Exact optimal control of photon blockade with weakly nonlinear coupled cavities

H. Z. Shen; Y. H. Zhou; H. D. Liu; G. C. Wang; X. X. Yi

doi:10.1364/OE.23.032835

1. Introduction

The generation of nonclassical states of photons [1, 2] represents a hot research topic in integrated quantum technologies and quantum optics and is vital for furture applications. In particular, the single-photon source plays an important role in quantum information technologies, and makes their production becoming an ultimate goal of defining a photonic-based architecture in quantum communication [3], quantum metrology [4], and quantum information technologies [5, 6]. The physics behind this mechanism is photon blockade effect, which has been demonstrated in an optical cavity coupled to a quantum emitter [7–10] and coupled single quantum dot-cavity system [11]. Subsequently, a sequence of experimental groups observed the strong antibunching behaviors in different systems, such as a quantum dot in a photonic crystal [12], circuit cavity quantum electrodynamics systems [13, 14].

It is well known that a coherent light beam enters a nonlinear system whose effective nonlinear response is able to produce a shift of the two-photon resonance that is larger than its linewidth [15], which is named as conventional photon blockade. Typical examples include quantum optomechanical systems [16–26] and cavity quantum electrodynamics [27–36]. A sequence of schemes have been proposed in nanostructured cavities and semiconductor microcavities with second-order nonlinear [37, 38]. Such photon blockade has been applied to transistors [39], interferometers [40], and optical rectifiers [41, 42], to mention a few.

The unconventional photon blockade effect was first predicted [43] in photon molecule system [44, 45], which shows that the presence of two photons in two coupled cavities with weak nonlinearities is prevented by destructive quantum interference between different paths in the nonlinear coupled cavity system [46–52]. Similar effects recently were considered in various models, such as coupled single-mode cavities with second- or third-order nonlinearity [53–58], double asymmetric cavities [59], Gaussian squeezed states [60], optical cavity with a quantum dot [61, 62], coupled optomechanical systems [63, 64], and bimodal coupled polaritonic cavities [65, 66], and so on.

Here, we show that an unconventional photon blockade can be engineered in two coupled cavities filled with nonlinear Kerr mediums, where two cavities are driven and dissipated. The approximate optimal conditions for the detuning Δ and Kerr nonlinearity U have been derived when J ≫ κ and φ = 0 in [47]. Here we consider a more general scenario, in which we remove these two constraints, and derive the exact optimal conditions in order to generate strong antibunching photons in the system for all the regime of parameters. There are a total of eight sets of solutions for the optimal detuning since the discriminant changes its sign as the strength ratio and the relative phase vary between two complex driving fields, which can be controlled precisely [67]. Hence exact optimal conditions give a fully description of photon blockade, and holds for all range of parameters. On the contrary, if there are no exact optimal conditions, it is very difficult to experimentally find the all eight optimal regimes of photon blockade. Our results provide a promising platform for the coherent manipulation of photon blockade, which has potential applications for quantum information processing and quantum optical devices.

2. Physical model and optimal antibunching conditions

The system under consideration is composed of two tunnel-coupled cavities filled with a Kerrtype nonlinear material. We assume that each cavity to be single mode, described by Bose operator â and $\hat{b}$ , respectively. The nonlinear second-quantized Hamiltonian reads [43, 46, 68]

{\hat{H}}_{0} = \bar{h} ω_{a} {\hat{a}}^{†} \hat{a} + \bar{h} ω_{b} {\hat{b}}^{†} \hat{b} + \bar{h} J (\hat{a} {\hat{b}}^{†} + \hat{b} {\hat{a}}^{†}) + U_{a} {\hat{a}}^{†} {\hat{a}}^{†} \hat{a} \hat{a} + U_{b} {\hat{b}}^{†} {\hat{b}}^{†} \hat{b} \hat{b},

where J denotes the tunnel coupling rate, ω_j and U_j eigenfrequency and Kerr nonlinearity strength for cavity j (j = a,b), respectively. The setup is illustrated in Fig. 1(a), where the driving frequency is denoted by ω_L, and the complex driving strength denoted by

F_{j} e^{i φ_{j}}

. The total Hamiltonian including the driving terms is then,

{\hat{H}}_{d r} (t) = {\hat{H}}_{0} + (\bar{h} F_{a} {\hat{a}}^{†} e^{i φ_{a} - i ω_{L} t} + \bar{h} F_{b} {\hat{b}}^{†} e^{i φ_{b} - i ω_{L} t} + H . c .),

which can be rotated with respect to the external laser frequency ω_L

\begin{array}{l} \hat{H} = \bar{h} Δ_{a} {\hat{a}}^{†} \hat{a} + \bar{h} Δ_{b} {\hat{b}}^{†} \hat{b} + \bar{h} J (\hat{a} {\hat{b}}^{†} + {\hat{a}}^{†} \hat{b}) \\ + U_{a} {\hat{a}}^{†} {\hat{a}}^{†} \hat{a} \hat{a} + U_{b} {\hat{b}}^{†} {\hat{b}}^{†} \hat{b} \hat{b} \\ + (\bar{h} F_{a} e^{i φ_{a}} {\hat{a}}^{†} + \bar{h} F_{b} e^{i φ_{b}} {\hat{b}}^{†} + H . c .), \end{array}

where Δa = ωa −ω_L and Δb = ωb −ω_L denote the detunings of cavity a and b from two driving fields, respectively. Here we note that the relative phase φ = φ_a−φ_b plays an importance role, which can be found by

\hat{a} \to \hat{a} e^{i φ_{b}}

\hat{b} \to \hat{b} e^{i φ_{b}}

. This also can be seen from the derived optimal conditions (21)–(27). Considering the dissipations to the system, the density matrix ρ of the system is governed by the master equation,

\dot{ρ} = - i [\hat{H}, ρ] + κ_{a} D (\hat{a}) ρ + κ_{b} D (\hat{b}) ρ,

where Ĥ is given by Eq. (3), κ_a and κ_b denote loss rates for the two cavities, respectively. The superoperator is defined by

D (\hat{o}) ρ = \hat{o} ρ {\hat{o}}^{†} - \frac{1}{2} {\hat{o}}^{†} \hat{o} ρ - \frac{1}{2} ρ {\hat{o}}^{†} \hat{o}

. The equal-time second-order correlation function for mode a can be given by [69, 70],

g_{a}^{(2)} (0) = \frac{〈 {\hat{a}}^{†}^{^{2}} {\hat{a}}^{2} 〉}{{〈 {\hat{a}}^{†} \hat{a} 〉}^{2}} .

Fig. 1 (a) Scheme diagram of the setup. Two cavities are driven by external fields with the complex Rabi frequency $F_{a} e^{i φ_{a}}$ and $F_{b} e^{i φ_{b}}$ , respectively. (b) The equal-time second-order correlation function $g_{a}^{(2)} (0)$ is plotted by numerically solving the master Eq. (4). The nearly perfect antibunching can be obtained at U = 1.593 × 10⁻³κ with φ = 1.6 rad. The other parameters are chosen as Δ_a = Δ_b = 0.1539κ, J = 7κ, U_a = 0, and F_a = 0.7κ, F_b = 0.028κ, φ = 1.6rad for blue-bold line, φ = 1.8rad for dashed-green line, φ = 1.4rad for red-dashed-dotted line. (c) The first (second) number in |mn〉 denotes the photon number in cavity a (cavity b). The transition paths (two dashed lines with arrows) lead to the quantum interference (indicated by the ellipse) for the strong photon antibunching.

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We present the variation of $g_{a}^{(2)} (0)$ defined by Eq. (5) as a function of U and Δ by numerically solving master equation (4) in Figs. 1 and 2. First we discuss the features of Fig. 1(b). We find there are three interesting points at U = 1.593 × 10⁻³κ, 2.03 × 10⁻³κ, and 1.18 10⁻³κ marked respectively by A, B,C. Points A, B, and C correspond to different relative phases between two complex driving fields, which are denoted by φ = 1.6 rad, 1.8 rad, and 1.4 rad, respectively.

Fig. 2 Dependence of logarithmic plot for $g_{a}^{(2)} (0)$ on U (U ≡ U_b) and Δ by numerically solving master Eq. (4). Defining F_b ≡ rF_a. Parameters chosen are J = 7κ, F_a = 0.7κ, U_a = 0, and r = 0.02, φ = 2rad for (a), and r = 0.2, φ = 1rad for(b)–(d). We have an interesting observation that there are two and four minimum values for two-order correlation function for (a) [see points P₁ and P₂ denoted by blue-star] and (b)–(d) [see points P₃-P₆ denoted by blue-star], respectively. Therefore points P₁-P₄ correspond to unconventional single-photon blockade, which is smaller than the mode broadening κ, while points P₅ and P₆ indicate conventional single-photon blockade, which is larger than the mode broadening κ.

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Interestingly, we find that there are two minimum values (see points P₁-P) for $g_{a}^{(2)} (0)$ in Fig. 2(a) for given strength ratio r = F_b/F_a, decay rate κ_a = κ_b ≡ κ, and the relative phase φ. However, $g_{a}^{(2)} (0)$ appears four minimum values (see points P₃-P₆) in Figs. 2(b)–2(d) as r and φ vary. Therein points P₁-P₄ correspond to unconventional single-photon blockade, where Kerr nonlinearity U is smaller than the mode broadening κ, while points P₅ and P₆ indicate conventional single-photon blockade, where Kerr nonlinearity U is larger than the mode broadening κ. One may wonder why there are four points in Figs. 2(b)–2(d) appearing for single-photon blockade under the same parameters? In the following, we will answer this question.

Assume that the system is initially prepared in |00〉, and only these levels are occupied due to two weak complex driving fields. Therefore the total sate of the system can be written as [46, 71],

\begin{array}{l} | Ψ 〉 = A_{00} | 00 〉 + A_{10} | 10 〉 + A_{01} | 01 〉 \\ + A_{20} | 20 〉 + A_{02} | 02 〉 + A_{11} | 11 〉, \end{array}

describes the dynamics by evolving |Ψ〉 under the action of the non-Hermitian Hamiltonian

\tilde{H} = \tilde{H} - i [κ_{a} {\hat{a}}^{†} \hat{a} + κ_{b} {\hat{b}}^{†} \hat{b}] / 2

. Substituting Eq. (6) into the Schrödinger equation

i \frac{\partial}{\partial t} | Ψ 〉 = \tilde{H} | Ψ 〉

(

\bar{h} = 1

, hereafter), we arrive at,

i \frac{\partial}{\partial t} A_{00} = F_{a} e^{- i φ_{a}} A_{10} + F_{b} e^{- i φ_{b}} A_{01},

i \frac{\partial}{\partial t} A_{10} = δ_{a} A_{10} + J A_{01} + F_{a} e^{i φ_{a}} A_{00} + A_{b} e^{- i φ_{b}} A_{11} + \sqrt{2} F_{a} e^{- i φ_{a}} A_{20},

i \frac{\partial}{\partial t} A_{01} = δ_{b} A_{01} + J A_{10} + F_{b} e^{i φ_{b}} A_{00} + F_{a} e^{- i φ_{a}} A_{11} + \sqrt{2} F_{b} e^{- i φ_{b}} A_{02},

i \frac{\partial}{\partial t} A_{11} = (δ_{a} + δ_{b}) A_{11} + \sqrt{2} J (A_{20} + A_{02}) + F_{b} e^{i φ_{b}} A_{10} + F_{a} e^{i φ_{a}} A_{01},

i \frac{\partial}{\partial t} A_{20} = 2 (δ_{a} + U_{a}) A_{20} + \sqrt{2} J A_{11} + \sqrt{2} F_{a} e^{i φ_{a}} A_{10},

i \frac{\partial}{\partial t} A_{02} = 2 (δ_{b} + U_{b}) A_{02} + \sqrt{2} J A_{11} + \sqrt{2} F_{b} e^{i φ_{b}} A_{01},

where δ_j = Δ_j − iκ_j/2. Under the weak driving condition, we have |A₀₀| ≫ |A₁₀|, |A₀₁| ≫ |A₂₀|, |A₁₁|, |A₀₂|. In steady state,

\frac{\partial}{\partial t} A_{j k} = 0

, the relations between A₁₀ and A₀₁ read,

\begin{array}{l} δ_{a} A_{10} + J A_{01} = - F_{a} e^{i φ_{a}} A_{00}, \\ J A_{10} + δ_{b} A_{01} = - F_{b} e^{i φ_{b}} A_{00}, \end{array}

and two-photon probability amplitude reduces to

\begin{array}{l} 0 = 2 (δ_{a} + U_{a}) A_{20} + \sqrt{2} J A_{11} + \sqrt{2} F_{a} e^{i φ_{a}} A_{10}, \\ 0 = 2 (δ_{b} + U_{b}) A_{02} + \sqrt{2} J A_{11} + \sqrt{2} F_{b} e^{i φ_{b}} A_{01}, \\ 0 = (δ_{a} + δ_{b}) A_{11} + \sqrt{2} J (A_{20} + A_{02}) + F_{b} e^{i φ_{b}} A_{10} + F_{a} e^{i φ_{a}} A_{01} . \end{array}

From Eq. (13)A₁₀ and A₀₁ can be written as,

\frac{A_{01}}{A_{10}} = \frac{e^{i φ_{a}} F_{a} J - e^{i φ_{b}} F_{b} δ_{a}}{e^{i φ_{b}} F_{b} J - e^{i φ_{a}} F_{a} δ_{b}} \equiv ζ .

The equal-time second-order correlation function $g_{a}^{(2)} (0)$ can be given by substituting Eq. (6) into Eq. (5) in weak driving limit.

g_{a}^{(2)} (0) ≃ \frac{2 {| A_{20} |}^{2}}{{| A_{10} |}^{4}} .

The conditions for $g_{a}^{(2)} (0) ≪ 1$ are derived from Eq. (14) by setting A₂₀ = 0. Hereafter, we will set U_b ≡ U, U_a = 0 because U_a does not appear in the coupled Eq. (14) after taking A₂₀ = 0.

For simplicity, in the following we mainly discuss the case of Δ_a = Δ_b ≡ Δ, κ_a = κ_b ≡ κ, F_b/F_a = r, φ_a− φ_b = φ. Complementally, in Appendix A we discuss the general situation of Δ_a ≠ Δ_b and κ_a ≠ κ_b, and give the optimal conditions for strong photon antibunching.

Noticing Eq. (15), the conditions for $g_{a}^{(2)} (0) \to 0$ are derived from Eq. (14) by setting A₂₀ = 0, so we arrive at,

\begin{array}{l} 0 = \sqrt{2} J A_{11} + \sqrt{2} F_{a} e^{i φ_{a}} A_{10}, \\ 0 = 2 (δ_{b} + U_{b}) A_{02} + \sqrt{2} J A_{11} + \sqrt{2} F_{b} e^{i φ_{b}} ζ A_{10}, \\ 0 = (δ_{a} + δ_{b}) A_{11} + \sqrt{2} J A_{02} + F_{b} e^{i φ_{b}} A_{10} + F_{a} e^{i φ_{a}} ζ A_{10} . \end{array}

The condition for A₀₂, A₁₁, and A₁₀ to have nontrivial solutions is that the determinant of the coefficient matrix of Eq. (17) is equal to zero, then we obtain

0 = 4 i r^{2} J^{2} [κ + i (U + 2 Δ)] + e^{2 i φ} [{(κ + 2 i Δ)}^{2} (2 U + 2 Δ - i κ) - 4 J^{2} U] - 4 r e^{i φ} J (κ + 2 i Δ) [κ + 2 i (U + Δ)] .

After a simple calculation, we obtain the equation for the detuning Δ:

Δ^{4} + 4 b Δ^{3} + 6 c Δ^{2} + 4 d Δ + e = 0,

where these coefficients take b = rJ cos(φ)[rJ sin(φ) − 2κ]/2κ,

c = \frac{J^{2}}{12} (3 + 8 r^{2}) + \frac{κ^{2}}{12} + \frac{r J}{12 κ} [r J κ \cos (2 φ) - 4 (J^{2} + r^{2} J^{2} + κ^{2}) \sin (φ)]

, d = rJ cos(φ)[rJ(4J² + 5κ²) sin(φ) − 4(1 + r²)J²κ − 2κ³]/8κ,

e = \frac{1}{2} r^{4} J^{4} + \frac{1}{8} (8 r^{2} - 1) J^{2} κ^{2} + \frac{1}{16} κ^{4} + \frac{1}{8} r J [r J (4 J^{2} - 3 κ^{2}) \cos (2 φ) - 4 κ (3 J^{2} r^{2} - J^{2} + κ^{2}) \sin (φ)]

. The nature of roots for Eq. (19) is mainly determined by its discriminant. When the discriminant

δ = m^{3} - 27 n^{2} < 0

with m = e−4bd+3c² and n = (4h³−hm−g²), Eq. (19) under this condition has two real roots,

Δ_{o p t}^{(1), (2)} = [- b - sgn (g)] \sqrt{λ} \pm \sqrt{| g | / \sqrt{λ} - λ + 3 h},

then the Kerr nonlinearity for optimal conditions is,

U_{o p t} = \frac{4 r J v_{2} + 2 Δ (3 κ^{2} - 4 Δ^{2}) \sin (2 φ) + v_{3}}{4 \cos (φ) v_{1} - 8 Δ [κ \cos (2 φ) + 2 r J \sin (φ)]},

where

Δ \equiv Δ_{o p t}^{(j)}

, j = 1,2 in Eq. (21). Other parameters take v₁ = 2rJκ + (2J² − κ² + 4Δ²)sin(φ),v₂ = rJκ − κ² sin(φ) + 4Δ² sin(φ),v₃ = −16rJκΔcos(φ) − κ(κ² − 12Δ²)cos(2φ), h = b² –c,.g = d – 3bc + 2b³,

λ = [\sqrt[3]{- n + \sqrt{- δ / 27}} + \sqrt[3]{- n - \sqrt{- δ / 27}}] / 2 + h

. Set η = 12h² – m.

When

δ = m^{3} - 27 n^{2} > 0,

if and only if

\begin{array}{l} h > 0, \\ η > 0, \end{array}

are simultaneously satisfied, Eq. (19) has four real roots,

Δ_{o p t}^{(3), (4)} = - b + s \sqrt{γ_{1}} \pm \sqrt{γ_{2}} \pm \sqrt{γ_{3}},

Δ_{o p t}^{(5), (6)} = - b - s \sqrt{γ_{1}} \pm \sqrt{γ_{2}} \mp \sqrt{γ_{3}} .

otherwise, Eq. (19) has four conjugate complex roots.

With Eqs. (25) and (26), the optimal Kerr nonlinearity is expressed as,

U_{o p t} = \frac{4 r J v_{2} + 2 Δ (3 κ^{2} - 4 Δ^{2}) \sin (2 φ) + v_{3}}{4 \cos (φ) v_{1} - 8 Δ [κ \cos (2 φ) + 2 r J \sin (φ)]},

where

Δ \equiv Δ_{o p t}^{(k)}

, k = 3,4,5,6 in Eqs. (25) and (26). Other parameters take v₁ = 2rJκ + (2J² − κ² + 4Δ²)sin(φ), v₂ = rJκ − κ² sin(φ) + 4Δ² sin(φ),v₃ = −16rJκΔcos(φ) − κ(κ² − 12Δ²)cos(2φ),

β = a r c \cos [- n / \sqrt{| m |^{3} / 27}]

,

y_{1} = \sqrt{| m | / 3} \cos (β / 3) + h

,

y_{2, 3} = \sqrt{| m | / 3} \cos (β / 3 \pm 2 π / 3) + h

. Here s = −sgn(g) when all y_j are positive, otherwise s = sgn(g).

Here we address that when the optimal conditions (21) and (22) [or (25), (26) and (27)] are simultaneously satisfied, strong antibunching can be obtained, otherwise the system is not in strong antibunching regimes.

Now we begin to discuss the features of Figs. 1(b) and 2 applying the optimal conditions Eqs. (21), (25) and (26). In Fig. 1(b), the discriminant δ is less than zero, Δ = 0.1539κ or −0.1664κ by Eq. (21) when φ = 1.6 rad, which explains the point A in Fig. 1. Δ = −0.2134κ or 0.1161κ at φ = 1.8κ rad explains point B when δ < 0. The explanation for the point C is the same as point B.

In Fig. 2, we plot $g_{a}^{(2)} (0)$ as a function of U and Δ by numerically solving the master Eq. (4). We find that these optimal points P₁-P₆ are in good agreement with those obtained by the effective Hamiltonian method. For unconventional single-photon blockade regimes, see Eqs. (21) and (22) [i.e., for (a), point $P_{1} : (Δ_{o p t}^{(2)}, U_{o p t}) = (- 0.276 κ, - 0.00436 κ)$ , point $P_{2} : (Δ_{o p t}^{(1)}, U_{o p t}) = (0.1945 κ, 0.004453 κ)$ when δ < 0], Eqs. (25)–(27) [i.e., in (b), point $P_{3} : (Δ_{o p t}^{(6)}, U_{o p t}) = (- 0.1614 κ, 0.02705 κ)$ , point $P_{4} : (Δ_{o p t}^{(4)}, U_{o p t}) = (0.4439 κ, 0.01518 κ)$ when δ > 0]. For the conventional single-photon blockade, Δ_opt and Δ_opt take $(Δ_{o p t}^{(3)}, U_{o p t}) = (6.9165 κ, - 4.222 κ)$ (denoted by point P₅), and $(Δ_{o p t}^{(5)}, U_{o p t}) (- 6.81 κ, 4.78 κ)$ (denoted by point P₆), which are shown in the Figs. 2(c) and 2(d), respectively. Although the optimal values can be found by numerically solving the master equation, the analytical optimal values given by the effective Hamiltonian have the following advantages. Firstly, the analytical optimal U and Δ (21)–(27) are convenient to analyze the regime of antibunching, this would help in experiments to find the required parameters. Secondly, by the analytical expressions for U_opt and Δ_opt, we can find the roles the system parameters playing. Hence exact optimal conditions (21)–(27) give a fully detailed description of photon blockade, and hold for all range of parameters. On the contrary, without these exact optimal conditions, it is very difficult to experimentally find all of the optimal regimes of photon blockade.

Physically, in order to understand the conventional and unconventional single-photon blockade, we sketch the corresponding bare energy levels in Fig. 1(c). This requires that two-photon probability amplitude A₂₀ must be suppressed via destructive quantum interference paths for A₂₀ (i) $| 0, 0 〉 \overset{F_{a} e^{i φ_{a}}}{\to} | 1, 0 〉 \overset{\sqrt{2} F_{a} e^{i φ_{a}}}{\to} | 2, 0 〉$ , (ii) $| 0, 0 〉 \overset{F_{a} e^{i φ_{a}}}{\to} | 1, 0 〉 \overset{F_{b} e^{i φ_{b}}}{\to} | 1, 1 〉 \overset{\sqrt{2} J}{\to} | 2, 0 〉$ , (iii), $| 0, 0 〉 \overset{F_{b} e^{i φ_{b}}}{\to} | 0, 1 〉 \overset{F_{a} e^{i φ_{a}}}{\to} | 1, 1 〉 \overset{\sqrt{2} J}{\to} | 2, 0 〉$ , (iv), $| 0, 0 〉 \overset{F_{b} e^{i φ_{a}}}{\to} | 0, 1 〉 \overset{\sqrt{2} F_{b} e^{i φ_{b}}}{\to} | 0, 2 〉 \overset{\sqrt{2} J}{\to} | 1, 1 〉 \overset{\sqrt{2} J}{\to} | 2, 0 〉$ . Above four paths result in the destructive quantum interference to obtain the strong antibunching in the cavity mode a. So the nonlinear interaction strength U needed for destructive quantum interference becomes smaller and exactly controlled by tuning the strength ratio and relative phase between two complex driving fields.

3. Unconventional single-photon blockade

3.1. Influence of the relative phase φ on photon blockade

In Fig. 3, we show $g_{a}^{(2)} (0)$ as a function of U and φ by numerically solving master Eq. (4), where Δ is given by Eq. (21), φ is plotted in units of π. In this case, the discriminant δ = m³ − 27n² in Eq. (20) is always less than zero when phase φ varies. We set $Δ_{o p t} = Δ_{o p t}^{(1)}$ for (a), whereas $Δ_{o p t} = Δ_{o p t}^{(2)}$ for (b). The red-dashed lines in (a)–(b) are plotted with U given by Eq. (22). We notice that the unconventional single-photon blockade occurs in Figs. 3(a)–3(b) due to the quantum interference between different pathways shown in Fig. 1(c), where the Kerr nonlinear strength U is smaller than the mode broadening κ. From Fig. 3, we find the analytical results show an excellent agreement with the numerical simulation.

Fig. 3 Dependence of logarithmic plot of $g_{a}^{(2)} (0)$ on U and φ (in units of π). The detuning Δ is given by Eq. (21). In this case, the discriminant δ = m³ − 27n² in Eq. (20) is always less than zero when phase φ varies. We set $Δ_{o p t} = Δ_{o p t}^{(1)}$ for (a), whereas $Δ_{o p t} = Δ_{o p t}^{(2)}$ for (b). The red-dashed lines in (a)–(b) are plotted with U given by Eq. (22). Other parameters are J = 7κ, F_a = 0.3κ, r = 0.02.

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However, when the discriminant δ = m³ − 27n² in Eq. (20) changes its sign as the strength ratio increases. How does it influence the single-photon blockade? In Figs. 4 and 5, we plot $g_{a}^{(2)} (0)$ as a function of U and φ, where the detuning Δ is given by Eqs. (21), (25) and (26). Figs. 4 and 5(a)–5(d) correspond to different Δ_opt. The red-dashed lines in (a)–(d) are plotted with U given by Eqs. (22) and (27). From Eqs. (21), (25) and (26), we find that there are a total of eight sets of solutions for the optimal detuning when the discriminant δ changes its sign, i.e., $(Δ_{o p t}^{(j)}, Δ_{o p t}^{(k)})$ , j = 1,2 and k = 3,4,5,6, which can be seen from Figs. 4 and 5. We notice that the conventional and unconventional single-photon blockade occurs in (a)–(d), where the Kerr nonlinear strength U is smaller and larger than the mode broadening κ, respectively.

Fig. 4 Dependence of logarithmic plot of $g_{a}^{(2)} (0)$ on U and φ (in units of π), where the detuning Δ is given by Eqs. (21), (25) and (26). The discriminant δ = m³ − 27n² in Eq. (20) changes its sign when phase φ varies, this results in the changes of solutions to Eq. (19). Figure (a)–(d) are for different Δ_opt. (a) $Δ_{o p t} = Δ_{o p t}^{(1)}$ when δ < 0, whereas $Δ_{o p t} = Δ_{o p t}^{(3)}$ when δ > 0. The same for (b), (c), and (d). (b) $Δ_{o p t}^{(1)}$ and $Δ_{o p t}^{(4)}$ , (c) $Δ_{o p t}^{(1)}$ and $Δ_{o p t}^{(5)}$ , (d) $Δ_{o p t}^{(1)}$ and $Δ_{o p t}^{(6)}$ . The red-dashed lines in (a)–(d) are plotted with U given by Eqs. (22) and (27). Other parameters are J = 7κ, r = 0.5, and F_a = 0.2κ.

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Fig. 5 The parameters of this figure are the same as Fig. 4, but Δ_opt is different. (a) $Δ_{o p t}^{(2)}$ and $Δ_{o p t}^{(3)}$ , (b) $Δ_{o p t}^{(2)}$ and $Δ_{o p t}^{(4)}$ , (c) $Δ_{o p t}^{(2)}$ and $Δ_{o p t}^{(5)}$ , (d) $Δ_{o p t}^{(2)}$ and $Δ_{o p t}^{(6)}$ . The red-dashed lines in (a) and (b) are plotted with U given by Eqs. (22) and (27).

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From Fig. 4(a), we find that the optimal regimes of $g_{a}^{(2)} (0)$ are not continuous, which derives from non-continuity of Δ_opt, i.e., it may take $Δ_{o p t}^{(1)}$ or $Δ_{o p t}^{(3)}$ through the change of the sign of δ. Similar observations can be found in Figs. 4 and 5. We find that U_opt is a periodic function of φ with period 2π, which can be seen from the expression of U in Eqs. (22) and (27), where U depends on φ via sin(φ), cos(φ), sin(2φ), and cos(2φ).

3.2. Comparison between exact and approximate optimal conditions

We now analyze the characteristics of photon-blockade in the coupled cavity systems by comparing the exact optimal conditions with that from the approximate one reported in [47]. The approximate optimal conditions given by [47] are obtained with these two assumptions

φ = 0,

J ≫ κ,

in which the optimal conditions are given by

Δ_{o p t} \approx J r,

U_{o p t} \approx \frac{κ^{2}}{2 J} \frac{r}{1 - r^{2}} .

Here we have set r = 1/η for comparison with [47] due to the different ratio defined in two papers.

Now we discuss the situation with the relative phase φ = 0. In Appendix B we analytically prove that three inequalities in Eqs. (23) and (24) can not hold simultaneously for all parameters. Therefore we conclude that the exact optimal detuning Δ has only two real roots, which corresponds to two optimal detuning $Δ_{o p t}^{(1), (2)}$ in Eq. (21). Examining Eqs. (28) and (29), we can summarize the comparison of exact and approximate optimal regimes in Table 1, which shows the validity of regimes for the approximate optimal conditions. In the following, we will discuss the three regimes:

When the tunnel strength J is close to κ, i.e., J ~ κ (regime I in Table 1), our exact analytical optimal conditions (21) and (22) are in good agreement with the numerical simulation, while the approximate optimal conditions (30) and (31) are not.
Even if J ≫ κ (regime II in Table 1) is satisfied, the approximate optimal conditions only give an optimal point (30) and (31), but it can not present the other optimal point, which has been given by the exact optimal values (21) and (22).
When φ ≠ 0 (regime III in Table 1), the approximate optimal conditions (30) and (31) are not available in this case. In section II, we have removed the constraint of two parameters φ and J in Eqs. (28) and (29) and derived the exact optimal conditions (21)–(27). Therefore there are a total of eight sets of solutions for the optimal detuning when the discriminant δ changes its sign as parameters vary, i.e., $(Δ_{o p t}^{(j)}, Δ_{o p t}^{(k)})$ , j = 1,2 and k = 3,4,5,6. When detunings Δ take its optimal values (21), (25), and (26), we can numerically find all eight optimal Kerr nonlinear strength U_opt, which can be seen from Figs. 4 and 5.

Table 1. Comparison of regimes of exact and approximate optimal parameters.

View Table

Based on above key results, we concretely turn to some new aspects of our work in the following.

In order to compare with the approximate results (30) and (31), we set φ = 0. In Fig. 6 we plot $g_{a}^{(2)} (0)$ as a function of U and Δ by solving numerically the master equation (4). We mark the minimum values of $g_{a}^{(2)} (0)$ by r₁ − r₄ (denoted by red-stars). See Eqs. (21) and (22) [for Fig. 6(a), point r₁: $r_{1} : (U_{o p t}, Δ_{o p t}^{(1)}) = (0.496 κ, 0.252 κ)$ ; point $r_{2} : (U_{o p t}, Δ_{o p t}^{(2)}) = (- 0.425 κ, - 0.1001 κ)$ ] for J = κ. When J = 20κ [Figs. 6(b) and 6(c)], point r₃ takes $(U_{o p t}, Δ_{o p t}^{(2)}) = (- 0.00861 κ, 0.612 κ)$ , and point r₄ takes $(U_{o p t}, Δ_{o p t}^{(1)}) = (- 0.00272 κ, 2.0597 κ)$ .

Fig. 6 Logarithmic plot for $g_{a}^{(2)} (0)$ by numerically solving master Eq. (4) as a function of U and Δ at φ = 0. Parameters chosen are r = 0.1, J = κ for (a), and J = 20κ for (b) and (c). Points r₁ − r₄ (denoted by red-star) are given by Eqs. (21) and (22). while points s₁ and s₂ (denoted by red-circle) are given by Eqs. (30) and (31). In (d), J = κ, red-dashed line: U_opt = −0.424κ given by $Δ_{o p t}^{(2)}$ in Eq. (22), blue-solid line: U_opt = 0.496κ given by $Δ_{o p t}^{(1)}$ in Eq. (22); pink-dashed-dotted line: U_opt = 0.051κ given by Eq. (31).

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For the approximate optimal points s₁ and s₂ (denoted by red-circles) given by (30) and (31) in Fig. 6, point s₁ takes (U_opt, Δ_opt)=(0.0505κ,0.1κ), point s₂ takes (U_opt, Δ_opt)=(0.00253κ,2κ).

We can see from Fig. 6(a) that these two optimal values r₁ and r₂ obtained from (21) and (22) are consistent with the numerical simulations, but the point s₁ obtained from the approximate optimal condition (30) and (31) has serious deviations with the numerical simulations. This difference comes from the assumptions J ≫ κ failing in this case with J = κ (regime I in Table 1). Second, examining the regime of large tunnel strength J ≫ κ, e.g., J = 20κ (regime II in Table 1), in Figs. 6(b) and 6(c) we find that the result of the approximate optimal point s₂ given by Eqs. (30) and (31) is close to the exact optimal point r₄. It is important to address that our exact methods give two optimal points r₁ and r₂ (or r₃ and r₄), while the approximate methods only show one point s₁ (or s₂). Furthermore, we plot $g_{a}^{(2)} (0)$ in Fig. 6(d) (the parameter takes J = κ) and find that two exact optimal detunings $Δ_{o p t}^{(1), (2)}$ (minimum values of blue and red-dashed lines, which correspond optimal points r₁ and r₂, respectively) given by (21) are consistent with numerical simulations, while the approximate solutions do not exactly predict optimal detuning (mininum value of pink dashed-dotted line, which corresponds the approximate optimal points s₁).

Correspondingly, when φ ≠ 0 (regime III in Table 1), in this case there are a total of six optimal points corresponding to single-photon blockade in Fig. 2. This also is a new result of our work.

Second, we plot $g_{a}^{(2)} (0)$ as a function of r and U with Δ taking its optimal values in Fig. 7, in this case J = 0.8κ (regime I in Table 1). We find that red-dashed lines given by (22) in Figs. 7(a) and 7(b) are in good agreement with those obtained by numerical simulations, but red-dashed line given by (31) in Fig. 7(c) is not consistent with those obtained by numerical simulations, which originates from the assumption of strong tunnel strength J ≫ κ. In addition, we plot $g_{a}^{(2)} (0)$ in Fig. 7(d) and find that two exact optimal detunings $Δ_{o p t}^{(1), (2)}$ given by (21) are consistent with numerical simulations at r = 0.303, while the approximate solutions do not exactly predict optimal point at r = 0.303. To compare with φ = 0, we plot $g_{a}^{(2)} (0)$ as a function of U and r in Fig. 8 by numerically solving master equation (4) when φ = 2.5rad (regime III in Table 1). We find the analytical results show an excellent agreement with the numerical simulations.

Fig. 7 Dependence of logarithmic plot of $g_{a}^{(2)} (0)$ given by numerically solving the master Eq. (4) on U and r at φ = 0. The detuning Δ satisfies $Δ_{o p t}^{(1)}$ in Eq. (21) in (a), $Δ_{o p t}^{(2)}$ in (b), and satisfies Eq. (30) in (c). The red-dashed lines in (a)–(b) are plotted with U given by Eq. (22), while in (c) U takes Eq. (31). Parameters chosen are J = 0.8κ, F_a = 0.1κ. In (d), blue-solid line: the parameters take $(U_{o p t}, Δ_{o p t}^{(1)}) = (2.244 κ, 0.291 κ)$ given by substituting r = 0.303 into Eqs. (21) and (22); simliarly red-dashed line: $(U_{o p t}, Δ_{o p t}^{(2)}) = (- 0.853 κ, 0.092 κ)$ ; green-dashed-dotted line: (U_opt,Δ_opt) = (0.208κ,0.242κ) given by substituting r = 0.303 into Eqs. (30) and (31).

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Fig. 8 Dependence of logarithmic plot $g_{a}^{(2)} (0)$ given by numerically solving the master Eq. (4) on U and the ratio r. The detuning Δ is given by Eq. (21): (a) $Δ_{o p t} = Δ_{o p t}^{(1)}$ , whereas (b) $Δ_{o p t} = Δ_{o p t}^{(2)}$ due to the discriminant δ < 0 when r varies. The red-dashed lines in (a)–(b) are plotted with U given by Eqs. (22) and (27). Other parameters are J = 0.8κ, F_a = 0.008κ, φ = 2.5rad.

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We set same quantity J = 10κ (regime II in Table 1) in Fig. 9. Clearly, black-dashed line given by the exact optimal Kerr nonlinearity U in (22) [Figs. 9(a) and 9(b)] and approximate optimal Kerr nonlinearity in (31) [Fig. 9(c)] are in good agreement with numerical simulations. Similar to Fig. 6(c), the approximate optimal Kerr nonlinearity U decided by Eq. (31) only gives one optimal value in Fig. 9(c), but it can not predict another optimal value given by exact optimal value in Fig. 9(a). Further observation can be found in Fig. 9(d).

Fig. 9 Dependence of logarithmic plot of $g_{a}^{(2)} (0)$ given by numerically solving the master Eq. (4) on U and r at φ = 0. The detuning Δ satisfies $Δ_{o p t}^{(1)}$ in Eq. (21) in (a), $Δ_{o p t}^{(2)}$ in (b), and satisfies Eq. (30) in (c). The black-dashed lines in (a)–(b) are plotted with U given by Eq. (22), while in (c) U takes Eq. (31). Parameters chosen are J = 10κ, F_a = 0.1κ. In (d), blue-solid line: the parameters take $(U_{o p t}, Δ_{o p t}^{(2)}) = (- 1.955 κ, 3.360 κ)$ given by substituting r = 0.803 into (21) and (22); similarly green-dashed-dotted line: $(U_{o p t}, Δ_{o p t}^{(1)}) = (0.115 κ, 8.045 κ)$ ; red-dashed line: (U_opt,Δ_opt) = (0.113κ,8.030κ) given by substituting r = 0.803 into Eqs. (30) and (31).

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Finally, we let J continuously varies with the strength ratio r = 0.2 in Fig. 10 (regimes I and II in Table 1). We find that good agreement between the two approaches at large tunnel strength J ≫ κ, but in this case the result obtained by the approximate optimal conditions (30) and (31) does not work well at small J, where the red-dashed in Fig. 10(c) does not follow the exact optimal conditions. Further observation can be found in Fig. 10(d).

Fig. 10 Dependence of logarithmic plot of $g_{a}^{(2)} (0)$ given by numerically solving the master Eq. (4) on U and J at φ = 0. The detuning Δ satisfies $Δ_{o p t}^{(1)}$ in Eq. (21) in (a) and $Δ_{o p t}^{(2)}$ in (b) or satisfies Eq. (30) in (c). The red-dashed lines in (a)–(b) are plotted with U given by Eq. (22), while in (c) U take Eq. (31). Parameters chosen are r = 0.2, F_a = 0.1κ. In (d), blue-dashed line: the parameters take $(U_{o p t}, Δ_{o p t}^{(1)}) = (0.0269 κ, 1.1055 κ)$ given by substituting J = 5κ into Eqs. (21) and (22); similarly, red-solid line: $(U_{o p t}, Δ_{o p t}^{(2)}) = (- 0.0369 κ, 0.247 κ)$ ; black-dashed-dotted line: (U_opt,Δ_opt) = (0.0208κ,κ) given by substituting J = 5κ into Eqs. (30) and (31).

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The same quantity φ = 2.5rad (regime III in Table 1) is taken in Figs. 11 and 12 comparing with Fig. 10, in which we study the influence of the tunneling rate J on the single-photon blockade effect as a function of U and J, where the detuning Δ is given by Eqs. (21), (25) and (26). The discriminant δ = m³ − 27n² in Eq. (20) changes its sign when the tunnel strength J varies, this results in the changes of solutions to Eq. (19). Figure (a)–(d) are for different Δ_opt. The white-dashed lines in (a)–(d) are plotted with U given by Eqs. (22) and (27). We find the analytical results show an excellent agreement with the numerical simulations.

Fig. 11 Dependence of logarithmic plot $g_{a}^{(2)} (0)$ given by numerically solving the master Eq. (4) on U and J. The detuning Δ is given by Eqs. (21), (25) and (26). The discriminant δ = m³ − 27n² in Eq. (20) changes its sign when phase φ varies, this results in the changes of solutions to Eq. (19). Figure (a)–(d) are for different Δ_opt. (a) $Δ_{o p t} = Δ_{o p t}^{(1)}$ when δ < 0, whereas $Δ_{o p t} = Δ_{o p t}^{(3)}$ when δ > 0. The same for (b), (c), and (d). (b) $Δ_{o p t}^{(1)}$ and $Δ_{o p t}^{(4)}$ , (c) $Δ_{o p t}^{(1)}$ and $Δ_{o p t}^{(5)}$ , (d) $Δ_{o p t}^{(1)}$ and $Δ_{o p t}^{(6)}$ . The white-dashed lines in (a)–(d) are plotted with U given by Eqs. (22) and (27). Other parameters are r = 0.2, φ = 2.5rad, and F_a = 0.1κ.

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Fig. 12 The parameters of this figure are the same as Fig. 11, but Δ_opt is different. (a) $Δ_{o p t}^{(2)}$ and $Δ_{o p t}^{(3)}$ , (b) $Δ_{o p t}^{(2)}$ and $Δ_{o p t}^{(4)}$ , (c) $Δ_{o p t}^{(2)}$ and $Δ_{o p t}^{(5)}$ , (b) $Δ_{o p t}^{(2)}$ and $Δ_{o p t}^{(6)}$ . The red-dashed lines in (a) and (b) are plotted with U given by Eqs. (22) and (27).

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By the analytical expressions (21)–(27) for U_opt and Δ_opt, we can find the roles the system parameters play. Also they are used as a bench mark to check the approximate methods and to define their range of validity. For example, applying the exact optimal analytical conditions (21)–(27), we can check the validity of the approximate optimal conditions (30) and (31).

Hence the exact optimal conditions (21)–(27) give a fully description of photon blockade, and holds for all range of parameters. On the contrary, if there are no exact optimal conditions (21)–(27), it is very difficult to experimentally find the all eight optimal regimes of photon blockade.

4. Analytical result of the equal-time second-order correlation function

Under the weak driving condition, we have |A₀₀| ≫ |A₁₀|, |A₀₁| ≫ |A₂₀|, |A₁₁|, |A₀₂|. Thus we assume that the vacuum state is approximately occupied with A₀₀ ≈ 1.

Therefore, we obtain the analytical expression of $g_{a}^{(2)} (0)$ by substituting Eq. (44) in Appendix C into Eq. (16)

g_{a}^{(2)} (0) = \frac{q_{2} + {[4 r J (κ^{2} - 4 Δ U - 4 Δ^{2}) \cos φ + q_{3}]}^{2}}{q_{1}^{- 1} t_{2} [t_{1} + κ^{2} {(4 J^{2} + κ^{2} - 8 Δ U - 12 Δ^{2})}^{2}]},

where q₁ = 16κ²Δ² + (4J² + κ² − 4Δ²)², q₂ = [4r²J²κ − 8rJκ(U + 2Δ)cosφ + κ(8ΔU + 12Δ² − κ²)cos(2φ) − 4rJ(κ² − 4ΔU − 4Δ²)sinφ + 2(κ²U + 3κ²Δ − 2J²U − 4Δ²U − 4Δ³)sin(2φ)]², q₃ = [4J²U + 8Δ²(U + Δ) − 2κ²(U + 3Δ)]cos(2φ) + 4rJ(U + 2Δ)(rJ − 2κsinφ) + κ[4Δ(2U + 3Δ) − κ²]sin(2φ), t₁ = [2κ² (U + 3Δ) − 8Δ²(U + Δ) + 4J²(U + 2Δ)]², and t₂ = [(2Δcosφ + κsinφ − 2rJ)² + (κcosφ − 2Δsinφ)²]². We plot

g_{a}^{(2)} (0)

as functions of the detuning Δ in Fig. 13. Noticing that the discriminant δ = m³ − 27n² in Eq. (20) changes its sign so that the solutions of Eq. (19) vary with φ. Therefore, there are a total of six optimal points corresponding to single-photon blockade, which are in good agreement with these points obtained by numerically solving the master Eq. (4) (see the minimum values in Figs. 13(a)–13(f).

Fig. 13 $g_{a}^{(2)} (0)$ versus Δ. The solid lines and dotted lines denote the analytical expression Eq. (32) and the numerical simulation, respectively. The parameters chosen are J = 3κ, r = 0.5, and F_a = 0.03κ, φ = 4.2rad, U = 0.409κ for (a), φ = 4.2rad, U = 1.387κ for (b), φ = 0.8rad, U = 0.221κ for (c), φ = 0.8rad, U = 14.766κ for (d), φ = 0.8rad, U = 0.408κ for (e), φ = 0.8rad, U = −8.963κ for (f). We find the minimum values of the correlation functions corresponding to completely single-photon blockade are consistent with these optimal values Eqs. (21)–(27), i.e., the minimum value of (a) versus $Δ_{o p t}^{(1)}$ , (b) versus $Δ_{o p t}^{(2)}$ ,(c) versus $Δ_{o p t}^{(3)}$ , (d) versus $Δ_{o p t}^{(4)}$ , (e) versus $Δ_{o p t}^{(5)}$ , (f) versus $Δ_{o p t}^{(6)}$ .

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5. Analytical expression for time-delayed second-order correlation function

In this section, we will investigate the dynamical evolution of the time-delayed second-order correlation function, which can be defined as [69, 72]

g_{a}^{(2)} (τ) = \frac{〈 {\hat{a}}^{†} (t) {\hat{a}}^{†} (t_{1}) \hat{a} (t_{1}) \hat{a} (t) 〉}{{〈 {\hat{a}}^{†} (t) \hat{a} (t) 〉}^{2}},

where τ = t₁ − t,t → ∞. We now derive the analytical expression for the time-delayed second-order correlation function

\begin{array}{l} g_{a}^{(2)} (τ) = \frac{1}{c_{3}} {ς_{3} + e^{- \frac{τ κ}{2}} (ς_{1} \sin [τ (J - Δ) + β_{1}] \\ {+ ς_{2} \sin [τ (J + Δ) + β_{2}])}}^{2} + \frac{1}{c_{3}} {ς_{4} \\ + e^{- \frac{τ κ}{2}} (ς_{1} \sin [τ (J - Δ) + β_{3}] \\ {+ ς_{2} \sin [τ (J + Δ) + β_{4}])}}^{2}, \end{array}

where the relevant coefficients can be found in Appendix D. For more detail derivations of Eq. (34), also see Appendix D.

In Fig. 14, $g_{a}^{(2)} (τ)$ is plotted as a function of the time-delay τ. We find $g_{a}^{(2)} (τ) = 0$ at τ = 0, which corresponds to single-photon blockade because U and Δ take its optimal values. As τ increases $g_{a}^{(2)} (τ) > 0$ , i.e., $g_{a}^{(2)} (τ)$ increase above its initial value at finite delay. We thus have $g_{a}^{(2)} (τ) > g_{a}^{(2)} (0)$ . This is a violation of the classical inequality [31, 71]. This indicates that photons emitted at different times prefer to stay together comparing with initial delayed point. We find from Eq. (34) that when |Δ − J| ≫ κ/2, a fast oscillation behavior occurs for $g_{a}^{(2)} (τ)$ as the delay time increases at short times. The magnitude of these osciations decreases as τ is increased and $g_{a}^{(2)} (τ)$ approaches unity as τ → ∞. This explains Figs. 14(a)–14(f).

Fig. 14 $g_{a}^{(2)} (τ)$ versus τ. U and Δ take its optimal values, i.e., $(Δ, U) = (Δ_{o p t}^{(1)}, U_{o p t}^{(1)})$ or $(Δ_{o p t}^{(4)}, U_{o p t}^{(4)})$ . The solid lines and dotted lines denote the analytical expression (34) and the numerical simulation respectively. The parameters chosen are J = 7κ, φ = 2rad, Δ_opt = −3.086κ, U_opt = 7.020κ, r = 5, and F_a = 0.084κ, for (a), J = 7κ, φ = 2rad, Δ_opt = −2.074κ, U_opt = 6.965κ, r = 2, and F_a = 0.21κ for (b), J = 0.4κ, φ = 2rad, Δ_opt = −0.26κ, U_opt = 44.83κ, r = 5, and F_a = 0.0167κ for (c), J = 0.4κ, φ = 2rad, Δ_opt = −0.0416κ, U_opt = −0.201κ, r = 10, and F_a = 0.0832κ for (d), J = 7κ, φ = 2.7rad, Δ_opt = −4.52κ, U_opt = 12.247κ, r = 5, and F_a = 0.05κ for (e), J = 7κ, φ = 3.1rad, Δ_opt = −11.892κ, U_opt = −1.162κ, r = 5, and F_a = 0.2κ for (c).

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Interestingly, in Figs. 15(a), 15(b), and 15(d), we observe that there exists so called “beat frequency” phenomenon for photon. Let’s demonstrate the simple case at ς₁ ≈ ς₂ = ς. Therefore from two sine waves of amplitude in Eq. (34), we can obtain,

ς \sin [τ (J - Δ) + β_{1}] + ς \sin [τ (J + Δ) + β_{2}] = 2 ς \sin (τ J + {\bar{β}}_{1}) \cos (τ Δ + {\bar{β}}_{2}),

where

{\bar{β}}_{1} = \frac{β_{1} + β_{2}}{2}

,

{\bar{β}}_{2} = \frac{β_{1} - β_{2}}{2}

. If one frequency is much larger than another, e.g., J ≫ Δ [see Figs. 15(a) and 15(b)], the frequency of the cosine of the expression above, that is Δ, is often too slow to be perceived as a pitch. Instead, it denotes a periodic variation of the sine, whose frequency is J. Therefore, the periodic of beat frequency for photon is

T_{b e a t} = \frac{π}{Δ} .

Similar, when J ≪ Δ [see Fig. 15(d)], beat frequency phenomenon also can appears, whose periodic can be written as

T_{b e a t} = \frac{π}{J} .

However, if the two frequencies are quite close, J ~ Δ, We can not observe beat frequency phenomenon [see line L₁ in Fig. 15(c)]. It is very important to point out that the suppression of the beat can also be realized even for weak Kerr nonlinearity U ≪ κ [see lines L₂ and L₃ in Fig. 15(c)]. Therefore, this may provide us a way to observe the phenomenon of photon blockade through time-delayed second-order correlation function measurements [13, 73]. In experiments we may clearly can measure time-delay second-order correlation function by measuring the beat frequency. Moreover, using this relation between beat frequency and correlation function, a bunch of other applications, such as observation [74] and control [75] of quantum beating, resonant light scattering [76], and so on, have been realized in recent years. Therefore, this scheme we obtained here is not only feasible but also extendable.

Fig. 15 $g_{a}^{(2)} (τ)$ versus τ. The parameters chosen are φ = 3rad, r = 2, U = 2.667κ, F_a = 0.386κ, J = 10κ, Δ = 0.2κ for (a), J = 10κ, F_a = 0.278κ, Δ = 0.5κ for (b) J = 0.53κ, F_a = 0.98κ, Δ = 10κ for (d). In (c), J = 10κ, F_a = 0.0015κ, Δ = 10κ, and U = 2.667κ for L₁, U = 0.1κ for L₂, U = 0.02κ for L₃. Interestingly, we find beats frequency for photon are observed, whose period is equal to $\frac{π}{Δ}$ or $\frac{π}{J}$ given by Eqs. (36) and (37).

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6. Conclusions

In summary, we have studied unconventional photon-blockade effects in two linear coupled nonlinear cavities filled with nonlinear Kerr mediums with cavity modes being driven and dissipated. By analytical calculations, we derive a condition for optimized photon antibunching. We find the optimal detuning for strong photon antibunching depends on the strength ratio and the relative phase between two complex driving fields. There are a total of eight sets of solutions for the optimal detuning since the discriminant changes its sign as parameters vary. This provides us with a way to control exactly the single-photon blockade and alter it from conventional to unconventional regimes. We find that the analytical expressions of the equal-time and time-delayed correlation function are consistent with numerical simulations. Under the optimal parameters, the system can be used to generate single-photon sources.

This work shows the usefulness of the nonlinear coupled cavities system as single-photon sources on demand, with potential applications in the coherent manipulation of photon blockade with quantum photonic circuits [77]. We thus believe it is worth exactly controlling the single-photon blockade as a promising alternative to ongoing efforts in developing integrated single-photon sources by making use of quantum emitters, such as single molecules, quantum dots, defects in solids and so on.

Appendix A The optimal conditions for the situation of Δ_a ≠ Δ_b and κ_a ≠ κ_b

In this section, we cancel the constraints of the parameters Δ_a = Δ_b and κ_a = κ_b, and analytically give the optimal conditions for the situation of Δ_a ≠ Δ_b and κ_a ≠ κ_b. Therefore the condition for A₀₂, A₁₁, and A₁₀ in Eq. (17) to have nontrivial solutions is that the determinant of the coefficient matrix of Eq. (17) is equal to zero, then Eq. (18) becomes

\begin{array}{l} 0 = 4 e^{i φ} i J r [2 i (U + Δ_{b}) + κ_{b}] [2 i (Δ_{a} + Δ_{b}) + κ_{a} + κ_{b}] \\ + 4 r^{2} J^{2} [2 i (U + Δ_{a} + Δ_{b}) + κ_{a} + κ_{b}] \\ + e^{2 i φ} {8 i J^{2} U - (2 i Δ_{b} + κ_{b}) [2 i (U + Δ_{b}) \\ + κ_{b}] [2 i (Δ_{a} + Δ_{b}) + κ_{a} + κ_{b}]} . \end{array}

First, we select Δ_a and U as optimal parameters. After a simple calculation to Eq. (38), we obtain a general quadratic equation for the optimal detuning Δ_a,

η_{1} Δ_{a}^{2} + η_{2} Δ_{a} + η_{3} = 0,

where the coefficients take

η_{1} = - 4 [- 16 J^{3} r^{3} \sin φ + 16 J r \cos (φ) Δ_{b} (J r \sin φ - κ_{b}) - 4 J^{2} r^{2} (- 4 + \cos 2 φ) κ_{b} - 8 J r \sin (φ) κ_{b}^{2} + κ_{b}^{3}]

,

η_{2} = 32 J r \cos (φ) [J^{2} (1 + r^{2}) + 4 Δ_{b}^{2}] κ_{b} + 64 J r \sin (φ) Δ_{b} [J^{2} (1 + r^{2}) + κ_{b}^{2}] - 8 Δ_{b} κ_{b} [4 J^{2} (1 + 4 r^{2}) + 4 Δ_{b}^{2} + κ_{b}^{2}] - 8 J^{2} r^{2} \sin (2 φ) [4 (J^{2} + Δ_{b}^{2}) + κ_{b}^{2}]

,

\begin{array}{l} η_{3} = 64 J r \cos (φ) Δ_{b}^{3} κ_{b} - 16 Δ_{b}^{4} κ_{b} - 4 Δ_{b}^{2} {4 J^{2} [- 4 J r \sin (φ) + κ_{a} + r^{2} \cos (2 φ) κ_{a}] + [4 J^{2} (3 + 4 r^{2} + \\ 2 r^{2} \cos 2 φ) + κ_{a}^{2}] κ_{b} + 2 (- 4 J r \sin φ + κ_{a}) κ_{b}^{2} + 2 κ_{b}^{3}} - (κ_{a} + κ_{b}) {16 J^{3} r^{2} [J (r^{2} + \cos 2 φ) - r \sin (φ) κ_{a}] - \\ 4 J^{2} r [4 J (- 1 + 2 r^{2}) \sin φ + r k_{a} (- 4 + \cos 2 φ)] κ_{b} - 4 J (J - 4 J r^{2} + 2 J r^{2} \cos 2 φ + 2 r k_{a} \sin φ) κ_{b}^{2} + \\ (- 8 J r \sin φ + κ_{a}) κ_{b}^{3} + κ_{b}^{3}} + 8 J r Δ_{b} {- J r \sin (2 φ) [4 J^{2} + {(κ_{a} + 2 κ_{b})}^{2}] + 2 \cos (φ) [4 J^{2} (1 + r^{2}) κ_{b} + κ_{a}^{2} κ_{b} + \\ κ_{b}^{3} + 2 κ_{a} (J^{2} (1 + r^{2}) + κ_{b}^{2})]} \end{array}

. The solution of Eq. (39) can be given by

Δ_{a, o p t}^{(1), (2)} = \frac{- η_{2} \pm \sqrt{η_{2}^{2} - 4 η_{1} η_{3}}}{2 η_{1}} .

Based on this optimal detuning Δ_a,opt, then the optimal Kerr nonlinearity can be expressed as

\begin{array}{l} U_{a, o p t} = {- 8 J^{2} r^{2} (Δ_{a} + Δ_{b}) + 2 \cos (2 φ) [- 4 Δ_{b}^{2} (Δ_{a} + Δ_{b}) + 2 Δ_{b} κ_{a} κ_{b} + (Δ_{a} + 3 Δ_{b}) κ_{b}^{2}] \\ - 4 J r \cos (φ) [- 4 Δ_{b} (Δ_{a} + Δ_{b}) + κ_{b} (κ_{a} + κ_{b})] + 8 J r \sin (φ) [Δ_{a} κ_{b} + Δ_{b} (κ_{a} + 2 κ_{b})] \\ + \sin (2 φ) [- 8 Δ_{a} Δ_{b} κ_{b} + κ_{b}^{2} (κ_{a} + κ_{b}) - 4 Δ_{b}^{2} (κ_{a} + 3 κ_{b})]} / {8 J^{2} r^{2} - 16 J r \cos (φ) (Δ_{a} \\ + Δ_{b}) - 8 J r \sin (φ) (κ_{a} + κ_{b}) + \cos (2 φ) [8 J^{2} + 8 Δ_{b} (Δ_{a} + Δ_{b}) - 2 κ_{b} (κ_{a} + κ_{b})] \\ 4 \sin (2 φ) [Δ_{a} κ_{b} + Δ_{b} (κ_{a} + 2 κ_{b})]}, \end{array}

where

Δ_{a} \equiv Δ_{a, o p t}^{(1) . (2)}

in Eq. (40).

Second, we select Δ_b and U as optimal parameters. Similarly, Eq. (38) becomes to,

Δ_{b}^{4} + b_{1} Δ_{b}^{3} + c_{1} Δ_{b}^{2} + d_{1} Δ_{b} + e_{1} = 0,

where the coefficients

b_{1} = \frac{1}{2} (Δ_{a} - 2 r J \cos φ)

,

\begin{array}{l} c_{1} = \frac{1}{24 κ_{b}} {4 J^{2} [- 4 J r \sin (φ) + κ_{a} + r^{2} \cos (2 φ) κ_{a}] + \\ 16 J r \cos (φ) Δ_{a} (J r \sin φ - 2 κ_{b}) + 4 Δ_{a}^{2} κ_{b} + [4 J^{2} (3 + 4 r^{2} + 2 r^{2} \cos 2 φ) + κ_{a}^{2}] κ_{b} + 2 (- 4 J r \sin φ + κ_{a}) κ_{b}^{2} + 2 κ_{b}^{3}} \end{array}

,

\begin{array}{l} d_{1} = \frac{1}{8 κ_{b}} {- 8 J r \sin (φ) Δ_{a} [J^{2} (1 + r^{2}) + κ_{b}^{2}] + Δ_{a} κ_{b} [4 J^{2} (1 + 4 r^{2}) + κ_{b}^{2}] + J^{2} r^{2} \sin (2 φ) [4 J^{2} + \\ 4 Δ_{a}^{2} + {(κ_{a} + 2 κ_{b})}^{2}] 2 + J r \cos (φ) [- 4 (J^{2} + J^{2} r^{2} Δ_{a}^{2}) κ_{b} - κ_{a}^{2} κ_{b} - κ_{b}^{3} - 2 κ_{a} (J^{2} + J^{2} r^{2} + κ_{b}^{2})]} \end{array}

,

\begin{array}{l} e_{1} = \frac{1}{16 κ_{b}} {4 Δ_{a}^{2} [- 16 J^{3} r^{3} \sin φ - 4 J^{2} r^{2} (- 4 + \cos 2 φ) κ_{b} - 8 J r \sin (φ) κ_{b}^{2} + κ_{b}^{3}] + (κ_{a} + κ_{b}) [16 J^{3} r^{2} [J (r^{2} + \\ \cos 2 φ) - r \sin (φ) κ_{a}] - 4 J^{2} r [4 J (- 1 + 2 r^{2}) \sin φ + r (- 4 + \cos 2 φ) κ_{a}] κ_{b} - 4 J [J - 4 J r^{2} + 2 J r^{2} \cos 2 φ + \\ 2 r \sin (φ) κ_{a}] κ_{b}^{2} + (- 8 J r \sin φ + κ_{a}) κ_{b}^{3} + κ_{b}^{4}] + 16 J^{2} r \cos (φ) Δ_{a} [- 2 J (1 + r^{2}) κ_{b} + r \sin φ (4 J^{2} + κ_{b}^{2})]} \end{array}

.

The solutions of for Δ_b in Eq. (42) have been given by Eqs. (21), (25), and (26) with the replacement b → b₁, c → c₁, d → d₁, and e → e₁. With the optimal detuning Δ_b,opt, the optimal Kerr nonlinearity U_b,opt can be given by Eq. (41), where Δ_b = Δ_b,opt. Therefore, for realistic experiment with the system parameters Δ_a ≠ Δ_b and κ_a ≠ κ_b, we can exactly control the single-photon blockade by using the optimal parameters from Eq. (39)–(42).

Appendix B The proof of Eq. (19) only having two real roots when φ = 0

When the relative phase equals zero, the coefficients δ, h and η in Eqs. (23) and (24) can be reduced to

\begin{array}{l} δ = \frac{J^{4} {(r^{2} - 1)}^{2} [N_{2} J^{8} + N_{1} + 16 J^{2} κ^{6} (1 + 3 r^{2})]}{4096}, \\ h = \frac{1}{12} [3 (r^{2} - 1) J^{2} - κ^{2}], \\ η = \frac{J^{2}}{16} (r^{2} - 1) [9 (r^{2} - 1) J^{2} - 8 κ^{2}], \end{array}

where the parameters take N₁ = −162J⁶k²(r²−1)² (1+3r²) − 207J⁴k⁴(r²−1)² + 64κ⁸, N2 = −216r²(r²−1)² (1+r²). When the discrimination δ > 0 in Eq. (23), if and only if two inequalities (24) are simultaneously satisfied, Eq. (19) has four real roots, otherwise it has four conjugate complex roots. We now solve the three inequalities in Eqs. (23) and (24), we find three inequalities can not hold simultaneously for all parameters. Therefore, there are nonphysical optimal values when discriminant δ > 0 at φ = 0. We conclude that the exact optimal detuning Δ has only real roots

Δ_{o p t}^{(1), (2)}

in Eq. (21) at φ = 0.

Appendix C Analytical expression of the probability amplitude

We obtain the analytical expression for the equal-time second-order correlation function by solving the steady state in Eqs. (7)–(12) in weak driving limit

\begin{array}{l} {\bar{A}}_{10} = - \frac{2 e^{i φ_{b}} F_{a} [2 r J + i e^{i φ} (κ + 2 i Δ)]}{4 J^{2} + {(κ + 2 i Δ)}^{2}}, \\ {\bar{A}}_{20} = \frac{2 \sqrt{2} e^{2 i φ_{b}} F_{a}^{2} (4 i r J w_{1} e^{i φ} + w_{2} + e^{2 i φ} w_{3})}{w_{4} [{(κ + 2 i Δ)}^{2} w_{5} + 4 J^{2} κ + 4 J^{2} (U + 2 Δ) i]}, \\ {\bar{A}}_{11} = \frac{4 e^{2 i φ_{b}} F_{a}^{2} (2 J e^{i φ} + i r κ - 2 r Δ) (i w_{1} e^{i φ} + \frac{w_{2}}{2 r J})}{w_{4} [{(κ + 2 i Δ)}^{2} w_{5} + 4 J^{2} κ + 4 J^{2} (U + 2 Δ) i]}, \end{array}

where w₁ = (κ + 2iΔ)[κ + 2i(U + Δ)], w₂ = 4r²J²[κ + i(U + 2Δ)], w₃ = 4iJ²U−(κ + 2iΔ)²[κ + 2i(U + Δ)], w₄ = 4J² + (κ + 2iΔ)², w₅ = κ + 2i(U + Δ).

Appendix D Analytical derivation of the time-delayed second-order correlation function

Applying the Born approximation and assuming the environment being in zero temperature to Eq. (33), we can obtain the analytical expression for time-delayed two-order correlation function [38],

g_{a}^{(2)} (τ) = {Tr}_{S} [{\hat{A}}^{†} \hat{A} \tilde{ρ} (τ)],

with

\tilde{ρ} (τ) = {Tr}_{E} [U_{T} (τ) \tilde{ρ} (0) \otimes ρ_{E} U_{T}^{†} (τ)],

where the new initial state is defined by

\tilde{ρ} (0) = \hat{a} \bar{ρ} {\hat{a}}^{†}

and

\hat{A} = \hat{a} / {\bar{n}}_{a}

,

\bar{ρ}

is the steady state for Eq. (4). The unitary evolution operator is given by

U_{T} (t) = \exp (- i {\hat{H}}_{T} t)

,

{\hat{H}}_{T} = \hat{H} + {\hat{H}}_{E} + {\hat{H}}_{I}

, and

\hat{H}

given by Eq. (3),

{\hat{H}}_{E}

and

{\hat{H}}_{I}

the Hamiltonians of the environment and the system-environment interaction, respectively. ρ_E is the vacuum state of the environment. The dynamical evolution for Eqs. (46) and (4) are equal, i.e.,

\dot{\tilde{ρ}} (τ) \equiv \dot{ρ} (τ)

but they have different initial state. Therefore under effective Hamiltonian approximation,

\tilde{ρ} (τ) = | \tilde{Ψ} (τ) 〉 〈 \tilde{Ψ} (τ) |

and ρ(τ) = |Ψ(τ)〉〈Ψ(τ)| are equivalent to

| \dot{\tilde{Ψ}} (τ) 〉 = | \dot{Ψ} (τ) 〉

. Therefore the time-delayed correlation function can be calculated by introducing a initial state

| \tilde{Ψ} (0) 〉 = \hat{a} | \bar{Ψ} 〉,

with

| \bar{Ψ} 〉

given by the steady state solutions of Eqs. (7)–(12). Under this initial state, we obtain the time-delayed second-order correlation function,

g_{a}^{(2)} (τ) = \frac{{| A_{10} (τ) |}^{2}}{{| {\bar{A}}_{10} |}^{4}} .

To be specific, the initial state after the annihilation of a photon in the left cavity a is

| \tilde{Ψ} (0) 〉 = {\bar{A}}_{10} | 00 〉 + \sqrt{2} {\bar{A}}_{20} | 10 〉 + {\bar{A}}_{11} | 01 〉 .

Equation (48) can be obtained by solving Eqs. (7)–(9) for these amplitudes with the initial state (49). Under the weak driving condition, we have $A_{00} = {\bar{A}}_{10}$ . Therefore, Eq. (7) decouples with Eqs. (8) and (9). Defining two quantities

\begin{array}{l} A_{10} (τ) + A_{01} (τ) = x (τ), \\ A_{10} (τ) - A_{01} (τ) = y (τ), \end{array}

and substituting Eq. (50) into Eqs. (8) and (9), we obtain

\begin{array}{l} \frac{\partial x}{\partial τ} + u_{1} x (τ) = λ_{1}, \\ \frac{\partial y}{\partial τ} + u_{2} y (τ) = λ_{2}, \end{array}

where we ignore the high-order terms A₁₁, A₂₀ and A₂₀, and the initial values

x (0) = \sqrt{2} {\bar{A}}_{20} + {\bar{A}}_{11}

,

y (0) = \sqrt{2} {\bar{A}}_{20} - {\bar{A}}_{11}

, decided by Eq. (44). The other parameters take u₁ = i(δ_a + J), u₂ = i(δ_a − J),

λ_{1} = - i F_{a} e^{i φ_{b}} (e^{i φ} + r) A_{00}

,

λ_{2} = - i F_{a} e^{i φ_{b}} (e^{i φ} - r) A_{00}

with A₀₀ ≡ Ā₁₀. The solutions of the first order ordinary differential equation (51) are

\begin{array}{l} x (τ) = x (0) e^{- u_{1} τ} + \frac{λ_{1}}{u_{1}} (1 - e^{- u_{1} τ}), \\ y (τ) = y (0) e^{- u_{2} τ} + \frac{λ_{2}}{u_{2}} (1 - e^{- u_{2} τ}) . \end{array}

Noticing Eq. (44), Eq. (34) can be obtained by substituting Eqs. (50) and (52) into Eq. (48), where these time-independent coefficients take c₃ = 64[(−2aJ + 2Δcosφ + κsinφ)² + (κcosφ − 2Δsinφ)²]²/[16κ²Δ² + (4J² + κ² − 4Δ²)²]², r₁ = 2[−(2A₀₀₁(J + Δ) − A₀₀₂κ)(r + cosφ) + (2A₀₀₂(J + Δ) + A₀₀₁κ)sinφ]/[4(J + Δ)² + κ²], r₂ = −2[(2A₀₀₂(J + Δ) + A₀₀₁κ)(r + cosφ) + (2A₀₀₁(J + Δ) − A₀₀₂κ)sinφ]/[4(J + Δ)² + κ²], r₃ = 2[(2A₀₀₁(J − Δ) + A₀₀₂κ)(cosφ − r) + (2A₀₀₂(Δ − J) + A₀₀₁κ)sinφ]/[4(J − Δ)² + κ²], r₄ = 2[(2A₀₀₂(J − Δ) − A₀₀₁κ)(cosφ − r) + (2A₀₀₁(J − Δ) + A₀₀₂κ)sinφ]/[4(J −Δ)² + κ²]. $x_{01} = Re {x (0) / [F_{a}^{2} e^{2 i φ_{b}}]}$ , $y_{01} = Im {x (0) / [F_{a}^{2} e^{2 i φ_{b}}]}$ , $x_{02} = Re {y (0) / [F_{a}^{2} e^{2 i φ_{b}}]}$ , $y_{02} = Im {y (0) / [F_{a}^{2} e^{2 i φ_{b}}]}$ , $A_{001} = Re {{\bar{A}}_{10} / [F_{a} e^{i φ_{b}}]}$ , $A_{002} = Im {{\bar{A}}_{10} / [F_{a} e^{i φ_{b}}]}$ , z₁ = y₀₂ – r₄, z₂ = y₀₁ – r₂, z₃ = x₀₂ – r₃, z₄ = r₁ – x₀₁, $ς_{1} = \sqrt{z_{1}^{2} + z_{3}^{2}}$ , $ς_{2} = \sqrt{z_{2}^{2} + z_{4}^{2}}$ , ς₃ = r₂ + r₄, ς₄ = r₁ + r₃, $β_{1} = \arctan (\frac{z_{1}}{z_{3}})$ , $β_{2} = \arctan (\frac{z_{2}}{z_{4}})$ , $β_{3} = \arctan (\frac{z_{3}}{z_{1}})$ , $β_{4} = \arctan (\frac{z_{4}}{z_{2}})$ .

Acknowledgments

We would like to thank Prof. Yong Li and Prof. Xiao-Qiang Shao for valuable discussions. This work is supported by National Natural Science Foundation of China (NSFC) under grant Nos 11175032, 61475033, 11405008, 11405026 and the Plan for Scientific and Technological Development of Jilin Province (No. 20150520083JH).

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Exact optimal control of photon blockade with weakly nonlinear coupled cavities

Abstract

1. Introduction

2. Physical model and optimal antibunching conditions

3. Unconventional single-photon blockade

3.1. Influence of the relative phase φ on photon blockade

3.2. Comparison between exact and approximate optimal conditions

4. Analytical result of the equal-time second-order correlation function

5. Analytical expression for time-delayed second-order correlation function

6. Conclusions

Appendix A The optimal conditions for the situation of Δ_a ≠ Δ_b and κ_a ≠ κ_b

Appendix B The proof of Eq. (19) only having two real roots when φ = 0

Appendix C Analytical expression of the probability amplitude

Appendix D Analytical derivation of the time-delayed second-order correlation function

Acknowledgments

References and links

Cited By

Figures (15)

Tables (1)

Equations (52)

Optics Express

Abstract

1. Introduction

2. Physical model and optimal antibunching conditions

3. Unconventional single-photon blockade

3.1. Influence of the relative phase φ on photon blockade

3.2. Comparison between exact and approximate optimal conditions

4. Analytical result of the equal-time second-order correlation function

5. Analytical expression for time-delayed second-order correlation function

6. Conclusions

Appendix A The optimal conditions for the situation of Δa ≠ Δb and κa ≠ κb

Appendix B The proof of Eq. (19) only having two real roots when φ = 0

Appendix C Analytical expression of the probability amplitude

Appendix D Analytical derivation of the time-delayed second-order correlation function

Acknowledgments

References and links

Cited By

Figures (15)

Tables (1)

Equations (52)

Optics Express

Appendix A The optimal conditions for the situation of Δ_a ≠ Δ_b and κ_a ≠ κ_b