[Paper Review] Non-Gaussian and Gaussian Entanglement in Coupled Leaky Waveguides

Amit Rai, Sumanta Das+1 more|arXiv (Cornell University)|Jul 14, 2009

Quantum Information and Cryptography2 references3 citations

TL;DR

This paper investigates quantum entanglement dynamics in coupled single-mode waveguides, modeling both Gaussian (squeezed states) and non-Gaussian (Fock states, NOON states) inputs. Using logarithmic negativity as a measure, it analytically derives entanglement evolution and demonstrates robustness against waveguide loss, showing that entanglement persists even under significant decay, making such systems viable for quantum circuits and coherent phenomena like quantum walks.

ABSTRACT

We investigate the viability of coupled waveguides as basic units of quantum circuits. In particular, we study the dynamics of entanglement for the single photon state, and single mode squeezed vacuum state. We further consider the case of entangled inputs in terms of the two mode mode squeezed vacuum states and the two photon NOON state. We present explicit analytical results for the measure of entanglement in terms of the logarithmic negativity. We also address the effect of loss on entanglement dynamics of waveguide modes. Our results indicate that the waveguide structures are reasonably robust against the effect of loss and thus quite appropriate for quantum architectures as well as for the study of coherent phenomena like random walks. Our analysis is based on realistic structures used currently.

Motivation & Objective

• To assess the viability of coupled waveguides as quantum circuit elements by analyzing entanglement dynamics.
• To quantify entanglement evolution for both Gaussian (squeezed states) and non-Gaussian (Fock states, NOON states) input states.
• To investigate the impact of waveguide loss on entanglement robustness using realistic experimental parameters.
• To provide explicit analytical expressions for logarithmic negativity as a measure of entanglement in lossy systems.
• To guide experimental design by identifying optimal operating parameters (e.g., propagation distance) that avoid entanglement minima.

Proposed method

• Models two coupled single-mode waveguides using coupled-mode theory with evanescent coupling, described by a Hamiltonian with tunneling amplitude $ J $.
• Derives time evolution of field operators $ a(t), b(t) $ using unitary evolution under the Hamiltonian, leading to coherent superpositions of mode excitations.
• Applies the logarithmic negativity $ E_{\mathcal{N}} $ as a quantitative measure of entanglement, derived from the covariance matrix of Gaussian states.
• For non-Gaussian states (e.g., NOON state), uses exact time evolution via binomial expansion of the output state in the Fock basis.
• Incorporates loss via a decay rate $ \gamma $, modifying the time evolution of the covariance matrix and symplectic eigenvalues.
• Calculates symplectic eigenvalues $ \tilde{\nu}_{\pm} $ from the modified covariance matrix to evaluate $ E_{\mathcal{N}} $ under loss.

Experimental results

Research questions

• RQ1 How does entanglement evolve over time in coupled waveguides for initially separable single-photon and two-mode squeezed states?
• RQ2 What is the behavior of entanglement for initially entangled states such as the NOON state and two-mode squeezed vacuum state?
• RQ3 How does waveguide loss (mode decay rate $ \gamma $) affect the persistence of entanglement in both Gaussian and non-Gaussian input states?
• RQ4 To what extent is entanglement robust against decoherence in realistic waveguide systems with material absorption and leakage?
• RQ5 Can analytical expressions for logarithmic negativity be derived for both lossy and lossless cases, and how do they compare?

Key findings

• The logarithmic negativity for the separable two-mode squeezed state $ |\zeta\rangle $ shows periodic entanglement oscillations that decay slowly with increasing loss rate $ \gamma $.
• For the initial entangled two-mode squeezed state $ |\xi\rangle $, entanglement decreases gradually with increasing $ \theta = Jt $ under loss, with periodic disentanglement events observable.
• Entanglement robustness is maintained even for non-zero $ \gamma/J $, indicating that waveguide systems are resilient to realistic loss mechanisms.
• The symplectic eigenvalues $ \tilde{\nu}_{\pm} $ depend explicitly on the loss rate $ \gamma $, and the resulting $ E_{\mathcal{N}} $ is computed analytically using the covariance matrix formalism.
• For the NOON state, the output state evolves into a superposition of Fock states $ |k,N-k\rangle $, with coefficients $ \beta_k $ depending on $ \cos(Jt) $ and $ \sin(Jt) $, preserving non-classical interference.
• Numerical plots (e.g., Fig. 10) confirm that entanglement persists over time even with loss, especially when $ \gamma $ is small, suggesting practical utility in quantum information processing.