A stimulated Raman process between two momentum states

In this new basis, the non-resonant Rabi frequency has to be calculated taking the Doppler shift ~p·~k/m and the photon recoil ¯h|~k|²/2m into account. This leads to an effective detuningǫeff, which is now to be calculated as

ǫ_eff=ω_L−

"

~ p·~k

m +¯h|~k|² 2m

#

≡ω_L−ω.¯

(3.23)

3.2.2 A stimulated Raman process between two momentum states

The derived expressions for the two-level dynamics in the previous section are based on the assumption of negligible spontaneous processes. The involved states|1iand |2i have to have sufficiently long lifetimesτ, which at least are longer than the interaction time with the driving laser fields. However, if|2i is associated with an excited state, the population will rapidly decay (τ ∼ns) into the ground state by randomly emitting photons.

We now want to adapt this treatment to derive important quantities of a two-photon process between long-living ground states. When a coherent transition between two momentum states of the same ground states is realized, we call it Bragg diffraction

Figure 3.3: Schematic of first-order Bragg diffraction as a two-photon Raman process coher-ently coupling two momentum states |0¯hki and |2¯hki of the same ground state

|gi. Two light fields (ω1andω2) are detuned from the excited state|eiand effec-tively couple the momentum states via a virtual level|ii. Both ground states lie on the dispersion relation curve of a free particle, hence determining¯hδ=p²_r/2m.

[172, 160]. Transitions between two different atomic states (e.g., hyperfine ground states of an alkali atom) will be referred to as Raman diffraction [59, 175].

In the QUANTUS-I experiment, Bragg diffraction will be implemented, a basic schematic of which is given in Fig. 3.3. Here, one lattice beam couples the ground state |g,0¯hki with a virtual level |i,1¯hki, the second laser beam couples that virtual level again with the ground state |g,2¯hki. Since the absolute detuning ∆ w.r.t the excited level is sufficiently large, the population of|ei as well as spontaneous emission and scattering processes leading to decoherence can both be neglected.

We treat our problem as an idealized atom with two energy levels|g,0¯hki ≡ |1iand

|g,2¯hki ≡ |2i, an effective two-level system whose energy difference is supposed to be

∆E = ¯hδ =p²_r/2m (see Sec. 3.1). The latter determines the dispersion relation to be of parabolic shape. Compared to the single-photon treatment, the atom now interacts with an electromagnetic field composed of two frequencies,

E(t) =~ E~1(t) +E~2(t) =αE~ 1cos(ω1t) +αE~ 2cos(ω2t). (3.24) For Bragg diffraction the ratio of relative detuning between the laser frequencies and their absolute frequency is negligibly small, δ/ω ≈ 0.01 ppb. To calculate the eigenenergies, we approximate the laser fields to have the same wave vector norm as

|k~1| ≈ |k~2| ≡k. This results in the following eigenenergies of the two momentum states E_|1i= ¯hω_g,

E_|2i= ¯hω_g+(p₀+ 2¯hk)²

2m ≡¯hω¯_g. (3.25)

The Hamiltonian has to consider interactions with both laser fields (see Fig. 3.3).

By neglecting three-photon processes [176], it can be written as Hˆ = ˆH₀+ ˆH_AL

= ¯hω_g|g,0i hg,0|+ ¯hω¯_g|g,2¯hki hg,2¯hk|

+ ¯hω₁|i,¯hk₁i hi,¯hk₁|+ ¯hω₂|i,¯hk₂i hi,¯hk₂| −d~·E.~

(3.26)

Analogously to Sec. 3.2.1, a superposition state can be used as an ansatz to solve the time-dependent Schrödinger equation,

|Ψ(t)i=c₁(t)|g,0i+c₂(t)|g,2¯hki+c_i,1(t)|i,¯hk₁i

+c_i,2(t)|i,¯hk₂i. (3.27)

The resulting system of rate equations obtained in RWA can again be transformed in a time-independent set of equations [174]. For the starting parametersc1(t= 0) = 1 andc₂(t= 0) = 0, the populations of both states can be expressed as

P₁(t) = Ω²_eff

Ω²_eff+ (ǫ_eff−δ_AC)² cos^2hqΩ²_eff+ (ǫeff−δ_AC)²·t/2ⁱ, P₂(t) = Ω²_eff

Ω²_eff+ (ǫeff−δ_AC)² sin^2hqΩ²_eff+ (ǫ_eff−δ_AC)²·t/2ⁱ.

(3.28)

We now discuss the important parameters of the equations given above.

• Effective Rabi frequency: Ωeff = ^Ω1,1_2∆^Ω2,2, which is the resulting coupling strength of the coherent two-photon transition |1,0¯hki → |i,1¯hki → |2,2¯hki. The detuning of the laser beams w.r.t to the excited state |ei is given by ∆.

• Single photon coupling strength: Ω_l,m = −2hi|d~|liE_m/¯h, with l ∈ (1,2) being one of the two ground states, ithe virtual level and m∈(1,2)indicating the two laser beams generating the optical lattice. Given the linewidth Γof the 5S_1/2 →5P_3/2 transition in⁸⁷Rb and laser beam intensity I_m, one can calculate the resonant single photon Rabi frequencies as Ω_l,m=^p6πc²ΓI_m/¯hω_m³.

• Effective two-photon transition detuning: ǫ_eff=ω_eff− p~·^~keff

m +^¯h|~k_2m^eff|2

≡ ω_eff−δ, with frequency difference of the lattice beams¯ ω_eff =ω₁−ω₂, effective wave vector~keff=~k1−~k2≈2k, initial momentum~p and massmof the atoms.

• AC Stark shift of |g,0¯hki ≡ |1i: Ω^AC_|1i = ^|Ω_4∆^1,1|2 + _4(∆−¯^|Ω1,2|_ω)² , caused by the interaction of state |1i with both laser beams,

• AC Stark shift of |g,2¯hki ≡ |2i: Ω^AC_|2i = _4(∆+¯^|Ω2,1|_ω)² + ^|Ω_4∆^2,2|2, caused by the interaction of state |2i with both laser beams,

Figure 3.4: Schematic of n-th order Bragg diffraction as a stimulated 2n-photon Raman pro-cess. Energy conservation requires¯hnδn = (2n¯hk)²/2mto coherently couple the momentum states|0¯hkiand|2n¯hki. The energy levels are labeled by their trans-verse momentum states which are displayed in units of ¯hk. The corresponding detunings to ground and excited state are given by∆i.

• differential AC Stark shift: δ_AC = Ω^AC_|2i −Ω^AC_|1i.

Two-photon Raman transitions will give rise to Rabi oscillations with a Rabi fre-quency of Ω_eff, for example a π-pulse will ideally transfer all of the population from one momentum state to the other. For Bragg diffraction ωeff/∆ ≈ 0.01 ppb, which allows to approximate the effective Rabi frequency to

Ωeff = Ω_1,1Ω_2,2 2∆ ≈ Ω²

2∆, (3.29)

with Ω_1,1 = Ω_2,2 ≡Ω as the single photon Rabi frequency. In the case of a resonant interaction (ǫeff= 0), Eq. 3.28 now simplify to the basic expressions

P₁(t) = cos²[Ω_eff·t/2] = 1

2[1 + cos(Ω_eff·t], P₂(t) = sin²[Ωeff·t/2] = 1

2[1−cos(Ωeff·t].

(3.30)

Higher-order Bragg diffraction

In a momentum picture, the above depicted process is described by an atom undergo-ing a two-photon Raman process with correspondundergo-ing momentum transfer ofp_r= 2¯hk.

We call this first-order Brag diffraction. It is also possible to transfer even higher mo-mentum, i.e. 2n¯hk, which is called n-th order Bragg diffraction. For this process, again

an effective Rabi frequency can be derived [177]

Ω^nth_eff = Ω²ⁿ

2¹ⁿ⁻¹∆₁∆₂...∆_n−1, (3.31) where Ω₀ is the single-photon Rabi frequency and ∆_i are the detunings from any virtual level as indicated in Fig. 3.4. Here, an atom prepared in momentum state

|g,0¯hki interacts via a 2n-photon transition, involving 2n-1 virtual levels, and will subsequently end up in |g,2n¯hki. Energy and momentum conservation of a moving atom with initial momentump₀ leads to the following resonance condition

∆E = (npr+p0)²−p²₀

2m = (2n¯hksin(ϑ/2) +mv0)²−m²v²₀

2m = ¯hnδn (3.32)

With the recoil frequency ω_r = ¯hk²/m, the frequency difference between the laser beams for n-th order Bragg diffraction results in

δ_n= 4nω_rsin²(ϑ/2) + 2kv·sin(ϑ/2), (3.33) thus scaling linearly with diffraction order n. To give an example, for ϑ = 180^◦ (counter-propagating laser beams), a third order Bragg pulse for resting⁸⁷Rb atoms (p₀ = 0) would need an optical lattice with a frequency difference of δ₃ = 12·ω_r ∼= 2π·45.24 kHz.