1. Introduction

Optical nonreciprocity, allowing the flow of light from one side but blocking it from the opposite side, is indispensable in designing directional devices. However, previous works mainly focused on the classical input-output nonreciprocity, i.e., one-way control of transmissivity. Recently, quantum nonreciprocity of light has been explored, including one-way quantum amplifier [1,2], router of thermal noises [3], and nonreciprocal photon blockade [4].

Photon blockade (PB) is a pure quantum effect which has drawn extensive attention due to its promising applications in quantum communication and quantum networking [5–12]. Until now, two kinds of PB with different physical mechanisms have been uncovered. One is the conventional photon blockade (CPB) which is induced by anharmonicity [13–17], and the other is the unconventional photon blockade (UCPB) which originates from quantum destruction interference [18–28]. UCPB provides a strong PB effect beyond strong coupling limitation, and conquers the weakness of CPB.

Recently, controllable UCPBs, which are essential to producing nonreciprocal UCPB, have been introduced in cavity-driven QED systems [29–31], and nonreciprocal CPBs and UCPBs have been reported in high-speed spinning nonlinear whispering-gallery resonators [4,32,33]. Anti-bunching or bunching light outcomes when the incident direction of light is alternated. However, a general theory of nonreciprocal UCPB and quantum nonreciprocity is still blurry.

As nonreciprocal PB is a kind of optical nonreciprocity, spatial symmetry and linearity breakings are essential to producing nonreciprocity [34–36]. In this sense, the rotating ring cavity or asymmetrical cavity played the role of breaking spatial symmetry in previous works.

In this work, we will introduce an all-optical nonreciprocal UCPB in an asymmetrical Fabry-Perot cavity embedded two two-level atoms (TLAs) with different coupling strength. In the atoms-cavity coupling model, time-reversal symmetry is broken naturally by taking atomic spontaneous decay into account, and weak optical nonlinearity is at hand when higher excitations are considered. Spatial symmetry breaking comes out from two aspects, one is from the asymmetrical cavity itself with different walls, and the other is from the two embedded atoms with different atom-cavity coupling strength. Therefore, such model provides us an ideal platform to check the contributions of time-reversal asymmetry, two kinds of spatial asymmetry and optical nonlinearity to nonreciprocal UCPB. In fact, the contribution of spatical asymmetry induced by asymmetrical arranged atoms to optical nonreciprocity has rarely been revealed until now.

This paper is organized as follows. In Sec.II, the model and its Hamiltonian are introduced. The wave functions and their steady solution have been obtained by Schrodinger equation and Cramer’s rule. In Sec.III, the contribution of two asymmetrical arranged atoms to controllable UCPB is discussed in detail, and the collaboration of time-reversal asymmetry, spatial asymmetry and optical nonlinearity is specified in producing nonreciprocal UCPB. As a result, an improving nonreciprocal UCPB is gotten in wider operating regime in Sec.IV. The conclusions are given in Sec. V.

2. Theoretical model

Recently, UCPB in symmetrical cavity-driven QED system has been discussed, showing a prospective UCPB effect in weak coupling regime [30]. Here, we modify such model and consider two noninteracting two-level atoms (TLAs) with transition frequency ω_a embedded in a single-mode asymmetrical cavity with frequency ω_c. The cavity is driven by a continuous-wave (CW) coherent field with frequency ω_L as the input field.

Different from previous works, we introduce two kinds of spatial symmetry breaking here. The first is the asymmetrical F-P cavity itself which is assembled two mirrors with different reflectivity. In general, cavity-loss rates $\kappa_1$ and $\kappa_2$ of the cavity walls M₁ and M₂ are different from each other, fulfilling $\kappa_i ={-} c\ln {R_i}/2L$ (i = 1,2), where c is the velocity of light, R_i the reflectivity of cavity wall M_i, L the effective length of the optical cavity. The cavity is symmetrical in case of κ₁ = κ₂. We present κ_ave = (κ₁ +κ₂)/2 as the average cavity-loss rate, and the total cavity-loss rate is κ = κ_ave +κ_loss where κ_loss represents the cavity mode dissipation. We fix κ_loss/κ <<1 as the high-quality and low dissipation cavity is considered in this work. The second is two asymmetrical arranged atoms with different atom-cavity coupling strengths. We note g_i = gcosϕ_i (i = 1,2) as the position-dependent atom-cavity coupling strength where g is the maximal atom-cavity coupling strength of single atom. Here ${\phi _i} = 2\pi {z_i}/{\lambda _c}$ is related to the position of atom ${z_i}$ and the wavelength of the cavity mode λ_c =2π/ω_c. Atomic arrangement is symmetric (or antisymmetric) only when ${\phi _1} = {\phi _2}$ (or ${\phi _1} = {\phi _2} + \pi $), but it is arranged asymmetrically in common cases.

Time-reversal symmetry breaking is gotten by the atomic spontaneous decay rate γ and cavity loss rate κ_loss in this work. Therefore, we present two kinds of spatial symmetry breaking and time-reversal symmetry breaking in the atoms-cavity coupling model. The schematic diagram is shown in Fig. 1(a).

 

Fig. 1. (a) Scheme of the asymmetrical cavity-atoms coupling system driven by CW on the left wall M₁ with driving strength η. Here, cavity-loss rate κ₁ (κ₂) represents the decay of cavity mode through M₁ (M₂), ω_a (ω_c and ω_L) the frequency of atom (cavity and light). The purple circles in the cavity represent two TLAs with different atom-cavity coupling strengths g₁ and g₂. Panel (b) shows a state-crystal diagram and the green nodes are the quantum states in two-exciton framework. By introducing the collective basis, each state only couples with the nearest state and the state-state interaction strength is specified.

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In a frame rotating with the frequency of input field ω_L, the Hamiltonian of the whole system is

Here, the first two items are the energies of the bare atoms and cavity field. ${\mathrm{\Delta }_a} = {\delta _a} - i\gamma /2$ (${\mathrm{\Delta }_c} = {\delta _c} - i\kappa /2)$ represents the complex detuning between the atom (cavity mode) and input field while δ_a=ω_a− ω_L (δ_c=ω_c− ω_L) is the real detuning. The third one is the atom-cavity coupling, and the last term in Eq. (1) $\; {H_d} = \eta ({a + {a^\dagger }} )$ is the cavity-driven term with η being the effective driving strength. Here, a is the annihilation operator of cavity mode, and σ_i = |g〉_ii〈e| (i =1,2) the atomic pseudo spin operators.

It needs to point out that the effective driving strength $\eta = \sqrt {{\kappa _i}} {b_{in}}$ (i =1 or 2) is relied on the input direction, where ${\; }{b_{in}} = \sqrt {{n_{in}}} $ and ${n_{in}} = {P_{in}}/\hbar {\omega _L}$ characterize the normalized amplitude and intensity of incoming field with power P_in respectively. If the external light field is incident on left wall (M₁) of the cavity and transmits through the right wall (M₂), we define it as the forward case, as shown in Fig. 1(a). Meanwhile the backward case is the opposite one, in which light propagates from M₂ to M₁. We focus on controllable UCPB in the forward case at first, and then the backward case can be easily obtained by alternating κ₁ with κ₂ and ϕ₁ with ϕ₂.

It is convenient to explore the physical mechanism of UCPB from the state-crystal-like energy diagram, as shown in Fig. 1(b). The wave function of the system can be expanded by the collective basis |gg, n+1〉, |±, n〉 = (|eg, n〉 ± |ge, n〉) /$\sqrt 2 \; $ and |ee, n−1〉. Here, the state |αβ, n〉 = |αβ〉⊗|n〉 is the product state of the atoms and cavity. We truncate the Hilbert space into the two-quantum manifold which is reasonable in the case of weak driving limitation. The wave function is reading

In Fig. 1(b), for n = 0, only ground state |gg, 0〉 occupies the lattice node of the bottom layer. For n = 1, available states |gg, 1〉, |+, 0〉 and |−, 0〉 occupy three nodes of the middle layer. For n = 2, there are four nodes in the up layer with states |gg, 2〉, |+, 1〉, |−, 1〉 and |ee, 0〉. The state only interacts with the nearest neighbor states in the energy diagram, and their interaction strengths are shown in Fig. 1(b). It should be noticed that such state-crystal-like energy diagram can be extended to higher excitation case if higher order UCPB effect is considered.

It can be seen that the bunching state |gg, 2〉 interacts directly with states |+, 1〉, |−, 1〉 and |gg, 1〉, which provides rich quantum interferential pathways to produce bunching photons. If all these quantum channels interfere destructively and completely, two-photon excitation will not be allowed. And then the probability of detecting bunching photons will be zero.

We followed a much standard procedure, which has been widely applied and been verified its validity in previous works, to get the steady solution of the coupled system. The evolution of the wave function is governed by Schrödinger equation. By setting column vectors as $C = {\left( {\begin{array}{{ccccccc}} {{c_{gg,1}}}&{{c_{ + ,0}}}&{{c_{ - ,0}}}&{{c_{gg,2}}}&{{c_{ + ,1}}}&{{c_{ - ,1}}}&{{c_{ee,0}}} \end{array}} \right)^T}$ and $A = {\left( {\begin{array}{{ccccccc}} { - \eta {c_{gg,0}}}&0&0&0&0&0&0 \end{array}} \right)^T}$, we can get the dynamical equations for the amplitudes of each state, reading

The 7 × 7 matrix M is the truncated Hamiltonian in the two-exciton framework, reading

Here, ${\mathrm{\Delta }_{ac}} = {\mathrm{\Delta }_a} + {\mathrm{\Delta }_c}$, and ${g_ \pm } = {{({{g_1} \pm {g_2}} )} / {\sqrt 2 }}$. Basically, the differential group of Eq. (3) can be numerically solved by using Runge-Kutta method. We reconstruct the column vector $C = {C / {\sqrt {\sum\limits_{\alpha \beta ,n} {{{|{{c_{\alpha \beta ,n}}} |}^2}} \textrm{ + }\sum\limits_{ {\pm} ,n} {{{|{{c_{ {\pm} ,n}}} |}^2}} } }}$ to make sure the total population equal to 1 in each step of numerical simulation. Meanwhile, according to the Cramer's rule, the steady solution can be obtained analytically by setting c_gg,0 ≡ 1, reading

Here, C_i is the i-th component of vector C, $D = \textrm{det}(M )$ and ${D_i} = \textrm{det}({{M_i}} )$ are the determinants of M and M_i where M_i is the transformation of matrix M by replacing the i-th column with column vector A.

Then we get the steady solution of the differential simultaneous equations with respect to the probability amplitude of each state. The average photon number in the cavity ${n_c} = \langle{a^\dagger }a\rangle$ is reading

It indicates that the cavity field is composed both bunching component, i.e., c_gg,2 and anti-bunching components, i.e., c_gg,1, c_+,1 and c_−,1. By introducing the output field operator b_out and applying the quantum input-output theory [37], we can get

The normalized output intensity from cavity wall M₂ is ${n_{out}} = \langle b_{out}^\dagger {b_{out}}\rangle = {\kappa _2}{n_c}$. The equal-time second-order correlation function for the output photons is

The fully antibunching light can be obtained by solving G⁽²⁾(0) = 0 or c_gg,2= 0 analytically. The result should satisfy the following equation

Here, $X = {\eta ^2} = {\kappa _1}{n_{in}}$, $B ={-} {{({g_\textrm{ + }^2 + g_ -^2} )} / 2} - 2{\Delta _a}{\Delta _{ac}}$, and $C ={-} 2g_ + ^2g_ - ^2 + {{{\Delta _a}{\Delta _{ac}}({g_\textrm{ + }^2 + g_ -^2} )} / 2} + \Delta _a^2\Delta _{ac}^2$. Mathematically, we can get restrictions for antibunching output light by solving Eq. (9), which is to be discussed in the following section. Without loss of generality, we focus on the asymmetrical cavity with ${\kappa _1} < {\kappa _2}$ to discuss the nonreciprocal UCPB. The case of backward input can be easily obtained by alternating κ₁ with κ₂ and ϕ₁ with ϕ₂ simultaneously.

3. Parameter analysis in the case of G⁽²⁾(0) = 0

In this work, we will discuss the parameters satisfying G⁽²⁾(0) = 0 which respects to UCPB. Especially, we will focus on the influence of two asymmetrical arranged atoms to controllable UCPB. After extracting and solving Eq. (9), two restrictions for UCPB can be obtained. The first one is the frequency restriction, reading

It shows that the detunings (δ_a, δ_c) and decay rates (γ, κ) are combined together to match the restriction in such cavity-driven case, which is exactly the same to that in single-atom case [38] and is different from that in the atom-driven case [30]. Especially, in the resonant case with ${\delta _a} = {\delta _c} = 0$, the frequency restriction is matched naturally.

The second one is the intensity restriction, reading

It shows that there are some optimal input intensities to generate UCPB, and they are relied on the input port of the cavity, i.e., ${\kappa _1}$, and the arrangement of atoms, i.e., g₊ and g₋.

It is known that state |+, n〉 evolves into dark state when two atoms are off-phase (g₁= −g₂ ≡ g₀ leading to g₊= 0), while |−, n〉 is the dark state when two atoms are in-phase (g₁= g₂ ≡ g₀ leading to g₋= 0). In the off-/ in-phase cases, the intensity restriction can be further simplified, reading

Such reduced intensity restriction is the same to that for single-atom case, which indicates that single-atom UCPB can be retrieved by exploiting two off-/in-phase atoms in the cavity. In general, both |+, n〉 and |−, n〉 are no longer dark states when |g₁| ≠ |g₂|.

On the other hand, for the fixed input power, we can get the optimal atom-cavity coupling strength g_opt-j (j = F or B represents the forward or backward case), which is defined as the maximal single atom-cavity coupling strength g, to achieve the UCPB. When g_opt-B is different from g_opt-F, nonreciprocal UCPB will happen.

Actually, nonreciprocal UCPB is also exist in the case of single atom in the asymmetric cavity [38]. The corresponding optimal coupling strength in single atom case g_opt-cav-j is derived, reading

Here i = 1 or 2 is for j = F or B, respectively. The direction-dependent optimal atom-cavity coupling strength g_opt-cav-j in case of single-atom originates only from asymmetrical cavity itself. As ${\kappa _1} < {\kappa _2}$, it leads to g_opt-cav-F < g_opt-cav-B.

Now return to the two atom case. g_opt-j is proportional to the direction-dependent coupling strength g_opt-cav-j but inverses to a direction-independent factor $\alpha ({{\phi_1},{\phi_2}} )$. Noticing g_i = gcosϕ_i, optimal atom-cavity coupling strength in two-atoms case is

Such optimal atom-cavity coupling strength obviously related to the input direction and locations of two atoms. The direction-dependent g_opt-cav-j originates only from asymmetrical cavity itself, while the direction-independent $\alpha ({{\phi_1},{\phi_2}} )$ originates from two asymmetrical arranged atoms.

The key point of the case of two atoms is that the asymmetrical arrangement of two atoms will enhance or weaken such direction-dependent UCPB, which is embodied by the atomic arrangement factor α, reading

The periodicity of α shows that it unchanges when the distance between atoms changes λ/2. In addition, it keeps exchange symmetry when we swap the two inside atoms, i.e., $\alpha ({{\phi_1},{\phi_2}} )= \alpha ({{\phi_2},{\phi_1}} )$. It is the reason why asymmetrical arrangement of atoms cannot split the optimal atom-cavity coupling strength g_opt-j, but it can promote the splitting effect as $\alpha \le 1$. Therefore the splitting of direction-dependent optimal atom-cavity coupling strength | g_opt-B − g_opt-F | is enlarged by a factor of 1/α. We call α the promote factor. It needs to point out that a higher coupling strength of g_opt-j may increase the difficulty to produce UCPB, but it indeed amplifies the splitting of | g_opt-B − g_opt-F |. In the special case of α = 0, in which ${\phi _1} = {\phi _2} = \mathrm{\pi }/2$ and g₁ = g₂ = 0, our model becomes into an empty cavity leading to the vanishment of any optical nonreciprocity. When α → 0, it is hard to generate UCPB as g_opt-j → ∞, and it is hard to produce such promoted nonreciprocal UCPB no doubt. The enlarged spliting does not mean an easier nonreciprocal UCPB.

In order to clarify the influence of asymmetrical arrangement of atoms, we plot promotion factor α as function of atomic phases ϕ₁ and ϕ₂ in Fig. 2. As predicted in previous discussion of dark states, promotion factor $\alpha = 1{\; }$ when atoms are off-/in-phase, corresponding those points of $({{\phi_1},{\phi_2}} )= ({0,0} ),({0,\pi } ),({\pi ,0} )$ and $({\pi ,\pi } )$. Also, $\alpha = 1{\; }$ when $({{\phi_1},{\phi_2}} )= ({0,\pi /2} ),\; \; ({\pi ,\pi /2} ),\; \; ({\pi /2,0} )$ and $({\pi /2,\pi } )$ since only one atom couples with the cavity.

 

Fig. 2. Spatial symmetry breaking induced by asymmetrical arrangement of two atoms. The pseudo color maps the value of α versus atomic phases ϕ₁ and ϕ₂, which presents a promoted nonreciprocal UCPB as α is less than the unit.

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Now, we can summarize the effects of these two spatial asymmetries, i.e., asymmetric cavity and asymmetrical arranged atoms, on controllable UCPB. The asymmetric cavity causes the splitting of optimal coupling strength, i.e., g_opt-cav-F ≠ g_opt-cav-B, which is the key to the nonreciprocal UCPB. Meanwhile, the asymmetrical arranged atoms can make the splitting 1/α times, leading to a promoted UCPB as $\alpha \le 1$. Therefore, asymmetrical arrangement of atoms in the cavity opens a new approach to manipulate UCPB.

4. Improvement of nonreciprocal UCPB

As ${\kappa _1} \ne {\kappa _2}$, the splitting of g_opt-F and g_opt-B can produce nonreciprocal UCPB. Under the fixed input power, the second-order correlation function of cavity field for forward input $G_F^{(2 )}(0 )$ would be different from that for backward input $G_B^{(2 )}(0 )$. But in general they may be both larger than 1 or less than 1. Here we focus on the promoted nonreciprocal UCPB, in which $G_F^{(2 )}(0 )> 1$ & $G_B^{(2 )}(0 )< 1$ or vise versa. To achieve it, the asymmetric arraged atoms play the key role.

Reminding the definition of coupling g_i = gcosϕ_i (i = 1, 2), we plot the second-order correlation function of cavity field $G_j^{(2 )}(0 )$ (j = F, B) as a function of the maximal single-atom-cavity coupling strength g in both forward and backward input cases in Figs. 3(a)-(d). For simplicity, we normalize all the parameters by setting $\kappa \equiv 1$& κ_loss = 0.02 and set the asymmetrical cavity with κ₁ = 0.36 & κ₂ = 1.60. When g = g_opt-j (j = F, B), it gets $G_j^{(2 )}(0 )\to 0$, corresponding to minimum point of $G_j^{(2 )}(0 )$ function.

We firstly focus on the resonant cases (${\delta _a} = {\delta _c} = 0$.) in Figs. 3(a) and (b), and set $\gamma = 0.1$, ${n_{in}} = 0.04$ in present numerical simulation. Figure 3(a) refers to the case of symmetric arranged atoms, while Fig. 3(b) to the asymmetrical arranged atoms with ${\phi _1} = \pi /3$ and ${\phi _2} = 2\pi /5$. As expected, optal atom-cavity coupling strengths g_opt-F (minimum points in red curves) and g_opt-B (minimum points in blue curves) are separated from each other, and opens an operation window (grey region) to perform promoted nonreciprocal UCPB, in which the output light for the forward input is bunching ($G_F^{(2 )}(0 )> 1$) and that for the backward input is antibunching ($G_B^{(2 )}(0 )< 1$), shown in Figs. 3(a) and (b).

 

Fig. 3. Promoted NONRECIPROCAL UCPB in asymmetrical cavity. In all panels, the red solid curves are G⁽²⁾(0) for the forward case ($G_F^{(2 )}(0 )$) and the blue ones are for the backward case ($G_B^{(2 )}(0 )$). Average cavity-loss rate is fixed and normalized as unit, and we set κ₁ = 0.36, κ₂ = 1.60, κ_loss = 0.02, γ = 0.1 and n_in= 0.04. Panels (a-b) are in resonant case with and without asymmetrical atomic arrangement, where (a) is ϕ₁ = ϕ₂ = 0 and (b) is ϕ₁ = π/3 & ϕ₂ = 2π/5. Normal NONRECIPROCAL UCPB regimes (defined by $G_F^{(2 )}(0 )> 1$ & $G_B^{(2 )}(0 )< 1$) are covered by the gray rectangles while the green lines note the points of giant NONRECIPROCAL UCPB (defined by $G_F^{(2 )}(0 )> 1$ & $G_B^{(2 )}(0 )= 0$). Panels (c) and (d) are in detuning case fulfilling the frequency restriction with asymmetrical atomic arrangement ϕ₁ = π/3 & ϕ₂ = 2π/5, where (a) is δ_a = 0.05 and (b) is δ_a = 0.1. The red cross notes the vanishing of the giant NONRECIPROCAL UCPB.

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The giant nonreciprocal UCPB, defined by $G_F^{(2 )}(0 )> 1$ & $G_B^{(2 )}(0 )\to 0$ and as specified by the green dashed lines in Fig. 3(a) and (b), can produce the strongest UCPB nonreciprocity. The contrast of the second-order correlation function of giant nonreciprocal UCPB is improved dramatically since $G_B^{(2 )}(0 )\to 0$ in the backward case.

Compared Fig. 3(a) with (b), the operation regime in the case of symmetric arranged atom is narrower than that of asymmetrical arranged atoms. It is the result of promoted factor α mentioned above. We call it an improving nonreciprocal UCPB as the regime of normal nonreciprocal UCPB is amplified while it does not mean an easier nonreciprocal UCPB.

We then discuss the case of detuning. Under the asymmetrical arranged atoms with ${\phi _1} = \pi /3$ and ${\phi _2} = 2\pi /5$, the cases of ${\delta _a} = 0.05$ and ${\delta _a} = 0.1$ are discussed in Figs. 3(c) and (d), respectively. Noticed that ${\delta _c}$ is set as ${\delta _c} ={-} ({2\gamma + \kappa } ){\delta _a}/\gamma $ to fulfill the frequency restriction Eq. (7).

In present asymmetrical system with κ₁ = 0.36, κ₂ = 1.60, κ_loss = 0.02, ϕ₁ = π/3, ϕ₂ = 2π/5, γ = 0.1 and n_in = 0.04, the optimal coupling strength is g = 0.637 for δ_a = 0, g = 0.726 for δ_a = 0.05 and g = 0.945 for δ_a = 0.1, shown in the green dashed lines in Figs. 3(b), (c) and (d), respectively. Comparing the resonant case (Fig. 3(b)) with the detuning cases (Figs. 3(c-d)), the regimes of promoted nonreciprocal UCPB shrink with the increasing of detuning ${\delta _a}$. More important, the giant nonreciprocal UCPB disappears when ${\delta _a} = 0.1$, see Fig. 3(d).

In order to show the influence of detuning to giant nonreciprocal UCPB clearly, we plot G⁽²⁾(0) against δ_a in present asymmetrical cavity, as shown in Fig. 4. When g = 0.637 and g = 0.726, giant nonreciprocal UCPBs ($G_F^{(2 )}(0 )> 1$ & $G_B^{(2 )}(0 )\approx 0$) can be obtained at the detuning of ${\delta _a} = 0$ and ${\delta _a} ={\pm} 0.05$, respectively (see green dashed lines in Figs. 4(a) and (b)). However, $G_B^{(2 )}(0 )\approx 0$ at the point of ${\delta _a} ={\pm} 0.1$ when g = 0.945, but $G_F^{(2 )}(0 )< 1$ which indicates the vanishing of giant nonreciprocal UCPB, see the green lines in Fig. 4(c). It is clear that detunings do weaken the regime of giant nonreciprocal UCPB.

 

Fig. 4. Giant NONRECIPROCAL UCPB in detuning case. In all panels, the red solid curves are G⁽²⁾(0) for the forward case and the blue ones are for the backward case. We set κ₁ = 0.36, κ₂ = 1.60, κ_loss = 0.02, γ = 0.1, n_in= 0.04, ϕ₁ = π/3, ϕ₂ = 2π/5 and ${\delta _c} ={-} ({2\gamma + \kappa } ){\delta _a}/\gamma $ to fulfill the frequency restriction . Panels (a) is for g = 0.637, (b) is for g = 0.726 and (c) is for g = 0.945. The green line presents the points of giant NONRECIPROCAL UCPB while the red cross denies it.

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Our model is all-optical without any mechanical rotation comparing to previous works of nonreciprocal UCPB within the rotating optomechanical systems [4,32,33], and it provides a method to manipulate UCPB by controlling asymmetrical arrangement of atoms comparing to single-atom nonreciprocal UCPB in asymmetrical cavity [38]. The physics underlying the nonreciprocal UCPB is controllable quantum destructive interference. From the state-crystal-like energy diagram, as shown in Fig. 1(b), interactions between crystal layers governed by weak driving strengths provide a set of cavity-spatial-symmetry-dependent interferential pathways, while couplings between the nodes on the same layer provide another set of atom-spatial-symmetry-dependent interferential pathways. Quantum destructive interference between two sets of interferential pathways produces controllable UCPB and improvement of nonreciprocal UCPB finally.

At last, we show the experimental possibility to realization of the model. By using the existing magneto-optical trapping technology, cold atoms can be captured and moved into the optical microcavity. Therefore, it is feasible to carry out our prediction under careful designing in cavity QED system. However, capturing two atoms and placing them on appointed positions in microcavity are still challenges. Recently, observation of simultaneous excitation of two noninteracting atoms by a pair of time-frequency correlated photons in a superconducting circuit has been reported [39], and it provides another candidate of experimental realization in superconducting circuits after careful designing.

5. Conclusion

In conclusion, we have proposed a cavity-driven QED model, which is made of an asymmetrical Fabry-Perot cavity containing two noninteracting atoms, to generate nonreciprocal UCPB. In such atoms-cavity coupling system, spatial symmetry is broken by the asymmetrical cavity itself and asymmetrical arrangement of atoms, and weak nonlinearity is at hand by taking higher excitations into account. By solving the motion of Schrodinger equation in weak driving limitation, the frequency and intensity restrictions of UCPB can be formulized. Spatial asymmetry induced by the asymmetrical cavity, cooperated with optical nonlinearity is the origin of nonreciprocal UCPB, while the asymmetrical arrangement of atoms can promote the nonreciprocal UCPB into a wider operating regime. This work reveals the incoordinate roles of two kinds of spatial symmetry breaking for the first time.