Degenerate quantum gases of strontium
Degenerate quantum gases of alkaline-earth-like elements open new opportunities in research areas ranging from molecular physics to the study of strongly correlated systems. These experiments exploit the rich electronic structure of these elements, which is markedly different from the one of other species for which quantum degeneracy has been attained. Specifically, alkaline-earth-like atoms, such as strontium, feature metastable triplet states, narrow intercombination lines, and a non-magnetic, closed-shell ground state. This review covers the creation of quantum degenerate gases of strontium and the first experiments performed with this new system. It focuses on laser-cooling and evaporation schemes, which enable the creation of Bose-Einstein condensates and degenerate Fermi gases of all strontium isotopes, and shows how they are used for the investigation of optical Feshbach resonances, the study of degenerate gases loaded into an optical lattice, as well as the coherent creation of Sr molecules.
All of the early experiments reaching Bose-Einstein condensation (BEC) Inguscio et al. (1999) and the Fermi-degenerate regime Inguscio et al. (2008) in ultracold gases were performed with alkali atoms. In recent years, degenerate samples of more complex atoms, such as the alkaline-earth (-like) species ytterbium Sugawa et al. (2013), calcium Kraft et al. (2009), and strontium became available. These samples bring us closer to the realization of intriguing experiments that are intimately connected to the properties of alkaline-earth elements, ranging from the creation of ultracold open-shell polar molecules to the study of novel, strongly correlated many-body systems. In this Chapter we review the creation of degenerate quantum gases of strontium and the first experiments based on these gases. We will start by introducing the properties of strontium most relevant to quantum gas experiments and some of the possibilities opened up by these properties.
There are four stable isotopes of strontium; three are bosonic and one is fermionic. The bosonic isotopes Sr, Sr, and Sr have zero nuclear spin, just as all other bosonic alkaline-earth (-like) elements. The reason for this zero spin is that bosonic isotopes of atoms with an even number of electrons must have an even-even nucleus, for which the proton and neutron spins pair up such that the total nuclear spin vanishes Fuller (1976). The absence of a nuclear spin in these isotopes precludes the appearance of hyperfine structure, as well as of Zeeman structure for the states, and thus leads to a simple electronic level scheme. The fermionic isotope Sr carries a nuclear spin of , which forms the basis of many proposed experiments.
Strontium features two valence electrons. The electronic level structure decomposes into singlet states, in which the spins of the two valence electrons are aligned anti-parallel, and triplet states with parallel spins. Transitions between singlet and triplet states are dipole-forbidden, leading to narrow linewidths and the emergence of metastable triplet states. Some of these intercombination transitions have linewidths on the order of kHz, ideally suited for narrow-line cooling Katori et al. (1999a), while others show linewidths well below 1 Hz and are employed in optical clocks Derevianko and Katori (2011).
A key property of atomic gases is the scattering behavior of its constituents. In the limit of low temperatures, we can express the scattering between two atoms by a single parameter: the scattering length , which is usually stated in units of the Bohr radius pm. The evaporation efficiency, the stability of a quantum gas, and the mean-field energy all depend on the scattering length. Magnetic Feshbach resonances are widely used in quantum gas experiments for interaction tuning Chin et al. (2010), but such resonances are absent in alkaline-earth species due to the nature of the ground state. It is thus fortunate that the scattering lengths of the various isotopes are very different; see Tab. 1. The isotope Sr has an extremely small scattering length of , making it ideally suited for certain precision measurements due to the almost vanishing mean-field shift. Experiments involving excited-state atoms, such as optical clocks, are suffering from additional shifts arising from interactions involving atoms in the excited state. The Sr isotope, on the other hand, exhibits a moderate scattering length of that allows for efficient evaporation and stable BECs. Optical Feshbach resonances Fedichev et al. (1996a); Theis et al. (2004); Ciuryło et al. (2005); Blatt et al. (2011); Yan et al. (2013), a means to vary the scattering length by a suitable light field, are discussed in Sec. VIII.
The unique combination of properties of alkaline-earth atoms are the long lifetime of the states, the associated clock transitions originating from the state, and the near-perfect decoupling of electronic and nuclear spin for the and states of the fermionic isotope. A certain set of elements, namely zinc, cadmium Brickman et al. (2007), mercury Hachisu et al. (2008), ytterbium Kuwamoto et al. (1999), and nobelium, share these features with the “true” alkaline-earth elements. For simplicity, we will refer to these species as alkaline-earth elements as well.
Proposals demanding some or all of these properties describe the creation of artificial gauge fields Dalibard et al. (2011); Gerbier and Dalibard (2010); Cooper (2011); Béri and Cooper (2011); Górecka et al. (2011), the implementation of sub-wavelength optical lattices Yi et al. (2008), the processing of quantum information Stock et al. (2008), or the study of many-body systems with dipolar or quadrupolar interaction Olmos et al. (2013); Bhongale et al. (2013). The large nuclear spin of the fermionic isotopes is at the heart of many recent proposals to study SU() magnetism Wu et al. (2003); Wu (2006); Cazalilla et al. (2009); Gorshkov et al. (2010); Xu (2010); Foss-Feig et al. (2010a, b); Hung et al. (2011); Manmana et al. (2011); Hazzard et al. (2012); Bonnes et al. (2012); Messio and Mila (2012), where various phases such as chiral spin liquids Hermele et al. (2009), algebraic spin liquids Corboz et al. (2012a), spatial symmetry breaking Corboz et al. (2012b) and spontaneous SU() symmetry breaking Läuchli et al. (2006); Tóth et al. (2010); Corboz et al. (2011) are predicted to occur. Further proposals suggest using alkaline-earth atoms to simulate lattice gauge theories Banerjee et al. (2013), or to robustly store quantum information and perform quantum information processing Hayes et al. (2007); Daley et al. (2008); Gorshkov et al. (2009); Daley et al. (2011). Quantum gas mixtures of alkaline-earth atoms with alkali atoms can be used as a basis for the production of ground-state open-shell molecules, such as RbSr Zuchowski et al. (2010); Guérout et al. (2010), which constitute a platform towards the simulation of lattice-spin models Micheli et al. (2006); Brennen et al. (2007). Bi-alkaline-earth molecules, such as Sr, are sensitive and model-independent probes for variations of the electron-to-proton mass ratio Zelevinsky et al. (2008); Kotochigova et al. (2009). Narrow optical transitions to states are useful, to create molecular condensates through coherent photoassociation Naidon et al. (2008), and to manipulate the scattering properties through optical Feshbach resonances Ciuryło et al. (2005); Blatt et al. (2011); Yan et al. (2013). Aside from degenerate gases, two-valence-electron atoms have been used for optical clocks Derevianko and Katori (2011) and other precision experiments Ferrari et al. (2006a); Poli et al. (2011), and the clock transition has recently been used to explore quantum many-body physics Martin et al. (2013). Other experiments investigate the coherent transport of light Bidel et al. (2012), as well as the production of ultracold plasmas Killian et al. (2007) and Rydberg gases Millen et al. (2010); McQuillen et al. (2013).
So far, two research groups have reported on the attainment of BECs and degenerate Fermi gases of strontium, and quite naturally, this Chapter is a joint effort of these two teams. We have already discussed the nuclear, electronic, and scattering properties of strontium, and we will show how they combine to form a very powerful platform for quantum gas experiments. In Sec. II we begin with a historical overview on work performed in various groups around the world. In Secs. III to VII we focus on the experimental procedure to generate quantum-degenerate samples of all stable isotopes, both bosonic (Sec. V) and fermionic (Sec. VII). In addition, we present a few experiments carried out with such samples: the study of optical Feshbach resonances (Sec. VIII), the observation of the Mott-insulator transition (Sec. IX), and the creation of Sr molecules (Sec. X). The work presented here has already been published by the Rice Nagel et al. (2005); Mickelson et al. (2005); Martinez de Escobar et al. (2008, 2009); Mickelson et al. (2010); DeSalvo et al. (2010); Yan et al. (2013) and Innsbruck groups Stellmer et al. (2009); Tey et al. (2010); Stellmer et al. (2010, 2011, 2012); Stellmer et al. (2013a, b); Stellmer (2013) and complements a review on similar work on ytterbium, which appeared in the preceding volume of this series Sugawa et al. (2013).
Ii Historical overview
ii.1 Laser cooling on the broad transition
Laser cooling of alkaline-earth atoms was pioneered by the Tokyo groups. The first cooling and trapping of various isotopes of calcium and strontium was reported in the beginning of the 1990s Kurosu and Shimizu (1990). These magneto-optical traps (MOTs) were operated on the blue singlet transitions, but the lifetimes were very short compared to typical alkali MOTs. As known from earlier experiments with calcium, the strontium MOT lifetimes are limited to a few 10 ms due to a weak decay channel from the state out of the cooling cycle into the state Lellouch and Hunter (1987); Beverini et al. (1989); see Fig. 1. Contrary to early assumptions, atoms do not remain in the state Uhlenberg et al. (2000), but decay further into the long-lived and the short-lived state Kurosu and Shimizu (1990, 1992). Repumping of strontium atoms from the state into the state using light at 717 nm allowed only for a small increase of the MOT atom number by about a factor of two Kurosu and Shimizu (1992); Vogel (1999); Bidel (2002) due to unfavorably large branching ratios from the state into long-lived metastable triplet states.
Substantial increase of the MOT atom number came about only when repumping of the and states was implemented Vogel et al. (1998), using the state as an intermediate state to transfer population into the state, which decays into the singlet ground state with a comparably short lifetime of s. Further studies performed by the Boulder group include the quantification of loss processes from excited-state collisions Dinneen et al. (1999), sub-Doppler cooling of the fermionic isotope Xu et al. (2003a), and simultaneous MOTs of two different isotopes Xu et al. (2003b).
ii.2 Laser cooling on the narrow transition
Alkaline-earth elements, cooled to mK temperatures on the broad transition, constitute an adequate starting point to probe the intercombination lines. This was first done for the transition in calcium Barger et al. (1979); Beverini et al. (1989); Kurosu et al. (1992). The first MOT operated on the intercombination line of the bosonic Sr isotope was presented by the Tokyo group Katori et al. (1999a) and showed remarkable features: the attainable temperature reached as low as about 400 nK, indeed close to the recoil temperature. This group observed the peculiar pancake shape of the atomic cloud and showed that the attainable temperature can be reduced by lowering the MOT light intensity. The temperature was found to be independent of the detuning over a large range; see Fig. 2(a). The “magic” wavelength for this cooling transition was calculated Katori et al. (1999b), and a dipole trap at this wavelength was used to confine atoms at a phase-space density of 0.1, just one order of magnitude away from quantum degeneracy Ido et al. (2000). The behavior of the narrow-line MOT was studied further by the Boulder group Loftus et al. (2004a, b).
The narrow-line MOT for fermionic isotopes is more involved due to the appearance of hyperfine structure, and was first described and implemented by the Tokyo group Mukaiyama et al. (2003). This experiment included already the loading into a 1D optical lattice, optical pumping into a single Zeeman substate, and cooling down to the recoil temperature; see Fig. 2(b). A value of was reached in this experiment.
ii.3 Optical clocks
The potential of optical clocks operated on the ultranarrow transition was appreciated very early; see Ref. Derevianko and Katori (2011) for a recent review. The narrow linewidth can only be exploited for sufficiently long interrogation times, thus the atoms would need to be trapped. The absence of both charge and sizable magnetic moment suggests an optical trap. The deployment of optical lattices even allows to reach the Lamb-Dicke regime Katori (2002), thus removing the Doppler broadening. These traps, however, are prone to induce a light shift onto the clock transition, thereby shifting and broadening the transition substantially. Spectroscopy of atoms in a lattice of carefully chosen magic wavelength was first proposed Katori (2002) and demonstrated Katori et al. (1999b); Ido and Katori (2003) by the Tokyo group using the cooling transition; see Ref. Ye et al. (2008) for a review.
The same concepts were also applied to the clock transition: calculations of the magic wavelength for the bosonic isotope Sr were followed by experimental realizations, e.g. in Paris Baillard et al. (2007). Optical clocks based on bosonic Sr suffer from interaction shifts due to the interaction between atoms in the and states Lisdat et al. (2009). This issue can be overcome by advancing from a one-dimensional to a three-dimensional optical lattice Akatsuka et al. (2010).
As another possible solution, it was proposed to use the fermionic isotope Sr with all atoms prepared in the same state, thus removing collisions of the identical particles through the Pauli exclusion principle. This concept was proposed Katori et al. (2003) and realized Takamoto and Katori (2003); Takamoto et al. (2005) by the Tokyo group, and soon after, optical clocks were operated by the Tokyo Takamoto et al. (2005), Boulder Boyd et al. (2006); Ludlow et al. (2006, 2008); Martin (2013), Paris Le Targat et al. (2006), and various other groups. Since 2007, strontium lattice clocks constitute the best agreed-upon frequency standard, and have recently almost drawn level with ion clocks Chou et al. (2010); Nicholson et al. (2012); Hinkley et al. (2013) in terms of the achieved accuracy. A set of measurements of various experiments around the world has been analyzed to calculate limits on possible drifts of fundamental constants Blatt et al. (2008); Blatt (2011).
These experiments were performed in 1D optical lattices, where interactions induced by inhomogeneous probing Campbell et al. (2009) can be observed despite the fermionic character of the atoms. These experiments opened the door towards the exploration of many-body phenomena in optical clocks Martin et al. (2013). To overcome the residual influence of interactions, the density of the sample can be reduced by a sufficient increase of the trap volume Nicholson et al. (2012), or a 3D optical lattice can be employed to separate the atoms from one another. Blue-detuned lattices are investigated as well Takamoto et al. (2009).
ii.4 Struggle to reach quantum degeneracy
The early experiments reached phase-space densities already very close to quantum degeneracy Ido et al. (2000); Poli et al. (2005), but it was understood that plain narrow-line cooling in a dipole trap could not yield phase-space densities substantially larger than 0.1 Ido et al. (2000); Stellmer et al. (2013a). This last order of magnitude called for evaporative cooling as an additional cooling stage, which seemed to pose an unsurmountable obstacle at that time. There are two main explanations: First, experiments involving strontium or calcium were primarily aiming for optical clocks, which typically operate at relatively fast cycle times and do not require a sophisticated vacuum. The lifetime of trapped samples in these experiments did not allow for accumulation of low-abundant isotopes or long evaporation times. Second, the scattering properties of the most abundant isotopes Ca, Sr, and Sr are not particularly favorable for evaporation. As a consequence, the first BECs and degenerate Fermi gases in alkali-earth systems were reached with ytterbium Takasu et al. (2003); Fukuhara et al. (2007a) in 2003 and with calcium Kraft et al. (2009) in 2009.
Attempts to reach BEC in the bosonic isotope Sr failed due to the small scattering length of Poli et al. (2005), which does not allow for efficient thermalization during evaporation. The scattering length of Sr amounts to about , leading to strong inelastic losses, which also impede evaporation in a mixture of Sr and Sr Ferrari et al. (2006b). At the time of these experiments (2006), the scattering properties of Sr, the third stable bosonic isotope of only 0.56% abundance, were not yeFt explored.
To circumvent the unfavorable scattering properties, some experiments aimed to increase the phase space density by laser cooling of atoms in the metastable state, which has a lifetime of about 500 s Xu et al. (2003b); Yasuda and Katori (2004); Santra et al. (2004). This state is naturally populated in the broad-transition MOT, it can be trapped in a magnetic trap, and the magnetic substructure allows for sub-Doppler cooling mechanisms. A variety of cooling transitions could be used, some of which have very low Doppler and recoil limits. So far, all of these attempts were spoiled by large inelastic two-body collisions. These have been quantified in calcium Hansen and Hemmerich (2006), ytterbium Yamaguchi et al. (2008); Uetake et al. (2012), and strontium Traverso et al. (2009), leaving little hope that laser or evaporative cooling towards quantum degeneracy will be successful in this state.
A number of experiments were instead performed with thermal samples of strontium. Indeed, the Sr isotope possesses remarkable properties: it combines a and ground state with a high natural abundance and a narrow cooling transition, constituting a B-field insensitive and easy-to-cool atomic species. In addition, this particular isotope is almost non-interacting, making it ideally suited for precision measurements besides optical clocks. The Florence group used this isotope to study Bloch oscillations Ferrari et al. (2006a) and measure the force of gravity Poli et al. (2011).
ii.5 Photoassociation measurements
Starting from about 2005, a series of photoassociation (PA) measurements was performed to explore both the ground- and excited molecular potentials. Knowledge of the ground-state energy levels would allow for a precise determination of all scattering lengths, while excited molecular states could be employed for optical Feshbach resonances Fedichev et al. (1996a); Theis et al. (2004); Ciuryło et al. (2005) to tune the scattering length. These PA measurements would therefore elucidate alternative approaches to evaporative cooling.
The first one-color PA measurements were performed near the broad singlet transition at 461 nm Nagel et al. (2005); Mickelson et al. (2005); Yasuda et al. (2006), quickly followed by measurements near the intercombination line at 689 nm Zelevinsky et al. (2006). Precise two-color PA near the intercombination line of ytterbium Tojo et al. (2006) had allowed for a determination of all intra- and interisotope scattering lengths Kitagawa et al. (2008). This approach was adopted to strontium Martinez de Escobar et al. (2008); Stellmer et al. (2012) and allowed for a calculation of all relevant scattering lengths in 2008 Ciu (); Martinez de Escobar et al. (2008); Stein et al. (2008).
From these calculations, it became immediately clear that Sr would be ideally suited for evaporative cooling: the scattering length of promises a favorable ratio of elastic to inelastic collisions Fedichev et al. (1996b); Bedaque et al. (2000). Provided that the low natural abundance could be overcome using the accumulation scheme introduced by the Florence group Katori et al. (2001); Nagel et al. (2003); Sorrentino et al. (2006), evaporative cooling into quantum degeneracy seemed within reach. About one year later, BEC of this isotope was reached by the Innsbruck and Rice groups, and within a few more months, BECs and Fermi gases of all stable isotopes were obtained as well. These experiments will be described in the following sections of this chapter.
ii.6 Proposals for quantum many-body simulations
In parallel to the experimental advances, an eagerly anticipated stream of theoretical proposals started to swell in 2008. These proposals, some of which were mentioned in the previous chapter, employ the specific properties of alkaline-earth elements, and are often worked out for strontium or ytterbium. They are centered around various flavors of many-body simulations, mostly using the states of Sr, as well as schemes of quantum computation. Experiments and theory have stimulated each other and continue to do so in a very fruitful way.
Iii Two-stage laser cooling
The rich level structure of strontium provides us with a variety of transitions Sansonetti and Nave (2010) that could be used for laser cooling; see Fig. 1. Specifically, these include the broad transition at 461 nm and the narrow intercombination line at 689 nm, which have linewidths of 30.5 MHz and 7.4 kHz respectively Kurosu and Shimizu (1990); Katori et al. (1999a); Mukaiyama et al. (2003); Sorrentino et al. (2006); Boyd (2007); Ludlow (2008); Stellmer (2013).
A sequence of three cooling stages is employed to bring strontium atoms into the regime of degeneracy. The first stage is a MOT operated on a broad transition, ideally suited to capture atoms from a thermal beam and cool them to mK temperatures. The second stage is a MOT operated on a narrow transition, capable of cooling the atoms to thousand-times lower temperatures at ten thousand-times higher densities. Such a sample is loaded into an optical dipole trap. The third stage, evaporative cooling, leads into quantum degeneracy. While the details of the last cooling stage depend on the respective isotope and the objective of the experiment, the first two cooling stages are rather similar for all experiments and will be described in the following. Further details can be found e.g. in an earlier review Sorrentino et al. (2006) and in Refs. Boyd (2007); Ludlow (2008); Martinez de Escobar (2010); Mickelson (2010); Stellmer (2013).
iii.1 The blue MOT
We will now describe the Innsbruck apparatus, to which the Rice experiment is similar. A stream of strontium atoms at about C is emitted from an oven and directed into a UHV chamber. The atomic beam has a divergence of order 10 mrad, which can be reduced by a 2D optical molasses, also known as transverse cooling. We use light red-detuned by from the transition at 461 nm. This light is split into two beams, propagating orthogonal to each other and to the atomic beam, intersecting with the atoms about 100 mm downstream from the oven. The interaction region is about 50 mm long. The beams are elliptically shaped, retro-reflected, and contain about 10 mW in each axis. Transverse cooling increases the number of atoms in the Sr MOT by a factor of three or even more, depending on the geometric design of the oven.
The broad 30-MHz transition allows for fast Zeeman-slowing and offers a high capture velocity of the MOT. The Zeeman-slower beam contains about 35 mW of power, it is slightly focussed onto the aperture of the oven and has a waist of about 8 mm at the position of the MOT.
Atoms in the MOT region are illuminated by three retro-reflected MOT beams, having waists of about 5 mm. They have intensities of and , corresponding to about 1 mW in the vertical and 4 mW in the horizontal beams. The saturation intensity of this transition is . Here, ns is the lifetime of the state. The detuning is , and the gradient of the quadrupole field is G/cm in the vertical direction. The Doppler temperature of this “blue” MOT is K, much higher than the recoil temperature nK. The recoil temperature is given by , where is the wave vector of the light field and the wavelength. Sub-Doppler cooling requires a magnetic substructure and has indeed been observed by the Boulder group Xu et al. (2003b) for the fermionic isotope, which has a nonzero nuclear spin . Repumping of hyperfine states as required for alkali atoms is not required due the lack of hyperfine structure in the state.
As we will see later, atoms from the upper MOT level can decay via the level into the metastable level, which possesses a magnetic moment. Atoms in weak-field-seeking states of this level can be trapped in a magnetic quadrupole field. This decay reduces the lifetime of the blue MOT to a few 10 ms. We do not optimize the MOT for fluorescence or atom number, but for the loading rate of the metastable reservoir, which we define as the container formed by the magnetic trap for atoms. The loading rate depends on various experimental parameters, among them the temperature (i.e. the flux) of the oven, the amount of light available at 461 nm (i.e. the slowing and capture efficiency), the natural abundance of the respective isotope, and the temperature of the blue MOT.
We usually operate the blue MOT until a few to atoms are accumulated in the reservoir. This takes between 50 ms and 10 s, depending mainly on the abundance of the isotope.
iii.2.1 General considerations
The electronic structures of calcium, strontium, barium, and radium share a common feature: a state appears below the state. Here, is the principal quantum number of the valence electrons, ranging from 4 to 7. The nonzero branching ratio between the and the states opens a decay channel from the blue MOT cycle. This branching ratio is roughly for strontium and calcium, and roughly for barium and radium. The atoms decay further into the metastable triplet states with a branching ratio of . Strontium atoms in the state have a lifetime of only s and decay back into the state. On the other hand, the state has a lifetime of 500 s in the absence of ambient black-body radiation Yasuda and Katori (2004). Atoms in this state have a magnetic moment and, provided they are in a low-field seeking state, can be trapped in the quadrupole field of the MOT.
There is an additional decay channel of the type , which is at least two orders of magnitude weaker Werij et al. (1992). Atoms in the very long-lived state are not trapped in the quadrupole field and might constitute an additional loss channel.
Atoms in the metastable state can be returned to the ground state either during ( and states) or after (weak-field seeking states) the blue MOT stage through optical pumping into the short-lived metastable state Katori et al. (2001). Generally, continuous repumping allows for a faster loading rate of the blue MOT, as most of the atoms falling into the state appear in non-trapped states and would otherwise be lost. This strategy is followed in optical clock experiments. If however large atom numbers are required, such as in BEC experiments, it is advantageous to accumulate atoms in the metastable reservoir and transfer them into the ground state after the blue MOT has been extinguished. The atom number in the blue MOT is limited by light-assisted collisions Dinneen et al. (1999), which are absent for atoms in the metastable state. Losses through inelastic collisions of atoms in the state Traverso et al. (2009) are negligible due to the low density of about cm in the large reservoir. The ambient black-body radiation reduces the lifetime of the state to about 20 s Yasuda and Katori (2004), which, together with the loading rate of atoms into the reservoir, sets the achievable atom number. For typical experimental parameters, this number is orders of magnitude larger than the blue MOT atom number, allowing for the accumulation of significantly more atoms. This accumulation stage plays a crucial role in generating mixtures and degenerate quantum gases of strontium.
A variety of transitions can be used for repumping. An early experiment tried to close the leakage of atoms into the triplet states by pumping them directly from the state into the state at 717 nm Kurosu and Shimizu (1992). This approach is inefficient due to a significant branching ratio from the state into the triplet states. Other experiments use a pair of repump lasers at 679 nm and 707 nm to pump both the and states into the state via the state Dinneen et al. (1999). This repumping approach rigorously collects atoms from all possible decay paths and facilitates blue MOT lifetimes of many seconds. A third strategy involves any of the states at m Mickelson et al. (2009), 497 nm Poli et al. (2005), or 403 nm Stellmer et al. (2013c) for , respectively. Repumping via the state at 481 nm is also efficient. Current quantum gas experiments repump only the state, thus loss through the state persists and limits the lifetime of a continuously repumped MOT to about 1 s. Population of the state originates both from decay via the pathway mentioned above, as well as a cascade of transitions from the via intermediate states into the state. The branching ratios of these indirect pathways are at most a few percent.
The experiments presented in this review follow the accumulation strategy, such that losses into the state are below the few-percent level and can be tolerated. The choice of the employed state involves a trade-off between repump efficiency and ease of laser operation: laser systems for the three wavelengths mentioned above tend to become simpler for shorter wavelengths. As one climbs up the ladder of states, however, more and more decay channels open up that lead atoms into the dark state Stellmer et al. (2013c). The experiments presented here employ lasers at either m, 497 nm, or 481 nm.
Repumping of the bosonic isotopes is straightforward even in isotopic mixtures, since the isotope shifts are at most 100 MHz. In the case of Sr, efficient repumping is complicated by the hyperfine structure of the states involved. We find that all five hyperfine states through of the level are populated during the blue MOT, however at different relative amounts: roughly 80% of the atoms populate the and states. The hyperfine splittings of the and states are on the order of GHz. In a typical experimental cycle, repumping is performed only on the and transitions, or the laser is rapidly scanned across all hyperfine transitions.
iii.2.3 Experimental parameters
The lifetime of metastable atoms in the reservoir is about 30 s in our experiment. This value is largely independent on the density of atoms in the reservoir. It is likely to be limited by decay along the pathway in presence of blue MOT light, and additional channels when the MOT light is turned off. Here, the first step is an excitation driven by the ambient black-body radiation Xu et al. (2003b); Yasuda and Katori (2004). Collisions with the background gas might limit the lifetime even further. The lifetime is certainly long enough to allow for sequential loading of different isotopes when working with mixtures.
The repumping flash lasts typically 50 ms and contains a few 100 W of light in a beam collimated to a diameter of 10 mm, corresponding to roughly , if light at a repump transition in the visible range is used. About 10 mW are used for the repumping at m.
iii.3 The red MOT
The availability of narrow intercombination lines in strontium offers the intriguing opportunity to add a second cooling stage after the blue MOT in order to reduce the temperature and increase the density of the ensemble further. The second cooling stage is often referred to as narrow-line MOT and frequently named “red” MOT, owing to the color of the transition wavelength at 689 nm.
The linewidth of this transition amounts to kHz, corresponding to a Doppler temperature of nK; a factor of 4300 smaller than for the blue transition. This impressively low temperature demonstrates the power of narrow-line cooling. For linewidths on the order of kHz, the Doppler temperature might become comparable to the recoil temperature. The recoil temperature is nK for the red transition, where the minimal attainable temperature is Castin et al. (1989). Thus, Doppler and recoil limit almost coincide for the red transition in strontium.
The atoms are repumped from the reservoir at temperatures set by the Doppler temperature of the blue MOT, roughly 1 mK. A single frequency of the red MOT light would not provide sufficient capture efficiency, and we frequency-broaden the MOT light to match its frequency spectrum to the velocity distribution of the atoms. We use an acousto-optical modulator (AOM) to scan the frequency of the MOT light with a rate of about 20 kHz, thereby creating a comb of lines extending from roughly kHz to MHz detuning. We have about 2.5 mW of laser power available on each MOT axis, collimated to a waist of about 3 mm. The saturation intensity of this transition is , yielding a maximum intensity of 2000 for our experimental setup. Considering a scan range of 5 MHz comprising 250 comb lines at a spacing of 20 kHz, the intensity per comb line is about 10 . We apply this broad-band red MOT already during the repumping process. The quadrupole field gradient along the vertical direction is ramped to G/cm within about a millisecond once the repumping light is applied. This capture phase lasts 50 ms and is rather robust: the lifetime of this MOT exceeds 1 s at this stage.
It is helpful to visualize the geometric region in which atoms interact with the MOT light: in the case of narrow-line cooling, the detuning is much larger than the natural linewidth, . The light is only resonant with the atomic transition in regions where the B-field induced Zeeman shift balances the detuning. Here, MHz/G is the Bohr magneton, and the Landé g-factor is for the state. This region is the surface of an ellipsoid, where the vertical radius of this ellipsoid is given by . Typical sizes are about 4 mm for a gradient of 1 G/cm and a detuning of 1 MHz, but only m for a detuning of 20 kHz. The thickness of such a shell is on the order of m for a small saturation parameter . As we apply a frequency comb that stretches from near-zero to about MHz, atoms can get into resonance with the light on 250 narrow, but overlapping shells, filling the entire volume of the ellipsoid. For each shell, , where is the natural and the intensity-broadened linewidth.
In a second phase, we narrow the scan range down to 2 MHz, where the comb line closest to resonance is 100 kHz red-detuned to the transition. During this phase of 200 ms, the total light intensity is reduced to about 100 (corresponding to about per comb line), and the magnetic field gradient remains unchanged. Afterwards, we jump to single-frequency operation with a detuning of kHz and an unchanged intensity of 100 .
In the third stage, which we call the single-frequency MOT, we shift the frequency very close to resonance while reducing the intensity dramatically to 0.5 . This stage lasts 200 ms and is concluded by a 50 ms wait at the final parameters. It is important to understand that the MOT is driven through very different regimes during this ramp: We begin in the condition . Atoms are in resonance with the light on a single shell, whose thickness is enlarged by the factor compared to the low-intensity case. The large intensity ensures that the scattering rate is high enough to keep the atoms in the MOT, and lifetimes are typically 400 ms. In this regime, the behavior of the atoms can be described semiclassically Loftus et al. (2004a), and the expected temperature is . Note that this temperature is independent of the detuning, and set only by the light intensity. In a simplified picture, the decrease of the detuning provides compression, and the decrease of the intensity provides cooling.
At the end of this stage, the detuning becomes comparable to the linewidth, and approaches unity: . The behavior of the atoms is determined by single photon recoils, and the system requires a full quantum treatment Castin et al. (1989); the temperature limit approaches . There is, however, a compromise between atom number and temperature. A temperature of is reached only for very low intensity, accompanied by a very low scattering rate. Atoms populate only on a very thin shell (the bottom of the ellipsoid) and interact predominantly with the upward propagating beam; see Fig. 3(a). At low intensity, an atom is at risk to fall through this shell without absorption of a photon, and be lost. This limits the lifetime at this stage to a few 10 ms. The attainable temperature is limited by heating due to the re-absorption of photons, and depends on the density and scattering properties of the atoms. We typically achieve temperatures around 800 nK with a few atoms of Sr, and temperatures as low as 400 nK for the non-interacting isotope Sr or equivalently Sr at very low densities. Note that the light is still far detuned from the bare atomic transition, such that the atoms occupy the shell of an ellipsoid, about m below the quadrupole center, where the diameter of the cloud is typically m.
The bosonic isotopes, for which we have discussed the red MOT dynamics in the previous section, have nuclear spin and therefore only one magnetic substate in the ground state. The fact that the magnetic moment is zero due to the singlet configuration of the two valence electrons () did not become apparent. This however changes as we consider the fermionic Sr isotope with and its ten magnetic states. The magnetic moment is now given by the nuclear moment, which is still orders of magnitude smaller than an electronic magnetic moment. The condition of and in the ground state is quite unusual for MOT operation, and is reflected by the fact that the Landé -factors of the ground- and excited state differ by about three orders of magnitude.
The experimental realization of a fermionic narrow-line MOT was pioneered by the Tokyo group Mukaiyama et al. (2003). Cooling is performed on the transition, where the large differential -factor leads to a position-dependent restoring force. As a consequence, atoms in certain states at certain locations even experience a force away from the trap center. An effective restoring force for all atoms is obtained by rapid randomization of states. This is achieved by adding a so-called stirring laser to the trapping laser, operating on the transition.
Just as in the bosonic case, we use the maximum available power on both the trapping and stirring beams to capture the atoms emerging from the metastable reservoir. Conditions for the broadband MOT are identical to the bosonic case described above. Final conditions of the red MOT are a gradient field of 1.15 G/cm, trapping and stirring beam intensity of a few , and detunings of only a few linewidths. We add a short wait time of 50 ms to ensure equilibration and attain typical temperatures of 800 nK with atoms. In contrast to the bosonic case, the fermionic MOT fills the entire volume of an ellipsoid; see Fig. 3(b).
iii.4 Design and loading of the dipole trap
Virtually all experiments using a narrow line for cooling towards quantum degeneracy choose a pancake-shaped dipole trap, or at least a trap that is elongated in the horizontal plane Takasu et al. (2003); Kraft et al. (2009); Stellmer et al. (2009); Martinez de Escobar et al. (2009).
There are two reasons for this choice. At first, the narrow-line MOT itself is pancake-shaped, and a dipole trap of similar shape provides improved mode-matching. The second reason refers to the evaporation efficiency: During evaporation, atoms will leave the trap predominantly vertically downwards, aided by gravity. The evaporation efficiency benefits from a high vertical trap frequency: once a high-energy atom is produced in a collision, it ought to escape the trap before colliding with another atom. The vertical trap frequency should thus be large compared to the scattering rate: this requirement suggests a pancake-shaped trap. It is fortunate that both the loading of the dipole trap and the evaporation efficiency are optimized with the same trap shape.
The trap can be formed by two intersecting horizontal beams Martinez de Escobar et al. (2009) or by an elliptic horizontal beam, intersecting with a rather large vertical beam that provides additional confinement in the horizontal plane Stellmer et al. (2013b). The ellipticity of the horizontal beams can be as extreme as , and the ratio between the vertical and the lowest horizontal trap frequency can reach .
The dipole trap is turned on from the beginning of the red MOT, and atoms are continuously loaded into the dipole trap once spatial overlap is achieved and the temperature drops below the trap depth. Taking great care to reduce and compensate the light shifts imposed by the dipole trap, we are able to transfer 50% of the atoms from the single-frequency MOT into the dipole trap while maintaining the temperature of the MOT Stellmer et al. (2013b, a). Once the atoms are loaded into the dipole trap, the MOT light is kept on for another 100 ms at an intensity of about . During this time, the atoms are pushed into the center of the dipole trap by the horizontal MOT beams, thereby increasing the density. For the bosonic case, the quadrupole center is placed about m above the horizontal dipole trap beam. The detuning of the cooling light from the Zeeman-shifted and light-shifted -resonance position is about . In the fermionic case, the quadrupole center is overlapped with the dipole trap. Working with a mixture of bosonic and fermionic isotopes requires a sequential loading scheme, in which we load the fermions first and then shift the quadrupole center upwards to load the bosons.
Iv Photoassociation of atomic strontium
Photoassociation (PA) spectroscopy is an important tool for determining and manipulating the scattering properties of ultracold atoms and for forming molecules. There has been a significant amount of work in this area with strontium for several reasons. The knowledge of atom-atom interactions gained through PA is critical for designing experiments to reach quantum degeneracy, and this is particularly important in strontium because some of the isotopes have scattering properties that are not ideal for evaporative cooling. The formation of ground-state molecules is now a major theme of ultracold physics research, and strontium offers efficient routes to achieve this through PA. Finally, narrow-line PA near the intercombination transition is different in many ways from traditional PA with broad, electric-dipole-allowed transitions, and it holds promise for optical Feshbach resonances with reduced losses.
iv.1 One-color photoassociation
For a PA measurement, a sufficiently cold and dense cloud of atoms is illuminated by light detuned from an atomic transition. The frequency of the light is varied, and whenever it comes into resonance with a transition between two free atoms and an excited molecular state, molecules are created. These excited molecules then quickly decay into deeply-bound states, which are invisible on absorption images.
The first PA experiments in strontium involved excitation to molecular states on the potential using 461 nm light Nagel et al. (2005); Mickelson et al. (2005); Yasuda et al. (2006). This allowed accurate determination of the coefficient and the atomic decay rate, MHz Yasuda et al. (2006). Measurement of the variation of the intensities of the transitions to different molecular levels allowed a preliminary determination of the ground-state -wave scattering lengths Mickelson et al. (2005).
Of more interest for the study and control of quantum degenerate gases is narrow-line one-color PA to bound states of the and molecular potentials to the red of the intercombination-line transition at 689 nm. The first experiments Zelevinsky et al. (2006) allowed accurate determination of atomic and molecular parameters, especially the atomic decay rate, kHz.
The small decay rate of the molecular states, , has several important implications. Narrow-line PA opens a new regime, also explored in ytterbium Tojo et al. (2006), in which the transition linewidth is much smaller than the level spacings even for the least-bound molecular levels. Small also implies a weak dipole-dipole interaction between and atoms during PA. This gives rise to similar ground and excited molecular potentials and very large Frank-Condon factors for bound-bound transitions.
Large free-bound matrix elements suggest that these transitions can be used to manipulate atomic interactions through an optical Feshbach resonance with reduced inelastic loss Ciuryło et al. (2005, 2006), which will be described in Sec. VIII. It has also been predicted that strong transitions and long molecular lifetimes can combine to yield atom-molecule Rabi frequencies that exceed decoherence rates to enable coherent single-photon PA Naidon et al. (2008), which is inaccessible with electric-dipole-allowed transitions. Large bound-bound matrix elements are important for creating ground-state molecules, either through spontaneous decay after one-color PA Reinaudi et al. (2012); Kato et al. (2012) or through two-color PA techniques Stellmer et al. (2012). In fact, near-unity Frank-Condon factors were found for several bound-bound transition Reinaudi et al. (2012), which enables very efficient, state-selective molecule production by spontaneous emission. Driving one or more additional Raman transitions should populate the absolute ground state of the Sr system.
Initial experiments with intercombination-line PA were performed with Sr Zelevinsky et al. (2006), but measurements have been extended to include all the bosonic isotopes Stellmer et al. (2012). Table 2 gives binding energies of excited molecular states that have been determined for Sr, Sr, and Sr, respectively.
|Sr (Ref. Stellmer et al. (2012))||Sr (Ref. Borkowski et al. (2013))||Sr (Ref. Zelevinsky et al. (2006))|
iv.2 Two-color photoassociation
In two-color PA, two laser fields couple colliding atoms to a weakly bound state of the ground molecular potential via a near-resonant intermediate state. In strontium, this was first performed in a thermal gas of Sr with the goal of measuring the binding energies of weakly bound levels of the ground-state potential Martinez de Escobar et al. (2008); see Fig. 4(a). It was also used in a stimulated Raman adiabatic passage (STIRAP) process Vitanov et al. (2001) to coherently produce molecules in the ground electronic state from quantum degenerate Sr in an optical lattice Stellmer et al. (2012). The STIRAP experiment will be described in Sec. X.
For experiments with a thermal gas of Sr, we closely follow the description in Ref. Martinez de Escobar et al. (2008). Atoms are held in an optical dipole trap, with a temperature of several K and peak densities on the order of cm. A dark resonance is used to determine the binding energy of molecular levels of the ground-state potential. The frequency of the free-bound laser is held fixed close to the one-color resonance, , while the bound-bound laser detuning is scanned. When , the system is in two-color resonance from state to , and one-color photoassociative loss is suppressed due to quantum interference. At this point, , so the spectrum allows accurate determination of . Averaging over is necessary in order to properly account for thermal shifts of the resonance.
At the low temperatures of atoms in the dipole trap, only -wave collisions occur so only intermediate levels and and 2 final states are populated. Figure 4(b) shows a series of spectra taken at various bound-bound intensities for equal to the , state; denoted as in Tab. 3. The detuning of the free-bound laser frequency from the free-bound resonance, which depends on the collision energy and AC Stark shift from the dipole trap, causes slight asymmetry in the lines and broadening, but this can be accounted for by a simple model Martinez de Escobar et al. (2008). We also measured the binding energy of the , state; see Tab. 3.
Knowledge of the binding energies in Sr allowed accurate determination of the -wave scattering lengths for all isotopic collision possibilities. This relied upon a relativistic many-body calculation of the dispersion coefficients for the long-range behavior of the ground-state molecular potential Porsev and Derevianko (2006). PA measurements were later combined with Fourier-transform spectroscopy of molecular levels of the potential to yield further improvements and the most accurate determination of the ground molecular potential and scattering lengths Stein et al. (2010).
According to the Wigner threshold law, the elastic cross-section for collisions between neutral particles approaches a constant as the collision energy goes to zero. Most experiments with ultracold atoms reach this limit, and the cross-section is well described by an energy-independent partial wave for distinguishable particles or indistinguishable bosons. However, this is not the case when there is a low-energy scattering resonance or when the scattering length is very small. Figure 5 demonstrates that SrSr and SrSr collision cross-sections vary significantly with collision energy, even at energies below K.
Table 3 lists all the binding energies that have been determined for molecular Sr in the ground electronic state.
V Bose-Einstein condensation of strontium
v.1 Bose-Einstein condensation of Sr
The early experiments towards quantum degeneracy in strontium were focused on the three relatively abundant isotopes Sr (9.9%), Sr (7.0%), and Sr (82.6%), the first and the last one being bosonic. The necessary phase-space density for BEC or Fermi degeneracy could not be achieved in spite of considerable efforts Katori et al. (2001); Ferrari et al. (2006b). For the two bosonic isotopes the scattering properties turned out to be unfavorable for evaporative cooling Ferrari et al. (2006b). The scattering length of Sr is close to zero, such that elastic collisions are almost absent. In contrast, the scattering length of Sr is very large, leading to detrimental three-body recombination losses. Magnetic Feshbach resonances are absent in the bosonic alkaline-earth systems, and optical Feshbach resonances are accompanied by strong losses on the timescales required for evaporation.
v.1.1 First attainment of BEC in strontium
The first BECs of strontium were attained in 2009 using the isotope Sr. This isotope has a natural abundance of only 0.56% and, apparently for this reason, had received little attention up to that time. The low abundance does not represent a serious disadvantage for BEC experiments, as it can be overcome by the accumulation scheme described in Sec. III. Because of the favorable scattering length of Ciu (); Stein et al. (2008); Martinez de Escobar et al. (2008), there is no need of Feshbach tuning. Ironically, this low-abundance isotope turned out to be the prime candidate among all alkaline-earth isotopes to obtain large BECs and might also allow for sympathetic cooling of other isotopes and elements.
In the following, we will state the experimental procedure of the early Innsbruck experiment Stellmer et al. (2009). The laser cooling stages were already described in Sec. III. To prepare the evaporative cooling stage, the atoms are transferred into a crossed-beam dipole trap, which is derived from a 16-W laser source operating at 1030 nm in a single longitudinal mode. Our trapping geometry follows the basic concept successfully applied in experiments on ytterbium and calcium BEC Takasu et al. (2003); Fukuhara et al. (2007b, 2009a); Kraft et al. (2009). The trap consists of a horizontal and a near-vertical beam with waists of m and m, respectively, thus creating a cigar-shaped geometry. Initially the horizontal beam has a power of 3 W, which corresponds to a potential depth of K and oscillation frequencies of 1 kHz radially and a few Hz axially. The vertical beam contains 6.6 W, which corresponds to a potential depth of K and a radial trap frequency of 250 Hz. Axially, the vertical beam does not provide any confinement against gravity. In the crossing region the resulting potential represents a nearly cylindrical trap. In addition the horizontal beam provides an outer trapping region of much larger volume, which is advantageous for the trap loading.
The dipole trap is switched on at the beginning of the red MOT compression phase. After switching off the red MOT, we observe atoms in the dipole trap with about of them residing in the crossing region. At this point we measure a temperature of K, which corresponds to roughly one tenth of the potential depth. We then apply forced evaporative cooling by exponentially reducing the power of both beams with a time constant of s. The evaporation process starts under excellent conditions, with a peak number density of cm, a peak phase-space density of , and an elastic collision rate of about 3500 s. During the evaporation process the density stays roughly constant and the elastic collision rate decreases to s before condensation. The evaporation efficiency is very large as we gain at least three orders of magnitude in phase-space density for a loss of atoms by a factor of ten.
The phase transition from a thermal cloud to BEC becomes evident in the appearance of a textbooklike bimodal distribution, as clearly visible in time-of-flight absorption images and the corresponding linear density profiles shown in Fig. 6. At higher temperatures the distribution is thermal, exhibiting a Gaussian shape. Cooling below the critical temperature leads to the appearance of an additional, narrower and denser, elliptically shaped component, representing the BEC. The phase transition occurs after 6.3 s of forced evaporation, when the power of the horizontal beam is 190 mW and the one of the vertical beam is 410 mW. At this point, with the effect of gravitational sag taken into account, the trap depth is K. The oscillation frequencies are 59 Hz in the horizontal axial direction, 260 Hz in the horizontal radial direction, and 245 Hz in the vertical direction.
For the critical temperature we obtain nK by analyzing profiles as displayed in Fig. 6. This agrees within 20%, i.e. well within the experimental uncertainties, with a calculation of based on the number of atoms and the trap frequencies at the transition point. Further evaporation leads to an increase of the condensate fraction and we obtain a nearly pure BEC without discernable thermal fraction after a total ramp time of 8 s. The pure BEC that we can routinely produce in this way contains atoms and its lifetime exceeds 10 s.
The expansion of the pure condensate after release from the trap clearly shows another hallmark of BEC. Figure 7 demonstrates the well-known inversion of the aspect ratio Anderson et al. (1995); Inguscio et al. (1999), which results from the hydrodynamic behavior of a BEC and the fact that the mean field energy is released predominantly in the more tightly confined directions.
Our images show that the cloud changes from an initial prolate shape with an aspect ratio of at least 2.6 (limited by the resolution of the in-situ images) to an oblate shape with aspect ratio 0.5 after 20 ms of free expansion. From the observed expansion we determine a chemical potential of nK for the conditions of Fig. 7, where the trap was recompressed to the setting at which the phase transition occurs in the evaporation ramp. Within the experimental uncertainties, this agrees with the calculated value of nK.
The corresponding Rice experiment reaching BEC in Sr Martinez de Escobar et al. (2009) resembles the Innsbruck experiment very closely. Nearly all experimental parameters of the sequence are almost identical to the one described above. The main difference lies in the fact that two near-horizontal dipole trap beams of m waist are used. The large trap allows for the loading of more atoms at a lower temperature, however at a lower density and smaller collision rate. To improve evaporative cooling, the trap is re-compressed after loading of atoms from the red MOT. The evaporation time of 4.5 s is slightly shorter than in the Innsbruck experiment and leads to pure BECs of typically atoms.
v.1.2 BECs of large atom number
The first BECs of Sr contained a few atoms, but were far from being optimized. A careful optimization of various parameters of the experimental sequence, most of all the transfer into the dipole trap and its geometric shape, allowed us to increase the BEC atom number into the range Stellmer et al. (2013b). To the best of our knowledge, these BECs are the largest ones ever created by evaporative cooling in an optical dipole trap. This experiment represents the current state of the art and will be described in the following.
To overcome the low natural abundance of Sr, we accumulate atoms in the metastable reservoir for 40 s. This time is slightly longer than the lifetime of the gas in the reservoir, and further loading does not increase the atom number significantly. The atoms are returned into the ground state, cooled and compressed by the red MOT, and transferred into the dipole trap. For this experiment, we use only the horizontal dipole trap beam, which has an initial depth of K and provides initial trapping frequencies of Hz and Hz in the horizontal and Hz in the vertical directions. This beam has an aspect ratio of , with waists of about m and m. After ramping the red MOT light off over 100 ms, the gas is allowed to thermalize in the dipole trap for 250 ms. At this point, about atoms reside in the dipole trap at a temperature of 1.5 K. The peak density of the gas is , the average elastic collision rate is , and the peak phase-space density is 0.3. The power of the dipole trap is reduced exponentially from its initial value of 2.4 W to 425 mW within 10 s.
After 7 s of evaporation a BEC is detected. At this time, atoms remain in the trap at a temperature of about 400 nK. The evaporation efficiency is high with four orders of magnitude gain in phase-space-density for a factor ten of atoms lost. After 10 s of evaporation, we obtain an almost pure BEC of atoms. The trap oscillation frequencies at this time are Hz, Hz, and Hz. The BEC has a peak density of and the shape of an elongated pancake with Thomas-Fermi radii of about m, m, and m. The lifetime of the BEC is 15 s, likely limited by three-body loss.
An increase of the BEC atom number towards the range of should be achievable by simple improvements. A larger volume of the dipole trap, facilitated by an increased ellipticity of the horizontal dipole trap beam, would allow us to support more atoms without a change to the density. The increase in atom number would be accomplished by an increased atomic flux of the oven, while the single-frequency red MOT would be operated at a larger detuning to avoid loss by light-assisted collisions. The larger detuning increases the size of the MOT, such that the peak density does not increase despite a larger atom number.
v.1.3 Short cycle times
In the previous section, we reported on experiments optimized for a large number of atoms in the BEC. We can also optimize our experimental sequence for a short cycle time. Nearly all experiments profit from the higher data rate made possible by a shorter cycle time. Precision measurement devices, such as atom interferometers, do require high repetition rates or a favorable ratio of probe time versus cycle time and might profit from the coherence of a BEC. Quantum gas experiments taking place in an environment of poor vacuum quality also require a short production time. Most quantum gas experiments have cycle times of a few ten seconds. Experiments that have been optimized for speed while using an all-optical approach achieve cycle times of 3 s for degenerate bosonic gases Kinoshita et al. (2005); Kraft et al. (2009). Cycle times down to 1 s can be reached by using magnetic trapping near the surface of a microchip Han ().
Making use of the very high phase-space density achieved already in the red MOT, as well as the excellent scattering properties of Sr, we are able to reduce the cycle time to 2 s Stellmer et al. (2013b). At the beginning of the cycle, we operate the blue MOT for 800 ms to load the metastable reservoir. A short flash of repump light returns the metastable atoms into the ground state, where they are trapped, compressed, and cooled to about 1.2 K by the red MOT. Close to atoms are loaded into a dipole trap, which is formed by the horizontal sheet and a vertical beam of 25 m -radius in the plane of the horizontal dipole trap. The atomic cloud is not only populating the cross of the dipole trap, but extends mm along the horizontal dipole trap. Forced evaporation reduces the trap depth over 550 ms with an exponential time constant of about 250 ms.
During evaporation, a large fraction of the atoms in the horizontal beam migrate into the crossing region. The phase transition occurs after about 270 ms of evaporation, and after 480 ms, the thermal fraction within the crossing region cannot be discerned, indicating an essentially pure BEC in this region. Further evaporation does not increase the BEC atom number, but efficiently removes the thermal atoms residing in the horizontal beam. The BEC is formed by about atoms at the end of evaporation.
The read-out of the charged-coupled device (CCD) chip used for imaging can be performed during the consecutive experimental cycle and is therefore not included in the 2 s period. The cycle time could be improved substantially if the reservoir loading time was reduced, e.g. by increasing the oven flux. It seems that cycle times approaching 1 s are within reach.
v.1.4 Laser cooling to quantum degeneracy
The remarkable conjunction of supreme laser cooling performance and the excellent scattering properties allow us to reach a phase space density of about 0.1 directly after loading into the dipole trap; just one order of magnitude shy of quantum degeneracy Ido et al. (2000). It is now a challenging and amusing task to bridge this last order of magnitude and create a BEC without the cooling stage of evaporation.
Reaching a high phase space density not only requires a low temperature and a high density, but also a mechanism to suppress the (re-)absorption of cooling light photons, which counteracts the advancement towards BEC by constituting an effective repulsion and leading to heating and loss.
Here, we present an experiment that overcomes these challenges and creates a BEC of strontium by laser cooling Stellmer et al. (2013a). Our scheme essentially relies on the combination of three techniques, favored by the properties of this element, and does not rely on evaporative cooling. The narrow 7.4-kHz cooling transition enables simple Doppler cooling down to temperatures of 350 nK Ido et al. (2000). Using this transition, we prepare a laser cooled sample of atoms of Sr in a large “reservoir” dipole trap. To avoid the detrimental effects of laser cooling photons, we render atoms transparent for these photons in a small spatial region within the laser cooled cloud. Transparency is induced by a light shift on the optically excited state of the laser cooling transition. In the region of transparency, we are able to increase the density of the gas, by accumulating atoms in an additional, small “dimple” dipole trap Stamper-Kurn et al. (1998a); Weber et al. (2003). Atoms in the dimple thermalize with the reservoir of laser cooled atoms by elastic collisions and form a BEC. A striking feature of our technique is that the BEC is created within a sample that is being continuously laser cooled.
The details of our scheme are shown in Fig. 8. The pre-cooling stages of two sequential MOTs and the dipole trap loading are identical to the protocol described in Sec. III. The trap consists of a 1065-nm laser beam propagating horizontally. The beam profile is strongly elliptic, with a beam waist of m in the transverse horizontal direction and m along the field of gravity. The depth of the reservoir trap is kept constant at K. After preparation of the sample, another laser cooling stage is performed on the narrow intercombination line, using a single laser beam propagating vertically upwards. The detuning of the laser cooling beam from resonance is about and the peak intensity is . These parameters result in a photon scattering rate of s. At this point, the ultracold gas contains atoms at a temperature of 900 nK.
To render the atoms transparent to cooling light in a central region of the laser cooled cloud, we induce a light shift on the state, using a “transparency” laser beam 15 GHz blue-detuned to the transition. This beam propagates downwards under a small angle of to vertical, it has a waist of 26 m in the plane of the reservoir trap and a peak intensity of kW/cm. It upshifts the state by more than 10 MHz and also influences the nearest molecular level tied to the state significantly. Related schemes of light-shift engineering were used to image the density distribution of atoms Thomas and Wang (1995); Brantut et al. (2008), to improve spectroscopy Kaplan et al. (2002), or to enhance loading of dipole traps Griffin et al. (2006); Clement et al. (2009). To demonstrate the effect of the transparency laser beam, we take absorption images of the cloud on the laser cooling transition. Figure 8(d) shows a reference image without the transparency beam. In presence of this laser beam, atoms in the central part of the cloud are transparent for the probe beam, as can be seen in Fig. 8(e).
To increase the density of the cloud, a dimple trap is added to the system. It consists of a 1065-nm laser beam propagating upwards under a small angle of to vertical and crossing the laser cooled cloud in the region of transparency. In the plane of the reservoir trap, the dimple beam has a waist of 22 m. The dimple is ramped to a depth of K, where it has trap oscillation frequencies of 250 Hz in the horizontal plane. Confinement in the vertical direction is only provided by the reservoir trap and results in a vertical trap oscillation frequency of 600 Hz. Figure 8(f) shows a demonstration of the dimple trap in absence of the transparency beam: the density in the region of the dimple increases substantially. However, with the dimple alone no BEC is formed because of photon reabsorption.
The combination of the transparency laser beam and the dimple trap leads to BEC. Starting from the laser cooled cloud held in the reservoir trap, we switch on the transparency laser beam and ramp the dimple trap within 10 ms to a depth of K. The potentials of the and states in this situation are shown in Fig. 8(c). About atoms accumulate in the dimple without being disturbed by photon scattering, and elastic collisions thermalize atoms in the dimple with the laser cooled reservoir during the next ms. The temperature of the reservoir gas is hereby not increased, since the energy transferred to it is dissipated by laser cooling. Figure 9(a) shows the momentum distribution 20 ms after switching on the transparency beam, which is well described by a thermal distribution. By contrast, we observe that 140 ms later, an additional, central elliptical feature has developed; see Fig. 9(b). This is the hallmark of the BEC, which appears about 60 ms after ramping up the dimple. Its atom number saturates at after 150 ms. The atom number in the reservoir decreases slightly, initially because of migration into the dimple and on longer timescales because of light assisted loss processes in the laser cooled cloud. We carefully check that evaporation of atoms out of the dimple region is negligible even for the highest temperatures of the gas.
Although clearly present, the BEC is not very well visible in Fig. 9(b), because it is shrouded by thermal atoms originating from the reservoir. To show the BEC with higher contrast, we have developed a background reduction technique. We remove the reservoir atoms by an intense flash of light on the transition applied for 10 ms. Atoms in the region of transparency remain unaffected by this flash. Only thermal atoms in the dimple remain and the BEC stands out clearly; see Fig. 9(c). This background reduction technique is used only for demonstration purposes, but not for measuring atom numbers or temperatures.
The ability to reach the quantum degenerate regime by laser cooling has many exciting prospects. This method can be applied to any element possessing a laser cooling transition with a linewidth in the kHz range and suitable collision properties. The technique can also cool fermions to quantum degeneracy and it can be extended to sympathetic cooling in mixtures of isotopes or elements. Another tantalizing prospect enabled by variations of our techniques is the realization of a continuous atom laser, which converts a thermal beam into a laser-like beam of atoms.
v.2 Bose-Einstein condensation of Sr
Some isotopes of alkaline-earth atoms feature large positive scattering lengths, such as Ca, Ca, Ca Dammalapati et al. (2011), and Sr. While scattering between atoms provides thermalization during evaporation, there is a downside of a very large scattering length : Inelastic three-body losses have an upper limit proportional to Fedichev et al. (1996b); Bedaque et al. (2000), and can reduce the evaporation efficiency drastically. Magnetic Feshbach resonances, a widely used means to tune the scattering length in ultracold samples, are absent in the alkaline-earth species, and a different strategy to reach degeneracy despite the large scattering length is needed.
Quantum degeneracy in Sr has been reached Stellmer et al. (2010); Stellmer et al. (2013b) despite the large scattering length of about 800 Martinez de Escobar et al. (2008). In this experiment, the crucial innovation is to perform evaporation at a comparatively low density in a dipole trap of large volume. Two-body collisions, vital for thermalization, scale proportional with the density , while detrimental three-body collisions scale as . At small enough densities, evaporation can be efficient even for large scattering lengths.
The dipole trap has an oblate shape with initial trap frequencies of about Hz, Hz, and Hz. Using a 500-ms reservoir loading stage, we load atoms at a temperature of about 1 K into the dipole trap. The initial density is about and the average elastic collision rate 200 s. The large vertical trap frequency allows us to perform evaporation very quickly, in just 800 ms, which helps to avoid strong atom loss from three-body collisions. The onset of BEC is observed after 600 ms of evaporation at a temperature of about 70 nK with atoms present. Further evaporation results in almost pure BECs of 25 000 atoms. The cycle time of this experiment is again short, just 2.1 s. Such a BEC with a large scattering length might constitute a good starting point for studies of optical Feshbach resonances.
v.3 Bose-Einstein condensation of Sr
The most abundant strontium isotope, Sr, presents significant challenges to reaching quantum degeneracy because of the small and negative -wave scattering length, . Fortunately, because of the good SrSr interspecies scattering length ( ), Sr can serve as an effective refrigerant for Sr for dual-species evaporative cooling. Use of an equal mixture of the 10 distinguishable ground states for Sr arising from the nuclear spin diminishes any limitation on fermion-fermion collisions due to Pauli blocking. Essentially pure condensate cans be created with up to 10 000 atoms, limited by the critical number for condensate collapse due to attractive interactions Ruprecht et al. (1995); Houbiers and Stoof (1996). This is adequate for many experiments that benefit from working with a BEC of a nearly ideal gas. Here we will describe results from the Rice group Mickelson et al. (2010); similar results were reported by the Innsbruck group Stellmer et al. (2013b). Sympathetic cooling of Sr with Sr can also produce quantum degenerate Sr Stellmer et al. (2013b), although this is less efficient.
We closely follow the presentation in Ref. Mickelson et al. (2010). Sr atoms are accumulated in the metastable state reservoir for 3 s, followed by 30 s of loading for Sr. atoms are returned to the ground state with 60 ms of excitation on the transition at 3.01 m. We typically recapture approximately Sr and Sr in the blue MOT at temperatures of a few mK.
The 461 nm light is then extinguished and 689 nm light is applied to drive the transitions and create intercombination-line MOTs for each isotope Katori et al. (1999a); Mukaiyama et al. (2003); Mickelson (2010). After 400 ms of laser cooling, an optical dipole trap consisting of two crossed beams is overlapped for 100 ms with the intercombination-line MOT with 3.9 W per beam and waists of approximately m in the trapping region. The dipole trap is formed by a single beam derived from a 20 W multimode, 1.06 m fiber laser that is recycled through the chamber in close to the horizontal plane.
After extinction of the 689 nm light, the sample is compressed by ramping the dipole trap power to 7.5 W in 30 ms, resulting in a trap depth of K. Typically the atom number, temperature, and peak density at this point for both Sr and Sr are , 7 K, and cm. The peak phase space density for Sr is 0.01.
We decrease the laser power according to , with time denoted by , , and s. This trajectory without was designed O’Hara et al. (2001) to yield efficient evaporation when gravity can be neglected. Gravity is a significant effect in this trap, and to avoid decreasing the potential depth too quickly at the end of the evaporation, we set W, which corresponds to the power at which gravity causes the trap depth to be close to zero. The Sr and Sr remain in equilibrium with each other during the evaporation, and we observe an increase of Sr phase space density by a factor of 100 for a loss of one order of magnitude in the number of atoms. Sr atoms are lost at a slightly faster rate, as expected because essentially every collision involves an Sr atom.
A Maxwell-Boltzmann distribution fits the momentum distribution well at 5 s of evaporation. At 6 s, however, a Boltzmann distribution fit to the high velocity wings underestimates the number of atoms at low velocity. A Bose-Einstein distribution matches the distribution well. This sample is close to the critical temperature for condensation and has a fit fugacity of . Further evaporation to 7.5 s produces a narrow peak at low velocity, which is a clear signature of the presence of a BEC. A pure condensate is observed near the end of the evaporation trajectory, which takes 9 s.
At the transition temperature, Sr atoms remain at a temperature of 200 nK. This corresponds to for an unpolarized sample, which is non-degenerate and above the point at which Pauli blocking significantly impedes evaporation efficiency DeMarco and Jin (1999).
Sr has a negative scattering length, so one expects a collapse of the condensate when the system reaches a critical number of condensed atoms given by Ruprecht et al. (1995)
for a spherically symmetric trap. Here is the harmonic oscillator length, where is the atom mass, is the reduced Planck constant, and is the trap oscillation frequency. Our initial studies Mickelson et al. (2010) showed significant fluctuation in condensate number, bounded by the critical number for the trap. Subsequent optimization showed that reducing the initial number of Sr atoms in the dipole trap to be about half that of Sr yields much less variation in condensate number. We reliably create condensates with about 90% of the critical number with a standard deviation of about 10% Stellmer et al. (2013b); Yan et al. (2013). The ability to make reproducible condensates is critical for experiments with Sr, such as investigation of an optical Feshbach resonance Yan et al. (2013).
v.4 Bose-Bose mixtures
Mixtures of two Bose-degenerate gases of different isotopes or elements allow the study of interesting phenomena, such as the miscibility and phase separation of two quantum fluids Hall et al. (1998); Riboli and Modugno (2002); Jezek and Capuzzi (2002). The many bosonic isotopes of alkaline-earth elements in principle allow the creation of many different Bose-Bose mixtures. Unfortunately, for many of these mixtures, the interaction properties are unfavorable to create large and stable BECs. To avoid rapid decay, the absolute value of the two intra- and the interspecies scattering length must not be too large, but it must be large enough for efficient thermalization. The intraspecies scattering lengths should not be strongly negative to permit the formation of detectably large BECs Fukuhara et al. (2009a). The scattering length of alkaline-earth-like atoms can only be tuned by optical Feshbach resonances, which introduce losses Ciuryło et al. (2005); Enomoto et al. (2008); Blatt et al. (2011). These limitations reduce the number of possible binary mixtures considerably. In particular, all combination of bosonic calcium isotopes seem unfavorable, since all intraspecies scattering lengths of the most abundant calcium isotopes are quite large Kraft et al. (2009); Dammalapati et al. (2011). In ytterbium, two out of five bosonic isotopes have large negative scattering lengths Kitagawa et al. (2008), excluding many possible combinations of isotopes. One remaining combination, Yb + Yb, has a large and negative interspecies scattering length. One of the two remaining combinations, Yb + Yb, has been brought to double degeneracy very recently, with 9000 atoms in the BEC of each species Sugawa et al. (2011a). The interspecies scattering length between these two isotopes is and provides only minuscule interaction between the two. The three bosonic isotopes of strontium give rise to three different two-isotope combinations; see Tab. 1. Of these the mixtures, Sr + Sr suffers from a large interspecies scattering length.
We will now present double-degenerate Bose-Bose mixtures of the combinations Sr + Sr and Sr + Sr, which have interspecies scattering lengths of and , respectively. The experimental realization is straightforward: We consecutively load the two isotopes into the reservoir, repump them simultaneously on their respective transitions, and operate two red MOTs simultaneously. The mixture is loaded into the dipole trap and subsequently evaporated to form two BECs. Imaging is performed on the blue transition, and we image only one isotope per experimental run. The frequency shift between the isotopes is only about 4.5 linewidths. To avoid a contribution of the unwanted isotope to the absorption image, we remove the unwanted species by an 8-ms pulse of resonant light on the very isotope selective intercombination transition. To avoid a momentum distribution change of the imaged species by interspecies collisions, the pulse of light is applied after 17 ms of free expansion, when the density of the sample has decreased sufficiently.
We will discuss the Sr + Sr combination first: () atoms of Sr (Sr) are loaded into the dipole trap, consisting of a horizontal beam and a weak vertical beam for additional axial confinement. The initial temperatures of the two species are quite different: 950 nK for Sr and 720 nK for Sr, which reflects the different intraspecies scattering behavior. The interspecies scattering length is around , and the two species clearly thermalize to reach equilibrium after 1 s of evaporation. As the trap depth is lowered further, we observe the onset of BEC in Sr (Sr) after 2.0 s (2.3 s). At the end of our evaporation ramp, which lasts 2.4 s, we obtain 10 000 (3000) atoms of Sr (Sr) in the condensate fraction. Further evaporation does not increase the BEC atom numbers.
In a second experiment, we investigate the Sr + Sr mixture with an interspecies scattering length of . Starting out with () atoms of Sr (Sr) in the dipole trap, we perform forced evaporation over 2 s, and the two species remain in perfect thermal equilibrium throughout this time. The phase transition of Sr is observed already after 1.3 s, with about atoms present at a temperature of 200 nK. After 1.9 s, the BEC is essentially pure and contains up to atoms. The atom number of Sr is kept considerably lower to avoid three-body loss. The phase transition occurs later: after 1.7 s, with atoms at a temperature of 130 nK. Till the end of evaporation, the BEC component grows to 8000 atoms but remains accompanied by a large thermal fraction.
We have here presented two binary Bose-Bose mixtures of alkaline-earth atoms with appreciable interaction between the two species. These mixtures enjoy the property that isotope-selective optical traps can be operated close to one of the intercombination lines. This might allow for an individual addressing of the isotopes by a dipole trap operated close to these transitions Yi et al. (2008), reminiscent of the case of rubidium in its hyperfine states and Mandel et al. (2003) or nuclear substates in ytterbium and strontium Taie et al. (2010); Stellmer et al. (2011).
Vi Spin state control in Sr
Fermionic Sr has a nuclear spin of . This large nuclear spin has many applications in quantum simulation and computation, for which preparation, manipulation, and detection of the spin state are requirements. For an ultracold Sr cloud, we show two complementary methods to characterize the spin-state mixture: optical Stern-Gerlach state separation and state-selective absorption imaging. We use these methods to optimize the preparation of a variety of spin-state mixtures by optical pumping and to measure an upper bound of the Sr spin relaxation rate.
vi.1 Optical Stern-Gerlach separation
Several alkaline-earth spin-state detection schemes have been demonstrated. The number of atoms in the highest state can be determined by selectively cooling Mukaiyama et al. (2003) or levitating Tey et al. (2010) atoms in this state. The number of atoms in an arbitrary state was determined using state-selective shelving of atoms in a metastable state Boyd et al. (2007). Recording the full -state distribution with this method is possible, but needs one experimental run per state. Determination of the -state distribution in only two experimental runs was shown for quantum-degenerate ytterbium gases, using optical Stern-Gerlach (OSG) separation Taie et al. (2010).
The Stern-Gerlach technique separates atoms in different internal states by applying a state-dependent force and letting the atomic density distribution evolve under this force Gerlach and Stern (1922). The implementation of this technique for alkali atoms is simple. Their single valence electron provides them with a -state dependent magnetic moment that, for easily achievable magnetic field gradients, results in -state dependent forces sufficient for state separation Stamper-Kurn et al. (1998b). By contrast, atoms with two valence electrons possess only a weak, nuclear magnetic moment in the electronic ground state, which would require the application of impractically steep magnetic field gradients. An alternative is OSG separation, where a state dependent dipole force is used. OSG separation was first shown for a beam of metastable helium Sleator et al. (1992), where orthogonal dressed states of the atoms were separated by a resonant laser field gradient. The case of interest here, OSG -state separation, has been realized as well for a quantum degenerate gas of ytterbium, by using -state dependent dipole forces Taie et al. (2010).
We first explain the basic operation principle of strontium OSG separation before discussing our experimental implementation. The experimental situation is shown in Fig. 10(a). An ultracold cloud of Sr atoms in a mixture of states is released from an optical dipole trap. The -state dependent force is the dipole force of two laser beams propagating in the plane of the pancake-shaped cloud, one polarized , the other . The diameter of these OSG laser beams is on the order of the diameter of the cloud in the -direction. The beams are displaced vertically by about half a beam radius to produce a force in the -direction on the atoms. To create a -state dependent force, the OSG beams are tuned close to the intercombination line, so that this line gives the dominant contribution to the dipole force. A guiding magnetic field is applied in the direction of the laser beams such that the beams couple only to or transitions, respectively. The line strength of these transitions varies greatly with the state Metcalf and van der Straten (1999), see Fig. 10(b), resulting in different forces on the states. For Yb, this variation, together with a beneficial summation of dipole forces from transitions to different hyperfine states, is sufficient to separate four of the six states using just one OSG beam Taie et al. (2010). The remaining two states could be analyzed by repeating the experiment with opposite circular polarization of the OSG beam.
Strontium, which has nearly twice as many nuclear spin states, requires an improved OSG technique to separate the states. The improvement consists of applying two OSG beams with opposite circular polarization at the same time. The -polarized beam produces dipole forces mainly on the positive states, the beam mainly on the negative states. By positioning the beams in the appropriate way (see below), the forces point in opposite directions and all states can be separated in a single experimental run. A second improvement is to enhance the difference in the dipole forces on neighboring states by tuning already strong transitions closer to the OSG beam frequency using a magnetic field, which splits the excited state states in energy. For our settings, the difference in forces on neighboring high states is enhanced by up to 25%, which helps to separate those states. This enhancement scheme requires the -polarized OSG beam to be tuned to the blue of the resonance, whereas the beam has to be tuned to the red of the resonance; see Fig. 10(b). Both beams are centered above the atomic cloud so that the repulsive blue detuned beam produces a force pointing downwards, whereas the attractive red detuned beam produces a force pointing upwards.
vi.2 Experimental demonstration
We demonstrate OSG separation of a cloud of Sr atoms in a mixture of states. To prepare the cloud, Zeeman slowed Sr atoms are laser cooled in two stages, first in a blue magneto-optical trap (MOT) on the broad-linewidth transition, then in a red MOT on the narrow-linewidth transition. Next, the atoms are transferred to a pancake-shaped optical dipole trap with strong confinement in the vertical direction. The sample is evaporatively cooled over 7 s. At the end of evaporation the trap oscillation frequencies are Hz, Hz, and Hz, where the coordinate system is defined in Fig. 10(a). The collision rate at this stage is only s, which is insufficient for complete thermalization. Since atoms are evaporated mainly downwards, along the -direction, the sample is not in cross-dimensional thermal equilibrium, having a temperature of 25 nK in the -direction and twice that value in the -plane. The sample is non-degenerate and the -widths of the Gaussian density distribution are m, m, and m.
The OSG beams propagate along the -direction. The power of the () beam is 4 mW (0.5 mW), the waist is m (m), and the beam center is displaced m (m) above the cloud. Both beams create dipole forces of similar magnitude since the reduced power of the beam compared to the beam is partially compensated by its decreased waist. At zero magnetic field, the beam is detuned MHz from resonance. To increase the difference in dipole potential on neighboring states, a magnetic field of 16 G is applied parallel to the OSG beams, which splits neighboring ) states by 6.1 MHz. With this field applied, the beam has a detuning of MHz to the , , ) transition and a detuning of MHz to the , , ) transition; see Fig. 10(b).
OSG separation is started by simultaneously releasing the atoms from the dipole trap and switching on the OSG beams. The atoms are accelerated for 1.6 ms by the OSG beams. Then the beams are switched off to avoid oscillations of atoms in the dipole trap formed by the red detuned OSG beam. The atoms freely expand for another 2.3 ms before an absorption image on the transition is taken. The result is shown in Fig. 11(a). All ten states are clearly distinguishable from each other.
To quantify the separation of the states, we fit ten Gaussian distributions to the density distribution integrated along the - and -directions, see Fig. 11(c). We obtain a separation of adjacent states very similar to the -widths of the distributions, which are between 24 and 36 m. From the Gaussian fits we also obtain an estimation of the atom number in each state. The -state dependence of the line strength of the blue imaging transition, as well as optical pumping processes during imaging, need to be taken into account to accurately determine the atom number in each spin state. Detailed simulations of classical atom trajectories describing the OSG separation process can be found in Ref. Stellmer et al. (2011). The simulations show very good agreement with the experiment, which can be appreciated by a comparison of Figs. 11(a) and (b).
OSG separation works only well for very cold samples. If the temperature is too high, the sample expands too fast and the individual -state distributions cannot be distinguished. For a density minimum to exist between two neighboring -state distributions of Gaussian shape, the -widths have to be smaller than times the distance between the maxima of the distributions. For our smallest separation of 24 m, this condition corresponds to samples with a temperature below 100 nK, which can only be obtained by evaporative cooling.
vi.3 Spin-state dependent absorption imaging
We also demonstrate a complementary method of -state detection: -state dependent absorption imaging. This method is often used for alkali atoms employing a broad linewidth transition Matthews et al. (1998). For strontium, -state resolved imaging on the broad transition is not possible since the magnetic field splitting of the exited state states is smaller than the linewidth of the transition Boyd et al. (2007). But -state dependent imaging can be realized using the narrow ) intercombination line. To achieve state selectivity, we apply a magnetic field of 0.5 G, which splits neighboring states by 200 kHz, which is 27 times more than the linewidth of the imaging transition. The advantages of this method compared to OSG separation is its applicability to samples that have not been evaporatively cooled, spatially resolved imaging, and a near perfect suppression of signal from undesired states. A disadvantage of this method is that it delivers a reduced signal compared to imaging on the transition, as done after OSG separation. The reduction comes from the narrower linewidth, optical pumping to dark states during imaging, and weak line strengths for some states.
To demonstrate absorption imaging on the intercombination line, we use a sample of atoms at a temperature of 500 nK in a trap with oscillation frequencies of Hz, Hz, and Hz, obtained after 1.4 s of evaporation. Figure 12 shows a spectroscopy scan and absorption images taken on the maxima of the absorption signal of this sample. The absorption is strongly -state dependent and to obtain the best signal, the polarization of the absorption imaging light has to be adapted to the state of interest: () for high (low) states and for low states. For our absorption imaging conditions (an intensity of 15 W/cm, which is five times the saturation intensity, and an exposure time of 40 s), even atoms in states corresponding to the strongest transition will on average scatter less than one photon. Therefore, for a sample with homogeneous -state distribution, the maximum absorption is expected to be nearly proportional to the -state dependent line strength of the transition, which we confirm using a simulation of the absorption imaging process. This proportionality is observed in the experimental data, indicating that the sample used has a nearly homogeneous -state distribution. The observed Lorentzian linewidth of the absorption lines is kHz. We expect a linewidth of kHz arising from power and interaction-time broadening. Doppler broadening and collisional broadening will contribute to the linewidth as well Ido et al. (2005).
vi.4 Preparation of spin-state mixtures
For applications of Sr to quantum simulation and computation, the -state mixture needs to be controlled. We produce a variety of different mixtures by optical pumping, making use of OSG separation to quickly optimize the optical pumping scheme and quantify the result. Optical pumping is performed on the intercombination line, before evaporative cooling. A field of 3 G splits neighboring excited state states by 255 kHz. This splitting is well beyond the linewidth of the transition of 7.4 kHz, allowing transfer of atoms from specific states to neighboring states using - or -polarized light, the choice depending on the desired state mixture. Sequences of pulses on different states can create a wide variety of state mixtures, of which three examples are shown in Fig. 13. The fidelity of state preparation can reach 99.9%, as confirmed by state-dependent absorption imaging.
vi.5 Determination of an upper bound of the spin-relaxation rate
A low nuclear spin-relaxation rate is an essential requirement to use Sr for quantum simulation and computation Cazalilla et al. (2009); Gorshkov et al. (2010). The rate is expected to be small since the nuclear spin does not couple to the electronic degrees of freedom in the ground state. Here, we use our nuclear spin state preparation and detection techniques to determine an upper bound for this spin relaxation rate. We start with a sample of atoms with near uniform -state distribution and a temperature of K, confined in a trap with oscillation frequencies Hz, Hz, and Hz, obtained after transferring the atoms from the MOT to the dipole trap and adiabatic compression of the trap. We optically pump all atoms from the state to neighboring states and look out for the reappearance of atoms in this state by spin relaxation during 10 s of hold. The atom number in the state and, as a reference, the state are determined from absorption images. During 10 s of hold at a magnetic field of either 5 G or 500 G the number of atoms remains below our detection threshold of about atoms, indicating a low spin-relaxation rate. To obtain a conservative upper bound for the spin-relaxation rate, we assume that the dominant process leading to the creation of -state atoms are collisions of - with -state atoms, forming two -state atoms. Since the second order Zeeman effect is negligible, no energy is released in such a collision and the resulting -state atoms will remain trapped. The number of atoms created in the state by spin relaxation after a hold time is , where is the atom number in each populated state, the spin-relaxation rate constant, cm the mean density and the factor 2 takes into account that two atoms are produced in the state per collision. From our measurement we know that , from which we obtain an upper bound of cms for the spin-relaxation rate constant. This bound for the rate constant corresponds for our sample to a spin relaxation rate which is 2000 times smaller than the elastic scattering rate. This value can be converted into a deviation of less than from an assumed SU() symmetry Bonnes et al. (2012). The rate constant could be even orders of magnitude smaller than the already low upper bound we obtained Jul ().
Vii Degenerate Fermi gases of Sr
Ground-breaking experiments with ultracold Fermi gases Inguscio et al. (2008); Giorgini et al. (2008) have opened possibilities to study fascinating phenomena, as the BEC-BCS crossover, with a high degree of control. Most experiments have been performed with the two alkali fermions K and Li. Fermions with two valence electrons, like Ca, Sr, Yb, and Yb, have a much richer internal state structure, which is at the heart of recent proposals for quantum computation and simulation; see Sec. I. Unlike bosonic isotopes of these elements, the fermions have a nuclear spin, which decouples from the electronic state in the ground state and the metastable state. This gives rise to a SU spin symmetry, where is the number of nuclear spin states, which is ten for Sr. A wealth of recent proposals suggest employing such atoms as a platform for the simulation of SU() magnetism Wu et al. (2003); Wu (2006); Cazalilla et al. (2009); Hermele et al. (2009); Gorshkov et al. (2010); Xu (2010); Foss-Feig et al. (2010a, b); Hung et al. (2011); Manmana et al. (2011); Hazzard et al. (2012); Bonnes et al. (2012), for the generation of non-Abelian artificial gauge fields Dalibard et al. (2011); Gerbier and Dalibard (2010), to simulate lattice gauge theories Banerjee et al. (2013), or for quantum computation schemes Hayes et al. (2007); Daley et al. (2008); Gorshkov et al. (2009); Daley et al. (2011).
Elements with a large nuclear spin are especially well suited for some of these proposals. They allow to encode several qubits in one atom Gorshkov et al. (2009), and could lead to exotic quantum phases, as chiral spin liquids, in the context of SU() magnetism Hermele et al. (2009). Furthermore, it has been shown that the temperature of a lattice gas is lower for a mixture containing a large number of nuclear spin states after loading the lattice from a bulk sample Hazzard et al. (2012); Taie et al. (2012); Bonnes et al. (2012). Pomeranchuk cooling Richardson (1997), which benefits from a large number of spins, has recently been observed in ytterbium Taie et al. (2012). The largest nuclear spin of any alkaline-earth-like atom is 9/2, and it occurs in the nuclei of Sr and of two radioactive nobelium isotopes. This fact makes Sr with its ten spin states an exceptional candidate for the studies mentioned.
The study of SU() magnetism in a lattice requires the temperature of the sample to be below the super-exchange scale, , where is the tunnel matrix element and the on-site interaction energy Bonnes et al. (2012). A high degree of degeneracy in the bulk would constitute a good starting point for subsequent loading of the lattice.
vii.1 A degenerate Fermi gas of ten spin states
The first degenerate Fermi gases of strontium were produced in a mixture of all ten spin states DeSalvo et al. (2010). The procedure of laser cooling the fermionic isotope is very similar to the bosonic case, but complicated by the hyperfine structure. In this experiment, atoms are loaded into an optical dipole trap consisting of two horizontal beams of m waist, powers of 3.9 W per beam, and a wavelength of 1064 nm.
The atoms are in a roughly even distribution of spin states. The trap depth is increased by about a factor of two to increase the rate of collisions required for thermalization. Forced evaporation increases the phase space density significantly and allows us to enter the quantum degenerate regime; see Fig.