intuitive atomic angular momentum operators and optical Stokes operators.
The dipole operator and the classical field can be broken up into their rotating and counter- rotating components as
dˆ = PˆgdˆPˆe+ ˆPedˆPˆg
= dˆ−+ ˆd+ (5.6)
E = X
σ
Eσe−σexp[−iωt] + h.c.
= E++E−. (5.7)
The rotating wave approximation (RWA) is best seen after going into the rotating frame but the simple result just amounts to leaving out the two sum terms rotating at optical fre- quencies and keeping the two difference terms, which rotate at the much smaller detunings.
This leads to Hˆi = −1
~
dˆ+·E−+ h.c.
= −1
~ X
f,f0,σ
Eexp[iωt]−σPˆg(ˆd·e−σ) ˆPe+ h.c.
= −1
~ X
f,f0,σ
X
mf,mf0
Eexp[iωt]−σhf||d||fˆ 0ihf, mf|f0, mf0; 1, σi|f, mfihf0, mf0|+ h.c.
= −Ω
2 exp[iωt] X
f,f0,σ
−σAˆf,f0,σ+ h.c. (5.8)
Here we have defined the Rabi frequency as
Ω = 2Ehj = 1/2||d||jˆ 0= 3/2i/~ (5.9) wherej is the spin-angular momentum plus the orbital angular momentum as discussed in chapter 3. We have also defined the jump operators as
Aˆf,f0,σ = X
mf,mf0
hf, mf|f0, mf0; 1, σi hf||d||fˆ 0i
hj= 1/2||d||jˆ 0 = 3/2i|f, mfihf0, mf0|. (5.10) See reference [84] for finding the ratio of dipole moments in terms of six-j and three-j symbols. Another indispensable reference for Clebsch-Gordan coefficients and quantum angular momentum is [95]. In Matlab, we can conveniently create the numerical jump
operators using
[ ˆAf,f0,−1,Aˆf,f0,0,Aˆf,f0,+1] = murelf(f,f0,j,j0,i) (5.11)
withj= 1/2,j0 = 3/2, andi= 7/2. Note that we have the useful identities X
f,σ
Aˆ†f,f0,σAˆf,f0,σ = Pˆf0 (5.12) X
f,f0,σ
Aˆ†f,f0,σAˆf,f0,σ = Pˆe, (5.13)
which are used below for simplifying expressions.
5.1.2 Full Master Equation
The unconditional master equation is given by
˙ˆ
ρ = −i[ ˆH,ρ] + Γˆ X
f,f0,σ
D[ ˆAf,f0,σ] ˆρ (5.14)
= −i[ ˆH,ρ] + Γˆ X
f,f0,σ
Aˆf,f0,σρˆAˆ†f,f0,σ−Γ
2( ˆPeρˆ+ ˆρPˆe) (5.15) where we have used the superoperator definition
D[ˆc] ˆρ = ˆcρˆˆc†−(ˆc†ˆcρˆ+ ˆρˆc†c)/2ˆ (5.16) along with the convenient identity of equation (5.13). Here we have simply asserted without derivation what the spontaneous emission decay terms are that correspond to the coupling of the atom with all of the three-dimensional free-space vacuum modes, following the treatment of [96]. We leave for future work how to derive a conditional master equation that would provide our description for the atomic state if we were to measure some fraction of the scattered light. Still, such a treatment should become the same as the case we consider here under the limit that the measured fraction of scattered light goes to zero.
5.1.3 Rotating Frame
Now our goal is to remove the explicit time dependence from the Hamiltonian, so we move into a frame rotating at the probe frequencyω. This is done by transforming all operators according to
Xˆ →exp[iPˆeωt] ˆXexp[−iPˆeωt]. (5.17)
Notice that the only effect this has is to remove the time dependence from the Hamiltonian and simultaneously turn the excited energy levels into detunings via the term derived from the transformation of ˙ˆρ itself. The dynamics are still given by equation (5.15), but with the Hamiltonian now given by the following (without bothering to change notation)
Hˆ = HˆB+ ˆHg+ ˆHe+ ˆHi (5.18) HˆB = X
f,mf
µBgf
~ B·ˆf (5.19)
Hˆg = −∆3Pˆ3 (5.20)
Hˆe = −X
f0
∆f0Pˆf0 (5.21)
Hˆi = −Ω 2
X
f,f0,σ
−σAˆf,f0,σ+ h.c. (5.22)
(notice that we chose the f = 4 state to be the zero energy reference) with the detunings defined as
∆f0 = ω−(ωf0 −ω4) (5.23)
∆3 = ω4−ω3 = 2π·9.192631770 GHz. (5.24) The last detuning is exact because this quantity currently defines the unit of time.
5.1.4 Adiabatic Elimination of Excited Levels
There are two reasons for not wanting to integrate the above master equation numerically.
First, there are terms proportional to the detunings, which can be quite large, meaning that the numerical integration would have to be done with unreasonably small timesteps.
Second, the total Hilbert space is somewhat big with 48 levels as opposed to the 16 levels describing only the ground states.
Fortunately, with a far-off resonant probe beam, the procedure of adiabatic elimination [96, 97, 98, 52] allows us to eliminate the excited state levels in our description while retaining their effect on the physics. The approximation assumes that there is little population in the excited states, i.e., the saturation parameter
s= Ω2
2(∆2+ Γ2/4) 1 (5.25)
should be small for the detuning under consideration.
Many references exist that partially explain the process of adiabatic elimination, but most are incomplete, therefore we will attempt to be more forthcoming in our presentation.
The following procedure is by no means rigorous, but at least every step is spelled out. We proceed as follows, ignoring the Hamiltonian ˆHB until the end. First, we decompose ˆρinto four parts
ˆ
ρ = ( ˆPg+ ˆPe) ˆρ( ˆPg+ ˆPe) (5.26)
= PˆgρˆPˆg+ ˆPgρˆPˆe+ ˆPeρˆPˆg+ ˆPeρˆPˆe (5.27)
= ρˆgg+ ˆρge+ ˆρeg+ ˆρee. (5.28) Now we find ˙ˆρij for each term, by judicious insertion of the identity ˆPe+ ˆPg into the above master equation
ρ˙ˆgg = −i[ ˆHg,ρˆgg] +iˆρgeHˆiPˆg−iPˆgHˆiρˆeg+ Γ X
f,f0,σ
Aˆf,f0,σρˆeeAˆ†f,f0,σ (5.29)
˙ˆ
ρge = iˆρggHˆiPˆe+iˆρgeHˆe−iHˆgρˆge−iPˆgHˆiρˆee−Γ ˆρge/2 (5.30)
˙ˆ
ρee = −iPˆeHˆiρˆge+iˆρegHˆiPˆe−i[ ˆHe,ρˆee]−Γ ˆρee (5.31) where ˙ˆρge= ( ˙ˆρeg)†.
According to the weak-coupling approximation, s1, the order of magnitude of each of these parts is given byhρˆeei ∼s,hρˆgei ∼√
s, andhρˆggi ∼1. Using these approximations
in the decay terms of equations (5.29, 5.30, 5.31) (the last term of each equation) we get hρ˙ˆggi/hρˆggi ∼ −Γs (5.32) hρ˙ˆegi/hˆρegi ∼ −Γ/2 (5.33) hρ˙ˆeei/hρˆeei ∼ −Γ. (5.34) Thus, we see that the decay rate for the ground states ˆρgg is much smaller than the decay rate for the cross ˆρeg and excited ˆρee components by a factor ofs. In other words, the cross and excited components reach equilibrium on a timescale much faster than the ground state component. As a result, we can set ˙ˆρeg = 0 and ˙ˆρee = 0, solve for ˆρee in terms of ˆρeg, solve ˆρeg in terms of ˆρgg, and substitute these results into equation (5.29) to get the evolution of ˆρgg in terms of itself. We say that ˆρee “adiabatically follows” ˆρeg, and in turn ˆρeg “adiabatically follows” ˆρgg. The entire process is thus referred to as “adiabatic elimination” and represents the core idea of this chapter.
There is one more step necessary to make the above procedure work. First, note that the Hamiltonians are approximately of orders ˆHg ∼Hˆe∼∆ and ˆHi ∼√
s∆. So all of the Hamiltonian (nondecay) terms in equation (5.30) are of order√
s∆ except for the ˆρee term, which is of order s√
s∆. Consequently, the latter can be ignored under the limits1.
We now follow this adiabatic elimination procedure in detail to produce the desired ground-state-only master equation. As mentioned above, we first set ˙ˆρee = 0 and solve for
ˆ
ρee in terms of ˆρeg and ˆρge. When doing this we run into an equation that looks like Lˆρˆee ≡ −Γ ˆρee−i[ ˆHe,ρˆee] =iPˆeHˆiρˆge−iˆρegHˆiPˆe (5.35)
ˆ
ρee = Lˆ−1(iPˆeHˆiρˆge−iˆρegHˆiPˆe). (5.36) Thus we need to somehow invert the superoperator ˆL. References [98, 52] give us a clue how to do this. Using the definition of the superoperator inversion (on arbitrary ˆX) we
have
Lˆ−1Xˆ = − Z ∞
0
exp[ ˆLt] ˆXdt (5.37)
= −
Z ∞ 0
exp[−Γt] exp[−iHˆet] ˆXexp[iHˆet]dt (5.38)
= −
Z ∞ 0
exp[−Γt]
1 +X
f0
(exp[i∆f0t]−1) ˆPf0
Xˆ
×
1 +X
f0
(exp[−i∆f0t]−1) ˆPf0
dt (5.39)
= −Xˆ
Γ −X
f0
1
Γ +i∆f0 − 1 Γ
XˆPˆf0 +h.c.
−X
f10,f20
1 i(∆f0
1 −∆f0
2) + Γ+ 1 i∆f0
1 −Γ + 1
−i∆f0
2 −Γ + 1 Γ
! Pf0
1
XˆPˆf0
2(5.40) where we have used the definition of ˆHe and the fact that ˆP2 = ˆP for the projectors.
Actually, what we care about in the end is only the simplification of equation (5.29) and we do not need to worry about the full inversion of the superoperator. We can rewrite the only term in equation (5.29) where ˆρee appears as
Γ X
f,f0,σ
Aˆf,f0,σρˆeeAˆ†f,f0,σ = Γ X
f,f0,σ
Aˆf,f0,σ
Pˆf0ρˆeePˆf0
Aˆ†f,f0,σ
= Γ X
f,f0,σ
Aˆf,f0,σ
Pˆf0Lˆ−1(iPˆeHˆiρˆge−iˆρegHˆiPˆe) ˆPf0
Aˆ†f,f0,σ
= Γ X
f,f0,σ
Aˆf,f0,σ
Pˆf0
−1 Γ
(iPˆeHˆiρˆge−iρˆegHˆiPˆe) ˆPf0
Aˆ†f,f0,σ
= − X
f,f0,σ
Aˆf,f0,σ
Pˆf0(iPˆeHˆiρˆge−iˆρegHˆiPˆe) ˆPf0
Aˆ†f,f0,σ (5.41)
where we have used equation (5.36) in the second line and
Pf0( ˆL−1X)Pˆ f0 =Pf0 −Xˆ Γ
!
Pf0 (5.42)
in the third line, which is evident from equation (5.40). In other words, we do not care about the coherences between different excited states and we are only concerned with the block diagonal components.
Next, as justified above, we set the ˆρeeterm in equation (5.30) to zero, set ˙ˆρgeto zero, and solve for ˆρge in terms of ˆρgg. We again use the inversion definition for a new superoperator, but this time we do not have to use any projectors to simplify the solution. This gives
ˆ
ρge = iX
f0
"
Pˆ4
Γ/2 +i∆f0 + Pˆ3
Γ/2 +i(∆f0−∆3)
# ˆ
ρggHˆiPˆf0 (5.43)
and we say that ˆρge adiabatically follows ˆρgg.
Finally, we use all of these results to solve for ˙ˆρgg in terms of itself in equation (5.29).
This results in what we have been seeking, a self-contained master equation involving only the ground states:
˙ˆ
ρgg = −i[ ˆHg+ ˆHB,ρˆgg] +iPˆ4ρˆgg
X
f0
HˆiPˆf0Hˆi
∆f0−iΓ/2+ h.c.
+iPˆ3ρˆgg
X
f0
HˆiPˆf0Hˆi
(∆f0 −∆3)−iΓ/2 + h.c.
+ X
f,f0,σ
Aˆf,f0,σHˆi Pˆ4
i∆f0+ Γ/2ρˆggHˆiAˆ†f,f0,σ+ h.c.
+ X
f,f0,σ
Aˆf,f0,σHˆi
Pˆ3
i(∆f0 −∆3) + Γ/2ρˆggHˆiAˆ†f,f0,σ+ h.c. (5.44) The key result here is that we have kept the state-dependent spontaneous emission terms (in the last two lines). Other treatments have found only the effective Hamiltonian for f = 4 (represented here in the second line) and treated spontaneous emission in a nonrigorous way [25, 29, 26]. Note that, whereas the ˆAoperators were the jump operators on the full Hilbert space connecting excited states to ground states, now the effective jump operators are of the form ˆAHˆi and serve to mix the ground states amongst themselves. We also see that, despite our approximations, the norm of the state is preserved, Tr[ ˙ˆρgg] = 0. This can be seen by using both the cyclic property of the trace and the jump-operator identity of equation (5.12). Finally, notice that in the limit ∆50 ∆3 (i.e., the detuning is much larger than the hyperfine ground state splitting) the probe does not distinguish ground states and we can group terms to get the same results as reference [96] forj = 1/2 to j0= 3/2 transition (for stationary atoms) as we should.
The second and third lines of the above master equation represent the polarizability Hamiltonian and describe the coherent, dispersive interaction of the beam with the atoms.
In the next chapter, we start from this point to analyze the evolution of the optical polar- ization due to the atomic spin-state. In the remaining part of this chapter, however, we continue to analyze the state dependence of the incoherent spontaneous emission terms.