spin ensembles. This supports protocols such as optimal collective sensing and hybrid entanglement generation (Mamaev et al., 30 Jun 2024).

Two-mode squeezing operators enable a universal gate set for bosonic Gaussian computation: the combination of $S_2(r)$ with single-mode squeezing, displacements, and beamsplitter operations suffices for arbitrary multimode Gaussian unitary transformations. Conditional implementation using discrete variables (qubit controls) integrates continuous-variable resources into hybrid quantum architectures (Blumenthal et al., 30 Jul 2025).