Preparing squeezed, cat and GKP states with parity measurements

TL;DR

This work presents a measurement-based protocol that uses displaced parity measurements to prepare nonclassical bosonic states, notably squeezed states, and extends naturally to multi-component cat and GKP states. By exploiting the fact that an ideal squeezed vacuum is even-parity and, in the infinite-squeezing limit, invariant under anti-squeezed displacements, the authors implement sequences of displaced parity projections to project into squeezed manifolds with high fidelity. Numerical results show squeezing up to about $S_{\text{dB}}\approx21.5$ dB achievable with $M=11$ measurements (and $p_{\text{suc}}\approx8.5\%$), and significant squeezing even for smaller $M$ under optimized parameters; the framework tolerates realistic losses. The approach is then extended to generate multi-component cat states and approximate Gottesman-Kitaev-Preskill (GKP) codewords by combining squeezed-state preparation with parity-driven lattice structure in phase space, offering a versatile, qubit-readout-compatible route to advanced bosonic encodings in platforms like cQED, cQAD, and trapped ions, with implications for metrology, communication, and quantum error correction.

Abstract

Bosonic modes constitute a central resource in a wide range of quantum technologies, providing long-lived degrees of freedom for the storage, processing, and transduction of quantum information. Such modes naturally arise in platforms including circuit quantum electrodynamics, quantum acoustodynamics, and trapped-ion systems. In these architectures, coherent control and high-fidelity readout of the bosonic degrees of freedom are achieved via coupling to an auxiliary qubit. When operated in the strong dispersive regime, this interaction enables parity measurements of the mode which, in combination with phase-space displacements, constitute a standard experimental tool for full Wigner-function tomography. Here, we propose a protocol based on displaced parity measurements that allows for the preparation of a variety of bosonic quantum states. As a first example, we demonstrate the generation of squeezed states, achieving up to ~9 dB of squeezing after only three parity measurements, and show that the protocol is robust against experimental imperfections. Finally, we generalize our approach to the preparation of other paradigmatic bosonic states, including cat and Gottesman-Kitaev-Preskill states.

Figures (6)

• Figure 1: Protocols for generating (a) squeezed states, (b) multi-component cat states, and (c) GKP states with parity measurements and displacements. Squares denote the displaced parity measurement $P_+$ as given in (\ref{['dispparity']}), where the location of the parity measurement is at $\alpha$. Circles and ovals represent coherent and squeezed states respectively. Large arrows denote displacements (\ref{['dispop']}), with the magnitude of the displacement indicated within the arrow, and the direction of the displacement is in the arrow direction. (a) The squeezed state measurement sequence occurs in the order indicated by the curved arrows. (b) In each measurement step, the displacement is made before the parity measurement, creating a square lattice with spacing $\delta$. (c) In first step, create a squeezed state using the sequence given in (a). Then use a sequence of displacements and parity measurements as shown to point reflect the squeezed state to form a comb separated by $\delta$.
• Figure 2: Wigner functions of the squeezed state generated according to (\ref{['claim']}). We compare two measurement sequences: (a) symmetric ordering (\ref{['symmetric']}) and (b) linear ordering (\ref{['linear']}). Wigner functions were calculated using (\ref{['wignerfunc']}). We use $t_{\max} = 2$ and $M = 11$ for both cases. Cutoffs of (a) $n_{\text{cut}} = 601$ and (b) $n_{\text{cut}} = 101$ were used.
• Figure 3: Convergence of the state (\ref{['claim']}) towards the infinitely squeezed state. We use the measurement sequence (\ref{['symmetric']}) with $M = 11$, $t_{\max} = 2$, $\theta = 0$. (a) Wavefunction $\psi_n = \langle n | {\cal P}_0 (\vec{t}) \rangle$. A comparison to the squeezed state with $\xi = 2.5$ is shown. (b) Success probability $p_n^{\text{suc}}$. (c) Fidelity with squeezed states $F = | \langle \xi = r | {\cal P}_0 (\vec{t}) \rangle |^2$. We use a Fock state truncation of $n_{\text{cut}} =601$.
• Figure 4: Effect of the amplitude of displacement $t_{\max}$ with a small number of measurements. We evaluate (\ref{['claim']}) with the measurement sequence (\ref{['symmetric']}) with $M = 3$ and $\theta = 0$. Wavefunction $\psi_n = \langle n | {\cal P}_0 (\vec{t}) \rangle$ for (a) $t_{\max} = 0.4$; (b) $t_{\max} = 0.8$; (c) $t_{\max} = 1.5$. A comparison to the squeezed state with (a) $\xi = 0.6$; (b) $\xi = 1.2$; (c) $\xi = 1.8$ is shown. The success probabilities of each of the sequences is indicated in each figure. (d) The squeezing (\ref{['dbsqueezing']}) as a function of $t_{\max}$. The Wigner distribution using the optimal $t_{\max} = 0.8$ for (e) $\epsilon = 0$ and (f) $\epsilon = 0.15$. (g) The total success probability (\ref{['totalsuccessprob']}) as a function of $t_{\max}$. The photon loss probability $\epsilon = 1 - \eta$ are shown for (d)(e). For (a)(b)(c)(e) the photon loss probability is $\epsilon = 0$. We use a Fock state truncation of $n_{\text{cut}} =201$ for the pure state calculations and $n_{\text{cut}} =51$ for the mixed state calculations.
• Figure 5: Effect of bosonic loss on a larger number of measurements $M = 11$. (a) The diagonal density matrix elements $\rho_{nn} = \langle n | \rho |n \rangle$ versus the Fock number for $t_{\max} = 1.1$, $\epsilon = 0.01$. The squeezed state that is the closest match to the obtained state with $\xi = 1.5$ is shown for comparison. (b) The attained squeezing for various $t_{\max}$ and the loss probabilities $\epsilon$ as shown. (c) The total success probability as a function of $t_{\max}$ for the loss probabilities $\epsilon$ as marked. (d) The Wigner function of the final state for $t_{\max} = 1.1$, $\epsilon = 0.01$. For pure state ($\epsilon = 0$) calculations $n_{\text{cut}} = 601$ is used, for mixed state ($\epsilon > 0$) calculations $n_{\text{cut}} = 51$ is used.