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Abstract
We investigated the transmission of single and two photons in a one-dimensional waveguide that is coupled with a Kerr micro-ring resonator and a polarized quantum emitter. In both cases, a phase shift occurs, and the non-reciprocal behavior of the system is attributed to the unbalanced coupling between the quantum emitter and the resonator. Our analytical solutions and numerical simulations demonstrate that the nonlinear resonator scattering causes the energy redistribution of the two photons through the bound state. When the system is in the two-photon resonance state, the polarization of the correlated two photons is locked to their propagation direction, leading to non-reciprocity. As a result, our configuration can act as an optical diode.
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1. Introduction
Nonreciprocity is a fundamental physical mechanism that signifies how symmetric operations generate opposite responses. In recent years, nonreciprocal devices have garnered considerable attention due to their potential applications in quantum information processing [1–14]. However, conventional devices that use the magneto-optical effect may experience significant losses in strong magnetic fields [15–20]. To address this issue and facilitate integrated applications, many physical systems, such as parity-time symmetry optics [21–24], optomechanics [25–30], parametric processes [31,32], and chiral quantum optics [4,33–37], have been attempted. Since the research objects are on a microscopic scale and differ from the classical physical studies mentioned above, nonreciprocal content has been extended to the quantum domain [38–42]. For example, routers of thermal noises [43], nonreciprocal quantum entanglement [44], one-way quantum amplifiers [45], and nonreciprocal photon blockade [46] are quantum physical processes that support the development of quantum technology.
Quantum correlations are an important phenomenon in studying the foundations of quantum theory [47–59]. Electromagnetically induced transparency and photon blockade are commonly used to achieve strong photon-photon interaction [60–63]. Recently, one-dimensional (1D) waveguides coupled to nonlinear systems have gained attention for the transmission of constrained correlated photons. The nonlinear system, acting as the scattering center, can take the form of a multi-level atom [64–71], a cavity filled with Kerr medium [72–74], or be present in optomechanical systems [75,76], which would form bound states with the transmitted photons when exchanging energy. To combine quantum correlation and nonreciprocity, chiral waveguides have been introduced [74,77–80]. By asymmetrically coupling the waveguide and the coupling body, photons incident from opposite directions produce unequal transmission probabilities. Despite this, the application of chiral waveguides is sometimes limited, and it is desirable to achieve the same function in conventional waveguides.
In this work, we propose a scheme for achieving nonreciprocal transmission of a few photons by using chiral quantum emitters (QEs) indirectly coupled with a 1D waveguide through a nonlinear microring resonator. By applying a magnetic field [81–84] or via the ac Stark effect [85–90], the QEs can be polarized and function as a quantum switch [91–94]. We construct two devices to transmit single and double photons, respectively. By using Laplace transform to solve the scattering process of photons, we obtain an analytic solution. For single photon conditions, the nonlinearity does not have any effect, and the photon incident from either port only produces a phase shift. This configuration is similar to a QE embedded with a waveguide [95–100]. For the two-photon case, the incident light collides with the mixture of the QEs and resonator and then exchanges energy to form a bound state. Effective nonreciprocal transmission occurs at two-photon resonance, i.e., when the photon energies match the resonant frequency shifted by the Kerr interaction. Otherwise, one-way transmission will not succeed, even at maximum polarization. This proposal transfers the asymmetric coupling from the waveguide to the QEs such that few-photon nonreciprocal transmissions can occur in conventional waveguides, and may lay the foundation for expanding applications and integration on chip.
This paper is organized as follows: In Section 2, we introduce two scattering models, which involve polarized QEs and nonlinear resonators. We provide simplified Hamiltonians for each of them. In Section 3., we analyze the scattering process of a single photon and derive analytical expressions for the transmission and reflection amplitudes. In Section 4, we investigate the scattering process of two photons and determine the conditions for achieving maximum nonreciprocal transmission of correlated photons. We also present numerical simulations of the transmission process. In Section 5, we discuss the experimental feasibility of our proposed scheme. Finally, we conclude with a summary in Section 6.
2. Physical model
Figure 1 illustrates the schematic of the proposed system, which comprises a nonlinear microring resonator coupled to QEs and an 1D waveguide located at the origin. Similar to the transverse spin and surface waves of acoustic systems [101], when photons are incident on port 1 (port 2), they excite the counterclockwise (CCW) [clockwise (CW)] mode, generating $\sigma _+(\sigma _-)$ polarized evanescent fields near the entire wall [4,102–105]. The QEs are chiral and driven by $\sigma _+$($\sigma _-$) polarization, coupling to the CCW resonator mode and the CW resonator mode with coupling strengths $g_{\circlearrowleft }$ and $g_{\circlearrowright }$, respectively. We can define $D$ to describe the degree of chirality of the QE, as proposed in [105]. Specifically, we have:
The conventional coupling strength is denoted by $g$, and $D=-1$ ($D=1$) corresponds to $\vert g_{\circlearrowleft }\vert = g$ ($\vert g_{\circlearrowright }\vert = g$), which means the QE exhibits perfect chirality. When $D=0$, the chirality disappears and corresponds to linear polarization.
Fig. 1. The system is schematically represented by a microring resonator filled with Kerr-type nonlinear medium, which couples to a nearby waveguide, with either a single negatively charged QE in (a) or two negatively charged QEs in (b). When photons are incident from port 1 (or port 2), they are scattered by the nonlinear resonator and QEs, which results in reflection or transmission of the photons in the waveguide.
In this section, we temporarily neglect the chirality of QEs for simplicity. Therefore, the Hamiltonian for single photon scattering process in Fig. 1(a) is given by Eq. (2) (set $\hbar =1$).
The first term in Eq. (2) is the nonlinear interaction of the system with interaction strength $U$. The creation (annihilation) operator of the resonator mode is denoted by $a^{\dagger }$ ($a$), and the resonance frequency is $\omega _c$. $r_k^{\dagger }(l_k^{\dagger })$ is a bosonic operator creating a right-going (left-going) photon in the waveguide with wave number $k$ and frequency $\omega _k$. In this paper, we assume $\omega _k=k$ (set $v_g=1$), where $v_g$ is the boson velocity in the waveguide. $V$ is the coupling constant, and $a_{e}^{\dagger }(a_{g}^{\dagger })$ is the creation operator of the excited (ground) state of the QE with transition frequency $\omega _q$. $S_{+}=a_{e}^{\dagger }a_{g}$ ($S_{-}=a_{g}^{\dagger }a_{e}$) is the raising (lowering) operator. We neglect the intrinsic decay rate of the resonator and the relaxation rate of the QE.
In the rotating frame with respective to $H_0=w_c [a^{\dagger }a+\int _0^\infty ( r_k^{\dagger } r_k+l_k^{\dagger } l_k) dk +a_{e}^{\dagger }a_{e}]$, there is
Through a similar process, we can obtain the Hamiltonian of the system in Fig. 1(b) for the two-photon scattering process:
The creation operator $a_{ne}^{\dagger }$ ($a_{ng}^{\dagger }$) corresponds to the excited (ground) state of the $n$-th QE with transition frequency $\omega _q$, and $S_{n+}=a_{ne}^{\dagger }a_{ng}$ ($S_{n-}=a_{ng}^{\dagger }a_{ne}$) is the raising (lowering) operator.
In this case, the cavity mode only couples to the $b$ mode with an effective coupling constant of $\overline {V}=\sqrt {2}V$, while photons in the $b'$ mode evolve freely in the waveguide. Therefore, we will only consider the excitations in the $b$ space in the following discussion.
3. Single photon scattering process
As there is no contribution from Kerr nonlinearity for single-photon states, the physical model is equivalent to the work of [105] which employs the method developed by Shen and Fan [109,110] to investigate the single-photon scattering process. In this work, we apply Laplace transform to solve the photon scattering problems [73,74], and the analytical solution will be utilized in the two-photon scenario in the next section.
For a single-photon excitation, a general time-dependent state for the system can be expressed as:
The states $b_k^{\dagger } \vert \varnothing \rangle$ and $a^{\dagger }\vert \varnothing \rangle$ correspond to single excitations in the $k$th $b$ mode of the waveguide and in the resonator, respectively. The operator $a_{e}^{\dagger }a_{g}$ represents the excitation of the QE. The vacuum state $\vert \varnothing \rangle$ denotes the absence of photons in the system, and the QE is in its ground state. The probability amplitudes $\phi _{k}(t)$, $e_{c}(t)$, and $e_{q}(t)$ are defined accordingly.
We use the Schrödinger equation $i\vert \dot {\psi }(t)\rangle =H_a\vert \psi (t)\rangle$ to derive the equations of motion for transmission and reflection:
To transform Eq. (7) into a system of algebraic equations, we perform the Laplace transform and introduce the initial probability amplitudes $\phi _{k}(0)$, $e_{c}(0)$, and $e_{q}(0)$:
The initial excitation of the system is only derived from the photon incident from port 1 (port 2) in the waveguide, and we assume that the incident single photon is prepared in a wave packet with a Lorentzian spectrum. Therefore, the initial condition of the system is given by:
$\delta$ and $\epsilon$ denote the detuning and spectral width of the incident photon, respectively. After performing some calculations, we obtain:Equation (11) indicates that the scattering process does not alter the properties of the incident photon, but only leads to a phase shift. We can define the transmission amplitude in the $b$ mode as follow:
Returning to the original left- and right-propagation modes, we assume that a single-photon wavepacket is incident on the system from port 1. Its propagation state can be described as:
Firstly, the right-moving photon is absorbed by the hybrid system consisting of the resonator and QE. Then, the excited system releases a photon back into the waveguide with a reflected and a transmitted part. The scattering amplitudes can be defined as follows:
If a single-photon wavepacket is incident on the system from port 2, its propagation state can be described as follow:
Since the expression of $r_k$ in Eq. (14) remains the same under the exchange of $\vert g_{\circlearrowleft }\vert$ and $\vert g_{\circlearrowright }\vert$, the reflection amplitude remains unchanged. However, due to the asymmetric coupling, the transmission amplitude changes to:
In general, the propagation of photons is mainly affected by the frequency detuning with the resonator. By introducing the chiral coupling of the QE, the balance of transmission for the original left- and right-moving emitted photons is broken, and the parameter $D$ must be taken into consideration. Figure 2 shows the transmittance (blue solid) and reflectance (red dotted) as a function of $\Delta _k$ for a given $g$ and $D$. Figures 2(b, d, f) are zoomed-in versions of Figs. 2(a, c, e) around the frequency detuning of [-0.1, 0.1]. The peaks and troughs of the scattering amplitude are generated at the resonance [Fig. 2(a)], which means that the scattering system acts as a mirror blocking the propagation of photons [33,110–113]. However, as $D$ increases, a sub-peak and sub-trough are generated with a narrow bandwidth around the resonance, and they reach the extremes at $D=-1$. At this point, previous blocking disappears. Throughout the whole process, the chirality of the QE controls the passage of the incident photon. In Fig. 3, we plot the isolation contrast $\eta =(|t_k|^2-|t_k^{'}|^2)/(|t_k|^2+|t_k^{'}|^2)$ [104,105,114] as a function of $\Delta _k$ for a given $g$ and $D$. We emphasize that $|t_k|^2$ dominates the transmittance when $\eta > 0$ (blue solid), and $|t_k^{'}|^2$ dominates the transmittance when $\eta < 0$ (red dotted). $D=0$ corresponds to $\eta = 0$, which means the system does not exhibit nonreciprocity property. For $D=-1(+1)$, there is $\eta \approx +1 (-1)$. The physical explanation is that only when photons incident to port 1 (port 2) will pass through the resonator, and our device becomes a nonreciprocal system.
Fig. 2. The transmittance (blue solid) and reflectance (red dotted) as a function of $\Delta _k$. (a,b),(c,d),(e,f) are shown for $D= 0, -0.5, -1$, respectively. (b),(d),(f) are the zoom in of (a),(c),(e), respectively. Other parameter is set as $g=0.1$, all parameters are in units of $\beta$.
Fig. 3. The isolation contrast $\eta$ as a function of $D$, for $g=0.1$, $\Delta _k=0.1$, all parameters are in units of $\beta$.
4. Two photon scattering process
Figure 1(b) illustrates the physical model of two-photon scattering. To enable scattering synchronization, we introduce an additional QE coupling in addition to the original hybrid system. The corresponding Hamiltonian is described by Eq. (5).
For double photon excitation, the system includes not only additional cross-combination terms of wave functions, but also Kerr nonlinearity coupling, leading to new physical phenomena. The time-dependent state of the system can be described as follows:
The amplitudes ${\phi _{{kk^{'}}}(t), e_{cc}(t), e_{qq}(t)}$ represent the states of double excitations generated in the waveguide, resonator, and QEs, respectively. $\phi _{{k}c}(t)$ corresponds to the state with one photon in the resonator and one photon with wave number $k$ in the waveguide, while $e_{cqn}(t)$ corresponds to the state with one photon in the resonator and the $n$-th QE excited.
The equations of motion are obtained by applying the Schrödinger equation $i\vert \dot {\Phi }(t)\rangle =H_b\vert \psi (t)\rangle$.
We assume that the incident signal takes the form of a two-photon Lorentz wave packet. Therefore, the initial condition of the system is given by:
To simplify the calculations, we assume that the two photons are initially independent of each other, i.e., $\delta _1=\delta _2=\delta$ and $\epsilon _1=\epsilon _2=\epsilon$. Therefore, Eq. (19) can be rewritten as
We are interested in the long-time limit scattering behavior of two photons in the waveguide. After some approximations (see Appendix), we obtain the asymptotic solution of $\phi _{{kk^{'}}}(t)$ for $t\gg \beta ^{-1}$ and $\epsilon ^{-1}$
The transmission amplitudes in $b$ space with wavenumbers $k$ and $k^{'}$ are denoted by $\overline {t}_k$ and $\overline {t}_{k^{'}}$, respectively. The additional term $C_{kk^{'}}$ is defined as
Equation (23) consists of two main parts. The first part results from the combined effect of the cavity and QE, while the second part arises from the individual function of the QE. Equations (22) and (23) demonstrate that the Kerr nonlinearity coupling contributes to the two-photon scattering process, as $C_{kk'}$ contains the parameter $U$. When $U=0$, the original nonlinear resonator degenerates to a normal one, and the scattered photons lose their correlation. When we set $g=0$, the QE effect disappears, and the result is consistent with [73,74].
Let us now consider two ways of photon incidence. First, assuming that two photons are injected from port 1, the initial state can be expressed as:
After scattering with the cavity and QEs, there are four possible outcomes when two photons are injected from port 1 with the initial state. These are: two photons are transmitted with amplitude $\phi r^{rr}_{kk^{'}}$; two photons are reflected with amplitude $\phi r^{ll}_{kk^{'}}$; two photons exit from port 1 and port 2, respectively, with amplitudes $\phi r^{rl}_{kk^{'}}$ and $\phi r^{lr}_{kk^{'}}$. Finally, in the long-time limit, the state can be expressed as:
The first part of each amplitude represents the free propagation of the two photons, while the second part describes the correlation of the output photons resulting from the scattering with the nonlinear cavity and QEs.
Second, assuming two photons are injected from port 2 with the initial state
Similar to the first situation, and there is
The amplitudes corresponding to the scattering of two photons injected from port 2 are given by $\phi l^{ll}_{kk^{'}}$ for the transmitted photons, $\phi l^{rr}_{kk^{'}}$ for the reflected photons, $\phi l^{rl}_{kk^{'}}$ and $\phi l^{lr}_{kk^{'}}$ for the case where one photon is transmitted and the other is reflected. The expressions for these amplitudes are given by:
Once we choose $D\neq 0$, the balance of the system is broken, meaning that $\phi r^{rr}_{kk^{'}}\neq \phi l^{ll}_{kk^{'}}$. In this sense, the system functions as an optical diode to some extent. For the sake of later discussion, we denote the two parts of $\phi r^{rr}_{kk^{'}}$ by NP and IP, respectively, i.e.,
In the frequency domain, we plot the norm square of the amplitudes of $\phi r^{rr}_{kk^{'}}$, $\phi l^{ll}_{kk^{'}}$, and $\phi r^{ll}_{kk^{'}}$ for $\delta =0$ and $\delta =5$, respectively. When $\delta =0$, the maximum frequency of the two-photon wave packet coincides with the resonator. Incident photons are mainly reflected and uncorrelated [see Fig. 4(e)], while the transmitted photons are strongly correlated, and the photon transmission probabilities are concentrated along the line $\Delta _k+\Delta _{k^{'}}=0$ [see Fig. 4(a) (c)]. It can be inferred that the transmission of incident photons is dominated by the $C_{kk^{'}}$ term since $t_\mu \approx 0$ $(\mu =k,k^{'})$ at zero detuning. Conversely, when $\delta =5$, the transmission parts are almost uncorrelated but dominant in probabilities [see Fig. 4(b) (d)]. Due to $r_\mu \approx 0$, the correlated photons mainly appear in the reflection, and the photon transmission probability is concentrated along the line $\Delta _k+\Delta _{k^{'}}=10$ [see Fig. 4(f)]. Comparing Fig. 4(a) and Fig. 4(c), we observe that in the same frequency domain, $|\phi r^{rr}_{kk^{'}}|^2$ is generally greater than $|\phi l^{ll}_{kk^{'}}|^2$. Specifically, for $\Delta _k=0,1,2$ with $\Delta _k+\Delta _{k^{'}}=0$, we have $|\phi r^{rr}_{kk^{'}}|^2=0.0975, 0.0244, 0.0039$ and $|\phi l^{ll}_{kk^{'}}|^2=0.0395, 0.0099, 0.0016$. This phenomenon is mainly caused by the asymmetric coupling of the QEs.
Fig. 4. (a), (c), (e) are plots of $|\phi r^{rr}_{kk^{'}}|^2$ , $|\phi l^{ll}_{kk^{'}}|^2$ and $|\phi r^{ll}_{kk^{'}}|^2$ of the two transmitted photons as functions of $\Delta _k$, $\Delta _k^{'}$, when $\delta =0$. (b), (d), (f) are plots of $|\phi r^{rr}_{kk^{'}}|^2$ , $|\phi l^{ll}_{kk^{'}}|^2$ and $|\phi r^{ll}_{kk^{'}}|^2$ of the two transmitted photons as functions of $\Delta _k$, $\Delta _k^{'}$, when $\delta =5$. Other parameters are set as $g=0.1$, $D=-0.1$, $U=10$ and $\epsilon =0.1$, all parameters are in units of $\beta$.
We define the contrast for two-photon isolation as $\eta ^2 = (|\phi r^{rr}_{kk^{'}}|^2 - |\phi l^{ll}_{kk^{'}}|^2) / (|\phi r^{rr}_{kk^{'}}|^2 + |\phi l^{ll}_{kk^{'}}|^2)$ and plot it as a function of $\Delta _\mu$ in Fig. 5 for given values of $\delta =5$ and $U=10$. It can be observed that $\eta ^2=1$ is mainly concentrated along the line $\Delta _k+\Delta _{k^{'}}=10$, which is shown as a black dotted line in the figure. However, the line of concentration for isolation contrast is not completely continuous, and is truncated around $(\Delta _k , \Delta _{k^{'}})=(5,5)$ as shown in Fig. 5(b). This is due to the incident photons being localized in this frequency region under the condition $\epsilon \ll \beta <\delta$. In Fig. 6, we plot the norm square of the amplitude of $\phi r^{rr}_{kk^{'}}$ as a function of $\delta$ and $U$, respectively. It is indicated that for fixed $U=10$, the peak value is achieved at $\delta =5$ [Fig. 6(a)], while for fixed $\delta =5$, the maximum exists in $U\in [10,11]$, which corresponds to the pink-shaded area in Fig. 6(b). In conclusion, to obtain preferable results for two-photon unidirectional or nonreciprocal transmission, we set $2\delta =U=10$ under the condition of $(\Delta _k , \Delta _{k^{'}})=(10,0)$ or $(0,10)$, $g = 0.1$, $D=-1$, and $\epsilon =0.1$.
Fig. 5. (a) The two photon isolation contrast $\eta ^2$ as a function of $\Delta _\mu$, for $\delta =5$, $g=0.1$, $D=-1$, $U=10$ and $\epsilon =0.1$, (b) is the zoom in of (a), all parameters are in units of $\beta$.
Fig. 6. (a) The norm square of the amplitude of $\phi r^{rr}_{kk^{'}}$ as a function of $\delta$ for $U=10$. (b) The norm square of the amplitude of $\phi r^{rr}_{kk^{'}}$ as a function of $U$ for $\delta =5$. Other parameter are set as $(\Delta _k , \Delta _{k^{'}})=(10,0)$ or $(0,10)$, $g = 0.1$, $D=-1$ and $\epsilon =0.1$, all parameters are in units of $\beta$.
To demonstrate the impact of asymmetric coupling of QEs on the scattering wave of two photons incident from port 1 and port 2, we plot the norm square of the amplitudes of $\phi r^{rr}_{kk^{'}}$ and $\phi l^{ll}_{kk^{'}}$ as functions of $D$ for $g=0.1, U=10$, and $\epsilon =0.1$ at single-photon resonance $\Delta _k=\Delta _{k^{'}}=\delta =0$ and two-photon resonance $(\Delta _k,\Delta _{k^{'}})=(10,0)$ or $(0,10),\delta =5$ in Fig. 7(a-b). At single-photon resonance in Fig. 7(a), when QEs are maximally polarized, i.e., $D=\pm 1$, the transmission amplitude of incident photons is greater in the weakly coupled direction. Moreover, the transmission probabilities monotonically decrease and increase as $D$ increases in the range of $[-1,0]$ and $[0,1]$, respectively. Figure 7(c) shows that the NP part dominates the transmission process, and its evolution is consistent with curves in Fig. 7(a). At two-photon resonance in Fig. 7(b), due to the dominance of the IP part in the transmission process [Fig. 7(d)], the transmission probabilities remain monotonous: $|\phi r^{rr}_{kk^{'}}|^2$ for two transmitted photons incident from port 1 decreases as $D$ increases, $|\phi l^{ll}_{kk^{'}}|^2$ for two transmitted photons incident from port 2 increases as $D$ increases, and they both reach a maximum in their respective strongly coupled point. Therefore, the system functions as a diode that realizes one-way transmission of few photons based on the chiral quantum coupling.
Fig. 7. The norm square of the amplitudes of $\phi r^{rr}_{kk^{'}}$ and $\phi l^{ll}_{kk^{'}}$ as functions of $D$ for (a) $\Delta _k=\Delta _{k^{'}}=\delta =0$, (b) $(\Delta _k , \Delta _{k^{'}})=(10,0)$ or $(0,10),\delta =5$. The norm square of the NP part and IP part as functions of $D$ for (c) $\Delta _k=\Delta _{k^{'}}=\delta =0$, (d) $(\Delta _k , \Delta _{k^{'}})=(10,0)$ or $(0,10),\delta =5$. Other parameter are set as $g = 0.1, U=10$ and $\epsilon =0.1$, all parameters are in units of $\beta$.
5. Experimental feasibility
In this section, we provide a detailed analysis of the experimental feasibility of our scheme. A Cs atom, Rb atom, or a quantum dot could be used as a quantum impurity to introduce the asymmetry in the coupling strength, with $g_{\circlearrowleft }/2\pi =\sqrt {45}g_{\circlearrowright }/2\pi =17-50$ MHz and the decay rate $\gamma _{\circlearrowleft }/2\pi =45\gamma _{\circlearrowright }/2\pi \approx 2.5$ MHz between the degenerate level and ground level [84,102,103,115]. Additionally, the NV center in diamond could also be applied to our solution and provide the required chirality [116–118]. In Fig. 8(a), the NV center, initially prepared in state $\vert A_2\rangle$, will decay into spin states $\vert \mp 1\rangle$ by emitting $\sigma _\pm$ polarized photons with a wavelength of 637 nm, while the ground state $\vert 0\rangle$ is connected to the spin states through 532 nm microwave signals. In Fig. 8(b), a strong homogeneous static magnetic field $B_{static}$ is applied to remove the degenerate states $\vert \mp 1\rangle$ with a Zeeman splitting $\delta _B=2g_e \mu _B B_{static}$, while the zero-field splitting between the degenerate spin sublevels and ground level is $D=2\pi \times 2.87$ GHz.
The fabrication techniques of optical microring resonators have been well developed and applied in many fields, such as an all-optical parity bit generator and checker circuit using GaAs–AlGaAs-based micro-ring resonators [119], the ultrasmall design of a universal linear circuit based on microring resonators [120], and the heterogeneously integrated III–V-on-Si microring resonator that works as a versatile building block [121]. Recently, nonlinear microring resonators have become a research hotspot [46,74,122]. In our scheme, the Kerr-type nonlinear interaction could be around $5\times 10^6$ rad/s [122], and the silicon waveguides have a width of 450 nm and a height of 250 nm. The radius of the ring is $R=5$ $\mu$m, and the spacing between the ring and the straight waveguide is 200 nm. The fiber-to-fiber insertion loss is measured to be 10.4 dB [123]. In a silicon-on-insulator (SOI) mesoscopic resonator, the quality factor $Q\sim 10^5$ at $\lambda _c \sim 1.55$ $\mu$m [124], and the mode volume is smaller than $V_m\sim 1.55$ $\mu$m$^3$ [125].
6. Summary
In conclusion, we have proposed two scattering systems for single photon and correlated two photons, respectively. By introducing QEs with polarization, the system exhibits nonreciprocal behavior under certain conditions. In the single photon scattering process, we obtain the transmission and reflection amplitudes, including the chiral degree $D$, and the incident light experiences a phase shift when coming out from the other port. The scattering of two photons is solved analytically in the long-time limit, and the asymptotic solution contains the NP part and IP part, where the former is the free propagation of photons, and the latter contains the nonlinear contribution of the resonator. Our work reveals the unique characteristic that the transmission amplitude only exhibits monotonicity at the two-photon resonances, which shows the possibility of studying few-photon nonreciprocity conduction in kerr nonlinear systems. Moreover, according to the integrated ring resonator theory [126,127], coupling another waveguide over the original resonator can combine it into a four-port circulator. By adjusting the diameter of the resonator or the frequency of the incident light, it is possible to transmit photons from the lower bus (throughput) waveguide to the upper drop waveguide subjectively. Nonreciprocal correlated photon transmission in higher-dimensional devices could meet more application requirements [128–136].
7. Appendix: solution of Eq. (18)
In this appendix, we use the Laplace transform to derive the solution of Eq. (18). Assuming the initial condition is given by (19), and choosing $\delta _1=\delta _2=\delta$ and $\epsilon _1=\epsilon _2=\epsilon$, we can rewrite Eq. (18) as:
If $\Delta _\mu \geqslant 0$, we can extend the lower limit of integration to $-\infty$ and use the approximation $\sum _0^{\infty } dk \dfrac {\overline {V}^2}{s+i(\Delta _k+\Delta _{k^{'}})}\approx \sum _{-\infty }^{\infty } d\Delta _k \dfrac {\overline {V}^2}{s+i(\Delta _k+\Delta _{k^{'}})}=\beta$. Solving the above equations yields:
Taking the inverse Laplace transform of $\widetilde {\phi }_{{kk^{'}}}(s)$ gives us the probability amplitude $\phi _{{kk^{'}}}(t)$ for the two photons. In the long-time limit, we obtain the asymptotic solution.
Funding
National Natural Science Foundation of China (92065105).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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