## Abstract

In this tutorial, the physical origins and mathematical analyses of laser linewidths are reviewed. The semi-classical model is based on an equation for the light-mode amplitude that includes random source terms, one term for each process that affects the amplitude (stimulated and spontaneous emission, stimulated absorption, and facet and material loss). Although the source terms are classical, their assigned strengths are consistent with the laws of quantum optics. Analysis of this equation shows that the laser linewidth is proportional to the sum of the (positive) source strengths for all gain and loss processes. Three-level and semiconductor lasers have broader linewidths than comparable four-level lasers, because stimulated absorption and the stimulated emission that compensates it both contribute to the linewidth.

© 2021 Optical Society of America

## 1. INTRODUCTION

Masing was demonstrated in 1954 [1] and lasing was demonstrated in 1960 [2]. Since then, lasers have become ubiquitous in science, engineering, and technology [3–6]. One aspect of lasers that sets them apart from other sources of light is their ability to emit waves with narrow frequency spectra. Many applications depend on this property. Hence, it is important to identify the processes that are responsible for linewidth and model them accurately.

Gordon *et al.* [7] modeled a maser that was driven by a beam of ammonia molecules. By using a classical power-balance argument, they showed that the maser linewidth

For microwaves, $kT \gg \hbar \omega$, whereas for light waves, $kT \ll \hbar \omega$. Schawlow and Townes [8] stated that one can apply formula (1) to lasers, provided that one replaces $kT$ by $\hbar \omega$, which is the spectral power density that corresponds to one photon per mode. This replacement leads to the Schawlow–Townes (ST) formula

which is a cornerstone of laser physics.Many papers on laser physics were written in the 1960s, by many different researchers. Some papers were based on quantum Langevin equations (stochastic differential equations for the amplitude operator), whereas others were based on density-operator equations. The calculations described in these papers are all complicated. We cite papers by Haken [9], Lax [10], and Scully [11], which are representative of this effort and whose results are consistent. The Haken–Lax–Scully (HLS) formula for the above-threshold linewidth of a four-level laser is

where $\Delta \omega$ is the smaller of the emission and cavity bandwidths ($1/\Delta \omega = 1/\Delta {\omega _e} + 1/\Delta {\omega _c}$). Notice that formula (3) differs from formula (2) by a factor of two. Measurements made by Manes [12] were consistent with a generalization of formula (3) that includes the effects of inhomogeneous broadening.The linewidth matter was settled until 1981, when Fleming [13] measured above-threshold linewidths of semiconductor lasers, which were an order-of-magnitude larger than those predicted by the HLS formula. Henry [14] made a first-principles analysis of spontaneous emission and showed that the HLS linewidth is increased by the factor $1 + {\epsilon ^2}$, where $\epsilon = \chi _r^{(3)}/\chi _{\imath}^{(3)}$ is the ratio of the real and imaginary parts of the third-order susceptibility. (The same enhancement factor had been predicted earlier by Haug [15].) Because the third-order response of the material is causal, there cannot be an imaginary response (gain or loss) without a related real response (frequency shifting). This result is called the Kramers–Kronig relation [16]. Shortly thereafter, several theoretical papers were published that confirmed the linewidth-enhancement factor and calculated other useful quantities, such as the power and phase spectra [17–22]. The calculations described in these papers are also complicated. Nonetheless, their predictions were verified by experiments [23–28].

In this tutorial, the physical origins and mathematical analyses of laser linewidths are reviewed. The semi-classical model described in Section 2 is based on a Langevin equation for the light-mode amplitude [29,30]. This equation includes random source terms, one term for each process that affects the amplitude (stimulated and spontaneous emission, stimulated absorption, and facet and material loss). Although the source terms are classical, their assigned strengths are consistent with the laws of quantum optics. Because the source terms vary randomly with time, so also does the driven amplitude. It is characterized by its two-time correlation and frequency spectrum. In Section 3, the amplitude equation is solved in the below-threshold regime, in which the amplitude is completely random. The solution is used to derive formulas for the temporal correlation and frequency spectrum of the complex amplitude. In Section 4, a simple generalization of the amplitude equation, which is called a van der Pol (vdP) equation [31,32], is solved in the above-threshold regime, in which the amplitude has both deterministic and random components. The solution is used to derive formulas for the temporal correlations and frequency spectra of the real and imaginary parts of the amplitude fluctuations. In Section 5, general formulas are derived for the below- and above-threshold linewidths, which are proportional to the sum of the source strengths for all gain and loss processes. The limits of these formulas for three- and four-level lasers are discussed. In Section 6, the main results of the tutorial are summarized. The tutorial also contains useful appendices on coherent and thermal states, the quantum fluctuations induced by gain and loss, the origin of the vdP equation, and the integral of colored noise. It is written at a level that is suitable for last-year undergraduate students or first-year graduate students.

## 2. AMPLITUDE EQUATIONS

Our semi-classical model of laser evolution is based on the stochastic amplitude equation

where $A$ is the complex wave amplitude, and $\alpha$ and $\beta$ are the complex gain and loss rates, respectively. The squared amplitude $|A{|^2}$ is the number of photons in the cavity. The parameter $\alpha$ represents stimulated emission, which depends on the number of upper-level (carrier) electrons, whereas $\beta = {\beta _a} + {\beta _f} + {\beta _m}$ represents stimulated absorption, which depends on the number of lower-level (valence) electrons, and facet and material loss, which do not depend on the electron numbers. We made the simplifying assumption that $\alpha$ and ${\beta _a}$ are constants, which requires the emission and absorption processes to have broad frequency bandwiths. The rate-equation model of a laser involves the electron and photon numbers, and the coefficients ${\alpha _r}$ and ${\beta _r}$, where the subscript $r$ denotes a real part. However, because gain and loss are causal processes, the Kramers–Kronig relation requires the amplitude gain and loss coefficients to have imaginary parts [16].The (Langevin) source function ${R_a}(t)$ mimics the effects of quantum fluctuations (including spontaneous emission). It is a random function of time, with the properties

Because the source functions vary randomly with time, so also does the driven amplitude, which is characterized by its moments in the time and frequency domains. The first time-domain moment $\langle A(t)\rangle$ is called the mean and the second moment $\langle {A^*}(t)A(t^\prime)\rangle$ is called the two-time correlation. Because noise is always present, the driven amplitude is not an integrable function of time. To avoid singularities in the theory, we use the finite-time Fourier transforms

In a previous tutorial [33], we discussed finite-time transforms and applied them to studies of simple harmonic oscillators driven by noise. Many of the mathematical results and physical insights described therein are also relevant to lasers. In this tutorial, $A(t)$ is dimensionless, so $A(\omega)$ has units of time. However, in some applications $|A(t{)|^2}$ has units of power, in which case $|A(\omega {)|^2}$ has units of energy multiplied by time (divided by frequency). For this reason, $|A(\omega {)|^2}$ is called the spectral energy density. We will use the descriptive terms power and energy whenever it is helpful to do so. It follows from Eq. (7) that the second frequency-domain moment## 3. BELOW-THRESHOLD LASER

First, consider a well-below-threshold laser (${\alpha _r} \ll {\beta _r}$). In this regime, the number of photons is too small to affect the numbers of upper- and lower-level electrons, which are determined by the pump power (electron current) and the relevant electron-decay processes. Let $A(t) = B(t){e^{i({\alpha _i} - {\beta _i})t/2}}$, where the subscript $i$ denotes an imaginary part. Then the transformed amplitude satisfies the equation

where ${\nu _r} = ({\beta _r} - {\alpha _r})/2$ is the net-damping rate and ${R_b}(t) = {R_a}(t){e^{i({\beta _i} - {\alpha _i})t/2}}$ is the transformed source function. It is easy to verify that ${R_b}$ also has properties (5), so the phase factor can be omitted (and ${\sigma _b} = {\sigma _a}$). Equation (10) has the same form as the equation for a strongly damped oscillator driven by noise, which was studied in Section 3 of [33].The solution of Eq. (10) is

where the subscript $r$ was omitted. Typically, the growth of the amplitude is initiated by noise, in which case $B(0) = 0$. One could mimic the fluctuations associated with an initial vacuum state by setting $\langle B(0)\rangle = 0$ and $\langle |B{(0)|^2}\rangle = 1/2$ (Appendix A), but this contribution to the amplitude is damped and does not affect the stochastic steady state, so we will omit it. The remaining (driven) part of solution (11) has the propertiesOne obtains a formula for the spectrum by combining Eqs. (9) and (15). The relevant integrals were done in Section 2 of [33], and the result is

Solution (11) shows that the amplitude is the sum of impulses, which have Gaussian statistics, multiplied by deterministic damping factors. Hence, the amplitude also has Gaussian statistics, and its probability distribution is specified completely by Eqs. (12) and (13). Let $x$ and $y$ denote the real and imaginary parts of $B$, respectively, and let $\sigma = {\sigma _b}/4\nu$. Then the probability distribution function

Now let $F(x,y)$ be an arbitrary function of its arguments. Then the average It is easy to verfy that $\langle {x^2}\rangle = \langle {y^2}\rangle = \sigma$, so the number mean $\langle {x^2} + {y^2}\rangle = 2\sigma = {\sigma _b}/2\nu$.It is instructive to consider the distribution in a different coordinate system. Let $u$ and $v$ be arbitrary functions of $x$ and $y$. Then

where $J$ is the Jacobian of the transformation. Hence, where the functions on the right side are written in terms of $u$ and $v$. For polar coordinates, $u = n = {x^2} + {y^2}$, $v = \phi = \mathop {\tan}\nolimits^{- 1} (y/x)$, and $J = 2$. It follows from Eqs. (20) and (23) that Distribution (24) is independent of phase, so the number distribution where ${\sigma _n} = 2\sigma$. A Gaussian amplitude distribution is equivalent to an exponential number distribution and a uniform phase distribution. It follows from Eq. (25) that the number mean $\langle n\rangle = {\sigma _n}$ and the number variance $\langle \delta {n^2}\rangle = \langle {n^2}\rangle - {\langle n\rangle ^2} = \sigma _n^2$. A below-threshold laser emits light for which the number deviation is as large as the number mean. This behavior is not useful. It contrasts markedly with the behavior of an above-threshold laser, for which the deviation is much smaller than the mean.## 4. ABOVE-THRESHOLD LASER

As one increases the pump power, the number of upper-level electrons increases. At some point (threshold), the gain ${\rm Re}({\alpha _e})$ equals the total loss ${\rm Re}({\beta _a}) + {\beta _f} + {\beta _m}$. A further increase in pump power causes photons to be generated, first by spontaneous emission and then by stimulated emission. Because the generation of a photon coincides with the lowering of an electron, the number of upper-level electrons decreases. There follows a sequence of (diminishing) relaxation oscillations in the electron and photon numbers. After some time, the laser reaches a self-consistent equilibrium, in which

In the vicinity of equilibrium, the net loss

where ${\nu _{0i}} = {\nu _i}(|{A_0}{|^2})$, and ${\nu _0^\prime} = d\nu /d|A{|^2}$, evaluated at ${A_0}$. Equation (27) is based on the assumption that the active medium responds instantaneously to changes in photon number. By combining it with Eq. (4), one finds that where ${R_a}$ has properties (5), which depend on the equilibrium gain and loss rates. The first term on the right side of Eq. (28) produces a frequency shift, which is of secondary importance. One can eliminate it by making the transformation $A(t) = B(t){e^{- i{\nu _{0i}}t}}$, in which case, where the source function ${R_b}(t) = {R_a}(t){e^{i{\nu _{0i}}t}}$ also has properties (5). In the optics literature, this (first-order) equation is called the vdP equation. Its relation to the (second-order) electrical–mechanical vdP equation is described in Appendix C.In equilibrium and in the absence of noise, $B(t) = {B_0}$ is a constant (which one can choose to be real and positive). To study small perturbations of this equilibrium in the presence of noise, let $B(t) = {B_0} + {B_1}(t)$, where $|{B_1}| \ll {B_0}$. Then, by making this substitution in Eq. (29), and splitting the resulting equation into real and imaginary parts, one obtains the linearized equations

where ${\mu _r} = 2\nu_{0r}^\prime |{B_0}{|^2}$ and ${\mu _i} = 2\nu_{0i}^\prime |{B_0}{|^2}$. The source terms ${R_r}(t)$ and ${R_i}(t)$ are real, and have the propertiesSecond, consider the imaginary amplitude, which is the integral of a combination of white noise (${R_i}$) and colored noise (${\mu _i}{B_{1r}}$). The mean amplitude $\langle {B_{1i}}(t)\rangle = 0$. The white-noise contribution to the correlation is

The colored-noise contribution is determined exactly in Appendix D. One can estimate it by observing that the right side of Eq. (35) can be rewritten as $({\sigma _r}/\mu _r^2){\delta _e}(\tau)$, where, for times that are longer than the correlation time, ${\delta _e}(\tau) = ({\mu _r}/2){e^{- {\mu _r}|\tau |}}$ is an effective $\delta$-function. (It has a narrow peak and an integral of 1.) For such times, it follows from Eqs. (34) and (36) that the colored-noise contribution is

One obtains formulas for the spectra of the real and imaginary amplitudes by combining Eq. (9) with Eqs. (35) and (38), respectively. The first calculation was described briefly in the previous section. It follows from Eq. (17) that the real spectrum

Although the below- and above-threshold (real) spectra are both Lorentzian, their strengths and widths differ significantly. The second calculation was done in Section 5 of [33]. The result is the imaginary spectrumPrevious researchers rewrote the vdP equation (or similar equations) in terms of the number $N = |B{|^2}$ and phase $\phi = \mathop {\tan}\nolimits^{- 1} ({B_i}/{B_r})$. However, this change of variables is nonlinear. It converts a pair of linear equations with additive noise to a pair of nonlinear equations with multiplicative noise. For such equations, one must follow the rules of stochastic calculus carefully [29]. In the context of this tutorial, we thought that stochastic calculus would be an unnecessary distraction, so we chose to continue working with the real and imaginary amplitudes. Previous researchers derived number spectra that are Lorentzian [19–21] and a phase spectrum that decreases as $1/{\omega ^2}$ [22]. Equations (39) and (40) are consistent with these results.

As long as the amplitude perturbations remain smaller than the equilibrium amplitude, the probability distribution function is a two-dimensional Gaussian. The real (in-phase) part of the amplitude is centered on ${B_0}$ and has the variance ${\sigma _r}/2{\mu _r}$, which is constant, whereas the imaginary (out-of-phase) part is centered on zero and has the variance ${\sigma _i}(1 + {\epsilon ^2})t$, which increases linearly with time. This distribution function resembles that of an amplitude-squeezed (phase-stretched) state, which is illustrated in [34].

The original amplitude $A$ is a phase-shifted version of the transformed amplitude $B$, so the original correlations are phase-shifted versions of correlations (35) and (38), and the original spectra are frequency-shifted versions of spectra (39) and (40). The center of the original amplitude distribution rotates.

## 5. LASER LINEWIDTHS

The laser linewidth $\delta \omega$ is defined to be the FWHM of the amplitude spectrum. In Section 3, it was shown that in the well-below-threshold regime, $\delta \omega = 2\nu$ is the (number) damping rate $\beta - \alpha$ [Eq. (17)], where $\alpha \ll \beta$. It was also shown that the photon number $N = \sigma /2\nu$, where the source strength $\sigma = (\alpha + \beta)/2$ [Eq. (13)]. Thus, the linewidth equals the damping rate, and the number is proportional to the source strength and inversely proportional to the damping rate. As the pump power increases, $\alpha$ increases and ${\beta _a}$ decreases, so $\nu$ decreases. This behavior is called line narrowing. As $\nu$ decreases, $N$ increases concomitantly. At some point, the approximation that $\alpha$ and ${\beta _a}$ are independent of $N$ ceases to be accurate, but by this point, the trends in linewidth and number are clear. For us, it is natural to characterize a below-threshold laser in terms of the damping rate and source strength. However, previous authors characterized it in terms of the output power. By using the facts that $2\nu = \sigma /N$ and $P = N\hbar \omega {\beta _f}$, one obtains the below-threshold linewidth formula

Equation (41) includes the effects of emission, absorption, and facet and material loss. The linewidth narrows as the power increases. Well below threshold, $\alpha \ll \beta$. For a four-level laser (without absorption or material loss), For a three-level laser (with absorption, but without material loss), Thus, three-level lasers have broader linewidths than four-level lasers (with the same ${\beta _f}$).In the above-threshold regime, the complex amplitude ${B_0} + {B_{1r}}(t) + i{B_{1i}}(t)$ is the sum of constant and fluctuating terms. In Section 4, it was shown that the real amplitude perturbation is damped and bounded, whereas the imaginary perturbation is undamped and unbounded. Under these circumstances, it is reasonable to model complex amplitude fluctuations as phase fluctuations, where the phase $\phi (t) \approx {B_{1i}}(t)/{B_0}$. It follows from Eq. (38) that the phase mean $\langle \phi (t)\rangle = 0$, and the phase variance

The amplitude correlation $\langle {B^*}(0)B(t)\rangle = |{B_0}{|^2}\langle {e^{i\phi (t)}}\rangle$. The phase is the time integral of impulses that have Gaussian probability distributions [Eq. (34)], so at any time, the phase also has a Gaussian distribution. For such a distribution,By converting from photon number to output power (as described above), one obtains the above-threshold linewidth formula

Equation (42) also includes the effects of emission, absorption, and facet and material loss. In equilibrium, $\alpha = \beta$, so addition (emission) and subtraction (absorption, and facet and material loss) contribute equally to the linewidth.The above-threshold formula (48) differs from the below-threshold formula (41) in three significant ways. First (as mentioned above), gain and loss contribute equally to the linewidth. Second, the linewidth is proportional to $1 + {\epsilon ^2}$, where $\epsilon = {\nu _{0i}}^\prime /{\nu _{0r}}^\prime = - \chi _r^{(3)}/\chi _{\imath}^{(3)}$ is the enhancement factor. For semiconductor lasers, this factor is signficant ($|\epsilon | \approx 5$). In the remainder of this section, the enhancement factor will be omitted (because it appears in all the formulas and previous researchers agree upon it). Third, the denominator in Eq. (48) is twice as large as the denominator in Eq. (41). The below- and above-threshold calculations are different, so there is no simple relation between them (in the context of a phenomenological vdP equation). However, the fact that only imaginary amplitude perturbations affect the above-threshold linewidth does contribute a factor of 1/2 to its final value.

For a four-level laser (without absorption or material loss), the linewidth

Emission and facet loss contribute equally to the linewidth, so $(\alpha + {\beta _f})/4 = {\beta _f}/2$. Hence, the above-threshold formula is identical to the well-below-threshold formula. For a three-level laser (with absorption, but without material loss), $\delta \omega = \hbar \omega {\beta _f}({\beta _a} + {\beta _f})/2P$. Once again, three-level lasers have broader linewidths than comparable four-level lasers, and the above-threshold formula is identical to the well-below-threshold formula. It follows from the equilibrium condition that ${\beta _a} + {\beta _f} = \alpha$ and $\alpha /{\beta _f} = \alpha /(\alpha - {\beta _a})$. By using these facts, one can rewrite the linewidth formula as By using the facts that $\alpha = g{N_2}$ and ${\beta _a} = g{N_1}$, where $g$ is the transition coefficient, and ${N_1}$ and ${N_2}$ are the lower- and upper-level electron numbers, respectively, one can rewrite the first factor in Eq. (50) as ${N_2}/({N_2} - {N_1})$. This factor is called the spontaneous noise factor (${n_{{\rm sp}}}$).Without the enhancement factor, formula (49) is equivalent to the short HLS formula, which also includes the effects of gain and loss fluctuations. With the enhancement factor, it has the same form as the Henry formula. However, Henry chose to include only gain fluctuations. To reproduce the spontaneous contribution to the photon-generation rate $\alpha (N + 1)$, he used the semi-classical strength ${\sigma _\alpha} = \alpha$ (which is twice our strength). This choice leads to the incremental quadrature variance $\langle |\delta A{|^2} + |\delta A{|^2}\rangle /2$, which is proportional to $2{\sigma _\alpha}/2 = \alpha$. When one does the same calculation quantum mechanically (Appendix B), one obtains the incremental variance $\langle \delta {a^\dagger}\delta a + \delta a\delta {a^\dagger}\rangle /2$, which is proportional to $\alpha /2$ (because only the second term contributes). The overestimation of quadrature gain fluctuations compensated the omission of loss fluctuations.

The preceding linewidth formulas are all based on the quantum fluctuations associated with gain (${\sigma _\alpha} = \alpha /2$) and loss (${\sigma _\beta} = \beta /2$). One should also consider thermal fluctuations. The properties of a thermal quantum state are described in Appendices A and B. One can mimic the effects of thermal fluctuations by adding to Eq. (4) a source of strength $\beta \langle {n_r}\rangle$, where $\langle {n_r}\rangle = 1/({e^{\hbar \omega /kT}} - 1)$ is the expected number of reservoir photons. In the absence of gain, this source allows the cavity mode to reach equilibrium with the reservoir, as required by the fluctuation–dissipation theorem. Thus, one can quantify the effects of thermal fluctuations on linewidth by making the substitution $\beta /2 \to \beta (1/2 + \langle {n_r}\rangle) = \beta (1 + 2\langle {n_r}\rangle)/2$ in Eq. (48). By including the thermal source associated with facet loss, which is the loss of a passive cavity (without the active medium), one obtains the generalized linewidth formula

For light waves, $\langle n\rangle \approx {e^{- \hbar \omega /kT}} \ll 1$, in which case formula (51) reduces to formulas (49) and (50) in the appropriate limits. In contrast, for microwaves, $\langle {n_r}\rangle \approx kT/\hbar \omega \gg 1$, in which case, formula (51) reduces to

Formula (52) is similar to formula (1), with the emission bandwidth $\Delta \omega$ replaced by the cavity bandwidth ${\beta _f}$.## 6. SUMMARY

In this tutorial, the physical origins and mathematical analyses of laser linewidths were reviewed. Our semi-classical model is based on an equation for the light-mode amplitude [Eq. (4)] that includes random source terms, one term for each process that affects the amplitude (stimulated and spontaneous emission, stimulated absorption, and facet and material loss). Although the source terms are classical, their assigned strengths [Eqs. (5)] are consistent with the laws of quantum optics [Eqs. (B20) and (B23)]. Analyses of this equation show that the below- and above-threshold laser linewidths [Eqs. (41) and (48)] are proportional to the sum of the (positive) source strengths for all gain and loss processes, and inversely proportional to the output power. Three-level and semiconductor lasers have broader linewidths than comparable four-level lasers, because stimulated absorption and the stimulated emission that compensates for it both contribute to the linewidths. In the below-threshold regime, the complex amplitude spectrum is Lorentzian [Eq. (17)]. In the above-threshold regime, the real amplitude (power) spectrum is Lorentzian [Eq. (39)], whereas the imaginary amplitude (phase) spectrum is Brownian [Eq. (40)]. Our above-threshold linewidth formula [Eq. (48)] is consistent with the Haken–Lax–Scully formula [9–11], and includes the linewidth-enhancement factor of Haug [15] and Henry [14].

## APPENDIX A: COHERENT AND THERMAL STATES

In this appendix, some properties of coherent and thermal states are derived and used to gauge the accuracy of the semi-classical model of amplitude fluctuations. Let $a$ be a mode-amplitude operator, which satisfies the boson commutation relations

where $[x,y] = xy - yx$ is a commutator and $\dagger$ denotes a hermitian conjugate. The quadrature operator $q(\phi) = ({a^\dagger}{e^{{i\phi}}} + a{e^{- i\phi}}{)/2^{1/2}}$, where $\phi$ is the phase of the local oscillator, and the number operator $n = {a^\dagger}a$.A coherent state is defined by the eigenvalue equation

where $|\alpha \rangle$ is the state vector and $\alpha$ is a complex number. It follows from Eq. (A2) that the expectation value $\langle \alpha |a|\alpha \rangle = \langle a\rangle = \alpha$, so $\alpha$ is the amplitude mean. The quadrature meanBy combining Eqs. (A1) and (A2), one finds that the second quadrature moment

A coherent state is pure, so its density matrix $\rho = |\alpha \rangle \langle \alpha |$. Let $o$ be an arbitrary operator. Then its expected value is the trace ${\rm tr}(\rho o)$. By using the identity ${\rm tr}(|x\rangle \langle y|) = \langle y|x\rangle$, one finds that ${\rm tr}(\rho o) = \langle \alpha |o|\alpha \rangle$. For a pure state, the definitions of expectation value are equivalent. Now let $\{|{\psi _n}\rangle \}$ be a complete set of state vectors, which are normalized, but are not necessarily orthogonal. Then a mixed state is specified by the density matrix $\rho = {\sum _n}{p_n}|{\psi _n}\rangle \langle {\psi _n}|$, where the total probability ${\sum _n}{p_n} = 1$. By using the trace identity, one finds that ${\rm tr}(\rho o) = {\sum _n}{p_n}\langle {\psi _n}|o|{\psi _n}\rangle$. The expectation value is the weighted sum of the expectation values associated with the basis states.

A thermal state is defined by the density matrix

where $|n\rangle$ is a number state. The probability ${p_n} = (1 - x){x^n}$, where $x = {e^{- \hbar \omega /kT}}$ is the Boltzmann factor. This probability distribution is called the Planck distribution. By using the fact that ${\sum _n}{x^n} = 1/(1 - x)$, one can verify that the total probability is one. The number meanDensity matrix (A6) is diagonal in the number-state basis. Hence, operators that change diagonal elements to non-diagonal ones (which do not appear in traces) have expectation values of zero. It follows from this observation that the quadrature mean

In this case, the quadrature variance equals the second quadrature moment. It follows from the first line of Eq. (A4) that Notice that the quadrature mean and variance are both phase independent. The quadrature variance of a thermal state exceeds that of a coherent state. In [34], the number variance of a thermal state is shown to be ${\langle n\rangle ^2} + \langle n\rangle$, which also exceeds that of a coherent state.In the semi-classical model of amplitude fluctuations, one replaces the mode operator $a$ by the amplitude $A + \delta A$, where $A = \langle a\rangle$ is the mean amplitude and $\delta A$ is a Gaussian random variable with the properties

In Section 2, we assumed that the amplitude (quadrature) fluctuations have Gaussian statistics. For coherent and thermal states, the quadrature distributions are Gaussian, with variances of 1/2 and $\langle n\rangle + 1/2$, respectively [35,36], so our assumption was valid.

## APPENDIX B: QUANTUM FLUCTUATIONS

Consider the evolution of a light mode in the presence of gain and loss. These processes involve auxiliary modes, which interact with the light mode. One can avoid modeling the interactions explicitly by using the stochastic amplitude-operator equation

where $\alpha$ and $\beta$ are the gain and loss coefficients (rates), respectively. The noise operators ${s_\alpha}$ and ${s_\beta}$ are random operator functions of time, which model the effects of auxiliary-mode fluctuations [30,37]. Like their classical counterparts, they are independent, and their first and second squared moments are where $\langle \,\rangle$ denotes an ensemble average. Their mixed second moments areLet $\tau$ be a short time interval. Then the differential equation (B1) is equivalent to the difference equation

Suppose, temporarily, that $\beta = 0$. Then, by combining Eq. (B4) with its conjugate and retaining the squares of the noise terms, which are of order $\tau$, one finds that

One can deduce the significance of the $\sigma$ coefficients by considering examples. First, consider parametric amplification [37,38], in which signal and idler photons are produced in pairs. The generation of a signal photon can be stimulated by a signal photon or an idler photon, at a rate that is proportional to the number of signal and idler photons. It also can be stimulated by the quantum fluctuations of the idler, in which case the process is called spontaneous emission. If the idler is in a vacuum state, there are no idler photons, so only spontaneous emission can occur. However, if the idler is in a thermal state, idler photons are present that can stimulate signal-photon emission. According to the second of Eqs. (B12), the number of signal photons generated in the time $\tau$ is $\alpha ({\sigma _\alpha} + 1)\tau$. Hence, ${\sigma _\alpha}$ is the mean number of idler photons.

Second, consider the beam splitting model of loss [34,38]. Signal photons are converted into loss-mode photons at a rate that is proportional to the number of signal photons. Likewise, loss-mode photons are converted into signal photons at a rate that is proportional to the number of loss-mode photons. If the loss mode is in a vacuum state, there are no loss-mode photons, so no signal photons are produced. However, if the idler is in a thermal state, loss-mode photons are present and can be converted. According to the first of Eqs. (B12), the number of signal photons generated in the time $\tau$ is $\beta {\sigma _\beta}\tau$. Hence, ${\sigma _\beta}$ is the mean number of loss-mode photons.

Third, consider a signal mode that is in a thermal state initially (before the pump is turned on). If only loss were present, the number of signal photons would decrease. Equilibrium is maintained by the interaction of the signal mode with a reservoir of other modes [39]. Signal photons can be converted to reservoir photons, but so also can reservoir photons be converted to signal photons. This interaction allows the signal and reservoir to reach an equilibrium, in which the number of signal photons equals the number of reservoir photons at the signal frequency. One can model this process by setting ${\sigma _\beta} = \langle {n_r}\rangle = 1/({e^{\hbar \omega /kT}} - 1)$. Then, according to Eq. (13), $\langle n\rangle \to {\sigma _b}/\beta = {\sigma _\beta} = \langle {n_r}\rangle$. In this example, signal loss (dissipation) and noise-driven signal fluctuations are different aspects of the same interaction. This relationship is called the fluctuation–dissipation theorem.

With the terms in Eq. (B1) specified completely, one can derive equations for the amplitude and quadrature moments. It follows from Eqs. (B1) and (B2) that

where $\langle\, \rangle$ denotes an expectation value. Gain increases the amplitude mean, whereas loss decreases it. The quadrature operator $q(\phi) = ({a^\dagger}{e^{{i\phi}}} + a{e^{- i\phi}}{)/2^{1/2}}$, where $\phi$ is the phase of the local oscillator. By combining this definition with Eq. (B13), one obtains the quadrature-mean equation Gain and loss change the quadrature mean in ways that are insensitive to the local-oscillator phase.It follows from Eqs. (B8)–(B10) and (B12) that

In the semi-classical model of gain and loss, one replaces the mode operator $a$ by the mode amplitude $A$, which has both deterministic and random components. It follows from Eq. (4) that

Equation (B21) is consistent with Eq. (B13). The semi-classical quadrature $Q$ is defined in the same way as the quantum one. By combining this definition with Eq. (B21), one obtains the quadrature-mean equation Equation (B22) is consistent with Eq. (B14). The semi-classical derivation of the quadrature-variance equation is similar to the quantum derivation. The only differences are that the 1s in Eqs. (B16) and (B17) are absent, and the second and third terms on the right side of Eq. (B18) contribute equally. Hence,## APPENDIX C: VAN DER POL OSCILLATOR

The textbook vdP equation [32] is

where $X$ is the dependent variable (displacement) and $\epsilon \gt 0$ is a parameter (not the enhancement factor). Equation (C1) describes an oscillator with a linear restoring force, a linear negative-damping (driving) term, and a nonlinear damping term. The net-damping rate is negative if $|X| \lt 1$ and positive if $|X| \gt 1$. In this appendix, the relation between Eqs. (29) and (C1) is determined.Suppose that the damping terms are small, so that perturbation theory can be used to model their effects. Let $X = {X_0} + \epsilon {X_1}$ be the perturbed displacement and let ${t_0} = t$ and ${t_1} = \epsilon t$ be fast and slow time variables, respectively. These variables are treated as independent, in which case the total time derivative $d/dt = d/d{t_0} + \epsilon d/d{t_1}$, which one can abbreviate as $D_0 + \epsilon D_1$. By making these substitutions in Eq. (C1), and retaining terms of order 1 and $\epsilon $, one obtains the perturbed equation

By collecting terms of order $\epsilon $, one obtains the first-order equation

Equation (C5) describes a driven first-order displacement. If the terms on the right side have a resonant component (which is proportional to ${e^{- i{t_0}}}$), the displacement will grow without bound. This behavior is unphysical, so the resonant component of the right side must be zero. The resonant components of the first and second terms are $2i{D_1}{A_0}{e^{- i{t_0}}}$ and $-2i{A_0}{e^{- i{t_0}}}$, respectively. The third term,## APPENDIX D: INTEGRAL OF COLORED NOISE

In [33], we derived formulas for the temporal correlation and frequency spectrum of an undamped oscillator driven by white noise. Such an oscillator is governed by the amplitude equation

where the source function has properties (5). In this appendix, analogous formulas are derived for an undamped oscillator driven by colored noise, which has the correlation These formulas are relevant to the study of phase fluctuations, which are driven by a combination of white and colored noise [Eq. (31)]. Equation (D1) has the solution In the same way that Brownian motion is the integral of white noise, solution (D3) is the integral of colored noise.It follows from the first of Eqs. (5) and Eq. (D3) that the amplitude mean $\langle B(t)\rangle = 0$. By combining Eqs. (D2) and (D3), one finds that the amplitude variance

For $t^\prime \gt t$, the temporal correlation

By combining Eqs. (9) and (D5), one obtains the spectrum equation

The second integrand is a separable function of $t$ and $t^\prime $, so

The third integrand does not depend on $t^\prime $, so

The fifth integral is split into two parts, the first of which is

It is convenient to write the frequency spectrum in terms of the coefficient $\sigma = 2\kappa /\nu$, which, as explained after Eq. (D4), is the strength of the equivalent white-noise source. By multiplying the preceding integrals by $\kappa /\nu = \sigma /2$, one obtains the spectrum

## Funding

Natural Sciences and Engineering Research Council of Canada.

## Acknowledgment

TJS and ASH were supported by NSERC.

## Disclosures

The authors declare no conflicts of interest.

## Data Availability

No data were generated or analyzed in this research.

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