Parity-time-symmetry-breaking gyroscopes: lasing without gain and subthreshold regimes

David D. Smith; Hongrok Chang; Luke Horstman; Jean-Claude Diels; Jean-Claude Diels

doi:10.1364/OE.27.034169

1. Introduction

Recent studies have demonstrated that the resonance frequencies of an optical cavity can be made more sensitive to changes in optical path length (OPL) by placing an anomalous dispersion, or fast-light (FL), medium inside the cavity [1–15]. The greatest boost in sensitivity occurs at the critical anomalous dispersion (CAD), a particular value of the group index corresponding to a system singularity. The increased sensitivity has been proposed as a means for enhancing the precision of active- and passive-cavity optical gyroscopes, as well as that of other types of sensors, because the perturbation (change in OPL) can be brought about by a variety of physical phenomena [16].

A sensitivity enhancement, by itself, is not sufficient to enhance precision, however. In particular, the enhancement in sensitivity is accompanied by an increase in measurement uncertainty. The crucial question for these applications is, therefore, whether the sensitivity increases faster than the uncertainty as the system approaches the singularity. Numerical simulations for passive FL cavities have shown that under ideal conditions the increase in uncertainty is not as large as that in sensitivity, so an overall enhancement in precision can occur as a result of FL [11]. A substantial advantage, however, may exist for active FL cavities [17]. Recent theoretical work has predicted that the major source of uncertainty, the active cavity laser linewidth, is unaffected by FL [10], which implies that the enhancement in precision in active FL cavities is larger than in corresponding passive cavities, being close to that of the enhancement in sensitivity itself.

A number of current approaches to active FL gyros (FLGs), however, rely on the use of nonlinear optical processes generated by added pump beams [4,6,7,9,12,15], which create a number of challenges such as difficulty of miniaturization, necessity for careful control of cavity and laser parameters via sophisticated control schemes, as well as added sensitivity to environmental effects. These nonlinear effects can be deleterious even when they are not used to generate the dispersion. In both passive and active FL cavities employing atomic vapors, for example, saturation and optical pumping can alter the resonance lineshape, couple the counterpropagating beams, and limit the achievable intensity and signal to noise. Even when nonlinear effects can be avoided, reliance on discrete material transitions has additional drawbacks. Transitions in atomic vapors, for example, are highly temperature dependent, requiring state-of-the-art temperature stabilization techniques to minimize the resultant noise. Use of these transitions also limits the wavelength of operation, which could inhibit wide adoption. Furthermore, experiments to date have not matured enough to reveal the predicted increase in measurement precision, in large part because this observation itself requires fresh development of a high-quality ring laser gyroscope (RLG) at a new wavelength, dictated by the discrete transitions of the FL medium.

A solution to some of the problems outlined above might be found in FLGs based on coupled resonators (CRs) [8,13,18–20]. These systems do not use a medium to generate the FL, and are therefore not as constrained in wavelength. Instead, the enhancement arises from the coherent coupling of resonant modes. Indeed, systems of non-Hermitian coupled oscillators, such as parity-time (PT) symmetric systems, have been of interest recently because of the potential for achieving enhanced sensitivity near an exceptional point (EP), where the system eigenvalues coalesce [19–27]. EPs are found in diverse areas of physics. Therefore, the enhancement could ultimately benefit applications far beyond those that rely on optical-cavity-based sensing. The equivalency between the two phenomena (FL- and EP-based enhancements) has been pointed out previously [8,11,18,28,29]. In both cases, the sensitivity to some perturbation is maximized by closely approaching the system singularity, i.e., the CAD. As we will see, for a FL laser gyro (often referred to as an active FLG) the CAD occurs at the EP. The same is not true for a passive FL gyro. In this case, the optimum enhancement can occur far away from the EP, because the spectrum of a passive cavity is not entirely determined by the system eigenvalues, but is also influenced by the input electric field. We demonstrate that a more universal description that applies for both passive and active systems, is that the singularity arises from spectral mode splitting.

In PT-symmetric CR systems one of the resonators has net gain and the other has an equivalent amount of loss. Recently, PT-symmetric gyros have been proposed [19,20,26,27] that would, in theory, display the same increase in sensitivity as active FLGs based on atomic media, but are far simpler, and can be microfabricated onto integrated optical chips. Thus far, results obtained from spectra taken from model PT-symmetric CR systems have confirmed the square-root dependency on the perturbation (or detuning between the resonators), the classic signature of an EP [25], but data is still sparse in the region where the enhancement in sensitivity should be greatest, owing to experimental difficulties that set in at small detunings. In particular, PT-symmetric gyros are unidirectional, i.e., they rely on generating two frequency modes for a single propagation/output direction (in contrast with conventional RLGs which are bidirectional, with each propagation direction supporting a single frequency). These modes can be difficult to resolve spectrally at small detunings (rotation rates). Mode competition can also eliminate one of the modes. More fundamentally, the sensitivity enhancement in PT-symmetric systems is only obtained very close to an EP. The slightest variation of the coupling away from the EP can result in a significant decrease, rather than an increase, in sensitivity, or cause one of the modes to disappear entirely. Moreover, PT-symmetric gyros in their current embodiments are not common path, and therefore do not possess the same level of common-mode noise rejection as typical RLGs or FLGs based on atomic media. As a result, small nonreciprocal phase fluctuations due to noise (that would be reciprocal in conventional RLGs) are amplified by the enhancement in sensitivity. Finally, as we will show, PT-symmetric CR lasers suffer from undamped oscillations in power at their outputs as a result of a coherent exchange of photons between the resonators. Similar power oscillations have been reported in other systems displaying PT-symmetry [30–35]. This amplitude modulation (AM) is essential to the operation of PT-symmetric gyros, in contrast with conventional RLGs which rely on pure phase modulation. The reliance on AM can make PT-symmetric gyros susceptible to Kerr effect and place limits on their range of measurable rotation rates.

We propose a new type of laser gyroscope based on PT-symmetry breaking in active CRs that could solve these problems. It turns out that when PT-symmetry is broken for a gyroscope at rest (no detuning), a sensitivity enhancement still occurs, but with just one frequency mode at each output port, and consequently no oscillations in power. Indeed, these symmetry-broken systems are overdamped, monotonically approaching a steady state. The process by which this occurs is called lasing without gain (LWG) [36]. The phenomenon is directly analogous to lasing without inversion (LWI) [37], which occurs even though there is no (bare-state) population inversion as a result of the coherent coupling of energy levels in atomic systems [38]. The resulting quantum interference gives rise to an asymmetry between absorption and emission as shown in Fig. 1(b). Similarly, in LWG, lasing occurs via coherent coupling between resonators, even though the net (incoherent) gain coefficient of the system is negative. Through classical interference, photons are localized in the resonator with gain, while being excluded from the lossy resonator [39], resulting in an asymmetry between gain and loss. We refer to this process, also shown in Fig. 1(b), as coherent photon trapping in analogy with coherent population trapping in atomic systems. As a result, in LWG gyros the red- and blue-shifted modes are distinguished by their directions at the output, just as in traditional RLGs. In addition, unlike PT-symmetric gyros, LWG gyros demonstrate enhanced sensitivity even for couplings far away from the EP, making observation of the enhancement easier to achieve, and reducing the effect of noise on the enhancement.

Fig. 1. (a) Two coupled resonators. (b) Analogy between LWG (left) and LWI (right). For LWI there is no inversion for the bare states, yet the probability for emission is larger than that for absorption owing to destructive interference of the two paths to the common excited state (transitions between the lower two levels are dipole disallowed). For LWG a similar asymmetry between gain and loss occurs by classical interference. This is evident in the intracavity spectra, which are shown at a gain level below threshold to emphasize the asymmetry, and allows lasing to occur even though the net gain coefficient of the system is negative, i.e., ${\hat{\gamma }_1} + {\hat{\gamma }_2} > 0.$ Thus, LWG requires one resonator to have net gain while the other has a larger amount of loss. In this case ${\hat{\gamma }_2} < 0$ (gain), ${\hat{\gamma }_1} > 0$ (loss), and $\,|{\hat{\gamma }_1}|\,\, > \,|{\hat{\gamma }_2}|$.

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It should be pointed out that LWG requires broken PT-symmetry, and in this sense is analogous to loss-induced transparency observed in PT-symmetric waveguides [30]. Moreover, though not identified as such, LWG has already been observed in CR phonon lasers [40], and in CR microring lasers that are forced to operate in a single mode through the process of PT-symmetry breaking [39–44]. In [44] it was recognized that the net gain coefficient of the system can be negative when PT-symmetry is broken. However, to our knowledge, use of the broken-symmetry phase has not been proposed previously for use in gyroscopes. Instead, previous proposals involving the PT-symmetric phase transition have focused on utilization of the symmetric phase to enhance the effect of the perturbation [19,20], a strategy that suffers from the attendant difficulties discussed above.

In Section 2, we review the problem of two CRs and demonstrate that when PT-symmetry is broken (either because the coupling is below the EP or the system is rotating), the eigenvalues at the lasing threshold correspond to the conditions for LWG. In Section 3 we derive the enhancement in scale-factor sensitivity under PT-symmetric and LWG conditions. We show that a pole occurs in the enhancement at the EP and that the enhancement for LWG gyros is always greater than unity, whereas for PT-symmetric gyros it quickly approaches zero as one moves away from the EP. Moreover, we demonstrate that the directional ambiguity for PT-symmetric gyros is removed for LWG gyros. In Section 4 we introduce subthreshold CR systems, and show that the scale-factor pole corresponds to spectral splitting. We further show that in these subthreshold systems there exist two regions of gain values where the pole is accessible that respond oppositely to the effect of rotation. In Section 5 we solve the equation of motion for CR systems at threshold and below threshold, showing that Rabi oscillations are generally expected above the EP. In Sections 6 and 7 we discuss the implementation of PT-symmetry-breaking CR gyroscopes in the LWG and subthreshold regimes, focusing in particular on how photon localization explains their behavior, and the advantages offered by the LWG scheme. Finally, in Section 8, we demonstrate the effects of gain saturation on CR gyroscopes.

2. Eigenvalues of two coupled resonators

Consider the problem of two coupled resonators, which in turn are coupled to two waveguides as shown in Fig. 1(a). A beam with electric field ${E_a}$ is incident at one of the input ports as shown. The resonators have resonant frequencies ${\omega _1}$ and ${\omega _2}$ when they are uncoupled from one another, $\kappa$ is the rate at which photons are coupled between the resonators, ${\gamma _a}$ and ${\gamma _b}$ are the rates at which photons are coupled to, or from, the external waveguides, and ${\gamma _1}$ and ${\gamma _2}$ represent additional losses not due to coupling. Therefore, the total loss rate for each resonator, apart from the coupling between the resonators, is ${\hat{\gamma }_{1,2}} = {\gamma _{1,2}} + {\gamma _{a, b}}$. Figure 1(b) shows the same two resonators with the loss for each resonator aggregated into the single parameter ${\hat{\gamma }_{1,2}}$.

For weak coupling, and provided these losses are small compared to the resonant frequencies, the dynamics of the coupled modes in this system are equivalent to those of a coherently-excited two-level atom [36,45,46]. To appreciate this, consider the coherent superposition

(1)$$|{\psi (t )} \rangle = {E_1}(t )|1 \rangle + {E_2}(t )|2 \rangle = {\left[ {\begin{array}{{cc}} {{E_1}(t )}&{{E_2}(t )} \end{array}} \right]^\textrm{T}},$$

where ${E_1}$ and ${E_2}$ are the normalized electric fields in each resonator. The basis vectors (or spinors) for this two-level system are $|1 \rangle = {({\begin{array}{{cc}} 1&0 \end{array}})^\textrm{T}}$ and $|2 \rangle = {({\begin{array}{{cc}} 0&1 \end{array}})^\textrm{T}}$, from which the Pauli matrices ${\sigma _z} = |1 \rangle \left\langle 1 \right|- |2 \rangle \left\langle 2 \right|$, ${\sigma _x} = |2 \rangle \left\langle 1 \right|+ |1 \rangle \left\langle 2 \right|$, and ${\sigma _y} = $ $i\left( {|2 \rangle \left\langle 1 \right|- |1 \rangle \left\langle 2 \right|} \right)$ can be formed. The coupled-mode equations [47,48] describing the field dynamics are:

(2)$$\begin{aligned} & {{\dot{E}}_1}(t )={-} i{{\tilde{\omega }}_1}{E_1}(t )+ i\frac{{{{\tilde{\kappa }}_1}}}{2}{E_2}(t )+ i\sqrt {\frac{{{\gamma _a}}}{{{\tau _1}}}} {E_a}(t )\\ & {{\dot{E}}_2}(t )={-} i{{\tilde{\omega }}_2}{E_2}(t )+ i\frac{{{{\tilde{\kappa }}_2}}}{2}{E_1}(t )\end{aligned}$$

where ${\tilde{\omega }_{1,2}} = {\omega _{1,2}} - i{\hat{\gamma }_{1,2}}/2$ are complex frequencies, ${\tau _1}$ is the round-trip time in the first resonator, and ${\tilde{\kappa }_{1,2}} = \kappa \exp (i{\theta _{1,2}})$ are complex coupling coefficients. We will assume energy is not lost in the coupling process (conservative coupling), so that ${\tilde{\kappa }_1} = \tilde{\kappa }_2^\ast{=} \tilde{\kappa }$ and ${\theta _1} ={-} {\theta _2}$ [49]. Therefore, either the coupling coefficients are out of phase in this manner or they are real.

For now, we will ignore the input field and set ${E_a} = 0$. We define a complex detuning between the resonators as $\tilde{\delta } \equiv {\tilde{\omega }_1} - {\tilde{\omega }_2} = \delta - i{\gamma \mathord{\left/ {\vphantom {\gamma 2}} \right.} 2}\textrm{,}$ where $\delta = {\omega _1} - {\omega _2}$ is the real-valued detuning and $\gamma = {\hat{\gamma }_1} - {\hat{\gamma }_2}$ is the difference in the resonator loss rates. Using the slowly-varying amplitudes ${A_{1,2}}(t )= {E_{1,2}}(t )\exp [{i({{{\tilde{\omega }}_{1,2}} \mp {{\tilde{\delta }} \mathord{\left/ {\vphantom {{\tilde{\delta }} 2}} \right.} 2}} )t} ]\textrm{,}$ the coupled-mode equations are then identical to the Schrödinger equation for a two-level system in the rotating-wave approximation

(3)$$i\hbar \dot{\alpha }(t )={-} \frac{\hbar }{2}\left( {\begin{array}{cc} { - \tilde{\delta }}&{\tilde{\kappa }}\\ {{{\tilde{\kappa }}^ \ast }}&{\tilde{\delta }} \end{array}} \right)\alpha (t )= \tilde{H}\alpha (t ),$$

where $\alpha (t )= {[{\begin{array}{{cc}} {{A_1}(t )}&{{A_2}(t )} \end{array}}]^\textrm{T}}$, and the Hamiltonian $\tilde{H} = \hbar \tilde{K} = (\hbar /2)(\tilde{\delta } \cdot {\sigma _z} - {\mathop{\textrm {Re}}\nolimits} (\tilde{\kappa }) \cdot {\sigma _x} + {\mathop{\textrm {Im}}\nolimits} (\tilde{\kappa }) \cdot {\sigma _y})$ is non-Hermitian owing to the loss rates in the diagonal components. The eigenvalues of $\tilde{K}$ are ${\pm} \,\tilde{\Omega }/2$, where $\tilde{\Omega } = {({\tilde{\delta }^2} + {\kappa ^2})^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}$ is the generalized Rabi frequency, which includes detuning and decay rates, and determines whether the resonance splits in the frequency domain and whether Rabi oscillations are observed in the time domain, as a result of the coupling. The transformation to α, however, eliminates a common term that we will need to establish laser threshold and PT-symmetry conditions. The full eigenvalues of Eq. (2) are

(4)$${\tilde{\omega }_ \pm } = {\tilde{\omega }_{{avg}}} \pm \frac{{\tilde{\Omega }}}{2} = {\omega _ \pm } - i\frac{{{\gamma _ \pm }}}{2},$$

where ${\tilde{\omega }_{{avg}}} = ({\tilde{\omega }_1} + {\tilde{\omega }_2})/2 = {\omega _{{avg}}} - i{{{\gamma _{{avg}}}} \mathord{\left/ {\vphantom {{{\gamma_{{avg}}}} 2}} \right.} 2}$ is the average complex frequency. The eigenvalues are in general complex, but can be decomposed into real-valued frequencies ${\omega _ \pm }$ and linewidths ${\gamma _ \pm }$, as shown.

This formal equivalency with the damped Rabi problem explains why a variety of familiar atomic coherence phenomena, ranging from simple effects such as optical nutation [46], to more elaborate ones such as electromagnetically-induced transparency, are also observed in systems of coupled optical resonators [36,50–52]. The difference between the two situations is that while in atomic systems the external field couples the energy levels, for coupled resonators the role of the external field is to populate the resonators with photons. The coupling in the latter situation is provided by the internal coupling between the resonators.

Note that the eigenvalues are fully degenerate when $\tilde{\Omega } = 0$, i.e., both the frequencies and the linewidths cross. First, consider the case where the resonators are not detuned from one another, i.e., δ =0. In this case, $\tilde{\Omega }$ is purely real when $\kappa \ge |{\gamma /2} |$ and the linewidths and frequencies of the eigenmodes are

(5)$$\begin{array}{l} {\gamma _ \pm } = {\gamma _{{avg}}}\\ {\omega _ \pm } = {\omega _{avg}} \pm {[{{\kappa^2} - {{(\gamma } \mathord{\left/ {\vphantom {{(\gamma } 2}} \right.} 2}{)^2}} ]^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}/2, \end{array}$$

respectively. The equation for the frequencies defines a hyperbolic cone of circular cross section in parameter space, ${\mathop{\textrm {Re}}\nolimits} {(\tilde{\kappa })^2} + {\mathop{\textrm {Im}}\nolimits} {(\tilde{\kappa })^2} - 4{({\omega _ \pm } - {\omega _{avg}})^2} = {(\gamma /2)^2}$, where the upper and lower cones represent the two real-valued frequencies ${\omega _ + }$ and ${\omega _ - }$, respectively [53]. On the other hand, $\tilde{\Omega }$ is purely imaginary when $\kappa < |{\gamma /2} |$ and the equations for the linewidths and frequencies become:

(6)$$\begin{array}{l} {\gamma _ \pm } = {\gamma _{{avg}}} \mp {[{{{(\gamma /2)}^2} - {\kappa^2}} ]^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}\\ {\omega _ \pm } = {\omega _{{avg}}}. \end{array}$$

The equation for the linewidths defines a sphere of radius $\gamma /2$ in parameter space, ${\mathop{\textrm {Re}}\nolimits} {(\tilde{\kappa })^2} + {\mathop{\textrm {Im}}\nolimits} {(\tilde{\kappa })^2} + {({\gamma _ \pm } - {\gamma _{avg}})^2} = {(\gamma /2)^2}$, where again the upper and lower hemispheres represent the real-valued linewidths ${\gamma _ - }$ and ${\gamma _ + }$, respectively. These shapes are plotted in two dimensions in Fig. 2. It is apparent that there exists a set of degeneracy points that form a closed loop along the waist of the cone, and the equator of the sphere, where the two complex eigenmodes coalesce. This circle of degeneracy is described by the equation

(7)$${\kappa _{EP}} = {[{{\mathop{\textrm {Re}}\nolimits} {{(\tilde{\kappa })}^2} + {\mathop{\textrm {Im}}\nolimits} {{(\tilde{\kappa })}^2}} ]^{1/2}} = |{\gamma /2} |,$$

where ${\kappa _{EP}}$ is the exceptional point (EP) coupling. When the coupling is real and positive, the circle of degeneracy collapses to a single point, known as the EP.

Fig. 2. Real and Imaginary parts of the eigenvalues for two coupled resonators at δ =0. The quantities plotted are $({\omega _ \pm } - {\omega _{avg}})/{\kappa _{EP}}$ (solid curve), and $({\gamma _ \pm } - {\gamma _{avg}})/{\kappa _{EP}}$ (dotted curve).

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As a function of the detuning, δ, between the resonators, the eigenvalues display crossings and avoided crossings upon passing through δ =0, as shown in Fig. 3. For superexceptional couplings $(\kappa > {\kappa _{EP}})$, the linewidths cross but the frequencies anti-cross at δ =0, which prevents a discontinuity in frequency. On the other hand, for sub-exceptional couplings $(\kappa < {\kappa _{EP}})$, the frequencies cross but the linewidths anti-cross, which prevents a discontinuity in linewidth. When $\kappa = {\kappa _{EP}}$, the two eigenmodes cross in both frequency and linewidth. Note, when there is no difference in the loss rates $(\gamma = 0)$, the EP is at ${\kappa _{EP}} = 0$. In this case, the eigenmodes anti-cross in frequency and have equal linewidths for all values of the coupling, and the EP is instead referred to as a diabolic point (DP) [54]. Conventional RLGs operate around DPs; providing a measure of the detuning-induced splitting that results when the degeneracy of the DP is lifted by rotation.

Fig. 3. Crossings and avoided crossings in the eigenmode (a) frequencies and (b) linewidths for two coupled resonators. The couplings are: superexceptional $\kappa = 1.2{\kappa _{EP}}$ (red curves), sub- exceptional $\kappa = 0.95{\kappa _{EP}}$ (blue curves), and uncoupled κ =0 (black curves). The separation of each set of curves into dotted and solid portions emphasizes the crossings and avoided crossings in the real and imaginary parts of the eigenmodes upon passing through δ =0.

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Now, let’s assume the second resonator contains an amplifying medium $({\gamma _2} < 0)$ and there is a knob that we can use to vary its gain. The lasing threshold corresponds to infinitely narrow linewidths in this formalism, and is therefore obtained by setting ${\gamma _ \pm } = 0$ in Eqs. (5) and (6). For $\kappa \ge {\kappa _{EP}}$, $\tilde{\Omega }$ is real, which implies that ${\gamma _{avg}} = 0$ at threshold. Therefore,

(8)$$\hat{\gamma }_2^{PT} ={-} {\hat{\gamma }_1},$$

which corresponds to the condition for PT-symmetry. Because the coupling is above the EP, two frequencies ${\omega _ + }$ and ${\omega _ - }$ appear at the output. Note that Eq. (8) is just the usual threshold condition one would expect to obtain, where the net gain coefficient of the system is zero at threshold.

For $\kappa < {\kappa _{EP}}$ on the other hand, $\tilde{\Omega }$ is imaginary, and the threshold condition is

(9)$$\hat{\gamma }_2^{LWG} ={-} {\kappa ^2}/{\hat{\gamma }_1}.$$

In this case, only one frequency appears at the output. Furthermore, substituting Eq. (7) into Eq. (9) shows that $\hat{\gamma }_2^{LWG} = \hat{\gamma }_2^{PT}$ when $\kappa = {\kappa _{EP}}\textrm{,}$ which is the smallest value that $\hat{\gamma }_2^{LWG}$ can take. At lower coupling values $\hat{\gamma }_2^{LWG} > \hat{\gamma }_2^{PT}\textrm{,}$ which means that the threshold gain in this regime $(\kappa < {\kappa _{EP}})$ is lower than in the PT-symmetric $(\kappa > {\kappa _{EP}})$ regime. As a result, ${\gamma _1} + \,\gamma _2^{LWG} > 0.$ The net gain coefficient of the system is, therefore, negative yet lasing still occurs. We refer to this phenomenon as lasing without gain (LWG). For LWG to occur, there must be gain in at least one of the resonators, i.e., ${\gamma _1} < 0$ or ${\gamma _2} < 0$. The net gain coefficient of the system in the bare-state basis $- ({\hat{\gamma }_1} + {\hat{\gamma }_2})$ can be negative, as long as it’s positive in the basis representing the eigenstates of the coherently coupled system, i.e., provided $- {\gamma _ \pm } \ge 0$.

In Fig. 4(a) the coupling is plotted vs. the lasing threshold for the two cases. The particular coupling at which Eqs. (8) and (9) meet also corresponds to the EP, and is given by ${\kappa _{PT}} = {\hat{\gamma }_1}\textrm{.}$ The point $( - \hat{\gamma }_2^{PT},{\kappa _{PT}}\textrm{)}$ is known as the PT-symmetry breaking phase transition. To summarize the results at δ = 0, for superexceptional couplings $(\kappa > {\kappa _{EP}})$ the lasing threshold occurs under conditions of PT-symmetry, whereas PT-symmetry is broken for subexceptional couplings $(\kappa < {\kappa _{EP}})$ corresponding to LWG. These results are equivalent to the linear analysis presented in [44].

Fig. 4. (a) The splitting-point coupling, κ_pole, corresponding to the pole in the scale-factor sensitivity (blue curve) for two coupled resonators at $\delta = 0$ (no detuning, i.e., rotation). The independent variable is the gain in the second resonator, whereas the loss in the first resonator is held constant. The lasing threshold, including PT-symmetric and LWG branches, is also shown (red curves). The dashed line represents the EP coupling. (b) The lasing threshold for $\delta = 0$ (red curve) and $|\delta |\, > 0$ (black curve). Upon rotation, the threshold decreases, pushing the hashed area above threshold. In (a) all three curves meet $({\kappa _{pole}} = {\kappa _{th}} = {\kappa _{EP}})$ at the PT-symmetry breaking transition, because $\delta = 0\,\textrm{.}$ In (b), on the other hand, the pole (not shown) crosses threshold above the EP when $|\delta |\, > 0\,\textrm{.}$ The constant parameters in (a) and (b) are ${\gamma _1} = 0.9\,,\,\,{\gamma _a} = 0.5\,,\,\,{\gamma _b} = 0\,.$ All quantities are chosen to be dimensionless.

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Next, we will examine the case where the resonators are detuned from one another, i.e., $\delta \ne 0$. Equating the real parts of Eq. (4) yields the real eigenfrequencies ${\omega _ \pm } = {\omega _{avg}} \pm {\mathop{\textrm {Re}}\nolimits} \tilde{\Omega }/2$, which can be rewritten as

(10)$${\omega _ \pm } = {\omega _{avg}} \pm \frac{1}{2}{\left[ {\frac{{{\kappa^2} - \hat{\kappa }_{EP}^2}}{2} + \sqrt {{{\left( {\frac{{{\kappa^2} - \hat{\kappa }_{EP}^2}}{2}} \right)}^2} + {{({\delta {\kappa_{EP}}} )}^2}} } \right]^{1/2}},$$

where $\hat{\kappa }_{EP}^2 = \kappa _{EP}^2 - {\delta ^2}$. Note, when $\delta = 0$ the results obtained in Eqs. (5) and (6) for the two cases, $\kappa > {\kappa _{EP}}$ and $\kappa < {\kappa _{EP}}$, are recovered. At $\kappa = {\kappa _{EP}}$ in the limit of large detunings, $|\delta |> > 2{\kappa _{EP}}$, Eq. (10) reduces to ${\omega _ \pm } = {\omega _{avg}} \pm |\delta |/2$, therefore the expected result ${\omega _ + } - {\omega _ - } = |\delta |$ is obtained. In the opposite limit, on the other hand, $\kappa \approx {\hat{\kappa }_{EP}}$ and ${\omega _ \pm } = {\omega _{avg}} \pm \sqrt {|\delta |{\kappa _{EP}}} /2$, i.e., ${\omega _ + } - {\omega _ - } \approx \sqrt {|{\delta \gamma } |/2}$, the same square-root dependency found in [19]. On the other hand, the result obtained from Eq. (10) for a DP $({\kappa _{EP}} = 0)$ with no coupling is ${\omega _ + } - {\omega _ - } = |\delta |$ for all detunings. It is, therefore, apparent that for small values of $|\delta |$, the frequency difference is enhanced by the square-root relation. The sensitivity of this frequency difference to changes in detuning is also enhanced, and is obtained from the derivative [55]

(11)$$\left|{\frac{{d({{\omega_ + } - {\omega_ - }} )}}{{d\delta }}} \right|= \frac{1}{2}{\left|{\frac{\gamma }{{2\delta }}} \right|^{1/2}} = \,\,\frac{1}{2}\,\left|{\frac{{{\omega_ + } - {\omega_ - }}}{\delta }} \right|\,\,\,\,\{{\kappa = {\kappa_{EP}}} \}.$$

Equation (11) applies for both sub and superexceptional systems near the EP. Moreover, because we did not assume threshold conditions in the derivation of Eq. (10), the square-root relation applies for both lasers and subthreshold systems (Chen et al. [24] have confirmed the square-root relationship in passive microresonators), with the caveat that for subthreshold systems the eigenfrequencies found in Eq. (10) do not generally coincide with the peak or dip observed in the reflection or transmission spectrum. Nevertheless, there does exist a deterministic relationship between the frequencies of these extrema and the eigenfrequencies, so we expect a similar square-root relationship to hold for the observed extrema in subthreshold CR systems.

Equating the imaginary parts of Eq. (4) yields the real linewidths ${\gamma _ \pm } = {\gamma _{{avg}}} \mp {\mathop{\textrm {Im}}\nolimits} \tilde{\Omega }$. Setting ${\gamma _ \pm } = 0$ yields the threshold condition

(12)$$\kappa _{th}^2 ={-} {\hat{\gamma }_1}{\hat{\gamma }_2} + {\delta ^2}({\kappa_{EP}^2/\gamma_{avg}^2 - 1} ).$$

Note that Eq. (9) can alternatively be written as a threshold condition for the coupling $\kappa _{th}^2 ={-} {\hat{\gamma }_2}{\hat{\gamma }_1}$, which is recovered from Eq. (12) when $\delta = 0$. These results are plotted in Fig. 4(b). Note that the effect of the detuning is to lower the threshold value of the gain, $- {\hat{\gamma }_2}$, for all values of $\kappa$, and that there is a pole in Eq. (12) at the condition of PT-symmetry, i.e., when ${\hat{\gamma }_2} ={-} {\hat{\gamma }_1}.$ Thus, in a CR gyroscope, the lasing gain threshold will decrease with rotation rate. Moreover, for any nonzero detuning, the system will not be PT-symmetric at the lasing threshold, even when $\kappa > {\kappa _{EP}}$, but will instead correspond to LWG. And because the gain typically saturates to the threshold value, this means that PT-symmetric gyros will in fact only satisfy PT-symmetry at δ =0 [56]. This is apparent from the Hamiltonian H associated with Eq. (2) (before the transformation to Eq. (3)). PT-symmetry requires that the operators PT and H commute (P switches the roles of the resonators, i.e., 1 ↔ 2 and T is equivalent to complex conjugation), which only occurs when the coupling is conservative, ${\gamma _{avg}} = 0$, and δ =0. (An additional criterion for exact or unbroken PT-symmetry is that H and PT have the same eigenvalues [30], which corresponds to $\kappa \ge {\kappa _{PT}}\textrm{)}\textrm{.}$ Indeed, Fig. 4 shows that the parameter space over which LWG occurs is much larger than that for PT-symmetry, which is the more precarious of these two mutually-exclusive situations. Accordingly, LWG will occur in any rotating CR gyroscope, whether or not it’s PT-symmetric when it’s not rotating.

There is a slight but important distinction between sub and superexceptional systems that is not apparent in Eq. (11). For subexceptional couplings, the system must switch from one eigenstate to the other upon passing through δ =0, to produce the crossings in frequency and avoided crossings in linewidth shown in Fig. 3. This is evident by inspection of Eq. (10), because ${\omega _ + }$ is positive for both signs of the detuning. Therefore, the roles of the eigenvalues ${\tilde{\omega }_ + }$ and ${\tilde{\omega }_ - }$ are reversed when the system passes through δ =0. The same switch does not occur for superexceptional couplings. The switch in eigenvalues when $\kappa < {\kappa _{EP}}$ leads to an important difference between PT-symmetric and LWG systems in the sign of the scale-factor sensitivity, which we discuss in the next section.

3. Scale-factor sensitivity

Let us now define more carefully the enhancement in the scale-factor sensitivity of the coupled system as

(13)$$S(\delta ) \equiv \frac{{d\Delta }}{{d\delta }} = \frac{{\tilde{\delta }}}{{2{{[{{{\tilde{\delta }}^2} + {{\tilde{\kappa }}^2}} ]}^{1/2}}}} + c.c..$$

where $\Delta \equiv {\omega _ + } - {\omega _ - } = {\mathop{\textrm {Re}}\nolimits} \tilde{\Omega } = (\tilde{\Omega } + {\tilde{\Omega }^\ast })/2$ is the relative detuning between the eigenfrequencies. For subexceptional couplings, the roles of ${\omega _ + }$ and ${\omega _ - }$ (as well as ${\gamma _ + }$ and ${\gamma _ - }$, although this role reversal is of less importance here) are reversed upon passing through δ =0, as discussed above. Thus, we define $\Delta \equiv {\omega _ - } - {\omega _ + }$ when $\delta < 0$ and $\kappa < {\kappa _{EP}}$.

In Fig. 5, Δ is plotted as the detuning between the individual resonators as δ is varied, under conditions of LWG and PT-symmetry, both at and far away from the EP. To make a fair comparison of the various cases for their use in gyroscopes requires an additional assumption that the area (and wavelength) are the same in each case. This is because δ depends on the rotation rate, as determined by the Sagnac effect. Therefore, the slope of the curves, S, represents the enhancement in scale-factor over that of a conventional (single-resonator) gyroscope whose area is equal to the sum of the areas for the two coupled resonators. The dashed blue line in the figure has a slope of unity, which corresponds to the conventional RLG. Note that the absolute value of S is the same for both systems near the EP as predicted by Eq. (11). In the limit of large detunings ${\delta ^2} > > |{{\kappa^2} - \kappa_{EP}^2} |$, evaluation of Eq. (10) yields $\Delta = |\delta |$, i.e., $|S |= 1$, for PT-symmetric systems and $\Delta = \delta$, i.e., $S = 1$, for LWG systems, as expected from the figure. In the opposite limit ${\delta ^2} < < |{{\kappa^2} - \kappa_{EP}^2} |$, differentiation of Eq. (10) results in

(14)$$S = \left\{ {\begin{array}{cc} {\delta {\kappa^2}/{{({\kappa^2} - \kappa_{EP}^2)}^{3/2}}}&{\kappa > {\kappa_{EP}}}\\ {\kappa_{EP}^2/(\kappa_{EP}^2 - {\kappa^2})}&{\kappa < {\kappa_{EP}}} \end{array}} \right\}.$$

Equation (14) applies for any value of the coupling except at the EP, and therefore reveals important differences between the two systems that are not apparent from Eq. (11), which only applies at the EP. First, when $\kappa > {\kappa _{EP}}$, S changes sign with δ. In this case, there is an ambiguity that arises because the eigenfrequencies ${\omega _ \pm }$ and the difference between them Δ, are symmetric about $\delta = 0$. Therefore, if the detuning arises from rotation of the resonators, there will be no way to distinguish the direction of rotation without some additional means. On, the other hand, when $\kappa < {\kappa _{EP}}$ this ambiguity is removed because ${\omega _ \pm }$ and Δ are antisymmetric about $\delta = 0$. Second, when $\kappa < {\kappa _{EP}}$, the minimum value of the scale factor is $S(0) = 1$ (dashed blue curve in Fig. 5). On the other hand, when $\kappa > {\kappa _{EP}}$ the scale factor is $S(0) = 0$ at $\delta = 0$ for all couplings except directly at $\kappa = {\kappa _{EP}}$ (the cusp in the red curve in Fig. 5 flattens as the coupling is increased; see the dashed red curve). This can be alleviated to some degree by operating around a nonzero detuning. The fact remains, however, that the sensitivity drops off quickly to zero as the coupling increases away from the EP. Thus, for PT-symmetric systems it is only when operating close to the EP, where a cusp forms in Δ, and one operates at a slight nonzero detuning, that the sensitivity is enhanced. This limitation does not apply to LWG. In this case, the sensitivity increases, i.e, $S(0) > 1$, for all values of the coupling $\kappa$, and drops off more slowly as the coupling decreases away from the EP. This is an inherent advantage of the LWG rotation-sensing scheme shown in Section 7.

Fig. 5. Eigenmode detuning Δ as the detuning between the individual resonators δ is varied, under conditions of LWG (blue curves) and PT-symmetry (red curves). Solid curves are just above and just below the EP $({\gamma _2} ={-} 1.4,\,\,\kappa = 1.4)$, while dashed curves are far from the EP $(\textrm{red:}\,\,{\gamma _2} ={-} 1.4,\,\,\kappa = 1.7)\,\textrm{,}\,\,(\textrm{blue:}\,\,{\gamma _2} ={-} 0.1,\,\,\kappa = 0.374)\textrm{.}$ The values of the constant parameters ${\gamma _1}, \,{\gamma _a}, \,\textrm{and}\,{\gamma _b}$ are the same as in Fig. 4.

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4. Subthreshold CR systems

CR systems below threshold may be entirely passive, or may contain one or more amplifying resonators, but in all cases require an external light source. To treat this case, we therefore return to Eq. (2) and assume a monochromatic input field ${E_a}(t) = {a_a}(\omega )\exp ( - i\omega t)$, where $\omega $ is the input frequency. The field inside the resonators will be of the same form i.e., ${E_{1,2}}(t) = {a_{1,2}}(t)\exp ( - i\omega t)$. One can then solve for the steady-state amplitude ${a_1}(\omega )$ in the first resonator, and apply the boundary condition for the reflected amplitude ${a_r}(\omega ) = {a_a}(\omega ) + i\sqrt {{\gamma _a}{\tau _1}} {a_1}(\omega )$, to obtain the field transfer function for reflection

(15)$$\tilde{\rho }(\omega ) = \frac{{{a_r}(\omega )}}{{{a_a}(\omega )}} = \frac{{{{(\kappa /2)}^2} - {{\tilde{\delta }}_1}{{\tilde{\delta }}_2} + i{\gamma _a}{{\tilde{\delta }}_2}}}{{{{(\kappa /2)}^2} - {{\tilde{\delta }}_1}{{\tilde{\delta }}_2}}}.$$

where ${\tilde{\delta }_{1,2}} \equiv \omega - {\tilde{\omega }_{1,2}} = {\delta _{1,2}} + i{\hat{\gamma }_{1,2}}/2$ and ${\delta _{1,2}} = \omega - {\omega _{1,2}}$ are the complex and real-valued detunings of the incident beam, and the reflectance is $R(\omega ) = {|{\tilde{\rho }(\omega )} |^2}$. Note that if $\omega$ is allowed to be complex in Eq. (15) and the poles are found, Eq. (4) is recovered for the system eigenvalues. In fact, Eq. (12) can be more directly obtained from the pole in $R(\omega )$ than from the procedure presented earlier of setting ${\gamma _ \pm } = 0$.

For sufficiently large couplings, rather than the single extremum observed (per free spectral range) in single resonators, the reflectance spectrum for the CR structure splits into two minima (or maxima). The frequencies of these extrema do not correspond to the eigenfrequencies calculated above, however, because for a subthreshold system the presence of the input (which is not generally itself an eigenmode of the system) can substantially influence the resulting spectrum. Therefore, for subthreshold systems the frequencies of the extrema are used to define S, and can be determined by setting $dR(\omega )/d\omega = 0\textrm{,}$ which yields ${({\omega _ \pm } - {\omega _{avg}})^2} = [ - b \pm {({b^2} - 4ac)^{1/2}}]/2a\textrm{,}$ where $a ={-} 2{\gamma _1}\textrm{,}\thinspace b = {\hat{\gamma }_2}({\gamma _a}{\hat{\gamma }_2} - 4{\hat{\kappa }^2})\textrm{,}\thinspace c = \alpha {\hat{\kappa }^4} + \beta {\hat{\kappa }^2} + \chi \textrm{,}$ $\alpha = 2({\gamma _1} + 2{\hat{\gamma }_2}),$ $\beta ={-} {\hat{\gamma }_2}({\gamma _a}{\hat{\gamma }_2} + 2\gamma _{avg}^2)\,, $ $\chi = {\gamma _a}\hat{\gamma }_2^2\gamma _{avg}^2/2\,\textrm{,}$ and ${\hat{\kappa }^2} = ({\hat{\gamma }_1}{\hat{\gamma }_2} + {\kappa ^2})/4.$ The splitting-point coupling, ${\kappa _{pole}}$, is the largest coupling that does not result in spectral splitting, and corresponds to the scale-factor pole $(S \to \infty )\,\textrm{.}$ The value of $\hat{\kappa }$ resulting in ${({\omega _ \pm } - {\omega _{avg}})^2} = 0$ is found by setting $c = \alpha {\hat{\kappa }^4} + \beta {\hat{\kappa }^2} + \chi = 0$, which results in

(16)$${\kappa _{pole}} = {\left[ {\frac{{ - {{\hat{\gamma }}_1}{{\hat{\gamma }}_2}\alpha - 4\beta \pm 4\sqrt {{\beta^2} - 4\alpha \chi } }}{\alpha }} \right]^{1/2}}.$$

The positive root is retained when $\alpha > 0$ $({\hat{\gamma }_2} < - {\gamma _1}/2)$, and the negative otherwise. This result is included in Fig. 4(a), which reveals several notable features. First, when ${\hat{\gamma }_2} ={-} {\hat{\gamma }_1}$, i.e, under conditions of PT-symmetry, only the first term in Eq. (16) is retained (because ${\gamma _{avg}} = 0$ and $\alpha > 0)\,\textrm{,}$ so the pole occurs at the PT-symmetry-breaking transition, ${\kappa _{pole}} = {\kappa _{th}} = {\kappa _{EP}}$, and only at this point. Second, note that in the limit $\alpha \to 0$, the splitting-point coupling approaches ${\kappa _{pole}} = \infty $. A sign change in the scale-factor occurs as the gain is varied through this point, i.e., the modes move in the opposite direction under the influence of a detuning $\delta \,\textrm{.}$ Third, in the range $- {\gamma _1}/2 > {\hat{\gamma }_2} > 0$ the solution to Eq. (16) is above the lasing threshold. Thus, the pole is only accessible to cavities above threshold in this region, and to subthreshold cavities outside this region (except at the PT-symmetry breaking transition which coincides with the laser threshold). While our treatment is not strictly valid in the region above threshold, we include this portion of the curve to demonstrate this point. The region where the pole is accessible to subthreshold cavities can be further divided into two regions as shown in the figure. In region I, $\alpha < 0\,\textrm{,}$ and ${\hat{\gamma }_2} > 0\,\textrm{,}$ whereas in region II, $\alpha > 0\,\textrm{.}$ Region I corresponds with dips in the spectrum, whereas Region II corresponds with peaks, as shown in Section 6. Finally, note that it is not required that the system be near an EP to approach the pole. Indeed, the pole only coincides with the EP at the PT-symmetry breaking transition. At other gain values, the pole occurs when the reflection from the structure splits, rather than at the EP. Again, the explanation is that the input itself is not an eigenmode of the system, and therefore has an effect on the spectrum, essentially biasing the splitting point away from the EP. This effect becomes more and more pronounced as the gain is decreased further below the PT-symmetry breaking transition where the threshold and EP curves cross in Fig. 4.

5. Photon dynamics

The equation of motion for the field amplitudes in the presence of the input is

(17)$$\left( {\begin{array}{{c}} {{{\dot{a}}_1}}\\ {{{\dot{a}}_2}} \end{array}} \right) = i\left( {\begin{array}{{cc}} {{{\tilde{\delta }}_1}}&{\tilde{\kappa }/2}\\ {{{\tilde{\kappa }}^ \ast }/2}&{{{\tilde{\delta }}_2}} \end{array}} \right)\left( {\begin{array}{{c}} {{a_1}}\\ {{a_2}} \end{array}} \right) + i\sqrt {\frac{{{\gamma _a}}}{{{\tau _1}}}} \left( {\begin{array}{{c}} {{a_a}}\\ 0 \end{array}} \right).$$

From this equation the following second-order equations can be formed

(18)$${\ddot{a}_{1,2}} - i({\tilde{\delta }_1} + {\tilde{\delta }_2}){\dot{a}_{1,2}} + [{{{(\kappa /2)}^2} - {{\tilde{\delta }}_1}{{\tilde{\delta }}_2}} ]{a_{1,2}} = {C_{1,2}},$$

where ${C_1} = (i + {\tilde{\delta }_2}){a_a}\sqrt {{\gamma _a}/{\tau _1}}$ and ${C_2} ={-} {\tilde{\kappa }^\ast }{a_a}\sqrt {{\gamma _a}/{\tau _1}} /2$ are driving terms [57]. The solutions consist of a homogeneous solution and a particular solution, i.e., $\alpha (t )= {\alpha ^h}(t) + {\alpha ^p}(t)$. For arbitrary initial conditions ${\alpha ^h}(t) = \chi (t)\alpha (0)\exp [i(\omega - {\tilde{\omega }_{avg}})t]$ where

(19)$$\chi (t) = \left[ {\begin{array}{{cc}} {\cos \frac{{\tilde{\Omega }t}}{2} - i\frac{{\tilde{\delta }}}{{\tilde{\Omega }}}\sin \frac{{\tilde{\Omega }t}}{2}}&{i\frac{{\tilde{\kappa }}}{{\tilde{\Omega }}}\sin \frac{{\tilde{\Omega }t}}{2}}\\ {i\frac{{{{\tilde{\kappa }}^\ast }}}{{\tilde{\Omega }}}\sin \frac{{\tilde{\Omega }t}}{2}}&{\cos \frac{{\tilde{\Omega }t}}{2} + i\frac{{\tilde{\delta }}}{{\tilde{\Omega }}}\sin \frac{{\tilde{\Omega }t}}{2}} \end{array}} \right],$$

and

(20)$${\alpha ^p}(t) = \left[ {\begin{array}{{c}} {a_1^{ss}\{{1 - ({{{\tilde{\chi }}_{11}} - {{\tilde{\kappa }}^\ast }{{\tilde{\chi }}_{12}}/2{{\tilde{\delta }}_2}} )\exp [{i(\omega - {{\tilde{\omega }}_{avg}})t} ]} \}}\\ {a_2^{ss}\{{1 - ({{{\tilde{\chi }}_{22}} - 2{{\tilde{\delta }}_2}{{\tilde{\chi }}_{21}}/{{\tilde{\kappa }}^\ast }} )\exp [{i(\omega - {{\tilde{\omega }}_{avg}})t} ]} \}} \end{array}} \right].$$

The steady-state solutions are

(21)$$\left[ {\begin{array}{{c}} {a_1^{ss}}\\ {a_2^{ss}} \end{array}} \right] = \frac{{{a_a}\sqrt {{\gamma _a}/{\tau _1}} }}{{{{(\kappa /2)}^2} - {{\tilde{\delta }}_1}{{\tilde{\delta }}_2}}}\left[ {\begin{array}{{c}} {{{\tilde{\delta }}_2}}\\ { - {{\tilde{\kappa }}^\ast }/2} \end{array}} \right].$$

The intracavity intensity in each resonator is then ${I_{1,2}}(t) = {|{{a_{1,2}}(t)} |^2}$.

Let us now examine the dynamics at δ =0. If there is no input field present, either because it has been turned off or because the gain of the system populates the resonators, the homogenous solution is obtained. The system is critically damped when $\tilde{\Omega } = 0$, i.e., at the EP. When $\kappa > {\kappa _{EP}}$ (superexceptional couplings) the system is underdamped and Rabi oscillations occur at the frequency $|\tilde{\Omega }|$ [46]. Furthermore, these oscillations are distinguishable, i.e., observable, when $|\tilde{\Omega }|\, \ge {\gamma _{avg}}$. When the system also lases along the PT-symmetric branch, such that ${\gamma _{{avg}}} = 0$, these oscillations are undamped and extend to all times. On the other hand, when $\kappa < {\kappa _{EP}}$ (subexceptional couplings) the system is overdamped and Rabi oscillations are precluded. When the system also lases along the LWG branch, the lasing threshold occurs at $|\tilde{\Omega }|\, = {\gamma _{avg}}$.

These results are shown on the left hand side of Fig. 6, where the intracavity intensities are plotted vs. time for CR systems at threshold for various values of the coupling coefficient κ (points along the red curve in Fig. 4(a)). For initial conditions we assume that resonator #1 is empty, i.e., ${a_1}(0) = 0$, and that photons spontaneously appear in resonator #2 as a result of the gain, i.e., ${a_2}(0) = \textrm{constant}$. Figures 6(a) and 6(b) are for LWG conditions, with Fig. 6(a) farther below the EP than Fig. 6(b). One can confirm that these systems are at threshold because the intensities converge to a steady state, and do not diverge to infinity (above threshold) or tail off to zero (below threshold). Note that the intensity is higher in resonator #2 owing to the higher gain in that resonator. As the coupling increases toward the EP, the intensities increase and equalize, until at the EP they become infinite.

Fig. 6. Relative intensity in each resonator I₁ and I₂ vs. time for CR systems at δ = 0. Left: at threshold for various values of the coupling coefficient κ (along the red curve in Fig. 4(a)). Right: below threshold at different values of the gain coefficient $- {\gamma _2}$ (along the blue curve in Fig. 4(a)). For the top sets of curves, the coupling increases from top to bottom with the constant parameters ${\gamma _1}, \,{\gamma _a}, \,\textrm{and}\,{\gamma _b}$ set to those used in Fig. 4, whereas for the bottom curves (e and h) these parameters are set to ${\gamma _1} = 0.05, \,{\gamma _a} = 0.001, \,\textrm{and}\,{\gamma _b} = 0.001$ to reduce the gain-loss contrast. On the left the parameters are (a) ${\gamma _2} ={-} 1.2, \,\kappa = 1.29615$, (b) ${\gamma _2} ={-} 1.3, \,\kappa = 1.34907$, (c and e) ${\gamma _2} ={-} 1.4, \,\kappa = 1.4014$, and (d) ${\gamma _2} ={-} 1.4, \,\kappa = 1.4495.$ On the right the parameters are (f) ${\gamma _2} = 0.5, \,\kappa = 0.274, $ (g) ${\gamma _2} ={-} 0.46, \,\kappa = 3.293$, and (h) ${\gamma _2} ={-} 0.02605, \,\kappa = 0.397.$ On the left the relative intensity is defined with respect to the initial intensity in resonator #2, whereas on the right it’s defined with respect to the input intensity. Dotted lines indicate steady-state solutions.

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Figures 6(c) and (d), on the other hand, are for PT-symmetric conditions, with Fig. 6(d) farther above the EP than Fig. 6(c). In this case Rabi oscillations with equal peak intensities occur. As the coupling decreases towards the EP the period and intensity of the oscillations increases, until at the EP they become infinite. Therefore, the particular EP where lasing occurs corresponds with critically-damped zero-frequency Rabi oscillations. The oscillations in Figs. 6(c) and 6(d) are not 180 degrees out of phase, but rather are nearly in phase because of the large gain-loss contrast between the two resonators. In Fig. 6(e) Rabi oscillations are shown for a system having the same oscillation period as in Fig. 6(d) but with a smaller gain-loss contrast. In this case the oscillations are almost completely out of phase.

In the presence of the input, if we assume initial conditions $\alpha (0) = 0$ so that both resonators are initially empty, the particular solution is obtained. Again, Rabi oscillations can result when $\kappa > {\kappa _{EP}}$, and they are distinguishable provided $|\tilde{\Omega }|$ is sufficiently large in comparison with ${\gamma _{avg}}$. These results are shown on the right hand side of Fig. 6, where the intracavity intensities are plotted for CR systems below threshold at different values of the gain coefficient (points along the blue curve in Fig. 4(a)). Figure 6(f) is in region I. Note that the intensity is now higher in resonator #1, owing to its proximity to the input waveguide. In contrast, Fig. 6(g) is in region II. In this case, destructive interference cancels out most of the light in resonator #1, resulting in photons being localized in resonator #2. This interference is also apparent in the doubling of the quasi-frequency for resonator #2, and gives rise to coupled-resonator-induced transparency [50,52] in the frequency domain (see Fig. 7). Rabi oscillations are not easily observed in region I because they are strongly damped in comparison to the coupling. In region II, on the other hand, they are underdamped and distinguishable. In Fig. 6(h) the gain-loss contrast is reduced to the same level as in Fig. 6(e), resulting in reduced damping and a decrease in the quasi-frequency of the oscillation.

Fig. 7. Calculated spectra for CRs: (a) at threshold and $\kappa > {\kappa _{EP}}$, (b) at threshold and $\kappa < {\kappa _{EP}}$, (c) below threshold in region I, and (d) below threshold in region II. In (a) the detuning between the uncoupled resonators is δ =0 (black curve) and δ =0.001 (green curve). In (b) δ =0.01. In (c) and (d) δ =0.6. The dotted curves show the response of resonator #1 by itself, i.e., when the coupling is set to κ =0 (in (b) the gain in the isolated resonator is also boosted to threshold). Δ_L is the laser detuning.

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6. Frequency domain response

In Fig. 7 the reflectance spectrum of CRs in different regimes is shown when their resonance frequencies are detuned from each other by δ. In the top two graphs the system is at threshold, i.e., lasing, whereas in the bottom two graphs the system is below threshold. The dashed curves in Figs. 7(b)–7(d) show the reflectance from resonator #1 by itself, i.e., when the coupling is set to κ =0. In each case the extrema of the dashed curves are shifted from zero by δ / 2, whereas the extrema of the solid curves are shifted by Δ / 2. Note that in Fig. 7(c) the minimum in the spectrum for the CR system moves in the same direction as that for resonator #1 by itself. This is because in region I the intensity is higher in resonator #1. (Note, however, there is a small bump in the spectrum that moves in the opposite direction as a result of the smaller amount of light in resonator #2.) This situation is akin to previous experiments on passive cavities containing an anomalous-dispersion medium. In Figs. 7(b) and 7(d), on the other hand, the maximum moves in the opposite direction from that of resonator #1 by itself, because the intensity is higher in resonator #2 in LWG and in region II, and the light in resonator #2 rotates in the opposite sense to that in resonator #1. In all these cases, the magnitude of the shift is larger than it is for the single resonator, demonstrating the enhancement in scale factor sensitivity that results from the coupling. In Fig. 7(a) two peaks are observed in the spectrum due to the frequency splitting described in Section 2. The system is PT-symmetric when δ =0 (black curve) and the peaks have the same amplitude, but the amplitudes are unequal when |δ | > 0 (green curve). To maintain PT-symmetry when |δ | > 0 would require that the system be above threshold according to Fig. 4(b), where the simple theory presented so far does not apply [58].

7. Implementation of CR gyroscopes

In this section we discuss how each of the regimes of operation shown in Fig. 7 can be used in a CR gyroscope. In Fig. 4 we observed that there are two branches where the EP corresponds to the lasing threshold: the PT-symmetric and LWG branches. Figures 8(a) and 8(b) show these two lasing regimes, respectively. In Fig. 8(a) the rings are PT-symmetric and at threshold: the upper ring has net gain whereas the lower ring has an equivalent amount of loss so that $\hat{\gamma }_2^{PT} ={-} {\hat{\gamma }_1}$, and δ =0. Because $\kappa > {\kappa _{EP}}$, photons are not localized in either ring, but oscillate back and forth as shown in Figs. 6(c)–6(e) with equal peak intensities. Moreover, light that copropagates with the rotation in one ring will counterpropagate with it after coupling into the adjoining ring. Consequently, two equal-amplitude modes exit the system in each direction, one blue shifted and one red shifted, as shown in the black curve of Fig. 7(a). This can be verified by inspection of Fig. 3, where the frequencies are observed to anticross $({\omega _ + } \ne {\omega _ - })$, while the linewidths cross $({\gamma _ + } = {\gamma _ - })$, at δ =0, which means the threshold is the same for the two modes. These modes interfere and a beat note is observed at the detector. In this case, one only needs to collect light that exits the rings in one direction. What happens when the gyro rotates and/or the system is above threshold depends on how the gain saturates, as we will discuss in the next section.

Fig. 8. Possible schemes for the use of two CRs as a gyroscope: (a) PT-symmetric gyro, (b) LWG gyro, (c) a subthreshold passive (region I) gyro, and (d) a subthreshold active (region II) gyro. The gain is assumed to be in the upper resonator. PD = photodetector. The red and blue curved arrows inside the rings represent the eigenfrequencies of the individual uncoupled rings.

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In Fig. 7(b), on the other hand, $\kappa < {\kappa _{EP}}$. By interference, photons are trapped in the (upper) resonator with gain, while being excluded from the lossy (lower) resonator, as shown in Fig. 6(a) and Fig. 8(b). This process of coherent photon trapping is analogous to coherent population trapping in atomic systems. In this case the frequencies cross in Fig. 3, whereas the linewidths anticross $({\gamma _ + } \ne {\gamma _ - })$ at δ =0, so the threshold is different for the two modes and only one of them lases. As a result, a single mode exits the system in each output direction. The direction of the frequency shift of this mode is determined by the direction of light propagation in the ring containing the gain, the upper ring in the figure. LWG gyros, therefore, function similar to conventional laser gyroscopes; light exiting the system in both directions must be collected and made to interfere on a detector to measure the beat note. The coupling to the lower ring simply amplifies the magnitude of the frequency shift, boosting the sensitivity to rotation, particularly near the EP.

Figures 8(c) and 8(d) depict subthreshold CR gyroscopes. These gyros may be entirely passive, Fig. 8(c), or may contain gain, Fig. 8(d), but not so much gain that lasing occurs. As discussed in Section 4, it is not required that the system be near an EP. Indeed, as shown in Fig. 4, the $S = \infty$ splitting-point coupling only occurs at the EP when the system is at threshold. Below threshold, the splitting-point coupling is that which is just sufficient to cause the reflectance spectrum to split when δ =0. Similar to the operation of conventional passive cavity gyroscopes, light is injected into the rings from both directions, and is held on resonance with the structure due to a feedback mechanism (not shown). In Fig. 8(c) the system is entirely passive and located in region I. Thus, the photons are localized in the bottom resonator, as shown in Fig. 6(f), and the spectrum is as shown in Fig. 7(c). This situation describes previous experiments on passive cavities containing an absorbing alkali vapor cell as the anomalous dispersive medium [3,5,8,11,14]. The absorbing medium plays the role of the upper resonator. For Fig. 8(d), on the other hand, there is a small amount of gain in the upper resonator such that the system is in region II. Therefore, similar to the LWG CR gyro, photons are localized in the upper resonator as shown in Fig. 6(g), and the spectrum is as shown in Fig. 7(d). Hence the frequency shifts of the inputs are reversed from those of Fig. 8(c).

8. Effect of gain saturation

Thus far we’ve neglected the nonlinear dynamics that occur above threshold, where the simple coupled-mode theory does not apply. We showed in Fig. 4(b) that the threshold (saturated) gain decreases with increasing rotation rate, i.e., $\gamma$ is a function of $\delta$. Therefore, the dependent variables in $\tilde{\Omega }\,(\delta ,\gamma ,\kappa )$ cannot be considered independent, and it stands to reason that gain saturation will modify the gyro scale factor, $d({\mathop{\textrm {Re}}\nolimits} \tilde{\Omega })/d\delta$. Below we discuss the effect of gain saturation with homogeneous and inhomogeneous broadening on CR gyros.

8.1 Gain saturation in homogeneously-broadened CR systems

To account for gain saturation in homogeneously-broadened CR systems, we follow the procedure described in [59]. The loss coefficient in the second resonator is separated into saturable and non-saturable parts, i.e., ${\hat{\gamma }_2} = {\gamma _{02}}/(1 + \beta {I_2}) + {\gamma _L}$, where $- {\gamma _{02}}$ is the unsaturated gain coefficient, $\beta$ is the self-saturation coefficient, and ${\gamma _L}$ is a constant loss coefficient. The coupled-mode equations are then solved numerically. The results of these calculations show that gain saturation eliminates the frequency splitting in PT-symmetric gyros, which in turn eliminates the power oscillations. Note that this is much different from the situation described in [59], where gain saturation distorts the oscillations that occur near the EP at the deadband edge (see Appendix), which in turn broadens the beat note. Here, the broadening does not occur, because the Rabi oscillations are completely damped by the gain saturation, leaving only one frequency peak at the output. This is because when a PT-symmetric system is detuned, one mode is stronger than the other as shown in Fig. 7(a). Thus, for homogeneously-broadened systems, gain saturation ensures that only one frequency lases, with the other frequency being below threshold. In this case, PT-symmetric gyros become more like LWG and conventional gyros, and must operate with only one mode for a given output direction, for example as shown in Fig. 8(b). In our calculations we assume the bidirectional scheme shown in Fig. 8(b), but ignore any dissipative coupling between the two directions, i.e., backscattering, which could lead to a deadband. We calculate the beat frequency by adding together the numerically-determined output fields for the two directions, taking the fast Fourier transform of the resulting intensity, and then finding the average or peak value of this beat-note spectrum.

In Fig. 9 the beat frequency ${\mathop{\textrm {Re}}\nolimits} \tilde{\Omega }$ and saturated gain coefficient $- {\hat{\gamma }_2}$ are plotted vs. the rotation-induced detuning δ for $\kappa = {\kappa _{EP}}$ (black curves), $\kappa > {\kappa _{EP}}$ (red curves), and $\kappa < {\kappa _{EP}}$ (blue curves), for cases above and below threshold. For systems initially below threshold at δ =0, the gain does not saturate, remaining constant as δ increases (see horizontal branch in Fig. 9(c)), until threshold is reached, after which the gain saturates to the threshold value predicted by Eq. (12). The beat frequency increases approximately linearly in this subthreshold region with a scale-factor enhancement on the order of unity (see horizontal branch in Fig. 9(a)). For systems starting out above threshold at δ =0, on the other hand, the gain immediately saturates down to the threshold value, and continues to decrease as the threshold gain value decreases with increasing δ (see main curve in Fig. 9(c) and red and blue curves in Fig. 9(d)). A helpful way to visualize the gain saturation process is to imagine points on Fig. 4(b). Points to the right of the threshold line will immediately move left to the line due to the saturation, whereas points that are to the left of the line will remain where they are, but the line will move left as the detuning is increased until these points reach threshold. Thus, ${\kappa _{EP}}$ must be determined using the saturated gain value at δ =0.

Fig. 9. Beat frequency and saturated gain coefficient near the EP below and above threshold. The couplings are: $\kappa = 1.4 = {\kappa _{EP}}\,\,({\textrm{black curves}} ),\,\,\kappa = 1.45\,\,({\textrm{red curves}} ),\,$ and $\kappa = 1.25$ (blue curves). The second resonator has a saturable gain with $- {\gamma _{02}} = 3.5\,,\,\,{\gamma _L} = 1.4\,,$ and $\beta = 1$, while the first resonator has a constant loss, ${\hat{\gamma }_1} = {\gamma _L} = 1.4\,\textrm{.}$ The shorter horizontal branches off the main curves in (a) and (c) are the results when the unsaturated gain is changed to $- {\gamma _{02}} = 2.65$. In this case the system is below threshold at small detunings. Circles represent numerical solutions. Solid curves on the right are the threshold values of the gain predicted by Eq. (12). Solid curves on the left are the theoretical prediction for the beat frequency, using the saturated gain values (on the right) to compute ${\mathop{\textrm {Re}}\nolimits} \tilde{\Omega }$. The dashed curves are the results with no gain saturation $(\beta = 0)$ where the gain is a constant, set to its saturated value at $\delta = 0$, i.e., $- {\hat{\gamma }_2} = 1.4$ (black and red curves) and $- {\hat{\gamma }_2} = 1.116$ (blue curve). Dotted lines in (c) and (d) indicate PT symmetry.

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We note that our nonlinear model is similar to that in [44], which found that systems initially in a broken-symmetry regime can saturate into a regime with unbroken symmetry for sufficiently high pumping rates. Our model does not predict this transition because it ignores saturable loss, and therefore the phase transition point shown in Fig. 4(a), $- \hat{\gamma }_2^{PT} = {\kappa _{PT}} = {\hat{\gamma }_1}$ is a constant. However, it is apparent that if we allowed the loss to saturate, $- \hat{\gamma }_2^{PT}$ and ${\kappa _{PT}}$ would both decrease in the same proportion, translating the phase transition down and to the left in the figure. At the same time gain saturation would move the gain $- {\hat{\gamma }_2}$ left to the threshold level. Thus, for sufficient pumping the system can end up on the PT-symmetric line rather than on the LWG curve. Importantly, however, these results only apply at δ =0. The nonlinear model in [44] neglects the detuning, δ, and so does not determine the effect of the saturation on the beat frequency, which we have calculated in Fig. 9.

Indeed, although we have been referring to them as PT-symmetric gyros, systems with $\kappa \ge {\kappa _{EP}}$ are in fact only PT-symmetric at δ =0, as we pointed out near the end of Section 2. At all other values of δ, the threshold gain level falls below the horizontal lines marking PT symmetry shown in Figs. 9(c) and 9(d), into a symmetry-broken LWG regime. Indeed, the steady-state intensities in the two resonators are equal only when δ =0, as depicted in Figs. 7(a) and 8(a). For |δ | > 0 the intensity in the amplifying resonator grows larger than that in the lossy resonator. This photon localization is indicative of LWG.

The decreasing threshold gain value has important consequences for the beat frequency. First, the beat frequency is much larger than it would be if the gain were held constant (dashed curves), and this effect is particularly pronounced for the so-called PT-symmetric gyros where $\kappa \ge {\kappa _{EP}}$ (black and red curves). In fact, without gain saturation the prediction for $\kappa > {\kappa _{EP}}$ (red dashed curve) is that the scale factor should decrease. This is because the cusp that occurs at the EP in Fig. 5 should be transformed above the EP into a shallow curve. Instead, gain saturation maintains the square-root dependency found at the EP, leading to an increase in scale factor, even for couplings well above the EP. Thus, gain saturation allows operation farther from the EP as well as dramatically increasing the size of the sensitivity enhancement. The inclusion of loss saturation in the model could modify these findings by shifting the phase transition point as mentioned above.

8.2 Gain saturation in inhomogeneously-broadened systems

For homogeneously-broadened systems the localization of photons into the amplifying resonator when the CR system is detuned, means that PT-symmetric gyros $(\kappa > {\kappa _{EP}})$ cannot work with the beat frequency measured from a single output direction, because gain saturation ensures that only one mode lases, while the other mode is below threshold. The same is not true when the gain is inhomogeneously-broadened, such as in [25,41]. To appreciate this, consider that in Fig. 4(b) only the threshold level for the stronger mode is shown (the black curve to the left of the PT-symmetric line). There is another threshold for the weaker mode, however, which lies to the right of this line. Therefore, given sufficient pumping and gain, both of the modes shown in Fig. 7(a) can reach threshold provided the gain is inhomogeneously broadened. The Rabi oscillations can then be maintained over a large range of detunings. This allows PT-symmetric gyros to operate unidirectionally, as originally envisioned, with the beat frequency measured from a single output direction. In fact, LWG gyros can also operate unidirectionally when the broadening is inhomogeneous, with two modes appearing upon rotation, but generally must be pumped harder and require more gain to do so (see Fig. 3). At the EP coupling, of course, the difference between the PT-symmetric and LWG systems disappears with respect to the pumping required to sustain this dual-mode unidirectional operation. The oscillations in power that accompany the simultaneous appearance of these two modes may still be a concern, however, as they can lead to a false rotation signal in nonlinear media via the Kerr effect. And, for sufficiently large rotation rates the weaker mode will eventually fall below threshold, precluding unidirectional operation. This is required because the eigenvalues of CRs approach those of the uncoupled resonators at large δ, and one of the resonators is lossy. On the other hand, CRs operating bidirectionally will be subject to the same backscattering-induced lock-in problem that plagues conventional RLGs (see Appendix). It is not clear whether a similar dissipative coupling mechanism between the two modes will lead to lock-in in unidirectional CR systems, but it’s generally thought that unidirectional operation removes lock-in [26].

Note that the increase in the size of the sensitivity enhancement shown in Fig. 9 would in fact be difficult to obtain in homogeneously-broadened CR systems because they are single-mode and bidirectional operation is generally precluded, but should be possible with inhomogeneously-broadened CR systems, operating either in the dual-mode unidirectional or single-mode bidirectional configuration. Again, when operating bidirectionally one mode will be below threshold either because the system is operating well into the LWG regime or because it’s in the PT-symmetric regime but the pumping or gain is too weak to support both modes.

9. Summary and conclusions

We have examined the use of PT-symmetry breaking in lasing and subthreshold systems composed of two CRs, to improve the sensitivity of optical-cavity-based gyroscopes. While PT-symmetric and FL passive-cavity gyroscopes have been discussed previously, to our knowledge there are no examples of the use, or proposed use, of the schemes in Figs. 8(b) and 8(d) for gyroscopes. LWG gyros, in particular, are promising due to their narrow linewidth. Moreover, unlike PT-symmetric gyros, the sensitivity of LWG gyros is enhanced even at couplings far from the EP (see Fig. 5), and their outputs do not oscillate in intensity (see Fig. 6). Although their effect can be minimized by operating close to the EP, these oscillations can broaden the beat note and lead to a time-varying nonreciprocal phase shift via the Kerr effect, which could make the use of the PT-symmetric phase problematic for some gyroscopes.

Indeed, the beat note in PT-symmetric gyros relies on AM, whereas in conventional and LWG gyros it’s pure phase modulation. Therefore, anything that disturbs these oscillations will impact the beat note. As a result, PT-symmetric gyros will not work as originally envisioned (with the beat frequency measured from a single output direction) when: (i) the gain is homogeneously broadened, (ii) the pumping or gain is too low, or (iii) the rotation rate is too high. Under these conditions the power oscillations are eliminated because one of the modes falls below threshold as the rotation rate increases (just as sufficiently large detunings or saturation can reduce or eliminate Rabi oscillations in two-level atoms), and CR gyros must operate bidirectionally, similar to conventional RLGs. On the other hand, gain saturation benefits both LWG and PT-symmetric gyros by dramatically boosting the size of the sensitivity enhancement, which also enlarges the parameter space around the EP over which CR gyros can operate.

A critical problem for all CR gyros is that they are not common path and therefore do not benefit from the same level of common-mode noise rejection as in typical ring-laser gyroscopes. Any fluctuations in the detuning between the resonators will show up as a rotation signal. Therefore, some independent means of maintaining the relative detuning between the resonators constant is required. Indeed, the spectral splitting point (which is equivalent to the EP at threshold) is not a stable operating point [14,60]. Therefore, the ability of LWG gyros to operate farther away from the EP should make observation of the enhancement more straightforward by reducing the effect of noise.

We end with a caveat: the fundamental question of whether precision can be improved by FL- or EP-based sensing remains unresolved [11,14,61–64], and we have not addressed it here except in the following sense. It has long been established that in non-Hermitian systems the fundamental laser linewidth increases above the Schawlow-Townes limit as a result of the non-orthogonality of the resonant modes [65–67]. This broadening is characterized by the Petermann excess-noise factor, which has been predicted to diverge at an EP [40,68,69]. If, as we claim, the superluminal laser is simply one that operates at an EP, then it cannot be simultaneously true that the linewidth of such a laser is unaffected by FL [10] and that the Petermann factor diverges. These contradictory predictions highlight the importance of obtaining measurements of the intrinsic linewidth of such systems. In particular, these measurements should attempt to determine how fast the linewidth diverges compared to the sensitivity.

Appendix A: EPs in dissipative CR systems

The treatment can be generalized to account for dissipative coupling by setting ${\tilde{\kappa }_1} = {\tilde{\kappa }_2} = \tilde{\kappa }$, such that the coefficients are in phase, i.e., ${\theta _1} = {\theta _2}$, and the Hamiltonian is symmetric. In this case, the Rabi frequency is $\tilde{\Omega } = {({\tilde{\delta }^2} + {\tilde{\kappa }^2})^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}$, i.e.,

(22)$$\tilde{\Omega } = {[{{\delta^2} - {{({\gamma /2} )}^2} - i\delta \gamma + {\mathop{\textrm {Re}}\nolimits} {{(\tilde{\kappa })}^2} - {\mathop{\textrm {Im}}\nolimits} {{(\tilde{\kappa })}^2} + 2i{\mathop{\textrm {Re}}\nolimits} (\tilde{\kappa }){\mathop{\textrm {Im}}\nolimits} (\tilde{\kappa })} ]^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}.$$

The EP is then found by setting $\tilde{\Omega } = 0$. When $\tilde{\kappa }$ is real (conservative coupling) this condition is satisfied when $\gamma \ne 0,\delta = 0$, and $|{{\mathop{\textrm {Re}}\nolimits} \tilde{\kappa }} |= |{\gamma /2} |$, just as we obtained previously (see Eq. (7)). There are two additional dissipative-coupling cases to consider: (a) $\tilde{\kappa }$ is imaginary, i.e., the system is maximally dissipative. The condition cannot be satisfied at $\delta = 0.$ An EP still exists, however, when $\gamma = 0$ and $|\delta |= |{{\mathop{\textrm {Im}}\nolimits} \tilde{\kappa }} |{.}$ This detuning corresponds to the edge of the deadband, the region of zero sensitivity that occurs in conventional RLGs [45,70] and in other situations [71]; (b) $\tilde{\kappa }$ is fully complex. An EP forms when $\gamma \ne 0, $ $|{{\mathop{\textrm {Re}}\nolimits} \tilde{\kappa }} |= |{\gamma /2} |$, and $|\delta |= |{{\mathop{\textrm {Im}}\nolimits} \tilde{\kappa }} |$. Therefore, whenever there is dissipative coupling, an EP exists at the detuning corresponding to where the deadband edge would be located if one were to exist. The EPs for the various cases considered above are shown below in Fig. 10. Note that the only EPs that can be PT-symmetric are along the vertical axis (conservative coupling) and that these two points are only PT-symmetric when ${\gamma _{avg}} = 0$ (at threshold). Finally, we remark that other EP locations are possible if one relaxes the assumption that the off-diagonal components ${\tilde{\kappa }_1}$ and ${\tilde{\kappa }_2}$ are equal in magnitude, as in [59].

Fig. 10. Parameters values for EPs (magenta dots). Broken (blue line) and unbroken (red line) PT-symmetry occurs along the vertical axis when ${\gamma _{avg}} = 0.$ The other EPs are not PT-symmetric. The grey line indicates the deadband.

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The scale factor at these EPs can be found by generalizing Eq. (12), the expression for the real-valued frequencies, i.e.,

(23)$${\omega _ \pm } = {\omega _{avg}} \pm \frac{1}{{2\sqrt 2 }}{\left[ {{\mathop{\textrm {Re}}\nolimits} ({{\tilde{\Omega }}^2}) \pm \sqrt {{\mathop{\textrm {Re}}\nolimits} {{({{\tilde{\Omega }}^2})}^2} + {\mathop{\textrm {Im}}\nolimits} {{({{\tilde{\Omega }}^2})}^2}} } \right]^{1/2}},$$

where ${\mathop{\textrm {Re}}\nolimits} ({\tilde{\Omega }^2}) = {\mathop{\textrm {Re}}\nolimits} {(\tilde{\kappa })^2} - {(\gamma /2)^2} + {\delta ^2} - {\mathop{\textrm {Im}}\nolimits} {(\tilde{\kappa })^2}$ and ${\mathop{\textrm {Im}}\nolimits} ({\tilde{\Omega }^2}) = 2{\mathop{\textrm {Re}}\nolimits} (\tilde{\kappa }){\mathop{\textrm {Im}}\nolimits} (\tilde{\kappa }) - \delta \gamma \textrm{.}$ The frequency difference $\Delta \equiv {\omega _ + } - {\omega _ - }$ is plotted versus the detuning δ for the two dissipative cases in Fig. 11. Taking the derivative with respect to the detuning and evaluating the result in the vicinity of the EP results in

(24)$$|S |= \left\{ {\begin{array}{{cc}} {|\delta |/{{({{\delta^2} - {\mathop{\textrm {Im}}\nolimits} {{(\tilde{\kappa })}^2}} )}^{1/2}}}&{\; \tilde{\kappa }\textrm{ imaginary}}\\ {1/|{{\mathop{\textrm {Re}}\nolimits} (\tilde{\kappa }) - (\gamma /2)} |}&{\tilde{\kappa }\,\textrm{ complex}\textrm{.}} \end{array}} \right.$$

Fig. 11. Dissipative coupling shifts EP to (a) $|\delta |= |{{\mathop{\textrm {Im}}\nolimits} \tilde{\kappa }} |$ and (b) $\delta = {\mathop{\textrm {Im}}\nolimits} \tilde{\kappa }$. ${\mathop{\textrm {Im}}\nolimits} \tilde{\kappa }$ is the same constant value for each curve. In (a) a deadband is formed and an EP occurs at each edge, whereas in (b) the curve is still distorted but less asymmetric about the EP, which is shifted by the same amount as in (a) but in only one direction. There is no EP for the black curve, but this is included to show the effect of increasing the differential loss γ. In (a) ${\mathop{\textrm {Re}}\nolimits} \tilde{\kappa } = 0$, and the loss differences are γ =0 (red curve) and $\gamma = 2{\mathop{\textrm {Im}}\nolimits} (\tilde{\kappa })/3$ (black curve), whereas in (b) ${\mathop{\textrm {Re}}\nolimits} \tilde{\kappa } = {\mathop{\textrm {Im}}\nolimits} \tilde{\kappa }$ and $\gamma = 2{\mathop{\textrm {Im}}\nolimits} \tilde{\kappa }$.

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The first expression only applies outside the deadband (|S|=0 inside the band), while the second only applies at the EP. For both expressions $|S |\to \infty$ at the EP. The effect of dissipative coupling in either case is simply to detune the EP away from $\delta = 0$ as shown in Fig. 11.

There has been some work that has attempted to exploit the sharp increase in sensitivity at the deadband edge (see the red curve in Fig. 11(a)) for the enhancement of gyroscopes [72]. This is not only convenient because it allows use of a single cavity, which preserves common-mode noise rejection, but is also the only option to approach an EP for a gyro with equal counterpropagating intensities (the EP around zero detuning requires a loss difference). A disadvantage of this approach is that gain saturation distorts the Rabi intensity oscillations that occur near the EP, which broadens the beat note [59]. Another drawback is that once the sensitivity is optimized by closely approaching the EP, the detuning cannot then be decreased in magnitude without ending up inside the deadband where |S|=0. Indeed, whenever ${\mathop{\textrm {Im}}\nolimits} \tilde{\kappa }$ is large compared to ${\mathop{\textrm {Re}}\nolimits} \tilde{\kappa }$ and γ, the frequency versus detuning curves will be distorted, resulting in an asymmetry as can be observed in all the curves in Fig. 11, with larger values of |S| on one side of the EP and smaller values of |S| on the other. This asymmetry is maximized for the maximally dissipative case when $\gamma = 0$ as seen in the red curve. The black curve shows the effect of increasing the differential loss γ. There is no EP in this case, but the asymmetry remains to some extent, resulting in decreased sensitivity for small detunings about $\delta = 0$, and increased sensitivity at larger detunings. On the other hand, for sufficiently small values of ${\mathop{\textrm {Im}}\nolimits} \tilde{\kappa }$ relative to ${\mathop{\textrm {Re}}\nolimits} \tilde{\kappa }$ and γ, the distortion is minimal, and the effect of the dissipation amounts to little more than a shift of the EP, such as that seen in the blue curve. Experimental evidence of this sort of shift and distortion can be found in [14].

Funding

Marshall Space Flight Center (80MSFC19M0042).

Acknowledgments

D. D. Smith was supported internally through the Center Innovation Fund of the Marshall Space Flight center under NASA’s Space Technology Mission Directorate and the Office of the Chief Technologist.

Disclosures

The authors declare no conflicts of interest.

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56. However, we will continue to refer to systems as being either PT-symmetric or LWG for the remainder of the discussion, provided they have those characteristics at δ = 0.

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Parity-time-symmetry-breaking gyroscopes: lasing without gain and subthreshold regimes

Abstract

1. Introduction

2. Eigenvalues of two coupled resonators

3. Scale-factor sensitivity

4. Subthreshold CR systems

5. Photon dynamics

6. Frequency domain response

7. Implementation of CR gyroscopes

8. Effect of gain saturation

8.1 Gain saturation in homogeneously-broadened CR systems

8.2 Gain saturation in inhomogeneously-broadened systems

9. Summary and conclusions

Appendix A: EPs in dissipative CR systems

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (11)

Equations (24)

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