Atomic source selection in space-borne gravitational wave detection

In this section we define and apply the criteria to identify an optimal species choice for the envisioned experiment. Desired properties can be summarized in the following three categories.

• (i)  
  Electronic structure and narrow line transitions—As the sensitivity of the proposed gravitational wave detector scales linearly with the effective wave number linked to the momentum (∝Nω_a) transferred onto the atomic wave packet, large transition frequencies are desired. Unlike the case of a small-scale experiment, the proposed single-photon beam splitting scheme studied here implies that the wave packets spend a non-negligible time, on the order of seconds, in the excited state (see figure 1). Consequently, this state has to have a lifetime significantly larger than 2L/c to overcome spontaneous emission, loss of coherence and deterioration of the output signal [22]. The obvious species considered here are typical optical clock atoms. Their two valence electrons can align parallel or anti-parallel, thus giving rise to a singlet and a triplet system. Naturally, dipole selection rules render electronic intercombination transitions forbidden and these transitions have narrow linewidths. In strontium e.g. this makes ¹S₀$\to$ ³P₁a favorable cooling transition due to the intimately related low Doppler cooling limit. The even further suppressed ¹S₀$\to$ ³P₀ transition is consequently used in many optical atomic clocks, where spectroscopy on a mHz or narrower transition at a THz frequency is performed.
• (ii)  
  Coherent excitation and ultra-low expansion rates—Efficiently addressing an optical transition implies maintaining a good spatial mode overlap of the driving laser beam with the corresponding atomic ensemble. The Rabi frequency when driving a transition with linewidth Γ and saturation intensity ${I}_{\mathrm{sat}}=2{\pi }^{2}{\hslash }c{\rm{\Gamma }}/3{\lambda }^{3}$ reads

  $$\begin{eqnarray}&&{\rm{\Omega }}={\rm{\Gamma }}\,\sqrt{\displaystyle \frac{I}{2{I}_{\mathrm{sat}}}}.\end{eqnarray} \tag{ 2 }$$

  Since the available laser intensity I is always finite, and especially limited on a spacecraft, small laser mode diameters and correspondingly even smaller atomic wave packet diameters are desired. The detector's frequency band of interest lies in the range of tens of millihertz, and hence the resulting evolution time T for maximum sensitivity is on the order of hundreds of seconds (equation (1)). During an interferometer time scale 2T, the thermal expansion of an ensemble of strontium atoms at a temperature of 1 μK yields a cloud radius on the order of meters. As a direct consequence, cooling techniques to prepare atomic ensembles with the lowest possible expansion rates are required and heavier nuclei are in favor. Moreover, matter-wave collimation as realized in [23–25] is an indispensable tool to engineer the required weak expansion energies. Throughout the manuscript, we express this expansion energy in units of temperature and refer to it as the effective temperature ${T}_{\mathrm{eff}}$. For the purpose of this study, it lies typically in the picokelvin regime, which corresponds to few tens of μm s⁻¹ of expansion velocity.
• (iii)  
  Available technology and demonstration experiments—Finally, any heritage from demonstration experiments is of importance when designing the sensor, especially in the scope of a space mission. Similarly, the availability of easy-to-handle reliable high-power laser sources with perspectives to develop space-proof systems are important criteria in the selection of an atomic species. As an example, laser wavelengths far-off the visible range should be avoided for the sake of simplicity, robustness, and mission lifetime.

In table 1, we provide an overview of available atomic species. While usually not occurring in atomic clocks, the proposed experimental arrangement requires the metastable state to be populated over time scales on the order of seconds or more. Within a single pair of sequential single-photon beam splitters, the time an atom spends in the excited state is ∼2L/c (dashed lines in figure 1), dominated by the light travel time between the two spacecraft. With an excited clock state decay rate Γ₀, a baseline L, and diffraction order N the remaining fraction of atoms in the interferometer reads

$$\begin{eqnarray}&&{P}_{r}=\exp \left[-\displaystyle \frac{4\,L\cdot N}{c}\cdot {{\rm{\Gamma }}}_{0}\right].\end{eqnarray} \tag{ 3 }$$

This loss of atoms by spontaneous emission³ causes an increase in quantum projection noise by a factor of $1/\sqrt{{P}_{r}}$. In order to keep up the device's single-shot sensitivity, the atomic flux has to be increased accordingly or non-classical correlations have to be utilized to compensate for these losses. Similarly, when mitigating spontaneous losses via reduction of the instrument baseline or the beam splitting order, the linearly reduced sensitivity needs to be recovered with a quadratically larger atomic flux. As a result of their nuclear spins ($I\ne 0$), the electronic structure of fermionic species is subject to hyperfine interactions and has significantly larger clock linewidths than their bosonic counterparts [39]. Consequently, losses due to finite excited state lifetimes can significantly attenuate the signal for some species. Remaining atomic fractions after a full interferometer cycle for several fermionic isotopes are stated in table 2.

Table 1.  Overview of possible two-electron systems featuring clock transitions. The isotopes treated in detail in this article are printed in boldface.

	Mass	1S0$\to$3P0	Nat.	1S0$\to$			References
	in u	linewidth	abund.	1P1	3P1	3P0
		Γ0/2π in Hz		in nm
Fermions
Mg	25	70 × 10−6	10%	285	457	458	[26]
Ca	43	350 × 10−6	0.1%	423	657	659	[26, 27]
Sr	87	1.5 × 10−3	7%	461	689	698	[28]
Cd	111	5 × 10−3a	13%	228	325	332	[29]
Yb	171	8 × 10−3	14%	399	556	578	[30]
Hg	199	100 × 10−3	17%	185	254	266	[31]
Bosons
Mg	24	403 × 10−9b	79%	285	458	457	[32]
Ca	40	355 × 10−9b	97%	423	657	659	[33]
Sr	84	459 × 10−9b	0.6%	461	689	698	[34]
Cd	114	c	29%	228	325	332	[35]
Yb	174	833 × 10−9b	32%	399	556	578	[36]
Hg	202	c	30%	185	254	266	[37]

Notes.

^aLinewidth estimation [29]. ^bLinewidth achievable with external magnetic field as described below; Calculated using equation (5) and [38] assuming B = 100 G, P = 1 W, laser waist w = 4σ_r, atomic ensemble radius σ_r = 6 mm and expansion rate ${T}_{\mathrm{eff}}=10\,\mathrm{pK}$. ^cNecessary coefficients for the calculation unknown to the authors.

Using bosonic isotopes with theoretical lifetimes of thousands of years in the metastable state ³P₀ circumvents the losses described above but requires different experimental efforts. Indeed, unlike fermionic candidates, single-photon clock transitions in bosonic species are not weakly allowed through spin-orbit-induced and hyperfine interaction mixing [39] and the excited state lifetime is limited by two-photon $E1M1$-decay processes, hence typically lying in the range of picohertz [34, 39].

Accordingly, efficient manipulation on the clock transition for beam splitting depends on induced state-mixing by magnetic-field induced spectroscopy [38]. For example, such a magnetic quench allows to weakly mix the triplet states ³P₀ and ³P₁and thus increases the clock transition probability. Using the formalism described in [38], which holds for linear polarizations, it is possible to infer Rabi frequencies

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{0}=\alpha \cdot \sqrt{I}\cdot B,\end{eqnarray} \tag{ 4 }$$

and corresponding effective clock linewidths

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{0,\mathrm{eff}}=\gamma \,\displaystyle \frac{{{\rm{\Omega }}}_{L}^{2}/4+{{\rm{\Omega }}}_{B}^{2}}{{{\rm{\Delta }}}_{32}^{2}},\end{eqnarray} \tag{ 5 }$$

under the assumption that the external magnetic field is colinear to the laser polarization⁴ . Here, γ denotes the decay rate of ³P₁, Δ₃₂ is the splitting between the triplet states and Ω_L and Ω_B are the coupling Rabi frequencies induced by the laser and the static magnetic field, respectively. Supporting the concept of concurrent operation of multiple interferometers [17], the external fields can be limited in terms of spatial extent to distinct interaction zones.

In order to induce homogeneous Rabi frequencies over the spatial extent of the atomic ensemble, a reasonable spatial overlap between the exciting beam and the atomic cloud is required. Given the long drift times in the order of seconds, the clouds reach sizes in the order of millimeters, necessitating even larger beam waists. In view of limited laser power in a space mission, the resulting low intensities lead to Rabi frequencies in the few hundred Hz range for fermions. Assuming a magnetic field of 100 G, the corresponding Rabi frequencies are in the order of a few Hz for bosons. Table 3 illustrates the orders of magnitude for the two isotopes of strontium. Generally, smaller cloud sizes are advantageous, favoring the use of colder, i.e. slowly expanding sources.

The excitation probability is intimately connected to the phase space properties of the atomic cloud. An intensity profile of the exciting beam that varies over the spatial extent of the ensemble induces a space-dependent Rabi frequency except when the laser beam is shaped to be spatially uniform [40, 41]. One can overcome it by an increased beam waist leading to a homogeneous but smaller Rabi frequency. On the other hand, the effective Rabi frequency associated to a beam splitting light of wave number k being ${{\rm{\Omega }}}_{\mathrm{eff}}(r,v)\,=\sqrt{{{\rm{\Omega }}}_{0}^{2}(r)+{\left(k\cdot v\right)}^{2}}$, large waists (at limited power) would cause the Doppler detuning ${\left(k\cdot v\right)}^{2}$ to become the dominant term in Ω_eff(r, v) thereby making the process very sensitive to the velocity distribution of the atomic ensemble. A trade-off to find the optimal waist maximizing the number of excited atoms throughout the full sequence is made in each scenario presented in this study. The respective excitation probability is calculated [42] as

$$\begin{eqnarray}&&{P}_{{exc}}=2\pi \int \int r\,f(v)n(r,t){\left(\displaystyle \frac{{{\rm{\Omega }}}_{0}(r)}{{{\rm{\Omega }}}_{\mathrm{eff}}(r,v)}\right)}^{2}{\sin }^{2}\left(\displaystyle \frac{{{\rm{\Omega }}}_{\mathrm{eff}}(r,v)}{2}\,t\right){\rm{d}}r{\rm{d}}v,\end{eqnarray} \tag{ 6 }$$

where f(v) is the longitudinal velocity distribution, Ω₀(r) is the spatially-dependent Rabi frequency and n(r, t) is the transverse atomic density distribution. The resulting excited fraction for typical parameters of this study and for one pulse can be found in table 3.

In order to calculate the total fraction of atoms left at the detected state at the end of the interferometric sequence, one has to successively evaluate the integral (6) for each pulse. Indeed, the first light pulse selects a certain area in the ensemble's phase space distribution. The resulting longitudinal velocity distribution f_new(v) is computed and will constitute the input of the integral (6) relative to the next pulse. This treatment is iterated over the full pulses sequence of the considered scenarios. The long baseline scenario comprises n_p = 7 pulses while the short baseline scenario is realized by a sequence of n_p = 47 pulses. We illustrate, in figure 2, the short baseline case by showing, after each pulse, the new effective expansion temperature calculated after the new velocity width ${\sigma }_{{v}_{i}}$, the individual-pulse excitation rate ${P}_{{\rm{exc}},i}$ and the overall excitation probability at that point, given by the product of all previous pulses.

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The total number of atoms detected at the interferometer ports is given, for each isotope, by evaluating the product of the excitation and the lifetime probabilities. In figure 3, we compile the outcome of these two studied aspects for the species considered in table 1. Assuming parameters that are well in line with state-of-the-art technology (filled symbols), i.e. an excitation field with B = 100 G, P = 1 W as well as an effective expansion temperature ${T}_{\mathrm{eff}}=10\,\mathrm{pK}$ and σ_r = 6 mm at the time of the matter wave lens, the plot suggests a preliminary trade-off. Although the bosons benefit from their small transition linewidths rendering them resilient to spontaneous decay, they all can only be weakly excited in the order of a few % or less (lower right corner of the figure). For clarity reasons, the isotopes that lie under an excitation probability of less than 0.5% are not represented. Heavier fermions, such as cadmium, mercury and ytterbium are subject to particularly large losses due to their broad linewidths (see table 2) in spite of very promising previous demonstration work in the case of ¹⁷¹Yb [43]. It turns out that fermionic strontium and ytterbium are the most promising candidates, with a total fraction of around 12% of the atoms contributing to the interferometric signal in the long baseline scenario (circles), and around 10% in the case of strontium in the short baseline scenario (squares). Pushing the parameters to more ambitious values of B = 500 G, P = 2 W and ${T}_{\mathrm{eff}}=1\,\mathrm{pK}$, improves the results significantly. In bosonic ytterbium and both isotopes of strontium, more than half of the atoms are left at the end of the pulse sequence of the long baseline scenario, and decent excitation rates are reached even in the short baseline configuration. Overall, ⁸⁷Sr turns out to be the most favorable isotope in this comparison.

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The worldwide efforts on demonstration experiments towards using the narrow clock transitions in Sr as a future frequency standard [28, 44] promises additional advantages of this choice through technological advances and research. In contrast, fermionic magnesium is difficult to address due to the ultraviolet singlet line, the weak cooling force of the ¹S₀ $\to$ ³P₁transition [45] and quantum degeneracy not being demonstrated thus far. Likewise, trapping of fermionic calcium has only sparsely been demonstrated [27] and the intercombination cooling force is almost as weak as in the case of magnesium. Cooling techniques can be applied to all candidate bosons and finite clock transition linewidths can be achieved through magnetic field induced state mixing. A selection of a bosonic species would thus be motivated by previous demonstration experiments despite the weak excitation probability. In contrast, magnesium and calcium isotopes are missing simple paths to quantum degeneracy as a starting point for picokelvin kinetic energies. Although Bose–Einstein condensation has been shown for ⁴⁰Ca [33], the scheme is not particularly robust and the scattering length of 440$\,{a}_{0}$ inhibits long-lived Bose–Einstein condensates (BEC). For cadmium, only magneto-optical trapping has been demonstrated [35]. Next to missing pathways to quantum degeneracy, its transition lines lie in the ultraviolet range. Mercury atoms can be ruled out for the same reason, although significant experience is available [37]. Additional candidates with convincing heritage are ¹⁷⁴Yb [46, 47] and ⁸⁴Sr, which has been brought to quantum degeneracy with large atom numbers in spite of its low abundance [48].

To conclude this section, we remark that the bosonic and fermionic isotopes of Yb and Sr are the most favorable when it comes to the number of atoms involved in the interferometric measurement. This holds for all scenarii considered: short and long baselines, modest and ambitious parameters and their combinations. We, therefore, pursue this trade-off focusing on these elements. We analyze their suitability for the use in the proposed gravitational wave detector by considering the respective laser sources requirements and the necessary environmental control to constrain systematic effects.

The error model devised in the previous section assumes a different size of the atomic cloud at different steps of the experimental sequence. The expansion dynamics relies decisively on the temperature and densities considered. Depending on these parameters, bosonic gases, assumed to be confined in harmonic trapping potentials, are found in different possible regimes. Here, we treat Bose–Einstein condensed gases as well as non-degenerate ensembles in all collisional regimes ranging from the collisionless (thermal) to the hydrodynamic limit. We comment on the analogy with fermions later in this section.

The phase-space behavior of ensembles above the critical temperature of condensation is well described by the Boltzmann–Vlasov equation in the collisionless and hydrodynamic regimes [69, 70], whereas the mean-field dynamics of a degenerate gas are captured by the time-dependent Gross–Pitaevskii equation [71]. However, gases released from a harmonic confinement, experience a free expansion that can conveniently be rendered by simple scaling theories. In this approach, the gas is assumed to merely experience a dilation after release with an unchanged shape but a size L_i(t) evolving according to

$$\begin{eqnarray}&&{L}_{i}(t)={b}_{i}(t){L}_{i}(0),\end{eqnarray} \tag{ 7 }$$

with L_i(0) being the initial (in-trap) size and i denoting the spatial coordinate x, y or z. The dynamics in time are accounted for by the scaling parameters b_i(t), which interpolate between all collisional regimes of non-degenerate (bosonic⁶ ) gases in reference [70] and for degenerate gases of bosons in [72, 73]. The initial size L_i(0) depends on the interaction and temperature regime of the gas.

In the thermal non-interacting case, the initial size corresponds to the rms-width ${\sigma }_{i}^{\text{th}}(0)=\sqrt{{k}_{{\rm{B}}}{T}_{a}/m{\omega }_{i}^{2}}$ of the Gaussian density distribution trapped with the angular frequency ω_i in the direction i at a temperature T_a [74], the atomic mass m and the Boltzmann constant k_B. Considering elastic interactions, the initial size is a correction of the collisionless rms-width with a modified trapping frequency ${\tilde{\omega }}_{i}^{2}={\omega }_{i}^{2}(1-\xi )$ accounting for the mean-field E_mf via the parameter $\xi ={E}_{\mathrm{mf}}/({E}_{\mathrm{mf}}+{k}_{{\rm{B}}}{T}_{a})$ [75]. In the bosonic case, E_mf equals $2{gn}$, with the density of the cloud n and the interaction strength $g=4\pi {{\hslash }}^{2}{a}_{s}/m$ for an s-wave scattering length a_s and the the modified Planck constant ℏ. BECs are, on the other hand, well represented with a parabolic shape in the Thomas–Fermi regime for a large number of particles (the study case here). Their size is hence parametrized with the Thomas–Fermi-radius ${R}_{i}(0)=\sqrt{2\mu /m{\omega }_{i}^{2}}$, where μ is the chemical potential of the degenerate gas [71]. Although the physical origin is different, trapped Fermions display a similar density distribution as the interacting bosons. The Thomas–Fermi radii ${R}_{i}(0)=\sqrt{2{E}_{{\rm{F}}}/m{\omega }_{i}^{2}}$ are determined by the Fermi-energy E_F [76].

Having defined the initial sizes for the different regimes of interest, we obtain the size at time t by solving the differential equations for the scaling parameters b_i(t) following the treatment in [72, 73] for condensed and in [70] for non-degenerate gases in all collisional regimes. The result is illustrated in figures 4(a) and (c) in the case of ⁸⁴Sr and ⁸⁷Sr. The free expansion of the cloud in the different regimes is in each case plotted for times smaller than t_DKC denoting the application time of a delta-kick collimation (DKC) pulse. This pulse consists in re-flashing the initial trap causing a collimation of the atomic cloud [23, 24]. In the case of fermionic atoms populating a single-spin state, the cloud's expansion behavior is similar to that of a non-interacting (thermal) bosonic ensemble [76]. However, for a superposition of hyperfine states, s-wave scattering interactions are possible and the phase diagram of such gases is very rich leading to different expansion laws ranging from collisionless to hydrodynamic, BCS or unitary behavior [77]. DKC of molecular BECs [78] would give results similar to the atomic BEC case. For simplicity, we restrict the dynamics study (expansion and DKC) to the bosonic and single-spin-component fermionic cases keeping in mind that similar results can be retrieved for a superposition of hyperfine states in a fermionic ensemble. Different considerations in this study would therefore be more decisive for the bosons/fermions trade-off.

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In the absence of interactions, the physics of an expanding cloud is captured by the Liouville's theorem (phase-space density conservation) and reads

$$\begin{eqnarray}&&{\sigma }_{{v}_{f,i}}{\sigma }_{f,i}={\sigma }_{{v}_{0,i}}{\sigma }_{0,i},\end{eqnarray} \tag{ 8 }$$

${\sigma }_{0,i}={{\sigma }_{i}^{\text{th}}}_{i}(0)$ and ${\sigma }_{{v}_{0,i}}=\sqrt{{k}_{{\rm{B}}}{T}_{a}/m}$ being the initial size and velocity widths of a thermal cloud, respectively, and ${\sigma }_{f,i}={{\sigma }_{i}^{\text{th}}}_{i}({t}_{{\rm{D}}{\rm{K}}{\rm{C}}})$ is the size when the lens is applied. Evaluating this expression thus yields the minimum cloud size required at the delta-kick to achieve a certain target temperature performance ${T}_{\mathrm{eff}}$. However, interactions affect the free expansion of the cloud (hence the time of free expansion needed to reach the required size at the kick) and the residual velocity width after application of the lens. For non-degenerate gases we account for this by choosing the following ansatz for the phase-space distribution f of the ensemble:

$$\begin{eqnarray}&&f({t}_{\mathrm{DKC}}+\tau ,{x}_{i},{v}_{i})=f({t}_{\mathrm{DKC}},{x}_{i},{v}_{i}-{\omega }_{i}^{2}\tau {x}_{i}).\end{eqnarray} \tag{ 9 }$$

This approach, which is inspired by the treatment in [79], assumes that the duration τ of the lens is very small compared to the time of free expansion, such that the spatial distribution is left unchanged while the momentum is changed instantaneously by $\delta {p}_{i}=-m{\omega }_{i}^{2}\tau {x}_{i}$ when the harmonic lens potential is applied. This, combined with the free expansion of interacting, non-degenerate gases [70], gives rise to the momentum width

$$\begin{eqnarray}&&{\sigma }_{{v}_{f,i}}={\sigma }_{{v}_{0,i}}{\theta }_{i}^{1/2}({t}_{\mathrm{DKC}})\end{eqnarray} \tag{ 10 }$$

after a lens which satisfies the condition ${\dot{b}}_{i}({t}_{\mathrm{DKC}})=\tau {\omega }_{i}^{2}{b}_{i}({t}_{\mathrm{DKC}})$. The scaling parameters θ_i are the time-evolved effective temperatures in each direction and are determined, similarly to the spatial scaling parameters b_i, by solving the differential equations in [70]. It is worth noticing that this general treatment leads to equation (8) in the non-interacting case, which we also use to assess the delta-kick performance of a degenerate Fermi gas in one spin state (where interactions are absent [76]).

For BECs at zero temperature, the previous models can not be applied anymore. We employ, instead, an energy conservation model which assumes that the energy due to repulsive atomic interactions converts into kinetic energy during free expansion at a first stage. The asymptotic three-dimensional expansion rate Δ v_f after the delta-kick, in this model, stems from the residual mean-field energy and a Heisenberg term $\propto {{\hslash }}^{2}/{{mR}}_{f}^{2}$, which dominates for larger time of flights when the mean-field energy has dissipated. It reads

$$\begin{eqnarray}&&{\rm{\Delta }}{v}_{f}={\left(\displaystyle \frac{5{N}_{a}g}{2m\pi {R}_{f}^{3}}+\displaystyle \frac{14{{\hslash }}^{2}}{3{{mR}}_{f}^{2}}\right)}^{1/2},\end{eqnarray} \tag{ 11 }$$

with N_a being the number of atoms and ${R}_{f}=R({t}_{\mathrm{DKC}})$ being the size at lens [71]. We relate this expansion rate to an effective temperature via $\displaystyle \frac{3}{2}{k}_{{\rm{B}}}{T}_{\mathrm{eff}}=\tfrac{m}{2}{\left({\rm{\Delta }}{v}_{f}/\sqrt{7}\right)}^{2}$ [25, 80] and restrict ourselves to the isotropic case for simplicity.

After the application of the delta-kick pulse, we assume a linear expansion during the interferometry sequence lasting 2T. The full size L(2T) of the cloud at the end of the sequence is then given in all regimes by

$$\begin{eqnarray}&&L(2T)=2\sqrt{{L}_{f}^{2}+{\left(2T{\rm{\Delta }}v\right)}^{2}},\end{eqnarray} \tag{ 12 }$$

with L_f = σ_f, ${\rm{\Delta }}v={\sigma }_{{v}_{f}}$ in the non-degenerate regimes and L_f = R_f, ${\rm{\Delta }}v={\rm{\Delta }}{v}_{f}$ for condensed ensembles. In what follows, indices relative to spatial directions are left since we, for simplicity, chose to treat isotropic cases. With the models adopted above, we show in the tables 6 (non-degenerate gases) and 7 (quantum degenerate ensembles) the characteristic figures for the various regimes for a given asymptotic target expansion temperature of 10 pK. The minimum required cloud sizes are printed in bold and are depicted in figures 4(b) and (d), along with the size at the end of the interferometric sequence. The extent over which state-of-the art magnetic and optical potentials can be considered harmonic is typically limited to a few mm in the best case. This operating range is a decisive criterion for the choice of the initial cloud temperature and density configuration.

Table 6.  Ensembles sizes compatible with the 10 pK expansion rate requirement for classical gases in the collisionless and hydrodynamic regimes. The characteristics of the considered experimental arrangement are stated in the six first rows of the table. The computed resulting sizes are given, after the treatment of section 6.2, in the next rows. Of particular importance for the trade-off performed in this paper, are the sizes at lens (bold) and the final detected sizes for several interferometry times 2T. The larger these sizes, the harsher the requirements are on the DKC and interferometry pulses.

3D expansion rate ${T}_{\mathrm{eff}}=10$ pK	Collisionless		Hydrodynamic
	174Yb	84Sr	174Yb	84Sr
Number of atoms	5 × 108		5 × 107
Trapping frequency (2π Hz)	25		50
Initial temperature (μK)	10		0.83
Initial size 2σ0 (μm)	393.77	566.91	58.03	83.22
Knudsen number [75]	0.28	0.42	0.06	0.09
Phase space density	$\lt {10}^{-3}$		0.6
Pre-DKC expansion time (tDKC) (ms)	6359		924
Size at lens 2σ(tDKC) (mm)	393.32	566.27	16.73	23.99
Final size $2\sigma ({t}_{{\rm{D}}{\rm{K}}{\rm{C}}}+2T)$ (mm)
T = 40 s	393.34	566.29	17.09	24.51
T = 100 s	393.42	566.41	18.88	27.09
T = 160 s	393.57	566.63	21.81	31.33

Table 7.  Ensembles sizes compatible with the 10 pK expansion rate requirement for quantum degenerate regimes. The entries of the table are the same than 6. For BECs and DFGs the computed sizes are dramatically smaller than the thermal counterparts.

3D expansion rate ${T}_{\mathrm{eff}}=10$ pK	BEC		DFG
	174Yb	84Sr	171Yb	87Sr
Number of atoms	7 × 106		7 × 106
Trapping frequency (2π Hz)	50		50
Critical temperature (μK)	0.431		0.834
Initial size 2R0 (μm)	30.2	41.8	56.86	81.86
Pre-DKC expansion time (tDKC) (ms)	63	61	460	460
Size at lens $2R({t}_{\mathrm{DKC}})$ (mm)	0.50	0.67	8.21	11.82
Final size $2R({t}_{\mathrm{DKC}}+2T$) (mm)
T = 40 s	9.27	13.34	12.86	18.51
T = 100 s	23.15	33.32	26.07	37.53
T = 160 s	37.03	53.31	40.43	58.20

As a conclusion to this section with respect to the required size at lens, we find out that degenerate ensembles thanks to their point-like initial extension are largely favored. The availability of traps with significantly larger harmonic extent could eventually make the use of a non-degenerate gas in the hydrodynamic regime feasible in the future. Another possibility in to use classical gases through a velocity selection stage, which, however, is always accompanied by a substantial loss of atoms and typically reduces the velocity spread in one dimension only.