Photon blockade via quantum interference in a strong coupling qubit-cavity system

Wen-Wu Deng; Gao-Xiang Li; Hong Qin

doi:10.1364/OE.25.006767

1. Introduction

In quantum information processing, it requires a more powerful mechanism to realize a generation and manipulation of photon. This is because the generation and manipulation of photon is crucial for the exchange and handle of photonic quantum information [1–5]. Photon blockade effects, which have been studied in various systems [6–17], provide an effective mechanism to meet these requirements. Generally, photon blockade means that a single photon in a cavity can block the transmission of other photons, thus, a sequence of single photon can be generated one by one, and such system can act as a one-photon turnstile device and a one-photon quantum switch [18, 19]. The quantum statistical properties of photon blockade exhibit an antibunching effect characterized by second-order correlation function [20]. Recently, the concept of photon blockade has been extended from one-photon blockade to multi-photon blockade [21]. Under strong antibunching of photons, multi-photon blockade systems can generate multi-photon groups, provide ideal entangled sources, and thus most importantly achieve the key technologies in quantum information. For a convenient way to obtain photon blockade, strong quantum nonlinear source is one of the necessary prerequisites. And the realizing of photon blockade through the quantum nonlinear source has already been studied, such as by using a highly nonlinear quantum system with an intrinsic nonlinear susceptibility [22–26] or nonlinear quantum truncation [27]. However, experimental realization of the strong quantum nonlinear source is still a challenge in most systems, because the strong quantum nonlinear source largely relies on a high-quality nanocavity, a very strong nonlinear susceptibility or a strong coupling condition. Meantime, the number of output photon is also low under such a strong photon blockade regime. Recently, a new mechanism called unconventional photon blockade was proposed, essentially relying on the quantum interference under suitable pump or detection conditions [28–31]. This new mechanism relaxes the requirements of the strong quantum nonlinear source, what’s more, a large number of output photon is achieved under a strong photon blockade regime. Even in a weak nonlinear source, effects of strong photon blockade have also been observed, which provide another way to realize tunable photon sources in quantum information processing, and it is more realizable in experiments.

More recently, with the technological developments in cavity quantum electrodynamics such as Josephson charge qubit [32–34], semiconducting dots [35–37] and circuit QED [38–41], the coupling strength between a transmission and a cavity mode is comparable to the transition frequency or the resonance frequency of the cavity mode, and the coupling strength has reached the strong coupling regime [42, 43]. These technological developments greatly improve the level of generation and manipulation of photon in the quantum information science. More importantly, the dynamics of these strong coupling system can also be illustrated by the Rabi model [44]. In these strong coupling systems illustrated by the Rabi model, the conventional rotating-wave approximation (RWA) corresponding to Jaynes-Cummings (JC) model is no longer applicable, and thus the effects of the counter-rotating wave terms (CRTs) in the Rabi model should be considered [45–54]. As a consequence, it not only becomes very difficult to get the analytical solution including the CRTs [55], but also leads to some novel features. For example, the ground state of the system is no longer a standard vacuum, but a quasidegenerate with an entangled structure, photons can be created from the quantum vacuum [56], and an intercrossing of energy levels may exist in the regime of the strong interaction [57]. In order to explain and study these novel physical phenomena, it requires a proper modification of the related theory such as dissipation, input-output relation, standard Lindblad master equation and correlation functions [58–61]. So, we are interested, in a strong coupling system, whether a strong photon blockade and a large output photon can be realized through unconventional mechanism.

In this paper, we provide a coherently-driven nanocavity QED system to achieve a strong one-photon blockade based on complete elimination of the two-photon excitation with multichannel quantum interference. In this system, there exists a strong coupling qubit-cavity subsystem illustrated by the Rabi model, the coupling strength between qubit and cavity mode is strong, and the effect of CRTs in Rabi model is considered. In order to realize the multichannel quantum interference, there exist two external classical driving fields in our system, and the Rabi coupling strengths {ε, Ω} of the two external classical driving field are weaker than the coupling strength g between the qubit and the single cavity mode, i.e., {ε, Ω} ≪ g. Based on the dressed bases of the Rabi Hamiltonian in our previous paper [57, 62–64], a modified Lindblad master equation and a generalized second-order correlation function valid for any arbitrary degree of the qubit-cavity interaction are obtained. Using the generalized second-order correlation function and the second-order perturbation, how to enhance the one-photon blockade by using the quantum interference effect is discussed. It is found that, under suitable pump or detection conditions for the two external classical driving field, a very strong one-photon blockade can be realized. Even for a strong cavity damping rate, a large number of cavity photons can also be realized, which avoids the challenges of fabrication for preparing nanocavities with high quality factors. Thus, the scheme with the multichannel quantum interference can provide an effective way to realize an ideal single photon source and the manipulation of photon. And this scheme is more realizable in expeiriments. The rest of this paper is organized as follows. In section 2, the coherently-driven nanocavity QED system is introduced and the modified Lindblad master equation in the Rabi dressed picture without using the secular approximation is obtained. In section 3, the generalized second-order correlation function validing for any arbitrary degree of the qubit-cavity interaction is given, and quantum interference mechanism with a very strong one-photon blockade is discussed. Finally, some conclusions are given in section 4.

2. Model and modified Lindblad master equation in the dressed picture

Here a coherently-driven nanocavity QED system is considered as depicted in Fig. 1. In this system, there exists a qubit-cavity subsystem with the most general linear coupling between a qubit (two-level system) and a single cavity mode, in which the coupling strength g between the qubit and the single cavity mode is comparable to the qubit-cavity resonant frequency ω, i.e., the coupling strength g is in the strong regime of the interaction. Generally, this strong coupling qubit-cavity subsystem inside the nanocavity QED system is illustrated by Rabi model. Additionally, the nanocavity with a strong cavity damping rate of κ is driven by a weak laser field with a frequency of ω_p and a weak coupling strength of ε. And there exists a pump field applied to pump the single qubit directly with a frequency of ω_L. The Rabi coupling strength Ω between the pump field and the single qubit is quite weak. Then, the whole Hamiltonian of the nanocavity QED system is given by (ħ = 1)

H_{S} = H_{R} + H_{p} + H_{L},

H_R is the Rabi Hamiltonian of the strong coupling qubit-cavity subsystem, which in general reads

H_{R} = ω b^{†} b + \frac{1}{2} ω_{0} σ_{z} + g (b^{†} + b) (S_{+} + S_{-}),

Fig. 1 Scheme of the considered coherently-driven nanocavity QED system. There exists a strong coupling qubit-cavity subsystem illustrated by Rabi model, in which the coupling strength g between the qubit and the single cavity mode is comparable to the transition frequency ω₀ of qubit, the frequency ω of cavity mode, and the cavity damping rate κ. The cavity mode and qubit are driven simultaneously by driving laser with frequency ω_p and pumping laser with frequency ω_L, respectively. The coupling strengths from the pumping and driving fields are weak, and satisfy {ε, Ω} ≪ {g, ω, ω₀, κ}.

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The interactive Hamiltonian between the external classical laser driving field and the cavity mode can be written as

H_{p} = 2 ε \cos (ω_{p} t) (b^{†} + b),

additionally, the interactive Hamiltonian between the pump field and the single qubit is

H_{L} = 2 Ω \cos (ω_{L} t + θ) (S_{+} + S_{-}),

here S₊ (S₋) is the dipole raising (lowering) operator and σ_z is the Pauli operator of the qubit with a transition frequency ω₀. b^† and b are the creation and annihilation operators of the cavity mode with a frequency of ω, respectively. Δ(= ω₀ − ω) denotes the detuning between the qubit and the cavity mode, and θ is the relative phase between the qubit pumping field and the cavity driving field. In this nanocavity QED system, the corresponding coupling strengths from the pumping and driving fields are weak, and the coupling strengths satisfy {ε, Ω} ≪ {g, ω, ω₀, κ}.

In Hamiltonian H_R (2) of the qubit-cavity subsystem, as the coupling strength of the qubit and the cavity mode is so strong that the well-known RWA breaks down, the effects of the CRTs (b^†S₊ + bS₋) should be considered. But the analytical solution becomes very difficult to get when the CRTs are included. According to Appendix A, the solution of the Rabi model (2) is obtained exactly when the effects of the CRTs are included, and the Hamiltonian of the Rabi model (2) can be approximately expressed as

ℋ_{R} = \sum_{n = 0}^{\infty} E_{n} | ψ_{n} 〉 〈 ψ_{n} |,

where the dressed bases |ψ_n〉 of the Rabi Hamiltonian satisfy stationary Schrödinger equation ℋ_R|ψ_n〉 = E_n|Ψ_n〉. The analytical expressions of dressed bases are valid for any arbitrary degree of the qubit-cavity interaction.

Since the applied pumping and driving fields are weak, i.e., {ε, Ω} ≪ {g, ω, ω₀}, which are different from the case of the strong coupling between the qubit and the cavity mode, the energy spectrum of the nanocavity QED system is almost the same with the Rabi model (2), thus, the interactive Hamiltonian H_p (3) and H_L (4) can be expanded by the dressed bases |ψ_n〉 (5) of the Rabi Hamiltonian H_R as shown in Fig. 2, and can be written as

ℋ_{p} = 2 ε \cos (ω_{p} t) [\sum_{m, n > m} Z_{m n} σ_{m n} + H . c .],

ℋ_{L} = 2 Ω \cos (ω_{L} t + θ) [\sum_{m, n > m} S_{m n} σ_{m n} + H . c .],

here σ_mn = |ψ_m〉〈ψ_n|, and

Z_{m n} = 〈 ψ_{m} | (b^{†} + b) | ψ_{n} 〉, S_{m n} = 〈 ψ_{m} | (S_{+} + S_{-}) | ψ_{n} 〉 .

Fig. 2 Energy level diagram of Rabi model with the five-state truncation without the inter-crossing of the energy levels.

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According to the dressed bases (34) in Appendix A, the specific expressions of Z_mn and S_mn satisfy

Z_{m n} = 2 \sum_{l = 0}^{\infty} [\sqrt{l + 1} (c_{m}^{l} c_{n}^{l + 1} + c_{m}^{l + 1} c_{n}^{l})],

S_{m n} = 2 \sum_{l = 0}^{\infty} (c_{m}^{2 l} c_{n}^{2 l} + c_{m}^{2 l + 1} c_{n}^{2 l + 1}) (n > m) .

For a given E_n, the coefficients $c_{n}^{k} (k = l, l + 1)$ of the dressed bases |ψ_n〉 can be determined by the recurrence relation (35). According to the parity of dressed bases |ψ_n〉 in Eq. (5), the orthogonal features with Z(S)_n,n = 0 and Z(S)₂_n_,2_n₋₁ = 0 can be found.

Meantime, the corresponding coupling strengths from the pumping and driving fields are weak, and the coupling strengths satisfy {ε, Ω} ≪ {ω, ω₀, g, E_nm}, so the RWA can be applied reasonably. The Hamiltonian ℋ_p (6) and ℋ_L (7) can be rewritten as

{ℋ *}_{p} (t) = ε [\sum_{m, n > m} Z_{m n} σ_{m n} e^{i ω_{p} t} + H . c .],

{ℋ *}_{L} (t) = Ω [\sum_{m, n > m} S_{m n} σ_{m n} e^{i (ω_{L} t + θ)} + H . c .] .

To simplify it, we assume ω_p = ω_L. Then the Hamiltonian from the pumping and driving field can be expressed as

H^{'} (t) = {ℋ *}_{p} (t) + {ℋ *}_{L} (t) = \sum_{m, n > m} K_{m n} σ_{m n} e^{i ω_{p} t} + H . c .,

with

K_{m n} = ε Z_{m n} + Ω S_{m n} e^{i θ} .

Thus, the Hamiltonian of the whole nanocavity QED system can be effectively replaced by

ℋ_{S} (t) = \sum_{n = 0}^{\infty} E_{n} σ_{n n} + [\sum_{m, n > m} K_{m n} σ_{m n} e^{i ω_{p} t} + H . c .] .

In this paper, we will focus on the one-photon blockade. To get the one-photon blockade, it is required to excite the one-photon near-resonant transition by the external driving field and to suppress the two-photon near-resonant excitation. So, we focus on that both the two external deriving fields should satisfy

ω_{p} = ω_{L} ≃ E_{10} (E_{20}),

i.e., the one-photon near-resonant excitation is from |ψ₀〉 to |ψ₁〉 or |ψ₂〉 as depicted in Fig. 2. Here E_nm = E_n − E_m. In this case, the off-resonance Δ′ should be larger and change with the increase of the coupling strength g between the qubit and the single cavity mode as shown in Fig. 2. Under this condition of one-photon near-resonant excitation, we can cut off the photons into the two-photon excitation subspace with n = 2 and eliminate all higher photon excitations with n ≥ 3 [65]. As a result, we can consider only the five-state Hilbert space, namely, the ground state |ψ₀〉, and excited states |ψ₁〉, |ψ₂〉, |ψ₃〉, |ψ₄〉 as depicted in Fig. 2. So, the quantum state of the whole nanocavity QED system in the five-state Hilbert space can be written by

| Ψ (t) 〉 = \sum_{n = 0}^{4} C_{n} | ψ_{n} 〉,

the coefficients C_n describe the probability amplitudes of the corresponding dressed bases, and the probability amplitudes satisfy

\sum_{n = 0}^{4} {| C_{n} |}^{2} = 1

. Consequently, this provides the simplest model to capture the physics, and the five dressed bases contain the effects of the CRTs.

In this five-state Hilbert space and in the rotating frame by utilizing the unitary transformation U,

U = \exp {- i t [E_{0} σ_{00} + (E_{0} + ω_{p}) (σ_{11} + σ_{22}) + (E_{0} + 2 ω_{p}) (σ_{33} + σ_{44})]},

the whole Hamiltonian (1) of the nanocavity QED system in the five-state Hilbert space will be time-independent and can be given by

\begin{array}{l} H_{T} = (E_{10} - ω_{p}) σ_{11} + (E_{20} - ω_{p}) σ_{22} + (E_{30} - 2 ω_{p}) σ_{33} + (E_{40} - 2 ω_{p}) σ_{44} \\ + [\sum_{m, n > m}^{4} K_{m n} σ_{m n} + H . c .] . \end{array}

It is worth notice that, in our system, the coupling strength g of the qubit-cavity subsystem is so strong that it is comparable to the qubit-cavity resonant frequency ω. The usual RWA employed the weak coupling qubit-cavity system is no longer applicable, and the effects of the CRTs should be considered. As a consequence, the ground state |ψ₀〉 of the Rabi model now contains the qubit-cavity excitations with an entangled structure [62, 63]. The number of excitations in the qubit-cavity system is no longer conserved, and photon can be created from the quantum vacuum. With the increasing of the interaction strength g, there may exist the intercrossing of energy levels. Then, the standard Lindblad master equation employing the RWA fails. In order to avoid some unphysical contributions due to the effects of the CRTs, a viable description of the strong coupling system with dissipation requires a modified Lindblad master equation [58, 59]. According to the Appendix B, the modified Lindblad master equation can be expressed in the dressed bases |ψ_n〉 of the Rabi model (2), and is given by

\dot{ρ} (t) = - i [H_{T}, ρ (t)] + ℒ_{c} ρ,

where the Lindbladian ℒ_cρ describes the spontaneous damping of the qubit, the cavity mode and the cross coupling damping. The modified Lindblad master equation is valid for any arbitrary degree of the qubit-cavity interaction. In the meantime, because the coupling strength g is comparable to the cavity damping rate κ, the convenient secular approximation breaks down. The specific expressions of the Lindbladian ℒ_cρ without using the secular approximation have been given in Appendix B.

3. Generalized second-order correlation function and quantum interference mechanism

The one-photon near-resonant excitation can happen when the driven frequency of the cavity driving field satisfies ω_p ≃ E₁₀ or ω_p ≃ E₂₀ as shown in Fig. 2. As discussed below, we will focus on how to enhance the one-photon blockade, especially by using the quantum interference effect of the two external deriving fields. The relative phase θ of the two external deriving fields and the Rabi coupling strength Ω of the qubit pumping field play a significant role in the photon blockade effect. In what follows, with the quantum interference mechanism, we use the generalized second-order correlation function in the regime of the strong interaction to discuss the optimal condition of the strong one-photon blockade.

3.1. Generalized second-order correlation function in the regime of the strong interaction

Similarly, in the regime of the strong interaction between the qubit and the cavity mode, the standard second-order correlation function also requires a proper generalization with the modified input-output relations [58,59], and the normalized and generalized second-order correlation function for output fields in vacuum is equal to [66]

g^{(2)} (0) = \frac{〈 {\dot{X}}^{-} {\dot{X}}^{-} {\dot{X}}^{+} {\dot{X}}^{+} 〉}{{〈 {\dot{X}}^{-} {\dot{X}}^{+} 〉}^{2}} .

This stationary second-order correlation function (21) is used to benchmark the quantum statistic characteristics of the photons, and is also valid for any arbitrary degree of the qubit-cavity interaction. When g⁽²⁾(0) ≪ 1, it indicates that a strong photon blockade with antibunching effect can be achieved [20]. Meantime, the mean cavity photon number n is proportional to n = 〈Ẋ⁻ Ẋ⁺〉. The operator Ẋ⁺ (t) can be expanded in terms of the qubit-cavity dressed bases |ψ_n〉 and be given by [58, 59]

{\dot{X}}^{+} = \sum_{m, n > m} Y_{m n} | m 〉 〈 n |,

where Y_mn = X₀E_nm Z_mn, and Ẋ⁻ = (Ẋ⁺)^†. Because there may exist the intercrossing of energy levels in the strong coupling regime between the qubit and the single cavity mode, and a photon emission from the cavity is associated with a transition in physical consideration from a high-energy state to a low-energy state, the dressed bases |ψ_n〉 are labeled according to E_n > E_m for n > m in order to avoid some unphysical contributions due to the effects of the CRTs.

In the five-state Hilbert space, we can introduce two parameters Y₃ = Y₃₁Y₁₀ + Y₃₂Y₂₀ and Y₄ = Y₄₁Y₁₀ + Y₄₂Y₂₀, so the stationary second-order correlation function of the Eq. (21) is given by

g^{(2)} (0) = \frac{{| Y_{3} C_{3} + Y_{4} C_{4} |}^{2}}{[{| Y_{10} C_{1} + Y_{20} C_{2} |}^{2} + {| Y_{31} C_{3} + Y_{41} C_{4} |}^{2} + {| Y_{32} C_{3} + Y_{42} C_{4} |}^{2}]} .

When |Y₃C₃ + Y₄C₄ |² ≈ 0, a strong photon antibunching indicated by g⁽²⁾ (0) ≪ 1 will be the signature of the strong photon blockade effect [21]. Then, using the quantum interference mechanism according to the Eq. (23), we now turn our attention to how to enhance the one-photon blockade in the coherently-driven nanocavity QED system, in which the coupling strength of the qubit-cavity subsystem is so strong that the effects of the CRTs should be considered.

3.2. Quantum interference mechanism and optimal condition of the strong one-photon blockade

In what follows, in order to discuss how to enhance the one-photon blockade, we now analyze the quantum interference mechanism and optimal condition of the strong one-photon blockade according to the Eq. (23). The relative phase θ of the two external deriving fields and the Rabi coupling strength Ω of the qubit pumping field play significant roles in the photon blockade effect.

In this nanocavity QED system, the strong coupling strength between the qubit and the single cavity mode is comparable to the cavity damping rate κ, and the two external deriving fields are weak field, i.e.,

{g, κ} ≫ {ε, Ω} .

When the external driving field is tuned to near-resonant with the one-photon excitation, i.e., ω_p ≃ E₁₀ or ω_p ≃ E₂₀, the two-photon near-resonant excitation is strongly suppressed because the energy levels of the second excited states are far-off resonance with an energy gap Δ′ as shown in Fig. 2. Furthermore, the off-resonance energy gap Δ′ should be larger and change with the increase of the coupling strength g between the qubit and the single cavity mode. These conditions are also necessary to ensure the one-photon blockade. In these weak-driving and one-photon excitation limits [30, 67, 68], the probability amplitudes satisfy the relations

C_{0} ≫ {C_{1}, C_{2}} ≫ {C_{3}, C_{4}} .

So, we can calculate the coefficients C_n iteratively with the perturbation theory in the five-state truncation of the Hilbert space in order to get the optimal conditions for the occurrence of the strong one-photon blockade via quantum interference mechanism.

For the probability amplitudes {C₁, C₂} of one-photon states, i.e., the first-order perturbation, we can make a three-state truncation of the Hilbert space, keeping only the ground state |ψ₀〉 and excited states |ψ₁〉, |ψ₂〉. In this three-state Hilbert space, the quantum state for the nanocavity QED system can be expressed as $| Ψ (t) 〉 = \sum_{n = 0}^{2} C_{n} | ψ_{n} 〉$ . Substituting of the wave function into the Schrödinger equation $i \frac{\partial | Ψ (t) 〉}{\partial t} = H_{T} | Ψ (t) 〉$ , we can get the coupled differential equations of the probability amplitudes. But the coupled differential equations of the probability amplitudes should include the dissipations, which are calculated here by using a modified Lindblad master equation adopted to an arbitrary coupling regime between the qubit and the cavity mode in the dressed bases, as discussed in Appendix B. Furthermore, when the coupling strength g and the cavity damping rate κ of the system satisfy g ≪ κ, the secular approximation can be adopted. As mentioned in this paper, in this nanocavity QED system, the coupling strength g is comparable to the cavity damping rate κ, the convenient secular approximation breaks down, and the relaxation in these equations for the probability amplitudes is discussed in Appendix B in the regime of the strong interaction between the qubit and the cavity mode. Then, according to Appendix B, the dynamic equations for the probability amplitudes {C₁, C₂} of one-photon states are given as

\begin{array}{l} i {\dot{C}}_{1} = C_{0} K_{10} + C_{1} Λ_{10} - i C_{2} Γ_{21}^{0}, \\ i {\dot{C}}_{2} = C_{0} K_{20} + C_{2} Λ_{20} - i C_{1} Γ_{12}^{0}, \end{array}

where Λ_n₀ = E_n₀ − ω_p − iΓ_n₀ (n = 1, 2), and the damping constants Γ_nm and cross-damping terms

Γ_{n m}^{a}

are given in Appendix B. Then, the first-order solution of steady-state coefficients {C₁, C₂} can be obtained through Eq. (26), which have the forms

\begin{matrix} C_{1} = - \frac{Λ_{20} K_{10} + i Γ_{21}^{0} K_{20}}{Λ_{20} Λ_{10} + Γ_{12}^{0} Γ_{21}^{0}} C_{0}, & C_{2} = - \frac{Λ_{10} K_{20} + i Γ_{12}^{0} K_{10}}{Λ_{20} Λ_{10} + Γ_{12}^{0} Γ_{21}^{0}} C_{0}, \end{matrix}

For the second-order perturbation in the five-state truncation of the Hilbert space as shown in Fig. 2, the dynamic equations for the probability amplitudes {C₃, C₄} of two-photon states can be given by

\begin{array}{l} i {\dot{C}}_{3} = C_{1} K_{31} + C_{2} K_{32} + C_{3} Λ_{31} - i C_{4} Γ_{4}^{'}, \\ i {\dot{C}}_{4} = C_{1} K_{41} + C_{2} K_{42} + C_{4} Λ_{41} - i C_{3} {Γ^{'}}_{3}, \end{array}

where Λ_n₁ = E_n₀ − 2ω_p − i(Γ_n₁ + Γ_n₂) (n = 3, 4),

Γ_{3}^{'} = Γ_{34}^{1} + Γ_{34}^{2}

,

Γ_{4}^{'} = Γ_{43}^{1} + Γ_{43}^{2}

.

By substituting the steady-state coefficients {C₁, C₂} into Eq. (28), the second-order solution of steady-state coefficients {C₃, C₄} can be obtained approximately

\begin{array}{l} C_{3} = - \frac{(C_{1} K_{31} + C_{2} K_{32}) Λ_{41} + i Γ_{4}^{'} (C_{1} K_{41} + C_{2} K_{42})}{Λ_{41} Λ_{31} + {Γ^{'}}_{3} Γ_{4}^{'}}, \\ C_{4} = - \frac{(C_{1} K_{41} + C_{2} K_{42}) Λ_{31} + i {Γ^{'}}_{3} (C_{1} K_{31} + C_{2} K_{32})}{Λ_{41} Λ_{31} + {Γ^{'}}_{3} Γ_{4}^{'}} . \end{array}

In this paper, a more important matter is about how to enhance the one-photon blockade by using the quantum interference effect. This blockade mechanism is different from the strong coupling mechanism (g ≪ κ), where the higher photon excitations are far off-resonance due to anharmonicity of energy spectrum. In this scheme, the strong photon blockade can be achieved even at a moderate qubit-cavity coupling regime (g ≅ κ), i.e., the coupling strength between the qubit and the single cavity mode is comparable to the cavity damping rate κ. In general, the one-photon near-resonant excitation of the external driven field is the requirement for the one-photon blockade, and the two-photon near-resonant excitation should be suppressed. There are two methods to suppress the two-photon near-resonant excitation. The first method is generally to use the far off-resonance between the energy spectrum and the external classical field, as shown in Fig. 2. The off-resonance Δ′ should be larger by increasing the coupling strength between the qubit and the single cavity mode (g ≪ κ). Then, the process of two-photon excitation is strongly suppressed and one-photon blockade effect is realized. Another method, based on the quantum interference effect under suitable pump or detection conditions, is to suppress the two-photon emission. Obviously, according to the Eq. (23), a strong one-photon blockade can be realized by completely eliminating the two-photon excited states with g⁽²⁾(0) ∼ 0 when

Y_{3} C_{3} + Y_{4} C_{4} \approx 0.

Comparing these two methods, we see that this quantum interference method can relax the requirement of the larger coupling strength between the qubit and the single cavity mode, and the strong photon blockade can be achieved even at a moderate qubit-cavity coupling regime (g ≅ κ). When the parameters (Ω, θ) are chosen appropriately by using the quantum interference mechanism with optimized condition of Eq. (30), it can always realize the strong photon blockade by the quantum interference between the transition paths |ψ₁〉 → |ψ₃〉 and |ψ₂〉 → |ψ₃〉, or by the quantum interference between the transition paths |ψ₁〉 → |ψ₄〉 and |ψ₂〉 → |ψ₄〉, as shown in Fig. 2. The optimized coupling strength and relative phase (Ω_opt, θ_opt) satisfy the following condition

\begin{array}{l} T (Ω_{o p t}, θ_{o p t}) = Y_{3} C_{3} + Y_{4} C_{4} \\ = K_{31} (K_{10} Λ_{20} + i K_{20} Γ_{21}^{0}) (Y_{3} Λ_{41} + i Y_{4} Γ_{3}^{^{'}}) \\ + K_{32} (K_{20} Λ_{10} + i K_{10} Γ_{12}^{0}) (Y_{3} Λ_{41} + i Y_{4} Γ_{3}^{^{'}}) \\ + K_{41} (K_{10} Λ_{20} + i K_{20} Γ_{21}^{0}) (Y_{4} Λ_{31} + i Y_{3} Γ_{4}^{^{'}}) \\ + K_{42} (K_{20} Λ_{10} + i K_{10} Γ_{12}^{0}) (Y_{4} Λ_{31} + i Y_{3} Γ_{4}^{^{'}}) \\ \approx 0. \end{array}

Making the imaginary and real parts of this optimized Eq. (31) equal to zero, we can get the optimized coupling strength and relative phase (Ω_opt, θ_opt). The relative phase between the two driving fields can be controlled precisely by using piezoelectric transducers in experiments [69]. With an adjustment to the coupling strength Ω and relative phase θ, the strong photon blockade can be achieved even at a moderate qubit-cavity coupling regime (g ≅ κ) with the quantum interference mechanism. But, in this strong coupling regime with the CRTs and without using the secular approximation, the optimized coupling strength and relative phase (Ω_opt, θ_opt) are very difficult to solve analytically; however, we can get the numerical solution of the optimized conditions (Ω_opt, θ_opt) for g⁽²⁾(0) ≪ 1 which is the signature of the strong one-photon blockade effect.

Having achieved the optimized conditions (Ω_opt, θ_opt) of the one-photon blockade in the strong coupling regime between the qubit and the cavity mode with the CRTs and without using the secular approximation, we now discuss the one-photon blockade. In Fig. 3, we show the second-order correlation function g⁽²⁾(0) (a) and the mean output photon number n (b) as a function of cavity-light detuning Δ_c = ω₀ − ω_p, respectively. The black solid lines show the results of Rabi model with Ω = 0, the blue dashed lines represent the results in the optimized conditions (Ω_opt, θ_opt) = (0.034, −0.396) for ω_p = E₁₀, and the red dash-dotted lines represent the results in the optimized conditions (Ω_opt, θ_opt) = (0.035, 0.373) for ω_p = E₂₀, respectively. Comparing the second-order correlation function g⁽²⁾(0) with (Ω_opt, θ_opt) and without (Ω = 0) the quantum interference mechanism, we can see that, based on the quantum interference mechanism under optimized conditions, strong antibunching g⁽²⁾(0) ≪ 1 is obtained in the cases of Δ_c = ω₀ − E₁₀ and Δ_c = ω₀ − E₂₀ when g = κ. This indicates that a strong one-photon blockade can be achieved in our scheme under the optimized conditions by using the quantum interference. Physically, the strong one-photon blockade and the strong antibunching are originated from the quantum interference between the transition paths |ψ₁〉 → |ψ₃〉 and |ψ₂〉 → |ψ₄〉, or the quantum interference between the transition paths |ψ₁〉 → |ψ₄〉 and |ψ₂〉 → |ψ₄〉, as shown in Fig. 2. Meantime, comparing with Ω = 0, a large output photon number is achieved under the optimized conditions even in the case of g ≃ κ. So, in reality, this avoids the fabrication challenges for preparing nanocavities with high quality factors to get a large output photon number.

Fig. 3 (a) The second-order correlation function g⁽²⁾ (0) and (b) the mean cavity photon number n are functions of cavity-light detuning Δ_c = ω₀ − ω_p in the regime of strong coupling between the qubit and the cavity mode, respectively. The black solid lines show the results of Rabi model with Ω = 0, the blue dashed lines represent the results in the optimized conditions (Ω_opt, θ_opt) = (0.034, − 0.396) for ω_p = E₁₀, and the red dash-dotted lines represent the results in the optimized conditions (Ω_opt, θ_opt) = (0.035, 0.373) for ω_p = E₂₀. Here g = κ, Δ = ω₀ − ω = 0, ε = 10⁻²κ, κ = 0.2GHz, γ = 10⁻³κ and ω₀ = 1GHz.

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Notice that, in this strong coupling regime between the qubit and the cavity mode, the effects of the CRTs are included. It can lead to the asymmetry of the second-order correlation function g⁽²⁾(0) and the mean cavity photon number n as depicted in Fig. 3, including the asymmetry of the height and position of the peak. This is very different from the J-C model employing the RWA. The physical origin of this phenomenon is that the damping coefficients $Γ_{n m}^{c}$ , $Γ_{n^{'} m^{'}}^{c}$ depend on the respective transition frequencies E_nm = E_n − E_m, i.e., these damping coefficients are very sensitive to interaction strength g, and they change with the interaction strength g between the qubit and the cavity mode, which means that all these damping coefficients $Γ_{n m}^{c}$ , $Γ_{n^{'} m^{'}}^{c}$ have different damping strength. This physical origin contributes to the asymmetry of the height of the second-order correlation function. Meantime, the effect of the CRTs is considered, which leads to the asymmetry of the energy level gaps, i.e., ω₀ − E₁₀ ≠ E₂₀ − ω₀. Consequently, this means the asymmetry of the position of the peak.

Under this strong one-photon blockade g⁽²⁾(0) ≪ 1 by using the quantum interference, we are also interested in the mean cavity photon number n. This is because that, in our system, the cavity damping rate κ is strong and comparable to the strong coupling strength between the qubit and the single cavity mode, i.e., κ ≠ g, and this condition is unfavorable to realize an ideal single photon source. Here, the mean cavity photon number n, which is function of cavity-light detuning Δ_c and cavity-qubit detuning Δ, is plotted in Fig. 4 under the optimized conditions (Ω_opt, θ_opt). Even for a strong cavity damping rate, a large mean cavity photon number (cavity output) is achieved with optimized parameters when the cavity-light detuning Δ_c satisfies Δ_c = ω₀ − E₁₀ or Δ_c = ω₀ − E₂₀, and the second-order correlation function still satisfies g⁽²⁾(0) 1. Consequently, it avoids the fabrication challenges for preparing nanocavities with high quality factors. Thus the scheme with the quantum interference can be used to obtain an ideal single photon source, which is more feasible in experiments.

Fig. 4 The mean cavity photon number n under the optimized conditions (Ω_opt, θ_opt) is a function of cavity-light detuning Δ_c (= ω₀ − ω_p) and cavity-qubit detuning Δ = ω₀ − ω. Here, the other parameters are the same as in Fig. 3.

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In theory, with the increase of the coupling strength between the qubit and the single cavity mode, one-photon blockade effect can be enhanced with g⁽²⁾(0) ≪ 1. This blockade mechanism is the strong coupling mechanism. The physical origin of this blockade mechanism is that, under the one-photon near-resonant excitation, i.e., ω_p = ω_L ≃ E₁₀ (E₂₀), the two-photon near-resonant excitation are strongly suppressed because the energy levels of the second excited states are far-off resonance with an energy gap Δ′ as shown in Fig. 2. With the increase of the coupling strength between the qubit and the single cavity mode, the far-off resonance is enhanced, the process of two-photon excitation is strongly suppressed, and one-photon blockade effect is enhanced as shown in Fig. 5, even under Ω = 0. Meantime, it can be seen that, under the quantum interference mechanism with optimized condition (Ω_opt, θ_opt), the one-photon blockade effect is tremendously enhanced to those without using the quantum interference mechanism. And the mean cavity photon number shows an obvious increase under the strong one-photon blockade as illustrated in Fig. 5(b). This effect provides an effective way to realize an ideal single photon source.

Fig. 5 (a) The second-order correlation function g⁽²⁾(0) and (b) the mean cavity photon number n are functions of the coupling strength g between the qubit and the single cavity mode in the optimized conditions (Ω_opt, θ_opt). The black solid lines show the results of Rabi model with Ω = 0, The blue dashed lines represent the results of the driving frequency ω_p = E₂₀, and the red dash-dotted lines represent the results of the driving frequency ω_p = E₁₀. Here ε = 10⁻¹κ, κ = 0.1GHz, γ = 10⁻²κ, Δ = ω₀ − ω = 0 and ω₀ = 1GHz.

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4. Conclusions

In the nanocavity QED system, the one-photon blockade via quantum interference is investigated by the modified Lindblad master equation and without using the secular approximation. Based on the dressed bases of the Rabi Hamiltonian, the Hamiltonian of the external weak pumping and driving fields is expanded by the dressed bases of the Rabi Hamiltonian. Then, we can get the Hamiltonian of the total system with the dressed bases of the Rabi Hamiltonian. By considering the effects of the CRTs in the strong coupling regime between the qubit and the single cavity mode, the standard Lindblad master equation employing the RWA fails, so a modified Lindblad master equation valid for any arbitrary degree of the qubit-cavity interaction is obtained by using the dressed bases of the Rabi Hamiltonian. It is found that the damping coefficients are very sensitive to interaction strength between the qubit and the cavity mode. In the meantime, the generalized second-order correlation function in the regime of the strong interaction is also obtained. Using the generalized second-order correlation function and the second-order perturbation in the five-state truncation of the Hilbert space, how to enhance the one-photon blockade by using the quantum interference effect is discussed, it is found that, under suitable pump or detection conditions, a strong one-photon blockade can be realized by completely eliminating the two-photon emission. Moreover, even for a strong cavity damping rate, the mean cavity photon number shows an obvious increase under the strong one-photon blockade. It avoids the fabrication challenges for preparing nanocavities with high quality factors. Thus the scheme with the quantum interference can provide an effective way to realize an ideal single photon source, which is more feasible in experiments.

Appendix A: Exact solution of the Rabi model

In order to get the exact solution of the Rabi model (2) when the effects of the CRTs are included, we can write a transformed Hamiltonian with a rotation around the y axis by an angle of π/2. Consequently, the Hamiltonian of the Rabi model (2) can be given by a matrix form, and it is

ℋ = (\begin{matrix} ω b^{†} b + g (b^{†} + b) & - \frac{ω_{0}}{2} \\ - \frac{ω_{0}}{2} & ω b^{†} b - g (b^{†} + b) \end{matrix}) .

For this transformed Hamiltonian of the Rabi model, it is a conserved quantity about $Π = - σ_{z} e^{i π b^{†} b}$ , and is satisfies [ℋ, Π] = 0. The dressed bases of Eq. (32) can be written as

| Ψ_{n} 〉 = (\begin{matrix} Σ_{l = 0}^{m} c_{n}^{l} | l 〉 \\ \pm Σ_{l = 0}^{m} {(- 1)}^{l} c_{n}^{l} | l 〉 \end{matrix}),

where +(−) stands for even (odd) parity, and m is the truncated number. Transforming back to the original frame, the dressed bases of Hamiltonian (2) can be expressed as

| ψ_{n} 〉 = \frac{1}{\sqrt{2}} \sum_{l = 0}^{m} c_{n}^{l} {[1 \mp {(- 1)}^{l}] | e, l 〉 + [1 \pm {(- 1)}^{l}] | g, l 〉} .

The dressed bases of Eq. (34) satisfy stationary Schrödinger equation ℋ|ψ_n〉= E_n|Ψ_n〉. Then, the coefficients $c_{n}^{l}$ meet the following recurrence relation [48,57]

c_{n}^{l + 1} = \frac{E - ω l \pm \frac{ω_{0}}{2} {(- 1)}^{l}}{g \sqrt{l + 1}} c_{n}^{l} - \sqrt{\frac{l}{l + 1}} c_{n}^{l - 1} .

The roots of Eq. (35) will give the high-accuracy numerical eigenenergies E_n of the Rabi model without the RWA if l (or m) is large enough so that we can set $c_{n}^{l + 1} = 0$ . Using the high-accuracy energy levels E_n and the recurrence relation (35) in the truncated state space, we can get the high-accuracy coefficients $c_{n}^{l}$ of the dressed bases |ψ_n〉 corresponding to eigenenergy E_n [57]. Here the high-accuracy coefficients $c_{n}^{l}$ should meet the normalization relation $2 Σ_{l = 0}^{m} | c_{n}^{l} |^{2} = 1$ for given a E_n.

After obtaining the dressed bases and eigenenergies, the Hamiltonian of the Rabi model (2) can be expressed as Eq. (5).

Appendix B: Dissipations in the dressed bases for the strong coupling system

Likewise, in the regime of the strong interaction in qubit-cavity system, the standard Lindblad master equation employing the RWA fails. In this Appendix B, we derive a modified Lindblad master equation adopted to an arbitrary coupling regime between the qubit and the cavity mode in the dressed bases.

Expressing the system Hamiltonian in the dressed bases H_s = Σ_n E_n|n〉〈n| and a free Hamiltonian of bath of harmonic oscillators $H_{B} = Σ_{l} ν_{l} b_{l}^{†} b_{l}$ the coupling in the interaction picture takes the form [61]

H_{s B} = \sum_{m n l} α_{l} C_{m n} | m 〉 〈 n | (b_{l}^{†} e^{- i ν_{l} t} + b_{l} e^{i ν_{l} t}) e^{- i E_{k j} t},

where α_l is the coupling strength between the system and the bath of quantum harmonic oscillators, C_mn = 〈m|(c^† + c)|n〉 (c = a, S₋) and E_nm = E_n − E_m.

For the Hamiltonian H_R of the Rabi model (2), the quantity $Π = {(- 1)}^{a^{†} a + S_{+} S_{-}}$ is a conserved quantity and satisfies [H_R, Π] = 0, system dressed bases |j〉 have a well-defined parity, and the orthogonal features with C_n,n = 0 and C₂_n_,2_n₋₁ = 0 can be found. Meantime, we assume that the coupling strength α_l between the system and the bath of quantum harmonic oscillators is weak, the RWA can still be adopted, then the Eq. (36) can be rewritten as

H_{s B} = s (t) B^{†} (t) + s^{†} (t) B (t)

with

s (t) = \sum_{m, n > m} C_{m n} | m 〉 〈 n | e^{- i E_{n m} t}, B (t) = \sum_{t} α_{l} b_{l} e^{- i v_{l} t} .

where the dressed bases are labeled according to E_nm for n > m.

Assuming that the dissipation baths are still treated in the Born-Markov approximation, we can obtain the master equation in the dressed picture when following the standard procedure. For a T = 0 reservoir, the master equation in the dressed picture takes the form

{\dot{ρ}}_{I} (t) = \int_{0}^{t} d t^{'} 〈 B (t) B^{†} (t^{'}) 〉 [s (t^{'}) ρ_{I} (t^{'}), s^{†} (t)] + H . c .,

where the reservoir correlation functions take the form

〈 B (t) B^{†} (t^{'}) 〉 = \int_{0}^{\infty} D (ω) {| α_{l} |}^{2} e^{- i v_{l} (t - t^{'})} d ω, (T = 0)

the spectral density D(ω) of the bath and the system-bath coupling strength α_l depend on the respective transition frequency E_nm. Considering the cavity that couples to the momentum quadratures of fields in one-dimensional output waveguides, we can assume the spectral densities D(ω) to be constant and

α_{l}^{2} \propto E_{n m}

.

Following the standard procedure and neglecting the Lamb shifts, we can get the damping terms ℒ_cρ(T = 0) of the master equation as the form [70,71]

ℒ_{c} ρ = - \frac{1}{2} \sum_{n > m} Γ_{n m}^{c} {[| n 〉 〈 m |, | m 〉 〈 n | ρ] + H . c .} - \frac{1}{2} \sum_{n^{'} > m^{'}} Γ_{n^{'} m^{'}}^{c} {[| n 〉 〈 m |, | m^{'} 〉 〈 n^{'} | ρ] + H . c .},

where

Γ_{n m}^{c}

are the spontaneous damping constant, and

Γ_{n^{'} m^{'}}^{c}

represent the cross-damping terms from the cross coupling without using the secular approximation. These constants take the form [58,59,66]

\begin{matrix} Γ_{n m}^{c} = Γ \frac{E_{n m}}{ω_{0}} {| C_{m n} |}^{2}, & Γ_{n^{'} m^{'}}^{c} = Γ \frac{E_{n^{'} m^{'}}}{ω_{0}} C_{m n} C_{n^{'} m^{'}}^{*}, \end{matrix}

where Γ = κ, γ. γ and κ are the standard damping rates of the qubit and the cavity mode in weak coupling scenario (g/ω₀≪1) in which the RWA can be adopted. Note that these damping coefficients

Γ_{n m}^{c}

,

Γ_{n^{'} m^{'}}^{c}

depend on the respective transition frequencies E_nm = E_n−E_m, i.e., these damping coefficients are very sensitive to the interaction strength g, and they change with the interaction strength g between the qubit and the cavity mode. Furthermore, we can find that, when ω₀≫g, the coupling between the qubit and the cavity mode is very weak, which means that the RWA can be adopted. Then, E_nm/ω₀ ≈ 1 and the dressed bases |m〉, 〈n| are simplified as the dressed bases in J-C model; these damping constants

Γ_{n m}^{c}

and cross-damping terms

Γ_{n^{'} m^{'}}^{c}

agree with the results in J-C model, i.e., these constants do not depend on the interaction strength g.

Then the density-matrix element can be obtained when following the standard procedure. Furthermore, when the coupling strength g and the damping constant Γ of the system satisfy g≫Γ, the secular approximation can be adopted. As mentioned in this paper, the the coupling strength g is comparable to the damping constant Γ, then the convenient secular approximation breaks down. In the Schrödinger picture and in the three-state Hilbert space, the density-matrix elements about the part of dissipation are given by

\begin{array}{l} {\dot{ρ}}_{11} = - 2 Γ_{10} ρ_{11} - (Γ_{21}^{0} ρ_{21} + c . c .), \\ {\dot{ρ}}_{22} = - 2 Γ_{20} ρ_{22} - (Γ_{12}^{0} ρ_{12} + c . c .) . \end{array}

Meantime, in the five-state Hilbert space, the density-matrix elements about the part of dissipation take the form

\begin{array}{l} {\dot{ρ}}_{33} = - 2 (Γ_{31} + Γ_{32}) ρ_{33} - [(Γ_{43}^{1} + Γ_{43}^{2}) ρ_{43} + c . c .], \\ {\dot{ρ}}_{44} = - 2 (Γ_{41} + Γ_{42}) ρ_{44} - [(Γ_{34}^{1} + Γ_{34}^{2}) ρ_{34} + c . c .] . \end{array}

The damping coefficients including the qubit and the cavity mode can be written as

Γ_{n m} = \frac{γ}{2} \frac{E_{n m}}{ω_{0}} {| S_{m n} |}^{2} + \frac{κ}{2} \frac{E_{n m}}{ω_{0}} {| Z_{m n} |}^{2},

Γ_{n m}^{a} = \frac{γ}{2} \frac{E_{n a}}{ω_{0}} S_{a n} S_{a m}^{*} + \frac{κ}{2} \frac{E_{n a}}{ω_{0}} Z_{a n} Z_{a m}^{*} .

Using the probability amplitudes Ċ_n = −AC_n − BC_m,we can get

{\dot{ρ}}_{n n} = \frac{d C_{n} C_{n}^{*}}{d t} = C_{n}^{*} \frac{d C_{n}}{d t} + C_{n} \frac{d C_{n}^{*}}{d t} = - 2 A ρ_{n n} - B (ρ_{n m} + ρ_{m n}) .

In a comparison between the expressions of the density-matrix elements ${{\dot{ρ}}_{n n}}$ and the probability amplitudes {Ċ_n}, the probability amplitudes {Ċ_n} can be written as

\begin{array}{l} {\dot{C}}_{1} = - C_{1} Γ_{10} - C_{2} Γ_{21}^{0}, & {\dot{C}}_{2} = - C_{2} Γ_{20} - C_{1} Γ_{12}^{0}, \\ {\dot{C}}_{3} = - C_{3} (Γ_{31} + Γ_{32}) - C_{4} (Γ_{43}^{1} + Γ_{43}^{2}), & {\dot{C}}_{4} = - C_{4} (Γ_{41} + Γ_{42}) - C_{3} (Γ_{34}^{1} + Γ_{34}^{2}), \end{array}

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grants No. 61275123, No. 11271147, and No. 11474119), the China Postdoctoral Science Foundation (Grant No. 2015M570652), and the Natural Science Foundation of Hubei Province of China (Grant No. 2015CFB661).

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Photon blockade via quantum interference in a strong coupling qubit-cavity system

Abstract

1. Introduction

2. Model and modified Lindblad master equation in the dressed picture

3. Generalized second-order correlation function and quantum interference mechanism

3.1. Generalized second-order correlation function in the regime of the strong interaction

3.2. Quantum interference mechanism and optimal condition of the strong one-photon blockade

4. Conclusions

Appendix A: Exact solution of the Rabi model

Appendix B: Dissipations in the dressed bases for the strong coupling system

Acknowledgments

References and links

Cited By

Figures (5)

Equations (48)

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