Non-Abelian adiabatic geometric transformations in a cold Strontium gas

# Non-Abelian adiabatic geometric transformations in a cold Strontium gas

## Abstract

Topology, geometry, and gauge fields play key roles in quantum physics as exemplified by fundamental phenomena such as the Aharonov-Bohm effect, the integer quantum Hall effect, the spin Hall, and topological insulators. The concept of topological protection has also become a salient ingredient in many schemes for quantum information processing and fault-tolerant quantum computation. The physical properties of such systems crucially depend on the symmetry group of the underlying holonomy. We study here a laser-cooled gas of strontium atoms coupled to laser fields through a 4-level resonant tripod scheme. By cycling the relative phases of the tripod beams, we realize non-Abelian SU(2) geometrical transformations acting on the dark-states of the system and demonstrate their non-Abelian character. We also reveal how the gauge field imprinted on the atoms impact their internal state dynamics. It leads to a new thermometry method based on the interferometric displacement of atoms in the tripod beams.

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Introduction. In 1984, M. V. Berry published the remarkable discovery that cyclic parallel transport of quantum states causes the appearance of geometrical phases factors Berry (1984). His discovery, along with precursor works Pancharatnam (1956); Aharonov and Bohm (1959), unified seemingly different phenomena within the framework of gauge theories Simon (1983); Wilczek and Shapere (1989). This seminal work was rapidly generalized to non-adiabatic and noncyclic evolutions Wilczek and Shapere (1989) and, most saliently for our concern here, to degenerate states by F. Wilczek and A. Zee Wilczek and Zee (1984). In this case, the underlying symmetry of the degenerate subspace leads to a non-Abelian gauge field structure. These early works on topology in quantum physics have opened up tremendous interest in condensed matter Baibich et al. (1988); Kato et al. (2004); König et al. (2007); Hsieh et al. (2008); Chang et al. (2013) and more recently in ultracold gases Hadzibabic et al. (2006); Lin et al. (2011); Aidelsburger et al. (2015); Jotzu et al. (2014); Mancini et al. (2015); Phuc et al. (2015); Wu et al. (2016); Song et al. (2016); Li et al. (2017) and photonic devices Wang et al. (2009); Kuhl et al. (2010); Schine et al. (2016).

Moreover, it has been noted that geometrical qubits are resilient to certain noises, making them potential candidates for fault-tolerant quantum computing Zanardi and Rasetti (1999); Jones et al. (2000); Duan et al. (2001); Solinas et al. (2012). So far, beside some recent proposals Kowarsky et al. (2014); Sjöqvist (2016), experimental implementations have been performed for a 2-qubit gate on NV-centers in diamond Zu et al. (2014) and for a non-Abelian single qubit gate in superconducting circuits Abdumalikov Jr et al. (2013). These experiments were performed following a non-adiabatic protocol allowing for high-speed manipulation Zhu and Wang (2002); Sjöqvist et al. (2012); Sjöqvist (2016). Recently, coherent control of ultracold spin-1 atoms confined in optical dipole traps was used to study the geometric phases associated with singular loops in a quantum system Bharath et al. (). If non-adiabatic manipulation are promising methods for quantum computing, they prevent the study of external dynamic of quantum system in a non-abelian gauge field, where non-trivial coupling occurs between the internal qubit state and the center-of-mass wave function of the particle.

We report here on non-Abelian adiabatic geometric transformations implemented on a noninteracting cold fermionic gas of Strontium-87 atoms by using a 4-level resonant tripod scheme set on the intercombination line at nm (linewidth: kHz). About atoms are loaded in a crossed optical dipole trap, optically pumped in the stretched Zeeman state and Doppler-cooled down to temperatures Chalony et al. (2011); Yang et al. (2015), see Methods. A magnetic bias field isolates a particular tripod scheme in the excited and ground Zeeman substate manifolds, see Fig. 1. Three coplanar coupling laser beams are set on resonance with their common excited state . Their relative phases can be varied using two electro-optic modulators (EOM), to operate the non-Abelian transformation in the system.

For any value of the amplitude and phase of the lasers beams, the effective Hilbert space defined by the four coupled bare levels contains two degenerate dark-states and which do not couple to and are thus protected from spontaneous emission decay by quantum interference. For equal Rabi transition frequencies,

 |D1⟩ =e−iΦ13(r)|1⟩−e−iΦ23(r)|2⟩√2 |D2⟩ =e−iΦ13(r)|1⟩+e−iΦ23(r)|2⟩−2|3⟩√6, (1)

where (). , where the space-dependent laser phases read . is the wavevector of the beam coupling state to and its phase at origin. The two independent offset phases tuned by the EOM shown in Fig. 1a are ().

In a first set of experiments, we probe and quantify the thermal decoherence of the dark-states due to the finite temperature of our atomic sample. In a second set of experiments, we analyze the non-Abelian character of a cyclic sequence of geometric transformations within the dark-state manifold by changing their relative order. In all experiments, we monitor the subsequent manipulation and evolution of the atomic system in the dark-state manifold by measuring the bare ground-state populations with a nuclear spin-sensitive shadow imaging technique, see Methods.

Thermal decoherence. Starting from state , we prepare the atoms in dark-state after a suitable adiabatic laser ignition sequence, see Methods. We assume that the atoms do not move significantly during the time duration of this sequence, see Fig. 2. Following Ruseckas et al. (2005); Juzeliūnas et al. (2008), the subsequent evolution of the atoms is described by the Hamiltonian

 H=12M(^p\openone−A)2+W (2)

where is the momentum operator, is the identity operator in the internal dark-state subspace, the atom mass, the geometrical vector potential with matrix entries and the geometrical scalar potential with matrix entries

 Wjk=ℏ2⟨∇Dj|∇Dk⟩−A2jk2M. (3)

With our laser beams geometry, , , and have the same matrix form, and are uniform and time-independent, see Methods. Since is Abelian here, we can look for states in the form where is the initial momentum of the atoms and some combination of dark-states. Denoting by the initial atomic velocity distribution, we find that the population of state remains constant while the two others display an out-of-phase oscillatory behavior at a velocity-dependent frequency :

 P1(v,t) =5P0(v)12(1−35cosωvt), (4) P2(\emph{v},t) =P0(\emph{v})6, P3(\emph{v},t) =5P0(v)12(1+35cosωvt),

where is the atomic recoil frequency and is the laser wavenumber. The frequency component proportional to comes from the momentum-dependent coupling term in equation 2 (Doppler effect) whereas the other frequency component proportional to comes from the scalar term .

Averaging over the Maxwellian velocity distribution of the atoms, the bare state populations of the thermal gas read

 ¯P1(t) =512−14cos(43ωRt)exp[−49(k¯vt)2], ¯P2(t) =16, ¯P3(t) =512+14cos(43ωRt)exp[−49(k¯vt)2], (5)

where is the thermal velocity of the gas at temperature . We see that and converge to the same value at long times. Because of the orientation of the laser beams, this means that the thermal average breaks the tripod scheme into a -scheme coupled to the two circularly-polarized beams and a single leg coupled to the linearly-polarized beam. As a consequence, quantum coherence partially survives the thermal average.

Our experimental results confirm this behavior even if and do not merge perfectly, see Fig. 2. This discrepancy can be lifted by introducing a imbalance between the Rabi transition frequencies in our calculation. The population difference measures in fact the Fourier transform of the velocity distribution along the diagonal direction . It decays with a Gaussian envelope characterized by the time constant . This interferometric thermometry is similar to some spectroscopic ones such as recoil-induced resonance Courtois et al. (1994); Meacher et al. (1994) or stimulated two-photons transition Kasevich et al. (1991); Peters et al. (2012). From our measurements, we get K, s and mm/s.

Non-Abelian transformations. We now investigate the geometric non-Abelian unitary operator acting on the dark-state manifold when the relative phases of the tripod beams are adiabatically swept along some closed loop in parameter space. For a pinned atom (), is given by the loop integral along of the Mead-Berry 1-form

 U=Pexp(iℏ∮Cω), (6)

where is the path-ordering operator Wilczek and Zee (1984).

As before, the system is initially prepared in dark-state . Then, starting from the origin, the phase loop is cycled counterclockwise, see Fig. 3a. Each segment is linearly swept in s and the phase excursion is . The total duration of the loop is thus less than the thermal decoherence time discussed above. In Fig. 3b, we plot the bare state populations measured right after the phase loop as a function of and their comparison with our theoretical predictions for a pinned atom and for atoms at finite temperature, see Methods. The very good agreement with the finite temperature case shows that we capture most of the mechanisms at play and that the adiabatic approximation is indeed valid. Note that the mismatch with a pinned atoms decreases with increasing . This is because the thermal decoherence is quenched by the increasing geometrical coupling among the dark-states when the sweep rate . As a further approximation, we now disregard thermal decoherence and consider that the system after the phase loop is described by a pure quantum state rho (). As shown in Figs. 3b and 3c, one can easily extract the dark-state populations and the absolute value of the azimuthal angle from the measured bare state populations, see Methods. When , the values for match well with the prediction for a pinned atom confirming the quenching of thermal decoherence. At , the two dark-state populations are almost equal. In the language of the Bloch sphere representation, this corresponds to a rotation of the initial south pole state to the equatorial plane.

We now reconstruct the full geometric unitary operator for . Up to an unobservable global phase, we write:

 U=[αβ−β∗α∗] (7)

with . The previous dark-state reconstruction, done after the phase loop is applied on , gives access to , and , see Methods. To obtain and and fully determine , we start from a linear combination of dark-states and , perform the phase loop and process the new data. The results are shown in Fig. 4a and compared to the theoretical predictions for a pinned atom and a gas at finite temperature. The good agreement with our data validates the expected small impact of temperature for .

Probing non-Abelianity. With the previous phase loop protocol, we have where , , and label the edges of the loop, see Fig. 3a. To illustrate the non-commutative nature of the transformation group, we will cycle the phase loop counterclockwise starting from the upper corner. We then reconstruct the corresponding unitary operator like done for . The results are depicted in Fig. 4b and show that and are indeed different, though unitarily related, confirming the sensitivity of these geometric transformations to path ordering. The Frobenius distance between the two unitaries is and is in agreement with the theoretical result for a pinned atom () and for a finite-temperature gas (). These values have to be contrasted with the maximum possible Frobenius distance .

Conclusion and discussion. Using a tripod scheme on Strontium-87 atoms, we have implemented adiabatic geometric transformations acting on two degenerate dark-states. This system realizes a universal geometric single-qubit gate. We have studied SU(2) transformations associated to laser beams phase loop sequences and shown their non-Abelian character. In contrast to recent works done in optical lattices Aidelsburger et al. (2015); Jotzu et al. (2014); Mancini et al. (2015); Phuc et al. (2015); Wu et al. (2016); Song et al. (2016); Li et al. (2017), our system realizes an artificial gauge field in continuous space. Indeed, if the phases of the lasers remain constant in time, a gauge field emerges and governs the center-of-mass dynamics of the spinor-like wave function. Depending on the laser field configuration, different gauge fields can be engineered such as spin-orbit coupling Jacob et al. (2007); Juzeliūnas et al. (2008), Zitterbewegung Juzeliūnas et al. (2008), magnetic monopole Ruseckas et al. (2005) or non-Abelian Aharomov-Bohm effect Jacob et al. (2007) (see Dalibard et al. (2011) for a review). A generalization to the SU(3) symmetry is also discussed in Hu et al. (2014). Some of these schemes might be difficult to implement in optical lattices. Gauge fields generated by optical fields come from a redistribution of photons among the different plane waves modes and involve momenta transfer comparable to the photon recoil. Observing mechanical effects of gauge fields would thus require atomic gases colder than the recoil temperature and thus cooling techniques beyond the mere Doppler cooling done here DeSalvo et al. (2010); Tey et al. (2010). However, the gauge field is still driving the internal state dynamics regardless of the temperature of the gas provided the adiabatic condition is fulfilled. This led us to a new interferometric thermometry based on the Fourier transform of the velocity distribution of the gas.

Acknowledgements F. C. thanks CQT and UMI MajuLab for their hospitality. The Authors thank Mehedi Hasan for his critical reading of the manuscript. C. M. is a Fellow of the Institute of Advanced Studies at Nanyang Technological University (Singapore). This work was supported by the CQT/MoE funding Grant No. R-710-002-016-271.

Author contributions F. L., K. P., and R. R. have developed the experimental platform and performed the experiments. F. L. and D. W. have analyzed the experimental data. B. G., F. C., C. M., and D. W. have developed the models. All authors have contributed to the writing of the manuscript.

Competing financial interests The authors declare no competing financial interests.

## Methods

Cold sample preparation and implementation of the tripod scheme. The cold gas is obtained by laser cooling on the intercombination line at nm (linewidth kHz). Atoms are first laser cooled in a magneto-optical trap and then transferred into an elliptical crossed optical dipole trap at  nm (trapping frequencies , , and Hz) where they are held against gravity. Atoms are then optically pumped in the stretched magnetic substate and subsequently Doppler cooled in the optical trap using the close transition, see Fig. 5. The atomic cloud contains about atoms at a temperature K (recoil energy K, is the Boltzmann constant). A magnetic field bias of  G is applied to lift the degeneracy of the Zeeman excited states. It has essentially no effect on the ground-state levels since their spin is of nuclear origin (Landé factor ). Because the Zeeman shift between levels in the excited manifold is large, one can isolate a tripod scheme between three ground-state levels and a single excited state, namely , as indicated in Fig. 5. The three laser beams are tuned at resonance and their polarizations are chosen according to the electrical dipole transition selection rules. In practice, the two laser beams with right and left circular polarizations, respectively addressing the and transitions, are co-propagating. The laser beam with linear polarization, aligned with the magnetic bias field, addressing the , is orthogonal to the circularly polarized beams, see Fig. 1b. The plane of the lasers is chosen orthogonal to the direction of gravity. The two independent laser offset phases and (see main text) can be tuned by using two electro-optic modulators.

Initial dark-state preparation: Starting with atoms in the stretched state, the tripod beams are turned on following two different sequences. The first sequence prepares dark-state , see equation Non-Abelian adiabatic geometric transformations in a cold Strontium gas. More precisely, we first turn on the two laser beams connecting the empty states and to the excited state and then adiabatically ramp on the last laser beam. This projects state onto with a fidelity of . A different ignition sequence is used to prepare a combination of dark-states and . By turning on sequentially abruptly the left-circular beam, and adiabatically the right-circular beam, we create a coherent (dark) superposition of the state and . Finally, we turn on abruptly the linearly-polarized beam and we expect to produce the linear combination . In practice, a systematic phase rotation occurs once the last beam is turned on which adds an extra mixing among the dark-states. Performing the bare state population analysis, we find that this initial state corresponds in fact to .

Spin sensitive imaging system: The bare state populations in the ground-state are obtained with a nuclear spin-sensitive shadow imaging technique on the line, see Fig. 5. First we measure the population of state with a shadow laser tuned on the closed transition. Then, using the same atomic ensemble, we measure the population of state by tuning the shadow laser on the transition. This transition is open but its large enough Clebsch-Gordan coefficient () ensures a good coupling with the shadow laser. The population of state is measured in the same way ( open transition, its Clebsch-Gordan coefficient being still large enough). The shadow laser beam shines the atoms during s with an on-resonance saturation parameter (saturation intensity ). With such values, the average number of balistic photons scattered per atom is less than one and optical pumping can be safely ignored, ensuring an accurate measurement of the ground-state populations.

Dark state and unitary matrix reconstruction: A state in the dark-state manifold takes the form with and the populations of states and and the azimuthal angle. Using equation Non-Abelian adiabatic geometric transformations in a cold Strontium gas, we immediately find

 |d2|=√3¯P32cosφ=¯P1−¯P2√¯P3(2−3¯P3). (8)

Do note that the normalization of restricts the possible values of the summing up to 1. The sign of is determined using the prediction of equation 6 for a pinned atom ().

To reconstruct the unitary matrix , as expressed in equation 7, we perform the phase loop sequence on two different initial dark-states (their representative points on the Bloch sphere should not be opposite) and perform the dark-state reconstruction for each of them. The two phase terms in are reconstructed up to a sign. As for the dark-state reconstruction, we rely on the prediction for a pinned atom to lift this sign ambiguity.

Gauge fields and adiabatic Schrödinger equation: The time-dependent interaction operator for the resonant tripod scheme, in the rotating-wave approximation, has the following expression:

 H(t)=ℏΩ(r,t)23∑i=1|e⟩⟨i|+H.c. (9)

We assume here that the laser Rabi frequencies coupling the ground-states to the excited state have all the same amplitude denoted by . The time dependency comes from the cyclic ramping sequence of the two offset laser phases (). Neglecting transitions outside the dark-state manifold (adiabatic approximation), the system is described by a quantum state , where is the wave function of the centre-of-mass of the atom in an internal state . In this basis, the adiabatic Schrödinger equation for the column vector reads:

 iℏ˙Ψ––=⎡⎣(^p\openone−A)22M+W−ωt⎤⎦Ψ––, (10)

where the dot denotes time derivative. The first two terms on the right-hand side describe the dynamics of an atom subjected to the synthetic gauge field. The last term is due to the cyclic ramping sequence of the laser phases. Only this term remains for a pinned atom (), in which case one recovers equation 6. The general expressions of A and are given in the main text. With equal and constant Rabi frequencies amplitude, and for the orientation of our laser beams, one finds:

 Missing dimension or its units for \hskip (11)

where is the atomic recoil energy and is the wavevector of laser beam (see main text). As one can see, all these operators have the same matrix form. The matrix reads:

 M=\openone+s⋅σ2=(3/4−√3/4−√3/41/4) (12)

and satisfies , its unit Bloch vector being . As a consequence, all these operators can be diagonalized at once by the same transformation and amenable to the simple projector matrix form:

 M→MD=(1000). (13)

Because of our laser beams geometry, the vector potential A is in fact Abelian since its only non-zero matrix component is along .

In contrast, the operator has a different matrix form. Indeed, the two offset phases of the lasers (see main text) can be addressed at will. Following Dalibard et al. (2011), we get:

 ωt=ℏ2(˙ϕ1+˙ϕ2(˙ϕ1−˙ϕ2)/√3(˙ϕ1−˙ϕ2)/√3(˙ϕ1+˙ϕ2)/3). (14)

In particular, we note that leads to non-Abelian transformations. An immediate consequence is that, for a given phase loop in parameter space, the geometric unitary operator associated with a cycle of phase ramps depends on the starting point of the cycle on the loop. Different starting points lead to different, though unitarily related, non-commuting geometric unitary operators.

### References

1. Michael V Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A 392, 45–57 (1984).
2. Shivaramakrishnan Pancharatnam, “Generalized theory of interference and its applications,” Proceedings of the Indian Academy of Sciences-Section A,  44, 398–417 (1956).
3. Yakir Aharonov and David Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959).
4. Barry Simon, “Holonomy, the quantum adiabatic theorem, and berry’s phase,” Phys. Rev. Lett. 51, 2167 (1983).
5. Frank Wilczek and Alfred Shapere, eds., Geometric Phases in Physics (World Scientific, Singapore, 1989).
6. Franck Wilczek and Anthony Zee, “Appearance of gauge structure in simple dynamical systems,” Phys. Rev. Lett. 52, 2111 (1984).
7. Mario Norberto Baibich, Jean-Marc Broto, Albert Fert, F Nguyen Van Dau, Frédéric Petroff, P Etienne, G Creuzet, A Friederich,  and J Chazelas, “Giant magnetoresistance of (001) fe/(001) cr magnetic superlattices,” Phys. Rev. Lett. 61, 2472 (1988).
8. Yuichiro K Kato, Roberto C Myers, Arthur C Gossard,  and David D Awschalom, “Observation of the spin hall effect in semiconductors,” Science 306, 1910–1913 (2004).
9. Markus König, Steffen Wiedmann, Christoph Brüne, Andreas Roth, Hartmut Buhmann, Laurens W Molenkamp, Xiao-Liang Qi,  and Shou-Cheng Zhang, “Quantum spin hall insulator state in hgte quantum wells,” Science 318, 766–770 (2007).
10. David Hsieh, D Qian, L Wray, Y Xia, Y S Hor, R J Cava,  and M Zahid Hasan, “A topological dirac insulator in a quantum spin hall phase,” Nature 452, 970–974 (2008).
11. Cui-Zu Chang, Jinsong Zhang, Xiao Feng, Jie Shen, Zuocheng Zhang, Minghua Guo, Kang Li, Yunbo Ou, Pang Wei, Li-Li Wang, et al., “Experimental observation of the quantum anomalous hall effect in a magnetic topological insulator,” Science 340, 167–170 (2013).
12. Zoran Hadzibabic, Peter Krüger, Marc Cheneau, Baptiste Battelier,  and Jean Dalibard, “Berezinskii-kosterlitz-thouless crossover in a trapped atomic gas.” Nature 441, 1118–1121 (2006).
13. Yu-Ju Lin, Karina Jiménez-García,  and Ian B Spielman, “A spin-orbit coupled bose-einstein condensate,” Nature 471, 83–86 (2011).
14. Monika Aidelsburger, Michael Lohse, C Schweizer, Marcos Atala, Julio T Barreiro, S Nascimbène, NR Cooper, Immanuel Bloch,  and Nathan Goldman, “Measuring the chern number of hofstadter bands with ultracold bosonic atoms,” Nature Phys. 11, 162–166 (2015).
15. Gregor Jotzu, Michael Messer, Rémi Desbuquois, Martin Lebrat, Thomas Uehlinger, Daniel Greif,  and Tilman Esslinger, “Experimental realization of the topological haldane model with ultracold fermions,” Nature 515, 237–240 (2014).
16. Marco Mancini, Guido Pagano, Giacomo Cappellini, Lorenzo Livi, Marie Rider, Jacopo Catani, Carlo Sias, Peter Zoller, Massimo Inguscio, Marcello Dalmonte,  and Leonardo Fallani, “Observation of chiral edge states with neutral fermions in synthetic hall ribbons,” Science 349, 1510–1513 (2015).
17. Nguyen Thanh Phuc, Gen Tatara, Yuki Kawaguchi,  and Masahito Ueda, “Controlling and probing non-abelian emergent gauge potentials in spinor bose-fermi mixtures,” Nature Comm. 6 (2015).
18. Zhan Wu, Long Zhang, Wei Sun, Xiao-Tian Xu, Bao-Zong Wang, Si-Cong Ji, Youjin Deng, Shuai Chen, Xiong-Jun Liu,  and Jian-Wei Pan, “Realization of two-dimensional spin-orbit coupling for bose-einstein condensates,” Science 354, 83–88 (2016).
19. Bo Song, Chengdong He, Shanchao Zhang, Elnur Hajiyev, Wei Huang, Xiong-Jun Liu,  and Gyu-Boong Jo, “Spin-orbit-coupled two-electron fermi gases of ytterbium atoms,” Phys. Rev. A 94, 061604 (2016).
20. Jun-Ru Li, Jeongwon Lee, Wujie Huang, Sean Burchesky, Boris Shteynas, Furkan Çağrı Top, Alan O Jamison,  and Wolfgang Ketterle, “A stripe phase with supersolid properties in spin–orbit-coupled bose-einstein condensates,” Nature 543, 91–94 (2017).
21. Zheng Wang, Yidong Chong, John D Joannopoulos,  and Marin Soljacic, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772 (2009).
22. Ulrich Kuhl, Sonja Barkhofen, Timur Tudorovskiy, Hans-Jürgen Stöckmann, Tasnia Hossain, Laurent de Forges de Parny,  and Fabrice Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B 82, 094308 (2010).
23. Nathan Schine, Albert Ryou, Andrey Gromov, Ariel Sommer,  and Jonathan Simon, “Synthetic landau levels for photons,” Nature 534, 671–675 (2016).
24. Paolo Zanardi and Mario Rasetti, “Holonomic quantum computation,” Phys. Lett. A 264, 94–99 (1999).
25. Jonathan A Jones, Vlatko Vedral, Artur Ekert,  and Giuseppe Castagnoli, “Geometric quantum computation using nuclear magnetic resonance,” Nature 403, 869–871 (2000).
26. Lu-Ming Duan, Juan I Cirac,  and Peter Zoller, “Geometric manipulation of trapped ions for quantum computation,” Science 292, 1695–1697 (2001).
27. Paolo Solinas, Maura Sassetti, Piero Truini,  and Nino Zanghì, “On the stability of quantum holonomic gates,” New J. Phys. 14, 093006 (2012).
28. Mark A Kowarsky, Lloyd C L Hollenberg,  and Andrew M Martin, “Non-abelian geometric phase in the diamond nitrogen-vacancy center,” Phys. Rev. A 90, 042116 (2014).
29. Erik Sjöqvist, “Nonadiabatic holonomic single-qubit gates in off-resonant systems,” Phys. Lett. A 380, 65–67 (2016).
30. Chong Zu, W-B Wang, Li He, W-G Zhang, C-Y Dai, F Wang,  and Lu-Ming Duan, “Experimental realization of universal geometric quantum gates with solid-state spins,” Nature 514, 72–75 (2014).
31. Abdufarrukh A Abdumalikov Jr, Johannes M Fink, Kristin Juliusson, Marek Pechal, Simon Berger, Andreas Wallraff,  and Stefan Filipp, “Experimental realization of non-abelian non-adiabatic geometric gates,” Nature 496, 482–485 (2013).
32. Shi-Liang Zhu and Zidan Wang, “Implementation of universal quantum gates based on nonadiabatic geometric phases,” Phys. Rev. Lett. 89, 097902 (2002).
33. Erik Sjöqvist, Dian-Min Tong, L Mauritz Andersson, Björn Hessmo, Markus Johansson,  and Kuldip Singh, “Non-adiabatic holonomic quantum computation,” New J. Phys. 14, 103035 (2012).
34. Hebbe M. Bharath, Matthew Boguslawski, Maryrose Barrios, Xin Lin,  and Michael S. Chapman, “Singular loops and their non-abelian geometric phases in spin-1 ultracold atoms,” ArXiv:1801.00586 [Quantum Gases (cond-mat.quant-gas)] .
35. Maryvonne Chalony, Anders Kastberg, Bruce Klappauf,  and David Wilkowski, “Doppler cooling to the quantum limit,” Phys. Rev. Lett. 107, 243002 (2011).
36. Tao Yang, Kanhaiya Pandey, Mysore Srinivas Pramod, Frederic Leroux, Chang Chi Kwong, Elnur Hajiyev, Zhong Yi Chia, Bess Fang,  and David Wilkowski, “A high flux source of cold strontium atoms,” Eur. Phys. J. D. 69, 226 (2015).
37. Julius Ruseckas, Gediminas Juzeliūnas, Patrik Öhberg,  and Michael Fleischhauer, “Non-abelian gauge potentials for ultracold atoms with degenerate dark states,” Phys. Rev. Lett. 95, 010404 (2005).
38. Gediminas Juzeliūnas, Julius Ruseckas, Andreas Jacob, Luis Santos,  and Patrik Öhberg, “Double and negative reflection of cold atoms in non-abelian gauge potentials,” Phys. Rev. Lett. 100, 200405 (2008).
39. Jean-Yves Courtois, Gilbert Grynberg, Brahim Lounis,  and Philippe Verkerk, “Recoil-induced resonances in cesium: An atomic analog to the free-electron laser,” Phys. Rev. Lett. 72, 3017 (1994).
40. D R Meacher, Denis Boiron, Harold Metcalf, Christophe Salomon,  and Gilbert Grynberg, “Method for velocimetry of cold atoms,” Phys. Rev. A 50, R1992–R1994 (1994).
41. Mark Kasevich, David S. Weiss, Erling Riis, Kathryn Moler, Steven Kasapi,  and Steven Chu, “Atomic velocity selection using stimulated raman transitions,” Phys. Rev. Lett. 66, 2297–2300 (1991).
42. Thorsten Peters, Benjamin Wittrock, Frank Blatt, Thomas Halfmann,  and Leonid P Yatsenko, “Thermometry of ultracold atoms by electromagnetically induced transparency,” Phys. Rev. A 85, 063416 (2012).
43. A pure state is denoted by a density matrix fulfilling Tr. In our system, we find Tr, the exact value depending on .
44. A. Jacob, Patrik Öhberg, Gedeminas Juzeliūnas,  and Luis Santos, “Cold atom dynamics in non-abelian gauge fields,” Appl. Phys. B 89, 439–445 (2007).
45. Jean Dalibard, Fabrice Gerbier, Gediminas Juzeliūnas,  and Patrik Öhberg, “Colloquium: Artificial gauge potentials for neutral atoms,” Rev. Mod. Phys. 83, 1523 (2011).
46. Yu-Xin Hu, Christian Miniatura, David Wilkowski,  and Benoît Grémaud, “U (3) artificial gauge fields for cold atoms,” Phys. Rev. A 90, 023601 (2014).
47. Brian J DeSalvo, Mi Yan, Pascal G Mickelson, Y N Martinez de Escobar,  and Thomas C Killian, “Degenerate fermi gas of sr 87,” Phys. Rev. Lett. 105, 030402 (2010).
48. Meng Khoon Tey, Simon Stellmer, Rudolf Grimm,  and Florian Schreck, “Double-degenerate bose-fermi mixture of strontium,” Phys. Rev. A 82, 011608 (2010).
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