Actividad 2: Utilizando los sentidos como instrumentos para organizar lo caliente y frío

In the following, an eventual particle loss during imaging of the two photoionisation fragments into the corresponding CEM detectors is investigated. From this, the collection efficiencyηcol of the CEM detection system is estimated (see section 2.4). In order to quantifyηcol (eq. 2.8; subsection 2.4.1), three possible contributions of an eventual particle loss after photoionisation until primary particle impact in the CEM are analysed.

First, after photoionisation, there is a nonvanishing probability in the ionisation volume for charge recombination of the photoionisation fragments with adjacent atoms and molecules, or with simultaneously generated charged particles. However, due to the diluteness of the background vapour at typical background pressures ofp∼10−9mbar, the probability of, e.g., a simultaneous second photoionisation event is very low due to the low particle density of the background (typicallynback ∼106atoms cm−3) in the photoionisation volume. Even in a more dense MOT environment, any significant charge recombination of ionisation fragments after photoionisation due to interaction of the particles with the surrounding MOT environ- ment has not been observed [216]. Note, however, that also a mutual charge recapture of the

17_{The actual flight times scale by a factor oft} e/ti∼

p

me/mRbdue to the much larger mass of the87Rb-ion

4.4. Photoionisation fragment flight time model ionisation plane r(x,y,t) y x z copper aperture E(z)

Figure 4.14: Schematic illustration of the propagation of the ionisation fragment through the CEM copper aperture.

two generated fragments immediately after photoionisation can be excluded due to the consi- derable excess energyEexcess for the photoionisation transition scheme used in the context of this thesis.

Second, homonuclear or heteronuclear collisions with the vacuum background are a pro- bable source of particle loss. The mean free path of a particle in a thermal ensemble is l=kBT /(

√

2πd2

Rbp). For thermal atoms (T = 25◦C) with a diameter ofdRb∼235 pm at an UHV background ofp≈10−9mbar, the mean free path length isl= 167 km. This comparable huge distance leaves a collision of an ionisation fragment with the vacuum background very unlikely if one considers a covered path length of onlylz≈20 mm up to the primary particle impact in the active surface of the CEM detector. Moreover, this will also hold for the single atom trap immersed in a cloud of trapped atoms of a MOT environment. Even the enhanced atom density in the MOT region with typical particle densities ofnRb = 109−1010atoms cm−3 will still yield a negligible collision probability at these densities (nRb ∼p ≈4×10−7mbar, withl≈400 m).

Consequently, the main contribution of an eventual particle loss will occur due to the imaging of the particles into the corresponding CEM detector. In fig. 4.14, the imaging of an ionisation fragments out of the ionisation volume at z = 0 until propagation through the CEM aperture is illustrated. In particular, the lateral translation ~r = ~r(x, y, t) of the fragments in thex−y plane perpendicular to thez-axis is investigated while the fragment is accelerated up to the CEM cone entrance at z = d/2. If the accumulated lateral deflection ~r(x, y, t = tA) at the entrance is too large, the ionisation fragment will hit the bulk copper instead of propagating through the entrance of the CEM aperture (fig. 4.14). As the aperture diameter isdaperture= 2 mm, the maximum lateral deflection~r(x, y, tA) at aperture entrance should thus not exceed1 mmfor both ionisation fragments.

To calculate the lateral deflection after a particle propagation for a timetA, the initial velo- city components perpendicular to thez-axis are considered in accordance with the linear field approximation −→Eacc(x, y, z) = Eacc(z). Moreover, the vector of the initial particle velocity is chosen to be situated entirely in the radial x−y plane in order to obtain the maximum attainable deflection of the particles (fig. 4.14). For the calculation, a cylindrical symmetry

(ρ2 =x2+y2; z)is chosen. Therefore, thez-axis is situated in the center of the CEM aperture

(~r(x, y, t) =r(ρ, t))and represents the origin of the radial displacement.

The initial velocity vion,ele of the fragments is composed of the kinetic energy Etherm of the neutral87Rb-atom prior to photoionisation as well as the excess energy Eexess from the photoionisation process. In the first case, the kinetic energy will correspond to Etherm =

temperature(T = 25◦C). In the second case, the individual particle velocity v∗_ion,ele follows from the excess energyEexess of the photoionisation process18. As energy and momentum are conserved during photoionisation corresponding to an elastic collision, the generated87Rb-ion and its photoelectron gain a velocity component in the center-of-mass frame of the neutral 87_{Rb-atom of} v_ion∗ = v u u t 2Eexcess mRb mRb me + 1 (4.8)

and v∗_ele=mRbvion∗ /me. For an excess energy of Eexess = 33 meV, the actual velocities are v_ion∗ = 0.68 m s−1 _andv∗

ele = 1.08×105m s−1, respectively. In the moving frame of the 87Rb- atom, this velocity adds linearly to the thermal velocity19 of the neutral atom for maximum deflection, yielding an initial particle velocity ofvion,ele=vele∗ +vtherm after photoionisation. Therefore, the maximum attainable deflection in thex−y plane at z=d/2 is

ri,e(ρ, t) =ri,e(t) =vion,ele·t. (4.9) For acceleration voltages of∆Uacc= 3.8 kV, the calculated maximum displacement for87Rb- ions in the x −y plane from the central z-axis is ri(tA) ≈ 61µm (using a CEM entrance arrival time oftA= 256ns as calculated in eq. 4.5). Due to the small mass of the photoelectron compared to the87Rb-ion, the electron takes away almost all the excess energyEexcessfrom the photoionisation, yielding a comparably large initial velocity ofvele≈v∗ele= 1.08×105m s−1. Thus the thermal component of the velocityvthermbased on the motion of the former neutral 87_{Rb-atom becomes insignificant compared to the gained velocity out of the photoionisation.} The corresponding displacement for the photoelectron is re(tA = 0.85ns) ≈ 92µm with an estimated flight time of tA ≈ 0.85ns until entrance into the detector through the copper aperture.

Even for neutral atoms photoionised at the edges of the photoionisation volume correspond- ing to an initial radial position of ρ ≈100µm in the x−y plane (see subsection 4.5.2), the overall displacement of the ionisation fragments will still be smaller20 than200µm. As this value is by a factor of ten smaller than the diameter of the entrance of the aperture, all

incident ionisation fragments should be properly imaged into the CEMs and subsequently detected.

However, for the future application of a trapped atom in an optical dipole trap, the lateral displacement of the 87_{Rb-ion will become entirely insignificant due to the ultra-} cold temperature of the atom. The thermal component of vion will only contribute with vtherm = 0.12m s−1, considering a measured trapped atom temperature of T = 70µK [45]. With the full initial particle velocityvion, the ion21will experience only a lateral displacement of ri(tA= 256ns) ≈205nm. Interestingly, from a kinetic energy point of view, already the

18

In this thesis, a two-step process is employed (fig. 4.1(a)), with87Rb + (~ω12+~ω2i)−→87Rb++ e−, and

λ12= 780.241 nmandλ2i= 473 nm, correspondingly.

19

The corresponding most probable velocity for a thermal neutral 87Rb-atom is vtherm = 239m s−1, where

thermal denotes room temperature of the ensemble(25◦C).

20

The displacement is ri,e(ρ, tA) =ρ+ri,e(tA), with values ofri(256ns)≈161µm andre(0.85ns)≈192µm, respectively.

21_{For the photoelectron, the situation is identical to the initial thermal example due to the huge gain in kinetic}

4.5. Beam overlap and photoionisation fragment imaging