Abstract
We investigate chiral emission and the single-photon scattering of spinning cavities coupled to a meandering waveguide at multiple coupling points. It is shown that nonreciprocal photon transmissions occur in the cavities-waveguide system, which stems from interference effects among different coupling points, and frequency shifts induced by the Sagnac effect. The nonlocal interference is akin to the mechanism in giant atoms. In the single-cavity setup, by optimizing the spinning velocity and number of coupling points, the chiral factor can approach 1, and the chiral direction can be freely switched. Moreover, destructive interference gives rise to the complete photon transmission in one direction over the whole optical frequency band, with no analogy in other quantum setups. In the multiple-cavity system, we also investigate the photon transport properties. The results indicate a directional information flow between different nodes. Our proposal provides a novel way to achieve quantum nonreciprocal devices, which can be applied in large-scale quantum chiral networks with optical waveguides.
1 Introduction
Waveguide quantum electrodynamics (QED) has emerged as an excellent platform for studying the interactions between atoms and itinerant photons in the past 2 decades [1–3]. A one-dimensional waveguide supports a continuum of photon modes with a strong transverse confinement, and is applicable to significantly enhance light-matter interactions [3]. Moreover, waveguide-QED systems serve as quantum channels in quantum networks, which can be realized in both natural and artificial systems, such as trapped atoms (quantum dots) interacting with nanofibers [4–8] and superconducting qubits coupled with transmission lines [9–11]. To date, a great deal of quantum optical effects have been revealed in waveguide-QED systems, including controlling single-photon scattering [12–16], photon-mediated long-range interactions [17–20] and directional photon emission [21, 22].
In traditional waveguide QED, atoms are commonly considered as point-like dipoles and coupled to the waveguide at a single point. However, an emergent class of artificial atoms, called giant atoms, break down this dipole approximation. Their sizes are comparable to the wavelength of photons (phonons) interacted [23–35]. Recent experiments have demonstrated that superconducting artificial atoms can be successfully coupled with propagating surface acoustic waves at several points [36–38]. The self-interference effects among multiple points dramatically modify the emission behaviors of giant atoms, such as frequency-dependent decay rates [23, 24], decoherence-free dipole-dipole interactions [25, 26], and nonreciprocal photon transport [30, 31]. All the above achievements indicate potential applications in quantum information processing.
Optical nonreciprocity allows photons to pass through from one side but blocks it from the opposite direction, which is requisite for preventing the information back flow in quantum network. At optical frequencies, magneto-optical Faraday effect is often applied to achieve optical nonreciprocity, which is lossy and cannot be integrated effectively on a chip [39, 40]. Therefore, several magnetic-free nonreciprocal proposals were developed. Their mechanisms include optical nonlinearity [41, 42], dynamic spatiotemporal modulation [43–45], and atomic reservoir engineering [46]. Recently, the whispering-gallery-mode resonators with mechanical rotation provide another approach to study many quantum nonreciprocal phenomena [47–50]. The simplest implementation contains a spinning resonator and a stationary tapered fiber. The rotation leads to Sagnac effect and shifts the frequency of the optical mode. Compared with previous studies, the nonreciprocal transmission of light has been achieved in experiment with very high isolation (about 99.6%) [51]. In early studies, spinning resonators, similar to small atoms, typically couple to waveguides at a single point. Nevertheless, multiple-point coupling in spinning resonator-waveguide systems has not been considered, and the photon emission and transport properties in this system are worth being explored.
In this work, we address this issue by considering spinning resonators interacting with a meandering waveguide at multiple coupling points. Such resonators are akin to the “giant atoms,” but with mechanical rotation. First, in the single-cavity setup, the complete unidirectional transparency over the whole optical frequency band is observed, which can be realized by considering the spinning resonator and multiple-point coupling simultaneously, with no analogy in other quantum setups. This phenomenon results from the interference effects among different coupling points and mode frequency shifts led by the Sagnac effect. Additionally, the chiral emission direction is switchable by simply changing the rotation direction and speed. Afterward, we extend to two-cavity system, where each resonator interacts with two separate points. The phase factors and the coupling strengths between the CW and CCW modes can significantly modulate the nonreciprocal transmission behaviors, which implies chiral photon transfer among different points. Employing spinning resonators as quantum nodes, those results obtained in this paper might have potential applications in large-scale chiral quantum networks.
The paper is organized as follows: in Section 2, we present the single-spinning-resonator model and give the motional equations. The chiral emission and nonreciprocal transmission by tuning spinning velocity or number of coupling points are also discussed. In Section 3, we extend to two separate spinning resonators interacting with several coupling points. Both analytical and numerical results for the weak-field transmission are obtained. Finally, the conclusions are given in Section 4.
2 A Spinning Resonator Interacting With Multiple Points
2.1 Hamilton and Motional Equations
Here we first consider a spinning optical resonator evanescently coupled to a meandering optical waveguide at N coupling points, as shown in Figure 1. The resonator is rotated and the waveguide is stationary. The separation distance between different coupling points on the waveguide is denoted by L = xm − xn. We assume the coherence length of photons in the waveguide is larger than the smallest distance Lmin, and therefore we can ignore the non-Markovian retarded effects [19, 52]. The nonspinning resonator, for example, a whispering-gallery-mode resonator with a resonant frequency ωc, simultaneously supports both clockwise (CW) and counter-clockwise (CCW) travelling modes. The CW and CCW modes couple to each other through a scatterer or induced by surface roughness [53, 54], which results in an optical mode splitting. When the optical resonator rotates in one direction at an angular velocity Ω, the propagating effects of the CW and CCW modes are different, leading to an opposite Sagnac-Fizeau shift in resonant frequencies, i.e., ωc → ωc + ΔF, with [55].where n is the refractive index of the dielectric material, R is the radius of the optical resonator, and c (λ) is the velocity (wavelength) of light in vacuum. The dispersion term λdn/ndλ, denoting the relativistic origin of the Sagnac effect [51, 55], is very small in typical materials compared to the value of (1–1/n2). In the following we assume the resonator rotates along the CCW direction, hence ΔF > 0 (ΔF < 0) represents the case of the driving field coming from the left-hand (right-hand) side. The resonant frequencies of the CW and CCW modes in this situation are ωcw = ωc + ΔF and ωccw = ωc − ΔF, respectively.
FIGURE 1
In our consideration, the Hamiltonian of the spinning resonator can be written as (ℏ = 1)Here ccw and cccw ( and ) are the annihilation (creation) operators of the CW and CCW modes, respectively. The coupling strength J denotes the interaction between these two modes induced by optical backscattering. The CW (CCW) mode can only be driven by an optical field coming from the left (right) side of the waveguide, own to the directionality of travelling wave modes in the resonator. The driving Hamiltonian iswhere cm,in and are the input fields coming from the left and right sides at coupling point xm, respectively. According to Fermi’s golden rule [56], describes the spontaneous emission of the resonator modes into the waveguide at coupling point xm, with gm being the resonator-waveguide coupling strength and being the photon density of states in the waveguide. In the presence of decay channels, the effective non-Hermitian Hamiltonian of the whole system is given bywithwhere Γc is the total decay rate of the resonator mode, and κc is the intrinsic decay rate of the resonator.
According to the Heisenberg motional equations, the dynamic equations of the CW and CCW modes are yielded byNote that kcw = (ωc + ΔF)/c and kccw = (ωc − ΔF)/c are approximately regarded as the central mode vector of right-going and left-going photon in the waveguide emitted by the resonator [23], respectively. Different from the case without rotation, the accumulated phase shifts between neighbor coupling points for opposite propagation directions of the photons are distinct. As given in Refs. [17, 57–59], the local input-output relations for the CW and CCW modes at each coupling point xm are written asSubstituting Eq. 7 into Eq. 6, we obtain the effective dynamic equationsThe total input-output relations of this system take the form
Eq. 8 exhibits a self-coupling in the CW or CCW mode, which arises from the self interference effects of reemitted photons between different connection points. Moreover, the Sagnac effect and the self-interference effects may significantly affect the optical properties of the system. We note that only when the resonator is nonspinning, the system is reciprocal. Based on these derivations, we will investigate the photon emission and transport properties in this system.
2.2 Phase Controlled Chiral Emission
In the giant-atom waveguide-QED systems, the multiple coupling points result in a frequency-dependent decay rate and Lamb shift for a giant atom [23, 60]. Similarly, the interference effects induced by multiple coupling points in our system also give a modification of the frequency shift Δj and decay rate Γj for the CW and CCW mode. According to Eq. 8, we havewhere with j = cw, ccw.
Here we consider the maximally symmetric case, in which decay rates of the resonator modes into the waveguide are the same at each coupling point with κm,e = κe and the distance between neighboring coupling points is identical with xm+1 − xm = d. Then we can set xm − xn = (m − n)d and θj = kjd. Similar to the Lamb shift and decay rate in atomic physics, Eq. 10 becomesWe begin to discuss the effects of the rotation speed and number of coupling points on the emission properties under the condition of κc = 0. When the resonator is nonspinning with Ω = 0, the CW and CCW modes are degenerate with the Fizeau drag ΔF = 0 and ωccw = ωcw = ωc. As increasing the rotation speed Ω, the Sagnac-Fizeau shift described by Eq. 1 linearly increases, as given in Supplementary Material. In our calculations, we choose the related parameters as follows: λ = 1,550 nm, R = 4.73 mm, and n = 1.4. For Ω = 0.97 GHz, we have ΔF/ωc = ±0.05 and (RΩ)/c ≈ 0.015. For the spinning resonator with a single coupling point (N = 1), Eq. 11 gives the results of Δcw = Δccw = 0 and Γcw = Γccw = (κc + κe)/2. When increasing the number of coupling points, the frequency shifts and decay rates for the CW and CCW modes have an opposite shift due to the rotation.
In Figures 2A,B, frequency shifts Δj and decay rates Γj are plotted as a function of the phase θc = ωcd/c with N = 10 and Ω = 0.97 GHz. The frequency shifts Δcw and Δccw take negative and positive values with the maximum at about 0.6Γmax. Given that Δcw (Δccw) is zero, the decay rate Γcw (Γccw) reaches its highest magnitude at θc = 0.95 × 2π (θc = 1.05 × 2π). For θc = 0.95 × 2π, the accumulated phase of photons propagating along the CCW direction leads to Γccw = 0, which arises from the destructive interference effects among the coupling points. In this case, the CCW mode of the resonator is decoupled from the waveguide. Moreover, there are a lot of additional lower and local maximum values in the decay rates. The phase θc of the local minima between these maxima scales with (1/N + ΔF/ωc). Note that the rotation speed and number of coupling points make a big difference in the values of Γcw and Γccw. Narrower resonances can be found in the decay rates when we consider more coupling points.
FIGURE 2
In order to study the emission properties more clearly, for a special frequency we define the chirality parameter aswhere implies a truly unidirectional excitation of the right-going (left-going) photon, and denotes the photon coupling into the waveguide without preference in both propagating directions. Figure 2C depicts the chiral factor changing with number of coupling points N. When N = 1, the chiral factor is . For θc = 0.95 × 2π, as increasing number of coupling points N, the chirality factor first goes up and then oscillates slowly with a relative larger value around 1. Note that is obtained for N = 10, corresponding to Γcw = 50κe and Γccw = 0. The essence of the chirality is that accumulated phases for photons propagating in CW and CCW directions are different. By tuning the phase shift θc, for example, θc = 1.05 × 2π, the photon emission direction is totally switched. Figure 2D shows the chiral factor as functions of number of coupling points N and rotation speed Ω for θc = 0.95 × 2π. By optimizing the rotation speed and number of coupling points, the chiral factor can approach 1, and the chiral direction can be freely switched. Moreover, the directional emission will be realized in a large parameter regime.
2.3 Nonreciprocal Photon Transmission
Now we study how the rotation velocity and number of coupling points affect the optical response of the spinning resonator. We consider the resonator is excited by an external input signal in the CW direction with frequency ωl and amplitude ɛ. In this case, the input signal from the left side is given by , with c1,in being the vacuum input signal, while the input signal from the right side only contains the vacuum input field . In the rotating frame at the driving frequency ωl, the steady-state solutions of Eq. 8 can be written asHere Δc = ωc − ωl is the detuning between the resonator without rotation and the driving field. The transmission rate of the input signal is given by
Similarly, we also consider the case of an external input signal coming from the right side of the waveguide with . By solving the steady-state solutions of Eq. 8, we obtainThe transmission rate of the input signal is written as
A nonreciprocal photon transmission with TR ≠ TL can be observed when the resonator is spinning. This fact is due to the different numerators in Eqs 13, 15. For the maximally symmetric case, we haveFor J = 0, the incident photon will be transmitted and absorbed with reflection being zero. In this scenario, the transmission curve TL represents a Lorentzian line shape centered at Δc = −(ΔF + Δcw) with a linewidth Γcw. For N = 1, we obtain Δc = −ΔF and Γcw = (κc + κe)/2. The transmission dip is around 0. For multiple coupling points, as discussed above, Δcw and Γcw vary periodically with phase θc. The transmission rate TL versus the detuning Δc and the phase θc are plotted in Figure 3A. It shows that θc will dramatically modify the transmission window. As we increase θc, the position of the transmission dip has a red-shift. When the phase θcw is 2π/N, the transmission dip disappears totally with T = 1, which means the resonator cannot be excited by the external field and corresponds to the optical dark state. This phenomenon arises from the destructive interferences in the multiple coupling points, which can be explained by Eq. 17. Moreover, the mode splitting is observed in some parameter range in Figure 3B when J = 5κe. The asymmetry of the two dips results from different decay rates and frequency shifts of these two modes.
FIGURE 3
In Figures 3C,D, we plot the transmission rates TL and TR when the incident photon coming from the left side and right sides versus the detuning Δc for θ = 0.95 × 2π. It shows that TL can be larger or smaller than TR for N = 5. In other words, the nonreciprocal transmission is clearly observed due to the rotation. The interference effects between coupling points enable the transmission dips asymmetric with different linewidths. For N = 10, the decay rate of the CCW mode is very small, which leads to the complete photon transmission with TR = 1. Moreover, a sharp dip appears in the transmission spectra TL for J = 5κe. Note that the phase θc can also be used to adjust the nonreciprocal transmission behavior.
3 Two Spinning Resonators Interacting With Multiple Points
3.1 Hamiltonian and Dynamic Equations
The single-photon transport properties in a one-dimensional waveguide interacted with two giant atoms for three distinct topologies have been discussed in Ref. [61]. To study potential applications of the spinning resonator with multiple coupling points in large-scale quantum chiral networks, we now consider two separate spinning resonators evanescently coupled to a meandering waveguide at several different connection points. As shown in Figure 4, the optical resonator a (b) simultaneously supports both clockwise and counter-clockwise travelling optical modes. The creation operators of the CW and CCW modes are denoted by and ( and ), respectively. The optical resonator a (b), with stationary resonant frequency ωa (ωb) and intrinsic decay rate κa (κb), rotates along the CCW direction by an angular velocity Ωa (Ωb). Owing to the rotation, the resonant frequencies of the CW and CCW modes in the resonator become ωi,cw = ωi + ΔF,i and ωi,ccw = ωi − ΔF,i with the subscript i = a, b, where ΔF,i is given by Eq. 1. The resonator a (b) is coupled to the bent waveguide at connection points and ( and ). The phase factor ϕi is calculated as when an optical signal travelling between them, and the phase factor when photons travelling from resonator a to resonator b is . Here we note that there is no direct coupling between cavity a and cavity b due to the absence of the modal overlap.
FIGURE 4
The Hamiltonian of these two spinning resonators are given byHere Ja (Jb) is the coupling strength between the CW and CCW modes of the resonator a (b). The CCW (CW) modes in the resonators can only be driven by an optical field coming from the left (right) side of the waveguide. The amplitudes of the input fields at different coupling points are denoted by am,in, bm,in, , and with m = 1, 2. The driving fields give the HamiltonianThe non-Hermitian Hamiltonian of the whole system can be given bywhere and i = a, b. Note that κa (κb) is the intrinsic optical loss of the resonator a (b), and are the waveguide-resonator coupling rates at coupling points and , respectively.
The effective dynamic evolution equations of the cavity modes can be written aswhereNote that Fcw (Fccw) denotes the effective unidirectional coupling strength between the CW (CCW) modes of these two resonators. The total input-output relations of this system take the formBy using Eqs 21, 23, we can investigate the photon transport properties of this system in the steady state.
3.2 Nonreciprocal Photon Transmission
In the following, we consider the input signal only comes from one side of the waveguide. Supposed that an external input signal b2,in is injected from the right side of the waveguide with , where ɛ and ωl are the amplitude and frequency of the driving field, respectively. In the rotating frame at the driving frequency ωl, the steady-state solutions of the CW resonator modes in Eq. 21 are solved aswhereHere, Δa = ωa − ωl (Δb = ωb − ωl) is the detuning between the resonator a (b) without rotation and the driving field. According to Eq. 23, the transmission rate of the output port a1,out for the input signal b2,in can be defined as .
Similarly, when an external input signal is injected from the left side of the waveguide with , the steady-state solutions of the CCW resonator modes in Eq. 21 are also solved asOnce again, the transmission rate of the ouput port is given by .
In the following, we choose the related parameters as follows: ωa = ωb = ωc, Ωa = Ωb = 0.97 GHz, , κa = κb = 0.5κ, κ = 5 × 10−3ωc and ϕL,cw = π. Thus, Δa = Δb = Δc and ΔF,a = ΔF,b. We first consider the CW and CCW modes decoupling, i.e., Ja = Jb = 0. In Figures 5A,B, we plot the transmission rates TR and TL versus the detuning Δc/κ and the phase ϕb,cw/π for ϕa,cw = π. According to Eqs 23, 24, the transmission rate TR represents a Lorentzian line shape centered at Δc = −ΔF − κ sin (ϕb,cw) with a linewidth Γb + κ cos (ϕb,cw). However, the behavior of transmission rate TL is different. A mode splitting may appear around Δc = ΔF, which implies indirect coherent coupling between the CCW modes of these two resonators is achieved. The reason behind this phenomenon is that the phase ϕa,ccw is not equal to π own to the rotation. Moreover, the phase ϕb,cw can significantly change the transmission windows with a period 2π. To give more details, in Figures 5C,D we plot the profiles of TR and TL changing with Δc/κ for ϕb,cw = π and ϕb,cw = 1.5π. By contrast, one finds that for ϕb,cw = π, the CW modes decouple to the waveguide corresponding to an optical dark state with TR = 1, while the CCW modes are excited with a transmission dip in TL. For ϕb,cw = 1.5π, strong coupling with a double-dip-type curve in TL can be realized. The photon nonreciprocal transmission behavior is observed due to the Sagnac effects and the interference effects among multiple coupling points. Note that for ϕb,cw = π, similar results are obtained by tuning the phase ϕa,cw.
FIGURE 5
In Figures 6A,B, we plot the transmission rates TR and TL versus the detuning Δc/κ for different Jb. For Jb = 10κ, the transmission spectra display an asymmetric four-dips structure. When decreasing Jb, the transmission dips can be suppressed. Moreover, TL is always larger (smaller) than TR in the region of Δc < 0 (Δc > 0). In order to describe the nonreciprocity clearly, we define the isolation ratio asIn Figure 7, the isolation ratio changing with the detuning Δc/κ and the coupling strength Jb is plotted. It shows that for Jb = 0 the ratio achieves when fixing Δc = 11κ (Δc = −11.5κ). As we increase Jb, a larger mode splitting for Δc > 0 is observed. For Jb = 10κ, the ratio reaches when Δc is set as 15κ. In this case, the photons coming from the left side are blocked, which implies a directional photon transfer between different coupling points. Therefore, the nonreciprocal transmission behavior is also controlled by adjusting the coupling strengths between the CW and CCW modes and the detuning Δc.
FIGURE 6
FIGURE 7
4 Conclusion
In conclusion, we have explored the photon emission and transport properties of spinning resonators coupled to a meandering waveguide at multiple coupling points. We demonstrate that the accumulated phases between multiple coupling points for photons propagating in CW and CCW directions are different. Both “giant-atoms” induced interference effects and mode frequency shifts led by the Sagnac effect dramatically modify photon transport properties. The emission direction and rates can be tuned by changing the spinning speed or number of coupling points. Moreover, the complete photon transmission over the whole optical frequency band led by destructive interference is observed, when photons coming from the right hand of the waveguide. This nonreciprocal phenomenon is very different from that observed in other optical systems. We have also studied the extended two-cavity system. The nonreciprocal photon transmission is controlled by changing the phases among adjacent coupling points or coupling strengths between the CW and CCW modes. By extending our proposal to multiple cavities interacting with multiple points, one can implement a multi-node chiral quantum network. In experiment, such a system with a spinning spherical resonator coupling to a stationary taper has been realized, where the angular speed is about 6.6 kHz [51]. The silica nanoparticle rotating with frequency exceeding 1 GHz has also been reported [62]. Therefore, we believe our theoretical proposals can be realized under current experimental approach. Those results in our paper provide a novel way to engineer rotatable nonreciprocal optical devices, which can be exploited for the realization of large-scale quantum networks and quantum information processing.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Author contributions
XW contributed to conception. WL performed the numerical simulations and produced the first draft. All authors contributed to manuscript revision, read, and approved the submitted version.
Funding
WL was supported by the Natural Science Foundation of Henan Province (No. 222300420233). XW was supported by the National Natural Science Foundation of China (NSFC) (Nos. 12174303 and 11804270) and the China Postdoctoral Science Foundation (No. 2018M631136).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2022.894115/full#supplementary-material
Supplementary Figure S1The Sagnac-Fizeu shift ΔF as a function of the spinning angular velocity Ω for the CW (red line) and CCW (blue line) mode. Other related parameters are Λ = 1550∼nm, R = 4.73∼nm, and n = 1.4.
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Summary
Keywords
chiral emission, nonreciporal transmission, quantum interference, spinning cavity, waveguide-QED
Citation
Liu W, Lin Y, Li J and Wang X (2022) Nonreciprocal Waveguide-QED for Spinning Cavities with Multiple Coupling Points. Front. Phys. 10:894115. doi: 10.3389/fphy.2022.894115
Received
11 March 2022
Accepted
25 March 2022
Published
25 April 2022
Volume
10 - 2022
Edited by
Zhihai Wang, Northeast Normal University, China
Reviewed by
Yan Zhang, Northeast Normal University, China
Wenzhi Jia, Southwest Jiaotong University, China
Updates
Copyright
© 2022 Liu, Lin, Li and Wang.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Xin Wang, wangxin.phy@xjtu.edu.cn
This article was submitted to Optics and Photonics, a section of the journal Frontiers in Physics
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