We model a laser by a laser mode in a cavity that is resonant with the lasing transition of the atoms in the gain medium. We use three-level atoms in theΛconfiguration, as depicted in Fig. 9.1. The lasing transition 2-1 is coherently driven by the laser mode. Spontaneous emission occurs from state 2 to state 1, and from state 1 to state 0, and the corresponding rates areγ2andγ1. The lower state 0 is incoherently pumped to the upper level 2 at the rate
γ0. For the number of photons in the laser mode we use the following rate equation
˙
n=−2κn+Gat(n) +fn, (9.1)
whereκis the cavity decay rate. The functionGat(n)models the effect the atoms in the gain medium have on the number of photons in the laser mode. It can be obtained from a detailed description of the interaction of the atoms in the gain medium with the photons in the laser mode. Becausenis not a continuous but an integer number, the creation and annihilation of photons in the cavity introduces noise, which is accounted for by the noise source fn.
The evolution of the density matrix of a three-level atom coherently coupled to the laser mode is described by the optical Bloch equations. We have
d dtρ22=−γ2ρ22+γ0ρ00+ i 2Ω(ρ12−ρ21), d dtρ11=−γ1ρ11+γ2ρ22+ i 2Ω(ρ21−ρ12), d dtρ21=−γ⊥ρ21+ i 2Ω(ρ11−ρ22) = d dtρ ∗ 12, d dtρ00=− d dtρ22− d dtρ11. (9.2)
We neglect collisional dephasing, so that the optical coherence on the lasing transition decays at the transverse rate
γ⊥=1
9. Quantum-trajectory description of laser noise with pump depletion
No optical coherence involving the state 0 is created. This scheme can be expected to give rise to laser action only when the lower state of the lasing transition decays much faster than the upper state, so thatγ1γ2. The coupling of the lasing transition to the laser mode is
modeled by an effective Rabi frequencyΩ. Adiabatic elimination of the optical coherence ρ21shows that the effective rate of stimulated emission is
γst=
Ω2
2γ⊥ . (9.4)
This rate is related to the average photon numbernin the laser mode by the equalityγst=
γ2βn, withβ being the fraction of spontaneously emitted photons on the laser transition that
go into the laser mode. This connects the Rabi frequency to the properties of the laser cavity.
9.2.2
The fraction of atoms needed for lasing
In a laser the stimulated emission rate by the whole gain medium is equal toγ2βnN2, where
N2is the number of atoms in level 2. The absorption rate isγ2βnN1, whereN1is the number
of atoms in level 1. The decay rate of photons from the cavity is 2κn. In the steady state above threshold we find
γ2βn N2−N1
=2κn, (9.5)
where the quantities with a bar refer to the steady-state values. For the inversion we find
D=N2−N1=
2κ γ2β
=Nthr. (9.6)
The number of atoms that is at least needed for the laser to operate is then equal toNthr.We defineε as the fraction of the total number of atoms that is at least needed for the laser to operate, that is
ε=Nthr
N , (9.7)
whereNis the total number of atoms in the gain medium.
In the laser we distinguish a limited number of uncorrelated sources of noise in the photon number. We use fst for stimulated emission, fabs for absorption and fvacfor the fluctuations related to the emission of photons from the cavity. The relation between the noise term fnin Eq. (9.1) and the uncorrelated noise terms is given by
fn=fst−fabs−fvac, (9.8) where the signs reflect the gain and loss nature of the noise process for the variable involved. We assume that the noise sources are Gaussian andδ-correlated random variables with zero average. For the correlations between the noise sources we have
fi(t)fj t0 =Di jδ t−t0 , (9.9)
whereDi j are the diffusion coefficients. Since the noise sources are uncorrelated, the only non-zero diffusion coefficients are the diagonal ones, which we indicate asDst,DabsandDvac. According to Lax [91] these diffusion terms are equal to the corresponding rate, so that
9.2.3
Spectrum of intensity fluctuations
We are interested in the spectrumV(ω)of the intensity fluctuations at the frequencyω=0. The spectrumV(ω)is defined as follows
D e ∆I(ω)∆Ie ω0∗E=2κnV(ω)δ ω−ω0, (9.11) where e ∆I(ω) =√1 2π Z ∞ −∞ dtexp(−iωt)∆I(t) (9.12)
is the Fourier transform of the fluctuations∆I(t)of the output intensity of the laser around the steady-state valueI=2κn. To find∆I(t)we linearise Eq. (9.1) around the steady state by substitutingn(t) =n+∆n(t)and neglecting terms that are second order in∆nor higher. The next step is to relate the internal fluctuations in the photon number to the fluctuations in the output intensity. Therefore we use the input-output formalism of Gardiner and Collett [92]. In their paper they use the standard model of a system coupled to a heat bath. They derive a boundary condition that relates the input and the output to the internal modes of the system. In our case the input to the system is the vacuum state and the output is the intensity of the laser. We find
∆I(t) =2κ∆n(t) +fvac(t). (9.13)
We study the dependence of the intensity fluctuations on the parameterεby using, respec- tively, a microscopic and a macroscopic approach to describe the effect of the gain medium on the number of photons in the laser mode. We determine the functionGat(n)that accounts for the gain medium in the rate equation forn Eq. (9.1) and use the approach given above to find an expression for the spectrum of the intensity fluctuations at frequencyω=0 as a function ofε. In both cases the optical Bloch equations (9.2) are the starting point.