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Optical tuning of the scattering length of cold alkaline earth atoms

It is possible to tune the scattering length for the collision of ultra-cold 1S0 ground state alkaline-earth atoms using an optical Feshbach resonance. This is achieved with a laser far detuned from an excited molecular level near the frequency of the atomic intercombination 1S0--3P1 transition. Simple resonant scattering theory, illustrated by the example of 40Ca, allows an estimate of the magnitude of the effect. Unlike alkali metal species, large changes of the scattering length are possible while atom loss remains small, because of the very narrow line width of the molecular photoassociation transition. This raises prospects for control of atomic interactions for a system without magnetically tunable Feshbach resonance levels.

It is possible to tune the scattering length for the collision of ultra-cold 1 S0 ground state alkaline-earth atoms using an optical Feshbach resonance. This is achieved with a laser far detuned from an excited molecular level near the frequency of the atomic intercombination 1 S0-3 P1 transition. Simple resonant scattering theory, illustrated by the example of 40 Ca, allows an estimate of the magnitude of the effect. Unlike alkali metal species, large changes of the scattering length are possible while atom loss remains small, because of the very narrow line width of the molecular photoassociation transition. This raises prospects for control of atomic interactions for a system without magnetically tunable Feshbach resonance levels.
PACS numbers: 34.50.Rk, 34. 10.+x,32.80.Pj In recent years the ability to change the interaction between ultra-cold colliding atoms has opened the way for new and exciting experiments with ultra-cold atomic gases. The observation of a Bose-Einstein condensate (BEC) in atomic cesium [1] would have been impossible without the ability to change the interaction from being attractive to being repulsive. More impressively, time-varying interactions have allowed the creation of condensates of two-atom molecules starting from an atomic Bose condensate [2,3]. Most recently, the observation of the condensation of pairs of fermionic [4] atoms has started investigations into the so-called BEC-BCS crossover [5], where BCS is the abbreviation for the Bardeen-Cooper-Schrieffer phase transition in a fermionic gas [6].
The key to these developments has been the ability to change the interaction between atoms by a magnetically-tuned Feshbach resonance [7]. Theoretical discussion of the properties of these resonances can be found in Refs. [8,9,10,11]. The interaction between the atoms at ultra-cold temperature can be characterized by a single parameter, the scattering length [12,13], which can be controlled in sign and magnitude using these resonances.
Another way to change the scattering length of two colliding atoms is to optically couple the ground scattering state with an excited bound state [14]. These optical Feshbach resonances are theoretically analyzed in Refs. [15,16] and implemented experimentally in Refs. [17,18]. The recent experiment of Theis et al. [18] with 87 Rb atoms showed, however, that a significant change of the scattering length is accompanied with substantial loss of atoms. The same is true if a twocolor Raman process is used [19].
In this paper we discuss the optical tuning of scattering lengths in ultra-cold alkaline-earth atom vapors. To do so we will assume that the laser is far detuned from excited molecular states near the intercombination transition, 1 S 0 -3 P 1 , as recently analyzed in Ref. [20]. We show that significant changes of the scattering strength can be achieved without the excessive atom loss that plagues experiments with alkali-metal gases [18]. Prospects for an optically tuned scattering length in ultra-cold alkaline-earth vapors seems to be particularly attractive, since magnetically tuned Feshbach resonances do not occur between the isotopes of ground-state 1 S 0 alkaline-earth atoms with zero nuclear spin. Several alkaline earth metal atoms also have isotopes with nonzero integer or half-integer nuclear spin. Optical methods could be used to tune their scattering lengths, although we will not consider these specifically in this paper. Optical Feshbach control can be also applied to other atomic systems having a similar electronic structure. The recent Bose-Einstein condensation of ytterbium [21] makes this system especially interesting.
The theoretical description of optical Feshbach resonances and the closely related photoassociation (PA) process is well established [14,15,16]. The elastic and inelastic scattering rates due to a single photoassociation resonance level depends on (1) the natural line width Γ e,nat ≈ 2Γ A of the excited molecular level e, (2) the stimulated width Γ eg (ε r ) that couples the excited level to the s-wave collision of the ground state g at relative kinetic energy ε r , and (3) the detuning ∆ − ∆ e from optical resonance. Here, following the notation of Ref. [20] for PA near the 1 S 0 -3 P 1 intercombination line of a Group II atomic species, Γ A is the natural decay width of the atomic transition, ∆ =hω − E A ∆ e = E e − E A , where E e is the energy of an isolated excited molecular bound level, E A is the energy of the 3 P 1 atom, and ω is the frequency of the light driving the transition. The stimulated width is where V las is the optical coupling proportional to the laser intensity I, λ A is the wavelength of the atomic transition, c is the speed of light, f rot is a dimensionless rotational line strength factor of order unity, and f FC is the Franck-Condon factor per unit energy for the free-bound PA transition: Here, F e is the unit normalized excited state wave vibrational function and F g (ε r , R) is the energy normalized ground state scattering wave function. The low energy s-wave form of the latter is (2µ/πh 2 k r ) 1/2 sin(k r (R − a bg )) at large R , where µ is reduced mass of the atom pair,h is Planck's constant divided by 2π, k r = √ 2µε r /h, and a bg is the ground state s-wave scattering length in the absence of light. It should be noted that the details of the molecule structure are hidden in Γ eg (ε r ), ∆ e , and a bg . The rate constant for inelastic collisions that lead to atom loss is typically large when the detuning from molecular resonance is small. Consequently, we will only consider the case of unsaturated transitions at large detuning, defined by the condition We also require that |∆ − ∆ e | be much larger than other contributions to the width of the photoassociation line such as the thermal width, Doppler width [20], light induced shift [16,22], and the mean-field shift in the case of BEC [23]. The theoretical description can be framed using the definition of a complex scattering length A based on the elastic scattering S-matrix element as k r → 0 [15] The length A is complex and in the presence of light can be written as where the dependence on ∆ and I are made explicit. The optically induced a opt (∆, I) and b opt (∆, I) vanish for I = 0 and are linear in I for the limit in Eq. (3). The length a scat (∆, I) = a bg + a opt (∆, I) is interpreted as the usual scattering length, and b opt is related to the atom loss rate coefficient K(∆, I) determined by the unitary S-matrix: The real and imaginary parts of A are directly related to the mean field energy and the lifetime of a Bose-Einstein condensate. For the case of a condensate at density n, the requirement |a scat | ≫ b opt ensures that the mean field energy per atom pair 4πh 2 a scat n/µ is large compared to the decay width hKn = 4πh 2 b opt n/µ. The time scale for decay is (Kn) −1 for a condensate and (2Kn) −1 for a noncondensed thermal gas [24]. In the noncondensed thermal gasṅ = −2Kn 2 if other processes are neglected. Given the large detuning condition of Eq. (3), the expressions in Refs. [15,16] reduce to where a bg is the background scattering length defined previously and Since the threshold properties of low energy scattering ensure that Γ eg (ε r ) ∝ k r as ε r → 0, we see that δ eg , a opt (∆, I), and b opt (∆, I) are independent of collision energy for ε r → 0. Equation (8) shows that the change in scattering length for an optically induced Feshbach resonance has the same form as that for a magnetically tunable Feshbach resonance when the width δ eg is used. With our definitions, ∆ − ∆ e > 0 corresponds to blue detuning and a positive change in scattering length.
For optical Feshbach resonances it is also convenient to express the detuning dependence in terms of the natural linewidth, namely where the "optical length" is defined as This length depends on the molecular physics parameters of the ground and excited states but is independent of collision energy in the low energy threshold regime and is proportional to the laser intensity I, given our large detuning assumption. The optical length is the same as the radius introduced in Eq. (12) of Ref. [15]. In order to make useful changes in the scattering length, the change has to be large while the losses remain small. The former criterion requires that |a opt | ≫ |a bg |, whereas the latter requires that |a opt | ≫ b opt . Equation (7) shows that the first criterion is satisfied as long as |∆ − ∆ e | ≪ δ eg , whereas Eq. (9) shows that the second is satisfied if |∆−∆ e | ≫ Γ e,nat . The condition in Eq. (3) requires a more stringent condition on the detuning, which we can state in terms of l opt by combining Eqs. (3) and (13): |∆ − ∆ e | ≫ Γ e,nat (1 + 2k r l opt ), where we may take k r typical of a collision energy in the system. Combining these criteria, they may be stated either in terms of the detuning: or the length parameters: In general it will be difficult to satisfy these criteria for the case of strongly allowed molecular transitions with large Γ e,nat since they typically have a relatively small l opt at the large detunings that are necessary. However, using the example of Ca atoms, we now will show that these criteria can be satisfied by an intercombination line transition, for which the needed large detuning can be achieved close enough to atomic resonance that the Franck-Condon factor is large enough that l opt is not too small.
The first experimental demonstration of the optical tuning of the 87 Rb scattering length were presented by Theis et The short range of the 0 + u potential is varied to change ∆e. The arrows show the position of the last two unperturbed 1u levels, which are held fixed in the calculation. The vertical dotted lines mark the edges of the energy ranges ("bins") within which one and only one level of 0 + u symmetry will lie, given some 0 + u potential curve (the first "bin" starts at ∆e = 0). The actual 0 + u and 1u line positions are not known and will need to be determined experimentally.
al. [18]. Measurements were done with a moderate laser intensity of about 500 W/cm 2 . The authors observed a tuning range of the scattering length of about 200 a 0 accompanied by a trap loss coefficient as large as 2 × 10 −10 cm 3 /s (a 0 = 0.05291772 nm). The optical length for this strongly allowed 87 Rb 2 transition is l opt ≈ 100 a 0 , which is relatively small and comparable to a bg = 103 a 0 . Therefore, a significant change of the scattering length, i.e. a opt ∼ a bg , can only be induced close to resonance and is accompanied by a large trap loss.
We have calculated properties of optical Feshbach resonances for calcium near the intercombination line. The molecular structure and transition dipole moments are evaluated as in Ref. [20]. The molecular structure is insufficiently known to quantitatively predict the absolute positions of the excited bound vibrational levels. It is necessary to measure these positions experimentally. Moreover, the background scattering length of the ground state is not precisely known but is believed to be positive and on the order of a few hundred a 0 [25]. Nevertheless, we can map out the values of l opt , a opt , and other Feshbach properties as a function of the background scattering length, binding energy, and laser detuning and intensity. Figure 1 shows l opt as a function of both background scattering length a bg and ∆ e (The positive binding energy relative to 3 P 1 + 1 S 0 atoms is −∆ e ). Details of the calculation of the stimulated width Γ eg (ε r ), necessary for the evaluation of l opt , are described in the figure caption and Ref. [20]. length at the maxima ranges between 10 a 0 and 10 5 a 0 . The interference minima to the right of the arrows are due to the mixing of 0 + u and 1 u bound states . The third minimum is due to vanishing overlap between excited 0 + u and ground state wave functions. The nature and properties of these features are discussed in detail in Ref. [20].
The figure also shows that the envelop of l opt (i.e., ignoring oscillations) increases when |∆ e | decreases or a bg increases. In fact, for a laser intensity of 1 W/cm 2 and binding energies on the order of 1 GHz the optical length can be bigger than 10 3 a 0 , while it is 0.5 × 10 6 a 0 for ∆ e ≈ −1 MHz and a bg ≈ 1000 a 0 .
In order to provide a specific example of a opt (∆, I) and b opt (∆, I), we assume a bound state with −∆ e /h = 150 MHz and a bg = 389.8. (This optical resonance corresponds to line number 1 of the 0 + u band as defined in Ref. [20].) For this case l opt = 0.9 × 10 6 a 0 at a laser intensity of 500 W/cm 2 . This is the same laser intensity as used in Ref. [18], but l opt is four orders of magnitude larger than that for Ref. [18]. Figure 2(a) and (b) show the optically-induced scattering length a opt (∆, I) and loss rate coefficient K(∆, I) as a func-tion of blue detuning ∆−∆ e for the parameters defined above. The laser detunings shown in the figure are orders of magnitude larger than the natural linewidth of 0.663 kHz. For these parameters the stimulated linewidth is 18.2 kHz at a collision energy of 1 nK, for which 2k r l opt = 27. The requirement that |∆ − ∆ e |/h ≫ (1 + 2k r l opt )Γ e,nat /h = 19 kHz is easily satisfied. However, for collision energies on the order of of 1 µK, Γ e,nat (1 + 2k r l opt ) is of the same order as the frequency range in Fig. 2 and our assumptions are broken. Consequently, for these collision energies we need a detuning ∆ − ∆ e that is larger than 5 MHz.
In the figure both a opt and K(∆, I) decrease with increasing detuning. The optically-induced scattering length is ≈ 1200 a 0 for (∆ − ∆ e )/h = 500 kHz, while simultaneously the trap loss rate is relatively small with K(∆, I) = 1.7 × 10 −12 cm 3 /s. This loss rate coefficient is two orders of magnitude less than in the case of rubidium. The time scale parameter (Kn) −1 ≈ 6 ms, assuming an initial atom density n = 10 14 atom/cm 3 .
Equations (11) and (12) apply to red as well as blue detunings so that negative ∆ − ∆ e gives a negative a opt (∆, I). Using the parameters in Fig. 2, it should be possible to tune the scattering length a scat from ≈ −800 a 0 to zero to ≈ +400 a 0 while the loss rate remains below 1.7 × 10 −12 cm 3 /s.
The essential molecular physics that sets the magnitude of l opt depends on the molecular Franck-Condon factor in Eqs. (1) and (2). The transition dipole moment cancels in the definition of l opt in Eq. (13) due to the relation Γ e,nat ≈ 2Γ A . However, the small natural line width of the 1 S 0 -3 P 1 transition for alkaline-earth atoms has an important role to play, since much smaller detuning from atomic resonance can be used than in the case of alkali-metal atoms. Both numeri-cal and analytic calculations, similar to those used in the Appendix of Ref. [16], show that the magnitude of the Franck-Condon factor (ignoring an oscillating phase factor) increases as the binding energy of the excited level decreases. To operate at sufficiently large detuning to satisfy Eqs. (14) and (15) requires a quite large binding energy for strongly allowed transitions, i.e, many GHz to more than a THz. On the other hand, binding energies in the MHz range can be used in the case of weak intercombination line transitions. Consequently we find that optical lengths can be orders of magnitude larger for weak intercombination lines than allowed transitions, since the Franck-Condon factors can be intrinsically much larger once the necessary conditions are satisfied for changing scattering length without major losses.
In summary, we have shown that molecular energy levels close to the 1 S 0 + 3 P 1 dissociation limit of alkaline-earth metal atoms provide optical Feshbach resonance states that allow for a significant change of scattering length even for moderate laser intensities and laser frequencies far detuned from optical resonance, with a relatively small trap loss rate coefficient. A small loss rate coefficient leads to longer observation times. Although we have used calcium as an example system, we expect our conclusions to remain valid for other alkalineearth atoms as well as atoms with similar electronic structure like ytterbium [21]. Optical Feshbach resonances seem to be a promising tool that will allow a tunable interaction strength in atomic systems which do not have magnetic Feshbach resonances.