Magnetic microtraps for ultracold atoms

Trapping and manipulating ultracold atoms and degenerate quantum gases in magnetic micropotentials is reviewed. Starting with a comprehensive description of the basic concepts and fabrication techniques of microtraps together with early pioneering experiments, emphasis is placed on current experiments on degenerate quantum gases. This includes the loading of quantum gases in microtraps, coherent manipulation, and transport of condensates together with recently reported experiments on matter-wave interferometry on a chip. Theoretical approaches for describing atoms in waveguides and beam splitters are briefly summarized, and, finally, the interaction between atoms and the surface of microtraps is covered in some detail.

Figure 1

A simple magnetic waveguide (side guide). A current conductor is placed in a homogeneous bias field resulting in a line of vanishing magnetic field (dashed line). This forms the center of a quadrupole waveguide for paramagnetic atoms.

Figure 2

Wire layout for a U trap (a) with a quadrupole potential and a $\mathrm{Z}$ trap (b) with a harmonic potential (Ioffe trap).

Figure 3

Atom guide using two current carrying wires and a bias field. With the bias field increasing, two guiding regions move towards each other along $y$ (dotted line) until they coalesce at $y=A=d∕2$ with $d$ being the distance between the two wires. They then separate along the dashed semicircle. Adapted from 102.

Figure 4

Two wires with opposite currents in a vertical bias field.

Figure 5

Transformation of a quadrupole trap into an Ioffe trap. (a) The setup consists of a pair of coils operated in the anti-Helmholtz configuration and a Ioffe wire. (b) The spherical quadrupole field generated by the coils is superimposed with the circular field of the wire. (c) In the symmetry plane between the coils two spherical quadrupole minima appear (QP1 and QP2). With increasing current in the wire the two minima approach each other on a semicircular trajectory and merge eventually to form an Ioffe trap.

Figure 6

Position of the spherical quadrupole minima for the geometry in Fig. 5. The center of the quadrupole generated by the coils is at $y=0$ and $x=-d∕2$. The wire is at $y=0$ and $x=d∕2$. Adapted from 61.

Figure 7

Absorption images at various stages of the adiabatic transfer of atoms into the microtrap. The microtrap is generated at the surface of a thin wire ($90\mathrm{\mu }\mathrm{m}$ diameter) carrying a current of $1.4\mathrm{A}$. The thin wire (indicated by the dashed line) is attached to the surface of the Ioffe wire ($1.4\mathrm{mm}$ diameter, to the right of the dashed line). Atoms are initially loaded into the spherical quadrupole trap QP1 with zero current in the Ioffe wire. Within the next $320\mathrm{ms}$, the trap is adiabatically transferred into the Ioffe trap $(I=9.5\mathrm{A})$. The microtrap is loaded from QP2 within the following $320\mathrm{ms}$ when the current in the Ioffe wire is reduced to zero. Here 14% of the atoms is loaded into the microtrap. The rest goes back to QP1 and is visible to the left. From 58.

Figure 8

Configurations for atom guides with internal bias field. From 25.

Figure 9

Y junctions based on (a) a single-wire and (b) a double-wire geometry. From 25.

Figure 10

(Color online) Equipotential surface of a Y junction made from a single-wire splitting. From 25.

Figure 11

(Color online) Y junction on a chip: (a) Chip schematic. The two large U-shaped, $200\mathrm{\mu }\mathrm{m}$ wide wires are used to load atoms into the $10\mathrm{\mu }\mathrm{m}$ wide Y-shaped guide. (b) Fluorescence images of guided atoms. In the first two images, the current is driven through a single arm of the Y junction, thereby guiding atoms either to the left or to the right. The last two images were taken for two different radial confinements with equal currents in the two arms of the Y junction. The expansion in the waveguide was driven by the thermal velocity distribution of the atoms. From 26.

Figure 12

X junctions and optical couplers for matter-waveguides. Depending on the strength of the bias field two approaching guides can completely merge their potentials. If the bias field is sufficiently large and a residual barrier is left, the atoms may escape the waveguide by quantum tunneling. From 25.

Figure 13

Switch schematic (not to scale). Insets (a) and (b) show the magnetic field contour lines for each region. The switch starts and ends with a center guide (a). In the middle of the switch, atoms are transferred from a center guide to a side guide (b). Depending on which one of the two auxiliary wires 1 and 2 is turned on, the atoms exit port 1 or 2, respectively. From 157.

Figure 14

Magnetic lattice with alternating currents in neighboring wires.

Figure 15

Schematic of field-induced adiabatic potentials $V_{\pm }$ for $({\omega }_{\mathrm{rf}}>{\omega }_L)$. Dashed curves show the harmonic bare potentials crossing at $r_c$. Inset: coupling of bare potentials at $\pm r_c$ by rf radiation (arrows). From 240.

Figure 16

(Color online) Wire layout of the atomic conveyor belt by 177 and 84. Inset: The relevant part of the conductor pattern. $I_0$, $I_1$, $I_{M1}$, and $I_{M2}$ create the various magnetic potentials for trapping and transport. Adapted from 84.

Figure 17

Chip for three-dimensional positioning of atomic clouds. (a) Schematic plot of the dual-layer microstructure with conductors at both surfaces of the substrate and contact pads on the top. (b) Central part of the chip. Indicated are the conductors QP1, QP2, and QP3 on the top producing the three-wire guide and the transport wires $T1–T8$ on the back surface of the chip. The eight transport wires are periodically repeated underneath the chip. Electrical connection between the separate blocks of $T1–T8$ is achieved by running the wires through laser cut holes from underneath to the top surface of the chip and back. The additional wire pattern between QP1 and QP2 as well as between QP2 and QP3 can be used for generating additional microtraps as described by 126. From 76.

Figure 18

Temperature evolution of a $5\mathrm{\mu }\mathrm{m}$ wide and $1.4\mathrm{\mu }\mathrm{m}$ thick wire mounted on a $700\mathrm{\mu }\mathrm{m}$ thick Si substrate with a $500\mathrm{nm}$ ${\mathrm{SiO}}_2$ insulating layer (interface). The thick solid curves show measured data for 0.6, 0.5, and $0.3\mathrm{A}$ current pulses. In one case $\left(0.5\mathrm{A}\right)$ the theoretical predictions (without fitting parameters) are also shown. The initial fast temperature increase (thin dash-dotted curve) occurs on a microsecond time scale. The analytical model for the heat transport through the substrate (thin solid curve) holds only as long as the approximation of an infinite plane substrate is valid. A two-dimensional numerical model (dashed curve) accurately reproduces the measurements. Adapted from 75.

Figure 19

Scanning electron microscopy images of microconductors with geometries for trapping atoms as suggested by 228 and fabricated by 44. The gold wires ($w=3\mathrm{\mu }\mathrm{m}$ and $h=1\mathrm{\mu }\mathrm{m}$) are fabricated on sapphire substrates. (a) Two half loops. (b) Three concentric half loops. From 44.

Figure 20

Electron micrograph of a typical microstructure, showing a segment of the electroplated wire geometry. The wire widths are 30, 11, and $3\mathrm{\mu }\mathrm{m}$, and the nominal separation between wires is $1\mathrm{\mu }\mathrm{m}$. The image shows the grained structure of the electroplated wires, which is here partially due to the roughness of the sapphire substrate $(\sim 500\mathrm{nm})$. From 64.

Figure 21

Imperfections of the geometry of an electroplated wire. (a) Scanning electron microscopy (SEM) image from the side and (b) from the top. (c) $J_f$ is the spectral density of the spatial fluctuation of the wire edge as derived from the SEM image. From 51.

Figure 22

Microscope images of chip details, during and after the fabrication. (a) Scanning electron microscopy image of the photoresist structure. Its thickness is $4.5\mathrm{\mu }\mathrm{m}$; the undercut is $0.6\mathrm{\mu }\mathrm{m}$. (b) Scanning electron microscopy image of a typical fabricated wire. (c) 1, 5, and $10\mathrm{\mu }\mathrm{m}$ wide gold wires on a fully fabricated chip. (d) Atomic force microscopy picture of the gold surface. The grain size is $50–80\mathrm{nm}$. From 75.

Figure 23

Schematic view of the splitting region. Parameter ${\beta }_x$ increases from the lower left to the upper right passing through ${\beta }_x=1$ in the center when two waveguides merge into one. From 214.

Figure 24

Several lowest eigenfrequencies of the splitting region versus transverse bias field for $\epsilon =0.01$ and ${\beta }_z=0.1$. From 214.

Figure 25

Output population of the lowest mode $(0,0{)}_u$ as a function of the relative phase shift for several values of the nonlinearity parameter $P$. From 214.

Figure 26

Schematic view of the splitting region for an optical coupling case. From 214.

Figure 27

Input and output intensities in the left and right arms of the beam splitter. From 214.

Figure 28

Probability for exit in the various arms of the beam splitter vs incident atom velocity. This assumes that the atoms have entered in the lowest mode of the left arm. The velocities shown correspond to atom energies below the second threshold, thus suppressing higher-mode excitations. The bottom panel is a $26\mathrm{\mu }\mathrm{K}$ Gaussian convolution of the one above. From 17.

Figure 29

Guided matter-wave Mach-Zehnder interferometer. (a) Two beam splitters are combined to form the interferometer. (b) Transverse eigenfunctions of the guiding potentials in various places along the first beam splitter. When the two outgoing waveguides are separated far enough so that no tunneling between left and right occurs, the symmetric and antisymmetric states become pairwise degenerate. (c) Energy eigenvalues for the lowest transverse modes as they evolve along the interferometer. From 3.

Figure 30

The eigenfrequencies of the local modes as functions of the distance between the two wells of a double-well trap. An avoided crossing is shown inside the dashed box. Two corresponding eigenmodes of a single-well potential are shown on the left. The eigenmodes, in the limit when the wells are far apart, are shown on the right. These eigenmodes are localized in either the left or right potential wells. From 215.

Figure 31

Trap setup for an optimized adiabatic transfer of atoms into the microtrap (to scale). The compression wire and microstructure are rigidly connected to a heat sink between the transfer coils. The photograph shows the trap setup and dispenser source behind the MOT coils. The complete setup is mounted to a vacuum flange and operated in a vacuum chamber at a pressure of ${10}^{-11}\mathrm{mbar}$. From 63.

Figure 32

Absorption images of the adiabatic magnetic transfer into the microtrap and cooling towards Bose-Einstein condensation. The dashed line indicates the surface of the microstructure. (a) Transfer and compression of the Ioffe trap into the microtrap. (b) rf cooling in the microtrap. The image on the right side shows the condensate in the trap. (c) Release of the condensate. Images taken after 5, 10, and $15\mathrm{ms}$ time of flight (left to right). From 165.

Figure 33

(Color online) Observation of Fermi statistics. The quantity $r_K^2$ is proportional to expansion energy and is plotted versus the temperature of the ${}^{87}\mathrm{Rb}$ reservoir. The vertical axis is scaled using $R_F^2=E_F({\omega }_r^{-2}+{\tau }^2)∕m_K$, where $\tau$ is the expansion time before imaging. The temperature is also scaled by the Fermi energy $E_F$ (typically $k_B\times 1.1\mathrm{\mu }\mathrm{K}$) of each ${}^{40}\mathrm{K}$ cloud. Gaussian fits of data taken with both thermal (diamonds) and Bose-condensed (circles) ${}^{87}\mathrm{Rb}$ are compared to Gaussian fits of an ideal Fermi distribution (solid line) and a Boltzmann distribution valid for classical particles (dashed line). Absorption images are shown for (a) $k_{B}T∕E_F=0.35$ and (b) 0.95, including a white circle indicating the Fermi energy $E_F$. (c) The fit residuals of a radially averaged cloud profile show a strong systematic deviation when assuming Boltzmann (circles) instead of Fermi (diamonds) statistics. A degenerate Fermi cloud is flatter at its center than a Boltzmann distribution and drops off more sharply near its edge. Adapted from 8.

Figure 34

The mirror MOT. Left: perspective view indicating circular polarization of the diagonal beams incident on the mirror $M$ and the orientation of the quadrupole coils $Q1$ and $Q2$. The horizontal beams (perpendicular to the drawing plane) are not shown. Right: Projections on the $y\text{−}z$ and $x\text{−}z$ planes, illustrating the geometry of the laser beams. From 178.

Figure 35

Absorption images of condensates in the science chamber, side view. All images have the same scale. Condensates of $\sim 6\times {10}^5$ atoms are shown in (a) optical trap and (b) wire trap. The center segment of the Z-shaped wire is visible as a dark speckled horizontal stripe and is located $740\mathrm{\mu }\mathrm{m}$ above the trapped atoms. The condensate was released from (c) an optical trap after $10\mathrm{ms}$ time of flight and (d) wire trap after $23\mathrm{ms}$ time of flight. (e) Schematic of the wire trap, top view. $I_w=2\mathrm{A}$ is the current through the wire, and $B_0=2.9\mathrm{G}$ is the bias field. Atoms are trapped below the $5\mathrm{mm}$ long central segment of the wire, which is aligned with the optical trap axis. The supporting end segments, which provide field curvature, are truncated. From 79.

Figure 36

Microfabricated magnetic trap and waveguide. Optical tweezers loaded a Bose-Einstein condensate into the Z-wire trap formed by currents $I_1$ and $I_2$ in conjunction with a magnetic bias field $B_{\perp }$. Lowering $I_2$ to zero released the condensate into a single-wire magnetic waveguide. Atom flow was from left to right. Inset: The widening of the waveguide wire in the region where another wire merges with it at a small angle. The only current flowing in the inset is $I_1$. The condensate was trapped above the plane of the page, and the gravitational acceleration $\vec{g}$ points out of the page. All microfabricated features are drawn to scale. From 131.

Figure 37

Oscillation of the condensate in an anharmonic waveguide. The condensate is trapped below the microstructure that generates the magnetic field for the trapping potential. (a) Magnetic field in the trap minimum along the axial direction. The black dots indicate the starting positions $A$, $B$, and $C$ of the oscillation for three different experimental series. The circle marks the initial position without displacement. (b) Absorption images of the condensate after $20\mathrm{ms}$ time of flight (series $C$, $10\mathrm{ms}$ intervals). In the time-of-flight image, the amplitude of the oscillation is enhanced with respect to oscillation in the trap (also a phase shift occurs). For condensates marked with an asterisk the size of the condensate cannot be properly determined due to the limited resolution of the absorption imaging. (c) Center-of-mass motion of the condensate (series B): experimental data (dots) and fit with three sinusoidal functions (line). From 164.

Figure 38

Evolution of the aspect ratio of the condensate for the data series (a) $A$, (b) $B$, and (c) $C$ (dots) and theoretical model (solid line). (d) Plot (c) with a logarithmic scale. (e) All data points of series $B$. From 164.

Figure 39

Schematic drawing of the atom Michelson interferometer (not to scale). The dimensions of the chip are $5\mathrm{cm}\times 2\mathrm{cm}$. From 226.

Figure 40

Atom Michelson interferometer. Interference fringes after $1\mathrm{ms}$ propagation time in the waveguide with an axial confinement of ${\omega }_a=2\pi \times 5{\mathrm{s}}^{-1}$. The differential phase shift between counterpropagating condensates is introduced by a magnetic field gradient, which is turned on for $500\mathrm{\mu }\mathrm{s}$. The average separation of clouds during the magnetic gradient pulse is $8.82\mathrm{\mu }\mathrm{m}$. From 226.

Figure 41

Fringe visibility vs duration of the interferometric cycle corresponding to the parameters of the JILA experiment with a Michelson interferometer on a chip, using magnetic gradient as the phase element. (a) The theoretical curves correspond to $N_a=3\times {10}^3$ atoms, according to the $4\mathrm{ms}$ point in the experiment. The theoretical predictions for the actual experimental numbers of atoms at every run are also shown. (b) The same as (a), but for $N_a=4.5\times {10}^3$ atoms, where a reduction of the fringe contrast to 50% is expected. From 162.

Figure 42

Sketch of the experimental situation. (a) Current scheme of the magnetic lattice. The currents in neighboring conductors are equal and oppositely poled. (b) The condensate approaches the lattice during a vertical oscillation ($y$ direction) in an elongated harmonic trap. After phase imprinting the condensate is released from the trapping potential. From 77.

Figure 43

Diffraction of a condensate from a magnetic lattice. (a) Absorption images and vertically integrated density profiles after $\tau =20\mathrm{ms}$ time of ballistic expansion for three different displacements $d=13$, 14, and $14.6\mathrm{\mu }\mathrm{m}$. The distance $z_0=v_q\tau$ corresponds to one reciprocal lattice velocity $v_q$. The density profiles are described by the sum of up to five overlapping diffraction orders (solid line). While the relative strength of each diffraction order is given by the phase imprint parameter $S$, an additional Gaussian density distribution (dashed line) is used to take into account the fraction of thermal atoms. (b) The strength of phase imprinting $S$ can be controlled, e.g., by the displacement $d$, initiating the oscillation towards the lattice. From 77.

Figure 44

(Color online) Splitting a Bose-Einstein condensate. A magnetic double-well potential is generated by two chip wires with a current $I_C$ in combination with an external magnetic bias field. The distance between the two chip wires is $300\mathrm{\mu }\mathrm{m}$. A pair of extra wires with $I_B$ provides the axial confinement along the $y$ direction. A second pair with $I_T$ is used for compensating asymmetry effects. Gravity points along the $+z$ direction. From 206.

Figure 45

Splitting of condensates. (a) Condensates were initially loaded and prepared in the bottom well of the two-wire microtrap and (b) split into two parts by increasing the external magnetic field $B_x$. For clarity, two condensates were split by $80\mathrm{\mu }\mathrm{m}$. The dashed line indicates the chip surface position. The currents in the chip wires flow into the page and $B_x$ is parallel to the wire separation. Two condensates were released from the magnetic double-well potential and the matter-wave interference pattern of two condensates formed after time of flight. (c) Typical absorption image of interference fringes taken after $22\mathrm{ms}$ time of flight. The fringe spacing was $14.8\mathrm{\mu }\mathrm{m}$, corresponding to a condensate separation of $25.8\mathrm{\mu }\mathrm{m}$. From 206.

Figure 46

Spatial phase of interference fringes. For potential barriers smaller than the chemical potential $(\sim 1.4\mathrm{kHz})$, two condensates are phase locked and the interference pattern exhibits a predictable spatial position. Fully separated condensates produced an interference pattern with random spatial phase. The separation of two condensates $d$ was determined from the spacing of interference fringes $\lambda$, according to the asymptotic expression $d=h\tau ∕m\lambda$ (33). Fifty repetitions of the same experiment are plotted, with the external magnetic field $B_x$ held fixed when atoms were released. The three dashed lines indicate the separations of two wells with a barrier height of 1, 2, and $3\mathrm{kHz}$, respectively. From 206.

Figure 47

Double-well potential generated by an oscillating magnetic field. (a) A straight wire carrying a static (dc) current $(\sim 1\mathrm{A})$ is used to trap a BEC directly below a second wire carrying an rf current $(\sim 60\mathrm{mA}$ at $500\mathrm{kHz})$. The dc wire has a width of $50\mathrm{\mu }\mathrm{m}$ and is separated by $80\mathrm{\mu }\mathrm{m}$ from the rf wire (width $10\mathrm{\mu }\mathrm{m}$). Placing the trap $80\mathrm{\mu }\mathrm{m}$ below the chip surface at the indicated position allows for symmetric horizontal splitting. (b) Top view of the chip: An elongated BEC is transversely split. (c) Left: the rf magnetic field couples different atomic spin states (only two are shown for simplicity). Right: Under the influence of the rf field with a frequency below the Larmor frequency at the trap minimum $(\sim 1\mathrm{G})$ the initial dc trapping potential is deformed into a double-well potential. Also shown is the slight relaxation of the trap in the vertical $\left(y\right)$ direction (dashed line), and the splitting along the horizontal $\left(x\right)$ direction, with a well separation $d$ and potential barrier height $V_{\mathrm{bar}}$ (solid line). Adapted from 195.

Figure 48

(Color online) Coherence of the splitting is examined by analyzing matter-wave interference. (a) Interference pattern $0.1\mathrm{ms}$ after the splitting for a cloud separation of $d=3.4\mathrm{\mu }\mathrm{m}$. Tunneling through the barrier is inhibited. (b) Interference pattern $0.8\mathrm{ms}$ after the splitting for a cloud separation of $d=3.85\mathrm{\mu }\mathrm{m}$. Left: A cosine function with a Gaussian envelope is fitted to the profiles derived from the two-dimensional images (insets). Right: Contrast and relative phase for 40 realizations of the same experiment are plotted in a polar diagram (inset). A histogram of the same data shows a narrow distribution of the differential phase $(\sigma =13°)$ directly after separating the clouds and a slightly broadened distribution $(\sigma =28°)$ some time later. (c) The spread of the spatial phase is shown for different stages of the splitting process. The relative phase of the condensates shows a nonrandom behavior in the first $2\mathrm{ms}$ of the splitting. Adapted from 195.

Figure 49

Schematic of the geometry of microstructures as used for calculations of the lifetime by 186. A plane metallic layer of thickness $h$ lies parallel to the $(x,z)$ plane above a nonmetallic substrate. The atom is located in vacuum at a distance $d$ from the surface. From 186.

Figure 50

Lifetime $\tau$ as a function of skin depth $\delta$ with the atom-surface distance fixed at $50\mathrm{\mu }\mathrm{m}$. Solid line, infinitely thick surface. Dotted line, $1\mathrm{\mu }\mathrm{m}$ thick surface. The calculation was done for a spin-flip frequency of $560\mathrm{kHz}$ and a temperature of $300\mathrm{K}$. From 186.

Figure 51

Lifetime of trapped atoms versus distance from the Al surface of the cylindrical wire with $500\mathrm{\mu }\mathrm{m}$ diameter. Solid squares (open circles), measurements with spin-flip frequency $\omega ∕2\pi =560\mathrm{kHz}$ $(\omega ∕2\pi =1.8\mathrm{MHz})$. Solid (dashed) lines, calculated lifetimes above a thick slab of copper (aluminum) for these two spin-flip frequencies. Dotted line, expected scaling for technical noise. From 114. See also reanalysis of the exact conductor geometry by 180.

Figure 52

Inferred lifetimes near the surfaces after subtraction of the background loss. (a) Resistivity dependence: Copper (circles), titanium (triangles), and silicon (squares) at $\omega ∕2\pi =1.80\mathrm{MHz}$. Open and solid symbols represent measurements made with condensates and normal clouds, respectively. Solid (dashed) line indicates lifetimes predicted for copper (titanium), based on numerical integration of Eq. (22) of 94. The dotted line, plotted for copper, shows the closed-form interpolation suggested by 94. (b) Larmor frequency dependence near copper: $\omega ∕2\pi =1.80\mathrm{MHz}$ (circle) and $\omega ∕2\pi =6.24\mathrm{MHz}$ (triangle). Solid (dashed) line indicates lifetimes predicted for $\omega ∕2\pi =1.80\mathrm{MHz}$ $(\omega ∕2\pi =6.24\mathrm{MHz})$. From 87.

Figure 53

Lifetime $\tau$ as a function of distance from the surface of a thin metal layer, for $B_{\mathrm{off}}=0.57\mathrm{G}$ (145). Line: Calculated lifetime using a skin depth of $103\mathrm{\mu }\mathrm{m}$, a temperature of $400\mathrm{K}$, and a spin-flip frequency of $400\mathrm{kHz}$. For distances larger than $10\mathrm{\mu }\mathrm{m}$, the lifetime is limited by background gas collisions. Adapted from 145 and 186.

Figure 54

Ramsey spectroscopy of the $\mid 0⟩\leftrightarrow \mid 1⟩$ transition with atoms held at a distance $d=9\mathrm{\mu }\mathrm{m}$ from the surface of a silver foil of $250\mathrm{nm}$ thickness. An exponentially damped sine fit to the Ramsey fringes yields a $1∕e$ coherence lifetime of ${\tau }_c=2.8\pm 1.6\mathrm{s}$. Each data point corresponds to a single shot of the experiment. From 220.

Figure 55

(Color online) Remaining atom fraction $\chi$ in a trap at distance $d$ from the dielectric surface for a condensate (solid squares), and for thermal clouds at $2.1\mathrm{\mu }\mathrm{K}$ (open squares) and $4.6\mathrm{\mu }\mathrm{K}$ (triangles). The solid (dashed) lines are calculated with (without) Casimir-Polder potential for the condensate, the $2.1\mathrm{\mu }\mathrm{K}$ and $4.6\mathrm{\mu }\mathrm{K}$ clouds (left to right). Inset: The trapping potential for $C_4=8.2\times {10}^{-56}\mathrm{J}{\mathrm{m}}^4$ (solid line) and $C_4=0$ (dotted line). From 145.

Figure 56

Reflection probability vs incident velocity. Data were collected in a magnetic trap with trap frequencies $2\pi \times (3.3,2.5,6.5){\mathrm{s}}^{-1}$. Incident and reflected atoms were averaged over several shots. Vertical error bars show the standard deviation of the mean of six measurements. Horizontal error bars reflect the uncertainty in deducing $v_{\perp }$. The solid curve is a numerical calculation for individual atoms incident on a conducting surface. From 169.

Figure 57

Lifetime in the Si surface trap. Solid (open) circles show the remaining atom fraction vs time for a $2\pi \times (9,9,6.5){\mathrm{s}}^{-1}$ $[2\pi \times (3.3,2.5,6.5\left){\mathrm{s}}^{-1}\right]$ trapping potential with an initial atom number $3\times {10}^4$ $[9\times {10}^4]$. The solid line exponential fit gives a lifetime of $23\mathrm{ms}$ $\left(170\mathrm{ms}\right)$ for the high- (low-) frequency trap geometry. The lifetime for atoms confined far from the surface exceeded $20\mathrm{s}$ for either geometry. From 169.

Figure 58

(Color online) Surface-based electric field $E_{\mathrm{dip}}$ $(=E_{\mathrm{surface}})$ as a function of atoms stuck to the surface. Si (squares) and Ti (triangles) exhibit similar behavior, roughly increasing linearly with adsorbate number, while the glass substrate (circles) shows only a small effect of adsorbates. The distance between the center of the magnetic trap and the surface is fixed to $10\mathrm{\mu }\mathrm{m}$. Vertical error bars denote statistical errors only and do not represent systematic uncertainties, notably uncertainties in $E_{\mathrm{app}}$ and in the power law of $E_{\mathrm{dip}}$. Inset: A typical plot of frequency vs voltage before (circles) and after (squares) one condensate has deposited on Ti $({\nu }_0=216.5\mathrm{Hz})$. There is a measurable effect from the adsorption of just one condensate. The lines are weighted fits to extract the frequency shift per applied voltage. From 152.

Figure 59

Fragmentation of an ultracold cloud of ${}^{87}\mathrm{Rb}$ atoms in a magnetic waveguide potential of an electroplated conductor ($30\mathrm{\mu }\mathrm{m}$ width, $2.5\mathrm{\mu }\mathrm{m}$ height, and $25\mathrm{mm}$ length). The conductor is indicated by the horizontal dashed line. (a) The atoms were prepared in a waveguide by ramping down the axial confinement of the magnetic trap within $400\mathrm{ms}$. The current in the microfabricated conductor was $0.045\mathrm{A}$. With a bias field of $B_{\mathrm{bias}}=2\mathrm{G}$, the waveguide is formed at a distance of $45\mathrm{\mu }\mathrm{m}$ from the surface. The offset field $B_{\mathrm{off}}$ parallel to the conductor was $1.3\mathrm{G}$. The vertical dashed lines mark the position of the potential maxima. In (b) the atoms were initially prepared in the same trap as in (a). Next, the orientation of the offset field was flipped within $1\mathrm{ms}$. Further $50\mathrm{ms}$ later, the atoms were located at the positions of previous potential maxima. The position of the potential minima changed with the direction of the offset which proves that the axial potential modulation is due to an axial magnetic field modulation. From 126.

Figure 60

Magnetic fields vs height $h$ above the conductor surface ($h=y-\delta$, with the radius of the conductor $\delta =250\mathrm{\mu }\mathrm{m}$) above a wire carrying $3.7\mathrm{A}$. Data points, amplitude of the anomalous magnetic field variation $\mathrm{\Delta }{B}_z$ ($B_{\mathrm{mod}}$ in the text). Solid curve, best fit of the modified Bessel function $a{K}_1$ $\left[2\pi \right(h+\delta \left)\right]∕\lambda$, in which a, $\delta$, and $\lambda$ were allowed to vary freely as fitting parameters. This has $\lambda =217\pm 10\mathrm{\mu }\mathrm{m}$ and $\delta =251\pm 12\mathrm{\mu }\mathrm{m}$. Dashed curve, the (usual) azimuthal field referred to the auxiliary ordinate on the right. From 113.

Figure 61

Horizontally meandering current path and axial field modulation $B_{\mathrm{off}}+B_{\mathrm{mod}}$.

Figure 62

Correlation function of random magnetic potentials and calculated condensate density profiles. (a) Correlation functions of the random magnetic potential in a microtrap ${\Delta }_k$, assuming white noise fluctuations of the wire position. Inset, ${\Delta }_k$ when the wire fluctuations have an intrinsic length scale: $F_k\propto (e^{-(k-k1{)}^2{\eta }_1^2}+e^{-(k+k1{)}^2{\eta }_1^2})$. Here $2\pi ∕k_1=200\mathrm{\mu }\mathrm{m}$, ${\eta }_1=100\mathrm{\mu }\mathrm{m}$, and the disorder average over the wire position fluctuations $f^{+}$ is $⟨f^{+}\left(z\right)f^{+}\left(z\right){⟩}_{\mathrm{dis}}=(0.1\mathrm{\mu }\mathrm{m}{)}^2$. Solid to dash-dotted lines correspond to $d=50$, 100, 150, and $200\mathrm{\mu }\mathrm{m}$, respectively. (b) Condensate density profiles with parameters chosen to be close to the values used in the experiments of 131; 134. Dotted lines are the results without random potential. From 225.

Figure 63

Rough potentials normalized to the current in the wire for different heights from the wire. Solid lines, potentials measured using cold atomic clouds. Dashed lines, potentials calculated from the measured geometric roughness of the edges of the wire. The different curves have been shifted by $6\mathrm{\mu }\mathrm{K}∕\mathrm{A}$ for clarity. From 51.