The UV dissociation with circularly polarized light of some common hydro-halide molecules HX, such as HCl and HBr, follows the reaction: H X → H S 1 / 2 2 + X P J 2 (1) with J = 3 / 2 corresponding to the ground and J = 1 / 2 to the excited spin-orbit state of the halide atom. Despite the presence of multiple dissociation channels, which can in general produce different dissociation recoil distributions, often dissociation proceeds mainly through a perpendicular transition and results in a simple distribution of recoil velocity and spin polarization along the z-axis [6, 8, 15, 16]: I θ = N 1 + β 2 P 2 c o s θ (2) P θ = c o s 2 θ (3) with I ( θ ) being the SPH recoil distribution and P ( θ ) the spin polarization, P 2 ( x ) is the second order Legendre polynomial and β 2 an experimentally determined parameter, whose values lies close to 0.5 for both HCl and HBr [15], N is a normalization factor, and θ = 0 is defined by the propagation direction of the circularly polarized photodissociation laser. The ns laser dissociation process results in narrow velocity distributions while the mean speed of the atomic fragments can easily be derived using energy and momentum conservation.

In Figure 1A we show the speeds and angles associated with UV dissociation and how these can be used to create isolated SPH samples. A first Dissociation Pulse (PD1) dissociates molecules in the vicinity of its focus. The velocity acquired by the fragments leads to them exiting the volume around the focus, hereby referred to as “hole”. A second Dissociation Pulse (PD2), with larger dimensions, dissociates molecules in a volume around the hole, hereby referred to as “reservoir”. The large difference between the Hydrogen and the Halogen atom speeds means that there are times in which only H atoms will have filled the hole volume. Using circularly polarized light for PD2 and well selected dimensions for the hole and reservoir allows the accumulation of highly-polarized, high density SPH atoms in the volume of the hole.

Creating a hole devoid of atoms inside a gas-phase sample after dissociation is only viable if the (random) thermal velocities of the parent molecules are small compared to the recoil velocities of the molecular photofragments produced in the hole. In a molecular beam, the molecules can move with speeds of hundreds of meters per second, however, their thermal motion in the molecular rest frame can be cooled down to the 0.1–10 K range [17, 18]. For example, the average thermal speed of HCl molecules at 3 K (46 m/s) is much smaller than the recoil speeds of the H and Cl photofragments from photodissociation at 213 nm.

We now consider the geometry of the focused laser pulses, needed to create the hole and reservoir. The intensity distribution of a laser pulse follows Gaussian optics: I r , z = I 0 w 0 2 w z 2   e x p − 2 r 2 w z 2 (4) with r 2 = x 2 + y 2 , and w ( z ) = w 0 1 + z z R 2 the focus radius with w 0 minimum focus radius and z R = π w 0 2 n λ the Rayleigh length, n the sample refraction index and λ the laser wavelength. It is possible to engineer lasers beams with geometrical characteristics which deviate from these Gaussian beams; we will see some examples in Section 3.

When illuminating a molecular beam with UV laser light, molecular photodissociation can be initiated with a probability which depends on σ d the dissociation cross section. As a rule of thump, this probability approaches unity when P c d σ d ≈ 2, with P c d being the photon column density of the dissociating pulse. Let us denote as I sat the intensity when this condition is met. Since the laser focus is a function of z (the distance in the laser propagation direction), the condition I ( x , y , z ) = I sat can be satisfied for various values, as long as I sat < I 0 . Since the dissociation laser intensity is an experimentally controlled variable, we can choose it so that I sat / I 0 = R , which will result in a specific geometry for the hole: in Figure 1B, we see a two-dimensional cut of the volume which results from this condition when R is chosen to be 0.1, 0.5 and 0.9. We see that for choices of R > 0.5 , the resulting shape is relatively simple and can be approximated by an ellipsoid. From now on, we will assume an ellipsoidal shape for our hole to simplify our demonstrations, however, any three-dimensional continuous shape can be easily simulated.

A realistic geometry should include gradual descending borders, since the probability for dissociation is non-zero for intensities lower, yet close to I sat . We can use a generic sigmoidal function S ( r , r 0 , Δ r ) = 1 1 + E x p r − r 0 Δ r to simulate the geometry of such a hole H ( x , y , z ) as follows: H r ̂ = H x , y , z = S x 2 + y 2 , r sat z , σ x , y (5) with σ x , y being an adjustable parameter to account for the finite width of the hole border (see Section 3). The point in space for which the condition I = I sat is met, has coordinates x , y and z which satisfy the condition: x 2 + y 2 = r sat z = w z 2 2 − l n R w z 2 w 0 2 (6)

If we choose a single recoil direction, characterized by θ and ϕ , the polar and the azimuthal angle respectively, its contribution to the time evolution of the destiny inside the hole will be (step 1: emptying the hole): d e m θ , ϕ , t = d 0 V 0 ∫ H r ̂ H r ̂ − v ⃗ t d V r ̂ (7) with d 0 being the initial density of the parent molecules, and V 0 the total volume of the hole, and v ⃗ the velocity of the halogen atom (the H atoms move faster and leave the hole within much shorter timescales). The spatial integration is over the volume of the hole.

The total density inside the hole, as a function of time will be: d tot e m t = 1 N θ ∫ 0 π ∫ 0 2 π I θ d e m θ , ϕ , t s i n θ d θ d ϕ (8) with N θ being the normalization factor N θ = ∫ 0 π ∫ 0 2 π I ( θ ) s i n θ d θ d ϕ . In Figure 1C we see the density as a function of time for the dissociation of HBr (solid line) and HCl (dashed line) at 213 nm (hole dimensions visible in b).

The hole can be reloaded by illuminating a larger volume using a second laser pulse. We can define a “reservoir” function R ( x , y , z ) (similar to H ( x , y , z ) ), which describes the density of the molecules dissociated in the vicinity of the hole. The reservoir function R can be approximated by: R r ̂ = R x , y , z = H x / q , y / q , z / q − H x , y , z (9)

The density of the hydrogen atoms entering the hole as a function of time, for a given recoil direction is (step 2: reloading the hole): d load θ , ϕ , t = d 0 V 0 ∫ H r ̂ R r ̂ + v ⃗ H t d V r ̂ (10) with v ⃗ H being the recoil velocity of the hydrogen atoms. The total density inside the hole as a function of time will be: d tot load t = 1 N θ ∫ 0 π ∫ 0 2 π I θ d load θ , ϕ , t s i n θ d θ d ϕ (11)

The polarization of the hydrogen atoms depends on both their recoil direction and the time after dissociation. The temporal dependence results from the hyperfine interaction [7]: P H F = 1 − s i n 2 ω H F t / 2 (12) with ω H F = 2 π 1.42 × 1 0 9 ) rad/s. We can separate these two dependencies: let’s call the total polarization as P tot = P G P H F , the product of the hyperfine polarization and P G , what we will refer to as “geometrical” polarization. The evolution of P G can be understood using a qualitative model, which makes use of the general integral: P G = ∫ θ min θ max P θ I θ s i n θ d θ ∫ θ min θ max I θ s i n θ d θ (13) where θ min and θ max are the limiting angles for the integration ( 0 ≤ θ max , θ min ≤ π ) . By setting θ min = 0 and θ max = π for example, one easily obtains the free space value of 40%.

Numerically, the polarization inside the hole will be given by the similar formula: P G num = 1 N θ d tot load t ∫ 0 π ∫ 0 2 π I θ P θ d load θ , ϕ , t d θ d ϕ (14)

In Figure 2A, we see the evolution of P G for three different geometries. The brown line corresponds to a hole shaped as a prolate ellipsoid, with axes a = b, c = 3a. The blue line corresponds to a spherical hole while the green line corresponds to a hole with the shape of an oblate ellipsoid (a = b, c = 0.5a). Choosing a spherical geometry for our hole (note that this is impossible to achieve experimentally using lasers), will result in P G equal to the free-space value of 40% (blue dashed line). In contrast, choosing an elliptical geometry (feasible with lasers) results in P G different from the free space value: a prolate geometry (orange dashed line) maximizes whereas an oblate geometry (green dashed line) minimizes P G as time increases.

The brown and green points are values obtained for different choices for θ min and θ max in Equation 13. Let’s first discuss the case of a prolate ellipsoid considered in Figure 2B. The red ellipse is a slice of the hole, and the dashed brown a slice of the reservoir, in the xz plane. “a” and “A” are either of the small axes of the hole and reservoir ellipsoids. Consider the first quadrant; the complete results can be obtained by reflecting the resulting angles over the x and z-axes respectively. In the x direction, the last atoms will exit the ellipse at time t = t 0 = a + A v H . After this time, no atoms will be present in the trap coming from this direction, thus their polarization will no longer contribute to the overall geometric polarization inside the hole. In this direction ( θ = π /2), the polarization is zero (since P ( θ ) = cos 2 θ ), thus, the overall P G inside the hole will increase (at the cost of reduced density). At a later time t, more atoms found in the outer limits of the reservoir will be exiting the hole, and in particular those for which r h + r R = t v H . Using simple geometry, one can deduce a limiting value for θ max used in the integration of Equation 13 ( θ min = 0) as a function of time, from which one can obtain the brown points in Figure 2A.

The situation is similar in the case of an oblate ellipsoidal geometry shown in Figure 2C. The light green ellipse is a slice of the hole and the dashed dark green ellipse a slice of the reservoir, in the xz plane. Here, the first atoms leaving the hole are moving in the z direction. Thus, at later times, a similar integration angle can be found as a function of time, however, now it will be the angle θ min of Equation 13 ( θ max = π / 2 ), which results in the green points of Figure 2A.

We apply this model for some characteristic geometries for the dissociation pulses. We will compare holes created using a w 0 = 4.5 and 10 μ m spotsize lasers. In Figure 3A we plot the density inside the hole as a function of time after firing PD2 for w 0 = 4.5 μ m spotsize (gray line) and a w 0 = 10 μ m spotsize (black line) and R = 0.55. We see that the overall dynamics are faster for the smaller hole, which is to be expected. In Figure 3B we see the geometric polarization P G and in Figure 3C the total polarization as a function of time after firing PD2; we see that the hyperfine polarization evolution allows reaching high values for the total polarization in times that are integer multiples of the hyperfine period. This means that when a ∼ 10 m u m hole is considered, very low values of polarization can be achieved within the first hyperfine oscillation period, in contrast to the case of the smaller ∼ 4.5   μ m hole.