Realising a species-selective double well with multiple-radiofrequency-dressed potentials

Techniques to manipulate the individual constituents of an ultracold mixture are key to investigating impurity physics. In this work, we confine a mixture of the hyperfine ground states of Rb-87 in a double-well potential. The potential is produced by dressing the atoms with multiple radiofrequencies. The amplitude and phase of each frequency component of the dressing field are individually controlled to independently manipulate each species. Furthermore, we verify that our mixture of hyperfine states is collisionally stable, with no observable inelastic loss.


Introduction
Cold atom experiments have emerged as a valuable tool to engineer many-body quantum systems [1,2]. For instance, mixtures of cold atoms have been used to study the superfluid properties of bosons and fermions [3][4][5][6][7], and the immiscibility of quantum fluids [8,9]. Impurity physics, in which a minority species interacts with a large reservoir of a second species, can be studied by immersing probes into a larger quantum system, which necessitates a means of control for separate constituents of the mixture [10]. Furthermore, the species-selective control of mixtures in double-well potentials can be used to probe several physical effects, including the excitation spectrum in a Bose-Einstein condensate (BEC) [11], the decoherence of impurities coupled to open quantum systems [12] and to realise sub-nK thermometry techniques [13][14][15]. Many optical methods of species-selective control have been investigated, including optical tweezers [16,17], holographic light fields and lattices [18][19][20]. These methods are well established but are unsuitable when the optical frequencies required to manipulate the separate components are similar, where scattering induces significant heating of the trapped mixture.
Atoms in a static magnetic field can be confined by the application of a strong radio-frequency (RF) field, forming an RF-dressed potential [21,22]. In contrast to optical traps, these potentials are smooth, have low heating rates and are free from defects; these features make them well suited to investigations of out-of-equilbrium phenomena [23][24][25][26]. For specific choices of magnetic and RF fields, these traps can also reduce the effective dimensionality of a trapped gas to one or two dimensions [26][27][28]. Additionally, irradiating atoms with multiple RF (MRF) fields significantly extends the range of possible trapping geometries, for example double-well potentials [29,30], ring traps [31,32], toroids [33][34][35] and lattices [36].
RF-dressed potentials are species-selective for mixtures of atoms with differing magnetic moments, for instance where the magnitude or sign of the Landé g-factor for each constituent is distinct [37][38][39][40]. Exploiting this feature, we implement a species-selective double well using MRF-dressed potentials and manipulate the spatial distribution of the individual mixture constituents. Recent theoretical [41] and experimental [38] studies have shown that not all mixtures are collisionally stable when confined in RF-dressed potentials. In this work, we demonstrate the merit of an RF-dressed mixture of 87 Rb F = 1 and F = 2, which we find to be long-lived.
This paper is structured as follows. In section 2 we explain the dressed-atom picture and RF-dressed potentials as a method of confinement for ultracold atoms. In section 3 we outline the experimental procedure used to produce ultracold mixtures of hyperfine states of 87 Rb. In section 4 we present species-selective manipulations of a mixture of 87 Rb F = 1 and F = 2 via control of the MRF-dressing fields amplitudes and polarisations. In section 5 we show that this mixture is collisionally stable against inelastic loss.

Potentials of RF-dressed atoms
In the low-field regime, atoms in a static magnetic field B have eigenenergies m F g F µ B B, corresponding to the Zeeman substates |m F [42], where µ B is the Bohr magneton, g F is the Landé g-factor and B = |B| [43]. The weak-field seeking states, for which g F m F > 0, can be trapped at local minima of B. For example, in the magnetic quadrupole field with field gradient B , atoms are confined around the node at the origin. When an atom is irradiated by a strong photon field, the eigenstates of the system are conveniently described using the dressed-atom formalism [44]. The eigenenergies depend intrinsically on the magnetic nature of the atom, via the sign and magnitude of the g F factor. In particular, sgn(g F ) determines the handedness of the circularly polarised RF field which couples the Zeeman substates. Atoms with negative g F couple to the σ + components of the RF field, and those with positive g F couple to the σ − components. If a mixture contains atoms of differing sgn(g F ), each species couples to different polarisation components in an applied RF field, providing a means of manipulating each species separately. For instance, for a mixture of the hyperfine ground states of 87 Rb with F = 1 and F = 2, which have g F = −1/2 and 1/2, respectively, can be separately manipulated by controlling the amplitudes of the σ − and σ + components of the applied RF field. The interaction between the atom and the RF field couples states within manifolds of constantÑ = sgn (g F ) m F + N , where N indicates the Fock state |N of the RF field.
In the rotating-wave approximation, and incorporating gravity, the eigenenergies of a dressed atom at position r = (x, y, z) take the form wherem F labels each dressed eigenstate, Ω(r) is the Rabi frequency of the dressing field, M is atomic mass, g is gravitational acceleration, z is vertical position and δ(r) = ω − g F µ B B/ is the detuning between the applied RF field with angular frequency ω and the Zeeman splitting of the atoms. The avoided crossings between eigenstates occur when the energy of an RF photon is equal to the Zeeman splitting, fulfilling the resonance condition Figure 1 a) and b) illustrate the dressed eigenenergies for the cases of an atom with angular momentum F = 1 and F = 2, respectively, for a dressing field of frequency 2.0 MHz and amplitude 2 Ω/g F µ B = 570 mG, with a magnetic quadrupole field of gradient B = 139.5 G cm −1 . States withm F = 1 are labelled in bold and are discussed in section 3. The spatial variation of the eigenenergies withm F > 0 creates a trapping potential [22,45] where atoms are trapped in states for which U has a local minimum near δ(r) = 0, as indicated by the dashed lines in figure 1. For atoms in the static quadrupole field of equation (1), the resonance condition equation (3) is satisfied on the surface of an oblate spheroid centered on the origin, forming a 'shell' on which atoms are confined. Under the influence of gravity, atoms collect around the lowest point on this shell. More complex potentials, such as double-wells, can be engineered by increasing the number of dressing RF frequencies, as shown in figure 1 c) [29]. In section 4, we present the realisation of a species-selective double-well potential.

Experimental methods
Our experimental sequence follows that described in [29]. We produce cold atomic gases of 87 Rb atoms in the state F = 1,m F = 1, which are trapped in a single-RFdressed potential and have been evaporatively cooled to approximately 0.4 µK. We implement our double-well potential by applying multiple RF fields using currentcarrying coil pairs, illustrated in figure 2 a) (purple coils), which surround the ultrahigh-vacuum glass cell.
To produce a mixture of hyperfine states, we apply a pulse of microwave (MW) radiation using a patch antenna which is resonant with the hyperfine splitting of 87 Rb at approximately 6.8 GHz. The MW radiation is sourced from a commercial synthesizer ‡ in series with a 20 W amplifier §. A pulse at 6.83468 GHz transfers a fraction of atoms from the F = 1,m F = 1 state to the F = 2,m F = 1 state. The fraction of atoms transferred, and hence the relative densities of the two states, can be controlled via the MW pulse duration. We identify the relevant transition from a spectrum of MW transitions, as shown in figure 2 e), which is produced by applying a 40 ms MW pulse at a given frequency and measuring the resulting atom number in the F = 2 state. The features of this spectrum have been investigated extensively [46].
We image both states in the same experimental sequence and, unless stated otherwise, while they are confined by the trapping potential (in situ). First, we image the F = 2 cloud using light resonant with the F = 2 to F = 3 transition in the D2 manifold (cooling light) [47], where F denotes the angular momentum of the excited state. F = 1 atoms are dark to this transition and are unaffected during this first image. F = 1 atoms are then optically pumped to F = 2 using light resonant with the F = 1 to F = 2 transition in the D2 manifold (repumping light) immediately before they are imaged with cooling light. The two images are taken approximately 6 ms apart.

A species-selective double-well potential
We now describe our species-selective double-well potential, which we demonstrate for a mixture of the 87 Rb hyperfine ground states with F = 1 and F = 2. We express the multiple-RF dressing field as with frequency components i = 1, 2, ...N , at frequencies ω i . A i and B i correspond to the circularly polarised RF field amplitudes that couple states with F = 1 and F = 2, respectively. The sum and difference of A i and B i correspond to the magnitudes of the linearly polarised fields aligned along the e x and e y Cartesian axes, respectively, which are produced by current-carrying coils, as illustrated in figure 2 a). ‡ DS Instruments SG12000PRO § Microwave Amps, AM53-6.4-7-43-43 To create our double-well potential, we apply 3 RF components with frequencies ω i /2π = {1.8 MHz, 2 MHz and 2.2 MHz} and amplitudes (A i +B i ) = {330 mG, 370 mG and 440 mG}, which are linearly polarised along e x . Each RF field component i creates an avoided crossing near the position r where the species-dependent resonance condition ω i = g F µ B B(r) is satisfied, as shown in figure 1 c). The RF field at 2 MHz defines the 'barrier' of the double-well potential. Changing the polarisation of this field enables the barrier height to be independently controlled for each species, for instance, to raise the barrier for one species while simultaneously lowering it for the other. We achieve this by the addition of an RF field component at 2 MHz, linearly polarised along e y and with amplitude (B 2 − A 2 ). Figure 3 a-c) show the deformation of the potentials as the polarisation is changed. Changing the balance of (A 2 + B 2 ) and (A 2 − B 2 ) alters the amplitude of the σ + and σ − components, independently modifying the potential energies for F = 1 and F = 2. Figure 3 d) and e) show vertical slices of absorption images of the trapped F = 1 and F = 2 clouds, respectively, as the barrier polarisation is controlled. Both species display the expected spatial distribution as the central RF component is controlled, with the separation in position clearly illustrated. When no RF field component at the frequency of the barrier is applied along e y , the polarisation is linear and the coupling strength for F = 1 and F = 2 states is equal. As a consequence, raising or lowering the barrier via the amplitude of the linearly polarised 2 MHz RF field along e x modifies the eigenstates equally for both species. Figure 3 f-h) show the deformation of the potentials as the barrier field amplitude is reduced from 500 mG to 180 mG. Figure 3 i) and j) show vertical slices of absorption images of the trapped F = 1 and F = 2 clouds, respectively, as a function of the barrier amplitude A 2 + B 2 , which demonstrates the equivalence of potentials for both species. In this section we have demonstrated a species-selective potential which is engineered via the amplitude and polarisation of dressing RF fields. In particular, we demonstrated combinations of single-and double-well potentials for the two mixture constituents. Furthermore, these methods are applicable to any mixture for which the magnetic nature of the constituents, as described by the Landé g-factors, is distinct.

Collisional stability of the RF-dressed mixture
The species-selective manipulations we have demonstrated are only of practical use if the mixture does not suffer large inelastic losses. Recent work concerning a mixture of 85 Rb and 87 Rb indicates that most RF-dressed mixtures are expected to undergo fast inelastic loss [38]; the atom-photon coupling allows inelastic collisions to conserve angular momentum through absorption or emission of an RF photon, allowing spin exchange to occur with large rate coefficients [38,41]. Fortunately, for collisions of 87 Rb atoms, the rate coefficients for spin exchange are small, and it proceeds slowly even for cases that are allowed by angular momentum conservation rules. This special property of this particular isotope has previously been credited to the similarity of the singlet and triplet scattering lengths [48]. We note that mixtures of 87 Rb hyperfine states have been used for interferometry in an RF-dressed potential [40,49,50]. Inelastic collisions of this mixture of RF-dressed atoms have also been investigated in the context of RF-dressed Feshbach resonances [51], however there has been no experimental study of the dependence of inelastic loss processes on the strength of the dressing RF field.
To verify the collisional stability of this mixture when RF-dressed, we have measured the loss rates of F = 1 and F = 2 atoms when confined in an RF-dressed potential, first separately and then together. For a pure cloud of atoms in the F = 1 state, the atom number evolves as where k i,j 2 and k i,j,k 3 are the rate coefficients for two-body and three-body inelastic collisions of atoms in states {i, j, k} ∈ {1, 2}, where {1, 2} represent the dressed states F = 1,m F = 1 and F = 2,m F = 1, respectively. For the decay of F = 1 alone described above, i, j, k = 1. When inelastic loss is negligible, the terms with k 2 and k 3 can be discarded, and the atom number decays exponentially. Figure 4 a) (inset) illustrates the decay of F = 1 alone in the trap for three dressing RF amplitudes. We observe an exponential decay of atom number which shows that inelastic loss is negligible at our densities of 1.6 × 10 11 cm −3 , and places bounds on the rate coefficients of k 2 < 1.6 × 10 −14 cm 3 s −1 and k 3 < 7.7 × 10 −27 cm 6 s −1 . An imperfect vacuum in the chamber and technical noise in the apparatus are the dominant causes of the observed decay for this single species.
For a mixture of atoms in the states F = 1 and F = 2, the number of F = 1 atoms evolves as: This choice of states ensures both clouds have approximately the same gravitational sag and spatially overlap [37]. Figure 4 a) & b) show the atom numbers of F = 1 and F = 2, respectively, as a function of hold time, when both species are present in the mixture. The initial densities of atom clouds with F = 1,m F = 1 and F = 2,m F = 1 are controlled via the MW pulse duration, which determines the number of atoms transferred from the state with F = 1 to F = 2. In this investigation, we vary the initial densities of atoms with F = 1 and F = 2 between 4 − 16 × 10 11 cm −3 and 1 − 2 × 10 11 cm −3 , respectively. To measure the inter-species rate constant it is sufficient to consider either loss of a small number of F = 1 atoms in a F = 2 bath, or to reverse the role of the species. As RF field strength has been predicted to modify the rate coefficient [41], we also repeat the measurements for three dressing field amplitudes, namely 290 mG, 570 mG and 940 mG. The decay is clearly exponential for all measurements, for both values of F and for all three field strengths. This implies that inelastic loss remains negligible for these species over our range of initial densities and within the upper bounds of k 2 < 2.2 × 10 −14 cm 3 s −1 and k 3 < 1.2 × 10 −25 cm 6 s −1 .
To further illustrate the lack of dependence on density, figure 4 c) shows rate coefficents for each dressing RF field amplitude and for a range of F = 2 densities. Coloured boxes illustrate the 1/e lifetimes and their uncertainties, which are extracted from the fitted rate constants 1/α, for the F = 1 population in the absence of a coexisting F = 2 cloud (figure 4 a) (inset)). A surprising feature is that the lifetime of F = 2 vs. the density of F = 1, as shown in figure 4 d), rises as the density of F = 1 is increased. We speculate this is due to a temperature dependence of the lifetime 1/α; longer microwave pulses are required to transfer more atoms from F = 1 to F = 2 and these pulses induce heating in the sample. As we begin with F = 1 and transfer a fraction to F = 2, rate coefficients for F = 2 only in the RF-dressed potential are not determined. Additionally, lifetimes and initial densities of atoms with F = 2 have a larger standard error than those for F = 1; this occurs because the number of atoms with F = 2 is small compared to those with F = 1, even at maximum, as the imperfect MW transfer leads to atom loss and heating.

Conclusion and Outlook
We have experimentally realised species-selective manipulations of an atomic mixture using MRF-dressed potentials, which are made possible by the different g F factors of the constituents. We have demonstrated that combinations of single-and doublewell potentials for mixture constituents are made possible through control of the polarisations and amplitudes of dressing RF fields. Furthermore, the scales of spatial variation can be easily tuned by the choice of dressing field frequencies and magnetic field gradient. In a previous experiment, MRF-dressed potentials were used to realise a double-well potential with well separations of tens of µm [29], and we have since reduced this separation to 2.5 µm [52].
We do not observe inelastic loss for a mixture of 87 Rb atoms in RF-dressed states with F = 1 and F = 2 over the range of densities and dressing field strengths used in these experiments. The long lifetimes of RF-dressed mixtures of 87 Rb hyperfine states make this mixture highly promising for future experiments, such as probing impurity physics or non-equilibrium dynamics. In addition, the lifetime remains favourable for There exists a small difference in |g F | between these species and, consequently, there is a minor separation between the spatial locations of the potential minima [40]. In our context, however, this separation is negligible compared to the spatial extent of the trapped clouds. Decay of the number of atoms in the F = 1 and F = 2 states. Results are shown for clouds held in a single-frequency, linearly polarised RFdressed potential, for Rabi frequencies of 290 mG ( ), 570 mG ( ) and 940 mG ( ). a) Measurements of F = 1 atom number against hold time, in contact with atoms of F = 2. The lines indicate best-fit curves for exponential decay. Inset: exponential fits of F = 1 atom number vs. hold time for three Rabi frequency shells for atoms with F = 1 alone. b) Measurements of F = 2 atom number against hold time, in contact with atoms of F = 1. c) Measured 1/e lifetimes for F = 1 as a function of the initial number density of F = 2. Shaded regions represent the lifetime of F = 1 atoms alone when trapped in an identical potential. d) Measured 1/e lifetimes for F = 2 as a function of the initial number density of F = 1. The vertical error bars in both plots correspond to the uncertainty in the fitted rate coefficient, while horizontal error bars indicate the uncertainty in the initial number density.
clouds of higher atomic number density. A preliminary measurement has shown that the lifetime of F = 1 and F = 2 BECs is of order seconds, although immiscibility may reduce the effective overlap of the species.