Inelastic collisions in radiofrequency-dressed mixtures of ultracold atoms

Radiofrequency (RF)-dressed potentials are a promising technique for manipulating atomic mixtures, but so far little work has been undertaken to understand the collisions of atoms held within these traps. In this work, we dress a mixture of 85Rb and 87Rb with RF radiation, characterize the inelastic loss that occurs, and demonstrate species-selective manipulations. Our measurements show the loss is caused by two-body 87Rb+85Rb collisions, and we show the inelastic rate coefficient varies with detuning from the RF resonance. We explain our observations using quantum scattering calculations, which give reasonable agreement with the measurements. The calculations consider magnetic fields both perpendicular to the plane of RF polarization and tilted with respect to it. Our findings have important consequences for future experiments that dress mixtures with RF fields.

Experiments that use mixtures of ultracold atoms are now established as versatile quantum simulators for a range of physical phenomena.For systems of many particles, recent studies have examined superfluidity [1], non-equilibrium dynamics in many-body quantum systems [2], and the interactions mediated by a bath [3].At the single-particle level, experiments have observed diffusion [4], chemical reactions [5] and ultralow-energy collisions [6].Mixtures of ultracold atoms are also used as a starting point for the production of ultracold molecules [7,8].These experiments have been made possible by techniques to manipulate ultracold mixtures and their constituents, and new investigations will become possible as laboratory methods evolve.
A number of different techniques are used to trap and manipulate cold atoms.In this paper we consider RF-dressed potentials, which confine cold atoms through a combination of static and radiofrequency magnetic fields [9,10].Notable advantages of this technique include smooth, defect-free traps and low heating rates [11].The potential can be shaped by controlling the RF-dressing field [12], or by adding additional RF components [13].Furthermore, the potential may be combined with additional time-averaging fields to produce trap geometries such as rings or double wells [14][15][16].The confining forces depend only on an atom's magnetic structure, unlike optical methods which depend on electronic structure.Therefore, RF-dressed potentials permit species-selective manipulations of mixtures that have similar confinement in dipole traps, such as mixtures of hyperfine states or isotopes [17][18][19].
In spite of the advantages of RF-dressed potentials, there has so far been little consideration of the collisional stability of mixtures that are trapped using them.In this work, we investigate collisions in an RF-dressed mixture of 85 Rb and 87 Rb atoms in their lower hyperfine states.We observe a rapid loss of 85 Rb atoms from the trap due to two-body 87 Rb+ 85 Rb inelastic collisions, which occur through a spin-exchange mechanism.We use a theoretical model to explain the inelastic collisions, and compare predictions from quantum scattering calculations to our measurements of the two-body rate coefficients.Our results suggest that spin-exchange collisions will occur for most combinations of alkali-metal atoms in RF-dressed potentials.Furthermore, our results verify our understanding of ultracold collisions in the presence of strong, resonant dressing fields.
This paper is structured as follows.In Section I we explain the dressed-atom picture.In Section II we describe the experimental procedure used to produce cold clouds of 85 Rb and 87 Rb, and we demonstrate speciesselective manipulations.In Section III we present measurements of the inelastic loss, which we show is dominated by two-body 87 Rb+ 85 Rb collisions, and we measure the two-body rate coefficient k 85,87  convention that lower-case quantum numbers refer to individual atoms and upper-case quantum numbers refer to a colliding pair.An atom is described by its electron spin s = 1/2 and nuclear spin i, which couple to form a resultant spin f .In a static magnetic field B 0 along the z axis, the Hamiltonian for each atom is where ζ is the hyperfine coupling constant, µ B is the Bohr magneton and g S and g i are electron and nuclear spin gfactors with the sign convention of Arimondo et al. [20].
At the low magnetic fields considered here, each atomic state splits into substates with a well-defined projection m f of the total angular momentum f along B 0 .In this regime, f is nearly conserved but the individual projections m s and m i of s and i are not.At low fields, the field-dependent terms in Eq. ( 1) may be approximated in the coupled basis |f, m f to where substates m f are separated in energy by the Zeeman splitting and g f is the Landé g-factor.
In addition to the static magnetic field, we consider a radiofrequency field with angular frequency ω that is σ − polarized about the z axis, with B RF (t) = B RF [ e x cos ωt − e y sin ωt].The Hamiltonian of the RF field is where N = â † − â− − â † − â− is the photon number with respect to the average photon number N 0 = â † − â− , and â † − and â− are photon creation and annihilation operators for σ − photons.For σ − polarization, the N photons have angular momentum projection M N = −N onto the z axis.
The Hamiltonian for the interaction of the field with an atom is where ŝ+ and ŝ− are raising and lowering operators for the electron spin and î+ and î− are the corresponding operators for the nuclear spin.
The atom-photon interaction for σ − polarization conserves m tot = m f + M N = m f − N .If the couplings are neglected, states with the same m tot cross as a function of magnetic field at the radiofrequency resonance ω = g f µ B B 0 , as shown by the dashed lines in Fig. 1.For the RF frequency of 3.6 MHz used in this work, the states (f, m f , N ) = (1, +1, 1), (1, 0, 0) and (1, −1, −1) of 87 Rb all cross near B = 5.12 G, and the states (f, m f , N ) = (2, +2, 2), (2, +1, 1), (2, 0, 0), (2, −1, −1) and (2, −2, −2) of 85 Rb all cross near B = 7.70 G.These crossings become avoided crossings when the couplings of Eq. ( 4) are included.The eigenstates within each manifold of constant m tot are labelled by the quantum number m, which takes values in the range −f to f .The corresponding eigenenergies are where δ = (g f µ B B 0 / − ω) is the angular frequency detuning from resonance and Ω is the Rabi frequency on resonance.In an inhomogeneous field, atoms in states for which m > 0 may be trapped in the resulting potential minimum [9,10].

II. EXPERIMENTAL METHODS
In this section, we describe the methods used to cool and trap mixtures of 85 Rb and 87 Rb.Our apparatus was described previously [13], and has since been modified to allow the trapping of two species.
An experimental sequence begins by collecting atoms of 85 Rb and 87 Rb into a dual-isotope magneto-optical trap (MOT).The cooling and repumping light for 87 Rb is generated by two external-cavity diode lasers.Each laser is locked to one of the transitions, and injection-locks a laser diode that is current-modulated at a frequency of 1.1 GHz (cooling) or 2.5 GHz (repumper).The modulation generates sidebands at the frequencies required to laser cool and repump 85 Rb atoms.Light from the two injection-locked diodes is combined and passed through a tapered amplifier, before illuminating a 3D pyramid MOT, which collects 4 × 10 9 atoms of 87 Rb and 1 × 10 8 atoms of 85 Rb.These atoms are optically pumped into their lower hyperfine levels, with f = 1 and 2 respectively, and the low-field-seeking states are loaded into a magnetic quadrupole trap.
The trapped mixture of isotopes is transported to an ultra-high-vacuum region where it is evaporatively cooled using a weak RF field, first within a quadrupole trap and then in a Time-Orbiting Potential (TOP) trap, to a temperature of ∼ 0.5 µK.This process predominantly ejects 87 Rb atoms from the trap and the 85 Rb atoms are sympathetically cooled with minimal loss [21].The final atom numbers of each species are controlled by adjusting the power of the cooling light that is resonant with each isotope during the MOT loading stage, which determines the number of atoms initially collected.We observe no evidence of interspecies inelastic loss in these magnetic traps, imposing a bound of k 85,87 2 10 −14 cm 3 s −1 on the two-body rate coefficient; this is expected because spin exchange is forbidden between the (f , m f ) = (2, −2) and (1, −1) states of 85 Rb and 87 Rb respectively.

A. Species-selective manipulations
After evaporation, the atoms are loaded into a timeaveraged adiabatic potential (TAAP) [14,15].This potential is formed by combining a spherical quadrupole field B quad , a slow time-averaging field B TA , and an RFdressing field B RF that is σ − polarized in a plane perpendicular to z: The RF field, with ω = 3.6 MHz, drives transitions between the Zeeman substates so that the atoms are in RF-dressed eigenstates.The RF field is resonant with the atomic Zeeman splitting at points on the surface of a spheroid, centered on the quadrupole node, with semi-axes of length ω/g f µ B B × {1, 1, 0.5} along the { e x , e y , e z } axes.The time-averaging field sweeps this resonant surface in a circular orbit of radius r orbit = B TA /B around the z axis.The frequency of the timeaveraging field, ω TA = 7 kHz, is slow compared to the Larmor, RF and Rabi frequencies, so atoms adiabatically follow the RF-dressed eigenstates as the potential is swept.For a single species, the TAAP operates in two modes, depending on the value of B TA .When B TA > ω/g f µ B , the resonant spheroid orbits far from the atoms, which are confined near the origin by the rotating field B TA , as in a TOP trap.When B TA < ω/g f µ B , the resonant spheroid intersects the z axis, forming a double-well potential with minima at positions [14] In this work we load atoms only into the lower well, and henceforth neglect the upper well.The vertical position of the lower well is controlled by changing B TA , which determines the radius of orbit and thus the point of intersection of the resonant spheroid and the z axis.
The TAAP differs for species with g-factors of different magnitude, such as 85 Rb and 87 Rb in their lower hyperfine states, which have g 85 f = −1/3 and g 87 f = −1/2.Fig. 2 shows how the positions of each species change as a function of the time-averaging field B TA .With B TA > ω/g 85 f µ B , the resonant spheroids for both species orbit far from the atoms, which are confined near the origin.This scheme is illustrated in Fig. 2a.For g 85 f µ B B TA < ω < g 87 f µ B B TA , only the resonant spheroid for 85 Rb intersects the rotation axis, confining 85 Rb in the lower well of the TAAP but keeping 87 Rb confined near the origin by the TOP-like trap, as in Fig. 2b.In this configuration, the vertical position of the 85 Rb potential minimum is strongly affected by B TA , while that of 87 Rb is not.When B TA < ω/g 87 f µ B , the resonant spheroids for both species intersect the rotation axis, and both are loaded into the lower well of their respective TAAPs, as in Fig. 2c.

B. Measuring inelastic loss
To observe inelastic loss between 85 Rb and 87 Rb, we work in the regime in which the clouds of the two species are spatially overlapped, which requires that g 85 f µ B B TA > ω, as shown in Fig. 2. The two isotopes are held in contact for a specified duration, then the remaining atom numbers N i of both are measured using absorption imaging.The raw images are processed using the fringe-removal algorithm developed by Ockeloen et al. [22].The temperatures T i of both species are measured using time-of-flight expansion.
The mixtures used in this work have atom numbers N 85 , N 87 of the two species, with N 85 N 87 ; the decrease in N 85 over time provides a clear signal to measure the inelastic loss rate.The fractional decrease in N 87 is negligible and cannot be distinguished above shot-toshot variations.The inelastic collisions have a negligible effect on the temperature of 87 Rb, and the 87 Rb atoms thus provide a large bath of nearly constant density n 87 .
Our RF-dressed trap confines two states of 85 Rb, with m = 1, 2. The two separate clouds that correspond to these states are discernible in absorption images, but their overlap means that only the total atom number N 85 (t) = N 1 + N 2 can be measured accurately.For our experiments, initially N 2 N 1 because the method used to load the RF-dressed trap favours projection from

III. INELASTIC LOSS
Including up to 3-body collision processes, N 85 (t) de where n i (t) are atom number densities, and τ 85 is the lifetime of trapped 85 Rb atoms from one-body losses and collisions with the background gas.The coefficients k j are j-body rate coefficients, with the colliding species indicated by the superscript.For pure 85 Rb samples, Eq. ( 10) reduces to When both species are present, inelastic collisions cause a rapid loss of 85 Rb, with almost all atoms lost after a few hundred milliseconds.We argue that the loss occurs through two-body 87 Rb+ 85 Rb collisions, as follows.Neglecting both the intraspecies loss and one-body loss, Eq. (10) Thus, depending on which term dominates, dN 85 /dt is proportional to either n 85 or n 2 85 .For an atomic cloud at constant temperature, n i ∝ N i .
In Fig. 3a, we show measurements of N 85 against hold time.The total 85 Rb atom number is well described by a model where the two trapped states of 85 Rb each decay exponentially, N 85 = N 1 + N 2 with N i = A i e −βit .The different decay constants β i arise from the differing overlap of each state's density distribution with that of 87 Rb.For the measurements in Fig. 3 the overlap of 85 Rb atoms in the m = 2 state with 87 Rb is optimized, thus β 2 β 1 .A model in which dN i (t)/dt ∝ N 2 i shows poor agreement with the data, as shown in Fig. 3a.From dN i /dt ∝ N i , it follows that one 85 Rb atom is involved in each inelastic collision.
For short hold times, t 1/β 1 , the atom number decays exponentially as Fig. 3b shows N 85 as a function of time during the initial fast exponential decay, for different densities of 87 Rb.We fit Eq. ( 13) to the data in Fig. 3b, and in the inset plot β 2 against n max 87 , the maximum atom number density of 87 Rb, which occurs at the centre of the trap.The measured decay rate is proportional to n max 87 , indicating that the inelastic collisions involve a single 87 Rb atom.Thus we deduce that the inelastic loss arises from a mechanism involving two-body 87 Rb+ 85 Rb collisions.

Measuring the two-body rate coefficient
Having determined that two-body 87 Rb+ 85 Rb inelastic collisions are the dominant loss mechanism for 85 Rb in the trapped mixture, we now measure the two-body rate coefficient.Eq. ( 12) further approximates to We measure the inelastic loss rate by fitting Eq. ( 13 where the term n 85 n 87 dV is the overlap integral that quantifies the spatial overlap.Hence, Determining the overlap integral requires knowledge of the atom number densities n i .We calculate these densities using measured values of the cloud temperatures, quadrupole field gradient, rotating bias field amplitude, and RF field.The temperatures are determined from time-of-flight expansion of the clouds, and we find that T 87 is independent of hold time.However, it is not possible to determine T 85 at arbitrary hold times; the significant 85 Rb atom loss results in weak absorption imaging signals, which cannot be reliably fitted with Gaussian profiles.Instead, we determine T 85 at t = 0 and assume it is constant thereafter.A Monte-Carlo method is used to determine the uncertainties in n 85 n 87 dV , which incorporate the individual uncertainties (including systematic errors) of all independent parameters.The uncertainties in n 85 n 87 dV are combined in quadrature with those of the fitted decay rates to determine the uncertainty of k 85,87 2 .We explore the dependence of k 85,87 2 on the static magnetic field by adjusting B TA , which is akin to a bias field in our setup.This is possible provided the two species remain overlapped, which requires that B TA > ω/(g 85 f µ B ), as described in Section II A. For any given value of B TA , collisions occur over a range of different static magnetic fields because of the field gradient that is required to confine the atoms.As such, we compare our measured rate coefficients as a function of the overlapweighted average B 0 , defined analogously to k 85,87 2 .Our measurements are shown in Fig. 4, for three different amplitudes of the RF-dressing field.We observe that the two-body rate coefficient increases with decreasing B 0 , and within the uncertainties observe no clear dependence on RF amplitude.We also plot the values of k 85,87 2 predicted from our scattering calculations, which are described in the next section.

IV. QUANTUM SCATTERING CALCULATIONS
We model the collisional losses by carrying out quantum-mechanical scattering calculations using the MOLSCAT program [23,24].The method used was described in ref. 25 for RF polarization in the plane perpendicular to the magnetic field, and is summarized in Appendix A. It has antecedents in refs.26-28.The wave function for a colliding pair of atoms is expanded in an uncoupled RF-dressed basis set, where the indices (1, 2) label quantities associated with the first and second atoms, L is the angular momentum relative motion of two and M L is its proonto the axis.
To collisions between trapped atoms, it is useful to consider the thresholds (i.e., the energies of separated atomic pairs) as a function of magnetic field.Figure 5 compares the thresholds for 87 Rb+ 85 Rb with those for 87 Rb+ 87 Rb and 85 Rb+ 85 Rb for an RF field strength B RF = 0.5 G.Only states with m f 1 + m f 2 + M N = 0 are shown; as discussed below, this quantity is conserved in spin-exchange collisions (though not in spin-relaxation collisions).

A. Homonuclear systems
The thresholds for the homonuclear systems are shown in the end panels of Fig. 5.They show simple maxima or minima at a single magnetic field, B = 5.12 G for 87 Rb and B = 7.70 G for 85 Rb.In all cases the trapped states correspond to the uppermost threshold of those shown, though other thresholds exist for different photon numbers or higher hyperfine states.For 87 Rb+ 87 Rb, the uppermost state has character (m f 1 , m f 2 , N ) = (1, 1, 2) at fields below the crossing and (−1, −1, −2) above it.For 85 Rb+ 85 Rb, the uppermost state has character (2, 2, 4) below the crossing and (−2, −2, −4) above it.
In both homonuclear cases, there are no lower-energy states in the same multiplet with the same photon number, so collisional decay can occur in only two ways [25]: (1) Close to the crossing, the m f quantum numbers are mixed by the photon couplings, so that RF-induced spinexchange collisions can transfer atoms to lower thresholds without changing L from 0.
(2) At fields above the crossing, the m f = −1 state for 87 Rb (or m f = −2 state for 85 Rb) is not the ground state.Even in the absence of RF radiation, two m f = −1 or −2 atoms can undergo spin-relaxation collisions that change both M F = m f 1 + m f 2 and M L (and thus must change L from 0 to 2) but conserve M F + M L .Spin relaxation is usually very slow, both because the spin-dipolar coupling V d (R) is very weak and because there is a centrifugal barrier higher than the kinetic energy in the outgoing channel with L = 2.
87 Rb+ 87 Rb is a special case, with very similar singlet and triplet scattering lengths a s = 90.6 bohr and a t = 98.96 bohr.This is known to suppress spin-exchange collisions dramatically [29][30][31], and ref. 25 showed that it also suppresses RF-induced spin-exchange collisions.Thus collisional losses in RF-dressed traps for pure 87 Rb are dominated by spin relaxation, somewhat modified by the RF radiation [25]. 85Rb+ 85 Rb has a s = 2,735 bohr and a t = −386 bohr [32]; although superficially very different, these give similar values of the low-energy s-wave scattering phase.As a result, RF-free and RF-induced spin-exchange collisions are suppressed for pure 85 Rb as well, though not as strongly as for 87 Rb.

B. Heteronuclear systems
The thresholds for 87 Rb+ 85 Rb are very different from those for the homonuclear systems.The uppermost state has character (m f 1 , m f 2 , N ) = (1, 2, 3) at fields below the 87 Rb resonance at 5.12 G, and (−1, −2, −3) above the 85 Rb resonance at 7.70 G. RF-induced spin exchange is possible close to the crossings and RF-modified spin relaxation is possible above 5.12 G, as for the homonuclear systems.However, at magnetic fields between the two crossings the uppermost state has predominantly (−1, 2, 1) character, and there are lower pair states that have predominantly (0, 1, 1) and (1,0,1) character, with the same photon number and value of M F , as shown by dashed lines in Fig. 5. Spin-exchange collisions that transfer atoms to these lower thresholds are thus allowed in this intermediate region, even without the couplings due to RF radiation.The scattering lengths for 87 Rb+ 85 Rb are a s = 202 bohr and a t = 12 bohr, so spin exchange is not suppressed in the mixture and fast losses are expected at these intermediate fields.

C. Calculated rates and comparison
Fig. 6a shows the calculated inelastic rate coefficients for collisions between RF-dressed 87 Rb atoms in f = 1, m = 1 and 85 Rb atoms in f = 2, m = 2 as a function of magnetic field, for several RF field strengths.The calculations were carried out for σ − polarization in the plane perpendicular to B 0 .In these calculations L max = 0, so spin-relaxation collisions are excluded.At the lowest RF field strength, B RF = 50 mG, the avoided crossings between the thresholds are very sharp and the states are well described by the quantum numbers (m f 1 , m f 2 , N ) introduced at the start of Section IV.In this regime the spin-exchange losses are forbidden below the 87 Rb radiofrequency resonance at 5.12 G and above the 85 Rb radiofrequency resonance at 7.70 G, but at intermediate fields they occur at almost the full RFfree rate for (f, m f ) = (1, −1) + (2, 2) collisions, shown as the black dashed line.At higher RF field strengths, the avoided crossings extend further into the intermediate field region; the uppermost state is a mixture of (m f 1 , m f 2 , N ) = (−1, 2, 1) and other pair states that do not decay as fast.The effect is to broaden the edges of the flat-topped peak that exists for B RF = 0.05 G and depress the height of the peak in the central region.
In the experiment, the atoms are trapped at locations where the magnetic field is not perpendicular to the plane of circular polarization.To explore the effects of this, we carried out additional calculations where the radiation is still σ − polarized in the plane perpendicular to z, but the static magnetic field B 0 is tilted by an angle θ from the z axis.In this case M tot is no longer conserved, resulting in an increase in the number of open channels.For L max = 0, the number of open channels increases from 15 to 56.The calculated loss profiles are shown in Fig. 6b for B RF = 0.5 G and different values of the tilt angle θ; the profile remains qualitatively similar to that at θ = 0, especially far from the avoided crossings in Fig. 5, where the magnetic field dominates.However, the onset of loss is sharper for tilted fields, resembling that at smaller values of B RF in Fig. 6a.
At fields above the RF resonance at 5.12 G, spin relaxation can also occur.
Overlap-weighted averages k 85,87 2 of the calculated rate coefficients k 85,87 2 are plotted as solid lines alongside the experimental data in Fig. 4. To perform the overlap-weighted averaging, we calculate the spatial distributions n 85 , n 87 using the average temperature, atom number and trapping fields for each particular value of B RF shown.We numerically integrate these density distributions to determine k 85,87 2 = n 85 n 87 k 85,87 2 dV , taking into account the variation of k 85,87 2 with B 0 .The tilt angle θ varies by only a few degrees across the region where the two species overlap, and so we use a constant θ = 80 • for the calculations.
Our calculated values of k 85,87 2 are in reasonable agreement with the experimental measurements shown in Fig. 4. The measurements clearly demonstrate that k 85,87 2 increases with magnetic field as the 85 Rb resonance is approached from the high-field side, which is consistent with the predicted rate coefficients.No clear trend with RF amplitude is discernible in our measured data, although this would be difficult to observe given our uncertainties.In general, the measured rates are slightly higher than the predicted values.This discrepancy could be caused by a systematic error that underestimates the atom numbers N 85 and N 87 .

V. SEMICLASSICAL INTERPRETATION
Low inelastic loss rates in RF-dressed potentials were previously measured in experiments using 87 Rb.Those results were interpreted using a semiclassical model [9,12], which we now revisit in light of our work.The model was first introduced in the context of microwave dressing [33], and it has also been applied to collisions during RF evaporative cooling [34].
Before we discuss the collision model, we first recap the semiclassical picture of an atom in an RF field.The Hamiltonian of a single atom interacting with a magnetic field is The time-dependence is removed by transforming into a frame that rotates with the RF field, with coordinate axes e x = e x cos(ωt) + e y sin(ωt), e z = e z , followed by making the rotating-wave approximation.
The resulting time-independent Hamiltonian is where δ is the angular frequency detuning and Ω the resonant Rabi frequency, defined previously.Diagonalising this semi-classical Hamiltonian gives the eigenenergies of an atom in the applied magnetic fields.
H RWA is proportional to the dot product of f with the vector V , V = (Ω e x + δ e z ) .
Consequently, the eigenstates of H RWA have a welldefined projection m of f in the direction of V .Fig. 7a shows V observed from the laboratory frame, in which it precesses about the field B 0 at the angular frequency ω and with angle Θ = arctan(δ/Ω).
The semiclassical model of RF-dressed collisions posits that spin exchange does not occur between identical atoms that are in eigenstates of extreme m [9,12,34].When two such atoms collide, their total angular momentum also has a maximum projection along V , with M = m 1 + m 2 .In the semiclassical picture, there are no other open channels with the same value of M , thus spin-exchange collisions are forbidden by violation of angular momentum conservation.We stress that these conclusions are incorrect; the semiclassical picture neglects couplings to the RF field during collisions, and therefore fails to predict the RF-induced spin exchange described earlier.The rate coefficient for RF-induced spin exchange is usually large, but this is not the case for either 87 Rb or 85 Rb; it appears that the low inelastic loss rates observed previously for RF-dressed 87 Rb atoms are in agreement with the semiclassical model's predictions only by coincidence.
The same semiclassical model predicts that spin exchange can occur when two atoms with different values of |g f | collide, even if they are in states of maximum m.The vectors V 85 , V 87 of the two species are in general not parallel, due to the different Rabi frequencies and detunings from the RF resonance.They precess around B 0 at the same angular frequency ω, but with different angles Θ 85 , Θ 87 .These vectors are illustrated for different B 0 in Fig. 7b, with the associated angles Θ 85 and Θ 87 shown in Fig. 7c.
At fields much greater than 7.70 G, the detunings of both species are large and positive.Both angles tend to π/2, and the vectors V 85 and V 87 are nearly parallel.In this case, a 87 Rb + 85 Rb pair has an extreme value of the total angular momentum M when each atom is in an eigenstate of maximum m.Spin exchange is forbidden on the grounds of angular momentum conservation, as for identical atoms, and the inelastic rate coefficient k 85,87 2 is small.A similar argument follows for very weak fields below 5.12 G, where the detunings are large and negative and both Θ 85 and Θ 87 tend to −π/2.
At intermediate fields, the angles Θ 85 and Θ 87 are very dissimilar.The vectors V 85 , V 87 are misaligned, and spin exchange is not forbidden on grounds of angular momentum conservation.The rate coefficient increases as the angles diverge, and peaks at the midpoint between the two RF resonances, where V 85 , V 87 are almost antiparallel to each other.The RF amplitude determines how slowly Θ 85 and Θ 87 change with respect to magnetic field Finally, we remark that the semiclassical model predicts that spin exchange occurs for collisions between atoms with g-factors of different sign, even when the magnitudes of the g-factors are the same; although angles Θ f=1 and Θ f=2 are matched, the V s precesses with different handedness around the static field for each species.Further work is required to compare the semiclassical and quantal pictures with experimental data of different hyperfine states.

VI. CONCLUSION
In this paper we have investigated the inelastic collisions that occur in an RF-dressed mixture of 85 Rb and 87 Rb.We measured the loss of a small population of 85 Rb atoms in the presence of a larger 87 Rb bath, and identified the dominant mechanism as two-body 87 Rb+ 85 Rb inelastic collisions.The inelastic rate coefficient k 85,87 2 was shown to vary as a function of magnetic field, with k 85,87 2 increasing as the atomic RF resonance was ap- proached from the high-field side.We used a theoretical model of RF-dressed collisions to predict values of k 85,87 2 , and find they are in reasonable agreement with the measured values given that no free parameters were used to fit.
When RF-dressed potentials are used to confine atoms, the atoms are in states with a potential energy minimum at the atomic RF resonance.If two atoms have different magnitudes of g f , they are resonant with an applied RF field at different values of the static field.At fields between these two values, the atoms are predominantly in states where spin-exchange collisions are allowed, even in the absence of coupling to the RF field.Unless the singlet and triplet scattering lengths are similar, or the magnitudes of both are large, this spin exchange is expected to be fast.This contrasts with the situation when two atoms very similar values of g f , and are thus resonant at the same value of the static field.In this case spin-exchange collisions are forbidden except close to the trap center, where mixing of the Zeeman states by photon interactions permits RF-induced spin exchange.This RF-induced spin exchange can also be moderately fast unless the singlet and triplet scattering lengths are similar [25].
Table I shows the singlet and triplet scattering lengths for different pairs of alkali metal atoms.These values demonstrate that 87 Rb+ 87 Rb is a special case.For most other combinations of alkali-metal isotopes, the singlet and triplet scattering lengths are very different and the rate coefficients for both RF-induced and RF-free spin exchange will be large.Although RF-dressed potentials may enable the manipulation of different isotopes in a mixture [19], this paper finds that the RF dressing will generally cause high rates of inelastic collisions.Nonetheless, there may be some mixtures for which inelastic losses are low.For instance, the combinations 6 Li+ 23 Na, 6 Li+ 39 K and 6 Li+ 40 K have similar singlet and triplet scattering lengths, which may suppress interspecies spin exchange.Unfortunately, the singlet and triplet scattering lengths are dissimilar in 6 Li+ 6 Li, 23 Na+ 23 Na, 39 K+ 39 K and 40 K+ 40 K, hence RF-induced intraspecies spin exchange may be fast for these species. 87Rb+ 133 Cs may also be interesting; although the interspecies scattering lengths are dissimilar, they are both large and may give rise to similar phase shifts.Furthermore, 87 Rb and 133 Cs have different magnitudes of g f , allowing species-selective manipulations.It is also well established that 87 Rb+ 87 Rb has low inelastic loss rates when RF dressed.Further calculations would be required to predict the rate coefficients for a 87 Rb+ 133 Cs mixture.This paper has not considered inelastic collisions between different hyperfine states of the same isotope, which can also be independently manipulated using RFdressed potentials, as was shown for 87 Rb [18].The choice of 87 Rb was fortunate, as the similarity of singlet and triplet scattering lengths suppresses spin-exchange collisions even when such collisions are otherwise allowed [30].RF-dressed potentials have found use for this specific mixture, but our work suggests that this promising technique may be more limited in scope than was previously realized.

FIG. 1 .
FIG.1.The RF-dressed eigenstates (solid lines) for 87 Rb (top) and 85 Rb (bottom) as a function of magnetic field B0, for BRF = 0.86 G and ω = 3.6 MHz.A single manifold with mtot = 0 is emphasized in bold.Dashed lines show the dressed states of this manifold in the limit of zero atom-photon interaction.

FIG. 2 .
FIG. 2. (a-c) The different operating regimes of the dualspecies TAAP.Filled circles show the locations of potential minima for 85 Rb (blue) and 87 Rb (purple).The ellipses show the resonant spheroids at phases ωTAt = 0 (solid lines) and π (dotted lines) of the rotating field BTA.Three distinct regimes are shown: (a) ω < g 85 f µBBTA, (b) g 85 f µBBTA < ω < g 87 f µBBTA, (c) g 87 f µBBTA < ω.(d) Measurements of the vertical position of 85 Rb and 87 Rb clouds as a function of BTA.The observed density distribution for each species along the vertical direction is shown as a vertical slice for each unique value of BTA.The dotted vertical lines show the two RF resonances, where g i f µBBTA = ω.For each species, the colored lines show the value of z from Eq. (9) (dashed), and the numerically calculated TAAP trap minimum (solid).

2 < 3 × 3 <
When only85 Rb is present, we observe an exponential decay of N 85 (t) with lifetime τ 85 = 43 s, and the trapped atoms heating at a rate of 74 nK s −1 from an initial temperature of 1 µK.The fitted rate coefficients k 85,85 2 and k 85,85,85 3 are consistent with zero, with upper bounds of k 85,85 10 −12 cm 3 s −1 and k 85,85,85 10 −22 cm 6 s −1 in the RF-dressed trap.These bounds are sufficiently low that the intraspecies inelastic loss is negligible for all experiments discussed in this work.

10 FIG. 3 .
FIG. 3. (a) The measured total 85 Rb atom number N85 as a function of hold time in the trap.The solid black line shows the best fit of a model in which the population of both trapped states of 85 Rb decays exponentially.The solid blue line shows an alternative model in which each population decays with a rate proportional to N 2 i .(b) At short hold times the change in atom number is dominated by the exponential atom loss from the state with m = 2.The decay rates in the presence of three different atom number densities of 87 Rb are shown.Inset: The fitted rate coefficients β2 are linearly proportional to the peak 87 Rb atom number density, n max 87 .Arrows indicate the three sets that are plotted in the outer panel.

FIG. 5 .
FIG.5.The RF-dressed atomic thresholds (black, solid) for 87 Rb+ 87 Rb (left), 87 Rb+ 85 Rb (center) and 85 Rb+ 85 Rb (right) for m f 1 + m f 2 + MN = 0, with BRF = 0.5 G at a frequency of 3.6 MHz.Both atoms are in their lower hyperfine state.Selected thresholds are also shown for zero RF intensity (dashed lines), labelled with quantum numbers f 1 m f , N ).The RF resonances for each species are indicated by vertical dotted lines.The thresholds corresponding to collisions of trapped atoms in this work are indicated in bold.

FIG. 6 .
FIG.6.Rate coefficients for inelastic collisions between RFdressed 87 Rb and 85 Rb atoms in their hyperfine ground states, as a function of magnetic field, at a collision energy of 0.4 µK.Results are shown for RF radiation with σ− polarization at a frequency of 3.6 MHz.Dependence on BRF with RF polarization in the plane perpendicular to B0.The dashed black line shows the rate coefficient for RF-free spin exchange for (f , m f ) = 87 Rb (1, −1) + 85 Rb (2, 2).The dotted lines indicate the magnetic field at which the RF is resonant for 85 Rb and 87 Rb.(b) Dependence on the tilt angle θ, for BRF fixed at 0.5 G.

FIG. 7 .
FIG. 7. The semiclassical picture.(a) V precesses around B0 at the RF frequency, and is coplanar with both BRF and B0.The angle Θ is defined in the text.(b) Illustrations of V for 85 Rb (top row) and 87 Rb (bottom row).Each frame illustrates the fields at different magnetic fields B0.(c) The angle Θi is shown for each species as a function of magnetic field for a 3.6 MHz RF dressing field.Dotted vertical lines mark the RF resonances for each species.The rate coefficient k 85,87 2

TABLE I .
Singlet as and triplet at scattering lengths for isotopic mixtures of alkali-metal atoms.The uncertainties have been added where possible.