Linear response theory for superradiant lasers

Linear response theory for superradiant lasers

Justin G. Bohnet    Zilong Chen    Joshua M. Weiner    Kevin C. Cox    James K. Thompson JILA, NIST and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440, USA
July 7, 2019

We theoretically study a superradiant laser, deriving both the steady-state behaviors and small-amplitude responses of the laser’s atomic inversion, atomic polarization, and light field amplitude. Our minimum model for a three-level laser includes atomic population accumulating outside of the lasing transition and dynamics of the atomic population distribution causing cavity frequency tuning, as can occur in realistic experimental systems. We show that the population dynamics can act as real-time feedback to stabilize or de-stabilize the laser’s output power, and we derive the cavity frequency tuning for a Raman laser. We extend the minimal model to describe a cold-atom Raman laser using Rb, showing that the minimal model qualitatively captures the essential features of the more complex system Bohnet et al. (2012b). This work informs our understanding of the stability of proposed millihertz linewidth lasers based on ultranarrow optical atomic transitions and will guide the design and development of these next-generation optical frequency references.


I I. Introduction

Steady-state, superradiant lasers based on narrow optical atomic transitions have the potential to be highly stable optical frequency references, with unprecedentedly narrow quantum-limited linewidths below 1 millihertz Meiser et al. (2009); Chen (2009); Bohnet et al. (2012a). These lasers may achieve such high frequency stability because the laser linewidth and frequency are determined primarily by the atomic transition rather than the cavity properties. As a result, the lasing frequency is predicted to be many orders of magnitude less sensitive to both the thermal and technical mirror motion that currently limits the frequency stability of passive optical reference cavities Kessler et al. (2012); Hinkley et al. (2013). The insensitivity to vibration means that superradiant lasers may be able to stably operate outside carefully engineered, low-vibration laboratory environments for both practical and fundamental applications Leibrandt et al. (2011); Argence et al. (2012).

To minimize inhomogeneous broadening of the atomic transition, proposed narrow-linewidth superradiant lasers would use trapped, laser-cooled atoms as the gain medium Meiser et al. (2009); Chen (2009). The first use of cold atoms as a laser gain medium was reported in Ref. Hilico et al. (1992). Recently, the spectral properties of a cold-atom Raman laser were studied in a high finesse cavity, deep into the so-called good-cavity regime Vrijsen et al. (2011). Clouds of cold atoms can also simultaneously provide gain and feedback for distributed feedback lasing Schilke et al. (2012) and random lasing Baudouin et al. (2013). Cold atoms have also been used as the gain medium in four-wave mixing experiments Greenberg and Gauthier (2012); Baumann et al. (2010); Black et al. (2003) and in collective atomic recoil lasing Kruse et al. (2003).

Beyond the technical applications, superradiant lasers are of fundamental interest. The narrow natural and inhomogenous linewidths provided by laser trapped and cooled atoms means that proposed superradiant lasers are bad-cavity lasers. This unusual regime of laser physics is accessed when the cavity linewidth is much larger than the linewidth of the gain medium. The quantum-limited linewidth of a bad-cavity laser follows the Schawlow-Townes linewidth Schawlow and Townes (1958) usually applied to microwave masers Haken (1984); Kuppens et al. (1994); Meiser et al. (2009), instead of the linewidth applied to optical lasers that typically operate in the opposite good-cavity limit. Bad-cavity lasers near the cross-over regime (i.e. where the cavity linewidth is approximately equal to the linewidth of the gain medium) have yielded signatures of chaos, demonstrating the predicted equivalence to the Lorenz model Haken (1975); Weiss and Brock (1986). Operation of a laser deep into the bad-cavity regime has only recently begun to be studied in detail using laser cooled atoms as the gain medium Bohnet et al. (2012a, b); Weiner et al. (2012); Bohnet et al. (). The first analysis of a cold-atom, superradiant Raman laser focused on the laser phase noise Bohnet et al. (2012a).

This paper presents theoretical studies of both the steady-state and amplitude stability properties of a superradiant laser. Our work both guides the future implementation of proposed superradiant optical lasers, and explains already experimentally-realized superradiant Raman lasers. This work directly supports the experimental efforts using Raman transitions in Rb of Refs. Bohnet et al. (2012a, b, ); Weiner et al. (2012).

The key new results presented here are a simple minimum model that nonetheless captures the qualitative features in recent experimental demonstrations, a derivation of crucial laser emission frequency tuning effects in a good or bad cavity cold-atom Raman laser, and an investigation of laser amplitude instabilities caused by frequency tuning effects in the case of a bad-cavity laser.

We begin in Sec. II by constructing a model of a steady-state Raman laser that makes three extensions to the two-level superradiant laser model presented in Ref. Meiser et al. (2009). These extensions are motivated by pumping and cavity tuning effects present in the experimental work of Refs. Bohnet et al. (2012a, b, ); Weiner et al. (2012). The extensions include: (1) an imperfect atomic repumping scheme in which some population remains in an intermediate third level, (2) additional decoherence caused by Rayleigh scattering during pumping, and (3) a tuning of the cavity mode frequency in response to the distribution of atomic populations among the available atomic states. The model makes no assumptions about operation in the good or bad cavity regime.

Figure 1: (color online) Energy level diagram of a three-level superradiant laser using the optical transition from to . The emitted optical laser light (blue) is nearly resonant with the cavity mode (dashed lines), detuned from by . The atoms are incoherently pumped to a third state at a rate . Atoms in also Rayleigh scatter at a rate , but leaves them in state . The incoherent pumping from to at rate completes the cycle.

We then restrict ourselves to considering only bad-cavity regime and linearize the coupled atom-field equations about steady-state to study the small signal response of the laser to external perturbations. We identify relaxation oscillations and dynamic cavity feedback that can serve to damp or enhance oscillatory behavior of the laser amplitude.

In Sec. III, we explicitly show that a Raman lasing transition involving three levels can be reduced to the previous section’s two level lasing transition. The formalism directly produces the cavity tuning in response to atomic populations that was introduced by hand in Section II.

Finally, in Sec. IV, we model the experimental Rb Raman system of Refs. [3,21], incorporating all eight atomic ground hyperfine states. We derive the steady-state behavior and linear response to small perturbations of this more realistic system and compare the qualitative features to the results of the more simple model introduced in Sec. II and III.

Ii II. Three-Level Model

ii.1 A. Deriving the laser equations

We begin by equations for a general three-level laser, making no assumptions about a good cavity or bad cavity regime. The three-level model for the laser presented here is pictured in Fig. 1. It consists of two lasing levels denoted by excited state and ground state separated by optical frequency , a third state which the atoms must be optically pumped to before they can be optically pumped back to , and a single optical cavity mode with resonance frequency . The cavity resonance is near the transition frequency, with . We describe the atoms-cavity system using the Jaynes-Cummings Hamiltonian Jaynes and Cummings (1963)


Here is the single atom vacuum Rabi frequency that describes the strength of the coupling of the atoms to the cavity mode, set by the atomic dipole matrix element. The operators and are the bosonic annihilation and creation operators for photons in the cavity mode. We have introduced the collective spin operators , and for the to transition, assuming uniform coupling to the cavity for each atom. The index labels the sum over individual atoms. We also define the number operator for atoms in the state , , as and the collective spin projection operator .

The density matrix for the atom cavity system is where the second sum is over the atomic basis states , and the third sum is over the cavity field basis of Fock states. The time evolution of is determined by a master equation for the atom cavity system


Dissipation is introduced through the Liouvillian Meiser et al. (2009). Sources of dissipation and associated characteristic rates include the power decay rate of the cavity mode at rate , the spontaneous decay from to at rate , the spontaneous decay from to at rate , and Rayleigh scattering from state at rate . The repumping, usually just called pumping in other laser literature, is treated as “spontaneous absorption” at rate , analogous to spontaneous decay, but from a lower to higher energy level. Physically, this is achieved by coupling to a very short lived excited state that decays to . The Liouvillian is written as a sum of contributions from the processes above respectively as . The individual Liouvillians are given in Appendix A.

Figure 2: Representing the state of the , two level system using a collective Bloch vector.

We obtain equations of motion for the relevant expectation values of the atomic and field operators using . Complex expectation values are indicated with script notation, while real definite expectation values are standard font, so and . We assume the unknown emitted light frequency is and factor this frequency from the expectation values for the cavity field and the atomic polarization, and . The symbol indicates a quantity in a frame rotating at the laser frequency. The set of coupled atom-field equations is then


In the above equations, we have combined the broadening of the atomic transition into a single transverse decay . We have assumed no entanglement between the atomic degrees of freedom and the cavity field in order to factorize expectation values of the form . The equations make no assumptions about the relative sizes of the various rates, making them general equations for a three level laser, but one of the distinct differences in cold-atom lasers versus typical lasers is that the transverse decay rate is often dominated by the repumping rate .

It is useful to represent the two-level system formed by and as a collective Bloch vector (Fig. 2). The vertical projection of the Bloch vector is given by the value of , and is proportional to the laser inversion. The projection of the Bloch vector onto the equatorial plane is given by the magnitude of atomic polarization , with . We refer to as the collective transverse coherence of the atomic ensemble.

ii.2 B. Steady-state solutions

To understand how extending to this three-level model affects the fundamental operation of the laser, we now study the steady-state solutions with respect to repumping rates, cavity detuning, and Rayleigh scattering rates. The steady-state solutions assume , the regime of operation for proposed superradiant light sources Meiser et al. (2009); Chen (2009) and the experiments of Refs. Bohnet et al. (2012a, b), but make no approximations based on the relative magnitudes of and . Thus , but otherwise the results in the section hold for both good-cavity () and bad-cavity () lasers. We first determine the steady-state oscillation frequency, starting by setting the time derivatives in Eqn. 3 and Eqn. 4 to zero. After solving for 


where denotes the cavity detuning from the laser emission frequency . Substituting the result into Eqn. 4, we have


Since is always real, the imaginary part of Eqn. 8 must be zero. This constrains the frequency of oscillation to


a weighted average of the cavity frequency and the atomic transition frequency.

We solve for the steady-state solutions of Eqns. 3-5 by setting the remaining time derivatives to zero and substituting Eqn. 7 for in all the equations. In this work, the amplitude properties are our primary interest (as compared to the phase properties studied in Refs. Meiser et al. (2009); Bohnet et al. (2012a)), so we further simplify the equations, at the expense of losing phase information, by considering the magnitude of the atomic polarization . The equation for the time derivative of is


The steady-state output photon flux is just proportional to the square of the equatorial projection


Here we have also defined a normalized detuning and a single particle cavity cooperativity parameter


that gives the ratio of single-particle decay rate from to for which the resulting photon is emitted into the cavity mode, making equivalent to the Purcell factor Tanji-Suzuki et al. (2011).

Figure 3: (a) Steady-state photon flux versus ground state repumping rate , with a series of curves showing the effects of repumping through the additional state .The case of is the two level model of Ref. Meiser et al. (2009) (black). (b) Photon flux versus with . For all curves, and , and the photon flux is plotted in units of .

After these substitutions and simplifications, the steady-state solutions (denoted with a bar) are


written in terms of the repumping ratio . Note that also determines the steady-state build up of population in as . To succinctly express the modification of and due to inefficient repumping, we define the reduction factor


which appears in Eqns. 14 and 16 above.

Next we discuss the behavior of these solutions for the characteristic parameters of the three-level model: , , , and . The results are illustrated in Figs. 3-5.

First, we focus on the impact of repumping on the steady-state behavior. The photon flux follows a parabolic curve versus the ground state repumping rate (Fig. 3). In the limit , , and , Eqn. 16 reduces to the result for the simple two-level model of Ref. Meiser et al. (2009). This limit is shown as the black curve in part (a) of Figs. 3-5. At low , the photon flux is limited by the rate at which the laser recycles atoms that have decayed to back to . At high , the photon flux becomes limited by the decoherence from the repumping, causing the output power to decrease with increasing . When the atomic coherence decays faster than the collective emission can re-establish it, the output power goes to zero. This decoherence limit is expressed in the condition for the maximum repumping threshold, above which lasing ceases:


The output photon flux is optimized at . Notice that the maximum repumping rate is not affected by . However, the additional decoherence (here in the form of Rayleigh scattering) lowers the turn-off threshold. If , the decoherence will prevent the laser from reaching superradiant threshold regardless of .

Figure 4: (a) Steady-state photon flux versus ground state repumping rate , with a series of curves showing the effects of detuning of the cavity resonance frequency from the emitted light frequency . (b) Photon flux versus with . The photon flux is plotted in units of . For all curves, and .

In Figs. 3-5, we plot Eqn. 16 emphasizing (a) the modification to the photon flux parabola, and (b) the optimum photon flux as a function of the population in the third state (as parameterized by the repumping ratio ), detuning of the cavity resonance from the emission frequency , and additional decoherence from Rayleigh scattering . The photon flux is plotted in units of the optimum photon flux in the two-level model of Refs. Meiser et al. (2009); Meiser and Holland (2010), .

As the repumping process becomes more inefficient and population builds up in , parameterized by as , we see from Eqns. 14 and 16 that the photon flux decreases (Fig. 3). A repumping ratio ensures the laser operates within a few percent of its maximum output power. Notice that saturates after is greater than . Although inefficient repumping suppresses , the optimum and maximum repumping rates and are not modified.

The preservation of the operating range can be important, as in practice large values of can lead to added decoherence (due to intense repumping lasers for example), which does reduce the operating range. Lowering the value of allows some flexibility as some output power can be sacrificed to keep the laser operating over a wider range of   .

Figure 5: Steady-state photon flux versus ground state repumping rate , with a series of curves showing the effects of decoherence in the form of Rayleigh scattering from the ground state . (b) Photon flux versus with . The photon flux is plotted in units of . For all curves, and .

Cavity detuning modifies both the and (Fig. 4). The modification arises from the dependent cavity cooperativity


The modified cooperativity originates from the atomic polarization radiating light at , which non-resonantly drives the cavity mode with the usual Lorentzian-like frequency response. Thus, the output photon flux , turn-off threshold , and optimum repumping rate all scale like . This effect is symmetric with respect to the sign of . Physically, the rate a single atom spontaneously decays from to by emitting a photon into the cavity mode is , which we use to simplify some later expressions.

Finally, we examine the effect of additional atomic broadening through in Fig. 5. Additional broadening linearly reduces and , but because we require the repumping rate to remain at in Fig 5b, has a dependence.

The key insight from the steady-state solutions for our three-level model is that imperfections in the lasing scheme can quickly add up, greatly reducing the expected output power of the laser. A repumping scheme should be chosen to minimize Rayleigh scattering and maximize the repumping ratio . Added decoherence, as well as the detuning are especially problematic because they restrict the possible range of for continuous operation.

ii.3 C. Linear expansion of uncoupled equations

For future applications of steady-state superradiant light sources as precision measurement tools, we are interested in the system’s robustness to external perturbations. As is common in laser theoryMcCumber (1966); Siegman (1986); Kolobov et al. (1993), here we analyze the system’s linear response to perturbations by considering small deviations from the steady-state solutions. While all previous expressions are valid for both the good-cavity and bad-cavity limit, as no assumptions were made about the relative magnitudes of and , it is convenient now to simplify to two equations for the dynamics by assuming that the laser is operating deep in the bad-cavity regime, where . In this regime, the cavity field adiabatically follows the atomic polarization, providing the physical motivation to eliminate the field from Eqns. 3-6 Meiser et al. (2009); Kuppens et al. (1994).

The cavity field is eliminated by assuming that the first time derivative of the complex field amplitude in Eqn. 3 is negligible compared to . This effectively results in Eqn. 7 being the equation for the cavity field. After substituting Eqn. 7 into Eqns. 4-6, we only concern ourselves with the amplitude responses, simplifying the equations by using Eqn. 10 and substituting with . With these simplifications, the dynamical equations for , , and are


We perform the linear expansion by re-parameterizing the degrees of freedom in terms of fractionally small perturbations about steady-state: , , and . We also define the response of cavity field through the relationship . Since from Eqn. 7, follows the atomic polarization, except for the modification from dynamic cavity detuning as will be discussed below. We analyze the response in the presence of a specific form of external perturbation – the modulation of the repumping rate with , where is a real number much less than 1. The quantities , , , , and are unitless fractional perturbations around the steady-state values that we assume are much less than .

We also include, by hand, an inversion-dependent term in the detuning . The cavity mode’s frequency is tuned by the presence of atoms coupled to the cavity mode. The tuning is equal but opposite for atoms in the two different quantum states and . The detuning is the steady-state value of the detuning of the dressed cavity from the emitted light frequency. The variation about this steady-state detuning is governed by the second contribution . Effects such as off-resonant dispersive shifts due to coupling to other states can lead to this dependent detuning in real experiments. We derive this cavity tuning in Sec. III.

To linearize the resulting equations, we substitute the expansions around steady-state into Eqns. 6, 20, and 21. We neglect terms beyond first order in the small quantities , , , , and . For ease of solving the equations, we treat , , , and as complex numbers where the real part gives the physical value. After eliminating the steady-state part of the equations, the equations for small signal responses and can be reduced to two uncoupled, third order differential equations


We have written the uncoupled differential equations in a form that suggests a driven harmonic oscillator, with damping rate , natural frequency and a drive unique to the or equation or . The drives contain derivatives of the repumping modulation , resulting in frequency dependence. The third derivative term is a modification to the harmonic oscillator response from the third level, characterized by the factor that goes to zero in the two-level limit (). To preserve the readability of the text, we have included the full expressions for the coefficients as Appendix A. Each of the terms will be discussed subsequently in physically illuminating limits.

The drive of this harmonic oscillator-like system varies with the modulation frequency and other system parameters. In the case of the two-level model of Ref. Meiser et al. (2009), with , , and , the drive terms are


The modulation-frequency-dependent terms add an extra 90 of phase shift at high modulation frequencies to the observed response. Additionally, the cancellation in results in an insensitivity of the output photon flux to the ground state repumping rate at . The cancellation agrees with the parabolic dependence of on , as seen in the steady-state solutions.

The frequency dependent terms in also cause a growing drive magnitude versus . This is canceled out in the responses and by the roll-off from the oscillator, keeping the response finite versus modulation frequency. These characteristic features remain in the response, even as the complexity of the model increases as additional effects are included.

To proceed, we solve the equations for the complex, steady-state response to a single modulation frequency , (e.g. ). The complex response of the cavity field amplitude results from these solutions,


In contrast to Eqn. 7, where depends only on , including dispersive cavity tuning from the inversion couples the cavity output power to as well.

ii.4 D. Transfer function analysis

We analyze the response of the cavity field amplitude to an applied modulation of the repumping rates by plotting the amplitude transfer function and the phase transfer function versus the modulation frequency , defined as and respectively. We consider the maximum of the transfer function to define the resonant frequency . The calculated variation in the transfer functions versus various experimental parameters is shown in Figs. 6 - 10. All results are given as a series of transfer functions varying a single specified system parameter, with other unspecified parameters set to , , , and .

Figure 6: Output photon flux transfer function for different ground state repumping rates, with , , and .
Figure 7: Output photon flux transfer function for different repumping ratios , with , , and .

The expressions for the damping and the natural frequency guide our understanding of the transfer functions. Holding , , and , the damping reduces to . Physically, the damping enters through the decay of at a rate proportional to . The natural frequency is set by the steady-state rate of converting collective transverse coherence into atoms in the ground state, , normalized by the steady-state transverse coherence .

To examine the effect of the steady-state repumping rate   on the response, we plot the transfer functions and for different values of   in Fig. 6. For , we see a narrow resonance feature in the response(blue curve). The frequency of the resonance increases until (green curve). Also at , the dc amplitude response , because the drive goes to zero (Eqn. 25), consistent with the maximum in at . For , the phase of the response near dc sharply changes sign, as understood from the parabolic response of versus   ; on the side of the parabola, the same change in produces the opposite change in the output photon flux compared to the side of the parabola. Meanwhile, the natural frequency has decreased with the increase in when . As   approaches , the response has essentially become that of a single-pole, low pass filter with an additional phase shift.

Figure 8: Output photon flux transfer function for different dressed cavity detuning from emitted light frequency , with , , and . The solid (dashed) lines show () to demonstrate the effects of increased cavity feedback.

To examine the effect of population in the third state , we now hold and show and for different in Fig. 7. The black curve shows the result for , which is the two-level model of Ref. Meiser et al. (2009), as no population accumulates in (recall that ). For smaller , the relaxation oscillations grow, shown by the increasing maximum in . This response is consistent with the reduced damping rate and increased drive seen in the following expressions.

The damping is . The additional dependence, associated with the repumping delay from atoms spending time in , results from the third derivative term that scales with in Eqns. 23 and 24.

The complex drive in this limit is . The term proportional to in arises from modulating the rate out of the state . Although the term in the damping would introduce a roll off in the transfer function with the form , the frequency dependence is canceled. The final transfer function maintains a frequency dependence of for , similar to that of the two-level system.

Figure 9: Stability plot using stability condition of Eqn. 28 as a function of detuning and the cavity shift parameter , assuming . The stability condition also assumes . The region of stability is exact for the two-level model (), and a good approximation for all values of . The blue region shows where the real part of all the poles of the solution are negative, indicating a damped return to steady-state conditions for a perturbation. The red region shows where any of the real parts of the poles become positive, making unstable, with no steady-state solutions.

Next we consider the effect of the dynamically tunable cavity mode. The cavity mode response can strongly modify the damping of the oscillator and even lead to instabilities in the cavity light field, eliminating steady-state solutions. We first consider the damping rate of the two-level model () with cavity tuning, where . The damping is modified by a detuning dependent feedback factor that is positive or negative depending on the sign of . Because to meet superradiant threshold, has the same sign as . Applying negative cavity feedback, when , increases the damping and may be useful for reducing relaxation oscillations and suppressing the effect of external perturbations. When , positive feedback decreases and amplifies the effect of perturbations.

We show the effect of this cavity feedback on the transfer functions in Fig. 8 for the conditions , , and . The red (blue) curves show positive (negative) feedback, with the black curve serving again as a reference to the model of Ref. Meiser et al. (2009) with no cavity feedback.

Fig. 8 also shows the effect of increasing the cavity shift parameter . The solid lines result from , a cavity shift similar in magnitude to experiments performed in Refs. Bohnet et al. (2012a, b, ); Weiner et al. (2012). The dashed lines result when is increased by a factor of two.

With enough positive feedback, the system can become unstable, with any perturbations exponentially growing instead of damping, which eliminates steady-state solutions. For a driven harmonic oscillator, the condition for steady-state solutions is . Again assuming , and remaining in the two level limit () the stability condition reduces to


In Fig. 9, we plot the stability condition as a red line.

Figure 10: Output photon flux transfer function for different Rayleigh scattering rates , with , , and . The dashed line shows the transfer functions when   is held to , not varied to remain at , and . A dashed red curve is not shown, as with and the maximum repumping rate threshold has been exceeded and the output photon flux is zero.

In general, the stability of a linear system can be determined by examining the poles of the solution. If any pole crosses into the right half of the complex plane, the system is unstable with an oscillating solution that grows exponentially. In the two-level limit (), this condition on the solutions and is mathematically equivalent to the condition on , Eqn. 28. As the level structure becomes more complex, e.g. or in the full Rb model in Sec. IV, we use the pole analysis to examine the regions of stable operation. For the model here, as changes, the pole analysis shows that the stability condition in Eqn. 28 is no longer exactly correct. However, the change is small enough that Eqn. 28 remains a good approximation of the stability condition for all values of .

Figure 11: Response of the 2D Bloch vector to external modulation of the repumping rate. The steady-state Bloch vector, i.e. and from Eqns. 13-14, is indicated by the blue line, plotted on the axis with units of , so is the maximum value. The ellipse is the trajectory of the Bloch vector responding to the modulation of the repumping rate , described by the small signal responses and in Eqns. 23 and 24. The parameters are , , , , and . The black arrow indicates the direction of the trajectory, starting from the blue dot at . The values of are chosen to show , , and .
Figure 12: Parametric plots of response of the three degrees of freedom , , and , highlighting effect of cavity frequency tuning on response of atomic coherence and output light field. The blue line represents the steady-state atomic Bloch vector, and , from Eqns. 13-14. The blue ellipse shows small single response of the Bloch vector to a modulation of the repumping rate , given by and from Eqns. 23 and 24. The parametric response is plotted with units of , so is the maximum value. The red dashed line is the trajectory formed by the response of the cavity field (Eqn. 27) and . The cavity field is plotted as a fraction of the average field, then centered on the steady-state Bloch vector to compare with the atomic response. The arrows indicate the direction of the response with respect to a modulation . Here s, , , , , and , chosen to show the stable response. (a) When , the cavity feedback can be positive, leading to larger oscillations compared to the case of no feedback shown in (b). Because of the coupling of to the cavity mode frequency, is not locked to the response, as in (b), but is anti-correlated with . In (c), where , the negative feedback reduces the response amplitudes in all quadratures. The cavity tuning again shifts the cavity amplitude response, but with the opposite phase relationship due the change in sign of the slope of the Lorentzian, so follows . The inset shows a close up of the response.

Finally, in Fig. 10 we show the effect of additional decoherence by plotting and for different values of . Here , , and . As a reference, the black curve shows the transfer function with . For the solid curves, the ground state repumping rate is varied with to remain at the point of maximum output power (Fig. 5) which amounts to holding constant. Thus, as the decoherence increases by increasing the rate of Rayleigh scattering from the ground state, the resonance frequency only moves because   is changing, as seen in the expression for the natural frequency . Notice that additional decoherence does not affect the peak size of the relaxation oscillations. Although the damping rate decreases because , this effect is canceled by the drive decreasing with   as well, with when .

If we hold   constant at , the resulting transfer function is the dashed line in Fig. 10. With   constant, the coherence damping rate varies with , and the response actually behaves similar to the case where   is increased (Fig. 6) because of the symmetric roles and have in the natural frequency and the drive.

The main conclusion from our examination of the linear response theory of the three-level, bad cavity laser is that most conditions for optimizing the output power are compatible with an amplitude stable laser. Operating at the optimum repumping rate in particular suppresses the impact of low frequency noise on the amplitude stability. However, we also find that because the cavity detuning couples to the population of the laser levels, cavity feedback can act to suppress perturbations, or cause unstable operation, depending on the sign of . A simple relationship between , , and gives the condition for stable operation at .

ii.5 E. Bloch vector analysis of response

Relaxation oscillations in a good-cavity laser arise from two coupled degrees of freedom, the intracavity field and the atomic inversion , responding to perturbations at comparable rates. Parametric plots of the amplitude and inversion response provide more insight into the nature of the relaxation oscillations than looking at the laser field amplitude response aloneSiegman (1986). In the bad-cavity regime, the cavity-field adiabatically follows the atomic coherence , and the oscillations arise from a coupling of and the inversion . Thus the relevant parametric plot is the 2D projection of the 3D Bloch vector in the rotating frame of the azimuthal angle. In this section, we study this response of the Bloch vector to better understand the stability of the bad-cavity laser.

The individual plots of Fig. 11 show the trajectory of the Bloch vector for the small signal response at different applied modulation frequencies and different repumping rates   . The trajectory is calculated using the amplitude and phase quadratures of the responses and to define the sinusoidal variation of each quadrature with respect to a sinusoidal modulation of . The series of plots show the trend in the responses versus the ground state repumping rate   and modulation frequency , with , , and . Although the oscillator characteristics of the two quadratures are identical, they display a differing phase in their response due to the differences in the drives , on the two quadratures.

At high repumping rates and high modulation frequencies , the perturbation modulates the polar angle of the Bloch vector, leaving the length largely unchanged. Near , the two quadratures have large amplitudes and oscillate close to 90 out of phase, leading to the trajectories that encloses a large area. When and with   near , the cancellation in the drive term leads to almost no amplitude of oscillation in the quadrature, making the modulation predominately -like. For or , this means the cavity field amplitude will also be stabilized, as it is locked to the transverse coherence (Eqn. 27).

Figure 13: Cavity tuning stabilizing the cavity field amplitude. By changing the average repumping rate   from to for the same parameters as Fig. 12c ( s, , , , , and ), the response of the Bloch vector (blue ellipse) becomes primarily perpendicular to the steady state Bloch vector (blue line). Under these conditions, the cavity field response (red dashed ellipse) has the smallest fractional deviation of the three degrees of freedom. The inset shows a close up of the response.

However, dynamic cavity tuning creates a coupling of the inversion to the cavity field as well, breaking the simple time-independent proportionality of the cavity field amplitude and the atomic coherence , as expected from Eqn. 27. Fig. 12a show the case of . Because of the coupling to the inversion, the cavity field response has a larger amplitude than response in addition to a phase shift. It is also nearly out of phase with the response of the inversion.We include the case of (Fig. 12b) as a reference. The cavity field is locked to the coherence, even for , due to the second order insensitivity in the cavity coupling. For the case of negative feedback , shown in Fig. 12c, all the response amplitudes are reduced due to the increased damping. Notice that the inversion and cavity field are now responding in phase.

Because of the coupling between all three degrees of freedom, it is possible to choose parameters that lead to a stabilization of the cavity field. Operating away from , the response of the Bloch vector becomes primarily a modulation of the polar angle as the inversion and coherence respond 180 out of phase. Combined with the cavity tuning, the cavity field is stabilized, as shown in Fig. 13, where the parametric plot of and (dashed red ellipse) shows a response that is primarily -like. The response of the cavity field has the smallest fractional variation among the three degrees of freedom.

To conclude our discussion of linear response theory in the three-level model, we point out that the parametric plot analysis highlights the role that the dispersive cavity frequency tuning plays in amplifying or suppressing perturbations in both the atomic degrees of freedom and the cavity field. Crucially, frequency stable lasers may need to seek a configuration that suppresses fluctuations in the degree of freedom to minimize the impact of cavity pulling on the frequency of the laser. We also see that the dispersive tuning breaks the exact proportionality of the cavity field and the transverse atomic coherence, restoring an additional degree of freedom that may be crucial for observing chaotic dynamics in lasers operating deep into the bad-cavity regime Haken (1975).

Iii III. Raman laser system

In the previous section, we presented a model for a three level laser for qualitatively describing the results from recent experiments that use laser cooled Rb as the gain medium Bohnet et al. (2012a, b, ). However, the Rb system also relies on a two-photon Raman lasing transition between hyperfine ground states, instead of a single optical transition. To address this difference, here we provide a model that has a two-photon Raman lasing transition, but a simple one-step repumping scheme directly from to . Then in Sec. IV, we present a full model of the bad-cavity laser in Rb that has both the two-photon Raman transition and a more complex repumping scheme.

In the first subsection, we derive equations of motion for the expectation values in the Raman model, then explicitly adiabatically eliminate the optically excited intermediate state in the Raman transition. In the second subsection, will establish the equivalences (and differences) between the Raman and non-Raman models. We will find that the Raman transition is well described as a one-photon transition with a spontaneous decay rate , an effective atom-cavity coupling , and with a two-photon cooperativity parameter equal to the original one-photon cooperativity parameter. The Raman system differs in the appearance of two new phenomena: differential light shifts between ground states and cavity frequency tuning in response to atomic population changes. The latter effect was inserted by hand in Sec II. As in Sec. II, we first derive equations without assuming a good-cavity or bad-cavity laser, only specializing to the bad-cavity limit at the end of the section.

Figure 14: Energy level diagram for a superradiant laser enabled by an induced Raman transition. States and are two metastable states separated by a non-optical frequency . They share an optically excited state that has a linewidth . Using a Raman dressing laser (green), detuned from by , we can induced a optical decay to , which, in absence of collective effects, would proceed at rate . Including a single optical cavity mode, coupled to the to transition with coupling constant , gives rise to collective emission. The cavity mode frequency is , detuned from by , making the two-photon detuning . To complete the laser cycle, the atoms are incoherently repumped from to at a rate .

iii.1 A. Adiabatic elimination of the intermediate state

To establish the connection between two-photon Raman lasing and one-photon lasing, we start by defining the Hilbert space for a three-level Raman system with two ground states denoted and (separated by only 6.834 GHz in Rb) and an optically excited intermediate state (Fig. 14). The Hilbert space also includes a single cavity mode that couples to . The density operator for the Hilbert space is . The first sum is over individual atoms, the second is over the atomic basis states , and the third sum is over cavity Fock, or photon-number, states. Raising and lowering operators for the cavity field and atoms are defined as in Sec. II. The state occupation operators for atoms in the state are again , where the index denotes a sum over individual atoms. We also define collective atomic raising and lower operators .

We describe the system via the semi-classical Hamiltonian


The Raman dressing laser at frequency is described by the coupling , and the atoms are uniformly coupled to the dressing laser. The rotating wave approximation will be applied so that only near-resonant interactions will be considered. The dressing field is externally applied, and we assume it is unaffected by the system dynamics (i.e., there is no depletion of the field).

To reduce the Raman transition to an effective two-level system, we derive the equations of motion for expectation values of the operators that describe the field and the atomic degrees of freedom. As was done in Sec. II, we use the time evolution of the density matrix obtained from the master equation (Eqn. 2) to derive the equations of motion . The details are included in Appendix B.

After adiabatic elimination of the optically excited state, we have the set of three coupled equations analogous to Eqns. 3-5:


Here and , which is also the average of and . The Eqns. 30-32 are general equations, valid without assuming a good-cavity or bad-cavity laser.

iii.2 B. Defining effective two level parameters for the Raman system

We can now identify the effective two-photon atom-cavity coupling constant


The effective Rabi flopping frequency between and is just .

Using this coupling constant, we can also construct an effective cooperatively parameter for the two-photon transition using , where


is the decay rate for an atom in to induced by the dressing laser, calculated for large detunings. Substituting Eqns. 33 and 34 into the above expression for , one finds that the two-photon cooperatively parameter and the one-photon cooperatively parameter (Eqn. 12) are identical . This is explained by the geometric interpretation of , a ratio which is determined by the fractional spatial solid angle subtended by the cavity mode and the enhancement provided by the cavity finesse which enters through the value of Tanji-Suzuki et al. (2011).

The adiabatic elimination yields the two-photon differential ac Stark shift of the frequency difference between and