Building of one-way Hadamard gate for squeezed coherent states

Sergey A. Podoshvedov

Phys. Rev. A 87, 012307 – Published 10 January 2013

Abstract

We present an optical scheme to conditionally generate even or odd squeezed superpositions of coherent states (SSCSs). The optical setup consists of an unbalanced beam splitter whose transmittance tends to unity, and additional balanced beam splitters and photodetectors in auxiliary modes. Squeezed coherent states with different amplitudes are the input states in the optical scheme. The single-qubit operations are probabilistic and employ two- and three-photon subtractions from initial beams as the driving force. Generation of the even or odd SSCSs is observed in a wide diapason of values of used parameters. We consider a possibility to realize a one-way Hadamard gate for the squeezed coherent states when the base states are transformed into superposition states. States approximating the output states of a Hadamard gate with high fidelity can be realized by imposing restrictions on the values of used parameters. Higher-order subtractions from input beams are necessary to generate the SSCSs with larger amplitudes and higher fidelities. The problem is resolved in a Wigner representation to take into account imperfections of the optical devices.

Received 9 July 2012

DOI:https://doi.org/10.1103/PhysRevA.87.012307

Authors & Affiliations

Sergey A. Podoshvedov

Department of General and Theoretical Physics, Southern Ural State University, Lenin Av. 76, Chelyabinsk, Russia

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Issue

Vol. 87, Iss. 1 — January 2013

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Images

Figure 1
Optical scheme to generate two-photon subtracted squeezed coherent states that can approximate displaced SSCSs under a certain choice of values of used parameters. Only the amplitude of squeezed coherent states is varied to generate the states that approximate the output states of one-way Hadamard gate, provided that two photodetectors fixed two simultaneous clicks. Mode 1 is principal while modes 2 and 3 are auxiliary. (BBS) balanced beam splitter, (UBS) unbalanced beam splitter, (PD) photodetector.Reuse & Permissions
Figure 2
Dependencies of maximal possible fidelities (a) $F_{+ \max}$ , (b) $F_{- \max}$ on absolute values of the parameters $α_{I n \pm} = |α_{I n \pm}|$ and $α_{\pm} = |α_{\pm}|$ , respectively, in scheme with two-photon subtraction in the case of $α_{SCS} = 1.5$ .Reuse & Permissions
Figure 3
(a) Fidelities $F_{\pm}$ between generated states and output states of the one-way Hadamard gate and parameters: (b) squeezing amplitudes $r$ , $r_{1}$ , (c) absolute amplitudes of the shift $α_{\pm} = |α_{\pm}|$ , and (d) absolute amplitudes of seed coherent states $α_{I n \pm} = |α_{I n \pm}|$ which provide the fidelities, against an amplitude $α_{SCS}$ of the SSCSs.Reuse & Permissions
Figure 4
Dependencies of fidelities between generated states and output states of a one-way Hadamard gate on $α_{SCS}$ in a realistic case. Values of the parameters for calculation of the fidelities are taken from Figs. 3b, 3c, 3d, where a simplified model of the optical scheme in Fig. 1 is used, in which the BS operator is replaced by the operator $a_{1}^{2} a_{2}^{† 2}$ followed by projective measurement onto the state $|2〉$ in the second auxiliary mode. The beam splitter parameter is taken to be $Q = 0.01$ (curves 1 and 2 are for $F_{+}$ and $F_{-}$ , respectively) and $Q = 0.1$ (curves 3 and 4 are for $F_{+}$ and $F_{-}$ ). The quantum efficiency of detectors is $η = 0.8$ for all curves. The fidelities decrease when BS's parameter $Q$ grows, while the influence of the quantum efficiency of detectors $η$ on the quality of the generated states is negligible.Reuse & Permissions
Figure 5
Optical scheme to realize three-photon subtraction from input beam. The optical scheme enables one to realize one-way Hadamard transformation under a certain choice of values of used parameters. Generated states approximate even or odd SSCSs with larger amplitude $α_{SCS}$ and higher fidelity. The same notations are used as in Fig. 1.Reuse & Permissions
Figure 6
(a) Fidelities $F_{\pm}$ between generated states and output states of the one-way Hadamard gate and parameters: (b) squeezing amplitudes $r$ , $r_{1}$ , (c) absolute amplitudes of the shift $α_{\pm} = |α_{\pm}|$ , and (d) absolute amplitudes of seed coherent states $α_{I n \pm} = |α_{I n \pm}|$ which provide the fidelities, against an amplitude $α_{SCS}$ of the SSCSs. (e) Dependency of absolute difference of input amplitudes of coherent states $|α_{d I n}| = α_{I n +} - α_{I n -}$ on $α_{SCS}$ .Reuse & Permissions

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