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Quantum Gases in Optical Boxes

Advances in light shaping for optical trapping of neutral particles have led to the development of box traps for ultracold atoms and molecules. These traps have allowed the creation of homogeneous quantum gases and opened new possibilities for studies of many-body physics. They simplify the interpretation of experimental results, provide more direct connections with theory, and in some cases allow qualitatively new, hitherto impossible experiments. Here we review progress in this emerging field.

Since the earliest days of ultracold atomic gases, the success in using them to study many-body physics [1][2][3] has owed a lot to the possibility to trap the atoms in versatile potentials, including low-dimensional traps [4], double wells [5], and optical lattices [6][7][8]. The electromagnetic trapping potentials are often also dynamically tuneable, which has allowed experiments ranging from studies of elementary excitations [9][10][11] to reversible crossing of phase transitions [12,13].
More recent advances in the shaping of optical potentials have opened many new possibilities. One major development is the increasingly popular use of the uniform (flat-bottom) optical-box traps [14][15][16][17][18][19], as opposed to the traditionally used (optical or magnetic) harmonic ones. Making a quantum gas homogeneous almost invariably makes the interpretation of the experiments and the comparisons with theory more direct and easier. Moreover, in some cases it allows qualitatively different experiments and discoveries of new physics.
In this brief review we outline the developments that have allowed the creation of box traps and highlight some of the scientific successes in this new and growing field.
Before starting, we also draw the reader's attention to contemporary reviews on two related emerging fields -(i) the creation of 'atomtronics' circuits for coherent matter waves, such as ring traps that support persistent currents [20], and (ii) the trapping of individual atoms or molecules in arrays of optical tweezers [21]. All three fields take advantage of the advances in light shaping, and there are also scientific connections; as a prominent example, the dynamics of phase transitions in a homogeneous system have been studied in ring traps [22], 2D and 3D box traps [15,23], and a 1D tweezers-array [24].

MAKING BOX TRAPS
In this section we briefly introduce the key tools (Fig. 1a) and ideas involved in the making of optical box traps. These methods have been used to create boxes for Bose and Fermi atomic and molecular gases in various dimensionalities [14][15][16][17][18][19], as illustrated in Fig. 1b; see also Refs. [26,27] for early prototypes of 1D box traps, and Ref. [28] for a comprehensive review of recent advances in light shaping.
Optical box traps can be made using either red-detuned (attractive) or blue-detuned (repulsive) far-off-resonant optical fields, but the latter approach is generally simpler. In 'red traps', to create a uniform trap floor one needs to sculpt high intensity light such that the variations in the optical potential are smaller than all the relevant energy scales in the gas. On the other hand, in 'blue traps', the floor is dark and the sharp walls confining the cloud just need to be high enough. To make a 3D box potential, one must also levitate the particles against gravity. This can be done using a light field of linearly varying intensity [29], but usually atoms are levitated using a static magnetic field gradient [14], while polar molecules can be levitated using a static electric field gradient [19].
Regarding light-shaping, many key developments have been enabled by two complementary types of programmable spatial light modulators (SLMs) -the liquid-crystal SLMs that modulate the phase of laser beams and the Digital Micromirror Devices (DMDs) that modulate their amplitude.
A liquid-crystal SLM is a rectangular array of ∼ 10 6 pixel elements (each ∼ 10 µm in size) with individually controllable indices of refraction. Using it to imprint a spatially-modulated phase delay on a laser beam, one controls the intensity pattern in the conjugate (Fourier) plane. To create an arbitrary pattern in the conjugate plane, one should in general modulate both the phase and the amplitude of the beam, but various numerical algorithms have been developed for creating good approximations of arbitrary trapping potentials using only phase control [28]. Moreover, in some cases we have analytical solutions for the required phase modulations. An example of this is the often-used cylindrical 3D box [14] 'assembled' using one hollow-tube beam and two end-caps sheet beams (see Fig. 1a,b) -in this case all three beams can be created starting with a single Gaussian beam and using a single SLM to imprint an analytically calculated phase pattern [30].
Similarly, a DMD is a rectangular array of ∼ 10 6 mirrors (each ∼ 10 µm in size) that can individually be turned 'on' or The first 3D box trap [14] was generated using a phase-controlling spatial light modulator (SLM) and a blue-detuned laser to create the box walls that repel and confine the atoms. As illustrated in the cartoon, a phase pattern imprinted by the SLM onto a single Gaussian beam creates three separate beams, a tube and two sheets; the three beams are then 'folded' to form a cylindrical box trap (such as shown for a 3D Fermi gas in (b)). Other experiments have used Digital Micromirror Devices (DMDs) that spatially modulate the amplitude of a laser beam. While these versatile spatial modulators were instrumental for the development of first box traps, in many cases one can also create them using more traditional tools, such as specialised fixed (non-tuneable) optics. (b) Box-trapped gases have been realised for Bose and Fermi atoms in various dimensionalities, as well as for (fermionic) ground-state diatomic molecules in 3D [14][15][16][17][18][19]. One can also create other geometries, such as the two-trap configuration shown in the left panel for the 3D Bose gas; in this first experiment, this was used to demonstrate the coherence of the gas through matter-wave interference (after release and free expansion) seen in the right panel. For the 1D Bose gas the figure directly compares the line density distribution in a box trap (blue) and a harmonic one (orange). (a) One simple measure of the gas homogeneity is the distribution of the real-space densities. Here we show the distribution of column densities, n 2D , extracted from in situ images of 3D clouds [16]. In a box trap (blue) the probability distribution P (n 2D ) is narrow, strongly peaked near the average value n 2D . This means, for example, that most atoms experience essentially the same mean-field potential. For comparison, the corresponding distribution in a nondegenerate harmonically trapped gas (red) is very broad; the expected distribution is uniform between 0 and 2n 2D . (b) A complementary way to characterise box traps is to look at the single-particle density of states. This is seen, for example, in the dependence of the critical atom number for Bose-Einstein condensation, N c , on the temperature T [25]. In a 3D harmonic trap N c ∝ T 3 , while in a perfect 3D box N c ∝ T 3/2 . Here the experimental data is captured by N c ∝ T α with α = 1.65 (red line). A common way [14] to characterise (imperfect) box traps is to model them by an isotropic power-law potential V (r) ∝ r p with p 1. Then α = 3/2 + 3/p, and for a Fermi gas one similarly gets that the Fermi energy is [16] E F ∝ N 1/α . In current experiments p > 10 is readily achieved, and there are indications that values of p up to ∼ 100 are feasible [17].

Box 1: Characterizing box traps
'off' (by changing their tilt angle) to spatially modulate the amplitude of a laser beam. An arbitrary intensity pattern can then be imaged onto the cloud [31], which is convenient for making rectangular traps with sharp corners in 2D (Fig. 1b) or cuboid traps in 3D. The two approaches have different advantages. The liquidcrystal one is more power-efficient, because the light is steered rather than partially dumped (by the 'off' DMD mirrors). On the other hand, DMDs are much better for creating dynamical potentials. Due to their (currently) sub-kHz refresh rate, the liquid-crystal SLMs cannot be reprogrammed and updated with a different phase pattern during an experimental run, since this would temporarily turn off the trap and release the cloud. Meanwhile, the ∼ 10 kHz refresh rate of DMDs is sufficiently high for them to be reprogrammed and the trapping pattern dynamically changed without the ultracold particles moving significantly during the updates.
The versatility of SLMs has been essential for experimental exploration [30], but box traps can also be made with nontuneable tools [28] such as axicons [16,17,32] and custommanufactured masks [15]. Another option is to use 'painted', time-averaged potentials created by fast spatial scanning of a laser beam [28,33].
As a final point in this section, optical box traps are not perfect. The sharpness of their walls is limited by the optical wavelength of ∼ 1 µm, which is not negligible compared to the typical box dimensions 10−100 µm. Moreover, additional fields used for levitation, tight out-of-plane confinement of 2D samples [15,34], or tuning of interactions via Feshbach resonances [35], can lead to further imperfections. In Box 1, we discuss two complementary methods used to quantify the uni-

FIG. 2. Quantum statistics in homogeneous gases. (a)
In situ absorption images of homogeneous quantum-degenerate 3D gases with different phase space densities D > 1. The gas with lowest density (lowest optical density OD) is the coldest and actually has the highest D, but one cannot tell. One cannot even tell whether the gases are fermionic or bosonic (here they are fermionic 6 Li gases). (b) While no effects of quantum statistics and degeneracy are seen in real space, they are striking in momentum space. For a Bose gas, the statistical nature of the BEC phase transition, driven by the saturation of the excited states, is revealed more clearly than in the corresponding harmonic-trap experiments [36]. Here the plot shows the total number of atoms in the thermal cloud (N ) and the condensate (N 0 ), as the number of atoms in the cloud (N ) is varied at constant temperature [25]; the inset shows an image of a partially condensed gas after release from the box and free expansion. For a Fermi gas below the Fermi temperature T F , the effects of the Pauli exclusion principle are clearly observed -the occupation of individual momentum states are limited to n k ≤ 1, and the Fermi surface forms at the Fermi wavevector k F (adapted from [16]).
formity of box-trapped gases. The currently achieved uniformity has been good enough for the success stories we discuss in the next section, but how close to perfect a box needs to be ultimately depends on the specific scientific problem.

SUCCESS STORIES
Here we outline the scientific advances afforded by homogeneous atomic gases, which also illustrate the general types of problems for which the box traps are advantageous.
We start with experiments on purely quantum-statistical phenomena (Fig. 2). In harmonic traps the real and momentum space are coupled, so one can for example observe realspace effects of Bose-Einstein condensation (BEC) [37] and Fermi pressure [38]. In a box trap, the two spaces are disentangled and there are no clear signatures of quantum statistics in real space (see Fig. 2a), but in momentum-space they are revealed more cleanly (Fig. 2b).
For bosons, one clearly observes the statistical nature of the BEC transition, which is driven by the saturation of the total occupation of all excited (momentum) states [25]. As the total atom number in the gas is increased at a fixed temperature, the number of atoms in the thermal cloud saturates at the critical value for condensation, and all the extra atoms accumulate in the condensate. For fermions, one instead observes the occupation-number saturation at the level of individual singleparticle states, as prescribed by the Pauli exclusion principle. As the temperature (normalised to the Fermi temperature T F ) is reduced, the occupation of all momentum states remains bound to n k ≤ 1 and the Fermi surface forms [16].
Experiments on the thermodynamics of nearly-ideal boxtrapped gases also lead to the unexpected observation of the Joule-Thomson effect that arises solely from quantum correlations [25]. While the ideal classical gas does not change temperature under isoenthalpic rarefaction, the ideal Bose gas cools and the ideal Fermi gas is expected to heat [39].
We now move on to the experiments on the many-body physics of interacting gases, starting with the broad class of spectroscopic and transport measurements in which one weakly perturbs a system in order to extract information on its equilibrium properties (Fig. 3).
A lot of attention has been given to long-wavelength sound waves [40][41][42][46][47][48][49][50]. In Fig. 3a we show examples of sound propagation in 3D Fermi [40] and 2D Bose [41] gases, and in Fig. 3b we illustrate two scientific highlights of such experiments. In both 3D and 2D unitary Fermi gases the quantum limit of sound diffusivity, set by /m (where m is the atom mass), was demonstrated [40,49]; this universal limit should also be relevant for other strongly interacting Fermi systems such as neutron stars. In 2D, superfluidity emerges via the infinite-order Berezinskii-Kosterlitz-Thouless (BKT) phase transition [51,52] instead of the BEC one; the 2D box trap allowed the first observation of first and second sound in a BKT superfluid [42], and the measurements of the two sound speeds revealed the universal superfluid-density jump at the BKT transition [53]. Related experiments on superfluidity have investigated the Josephson effect [54] and the critical velocity [55] in a 2D Fermi gas, while a more complex trapping geometry allowed studies of a Bose-gas superfluid flow through a constriction between two reservoirs [56]. Fig. 3b illustrates benefits of gas homogeneity for spectroscopic measurements that globally probe the system [43][44][45][57][58][59][60][61]; for experiments on extraction of homogeneousgas properties by local probing of harmonically trapped gases see [62][63][64][65]. The Rabi radio-frequency (rf) spectra shown in Fig. 3b(i) measure the energy cost of removing a particle from a spin−1/2 3D Fermi gas at different reduced temperatures T /T F [43]. Measurements were performed on the whole sample and the spectra taken at closely-spaced T /T F values are clearly distinct only thanks to the lack of inhomogeneous broadening; in a harmonic trap global measurements would mix signals for a wide range of T /T F (r) ∝ n(r) −2/3 , where n(r) is the local density. Rabi rf spectroscopy of 3D Fermi gases [43,60] has, for example, provided an observation of non-Fermi-liquid behaviour in a normal strongly interacting gas [43], while Ramsey rf spectroscopy of 2D Bose gases has provided measurements of short-range correlations across the BKT transition (Fig. 3b(ii)) and an observation of magnetic-

FIG. 3.
Sound and spectroscopy measurements on box-trapped gases (a) Sound: (i) Low-energy sound modes can be probed by perturbing the gas with an external potential and observing the evolution of the resulting density modulations in time and space. Images show examples of measurements in a 3D Fermi gas [40] (top) and a 2D Bose gas [41] (bottom). (ii) Top: the sound diffusivity D (seen in the attenuation of the wave) in the low-temperature unitary Fermi gas approaches the universal quantum limit, D ≈ /m; the red line indicates the critical temperature for superfluidity (adapted from [40]). Bottom: the superfluid phase-space density D s in a 2D Bose gas, which was deduced from the measured speeds of first and second sound, undergoes a universal jump from 0 to 4 at the BKT phase transition; here D is the total phase-space density and D c its critical value [42]. (b) Spectroscopy: (i) Particle-ejection spectroscopy on the unitary 3D Fermi gas [43]. Rabi rf spectroscopy was performed on the whole cloud and the differences induced by small changes in T /T F are observable only due to the lack of inhomogeneous broadening. Such measurements probe short-range correlations and have also revealed non-Fermi-liquid behaviour in the normal unitary gas, at T > T c . (ii) Ramsey rf spectroscopy on a 2D Bose gas was used to determine the two-body contact (C ) as a function of D; the shaded region indicates the non-superfluid phase (adapted from [44]). (iii) Bragg spectroscopy was used to measure the excitation spectrum in a strongly interacting 3D Fermi gas, here shown on the BEC (top) and the BCS (bottom) side of the BEC-BCS crossover [45]. dipole interactions [44,61].
Gas homogeneity is similarly beneficial for Bragg spectroscopy [66,67]. Fig. 3b(iii) shows measurements of the excitation spectrum of a strongly interacting 3D Fermi gas, which were used to extract the concavity of the dispersion relation and the pairing gap in the BEC-BCS crossover [45]. Bragg-spectroscopy experiments on condensed homogeneous 3D Bose gases have provided an observation of Heisenberglimited long-range coherence [57], the confirmation of Bogoliubov's theory of quantum depletion [59], and an observation of the breakdown of Bogoliubov's theory of the excitation spectrum for sufficiently strong interactions [58].
In another large class of experiments, interacting gases have been driven or quenched far from equilibrium (Fig. 4).
One paradigmatic topic in non-equilibrium physics is turbulence in strongly driven systems. Depending on the system and the excitation protocol, turbulent dynamics can be dominated by either waves or vortices, and the advent of box traps has led to new insights in both cases (Fig. 4a). In 'shaken' 3D Bose gases the power-law momentum distribution characteristic of a direct turbulent cascade was observed [46], and the elusive particle and energy fluxes through the cascade have been measured [68]. In (quasi-)2D Bose gases vortex clustering corresponding to negative temperatures was observed [69,70] (see also [73,74]), as predicted by Onsager in 1949 [75]. In another recent experiment the interplay of vortices and waves has also been observed [76].
Another major topic for which homogeneous gases have distinct advantages is the study of the diverging-range correlations associated with the critical behavior near second-order phase transitions (Fig. 4b and Box 2). In a non-equilibrium context, dynamic crossing of such a transition results in causally disconnected domains that display different choices of the symmetry-breaking order parameter. The Kibble-Zurek (KZ) theory [77,78] that describes these dynamics was originally developed specifically for homogeneous systems and its key assumption is that when it comes to the choice of the order parameter all parts of the system have 'equal voting rights' [79,80]. Box-trap experiments have provided quantitative tests of the KZ predictions for how the domain size [23] and the resulting density of defects in the ordered state [15] depend on the quench rate (see also [22,24,81] for ring-trap and optical-tweezers experiments).
In other non-equilibrium experiments, the excitation spectrum of a homogeneous 1D Bose gas allowed the observation of recurrences in a closed quantum system [71,86] (Fig. 4c), while homogeneous 3D Bose gases allowed observations of the universal loss and prethermalization dynamics following a quench to unitarity [87,88] and the universal dynamic scaling in a far-from-equilibrium weakly interacting gas [89]. 4. Non-equilibrium phenomena (a) Wave and vortex turbulence. Top: power-law momentum distribution,ñ k ∝ k −γ , in a 3D Bose gas driven on a large lengthscale reveals a wave-turbulence cascade towards smaller lengthscales [46,68]. Bottom: experiments on vortex turbulence in quasi-2D Bose gases showed large-scale vortex clustering, which was observed using direct absorption imaging [69] (left) and circulation-sensitive imaging [70] (right). (b) Critical dynamics. (i) Critical slowing down results in nonadiabatic crossing of the BEC transition and formation of domains with different spontaneously chosen condensate phases (indicated by the arrows). (ii) Topological defects form at the domain boundaries; here the image shows vortices spontaneously generated in a quench-cooled gas [15]. (iii) The power-law dependence of the average domain size, , on the quench time, τ Q , is in agreement with the Kibble-Zurek theory [23]. (c) Recurrences of phase correlations were observed in a 1D Bose gas [71]. (d) A novel breather was seen in a 2D Bose gas; for particular initial density distributions, such as a uniform triangle prepared in a box trap, the cloud evolving in a harmonic potential periodically returns to its initial state [72].
Finally, box-trapping and related technologies have also facilitated many experiments that are less directly related to the physics of homogeneous gases. One example of this is the discovery of a novel breather in a 2D Bose gas [72]; the breather shown in Fig. 4d is observed in a harmonic potential, but the initial state had to be prepared in a box trap [72]. Another similar example is the deterministic preparation of a Townes soliton in Ref. [90] (see also Refs. [91,92] for other observations of Townes solitons in box traps); this 2D soliton is an inhomogeneous ground state of the system, but its deterministic preparation was based on starting with a homogeneous gas and imprinting arbitrary density profiles using a DMD [93]. A different example of a practical advantage of box traps is the observation of the transition from an atomic to a molecular condensate [94]; in this case the creation of a (quasi-)equilibrium condensed gas of unstable molecules was facilitated by the use of a 2D box trap to minimise losses and heating. Further examples of box-enabled experiments include observation of the weak collapse of a condensate with attractive interactions [95] and the studies of the modulation instabilities that lead to emission of matter-wave jets, pattern formation and quasi-particle pair-production [96][97][98][99].

OUTLOOK
The scientific exploitation of box-trapped quantum gases is still in its infancy. Here we highlight some exciting possibilities for the future, and also some open technical challenges.
The successful studies of phase-transition dynamics could be extended to the infinite-order BKT transition [100][101][102][103][104] and the bubble-nucleation dynamics associated with firstorder transitions, including some believed to be relevant for the physics of the early universe [105][106][107][108]. Another general area where box traps could offer great advantages is topological physics [109,110]; sharp boundaries could allow realspace studies of edge states [111], which have so far been observed in cold-atom systems exploiting synthetic dimensions associated with internal (spin) degrees of freedom [112][113][114]. It has also been predicted [115] that the supersolid phases of gases with strong dipolar interactions [116][117][118][119][120][121] should be qualitatively different in a box trap.
Further possibilities are offered by combining box traps and other trapping methods. For example, combining box traps and optical lattices has already facilitated the observation of a low-entropy homogeneous state with long-range antiferromagnetic correlations [122] and studies of competing magnetic orders in the bilayer Hubbard model [123]. Further possibilities are suggested by the hybrid trap of Ref. [16], which is box-like along two directions and harmonic along the third. In this case the harmonic direction provides tuning of the local chemical potential, while probing the system along a perpendicular direction retains many of the advantages of box traps, at least as long as the local density approximation (LDA) is valid. Such a setup could be used to study interfaces between different phases of matter, and could also facilitate searches for exotic states that are expected to occur only in narrow regions of bulk phase diagrams; an important example of such a still-sought-for phase is the Fulde-Ferrell-Larkin-Ovchinikov (FFLO) superfluid [124][125][126].
While the range of scientific possibilities is broad and exciting, we can already anticipate that some of them will also require further technological developments.
The first issue is that many interesting experiments are likely to require increasingly larger and closer to perfect box traps. This is particularly true for studies of critical phenomena (see Box 2), and more generally long-range correlations.

Box 2: Critical phenomena in harmonically and box trapped gases
Critical behaviour near a second-order phase transition is a paradigmatic problem for which using a homogeneous system is advantageous, because the local density approximation (LDA), used to interpret harmonic-trap experiments, fundamentally breaks down. Here we illustrate this with a simple ideal-gas calculation for the BEC phase transition. In a homogeneous system, near the critical point the correlation length ξ diverges as illustrated in panel (a). At a fixed temperature, ξ/λ = A(|n − n c |/n c ) −ν , where λ is the thermal wavelength, n c is the critical density, ν the critical exponent, and A is a non-universal prefactor; for the ideal-gas BEC transition n c = 2.612/λ 3 , ν = 1, and A = 1/2.612.
For a harmonically trapped gas with spatially uniform T , as n(r = 0) → n c , the spatially varying ξ(r ) evaluated within the LDA exceeds the lengthscale over which it changes. Relaxing the usual LDA assumptions, one can still, at the cost of reducing the experimental signal, focus on the central part of the cloud and assume that n is constant within some non-infinitesimal volume (similar to Refs. [62,63,65,82]). In reality n and ξ vary within this region, but one can still directly probe correlations on a lengthscale if ξ(r ) > for all r < /2; this approach was successfully used in Ref. [83]. Setting n(r = 0) = n c , assuming an isotropic trap of frequency ω, and expanding the ideal-gas distribution [1] near r = 0 gives ξ(r )/λ ≈ k B T /( ω) × [2πr /λ] −1 . Noting that k B T /( ω) = (N /1.202) 1/3 , where N is the total number of atoms in the trap, in panel (b) we plot ξ/λ versus r /λ for different N (black lines). The intersects of these curves with the line ξ = 2r (red) then give the achievable = 2r for a given N , irrespective of the choices of ω and corresponding T . Conversely, the atom number needed to directly observe correlations on a lengthscale in a harmonic trap is shown by the red line in panel (c). With a typical N ∼ 10 6 , one can reach only ∼ 5 λ.
On the other hand, working with a box trap, one just needs a box of size and the corresponding atom number shown by the green line in panel (c). In this case the same N ∼ 10 6 is sufficient for measurements up to ∼ 70 λ.
As an illustration, a paradigmatic problem for which the current technology is still marginal is that of the critical temperature for Bose-Einstein condensation in an interacting homogeneous gas [127][128][129][130][131][132]. Critical fluctuations in a repulsivelyinteracting gas are predicted to raise T c above the ideal-gas value. However, in a harmonic trap one observes the opposite [133][134][135][136][137] -the beyond-mean-field correlation shift of T c is diminished because only a small fraction of the cloud is critical at T c , and it is overpowered by a geometric mean-field (MF) effect [138] that reduces T c . For a general power-law trap, V (r ) ∝ r p (see Box 1), with increasing p the beyond-MF shift should be more pronounced and the MF one should diminish. Based on an LDA estimate, we find that for the currently typical p ∼ 10 the two effects are still comparable, and that one needs p 100 to cleanly observe the beyond-MF correlation shift of T c . We expect other fundamental correlation-physics problems to similarly create a moving target for the tolerable box imperfections.
The second issue is that many exciting possibilities rely on specific features of different atomic species and their mixtures, but the methods for the levitation of gases in 3D box traps are generally species specific. In current experiments one can study mixtures of species that have very similar ratios of mass and magnetic moment [16,139], but creating arbitrary homogeneous mixtures of different chemical elements, different isotopes, or even just different spin states of the same isotope, is an open challenge.
These two issues are in fact related, since even in singlespecies experiments the limitations for making the box traps larger and closer to uniform (with larger p values) are often related to the need to levitate particles with additional fields. An exciting possibility for the future would therefore be to send box-trap setups to space and perform many-body experiments in a microgravity environment [140][141][142].
We thank C. Eigen for his help in the preparation of the figures, and M. Zwierlein, P. Patel, B. Mukherjee, J. Beugnon, J. Dalibard, R. Saint-Jalm, H. Biss, T. Lompe, and H. Moritz for sharing their data. We thank ♣ for comments on the manuscript. This work was supported by the EPSRC (grant nos EP/N011759/1, EP/P009565/1, and EP/T019913/1), ERC (QBox), QuantERA (NAQUAS, EP-SRC grant no. EP/R043396/1), NSF CAREER (grant no. 1945324), and DARPA (grant no. 00010372). N. N. acknowledges support from the David and Lucile Packard Foundation, and the Alfred P. Sloan Foundation. R.P.S. acknowledges support from the Royal Society. Z.H. acknowledges support from the Royal Society Wolfson Fellowship.