Mott-Insulator-Aided Detection of Ultra-Narrow Feshbach Resonances
Manfred J. Mark, Florian Meinert, Katharina Lauber, Hanns-Christoph Nägerl
SSciPost Physics Submission
Mott-Insulator-Aided Detection of Ultra-Narrow FeshbachResonances
M. J. Mark , F. Meinert , K. Lauber , H.-C. N¨agerl Institut f¨ur Experimentalphysik und Zentrum f¨ur Quantenphysik, Universit¨at Innsbruck,6020 Innsbruck, Austria Institut f¨ur Quantenoptik und Quanteninformation, ¨Osterreichische Akademie derWissenschaften, 6020 Innsbruck, Austria
5. Physikalisches Institut and Center for Integrated Quantum Science and Technology,Universit¨at Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany* [email protected] 17, 2018
Abstract
We report on the detection of extremely narrow Feshbach resonances by em-ploying a Mott-insulating state for cesium atoms in a three-dimensional opticallattice. The Mott insulator protects the atomic ensemble from high backgroundthree-body losses in a magnetic field region where a broad Efimov resonance oth-erwise dominates the atom loss in bulk samples. Our technique reveals threeultra-narrow and previously unobserved Feshbach resonances in this region withwidths below ≈ µ G, measured via Landau-Zener-type molecule formation andconfirmed by theoretical predictions. For comparatively broader resonances wefind a lattice-induced substructure in the respective atom-loss feature due to theinterplay of tunneling and strong particle interactions. Our results provide a pow-erful tool to identify and characterize narrow scattering resonances, particularlyin systems with complex Feshbach spectra. The observed ultra-narrow Feshbachresonances could be interesting candidates for precision measurements.
Contents a r X i v : . [ qu a n t - ph ] A ug ciPost Physics Submission Ultracold atomic gases provide an ideal platform for precise studies of atom-atom interac-tions in the low-temperature regime of quantum scattering. As such, they allow to gatherexperimental information on the fine details of interaction potentials for benchmarking the-oretical models that are in most cases beyond the scope of ab initio calculations. Accurateknowledge of atom-atom interactions, in turn, forms the basis for exploiting quantum gases astunable quantum simulators [1], examples being the celebrated Fermi- and Bose-Hubbard sys-tem [2–5], interacting dipolar gases [6], or charge-neutral hybrids [7]. Precision spectroscopy,such as performed in optical lattice clocks, also depends on an accurate determination ofparticle interactions [8, 9]. An important experimental contribution to pin down interactionpotentials with highest accuracy constitutes the characterization of Feshbach scattering reso-nances, i.e. their positions and widths [10–13]. Moreover, very narrow resonances have evenbeen proposed for ultracold atom-based detection of changes in fundamental constants [14–16].In systems with a rich Feshbach spectrum, one often encounters broad and narrow over-lapping Feshbach resonances. This can lead to situations where narrow resonance featuresare completely disguised by broader ones and are hard to detect. Here, we report on anoptical-lattice-based method that allows us to observe such hidden Feshbach resonances forthe example of ultracold cesium atoms in the vicinity of a broad Efimov three-body lossresonance [17]. Our approach leads to strong suppression of the Efimov loss feature, whichfully dominates the loss in the absence of the lattice, by protecting the sample in a Mott-insulating state. This provides access to previously unobserved ultra-narrow g -wave Feshbachresonances with widths on the order of ≈ µ G in the low magnetic field region
B <
10 G.We measure the resonance positions via Feshbach spectroscopy in the strongly interactingsystem and provide a detailed characterization of their widths by analyzing the efficiency ofFeshbach-molecule production. Finally, we study in detail the loss features caused by theresonances, which in the lattice result from resonant tunneling processes [18, 19].
Experimental detection of Feshbach resonances in trapped atomic ensembles typically relieson the observation of enhanced atom loss in the vicinity of the resonance pole as a result ofthe rapid increase of the three-body recombination rate with the diverging scattering length a s . Consequently, the overlap of multiple recombination-induced loss features can hinder theidentification of Feshbach resonances, in particular, when they are narrow and arise from acomparatively weak coupling to bound molecular states. In a first set of measurements, westudy such a scenario for an ultracold cesium sample and explore Feshbach spectroscopy for astrongly correlated lattice gas as a pathway to observe otherwise hidden Feshbach resonances.We start the discussion with standard Feshbach spectroscopy on a weakly confined ther-mal, ultracold ensemble by measuring magnetic-field dependent atom loss. Specifically, ourexperiment starts with ≈ × cesium atoms in the lowest hyperfine state | F = 3; m F = 3 (cid:105) trapped in a crossed optical dipole trap and prepared at a temperature T = 137(10) nK. Dur-ing sample preparation, we apply a magnetic offset field B of about 21 G, for which a s = 210 a .Additionally, a magnetic field gradient of 31 G/cm along the vertical z -axis compensates thegravitational force and thus levitates the trapped ensemble [20]. We ramp B with a speed of2 ciPost Physics Submission A t o m n u m b e r n A ( x ) Magnetic field (G)2 4 6 8 10 12 14 161015202530 A t o m n u m b e r n A ( x ) Magnetic field (G) (b)(a)
Magnetic field (G)10 12 14 16 a S ( x ) -25-20-15-10-505 8642 Figure 1: Feshbach atom-loss spectroscopy with and without the optical lattice. Remainingatom number n A as a function of magnetic field B for a thermal ensemble ( T = 137(10) nK)after a hold time t H = 50 ms (a) and for a Mott insulator prepared at V x,y,z = 20 E R after t H = 200 ms (b). Three Feshbach resonances above 10 G are evident in both spectra. Ultra-narrow resonances below 10 G are only observed in the optical lattice. They are masked by abroad Efimov resonance around 8 G [17] in the bulk system. Weakly increased loss at 8 . . a s as a function of B showing the position of predicted Feshbach resonances in the magneticfield region of interest. Error bars indicate one standard deviation and solid lines connect thedata points to guide the eye. 3 ciPost Physics Submission ∼ . n A after a fixed hold time of t H = 50 ms.The measurement (Fig.1(a)) reveals multiple loss features. Three resonances at 15 , 14 . f l ( m f ) = 6g(5), 4g(3), and 4g(2), respectively[10]. Here, f is the total internal angular momentum with its projection m f , and l denotesthe nuclear rotational angular momentum. Note that M m = m f + m l = 6 for the projection M m of the molecule’s total angular momentum (cid:126)F m = (cid:126)f + (cid:126)l . For decreasing values of B themeasurement is dominated by a broad loss feature centered around 8 G, which is connected toan Efimov trimer resonance [17]. Weak signatures of four-body Efimov resonances at 8 . . . × atoms [20].The sample is then adiabatically loaded into a cubic optical lattice with a final lattice depthof V x,y,z = 20 E R , where E R = h / (2 mλ ) denotes the recoil energy with the mass m of the Csatom and λ = 1064 . ∼
10 to 20 %of the total atom number) double occupancy in the center. Subsequently, we ramp B to thedesired value as before and hold the sample for 200 ms. To avoid any loss or heating whenramping over the already known Feshbach resonances, we use ultrafast ramps with a speed of2 · G/ms across those field regions [23]. As the initial Mott insulator is always preparedat repulsive interaction and subsequently quenched to the vicinity of the Feshbach resonanceof interest for the loss spectroscopy, the correlated many-body state features lifetimes above10 s also for large negative values of a s [24].After the hold time we ramp the field quickly (ramp speed ∼
25 G/ms) to ≈ . . . . f l ( m f ) = 6g(4), 6g(3), and 6g(2), respectively [10]. Before investigating the loss features in the lattice in more detail, we first measure the widthsof the narrow Feshbach resonances in a way that does not rely on the shape of the loss featureand is largely insensitive to residual magnetic field fluctuations. For this, the efficiency of4 ciPost Physics Submission (b)(a) . ≈ R e l a t i v e a t o m n u m b e r Ramp speed B (G/s) 7.65 7.75 14.30 14.404567 A t o m n u m b e r n A x Magnetic field B (G) Figure 2: (a) Relative atom number after ramping across the Feshbach resonance at 14 . . . B . Here, thelattice depth V x,y,z = 20 E R ( V x,y,z = 30 E R ) for the resonance at 14 . . . B as free parameter, giving ∆ B = 14 . .
9) mG(circles), ∆ B = 8 . . µ G (squares), and ∆ B = 1 . . µ G (diamonds). The shaded areaindicates the uncertainty region 0 µ G < ∆ B < µ G for the 7 . n A starting from an initially singly occupied Mottinsulator state at V x,y,z = 20 E R as a function of B around 7 . t H = 5000 ms (circles)and around 14 . t H = 50 ms (squares). Solid lines connect the data points and areguides to the eye.Feshbach molecule association via a magnetic-field sweep across the resonance is monitoredas a function of the sweep rate ˙ B [25]. For an initially doubly occupied lattice site, themolecule formation probability depends on ˙ B and is described via the well-known Landau-Zener formula [26] p = p + (1 − p ) e − πδ LZ . (1)Here, p denotes the probability for the initial atom pair to remain unbound after the fieldsweep. The dependence on the sweep rate is encoded in δ LZ = √ hπma (cid:12)(cid:12)(cid:12)(cid:12) a bg ∆ B ˙ B (cid:12)(cid:12)(cid:12)(cid:12) (2)with ∆ B = B − B ∗ the width of the Feshbach resonance, defined by the magnetic field valuesof the resonance pole B and the zero crossing B ∗ of the scattering length a s . Further, a bg is the background scattering length, a ho the harmonic oscillator length for atoms confined ata lattice site, and p a constant offset that accounts for residual single occupancy and finitemaximum conversion efficiency.Our measurements of ∆ B for the individual resonances start with the preparation of aMott insulator with predominant double occupancy. Briefly, this is achieved by applyinga stronger external harmonic confinement during lattice loading to maximize the number5 ciPost Physics Submission of doubly occupied lattice sites. The sample is then purified by selective removal of singlyoccupied sites. This cleaning sequence exploits a Feshbach resonance at 19.9 G to transferdoublons into molecules, before a combined radio-frequency and resonant-light pulse is appliedto remove the singly occupied sites. Finally, the molecules are transferred back into unboundatom pairs (for details see Ref. [27]). Note that during the whole sequence the magneticgradient field is switched off to achieve maximum conversion efficiency.After sample preparation, the magnetic field is quickly set to a value ∼ . B , whichstops ∼ . B . Exemplarydatasets for the resonances located at 14 . . . B for the comparatively broad Feshbachresonances at 15 , 14 . V x,y,z = 20 E R . Forthe narrow Feshbach resonances below 10 G the situation is more delicate, as we will discussin the following paragraph.A smaller resonance width requires lower sweep rates for efficient molecule formation.In our experiment, the minimum useable ramp speed is limited by low-frequency magneticfield noise on the order of 10 mG (peak-to-peak value) with main spectral components at50 Hz and 150 Hz. Specifically, this leads to shot-to-shot fluctuations of the sweep rate overthe resonance on the order of ≈ a ho . For the measurements ofthe narrow resonances below 10 G, we thus increased the lattice depth to V x,y,z = 30 E R toshift the signal to faster ramp speeds. As shot-to-shot fluctuations are still relatively strongfor low ramp speeds ( cf. Fig. 2(a)), we account for this by an additional uncertainty regionof 0 − µ G, exemplary indicated by the shaded area in Fig. 2(a) for the case of the 7 . . µ G the experimental value of 1 . . µ G obtained from the fitto the data based on Eq. 1 is in good agreement.
We now perform a more detailed analysis of the atom-loss structure at the Feshbach resonanceswhen measured in the correlated lattice system. This is not only of relevance for a precisedetermination of their positions, but also reveals fine details on the loss mechanism in thelattice. First, we note that the loss features in Fig. 1 are inhomogeneously broadened bythe applied magnetic-field gradient that counteracts gravity. Once the sample is prepareddeep in the Mott-insulating phase, however, the gradient field can be removed during the lossmeasurements as the lattice is strong enough to hold the atoms against gravity. Second, weobserve that the residual double occupancy leads to a broadening and enhancement of the lossfeatures. Possible mechanisms leading to stronger heating and losses in this situation include6 ciPost Physics Submission on-site molecule formation, coupling of atom pairs to higher lattice bands in the vicinity ofthe Feshbach resonance, and comparably larger off-site three-body losses [29]. Therefore,we modify our loading procedure into the Mott insulator state such that we end up withessentially pure single occupancy as detailed in Ref. [30].This allows us to perform high-resolution atom-loss spectroscopy in the close vicinity ofall the observed resonances that are shown in Fig. 1. In Fig. 2(b), we show two examples formagnetic-field scans around the resonances at 7 . . V x,y,z = 20 E R .The hold time for the Feshbach resonance at 7 . . ≈ . V x,y,z at which the loss measurements in the Mott insulator state are carried out. Measurementsof the split loss feature for V x,y,z = 20 E R and V x,y,z = 30 E R are shown in Fig. 3 for theexample of the Feshbach resonance at 19 . B for increasing V x,y,z . These observations are a resultof the variation of the on-site interaction U near the Feshbach resonance. Starting from aone-atom-per-site Mott insulator, particle loss requires atom tunneling in combination withon-site three-body recombination. To first order, this is possible when U is sufficiently small,as it is the case in the vicinity of the zero crossing of the Feshbach resonance B ∗ . Accordingly,a particle-loss feature arising from such a mechanism is expected to be rather independentof the lattice depth. Alternatively, the removal of the magnetic gradient field subjects thesample to gravity, which introduces an energy offset E between neighboring lattice sites alongthe vertical z -direction. This provides a second loss channel, which occurs at a magnetic fieldwhere | U | ≈ E . Here, resonant tunneling along z becomes energetically allowed [18, 19], andthus opens up the system again to three-body recombination. Evidently, the associated lossfeatures should shift with varying lattice depth as U increases for increasing V x,y,z [31].To test our reasoning, we plot the calculated scattering length around the Feshbach reso-nance according to its functional dependence on the magnetic field a s ( B ) = a bg · (1 − ∆ B/ ( B − B ))in Fig. 3(b) [11]. For the modeling we neglect corrections due to finite range and finite colli-sion energy [32,33] as well as corrections due to higher bands [30]. The background scatteringlength a bg is known from multi-channel calculations [22] and the resonance width (in thiscase ∆ B = 11 . B = 19 .
874 G for the pole of the resonance the observed loss featuresare in good agreement with the tunneling-induced decay mechanism discussed above. This isindicated by the symbols in Fig. 3(b), which denote the magnetic field values where U = 0(diamond) or | U | ≈ E (circles and squares) for the two different lattice depths. Additionally,we mark the magnetic field values where | U | = E by the shaded areas in Fig. 3(a) with a widththat indicates our typical magnetic field noise. Note that the two expected loss channels forwhich U = − E and U = 0 are not resolvable given our experimental resolution and hencelead to a single loss feature. 7 ciPost Physics Submission A t o m n u m b e r n A ( x ) S c a tt e r i n g l e n g t h a S ( a x ) Magnetic field B (G)468468 19.8 19.85 19.9 19.95 (a) -4-2024 (b)
Figure 3: (a) Remaining atom number n A for a singly occupied Mott insulator as a functionof B in the vicinity of the Feshbach resonance at 19 . t H = 500 ms and V x,y,z = 30 E R (top circles) and t H = 50 ms and V x,y,z = 20 E R (bottom squares). The shaded areas mark theregions for which | U | = E , incorporating a width of 8 mG (see (b)). The solid lines connectthe data points are guides to the eye. (b) Calculated scattering length around the Feshbachresonance using B = 19 .
874 G and ∆ B = 11 mG. The local background scattering length inthis range is ≈
160 a . Magnetic field values where | U | = E for V x,y,z = 30 E R (circles) and V x,y,z = 20 E R (squares), and where | U | = 0 (diamond) are indicated.8 ciPost Physics Submission Table 1: Experimentally determined and theoretically predicted Feshbach resonance positionsand widths [28]. Note that the values and error bars for resonances <
10 G are the bare fitvalues. We estimate an uncertainty region of 0 − µ G for those resonances.Experiment TheoryResonance B ( G ) ∆ B ( mG ) B ( G ) ∆ B ( mG )4g(4) 19.874(0.01) 11.1(0.7) 19.682 9.76g(5) 15.014(0.01) 3.4(1.0) 14.761 3.64g(3) 14.345(0.01) 14.0(0.9) 14.195 13.44g(2) 10.994(0.01) 3.2(0.3) 10.893 6.06g(4) 7.704(0.01) 0.0080(0.0016) 7.555 0.0166g(3) 5.122(0.01) 0.0014(0.0008) 5.038 0.0066g(2) 3.753(0.01) 0.0012(0.0004) 3.703 0.002A similar analysis is performed for the narrow Feshbach resonances at 15 G, 14 . a bg takes negative values( c.f. inset to Fig. 1(a)). Here the zero crossing is located at magnetic fields below the pole.For the ultra-narrow Feshbach resonances, which are too narrow to experimentally resolve asubstructure, we identify the center of the single loss feature with the resonance pole. Finally,our results for all extracted positions B and widths ∆ B are summarized in Table 1, includingthe statistical error for the widths and the expected experimental error due to uncertaintiesin magnetic-field calibration and magnetic-field noise. A comparison to the theoretical valuesbased on latest multichannel calculations [28] shows again a very good overall agreement whentaking into account the uncertainty on the theory positions on the order of 0 . In summary, we have explored an alternative technique to detect and characterize Feshbachresonances employing strong correlations of a Mott-insulating many-body state. This allowsfor the observation of previously unexplored ultra-narrow resonances, which are fully maskedby strong three-body loss in more common bulk measurements. The approach is of interest forvarious elements with complex and overlapping Feshbach spectra e.g. lanthanides like Er andDy [34–37]. We have provided a detailed analysis of the resonant loss features in the lattice,which show substructure due to the interplay of particle tunneling, interactions, and externallyapplied energy offsets. Combined with independent measurements of the resonance widths viaLandau-Zener type Feshbach molecule formation, this allows for a full parametrization of theresonances, which is important for improving the accuracy of coupled-channels calculations ofmolecular potentials [22]. The observed ultra-narrow Feshbach resonances could be suited torealize several proposals in high precision measurements [14–16]. Their location at compara-tively low magnetic fields should allow to reach the necessary accuracy, precision, and noiserequirements of magnetic fields using passive and active stabilization techniques. Moreover,9 ciPost Physics Submission the substructure observed in the lattice calls for future studies at broader resonances, forwhich we expect to resolve resonant coupling to higher lattice bands [38, 39].
Acknowledgements
We are indebted to R. Grimm for generous support. We thank P. S. Julienne for fruitfuldiscussions.
Funding information
F. M. acknowledges support from the Carl-Zeiss foundation and isindebted to the Baden-W¨urttemberg-Stiftung for the financial support by the Eliteprogrammfor Postdocs. We gratefully acknowledge funding by the European Research Council (ERC)under Project No. 278417 and the Austrian Science Foundation (FWF) under Projects No.I2922-N36 and No. Z336-N36.
References [1] C. Gross and I. Bloch,
Quantum simulations with ultracold atoms in optical lattices ,Science , 995 (2017). doi:10.1126/science.aal3837[2] M. Greiner, O. Mandel, T. Esslinger, T. W. H¨ansch, and I. Bloch,
Quantum phasetransition from a superfluid to a Mott insulator in a gas of ultracold atoms , Nature ,39 (2002). doi:10.1038/415039a[3] R. J¨ordens, N. Strohmaier, K. G¨unter, H. Moritz, and T. Esslinger,
A Mott insulator offermionic atoms in an optical lattice , Nature , 204 (2008). doi:10.1038/nature07244[4] U. Schneider, L. Hackerm¨uller, S. Will, Th. Best, I. Bloch, T. A. Costi, R. W. Helmes, D.Rasch, and A. Rosch,
Metallic and Insulating Phases of Repulsively Interacting Fermionsin a 3D Optical Lattice , Science , 1520 (2008). doi:10.1126/science.1165449[5] T. Fukuhara, S. Sugawa, M. Sugimoto, S. Taie, and Y. Takahashi,
Mott insula-tor of ultracold alkaline-earth-metal-like atoms , Phys. Rev. A , 041604 (2009).doi:10.1103/PhysRevA.79.041604[6] T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, The physics of dipo-lar bosonic quantum gases , Rep. Prog. Phys. , 126401 (2009). doi:10.1088/0034-4885/72/12/126401[7] M. Tomza, K. Jachymski, R. Gerritsma, A. Negretti, T. Calarco, Z. Idziaszek, and P. S.Julienne, Cold hybrid ion-atom systems , arXiv:1708.07832 (2017).[8] M. Takamoto, F.-L. Hong, R. Higashi, and H. Katori,
An optical lattice clock , Nature , 321-324 (2005). doi:10.1038/nature03541[9] M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin, A. M. Rey, J. Ye,
Suppres-sion of Collisional Shifts in a Strongly Interacting Lattice Clock , Science , 1043-1046(2011). doi:10.1126/science.1196442 10 ciPost Physics Submission [10] C. Chin, V. Vuleti´c, A. J. Kerman, S. Chu, E. Tiesinga, P.J. Leo and C.J. Williams,
Precision Feshbach spectroscopy of ultracold Cs , Phys. Rev. A , 032701 (2004).doi:10.1103/PhysRevA.70.032701[11] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases ,Rev. Mod. Phys. , 1225 (2010). doi:10.1103/RevModPhys.82.1225[12] S. Knoop, T. Schuster, R. Scelle, A. Trautmann, J. Appmeier, M. K. Oberthaler,E. Tiesinga, and E. Tiemann, Feshbach spectroscopy and analysis of the in-teraction potentials of ultracold sodium , Phys. Rev. A , 042704 (2011).doi:10.1103/PhysRevA.83.042704[13] M. Gr¨obner, P. Weinmann, E. Kirilov, H.-C. N¨agerl, P. S. Julienne, C. Ruth Le Sueur,and J. M. Hutson, Observation of interspecies Feshbach resonances in an ultracold K- Cs mixture and refinement of interaction potentials , Phys. Rev. A , 022715 (2017).doi:10.1103/PhysRevA.95.022715[14] C. Chin and V. V. Flambaum, Enhanced Sensitivity to Fundamental Constants In Ul-tracold Atomic and Molecular Systems near Feshbach Resonances , Phys. Rev. Lett. ,230801 (2006). doi:10.1103/PhysRevLett.96.230801[15] T. Zelevinsky, S. Kotochigova, and J. Ye, Precision Test of Mass-Ratio Varia-tions with Lattice-Confined Ultracold Molecules , Phys. Rev. Lett. , 043201 (2008).doi:10.1103/PhysRevLett.100.043201[16] A. Borschevsky, K. Beloy, V. V. Flambaum, and P. Schwerdtfeger,
Sensitivity of ultracold-atom scattering experiments to variation of the fine-structure constant , Phys. Rev. A ,052706 (2011). doi:10.1103/PhysRevA.83.052706[17] T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange,K. Pilch, A. Jaakkola, H.-C. N¨agerl, and R. Grimm, Evidence for Efimov quantum statesin an ultracold gas of caesium atoms , Nature , 315 (2006). doi:10.1038/nature04626[18] F. Meinert, M. J. Mark, E. Kirilov, K. Lauber, P. Weinmann, A. J. Daley, and H.-C.N¨agerl,
Quantum Quench in an Atomic One-Dimensional Ising Chain , Phys. Rev. Lett. , 053003 (2013). doi:10.1103/PhysRevLett.111.053003[19] F. Meinert, M. J. Mark, E. Kirilov, K. Lauber, P. Weinmann, M. Gr¨obner, A. J. Daley,and H.-C. N¨agerl,
Observation of many-body dynamics in long-range tunneling after aquantum quench , Science , 1259 (2014). doi:10.1126/science.1248402[20] T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H.-C. N¨agerl, and R. Grimm,
Optimized production of a cesium Bose–Einstein condensate , Appl. Phys. B , 1013(2004). doi:10.1007/s00340-004-1657-5[21] F. Ferlaino, S. Knoop, M. Berninger, W. Harm, J.P. D’Incao, H.-C. N¨agerl, and R.Grimm, Evidence for Universal Four-Body States Tied to an Efimov Trimer , Phys. Rev.Lett. , 140401 (2009). doi:10.1103/PhysRevLett.102.140401[22] M. Berninger, A. Zenesini, B. Huang, W. Harm, H.-C. N¨agerl, F. Ferlaino, R. Grimm, P.S. Julienne, and J. M. Hutson,
Feshbach resonances, weakly bound molecular states, and ciPost Physics Submission coupled-channel potentials for cesium at high magnetic fields , Phys. Rev. A , 032517(2013). doi:10.1103/PhysRevA.87.032517[23] J. G. Danzl, M. J. Mark, E. Haller, M. Gustavsson, R. Hart, A. Liem, H. Zellmer, andH.-C. N¨agerl, Deeply bound ultracold molecules in an optical lattice , New J. Phys. ,055036 (2009). doi:10.1088/1367-2630/11/5/055036[24] M. J. Mark, E. Haller, K. Lauber, J. G. Danzl, A. Janisch, H. P. B¨uchler, A.J. Daley, and H.-C. N¨agerl, Preparation and Spectroscopy of a Metastable Mott-Insulator State with Attractive Interactions , Phys. Rev. Lett. , 215302 (2012).doi:10.1103/PhysRevLett.108.215302[25] G. Thalhammer, K. Winkler, F. Lang, S. Schmid, R. Grimm, and J. Hecker Denschlag,
Long-Lived Feshbach Molecules in a Three-Dimensional Optical Lattice , Phys. Rev. Lett. , 050402 (2006). doi:10.1103/PhysRevLett.96.050402[26] P. S. Julienne, E. Tiesinga, and T. K¨ohler, Making cold molecules by time-dependent Feshbach resonances , Journal of Modern Optics , 1787-1806 (2004)doi:10.1080/09500340408232491[27] J. G. Danzl, M. J. Mark, E. Haller, M. Gustavsson, R. Hart, J. Aldegunde, J. M. Hutson,and H.-C. N¨agerl, An ultracold high-density sample of rovibronic ground-state moleculesin an optical lattice , Nat. Phys. , 265 (2010). doi:10.1038/nphys1533[28] P. S. Julienne, private communication (2014).[29] A. J. Daley, J. M. Taylor, S. Diehl, M. Baranov, and P. Zoller, Atomic Three-BodyLoss as a Dynamical Three-Body Interaction , Phys. Rev. Lett. , 040402 (2009).doi:10.1103/PhysRevLett.102.040402[30] M. J. Mark, E. Haller, K. Lauber, J. G. Danzl, A. J. Daley, and H.-C. N¨agerl,
PrecisionMeasurements on a Tunable Mott Insulator of Ultracold Atoms , Phys. Rev. Lett. ,175301 (2011). doi:10.1103/PhysRevLett.107.175301[31] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller,
Cold Bosonic Atoms inOptical Lattices , Phys. Rev. Lett. , 3108 (1998). doi:10.1103/PhysRevLett.81.3108[32] P. Naidon, E. Tiesinga, W. F Mitchell, and P. S. Julienne, Effective-range descriptionof a Bose gas under strong one- or two-dimensional confinement , New J. Phys. , 19(2007). doi:10.1088/1367-2630/9/1/019[33] C. L. Blackley, P. S. Julienne, and J. M. Hutson, Effective-range approxima-tions for resonant scattering of cold atoms , Phys. Rev. A , 042701 (2014).doi:10.1103/PhysRevA.89.042701[34] K. Baumann, N. Q. Burdick, M. Lu, and B. L. Lev, Observation of low-field Fano-Feshbach resonances in ultracold gases of dysprosium , Phys. Rev. A 89, 020701(R) (2014).doi:10.1103/PhysRevA.89.020701[35] A. Frisch, M. Mark, K. Aikawa, F. Ferlaino, J. L. Bohn, C. Makrides, A. Petrov, and S.Kotochigova,
Quantum chaos in ultracold collisions of gas-phase erbium atoms , Nature , 475 (2014). doi:10.1038/nature1313712 ciPost Physics Submission [36] T. Maier, H. Kadau, M. Schmitt, M. Wenzel, I. Ferrier-Barbut, T. Pfau, A. Frisch, S.Baier, K. Aikawa, L. Chomaz, M. J. Mark, F. Ferlaino, C. Makrides, E. Tiesinga, A.Petrov, and S. Kotochigova,
Emergence of Chaotic Scattering in Ultracold Er and Dy ,Phys. Rev. X , 041029 (2015). doi:10.1103/PhysRevX.5.041029[37] S. Baier, D. Petter, J. H. Becher, A. Patscheider, G. Natale, L. Chomaz, M. J. Mark,and F. Ferlaino, Realization of a Strongly Interacting Fermi Gas of Dipolar Atoms ,arxiv:1803.11445 (2018).[38] S. Sala, P.-I. Schneider, and A. Saenz,
Inelastic Confinement-Induced Resonancesin Low-Dimensional Quantum Systems , Phys. Rev. Lett. , 073201 (2012).doi:10.1103/PhysRevLett.109.073201[39] S. Sala, G. Z¨urn, T. Lompe, A. Wenz, S. Murmann, F. Serwane, S. Jochim, and A. Saenz,
Coherent Molecule Formation in Anharmonic Potentials Near Confinement-Induced Res-onances , Phys. Rev. Lett.110