Precision measurement of spin-dependent inter-hyperfine scattering lengths in Rb
We present precision measurements of the to inter-hyperfine scattering lengths in a single-domain Rb spinor Bose-Einstein condensate. The inter-hyperfine interaction leads to a strong and state-dependent modification of the spin-mixing dynamics with respect to a non-interacting description. We employ hyperfine-specific Faraday-rotation probing to reveal the evolution of the transverse magnetization in each hyperfine manifold for different state preparations, and a co-magnetometer strategy to cancel laboratory magnetic noise. We find the scattering length ratios and , limited by atom number fluctuations. Using better control of atom number, we estimate precisions of should be possible with this technique.
Since the advent of Bose-Einstein condensation (BEC) in ultracold quantum gases, experimental access to the spin degrees of freedom and resulting spin-dependent interactions have expanded greatly. The pioneering Rb, Na and Li experiments Anderson et al. (1995); Davis et al. (1995); Bradley et al. (1995) used magnetic trapping that restricted their studies to scalar BECs in low field seeking Zeeman sublevels. By introducing optical trapping techniques Stenger et al. (1998); Barrett et al. (2001), the spin degree of freedom became accessible, enabling the study of spin-mixing dynamics Chang et al. (2004); Schmaljohann et al. (2004); Chang et al. (2005); Jacob et al. (2012), spontaneous magnetic symmetry breaking Sadler et al. (2006); Vengalattore et al. (2010); Scherer et al. (2013), domain formation Stenger et al. (1998); Sadler et al. (2006); De et al. (2014) and exotic topological spin excitations Choi et al. (2012); Ray et al. (2014).
These rich dynamics arise from the interplay between superfluidity and magnetism, which for a single, spin- species and s-wave binary contact interactions are described by parameters, the intra-hyperfine scattering lengths. In the case of Rb, these have been separately determined for the and ground-state manifolds van Kempen et al. (2002); Chang et al. (2005); Widera et al. (2006). Inter-hyperfine interactions are less well studied, but nonetheless play an important role in determining the miscibility of multiple BEC species Papp et al. (2008); Thalhammer et al. (2008); McCarron et al. (2011), and have been used to produce spin-squeezing with its attendant entanglement, and Bell-type correlations Muessel et al. (2014, 2015); Schmied et al. (2016); Fadel et al. (2018); Anders et al. (2018). For Rb, the full set of inter-hyperfine spin interaction parameters has recently been measured Eto et al. (2018) with intriguing results. The current best values indicate that in an equal , ground-state mixture, the component manifests a polar ground state at zero magnetic field Irikura et al. (2018) even though the component alone is ferromagnetic Chang et al. (2004).
In this work we report precision measurements on the Rb inter-hyperfine scattering lengths, using a novel co-magnetometer strategy. We use a single-domain spinor BEC Palacios et al. (2018), with non-destructive Faraday probing Koschorreck et al. (2010) for simultaneous readout of amplitude and phase of the transverse magnetization in and . The observed dynamics are compared to mean field simulations under the single-mode approximation (SMA) Pu et al. (1999); Kawaguchi and Ueda (2012), yielding the two spin-dependent inter-hypefine interaction parameters Irikura et al. (2018). The presentation is organized as follows: Section II describes the inter-hyperfine interaction for Rb. It discusses the simplifications under the rotating wave approximation (section II.1) and the implementation of the numerical simulations (section II.2). Data interpretation and error sources are detailed in section II.3. Section III and section IV introduce the experimental setup and required classical calibrations. Section V describes the measurement of the spin-dependent interaction parameters. In section VI we present the resulting inter-hyperfine scattering lengths and compare against literature values.
Ii Mean-field description
A spinor BEC can be described by a vectorial order parameter, which in the SMA can be written Kawaguchi and Ueda (2012)
where and . The spin-independent spatial wavefunction and the relative spin amplitudes are normalized as follows: \cref@addtoresetequationparentequation
where is the number of atoms. For BECs significantly larger than the density healing length, the kinetic contribution to the total energy is negligible and the density distribution is described by a Thomas-Fermi profile Baym and Pethick (1996); Dalfovo et al. (1996); Lundh et al. (1997):
where is the underlying trapping potential and the spin-independent interaction coefficient for (see eq. (6a) below). The chemical potential is obtained by normalizing the spatial wavefunction as defined in eq. 2a.
In the SMA, the spatial dependence of the wavefunction is integrated out and only contributes through the effective volume . For the density profile in eq. 3 and a harmonic trapping potential with mean trapping frequency , the effective trapping volume becomes:
where is the mean Thomas-Fermi radius and the atomic mass.
We write s-wave scattering lengths , where is the scattering channel, i.e., the total spin quantum number of the colliding atoms, and indicates or intra-, or inter-hyperfine scattering, respectively. Scattering length differences we write as . In terms of these are defined the interaction coefficients that appear in the single-mode Hamiltonian:
The manifold contributes an energy per particle
where and describe the linear and quadratic Zeeman shifts (LZS and QZS, respectively), and
is the mean spin vector with cartesian components where are spin- matrices.
The manifold contributes an energy per particle
where and describe the LZS and QZS of the manifold, and is the spin-singlet scalar
The inter-hyperfine scattering contribution has been recently described Irikura et al. (2018) and can be written
results from inter-hyperfine scattering via the total channel.
ii.1 Rotating wave approximation
The LZS terms and induce Larmor precession of the spins about the magnetic field direction, assumed to be , the same as the quantization axis. and precess in opposite senses and with nearly equal angular frequency: , , where in Rb and . It is natural to work in a dual-rotating frame defined by , , with the consequence , . We note that rotation-invariant terms such as are unaffected by this change of frame. In contrast, many inter-hyperfine interaction terms like acquire an oscillating factor, e.g. .
In the experiments described below, the precession frequency is much faster than the collisional spin dynamics, e.g. . This motivates the rotating wave approximation (RWA), i.e. dropping the rapidly oscillating terms. From perturbation theory we expect the RWA to introduce a fractional error at the level, which is negligible in this context.
Under this simplification, and excluding the constant term , the inter-hyperfine energy per particle becomes:
|see sec. IV, V|
|van Kempen et al. (2002)|
|Widera et al. (2006)|
|Widera et al. (2006)|
|Widera et al. (2006)|
ii.2 Numerical integration
Once the intra- and inter-hyperfine contributions have been obtained, the dynamical evolution of the spin amplitudes are computed by differentiating the total energy:
where . The right-hand side of eq. 14 is computed analytically and numerical integration (via the ODEPACK routine LSODA) is used to solve the resulting set of 8 coupled differential equations Gómez (2018).
ii.3 Data interpretation and error estimates
In Sections IV, V.1 and V.2 we fit the model dynamics of eq. 14 to observed data, with the intent to determine (a global factor in the spin-dependent contributions to the energy per particle), and , or equivalently and . The atom number and intra-hyperfine scattering lengths , and also appear as parameters in eq. 14. We use literature values for the scattering lengths, which introduces a systematic uncertainty into the fit results. See table 1 for literature values, uncertainties and references, but note that theory and experiment are at present discrepant for Chang et al. (2005); Widera et al. (2006); van Kempen et al. (2002).
A numerical exploration of the dependence of the fitted values for , and finds that in each case only contributes an uncertainty that is significant on the scale of the experimental precision. For the inter-hyperfine interaction terms, that dependence is linear and we report ratios of the form , where the numerator is the fit result and the denominator is a fixed parameter in eq. 14. Similarly, we report , , and . These ratios, unlike the fit result itself, are insensitive to the value of , again at the level of precision of the experimental results. The dependence of the fitted on is described in section IV.
A remaining uncertainty arises from atom numbers fluctuations and drifts. Atom numbers and their fluctuations were estimated by repeated trap loading, state preparation, and destructive absorption imaging prior to acquiring data runs such as the one reported in fig. 2. Despite this, a significant uncertainty in atom number accrues due to drifts in the Rb background pressure. We account for this with a systematic uncertainty of rms deviation around the measured atom numbers. The value describes the observed drifts from run to run, as well as the observed fluctuations of population shown in fig. 3.
For most quantities derived from this analysis, we report the statistical averages and standard deviations of the corresponding fit parameters. For the transverse magnetization we report the median and 90% confidence interval, which is more meaningful as is intrinsically positive-valued and asymmetrically distributed.
Iii Experimental setup
The experiments have been performed in a single-domain spinor BEC of Rb Palacios et al. (2018). The spinor BEC is achieved after of all-optical evaporation in a crossed-beam optical dipole trap with typically atoms in a pure BEC. The spin state of the atoms can be probed by Stern-Gerlach imaging or non-destructive Faraday probing, shown in fig. 1. In the later case, a probe beam focused to a few times the Thomas-Fermi radius separately probes the transverse magnetization in and . By alternating between light closely detuned to or ( line transitions), we either interrogate the or the manifold. The probing pulses are linearly polarized and experience a rotation , which is proportional to the atomic spin projection along their propagation direction ( axis). Under the influence of externally applied magnetic fields (along axis), the LZS terms in eq. 7 and section II induce rapid Larmor precessions of the transverse spin and the Faraday rotation signal is of the form Palacios et al. (2018):
where the vector atom-light coupling factor depends on the detuning to the above mentioned transitions and will be specified for the different experimental sequencs of this work. The polarization rotation is continuosly monitored for several Larmor periods and recorded on a balanced differential photodetector Ciurana et al. (2016). The amplitude and initial phase of the obtained oscillatory traces reveal the amplitude and precession angle of the transverse magnetization in and . The damage to the atomic state caused by off-resonant photon absorption events is modeled via the exponential decay term, which depends on the number of photons sent and the characteristic damage factor . The power of the Faraday probe beam is monitored on a separate photodetector, from which the photon number is obtained:
where is the energy of a probe photon. In eq. 15 and eq. 17, is the elapsed time since the start of the corresponding Faraday probe.
The above described Faraday probing setup is operated in the photon shot-noise limited regime Ciurana et al. (2016). In this regime, the readout noise of the transverse spin components, and is
For the different experimental sequences of this work, the readout noise ranges from to spins.
Iv Calibration of trap conditions
For a precise determination of the inter-hyperfine scattering parameters, we require best-estimate values and uncertainties for the experimental parameters that appear in section II. These are the QZS , the mean trapping frequency and the atom number . In what follows, precise knowledge of the LZS will not be required, because the signals are either insensitive to the Larmor precession angles and , or only to their sum, to which the net LZS contribution is small. The LZS must, however, be large enough that the RWA is valid. We note that and contribute to the intra- and inter-hyperfine interaction energies only through a common prefactor (see Eqs. (4-7)). Calibration of this prefactor is a key element of the work we report here: it simultaneously accounts for systematic effects in both the atom number measurement and in the trap conditions. For simplicity of description, we define an effective trapping frequency via , where is the atom number as measured by absorption imaging or Faraday rotation. A comparison of observed spin dynamics against simulations then fixes .
To calibrate and , we first create a spinor BEC in the non-magnetic state, in the presence of a constant field . A RF pulse rotates the spin state to . After a variable hold time the transverse magnetization is obtained by fitting eqs. (15-16) to the measured Faraday rotation signals. Results are shown in fig. 2, exhibiting the expected oscillation of produced by competition between the QZS and the ferromagnetic interaction. These data are fitted with SMA mean field simulations as per section II.2, with the and as fit parameters. We find and . The atom number is estimated by destructive absorption imaging prior to the measurements of this section, yielding atoms.
Through eq. 7 the estimated value of depends on the ferromagnetic interaction coefficient and thus on . As mentioned in section II.3, this dependence is undesirable and our preferred quantity to report is the re-scaled frequency , which is -independent.
The value is consistent with independent measurements of the trap characteristics, and thus provides secondary support for the estimate of . The obtained value for is close to the theoretically expected at the given applied magnetic field. We attribute the discrepancy to tensorial light-shifts induced by the optical trapping beams. We note that, to the precision of this work, the and hyperfine manifolds feature opposite QZS and tensorial light shifts Coop et al. (2017), so that .
V Measurement of inter-hyperfine interaction parameters
We now describe our strategies for measuring and , the inter-hyperfine interaction parameters that appear in eq. 12. First, we note that , which in general is quite complicated, greatly simplifies in the case of stretched spin states in the manifold, for which all elements are zero except for either or . For these states reduces to a single term, which describes an effective LZS plus an effective QZS acting upon the manifold. As already seen in fig. 2, the QZS causes oscillations of . Because the QZS-ferromagnetic competition is the only source of such oscillations, this provides an unambiguous signal by which to measure .
To measure , we note that describes an effective LZS of the levels, with a strength proportional to , the magnetization along the B-field. The magnetization similarly produces a LZS in the manifold. The resulting modification of the Larmor frequency of the order of , a tiny fraction of the LZS due to the external magnetic field . To accurately resolve this shift and decouple the measurements from external magnetic field noise, we operate our spinor BEC as a co-magnetometer. This technique, in which a signal is simultaneously acquired from distinct but co-located sensors, can efficiently reject magnetic field noise while retaining sensitivity to other effects. In hot vapors, co-magnetometer techniques have been used for sensing rotation Kornack et al. (2005); Limes et al. (2018) and searches for physics beyond the standard model Smiciklas et al. (2011).
Here, the two sensors are the and manifolds of the SBEC. Their precession angles have opposite dependences on , and the summed precession angle is sensitive to with vanishing contribution. We note that in the SMA the magnetic dipole-dipole interaction produces a field within the condensate that is equally experienced by the and manifolds, and thus with no effect on .
v.1 Interaction parameter
To measure , we first prepare the state
which describes and equal superposition of in an aligned () state and in a stretched state. After a variable wait time the transverse magnetization is measured by Faraday rotation, as in section IV. Note that the state is unchanged by the evolution and readout of the state. A RF pulse then rotates the stretched state into the transverse plane and is measured by Faraday rotation. This provides a measure of the atom number atoms. The procedure is described in detail in section A.1.
v.2 Interaction parameter
To measure , we first prepare one of the following two states \cref@addtoresetequationparentequation
where is a rotation about the axis by angle . The rotation angle was chosen as a compromise between a strong spin component parallel to the external magnetic field (to have a large signal from the term in eq. 12) and a strong transverse magnetization. After a variable wait time the and precession angles are measured by Faraday rotation. A detailed description is given in section A.2.
For an initial state the co-magnetometer signal contains contributions from the differential LZS between and , the QZS and the spin-dependent inter-hyperfine interaction, i.e., the and contributions. We analyze the difference in co-magnetometer readouts , in which also the differential LZS contribution cancels. The QZS is known from the calibration of section IV. The results are shown in fig. 4, where the experimental are fitted to SMA mean field simulations in which is a free fit parameter whereas is fixed at the value found in section V.1. We obtain .
Vi Comparison with prior work and outlook
Using eq. 6f and eq. 6g for the above values of and , we find and , i.e., with relative uncertainties of 12% and 10%, respectively. As noted above, these ratios are insensitive to the exact value of , which serves as an input parameter in the modeling and fits. The same sensitivity to applies also to the prior measurements of Eto et al. (2018), which found and . These differ by and combined uncertainty from the result presented here.
Our accuracy is presently limited by uncertainty in the SBEC atom numbers, which reflect loading fluctuations and atom loss during the experiment. Active control schemes can stabilize the atom numbers of cold atomic ensembles below shot noise by using dispersive probing Gajdacz et al. (2016). Applied to the current experiment, such stabilization is foreseen to reduce the relative uncertainties in the results bellow .
We have demonstrated an interferometric method to precisely measure the inter-hyperfine collisional interactions in , mixtures of ultracold atoms. The method employs a single-domain spinor BEC and hyperfine-state-specific Faraday rotation to measure spin evolution. Two new multi-pulse RF and W state preparations are used. Each one generates a hyperfine-state mixture that gives high-visibility spin dynamics, that sensitively depends on one or more inter-hyperfine scattering lengths. We also describe a new calibration, based on Faraday-rotation measurement of collision-induced alignment-to-orientation conversion, to determine effective trapping frequency and quadratic Zeeman shifts. This new calibration substitutes for an absolute calibration of the atom number, typically one of the larger uncertainties in ultracold gas experiments. Applying these techniques to Rb, we measure and with relative uncertainties of 12% and 10%, respectively, limited by atom number drift between calibration and measurement. A relative uncertainty of is projected for experiments with nondestructive monitoring of atom number. The methods are directly applicable to other commonly-used alkali species Li, Na and K, in addition to Rb.
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Appendix A Experimental sequences
a.1 Experimental sequence for measuring
[See fig. 5 top] After all-optical evaporation, a pure SBEC is obtained in the state. The ensemble is coherently transferred into an equal superposition by means of a resonant RF rotation around the x-axis and a sequence of W pulses (I, II and III). Thereafter, the magnetic field is ramped up to in order to raise the differential LZS to . A Ramsey-like sequence, consisting of two RF pulses (rotating about ) separated by is used to produce a net rotation of the manifold, and zero net rotation of the manifold. The resulting state is given in eq. 19. The magnetic field is rapidly ramped down to , ensuring a modest QZS during the subsequent many-body (MB) evolution. After a variable hold time , the magnetizations in and are detected by Faraday rotation. A first pulse ( red detuned from , ) probes the transverse magnetization. A RF pulse is then applied to rotate the stretched state into the transverse plane for detection with a second pulse ( blue detuned from , ). The damped oscillatory signals illustrate the recorded Faraday signals described in eq. (15-16).
a.2 Experimental sequence for measuring
[See fig. 5 bottom] The sequence starts with a pure SBEC in which is coherently split by a RF pulse into . Subsequently, either the initial state or is prepared via W pulses (I, II and III) and a RF rotation around the axis. Hereafter, the many-body (MB) evolution begins. For the applied constant magnetic field of the LZS is with a differential frequency of . The insets illustrate how the and transverse spin orientations ( and or and ) rapidly evolve due to the LZS. The differential is represented by the green co-magnetometer readouts, which depending on the state preparation are labeled by and . After a variable hold time of up to , the transverse magnetization is interrogated . First the Faraday probe of is applied, from which, depending on the state preparation, the spin orientation or is obtained. Next, and without any additional RF pulse, the manifold is probed, yielding or . The co-magnetometer readout is obtained by , where . Faraday probing frequencies and atom-light coupling factors are identical to the previous section.