Single photon polarization conversion via scattering by a pair of atoms

Mu-Tian Cheng; Xiuwen Xia; Jingping Xu; Chengjie Zhu; Bing Wang; Xiao-San Ma

doi:10.1364/OE.26.028872

1. Introduction

It is well known that single photons are ideal carriers for quantum information. Waveguide can guide the photon transport naturally. Furthermore, the strong coupling between the waveguide and the atoms has been realized [1–4]. Thus, coherent controlling single-photon and multiphoton transport in one-dimensional waveguide has been investigated widely [5–19]. Many quantum devices, such as quantum routing [20], diode [21] and circulator [22] were proposed or realized experimentally. The quantum information processing based on the interaction between the waveguide and atoms is also investigated widely [23, 24]. Based on these advances in this field, the concept of waveguide quantum electrodynamics (WQED) is proposed. Some of the significant recent efforts in WQED have been reviewed in [25, 26].

Polarization is a very important property of single photon since polarization state of single photon can be used to code quantum information. There have been many significant advances in controlling polarization conversion of light using various photonic structures, such as diffraction gratings [27], metamaterials [28–32], and topological photonics [33]. Recently, Tsoi and Law investigated single photon scattering by an array of Λ-type three-level atoms in a one-dimensional waveguide [34]. They showed that an incident photon with an unknown polarization can be converted into a specified polarization. In their study, the dispersion-relation of the waveguide is linear. Obi and Shen further showed that the polarization conversion efficiency is no more than 0.5 when just only one Λ system couples to the one-dimensional waveguide [35]. Later, the polarization-dependent single photon scattering in coupled-resonant waveguide with nonlinear dispersion relation has been discussed [36]. Very recently, Li et al. investigated the single photon polarization conversion between the two linear polarizations via scattering by a Λ-type system assisted by an additional cavity [37].

In this paper, we further investigate single photon polarization-dependent scattering in one-dimensional waveguide with linear dispersion-relation. We show that polarization of the incident single photon can be converted from linearity to circularity and the polarization conversion fidelity approaches unit in the ideal case. In our model, one can also choose the initial state of the atom to control whether to realize the polarization conversion. These properties make our model useful in designing quantum devices at single-photon level and quantum information processing.

2. Theoretical model

The configuration of the single photon converter is shown in Fig. 1. A waveguide couples to two atoms. The first atom is considered as a Λ system with excited state |e〉 and two ground states |g_h〉 and |g_v〉. Here, without loss of generality, we suppose that the polarization of the incident photon is horizontal. The transitions |e〉 ↔ |g_h〉 and |e〉 ↔ |g_v〉 couple to the photons with horizontal (p -) and vertical (v-) polarization, respectively. The corresponding coupling strengths are denoted by J_h and J_v. The initial state of the Λ-system is |g_h〉. The energy-level configuration of the second atom is a V -type system. The two excited states are |e_h〉 and |e_v〉 and the ground state is |g〉. The transition between |e_h〉 (|e_v〉) and |g〉 interacts with photon with horizontal (vertical) -polarization and the coupling strength is J_ch (J_cv).

Fig. 1 The system considered in the manuscript. Two atoms couple to an one-dimensional waveguide. The energy-level configurations of the two atoms are taken as Λ and V -type systems, respectively. The distance between the two atoms is L.

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The Hamiltonian describing the system is given as in [5]

H = H_{w} + H_{a} + H_{i n t},

H_{w} = \sum_{p = h, v} [- i v_{g} \int d x C_{R p}^{†} (x) \frac{\partial}{\partial x} C_{R p} (x) + i v_{g} \int d x C_{L p}^{†} (x) \frac{\partial}{\partial x} C_{L p} (x)]

H_{a} = (ω_{e} - i γ_{e} / 2) σ_{e e} + \sum_{p = h, v} (ω_{e_{p}} - i γ_{e_{p}} / 2) σ_{e_{p} e_{p}},

\begin{matrix} H_{i n t} = \sum_{p = h} \sum_{d = R, L} {J_{p} \int d x δ (x) [C_{d p}^{†} (x) σ_{g_{p} e} + H . c .] \\ + J_{c p} \int d x δ (x - L) [C_{d p}^{†} (x) σ_{g e_{p}} + H . C .]}, \end{matrix}

where v_g is the group velocity of the photon in the waveguide.

C_{R p}^{†} (x) (C_{L p}^{†} (x))

represents creating a right (left) -prorogation photon at x with horizontal (p = h) or vertical (p = v) polarization. In derivations of Eq.(2) and Eq.(4), the dispersion relation of the waveguide is linearized. The details about the derivations can be found in [38]. ω_e and

ω_{e_{p}}

are the transition frequencies from excited states to the ground states. Here, we have set the energy scale such that

ω_{g_{h}} = ω_{g_{v}} = ω_{g} = 0

. γ_e and

γ_{e_{p}}

are the energy loss rates from the |e〉 and |e_p〉, respectively. Here, the decay terms are added phenomenally. However, the dynamics of the system can be accurately described by this effective Hamiltonian provided that the thermal energy much less than the atomic transition energy [6]. For simplification, we also supposed that

ω_{e} = ω_{e_{h}} = ω_{e_{v}}

in the following discussions. H.c. in [4] denotes Hermitian conjugation.

Since we consider the single photon case and the number of excitations is conserved in this system, the wavefunction of the Hamiltonian can be expressed as [5]

| Ψ 〉 = \sum_{p = h, v} \sum_{d = R, L} [\int d x ϕ_{d p} (x) C_{d p}^{†} | 0, g, g_{p} 〉 + u_{p} | 0, e_{p} 〉] + u_{e} | 0, e 〉,

where |0, g, g_p〉 denotes that there is no photon in the waveguide and both of the atoms are in the ground states. ϕ_dp(x ) represents the single photon wavefuction in the p mode at x. |0, e〉 (|0, e_p〉) denotes no photon in the waveguide and the atom in state |e〉 (|e_p〉). Supposing that the initial state of the first atom is |g_h〉 and the single photon incidents from the left of the waveguide with horizontal-polarization. we can get the expressions of ϕ_dp(x) as in

ϕ_{R h} (x) = e^{i k x} [θ (- x) + a_{h} θ (x) θ (L - x) + t_{h} θ (x - L)],

ϕ_{L h} (x) = e^{- i k x} [r_{h} θ (- x) + b_{h} θ (x) θ (L - x)],

ϕ_{R v} (x) = e^{i k x} [a_{v} θ (x) θ (L - x) + t_{v} θ (x - L)],

ϕ_{L v} (x) = e^{- i k x} [r_{v} θ (- x) + b_{v} θ (- x) θ (L - x)],

where θ(x ) is the Heaviside step function with θ(0) = 1/2. t_p and r_p denote the transmission and reflection amplitudes of the single photon with p-polarization.

a_{p} θ (x) θ (L - x) e^{i k x} (b_{p} θ (x) θ (L - x) e^{- i k x})

denotes the wavefunction of right (left) propagation photon with p-polarization between x = 0 and x = L.

3. Single photon polarization conversion

From the Schrödinger equation H |Ψ〉 = ω|Ψ〉, one can obtain the analytical expressions of t_p and r_p by following the derivations given in [38]. However, they are very cumbersome. To simplify the analytical analysis, we neglect the dissipations temporarily. The coupling strengths are also set as J_h = J_v = J, J_ch = J_cv = J_c. Then we can obtain

t_{h} = \frac{(- 1 + e^{2 i θ}) Δ G G_{c} + i Δ^{2} (G + G_{c}) + Δ^{3}}{2 i (- 1 + e^{2 i θ}) G G_{c}^{2} + [2 (- 1 + e^{2 i θ}) G G_{c} - 2 G G_{c} - G_{c}^{2}] Δ + 2 i Δ^{2} (G + G_{c}) + Δ^{3}},

t_{v} = \frac{(- 1 + e^{2 i θ}) Δ G G_{c} + i G Δ^{2}}{2 i (- 1 + e^{2 i θ}) G G_{c}^{2} + [2 (- 1 + e^{2 i θ}) G G_{c} - 2 G G_{c} - G_{c}^{2}] Δ + 2 i Δ^{2} (G + G_{c}) + Δ^{3}},

r_{h} = \frac{- (- 1 + e^{4 i θ}) G G_{c}^{2} - 2 i G G_{c} Δ - i e^{2 i θ} G_{c}^{2} Δ - G Δ^{2} - e^{2 i θ} G_{c} Δ^{2}}{2 i (- 1 + e^{2 i θ}) G G_{c}^{2} + [2 (- 1 + e^{2 i θ}) G G_{c} - 2 G G_{c} - G_{c}^{2}] Δ + 2 i Δ^{2} (G + G_{c}) + Δ^{3}},

r_{v} = \frac{i {(- 1 + e^{2 i θ})}^{2} G G_{c}^{2} - 2 (- 1 + e^{2 i θ}) G G_{c} Δ - i G Δ^{2}}{2 i (- 1 + e^{2 i θ}) G G_{c}^{2} + [2 (- 1 + e^{2 i θ}) G G_{c} - 2 G G_{c} - G_{c}^{2}] Δ + 2 i Δ^{2} (G + G_{c}) + Δ^{3}},

where θ = kL, G = J²/v_g,

G_{c} = J_{c}^{2} / v_{g}

, Δ = ω − ω_e.

Figure 2 exhibits T_p ≡ |t_p|² and R_p ≡ |r_p|² as a function of Δ and θ. It exhibits that T_v can change between 0 and 0.25 but R_v can vary between 0 and 1, which implies that the polarization conversion of the single photon can be realized since we have set that the polarization of the single photon is h − polarized. To understand this property more clearly, we set Δ = 0, which means that the incident single photon is resonant with the two atoms. Then, Eqs. (10) to (13) degenerate into t_h = t_v = 0 and

r_{h} = (- 1 - e^{2 i θ}) / 2,

r_{v} = (- 1 + e^{2 i θ}) / 2.

Fig. 2 Single photon scattering probabilities varying with θ and Δ without dissipations. Panels (a), (b), (c) and (d) denote T_h, T_v, R_h and R_v, respectively. In the calculations, G = G_c = 10⁻⁵ω_e.

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These analytical expressions show that R_h = 1 when θ = mπ, and R_v = 1 if θ = (n + 1/2)π. Here, m and n are integers. Fig. 3 shows R_h and R_v as a function of θ for the resonant photon. It exhibits that the polarization of the reflected resonant photon can be controlled by the distance between the two atoms.

Fig. 3 The reflected probabilities of the single photon with h -polarization and v-polarization as a function of θ. In the calculations, Δ = 0, G = G_c = 10⁻⁵ω_e.

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To investigate how to convert the polarization of the single photon from linearity to circularity, we rewrite the wavefunction of the reflected photon as |φ〉_r = rh|h〉 + r_v|v〉. From Eqs. (14, 15), one can obtain that when $θ = (k + 1 / 4) π,$ ${| φ 〉}_{r} = (- 1 - i) | σ_{-} 〉 / \sqrt{2}$ where $| σ_{-} 〉 \equiv (| h 〉 - i | v 〉) / \sqrt{2}$ . The h–polarized photon is converted into left circularly polarized photon. However, when θ is chosen as (k + 3/4)π, the wavefunction of the reflected photon is ${| φ 〉}_{r} = (- 1 + i) | σ_{+} 〉 / \sqrt{2}$ , where $| σ_{+} 〉 \equiv (| h 〉 + i | v 〉) / \sqrt{2}$ . The h–polarized photon is converted into right circularly polarized photon. Thus, one can also convert the linear polarized photon into circularly polarized photon by controlling the phase θ.

We now show the influences of G_c and G on the single photon polarization conversion. To calculate the fidelity of polarization conversion, the wavefunction of the photon after scattering is rewritten as $| ξ 〉 = t_{h} {| h 〉}_{R} + t_{v} {| v 〉}_{R} + r_{h} {| h 〉}_{L} + r_{v} {| v 〉}_{L}$ , where |p〉_d denotes the photon propagation in d-direction with p-polarization. The fidelity of the polarization conversion is defined as $F_{p} \equiv {| {〈 ξ | ξ 〉}_{d e s} |}^{2}$ , where |ξ〉_des is the target state, which is given as |h〉_L, |v〉_L, |φ〉_l and |φ〉_r for the reflected photon with h−, v−, left circularly, and right circularly polarization photon, respectively. Eqs. (10) to (13) show that F_p is independent of G_c and G when Δ = 0. However, G and G_c can affect F_p strongly when Δ ≠ 0. Fig. 4 shows F_p as a function of Δ and G_c/G. It exhibits that F_p increases with increasing of the ratio of G_c/G when Δ deviates from the ideal value of zero.

Fig. 4 F_h (a), F_v (b), F_r (c) and F_l (d) as a function of Δ and G_c/G. In (a), (b), (c) and (d), θ is taken as π, 0.5π, 3π/4 and π/4, respectively. G = 10⁻⁵ω_e in all the calculations.

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Finally, we show the influences of dissipations on the resonant single photon polarization conversion. This can be done by replacing ω_e_,_h_,_v by ω_e_,_h_,_v − iγ_e_,_h_,_v/2 in Eqs. (10) to (13). Fig. 5 exhibits F_h, F_v, F_l and F_r as a function of θ and γ_e = γ_h = γ_v = γ for the resonant photon. It shows that F_h and F_v can still reach about 0.9 when γ reaches 0.1G. However, F_l and F_r are more sensitive to γ. Furthermore, F_p depends strongly on θ, thus, one should control the distance between the two atoms finely to obtain high fidelity.

Fig. 5 F_h (a), F_v (b), F_l (c) and F_r (d) as a function of θ and γ/G. In the calculations, G = G_c = 10⁻⁵ω_e, Δ = 0.

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4. Discussions and summary

The resonant single photon polarization conversion proposed in this paper depends on the initial state of the first atom. When the initial state of the first atom is |g_v〉, the photon with h–polarization will not interact with the first atom. Thus, it can freely pass through the first atom and interacts with the second atom. It will be reflected back via scattering by the second atom if the photon is resonant with the atom. However, the polarization of the photon always keeps h – polarization. Thus, the polarization conversion can also be controlled by the initial state of the first atom, which may play important roles in generation of entanglement and quantum information processing.

In summary, we obtain the analytical expressions of the scattering properties of a single photon scattered by a pair of atoms. We take the incident photon with h–polarization for example. The calculated results show that the polarization of a resonant reflected photon is converted to v–polarization when θ = (n + 1/2)π. While when θ = m π, the polarization of the resonant reflected photon keeps the same as the incident photon. The analytical results also show that the polarization of the reflected photon becomes left (right) circularly polarized when θ = (k + 1/4)π (θ = (k + 3/4)π). Furthermore, one can also control whether to realize the single photon polarization conversion by controlling the initial state of the first atom. Our results are useful in designing all-optical quantum devices at single-photon level and quantum information processing.

Funding

Natural Science Foundation of Anhui Province (1608085MA09); National Natural Science Foundation of China (NSFC) (11774262, 11474003).

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Single photon polarization conversion via scattering by a pair of atoms

Abstract

1. Introduction

2. Theoretical model

3. Single photon polarization conversion

4. Discussions and summary

Funding

References

Cited By

Figures (5)

Equations (15)

Optics Express