Investigation of Feshbach resonances in ultracold ⁴⁰K spin mixtures

Abstract

Magnetically tunable Feshbach resonances are an indispensable tool for experiments with atomic quantum gases. We report on 37 thus far unpublished Feshbach resonances and four further probable Feshbach resonances in spin mixtures of ultracold fermionic ⁴⁰K with temperatures well below 100 nK. In particular, we locate a broad resonance at B = 389.7G with a magnetic width of 26.7 G. Here 1 G = 10⁻⁴ T. Furthermore, by exciting low-energy spin waves, we demonstrate a means to precisely determine the zero crossing of the scattering length for this broad Feshbach resonance. Our findings allow for further tunability in experiments with ultracold ⁴⁰K quantum gases.

I. INTRODUCTION

Ultracold fermionic atomic gases are ideally suited for the study of many-body quantum phenomena owing to the unrivaled control over experimental parameters such as the spatial geometry of confining potentials and the interaction strength between the atoms. The interaction strength is controlled using magnetically tunable Feshbach resonance and typically characterized by the s-wave scattering length, which can be set to a wide range of values, either negative or positive. Feshbach resonances have been found in many bosonic, as well as fermionic, atomic systems (see [1–5] and references therein). The isotope ⁴⁰K constitutes one of the workhorses in current experiments with ultracold fermions and provides a rich ground-state structure allowing for the realization of binary and multicomponent spin mixtures [6–23], as well as several Bose-Fermi [24–30] and Fermi-Fermi mixtures [31–35]. In the energetically lowest hyperfine manifold with total angular momentum f = 9/2 ten magnetic spin states are available ranging from m = −9/2, …,+9/2, and 45 binary spin mixtures can be realized [36]. So far, only three Feshbach resonances have been reported, one for each of collision channels {m₁, m₂} = {−9/2,−7/2} [37], {−7/2,−7/2} [38,39], and {−9/2,−5/2} [40].

Here, we report on the experimental observation of 37 theoretically confirmed and 4 further probable magnetic Feshbach resonances in different spin mixtures of ultracold ⁴⁰K. Their positions are determined from the enhanced, resonant loss of atoms near the resonance, either caused by three-body recombination or two-body inelastic spin flips. In addition, we introduce a method for precisely determining the sign changes of the scattering length around the Feshbach resonance by exciting low-energy spin waves. In particular, this approach enables us to measure the zero crossing with high accuracy. We also find that the measured positions of the assigned Feshbach resonances agree well with theoretical calculations based on multichannel coupled-channels and quantum-defect theory using the best available Born-Oppenheimer potentials for ⁴⁰K [41].

A Feshbach resonance occurs when two atoms in well-defined spin states collide and couple to a virtual molecular state with a different spin configuration [1–5]. Because these configurations have different magnetic moments their relative Zeeman energy can be tuned with a magnetic field. This leads to a magnetic-field-dependent complex scattering length ã(B) = a(B) − ib(B), where real-valued a(B) and b(B) > 0 describe elastic and inelastic two-body processes, respectively. Here, we have allowed for inelastic transitions to spin configurations whose Zeeman energy is below that of the entrance configuration. In fact, near a resonance and in the limit of zero collision energy,

$$a(B)=a_{\text{bg}}\left[1-\frac{\mathrm{\Delta }B(B-B_{\text{res}})}{{(B-B_{\text{res}})}^2+{({\gamma }_2/2)}^2}\right],$$ (1)

with resonance position B_res, magnetic width ΔB, and background scattering length a_bg. Finally, γ₂ describes two-body decay to other spin-channels (expressed in units of the magnetic field). Similarly, we have

$$b(B)=2a_{\text{res}}\frac{{({\gamma }_2/2)}^2}{{(B-B_{\text{res}})}^2+{({\gamma }_2/2)}^2},$$ (2)

with resonance length a_res = a_bgΔB/γ₂. Atom loss, quantified by the two-body rate coefficient K₂(B) = 4πħb(B)/μ, is largest in the vicinity of the resonance position B_res. Here, ħ is the reduced Planck constant, μ = m/2, and m is the atomic mass.

II. LOSS SPECTROSCOPY OF FESHBACH RESONANCES

We begin our experiments by preparing a spin mixture of m₁ = +9/2 and m₂ = +7/2 atoms with about N = 5×10⁴ atoms per spin state in an optical dipole trap. The trap is harmonic and nearly isotropic with mean trapping frequency ω̄ = 2π × 50 Hz and temperature T ≈ 0.3T_F, where T_F = ħω̄(6N)^1/3/k ≈ 170 nK is the Fermi temperature and k is the Boltzmann constant. For the investigation of different collision channels, the corresponding two-component 50:50 spin mixture is prepared at a magnetic field of B =45 G using radio-frequency sweep and pulse protocols optimized for each mixture. After the preparation we do not observe any atoms in undesired spin states within our detection sensitivity of N_min ≈ 200. After this preparation, the magnetic field is ramped to its final value and the ensemble is held for a time of 100 ms. The magnetic field is calibrated by radio-frequency spectroscopy, resulting in an uncertainty of ΔB ≤ 0.2 G. Subsequently, the magnetic field is switched off and the remaining atoms are counted after a time of flight in a Stern-Gerlach gradient field. The field value where atom loss is maximal, B^exp, is assigned as the resonance position B_res. The full-width-half-maximum magnetic width of the experimental loss feature is denoted by σ^exp.

We have located 41 resonant experimental loss features in this manner and assigned 37 based on two complementary theoretical approaches for collisions between two fermionic ⁴⁰K atoms. The first approach corresponds to a coupled-channels calculation based on the spectroscopically accurate $X{{}^1\mathrm{\sum }}_g^{+}$ and $a{{}^3\mathrm{\sum }}_u^{+}$ Born-Oppenheimer potentials [41] mixed by the atomic Fermi-contact, atomic Zeeman, and magnetic dipole-dipole interactions. The second approach corresponds to a multichannel quantum-defect theory [4,42], where the Born-Oppenheimer potentials are described by their scattering length and common van der Waals dispersion coefficient and mixing is due only to the Fermi-contact and Zeeman interactions. For our ultracold ⁴⁰K collisions it is sufficient to include collisional channels with ℓ = 0 or ℓ = 1 mechanical orbital angular momentum, i.e., s or p partial waves. (Parity reflection symmetry ensures that there is no mixing between even and odd ℓ.)

We find quantitative agreement between the location of experimental loss maxima and theoretical resonance locations obtained from either theory. Table I lists the 37 assigned resonances as well as predictions for 17 additional resonances. The theoretical numbers are those extracted from the coupled-channels calculation at a collision energy of E/k = 60 nK. The resonance width ΔB^calc is the difference in the magnetic field of the elastic rate coefficient minimum and maximum. For our mainly lossy resonances these two fields correspond to good approximation to a scattering length that is zero and infinite, respectively.

TABLE I.

Thirty-seven measured and theoretically assigned Feshbach resonances of ⁴⁰K and the prediction of a further 17 resonances. The first four columns represent the collision channel {m₁, m₂} in the f = 9/2 manifold, the total spin quantum number M = m₁ + m₂, and the partial wave ℓ, with its projection m_ℓ. The next two columns are the experimental maximum loss position B^exp and the observed magnetic width σ^exp. The uncertainty of B^exp, given in parentheses, is a 1 standard deviation uncertainty with combined systematic and statistical error. The last four columns show resonance data from coupled-channels calculations at a collision energy of E/k = 60 nK. Here, $B_{\text{res}}^{\text{calc}}$ is the resonance position, where the elastic rate coefficient K_elas has its maximum, ΔB^calc is the (signed) magnetic width defined as the difference between the field locations, where K_elas has a minimum and maximum, respectively. γ₂ is the two-body decay width, as defined in Eq. (1). Some resonances only exhibit resonant behavior in the calculated losses. We give the corresponding resonance position in the $B_{\text{res}}^{\text{calc}}$ column and mark them with a superscript star. Missing numbers in one or more of the last two columns imply the absence of a minimum in K_elas and/or that the collision has no losses. Magnetic fields and widths are in Gauss.

{m1, m2}	M	ℓ	mℓ	Bexp	σexp	$B_{\text{res}}^{\text{calc}}$	ΔBcalc	γ2
{+5/2,−9/2}	−2	s	0	24.0(0.5)	4.2	24.5	2.1	0.24
	−2	s	0			33.2	0.7	1.7
{+9/2,−9/2}	0	s	0	17.6(0.3)	5.4	18.7	1.8	0.30
	0	s	0	35.9(0.4)	5.6	34.9	4.2	5.1
	0	p	−1	93.6(1.3)	17.3	89.1★		20
	0	p	1	ibid.		88.2		20
	0	p	0	ibid.		98.1	−27.4	21
{+7/2,−7/2}	0	s	0			18.0	0.2	0.17
	0	s	0	34.3(0.8)	10.8	35.2	3.4	0.77
	0	s	0	65.5(1.3)	11	64.2	7.0	9.1
	0	s	0	147.1(3.0)	0.8	145.8	0.2	0.01
{+9/2,−7/2}	1	s	0	13.9(0.2)	1.3	14.4	0.4	0.16
	1	s	0	28.4(0.3)	6.1	30.2	2.9	0.39
	1	s	0	66.3(0.7)	14.5	62.2	11.9	15
	1	p	1	139(1)	20	131.6		25
	1	p	−1	ibid.		132.5★		25
	1	p	0	ibid.		143.0	−41.4	26
{+5/2,−5/2}	0	s	0			17.6	0.1	0.08
	0	s	0	31(4)	6	32.3	0.4	0.38
	0	s	0	61(4)	21	61.6	4.0	1.0
	0	s	0			111.8	0.02	0.001
	0	s	0			156.7	1.0	0.04
{+7/2,−5/2}	1	s	0			14.2	0.2	0.04
	1	s	0			28.1	0.2	0.22
	1	s	0	61(8)	31	60.0	6.0	0.79
	1	s	0			149.4	0.3	0.13
{+9/2,−5/2}	2	s	0	27.3(0.3)	4.8	26.8	1.9	1.2
	2	s	0	63.4(0.7)	30	62.1	8.4	6.3
	2	p	1	159(3)	40	157.1		23
	2	p	−1	ibid.		157.1		23
	2	p	0	ibid.		166.1	−46.5	25
{+3/2,−3/2}	0	s	0			17.4	0.08	0.02
	0	s	0	31(4)	6	31.4	0.2	0.13
	0	p	0	46(4)	4	45.6		0.11
	0	p	1	ibid.		46.4		0.07
	0	p	−1	ibid.		46.4		0.13
	0	s	0	53(4)	4	55.3	0.3	0.35
	0	s	0	95(4)	23	93.6	2.3	1.2
	0	s	0			98.8★		0.009
	0	s	0			120.9	0.1	0.0003
	0	s	0	182(4)	12	181.4	2.3	0.25
{+5/2,−3/2}	1	s	0			14.1	0.06	0.007
	1	s	0	23(8)	8	27.3	0.1	0.04
	1	s	0	53(8)	8	51.8	0.1	0.07
	1	s	0			115.4	0.8	0.55
	1	s	0	122(8)	>40	120.2	7.7	2.5
	1	s	0			168.8	1.1	0.13
	1	s	0			288.4	5.9	6.0
{+9/2,−3/2}	3	s	0	53(4)	14	51.9	4.4	2.6
	3	s	0	137(8)	53	140.4	14.4	5.5
{+1/2,−1/2}	0	s	0	15(4)	4	17.3	0.06
	0	s	0	31(4)	5	30.9	0.2
	0	s	0	61(8)	8	53.4	0.4
	0	s	0	88(4)	4	87.1	0.4
	0	s	0			93.6
	0	s	0			109.9	0.02
	0	s	0			145.1	0.03
	0	s	0	246(0.8)	2.4	246.6	2.0
	0	s	0	389(1)	5.5	389.7	26.7
{+9/2,−1/2}	4	s	0	114(8)	>40	112.2	11.7	7.6
{−1/2,−1/2}	−1	p	1	373(2)	2	372.4	30.1
	−1	p	−1	ibid.		372.4	30.1	0.001
	−1	p	0	ibid.		373.4	−11.5	0.001
	−1	p	1			418.9	8.1
	−1	p	0			418.9	−0.8	0.003
	−1	p	−1			418.9	8.1	0.003
{+3/2,+3/2}	3	p	1	140(4)	4	139.6	−37.1
	3	p	−1	ibid.		139.8	−37.0	0.03
	3	p	0	ibid.		141.5	−30.5	0.009

Most of the experimental features are Feshbach resonances with s partial-wave character, while a handful have p-wave character. In fact, for each resonance the partial wave of the incoming collision channel and the corresponding resonant bound state are the same. Interestingly, resonances can overlap within their widths. For example, the experimental loss feature at B = 122G in {+5/3,−3/2} collisions are two overlapping s-wave resonances according to our coupled-channels calculations, one at ≈115.5G and one at ≈120 G. (We assign the experiment with the latter in the table.) Overlapping p-wave resonances can also occur and correspond to its three rotational projection quantum numbers m_ℓ = ±1,0. Their degeneracy is lifted only by the magnetic dipole-dipole interaction resulting in B-field splittings that are smaller than our corresponding experimental σ^exp, as verified by our coupled-channels calculations. In fact, the lines are not resolved.

The remaining four loss features are probable Feshbach resonances and listed in Table II. For these features we found no corresponding theoretical resonances. Parallel to this work, groups in Munich and Amsterdam have measured other ⁴⁰K Feshbach resonances in different collision channels [43].

TABLE II.

Four further measured loss resonances in ⁴⁰K. Columns represent collision channel {m₁, m₂} in the f = 9/2 manifold, quantum number M = m₁ + m₂, the partial wave ℓ, the maximum loss position B^exp, and the magnetic width σ^exp. For these loss features no corresponding theoretical value exists. Magnetic fields and widths are in Gauss. The partial wave for each resonance has been assigned s wave for equal losses in both components and p wave if losses occur in only one component.

{m1, m2}	M	ℓ	Bexp	σexp
{+9/2,−1/2}	4	s	61(8)	9
{+3/2,+3/2}	3	p	76(4)	4
{+7/2,+7/2}	7	p	105(3)	11
	7	p	182(2)	12

Note that the position of a resonance determined from atom loss measurements contains systematic deviations as reported previously; i.e., B_res ≠ B^exp [16]. Similarly, σ^exp does not coincide with the calculated γ₂. Atom loss is not only due to two-body collisions; it is also caused by three-body recombination, where three colliding atoms react to produce a hot molecule. The field-dependent recombination rate coefficient K₃(B), does not need to peak at the same B field or have the same width as K₂(B). In addition, for quantum degenerate Fermi gases of ⁴⁰K atoms, collective phenomena can modify the resonance feature especially when the scattering length a(B) is large compared to 1/k_F, where the Fermi wave vector k_F is defined by ${\hslash }^2{k}_{\mathrm{F}}^2/(2m)=k{T}_{\mathrm{F}}$ [44]. Finally, line shapes can also be distorted when a large fraction of the atoms is lost.

The collision channel {+1/2,−1/2} is of particular interest. It is the magnetic ground state of the spin subspace with zero magnetization M = m₁ + m₂ = 0 and, hence, losses due to inelastic two-body collisions can occur only by spin-relaxation from the weak magnetic dipole-dipole interactions [23]. In this mixture, we have located a Feshbach resonance at $B_{\text{res}}^{\text{calc}}=389.7\mathrm{G}$ with a width of ΔB^calc = 26.7G, which is about three times larger than the width of the commonly used Feshbach resonances in the channels {−9/2,−7/2} and {−9/2,−5/2}.

III. PRECISION DETERMINATION OF FESHBACH RESONANCE PARAMETERS VIA SPINWAVE ANALYSIS

From an experimental point of view, a broad resonance is desirable as it lowers the technical demands for setting a stable value of the interaction strength close to resonance. Hence, the accurate determination of the resonance position B_res and the zero crossing, where a(B) has a node as a function of B, are of particular importance. The latter occurs when B_zero = B_res + ΔB, as can be seen from Eq. (1) when γ₂ → 0. At this zero crossing atom loss tends to be small. Recently, detection methods such as radio-frequency spectroscopy [7,45], collective excitations [46,47], Bloch oscillations [48], and spin segregation [49] have been used to determine the resonance position or the zero crossing. Here, we report on a method based on creating spin-wave excitations near a Feshbach resonance. These excitations are sensitive to the sign of the scattering length a(B) [50] and, in particular, the phase of the spin-wave changes sign as the sign of a(B) changes. Spin waves are an interaction-induced phenomenon and cannot be excited at the zero crossing. For strong interactions close to the pole in a(B), where the sign of a(B) also changes, many-body effects induce additional corrections [44]. Therefore, spin waves are particularly suited for finding the zero crossing of Feshbach resonances.

We excite spin waves with spatially dependent magnetic fields that induce spatially dependent relative phase evolution between the two spin components (for details, see [21]). For this purpose, we first prepare a single-component Fermi gas in spin state m = +1/2 in an elongated dipole trap with trapping frequencies ω = 2π × (70,70,12) Hz along the three independent spatial directions. At a magnetic field near the {+1/2,−1/2} Feshbach resonance at $B_{\text{res}}^{\text{calc}}=389.7\mathrm{G}$, we subsequently apply a radio-frequency pulse with a duration of 10 μs to create a coherent and equal superposition of the spin states m = +1/2 and −1/2. We then excite a spin wave by using one of two types of field inhomogeneities along the weakest trapping direction. Close to the zero crossing, where the interaction strength is small, we apply a linear magnetic gradient and excite linear spin waves leading to dipole oscillations. For larger interaction strengths small field inhomogeneities are sufficient. Here, even the small residual magnetic quadrupole component originating from the Helmholtz coils excite quadrupole spin waves and spatial breathing modes. Examples of the spatial breathing and dipole modes are shown in Figs. 1(a) and 1(b). In both cases counterflow spin currents between the two spin components are induced, even though the overall density remains constant. While the dipole mode induced near the zero crossing of the resonance is long lived [21], the breathing mode quickly decays due to incoherent collisions in the vicinity of B_res [19].

FIG. 1.

Breathing and dipole mode resulting from a quadrupole and linear spin wave. Time evolution of the density of the spin components m = +1/2 (top row) and m = −1/2 (bottom row), obtained by integrating over the two spatial directions orthogonal to the spin-wave excitation. Panel (a) shows quadrupole oscillations induced at B = 401G for strong interactions, and panel (b) shows dipole oscillations induced at B = 422 G for weak interactions. Counterflow dynamics between the spin components can be observed in both panels.

The initial phase and amplitude of the breathing and dipole oscillation depends on the magnetic-field strength. To extract this behavior from our data we analyze the time dependence of the variance of the spatial density profile along the weak trapping direction for the breathing mode and of the displacement of each of the spin components for the dipole mode. The time evolution of the difference in the variance of the two spin clouds is shown in Fig. 2(a) for two magnetic fields on either side of the B_res = 389.7G resonance position. Initially, the differential width grows up to a maximum value and then slowly decays to zero, indicative of strongly damped motion. Figure 2(b) depicts the time evolution of the difference in the displacement of the two spin components for two magnetic fields on either side of the zero crossing. The dipole oscillations remain visible over several periods. We repeat measurements as shown in Figs. 2(a) and 2(b) for various magnetic fields and find that the observed amplitudes reveal a strong magnetic-field dependence. Figures 2(c) and 2(d) summarize this dependence as a function of magnetic field. Using a linear fit, we can accurately determine the magnetic field at which the spin waves change their oscillation phase. This yields $B_{\text{sw}}^{\text{res}}=389.5(0.1)\mathrm{G}$ and $B_{\text{sw}}^{\text{zero}}=416.1(0.1)\mathrm{G}$, respectively. The quoted 1 standard deviation uncertainty follows from the fit. We estimate a systematic error due to an uncertainty in themagnetic-field calibration of ΔB_sys = 0.2 G.

FIG. 2.

Spin waves for different magnetic fields. (a) Time evolution of the differential variance for a breathing-mode spin wave at two magnetic fields close the broad {+1/2,−1/2} Feshbach resonance. (b) Time evolution of the differential displacement for a dipole-mode spin wave at two magnetic fields close to the zero crossing of this same resonance. In (a) and (b) the solid lines are only guides to the eye. (c) Maximum amplitude of the differential variance as a function of magnetic field around the resonance position. (d) Amplitude of the differential displacement as a function of magnetic field near the zero crossing. The amplitude is extracted by fitting a damped sine oscillation to data similar to that in panel (b). The positions of the sign change of the scattering length are determined from a linear fit.

The measured value $B_{\text{sw}}^{\text{zero}}=416.1(0.1)\mathrm{G}$ should coincide with the zero crossing of the scattering length. In fact, our measured value is in very good agreement with the theoretical value of 416.4 G. In contrast, the field value $B_{\text{sw}}^{\text{res}}$ is affected by many-body effects and does not serve as a precise measure for the Feshbach resonance position. In future experiments, this could be overcome by using thermal gases, where these effects are negligible [44].

IV. CONCLUSION

In conclusion, we have observed 37 Feshbach resonances in ⁴⁰K over a broad range of magnetic-field values, as well as four further loss resonances whose origin has not yet been theoretically determined. Thirty-one of the theoretically confirmed resonances have s-wave character and six are p-wave resonances. Most of these resonances are accompanied by losses. In fact, these losses as well as the elastic interactions can be tuned for each Feshbach resonance. This allows for various future applications, such as the study of a quantum Zeno insulator in optical lattices [51,52]. Furthermore, a broad Feshbach resonance in the collision channel {+1/2,−1/2} has been identified at a magnetic field of B = 389.7 G with a width of 26.7 G, which constitutes an ideal candidate for two-component studies with accurate control over the interaction strength. In addition, we demonstrated the creation of spin waves around this Feshbach resonance, which allowed for a precise determination of the zero crossing. Furthermore, we observed a phase shift of the spin waves near the Feshbach resonance position, which might allow for the study of many-body effects in strongly interacting Fermi gases in the future.