Four wave mixing processes - Superfluidity of light in a nonlinear atomic medium

Figure 5.4. Images of the shift for big (on the left) and small (on the right) relative angles between pump and probe. Above the HI beam, whose experiences the nonlinear effects, on the bottom the reference LI beam. The yellow line, centered on top of the Gaussian profile of the reference, put in evidence the shift between the two beams, that increases with the angle.

since the two patterns are disturbed in the same way.

An important condition is to align the central fringes of the two patterns along the x-axis. This can be done moving the probe with the automatized mirror in Fig. 5.2.

A more precise way to check if the position of the reference is good is to check in the k-space where, if thex-coordinate of the two patterns is the same, we observe a vertical interference as shown in figure 5.5. Adding the probe, if the pumps are well superposed, we observe the same k-space interference.

Using a strong probe beam this platform may also be used in the future to test the interaction between two superfluids.

In the next section it will be analyzed the origin of the FWM processes that comes from our phase matching condition.

Such a discussion is important because, without FWM, we could be able to extend our measurement for angles way smaller than the ones that we are able to reach experimentally.

5.2 Four wave mixing processes

As explained in the preceding section the conjugate beam generated by FWM is the main limit of this experiment and it is worth to understand its origin.

From a quantum-mechanical point of view, FWM occurs when photons from one or more waves are annihilated and new photons are created at a frequency such that the energy and momentum are conserved during the light-matter interaction.

It is possible to give a classical description of this phenomenon. Classically the first input field causes an oscillating polarization in the dielectric which re-radiates

Figure 5.5. Interference in the k-space between the 2 pump-only beams.

with a phase shift determined by the damping of the individual dipoles: this is just traditional Rayleigh scattering described by linear optics.

The application of a second field will also drive the polarization of the dielectric, and the interference of the two waves will cause harmonics in the polarization at the sum and difference frequencies.

Now, application of a third field will also drive the polarization, and this will beat with both the other input fields as well as the sum and difference frequencies. This beating with the sum and difference frequencies is what gives rise to the fourth field in four-wave mixing. Since each of the beat frequencies produced can also act as new source fields, a lot of interactions and fields may be produced from this basic process.

In general, the third order nonlinear susceptibility χ⁽³⁾ is a fourth rank tensor with 3⁴ = 81 elements and each of these elements consists of a sum of 48 terms. Fortu-nately this huge number of terms is drastically reduced through material symmetries and resonance, but unlike χ⁽²⁾, the suceptibility χ⁽³⁾ may have nonzero elements for any symmetry.

Each term has a typical form with three resonant factors in the denominator:

χ⁽³⁾= N L 6~³

g,k,n,j

µ_gkµ_knµnjµjgρgg

[ω_kg−ω₁−iΓ_kg][ω_ng−iΓ_ng−(ω₁−ω₂)][ω_jg−iΓ_jg−(ω₁−ω₂+ω₃)]+47terms . (5.3)

Here N is the oscillator density,µgk is the electric dipole matrix element between states g andkof a two level system,ω_kg is the frequency of the transition fromg to k, Γ_kg is the damping of the off-diagonal element of the density matrix that connects g tok, and ω1,2,3 are the frequencies of the fields.

The main difference between the 48 terms is the ordering of the frequencies involved in the summation. The susceptibility can be simplified further by considering terms which have small factors in the denominators due to resonance with oscillator frequencies. For example, Raman processes are described by the terms which contain

5.2 Four wave mixing processes 71

ω1−ω2 andω3−ω2, while two-photon absorption is described by terms that contain ω₁+ω₃.

The main features of FWM can be understood from the third order polarization term:

P~_{N L}=₀χ⁽³⁾E ~~E ~E . (5.4) In general FWM is polarization dependent and one must develop a full vectorial theory for it.

We will consider the scalar case, in which all the four field are linearly polarized along the ˆxaxis.

Assuming the quasi-CW condition it is possible to neglect the time dependence of the field componentAj (^∂A_∂t^j = 0).

The total electric field can be written as the sum of four field:

E~tot = 1

Substituting E~tot in the expression of the nonlinear polarization 5.4 : P~N L = ˆx1 All the beams involved in the process are polarized along the same axis and from now the vectorial notation will be omitted.

Considering two field ω1,ω2 entering in a nonlinear medium (with length L), two (in general) new frequency ω3,ω4 are generated in output.

One generic component of the polarization, as for example P₄ can be expressed as:

P₄= 3

The first four terms containing A₄ are responsible for phenomena such as self phase modulation or cross phase modulation.

Several kind of FWM processes are included in the expression of the fourth compo-nent of the polarization, and the effectiveness of every situation depends strongly on the phase mismatch betweenE₄ andP₄.

Let’s consider the case where 2 photons (ω1 and ω2) are annihilated, while two photons (ω₃ and ω₄) are generated simultaneously satisfying the conditions:

ω₃+ω₄−ω₁−ω₂= 0 , (5.8)

In the caseω₁ =ω₂, under the condition above, the FWM can be initiated also with a single pump beam [28]. This explains the two sidebands visible in the k-space, shown in Fig. 5.6.

Figure 5.6. Side-bands of the pump due to FWM processes. The interference on the three lines is due to the experimental conditions, not to FWM. The different intensities of the two side-bands are probably due to the way the pump is spatially filtered.

In our experiment we have FWM coming from the pump itself and also from the interference between pump and probe (at the same frequency ω).

In our situation, i.e. two photon of the pump are absorbed while the probe is re-emitted, we can identify our pump and probe, respectively with the index 1,3 and 2.

The energy and momentum conservation condition gives:

ω_p−ω_pr+ω_p =ω₄ . (5.9)

~k_p−~k_pr+~k_p =~k₄= 0 . (5.10) The second condition can be more clear with a graphical representation as shown in Fig. 5.7.

Neglecting all the situations with a different energy conservation the four component of the nonlinear polarization can be simplified. Here we express every ω_j or k_j respectively in function of the other three frequencies or momentum. The amplitude of the field are choosen in order to match the right phase factor.











P1 = ³⁰₈^χ⁽³⁾AprA^∗₃A4eî[(k^pr^−k³^+k⁴^)z−(ω^pr^−ω³^+ω⁴^)t]+c.c P2 = ³⁰₈^χ⁽³⁾ApA3A^∗₄eî[(k^p^+k³^−k⁴^)z−(ω^p^+ω³^−ω⁴^)t]+c.c P₃ = ³⁰₈^χ⁽³⁾A^∗_pA_prA₄eî[(−k^p^+k^pr^−k⁴^)z−(−ω^p^+ω^pr^−ω⁴^)t]+c.c P₄ = ³⁰₈^χ⁽³⁾A_pA^∗_prA₃eî[(k^p^−k^pr^+k³^)z−(ω^p^−ω^pr^+ω³^)t−]+c.c

(5.11)

In order to find the amplitude of the fourth field we have to substitute the nonlinear polarization into the Helmholtz equation 3.8 for each frequency ωj. For example we

5.2 Four wave mixing processes 73

Figure 5.7. On the left it is shown that in a two-level system, under the condition 5.9, a fourth photon is re-emitted with the same frequency of the two preceding beams. On the right it is shown that the the momentum conservation 5.10 is such that it is generated a fourth beam with opposite direction respect to the probe.

consider the fieldA_j=4, and we expand its double derivatives in time and space :

∂_t²[A4e^i(k⁴^z−ω⁴^t)] =−ω₄²A4e^i(k⁴^z−ω⁴^t) . (5.12)

∂_z²[A4e^i(k⁴^z−ω⁴^t)] = (∂_z²A4+ 2ik4∂zA4+k²₄A4)e^i(k⁴^z−ω⁴^t) . (5.13) This can be subsituted into Eq. 3.8. By defining d = ³₄χ⁽³⁾, and recalling that k_j²= ^ω

2 j

c²n² we find the following equation:

−∂_z²A₄−2ik₄∂zA₄ = 16dπω²₄

c² ApA^∗_prA₃e^i(k^p^−k^pr^+k³^−k⁴^)z . (5.14) Now this expression can be simplified furtherly by defining ∆k=kp−kpr+k3−k4

and applying the SVEA it becomes:

∂zA4 =iC4A1A^∗₂A3e^i(∆k) , (5.15) where C₄ = _c^8πd₂_k

Now it is possible to calculate the spatial evolution of the amplitude of the new fieldA₄ after a propagation distance L.

A₄(L) =iC₄A_pA^∗_prA₃ Z L

e^i∆kz dz= C₄

∆k(e^i∆kL−1) . (5.16) Another important assumption is to have no pump depletion, that here means A3 'Ap. In fact in the process are involved two photons from the pump.

Therefore we can write the intensity of the field:

I₄=|A₄|²n₄₀c

2 =C₄²|A_p|²|A_pr|²|A_p|²n₄₀c 2

e^i∆kL−1

∆k

. (5.17)

So the intensity of the conjugate beam in the shift experiment will have a sinc² profile.

I4=I_p²IprC₄^∗sinc²(∆kL/2) . (5.18) This can explain also second order side-bands observed in Fig. 5.8, where the pump is not spatially filtered with a pin-hole.

Figure 5.8. Image of the Fourier plane for a high intensity and not spatially filtered pump.

Here pump, probe and conjugate are clearly visible and each one generates multiple side-bands.

5.2.1 The problem of clearing the conjugate

Since in our experiment we need a strong χ⁽³⁾ nonlinearity we cannot avoid four-wave-mixing processes.

As seen in the preceeding section such phenomena can be induced with one or more beams.

For a single intense pump the FWM gives two side-band, visible in the k-space as shown in Fig. 5.6.

Adding the probe the phase matching and energy conservation conditions between these two waves (two photons are taken from the pump and one from the probe) are such that is generated a third beam with opposite k and same frequency of the others beam.

In this way the conjugate generates a shift in the interference pattern that is equal and opposite to the one generated by the probe.

In the Bogoliubov framework this can be understood as the fact that the conju-gate generates a sound wave with a direction opposite respect to the one of the probe.

Dans le document Superfluidity of light in a nonlinear atomic medium (Page 69-75)