Post-selected correlation function

The post-selection can be taken into account by choosing the dichotomization functions according to χ+(x) = 1 2 1+sgn cos m 2¯h 1 tm+τ− 1 tm x2 −cos h pπ 2 i , χ−(x) = 1 2 1−sgn cos m 2¯h 1 t_m+τ− 1 t_m x2 +cos h pπ 2 i . (4.30) The parameter p∈[0,1]denotes the post-selected fraction, hencep=1 corresponds to including all measurement outcomes, as was the case above. We can repeat the steps leading from (4.18) to (4.27) using (4.30), the validity of all assumptions is untouched by the exchange of the dichotomization functions. The only modification takes place when determining the periodic integrals in the interference term of (4.23), which now read 1 λ2 Z σ1λ/4+pλ/4 σ1λ/4−pλ/4 d∆x1 Z σ2λ/4+pλ/4 σ2λ/4−pλ/4 d∆x2cos 2π λ (∆x1−∆x2) +ϕ = 1 π2sin 2hπ 2p i cos h σ1 π 2 −σ2 π 2+ϕ i (4.31) =1 4sinc 2hπ 2p i σ1σ2cos[ϕ].

With this, the joint probability to detect Particle 1 at a position that has been assigned to the outcome σ1 and to detect Particle 2 at a position that has been assigned to the outcomeσ2reads P(σ1,σ2|t1,t2) = 1 4p 2n₁₊ σ1σ2E(t1,t2)sinc2 h_π 2p i cosω(∆t1+∆t2)−φ0 o . (4.32) By choosing p<1, we can increase the fringe contrast which is still limited by the envelope function. AssumingE(tm,tm)≈1 (as it was the case for the Gaussian envelope (4.29) when evaluated with the parameters (4.10)), a choice of p<0.63 supports the violation of a Bell inequality. Note that (4.32) is not normalized for p<1,

∑

σ1,σ2=±1

P(σ1,σ2|t1,t2) =p2<1. Normalizing to post-selected events, we get

P(ps)(σ1,σ2|t1,t2) = 1 4 n 1+σ1σ2E(t1,t2)sinc2 h_π 2p i cosω(∆t1+∆t2)−φ0 o . (4.33) This is our final result, showing that it is possible to regain Bell correlations in the HRE scenario that are strong enough to violate a Bell inequality, given the parameters are chosen appropriately (and accepting post-selection). In Chapter 8 I will demonstrate that the Feshbach dissociation scenario permits to generate HRE states that fulfill these requirements.

4.4.2 Summary

To conclude this part, the HRE scenario establishes an alternative for the demonstration of nonclassical correlations in the motion. In contrast to the DTE scenario, the corre- spondence to the spin-based Bell test is only revealed on the level of correlations. The

measurement times take the role of the “in plane” Bell control parameters, whereas there is no obvious analogue for “out of plane”-measurements. A post-selection procedure ensures sufficient fringe contrast as required for the violation of a Bell inequality. This takes the fair-sampling assumption as a basis, however, by redefining the dichotomiza- tion prescription all measurements can contribute to a Bell violation.

The idea of abandoning the interferometers simplifies the experimental setup consid- erably, but this comes at the cost of a required phase stability over the whole extent of the experiment and a high spatial and temporal measurement accuracy. In Chapter 8 I will argue that laser illumination of the dissociated atoms in the overlap regions indeed pro- vides the required resolution [91]. This way, the HRE state represents a complementary approach, interesting on its own, based on a genuine matter wave state with no photonic analogue.

41

Chapter 5

A proposed experiment based on

ultracold atoms

In the previous chapters I introduced two classes of dissociation states and exposed their potential to cause nonlocal correlations. Both the DTE and the HRE states share a very similar, simple generation protocol within the conceived dissociation scheme: a sequence of two dissociation pulses delocalizes each atom into a superposition of con- secutive wave packets, such that the wave packets are entangled in the dissociation times. Hence, choosing similar parameters, both states are amenable to the same experimental setup, which permits us to treat them at the same time.

In the following chapters I will argue that it is possible to realize these states within a scenario based on the dissociation of ultracold Feshbach molecules is possible and indeed seems to be favorable. The advantages are obvious: A Bose-Einstein conden- sate (BEC) as a source provides initial states of excellent reproducibility, and atoms propagating at velocities of cm/s can be optically guided and suitably detected by laser illumination. The resolution of the corresponding de Broglie wave lengths, which are on the order of micrometers, does not pose outstanding experimental challenges. And a time separationτon the order of seconds results in spatial separations between the early

and late wave packets on the order of centimeters, rendering their delocalization truly macroscopic. Apart from its fundamental relevance, such a centimeter-scale delocal- ization is also important for practical reasons in the DTE scenario, since the separation between the early and the late wave packets determines the size of the interferometers and should thus not be too small.

In this chapter, I work out a concrete experimental setup that provides the frame for the DTE and the HRE Bell test, respectively. The analysis of the Feshbach dissociation dynamics, which will be given in the following chapter, then enables us to investigate their viability in detail in the Chapters 7 and 8, respectively.

5.1

Experimental setup: overview

We consider a BEC of Feshbach molecules in an optical dipole trap, which can be es- tablished by two perpendicularly crossing laser beams, see Figure 5.1 a). Producing molecular BECs (mBEC) and tuning the interactions via Feshbach resonances is nowa- days routine [23–31, 34], and the use of their controlled dissociation for spectroscopic purposes has been demonstrated experimentally [32, 33], as well.

atom 1 atom 2 mBEC time- controlled magnetic field ~1 cm/s ~1 cm/s trap lase r a) b) guiding laser potential trapping laser potential guiding laser

Figure 5.1: a) Setup for generating pairs of motionally entangled pairs of atoms by dissociation of Feshbach molecules. Initially a BEC of approximately 102 Feshbach molecules resides in a trap constituted by two crossing laser beams. An externally con- trolled magnetic field induces dissociation of on average one molecule per trial and thus generates a pair of entangled atoms counter-propagating along the laser guide at a veloc- ity on the order of 1 cm/s. The asymptotic two-atom state in the laser guide is determined by the trap and guide geometry and by the shape of the dissociation pulse, see Chapter 6. When desired, the atoms can be detected conveniently by laser illumination; in the HRE scenario this would be in the overlap region, in the DTE scenario after the passage of Mach-Zehnder interferometers (not shown). b) The dissociation pulse promotes the initially trapped molecules to a pair of counter-propagating atoms. The lasers are chosen such that the trap laser potential can be overcome easily with the energy supply from the magnetic field sweep, whereas the atoms are transversally frozen in the ground state due to ¯hωG>Ekin.

5.1. EXPERIMENTAL SETUP: OVERVIEW 43

To be specific, I suggest to use a BEC produced from a 50:50 spin mixture of fermionic6Li. Such a fermionic mixture is favorable compared to bosonic ingredients, since huge lifetimes of more than 10s can be achieved due to the Pauli blocking of detri- mental 3-body collisions [25]. This makes truly macroscopic time separationsτbetween

the two dissociation pulses conceivable, and we may chooseτ=1s from now on. It has

been demonstrated that the molecular6Li BEC can be prepared efficiently and with near- perfect purity [25], and the comparatively small mass of lithium reconciles reasonable propagation velocities, on the order of 1cm/s, with resolvable de Broglie wave lengths, on the order of 10µm.

Extraction of an atom pair

The weak trap laser creates an elongated longitudinal trap within the wave guide pro- duced by the strong guiding laser. The preparation is arranged such that only a small number of molecules, on the order of 102, remains in the BEC at the end, such that one may neglect interactions between different molecules. The molecules can then be con- sidered to be in a product state with the center of mass motion given by the ground state of the trap, whereas the relative motion describes a bound molecular state. The latter can be turned into a Feshbach resonance by varying the external magnetic field, which allows one to dissociate the atoms in a controlled way. By applying one or several appropriately chosen dissociation pulses, a single molecule dissociates into two counter-propagating atoms on average, and we post-select the single-dissociation events.

The pulses are chosen to provide the atoms with a kinetic energy sufficiently large to overcome the trap potential in longitudinal direction, but still below the threshold to get beyond the ground state transversally, see Figure 5.1 b). This way we may end up with two dissociated atoms, counter-propagating with a relative velocity of 1cm/s along the guiding laser axis, whose two-particle state is determined by the initial state of the molecule and the dissociation pulse shape, as I will demonstrate in the next chapter. In particular, by applying a sequence of two appropriate dissociation pulses, one then can generate DTE and HRE states, respectively. Their detailed elaboration will be given in Chapters 6, 7 and 8.

DTE and HRE states

After the completion of the dissociation sequence, and for our chosen time separation ofτ=1 s between the early and the late dissociation pulse, the corresponding early and late wave packets of each particle are envisaged to propagate at a velocity of 5 mm/s (in the case of the HRE scenario, the early wave packets are slightly slower), separated by a distance of 5 mm on each side. This constitutes a truly macroscopic delocalization of each atom. Immediately after the dissociation process, the widths of the early and late single-particle wave packets turn out to be on the order of about 100µm, as we

will see in the Chapters 7 and 8. Their narrow momentum distributions guarantee that these wave packet extensions are not appreciably modified during the propagation to the measurement sites if the propagation time does not exceed about 10 s, which corresponds to the aspired distance on the order of 10 cm between the measurement sites. The early and late wave packets are thus spatially still sufficiently distinct when arriving at the interferometers for the switches to be applicable in the DTE scenario.

Detection

The final detection of the atoms can be achieved with laser illumination. In the HRE scenario this would be in the overlap region of the fast and the slow wave packet, in the DTE scenario the measurement would be done after passing the Mach-Zehnder inter- ferometers, whose implementation will be discussed in Chapter 7. Such a fluorescence detection of the slow, strongly confined atoms can be done with single particle resolu- tion [91], such that it is easy to disregard the cases of too many atom pairs in the process. For a start, we stick to the simple post-selection procedure; in a more refined setup it is conceivable to use a specially prepared optical lattice where each site is occupied by at most one molecule [95].

5.2

Trap and guide parameters

I now show that the above described laser setting is experimentally viable for an aspired relative velocity between the counter-propagating atoms ofvrel=1 cm/s and an overall extension of the setup on the order of 10 cm.1 The corresponding de Broglie wave length

λrel=12.4µm, which sets the scale for the fringe patterns, is in compliance with viable stability requirements for the setup.