Photon number projective measurement in Schrödinger cat quantum state preparation procedure after parametric down-conversion interaction

Abstract

Schrödinger-cat (SC) states are an important resource for continuous-variable quantum computing and quantum metrology. In our previous work [JOSA B, 42, 2 (2025)], we proposed a probabilistic protocol for generating bright squeezed SC states via degenerate spontaneous parametric down-conversion (SPDC) with pump depletion, followed by projective measurement of the pump mode. In the present work, we formulate a general theoretical description of SPDC with pump depletion, introduce an efficient numerical method for computing its dynamics, and develop a practical version of the protocol proposed in [JOSA B, 42, 2 (2025)].

Figures (8)

• Figure 1: Transition amplitude $A_{n, 0}(\tau)$ for the zero-photon pump outcome, evaluated at $\tau = \tau_{\rm opt}(\beta)$), together with the corresponding coherent-state probability amplitudes $b_n$ for $\lvert {\beta} \rangle$ (marked as CS - coherent state), for several values of $\beta$.
• Figure 2: Infidelity, $1 - F$ as a function of $\beta$ for $\lvert {\psi_{\rm res}^{(0)}} \rangle$ with constant $A_{n,0}$ (see Eq. \ref{['eq:prepared-state']}) and $\lvert {\psi_{\rm SSC}^{(0)}} \rangle$ with $e^{-2r} = 2$ and replacement $\alpha \rightarrow \beta$ (see Eq. \ref{['eq:cat-state-wavefunction-approx']}).
• Figure 3: First maximum of the achievable probability of successfully measuring $m$ photons in the pump mode as a function of the initial pump coherent-state amplitude $\beta$.
• Figure 4: Fidelity deviation from unity (infidelity) of the postselected output state $\lvert {\psi_{\rm res}^{(m)}} \rangle$ with the corresponding squeezed cat state $\lvert {\psi_{\rm SSC}^{(m)}} \rangle$, conditioned on detecting a small pump photon number $m$ after evolution for $\tau_{\rm opt}(\beta)$, plotted versus the initial pump coherent amplitude $\beta$.
• Figure 5: Deviation $\Delta$ incurred when representing the initial state $\lvert {0} \rangle_{\rm s} \otimes \lvert {n} \rangle_{\rm_p}$ using only eigenstates with eigenvalues within a radius $r$ of zero (see Eq. \ref{['eq:delta-def']}).