An array of square current-carrying wires is proposed to construct a two-dimensional (2D) array of surface magneto-optical traps and Ioffe magnetic ones for realizing a 2D lattice of Bose-Einstein condensations (BECs) on an atom chip. Our study shows that when a vertical homogeneous bias magnetic field is only used, the wires will form a 2D array of quadrupole magnetic traps, which can be used to construct 2D lattice of surface magneto-optical micro-traps. While another horizontal homogeneous bias field is added simultaneously, the above 2D array of quadrupole micro-traps will be changed as an array of surface Ioffe micro-traps, which can be used to form a 2D magnetic lattice and then realize a 2D array of BECs. The dynamic loading process of cold 87Rb atoms from each micro-MOT into Ioffe micro-trap is studied by using Monte-Carlo simulations, and our results show that the scheme can be used to realize a natural and effective loading of cold atoms from 2D array of micro-MOTs into 2D array of Ioffe micro-traps, and the loading efficiency can be reach ~65%. Moreover, the positions of each BEC (or MOT) in 2D array of magnetic micro-traps can be controlled by adjusting the currents in the wires or by changing the additional vertical bias field.
© 2006 Optical Society of America
Since the 87Rb and 23Na atomic Bose-Einstein condensations (BEC) were first realized [1–2] in 1995, a series of other atomic BECs (such as 7Li, 85Rb, 41K, 1H, and 4He,) in various macroscopic magnetic traps have been observed [3–8], and several atomic BECs (such as 52Cr, 133Cs and 174Yb) in far-resonance-detuned optical dipole traps (FORT) have also been reported [9–11]. In 2000, Schmiedmayer’s group first demonstrated the trapping and guiding of cold atoms in magnetic micro-structures formed by micro-sized current-carrying wires electroplated on an atom chip . Since then, “atom chip” has become one of hot research points in atom optics and led to the formation of a new field, so called “integrated atom optics”. Recently, the realization of BEC on an atom chip [13–15] has shown a promising future in the fields of quantum atom optics and quantum information science [16–17], and it has also shown some wide applications in manipulation and control of coherent, ultracold matter waves by using integrated atom-optical devices, such as foil-based atom chip , permanent-magnet atom chip, integrated atom interferometers , and magnetic lattice on a micro-chip , double-well mater-wave interferometry on an atom chip , and so on.
Atomic BECs have become a very useful tool to study many aspects of ultracold atomic and quantum physics. In particular, atomic BEC even a lattice of BECs on an atom chip can be used to realize quantum computing and quantum information process. In recent years, some quantum optical and nonlinear physical effects of BECs in 1D, 2D and 3D optical lattices have been studied [23,24]. In 2002, Yin’s group has proposed the first scheme to realize a 2D lattice of atomic BECs by using three-layer, planar current-carrying wires on a chip , but an effective loading question of cold atoms from an array of micro-MOTs into an array of Ioffe micro-traps was not solved well. In 2003, Pfau’s group first demonstrated 2×2 arrays of micro-MOTs by using two-layer, planar current-carrying wires on a chip , but so far no any lattice of BECs on a chip has been realized. Therefore it would be interesting and worthwhile to find a simple and suitable array of surface micro-traps and explore the possibility to prepare a lattice of BECs on an atom chip. It is clear that the loading of cold atoms from each micro-MOTs to each Ioffe micro-traps is a critical problem in such an experiment of 1D or 2D array of BECs.
To resolve well the loading of cold atoms mentioned above, in this paper, we propose a novel and practical scheme to form an array of magnetic quadrupole traps and Ioffe ones by using an array of square current-carrying wires on a substrate combined with one or two bias magnetic fields, and discuss the possibility to realize a lattice of BECs by using our scheme. This paper is organized as follows: in Section 2 a new scheme to realize 2D array of magneto-optical traps for neutral atoms is proposed, and the magnetic field and its gradient distributions of each magnetic quadrupole well are calculated, and the trapped number and temperature of atoms in each micro-MOT are estimated. In Section 3 the transformation of 2D array of micro-MOTs to 2D array of Ioffe micro-traps is present, and the magnetic field, gradient and curvature distributions of each Ioffe well are calculated and analyzed. In Section 4 the dynamic loading process of cold atoms from each micro-MOT into each Ioffe micro-trap is studied by Monte-Carlo simulations, and the corresponding loading efficiency and temperature change are estimated. In Section 5 the experimental practicability and the possibility to realize a 2D lattice of BECs are analyzed, and some potential applications of our chip scheme are briefly discussed. The main results and conclusions are summarized in the final section.
2. Scheme of 2D array of micro-MOTs
Figure 1 shows a new scheme to construct a 2D array of magneto-optical traps for cold atoms by using a 2D array of square (or circular) current-carrying wires. All the wires are fabricated on the surface of a substrate and carry the same current I, and the side length of each square wire is 2a , the distance between two adjacent square-wire (i.e., the spatial period of 2D array of the square wires) is c . The gap e between two wires at the inlet (or outlet) of each square wire is far smaller than a, such as setting e= a/10.
It is well known that the axial distribution of the magnetic field generated by a square current-carrying wire (CCW) is a Gaussian-like profile, so only a square CCW cannot be used to form a 3D magnetic trap for cold atoms. However, if an appropriate and homogeneous bias magnetic-field (B z0) along the -z direction is added, two magnetic quadrupole wells, located symmetrically above and below the wire plate, will be produced. In consideration of the thickness of the substrate (a few 5 mm), only one 2D array of the magnetic quadrupole wells above the substrate can be used to form a 2D array of micro-MOTs by using a three-dimensional (3D) molasses laser beams. In order to trap cold atoms as more as possible, and avoid the diffraction and scattering effects from the substrate, it is desirable to use three pairs of laser beams with flat-Gaussian intensity profiles , even flat-topped laser beams with a rectangle cross-section  to form the 3D optical molasses.
As shown in Fig. 1, when the geometric parameters of the wire array with an area of 30mm×30mm are chosen as 2a = 0.8 mm and c = 0.9 mm respectively, we can obtain a 2D array of square CCWs with 30×30 cells. The spatial distribution of the magnetic field from our wire layout combined with an additional bias magnetic field B z0 along the -z direction is given by
where 2N is the number of the square-wire cells in the x or y direction. From Eqs. (1)–(3), the contours of the magnetic field from our wire layout on the xoz and xoy planes are calculated for I=3A and BZ0=0.8G, and the results are shown in Figs. 2. Each white zone in Fig. 2 presents a magnetic-well center with a zero B value. It can be seen from Fig. 2 that there is a 2D array of magnetic quadrupole wells above the substrate, which can be used to prepare a 2D array of surface micro-MOTs by using the 3D molasses beams. The positions of the trapping centers are x0=0.9 k mm, y0= 0.9kmm, z0=1.62mm respectively, here k is the integer (i.e., k=0, ±1, ±2, ±3, ….).
Our scheme has an interesting characteristic, that is, the trapping center z0 of cold atoms above the surface of the substrate can be easily controlled by adjusting the current I in the wires or by changing the additional bias field Bz0. The dependence of the trapping center position z0 on the three parameters (a, I and Bz0) can be expressed as
From Eq. (4), we calculate the relationships between the trapping center z0 and the current I (or the bias field Bz0), and the result is shown in Fig. 3(a). We can find that with the increasing of the current I in the wire from about 0 to 17.5A, the position z0 of the trapping center will be increased from about 0 to 3mm as Bz0= 0.8 G. On the other hand, with the reduction of the bias field Bz0 from about 1.4 G to 0, the trapping center z0 of cold atoms will be moved upward from about 0 to 3mm as I= 3A. These results show that by adjusting the current I in the wires or by changing the bias field Bz0, the central position of each micro-MOT can be changed by a few millimeters, which is desirable to prepare a 2D array of micro-MOTs on the surface of the substrate by using 3D molasses beams.
To know the characteristics of each micro-MOT, we calculate the spatial distributions of each magnetic well and its field gradients in the x, y, and z directions, and the results are shown in Fig. 3(b). The field gradients near each trapping center for I= 3A and Bz0= 0.8G can be expressed approximately as
This equation shows an approximate dependence of the filed gradient on the parameters (a and I) of our wire layout, and presents an approximate relationship among the three field gradients as follows: which is one of the basic conditions to prepare a standard MOT for neutral atoms. When a= 0.4mm, c= 0.9mm, I= 3A and Bz0= 0.8G, we obtain the field gradient near each well center. It is clear from these results that the field gradients of our wire array are suitable for preparing a 2D array of micro-MOTs by using the 3D molasses beams.
Under the interaction of an atom with a weak light (i.e., I ≤IS, where Is is the saturation intensity of the atom), the steady-state number of cold atoms trapped in each MOT can be estimated by the following equation 
where V is the trapping volume of cold atoms in the trap, kB is the Boltzmann constant, T0 is the room temperature (300K), σ is the cross section due to the collision between the trapped atoms and background thermal atoms, m is the atomic mass, Vc is the maximum capture velocity of atoms in the trap. In our scheme, when I= 3A and Bz0= 0.8G the magnetic field gradients are respectively. For 87Rb atom, when δ = -Γ and choosing I ≈ Is, we obtainV = 0.4mm 3. By using the law of the energy equipartition, we obtain vc= 16.1 m/s. So we obtain the trapped atomic number N= 1.2×105 and atomic density n= 3×108 atoms/cm3 in each MOT, which are estimated by the theoretical model based on the interaction of an atom with a weak light. In fact, the trapped number of atoms in a standard MOT will be up to 107 ~108atoms as I/Is ≫ 1, and the corresponding atomic density will reach 3x1010~ 3x1011 atoms/cm3. From Doppler laser cooling, moreover, we also estimate the temperature of 85Rb atoms in each MOT by using the following Equation 
In our 3D MOT scheme, we should choose q= 3. From Eq. (10), when δ = -Γ and I≈IS, we obtain T ≈ 320μK .
3. Two-dimensional array of Ioffe microtraps
It is well known that in order to realize an atomic BEC, a very tight magnetic trap with a center of a nonzero B-value, so called Ioffe magnetic trap, should be needed. For our 2D array of current-carrying wires, when the original bias field Bz0 is remained and another horizontal homogeneous bias magnetic field Bx0 along the -x direction is added, our 2D array of magnetic quadrupole micro-traps will be evolved as an array of Ioffe magnetic ones. It should be worthy of note here that in the evolution of magnetic-well array, we don’t need to change any microstructure of our wire array, and just need to adjust the current I in the wire from 3A to 15A, and adiabatically change the bias magnetic fields to Bz0= 55G and Bx0= 4G respectively. In this case, the trap centers are located at x 0 = 0.0445+0.9k mm, y 0 = 0.9k mm, and z0= 0.535mm (here k is the integer, i.e., k= 0, ±1, ±2, ±3 …). Figure 4 shows the corresponding magnetic-field contours on the xoy (horizontal) and xoz (vertical) planes. We can find that there are a 2D array of Ioffe micro-traps with a well center of the minimum B-value (Bmin=1.54 G). In addition, we can also find from Fig. 4(b) that the magnetic wells on the xoz plane are slightly tilted relative to the z-axis, which results from the influence of the bias field Bx0.
To know the characteristics of each micro-well, we calculate the spatial distributions of the magnetic fields, gradients and their curvatures in the x, y and z directions, and the results are shown in Fig. 5(a)–(c), respectively. When I= 15A, Bz0= 55G and Bx0= 4G, we obtain Bmin= 1.54G at each trap center, and the corresponding depth of each magnetic well are ΔB(x)max☐ 24G within Δx= x-x0= ± 0.3mm, ΔB(y) max☐ 13G within Δy= y-y0= ± 0.3mm and ΔB(z) max☐ 8G within Δz= z-z0= 0.3mm, where (x0, y0, z0) are the position coordinates of each trap center. The maximal magnetic field gradients are respectively, and we have the curvatures at each well center, which are far greater than those of macroscopic BEC magnetic traps (usually the corresponding maximal gradient and curvature are ~200 G/cm and ~300 G/cm2 respectively, whereas the used typical currents I are about 25 ~300A), and similar to ones of micro-well BEC on an atom chip . So each magnetic micro-well in our array of surface micro-traps is tight enough to realize BEC by using rf-induced evaporative cooling.
Since the magnetic field B at each trap center is non-zero value, the atomic loss from the Majorana transitions can be neglected. In consideration of the interaction between a neutral atom with a magnetic dipole moment μ and an inhomogeneous magnetic field B, the position-dependent trapping potential for cold atoms is given by
where mF is the magnetic quantum number. When μ ∙ B <0, the potential is repulsive, and the atoms in the weak-field-seeking state will be repulsed to the minimum of the magnetic field. For 87Rb atoms in the |F=2, mF=3> state, the trapping potential is U Rb= 67.2|B| (μK), here the magnetic field B is in unit of Gauss. When I= 15A, Bz0= 55G, and Bx0= 4G, we obtain the shallowest depth of the magnetic well ΔB(z) = 8G and U Rb= 537.6μ K, which is far higher than the temperature of cold atoms in a 2D or 3D optical molasses (such as TOm~ 20μK for 87Rb atoms). So each micro-well in our array of surface Ioffe micro-traps is deep enough to collect and trap almost all of cold atoms from the 2D array of micro-MOTs.
4. Monte-Carlo simulations of loading process of cold atoms
One of the important and key steps to realize BEC is the efficient loading of cold atoms from 2D array of micro-MOTs into 2D array of Ioffe micro-traps. We use Monte-Carlo simulations (MCS) to study the dynamic loading process of cold 87Rb atoms from each micro-MOT into each Ioffe micro-trap. Since the process of loading cold atoms is adiabatic, the currents in the wire and the bias magnetic fields should be increased slowly and synchronously. In the simulations, the change of the current in the wire and the bias fields are chosen as
where the units of t, I, Bx0, and Bz0 are second, ampere and Gauss, respectively. Here we prepare five thousand cold 87Rb atoms with a temperature of 20μK in each micro-MOT, and the total loading time is 200ms and the collisions between cold atoms are ignored. Fig.6 shows the evolution of our micro-well magnetic-field contours (left column) from a micro-MOT to an Ioffe micro-trap and the corresponding spatial distribution (right column) of atomic clouds in the xoy and xoz planes at t= 0, 50ms, 100ms, 150ms and 200ms. During the total 200ms evolution time, the current in the wire and the bias fields are changed from I= 3A, Bx0= 0, Bz0= 0.8G to I= 15A, Bx0= 4 G, Bz0= 55G. We can see from Fig. 6 that with the increasing of both the currents I and the bias fields, each micro-well center is moved to near the surface of the substrate, and each micro-well become more and more deep and tight, so the atomic cloud in each micro-trap is greatly compressed during the loading process of cold atoms.
We assume that when t= 0, the cold atomic cloud has a Gaussian spatial distribution and its waist is about 0.4mm, and it is located above the wire plane by a vertical distance of h=1.6 mm when I= 3A, Bx0= 0, Bz0= 0.8G, and when the atomic number is N0= 5000, the atomic density is n0= 2.7 ×107 atoms / cm 3. Figure 7(a) shows the time evolution of the relative atomic number (N/N0) and its atomic density (n/n0) in each micro-well. We can see from Fig. 7(a) that after 200ms, about sixty-five percent of cold atoms are remained in each micro-trap (that is, the loading efficiency of cold atoms can reach 65%), and the corresponding density is increased by one hundred times. If each Ioffe micro-trap is assumed to be harmonic, the oscillation frequency along the ith eigenaxis of a harmonic potential V is given by and the corresponding temperature of cold atoms in the trap can be estimated by Then, the oscillation frequency of our each micro-trap in the three directions can be given by the numerical calculation as follows:
and the mean oscillation frequency of our each micro-trap is Fig. 7(b) shows the dependence of atomic temperatures (Tx, Ty ,TZ and T) on the loading time t. In which, the data points are the results of MCS, and the solid lines are the theoretical predicted results. It is clear from Fig. 7(b) that the MCS results are in good agreement with the theoretical calculated ones. At the first loading time of 20ms, due to the weak magnetic trapping potential, and the expansion rate of atomic clouds is larger than the compression rate of the magnetic well, the loss of outside hot atoms will result in the decrease of the temperature of cold atoms in the trap, which is similar to unforced evaporative cooling process. After that, the potential well becomes very tight, and then the cold atoms are compressed adiabatically and the atomic temperatures are greatly increased by about one order of magnitude.
5. Discussion and potential applications
5.1 Experimental feasibility
It is clear that our wire configuration can be fabricated on the surface of a substrate by using standard micro-fabrication techniques (such as photolithography and electroplating) [30–32]. After our 2D array of the wires is fabricated and inlayed inside the subsurface of the substrate, the surface should be polished and coated by a high-reflection metal film in order to prepare a mirror-MOT and avoid diffraction and scattering of light. In the experiments of atom chip, the frequently used materials are aluminum oxide for the substrate and gold for the wires. Other substrate materials, such as sapphire or diamond and so on, are also available. In 1998, the experiment of Prentiss’s group showed that, for Au wire at room temperature with the temperature difference ΔT= 100K between the wire and substrate, the maximum linear-current density was ~104A/cm. They also investigated the heating effect of Au, Cu and Ag wires on the sapphire substrate and the best results are obtained with Au, for which the area-current density and power dissipation at the temperature of liquid nitrogen (or liquid helium) are ~108 A/cm2 and ~10 kW/cm2, respectively [32,33]. For our scheme (i.e., a planar Au wire layout), we chose the wire diameter d= 50 μm, the current I= 3A for micro-MOTs and I= 15A for Ioffe micro-traps, the maximum current densities (I/d and I/d 2) in each wire are less than 1.5×103 A/cm and 6×105 A/cm 2 respectively, which are both far lower than the used current densities (1×104A/cm and 5 × 106 A/cm 2 ) in Prentiss’s experiment. So the cooling problem of our current-carrying wire can be resolved well by using cold cycled water.
5.2 Possibility to realize a 2D array of BECs
5.2.1 Preparation of 2D array of micro-MOTs
Similar to the most of BEC experiments using double MOT scheme, after a cold sample with about 109–1010 atoms is first prepared in a standard macro-MOT in an upper low-vacuum vapor-cell (10-9 Torr), a low-velocity intense atomic beam with a larger divergent angle is output along the gravity direction from the macro-MOT and propagated into a lower high-vacuum cell (10-11 Torr) by using Lu’s scheme , and the cold atoms are collected, trapped and cooled by our 2D array of surface quadrupole micro-wells (as I= 3A, BZ0= 0.8G, and Bx0= 0G) combined with 3D molasses beams, which is composed of 2D flat-topped beams with a rectangle cross-section (such as an area of 4mm×40mm) in the xoy plane and 1D flat-topped beams with a square cross-section (such as an area of 40mm×40mm) in the z direction, and then a 2D array of surface micro-MOTs located at z0= 1.6 mm above the substrate is prepared in the high vacuum cell (10-11 Torr). In addition, the 2D array of micro-MOTs on the surface of the substrate can also be prepared by using a 2D mirror-MOT technique . In this case, two pairs of σ + -σ - polarized Gaussian beams are impinged at ±45° degree onto the surface of atom chip in the x and y directions respectively, and then they will form 2D optical molasses, which can be used to prepare 2D array of mirror-MOTs by using our 2D array of magnetic quadrupole wells.
5.2.2 Effective loading of cold atoms and realization of 2D array of BECs
After the 2D array of micro-MOTs is prepared, cold atoms are loaded adiabatically from the 2D micro-MOTs into a 2D Ioffe micro-traps by increasing both the current in the wires and the bias fields from I= 3A, Bz0= 0.8G and Bx0= 0G to I= 15A, Bz0= 55G and Bx0= 4G Since this loading process is adiabatic and natural between 2D array of micro-MOTs and 2D array of Ioffe micro-traps, the loading efficiency of cold atoms should be higher, such as 65% (see Fig. 7(a)). After this, cold atoms can be further cooled by 2D or 3D polarization gradient cooling, and compressed adiabatically to a suitable density within n = 1012 ~ 1014 atoms / cm 3 by increasing the current in each wire, and then they will be further cooled to present BEC in each magnetic micro-trap by using the forced rf evaporative cooling.
5.3 Other potential applications
In addition to the realization of 2D array of BECs, our 2D array of magnetic quadrupole micro-wells can be used to form 2D magneto-optical lattice, which can be used to obtain a cold atomic sample with a large number of atoms by the efficient collection of a hollow-beam atomic funnel [35,36]. Such as, when the number of atoms in each micro-MOT is about 107 ~ 108, and there are 30×30MOTs in our 2D array of micro-MOTs, we can obtain a cold atomic sample with a number of about 1010 ~ 1011 atoms, which is very important and useful to prepare an atomic BEC with a larger condensed number, even to realize a real CW atom laser. In the same way, our 2D array of Ioffe micro-traps can also be used to form a 2D magnetic lattice. Since there is no spontaneous emission or no photon scattering effect of cold atoms in the 2D magnetic lattices, a 2D atomic magnetic lattice with a higher atomic density and a larger difference of refraction index can be obtained. So our magnetic lattice may be used to prepare a novel 2D crystal with cold atoms . In addition, the motion of cold atoms in the magnetic lattice may help us to study some nonlinear effects and wave-packet dynamics, quantum transport and tunneling effects, even to realize quantum computing  and entanglement with cold atoms , and so on.
In this paper, we have proposed a new scheme to form 2D arrays of magnetic quadrupole traps and Ioffe traps for cold atoms on a chip by using a 2D array of square (circular) current-carrying wires, and calculated the corresponding distributions of the magnetic field and its gradient and curvature distributions. The dynamic loading process of cold 87Rb atoms from each micro-MOT into Ioffe micro-trap is studied by means of Monte-Carlo simulations. We have also discussed the experimental practicability of our micro-trap array (including micro-fabrication and cooling of 2D array of current-carrying wires) and the possibility to realize a 2D array of BECs (including the preparation of 2D array of micro-MOTs and effectively loading of cold atoms and so on) as well as other potential application , and obtained some main results and conclusions as follows:
- Our 2D array of square (or circular) current-carrying wires, combined with an additional bias magnetic field Bz0 in the -z direction, can produce a 2D array of surface micro-MOTs for cold atoms on an atom chip, which can be used to construct a novel 2D magneto-optical lattice, even to obtain a cold atomic sample with a number of 1010 ~ 1011 atoms.
- If another bias field Bx0 in the -x direction is added, and the bias field Bz0 is remained at the same time, our 2D wire array will form a 2D array of surface Ioffe micro-traps for cold atoms, which can be used to construct a novel 2D magnetic lattice, or to prepare a novel 2D photon crystal with cold atoms, even to study some nonlinear and quantum optics effects, such as wave-packet dynamics, quantum transport and tunneling, and quantum computing and entanglement with cold atoms, and so on.
- Since cold atoms in the 2D array of micro-MOTs can be manipulated and controlled conveniently by changing the current I in the wires or the bias magnetic fields (Bz0 and Bx0), it is easy to realize an effective and natural loading of cold atoms from a 2D array of surface micro-MOTs into a 2D array of Ioffe micro-traps, which can be used to prepare a 2D lattice of BECs on an atom chip by using rf-induced evaporative cooling.
This work was supported by the National Natural Science Foundation of China under Grant Nos.10174050, 10374029 and 10434060, Shanghai Priority Academic Discipline and the 211 Foundation of the Educational Ministry of China.
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