Anomalous Longitudinal Magnetic Field near the Surface of Copper Conductors

We have used ultracold atoms to characterize the magnetic field near the surface of copper conductors at room temperature carrying currents between 0.045 A and 2 A. In addition to the usual circular field we find an additional, 1000 − 10000 times smaller longitudinal field. The field changes its strength periodically with a period of 200− 300 μm. Recently, several experiments have been successful in bringing clouds of ultracold alkali atoms close to the surface of copper current conductors [1]. The clouds have been prepared below and above the critical temperature for Bose-Einstein condensation and in all cases a fragmentation of the atomic distribution has been observed which suggests the presence of an as yet unexplained potential caused by the conductors. In this letter we show that this potential is of magnetic origin and is due to a longitudinal field component which is three to four orders of magnitude smaller than the usual circular field of the conductor. This result demonstrates that ultracold atoms, as provided by a Bose-Einstein condensate can be used as a sensitive probe for magnetic fields. The probing principle is based on the force that is acting on the atoms in a static magnetic field. It is proportional to the gradient of the magnetic field modulus [2] and results in a change of the atomic distribution which can be imaged by standard techniques. The spatial dependence of the magnetic field modulus can be probed by moving the atomic cloud within the magnetic field. This is possible either by using optical forces [3] or by means of magnetic potentials. In the latter case, the sample field, which is to be measured, and the trapping field may superimpose. This complicates the analysis, however, it can also be used to separate different components of the sample field. Fig. 1 shows the setup of our experiment. The field configuration is suitable for selectively observing a longitudinal magnetic field component. In addition to the field generated by the conductor, two homogenous fields are present, one oriented perpendicular to the wire (bias field) and the other oriented parallel to the wire (offset field). The bias field Bbias together with the usual circular field of the conductor form a linear magnetic quadrupole field with a vanishing magnetic field along a line parallel to the wire and separated from its centre by d = μ0 2π I Bbias . Here, I is the current in the conductor. Perpendicular to the line of vanishing magnetic field its modulus increases linearly with the distance and forms a waveguide like trapping potential for ultracold atoms [2]. While the transverse motion of the atoms is confined by the potential, the atoms can move freely along the longitudinal direction. A hypothetical Anomalous Longitudinal Magnetic Field near the Surface of Copper Conductors 2 Figure 1. Trap setup. Two conductors are used: a microfabricated copper conductor path with a width of 30 μm and an ordinary copper wire with a diameter of 90 μm. Together with a perpendicular bias field a waveguide is formed. The offset field generates a non vanishing homogeneous field component in the direction of the conductors. longitudinal field component would generate an additional longitudinal potential that can be detected by observing the atomic distribution along the wire. In fact, such an unexpected field component can be observed in our experiments. It is 3 to 4 orders of magnitude smaller than the usual circular field but still strong enough to trap the atoms in the longitudinal direction. The atomic distribution shows a pronounced periodicity of 200−300μmwhich corresponds to a similar modulation of the anomalous field component at the location of the atoms. We have used Rubidium atoms to study the magnetic field of two different types of copper conductors. The first conductor has a rectangular cross section with a height of 2μm and a width of 30μm. It has been electroplated onto an aluminium oxide substrate as described in detail elsewhere [4]. It is operated with a current between 45 mA and 0.5 A corresponding to a current density between 5·10 A cm and 5.6·10 A cm , respectively. The bias field amounts between 2 G and 22 G. The second conductor is an ordinary copper wire as used for electronic circuits with a diameter of 90μm. It carries 0.3 A to 2 A and is operated with the same bias field. To initially trap and cool the atomic cloud we apply an offset field along the longitudinal direction that is slightly inhomogeneous and forms a parabolic trapping potential in which the atoms oscillate with a frequency of 14 Hz [5]. Here, a cloud of 10 atoms is prepared at a temperature of about 1μK. The trapping and cooling procedure is described elsewhere [6]. By turning off the longitudinal trapping potential within 400 ms and keeping only a homogenous offset field of 1.3 G, the atoms are released into the wave guide for 100 ms where they are exposed to the anomalous longitudinal field generated by the wire or by the microstructure, respectively. Fig. 2a shows the density distribution of the atoms near the surface of the microstructure at a current of 0.045 A and a bias field of 2 G. Better imaging is possible 10 ms after all magnetic fields have been turned off (Fig. 2b). In this time the atoms have been separated from the surface while falling under gravity. We observe a spatial modulation of the density distribution with a periodicity of 260±15μm. Similar properties are found for the wire (Fig. 3.), however with less pronounced periodicity. Nevertheless, the fourier spectrum of the density distribution shows a clear peak at 220μm. The magnetic origin of the anomalous surface potential can be shown by changing the orientation of the offset field Boff . The atomic density distribution is now inverted with the maxima transformed into minima and vice versa (Fig. 4). To explain this Anomalous Longitudinal Magnetic Field near the Surface of Copper Conductors 3 Figure 2. In a waveguide near the copper conductor the density distribution of the atom cloud shows a periodic fragmentation. a) shows the cloud in the initial waveguide. The dashed line indicates the surface of the microstructure. To prepare the atoms in this trap the axial confinement was ramped down in 400 ms. b) shows the atoms after 10 ms time of flight. The periodic structure appeares more clearly. In c) a integrated scan of b) is plotted. Figure 3. The periodic structure at the 90 μm copper wire after 5 ms time of flight. The current in the wire, the bias field, and the offset field are 0.2 A, 6 G and 2.6 G respectively. effect, we assume an anomalous magnetic field Bz(z) and add it to Boff . This leads to a total field given by ±Boff +Bz(z) with the sign of Boff depending on the orientation of the offset field. The resulting potential is proportional to the modulus of the total field |±Boff +Bz(z)| which is equivalent to |Boff ±Bz(z)|. Therefore, flipping the offset field has the same effect as inverting the surface potential. In the case of a non magnetic potential, the potential would simply add to the magnetic potential that is generated by the offset field and inverting its orientation would have no effect. The same applies for any magnetic field component which does not have the same direction as the offset field. If Bz(z) consists of a constant field B0 and a modulated field Bmod(z), the constant part adds to Boff with opposite sign in the two cases. Anomalous Longitudinal Magnetic Field near the Surface of Copper Conductors 4 Figure 4. The position of the potential minima change when the direction of the offset field is changed. In a) the atoms are prepared in a waveguide by ramping down the axial confinement in 400 ms. The current in the microfabricated conductor is 0.045 A. Together with a bias field of 2 G a waveguide is formed at a distance of 45 μm from the surface. The offset field parallel to the conductor is 1.3 G. The vertical dashed lines mark the position of the potential maxima. In b) the atoms are initially prepared in the same trap as in a). Next the orientation of the offset field is flipped within 1 ms. After further 50 ms the atoms are located in the position of the former potential maxima. However, the change in sign of Bmod(z) still transforms minima into maxima and vice versa. To investigate the origin of the modulated field we have changed the orientation of the current in the conductor (and the bias field) while holding the offset field constant. Again the minima where transformed into maxima and vice versa. This shows, that the anomalous magnetic field is caused by the current in the wire rather than by permanent magnetic inhomogeneities of the conductor. To complete this type of experiment we flipped the orientation of all involved elements: the current orientation in the conductor, the bias field and the offset field. Then the original distribution was restored. The observed dependence on the current provides a simple method to eliminate the structuring effect of the surface potential on the cloud. By periodically inverting the current and the bias field on a fast time scale the atoms experience only a time avaraged potential with a strongly reduced spatial dependence. In order to estimate the strength of the anomalous magnetic field component we further cool the atomic cloud below the critical temperature where it undergoes transition into a Bose-Einstein condensate [7]. In contrast to the experiments described above we now keep the atoms trapped longitudinally. By changing the bias field the condensate can be shifted relative to the surface of the conductor. At large distances from the surface the anomalous surface potential is small and does Anomalous Longitudinal Magnetic Field near the Surface of Copper Conductors 5 not affect the condensate. However, at a critical distance the condensate splits into two components. This occurs if the centre of the external harmonic trap is placed at the position of a maximum of the surface potential. Then the total potential shows a double well structure with a barrier height that is comparable with the depth of the surface potential. In fact, for a sinosoidal surface field with a period of 220μm and an amplitude of Bmod(z), superimposed with the field of the harmonic trap of Btrap(z) = 84G/cm 2 · z one finds a ratio between the amplitude and the barrier hight of 2. The barrier can split the condensate if its height exceeds the chemical potential of the condensate. Since the chemical potential can easily be measured by standard time of flight methods [8] it provides a good estimate for the depth of the combined potential. This value represents a lower bound for the depth of the surface potential. A deviation of a pure sinosoidal shape can result in a change of the potential depth in one order of magnitude. For the wire with a current of 0.9 A, a bias field of 16.5 G and an offset field of 2.6 G we observe the splitting of the condensate at a distance of 109μm from the surface. The chemical potential is determined to be 130 nK resulting in an estimation for the minimum depth of the surface potential of 1.7 · 10 J. This corresponds to a magnetic field of 2 mG which is more than two orders of magnitude smaller than the field strength of the circular field component of 16 G at that distance. Qualitatively, the potential depth increases monotonically with the current, and decreases with the distance to the surface. We can further extend the analysis if we assume the Ansatz U = C̃ · I · d for the strength of surface potential U at a distance d form the surface and a current I in the conductor. Here, C̃ is a constant only depending on z and q is a real number. To find the value of q we have determined the distance at which the two condensate parts are separated by 80μm. For five currents in the wire ranging from 0.3 A to 1.2 A this distance increases from 80μm to 109μm. For each data pair we have also determined the chemical potential of the condensate and used it as a measure for the height of the potential barrier. Since, to a good approximation, the potential barrier is proportional to the depth of the surface potential we can use the above Ansatz to describe the data. The χ-parameter of a least square fit for q = 1, q = 2, and q = 3, with varying C̃ amounts to 140, 5.5, and 11, respectively. Best agreement is found for q = 2.2 with a χ-parameter of 5.49, giving a preference to a value around 2. Repeating the experiment with the microstructure leads to a different constant C̃. This difference can mostly be eliminated by replacing the current I with the product of the current density j and the width w of the conductor. In the case of the wire the radius is taken for the width. Then one arrives at a heuristic expression for U = C · j · w · d, with C ≃ 5 · 10mJ/A. This suggests that the field is generated at the surface rather than inside the conductor since the product j ·w may be interpreted as being proportional to the amount of surface current as “seen” by the atoms. Furthermore, the data recently reported from other experimental research groups seem to fit with the expression [9]. It is not possible to find a similar expression with d interpreted as the distance to the centre of the conductor. Such an approach does not allow a consistent description of all known observations. For the development of a model that describes the physical origin of the anomalous field it will be important to explain the observed periodicity of several 100μm of the anomalous field component. It is most clearly observed in the microfabricated wire and seems to be almost independent on the geometry of the conductor. One may speculate that the fabrication process plays a role. Imperfections in the conductor may inhibit or destroy the periodicity. Candidates for a possible explanation are kink Anomalous Longitudinal Magnetic Field near the Surface of Copper Conductors 6 instabilities of the current as known from plasma physics. Even more speculative are current induced spin orientations at the surface. Important experimental data are to be expected soon from experiments carried out with gold conductors [10].