Perfect chirality constructed by position-dependent backscattering in a whispering gallery mode microresonator

Unidirectional propagation of photons originated from perfect chirality meets the critical requirement for building a high-performance quantum network. However, it not only requires that the circular dipole emitter is precisely located at points of circularly polarized electric fields, which leads to non-reciprocal interactions for photons with opposite propagation directions, but also the light-emitter interaction strength should be strong enough to guarantee a π phase shift. Unfortunately, these perfect chirality points are scarce and accessible points with elliptically polarized fields result in non-ideal photon-emitter chiral interactions and emitters radiating photons bidirectionally. Meanwhile, reflection properties, phase shifts, and non-reciprocal interactions are sensitive to frequency detunings and dissipations. Here, without engineering the dipole and optimizing the distribution of the field, a scatter such as a nanotip placed at the evanescent field of a whispering gallery mode resonator (WGMR) is adopted to control the transporting properties of single photons under non-ideal chiral interactions. By properly adjusting the relative position between the nanotip and the atom or the overlap between the nanotip and the mode volume of the WGMR, amplitudes of reflected photons in different pathways are changed. Consequently, complete destructive interference appears and thus no photons are reflected. The corresponding phase shifts of π and non-reciprocal interactions are guaranteed simultaneously. Significantly, the perfect chirality reconstructed here is robust against frequency detunings and dissipations. Therefore, the atom-WGMR-nanotip structure can be regarded as a compound chiral atom with radiating photons in only one direction.


Introduction
Engineering photon emission and scattering at the single photon level is one of the central goals of modern photonic applications from single-photon device designing to quantum networks [1][2][3].To this end, photonic waveguides are well suited as they confine photons to a one-dimensional geometry and thereby increase the photon-emitter interactions [4][5][6][7][8][9][10][11].Generally, when a quantum emitter, such as an atom [12] or a quantum dot [13], is coupled to the waveguide, the excited emitter equally decays photons into the rightand left-propagation directions along the waveguide [14,15].This symmetry is violated in nanophotonic structures in which strong transverse light in such a waveguide can result in the presence of a locally circularly polarized electric field [16,17].The confinement introduces a link between local polarization and the propagation direction of light, which is a manifestation of optical spin-orbit coupling [18].If such spin-momentum-locked light is coupled to emitters with the corresponding polarization-dependent dipole transitions, then photon-emitter interaction with perfect chirality (i.e.direction-dependent emission, scattering, and absorption of photons) is obtained [19].
Perfect chirality directly leads to three fundamental characteristics: unidirectional propagation of photons with no reflections, phase shifts of π between the incident and transmission photons, and non-reciprocal ) is originally prepared on the ground state.Meanwhile, the local fields of the waveguide are elliptically polarized.Based on the Fermi's golden rule, the excited state decay rates via spontaneous emission are proportional to the dot product between the complex transition dipole and the complex electric field amplitudes [19].When the local polarization of the electric fields is elliptical, the circular dipole radiates in both directions of the waveguide with different amplitudes and thus the corresponding decay rates of the non-ideal chiral interactions are ΓR > ΓL > 0 [20].(b) The reflection probability R0 and transmission amplitude t0 for the resonant incident photon with ∆0 = 0 and 1/τq = 0. Perfect chirality (ΓL = 0) guarantees R0 = 0 and t0 = −1 simultaneously.In contrast, although the transmission amplitudes are still negative (i.e.phase shifts are π), non-ideal chiral interactions result in photon reflecting with R0 > 0. (c) R0 and t0 for the non-resonant photon with ∆ = −0.3ΓRand 1/τq = 0.It can be seen that the chiral interaction of ΓL = 0 leads to photons transporting with no reflection (i.e.R0 = 0), but the phase shift between the incident and transmission photons is not π.Non-ideal chiral interactions increase the reflection probabilities and changes the phase shifts.(d) R0 and t0 in dissipative case with 1/τq = 0.2ΓR.Even when the interactions between the resonant photon and the atom are ideal chiral with ΓL = 0 and R0 = 0, the phase shift is no longer guaranteed simultaneously to π as that in figure 1(b).(e) The phase shifts affected by frequency detunings and dissipations.The above results indicate that perfect chirality of achieving R0 = 0 and ϕ = π at the same time is accessible only when the system is under the ideal condition of chiral interactions, resonant incident photons, and without dissipations, i.e.ΓL = 0, ∆0 = 0, and 1/τq = 0. (f) Perfect chirality reconstructed by a nanotip placed at the evanescent field of a WGMR.The atom-waveguide interaction is non-ideal chiral.The local fields of the WGMR are also elliptically polarized.Therefore, the atom is asymmetrically coupled to WGMR modes a and b with coupling strengths ga > g b .The position of the nanotip can be adjusted by a nanopositioner.interactions for the opposite injected photons [19][20][21][22].These fundamental characteristics underpinning perfect chirality can be simply demonstrated in figure 1(a) [22].When the two-level quantum emitter (e.g. an atom with the right circularly polarized transmission σ + and transition frequency Ω) is excited by the incident photon (with frequency ω) from the left-side of the waveguide, the transporting properties can be obtained from the transmission and reflection amplitudes as being the dissipation rate of the atom, and Γ L (Γ R ) being the atom decaying rate of photons to the left (right ) direction.When the resonant incident photon is absorbed by the atom, perfect chirality with Γ L = 0 leads to unidirectional propagation of photons and thus the reflection probability R 0 = |r 0 | 2 = 0 and T 0 = |t 0 | 2 = 1.This unidirectional propagation of photons has many potential applications, such as transferring quantum states [23], designing optical isolators [24,25] or circulators [26], and constructing cascade quantum networks [27][28][29][30][31] without information backflow.
Additionally, perfect chirality is characterized not only by an excited atom radiating photons unidirectionally, but also by the single-photon transmission amplitude that equals to t 0 = −1 [20].These results can be seen in figure 1(b) that perfect chirality of Γ L = 0 directly leads to R 0 = 0 and t 0 = −1.It indicates that there is a phase shift of ϕ 0 = arctan(t 0,imag /t 0,real ) = π between the incident and the transmission photon [32], with t 0,imag and t 0,real being the imaginary and the real part of the transmission amplitudes.Actually, this phase shift of π has been proposed to perform quantum computing [28] and generate entangled photon sources [33].While the transmission properties for the photon incident from the right to the left of the waveguide can be calculated as t Consequently, the perfect chirality with Γ L = 0 results in t ′ 0 = 1 and r ′ 0 = 0.It means that the incident photon dose not interact with the atom with This non-reciprocal interaction for the opposite incident photon paves the way for designing non-reciprocal photonic elements [32].
However, the chiral photon-atom interaction crucially depends on both the distribution of the local electrical field and the polarization of the atom transition dipole moment [20].Perfect chirality is obtained by precisely placing the circular dipole at the point of the perfect circular polarization field.Unfortunately, these points are scarce.For example, when an atom is coupled to an optical nanofibre, the local spin density of the electric field is position dependent and strong varies as a function of the azimuthal position around the nanofibre [34].Thus, elliptical polarization is practically accessible in nanofibre waveguides [17].Although a quantum dot placed at the singular point of glide-plane photonic crystal waveguides (i.e.known as the C-point) can display a spin-dependent unidirectional emission, the decay rate at the C point is inherently half of that at a point of linear polarization [33,35].It indicates that the light field is elliptically polarized over the majority of the mode volume with strong photon-emitter interactions [20].This introduces difficulties to achieve the aforementioned contradicting requirements of pursuing strong photon-emitter interactions and making those interactions with high chiralities.Consequently, the local electric fields of the waveguides are generally elliptical and thus photon-emitter interactions are non-ideal chiral with Γ R > Γ L > 0 [35], which causes the reflections of resonant photons with R 0 > 0, as shown in figure 1(b).On the other hand, the chirality and the phase shift are drastically affected by the frequency detuning.As shown in figure 1(c), if ∆ 0 ̸ = 0, even when the photon radiated by the atom is in a single direction with Γ L = 0 and R 0 = 0, the phase shift is not π.The reflection probabilities and phase shifts change more obviously by increasing the Γ L .Actually, the unavoidable intrinsic dissipative processes also disturb the reflection properties and phase shifts, as shown in figure 1(d) with 1/τ q = 0.2Γ R .Although the atom radiating photons in a single direction with Γ L = 0, the phase shift is ϕ ̸ = π.Figure 1(e) demonstrates that the phase shifts are sensitive to frequency detunings and dissipations.Obviously, the third fundamental characteristics of non-reciprocal interactions demonstrated in t ′ 0 and ϕ ′ 0 are also perturbed by non-ideal chiral interactions, frequency detunings, and dissipations.In a word, the three fundamental characteristics of perfect chirality achieved simultaneously are limited to ideal chiral interactions, resonant incident photons and without dissipations.Therefore, in real experiments with non-ideal chiral interaction, frequency detunings of different nodes in the networks, and dissipations, the incident photon is inevitably reflected with R 0 > 0, ϕ 0 ̸ = π, and ϕ ′ 0 ̸ = 0, which directly suppresses the efficiency of information transfering in quantum networks [19].
Actually, two schemes for constructing perfect chirality are proposed: One is optimizing the distribution of the field to obtain the circular polarization of the local fields, such as designing the photonic crystal waveguides [33,35] and cavities [36], or carefully choosing materials of the WGMR and their ambient media [37].The other is engineering the circular dipole of the atom to an elliptical dipole and guaranteeing this elliptical dipole orthogonal to the corresponding elliptically polarized electric field of the waveguide [20].Specially, both the above two schemes need the atom to be precisely located at the selective points.Recently, unidirectional propagation of waveguide photons with no reflection is demonstrated by quantum interference among different pathways, such as interference between a V-type atom and a single photon in the superposition state of different frequencies [38] and an artificial molecule comprising two superconducting qubits with entangled states [39].We also find that when an external scatterer is coupled to the WGMM under non-ideal chiral interactions, unidirectional propagation of single photons with no reflections can be realized by applying the interplay between chirality and backscattering [40].However, the other two fundamental characteristics as phase shifts of π and non-reciprocal interactions are not exactly proved in these interference schemes.
In this work, without engineering the dipole or optimizing the distribution of the local electric field, a nanotip [41][42][43] coupled to a whispering gallery mode resonator (WGMR) is introduced to reconstruct perfect chirality as shown in figure 1(f).The nanotip placed in the evanescent field of the WGMR leads to coherent backscattering coupling between clockwise (CW) and counterclockwise (CCW) propagating modes.Given that the atom simultaneously interacts with the waveguide and the WGMR, the incident photon scattered by the waveguide-atom-WGMR structures can be controlled by the nanotip.By properly adjusting the relative position between the nanotip and the atom, the reflected photons in different pathways cause complete destructive interference and then no photons are reflected.Additionally, the phase shift of π is obtained at the same time even when the incident photon is detuned from the atom and the system includes dissipations.Therefore, the atom-WGMR-nanotip structure can be regarded as a compound chiral atom (CCA) and perfect chirality could be reconstructed by controlling this CCA.
This paper is organized as follows.The model is described in section 2 by directly calculating the response of a single injected photon scattered by the atom-WGMR-nanotip structure.Section 3 discusses the interplay between chirality and backscattering.The reconstruction of perfect chirality influenced by non-ideal chiral interactions, frequency detunings, and dissipations are investigated in details.Finally, we conclude our work and suggest experimental demonstrations of our proposal with current photonic techniques in section 4.

Model and solutions
In this paper, the investigation is under the case that the dipole transition of the atom is circularly polarized but the polarization of the local electric field is elliptical.The schematic of the system to reconstruct the perfect chirality is shown in figure 1 As the WGMR evanescent electric field at the position of the atom is also elliptically polarized, the atom is coupled to both the CW and CCW modes (i.e.modes b and a) [16,[44][45][46][47].When the transition frequency of the atom is far from the cutoff frequency of the dispersion relations of the WGMR and the waveguide, the effective Hamiltonian of the system in real space under the rotating wave approximation with modeling the single-excitation is given by (with h = 1) [14,40,46,48,49]: wherein, is the creation operator for right-propagating (left-propagating) photons of the frequency ω at the x position of the waveguide.ω 0 is a reference frequency, around which the waveguide dispersion relation is linearized.v g is the group velocity of the photons.a † (b † ) is the creation operator for the CCW (CW) mode with the frequency of ω a (ω b ) and the dissipation rates of 1/τ a (1/τ b ). a † g (a † e ) is the creation operator of the ground (excited) state with the atom transition of Ω = ω e − ω g and the dissipation rate of 1/τ q .V a and V b are the coupling strengths of different WGMR modes to the waveguide.Typically, the right-propagating (left-propagating) photon only couples to the CCW (CW) mode of the WGMR.V L (V R ) is the coupling strength between the atom and left (right) propagation photons along the waveguide.Because we are only interested in a narrow range in the vicinity of the atomic resonant frequency, V a , V b , V L and V R are safely assumed to be independent of frequencies [15,50,51].Such an assumption is equivalent to a Markovian approximation [48].The corresponding decay rates are defined as Γ m = V 2 m /2v g , m = a, b, L, and R. σ ge is the atomic operator with σ ge = a † g a e .The Dirac delta function δ(x) indicates that the WGMR and the atom are near the location x = 0 of the waveguide.
the coupling strengths between the atom and the WGMR modes [49].⃗ ϕ a and ⃗ ϕ b are the electric field profiles of modes a and b. ⃗ d is the atomic dipole vector.When the transition from the excited state to the ground state corresponds to the difference of magnetic quantum number ∆ mF = +1, ⃗ d is necessarily a complex vector.Consequently, the coupling strengths of g a and g b are complex numbers [46,49].As a resonator that supports a pair of degenerate WGMR modes, ⃗ ϕ a = ⃗ ϕ * b and thus |g a | ̸ = |g b | [49]. he iθ is the inter-mode backscattering strength.The phase θ depends on the relative position between the atom and the backscattering point [52].Experimentally, the effective size (denoted by h) of the scatter can be precisely controlled by a nanopositioner, while the quality factor of the WGMR is hardly affected by the scatter [42,43].
In this scheme, a single photon incident from the left side of the waveguide transmits to the right.After interacting with the system, the incident photon may be absorbed by the atom, may excite the WGMR modes, or may be scattered along the waveguide in the left or right direction.Initially the atom is prepared in the ground state, and the waveguide and cavity fields are in the zero-photon state, which is denoted as |0 w , a g ⟩.Therefore, the most general interacting eigenstates for the Hamiltonian H eff in steady states take the following form: where ϕ R (x) = e ikx [Θ(−x) + tΘ(x)] and ϕ L (x) = re −ikx Θ(−x) represent the wave functions of the single-photon waveguide modes propagating to the right and left directions respectively, and Θ(x) is a step function.T = |t| 2 and R = |r| 2 are the transmission and reflection probabilities of the input single photon, respectively.e a and e b are the excited amplitudes of modes a and b, and e q is the excitation amplitude of the atom.When ε is an eigenstate of frequency |Ψ⟩, the Schrödinger equation yields the time-independent eigenequation H eff |Ψ⟩ = ε|Ψ⟩.Bringing equations ( 1) and ( 2) into the eigenequation, the following equations of motion are obtained: with ε = ω + ω g , and ω = kv g + ω 0 .Solving equations (3a)-(3e) for r and t gives with with ).In the above equations ( 4) and ( 5), and ∆ e = ω − ω e + i τq .For simplicity, the cavity mode and the atom are assumed to in tune with When ∆ = 0 and without considering the dissipations, the expression of equation ( 4) looks like the transmission amplitude of an atom coupled to the waveguide [33], and thus if the transmission amplitude can be modulated to t = −1, then the atom-WGMR-nanotip structure can be regarded as a CCA.The corresponding results can be demonstrated subsequently.Because one of the fundamental characteristics of the perfect chirality is without reflection, how to suppress reflection as in equation ( 5) is the main goal of this work.As the atom couples to the waveguide and the WGMR, the incident single photon may be reflected by the emitter-WGMR-nanotip structure in different pathways, which are analysed in detail as follows: The term of √ Γ a Γ L (he iθ g b + g a ∆ cb ) corresponds to two pathways (with the right-moving waveguide photon driving the cavity mode a under the strength of √ Γ a ) [43].The first pathway is: Mode a photon is scattered to mode b photon with strength of he iθ , then mode b photon excites the atom with strength of g b , and eventually the excited atom decays to the left direction of the waveguide with strength of √ Γ L .While the second pathway is: Mode a photon drives the atom to the excited state with strength g a and then the excited atom decays to the left direction of the waveguide with strength of √ Γ L .When the incident photon is resonant with mode b as ∆ cb = 0, the procession of mode b photon exciting the atom is dominant; When ∆ cb ̸ = 0, both the two pathways take part in the reflection processions; When the frequency detuning is The last term of ) corresponds to the seventh pathway: The right-moving waveguide photon excites the atom under the strength of √ Γ R and then the excited atom decays to the left-direction of the waveguide with strength of √ Γ L .There is also a competition between the frequency detunings and the strengths of nanotip scatterings.
Given that the coupling strengths of Γ L , Γ R , Γ a , Γ b , g a , and g b are fixed in experiments, the reflection amplitudes originated from different pathways in equation ( 5) can be controlled by adjusting the parameters of h, θ, and ∆.Experimentally, the relative position between the nanotip and the atom (determining θ) and the overlap between the nanotip and the mode volume of the WGMR (determining h) can be precisely controlled by a nanopositioner, and thus the reflection photon in different pathways can be modulated by the nanotip [42,43].

Results
In this section, the transporting properties of the single incident photon along the waveguide scattered by the atom-WGMR-nanotip structure are studied.For simplicity, the phases of g a and g b are set to 0 by properly choosing the orientation of the atomic dipole polarization and the azimuthal origin, and then only the corresponding real values are concerned as g a > g b [49].The atom-WGMR and atom-waveguide interactions are non-ideal chiral.Therefore, by setting Γ R = Γ to be the unit, the general parameters are chosen as: First, the reflection properties of single photons without dissipations are investigated.As plotted in figure 2(a), when the atom-waveguide, atom-WGMR, WGMR-waveguide coupling strengths and the frequency detunings are fixed, the reflection probabilities can be modulated by the relative position between the nanotip and the atom (denoted as h) and the overlap between the nanotip and the mode volume of the WGMR (denoted as θ).The reflection probabilities controlled by h is shown in figure 2(b) in details, with θ = π and ∆ = 2.3Γ.The behaviors can be demonstrated as follows: Due to the non-ideal chiral interaction, the excited atom radiates photons bidirectionally, and thus the photons are reflected to the left side of the waveguide with R > 0 when h = 0.When the nanotip is placed at the evanescent field of the WGMR, the backscattering between mode a and mode b changes the amplitudes of photons from different pathways, and consequently the reflection probabilities are suppressed.Especially, as the backscattering strength is increased to h = 1.5Γ, the reflected photons from different pathways meet the requirements of the complete destructive interference and then no photon is reflected with R = 0. Further increasing h changes the amplitudes of reflected photons and consequently disturbs the complete destructive interference with R > 0. On the other hand, the reflection probabilities dependent on θ is shown in figure 2(c).It can be seen that complete destructive interference with R = 0 is realizable by adjusting the value of θ.Additionally, condition to reach R = 0 requires θ = nπ, n = ±1, ±2, ±3, . ...
Next, phase shifts between the incident and transmission photons are investigated in figure 3. The real parts of the transmission amplitudes modulated by frequency detunings and backscattering strengths are calculated in figure 3(a).The region of t = −1 can be found.Typically, the central problem of perfect chirality lies in whether the transmission amplitude at the point of R = 0 corresponds to t = −1 or not [20].As shown in figure 3(b), R = 0 and t = −1 are simultaneously obtained.Meanwhile, the corresponding phase shift of ϕ = arctan(t imag /t real ) = π can be found in figure 3(c).It means that two fundamental characteristics of perfect chirality, i.e. unidirectional propagation of photons and phase shifts of π can be achieved by modulating the nanotip.
The perfect chirality reconstructed by a nanotip coupled to the WGMR does not need all kinds of couplings being involved at the same time.As shown in figure 4(a), when the WGMR does not couple to the waveguide but couples to the atom with Γ a = Γ b = 0, g a = 2Γ, and g b = 0.5Γ, or inversely with parameters of Γ a = Γ b = 0.5Γ and g a = g b = 0, the unidirectional propagation of single photons with R = 0 can also be obtained by adjusting the position of the nanotip.Especially, if the WGMR does not couple to the atom, the nanotip backscattering the WGMR modes corresponds to a couple of mirrors and produces the Fano  resonance [53].Meanwhile, when the back scattering strengths are as large as h ≫ 10Γ, the mode shift introduced by the nanotip results in both the WGMR-atom and WGMR-waveguide decoupling.It indicates that this coupling limit reduces the system as the atom only couples with the waveguide under non-ideal chiral interaction.In this work, all the investigations are far from the coupling limit.Figure 4(b) shows that the mechanics to reconstruct perfect chirality with R = 0 is not sensitive to frequency detunings.
Then, the differences of transporting properties between the two opposite directions are compared.When the incident photon is injected from the right side of the waveguide, the corresponding wave functions are ϕ ′ R (x) = e ikx r ′ Θ(x) and ϕ ′ L (x) = e −ikx [Θ(x) + t ′ Θ(−x)].Based on the foregoing procedure by combining equations ( 1) and ( 2), the transmission and reflection amplitudes are calculated as: with Comparing the transmission and reflection amplitudes of the two opposite directions, only the numerators have differences.The corresponding reflection pathways in equation (7) can also be analyzed by following the same manners as in equation (5).For example, the term √ Γ a Γ L (he −iθ g * b + g * a ∆ cb ) describes the following two pathways (with the left-moving waveguide photon exciting the atom under the strength of √ Γ L ).One pathway: The excited atom decays to mode b photon with strength g * b , then mode b photon is scattered to mode a photon with strength he −iθ , and finally mode a photon decays to the right direction of the waveguide with strength √ Γ a .The other pathway: the excited atom decays to mode a photon with strength g * a and then mode a photon decays to the waveguide with strength √ Γ a .Next, we focus on the transmission difference between the two opposite directions.
These differences are shown in figure 5. Figures 5(a  ϕ ′ = arctan(t ′ imag /t ′ real ) = 0 with h = 3Γ.It means that the three fundamental characteristics as no reflections with T = 1, phase shifts of ϕ = π, and non-reciprocal interactions of T ′ = 1 and ϕ ′ = 0 can be achieved simultaneously.Therefore, the atom-WGMR-nanotip can be regarded as a CCA.The CCA with constructive interference in transmission and with destructive interference in reflection gives rise to radiate photons along the waveguide in only one direction, even when the interaction between the atom and the waveguide is non-ideal chiral.
Finally, the perfect chirality against dissipations is investigated in figure 6.It is well known that dissipations are unavoidable intrinsic processes and directly influence the interference properties [54].As shown in figure 6(a), the reflection spectra have the same responses to h and θ as those in figure 2(a), wherein the regions of R = 0 are accessible.Although the transmission amplitudes are suppressed to |t| < 1 by the dissipations, the phase shift of ϕ = π can still be guaranteed with R = 0 as shown in figures 6(b) and (c).It means that although the system involves photons dissipated into non-waveguide modes, the atom-WGMR-nanotip structure also radiates photons in a single direction.Therefore, difficulties encountered in non-ideal chiral interaction with bi-directional transportation and phase shifts of ϕ ̸ = π in dissipative system as shown in figures 1(d) and (e) can be solved by adjusting the nanotip to reconstruct perfect chirality with R = 0 and ϕ = π.

Conclusion and discussion
Due to the fact that chiral photon-atom interaction crucially depends on both the local electronic field and the polarization of the atomic transition dipole moment, there are two general ways to reconstruct perfect chirality under non-ideal chiral interactions: i.e. optimizing the distribution of the electric field [33,[35][36][37] and engineering the dipoles by tilting the applied magnetic field to the atoms or exploiting strain-tuning techniques to the quantum dot [20].In this work, a nanotip placed at the evanescent field of the WGMR is proposed to reconstruct perfect chirality.By properly adjusting the relative position between the nanotip and the atom or the overlap between the nanotip and the mode volume of the WGMR, amplitudes of reflected photons in different pathways are changed.Consequently, complete destructive interference appears and thus no photons are reflected.This unidirectional interaction between the waveguide and the atom-WGMR-nanotip structure results in phase shifts of π and non-reciprocal interactions [20].
Significantly, the perfect chirality reconstructed here is robust against frequency detunings and dissipations.Therefore, the atom-WGMR-nanotip structure can be regarded as a CCA.The CCA leads to interference which is constructive in transmission and destructive in reflection, and thus allows for chiral emission from the CCA [36].
Experimentally, silica nanotips as Rayleigh scattering engineered by wet etching of a tapered fiber are widely used to couple with the two WGMR modes [41][42][43], while the quality factor of the WGMR is unaffected by the nanotip.Moving the nanotip towards the WGMR increases the overlap of the tip with the mode volume and enhances the backscattering strength.The position of the tip can be controlled by a nano-positioning stage with a resolution of nanometers for each step [41][42][43].The WGMR can be fabricated at the edge of a separate chip placed on nanopositioning system to precisely control the distance and hence the coupling strength between the WGMR and the waveguide [55,56].Additionally, various approaches have been developed to trap single atoms at the surface of waveguide structures, such as using tightly focused optical tweezers to a nano photonic cavity [57], utilizing standing waves to nanofibers [58] or to WGMRs [59], and constructing evanescent-field trap potentials on a microring [60].Currently, experimental results show that the coupling strengths and dissipative rates (i.e.g a , g b , h, Γ L , Γ R , Γ a , Γ b , 1/τ a , 1/τ b , and 1/τ q ) of a single Rb atom and a nanotip coupled to a WGMR are in the megahertz regime [41][42][43][44][45][46]59].For example, strong couplings, i.e. the WGMR-atom coupling strengths (g ≈ 2π × 10 MHz) being larger than the dissipative rates of the WGMR and the atom (γ ≈ 2π × 3 MHz), are clearly observed with a symmetric Rabi splitting in the transmission spectra [59].Meanwhile, the chiral interaction between a Rb atom and a WGMR in the strong coupling regime (g ≈ 2π × 16 MHz and γ ≈ 2π × 3 MHz) is successfully obtained to construct a single photon router [44,46].Particularly, as analyzed in equations ( 4)- (7), although the transmission and reflection properties under fixed coupling strengths and dissipative rates can be modulated by adjusting the positions of the nanotips, numerical calculations show that the complete destructive interference among different pathways requires 1/τ q < Γ.Therefore, the linewidth of the atom needs to be suppressed to the megahertz regime and then the corresponding temperature should be lower than T k = ∆ 2 vD mc 2 /2ln2k B v 2 0 ≈ 10 −2 K, wherein ∆ vD is the Doppler width, m is the molecular mass, c is the speed of light, k B is the Boltzmann constant, and v 0 is the central frequency [61,62].

Figure 1 .
Figure 1.Transporting properties of single photons under non-ideal photon-atom chiral interactions.(a) The single photon incident from the left-side of the waveguide scattered by the atom.The atom (with energy Ω = ωe − ωg and the circularly polarized transition dipole σ + ) is originally prepared on the ground state.Meanwhile, the local fields of the waveguide are elliptically polarized.Based on the Fermi's golden rule, the excited state decay rates via spontaneous emission are proportional to the dot product between the complex transition dipole and the complex electric field amplitudes[19].When the local polarization of the electric fields is elliptical, the circular dipole radiates in both directions of the waveguide with different amplitudes and thus the corresponding decay rates of the non-ideal chiral interactions are ΓR > ΓL > 0[20].(b) The reflection probability R0 and transmission amplitude t0 for the resonant incident photon with ∆0 = 0 and 1/τq = 0. Perfect chirality (ΓL = 0) guarantees R0 = 0 and t0 = −1 simultaneously.In contrast, although the transmission amplitudes are still negative (i.e.phase shifts are π), non-ideal chiral interactions result in photon reflecting with R0 > 0. (c) R0 and t0 for the non-resonant photon with ∆ = −0.3ΓRand 1/τq = 0.It can be seen that the chiral interaction of ΓL = 0 leads to photons transporting with no reflection (i.e.R0 = 0), but the phase shift between the incident and transmission photons is not π.Non-ideal chiral interactions increase the reflection probabilities and changes the phase shifts.(d) R0 and t0 in dissipative case with 1/τq = 0.2ΓR.Even when the interactions between the resonant photon and the atom are ideal chiral with ΓL = 0 and R0 = 0, the phase shift is no longer guaranteed simultaneously to π as that in figure1(b).(e) The phase shifts affected by frequency detunings and dissipations.The above results indicate that perfect chirality of achieving R0 = 0 and ϕ = π at the same time is accessible only when the system is under the ideal condition of chiral interactions, resonant incident photons, and without dissipations, i.e.ΓL = 0, ∆0 = 0, and 1/τq = 0. (f) Perfect chirality reconstructed by a nanotip placed at the evanescent field of a WGMR.The atom-waveguide interaction is non-ideal chiral.The local fields of the WGMR are also elliptically polarized.Therefore, the atom is asymmetrically coupled to WGMR modes a and b with coupling strengths ga > g b .The position of the nanotip can be adjusted by a nanopositioner.
|he iθ g b |, the procession of mode a photon exciting the atom is dominant.The term of √ Γ b Γ R (he iθ g * a + g * b ∆ ca ) means another two pathways (with the right-moving waveguide photon exciting the atom under the strength of √ Γ R ).The third pathway: The excited atom decays to mode a photon with strength g * a , then mode a photon is scattered to mode b photon with strength he iθ , and consequently mode b photon decays to the left direction of the waveguide with strength √ Γ b .The fourth pathway: the excited atom decays to mode b photon with strength g * b and then mode b photon decays to the waveguide with strength √ Γ b .∆ ca = 0 means the procession of the excited atom decaying to mode a photon is dominant, while ∆ ca ̸ = 0 indicates the competition between pathways 3 and 4. √ Γ a Γ b (he iθ ∆ e + g * b g a ) presents pathways 5 and 6 (with the right-moving waveguide photon driving the cavity mode a under the strength of √ Γ a ).Pathway 5 is that mode a photon is scattered to mode b photon with strength he iθ and then mode b photon decays to the left-direction of the waveguide with strength √ Γ b .Pathway 6 is that mode a photon excites the atom with strength g a , then the excited atom decays to mode b with strength g * b , and eventually mode b photon couples to the waveguide with strength √ Γ b .∆ e = 0 means the procession of the atom excited by mode a photon is dominant, while ∆ e ̸ = 0 indicates the competition between pathways 5 and 6.Typically, |∆ e | ≫ |g * b g a | leads to the atom decoupling to the cavity and then the procession of mode a photon backscattering to mode b photon is dominant.

Figure 3 .
Figure 3.The phase shifts controlled by frequency detunings and nanotips, with θ = π.(a) The cross point of two dashed lines with t = −1 corresponds to h = 1.5Γ and ∆ = 2.3Γ.(b) R = 0 and t = −1 are simultaneously reached.(c) The corresponding phase shift of R = 0 is ϕ = π.It means that the first and the second fundamental characteristics of perfect chirality (i.e.R = 0 and ϕ = π) can be reconstructed by adjusting the nanotip.The other parameters are the same as those in figure 2.

Figure 4 .
Figure 4.The coupling limit and frequency dependence of reconstructing perfect chirality.(a) Reconstructing perfect chirality with different kinds of couplings.R = 0 is achieved even when the WGMR does not simultaneously coupled to the waveguide or the atom.(b) The perfect chirality reconstructed by the nanotip for different frequency detunings.The other parameters are the same as those in figure 2.

Figure 5 .
Figure 5. Non-reciprocal interactions reconstructed by a nanotip coupled to a WGMR.Comparing the transmission properties between photons transporting from left to right (a) and right to left (b) directions.Although the transmission probabilities of T and T ′ are the same, the corresponding phase shifts of ϕ and ϕ ′ are obviously different as plotted in (c).Especially, the three fundamental characteristics as no reflections with T = 1, phase shifts of ϕ = π, and non-reciprocal interactions of T ′ = 1 and ϕ ′ = 0 are achieved simultaneously.∆ = −0.46Γand θ = 0.The other parameters are the same as those in figure 2.

Figure 6 .
Figure 6.The reconstruction of perfect chirality including dissipations.(a) The regions of R = 0 can also be achieved with 1/τq = 1/τa = 1/τ b = 0.2Γ.Although the dissipations suppress the transmission amplitude to |t| < 1 as in (b), R = 0 and ϕ = π are still simultaneously reached as in (c).It means that in spite of non-ideal chiral interaction, non-resonant incident photons, and including dissipations, perfect chirality of R = 0 and ϕ = π can also be reconstructed by adjusting the nanotip.The other parameters are the same as those in figure 2.