It is possible, by changing the relative polarization between pump and probe, to attenuate the efficiency of the FWM. In the case limit of orthogonal pump-probe polarizations the conjugate disappear, but the interference as well. Hence the case of non orthogonal polarization does not solve the problem of clearing the conjugate because our ability to cut the conjugate is not linked to the intensities of the beam.
It is instead due to the range ofkof pump and conjugate, i.e. their extension in the k-space.
Since we have to avoid the unwanted effects coming from the conjugate we put a razor blade on a translation stage in the focus of the imaging lens, where we are able to cut it with a good precision.
The translation stage is necessary because, since the beam experiences also a self-defocusing effect, the focal point changes for different nonlinearities.
We stop the measurement when we are not able anymore to cut the conjugate without cutting the pump. If we go further with the measurement without properly cutting the conjugate we can clearly observe a decrease in the shift (see red circles in Fig. 5.10), due to the partial contribute of the conjugate.
Figure 5.9. On the left it is shown a limit condition where we are still able to fully cut the conjugate and the probe can be distinguished from the side-band of the pump.
On the right probe and conjugate start to superpose with the side-bands of the pump and is not possible to cut the conjugate without cutting the pump anymore.
In figure 5.9 it is shown the k-space relative to the measurement for small angles.
Here the razor blade is cutting from the left, as can be seen by the sharpness of the black line.
In the image on the left we are still able to fully cut the conjugate, in the right one we are not and we must stop the measurement.
5.3 Results
We tried this experiment several times using different Rb cells with different temperatures and different detuning. The results shown here are obtained for a
quasi-pure 85Rb (99.9% concentration) cell with lengthL= 7.5cm, a diameter of 1”
at an effective temperature of Tef f = 131 °C.
Here we lock the laser frequency at f = 384,2269THz. In this situation we are detuned of δ = −2.2 GHz from resonance (f0 = 384.2291), where we have a nonlinearity strong enough and relatively low absorption. In this condition we achieve a 70 % transmission and we can assume to be in a conservative regime.
We substract in post-processing the background with the pump only to reconstruct the shift with better accuracy. To improve the precisions of each measurement we acquire 10 consecutive images in less than 0.5 s1 and we average the shift over them.
Since we have a magnification of unity the error on each shift measurement in this way can be extimated asσ(∆S) =±1px=±6.5µm.
As shown in figure 3.2, for small k⊥ the dispersion relation is linear with a local nonlinearity. Instead, in presence of non-locality, it tends to converge to the single particle behaviour for k⊥ smaller than the ones expected in the case of a local nonlinearity. That is like to say that the non-locality tends to suppress the collective behaviour of our particles.
The expected shift is modified as well and it can be expressed in function of Λ as:
∆S = λ
First are reported the experimental datas of the shift ∆Svs probe in-plane wavelength Λ (Fig. 5.10). Thek of the pump is fixed perpendicularly to the cell and we move the angle of the probe (k⊥) in each measurement. The in-plane wavelength Λ can be extrapolated simply by measuring the distance between two fringes in the LI regime.
From the plot 5.10 it can be observed also that for small angles (big wavelength) the contribution of the conjugate effectively suppress the shift.
The shift is evaluated between the central fringes of the two interference patterns.
In presence of nonlinearity the fringes acquire a moon-like shape (Fig. 5.4), because the intensity has a gaussian distribution along the y-axis and ∆S∝√
I.
We measure the shift in the center of the x,y intensity distribution, where it is maximum.
From the elliptic gaussian profile shown in Fig. 5.4 it is possible to extrapolate the FWHM along the two axis.
Then we can extrapolate the waist of the intensity profile (w0 = F W HM
2√
ln2 ) along the two axis: wx = 2.3×10−3 m andwy = 0.31×10−3 m. Therefore the peak intensity can be extrapolated from the power of the pump Ppump=150mW :
I0= Ppump
πwxwy
= 6.7×104W/m2 . (5.20)
From the measurement of the shift it is possible to extrapolate a value for the nonlinearity of ∆n = −1.5 ×10−4, that for the intensity I0 corresponds to a nonlinear coefficient
n2 =−2.2×10−9m2
W . (5.21)
Such a value is in disagreement with the n2 extrapolated from the simulations (Fig. 4.2) of more than one order of magnitude, where n2(T = 404K, δ = −2.2
1This time depends on the exposure time we choose for the camera.
5.3 Results 77
Figure 5.10. The observed shift in function of the wavelength of the modulation (Λ = 2π/k⊥). The detuning is -2.2 GHz, the power of the probe isPprobe=1.5mW while the power of the pump isPpump=150mW.
GHz)=−1.6×10−11m2/W.
In the simulations has not been taken in account the contribution of multiple ex-citable hyperfine structures of the statesP3/2, that could enhance the nonlinearity.
It is instead in good agreement with the one found in the previous chapter measuring the SPM patterns (Eq. 4.45), wheren2 =−3.0×10−9 mW2 for I0 = 9.1×104 W/m2. In Fig. 5.11 it is shown the Bogoliubov dispersion relation in function ofk⊥, recon-structed from the shift using the formula 3.49.
The blue points are the experimental datas, that have to be fitted taking in account the non-local response function R(k⊥). From this fit we want to extrapolate the typical non-local lengthσnl.
We tried first to fit the Bogoliubov dispersion with a Lorentzian response func-tion, typical of the thermal diffusion as done in 3.8:
R(k⊥) = 1
1 +σnl2 k⊥2 . (5.22)
From which we extrapolate a non-local length ofσnl '400µm.
Our system is significantly different from the one of 3.8, where the nonlinearity has thermal origin and the nonlocal effects are due to thermal transport.
We consider then a ballistic non-local model ad hoc for a hot Rb vapour [50], described in section 3.6.1.
By fitting the dispersion with the response function 3.64, expressed in function of
Figure 5.11. The blue line is the fit of the experimental datas (blue point) reconstructed from the measurement of the shift. The red line represents the parabolic single particle behaviour Ω(k⊥) =2kk2⊥
0.
k⊥:
R(kˆ ⊥) = 2 Γ
e
1 (k⊥σnl)2
k⊥σnl
Z k 1
⊥σnl
0
e−y2dy , (5.23)
we find a value ofσnl '350µm.
Such a value is still way higher than the one expected with a simple model σnl = ˜vτ '10µm. and also of the one measured in Chap. 6 that is σnl = 66µm.
It is therefore necessary to find a better theory to evaluate the non-local effects in our system.