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The formation of Schrodinger cat-like states in the process of spontaneous parametric down-conversion

Ranjit Singh, Alexander E. Teretenkov

TL;DR

This work advances the generation of macroscopic quantum superpositions by treating spontaneous parametric down-conversion with a fully quantized pump, using a cubic interaction Hamiltonian and Lindblad dissipation to capture non-Gaussian, entangled Schrödinger cat-like states (SCLSs) in both fundamental and second-harmonic modes. Through QuTiP-based simulations, it demonstrates Wigner-function negativity, quadrature squeezing up to $-3.76$ dB, and super-Poissonian statistics that persist under moderate losses, with entanglement quantified by a Schmidt number near $1.93$ at key evolution times. Dissipation introduces parity mixing in photon-number distributions but does not destroy the non-Gaussian, entangled character of the states. The results highlight the potential of two-mode squeezed states as macroscopic quantum resources for metrology and CV quantum information processing, and suggest feasible experimental pathways with pump-conversion efficiencies around $64\%$ and homodyne-tomography verification.

Abstract

In recent years, there has been an increased interest in the generation of superpositions of coherent states with opposite phases, the so-called photonic Schrodinger cat states. These experiments are challenging, and, so far, cats involving only small photon numbers have been implemented. Here, we propose to consider two-mode squeezed states as examples of Schrodinger cat-like states. For this, we apply criteria that aim to identify macroscopic superpositions in a more general sense. We extend some of these criteria to the two-mode continuous variable regime. Furthermore, we compare the size of states obtained in several experiments and discuss experimental challenges for further improvements. Our results not only promote two-mode squeezed states for exploring quantum effects at the macroscopic level but also provide direct measures to evaluate their usefulness for quantum metrology.

The formation of Schrodinger cat-like states in the process of spontaneous parametric down-conversion

TL;DR

This work advances the generation of macroscopic quantum superpositions by treating spontaneous parametric down-conversion with a fully quantized pump, using a cubic interaction Hamiltonian and Lindblad dissipation to capture non-Gaussian, entangled Schrödinger cat-like states (SCLSs) in both fundamental and second-harmonic modes. Through QuTiP-based simulations, it demonstrates Wigner-function negativity, quadrature squeezing up to dB, and super-Poissonian statistics that persist under moderate losses, with entanglement quantified by a Schmidt number near at key evolution times. Dissipation introduces parity mixing in photon-number distributions but does not destroy the non-Gaussian, entangled character of the states. The results highlight the potential of two-mode squeezed states as macroscopic quantum resources for metrology and CV quantum information processing, and suggest feasible experimental pathways with pump-conversion efficiencies around and homodyne-tomography verification.

Abstract

In recent years, there has been an increased interest in the generation of superpositions of coherent states with opposite phases, the so-called photonic Schrodinger cat states. These experiments are challenging, and, so far, cats involving only small photon numbers have been implemented. Here, we propose to consider two-mode squeezed states as examples of Schrodinger cat-like states. For this, we apply criteria that aim to identify macroscopic superpositions in a more general sense. We extend some of these criteria to the two-mode continuous variable regime. Furthermore, we compare the size of states obtained in several experiments and discuss experimental challenges for further improvements. Our results not only promote two-mode squeezed states for exploring quantum effects at the macroscopic level but also provide direct measures to evaluate their usefulness for quantum metrology.
Paper Structure (9 sections, 8 equations, 3 figures)

This paper contains 9 sections, 8 equations, 3 figures.

Table of Contents

  1. Introduction
  2. SPDC under pump depletion and losses
  3. Formation of cat-like states in both modes
  4. Application-relevant statistical properties
  5. Mean number extrema as indicators of SCLSs formation. Squeezing in both modes
  6. Dissipation-induced parity mixing in SCLSs
  7. Robustness of super-Poissonian statistics to dissipation
  8. Entanglement of SCLSs
  9. Summary and conclusions

Figures (3)

  • Figure 1: Wigner functions $W_1(\alpha_1,\tau)$ and $W_2(\alpha_2,\tau)$ during SPDC at $\tau = 0.38$. Top row (a,b): Non-dissipative ($\gamma_{1,2} = 0$), $N_1 \approx 25.2$, $N_2 \approx 7.36$. Bottom row (c,d): Dissipative ($\gamma_{1,2} = 0.15$), $N_1 \approx 23.88$, $N_2 \approx 6.95$. Initial: $\hat{a}_1$ vacuum, $\hat{a}_2$ coherent state ($|\alpha_{20}|^2 = 20$, $\varphi_{20} = \pi/2$).
  • Figure 2: Evolution of (a) $N_j$ and (b) quadrature variances SPDC process. $N_1$ ($\hat{a}_1$, solid black, $-$) peaks while $N_2$ ($\hat{a}_2$, dashed black, $--$) minimizes at $\tau=0.38$. Variances: $\Delta^2x_1$ (blue $\cdot{\cdot}{\cdot}$), $\Delta^2p_1$ (blue $\cdot$-), $\Delta^2x_2$ (red $\cdot{\cdot}{\cdot}$), $\Delta^2p_2$ (red $\cdot$-). $\bigstar$ marks extrema.
  • Figure 3: Photon number distributions $P_j(n_j)$ at $\tau = 0.38$: (a,b) non-dissipative ($\gamma_{1,2}=0$) and (c,d) dissipative ($\gamma_{1,2}=0.15$) regimes for modes $\hat{a}_1$ (a,c) and $\hat{a}_2$ (b,d).