Optical continuous-variable quadratic phase gate via Faraday interaction

Ming-Feng Wang; Nian-Quan Jiang; Yi-Zhuang Zheng

doi:10.1364/OE.22.009182

1. Introduction

Quantum computation (QC), which strives to utilize the principles of quantum mechanics to realize efficient computation, is one of the most fascinating and fruitful area of research. In recent years, with the seminal discoveries of many quantum algorithms [1, 2], QC has been proved to be able to work faster than any known classic computation. Though originally based upon discrete variables, QC over CV has also recently been developed by Lloyd and Braunstein [3, 4].

CV QC generalizes the concept of QC to continuous degrees of freedom by the use of position $\hat{x} = (\hat{a} + a^{†}) / \sqrt{2}$ and momentum $\hat{p} = - i (\hat{a} - {\hat{a}}^{†}) / \sqrt{2}$ quadratures. A universal CV quantum computer is defined to be a device that can by local operations perform any desired unitary transformation over these quantum variables [3]. To realize such transformation, the canonical universal set of gates {D̂_k(s), R̂(s), Ĉ_Z} is used, where D̂_k(s) = exp(isx̂^k) for k = 1, 2, 3 and for all s ∈ R [5], with D̂₁(s) the displacement gate, D̂₂(s) the quadratic phase (QP) gate which maps the quadratures into

\begin{array}{r} {\hat{D}}_{2}^{†} (s) \hat{x} {\hat{D}}_{2} (s) = \hat{x}, \\ {\hat{D}}_{2}^{†} (s) \hat{p} {\hat{D}}_{2} (s) = \hat{p} + s \hat{x}, \end{array}

D̂₃(s) the cubic gate, R̂(s) = exp[is(x̂² + p̂²)] the rotation gate, and C_Z = exp(ix̂ ⊗ x̂) the two-mode controlled-Z gate. By repeated use of these gates, together with the approximation [3]

e^{i \hat{A} s} e^{i \hat{B} s} e^{- i \hat{A} s} e^{- i \hat{B} s} \approx e^{- [\hat{A}, \hat{B}] s^{2}} + O (s^{3})

in the limit s → 0 for arbitrary Hamiltonians Â and B̂ consisting of a general polynomials of x̂ and p̂, one is able to simulate any Hamiltonian transformations to any precision, and thus realize the universal CV QC.

The universal set without the cubic operation D̂₃(s) is still universal for arbitrary Gaussian transformations [5]. For quadrature amplitudes of light, operations D̂₁(s) and R̂(s) are the most readily available Gaussian transformations, which can be easily realized by using only coherent light resources and linear optics [5, 6]. Operation C_Z can be finitely decomposed into beam splitters and single-mode online squeezers [5, 7, 8]. Operation D̂₂(s) (together with operation R̂(s) providing online squeezing [9]), however, is the most challenging Gaussian transformation, as its realization usually involves nonlinear optical processes. To date, many methods have been proposed to perform optical QP operation or, equivalently, online squeezer [10–15]. A typical one is based on the nonlinear processes in the optical crystals or optical fibers. For these optical systems, since the nonlinear effect inside a nonlinear media is normally very weak, strong pumper power [10,11] or cavities [12] are usually required to obtain sufficient nonlinearities. Such enhancement, however, suffers several drawbacks. First, strongly optical pumping induced nonlinear process are usually not pure and introduces excess noise to the input signals; second, the use of cavity complicates the experimental realization and makes it difficult to inject the signals into the system. To overcome these drawbacks, an alternative approach using only linear optical elements, offline squeezing, and homodyne detection has also recently been proposed [13–15]. This approach, however, suffers the drawback that its gate fidelities in some extent are limited by the degree of squeezing of the input offline squeezed state.

In this paper, we propose a new and experimentally feasible method to realize optical QP gate in a free-space macroscopic atomic ensemble. We show that ideal optical QP transformation can be performed by simply sending a coherent light three times through a coherent-prepared atomic ensemble (see Fig. 1). In contrast to previous measurement-based schemes [13–15], our method requires neither non-classical offline squeezing nor homodyne detection, which greatly saves the resources needed and simplifies the experimental implementation. Moreover, we also checked the ability of the scheme to produce optical polarization squeezing, finding that substantial squeezing is obtainable within the presently experimentally available parameters.

Fig. 1 Scheme setup for realization of optical QP gate in an atomic ensemble. The light beam first enters an atomic ensemble at an angle α with respect to z to experience Ĥ₁ interaction. The outgoing beam then passes through a delay line. After the whole of the beam run through the atomic sample, it enters the ensemble again to experience Ĥ₂ interaction and subsequently Ĥ₃ interaction. When the beam exits, an optical QP gate is performed.

Download Full Size | PDF

The remainder of this paper is organized as follows. In section 2, we first give details of QP operation based on Faraday interaction with an atomic ensemble, and then calculate the optical squeezing generated. In section 3, we will consider the noise effects including atomic decay and light reflection. After that, the experimental feasibility of the scheme is also discussed. Finally, section 4 contains brief conclusions.

2. The protocol

2.1. Quadratic phase gate

The scheme is based on the Faraday interaction between light and atoms. Suppose that we have an atomic gas consisting of a large number of spin- $\frac{1}{2}$ atoms interacting with a light pulse traveling along the z direction. The atomic sample is initially prepared in a coherent spin state (CSS), i.e., a fully polarized state along the x axis. For such a state, the transverse collective spin components Ĵ_y and Ĵ_z maintain their quantum nature, while the macroscopic x component can be treated as classical constant number, that is Ĵ_x → J_x. The light pulse interacting with atoms is composed of a strong component polarized along the x axis and a weak component (quantum field) along y. For such a polarized state, we analogously have the relevant quantum variables Ŝ_y, Ŝ_z and the classical variable S_x, where Ŝ represents the Stokes operators characterizing the polarization properties of a light pulse. In the far off-resonant coupling regime, one obtains the Faraday interaction between light and atoms: Ĥ_eff ∝ Ĵ_zŜ_z [16–18], which leads to a rotation of the optical linear polarization due to the population difference between the atomic magnetic sublevels, and a rotation of spin around the z axis because of the different ac-Stark shift on the sublevels caused by the field intensity difference between σ⁺ and σ⁻ components [19]. This kind of interactions have been extensively studied to realize, e.g., polarization squeezing [19], spin squeezing [20–22], and quantum memory [23–26].

We here consider how to use this interaction to realize the optical QP gate. Supposing that a light pulse described above propagates through an atomic ensemble in the YZ plane at a certain angle α (0 ≤ α < 2π) with respect to direction z, (see Fig. 1). In this case, the light beam not only see the spin z component but also the spin y component. Consequently, the Hamiltonian is changed into [27]: Ĥ₁ = κx̂_L(x̂_A sinα + p̂_A cosα), where κ is the coupling strength with the dimension $1 / \sqrt{time}$ , and we have defined the new quantum variables $({\hat{x}}_{A}, {\hat{p}}_{A}) = ({\hat{J}}_{y}, {\hat{J}}_{z}) / \sqrt{J_{x}}$ for atoms, obeying the the canonical commutator relation [x̂_A, p̂_A] = i, and $({\hat{x}}_{L}, {\hat{p}}_{L}) = ({\hat{S}}_{z} (t), - {\hat{S}}_{y} (t)) / \sqrt{S_{x} (t)}$ for light, satisfying [x̂_L(t), p̂_L(t′)] = iδ (t − t′). Corresponding to this Hamiltonian, one may derive the relations between input and output state of light [19, 28]:

\begin{array}{l} {\hat{x}}_{L}^{out 1} (t) & = & {\hat{x}}_{L}^{in} (t), \\ {\hat{p}}_{L}^{out 1} (t) & = & {\hat{p}}_{L}^{in} (t) - κ [{\hat{x}}_{A} (t) sin α + {\hat{p}}_{A} (t) cos α] . \end{array}

Here, the field variables

{\hat{ϑ}}_{L}^{in} ({\hat{ϑ}}_{L}^{out 1}) [\hat{ϑ} \in {\hat{x}, \hat{p}}]

represents light before (after) interacting with atoms. In the Heisenberg picture, the spin variables evolve according to

\begin{array}{r} \frac{d}{d t} {\hat{x}}_{A} (t) = κ {\hat{x}}_{L}^{in} (t) cos α, \\ \frac{d}{d t} {\hat{p}}_{A} (t) = - κ {\hat{x}}_{L}^{in} (t) sin α . \end{array}

This set of equations can be easily solved and generates

\begin{array}{r} {\hat{x}}_{A} (t) = {\hat{x}}_{A} (0) + κ cos α \int_{0}^{t} {\hat{x}}_{L}^{in} (τ) d τ, \\ {\hat{p}}_{A} (t) = {\hat{p}}_{A} (0) - κ sin α \int_{0}^{t} {\hat{x}}_{L}^{in} (τ) d τ . \end{array}

We now define the dimensionless collective mode for light

{\hat{ϑ}}_{L}^{in, out 1} = \frac{1}{\sqrt{T}} \int_{0}^{T} {\hat{ϑ}}_{L}^{in, out 1} (t) d t

and the new variables for atoms

{\hat{ϑ}}_{A}^{in} = {\hat{ϑ}}_{A} (0)

,

{\hat{ϑ}}_{A}^{out 1} = {\hat{ϑ}}_{A} (T)

, where T denotes the duration of light pulse. With these definition, inserting (5) into (3), one obtains

y^{out 1} = S (\tilde{κ}, α) y^{in},

where we have collected quadratures into a vector y = (x̂_L, p̂_L, x̂_A, p̂_A)^T and defined the interaction matrix

S (\tilde{κ}, α) = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & - \tilde{κ} sin α & - \tilde{κ} cos α \\ \tilde{κ} cos α & 0 & 1 & 0 \\ - \tilde{κ} sin α & 0 & 0 & 1 \end{matrix}),

where the dimensionless coupling strength

\tilde{κ} = κ \sqrt{T}

. The off-diagonal matrix elements in (7) characterize the fact that information about the x̂_L and p̂_L quadratures is now stored in the atomic mode, and, simultaneously, information of x̂_A and p̂_A is also printed onto the light mode. While for QP operation, according to Eq. (1), the p̂_L quadrature contains no other information but the conjugate x̂_L. To pick up the information of x̂_L recorded in the atoms during the first pass, we suggest reflecting the pulse back into sample again to experience a second interaction Ĥ₂ = κx̂_L(x̂_A sinβ + p̂_A cosβ). Before the second interaction, the light is reflected by mirrors and propagates in a delay line (see Fig. 1), which ensures the two interactions do not overlap in the time line [29]. The second interaction transfers the atomic and light states into y^out2 = S (κ̃, β)y^out1 = S (α, β)yⁱⁿ, where the effective interaction matrix is given by

S (α, β) = (\begin{matrix} 1 & 0 & 0 & 0 \\ {\tilde{κ}}^{2} sin 2 ϕ_{-} & 1 & - 2 \tilde{κ} sin ϕ_{+} cos ϕ_{-} & - 2 \tilde{κ} cos ϕ_{+} cos ϕ_{-} \\ 2 \tilde{κ} cos ϕ_{+} cos ϕ_{-} & 0 & 1 & 0 \\ - 2 \tilde{κ} sin ϕ_{+} cos ϕ_{-} & 0 & 0 & 1 \end{matrix})

with ϕ_± = (α ± β)/2. The off-diagonal matrix element κ̃² sin2ϕ₋ in the first column indicates that the position x̂_L quadrature is now mixed into p̂_L as desired. Particularly, for a special value ϕ₋ = π/4 the output light state is exactly the same as the spin squeezed state created in [30]. As a result, the obtained light state is a polarization squeezed state. Such state, however, is not a pure state, as it still contains information about the atomic state, as can be seen from the off-diagonal elements in (8). This kind of information is unwanted, because they, on the one hand, reduce the amount of squeezing [31] and, on the other hand, prevent the realization of perfect QP gate. To eliminate them the pulse is sent back into the sample to experience a third interaction, Ĥ₃ = κx̂_L(x̂_A sinθ + p̂_A cosθ), which drives the system into y^out3 = S(κ̃, θ)y^out2 = S (α, β, θ,)yⁱⁿ, where the matrix is

S (α, β, θ) = (\begin{matrix} 1 & 0 & 0 & 0 \\ {\tilde{κ}}^{2} f_{α - β, α - θ, β - θ}^{s} & 1 & - \tilde{κ} f_{α, β, θ}^{s} & - \tilde{κ} f_{α, β, θ}^{c} \\ \tilde{κ} f_{α, β, θ}^{c} & 0 & 1 & 0 \\ - \tilde{κ} f_{α, β, θ}^{s} & 0 & 0 & 1 \end{matrix}) .

Here we have defined the new variables

f_{α, β, θ}^{s} = \sum_{φ = α, β, θ} sin φ

and

f_{α, β, θ}^{c} = \sum_{φ = α, β, θ} cos φ

. Obviously, to realize a perfect optical QP gate it is required that

f_{α, β, θ}^{s} = f_{α, β, θ}^{c} = 0,

f_{α - β, α - θ, β - θ}^{s} \neq 0 .

Equation (10) leads to the limitations: |α − β| = 2π/3, |α + β − 2θ| = 2π or |α − β| = 4π/3, |α + β − 2θ| = 0. According to these limitations one may choose a special set of angle parameters, e.g., α = π/6, β = 5π/6, θ = 3π/2, with which Eq. (11) can be easily calculated to give

f_{- \frac{2 π}{3}, - \frac{4 π}{3}, - \frac{2 π}{3}}^{s} = - \frac{\sqrt{3}}{2} {\tilde{κ}}^{2} \neq 0

. Finally, we obtain the input-output relations for light

{\hat{x}}_{L}^{out 3} = {\hat{x}}_{L}^{in}, {\hat{p}}_{L}^{out 3} = {\hat{p}}_{L}^{in} - \frac{\sqrt{3}}{2} {\tilde{κ}}^{2} {\hat{x}}_{L}^{in},

which is the main result of this paper. Apparently, this optical state is exactly the same as the state given in Eq. (1) with

s = - \sqrt{3} {\tilde{κ}}^{2} / 2

. Therefore, for arbitrary non-zero κ̃ a perfect optical QP gate is successfully performed. Moreover, it is worth noting here that, the sign of

f_{α - β, α - θ, β - θ}^{s}

can be flipped freely via suitable choice of α, β and θ, e.g., for α = 5π/6, β = π/6, θ = 3π/2 yielding

f_{\frac{2 π}{3}, - \frac{2 π}{3}, - \frac{4 π}{3}}^{s} = \frac{\sqrt{3}}{2} {\tilde{κ}}^{2}

. Such characteristic is especially valuable, since two conjugate QP operations D̂₂(s) and

{\hat{D}}_{2}^{†} (s)

together with the beam splitters enable the performance of perfect CV controlled-Z gate (see Appendix for details).

2.2. Optical polarization squeezing

As mentioned above, the performance of QP gate is always accompanied with the generation of optical squeezing. To find out the amount of squeezing generated, we perform a rotation of the final output state: ${\hat{ϑ}}_{L}^{out 3^{'}} \to {\hat{R}}^{†} (s) {\hat{ϑ}}_{L}^{out 3} \hat{R} (s)$ , leading to

(\begin{matrix} {\hat{x}}_{L}^{out 3^{'}} \\ {\hat{p}}_{L}^{out 3^{'}} \end{matrix}) = (\begin{matrix} cos (s) & sin (s) \\ - sin (s) & cos (s) \end{matrix}) (\begin{matrix} {\hat{x}}_{L}^{out 3} \\ {\hat{p}}_{L}^{out 3} \end{matrix}) .

Assuming that the input light field is in the vacuum state, the variance of

{\hat{x}}_{L}^{out 3^{'}}

can then be readily calculated to give

Δ^{2} {\hat{x}}_{L}^{out 3^{'}} = \frac{1}{2} [1 - \frac{\sqrt{3}}{2} {\tilde{κ}}^{2} sin (2 s) + \frac{3}{4} {\tilde{κ}}^{4} {cos}^{2} (s)],

which can be minimized at

s_{\min} = - \frac{1}{2} arctan (4 / \sqrt{3} {\tilde{κ}}^{2})

, resulting in the optimized squeezing parameter

ζ (s_{\min}) = 1 + 3 {\tilde{κ}}^{4} [1 - \sqrt{1 + 16 / (3 {\tilde{κ}}^{4})}] / 8 \Rightarrow {lim}_{\tilde{κ} \to \infty} 4 / (3 {\tilde{κ}}^{4})

. In contrast to the double-pass scheme [19] whose squeezing parameter is ζ_κ_̃_→∞ ≈ 3/(2κ̃²), our current scheme strongly enhances the degree of optical polarization squeezing.

3. Experimental considerations

3.1. Noise effects

So far, we have neglected the noise effects. While in reality, every photon on its way through the atomic media has a mall probability for being absorbed [32, 33], and thus cause photon losses. Such losses, however, can be modeled by a beam splitter type admixture of vacuum components [18,34], which transforms the output light quadratures as ${\hat{ϑ}}_{L}^{out} \to {\hat{ϑ}}_{L}^{{out}^{'}} = \sqrt{1 - ε} {\hat{ϑ}}_{L}^{out} + \sqrt{ε} {\hat{ϑ}}_{L}^{v}$ , where ε denotes the absorbtion coefficient and ${\hat{ϑ}}_{L}^{v}$ represents the vacuum noise quadrature with zero mean and 1/2 variance. Besides, absorption losses due to the propagation in delay lines can also be taken into account by replacing ε with ξ = ε + r, where r represents the the overall absorptivity of the delay line. On the other hand for atoms, due to collisional relaxation and weak excitation by light, they also undergo dissipation. When take the noise effects into account, as discussed in [17, 18, 32], the atomic state becomes ${\hat{ϑ}}_{A}^{out} \to {\hat{ϑ}}_{A}^{{out}^{'}} = \sqrt{1 - η} {\hat{ϑ}}_{A}^{out} + \sqrt{η} {\hat{ϑ}}_{A}^{v}$ where η is the atomic decay parameter and ${\hat{ϑ}}_{A}^{v}$ is the atomic vacuum noise. With these modifications, the atomic and light states after the first pass are then transformed into:

y^{out 1^{'}} = \bar{D} (ξ, η) S (\tilde{κ}, α) y^{in} + D (ξ, η) y^{v 1},

where y^vi denotes the noise vector of ith pass, and

D (ξ, η) = diag (\sqrt{ξ}, \sqrt{ξ}, \sqrt{η}, \sqrt{η})

,

\bar{D} (ξ, η) = \sqrt{1 - D {(ξ, η)}^{2}}

. Analogously, the final state created after three passages can then be calculated by iterating the map defined by the above equation. For the rest two interactions, however, one should take into account the fact that, the classical variables J_x and S_x also decay from pass to pass as: 〈J_x〉 → (1 − η)〈J_x〉, 〈S_x〉 → (1 − ξ)〈S_x〉, which reduces the coupling strength by a factor

τ = \sqrt{(1 - η) (1 - ξ)}

[32]. As a result, after three passages the atomic and light states become

\begin{array}{l} y^{out 3^{'}} & = & \bar{D} (ξ, η) S (τ^{2} \tilde{κ}, θ) [\bar{D} (ξ, η) S (τ \tilde{κ}, β) y^{out 1^{'}} \\ + D (ξ, η) y^{v 2}] + D (ξ, η) y^{v 3} . \end{array}

By inserting the expression for the interaction matrix given in Eq. (7) into this equation and using the set of special angle parameters chosen above, we finally arrive at the modified input-output relations for light

{\hat{x}}_{L}^{out 3^{'}} = {(1 - ξ)}^{\frac{3}{2}} {\hat{x}}_{L}^{in} + \sqrt{ξ} {\hat{f}}_{x},

\begin{array}{l} {\hat{p}}_{L}^{out 3^{'}} & = & {(1 - ξ)}^{\frac{3}{2}} ({\hat{p}}_{L}^{in} - \frac{\sqrt{3}}{2} {\tilde{κ}}^{'}^{2} {\hat{x}}_{L}^{in}) \\ + η {(1 - ξ)}^{\frac{3}{2}} \tilde{κ} [(η - \frac{3}{2}) {\hat{x}}_{A}^{in} + \frac{\sqrt{3}}{2} {\hat{p}}_{A}^{in}] + \sqrt{η} {\hat{f}}_{A}^{v} + \sqrt{ξ} {\hat{f}}_{L}^{v}, \end{array}

where we have defined

{\tilde{κ}}^{'} = \sqrt{(1 - η) (1 - ξ + ξ η)} \tilde{κ}

,

{\hat{f}}_{ϑ} = (1 - ξ) {\hat{ϑ}}_{L}^{v 1} + \sqrt{1 - ξ} {\hat{ϑ}}_{L}^{v 2} + {\hat{ϑ}}_{L}^{v 3}

, and the collective vacuum operator for atoms

{\hat{f}}_{A}^{v} = \sqrt{(1 - η)} {(1 - ξ)}^{3 / 2} \tilde{κ} [(1 / 2 - η) {\hat{x}}_{A}^{v 1} + \sqrt{3} {\hat{p}}_{A}^{v 1} / 2 + \sqrt{(1 - η)} {\hat{x}}_{A}^{v 2})

and for light

{\hat{f}}_{L}^{v} = - \sqrt{3} {(1 - η)}^{2} {(1 - ξ)}^{2} {\tilde{κ}}^{2} {\hat{x}}_{L}^{v 1} / 2 + {\hat{f}}_{p}

. The second line of Eq. (18) shows that, now the atomic quadratures can not be completely canceled due to the decay of the atomic system. As a result, compared to the ideal case of Eq. (12), the light state obtained here is somewhat distorted. In order to quantify the distortion of this state, we use the formula of fidelity

ℱ (ρ_{out 3}, ρ_{out 3^{'}}) = {[T r {(\sqrt{ρ_{out 3}} ρ_{out 3^{'}} \sqrt{ρ_{out 3}})}^{1 / 2}]}^{2}

which is a measure of how well the distorted output state ρ̂_out3′ of Eqs. (17) and (18) compares to the ideal output state ρ_out3 of Eq. (12). In principle, the input light state can be arbitrary. However, here for simplicity and concreteness (but without loss of generality), we assume the input light state is a Gaussian state, e.g. a coherent state or a squeezed state. In this case, the output state conserves the Gaussian character of the input state due to the fact that the interaction Hamiltonians involved here are bilinear in the canonical operators [4]. As a result, the fidelity becomes [35]

\begin{array}{l} ℱ (ρ_{out 3}, ρ_{out 3^{'}}) & = & \frac{1}{\sqrt{det (Γ_{out 3} + Γ_{out 3^{'}}) + δ} - \sqrt{δ}} \\ \times exp [- \frac{1}{2} 𝒜^{T} {(Γ_{out 3} + Γ_{out 3^{'}})}^{- 1} 𝒜], \end{array}

where Γ stands for the covariance matrix having its elements Γ_ij = 〈y_iy_j + y_jy_i〉 − 2〈y_i y_j〉, and where δ = (detΓ_out3 − 1)(detΓ_out3′ − 1), 𝒜 = m_out3 − m_out3′ with m the vector of mean values. If the ideal output state is a pure Gaussian state, we will have det(Γ_out3) = 1 and in turn δ = 0 [35]. Moreover, as an example we assume that the input Gaussian states center around zero, such that 𝒜 = 0. Equation (19) then reduces to a much simpler form

ℱ (ρ_{out 3}, ρ_{out 3^{'}}) = 2 / \sqrt{det (Γ_{out 3} + Γ_{out 3^{'}})}

.

From Eqs. (12), (17), and (18), Γ_out3 and Γ_out3′ can be respectively calculated, and finally one is able to show the gate fidelity ℱ in its dependence on κ̃, η, and ξ. In the case of coherent state input, the fidelity versus κ̃ is depicted in Fig. 2(a) for different loss parameters η, ξ. As can be seen from the figure that, the achievable fidelities are strongly dependent on the coupling strength κ̃. This result is reasonable since the larger the coupling strength is, the more “nonclassical” the output state become. Such (highly-nonclassical) output states are more fragile to noise effects, resulting in lower achievable fidelities. In the weak interaction regime [corresponding to small κ̃, where the CV QC works as shown in Eq. (2)], however, high fidelities are obtainable and the scheme is robust against the noises. Putting κ̃ = 2, the insert of Fig. 2(a) shows the fidelity in its dependence on light reflections ξ for different decay rate. One can see from the insert that, in contrast to the atomic decay the light losses affect the fidelities more significantly. For squeezed state input, in Fig. 2(b) we plot the fidelity ℱ vs κ̃ for different squeezing of the input state and for η = ξ = 0.10 (which corresponds to the experimental conditions of [36]). It clearly shows that the gate fidelity is very sensitive to the squeezing degree of the input state. The fidelity is greatly reduced for different input squeezing even when κ̃ = 0 (corresponding to no atom in the sample), which reflects the fact that the squeezed states are very sensitive to photon losses. The insert of Fig. 2(b) indicates that, in the weak interaction regime the scheme works quite well within certain range of squeezing.

Fig. 2 (a) Gate operation fidelity ℱ for coherent states vs coupling κ̃ in the presence of atomic decay and light reflection. Inset: the gate fidelity in its dependent on ξ for different decay rate and κ̃ = 2. (b) Gate operation fidelity ℱ for squeezed states vs coupling κ̃ including 10% atomic decay and 10% light losses. Inset: the gate fidelity in its dependent on input squeezing for decay parameters η = ξ = 0.10 and coupling κ̃ = 0.10. Choosing the light losses ξ = 0.05, the fidelity of a fixed QP operation with $s = \sqrt{3} / 2$ varies with the atomic decay (c) for ρ = 30 and with the optical depth (d). The inset of (d) shows how the optimal η varies with r.

Download Full Size | PDF

So far, the coupling strength κ̃ and the atomic decay η have been treated independently, while in fact they are linked to each other through κ̃² = ρη [17], where ρ is the optical depth of the atomic sample. For a realistic system such as room-temperature vapor, we typically have ρ ≈ 30 [37]. If one performs a fixed D̂₂(s) operation in such system with, e.g., $s = \sqrt{3} / 2$ (corresponding to κ̃ = 1), one is able to show in Fig. 2(c) the fidelity ℱ in its dependence on η (which can be tuned by choosing either the detuning from atomic resonance or the total photon number of the pulse [17]) for different squeezing of the input state including ξ = 0.05 absorption losses. In each case, there exist an optimal fidelity, ℱ = 0.98, ℱ = 0.93, and ℱ = 0.81, for η = 0.028, η = 0.032, and η = 0.034, respectively. The η-optimized fidelities vs ρ is also depicted in Fig. 2(d), showing that, the larger the optical depth is, the higher the fidelities one will obtain. Besides, it is worth noting in figure 2(d) that, for the region of 0 to 20, it is quite economical to tune the optical depth to improve the gate fidelity. Once the optical depth of the systems surpasses this region, it might be more efficient to use other ways (e.g., light-reflection reduction) to improve the performance of gate.

Unlike the gate performance, optical polarization squeezing usually works in the strong interaction regime. Along the lines presented in Sec. 2 above, one may directly calculate the amount of squeezing in the presence of noise effect based on Eqs. (17) and (18). Finally, in Fig. 3(a) we are able to plot the squeezing parameter ζ, optimized with respect to rotation s and direction θ, in its dependence on the coupling κ̃, which indicates the amount of achievable squeezing are bounded by the added noise. For the parameters κ̃² = 3 and η = 0.10 of room temperature atomic vapors [37], the achievable squeezing are 4.8 dB, 6.3 dB, 8.0 dB for the light losses ξ = 0.1, 0.05, 0.01, respectively. The s, θ -optimized squeezing vs ξ is also depicted in Fig. 3(b). One can see from the figure that, the atomic decay plays a minor role in the scheme, while the light losses are the major factor, which limits the creation of polarization squeezing quantitatively and qualitatively.

Fig. 3 (a) The optimized polarization squeezing ζ vs coupling κ̃ in the presence of atomic decay and light reflection. The inset shows how the optimal parameters s_min and θ vary with κ̃. (b) Optimized squeezing vs light losses ξ for different atomic decay.

Download Full Size | PDF

3.2. Experimental feasibility

To successfully and efficiently perform the CV QP gate, it is required that (i) T < T_DL and r ≪ 1, where T_DL represents a time in which the pulse pass through the delay lines, and (ii) η, ε ≪ 1. Condition (ii) can be fulfilled in both room-temperature [36, 37] and cold [38–40] atomic systems, since the optical depth in these systems usually satisfy ρ ≫ 1 (e.g., ρ ≥ 50 for cold atoms [40]). As a result, the decay parameter satisfies η = κ̃²/ρ ≪ 1 for small κ̃. If one chooses N_p ≈ N_a (N_p the total photon number and N_a the total atom number), we will have ε ∼ η ≪ 1 [18]. Condition (i), however, is the main challenge of current scheme. For previous experiments of quantum interface between light and atoms [28], the typical value of the width of a pulse is about T ≈ 1 ms. Correspondingly, hundreds of kilometer-long delay lines between the passages are normally required, which poses a challenge to the experimental realization. Fortunately, the situation has recently been changed dramatically. Faraday interaction using a very short pulse—the width of which is about T = 100 ns and Laser-cooled atoms has been successfully demonstrated [38, 39]. Besides, optical delay line using cylindrical mirrors with T_DL ≈ 1μs which conserves the polarization of light pulse has also been reported in [41], showing that reflective losses r less than 2% can be achieved with the aid of high reflective mirrors. Therefore, condition (i) is also feasible.

4. CONCLUSIONS

In conclusion, we have presented a novel and experimentally feasible scheme for realizing optical CV QP gate in an atomic ensemble. The process is based on off-resonant interaction between light and spin-polarized atomic ensembles. By sending a pulse three times through an coherent prepared atomic ensemble, we found that a QP operation is successfully performed, and its fidelities can ideally run up to one. We also found that the sign of the parameter s of D̂₂(s) can be freely flipped. This is a quite good characteristic, since it enables the performance of perfect two-mode CV controlled-Z gate. Such a gate is a key resource for CV QC, especially for CV one-way QC [29, 42, 43], while it is by now still experimentally challenging. We also studied the capacity of polarization squeezing of current scheme, showing that, in contrast to double-pass scheme, the amount of squeezing is strongly enhanced. Finally, the influences of the noise effects including the atomic decay and photon reflections losses are considered. For coherent input, the scheme works quite well in weak iteration regime, while for squeezed input, the scheme is sensitive to photon losses. When it comes to polarization squeezing, the situation is analogous to the double-pass scheme given in [19], that is, the attainable degree of squeezing of the scheme are mainly limited by light losses. This confirms our motivation to search for new techniques to reduce the losses in the loop, e.g. if one can use a little more shorter pulse than [38, 39], e.g. T = 10 ns, the delay line are no longer necessary, and light losses will be diminished significantly. We expect that our scheme can be beneficial in the context of CV QC.

5. Appendix

In this Appendix we present the details on how to realize the two-mode CV Controlled-Z operation using QP gates. Suppose that we want to perform a C_Z gate between two light beams i and j. An ideal controlled-Z gate C_Z = exp(ix̂_i ⊗ x̂_j) transfers the quadratures into [29, 42, 43]: x̂_i,j → x̂_i,j, p̂_i,j → p̂_i,j + x̂_j,i. Such transformation can be achieved by three steps. In the first step, the two beams are mixed on a beam splitter, which transfers the vector of quadratures v = (x̂_i, p̂_i, x̂_j, p̂_j)^T into v^out1 = B̂_ij(θ₁)vⁱⁿ, where the beam splitter operator can be written as [4]:

{\hat{B}}_{i j} (θ_{1}) = (\begin{matrix} cos θ_{1} & 0 & sin θ_{1} & 0 \\ 0 & cos θ_{1} & 0 & sin θ_{1} \\ - sin θ_{1} & 0 & cos θ_{1} & 0 \\ 0 & - sin θ_{1} & 0 & cos θ_{1} \end{matrix}) .

In the second step, the two output beams are then injected into two atomic ensembles described above to experience

{\hat{D}}_{2}^{i} (s)

and

{\hat{D}}_{2}^{j} (- s)

transformations, respectively. After this operation the light states become

v^{out 2} = {\hat{D}}_{2}^{i} (s) \oplus {\hat{D}}_{2}^{j} (- s) v^{out 1} = {\hat{D}}_{i j} (s) v^{out 1}

, where

{\hat{D}}_{i j} (s) = (\begin{matrix} 1 & 0 & 0 & 0 \\ s & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & - s & 1 \end{matrix}) .

In the last step, the two beams are mixed on a second beam splitter to generate the final output state v^out3 = B̂_ij(θ₂)v^out2 = B̂_ij(θ₂)D̂_ij(s)B̂_ij(θ₁)vⁱⁿ = ℳ_ij(s)vⁱⁿ. Taking the special parameters for beam splitters θ₁ = −θ₂ = π/4 and setting s = 1, we finally have

ℳ_{i j} (s) = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{matrix}) .

Consequently, a perfect CV controlled-Z gate between two light beams is successfully performed.

Acknowledgments

This work was supported by the Natural Science Foundation of Zhejiang province (Grants No. LY12A05001), the Natural Science Foundation of China (Grants No. 11074190), and the department of education of Zhejiang province ( Y201120838).

References and links

1. P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” SIAM J. Comput. 26, 1484–1509 (1997). [CrossRef]

2. S. Lloyd and S. L. Braunstein, “Quantum computation over continuous variables,” Phys. Rev. Lett. 82, 1784–1787 (1999). [CrossRef]

3. L. K. Grover, “Quantum mechanics helps in searching for a needle in a haystack,” Phys. Rev. Lett. 79, 325–328 (1997). [CrossRef]

4. S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005). [CrossRef]

5. M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009). [CrossRef]

6. M. G. Paris, “Displacement operator by beam splitter,” Physics Letters A 217, 78–80 (1996). [CrossRef]

7. S. L. Braunstein, “Squeezing as an irreducible resource,” Phys. Rev. A 71, 055801 (2005). [CrossRef]

8. M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994). [CrossRef] [PubMed]

9. S. Sefi, “Squeezing and photon counting with the cubic phase state,” arXiv:quant-ph/1310.2429 (2013).

10. J. A. Levenson, I. Abram, T. Rivera, and P. Grangier, “Reduction of quantum noise in optical parametric amplification,” J. Opt. Soc. Am. B 10, 2233–2238 (1993). [CrossRef]

11. J. A. Levenson, I. Abram, T. Rivera, P. Fayolle, J. C. Garreau, and P. Grangier, “Quantum optical cloning amplifier,” Phys. Rev. Lett. 70, 267–270 (1993). [CrossRef] [PubMed]

12. J. P. Poizat and P. Grangier, “Experimental realization of a quantum optical tap,” Phys. Rev. Lett. 70, 271–274 (1993). [CrossRef] [PubMed]

13. R. Filip, P. Marek, and U. L. Andersen, “Measurement-induced continuous-variable quantum interactions,” Phys. Rev. A 71, 042308 (2005). [CrossRef]

14. J.-i. Yoshikawa, T. Hayashi, T. Akiyama, N. Takei, A. Huck, U. L. Andersen, and A. Furusawa, “Demonstration of deterministic and high fidelity squeezing of quantum information,” Phys. Rev. A 76, 060301 (2007). [CrossRef]

15. Y. Miwa, J.-i. Yoshikawa, P. van Loock, and A. Furusawa, “Demonstration of a universal one-way quantum quadratic phase gate,” Phys. Rev. A 80, 050303 (2009). [CrossRef]

16. A. Kuzmich, N. P. Bigelow, and L. Mandel, “Atomic quantum non-demolition measurements and squeezing,” Europhys. Lett. 42, 481–486 (1998). [CrossRef]

17. K. Hammerer, A. S. Sørensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010). [CrossRef]

18. L.-M. Duan, J. I. Cirac, P. Zoller, and E. S. Polzik, “Quantum communication between atomic ensembles using coherent light,” Phys. Rev. Lett. 85, 5643–5646 (2000). [CrossRef]

19. J. F. Sherson and K. Mølmer, “Polarization squeezing by optical faraday rotation,” Phys. Rev. Lett. 97, 143602 (2006). [CrossRef] [PubMed]

20. Y. Takahashi, K. Honda, N. Tanaka, K. Toyoda, K. Ishikawa, and T. Yabuzaki, “Quantum nondemolition measurement of spin via the paramagnetic faraday rotation,” Phys. Rev. A 60, 4974–4979 (1999). [CrossRef]

21. A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N. P. Bigelow, “Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60, 2346–2350 (1999). [CrossRef]

22. R. Inoue, S.-I.-R. Tanaka, R. Namiki, T. Sagawa, and Y. Takahashi, “Unconditional quantum-noise suppression via measurement-based quantum feedback,” Phys. Rev. Lett. 110, 163602 (2013). [CrossRef] [PubMed]

23. B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature 432, 482–486 (2004). [CrossRef] [PubMed]

24. J. Fiurášek, J. Sherson, T. Opatrný, and E. S. Polzik, “Single-passage readout of atomic quantum memory,” Phys. Rev. A 73, 022331 (2006). [CrossRef]

25. C. A. Muschik, K. Hammerer, E. S. Polzik, and J. I. Cirac, “Efficient quantum memory and entanglement between light and an atomic ensemble using magnetic fields,” Phys. Rev. A 73, 062329 (2006). [CrossRef]

26. T. Takano, M. Fuyama, R. Namiki, and Y. Takahashi, “Continuous-variable quantum swapping gate between light and atoms,” Phys. Rev. A 78, 010307 (2008). [CrossRef]

27. J. Stasińska, C. Rodó, S. Paganelli, G. Birkl, and A. Sanpera, “Manipulating mesoscopic multipartite entanglement with atom-light interfaces,” Phys. Rev. A 80, 062304 (2009). [CrossRef]

28. B. Julsgaard, “Quantum memory and teleportation using macroscopic gas samples,” Ph.D. thesis, Aarhus University (2003).

29. M.-F. Wang, N.-Q. Jiang, Q.-L. Jin, and Y.-Z. Zheng, “Continuous-variable controlled-z gate using an atomic ensemble,” Phys. Rev. A 83, 062339 (2011). [CrossRef]

30. M. Takeuchi, S. Ichihara, T. Takano, M. Kumakura, T. Yabuzaki, and Y. Takahashi, “Spin squeezing via one-axis twisting with coherent light,” Phys. Rev. Lett. 94, 023003 (2005). [CrossRef] [PubMed]

31. C. M. Trail, P. S. Jessen, and I. H. Deutsch, “Strongly enhanced spin squeezing via quantum control,” Phys. Rev. Lett. 105, 193602 (2010). [CrossRef]

32. K. Hammerer, K. Mølmer, E. S. Polzik, and J. I. Cirac, “Light-matter quantum interface,” Phys. Rev. A 70, 044304 (2004). [CrossRef]

33. L. B. Madsen and K. Mølmer, “Spin squeezing and precision probing with light and samples of atoms in the gaussian description,” Phys. Rev. A 70, 052324 (2004). [CrossRef]

34. K. Hammerer, E. S. Polzik, and J. I. Cirac, “Teleportation and spin squeezing utilizing multimode entanglement of light with atoms,” Phys. Rev. A 72, 052313 (2005). [CrossRef]

35. H. Scutaru, “Fidelity for displaced squeezed thermal states and the oscillator semigroup,” J. Phys. A 31, 3659–3663 (1998). [CrossRef]

36. J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac, and E. S. Polzik, “Quantum teleportation between light and matter,” Nature 443, 557–560 (2006). [CrossRef] [PubMed]

37. J. F. Sherson, “Quantum memory and teleportation using macroscopic gas samples,” Ph.D. thesis, Aarhus University (2006).

38. M. Takeuchi, T. Takano, S. Ichihara, A. Yamaguchi, M. Kumakura, T. Yabuzaki, and Y. Takahashi, “Fast polarimetry system for the application to spin quantum non-demolition measurement,” Appl. Phys. B 83, 33–36 (2006). [CrossRef]

39. T. Takano, M. Fuyama, R. Namiki, and Y. Takahashi, “Spin squeezing of a cold atomic ensemble with the nuclear spin of one-half,” Phys. Rev. Lett. 102, 033601 (2009). [CrossRef] [PubMed]

40. M. Kubasik, M. Koschorreck, M. Napolitano, S. R. de Echaniz, H. Crepaz, J. Eschner, E. S. Polzik, and M. W. Mitchell, “Polarization-based light-atom quantum interface with an all-optical trap,” Phys. Rev. A 79, 043815 (2009). [CrossRef]

41. E. Jeffrey, M. Brenner, and P. Kwiat, “Spin squeezing of a cold atomic ensemble with the nuclear spin of one-half,” Proc. SPIE 5161, 269–279 (2004). [CrossRef]

42. N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, “Universal quantum computation with continuous-variable cluster states,” Phys. Rev. Lett. 97, 110501 (2006). [CrossRef] [PubMed]

43. N. C. Menicucci, X. Ma, and T. C. Ralph, “Arbitrarily large continuous-variable cluster states from a single quantum nondemolition gate,” Phys. Rev. Lett. 104, 250503 (2010). [CrossRef] [PubMed]

Optical continuous-variable quadratic phase gate via Faraday interaction

Abstract

1. Introduction

2. The protocol

2.1. Quadratic phase gate

2.2. Optical polarization squeezing

3. Experimental considerations

3.1. Noise effects

3.2. Experimental feasibility

4. CONCLUSIONS

5. Appendix

Acknowledgments

References and links

Cited By

Figures (3)

Equations (22)

Optics Express