Coherent frequency down-conversions and entanglement generation in a Sagnac interferometer

Yunning Lu; Shaoyan Gao; Aiping Fang; Pengbo Li; Fuli Li; M. Suhail Zubairy

doi:10.1364/OE.25.016151

1. Introduction

Quantum entanglement is a crucial resource for quantum information technology such as quantum communication, quantum computation, and quantum network [1]. Entanglement between qubits may be generated through their direct interactions (e.g., Coulomb interaction [2] and electric dipole-dipole interaction [3]), or photon-mediated interactions. For photon-mediated interactions, in cavity quantum electrodynamics (QED) and circuit-QED systems, entangled states of either cavity modes or atoms can be prepared based on either unitary dynamical evolution [4] or dissipative quantum dynamical process [5].

In the last decade, waveguide-QED systems have been demonstrated as a new platform to provide strong photon-matter coupling [6–32]. Compared to cavity-based systems, waveguide-QED systems support a continuum of electromagnetic field modes, and provide the input and output channels for photons in quantum networks. Besides, waveguide-QED systems provide strong and long-range photon-mediated interactions between emitters, and are suitable for studying quantum multi-body processes, such as superradiant and subradiant emission. Experimentally, several kinds of waveguide-QED systems have been proposed, such as quantum dots coupled with metallic nanowires [7–9], superconducting qubits coupled with one-dimensional (1D) open transmission lines [10–13], trapped atoms coupled with photonic crystal waveguides [14–16], and trapped cold atoms coupled with 1D nanofibers [17, 18]. Many applications of the waveguide-QED systems have been proposed, such as single-photon frequency shifter [19], switches [22,23], routers [24–26], diodes [27], transistors, single-photon frequency comb generator [28], preparing nonclassical electromagnetic field [29], and preparing entangled states [30–32].

Recently, several works have been reported on the generation of entangled states of emitters in waveguide-QED systems, mainly including generating transient and constant entanglement in spontaneous decay processes [33–38], steady entanglement by dissipation-driven dynamics [34–36,39–43], transient entanglement by effective unitary dynamics [44–47], and transient entanglement in photons scattering processes [48–51].

In [33–36,39], the authors considered to generate transient and steady entanglement between two two-level emitters in waveguide QED. First, transient entanglement can be obtained during the spontaneous decay of a special initial state. However, the concurrence cannot exceed 0.5, because the overlap between the target state and the initial state is 50%. Then, steady entanglement can be obtained by driving the emitters with constant coherent fields. Also, the concurrence cannot exceed 0.5. In order to obtain higher degree of entanglement, chiral waveguides are used to couple the emitters [37,40–42]. It is shown that in a spontaneous decay process, the maximum value of the concurrence reaches 0.73 [37]. In dissipation-driven process, with some detuning patterns on the lasers driving, the steady entanglement can be obtained with the concurrence closed to 1 [40–42]. In [44–47], the authors proposed schemes to prepare multi-qubit transient entanglement using effective unitary evolution within decoherence free subspaces (DFSs). These schemes have high fidelities and high success probabilities.

However, the above schemes may not be very convenient for scalable application in quantum network, because they require either preparing special initial states or using complex external driving fields. For the convenience in scalable implementation of quantum network, the authors of [48] considered to use only guided photon scattering processes to entangle two qubits which are coupled to a 1D line waveguide. The obtained maximal concurrence is about 0.3. In [49, 50], it is shown that in a chiral-waveguide-QED system, a larger concurrence of two-qubit entanglement (about 0.75) and multi-qubit entanglement can be obtained. The reason is that chiral waveguides can decrease the probability that the input photon is reflected back directly by the first qubit, and help the input photon propagate to the qubits behind and interact with them. However, in all the above schemes [48–50], the obtained concurrence values are less than 1, because when the photons are input from one side of a line waveguide, and the emitters are distinguishable to the input photons.

In recent publications, Sagnac interferometer was employed to realize single-photon frequency conversion [19–21, 52] and single-photon routing [24]. In those schemes, a Λ-type three-level emitter is coupled to a Sagnac interferometer. When coupling strengths of the two transition paths of the emitter to the Sagnac interferometer are equal, an incident photon can be completely absorbed from one transition path, due to quantum destructive interference between the re-emitted photon mode and the directly transmitted photon mode, and then a photon with a new frequency can be emitted through the other transition path and the frequency conversion can be realized. In addition, some other systems with geometry similar to those schemes are used as narrow band photon sources [53–55]. In the present investigation, we consider a system which consists of a Sagnac interferometer with a waveguide loop coupled to two external one-dimensional line waveguides via a 50/50 beam splitter and two Λ-type three-level quantum emitters located to the Sagnac loop. When a single photon in either of the right- or left-propagating mode of the line waveguides comes into the Sagnac interferometer, it is always in a superposition of the clockwise and counterclockwise modes of the waveguide loop. Since the two emitters are symmetrically located with respect to the incoming port of photon, the processes of photon scattering from the two emitters are symmetric and coherent. When the separation of the emitters and the coupling strengths of the emitters with the waveguide loop take some special values, we find that due to quantum interference a frequency down-conversion at one of the emitters can definitely happen during the photon scattering but one cannot know at which emitter the frequency down-conversion takes place. This indistinguishability of the coherent frequency down-conversion processes can lead to the generation of either the symmetric or antisymmetric two-qubit maximally entangled states of the emitters. In the present scheme, a single photon comes in and goes out of the waveguide loop, and no photon localization modes exists. In contrast to the previous schemes in which the entanglement results from photon exchange between the two-level emitters and depends on the localized photon mode, the present one has a different mechanism for the generation of entangled states. The quantum interference-induced frequency down-conversion processes in the Λ-type three-level emitters results in the entangled states. Thus, the resulting entangled states are stable if the two lower-lying states of the emitters have no decay. We believe that the present scheme opens a new way to generate steady-state strong entanglement and is suitable for the scalable implementation of quantum information processing with a single photon.

This paper is organized as follows. In section 2, we introduce our model and Hamiltonian of the system. In section 3, we derive the scattering eigenstates in the even-mode and the odd-mode subspaces. In section 4, we calculate the output states using Lippmann-Schwinger equations, and discuss under what condition, the target entangled states can be generated. In section 5, we discuss the influence of the finite bandwidth of photon wavepacket and the dissipation of the emitters in excited states on the success probability of generation of the target states. In section 6, the conclusion is presented.

2. Model

As shown in Fig. 1, we employ a Sagnac interferometer to couple two emitters [19, 24, 52]. The Sagnac interferometer is a waveguide loop coupled to two external line waveguides by a 50/50 beam splitter. A phase shifter is added in the waveguide loop. The beam splitter has four ports c, e, f, and g. The ports f and g are connected to the waveguide loop, and the ports c and e are connected to the external waveguides. On the left side of the system, a routing element is used to distinguish the input and output paths. It is assumed that the waveguide loop can support a continuum of photonic modes, and has an approximated linear dispersion relation ω = ck around the resonance transition frequencies of the emitters. The two identical quantum emitters A and B are coupled to the waveguide loop, and placed at symmetrical positions x_A = −d/2 and x_B = d/2 with respect to the point o.

Fig. 1 Schematic of the system. Two emitters A and B are coupled to a Sagmac interferometer. An external routing element is used to distinguish the input and output photons.

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The energy levels of the emitters are shown in Fig. 2. Each of the emitters has two metastable lower-lying states |1_α〉, |2_α〉, and an excited state |3_α〉 (α = A, B), with energies ħω₁, ħω₂, and ħω₃, respectively. The two metastable lower-lying states of each emitter are used to encode a qubit. The two transitions of the emitter A (B) are coupled to the waveguide modes with coupling strengths $V_{1}^{A (B)}$ , $V_{2}^{A (B)}$ , respectively.

Fig. 2 Energy levels of the Λ-type quantum emitters A and B. The coupling strengths corresponding to the transition paths are $V_{1}^{A}$ , $V_{2}^{A}$ , $V_{1}^{B}$ , and $V_{2}^{B}$ , respectively.

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The Sagnac interferometer is used to transform a photon unidirectionally propagating in the line waveguide into a symmetric or antisymmetric superposition of the clockwise (cw) and counterclockwise (ccw) modes in the Sagnac loop. In the following, we express the function of the Sagnac interferometer mathematically [52]. In the external line waveguides, we denote the photonic states propagating through the port e (including the right- and left-propagating photonic states) by the vector |ψ_e〉 = (1, 0)^T. Likewise, we denote the photonic states propagating through the port c by the vector |ψ_c〉 = (0, 1)^T. In the loop, we denote the cw and ccw propagating photonic states by the vectors |ψ_cw〉 = (1, 0)^T and |ψ_ccw〉 = (0, 1)^T, respectively. When a photon enters the loop from the external line waveguide, the function of the 50/50 beam splitter is described by the matrix

S_{bs - in} = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}) .

Without the emitters, the waveguide loop itself does not change the states of the cw and the ccw propagating photon, so the function of the waveguide loop is described by the matrix

S_{loop} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) .

When a photon leaves the waveguide loop and goes back into the external line waveguide, the function of the 50/50 beam splitter is described by the matrix

S_{bs - out} = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ - 1 & 1 \end{matrix}) .

With these notations, the output photonic state is related to the input photonic state by the unitary transformation |ψ_out〉 = S_bs−outS_loopS_bs−in |ψ_in〉.

It is clear that an input photon in the right (left)-propagating mode |ψ_R〉 (|ψ_L〉) in the external line waveguides will be transformed into a symmetric (antisymmetric) superposition of the cw and ccw modes in the waveguide loop $| ψ_{sym} 〉 = (| ψ_{cw} 〉 + | ψ_{ccw} 〉) / \sqrt{2}$ [or $| ψ_{anti} 〉 = (| ψ_{cw} 〉 - | ψ_{ccw} 〉) / \sqrt{2}$ ]. Conversely, an output photon in the symmetric (antisymmetric) superposition of the cw and ccw modes of the waveguide loop will be transformed into the right (left)-propagating mode in the external line waveguides. The phase shifter can change the relative phase between the cw and ccw modes in the Sagnac loop. When the two emitters are coupled to the waveguide loop, the effect of scattering by the emitters must be included into the matrix S_loop.

In the following sections of the paper, we will show that with such symmetric or antisymmetric photonic superposition states in the waveguide loop, the emitters can be controllably prepared into the entangled states with success probability 100%, in the ideal situation.

The Hamiltonian of the system in real space can be written in the form [19]

H = H_{wg} + H_{em} + H_{int} .

Here the free Hamiltonian of photons is given by

H_{wg} = \int d x c_{cw}^{†} (x) (- i ℏ c \frac{\partial}{\partial x}) c_{cw} (x) + \int d x c_{ccw}^{†} (x) (i ℏ c \frac{\partial}{\partial x}) c_{ccw} (x),

where

c_{cw}^{†} (x)

, c_cw(x),

c_{ccw}^{†} (x)

, and c_ccw(x) are creation and annihilation operators for photons in the cw and ccw modes at position x, respectively, and c is the group velocity of photons in the waveguide loop. The free Hamiltonian of the emitters is given by

H_{em} = \sum_{α = A, B} ℏ [ω_{1} σ_{11}^{α} + ω_{2} σ_{22}^{α} + (ω_{3} - i γ / 2) σ_{33}^{α}],

where

σ_{i i}^{α} = | i_{α} 〉 〈 i_{α} |

is the number operator for the state |i_α〉 (i = 1, 2, 3), and γ is the decay rate of the excited state |3_α〉 out of the waveguide modes. Here, the lower states |1_α〉 and |2_α〉 are assumed to be stable. The interaction Hamiltonian between the photons and the emitters is given by

\begin{array}{l} H_{int} & = \int d x ℏ δ (x - x_{A}) V_{1}^{A} {[c_{cw}^{†} (x) + c_{ccw}^{†} (x)] σ_{13}^{A} + σ_{31}^{A} [c_{cw} (x) + c_{ccw} (x)]} \\ + \int d x ℏ δ (x - x_{A}) V_{2}^{A} {[c_{cw}^{†} (x) + c_{ccw}^{†} (x)] σ_{23}^{A} + σ_{32}^{A} [c_{cw} (x) + c_{ccw} (x)]} \\ + \int d x ℏ δ (x - x_{B}) V_{1}^{B} {[c_{cw}^{†} (x) + c_{ccw}^{†} (x)] σ_{13}^{B} + σ_{31}^{B} [c_{cw} (x) + c_{ccw} (x)]} \\ + \int d x ℏ δ (x - x_{B}) V_{2}^{B} {[c_{cw}^{†} (x) + c_{ccw}^{†} (x)] σ_{23}^{B} + σ_{32}^{B} [c_{cw} (x) + c_{ccw} (x)]}, \end{array}

where

σ_{i j}^{α} = | i_{α} 〉 〈 j_{α} |

is the transition operators between the states |i_α〉 and |j_α〉.

3. Scattering eigenstates

Considering the symmetry of the Sagnac interferometer with respect to the spatial inversion, we introduce the even and odd modes of photons

c_{e} (x) = \frac{1}{\sqrt{2}} [c_{cw} (x) + c_{ccw} (- x)],

c_{o} (x) = \frac{1}{\sqrt{2}} [c_{cw} (x) - c_{ccw} (- x)] .

Suppose that the two emitters are initially in the states |1_A〉 and |1_B〉, and are fixed at the symmetric positions in the waveguide loop with respect to the beam splitter. When a single photon with a frequency that is identical to the resonance transition frequency between the states |1〉 and |3〉 is injected into the system, the emitters evolve into a state space which is expanded by the following state-vectors

| g 〉 = | 1_{A}, 1_{B} 〉,

| ϕ_{31 s} 〉 = \frac{1}{\sqrt{2}} (| 3_{A}, 1_{B} 〉 + | 1_{A}, 3_{B} 〉),

| ϕ_{31 a} 〉 = \frac{1}{\sqrt{2}} (| 3_{A}, 1_{B} 〉 - | 1_{A}, 3_{B} 〉),

| ϕ_{21 s} 〉 = \frac{1}{\sqrt{2}} (| 2_{A}, 1_{B} 〉 + | 1_{A}, 2_{B} 〉),

| ϕ_{21 a} 〉 = \frac{1}{\sqrt{2}} (| 2_{A}, 1_{B} 〉 - | 1_{A}, 2_{B} 〉) .

In terms of these new basis vectors, the Hamiltonian (4) can be rewritten as

H = H_{wg}^{e} + H_{wg}^{o} + H_{em} + H_{int},

with

H_{wg}^{e} = \int d x c_{e}^{†} (x) (- i c \frac{\partial}{\partial x}) c_{e} (x),

H_{wg}^{o} = \int d x c_{o}^{†} (x) (- i c \frac{\partial}{\partial x}) c_{o} (x),

\begin{array}{l} H_{em} & = & 2 ω_{1} | g 〉 〈 g | \\ + & (ω_{1} + ω_{2}) | ϕ_{21 s} 〉 〈 ϕ_{21 s} | \\ + & (ω_{1} + ω_{2}) | ϕ_{21 a} 〉 〈 ϕ_{21 a} | \\ + & [ω_{1} + (ω_{3} - i γ / 2)] | ϕ_{31 s} 〉 〈 ϕ_{31 s} | \\ + & [ω_{1} + (ω_{3} - i γ / 2)] | ϕ_{31 a} 〉 〈 ϕ_{31 a} |, \end{array}

and

\begin{array}{l} H_{int} & = & V_{1} \int d x [δ (x + d / 2) + δ (x - d / 2)] {c_{e}^{†} (x) σ_{13}^{s} + σ_{31}^{s} c_{e} (x)} \\ + & V_{1} \int d x [δ (x + d / 2) - δ (x - d / 2)] {c_{o}^{†} (x) σ_{13}^{a} + σ_{31}^{a} c_{o} (x)} \\ + & \frac{V_{2}}{\sqrt{2}} \int d x [δ (x + d / 2) + δ (x - d / 2)] {c_{e}^{†} (x) σ_{23}^{s} + σ_{32}^{s} c_{e} (x)} \\ + & \frac{V_{2}}{\sqrt{2}} \int d x [δ (x + d / 2) - δ (x - d / 2)] {c_{o}^{†} (x) σ_{23}^{a} + σ_{32}^{a} c_{o} (x)} . \end{array}

Here, we have set the coupling strengths

V_{1}^{A} = V_{1}^{B} = V_{1}

and

V_{2}^{A} = V_{2}^{B} = V_{2}

, and introduced a new set of operators as follows

\begin{array}{l} σ_{31}^{s} & = & \frac{1}{\sqrt{2}} (σ_{31}^{A} + σ_{31}^{B}) \\ = & | ϕ_{31 s} 〉 〈 g |, \end{array}

\begin{array}{l} σ_{31}^{a} & = & \frac{1}{\sqrt{2}} (σ_{31}^{A} - σ_{31}^{B}) \\ = & | ϕ_{31 a} 〉 〈 g |, \end{array}

\begin{array}{l} σ_{23}^{s} & = & σ_{23}^{A} + σ_{23}^{B} \\ = & | ϕ_{21 s} 〉 〈 ϕ_{31 s} | + | ϕ_{21 a} 〉 〈 ϕ_{31 a} |, \end{array}

\begin{array}{l} σ_{23}^{a} & = & σ_{23}^{A} - σ_{23}^{B} \\ = & | ϕ_{21 s} 〉 〈 ϕ_{31 s} | + | ϕ_{21 a} 〉 〈 ϕ_{31 a} |, \end{array}

and

σ_{i j}^{β} = σ_{j i}^{β †}

(β = s, a; i, j = 1, 2, 3).

Suppose that an input photon is in the right-propagating mode in the line waveguide and enters the Saganc loop from the port e. If the phase shifter in the waveguide loop gives a relative phase θ = 0 between the cw and ccw propagating modes, the photon will be in the even mode (8) before interacting with the emitters. We assume the emitters are initially in the ground state |g〉 (10). According to the interaction (19), the emitters can absorb the photon, and be excited to the state |ϕ_31s〉. Then, the emitters may decay through the following three possible paths: (1) the emitters may decay back into the ground state |g〉 and emit a photon in the even mode with the initial wave vector k; (2) the emitters may decay to the symmetric state |ϕ_21s〉, and emit a photon in the even mode with the wave vector k₂ (k₂ = k − ω₂/c + ω₁/c); (3) the emitters may decay to the antisymmetric state |ϕ_21a〉, and emit a photon in the odd mode with the wave vector k₂. Therefore, the scattering eigenstate can be written in the form

\begin{array}{l} | Ψ_{k}^{e} 〉 & = & \int d x f_{11}^{e} (k, x) c_{e}^{†} (x) | \emptyset, g 〉 \\ + & f_{31 s} | \emptyset, ϕ_{31 s} 〉 \\ + & \int d x f_{21 s}^{e} (k_{2}, x) c_{e}^{†} (x) | \emptyset, ϕ_{21 s} 〉 \\ + & \int d x f_{21 a}^{o} (k_{2}, x) c_{o}^{†} (x) | \emptyset, ϕ_{21 a} 〉, \end{array}

where |∅〉 is the vacuum state of photons and the amplitudes

f_{11}^{e} (k, x)

, f_31s,

f_{21 s}^{e} (k_{2}, x)

, and

f_{21 a}^{o} (k_{2}, x)

are to be determined. We call this subspace the even-mode subspace.

Similarly, if the phase shifter sets a relative phase θ = π between the cw and ccw propagating modes, the photon will be in the odd mode before interacting with the emitters. By use of the same analysis as above, the scattering eigenstate can be written in the form

\begin{array}{l} | Ψ_{k}^{o} 〉 & = & \int d x f_{11}^{o} (k, x) c_{o}^{†} (x) | \emptyset, g 〉 \\ + & f_{31 a} | \emptyset, ϕ_{31 a} 〉 \\ + & \int d x f_{21 s}^{o} (k_{2}, x) c_{o}^{†} (x) | \emptyset, ϕ_{21 s} 〉 \\ + & \int d x f_{21 a}^{e} (k_{2}, x) c_{e}^{†} (x) | \emptyset, ϕ_{21 a} 〉, \end{array}

where the amplitudes

f_{11}^{o} (k, x)

, f_31a,

f_{21 s}^{o} (k_{2}, x)

, and

f_{21 a}^{e} (k_{2}, x)

are to be determined. We call this subspace the odd-mode subspace.

In the following subsections, we will determine the amplitudes introduced in Eqs. (24) and (25), derive the input and output states, and find out the conditions for the emitters to evolve into entangled states.

3.1. The even-mode subspace

Upon substituting Eq. (24) into the Schrödinger equation $H | Ψ_{k}^{e} 〉 = c k | Ψ_{k}^{e} 〉$ , we obtain Eqs. (59)–(62) for the scattering amplitudes. We consider the initial situation where a single-photon plane wave with wave vector k propagates from left to right along the line waveguide and enters into the loop via the beam splitter, and the two emitters are in the ground state |g >. Thus we make the following ansatz on the amplitudes

f_{11}^{e} (k, x) = \frac{1}{\sqrt{2 π}} e^{i k x} [θ (- d / 2 - x) + t_{11 A B}^{e} θ (x + d / 2) θ (d / 2 - x) + t_{11}^{e} θ (x - d / 2)],

f_{21 s}^{e} (k_{2}, x) = \frac{1}{\sqrt{2 π}} e^{i k_{2} x} [t_{21 sAB}^{e} θ (x + d / 2) θ (d / 2 - x) + t_{21 s}^{e} θ (x - d / 2)],

f_{21 a}^{o} (k_{2}, x) = \frac{1}{\sqrt{2 π}} e^{i k_{2} x} [t_{21 aAB}^{o} θ (x + d / 2) θ (d / 2 - x) + t_{21 a}^{o} θ (x - d / 2)] .

These ansatz have a clear physics meaning. The amplitude

f_{11}^{e} (k, x)

describes that the input photon is elastically scattered, with the emitters in ground state |g〉. The elastically scattered photon has the initial wave vector k and is in the even mode. Note that both the incident and the scattered photon are coherent superposition of the cw and ccw modes. The coefficients

t_{11 AB}^{e}

and

t_{11}^{e}

can be understood as the transmission coefficients of the scattered photon in the regions between and out of the two emitters, respectively. The amplitude

f_{21 s}^{e} (k_{2}, x)

describes the case where the incident photon is inelastically scattered into even mode with the wave vector k₂, and the emitters evolve into the state |ϕ_21s〉. The amplitude

f_{21 a}^{o} (k_{2}, x)

describes the case where the photon is inelastically scattered into odd mode with the wave vector k₂, and the emitters evolve into the state |ϕ_21a〉. For these inelastic scattering situations,

t_{21 sAB}^{e}

,

t_{21 aAB}^{e}

and

t_{21 s}^{e}

,

t_{21 a}^{e}

can be understood as the transmission coefficients of the scattered photon in the regions between and out of the two emitters, respectively.

Inserting the ansatz (26)–(28) into Eqs. (59)–(62), we obtain Eqs. (63)–(69). Solving these equations, we obtain the coefficients f_31s, $t_{11 AB}^{e}$ , $t_{11}^{e}$ , $t_{21 sAB}^{e}$ , $t_{21 s}^{e}$ , $t_{21 aAB}^{o}$ , and $t_{21 a}^{o}$ as follows:

f_{31 s} = \frac{- 2 / \sqrt{2 π} icos (k d / 2) V_{1}}{(e^{i k d} + 1) Γ_{1} + Γ_{2} - i (Δ + i γ / 2)},

t_{11 AB}^{e} = \frac{Γ_{2} - i (Δ + i γ / 2)}{(e^{i k d} + 1) Γ_{1} + Γ_{2} - i (Δ + i γ / 2)},

t_{11}^{e} = \frac{- (e^{- i k d} + 1) Γ_{1} + Γ_{2} - i (Δ + i γ / 2)}{(e^{i k d} + 1) Γ_{1} + Γ_{2} - i (Δ + i γ / 2)},

t_{21 sAB}^{e} = \frac{- \sqrt{2} cos (k d / 2) e^{i k_{2} d / 2} \sqrt{Γ_{1} Γ_{2}}}{(e^{i k d} + 1) Γ_{1} + Γ_{2} - i (Δ + i γ / 2)},

t_{21 s}^{e} = \frac{- 2 \sqrt{2} cos (k d / 2) cos (k_{2} d / 2) \sqrt{Γ_{1} Γ_{2}}}{(e^{i k d} + 1) Γ_{1} + Γ_{2} - i (Δ + i γ / 2)},

t_{21 aAB}^{o} = \frac{- \sqrt{2} cos (k d / 2) e^{i k_{2} d / 2} \sqrt{Γ_{1} Γ_{2}}}{(e^{i k d} + 1) Γ_{1} + Γ_{2} - i (Δ + i γ / 2)},

t_{21 a}^{o} = \frac{- 2 \sqrt{2} icos (k d / 2) sin (k_{2} d / 2) \sqrt{Γ_{1} Γ_{2}}}{(e^{i k d} + 1) Γ_{1} + Γ_{2} - i (Δ + i γ / 2)} .

Here,

Γ_{1} = V_{1}^{2} / c

and

Γ_{2} = V_{2}^{2} / c

are the spontaneous emission rates of the emitters through the transition paths |3〉 → |1〉 and |3〉 → |2〉 to the waveguide modes, respectively. Δ = ck − (ω₃ − ω₁) is the detuning. When the system has no loss (i.e., γ = 0), it can be verified that

{| t_{11}^{e} |}^{2} + {| t_{21 s}^{e} |}^{2} + {| t_{21 a}^{o} |}^{2} = 1

. This probability conservation reflects the fact that there is no localized photon modes left in the system after the scattering. Therefore, the input photon certainly exits out of the waveguide loop.

3.2. Odd-mode subspace

Since the methods used in the odd-mode subspace are the same as that in the even-mode subspace, we here only give the main results. Making the following ansatz for the amplitudes in the state (25)

f_{11}^{o} (k, x) = \frac{1}{\sqrt{2 π}} e^{i k x} [θ (- d / 2 - x) + t_{11 AB}^{o} θ (x + d / 2) θ (d / 2 - x) + t_{11}^{o} θ (x - d / 2)],

f_{21 s}^{o} (k_{2}, x) = \frac{1}{\sqrt{2 π}} e^{i k_{2} x} [t_{21 sAB}^{o} θ (x + d / 2) θ (d / 2 - x) + t_{21 s}^{o} θ (x - d / 2)],

f_{21 a}^{e} (k_{2}, x) = \frac{1}{\sqrt{2 π}} e^{i k_{2} x} [t_{21 aAB}^{e} θ (x + d / 2) θ (d / 2 - x) + t_{21 a}^{e} θ (x - d / 2)],

and following the same approach as above, we can find out the coefficients appearing in equations (36)–(38). These coefficients are listed as follows:

f_{31 a} = \frac{- 2 \sqrt{2 π} sin (k d / 2) V_{1}}{(1 - e^{i k d}) Γ_{1} + Γ_{2} - i (Δ + i γ / 2)},

t_{11 AB}^{o} = \frac{Γ_{2} - i (Δ + i γ / 2)}{(1 - e^{i k d}) Γ_{1} + Γ_{2} - i (Δ + i γ / 2)},

t_{11}^{o} = \frac{- (1 - e^{- i k d}) Γ_{1} + Γ_{2} - i (Δ + i γ / 2)}{(1 - e^{i k d}) Γ_{1} + Γ_{2} - i (Δ + i γ / 2)},

t_{21 sAB}^{o} = \frac{\sqrt{2} i sin (k d / 2) e^{i k_{2} d / 2} \sqrt{Γ_{1} Γ_{2}}}{(1 - e^{i k d}) Γ_{1} + Γ_{2} - i (Δ + i γ / 2)},

t_{21 s}^{o} = \frac{- 2 \sqrt{2} sin (k d / 2) sin (k_{2} d / 2) \sqrt{Γ_{1} Γ_{2}}}{(1 - e^{i k d}) Γ_{1} + Γ_{2} - i (Δ + i γ / 2)},

t_{21 aAB}^{e} = \frac{\sqrt{2} i sin (k d / 2) e^{i k_{2} d / 2} \sqrt{Γ_{1} Γ_{2}}}{(1 - e^{i k d}) Γ_{1} + Γ_{2} - i (Δ + i γ / 2)},

t_{21 a}^{e} = \frac{2 \sqrt{2} i sin (k d / 2) cos (k_{2} d / 2) \sqrt{Γ_{1} Γ_{2}}}{(1 - e^{i k d}) Γ_{1} + Γ_{2} - i (Δ + i γ / 2)} .

It can be verified that when the system has no losses (i.e., γ = 0), the probability conservation condition

{| t_{11}^{o} |}^{2} + {| t_{21 e}^{o} |}^{2} + + {| t_{21 o}^{e} |}^{2} = 1

is satisfied.

4. Generation of the entangled states

In this section, we derive the output states from the scattering eigenstates (24) and (25) using the Lippmann-Schwinger equations. Then, we discuss the conditions under which the entangled emitters states |ϕ_21s〉 and |ϕ_21a〉 can be controllably generated.

4.1. Even-mode subspace

In the even-mode subspace, the input state and the output state are related to the scattering eigenstate (24) using Lippmann-Schwinger Eqs. [56,57]

| Ψ_{k}^{e} 〉 = {| Ψ_{k}^{e} 〉}_{in} + \frac{1}{E - H_{0} + i ∊} H_{int} | Ψ_{k}^{e} 〉,

| Ψ_{k}^{e} 〉 = {| Ψ_{k}^{e} 〉}_{out} + \frac{1}{E - H_{0} - i ∊} H_{int} | Ψ_{k}^{e} 〉 .

Here,

1 / (E - H_{0} + i ∊) \equiv G_{0}^{R}

is the free retarded Green’s function, and

1 / (E - H_{0} - i ∊) \equiv G_{0}^{A}

is the free advanced Green’s function. H₀ = H_wg + H_em is the free Hamiltonian. Upon inserting the state (24) into Eqs. (46) and (47), using the ansatz (26)–(28), we can extract the input state

{| Ψ_{k}^{e} 〉}_{in} = \int d x \frac{e^{i k x}}{\sqrt{2 π}} c_{e}^{†} (x) | \emptyset, g 〉,

and the output state

\begin{array}{l} {| Ψ_{k}^{e} 〉}_{out} & = & \int d x t_{11}^{e} \frac{e^{i k x}}{\sqrt{2 π}} c_{e}^{†} (x) | \emptyset, g 〉 \\ + & \int d x t_{21 s}^{e} \frac{e^{i k_{2} x}}{\sqrt{2 π}} c_{e}^{†} (x) | \emptyset, ϕ_{21 s} 〉 \\ + & \int d x t_{21 a}^{o} \frac{e^{i k_{2} x}}{\sqrt{2 π}} c_{o}^{†} (x) | \emptyset, ϕ_{21 a} 〉 . \end{array}

The derivation of Eqs. (48) and (49) is given in 7.2. Eq. (48) presents the photon-emitter initial state where a photon is in an even-mode plane wave state and the emitters are in the ground state. This state is just the state which we initially considered. After the input photon is scattered by the emitters, according to the interaction (19), the scattered photon may be in either the even-mode or the odd-mode state, and the emitters may correspongdingly be in either the state |ϕ_21s〉 or |ϕ_21a〉. Eq. (49) presents the required output scattered state. In addition, the probability conservation is hold because

{| t_{11}^{e} |}^{2} + {| t_{21 s}^{e} |}^{2} + {| t_{21 a}^{o} |}^{2} = 1

when γ = 0. Thus, based on the the ansatz (26)–(28), one can extract the correct forms of the input and output states from Lippmann-Schwinger equations. These points confirm that the ansatz (26)–(28) are correct.

In the following, we discuss the conditions when the entangled emitters states |ϕ_21s〉 and |ϕ_21a〉 can be generated in a controlled manner, using the expression of the output state (49). The output state (49) is a superposition of three states, with coefficients $t_{11}^{e}$ , $t_{21 s}^{e}$ , and $t_{21 a}^{o}$ , respectively. First, to make sure there is no initial photon mode in the output state, we require $t_{11}^{e} = 0$ . From Eq. (31), when Δ = 0 and γ = 0, we see that this can be realized by setting Γ₁/Γ₂ = 1/2 and kd = 2mπ. Second, to make sure that only the symmetric entangled state |ϕ_21s〉 is generated, we require $t_{21 s}^{e} = 1$ and $t_{21 a}^{o} = 0$ . From Eqs. (33) and (35), when Δ = 0 and γ = 0, these requirements can be satisfied by setting k₂d = 2nπ. Here, m and n are positive integers.

In the ideal situation with Δ = 0 and γ = 0, the success probability is 100%. However, in practical systems, some factors may decrease the success probability from 100%. For example, the photon wavepacket with finite width must contain off-resonant components (i.e., Δ ≠ 0), and the photon may be lost by the spontaneous decay of the emitters in excited states (i.e., γ ≠ 0). In our scheme, we herald the successful generation of the entangled states of the emitters by detecting the output photon at the exit ports. From the analysis in the above paragraph, we know that together with the generation of the symmetric entangled state of the emitters, an even-mode photon is generated. The photon passes the beam splitter, exits into the line waveguide through the port e, and arrives at the port b. When the detector at port b clicks, we know the entangled state |ϕ_21s〉 is generated successfully. In Section 5, we calculate the success probability with realistic parameters.

In order to show the generation of the target state |ϕ_21s〉 more intuitively, the evolution of the system is shown in Fig. 3. This state evolution can be physically understood as follows. In the first step, the transition $\int d x e^{i k x} / \sqrt{2 π} c_{e}^{†} (x) | \emptyset, g 〉 \to | \emptyset, ϕ_{31 s} 〉$ happens determinately, i.e., the input photon is absorbed completely, and the emitters are excited into the state |ϕ_31s〉. This is because the conditions Γ₁/Γ₂ = 1/2, kd = 2mπ lead to complete destructive interference between the initial incident photon and the absorbed-reemitted photon in the transition $\int d x e^{i k x} / \sqrt{2 π} c_{e}^{†} (x) | \emptyset, g 〉 \leftrightarrow | \emptyset, ϕ_{31 s} 〉$ . In the second step, the transition $| \emptyset, ϕ_{31 s} 〉 \to \int d x e^{i k_{2} x} / \sqrt{2 π} c_{e}^{†} (x) | \emptyset, ϕ_{21 s} 〉$ takes place, but the transition $| \emptyset, ϕ_{31 s} 〉 \to \int d x e^{i k_{2} x} / \sqrt{2 π} c_{o}^{†} (x) | \emptyset, ϕ_{21 a} 〉$ cannot happen. The above process of entangled-state generation can also be understood by means of subradiant / superradiant states. At first, as shown in Fig. 3, the incident photon is absorbed, and the emitters are excited into the state |ϕ_31s〉. Then, there are two possible decay channels for the emitters. In the first channel, the emitters emit an even-mode photon, and decay into the state |ϕ_21s〉. In the second channel, the emitters emit an odd-mode photon, and decay into the state |ϕ_21a〉. With the given parameters in Fig. 3, the first channel is superradiant because the emitted even-mode photon interferes constructively in the region out of the two emitters. The second channel is subradiant because the emitted odd-mode photon interferes destructively in the region out of the two emitters and then the photon emission to the odd mode is forbidden. In this way, after scattering, the photon exits out of the loop and the emitters decay to the state |ϕ_21s〉.

Fig. 3 Generation road of the state |ϕ_21s〉 in the even subspace. With the conditions shown in the figure, the system evolves from initial state, through state |ϕ_31s〉, to the target state, in which the emitters are in symmetric state |ϕ_21s〉.

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Similarly, if we want to generate the antisymmetric entangled state |ϕ_21a〉, the output state (49) shows that the conditions $t_{11}^{e} = 0$ , $t_{21 s}^{e} = 0$ , and $t_{21 a}^{o} = 1$ are required. From Eqs. (31), (33) and (35), when Δ = 0 and γ = 0, the conditions can be satisfied by setting Γ₁/Γ₂ = 1/2, kd = 2mπ, and k₂d = (2n + 1)π, where m and n are positive integers. The state evolution is shown in Fig. 4. After the scattering process, the emitters evolve into the entangled state |ϕ_21a〉, and the photon in the odd mode exits from the port c. When the detector at the port c clicks, we know the entangled state |ϕ_21a〉 is generated successfully.

Fig. 4 Generation road of the state |ϕ_21a〉 in the even subspace. With the conditions shown in the figure, the system evolves from initial state, through state |ϕ_31s〉, to the target state, in which the emitters are in antisymmetric state |ϕ_21a〉.

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4.2. Odd-mode subspace

We use the same method as in the even-mode subspace to derive the output state in the odd-mode subspace. Inserting the state (25) into the Lippmann-Schwinger equations

| Ψ_{k}^{o} 〉 = {| Ψ_{k}^{o} 〉}_{in} + \frac{1}{E - H_{0} + i ∊} H_{int} | Ψ_{k}^{o} 〉,

| Ψ_{k}^{o} 〉 = {| Ψ_{k}^{o} 〉}_{out} + \frac{1}{E - H_{0} - i ∊} H_{int} | Ψ_{k}^{o} 〉,

using the ansatz Eqs. (36)–(38), we can extract the input state

{| Ψ_{k}^{o} 〉}_{in} = \int d x \frac{e^{i k x}}{\sqrt{2 π}} c_{o}^{†} (x) | \emptyset, g 〉,

and the output state

\begin{array}{l} {| Ψ_{k}^{o} 〉}_{out} & = & \int d x t_{11}^{o} \frac{e^{i k x}}{\sqrt{2 π}} c_{o}^{†} (x) | \emptyset, g 〉 \\ + & \int d x t_{21 s}^{o} \frac{e^{i k_{2} x}}{\sqrt{2 π}} c_{o}^{†} (x) | \emptyset, ϕ_{21 s} 〉 \\ + & \int d x t_{21 a}^{e} \frac{e^{i k_{2} x}}{\sqrt{2 π}} c_{e}^{†} (x) | \emptyset, ϕ_{21 a} 〉 . \end{array}

If conditions Γ₁/Γ₂ = 1/2, kd = (2m + 1)π, and k₂d = (2n + 1)π in Eqs. (41), (43) and (45) are satisfied, where m and n are positive integers, we have $t_{11}^{o} = 0$ , $t_{21 s}^{o} = 1$ , and $t_{21 a}^{e} = 0$ , for the ideal situation with Δ = 0 and γ = 0. In this case, the evolution of the system follows the road shown in Fig. 5. After the interaction between the photon and the emitters, an odd-mode photon is generated. When the photon propagating in clockwise passes the phase shifter with θ = π, as shown in Fig. 1, it will acquire a phase shift π. In this way, the odd-mode photon becomes an even-mode photon. Then the photon passes the beam splitter, and exits from the port b. When the detector at the port b clicks, we know that the emitters have been prepared into the entangled state |ϕ_21s〉.

Fig. 5 The evolution to the state |ϕ_21s〉 in the odd subspace. With the conditions shown in the figure, the system evolves from initial state, through state |ϕ_31a〉, to the target state, in which the emitters are in symmetric state |ϕ_21s〉.

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If conditions Γ₁/Γ₂ = 1/2, kd = (2m + 1)π, and k₂d = 2nπ in Eqs. (41), (43) and (45) are satisfied, where m and n are positive integers, we have $t_{11}^{o} = 0$ , $t_{21 s}^{o} = 0$ , and $t_{21 a}^{e} = 1$ , for the ideal situation with Δ = 0 and γ = 0. In this case, the evolution of the system is shown in Fig. 6. Although the emitted photon is in an even mode, as discussed above, it will be transferred into an odd mode before exiting out of the loop because of the phase shifter with θ = π. As a result, the photon will arrive at the port c. When the detector at the port c clicks, we know that the emitters have been prepared into the entangled state |ϕ_21a〉.

Fig. 6 The evolution to the state |ϕ_21a〉 in the odd subspace. With the conditions shown in the figure, the system evolves from initial state, through state |ϕ_31a〉, to the target state, in which the emitters are in symmetric state |ϕ_21a〉.

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5. Effects of finite-bandwidth wavepacket and dissipation

In practice, an input photon must be a pulse with finite bandwidth, and the emitters must have intrinsic dissipation. The off-resonant components in a pulse and the dissipation of the emitters can do harm to the generation of the entangled states. We now discuss the influence of these factors on the success probability of obtaining the entangled states.

As an example, we consider the generation of the symmetric entangled state $| ϕ_{21 s} 〉 = 1 / \sqrt{2} (| 2_{A}, 1_{B} 〉 + | 1_{A}, 2_{B} 〉)$ in the even-mode subspace, which corresponds to the situation in Fig. 3. In our numerical calculation, we take some typical parameters [12, 13] as follows: ω₁₃ = ck₁₃ = 2π × 8 GHz, ω₂₃ = ck₂₃ = 2π × 4 GHz, d = 0.075 m, Γ₁ = 2π × 20 MHz, Γ₂ = 2π × 40 MHz, which are feasible in current experiments for the system of two three-level superconduction artificial atoms coupled with a 1D open transmission line. With the above parameters, the conditions Γ₁/Γ₂ = 1/2, k₁₃d = 4π, k₂₃d = 2π are satisfied, as is required in Fig. 3.

Without loss of generality, we consider a Gaussian-type input wave packet with intensity FWHM τ and central frequency resonant to the transition |1〉 → |3〉,

ϕ_{in} (x, t) = {(\frac{4 ln 2}{π c^{2} τ^{2}})}^{1 / 4} exp {- {[\sqrt{2 ln 2} (c t - x) / (c τ)]}^{2}} exp {- i k_{13} (c t - x)} .

Its spectral amplitude ϕ̃_in(k) is given by

{\tilde{ϕ}}_{in} (k) = {(\frac{c^{2} τ^{2}}{4 π ln 2})}^{1 / 4} exp {- {[c τ / (2 \sqrt{2 ln 2})]}^{2} {(k - k_{13})}^{2}} .

In previous publications, entanglement is measured by concurrence which is defined by $C = \sqrt{{(ρ_{+ +} - ρ_{- -})}^{2} + 4 Im {(ρ_{+ -})}^{2}}$ . Here, ρ₊₊ and ρ₋₋ are the probabilities that the system is in the symmetrical entangled state |+〉 of two qubits and the antisymmetrical entangled state |−〉, respectively, and ρ₊₋ represents the coherence between the states |+〉 and |−〉. Because the generated states are a superposition of the target state and other states, the concurrence is less than one. From discusses in above sections, we see that the emitters will be in either the symmetrical entangled state or antisymmetrical entangled state once the photon is detected. It means that our scheme always holds concurrence 1.0 if a photon is detected. However, the success probability in our scheme may not be 100%, due to some non-ideal factor such as dissipation of the emitters and the finite bandwidth of the input photon wavepacket. Instead of studying concurrence, therefore, we study the success probability.

The probability for the emitters to be prepared into the state |ϕ_21s〉 is given by

\begin{array}{l} p & = & \int d k {| t_{21 s}^{e} (k) {\tilde{ϕ}}_{in} (k) |}^{2} \\ \approx & \sqrt{\frac{π}{ln 2}} \frac{4 α Γ τ}{{(α + 1)}^{2} (\frac{2 α + 1}{α + 1} + \frac{1}{2 P})} exp [\frac{1}{4 ln 2} {(\frac{2 α + 1}{α + 1} + \frac{1}{2 P})}^{2} {(Γ τ)}^{2}] \\ \times erfc [\frac{1}{\sqrt{4 ln 2}} (\frac{2 α + 1}{α + 1} + \frac{1}{2 P}) Γ τ], \end{array}

where δ = Δ/c = (ω −ω₁₃)/c = (k − k₁₃)/c, α = Γ₁/Γ₂, Γ = Γ₁ + Γ₂, Purcell factor P = Γ/γ, and

erfc (z) = 2 / \sqrt{π} \int_{z}^{+ \infty} exp (- t^{2}) d t

is the complementary error function. Obviously, the success probability p is determined by parameters α, Γτ, and P.

In Fig. 7, we consider an ideal case where the spontaneous emission rate γ = 0, and show the success probability p as a function of the intensity FWHM τ (rescaled with Γ⁻¹) with different values of α. First, we notice that the probability p increases as the intensity FWHM τ increases for all the curves. The physical reason for this is clear. When the intensity FWHM τ is small, the non-resonant components of the wave packet prevents the system from evolving into the target state completely, and makes the system leave in the initial state with certain probability, which can be seen in Eqs. (31), (33) and (35). When the intensity FWHM τ becomes larger and larger, the wavepacket is closer and closer to a monochromatic plane wave and only the resonant component exists. Second, it can be observed that only for the curve with α = 0.5, the probability p can approach unity with increasing the FWHM τ. The reason is that only with the value α = 0.5, the condition $t_{11}^{e} = 0$ can be satisfied, as shown in Eq. (31). Otherwise, $t_{11}^{e} \neq 0$ , and the initial input photon mode cannot be absorbed completely after the scattering. Therefore, α = 0.5 is a necessary condition for preparing the maximally entangled state.

Fig. 7 Success probability p as a function of the parameter Γτ, for different values of α (α = Γ₁/Γ₂) with γ = 0.

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In Fig. 8, we set the intensity FWHM τ = 8 Γ⁻¹, and show the success probability p as a function of the Purcell factor P with different values of α. We find that the probability p increases as the Purcell factor P increases for all the curves. When the Purcell factor P is small, photons emitted from the spontaneous decay of the excited states of the emitters escape mostly into free space instead of coming back to the waveguide, and the probability is small. When the Purcell factor P is large, the coupling between the emitters and the waveguide modes is strong and the photons after interacting with the emitters are mostly back to the waveguide mode, and thus the probability p is large. We also find that the probability p changes slightly with the change of P in the range of value P = 10 ∼ 100. It means that the dissipation of the emitters do not affect the success probability seriously.

Fig. 8 Success probability p as a function of the Purcell factor P, for different values of α. The Intensity FWHM of the input photon is τ = 8 Γ⁻¹.

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In order to show the relation between the success probability p and the Purcell factor P analytically and intuitively, we derive a simplified analytical expression of p in the narrow-bandwidth limit. When the input wave packet has a very narrow bandwidth, and has a central frequency resonant with the emitters transition |1〉 → |3〉, the wave packet can approximately be seen as a plane wave with resonant frequency ω₁₃ = ck₁₃. In this situation, we have

\begin{array}{l} p & \approx & {| t_{21 s}^{e} (k_{13}) |}^{2} \\ = & \frac{8 α}{{(α + 1)}^{2}} {(\frac{2 α + 1}{α + 1} + \frac{1}{2 P})}^{- 2} . \end{array}

When we set α = 0.5, we find that p increases from 0 to 1 with P increasing from 0 to +∞. In waveguide QED systems, the Purcell factor usually has a large value P ≫ 1. In this situation, p can be further simplified as

p \approx 1 - \frac{3}{4 P} .

We find that the success probability is very closed to 1 with a large P.

6. Conclusion

A waveguide loop coupled to two external line waveguides by a 50/50 beam splitter forms a Sagnac interferometer. We consider the situation where two Λ-type three-level emitters are symmetrically coupled to the loop of a Sagnac interferometer and a single photon is input through one end of the line waveguides. Since the incoming photon is always in a superposition of the clockwise and counterclockwise modes of the loop and the two emitters are positioned symmetrically with respect to the input port of photon, the processes of photon scattering at the two emitters are symmetric and coherent. When the separation of the emitters and the coupling strengths of the emitters with the waveguide loop take some special values, we find that due to quantum interference the incoming photon absorption and the frequency down-conversion at the two emitters can certainly and coherently take place during the photon scattering but one cannot know at which emitter the frequency down-conversion happen. This indistinguishability of the coherent frequency down-conversion processes can lead to the generation of either the symmetric or antisymmetric two-qubit maximally entangled states of the emitters. In the present scheme, a single photon comes in and goes out of the waveguide loop, and no photon localization modes exists. The entangled states result from the coherent frequency down-conversion processes of the emitters. Thus, the resulting entangled states are stable if the two lower-lying states of the emitters have no decay. If the decay of the emitters to other modes except the loop modes can be neglected and the input photon wavepacket is a plane wave, the success probability for the generation of the entangled states is 100%. We also investigate the influence of the dissipation of the emitters and the finite bandwidth of an input photon wavepacket on the success probability of entanglement generation. We find that even if photon loss due to the decay of the emitters and the off-resonance components in the input photon wavepacket are taken into account, the success generation of the entangled states can still be heralded by detecting the output photon at the exit ports. Moreover, we show that the success probability can be maintained over 95% with the parameters available in current experiments. Thus the present scheme is robust to these inherent imperfect factors. We believe that the present scheme opens a new way for the generation of long-lived strong entanglement and is suitable for the scalable implementation of quantum information processing with single photons.

7. Appendix

7.1. Equations for the scattering amplification in the even-mode subspace

Upon substituting Eq. (24) into the Schrödinger equation $H | Ψ_{k}^{e} 〉 = c k | Ψ_{k}^{e} 〉$ , we can obtain the following equations for the scattering amplitudes

\begin{array}{l} - i c \frac{\partial}{\partial x} f_{11}^{e} (k, x) + 2 ω_{1} f_{11}^{e} (k, x) + V_{1} [δ (x + d / 2) + δ (x - d / 2)] f_{31 s} \\ = & (2 ω_{1} + c k) f_{11}^{e} (k, x), \end{array}

\begin{array}{l} (ω_{1} + ω_{3} - i γ / 2) f_{31 s} + V_{1} [f_{11}^{e} (k, - d / 2) + f_{11}^{e} (k, d / 2)] \\ + & \frac{1}{\sqrt{2}} V_{2} [f_{21 s}^{e} (k_{2}, - d / 2) + f_{21 s}^{e} (k_{2}, d / 2)] + \frac{1}{\sqrt{2}} V_{2} [f_{21 a}^{o} (k_{2}, - d / 2) - f_{21 a}^{o} (k_{2}, d / 2)] \\ = & (2 ω_{1} + c k) f_{31 s}, \end{array}

\begin{array}{l} - i c \frac{\partial}{\partial x} f_{21 s}^{e} (k_{2}, x) + (ω_{1} + ω_{2}) f_{21 s}^{e} (k_{2}, x) + \frac{1}{\sqrt{2}} V_{2} [δ (x + d / 2) + δ (x - d / 2)] f_{31 s} \\ = & (2 ω_{1} + c k) f_{21 s}^{e} (k_{2}, x), \end{array}

\begin{array}{l} - i c \frac{\partial}{\partial x} f_{21 a}^{o} (k_{2}, x) + (ω_{1} + ω_{2}) f_{21 a}^{o} (k_{2}, x) + \frac{1}{\sqrt{2}} V_{2} [δ (x + d / 2) - δ (x - d / 2)] f_{31 s} \\ = & (2 ω_{1} + c k) f_{21 a}^{o} (k_{2}, x) . \end{array}

Inserting the ansatz Eqs. (26)–(28) into Eqs. (59)–(62), we have the following equations for the coefficients $t_{11 AB}^{e}$ , $t_{11}^{e}$ , $t_{21 sAB}^{e}$ , $t_{21 s}^{e}$ , $t_{21 aAB}^{o}$ , $t_{21 a}^{o}$ , and f_31s,

- \frac{i c}{\sqrt{2 π}} e^{- i k d / 2} (- 1 / t_{11 AB}^{e}) + V_{1} f_{31 s} = 0,

- \frac{i c}{\sqrt{2 π}} e^{i k d / 2} (- t_{11 AB}^{e} + t_{11}^{e}) + V_{1} f_{31 s} = 0,

\begin{array}{l} \frac{V_{1}}{\sqrt{2 π}} e^{- i k d / 2} \frac{1 + t_{11 AB}^{e}}{2} + \frac{V_{1}}{\sqrt{2 π}} e^{i k d / 2} \frac{t_{11 AB}^{e} + t_{11}^{e}}{2} \\ + & \frac{V_{2}}{\sqrt{4 π}} e^{- i k_{2} d / 2} \frac{t_{21 sAB}^{e}}{2} + \frac{V_{2}}{\sqrt{4 π}} e^{i k_{2} d / 2} \frac{t_{21 s AB}^{e} + t_{21 s}^{e}}{2} \\ + & \frac{V_{2}}{\sqrt{4 π}} e^{- i k_{2} d / 2} \frac{t_{21 aAB}^{o}}{2} - \frac{V_{2}}{\sqrt{4 π}} e^{i k_{2} d / 2} \frac{t_{21 a AB}^{o} + t_{21 a}^{o}}{2} \\ = & (Δ + i \frac{γ}{2}) f_{31 s}, \end{array}

- \frac{i c}{\sqrt{2 π}} e^{- i k_{2} d / 2} t_{21 sAB}^{e} + \frac{V_{2}}{\sqrt{2}} f_{31 s} = 0,

\frac{i c}{\sqrt{2 π}} e^{i k_{2} d / 2} (t_{21 sAB}^{e} - t_{21 s}^{e}) + \frac{V_{2}}{\sqrt{2}} f_{31 s} = 0,

- \frac{i c}{\sqrt{2 π}} e^{- i k_{2} d / 2} t_{21 aAB}^{o} + \frac{V_{2}}{\sqrt{2}} f_{31 s} = 0,

\frac{i c}{\sqrt{2 π}} e^{i k_{2} d / 2} (t_{21 aAB}^{o} - t_{21 a}^{o}) - \frac{V_{2}}{\sqrt{2}} f_{31 s} = 0 .

Solving the above equations, we obtain the coefficients

t_{11 AB}^{e}

,

t_{11}^{e}

,

t_{21 sAB}^{e}

,

t_{21 s}^{e}

,

t_{21 aAB}^{o}

,

t_{21 a}^{o}

, and f_31s, as listed in Eqs. (29)–(35).

7.2. The input and output states in the even-mode subspace

Multiplying 〈∅, g|c_e(x) on the left of Eq. (46), we obtain

〈 \emptyset, g | c_{e} (x) | Ψ_{k}^{e} 〉 = 〈 \emptyset, g | c_{e} (x) {| Ψ_{k}^{e} 〉}_{in} + 〈 \emptyset, g | c_{e} (x) \frac{1}{E - H_{0} + i ∊} H_{int} | Ψ_{k}^{e} 〉 .

We can calculate

\begin{array}{l} 〈 \emptyset, g | c_{e} (x) | Ψ_{k}^{e} 〉 \\ = & f_{11}^{e} (k, x) \\ = & \frac{e^{i k x}}{\sqrt{2 π}} θ (- x - d / 2) + t_{11 AB}^{e} \frac{e^{i k x}}{\sqrt{2 π}} θ (x + d / 2) θ (- x + d / 2) + t_{11}^{e} \frac{e^{i k x}}{\sqrt{2 π}} θ (x - d / 2), \end{array}

\begin{array}{l} 〈 \emptyset, g | c_{e} (x) \frac{1}{E - H_{0} + i ∊} | Ψ_{k}^{e} 〉 \\ = & \frac{e^{i k x}}{\sqrt{2 π}} θ (- x - d / 2) + t_{11 AB}^{e} \frac{e^{i k x}}{\sqrt{2 π}} θ (x + d / 2) θ (- x + d / 2) + t_{11}^{e} \frac{e^{i k x}}{\sqrt{2 π}} θ (x - d / 2), \end{array}

Then we obtain

\begin{array}{l} 〈 \emptyset, g | c_{e} (x) {| Ψ_{k}^{e} 〉}_{in} \\ = & 〈 \emptyset, g | c_{e} (x) | Ψ_{k}^{e} 〉 - 〈 \emptyset, g | c_{e} (x) \frac{1}{E - H_{0} + i ∊} H_{int} | Ψ_{k}^{e} 〉 \\ = & \frac{e^{i k x}}{\sqrt{2 π}} . \end{array}

Likewise, we can derive the output state,

\begin{array}{l} 〈 \emptyset, g | c_{e} (x) {| Ψ_{k}^{e} 〉}_{out} \\ = & 〈 \emptyset, g | c_{e} (x) | Ψ_{k}^{e} 〉 - 〈 \emptyset, g | c_{e} (x) \frac{1}{E - H_{0} + i ∊} H_{int} | Ψ_{k}^{e} 〉 \\ = & t_{11}^{e} \frac{e^{i k x}}{\sqrt{2 π}}, \end{array}

\begin{array}{l} 〈 \emptyset, ϕ_{21 s} | c_{e} (x) {| Ψ_{k}^{e} 〉}_{out} \\ = & 〈 \emptyset, ϕ_{21 s} | c_{e} (x) | Ψ_{k}^{e} 〉 - 〈 \emptyset, ϕ_{21 s} | c_{e} (x) \frac{1}{E - H_{0} - i ∊} H_{int} | Ψ_{k}^{e} 〉 \\ = & t_{21 s}^{e} \frac{e^{i k_{2} x}}{\sqrt{2 π}}, \end{array}

\begin{array}{l} 〈 \emptyset, ϕ_{21 a} | c_{o} (x) {| Ψ_{k}^{e} 〉}_{out} \\ = & 〈 \emptyset, ϕ_{21 a} | c_{o} (x) | Ψ_{k}^{e} 〉 - 〈 \emptyset, ϕ_{21 a} | c_{o} (x) \frac{1}{E - H_{0} - i ∊} H_{int} | Ψ_{k}^{e} 〉 \\ = & t_{21 a}^{o} \frac{e^{i k_{2} x}}{\sqrt{2 π}} . \end{array}

From Eqs. (73)–(76), we can obtain the input state (48) and the output state (49).

Funding

National Natural Science Foundation of China (NSFC) (Nos.11534008, 11374239, 91536115, and 11604257); Ministry of Science and Technology of the People’s Republic of China (MOST) (2016YFA0301404); Natural Science Foundation of Shaanxi Province (No. 2016JM1005); NPRP Grant No. 8-352-1-074 from the Qatar National Research Fund.

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Coherent frequency down-conversions and entanglement generation in a Sagnac interferometer

Abstract

1. Introduction

2. Model

3. Scattering eigenstates

3.1. The even-mode subspace

3.2. Odd-mode subspace

4. Generation of the entangled states

4.1. Even-mode subspace

4.2. Odd-mode subspace

5. Effects of finite-bandwidth wavepacket and dissipation

6. Conclusion

7. Appendix

7.1. Equations for the scattering amplification in the even-mode subspace

7.2. The input and output states in the even-mode subspace

Funding

References and links

Cited By

Figures (8)

Equations (76)

Optics Express