On the stability of optical lattices

arXiv:physics/0412072v1 [physics.atom-ph] 13 Dec 2004 On the stability of optical lattices G. Di Domenico a,∗, N. Castagna a, M.D. Plimmer a, P. Thomann a, A.V. Taichenachev b,c, V.I. Yudin b,c a Observatoire cantonal, rue de l’Observatoire 58, 2000 Neuchâtel, Switzerland b Novosibirsk c Institute State University, Pirogova 2, Novosibirsk 630090, Russia of Laser Physics SB RAS, Lavrent’eva 13/3, Novosibirsk 630090, Russia Abstract In this article, we present an analysis of the stability of optical lattices. Starting with the study of an unstable optical lattice, we establish a necessary and sufficient condition for intrinsic phase stability, and discuss two practical solutions to fulfill this condition, namely minimal and folded optical lattices. We then present a particular example of two-dimensional folded optical lattice which has the advantage of being symmetric, power recycling and having a convenient geometry. We have used this lattice for laser collimation of a continuous cesium beam in a fountain geometry. Key words: Optical lattice, Laser cooling, Trapping PACS: 32.80.Pj 1 Introduction It is well known that the energy levels of an atom interacting with an electromagnetic field undergo an a.c. Stark or light shift [1,2], proportional to the local light intensity. In a standing wave, the ground state light shift gives rise to a periodic potential, called an optical lattice, which can be used to trap the atoms [3,4,5]. The first experiments demonstrating the trapping of atoms in the potential wells of an optical lattice and their localisation therein were performed by the groups of Grynberg [6] and Jessen [7]. Since then, studies of laser cooling, quantum state preparation, and Bose-Einstein condensates in optical lattices have intensified. ∗ Corresponding author. Email address: [email protected] (G. Di Domenico). Preprint submitted to Elsevier Science 20 August 2018 In this article, we present a general analysis of the stability of optical lattices. After a short presentation of the instability problem, we establish a necessary and sufficient condition for intrinsic phase stability. Then we discuss two practical solutions to guarantee intrinsic phase stability: the first of these, minimal optical lattices, is the solution proposed by Grynberg et al. in 1993 [8], while the second, folded optical lattices, was suggested by Rauschenbeutel et al. a few years later [9]. Finally, we discuss both approaches and describe a laser cooling experiment in which we have used a two-dimensional (2D) folded optical lattice for the collimation of a continuous atomic beam. In the following, we consider an optical lattice resulting from the superposition of l laser beams. The total electric field is given by EL (r, t) = Re [EL (r) exp(−iωL t)] where EL (r) = l X Ej ~εj exp [i(kj · r + φj )] , (1) j=1 φj being the phase of the jth laser beam. As explained in Refs. [3] and [10], the optical shift operator for atoms in the ground state is given by Û (r) = −E∗L (r) · α̂ · EL (r) (2) where α̂ is the atomic polarisability tensor operator given by α̂ = − e d̂ge d̂eg /~∆ge where ∆ge is the laser detuning with respect to the atomic transition |gi → |ei and d̂eg is the electric dipole operator between these levels. By inserting Eq. (1) into Eq. (2) we get P Û (r) = − X (~εi∗ · α̂ · ~εj )Ei∗ Ej i,j × exp [i(φj − φi )] exp [i(kj − ki ) · r] , (3) which is our starting point for the stability analysis. Although in the following stability analysis we concentrate on the optical shift operator (2), all other field dependent operators of the problem (e.g. the optical pumping rate operator) are also determined by quadratic combinations of the type (3). Therefore, the stability conditions will be the same for these operators too. 2 k4 k1 k2 y k3 x Fig. 1. Example of an unstable optical lattice of dimension 2. 2 Problem of stability From Eq. (3), it is clear that the relative phases φj − φi play a critical role. Any variation in one of these phases arising, for example, from the vibration of a mirror, may manifest itself as a dramatic change in the optical potential. To illustrate this problem, let us consider the 2D optical lattice shown in Fig. 1, where four laser beams intersect in a common plane, all linearly polarised within that plane. The total electric field is given by the superposition of the electric fields of the four laser beams (with identical amplitude E): EL (x, y) = E [ ~εy exp(ikx + iφ) + ~εy exp(−ikx) + ~εx exp(iky) + ~εx exp(−iky) ] (4) where we have introduced the variable φ to represent a change in the phase of the first laser beam. Grouping terms with identical polarisation, we obtain: EL (x, y) = 2E [ ~εx cos(ky) + ~εy exp(iφ/2) cos (kx + φ/2)] . (5) This expression shows that the total polarisation is critically dependent on the angle φ. Indeed, if we calculate the intensity of circular polarisation components I± (x, y) = | ~ε±∗ · EL (x, y)|2 where ~ε± = √12 (~εx ± i~εy ) we obtain, for φ = π: I± (x, y) = 2E 2 [cos(ky) ∓ sin(kx)]2 (6) and for φ = 0: h i I± (x, y) = 2E 2 cos2 (ky) + cos2 (kx) . (7) We have plotted these circular polarisation components in Fig. 2. These two situations are very different. 3 Fig. 2. Representation of the polarisation gradients for the 2D optical lattice shown in Fig. 1. (a) Case where the phase of the first laser beam is φ = π. We have plotted the intensity of circular polarisation components I± (x, y) from Eq. (6). In this case, sites of pure σ+ and σ− polarisation are in alternance every half-wavelength. (b) Case where the phase of the first laser beam is φ = 0. On the left we have plotted I± (x, y) from Eq. (7). In this case I+ (x, y) ≡ I− (x, y), therefore the polarisation is linear everywhere. However, there is still a strong polarisation gradient, as can be seen on the right graph where we have plotted the polarisation vector field EL (x, y) in the lattice plane. For φ = π, we observe alternating sites of pure σ+ and σ− polarisation with a period equal to the wavelength. This means the light shifts of mF = ±F Zeeman sub-levels are periodic in space with opposite phases, and the same is true for the optical pumping process which always populates the ground state mF sub-level of lowest energy. As a consequence, Sisyphus cooling [11] can take place if the laser beams are tuned correctly. On the contrary, for φ = 0 we have I+ (x, y) ≡ I− (x, y), both having a period equal to a half-wavelength. The polarisation is thus linear everywhere, which precludes Sisyphus cooling. However, there is still a strong polarisation gra4 dient, as can be seen in the right graph of Fig. 2(b) where we have plotted the polarisation vector as a function of position in the optical lattice plane. Thus, it is probable that another sub-Doppler cooling mechanism takes place in this situation, for example a mechanism similar to that operating in σ + -σ − molasses [11]. We conclude that the instability of the optical lattice leads to dramatic changes of the polarisation gradients. This is a crucial problem when one works with cooling mechanisms involving the spatial dependence of light polarisation. One solution to this problem is to stabilize mechanically the phase difference between laser beams, as was first implemented by Hemmerich et al. [12]. However, other approaches free of this mechanical constraint were proposed by the groups of Grynberg [8] and Meschede [9]. We shall discuss both of these approaches in sections 4 and 5, but first we start by defining intrinsic phase stability and by establishing a necessary and sufficient condition for it. 3 Necessary and sufficient condition for intrinsic phase stability Let us consider an optical lattice composed of l laser beams, and let us suppose that the phases of these laser beams change suddenly as follows φj → φj + ∆φj . (8) From equation (3), we see that it is possible to compensate the effect of this change by a translation r → r − ∆r, provided there exists a vector ∆r and an arbitrary phase φo (which has no effect on Û(r)) such that kj · ∆r = ∆φj + φo , ∀j = 1, . . . , l . (9) This brings us to the following definition, namely, we say that an optical lattice is intrinsically phase stable if any change in the phases of the laser beams can be compensated by a formal translation, as explained above. In practice, this means that every change in the phases of the laser beams manifests itself as a physical translation of the optical lattice in space. Such translations do not disturb the optical cooling mechanisms as long as the atoms’ internal variables evolve much more rapidly than the optical potential, which is usually the case. Let us state now an important result. We denote by K the matrix composed of the lines (kj − k1 )T for j = 2, . . . , l, and Φ the vector composed of the elements ∆φj − ∆φ1 for j = 2, . . . , l. We recall that the rank of a matrix is equal to the dimension of the vector space generated by its columns or by its 5 rows. Then, the optical lattice resulting from the superposition of the l laser beams is intrinsically phase stable if and only if rank(K) = rank (K|Φ) , ∀Φ (10) where (K|Φ) is the matrix obtained from K by adding a column composed of the elements of Φ. The proof of this result is as follows. It is clear that the optical lattice is intrinsically phase stable if and only if the linear system (9) always admits a solution, for any choice of the phase variations ∆φj . By subtraction of the first equation, we obtain the following linear system {(kj − k1 ) · ∆r = ∆φj − ∆φ1 | j = 2, . . . , l} (11) which is equivalent to the system (9). The system (11) can be written in matrix form as K∆r = Φ. For this equation to have at least one solution, it is both necessary and sufficient that rank(K) be equal to rank (K|Φ) (this is the Rouché-Capelli theorem for the existence of the solution of a linear system). Here ends the proof. 4 Minimal optical lattices Let us consider an optical lattice composed of l laser beams, and let us denote d its spatial dimension. We are going to demonstrate that d = l − 1 is a sufficient condition to guarantee the intrinsic phase stability of the optical lattice. Let us start by a demonstration of the following properties: (1) d = rank(K); (2) rank(K) ≤ rank (K|Φ); (3) rank (K|Φ) ≤ l − 1. As can be seen from Eq. (3), the optical lattice is generated by the vectors (kj − ki ), thus its spatial dimension d is equal to the dimension of the vector space generated by the vectors (kj −ki ). But this vector space is also generated by the vectors (kj − k1 ) which compose the matrix K. Therefore, by definition of the rank, d is equal to rank(K), which proves the first property. The second property comes from the trivial assertion that adding a column to a matrix cannot decrease the rank. Finally, the rank of a matrix cannot exceed the number of lines, and this proves the last inequality. 6 k4 k4 k3 k3 k2 k3 k2 k1 k2 k1 (a) k1 (b) (c) Fig. 3. Comparison of optical lattices of dimension d = 2: (a) folded optical lattice; (b) unstable optical lattice; (c) minimal optical lattice (with d = l − 1). Writing these three properties side by side, we get: d = rank(K) ≤ rank (K|Φ) ≤ l − 1 . (12) From this expression, it is obvious that by imposing the condition d = l − 1 we guarantee that rank(K) = rank (K|Φ) and therefore that the optical lattice is intrinsically phase stable. This is the solution proposed by Grynberg et al. in 1993 to build stable optical lattices [8]. Note that Eq. (12) implies l ≥ d + 1. Therefore, l = d + 1 is the minimum number of laser beams needed to create an optical lattice of dimension d, hence the term minimal optical lattice. At this point, it is important to note that the condition d = l − 1 is sufficient, but not necessary, to have rank(K) = rank (K|Φ). There is another method to obtain an optical lattice which is intrinsically phase stable. It is described in the next section. 5 Folded optical lattices Let us consider the 2D optical lattice geometry presented in Fig. 3(a). This optical lattice is intrinsically phase stable, even though it does not satisfy the condition d = l − 1. To explain this point, we start by considering the optical lattice of Fig. 3(c) which is composed of the first three laser beams k1 , k2 and k3 . This optical lattice is intrinsically phase stable since it satisfies the condition d = l − 1. Therefore, we have  T  k2 rank −kT1 kT3 −kT1     T  k2 = rank −kT1  ∆φ2 −∆φ1  kT3 −kT1 ∆φ3 −∆φ1  =2. (13) Let us now reconsider the optical lattice of Fig. 3(a). Since the retro-reflected beam follows the same path as the incident beam, the phases satisfy the rela7 tion φ4 − φ3 = φ2 − φ1 . Therefore, the differences φj − φ1 are linked via (φ4 − φ1 ) = (φ3 − φ1 ) + (φ2 − φ1 ) . (14) On the other hand, the differences kj − k1 are related by (k4 − k1 ) = (k3 − k1 ) + (k2 − k1 ) . (15) Since the linear combinations (14) and (15) are identical, we have  T  k2 −kT1   T T rank  k3 −k1   ∆φ2 −∆φ1    ∆φ3 −∆φ1   kT4 −kT1 ∆φ4 −∆φ1    T T  k2 −k1 ∆φ2 −∆φ1  = rank kT3 −kT1 ∆φ3 −∆φ1 (16)  and thus rank (K|Φ) = 2. Now, using Eq. (12) with d = 2, we can conclude that rank(K) = rank (K|Φ) is always satisfied, and therefore the optical lattice of Fig. 3(a) is intrinsically phase stable. The idea of using this type of intrinsically phase stable configuration was initially put forward by Rauschenbeutel et al. in a slightly different form [9]. Their explanation for stability is more intuitive and consists in observing that the optical lattice of Fig. 3(a) is created by folding a 1D lattice such that it intersects with itself. Since 1D lattices are instrinsically stable, as discussed above, folded ones must be too. One can also say that the stability is preserved while we add a fourth laser beam because the phase and wave vector of this laser beam are related to the phases and wave vectors of the other laser beams by the same linear combination, as shown by Eqs. (14) and (15). Indeed, if we consider the configuration of Fig. 3(b), the relation (15) is still satisfied, but the relation (14) is not since the phases are all independent. Therefore, we have rank(K) = 2 < rank (K|Φ) = 3 and the optical lattice is not intrinsically phase stable. 8 k6 k3 k3 k4 k1 k2 k5 k2 z y k4 x (a) k1 z y x (b) Fig. 4. Three dimensional optical lattices: (a) minimal (with d = l − 1); (b) folded. 6 Discussion 6.1 Optical lattices in 1D and 3D Although all the examples given above were 2D lattices, everything we have said is still true in other dimensions. For dimension d = 1, the minimal and folded optical lattices are degenerate and correspond to the usual optical molasses. In this case, the intrinsic phase stability is obvious, even without the above matrix analysis, because any displacement of the retro-reflecting mirror automatically shifts the phase of the standing wave by a corresponding amount since the electric field has a node on the mirror surface. For dimension d = 3, the minimal and folded optical lattices are generalizations of the two-dimensional case. An example of a minimal optical lattice in three dimensions is presented in Fig. 4(a). This configuration has been used by Treutlein et al. for degenerate Raman sideband cooling [13]. Although the geometry was not symmetrical, radiation pressure was reduced by using a large detuning. Other examples of minimal optical lattices are discussed in Ref. [14]. An example of a three-dimensional folded optical lattice is presented in Fig. 4(b). This configuration is obtained from configuration (a) by adding two laser beams, namely k5 and k6 . It is easy to show that the phases and wave vectors of these two beams are related to the other beams by the same linear relations: φ5 − φ1 = (φ4 − φ1 ) + (φ3 − φ1 ) − (φ2 − φ1 ) φ6 − φ1 = (φ4 − φ1 ) + (φ3 − φ1 ) (17) (18) k5 − k1 = (k4 − k1 ) + (k3 − k1 ) − (k2 − k1 ) k6 − k1 = (k4 − k1 ) + (k3 − k1 ) . (19) (20) and 9 Therefore the optical lattice is intrinsically phase stable. 6.2 Practical realisations of optical lattices Work on cold atoms usually requires one to employ a symmetrical beam configuration in order to avoid atoms being pushed aside by radiation pressure. Minimal optical lattices can be designed in a symmetrical geometry, but this requires a complex vacuum system. For the 2D case, this means using 3 beams intersecting at 120◦ and a hexagonal coplanar geometry for the vacuum system. To create a symmetrical 3D minimal lattice, the 4 beams should form a regular tetrahedron. This adds even further to the complexity of the vacuum apparatus (see Ref. [15] for an example of a tetrahedral magneto-optical trap). Folded lattices, on the other hand, involve more beams but have a more userfriendly geometry with beams intersecting at right-angles. They can be aligned by auto-collimation and they have the inherent advantage of balanced radiation pressure. In addition, it is straightforward to adapt them to a power recycling geometry, a tremendous advantage in many cases. 6.3 Atomic beam collimation with a folded lattice In a recent experiment, we have used the 2D folded optical lattice of Fig. 3(a) to perform the collimation of a continuous cesium beam in a fountain geometry [16]. In this folded lattice, we have realized Zeeman-shift degenerate-Ramansideband cooling in a continuous mode. This powerful cooling technique allowed us to reduce the atomic beam transverse temperature from 60 µK to 1.6 µK in a few milliseconds. Remark that in this context, power recycling is a big advantage since a high power is necessary to create a far off-resonance optical lattice. With the same experimental setup, we also realized collimation of the continuous cesium beam using Sisyphus-like cooling in a 2D optical lattice. We have experimented with the lattice configurations (a) and (b) of Fig. 3 and the results are summarized in table 1. The best collimation has been obtained with the folded optical lattice. 6.4 Multi-color optical lattices Before concluding, we should like to point out that fulfilling the condition (10) is by no means the only solution to get rid of the instability problem. Another 10 Table 1 Summary of transverse temperatures obtained in the collimation experiment with Sisyphus-like cooling in lattice configurations (a) and (b) of Fig. 3. See subsection 6.3 for details. Optical lattice configuration Transverse temperature (µK) (a) Folded 3.6(2) (b) Unstable 7.3(5) possibility is to average over the phase difference. To illustrate this, consider the unstable 2D optical lattice of Fig. 1. If the two molasses have different laser frequencies, the phase difference changes rapidly, and the atoms see an optical lattice which is the average of the optical shift over the phase variable. This solution has been used with success by other groups [17,18]. 7 Conclusion In this article we have established a necessary and sufficient condition for the intrinsic phase stability of an optical lattice. We have presented two practical solutions to fulfill this condition, namely minimal and folded optical lattices. We have shown that the minimal optical lattices, introduced for the first time by Grynberg et al. in 1993, are sufficient but not necessary for stability. 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