Intensity fluctuations

9.3.4

Dynamics of a laser

In the steady state the number of photons created per second by the gain medium ofNatoms is equal to the number of photons annihilated per second by cavity decay. Far above threshold we have

N f1(∞) =2κn, (9.39)

wherenis the steady-state number of photons. We make the following assumptions

βn1 , γ2βnγ1, (9.40)

which define a range of pump rates far above the threshold for which the number of photons in the laser mode is proportional to the pump rateγ0. Under these assumptions we have

1 f1(∞) =T0= 1 γ0 + 1 γ2βn . (9.41)

We solve fornin (9.39) and find

n= γ0

γ2β

1−ε

ε , (9.42)

whereε, defined in (9.7), is the fraction of atoms that is at least needed for the laser to operate. From (9.42) we find that in general the assumptions (9.40) are satisfied for the range of pump rates for which we haveγ2γ0/εγ1.

We calculate the average cycle time of an atom far above the threshold, which is equal to the average duration time of the gain trajectory T0 given in (9.37). Under the present

assumptions and using the steady-state value (9.42) forn, we find

T0=

1 γ0(1−ε)

. (9.43)

We see that the cycle time increases whenεis increased. This is because for a fixed pump rate the number of photons in the laser mode decreases ifεincreases. Then the Rabi frequency becomes smaller and it takes more time for the atoms to make the 2-1 transition. Each cycle results in the gain of one photon in the laser mode. Thus ifεincreases and thereby the cycle time, the atoms become less productive in creating photons.

9.3.5

Intensity fluctuations

The number of atoms in the gain medium is a macroscopically large number. If we assume that there is no correlation between the atoms, the creation of photons in the laser mode by the gain medium as a whole, is a random process. The emission rate at timetis then given

byN f1(∞)which is time dependent because it is a function ofn(t)as we see in (9.41). So in

the rate equation (9.1) fornwe substituteGat(n) =N f1(∞)and we obtain

˙ n=−2κn+N 1 γ0 + 1 γ2βn −1 +fn. (9.44)

As discussed in Section 9.2 we linearise around the steady state. From (9.41) we find

f1(∞) = 1 γ0 + 1 γ2βn ₋1 +2κε N ∆n+O ∆n 2 _{, (9.45)}

and Eq. (9.44) gives for the fluctuations∆n(t)in the photon number

∆n˙=−2κ(1−ε)∆n+fn, (9.46) with the solution

∆n(t) = Z t −∞dt 0_{exp−2κ(1−ε) t−t}0_f n t0 . (9.47)

According to (9.13) we have∆I(t) =2κ∆n(t) +fvac(t). The noise source fnis related to the uncorrelated noise sources in (9.8). The corresponding diffusion coefficients are given in (9.10). Because of the assumptions (9.40), the decay rateγ1of the lower lasing level is

much larger than the other rates in the model. For the evaluation of the diffusion coefficients it is, therefore, justified to neglect the number of atoms in level 1, and so thatN1=0 and

N2=D=Nthr. We use the Whiener-Khintchine theorem [99], that is

D e

∆I(ω)∆Ie ω0∗E=δ ω−ω0lim T→∞

Z ∞

−∞

dτexp(−iωτ)h∆I(T+τ)∆I(T)i , (9.48) and obtain the following expression for the spectrum of intensity fluctuations as defined in (9.11):

V(ω) =1+ 8κ

2_ε

4κ2_(1−ε)2_+ω2. (9.49)

For the spectrum at frequencyω=0 we have

V(0) =1+ 2ε

(1−ε)2 . (9.50)

The valueV(0) =1 corresponds to shot noise in our definition (9.11). We see that forε=0, which is the case of no depletion of the ground state at all, we have shot noise. If ε is increased, the fluctuations rise above shot noise.

We can see this also from a qualitative argument. The average cycle time of the atom is

T0=1/f1(∞). If we calculate the average fraction of the total cycle time that the atom spends

in the ground state we find, using (9.43),

T00

T0

= 1

γ0T0

9. Quantum-trajectory description of laser noise with pump depletion

We see that for smallεthe atom is in the ground state for most of the time. In that case the cycle time is not sensitive to fluctuations in the photon number. Ifεis increased, the atom spends less time in the ground state and a fluctuation in the photon number will cause more fluctuations in the cycle time.

In the Appendix the MandelQparameter [100] is calculated for the emission of a single atom in the gain medium that follows only the gain trajectory. It is defined by Q+1= ∆m2_{/hmi, wheremis the number of photon emissions. We haveQ=0 for a Poisson process.}

Under the assumptions (9.40) and using (9.42) it is found that

Q=−2ε(1−ε). (9.52)

We see that forε=0 we haveQ=0 and the emission of a single atom is Poissonian. This is expected from the fact that forε=0 the atom spends (almost) all its time in the ground state and thus the statistics of its emission are determined by the statistics of the pump, which are Poissonian. Forε=0 the output fluctuations are at the shot-noise level. Ifεis increased, the value ofQdecreases until a minimum value of−1

2is reached forε= 1

2. We can understand

this from the fact that forε=1

2 we haveT00=T21, so that the cycle contains two coherent

periods with the same average duration. According to Ritsch and Zoller [89] this regular recycling of the atom is accompanied by anti-bunching and sub-Poissonian emission. It is surprising that for non-zeroε the emission of a single atom is sub-Poissonian, while the output fluctuations are above shot noise.

9.4

Semi-classical rate equations