Solutions

Broadly, there are two pathways for obtaining the reflectivity. Heeg and Evers have opted to adiabatically eliminate the cavity mode on account of the fact that in a typical set up its linewidth is larger than the nuclear linewidth by some 10 orders of magnitude. A photon entering the cavity is therefore unlikely to be absorbed and emitted multiple times by the nuclear ensemble within the cavity - the system is far outside the strong-coupling regime. This approach delivers accurate results and is extremely convenient when dealing with Zeeman splitting of the lines. In that case, Heeg and Evers have shown that both cavity mode-induced self-interactions of individual nuclei resonances, as well as interactions between resonances of individual nuclei and interactions between different resonances of different nuclei take place. In this way, they could explain a range of observed effects, all due to interactions induced by the cavity vacuum. These were the collective Lamb shift of the ensemble [52], and spontaneously generated coherences [56] between the collective states of different sublevels, leading to the suppression of spontaneous emission at certain positions in the energy spectrum. However, in the context of a single-line system, it is straightforward to solve the system by solving the Heisenberg-Langevin equations in the steady

state. We will first reproduce Heeg’s and Evers’ derivation involving the adiabatic elimination, then take the second approach involving only the steady-state solution. Finally, we introduce Heeg’s and Evers’ generalized version of their first result, involving multiple cavity modes and layers.

The Heisenberg-Langevin equations for our model are

˙

a=i[H,a]−κa σ˙⁻=i[H,σ⁻]−γσ⁻

(84) To perform the adiabatic elimination of the cavity mode, we calculate the Heisenberg-Langevin equation of the cavity mode, and set the derivative to zero. Resolving fora, we get

a=

√2κrain−i∑ng^(n)∗σ₋⁽ⁿ⁾

κ+_i∆C ⁽⁸⁵⁾

Eq. (85) can be introduced into the Master equation

˙

ρ=i[H,ρ]− L[ρ] (86)

yielding the effective Hamiltonian which describes the system and its dynamics purely in terms of the nuclear raising and lowering operators. The effective Hamiltonian and effective Lindblad operators are:

H_Ω =_∑^N_n=1Ωgⁿσ₊⁽ⁿ⁾+h.c.

H_LS=_∑n,m=1^N δ_LSg⁽ⁿ⁾g^(m)∗σ₊⁽ⁿ⁾σ₋^(m) L^{e f f}_cav =−ξ∑^Nn,mg⁽ⁿ⁾g^(m)∗L[ρ,σ₊⁽ⁿ⁾,σ₋^(m)]

(87)

whereΩ=

√2κra_in

κ+i∆C andδ_LS=− ^∆C

κ²+∆²_C andξ= ^κ

κ²+∆²_C. The first Hamiltonian includes the effective driving of the ensemble of nuclei by the drive after elimination of the cavity. The second one includes the self interaction of nuclei due to the cavity vacuum (form=n), and the interaction of different nuclei due to re-emission and re-absorption form6=n. The former term results in an enhanced Lamb shift of a single nucleus due to the cavity vacuum; the second one results in a collective Lamb shift.

The expression foraarising from adiabatic elimination can also be introduced into Eq. (77) for the reflection coefficient:

R=−1+ ^2κr

κ+_i∆C − ⁱ

√2κr

a_in(κ+_i∆C)

∑

g^(n)∗hσ₋⁽ⁿ⁾i (88) Recall thathAi=Tr(ρA), where Ais any operator.

To proceed, we perform a change of basis. This is largely analogous to the Dicke model basis transformation, but since our sample exceeds the Dicke-limit in one dimension, we have to add

3. Quantum Optical Models 51 another phase factor. Moreover, in all calculations, we will use only one excited state, namely (−^N₂ +1,^N₂), which is the maximally symmetric Dicke state in the one-excitation subspace of Dicke states. The ground state will be denoted by|Gi.

In the old basis, the singly excited state of nucleusnis

|E⁽ⁿ⁾i=σ₊⁽ⁿ⁾|Gi (89) In the new basis, the excited state is a coherent superposition of all singly excited states, with an additional phase factor to account for spatial distribution. Again, this is the so-called ’timed Dicke state’.

|E⁺i= √¹ N

∑

e^i~k~rn|E⁽ⁿ⁾i (90) where~_kis the wave vector of the radiation exciting the timed Dicke state, and~rn is the position of then-th nucleus. This change of basis allows us to simplify the effective Hamiltonian and Lindblad terms:

H_Ω^{e f f} =_Ωg√

N|E⁺i hG|+h.c. (91)

H^{e f f}_LS =δ_LS|g|²N|E⁺i hE⁺| (92) L[ρ] =−ξ_s|g|²NL[ρ,|E⁺i hG|,|Gi hE⁺|] (93) Note that|E⁺i hG|is essentially a coherent phased superposition of raising operators∑^Nn e^−i~k~rnσ₊⁽ⁿ⁾. It can be interpreted as a new operator, which adds an excitation inside the symmetric subspaces of the many body system. An equivalent argument is valid for the lowering operator. If we restrict ourselves to the symmetric state of the one-excitation subspace, a single layer of ⁵⁷Fe can be regarded as a two-level system with an√

N-fold enhanced interaction with the environment. It is not entirely clear how largeNis; in principle it should account for all nuclei in a cavity. However, planar cavities, such as ours have a transverse quantum-correlation length [155], sometimes referred to as effective area [156], which limits this number. Briefly, it can be interpreted as the length over which resonant atoms in the cavity interact collectively with the mode. This is not necessarily the cavity length. We therefore always give the collective coupling strength, and do not calculate the strength for an individual nucleus or atom. We can insert the lowering operator into the term Eq.(88). Performing the trace, we get

R=−1+ ^2κr κ+i∆C

− ⁱ

√2κr

ain(κ+i∆C)

∑

g^(n)∗hE⁺|ρ|Gi. (94) It follows that we will have to solve the Master equation

˙

ρ=i[H^{e f f},ρ]− L^{e f f}[ρ] (95)

to calculate the matrix element appearing in Eq. (94). Once more, we solve the system of equations in the steady state. We also assume that hG|ρ|Gi = 1 and hE⁺|ρ|E⁺i = 0, neglecting the possibility of population redistributions. We end up with

hE⁺|ρ|Gi= ^iNΩg

∆+i^γ₂ +N|g|²(iξs−δ_LS) ⁽⁹⁶⁾ which can be inserted into Eq. (94) to calculate the reflectivity. A cursory inspection reveals that the reflectivity spectrum around resonance is given by a modified Lorentzian, which is shifted in energy by N|g|²δ_LS linewidths and has an additional decay term given by N|g|²ξ_s. These changes, Lamb shift and superradiant decay enhancement, are the result of the enhancement of the light-matter interaction due to the cavity. Keep in mind that their magnitude depends not only on the cavity mode characteristics, but also of the detuning both angular and energetic of the incoming beam from the resonance. Introducing additional resonances into this model results in additional coherences between the levels induced by the cavity mode interaction.