Abstract

Whispering-gallery-mode (WGM) microresonators provide a high-performance platform for measuring single nanoparticles and viruses, as well as large molecules. However, there is still room for further improving their sensitivity and detection limit, towards their theoretical limit. Here, we present a new method that enhances the performance of WGM sensors based on the mode-splitting method. We show that scatterer-induced mode splitting is significantly enhanced in a rotating resonator. This enhancement originates from the different Sagnac frequency shifts that the clockwise and counterclockwise optical fields in the resonator experience due to the rotation of the resonator. Our approach, combining Sagnac shift and mode splitting, provides a new route for enhancing the coherent optical sensing of nanoparticles with single-particle resolution. In addition, our results shed light on the studies of, e.g., topological or optoacoustic effects with rotating devices.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Whispering-gallery-mode (WGM) optical resonators with their high quality factors and small mode volumes, enabling strong light–matter interactions, have become a versatile platform for fundamental studies and technological applications of light and its interactions with matter [1,2]. A prominent example, closely related to the present study, is ultrasensitive detection of, e.g., weak forces [3], displacement [4], and magnetic field [5] with WGM resonators. In particular, label-free and self-referenced WGM sensing of nano-objects based on mode splitting (i.e., splitting of a resonance mode into two resonances with different frequencies and linewidths) induced by a nano-object entering the mode volume of the resonator has emerged as a promising new technology with widespread applications in, e.g., medical diagnosis and environmental monitoring [69]. This technique can accurately count and size nanoparticles as small as 10 nm in radius, with also superior suppression of noises (such as frequency fluctuation, thermal or back-action noise, and motional disturbance) [8,10]. To further improve the sensitivity of WGM sensors, considerable efforts have been made towards suppressing losses, decreasing the resonator mode volume, or utilizing topological degeneracies arising from the non-Hermiticity of the waveguide-coupled resonators, such as recent experiments utilizing plasmonic particles [1113], optical gain [10,14], or exceptional points [1519].

Here, we show that the performance of a WGM nanoparticle detector can be enhanced by utilizing relativistic motional effects of light, without the need for loss compensation or WGM–plasmon hybrid structures. Specifically, we utilize the Sagnac effect, which induces opposite frequency shifts to counterpropagating optical modes in a spinning resonator [20], to further enhance the scatterer-induced mode splitting. Sagnac- and rotation-induced effects have been theoretically and experimentally investigated for, e.g., sensitive measurements of angular velocity of the medium in optical gyroscopes [21,22], optical chiral symmetry breaking [23], rotational Doppler effect [24], rotary photon drag [25], nonreciprocal light or sound propagation [26,27], and magnetometry [28]. However, the possible role of a rotating WGM resonator in single-particle sensors has not yet been explored, to the best of our knowledge. Our scheme is experimentally accessible, as the required experimental techniques and platforms are readily available [8,18,22]. Also our method is compatible with existing WGM-sensing techniques and thus, besides an additional Sagnac-induced increase in mode splitting, advantages of the WGM sensors as demonstrated in Refs. [8,10], such as self-referencing, ultrasensitivity to perturbations, fast detection, and immunity to thermal drift and laser noise [10,26,29,30], can be integrated into our proposed method.

2. RESULTS AND DISCUSSION

A. Model and Working Principle of a Spinning Nanoparticle-Detector

We consider a waveguide-coupled WGM optical resonator with a single nanoparticle (or scatterer) in its mode volume, as shown in Fig. 1(a). This resonator, with optical resonance frequency ωa and intrinsic loss γa, is probed with an optical field at frequency ωl, which is input in the clockwise (CW) direction [i.e., no input field in the counterclockwise (CCW) direction]. Here, the resonator is rotated, and the waveguide is stationary. Such a system with a spinning resonator and stationary waveguide was experimentally realized in a recent work by Maayani et al. who also showed that such a system can be used for optical nonreciprocity [26]. In the experiment, a resonator with radius R=1.1mm was mounted on a turbine, which spins the resonator stably around its axis, reaching the rotation frequency 6.6 kHz [26]. By positioning the resonator near a single-mode fiber, the light can be evanescently coupled into or out of the resonator through the tapered region. The spinning resonator drags air into the region between the taper and the cavity, so that a boundary layer of air forms. Due to the air pressure on the surface of the taper that faces the resonator, the taper flies above the cavity, the position of which can be self-adjusted [26], leading to stable resonator–fiber coupling. The particle in the evanescent field of the resonator acts as a scatterer, inducing the coupling of CW and CCW propagating modes with strength J and thus leading to an optical mode splitting [8]. Mode splitting induced by other factors such as surface roughness or material inhomogeneity can be pre-detected and thus minimized [8,10,26], or modes and resonators without such intrinsic mode splitting can be used. We note that by directly revealing particle polarizability (a parameter depending on the size, shape, and refractive index of the particle) [8], this technique can discriminate between particles of the same size but different refractive indices or shapes [8,10].

 figure: Fig. 1.

Fig. 1. (a) Schematic of a single nanoparticle sensor with a spinning resonator. The resonator with rotation speed Ω is driven by an optical field. A nanoparicle placed in the mode volume of the resonator induces a coupling with strength J between the CW and CCW modes. (b) Schematic of the energy levels of the optical eigenmodes. The frequency splitting of 2J for a stationary resonator (Ω=0) is increased by the Sagnac effect to Δω>2J when the resonator is spinning (Ω0).

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The CW and CCW modes in the spinning resonator experience different refractive indices [26], i.e., n±=n[1±RΩ(n21)/c], where n and R denote, respectively, the refractive index and the radius of the resonator, Ω is the rotation speed, and c is the speed of light in vacuum. As a result, the frequencies of CW and CCW modes of a resonator experience Sagnac–Fizeau shifts [31], i.e., ωaωa±Δsag, with

Δsag=nRΩωac(11n2λndndλ),
where λ is the wavelength of the probe light. The dispersion term λdn/ndλ makes up 1% of the value of (11/n2), which characterizes the relativistic origin of the Sagnac effect [31]. The underlying physics is indeed well known [31]; however, the possibility of using it to enhance the performance of a particle sensor has not been explored previously.

In the rotating frame at the frequency ωl of the probing field, the effective Hamiltonian of the system in the traveling-wave basis can be written as [8,15]

Heff=(Δ+iγ)acwacw+(Δiγ)accwaccw+(Jiγc)acwaccw+(Jiγc)accwacw+iγexain(acwacw),
with
Δ±=Δa+J±Δsag,γ=(γa+γex)/2+γc,
where γ is the total optical loss, acw or accw denotes the intracavity field of the CW or CCW mode, Δa=ωaωl, γex is the waveguide–resonator coupling rate, γc denotes the optical loss induced by the particle, and ain is the amplitude of the probe field. The last term in Eq. (2) implies that the optical field is input only in the CW direction. The system studied here is fully classical, and acw,ccw can be treated as classical numbers. The eigenfrequencies of this system are readily derived as
ω1,2=ωa+J±ωi(γ±γ),
where
2ω=[2(D2+4J2γc2)1/2+2D]1/2,2γ=[2(D2+4J2γc2)1/22D]1/2,
and D=Δsag2+J2γc2. The complex optical frequency splitting is then found as Δω=ω1ω2=2ω2iγ, which is now strongly affected by rotation speed Ω.

B. Enhanced Mode-Splitting in a Spinning Resonator

Here, we consider that the scatterer falls onto the surface of the resonator and stays on the resonator, rotating with it, similar to the particles on a stationary WGM resonator [8,10]. In the following, we compare the particle-induced mode splitting in a spinning resonator with that in a stationary resonator, assuming that the particles falling onto the surface of the stationary and spinning resonators attach to the surface and stay at their place. Figure 1(b) shows the schematic of the energy levels of optical eigenmodes, which are strongly related to the detection limit. For a stationary cavity, two orthogonal standing wave modes (SWMs) are formed by the presence of the particle. The particle is detectable if the SMWs can be resolved in the transmission spectrum. The strict condition for the resolvability of mode splitting in the transmission spectrum was previously defined as J>γ [32], which can be generalized to ω>γ for the spinning case. Also, the upper limit of the detection is rλ [8], where r is the radius of the particle. To compare the frequency splitting in a rotating system and that in a stationary system, we define the enhancement factor

η=|Δω(Ω0)Δω(Ω=0)|=[D2+4J2γc2(J2+γc2)2]1/4>1.
Clearly, factor η depends on the rotation speed of the resonator and the frequency splitting induced by the particle. We note that for a stationary system, there is no observable splitting in the transmission spectrum if J<γc. Therefore, in evaluating η in Eq. (6). we restrict the discussion to the case J>γc (i.e., observable splitting for the stationary resonator). Under this condition, D2+4J2γc2>(J2+γc2)2, and hence D2>(J2γc2)2 should be satisfied to have η>1. It is easy to show that D2>(J2γc2)2 is always true, implying that η is always greater than one (i.e., η>1). Thus, mode splitting in a spinning resonator (i.e., Ω0) is always larger than mode splitting in a stationary resonator (i.e., Ω=0) for the same perturbation. We mention that particle position does not affect the Sagnac-induced enhancement (or change) in the splitting, but it does affect the splitting due to scatterer-induced modal coupling [8]. Thus enhancement of the Sagnac-based scheme over the stationary-resonator-based schemes utilizing mode splitting is not affected by the position of the particle. Similar to stationary sensors, the spinning sensor is also robust against additional (but detectable) noises or uncertainty of, e.g., rotation speed [22,26,28,33]. Moreover, the rotary uncertainty or the uniformity of the resonator can be optically detected and overcome via designing or calibrating the resonator and related systems [22,28,33].

Figure 2(a) shows enhancement factor η as a function of rotation speed for different particle-induced mode couplings. In our calculations, we choose experimentally available values [8,17,26], i.e., n=1.44, ωa=193.5THz, and γex=γa=6.43MHz. In Fig. 2(a), we choose R=1.1mm, J/γa=1.5, and γc/γa=0.3 for case A, while we choose R=0.5mm, J/γa=1.1, and γc/γa=0.065 for case B, as in experiments [17,18]. The values of J and γc depend on the radius or the refractive index of the particle, and the mode volume of the resonator [8,18]. We find that η increases with rotation speed Ω, leading to enhanced frequency splitting. We note that rotation speed Ω=6.6kHz for R=1.1mm has been realized experimentally [26]; for comparisons, we also plot the case with accessible parameters Ω=30kHz and R=0.5mm. We also note that much higher frequencies of rotation have been realized in recent experiments [34,35]. These results imply that spinning the resonator provides a potentially helpful way to further improve the performance of the particle sensor. Thus, particles that may go undetected with a stationary sensor may be detected with a spinning resonator.

 figure: Fig. 2.

Fig. 2. Sensitivity enhancement factor η (a) and relative photon number β=|a¯cw|2/|a¯ccw|2 (b) as a function of rotation speed Ω. We choose R=1.1mm, J/γa=1.5, and γc/γa=0.3 for case A, and R=0.5mm, J/γa=1.1, and γc/γa=0.065 for case B, as in recent experiments [8,17,18]. Also, J and γc depend on the properties of the particle and the resonator [8,17,18].

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C. Enhancement of Transmission in a Spinning Resonator

The rotation also induces resonance frequency shifts for optical modes, so that the intracavity fields can be strongly modified. To clearly see this, we derive the steady-state values of the fields in the CW and CCW directions as

a¯cw=γexain(γ+iΔ)(γ+iΔ+)(γ+iΔ)+(Jiγc)2,a¯ccw=γexain(iJ+γc)(γ+iΔ+)(γ+iΔ)+(Jiγc)2,
and define the relative photon number as
β=|a¯cw|2|a¯ccw|2=γ2+(Δa+JΔsag)2J2+γc2.
Figure 2(b) shows that β depends on both rotation speed Ω and optical detuning Δa. For Δa/J1, β always increases with Ω, while for Δa/J>1, β first decreases, arriving at a minimum at Δsag=Δa+J, and then increases with increasing Ω. We see from Eq. (7) that the intracavity photon numbers of both the CW and CCW modes decrease with Ω, as a result of the shifted optical resonance. However, for a high rotation speed Ω, the Sagnac shift Δsag exceeds Δa and J, leading to a larger value of β, and thus significantly affecting the properties of light propagation.

By using the input–output relation, aout=ainγexa¯cw, where aout is the amplitude of the output field, we obtain the optical transmission rate

T=|aoutain|2=|1γex(γ+iΔ)(γ+iΔ+)(γ+iΔ)+(Jiγc)2|2.
Figures 3(a) and 3(b) show the transmission rate T for the case γc/γa=0.3 (case A in Fig. 2) as a function of optical detuning. In the absence of a particle, a stationary resonator supports two degenerate modes with the same frequency but opposite propagation directions (CW and CCW traveling modes), exhibiting a resonance dip at Δa=0 in the transmission spectrum. When the same resonator spins at speed Ω, the resonance dip shifts to Δa=Δsag due to the Sagnac effect [blue curves in Figs. 3(a)3(b)]. When a nanoparticle enters the mode volume of the stationary resonator, scattering-induced modal coupling leads to two orthogonal standing-wave modes, which are reflected in the transmission spectrum as two spectrally different resonances. The asymmetry in the spectrum implies that the particle induces different losses to the two orthogonal modes. For a spinning resonator with a nanoparticle in the mode volume, the resonance dips of the split modes shift to Δa=J±ω. If Ω, for example, is tuned from 0 kHz to 6.6 kHz, the frequency splitting increases from 19.35 MHz to 20.51 MHz. The difference in linewidth, however, changes slightly. The resonance shift and the enhancement of frequency splitting due to Sagnac effect lead to changes in the transmission at fixed frequency detuning Δa=Δsag. For example, in the presence of a particle, the change in the transmission for Ω=6.6kHz is twice that for Ω=0, as shown in Figs. 3(a)3(b).

 figure: Fig. 3.

Fig. 3. Transmission rate T as a function of optical detuning Δa for Ω=0 (i.e., a stationary resonator) (a), Ω=6.6kHz (b), and Ω=30kHz (d). (c) Transmission difference ΔT as a function of optical detuning Δa. (e) Transmission rate T as a function of optical detuning Δa and rotation speed Ω. (f) Transmission difference ΔT as a function of detuning Δa and rotation speed Ω. We choose R=1.1mm, J/γa=1.5, and γc/γa=0.3 (i.e., case A in Fig. 2) in (a)–(c), and R=0.5mm, J/γa=1.1, and γc/γa=0.065 in (d)–(f) (i.e., case B in Fig. 2).

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We note that the location of the particle affects mode splitting due to the scatterer-induced modal coupling but not the enhancement (or change) in mode splitting due to the Sagnac effect (in a spinning resonator). In the case of mode splitting due to the scatterer-induced modal coupling, a particle located in a high-intensity field in the mode volume leads to a larger mode splitting than the same particle located in a low-intensity field in the mode volume. However, Sagnac-induced enhancement in mode splitting, depending on the spinning speed of the resonator, is the same for the particle located in the low or high field in the mode volume. Hence, in experiments it is possible to distinguish these two different kinds of effects. Also we note that the Sagnac effect can lead to an observable frequency shift of the output spectrum even for the case without any particle. Therefore, by analyzing the change of, e.g., the transmission spectra of the probe light, the additional effect induced by the scatter can also be identified by analyzing, e.g., the locations of the minima of the transmission curve.

To show the impact of the particle on light propagation in a spinning resonator, we define the transmission difference: ΔT=TT0, where T and T0 are the optical transmission with and without the particle, respectively. Figure 3(c) shows that for a high rotation rate, ΔT can be tuned in a large range. We note that ΔT can be changed even for a slow rotation speed. For example, for Δsagγa,J, we have

ΔT(Δa=0)J2+γc2+2JΔsag(γ+γc)2+4J2.
The term 2JΔsag in the numerator of Eq. (10) implies that the effect of frequency splitting on the transmission is enhanced by the rotation speed of the resonator. In other words, the transmission of a spinning resonator is very sensitive to perturbations induced by a particle.

Then we consider the transmission rate for the case γc/γa=0.065 (case B in Fig. 2) with much higher rotation speeds. Figure 3(d) shows the transmission rate T for Ω=30kHz. We find that compared with case A (i.e., γc/γa=0.065) with Ω=6.6kHz, the spectrum is strongly modified for higher rotation speed. If Ω is increased from 0 kHz to 30 kHz, the frequency splitting increases from 14.19 MHz to 20.13 MHz. Figure 3(e) shows the dependence of T on Δa and Ω. Clearly, the frequency splitting is broadened for a large value of Ω. In this regime,

a¯cwγexainγ+i(Δa+J+Δsag)
from which we find that the intracavity photon number of the CW mode for Δa<0 is larger than that for Δa>0. As a result, strong absorption emerges in the Δa<0 regime, which is in sharp contrast to the stationary system. Figure 3(f) shows that depending on both optical detuning and rotation speed, ΔT can be tuned to be positive or negative with a fixed optical detuning, indicating also a convenient way to observe the splitting induced by the particle.

D. Multiparticle Detection with a Spinning Resonator

The above discussion can be generalized to a more practical case with several particles falling into the mode volume one by one. In this situation, the effective Hamiltonian can be written as

Heff=(Δ+iγ)acwacw+(Δiγ)accwaccw+C1acwaccw+C2accwacw+iγexain(acwacw),
where
Δ±=Δa+i=1NJi±Δsag,γ=(γa+γex)/2+i=1Nγc,i,C1,2=i=1N(Jiiγc,i)exp(i2mβj),
where (Jiiγc,i) and βi are the complex frequency splitting and angular position of i-th particle, N is the particle number, and m is the azimuthal mode number. Figure 4 shows discrete up or down jumps of the frequency splitting with nanoparticles continuously deposited on the sensor, in which random values of (Jiγc,i) and βi are chosen. It is obvious that detection sensitivity is enhanced for each particle by spinning the resonator. We note that a higher rotation speed can be achieved for a smaller resonator, leading to further enhanced detection sensitivity, as shown in Fig. 4(b).

 figure: Fig. 4.

Fig. 4. Variation in frequency splitting for a spinning and a stationary WGM-based particle sensor as more than one particle is deposited on the resonators. The rotation speed is set as Ω=6.6kHz in (a) and 30 kHz in (b).

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3. CONCLUSION

In conclusion, we have showed that spinning a WGM resonator enhances the performance of a WGM-based nanoparticle detector. Compared to a stationary sensor, our scheme features both an enhanced mode splitting and a strongly modified transmission spectrum, providing a new way to enhance the sensitivity of WGM-based particle sensors, without the need for complex materials, hybrid systems, optical gain, or low temperatures. Our proposed method is complementary to other state-of-the-art sensing techniques already achieved experimentally, using, e.g., an active resonator [10] or a sensor operating at an exceptional point [18,19].

We note that rotating resonators or sensors have been pursued very recently in various experiments [22,26,28], with precisions limited by—as also in their stationary counterparts—optical absorption, optical recoil noise, or the motion of the particles. Future works will consider the effect of various losses and noises on the performance of these sensors and an in-depth study of time-varying interaction between WGM field and the flying-by nanoparticles [8]. Finally, besides sensing, our results may potentially lead to the use of spinning resonators in engineering of, e.g., topological or optoacoustic devices.

APPENDIX A: EXPERIMENTAL FEASIBILITY OF THE SPINNING SENSOR

In our proposed scheme, the ring cavity can be mounted on a turbine, which spins the resonator, as in the recent experiment performed by Maayani et al. [26]. In the experiment, the resonator with radius R=1.1mm can spin with the stability of its axis, reaching the rotation frequency 6.6 kHz. By positioning the resonator near a single-mode telecommunication fiber, the light can be coupled into or out of the resonator evanescently through the tapered region.

For such a spinning device [26], the aerodynamic process plays a key role in stable resonator–fiber coupling: a fast spinning resonator can drag air into the region between the taper and the cavity, so that a boundary layer of air forms. Due to the air pressure on the surface of the taper facing the resonator, the taper flies at a height above the cavity, which can be several nanometers. If some perturbation induces the taper rising higher than the stable equilibrium height, it floats back to its original position, which is called “self-adjustment” [26]. The self-adjustment of the taper separation from the spinning resonator enables critical coupling of light into the cavity, by which counter-circulating light experiences optical drag identical in size, but opposite in sign. Importantly, this experiment confirms that the taper did not touch or stick to the rotating resonator even if the taper is pushed towards it, which is in contrast to the situation for a stationary resonator (i.e., the taper may stick to the resonator through van der Waals forces and thus needs to be pulled back to break the connection). Other factors, such as intermolecular forces, lubricant compressibility, tapered-fiber stiffness, and wrap angle of the fiber, may affect resonator–waveguide coupling. However, these factors are confirmed to be negligible in the experiment, which are thus also neglected in our discussions on the spinning sensor.

We consider that the scatterer falls onto the surface of the resonator and stays on the resonator, rotating with it, similar to the particles on a stationary WGM resonator [8,10]. We compare the mode splitting in a spinning resonator with the splitting in a stationary resonator: if the particle is not stationary on the mode volume (e.g., particle diffuses on the surface), then we will observe changes in the amount of mode splitting. If the particle is detached from the resonator due to spinning or any other reasons, then mode splitting will disappear and the transmission will return back to the case when there is no particle.

The strength of the interaction between the cavity field and the particle is affected by the location of the particle. For example, a particle may be located on a position on the resonator where there is no field, or the field is very weak. This will lead to lower interaction strength and hence the amount of mode splitting will be small. If the particle sits on a position where the field is high, then the mode splitting will be larger as the light–particle interaction strength will be higher. This effect is valid in the Sagnac-based particle detector too. However, the Sagnac effect itself is not affected by the particle position. Thus the enhancement of mode splitting originating from the Sagnac effect will not be affected by the position of the particle. However, mode splitting that originates from the scatterer-induced modal coupling is affected by the position of the particle. Therefore, the fact that the Sagnac-based scheme performs better than a stationary-resonator-based scheme utilizing mode splitting does not change by the position of the particle. The complex optical mode coupling induced by the particle, related to the overlap between the particle and mode volume of the resonator, depends on the particle position. According to Ref. [8], the complex mode coupling induced by the particle can be expressed as

J=αf2(r)ωc2Vc,γc=2π2α2f2(r)ωc3λ3Vc,
where r is the particle position, Vc is the mode volume of the resonator, f(r) is the normalized mode distribution function, and λ is the light wavelength in the resonator. α is the particle polarizability, which depends on the size and refractive index of the particle. Then we can see that the particle position affects the values of J and γc, but does not change the ratio
γc/J=4π2α/(3λ3).

The particle position may affect detection efficiency, since the particle is undetectable if ω<γ is satisfied, where ω is half of the frequency splitting induced by the particle. However, in previous experiments [8,10], nanoparticles falling onto the stationary resonator randomly were detected and counted. Here we propose to enhance the performance of the WGM particle sensor by use of the Sagnac effect. We show that sensitivity enhancement factor η always exceeds 1 with the condition J>γc, which indicates an observable splitting for the stationary resonator. In Fig. 4, we also plot the frequency splitting variation for several nanoparticles, in which random values of J and γc are used. It is clear that the performance of a particle sensor can be enhanced by spinning the resonator.

The rotation of the resonator may lead to some mechanical effects on the particle. A fast spinning resonator drags a boundary layer of air past the region between the resonator and stationary taper, which may induce the particle diffusing in the air. This may change the overlap of the particle and the mode volume of the resonator, and then the values of J and γc. However, the ratio γc/J is not affected by the particle position. Our scheme is robust against the motion of the particle, since the sensitivity enhancement factor always exceeds 1 when there is an observable splitting in the transmission spectrum of a stationary resonator. In addition, the particle detection in current experimental WGM sensors can be very fast. We can safely say that the location of the particle and the rotation of the resonator will not change the fact that the spinning-resonator-based scheme performs better than the stationary-resonator scheme.

APPENDIX B: DERIVATION OF THE EFFECTIVE HAMILTONIAN

In consideration of the Sagnac effect, the Hamiltonian of the spinning resonator–nanoparticle system can be written as

H=(ωa+Δsag)acwacw+(ωaΔsag)accwaccw+J(acwaccw+accwacw)+γexain(acweiωltacweiωlt),
where acw or accw denotes the intracavity field of the CW or CCW mode, ωa is the resonance frequency of the stationary cavity, ωl is the frequency of the probe light, Δsag is the Sagnac shift, J is the optical coupling induced by the particle, γex is the resonator–waveguide coupling rate, and ain is the amplitude of the probe light. By using the unitary transformation
U=exp[iωlt(acwacw+accwaccw)],
the Hamiltonian can be transformed into the rotating frame, i.e.,
H=UHUiUUt,=(Δa+Δsag)acwacw+(ΔaΔsag)accwaccw+J(acwaccw+accwacw)+γexain(acwacw),
where Δa=ωaωl is the optical detuning. Notice that the total optical loss is γ=(γa+γex)/2+γc, where γa is the intrinsic optical loss of the resonator, and γc is the optical loss induced by the scatter. The effective Hamiltonian can be written as
Heff=(Δ+iγ)acwacw+(Δiγ)accwaccw+(Jiγc)acwaccw+(Jiγc)accwacw+iγexain(acwacw),
where Δ±=Δa±Δsag. Solving the characteristic equation of the system given by
(ωa+Δsagiγω)(ωaΔsagiγω)(Jiγc)2=0,
we find the eigenfrequencies of the system as
ω=ωaiγ±D2iJγc,D=Δsag2+J2γc2.
By setting D2iJγc=(ωiγ)2, where ω and γ are real numbers, we have
ω=[(D2+4J2γc2)1/2/2+D/2]1/2,γ=[(D2+4J2γc2)1/2/2D/2]1/2.
Then the eigenfrequencies of the system can be written as ω1,2=ωa+J±ωi(γ±γ), so that the complex mode splitting is Δω=2ω2iγ.

Funding

National Natural Science Foundation of China (NSFC) (11474087, 11774086); Asian Office of Aerospace Research and Development (AOARD) (FA2386-18-1-4045); Army Research Office (ARO) (W911NF-18-1-0358, W911NF-18-1-0043); MURI Center for Dynamic Magneto-Optics; Air Force Office of Scientific Research (AFOSR) (FA9550-18-1-0235, FA9550-14-1-0040); Core Research for Evolutional Science and Technology (CREST) (JPMJCR1676); Japan Society for the Promotion of Science (JSPS) (JSPS-RFBR 17-52-50023, JSPS-FWO VS.059.18N);National Science Foundation (NSF) (1807485); Pennsylvania State University (PSU), Materials Research Institute (MRI); Sir John Templeton Foundation; Israeli Centers for Research Excellence (I-CORE); RIKEN-National Institute of Advanced Industrial Science and Technology (AIST) (AIST) Challenge Research Fund.

Acknowledgment

H. J. thanks Ran Huang and Tian-Xiang Lu for their helpful discussions.

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6. A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,”Science 317, 783–787 (2007). [CrossRef]  

7. F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. USA 105, 20701–20704 (2008). [CrossRef]  

8. J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010). [CrossRef]  

9. L. He, Ş. K. Özdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6, 428–432 (2011). [CrossRef]  

10. Ş. K. Özdemir, J. Zhu, X. Yang, B. Peng, H. Yilmaz, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whisperinggallery Raman microlaser,” Proc. Natl. Acad. Sci. USA 111, E3836 (2014). [CrossRef]  

11. S. I. Shopova, R. Rajmangal, S. Holler, and S. Arnold, “Plasmonic enhancement of a whispering-gallery-mode biosensor for single nanoparticle detection,” Appl. Phys. Lett. 98, 243104 (2011). [CrossRef]  

12. V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Lett. 13, 3347–3351 (2013). [CrossRef]  

13. M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9, 933–939 (2014). [CrossRef]  

14. B.-B. Li, W. R. Clements, X.-C. Yu, K. Shi, Q. Gong, and Y.-F. Xiao, “Single nanoparticle detection using split-mode microcavity Raman lasers,” Proc. Natl. Acad. Sci. USA 111, 14657–14662 (2014). [CrossRef]  

15. J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014). [CrossRef]  

16. J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016). [CrossRef]  

17. B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. USA 113, 6845–6850 (2016). [CrossRef]  

18. W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017). [CrossRef]  

19. H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Nature 548, 187–191 (2017). [CrossRef]  

20. E. J. Post, “Sagnac effect,” Rev. Mod. Phys. 39, 475–481 (1967). [CrossRef]  

21. W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985). [CrossRef]  

22. J. Li, M.-G. Suh, and K. Vahala, “Microresonator Brillouin gyroscope,” Optica 4, 346–348 (2017). [CrossRef]  

23. R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015). [CrossRef]  

24. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013). [CrossRef]  

25. S. Franke-Arnold, G. Gibson, R. W. Boyd, and M. J. Padgett, “Rotary photon drag enhanced by a slow-light medium,” Science 333, 65–67 (2011). [CrossRef]  

26. S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–572 (2018). [CrossRef]  

27. R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Alù, “Sound isolation and giant linear nonreciprocity in a compact acoustic circulator,” Science 343, 516–519 (2014). [CrossRef]  

28. A. A. Wood, E. Lilette, Y. Y. Fein, A. Stacey, N. Tomek, L. P. McGuinness, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Quantum measurement of a rapidly rotating spin qubit in diamond,” Sci. Adv. 4, eaar7691 (2018). [CrossRef]  

29. J. D. Swaim, J. Knittel, and W. P. Bowen, “Detection of nanoparticles with a frequency locked whispering gallery mode microresonator,” Appl. Phys. Lett. 102, 183106 (2013). [CrossRef]  

30. J. Knittel, J. D. Swaim, D. L. McAuslan, G. A. Brawley, and W. P. Bowen, “Back-scatter based whispering gallery mode sensing,” Sci. Rep. 3, 2974 (2013). [CrossRef]  

31. G. B. Malykin, “The Sagnac effect: correct and incorrect explanations,” Phys. Usp. 43, 1229–1252 (2000). [CrossRef]  

32. Ş. K. Özdemir, J. Zhu, L. He, and L. Yang, “Estimation of Purcell factor from mode-splitting spectra in an optical microcavity,” Phys. Rev. A 83, 033817 (2011). [CrossRef]  

33. A. A. Wood, E. Lilette, Y. Y. Fein, V. S. Perunicic, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Magnetic pseudo-fields in a rotating electron–nuclear spin system,” Nat. Phys. 13, 1070–1073 (2017). [CrossRef]  

34. R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “Enhanced sensitivity at higher-order exceptional points,” Phys. Rev. Lett. 121, 033602 (2018). [CrossRef]  

35. J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033603 (2018). [CrossRef]  

References

  • View by:

  1. K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
    [Crossref]
  2. M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode,” Adv. Opt. Photon. 7, 168–240 (2015).
    [Crossref]
  3. E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nat. Nanotechnol. 7, 509–514 (2012).
    [Crossref]
  4. R. Schilling, H. Schütz, A. H. Ghadimi, V. Sudhir, D. J. Wilson, and T. J. Kippenberg, “Near-field integration of a SiN nanobeam and a SiO2 microcavity for Heisenberg-limited displacement sensing,” Phys. Rev. Appl. 5, 054019 (2016).
    [Crossref]
  5. S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108, 120801 (2012).
    [Crossref]
  6. A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,”Science 317, 783–787 (2007).
    [Crossref]
  7. F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. USA 105, 20701–20704 (2008).
    [Crossref]
  8. J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
    [Crossref]
  9. L. He, Ş. K. Özdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6, 428–432 (2011).
    [Crossref]
  10. Ş. K. Özdemir, J. Zhu, X. Yang, B. Peng, H. Yilmaz, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whisperinggallery Raman microlaser,” Proc. Natl. Acad. Sci. USA 111, E3836 (2014).
    [Crossref]
  11. S. I. Shopova, R. Rajmangal, S. Holler, and S. Arnold, “Plasmonic enhancement of a whispering-gallery-mode biosensor for single nanoparticle detection,” Appl. Phys. Lett. 98, 243104 (2011).
    [Crossref]
  12. V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Lett. 13, 3347–3351 (2013).
    [Crossref]
  13. M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9, 933–939 (2014).
    [Crossref]
  14. B.-B. Li, W. R. Clements, X.-C. Yu, K. Shi, Q. Gong, and Y.-F. Xiao, “Single nanoparticle detection using split-mode microcavity Raman lasers,” Proc. Natl. Acad. Sci. USA 111, 14657–14662 (2014).
    [Crossref]
  15. J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
    [Crossref]
  16. J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016).
    [Crossref]
  17. B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. USA 113, 6845–6850 (2016).
    [Crossref]
  18. W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
    [Crossref]
  19. H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Nature 548, 187–191 (2017).
    [Crossref]
  20. E. J. Post, “Sagnac effect,” Rev. Mod. Phys. 39, 475–481 (1967).
    [Crossref]
  21. W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
    [Crossref]
  22. J. Li, M.-G. Suh, and K. Vahala, “Microresonator Brillouin gyroscope,” Optica 4, 346–348 (2017).
    [Crossref]
  23. R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015).
    [Crossref]
  24. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
    [Crossref]
  25. S. Franke-Arnold, G. Gibson, R. W. Boyd, and M. J. Padgett, “Rotary photon drag enhanced by a slow-light medium,” Science 333, 65–67 (2011).
    [Crossref]
  26. S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–572 (2018).
    [Crossref]
  27. R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Alù, “Sound isolation and giant linear nonreciprocity in a compact acoustic circulator,” Science 343, 516–519 (2014).
    [Crossref]
  28. A. A. Wood, E. Lilette, Y. Y. Fein, A. Stacey, N. Tomek, L. P. McGuinness, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Quantum measurement of a rapidly rotating spin qubit in diamond,” Sci. Adv. 4, eaar7691 (2018).
    [Crossref]
  29. J. D. Swaim, J. Knittel, and W. P. Bowen, “Detection of nanoparticles with a frequency locked whispering gallery mode microresonator,” Appl. Phys. Lett. 102, 183106 (2013).
    [Crossref]
  30. J. Knittel, J. D. Swaim, D. L. McAuslan, G. A. Brawley, and W. P. Bowen, “Back-scatter based whispering gallery mode sensing,” Sci. Rep. 3, 2974 (2013).
    [Crossref]
  31. G. B. Malykin, “The Sagnac effect: correct and incorrect explanations,” Phys. Usp. 43, 1229–1252 (2000).
    [Crossref]
  32. Ş. K. Özdemir, J. Zhu, L. He, and L. Yang, “Estimation of Purcell factor from mode-splitting spectra in an optical microcavity,” Phys. Rev. A 83, 033817 (2011).
    [Crossref]
  33. A. A. Wood, E. Lilette, Y. Y. Fein, V. S. Perunicic, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Magnetic pseudo-fields in a rotating electron–nuclear spin system,” Nat. Phys. 13, 1070–1073 (2017).
    [Crossref]
  34. R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “Enhanced sensitivity at higher-order exceptional points,” Phys. Rev. Lett. 121, 033602 (2018).
    [Crossref]
  35. J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033603 (2018).
    [Crossref]

2018 (4)

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–572 (2018).
[Crossref]

A. A. Wood, E. Lilette, Y. Y. Fein, A. Stacey, N. Tomek, L. P. McGuinness, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Quantum measurement of a rapidly rotating spin qubit in diamond,” Sci. Adv. 4, eaar7691 (2018).
[Crossref]

R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “Enhanced sensitivity at higher-order exceptional points,” Phys. Rev. Lett. 121, 033602 (2018).
[Crossref]

J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033603 (2018).
[Crossref]

2017 (4)

A. A. Wood, E. Lilette, Y. Y. Fein, V. S. Perunicic, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Magnetic pseudo-fields in a rotating electron–nuclear spin system,” Nat. Phys. 13, 1070–1073 (2017).
[Crossref]

J. Li, M.-G. Suh, and K. Vahala, “Microresonator Brillouin gyroscope,” Optica 4, 346–348 (2017).
[Crossref]

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Nature 548, 187–191 (2017).
[Crossref]

2016 (3)

J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016).
[Crossref]

B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. USA 113, 6845–6850 (2016).
[Crossref]

R. Schilling, H. Schütz, A. H. Ghadimi, V. Sudhir, D. J. Wilson, and T. J. Kippenberg, “Near-field integration of a SiN nanobeam and a SiO2 microcavity for Heisenberg-limited displacement sensing,” Phys. Rev. Appl. 5, 054019 (2016).
[Crossref]

2015 (2)

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode,” Adv. Opt. Photon. 7, 168–240 (2015).
[Crossref]

R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015).
[Crossref]

2014 (5)

R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Alù, “Sound isolation and giant linear nonreciprocity in a compact acoustic circulator,” Science 343, 516–519 (2014).
[Crossref]

Ş. K. Özdemir, J. Zhu, X. Yang, B. Peng, H. Yilmaz, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whisperinggallery Raman microlaser,” Proc. Natl. Acad. Sci. USA 111, E3836 (2014).
[Crossref]

M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9, 933–939 (2014).
[Crossref]

B.-B. Li, W. R. Clements, X.-C. Yu, K. Shi, Q. Gong, and Y.-F. Xiao, “Single nanoparticle detection using split-mode microcavity Raman lasers,” Proc. Natl. Acad. Sci. USA 111, 14657–14662 (2014).
[Crossref]

J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
[Crossref]

2013 (4)

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Lett. 13, 3347–3351 (2013).
[Crossref]

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

J. D. Swaim, J. Knittel, and W. P. Bowen, “Detection of nanoparticles with a frequency locked whispering gallery mode microresonator,” Appl. Phys. Lett. 102, 183106 (2013).
[Crossref]

J. Knittel, J. D. Swaim, D. L. McAuslan, G. A. Brawley, and W. P. Bowen, “Back-scatter based whispering gallery mode sensing,” Sci. Rep. 3, 2974 (2013).
[Crossref]

2012 (2)

E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nat. Nanotechnol. 7, 509–514 (2012).
[Crossref]

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108, 120801 (2012).
[Crossref]

2011 (4)

L. He, Ş. K. Özdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6, 428–432 (2011).
[Crossref]

S. I. Shopova, R. Rajmangal, S. Holler, and S. Arnold, “Plasmonic enhancement of a whispering-gallery-mode biosensor for single nanoparticle detection,” Appl. Phys. Lett. 98, 243104 (2011).
[Crossref]

Ş. K. Özdemir, J. Zhu, L. He, and L. Yang, “Estimation of Purcell factor from mode-splitting spectra in an optical microcavity,” Phys. Rev. A 83, 033817 (2011).
[Crossref]

S. Franke-Arnold, G. Gibson, R. W. Boyd, and M. J. Padgett, “Rotary photon drag enhanced by a slow-light medium,” Science 333, 65–67 (2011).
[Crossref]

2010 (1)

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[Crossref]

2008 (1)

F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. USA 105, 20701–20704 (2008).
[Crossref]

2007 (1)

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,”Science 317, 783–787 (2007).
[Crossref]

2003 (1)

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref]

2000 (1)

G. B. Malykin, “The Sagnac effect: correct and incorrect explanations,” Phys. Usp. 43, 1229–1252 (2000).
[Crossref]

1985 (1)

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[Crossref]

1967 (1)

E. J. Post, “Sagnac effect,” Rev. Mod. Phys. 39, 475–481 (1967).
[Crossref]

Ahn, J.

J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033603 (2018).
[Crossref]

Alù, A.

R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Alù, “Sound isolation and giant linear nonreciprocity in a compact acoustic circulator,” Science 343, 516–519 (2014).
[Crossref]

Armani, A. M.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,”Science 317, 783–787 (2007).
[Crossref]

Arnold, S.

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Lett. 13, 3347–3351 (2013).
[Crossref]

S. I. Shopova, R. Rajmangal, S. Holler, and S. Arnold, “Plasmonic enhancement of a whispering-gallery-mode biosensor for single nanoparticle detection,” Appl. Phys. Lett. 98, 243104 (2011).
[Crossref]

F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. USA 105, 20701–20704 (2008).
[Crossref]

Baaske, M. D.

M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9, 933–939 (2014).
[Crossref]

Bang, J.

J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033603 (2018).
[Crossref]

Barbre, C.

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Lett. 13, 3347–3351 (2013).
[Crossref]

Barnett, S. M.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

Bowen, W. P.

J. D. Swaim, J. Knittel, and W. P. Bowen, “Detection of nanoparticles with a frequency locked whispering gallery mode microresonator,” Appl. Phys. Lett. 102, 183106 (2013).
[Crossref]

J. Knittel, J. D. Swaim, D. L. McAuslan, G. A. Brawley, and W. P. Bowen, “Back-scatter based whispering gallery mode sensing,” Sci. Rep. 3, 2974 (2013).
[Crossref]

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108, 120801 (2012).
[Crossref]

Boyd, R. W.

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Ş. K. Özdemir, J. Zhu, L. He, and L. Yang, “Estimation of Purcell factor from mode-splitting spectra in an optical microcavity,” Phys. Rev. A 83, 033817 (2011).
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J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033603 (2018).
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H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Nature 548, 187–191 (2017).
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A. A. Wood, E. Lilette, Y. Y. Fein, V. S. Perunicic, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Magnetic pseudo-fields in a rotating electron–nuclear spin system,” Nat. Phys. 13, 1070–1073 (2017).
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S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–572 (2018).
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V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Lett. 13, 3347–3351 (2013).
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[Crossref]

Kim, W.

L. He, Ş. K. Özdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6, 428–432 (2011).
[Crossref]

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R. Schilling, H. Schütz, A. H. Ghadimi, V. Sudhir, D. J. Wilson, and T. J. Kippenberg, “Near-field integration of a SiN nanobeam and a SiO2 microcavity for Heisenberg-limited displacement sensing,” Phys. Rev. Appl. 5, 054019 (2016).
[Crossref]

E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nat. Nanotechnol. 7, 509–514 (2012).
[Crossref]

Kligerman, Y.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–572 (2018).
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J. Knittel, J. D. Swaim, D. L. McAuslan, G. A. Brawley, and W. P. Bowen, “Back-scatter based whispering gallery mode sensing,” Sci. Rep. 3, 2974 (2013).
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J. D. Swaim, J. Knittel, and W. P. Bowen, “Detection of nanoparticles with a frequency locked whispering gallery mode microresonator,” Appl. Phys. Lett. 102, 183106 (2013).
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S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108, 120801 (2012).
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V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Lett. 13, 3347–3351 (2013).
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B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. USA 113, 6845–6850 (2016).
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A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,”Science 317, 783–787 (2007).
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B.-B. Li, W. R. Clements, X.-C. Yu, K. Shi, Q. Gong, and Y.-F. Xiao, “Single nanoparticle detection using split-mode microcavity Raman lasers,” Proc. Natl. Acad. Sci. USA 111, 14657–14662 (2014).
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Li, L.

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
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J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033603 (2018).
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B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. USA 113, 6845–6850 (2016).
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A. A. Wood, E. Lilette, Y. Y. Fein, V. S. Perunicic, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Magnetic pseudo-fields in a rotating electron–nuclear spin system,” Nat. Phys. 13, 1070–1073 (2017).
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Ş. K. Özdemir, J. Zhu, X. Yang, B. Peng, H. Yilmaz, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whisperinggallery Raman microlaser,” Proc. Natl. Acad. Sci. USA 111, E3836 (2014).
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J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033603 (2018).
[Crossref]

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S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–572 (2018).
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A. A. Wood, E. Lilette, Y. Y. Fein, A. Stacey, N. Tomek, L. P. McGuinness, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Quantum measurement of a rapidly rotating spin qubit in diamond,” Sci. Adv. 4, eaar7691 (2018).
[Crossref]

A. A. Wood, E. Lilette, Y. Y. Fein, V. S. Perunicic, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Magnetic pseudo-fields in a rotating electron–nuclear spin system,” Nat. Phys. 13, 1070–1073 (2017).
[Crossref]

McAuslan, D. L.

J. Knittel, J. D. Swaim, D. L. McAuslan, G. A. Brawley, and W. P. Bowen, “Back-scatter based whispering gallery mode sensing,” Sci. Rep. 3, 2974 (2013).
[Crossref]

McGuinness, L. P.

A. A. Wood, E. Lilette, Y. Y. Fein, A. Stacey, N. Tomek, L. P. McGuinness, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Quantum measurement of a rapidly rotating spin qubit in diamond,” Sci. Adv. 4, eaar7691 (2018).
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Ş. K. Özdemir, J. Zhu, X. Yang, B. Peng, H. Yilmaz, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whisperinggallery Raman microlaser,” Proc. Natl. Acad. Sci. USA 111, E3836 (2014).
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Moses, E.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–572 (2018).
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S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–572 (2018).
[Crossref]

Novotny, L.

R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “Enhanced sensitivity at higher-order exceptional points,” Phys. Rev. Lett. 121, 033602 (2018).
[Crossref]

Ozdemir, S. K.

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[Crossref]

Özdemir, S. K.

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. USA 113, 6845–6850 (2016).
[Crossref]

Ş. K. Özdemir, J. Zhu, X. Yang, B. Peng, H. Yilmaz, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whisperinggallery Raman microlaser,” Proc. Natl. Acad. Sci. USA 111, E3836 (2014).
[Crossref]

L. He, Ş. K. Özdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6, 428–432 (2011).
[Crossref]

Ş. K. Özdemir, J. Zhu, L. He, and L. Yang, “Estimation of Purcell factor from mode-splitting spectra in an optical microcavity,” Phys. Rev. A 83, 033817 (2011).
[Crossref]

Padgett, M. J.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

S. Franke-Arnold, G. Gibson, R. W. Boyd, and M. J. Padgett, “Rotary photon drag enhanced by a slow-light medium,” Science 333, 65–67 (2011).
[Crossref]

Pedrotti, L. M.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[Crossref]

Peng, B.

B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. USA 113, 6845–6850 (2016).
[Crossref]

Ş. K. Özdemir, J. Zhu, X. Yang, B. Peng, H. Yilmaz, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whisperinggallery Raman microlaser,” Proc. Natl. Acad. Sci. USA 111, E3836 (2014).
[Crossref]

Perunicic, V. S.

A. A. Wood, E. Lilette, Y. Y. Fein, V. S. Perunicic, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Magnetic pseudo-fields in a rotating electron–nuclear spin system,” Nat. Phys. 13, 1070–1073 (2017).
[Crossref]

Post, E. J.

E. J. Post, “Sagnac effect,” Rev. Mod. Phys. 39, 475–481 (1967).
[Crossref]

Prams, S.

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108, 120801 (2012).
[Crossref]

Rajmangal, R.

S. I. Shopova, R. Rajmangal, S. Holler, and S. Arnold, “Plasmonic enhancement of a whispering-gallery-mode biosensor for single nanoparticle detection,” Appl. Phys. Lett. 98, 243104 (2011).
[Crossref]

Reimann, R.

R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “Enhanced sensitivity at higher-order exceptional points,” Phys. Rev. Lett. 121, 033602 (2018).
[Crossref]

Rotter, S.

B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. USA 113, 6845–6850 (2016).
[Crossref]

Rubinsztein-Dunlop, H.

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108, 120801 (2012).
[Crossref]

Sanders, V. E.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[Crossref]

Sarma, R.

R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015).
[Crossref]

Schilling, R.

R. Schilling, H. Schütz, A. H. Ghadimi, V. Sudhir, D. J. Wilson, and T. J. Kippenberg, “Near-field integration of a SiN nanobeam and a SiO2 microcavity for Heisenberg-limited displacement sensing,” Phys. Rev. Appl. 5, 054019 (2016).
[Crossref]

Schleich, W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[Crossref]

Scholten, R. E.

A. A. Wood, E. Lilette, Y. Y. Fein, A. Stacey, N. Tomek, L. P. McGuinness, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Quantum measurement of a rapidly rotating spin qubit in diamond,” Sci. Adv. 4, eaar7691 (2018).
[Crossref]

A. A. Wood, E. Lilette, Y. Y. Fein, V. S. Perunicic, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Magnetic pseudo-fields in a rotating electron–nuclear spin system,” Nat. Phys. 13, 1070–1073 (2017).
[Crossref]

Schütz, H.

R. Schilling, H. Schütz, A. H. Ghadimi, V. Sudhir, D. J. Wilson, and T. J. Kippenberg, “Near-field integration of a SiN nanobeam and a SiO2 microcavity for Heisenberg-limited displacement sensing,” Phys. Rev. Appl. 5, 054019 (2016).
[Crossref]

Scully, M. O.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[Crossref]

Shi, K.

B.-B. Li, W. R. Clements, X.-C. Yu, K. Shi, Q. Gong, and Y.-F. Xiao, “Single nanoparticle detection using split-mode microcavity Raman lasers,” Proc. Natl. Acad. Sci. USA 111, 14657–14662 (2014).
[Crossref]

Shopova, S. I.

S. I. Shopova, R. Rajmangal, S. Holler, and S. Arnold, “Plasmonic enhancement of a whispering-gallery-mode biosensor for single nanoparticle detection,” Appl. Phys. Lett. 98, 243104 (2011).
[Crossref]

Sieck, C. F.

R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Alù, “Sound isolation and giant linear nonreciprocity in a compact acoustic circulator,” Science 343, 516–519 (2014).
[Crossref]

Sounas, D. L.

R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Alù, “Sound isolation and giant linear nonreciprocity in a compact acoustic circulator,” Science 343, 516–519 (2014).
[Crossref]

Speirits, F. C.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

Stacey, A.

A. A. Wood, E. Lilette, Y. Y. Fein, A. Stacey, N. Tomek, L. P. McGuinness, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Quantum measurement of a rapidly rotating spin qubit in diamond,” Sci. Adv. 4, eaar7691 (2018).
[Crossref]

Sudhir, V.

R. Schilling, H. Schütz, A. H. Ghadimi, V. Sudhir, D. J. Wilson, and T. J. Kippenberg, “Near-field integration of a SiN nanobeam and a SiO2 microcavity for Heisenberg-limited displacement sensing,” Phys. Rev. Appl. 5, 054019 (2016).
[Crossref]

Suh, M.-G.

Swaim, J. D.

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode,” Adv. Opt. Photon. 7, 168–240 (2015).
[Crossref]

J. D. Swaim, J. Knittel, and W. P. Bowen, “Detection of nanoparticles with a frequency locked whispering gallery mode microresonator,” Appl. Phys. Lett. 102, 183106 (2013).
[Crossref]

J. Knittel, J. D. Swaim, D. L. McAuslan, G. A. Brawley, and W. P. Bowen, “Back-scatter based whispering gallery mode sensing,” Sci. Rep. 3, 2974 (2013).
[Crossref]

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108, 120801 (2012).
[Crossref]

Szorkovszky, A.

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108, 120801 (2012).
[Crossref]

Tebbenjohanns, F.

R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “Enhanced sensitivity at higher-order exceptional points,” Phys. Rev. Lett. 121, 033602 (2018).
[Crossref]

Tomek, N.

A. A. Wood, E. Lilette, Y. Y. Fein, A. Stacey, N. Tomek, L. P. McGuinness, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Quantum measurement of a rapidly rotating spin qubit in diamond,” Sci. Adv. 4, eaar7691 (2018).
[Crossref]

Vahala, K.

Vahala, K. J.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,”Science 317, 783–787 (2007).
[Crossref]

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref]

van Ooijen, E. D.

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108, 120801 (2012).
[Crossref]

Verlot, P.

E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nat. Nanotechnol. 7, 509–514 (2012).
[Crossref]

Vollmer, F.

M. R. Foreman, J. D. Swaim, and F. Vollmer, “Whispering gallery mode,” Adv. Opt. Photon. 7, 168–240 (2015).
[Crossref]

M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9, 933–939 (2014).
[Crossref]

F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. USA 105, 20701–20704 (2008).
[Crossref]

Wiersig, J.

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. USA 113, 6845–6850 (2016).
[Crossref]

J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016).
[Crossref]

R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015).
[Crossref]

J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
[Crossref]

Wilson, D. J.

R. Schilling, H. Schütz, A. H. Ghadimi, V. Sudhir, D. J. Wilson, and T. J. Kippenberg, “Near-field integration of a SiN nanobeam and a SiO2 microcavity for Heisenberg-limited displacement sensing,” Phys. Rev. Appl. 5, 054019 (2016).
[Crossref]

Windey, D.

R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “Enhanced sensitivity at higher-order exceptional points,” Phys. Rev. Lett. 121, 033602 (2018).
[Crossref]

Wittek, S.

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Nature 548, 187–191 (2017).
[Crossref]

Wood, A. A.

A. A. Wood, E. Lilette, Y. Y. Fein, A. Stacey, N. Tomek, L. P. McGuinness, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Quantum measurement of a rapidly rotating spin qubit in diamond,” Sci. Adv. 4, eaar7691 (2018).
[Crossref]

A. A. Wood, E. Lilette, Y. Y. Fein, V. S. Perunicic, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Magnetic pseudo-fields in a rotating electron–nuclear spin system,” Nat. Phys. 13, 1070–1073 (2017).
[Crossref]

Xiao, Y.-F.

B.-B. Li, W. R. Clements, X.-C. Yu, K. Shi, Q. Gong, and Y.-F. Xiao, “Single nanoparticle detection using split-mode microcavity Raman lasers,” Proc. Natl. Acad. Sci. USA 111, 14657–14662 (2014).
[Crossref]

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[Crossref]

Xu, Z.

J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033603 (2018).
[Crossref]

Yang, L.

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. USA 113, 6845–6850 (2016).
[Crossref]

Ş. K. Özdemir, J. Zhu, X. Yang, B. Peng, H. Yilmaz, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whisperinggallery Raman microlaser,” Proc. Natl. Acad. Sci. USA 111, E3836 (2014).
[Crossref]

L. He, Ş. K. Özdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6, 428–432 (2011).
[Crossref]

Ş. K. Özdemir, J. Zhu, L. He, and L. Yang, “Estimation of Purcell factor from mode-splitting spectra in an optical microcavity,” Phys. Rev. A 83, 033817 (2011).
[Crossref]

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[Crossref]

Yang, X.

Ş. K. Özdemir, J. Zhu, X. Yang, B. Peng, H. Yilmaz, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whisperinggallery Raman microlaser,” Proc. Natl. Acad. Sci. USA 111, E3836 (2014).
[Crossref]

Yilmaz, H.

B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. USA 113, 6845–6850 (2016).
[Crossref]

Ş. K. Özdemir, J. Zhu, X. Yang, B. Peng, H. Yilmaz, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whisperinggallery Raman microlaser,” Proc. Natl. Acad. Sci. USA 111, E3836 (2014).
[Crossref]

Yu, X.-C.

B.-B. Li, W. R. Clements, X.-C. Yu, K. Shi, Q. Gong, and Y.-F. Xiao, “Single nanoparticle detection using split-mode microcavity Raman lasers,” Proc. Natl. Acad. Sci. USA 111, 14657–14662 (2014).
[Crossref]

Zhao, G.

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

Zhu, J.

Ş. K. Özdemir, J. Zhu, X. Yang, B. Peng, H. Yilmaz, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whisperinggallery Raman microlaser,” Proc. Natl. Acad. Sci. USA 111, E3836 (2014).
[Crossref]

L. He, Ş. K. Özdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6, 428–432 (2011).
[Crossref]

Ş. K. Özdemir, J. Zhu, L. He, and L. Yang, “Estimation of Purcell factor from mode-splitting spectra in an optical microcavity,” Phys. Rev. A 83, 033817 (2011).
[Crossref]

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[Crossref]

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (2)

S. I. Shopova, R. Rajmangal, S. Holler, and S. Arnold, “Plasmonic enhancement of a whispering-gallery-mode biosensor for single nanoparticle detection,” Appl. Phys. Lett. 98, 243104 (2011).
[Crossref]

J. D. Swaim, J. Knittel, and W. P. Bowen, “Detection of nanoparticles with a frequency locked whispering gallery mode microresonator,” Appl. Phys. Lett. 102, 183106 (2013).
[Crossref]

Nano Lett. (1)

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Lett. 13, 3347–3351 (2013).
[Crossref]

Nat. Nanotechnol. (3)

M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9, 933–939 (2014).
[Crossref]

E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nat. Nanotechnol. 7, 509–514 (2012).
[Crossref]

L. He, Ş. K. Özdemir, J. Zhu, W. Kim, and L. Yang, “Detecting single viruses and nanoparticles using whispering gallery microlasers,” Nat. Nanotechnol. 6, 428–432 (2011).
[Crossref]

Nat. Photonics (1)

J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4, 46–49 (2010).
[Crossref]

Nat. Phys. (1)

A. A. Wood, E. Lilette, Y. Y. Fein, V. S. Perunicic, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Magnetic pseudo-fields in a rotating electron–nuclear spin system,” Nat. Phys. 13, 1070–1073 (2017).
[Crossref]

Nature (4)

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558, 569–572 (2018).
[Crossref]

W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature 548, 192–196 (2017).
[Crossref]

H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, “Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor,” Nature 548, 187–191 (2017).
[Crossref]

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref]

Optica (1)

Phys. Rev. A (2)

J. Wiersig, “Sensors operating at exceptional points: general theory,” Phys. Rev. A 93, 033809 (2016).
[Crossref]

Ş. K. Özdemir, J. Zhu, L. He, and L. Yang, “Estimation of Purcell factor from mode-splitting spectra in an optical microcavity,” Phys. Rev. A 83, 033817 (2011).
[Crossref]

Phys. Rev. Appl. (1)

R. Schilling, H. Schütz, A. H. Ghadimi, V. Sudhir, D. J. Wilson, and T. J. Kippenberg, “Near-field integration of a SiN nanobeam and a SiO2 microcavity for Heisenberg-limited displacement sensing,” Phys. Rev. Appl. 5, 054019 (2016).
[Crossref]

Phys. Rev. Lett. (5)

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108, 120801 (2012).
[Crossref]

J. Wiersig, “Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: application to microcavity sensors for single-particle detection,” Phys. Rev. Lett. 112, 203901 (2014).
[Crossref]

R. Sarma, L. Ge, J. Wiersig, and H. Cao, “Rotating optical microcavities with broken chiral symmetry,” Phys. Rev. Lett. 114, 053903 (2015).
[Crossref]

R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, “Enhanced sensitivity at higher-order exceptional points,” Phys. Rev. Lett. 121, 033602 (2018).
[Crossref]

J. Ahn, Z. Xu, J. Bang, Y.-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li, “GHz rotation of an optically trapped nanoparticle in vacuum,” Phys. Rev. Lett. 121, 033603 (2018).
[Crossref]

Phys. Usp. (1)

G. B. Malykin, “The Sagnac effect: correct and incorrect explanations,” Phys. Usp. 43, 1229–1252 (2000).
[Crossref]

Proc. Natl. Acad. Sci. USA (4)

B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, “Chiral modes and directional lasing at exceptional points,” Proc. Natl. Acad. Sci. USA 113, 6845–6850 (2016).
[Crossref]

B.-B. Li, W. R. Clements, X.-C. Yu, K. Shi, Q. Gong, and Y.-F. Xiao, “Single nanoparticle detection using split-mode microcavity Raman lasers,” Proc. Natl. Acad. Sci. USA 111, 14657–14662 (2014).
[Crossref]

F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. USA 105, 20701–20704 (2008).
[Crossref]

Ş. K. Özdemir, J. Zhu, X. Yang, B. Peng, H. Yilmaz, L. He, F. Monifi, S. H. Huang, G. L. Long, and L. Yang, “Highly sensitive detection of nanoparticles with a self-referenced and self-heterodyned whisperinggallery Raman microlaser,” Proc. Natl. Acad. Sci. USA 111, E3836 (2014).
[Crossref]

Rev. Mod. Phys. (2)

E. J. Post, “Sagnac effect,” Rev. Mod. Phys. 39, 475–481 (1967).
[Crossref]

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. 57, 61–104 (1985).
[Crossref]

Sci. Adv. (1)

A. A. Wood, E. Lilette, Y. Y. Fein, A. Stacey, N. Tomek, L. P. McGuinness, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, “Quantum measurement of a rapidly rotating spin qubit in diamond,” Sci. Adv. 4, eaar7691 (2018).
[Crossref]

Sci. Rep. (1)

J. Knittel, J. D. Swaim, D. L. McAuslan, G. A. Brawley, and W. P. Bowen, “Back-scatter based whispering gallery mode sensing,” Sci. Rep. 3, 2974 (2013).
[Crossref]

Science (4)

R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Alù, “Sound isolation and giant linear nonreciprocity in a compact acoustic circulator,” Science 343, 516–519 (2014).
[Crossref]

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

S. Franke-Arnold, G. Gibson, R. W. Boyd, and M. J. Padgett, “Rotary photon drag enhanced by a slow-light medium,” Science 333, 65–67 (2011).
[Crossref]

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,”Science 317, 783–787 (2007).
[Crossref]

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Figures (4)

Fig. 1.
Fig. 1. (a) Schematic of a single nanoparticle sensor with a spinning resonator. The resonator with rotation speed Ω is driven by an optical field. A nanoparicle placed in the mode volume of the resonator induces a coupling with strength J between the CW and CCW modes. (b) Schematic of the energy levels of the optical eigenmodes. The frequency splitting of 2 J for a stationary resonator ( Ω = 0 ) is increased by the Sagnac effect to Δ ω > 2 J when the resonator is spinning ( Ω 0 ).
Fig. 2.
Fig. 2. Sensitivity enhancement factor η (a) and relative photon number β = | a ¯ cw | 2 / | a ¯ ccw | 2 (b) as a function of rotation speed Ω . We choose R = 1.1 mm , J / γ a = 1.5 , and γ c / γ a = 0.3 for case A, and R = 0.5 mm , J / γ a = 1.1 , and γ c / γ a = 0.065 for case B, as in recent experiments [8,17,18]. Also, J and γ c depend on the properties of the particle and the resonator [8,17,18].
Fig. 3.
Fig. 3. Transmission rate T as a function of optical detuning Δ a for Ω = 0 (i.e., a stationary resonator) (a), Ω = 6.6 kHz (b), and Ω = 30 kHz (d). (c) Transmission difference Δ T as a function of optical detuning Δ a . (e) Transmission rate T as a function of optical detuning Δ a and rotation speed Ω . (f) Transmission difference Δ T as a function of detuning Δ a and rotation speed Ω . We choose R = 1.1 mm , J / γ a = 1.5 , and γ c / γ a = 0.3 (i.e., case A in Fig. 2) in (a)–(c), and R = 0.5 mm , J / γ a = 1.1 , and γ c / γ a = 0.065 in (d)–(f) (i.e., case B in Fig. 2).
Fig. 4.
Fig. 4. Variation in frequency splitting for a spinning and a stationary WGM-based particle sensor as more than one particle is deposited on the resonators. The rotation speed is set as Ω = 6.6 kHz in (a) and 30 kHz in (b).

Equations (22)

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Δ sag = n R Ω ω a c ( 1 1 n 2 λ n d n d λ ) ,
H eff = ( Δ + i γ ) a cw a cw + ( Δ i γ ) a ccw a ccw + ( J i γ c ) a cw a ccw + ( J i γ c ) a ccw a cw + i γ ex a in ( a cw a cw ) ,
Δ ± = Δ a + J ± Δ sag , γ = ( γ a + γ ex ) / 2 + γ c ,
ω 1 , 2 = ω a + J ± ω i ( γ ± γ ) ,
2 ω = [ 2 ( D 2 + 4 J 2 γ c 2 ) 1 / 2 + 2 D ] 1 / 2 , 2 γ = [ 2 ( D 2 + 4 J 2 γ c 2 ) 1 / 2 2 D ] 1 / 2 ,
η = | Δ ω ( Ω 0 ) Δ ω ( Ω = 0 ) | = [ D 2 + 4 J 2 γ c 2 ( J 2 + γ c 2 ) 2 ] 1 / 4 > 1 .
a ¯ cw = γ ex a in ( γ + i Δ ) ( γ + i Δ + ) ( γ + i Δ ) + ( J i γ c ) 2 , a ¯ ccw = γ ex a in ( i J + γ c ) ( γ + i Δ + ) ( γ + i Δ ) + ( J i γ c ) 2 ,
β = | a ¯ cw | 2 | a ¯ ccw | 2 = γ 2 + ( Δ a + J Δ sag ) 2 J 2 + γ c 2 .
T = | a out a in | 2 = | 1 γ ex ( γ + i Δ ) ( γ + i Δ + ) ( γ + i Δ ) + ( J i γ c ) 2 | 2 .
Δ T ( Δ a = 0 ) J 2 + γ c 2 + 2 J Δ sag ( γ + γ c ) 2 + 4 J 2 .
a ¯ cw γ ex a in γ + i ( Δ a + J + Δ sag )
H eff = ( Δ + i γ ) a cw a cw + ( Δ i γ ) a ccw a ccw + C 1 a cw a ccw + C 2 a ccw a cw + i γ ex a in ( a cw a cw ) ,
Δ ± = Δ a + i = 1 N J i ± Δ sag , γ = ( γ a + γ ex ) / 2 + i = 1 N γ c , i , C 1 , 2 = i = 1 N ( J i i γ c , i ) exp ( i 2 m β j ) ,
J = α f 2 ( r ) ω c 2 V c , γ c = 2 π 2 α 2 f 2 ( r ) ω c 3 λ 3 V c ,
γ c / J = 4 π 2 α / ( 3 λ 3 ) .
H = ( ω a + Δ sag ) a cw a cw + ( ω a Δ sag ) a ccw a ccw + J ( a cw a ccw + a ccw a cw ) + γ ex a in ( a cw e i ω l t a cw e i ω l t ) ,
U = exp [ i ω l t ( a cw a cw + a ccw a ccw ) ] ,
H = U H U i U U t , = ( Δ a + Δ sag ) a cw a cw + ( Δ a Δ sag ) a ccw a ccw + J ( a cw a ccw + a ccw a cw ) + γ ex a in ( a cw a cw ) ,
H eff = ( Δ + i γ ) a cw a cw + ( Δ i γ ) a ccw a ccw + ( J i γ c ) a cw a ccw + ( J i γ c ) a ccw a cw + i γ ex a in ( a cw a cw ) ,
( ω a + Δ sag i γ ω ) ( ω a Δ sag i γ ω ) ( J i γ c ) 2 = 0 ,
ω = ω a i γ ± D 2 i J γ c , D = Δ sag 2 + J 2 γ c 2 .
ω = [ ( D 2 + 4 J 2 γ c 2 ) 1 / 2 / 2 + D / 2 ] 1 / 2 , γ = [ ( D 2 + 4 J 2 γ c 2 ) 1 / 2 / 2 D / 2 ] 1 / 2 .

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