4.1.1 Spontaneous parametric down conversion
When an electromagnetic field interacts with a dielectric medium it induces electric dipole moments. The macroscopic sum thereof results in a polarization P~ that is, for an anisotropic crystal, related to the electromagnetic field E~ via [145]:
Pi =²0
X
j
χ(1)ij Ej+X
j,k
χ(2)ijkEjEk+X
j,k,l
χ(3)ijklEjEkEl+...
(4.1)
where ²0 is the permittivity for a vacuum, χ are the nonlinear susceptibilities and the indices i, j, k, l∈ {1,2,3}.
For a strong pump field, the second order contribution cannot be neglected and two new electromagnetic fields are created. In second quantization this corresponds to the
spontaneous decay of a pump photon with energy ¯hωp and momentum ¯h~kp into two so-called signal and idler photons with energies ¯hωs,¯hωi and momentum ¯h~ks,¯h~ki. Due to energy and momentum conservation we have:
ωp = ωi+ωs (4.2)
~kp = ~ki+~ks (4.3)
For a more detailed quantum mechanical description of the process, refer to e.g. [146].
When a birefringent crystal is pumped by an extraordinarily polarized beam, there are two different types of SPDC:
Type I: The signal and idler photons are ordinarily polarized.
Type II: Signal and idler photons are orthogonally polarized, i.e. one ordinarily, the other extraordinarily.
Due to energy and momentum conservation the created photons are strongly correlated in these two degrees of freedom. In our experiments, the photons are distinguished by spatial mode and polarization, therefore we seek to achieve indistinguishability in all other degrees of freedom. Thus, we select only the degenerate SPDC emission, where the created photons have half the frequency of the pump beam.
We want to take advantage of the orthogonality of the polarizations of the signal and idler photons and use therefore SPDC of type II. The momentum conservation ensures that the photons are always emitted symmetrically around the pump beam. As they have different polarizations, the signal and the idler photon are emitted onto two emission cones, as can be seen in figure 4.1. The angle between these cones is varied by tilting the optical axes of the crystal relative to the pump beam. For the experiments presented in this thesis two configurations have been used. In figure 4.1 a) the most common configuration for the creation of polarization entangled photons is shown [147]. The two cones intersect and it is not possible to know which photon is emitted into which crossing mode. In figure 4.1 b) a collinear configuration is shown. The cones only touch in a mode collinear with the pump beam and the created photon pair is collected in one spatial mode, only.
These two configurations will be studied in a little more detail in the following.
Non Collinear Configuration
For the non-collinear configuration, entangled photon pairs can be emitted onto two spatial modesaandb, given by the directions where the two emission cones overlap (figure 4.1 a).
If, like in our case (see section 4.1.2), intense and short pump pulses are applied, then multiple emission events during a single pulse lead to the following state [148, 149]:
Z e−ic(a†Vb†H+a†Hb†V)|0i= Z
µ
ic(a†Vb†H +a†Hb†V)−c2
2(a†Vb†H +a†Hb†V)2+...
¶
|0i (4.4)
where Z is a normalization constant,cdepends on parameters of the crystal and is propor-tional to the pump beam intensity1 and a†V,b†H,a†H and b†V represent the photon creation operators for the modesaand b.
1Only valid forc¿1
4.1 Generation of photons
Figure 4.1: Two configurations to create photons with SPDC of type II. In the non-collinear configuration (a) the two characteristic emission cones intersect and entangled photon pairs are emitted onto the crossing modes. The collinear configuration (b) uses touching emission cones to obtain photon pairs of orthogonal polarization to be emitted into the same spatial mode.
The first order term is simply the emission of the entangled photon pairs into modes a and b. Thus with the probability 2Z2c2 a Bell state is created:
|ψ+i= 1
√2(|HaVbi+ |HaVbi). (4.5) The second order term corresponds to the emission of four photons resulting in the fol-lowing superposition of photon number states created with the probability 3Z2c4:
√1
3(|2Ha,2Vbi+|2Va,2Hbi+|1Ha,1Va,1Hb,1Vbi). (4.6) Note that the terms where equal polarizations are in the same mode have a higher ampli-tude than one would expect from a simple two-pair emission. This second order emission was already used to observe the three photon W3 state [88, 90, 91] and the four photon entangled state Ψ(4) [115, 138]. For the observation of the Cluster state in chapter 6 we use, however, twice the first order emission and the four photon emission enters only as an unwanted contribution to noise.
Collinear Configuration
The emission of the collinear configuration can be treated in a similar way. The only difference is that modes a and b collapse to one single spatial mode:
Z e−i√2c(a†Ha†V)|0i= Z
à i√
2c(a†Ha†V)−(√ 2c)2
2 (a†Ha†V)2−i(√ 2c)3
6 (a†Ha†V)3+...
!
|0i (4.7)
The three terms correspond to emission of the states |1H,1V ia,|2H,2V iaand |3H,3V ia into one mode with the probabilities 2Z2c2, 4Z2c4 and 8Z2c6, respectively2. In this type
2The parameters were chosen such that for the same value of c the same probability of a pair emission in collinear and non collinear configuration is obtained.
of source, the probability for a multiple emission is even stronger compared to the non collinear case, which is an advantage for the count rates obtained in chapter 7. The disadvantage is, however, that there is also a stronger contribution of the third order emission, which adds some additional mixed state in the experiment.
4.1.2 Experimental implementation
As we have seen, we can use the down conversion process to probabilistically create several photons. To do so it is necessary to use an extremely intense pump beam, as the probability to create four photons is proportional to the square of the intensity of the pump beam. Furthermore the photons need to be indistinguishable, independent of whether they are created in two independent sources or in one process. This means that their time of creation or detection needs to be much shorter than their coherence time. Both, a high intensity and short creation time window can be achieved by using a femtosecond pulsed laser system, guaranteeing a high peak intensity and a very short time window for the photons to be created. Here I will shortly describe our realization of such a pump laser system.
To create femtosecond UV pulses at a wavelength of 390 nm we frequency-double the output of a mode-locked Ti:sapphire laser. This combination of a Millenia3 Nd:Yag laser pumping a Tsunami3 Ti:Sa with 10 W cw pump power provides 82 MHz pulsed light (pulse length ≈130 fs) with an average power of 2.1 W at a wavelength of 780 nm. A 3 mm long Lithium-Borate crystal (LBO) is used for second harmonic generation (SHG) creating between 600 and 800 mW UV-pulses at 390 nm. Mirrors of high reflectivity for UV and high transmittivity for IR serve to separate the created UV beam from the residual IR light. Cylindric lenses serve to focus the pump beam to a circular spot onto the down conversion crystal, a 2 mm β-Barium Borate crystal (BBO) which is cut for collinear second harmonic generation (SHG). For the non-collinear down conversion the focus has a diameter of 200µm, for the collinear down conversion a spot size of 100µm was experimentally verified to be preferable. In the latter case, the UV pump beam is filtered from the down conversion emission by two mirrors with high transmission at IR and high reflection at UV under 0◦. Walk-off effects due to birefringence are compensated by a combination of a half wave plate, switching horizontal and vertical polarization, and a 1 mm BBO crystal. Coupling the photons into single mode fibers exactly defines the spatial mode desired to collect from the down conversion emission [150]. The spectral selection is achieved with a narrow bandwidth interference filters (usually ∆λ= 3 nm if not mentioned otherwise) that are located at convenient positions dependent on the setup.
The disadvantage of the femtosecond pulsed pump beam is the, in comparison with a cw pump beam, broad spectrum (∆λp,U V ≈ 1.2 nm). It leads to a broader spectrum of the down conversion emission, because the emitted light into the collected spatial modes also stems from a decay of photons that are not at the central wavelength. Further, the probability to detect both photons of an emitted pair is reduced, as we will see.
Usually, the coincidence to single ratio is used to quantify the probability for a created photon to be detected.
ηa= Cab
Ca, (4.8)
3Spectra Physics