ADVANCING NEUTRAL ATOM QUANTUM COMPUTING: STUDIES OF ONE-DIMENSIONAL AND TWO-DIMENSIONAL OPTICAL LATTICES ON A CHIP
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By
Katharina Christandl, M.S.
*****
The Ohio State University 2005
Dissertation Committee: Approved by Professor Gregory P. Lafyatis, Advisor
Professor Frank C. DeLucia ______
Professor Eric Herbst Advisor Graduate Program in Physics Professor Junko Shigemitsu
Copyright by
Katharina Christandl
2005
ABSTRACT
In this work we present the results of studies of one-dimensional (1D) and two- dimensional (2D) optical lattices on a chip. We have developed a method to create 1D and 2D optical lattices above an optical waveguide. The method is based on the destructive interference of the evanescent waves of two modes of the waveguide, resulting in a periodic array of nodes above the waveguide surface. For blue-detuned light, this constitutes an array of optical dipole traps for neutral atoms. One pair of modes coupled into a ridge waveguide creates a 1D array of nodes in this manner, and two orthogonal pairs of modes in a slab waveguide result in a 2D array of nodes.
Computational analysis of our method showed that these optical atom chips may be suitable for use as a quantum register, and for studies of atom-surface interactions between the trapped atoms and the waveguide surface. Using realistic waveguide materials and laboratory laser parameters we found that for a laser power of ~100 mW nodes can be created ~100 nm above the waveguide surface. The nodes constitute atom traps that are ~1mK deep, and have trap frequencies ~1MHz and ground state sizes of
~10 nm. The spacing between traps is ~1 µm enabling individual addressing with a focused laser beam.
ii With the implementation of two-qubit gates in mind, initial investigations of
extensions of the basic 2D optical lattice were conducted. Computational results for a
counter-propagating 2D optical lattice with very close trap spacing (~λ/4) in one dimension, and for a 2D optical lattice with adjustable spacing between pairs of traps in one dimension are presented.
An experimental setup for the realization of the proposed 2D optical lattices on a chip was designed and built. Several cooling and trapping steps necessary to prepare a cold atom sample for loading of the 2D optical lattice have been implemented successfully.
In the course of previous studies of atom-surface interactions, a novel surface- ionization detector for 2D imaging of neutral atom samples was built and characterized.
We find a detection efficiency ~50 %, a spatial resolution of 20 µm and a time response <
1/30 s.
iii
To my family.
iv
ACKNOWLEDGMENTS
First and foremost I would like to thank my advisor Greg Lafyatis for everything that he taught me. I also want to thank Frank DeLucia for always being there with his advice when I needed it. I appreciate Eric Herbst and Junko Shigemitsu for being willing to serve on my Graduate committee.
Next I want to express my thankfulness to the many people that have supported me and various aspects of my work: Reni Ayachitula – for being a great friend and labmate throughout my Graduate school years, making the long days and nights in the lab a bit more bearable. Andrei Modoran – for making sure to keep the balance in the lab.
Mike Chmutov gave the group a big boost of energy and has helped me much with his competent work and ample humor. Jenn Holt has not only managed to solve impossible problem upon problem with our experiments over the years, but also has become a close friend of mine – thank you for being there. Many thanks to others that have provided much technical support: Kent Ludwig, John Spaulding and all of the Machine Shop (Pete,
Brian, Jonathan, Fred, Scotty, Larry and Randy), Tom Kelch, Rita Rokhlin, John
Hoftiezer, the EE shop guys, and the computer staff. Without their patience and competence this work would not have been possible.
I also want to acknowledge my past and present friends that have made my Grad school years memorable: Jen Tate, Yuko Shiroyanagi, Chris Neu, Mike Schirber, Bob
v Nolan, Nathalie Dardare, Hannan Fersi, Christophe Buron, Babis Papachristou, Minako
Matsunaga, and Matt and Chastity Whitaker. I also am thankful to my friends in
Germany that have kept in touch over the years: Matthias Frink, Nils Benter, Richard and
Stana Willmann, and Jörg Dietrich.
Last but not least I want to acknowledge the infinite support of my family. I thank my Mom Angelika for raising me to be the person I am, giving me the confidence to go out and find my own path, and always being there for me. My Dad Herbert for stirring my interest in science and for pushing me to push myself to achieve my goals. My friend and brother Fabian for being there for me. Despite the physical distance I think we have grown closer over the last few years. And let’s not forget my new family: Chop
(Genevieve Gillen), Layton and Brooke, Laura, Mike, Jonathan and Chris, and Grant, who have taken me in as one of their own. Your support is deeply appreciated.
The last paragraph I saved for the person that deserves the most thanks of all:
Glen Gillen - for your infinite love and support, with my deepest appreciation.
vi
VITA
December 12, 1977 ...... Born – Bonn, Germany
1996...... Abitur, “Gymnasium der Franziskanerinnen”, Nonnenwerth im Rhein, Germany
1998...... Vordiplom Physics, Bonn University, Germany
2000...... M.S., Physics, The Ohio State University
1999...... Graduate Teaching and Research Associate, The Ohio State University
vii
PUBLICATIONS
1. Katharina Christandl, Gregory P. Lafyatis, Seung-Cheol Lee, and Jin-Fa Lee, “One- and two-dimensional optical lattices on a chip for quantum computing,” Physical Review A 70, 032302 (2004).
2. Katharina Christandl, Gregory P. Lafyatis, Andrei Modoran, and Tung-Hsiu Shih, “Two-dimensional imaging of neutral alkali atom samples using surface ionization,” Review of Scientific Instruments 73 (12), 4201 (2002).
FIELDS OF STUDY
Major Field: Physics
viii
TABLE OF CONTENTS
Page
Abstract...... ii
Dedication...... iv
Acknowledgments ...... v
Vita...... vii
List of Tables ...... xiii
List of Figures...... xv
Chapters
1. Introduction ...... 1
2. Review...... 6
2.1 Quantum Computing 101 – An Introduction...... 6
2.1.1 The Qubit ...... 6
2.1.2 Single Qubit Evolution ...... 9
2.1.3 Decoherence...... 11
2.1.4 Entanglement of Two Qubits...... 13
2.1.5 The Five Commandments of Quantum Computation...... 16
2.2 Trapping Atoms with Light ...... 17
2.3 Quantum Computing with Neutral Atoms in Optical Lattices ...... 23
ix 2.3.1 Qubits and Scalability...... 23
2.3.2 Properties of Free Space Optical Lattices...... 23
2.3.3 Initialization...... 27
2.3.4 The Rabi Two-Level Problem ...... 28
2.3.5 Single-Qubit Gates and Read-Out ...... 29
2.3.6 Two-Qubit Gates for Atoms in Optical Lattices...... 30
2.3.6.1 Two-Qubit Gates Using the Electric Dipole-Dipole Interaction .. 31
2.3.6.2 Two-Qubit Gates Using the Magnetic Dipole-Dipole Interaction 35
3. One- and Two-Dimensional Optical Lattices on a Chip ...... 46
3.1. How to Create a Two-Dimensional Optical Lattice on a Chip...... 48
3.2 Properties of the Basic Two-Dimensional Optical Lattice ...... 51
3.3 One-Dimensional Optical Lattices on a Chip...... 55
3.4 Variations of the Basic Two-Dimensional Optical Lattice...... 56
3.4.1 The Counter-Propagating 2D Optical Lattice...... 57
3.4.2 The Double-Well Optical Lattice ...... 59
4. Experimental Setup...... 73
4.1 Vacuum Chamber ...... 75
4.1.1 Rubidium Loading Chamber ...... 75
4.1.2 The Ultra-High Vacuum Chamber ...... 76
4.1.3 Chamber Construction and Bake-Out...... 79
4.2 Lasers...... 83
4.2.1 External Cavity Diode Lasers and Grating Feedback...... 84
4.2.2 Laser Construction...... 86
x 4.2.3 Frequency Stabilization ...... 88
4.2.3.1 Temperature and Current Control of the Free Running Diode..... 88
4.2.3.2 Grating Feedback...... 90
4.3 The Atom Trap...... 94
4.3.1 Magnetic and Magneto-Optical Traps ...... 95
4.3.2 The Setup of the MOT...... 100
4.3.3 MOT Optimization ...... 104
4.3.3.1 MOT Beam Polarization...... 105
4.3.3.2 MOT Beam Intensity ...... 107
4.3.3.3 Characterization of the MOT...... 108
4.3.4 Alignment of MOT and Magnetic Trap...... 110
4.3.5 Loading the Magnetic Trap...... 112
4.3.4 Transfer to the UHV chamber ...... 114
5. Neutral Atom Detector ...... 142
5.1 Introduction...... 142
5.2 Design ...... 145
5.3 Characterization...... 148
5.3.1 Efficiency...... 148
5.3.2 Background and Time Response ...... 150
5.3.3 Resolution ...... 152
5.3.4 Future Improvements...... 153
6. Conclusions ...... 157
xi Appendices
A. Derivation of the Average Spontaneous Emission Rate...... 161
B. List of Parts...... 165
C. Shop Drawings ...... 170
D. LabVIEW Codes...... 217
E. Circuit Diagrams...... 246
Bibliography ...... 256
xii
LIST OF TABLES
Table 1.1: List of quantum computing approaches. This list is not all-inclusive. For more information and additional references, see the Quantum Computing Roadmap [3]...... 5
Table 3.1: Results for trap parameters of the basic 2D optical lattice above a slab waveguide. The input parameters were: laser wavelength of 780 nm, detuning ∆ = 1000 Γ = 2π × 6 GHz to the blue of the 85Rb D2 line, waveguide material
As2 S3 ( nW = 2.4), substrate material NaF ( nS = 1.32), and waveguide thickness 230 nm. The traps were created using a co-propagating pair of TE0 and TE1 modes in each dimension (x and z), with a power of 100 mW (for a beam width of 100 µm) in each TE0 mode and 4.2 mW in each TE1 mode, resulting in a trap height of 150 nm above the waveguide surface...... 64
Table 3.2: Results for trap parameters of the 1D optical lattice above a ridge waveguide. The input parameters were: laser wavelength of 780 nm, detuning ∆ = 1000 Γ = 2π × 6 GHz to the blue of the 85Rb D2 line, waveguide material
As2 S3 ( nW = 2.4), substrate material NaF ( nS = 1.32), and waveguide dimensions of 1µm × 0.45 µm (see Figure 3.3 (a)). The traps were created 7 -1 using a co-propagating pair of two modes ( k1 = 1.304×10 m , 7 -1 k2 = 1.105×10 m ) along z, with a power of 1 mW in mode 1 and 0.081 mW in mode 2, resulting in a trap height of 150 nm above the waveguide surface...... 65
Table 3.3: Results for trap parameters of the counter-propagating 2D optical lattice above a slab waveguide. The input parameters were: laser wavelength of 780 nm, detuning ∆ = 1000 Γ = 2π × 6 GHz to the blue of the 85Rb D2 line, waveguide
material As2 S3 ( nW = 2.4), substrate material NaF ( nS = 1.32), and waveguide thickness 230 nm. The traps were created using a counter- propagating pair of TE0 modes along the z direction and co-propagating pair of TE0 and TE1 modes in the x direction, with a power of 100 mW (for a beam width of 100 µm) in each TE0 mode and 4.2 mW in the TE1 mode, resulting in a trap height of 150 nm above the waveguide surface...... 66
xiii Table 3.4: Results for trap parameters of the 2D double-well optical lattice above a slab waveguide. The input parameters were: laser wavelength of 780 nm, detuning ∆ = 1000 Γ = 2π × 6 GHz to the blue of the 85Rb D2 line, waveguide material
As2 S3 ( nW = 2.4), substrate material NaF ( nS = 1.2827), and waveguide thickness 663 nm. The double-wells were created along z using the TE0, TE2 and TE3 modes, which satisfy equation (3.2). Along the x direction, a co- propagating pair of TE2 and TE3 modes is used with a power of 100 mW (for a beam width of 100 µm) in the TE2 mode and 21 mW in the TE3 mode, resulting in a trap height of 100 nm above the waveguide surface. The second column shows the results for 100 mW (in a 100 µm beam) in the TE0 mode (along z), 0 mW in the TE2 mode and 0.9 mW in the TE3 mode. The third column shows the results for the closest possible double-well spacing, which we achieved by increasing the power in the TE2 mode up to 16 mW...... 67
Table 4.1: Magnetic trap loading sequence used in our experiment...... 117
Table B.1: List of parts for 2D optical lattice experiment...... 166
xiv
LIST OF FIGURES
Figure 2.1: Single qubit gates. (a) Visualization of a single qubit state on the Bloch sphere. (b) List of most common single qubit gates. (c) Illustration of a Hadamard gate on the Bloch sphere. (Figures taken from [16].)...... 38
Figure 2.2: Entanglement sequence for a cPHASE gate. The 11 component acquires an additional phase beyond the single qubit evolution, due to the electric dipole- dipole interaction. For the gate timing shown, the gate is a π cPHASE gate. 39
Figure 2.3: Dressed atom energy diagram (shown for ∆ > 0 ). (a) Without interaction, the energy levels of the laser-atom system are nearly degenerate manifolds
(splitting of ∆ ), separated by the energy of one laser photon, hωL . (b) The laser-atom interaction mixes the two states, creating the dressed states, with an energy splitting of Ω' . (c) Level shift in a position dependent laser field (e.g. Gaussian laser intensity profile). The energy splitting Ω'(r ) is now position dependent. (Figure adapted from [24].) ...... 40
Figure 2.4: Energy level structure of alkali atoms. (a) For very large laser detunings
( ∆ >> ∆'FS ) the fine structure is not resolved and the atom can be treated as a
two-level system. (b) For large detunings ( ∆'FS ≥ ∆ >> ∆ HFS ), the fine 2 2 structure becomes important. The two major transitions are D1 ( S1/ 2 → P1/ 2 ) 2 2 and D2 ( S1/ 2 → P3 / 2 ). (c) For sufficiently small laser detuning ( ∆ HFS > ∆ ), the light potential depends on the hyperfine state F of the atom. The hyperfine levels shown are for nuclear spin I = 3/2...... 41
Figure 2.5: One-dimensional optical lattice configurations. (a) Laser polarizations for different 1D optical lattice configurations. (b) σ + (blue dash) and σ − (red, solid) components of electric field for lin||lin and lin⊥lin configurations. (For lin||lin the σ + field was shifted up for clarity). (c) Light potential contributions from σ + (blue, long dash) and σ − (red, short dash) light and 87 total potential (solid) for an Rb atom in F = 1,mF = 1 . (d) Light potential
for atoms in mF = 1 (black) and mF = −1 (red/gray). All potentials are normalized to the maximum U0 of the total lin||lin potential...... 42 xv Figure 2.6: 3D optical lattices. (a) Three pairs of counter-propagating laser beams result in a 3D optical lattice. Two pairs are in lin||lin configuration, while the third has an adjustable polarization angle θ (“linθlin” configuration), allowing for adjustment of the relative positions of the σ + and σ − lattices. (b) Alternative beam configuration for a 3D optical lattice. This configuration does not allow for shifting of the σ + and σ − lattices. (Adapted from [36].) (c) 3D lattice structure. The cubic array of sites does not permit addressing of atoms on the inside of the lattice...... 43
Figure 2.7: Single qubit operations for a neutral alkali atom (F numbers for 87Rb). (a) Single qubit gates can be accomplished via laser pulses resonant with a Raman transition (light red/gray and solid, blue arrows), or with an RF pulse resonant with the 0 ↔ 1 transition (dotted arrow). (b) Read-out is accomplished by detecting fluorescence from a cycling transition. If the qubit is in state 0 no fluorescence is detected...... 44
Figure 2.8: Level diagrams for entanglement schemes using the electric dipole-dipole interaction. (a) Single qubit level diagram. (b) Sequence of entanglement pulses for first scheme in [32]. This is a cPHASE gate. For ϕ = 2π the entanglement phase is π . (c) Sequence of entanglement pulses for the second entanglement scheme from [32] (also a π cPHASE gate)...... 45
Figure 3.1: Creation of a 2D optical lattice above a chip. (a) TE mode in a slab waveguide. (b) Interference of two co-propagating modes leads to nodal lines away from the waveguide surface. (c) Two pairs of co-propagating modes create a 2D array of 3D traps – a 2D optical lattice...... 68
Figure 3.2: Trapping potential for realistic waveguide parameters. (a) Waveguide and substrate materials and thickness. (b) Trap potential along z. The trap potential along x is identical to that along z due to the geometry of the beam configuration. (c) Electric fields of TE0 (red) and TE1 (blue) modes at z* above the waveguide. The electric fields are out of phase. The relative intensities were chosen so that the fields cancel 150 nm above the waveguide surface, resulting in a 1.5 mK deep trapping potential along y (black)...... 69
Figure 3.3: 1D optical lattice above a ridge waveguide. (a) Illustration of waveguide materials and size. (b) Resulting electric fields and trap potential along x (at node z*). The blue (dotted) lines are the y-component of the electric field; the red (dashed) lines are the x-components of the electric field. The solid, black line is the resulting trap potential. (c) y-components of the electric field at node z* (red, long dash = mode 1; blue, short dash = mode 2) along y. The resulting field (purple, dotted line) for relative intensities as in the text cancels 150 nm above the waveguide surface, resulting in a 1.5 mK deep trapping
xvi potential (black, solid line) along y. The trap potential along z is periodic as in the basic 2D optical lattice...... 70
Figure 3.4: The counter-propagating 2D optical lattice. (a) Beam configuration of the counter-propagating optical lattice. (b) Trap potential along the counter- propagating direction z. As indicated, the trap spacing is approximately λ/4. The trap potential along x is identical to the potential for the basic 2D optical lattice. (c) Electric fields of the TE0 (red, long dash) and TE1 (blue, short dash) modes, total electric field (purple, dotted) and trap potential (black, solid) along y. The trapping is weaker than in the basic 2D lattice, because the counter-propagating mode pair does not contribute to the trapping along y. . 71
Figure 3.5: The double-well lattice. (a) Beam configuration for the double-well lattice. Three modes are necessary along z. (b) Trap potential for 860 nm spacing between wells (P(TE2)=0). (c) Trap potential for 136 nm spacing between wells (P(TE2)=16 mW). (d) The trap spacing varies with the power coupled into the TE2 mode. (e) The trap potential along y stays about the same as the power in the TE2 mode is adjusted from a trap (double-well) spacing of 860 nm to 136 nm...... 72
Figure 4.1: (a) Overview of the experimental system. (b) The Rb loading chamber is connected to the UHV chamber through a bellows with two 5 mm pinholes in it. Atoms are transferred from the loading chamber to the UHV chamber by translating the magnets along the servotrack...... 118
Figure 4.2: The Rb loading chamber. The Rb filaments are sticking into the vacuum chamber from an electrical feedthrough (off to the left of the picture). The quadrupole magnets (opposing magnetic coils) are also shown in this view...... 119
Figure 4.3: Close-up view of the UHV chamber. The straight-through, all-metal valve allowing for remote loading of our waveguide sample can be seen on the left. The top flange is holding a copper finger to which the sample will be attached. A tilt and translation stage will be holding the flange instead of the stainless steel sleeve shown in this picture...... 120
Figure 4.4: Alignment of copper finger with chamber. The boron nitride rod is attached to the copper finger, just as it would be after loading the actual waveguide sample...... 121
Figure 4.5: Setup for bake-out. A 60 l/s ion pump will take the place of the turbopump- roughing pump station during the bake-out once the pressure is low enough. The scaffolding outlines the position of the oven walls...... 122
xvii Figure 4.6: Grating feedback. (a) The extended cavity consists of a laser diode, a retro- reflecting and a wavelength selecting element. In our setup a grating fulfills both functions. (Adapted from [47].) (b) Littrow configuration. (c) Corner cube. While the lateral position of the outgoing beam shifts slightly as the grating is tilted, the angle of the outgoing beam remains steady...... 123
Figure 4.7: Grating-stabilized external feed-back laser...... 124
Figure 4.8: Illustration of the dichroic atomic vapor laser lock (top view)...... 125
Figure 4.9: Signals used for dichroic atomic vapor laser lock. (a) Absorption signal of each photodiode (magnetic field in solenoid is off). The frequency is scanned from higher to lower frequencies by applying a 100 ms voltage ramp to the laser grating. (b) Shifted absorption signals (σ + and σ − ) recorded by the two diodes when the magnetic field is on. (c) Error signal generated by subtracting the two photodiode signals. The signal shown has been shifted to move the zero voltage crossings of the error signal to the appropriate frequency...... 126
Figure 4.10: Saturated absorption setup. A powerful pump beam and a weak probe beam travel through a Rb vapor cell in opposite directions. The pump beam is amplitude modulated and the signal due to the saturation of atoms (probe beam strength) is detected with a lock-in amplifier...... 127
Figure 4.11: Saturated absorption (red/gray) and DAVLL error signal (blue/black) scans for frequency tuning of the lasers. (a) Trap transition. Note that the zero voltage crossing of the error signal is slightly to the right (i.e. red) of the first peak ( F = 2 → F'= 3 transition). (b) Pump transition...... 128
Figure 4.12: Overview of the elements of the experimental setup...... 129
Figure 4.13: Overview of control circuit of experiment...... 130
Figure 4.14: Illustration of magnetic quadrupole field. (a) Two magnetic coils with opposite currents create a quadrupole field. (b) The quadrupole field magnitude is zero at the center of the magnetic coils pair, and increases linearly in every direction. The field direction switches...... 131
Figure 4.15: Principle of the MOT (for the simplified case of an atom with J g = 0 and
J e = 1). The magnetic quadrupole field causes an energy level shift proportional to the distance from the trap center. In all points of the trap the net radiation pressure force is towards the trap center, because of the magnetic field induced detuning of the σ + and σ − transitions (for quantization axis + z ). (Figure adapted from [29].)...... 132
xviii Figure 4.16: Pump and trap transitions in 87Rb. The trap transition is a cycling transition. However, there is a small probability of exciting the F’=2 state, which can decay into the F=1 state, which does not interact with the trap laser. The pump laser lifts atoms in the F=1 state to F’=2 which can decay either to F=1 or F=2...... 133
Figure 4.17: Beam paths for our MOT. (a) Top view. Optics are arranged on two levels: 4” above the table and 13” above the table (except vertical beam optics). Optics on the lower level are indicated by dashed lines. (b) Side view (along chamber axis) of vertical beam optics (starting from “PBS 1”)...... 134
Figure 4.18: Rb hyperfine transitions in 85Rb and 87Rb. The Doppler-broadened Rb spectrum at the bottom shows four peaks for the four sets of hyperfine transistions...... 135
Figure 4.19: Saturated absorption spectra of the hyperfine structure of each of the four sets of hyperfine transitions in Rb...... 136
Figure 4.20: Polarization test setup. The orientation of the polarizing beamsplitter cube, quarter-wave plate and magnetic field polarity are matched to those of the actual MOT setup for each beam...... 137
Figure 4.21: Polarization optimization setup. (a) The polarizing beamsplitter cube is lined up with the laser polarization by maximizing the transmitted signal. (b) The angle of the cube is aligned so that the reflected beam is perpendicular to the incoming beam. (c) The mirror is aligned to retro-reflect the beam. The quarter-wave plate is adjusted to maximize the reflected signal...... 138
Figure 4.22: Illustration of photodetector and camera positions...... 139
Figure 4.23: Mechanical Shutter. Two of these shutters are used to shut off the pump and trap beams during the magnetic trap loading sequence...... 140
Figure 4.24: Chamber alignment. (a) The alignment of the first pinhole is very difficult because it requires sub-mm precision alignment of the steel post holding the 8 l/s ion pump. (b) The alignment of the second pinhole is achieved by adjusting this steel post, which does not bear any weight...... 141
Figure 5.1: Surface ionization detector for 2D imaging of neutral atom samples. The neutral atoms incident on the detector are indicated by the blue, narrow arrow along z. The ions boiled off the hot ribbon are accelerated onto the MCPs as indicated by the red, curved arrow...... 154
Figure 5.2: Detection efficiency measurement. (a) – (c) are three consecutive frames recorded at a frequency of 30 Hz by the frame grabber upon dropping a cold
xix atom sample onto the detector. Most of the atoms are detected within one single frame, (b). The ends of the detector ribbon are cooler, and therefore have a longer response time, spreading the detection of incident atoms over all three frames...... 155
Figure 5.3: Resolution measurement. The slits imaged here are 15 µm wide and separated by 500 µm. Analysis of this and similar images yields a resolution estimate of 20 µm along the y direction (see Figure 5.1)...... 156
Figure C.1: Glass cell for Rb loading chamber...... 171
Figure C.2: Glass cell for UHV chamber...... 172
Figure C.3: Resealable copper gasket for straight-through, all-metal valve. The sample is the original gasket (with clearance hole instead of solid copper cap)...... 173
Figure C.4: 1/8” copper gasket for bonnet seal of straight-through all-metal valve. Sample is a commercially available 3 3/8” copper gasket...... 174
Figure C.5: Waveguide mount. The sample (5 mm thick glass) is screwed to the end of the macor rod with alumina screws...... 175
Figure C.6: Stainless steel rod for attaching the waveguide sample mount to the copper finger...... 176
Figure C.7: Copper finger for mounting sample holder to chamber...... 177
Figure C.8: Modified rotatable flange to hold copper finger...... 178
Figure C.9: Tilt mount for sample holder flange (attaches to a translation stage). (a) Front piece. (b) Back piece...... 179
Figure C.10: Overview of laser assembly. Only home-built parts are listed here...... 181
Figure C.11: Laser heatsink...... 182
Figure C.12: Cover plate for laser heatsink and heater plate...... 183
Figure C.13: Laser heater plate...... 184
Figure C.14: Laser base plate...... 185
Figure C.15: Thermistor holder (mounts into base plate with 4-40 flathead screw)...... 186
Figure C.16: Laser head (precision machined to line up lens and laser diode)...... 187 xx Figure C.17: Laser diode adapters. (a) Front adapter. (b) Back adapter...... 188
Figure C.18: Laser diode mounting ring (attaches laser diode to laser head)...... 190
Figure C.19: Collimating lens casing. The laser beam is collimated by screwing this mount in and out of the laser head...... 191
Figure C.20: Grating mount. Laser frequency tuning is achieved by adjusting the adjustment screw or PZT voltage...... 192
Figure C.21: Mount for corner cube mirror...... 193
Figure C.22: Mount for 1” diameter MOT telescope lenses. When mounted to the optics track, the tilt and lateral position of the lens are fixed, simplifying and stabilizing the MOT laser beam alignment...... 194
Figure C.23: Mount for 3” diameter MOT telescope lenses. When mounted to the optics track, the tilt and lateral position of the lens are fixed, simplifying and stabilizing the MOT laser beam alignment...... 195
Figure C.24: Spacer for MOT mirrors. This spacer lines up the axis of rotation of the MOT mirror with the center of the precision optics track...... 196
Figure C.25: Precision optics track for MOT optics...... 197
Figure C.26: Legs for precision optics tracks. Each track sits on two of these legs. The horizontal position of each track can be fine-adjusted by sliding these legs along rulers fixed to the optics table...... 198
Figure C.27: Overview of the quadrupole magnet assembly...... 199
Figure C.28: Plate for mounting the quadrupole magnet assembly to the servotrack forcer...... 200
Figure C.29: Magnetic coil post. (a) Original design. (b) Modifications for magnetic coil position adjustment, copper tubing feed-through and counterweight attachment...... 201
Figure C.30: Coil holders. The plates for the two quadrupole plates are identical except for the orientation of the Nylon cap screw clearance holes for attaching the support brackets. (a) For top coil. (b) For bottom coil...... 203
Figure C.31: Brackets for magnetic coil support...... 205
Figure C.32: Overview of magnetic coil...... 206 xxi Figure C.33: Teflon spool (magnetic coil center)...... 207
Figure C.34: Magnetic coil cover plates (to hold copper tubing in place). (a) “Outside” cover plate (with feed-through for copper tubing). (b) “Inside” cover plate (groove cut out to increase space between coils near the trap center)...... 208
Figure C.35: Coil cover ring (to keep the tubing from springing outward). The ring is attached to the “inside” cover plate with 6-32 Nylon screws (not shown)... 210
Figure C.36: Cooling water feed-through. This piece ensures that the weight of the long water tubes providing the cooling water for the magnetic coils rests on the magnetic coil post rather than the 3/16” copper tubing...... 211
Figure C.37: Electrical terminal block. This block ensures that the weight from the 2/0 cables supplying the current for the magnetic coils rests on the magnetic coil post rather than the 3/16” copper tubing...... 212
Figure C.38: Electrical terminals for connecting the 2/0 cables to the copper tubing of the magnetic coils...... 213
Figure C.39: Overview of shutter assembly. A spring pulling up on the shutter arm, the BNC connector for the shutter signal and a protective cover over the solenoid and high voltage electric connections are not shown here...... 214
Figure C.40: Shutter base plate...... 215
Figure C.41: Shutter arm (made from PCB)...... 216
Figure D.1: Bake-out control program. (a) Front panel. (b) Block diagram...... 218
Figure D.2: Computer scan program used for tuning the lasers. (a) Front panel. (b) Block diagram (all cases are indicated; sub-vi “1” is “Assemblegraphclusters.vi”)...... 220
Figure D.3: Assemblegraphclusters.vi, called by the computer scan program. (a) Front panel. (b) Block diagram...... 224
Figure D.4: Trapping experiment control program. (a) Front panel. (b) Block diagram (all cases are indicated, the sub-vis Sequencer.vi (“trap”) and AI_read.vi (“1”) are called)...... 226
Figure D.5: Sequencer.vi (sub-vi of trapping experiment control program. (a) Front panel. (b) Block diagram...... 240
xxii Figure D.6: AI_Read.vi (sub-vi of trapping experiment control program). (a) Front panel. (b) Block diagram...... 242
Figure D.7: Frame grabber control program. (a) Front panel. (b) Block diagram (all cases are indicated, the sub-vis “Create path.vi” and Digital output.vi (“D”) are called)...... 243
Figure D.8: Sub-vis of frame grabber control program. (a) Create Path.vi, front panel (b) Create Path.vi, block diagram. (c) Digital output.vi, front panel. (d) Digital output.vi, block diagram...... 245
Figure E.1: Laser power supply. (a) Current control unit. (b) Temperature control unit...... 247
Figure E.2: Shorting box. An inductor in series and a capacitor in parallel with the laser diode protect the diode from sudden voltage spikes. The current lock signal (BNC input) bypasses the protective circuit...... 249
Figure E.3: Photodetector circuit (taken from manual). For atom number measurements we used RF=10 kΩ...... 250
Figure E.4: Laser locking circuit. This circuit provides both the current and grating feedback, allows ramping of the laser frequency manually or by computer, and allows setting the error signal offset manually or by computer...... 251
Figure E.5: Shutter supply circuit. The circuit can supply up to four shutters simultaneously...... 254
Figure E.6: Magnetic coil circuit. The computer controls the current through the magnetic coils by supplying a voltage to the gate of the IGBT Module (FZ600R12KE3)...... 255
xxiii
CHAPTER 1
INTRODUCTION
At the heart of physics lies quantum mechanics. Physics at the quantum level bears very unusual features from our everyday, classical point of view. This is very well illustrated by Schrödinger’s famous cat, which, amazingly so, is dead and alive at the same time. This phenomenon is called superposition. Another remarkable feature of quantum mechanics appears in multipartite systems. Quantum states for multipartite systems exist in which control of one part of the system affects the state of the other part even though they may be spatially separated, without any interaction between the two parts. This is called entanglement.
Since the discovery of quantum mechanics 100 years ago, more and more branches of physics have investigated quantum behavior and developed ways to harness the power of quantum physics. One of the current fields of investigation of quantum mechanics is the field of quantum computing, which makes use of the phenomena of superposition and entanglement to perform tasks impossible for a classical computer. It has been suggested by Feynman [1] that, in principle, a quantum system can be used to simulate another quantum system. This is the goal of quantum computing.
1 What is the difference between quantum computing and conventional, “classical” computing? In quantum computing, the classical bits of a conventional computer are replaced by quantum bits, which, following the laws of quantum mechanics, can be in any superposition of two states, rather than only in states “zero” or “one”. Because of this, an array of 100 quantum bits can encode 2100 ≈ 1030 bits as opposed to only 100 bits for a classical array of the same size.
So, what can quantum computers do that classical computers can’t? Certain computational problems are deemed to be “intractable” on classical computers.
“Intractable” means that solution of the problem requires resources (number of bits or time) that grow faster than any polynomial with the number of input bits. For some of these problems quantum algorithms have been developed that exploit superposition and entanglement to solve the problems exponentially faster on a quantum computer than a classical computer, making them tractable. One example of interest is Shor’s factoring algorithm [2], which allows finding the prime factors of very large numbers exponentially faster than any known classical algorithm. This has attracted wide interest by the government as well as the public because it would enable deciphering of the most common encryption system currently in use, RSA, which is based on the fact that it is difficult to factor large numbers. Similarly, simulation of a quantum system is intractable on a classical computer, but is deemed tractable with a quantum computer, which, in itself, is a quantum system.
Research in numerous fields of physics is being conducted towards finding a system that can be used as a quantum computer. Current approaches towards quantum computing and the suggested systems and interactions are listed in Table 1.1. Other ideas
2 for candidate systems are still being investigated. To date, not one of these approaches
has been able to demonstrate all requirements for quantum computing, so it is still uncertain whether or when a working quantum computer will be developed. An overview of progress made towards quantum computing within all the different fields can
be found in the Quantum Computing Roadmap [3].
The focus of this thesis lies in the field of quantum computing using neutral atoms
in optical lattices. An optical lattice is a periodic pattern of light intensity. The most basic example is a laser standing wave of two counter-propagating laser beams. All optical lattices suggested for use in a quantum computer to date are three-dimensional (3D) optical lattices (i.e. three pairs of counter-propagating laser beams) in free space. One atom is loaded into each site (intensity node or antinode, depending on laser frequency) of the optical lattice. The expression “optical lattice” was introduced to emphasize the similarity of these arrays of atoms to solid state lattices. Each atom stores one quantum bit of information. A major drawback of the 3D optical lattice approach is that atoms on the inside of this 3D array cannot be addressed by laser beams without disturbing other atoms (quantum bits) along the laser beam path, greatly limiting the quantum computing operations that can be performed on this system.
This thesis investigates a way to solve this problem by creating a one-dimensional
(1D) or two-dimensional (2D) optical lattice above the surface of an optical waveguide –
an optical atom chip. In this 1D or 2D geometry atoms can be addressed by laser beams
from above or below, or, in the 1D case, from the sides. In addition to its anticipated use
for quantum computing, this is the first array of microtraps this close to a dielectric surface and could be used to measure the interactions between atoms and surfaces. It also
3 provides the unique opportunity to investigate the quantum physics of 1D and 2D
systems.
In chapter 2 we will review the basics of quantum computing, trapping of atoms
using light forces, and state of the art of quantum computing with neutral atoms in an
optical lattice. In chapter 3 we will present the fundamental idea of our optical atom chip,
including relevant trap properties of the basic 2D lattice and some variations of it, based on results from computations. Chapter 4 contains a detailed description of the experimental setup built to demonstrate the 2D optical lattice in the laboratory. In chapter
5 we present the design and characterization of a neutral atom detector for 2D imaging of atom samples that was developed in the course of our studies. Chapter 6 summarizes the main conclusions of this work.
4
Quantum Computing Qubit System Entanglement References Approach Interaction Nuclear Magnetic Nuclear spin in a Spin-spin coupling [4, 5] Resonance (NMR) molecule (liquid state) or solid state Trapped Ions Internal spin Phonons, photons, [6, 7] states of trapped “head” ions ions Neutral Atoms in Internal or Electric or magnetic [8, 9] Optical Lattices motional states of dipole-dipole neutral atoms interaction, cold collisions Cavity Quantum Internal or Atom-photon [10] Electrodynamics motional states of interaction atoms in cavities, photons in cavities Optical Polarization Prior entanglement [11] modes of photons through parametric down conversion Solid State Spin or charge in Heisenberg [12, 13] quantum dots, P exchange interaction “impurities” in Si Superconducting Charge, flux, or NMR interaction [14, 15] energy levels in a superconducting circuit using a Josephson junction
Table 1.1: List of quantum computing approaches. This list is not all-inclusive. For more information and additional references, see the Quantum Computing Roadmap [3].
5
CHAPTER 2
REVIEW
2.1 Quantum Computing 101 – An Introduction
The following is a brief review of the most important aspects of quantum computing. A more detailed introduction can be found in the textbook by Nielsen and
Chuang [16] and in the lecture notes by John Preskill [17].
2.1.1 The Qubit
In a conventional computer, information is encoded in the state (zero or one) of a bit. The quantum equivalent of such a bit is called quantum bit (qubit) as introduced by
Ben Schumacher in 1993. A qubit is a two-state quantum system, with two orthonormal basis states, called 0 and 1 . Being a quantum system, a qubit can assume any superposition of its basis states:
ψ = a 0 + b 1 1−qubit where a and b are complex amplitudes satisfying the normalization condition
a 2 + b 2 = 1.
Any qubit state can thus be parameterized as follows:
6 θ θ ψ = eiγ (cos 0 + eiϕ sin 1 ) . 2 2
Since the global phase factor eiγ has no observable effects, we can ignore it. The single
qubit state can, thus, be written as
θ θ ψ = cos 0 + eiϕ sin 1 . 2 2
Here we have chosen to make the coefficient of 0 real and positive (by appropriate choice of global phase). The parameters θ and ϕ can assume values between 0 and π, and 0 and 2π, respectively. The two angles thus define a point on the three-dimensional
unit sphere, in this context called the Bloch sphere, depicted in Figure 2.1 (a). The Bloch
sphere representation is helpful for visualizing single qubit gates.
As governed by the rules of quantum mechanics, when measuring the state of a
qubit in state ψ , the result will be 0 with probability a 2 , or 1 with probability 1−qubit
b 2 . This means that the state of a qubit cannot be determined completely by a single
measurement. Many measurements on identical qubits need to be performed to get the
coefficients with high accuracy.
The wavefunction for a set of n qubits is:
ψ = c 0 ... 0 + c 0 ... 0 1 + ...+ c 1 ...1 , 0...0 1 n 0...01 1 n−1 n 1...1 1 n
where the complex amplitudes c n obey the normalization condition i∈{0,1}
2 ∑ ci = 1. i∈{0,1}n
7 There are 2n terms, so there are 2n complex amplitudes which carry information,
whereas in a classical array of n bits there can be only one input at a time, carrying n bits
of information. This exponential dependence of the number of bits encoded in an n qubit
quantum state allows for exponentially faster calculations. By the same means, though, it
would take many measurements to determine each of the 2n complex amplitudes with high accuracy. So, while we get the result of a calculation much faster on a quantum computer, we lose the exponential speed-up upon read-out of this result.
Through careful choice of an algorithm the result of a calculation may be
extracted without determining the full state of each qubit. Currently only very few
algorithms are known that can harness the tremendous speed-up of a quantum
computation without losing it at the read-out stage. The two main classes of quantum
algorithms known so far are algorithms based on Shor’s Fourier transform (exponential
speed-up over the fastest known classical algorithms) and algorithms based on Grover’s
quantum search algorithm [18] (quadratic speed-up). The former class includes Shor’s
factoring algorithm [2] for factoring large numbers, enabling the breaking of codes
encrypted with the RSA cryptosystem. Another algorithm known to be faster than the
best possible classical algorithm is Deutsch’s algorithm [19] for determining whether a
function of one qubit is constant or balanced, which can be extended to many qubits [20].
While this demonstrates the power of quantum computation, there are no known applications of these two algorithms to date.
8 2.1.2 Single Qubit Evolution
The time evolution of a quantum system acted on by a Hamiltonian Hˆ is
governed by the time-dependent Schrödinger equation:
ˆ r ∂ r H ψ (r,t) = ih ψ (r,t) ∂t
The formal solution of this equation written in the Schrödinger picture is:
ˆ ψ (r,t) = e −iHt / h ψ (r,t = 0)
The evolution of the quantum system is described completely by the time-dependent state
of the system.
ˆ Suppose the internal Hamiltonian of a qubit is H 0 . We choose our computational basis to be two eigenstates of this Hamiltonian, namely
ˆ H 0 0 = E0 0
and
ˆ H 0 1 = E1 1 .
The qubit state thus evolves as
ˆ ˆ ψ (0) = a 0 + b 1 → ψ (t) = ae −iH 0t / h 0 + be −iH 0t / h 1
= ae−iE0t / h 0 + be−iE1t / h 1
= e−iE0t / h (a 0 + be−i(E1 −E0 )t / h 1 ) .
Without any (externally driven) interaction the relative phase between the two basis states thus oscillates. During quantum computations, only relative phases acquired through interaction of qubits with the environment or interaction between qubits are
9 relevant. To simplify the notation we will absorb the above, internal phases into the basis
states, thus defining
0 ≡ e−iE0t / h 0
and
1 ≡ e−iE1t / h 1 .
Note that, while this is essentially the state as described in the Heisenberg (or
interaction) picture, we will be indicating some single qubit time-evolution explicitly below, in cases where the phase is relevant for a quantum computation operation.
For quantum computation, gates have to be performed on the qubits. It has been shown that any quantum algorithm can be implemented by a set of universal single and two-qubit gates [21, 22]. A list of frequently used single qubit gates is shown in Figure
2.1 (b). One particularly important single qubit gate is the Hadamard gate, which has the following truth table:
1 ( 0 + 1 ) 0 ⎯Hadamard⎯→⎯⎯ 2 1 1 ( 0 − 1 ) 2
When applied to each qubit in an array of n qubits initialized to state
ψ = 0 ... 0 0 1 n
the resulting state is an equal superposition of all possible input states:
1 Hadamard n ψ = ( 0 ... 0 + 0 ... 0 1 + ... + 1 ...1 ) , 0 n 1 n 1 n−1 n 1 n 2 an ideal starting point for calculations.
10 The action of the Hadamard gate can be visualized on the Bloch sphere, as shown
in Figure 2.1 (c) for an equal superposition of 0 and 1 . It is a rotation of the qubit state about the y axis by 90°, followed by a reflection through the x-y plane.
2.1.3 Decoherence
The ability to perform quantum algorithms on any of the suggested physical
systems is limited by decoherence. Any actual quantum system is not entirely closed.
Some interaction with the environment (i.e. any state or system outside the computational
basis) exists. This interaction can lead to decoherence (or “quantum noise”). During the
intended evolution of a system of n qubits during quantum computation, the phases of
each basis state will evolve as dictated by quantum mechanics:
ψ = c eiϕ0...0 (t) 0 ... 0 + c eiϕ0...01 (t) 0 ... 0 1 + ...+ c eiϕ1...1 (t) 1 ...1 0...0 1 n 0...01 1 n−1 n 1...1 1 n
Decoherence occurs when the relative phases of the basis states evolve randomly. A random change of the global phase of the quantum state does not affect the coherence of the state, since it has no observable effect on the quantum state. Random evolution of the relative phases may be introduced through an uncontrolled interaction (coupling) with the environment. Such interaction may entangle the n-qubit system with the environment.
Any evolution of the environment can then affect the relative phases of the basis states of
the n qubit system. Since the evolution of the environment can not be controlled, the
relative phases of the qubit system will change randomly. The coherence of the quantum
state is lost. The stronger the coupling to the environment, the faster decoherence occurs.
11 Two examples of importance in the systems discussed in this work shall be mentioned here. A neutral atom in an optical lattice potential can couple to the environment through several processes. One is absorption of a photon from the optical lattice lasers followed by spontaneous emission. After the emission the neutral atom may be in a state that is outside of the computational basis entirely, or, at least will have acquired a random phase due to the random nature of the spontaneous emission process.
Reducing the absorption probability (e.g. through increased detuning or lower intensity of the trapping lasers) reduces the spontaneous emission rate and, thus, the decoherence rate.
Another decoherence path of a neutral atom in an optical lattice is as follows.
Usually, the computational basis states suggested for quantum computing in optical lattices are internal states of the atom. In particular, most proposed qubit states are magnetic substates of the two hyperfine ground state levels of alkali atoms. While not part of the computational basis, the vibrational state of the atom in the optical lattice trapping potential can have an effect on the evolution of the qubit states. In general, the evolution of the qubit state will be different for different vibrational states. Random hopping between vibrational states of the trapping potential (“motional heating”) can therefore cause random phase evolution of the qubit state, i.e. decoherence. The tighter the trapping potential is, the larger the energy difference between the vibrational states.
Larger energy difference means smaller coupling between the vibrational states, and therefore a smaller decoherence rate.
12 2.1.4 Entanglement of Two Qubits
Implementation of general quantum algorithms requires a universal set of quantum gates. A universal set of quantum gates consists of single qubit operations, and a two-qubit quantum gate. To realize such two-qubit gates one needs to entangle two qubits. An entangled state is a two- or more qubit quantum state that cannot be decomposed into a product of single qubit states. A general two-qubit state can be written as
ψ = a 0 0 + b 0 1 + c 1 0 + d 1 1 , 1 2 1 2 1 2 1 2 where the complex amplitudes a, b, c and d satisfy the normalization relation a 2 + b 2 + c 2 + d 2 = 1, and the indices 1, 2 refer to qubit 1 or 2. For a = b = c = d = 1/2 we get the product (i.e. not entangled) state
ψ = 1 ( 0 0 + 0 1 + 1 0 + 1 1 ) initial 2 1 2 1 2 1 2 1 2
= 1 ( 0 + 1 ) × 1 ( 0 + 1 ) . 2 1 1 2 2 2
An example of an entangled state is
ψ = 1 ( 0 0 + 0 1 + 1 0 − 1 1 ) . final 2 1 2 1 2 1 2 1 2
This state cannot be written as a product of one-qubit states. It is a special case of the result of a controlled phase (cPHASE) gate on the initial state ψ . The truth table for initial the cPHASE gate is
00 00 01 01 ⎯cPHASE⎯→⎯⎯ , 10 10 11 eiϕ 11
13 with some phase ϕ . The special case above is a “π cPHASE gate” with ϕ = π .
This gate together with single qubit gates constitutes a set of universal quantum gates.
Other common two-qubit gates that can be used instead are the controlled NOT (cNOT)1 or the SWAP 2 gate. The cNOT gate can be created by applying a Hadamard gate to the target qubit before and after a π cPHASE gate. This shows that these two universal sets of gates are equivalent.
1 The cNOT gate is the quantum equivalent of the classical XOR gate (where the second qubit is the output).
Depending on the state of the first (“control”) qubit, the second (“target”) qubit is flipped (notation:
i j ≡ ij ): 1 2
00 00 01 01 ⎯cNOT⎯→⎯ 10 11 11 10
For appropriate input states this will lead to entanglement.
2 Applying the SWAP gate twice results in the switching of the states of qubits 1 and 2 (therefore the
name “SWAP”). The “truth table” for this two-qubit quantum gate is (notation: i j ≡ ij ): 1 2
00 00 1− i 1+ i 01 01 + 10 ⎯⎯→SWAP⎯ 2 2 1+ i 1− i 10 01 + 10 11 2 2 11
14 In general, to create entanglement, the evolution of one of the two qubits has to be conditional on the state of the other. This is achieved with an interaction between the qubits that depends on the states of both qubits.
For example, to implement a cPHASE gate, suppose such an interaction is
ˆ described by a Hamiltonian H int , which only has a non-zero matrix element for basis state 1 1 , such that: 1 2
Hˆ i j = E δ δ i j , int 1 2 int i1 j1 1 2
where i, j = {0,1} are the states of qubits 1 and 2,δ kl is the Kronecker symbol (0 for k ≠ l , 1 for k = l ), and Eint is the interaction energy due to the interaction between the
ˆ ˆ ˆ ˆ two qubits. The complete system Hamiltonian is H = H 0 + H int , where H 0 acts on each of the two qubits as described in section 2.1.2.
We assume the two-qubit interaction can be turned on and off at will. To perform a gate, we turn on the interaction for an appropriate amount of time tgate. The evolution of a two-qubit state during this gate is as follows (see Figure 2.2):
ψ = 1 ( 0 0 + 0 1 + 1 0 + 1 1 ) initial 2 1 2 1 2 1 2 1 2
ˆ ˆ → e −i( H 0 +H int )tgate / h 1 ( 0 0 + 0 1 + 1 0 + 1 1 ) 2 1 2 1 2 1 2 1 2
= 1 ( 0 0 + 0 1 + 1 0 + e−iEinttgate / h 1 1 ) . 2 1 2 1 2 1 2 1 2
π For a gate time of t gate = h , this results in a π phase gate: Eint
ψ = 1 ( 0 0 + 0 1 + 1 0 − 1 1 ) final 2 1 2 1 2 1 2 1 2
15 This is how an entangled state can be created through a conditional two qubit interaction.
As we can see, the stronger the interaction Eint between the qubits is, the shorter the gate
4 time will become. In order to do many operations (>10 ) before decoherence occurs, tgate has to be much shorter than the typical coherence time of the system. Examples of how entanglement can be achieved in actual physical systems are discussed below.
2.1.5 The Five Commandments of Quantum Computation
Five requirements for the successful implementation of a quantum computer have been identified by David DiVincenzo [23]:
1. A two-state quantum system to serve as a qubit, with the possibility of scaling
the system up to large numbers of qubits (~104)
2. The ability to initialize the state of the system of qubits to a well-defined input
state
3. The ability to perform a universal set of quantum gates, including (a) single
qubit operations and (b) two-qubit gates.
4. Long coherence times. For useful computations ~104 operations have to be
performed before decoherence occurs
5. The ability to make a projection measurement of the qubit state (often referred
to as “read-out”).
The fundamental difficulty is to find a way for two qubits to interact strongly with one another to perform fast gates, while suppressing the interactions with the environment that lead to decoherence.
16 2.2 Trapping Atoms with Light
Next, we briefly review how atoms can be trapped in intensity patterns of light.
The force responsible for the trapping of atoms in optical lattices is the dipole force. This force results from the interaction of the laser light field with the electric dipole moment of the atom induced by the laser light field, otherwise known as the “AC Stark effect”.
This interaction causes a shift in energy levels dependent on the intensity of the light. We show this here using the dressed atom approach [24]. We start out with the energy levels of the light-atom system, i.e. a single atom and a single mode of radiation field (laser mode), assuming no interaction between them. This system is described by the
ˆ ˆ ˆ ˆ ˆ Hamiltonian H 0 = H A + H L , where H A is the Hamiltonian of the atom and H L the
Hamiltonian of the laser field. Together the atom and light field form a system called the
“dressed atom”. We will treat the simplified case of a two-state atom (with ground state
g and excited state e ) in a single mode light field (wavelength λ, frequency ωL, n photon state n ). Note that g and e are not two states of a qubit ( 0 and 1 ).
For this simplified system the Hamitonian can be written as
ˆ ˆ ˆ H 0 = H A + H L
= (,Eg g g + Ee e e ) ⊗ I L + I A ⊗ ∑ En n n n
ˆ where the energy eigenvalues of the atomic Hamiltonian H A are E g = 0 and Ee = hω 0
(for an atomic transition frequency of ω0 ), the eigenvalues of the light field Hamiltonian
ˆ H L are the n photon energies En = nhω L (for a photon frequency of ωL ), and I L and
I A are the identity operators on the laser field and the atom, respectively. 17 The eigenstates of this system are the states g ⊗ n ≡ g,n and
e ⊗ n ≡ e,n . The energy eigenvalues (denoted E g ,n and Ee,n ) of the dressed atom
system are the sums of the eigenvalues of the atomic Hamiltonian, E g or Ee , and the
eigenvalues of the light field Hamiltonian, the n photon energies En . For a laser tuned
near resonance, i.e. a small detuning ∆ ≡ ωL − ω0 , this results in an energy level diagram with nearly degenerate pairs of levels (see Figure 2.3) with:
E g,n+1 = (n +1)hω L
Ee,n = hω 0 + nhω L
= (n +1)hωL − h∆ .
Now we introduce the electric dipole interaction between the atom and the laser
ˆ ˆ ˆ ˆ ˆ field H int = −d ⋅ E , where d and E are the dipole and electric field operators of the atom and the laser field, respectively:
ˆ t d = d eg e g + d eg g e ,
ˆ r ˆ ∗ r ˆ ∗ E = ∑(E0 (r,t)ε n +1 n n +1 + E0 (r,t)ε n +1 n +1 n ) , n
where d eg ≡ e e rˆ g is the dipole transition matrix element for the g → e transition,
r e is the elementary charge, E0 (r,t) is the electric field value, and εˆ is the electric field polarization.
Energy conservation permits coupling between the states g,n +1 and e,n , i.e. either an atom in the ground state absorbs a photon and goes to the excited state, or a
18 photon is emitted into the laser field by an excited atom through stimulated emission,
ˆ leaving the atom in the ground state. The full expression of H int is
ˆ ˆ ˆ H int = −d ⋅ E
ˆ t ˆ = −∑ (d eg e g ⊗ EL (r,t)ε n +1 n n +1 + d eg g e ⊗ EL (r,t)ε n n +1 n ) n
To characterize the interaction strength of the light-atom interaction, we define the Rabi frequency Ω by
Ω e,n Hˆ g,n +1 ≡ h . int 2
This yields the following expression for the Rabi frequency:
2e r Ω = − E0 (r,t) n +1 e rˆ ⋅εˆ g . h
Note that other definitions of the Rabi frequency exist in the literature.
ˆ ˆ ˆ ˆ The complete system Hamiltonian H = H 0 + H int , written in the H 0 eigenbasis
( g,n +1 , e,n ), is composed of blocks for the ( g,n +1 , e,n ) manifold for each n.
ˆ The blocks H n can be written as follows:
⎛ hΩ ⎞ ⎜(n +1)hωL ⎟ ˆ 2 H n = ⎜ ∗ ⎟ . ⎜ hΩ ⎟ ⎜ (n +1)hωL − h∆⎟ ⎝ 2 ⎠
Diagonalizing the complete system Hamiltonian Hˆ yields the new energy eigenvalues:
∆ Ω' E = (n +1) ω − h + h 1,n h L 2 2
h∆ hΩ' E2,n = (n +1)hω L − − , 2 2
19 with Ω'≡ Ω 2 + ∆2 . The eigenstates (“dressed states”) are superpositions of g and
e , which we will denote as 1,n and 2,n .
What is the shift in energy levels due to the laser field? For a laser light mode tuned above resonance ( ∆ > 0 ), Eg,n+1 is the higher level of the nearly degenerate pair, and turns into E1,n when the laser field and, therefore, the interaction is turned on. In this case, the energy level shift for atoms that are in the ground state (outside of the laser field) is
∆E g = E1,n − E g,n+1
= (Ω'−∆) h 2
= ( ∆2 + Ω 2 − ∆) . h 2
Throughout this work, we will be working in the low intensity, large detuning regime, so that ∆ >> Ω . In this case, the above expression can be approximated by:
Ω 2 ∆E = h (2.1) g 4∆
≡ U light .
This is the light potential seen by an atom in the ground state. Now we will put this expression into more familiar terms, involving well-known properties of the laser and atom. A common definition of the saturation intensity Isat of the atomic transition leads to the expression:
Γ 2 I Ω 2 = , 2I sat where Γ is the spontaneous decay rate of the atom and I the laser intensity. 20 This results in a light potential of: r r hΓ Γ I(r ) U light (r ) = , 8 ∆ I sat where we explicitly indicate the position dependence of the intensity of the light field.
The light potential Ulight is a function of position if the intensity is not constant; for example, in a Gaussian laser beam, or a laser standing wave, i.e. an optical lattice. The potential gradient results in a conservative force on the atoms towards the potential minima – the dipole force. This is how the atoms are trapped in a light field. For
∆ > 0 (blue detuning), the potential minima will be at minima (or nodes) of intensity – the atoms are weak-field seekers. For ∆ < 0 (red detuning) atoms are trapped at intensity maxima or antinodes – they are strong-field seekers.
Real atoms have a more complex level structure. The above expression found for the light potential can be extended to atoms with multiple levels by adding up the contributions of all levels of the atom (see [25]). We will explicitly give the relevant expressions for alkali atoms, since these are the atoms we use throughout this work. As can be seen in Figure 2.4 there is one fine structure (spin(S)-orbit(L) coupling) ground
2 3 2 2 state ( S1/ 2 ) and two fine structure excited states ( P1/ 2 and P3 / 2 ), also called the D
2 2 2 2 doublet with S1/ 2 → P1/ 2 commonly referred to as D1, and S1/ 2 → P3 / 2 referred to as
D2. These states, in turn, are split by the hyperfine interaction (coupling of the nuclear
3 This notation indicates quantum numbers for the total spin (S), total orbital angular momentum (L) and
2S +1 total angular momentum (J) of the state of the atom in the following manner: LJ , where
L = S, P, D, F,... ≡ 0,1,2,3,...
21 spin I to the total angular momentum J of the electron), characterized by the quantum number F. The hyperfine levels consist of a set of magnetic substates mF, which are degenerate in the absence of any external fields. Due to selection rules, transitions
between magnetic substates are dependent on the polarization of the laser light ( ∆mF = 0
+ − for π-polarized light, ∆mF = ±1 for right- (σ ) and left- (σ )circularly polarized light, respectively). Accounting for all of these factors the light potential becomes [25]:
Γ 2 ⎛ 2 + Pg m 1− Pg m ⎞ I(r ) U (r) = h ⎜ F F − F F ⎟ , light ⎜ ⎟ 8 ⎝ 3∆ 2,F 3∆1,F ⎠ I sat where P indicates the polarization of the laser light ( P = 0 for linear polarization, P = ±1
± for σ light), gF is the Landé factor, mF the magnetic substate, ∆1,F is the detuning from
2 2 2 the S1/ 2 , F→ P1/ 2 transition (using the center of the hyperfine split P1/ 2 level), and ∆2,F
2 2 is the detuning from the S1/ 2 , F→ P3 / 2 transition.
For what follows we assume that the laser light is tuned much closer to the D2
line than the D1 line ( ∆1,F >> ∆ 2,F ), and ignore the second term in the parentheses. The light potential can thus be written as:
2 r r hΓ ⎛ 2 + Pg F mF ⎞ I(r) U light (r) = ⎜ ⎟ , (2.2) 8 ⎝ 3∆ ⎠ I sat
where we defined ∆ ≡ ∆ 2,F .
In most of our calculations below we ignore the second (“magnetic”) term, and thus the light potential is independent of the light polarization.
22 2.3 Quantum Computing with Neutral Atoms in Optical Lattices
2.3.1 Qubits and Scalability
The qubit of the neutral atom approach for quantum computing is usually encoded into two internal states of an atom. Most cold atom experiments use alkali atoms. Their internal structure lends itself to efficient laser cooling and trapping. Cooling requires a cycling transition that allows to continuously run the dissipative absorption-spontaneous emission cycle. The qubit states chosen are usually magnetic substates of the two hyperfine ground states of alkali atoms.
A useful quantum computer requires many (>104) qubits. How can we create a suitable array of single atoms (i.e. qubits)? As was mentioned previously, atoms can be trapped in a periodic intensity pattern of light – an optical lattice. It is possible to create
3D optical lattices with millions of sites. Loading one atom into each of these sites yields a large array of atomic qubits. This system is a highly scalable approach to quantum computing.
2.3.2 Properties of Free Space Optical Lattices
How exactly is an optical lattice created, and what are its properties? Optical lattices are standing wave patterns created by interfering laser beams. The 3D optical lattices suggested for use in neutral atom quantum computing are created by overlapping three counter-propagating pairs of laser beams. With appropriate choice of laser polarizations it is possible to create two optical lattices simultaneously that each trap atoms in a particular magnetic substate. The two lattices (and therefore the two arrays of atoms) can be moved relative to each other by adjusting the laser polarizations. This
23 allows for massive parallelism during computation, speeding up the computation. In the following we will demonstrate step by step how such a double-lattice is created.
Let us begin with a one-dimensional optical lattice along z, which is simply a laser standing wave created by overlapping two counter-propagating laser beams. The electric field of one laser beam is:
r i(kz−ωt+ϕ ) E(z,t) = E0 ⋅εˆ ⋅ e , where E is the amplitude, εˆ is a unit polarization vector, k = 2π is the propagation 0 λ constant, λ is the laser wavelength, ω is the laser frequency, and ϕ is an additional phase.
We begin with the case of two counter-propagating laser beams with parallel polarizations along x, as shown in Figure 2.5. This optical lattice configuration is called
“lin||lin”.We find the total field to be:
r i(kz−ωt) −i(kz+ωt) Elin||lin (z,t) = E0 xˆe + E0 xˆe .
−iωt = 2E0 xˆe cos(kz)
This is a standing wave along z with linear polarization everywhere. Linear polarization is a superposition of left- (σ − ) and right- (σ + ) circularly polarized light. The same 1D linearly polarized standing wave is thus a superposition of a σ + - and a σ − - polarized optical lattice:
r −iωt −iωt Elin||lin = 2E0e eˆ+ cos(kz) + 2E0e eˆ− cos(kz)) ,
1 where we have introduced the circular polarization vectors eˆ± = (xˆ ± iyˆ) (for 2 quantization axis z). The (linear) standing wave pattern can be decomposed into two
24 overlapping standing waves with left and right circular polarization. The light potential
(see Figure 2.5 (c)) is obtained by plugging the electric field into equation (2.2).
Next we will treat the case of two counter-propagating electric fields with perpendicular polarizations (along x and y). This beam configuration is called “lin⊥lin”.
In order to simplify the expressions, we choose a relative phase of π 2 between the two fields. This has no effect on the resulting field except for an overall spatial shift along z.
The distance between the two optical lattices (σ − and σ + ), though, remains unaltered by adding this phase. The resulting electric field in this case is (see Figure 2.5):
r i(kz−ωt) −i(kz+ωt−π / 2) Elin⊥lin (z,t) = E0 xˆe + E0 yˆe )
−iωt ikz −i(kz−π / 2) = E0e (xˆe + yˆe )
−iωt ikz −ikz = E0e (xˆe + iyˆe )
−iωt = E0e ((xˆ + iyˆ)cos(kz) + i(xˆ − iyˆ)sin(kz))
−iωt = 2E0e (eˆ+ cos(kz) + ieˆ− sin(kz)) .
The σ + and σ − standing waves are thus shifted (spatially) by k∆z = 180°, or ∆z = λ / 2 with respect to one another. When we calculate the potential (equation (2.2)), the electric field is squared, and thus the two potentials (σ + and σ − ) are shifted by λ/4 with respect to one another.
Similar considerations show that by adjusting the polarization angles of the two laser polarizations between 90° and 0°, the two lattices (and therefore the two arrays of atoms) can be moved together and apart smoothly.
25 The two optical lattices trap atoms in different magnetic substates (due to their different polarizations). As can be seen in equation (2.2), the light potential depends on
+ the magnetic substate mF . Atoms in one substate will feel the potential due to the σ light more strongly, atoms in the other substate will feel the potential due to the σ − light more strongly, as can be seen in Figure 2.5 (c) and (d). This 1D optical lattice, however, is not suitable for quantum computing because its scalability is limited. The lattice is created along the focus of two counter-propagating Gaussian laser beams and the number of traps is limited by the Rayleigh range of these lasers. To make the Rayleigh range large, the beam needs to be as loosely focused as possible. On the other hand, the trapping in the radial dimensions of the laser beam is achieved solely by the Gaussian beam profile, and will only trap atoms for red-detuned laser beams. To get a tight trap
(i.e. small motional heating rate) in the radial direction the laser beam needs to be tightly focused, or have a large amount of laser power. Because of these constraints, 1D optical lattices are not suitable for quantum computing.
To solve this problem, three-dimensional (3D) optical lattices have been suggested for use for quantum computing. A three-dimensional (3D) optical lattice can be created by overlapping three counter-propagating pairs of laser beams. This has two advantages: The number of trapping sites is much larger, and the trapping sites are tightly confined in all three dimensions, as required to reduce the motional heating rate. Usually, two pairs are in lin||lin configuration, and the third pair has an adjustable polarization angle to allow changing the distance between the σ + and σ − lattices along this dimension (see Figure 2.6 (a)). A 3D optical lattice can also be achieved with only four laser beams at appropriate angles and polarizations, as shown in Figure 2.6 (b) (see [26]), 26 but this beam configuration does not allow for relative movement of the σ + and σ − lattices. For quantum computation, one can fill the two lattices (σ + , σ − ) with one atom per site (atoms are in different magnetic substates), bring the two atoms together by adjusting the polarization angles, and perform a fast two-qubit gate.
A major drawback of this approach is that atoms cannot be addressed individually, due to the 3D nature of the lattice, depicted in Figure 2.6 (c). The optical path to address sites on the inside will overlap with numerous other sites. Therefore, addressing a single site without disturbing others is difficult. Also, the site spacing is λ/2, too small to be able to focus a laser beam enough in order not to disturb the neighboring atoms. As mentioned below, there are ideas for performing quantum computations without being able to address individual atoms, but doing so certainly reduces the amount of control one has over the quantum system and limits the possibilities for two-qubit gates.
2.3.3 Initialization
For quantum computations, it is desirable to have one atom in each lattice site, in its motional ground state, and in a well-controlled initial internal state. Optical lattice cooling techniques exist that can cool atoms to their motional ground states [27], and the internal state can be controlled by laser light as described below for single-qubit gates.
The biggest challenge is the loading of single atoms into the lattice. Because of recent advances in cooling and trapping of atoms, it is now possible to load a Bose-Einstein condensate (BEC) of atoms into a 3D lattice. The atoms in the BEC all occupy the same state. Their wavefunctions are coherent. This means their phase is well-defined. Because 27 of Heisenberg’s uncertainty principle, when a BEC is placed in an optical lattice, the atom number per lattice site is unknown. It has been demonstrated [28] that increasing the strength of the lattice potential leads to a Mott-Insulator phase transition, i.e. a transition from a BEC to a Mott-insulator. The Mott-Insulator state has a well-defined atom number in each lattice site while the phase coherence of the atomic wavefunctions is lost completely. When loading with a sufficiently small number of atoms, there will be exactly one atom per lattice site. Because the atom sample was extremely cold (~0.1 nK), the atoms are automatically in the motional ground state, and additional cooling techniques can be applied to keep them there [27].
2.3.4 The Rabi Two-Level Problem
Because of its relevance for the implementation of single qubit gates in neutral atom quantum computing approaches, we will briefly review the Rabi two-level problem.
The underlying physics of this process is the same as described for the interaction of a two-state atom with a single mode of laser light in section 2.2. We can calculate the probability for an atom to be in state e after a time t (see [29] for derivation):
Ω 2 Ω't P (t) = sin 2 , e Ω' 2 where Ω is the Rabi frequency and Ω'= Ω 2 + ∆2 is the effective Rabi frequency.
The population oscillates back and forth between states g and e . This is called Rabi- flopping. The oscillation frequency of the population between ground and excited state is
Ω' . If the laser is in resonance with the transition g → e ( ∆ = 0 ), the population
28 oscillates between the ground and excited state (oscillation amplitude = 1) with the Rabi- frequency Ω. For increased detuning, the amplitude of the oscillation decreases, and the frequency increases. The further detuned the laser is from the atomic transition, the smaller the probability to transfer population to the excited state becomes.
By applying timed laser pulses to an atom it is thus possible to control its state.
For a laser on resonance ( ∆ = 0 ), applying a pulse for a time
t = π π 2 2Ω will put an atom initially in the ground state in an equal superposition of states g and
e . This is called a π 2 -pulse. Applying a resonant pulse of twice this duration to an atom in state g will drive the atom to state e . This is called a π-pulse. A 2π-pulse will drive one full oscillation and bring the atom back to state g , but the state will acquire a phase of π during the oscillation, leaving the atom in state − g . The full sequence, in steps of π 2 -pulses is:
g ⎯π⎯→2 g + e ⎯π⎯→2 e ⎯π⎯→2 e − g ⎯π⎯→2 − g
2.3.5 Single-Qubit Gates and Read-Out
Single qubit operations may be performed by driving the qubit with a laser pulse resonant with the 0 → 1 transition, as described in the previous section. For the two hyperfine ground states of an alkali atom this can be either a radio-frequency (rf) pulse, or a pair of laser beams tuned to drive a stimulated Raman transition from 0 to 1 (see
29 Figure 2.7 (a)). The advantage of using lasers is that one can focus them tightly, so that individual addressing may be possible for optical lattices, where the spacing of sites is large compared to the focal parameters (e.g. CO2 far-off-resonant lattice [30]).
Read-out may be accomplished by driving a cycling transition. For example, one can drive a transition from the upper hyperfine level to a short-lived excited state that can only decay back to the same hyperfine level (see Figure 2.7 (b)). Fluorescence will only occur if the qubit was in state 1 . Thus, after many absorption-spontaneous emission cycles, the qubit state can be determined by observing either fluorescence (qubit state
1 ) or no fluorescence (qubit state 0 ). This measurement projects the qubit state onto the computational basis. If the qubit was in a superposition of the two states the original quantum state of the qubit is destroyed by the measurement (i.e. the qubit is in state 0 or 1 after the measurement). To determine the amplitudes of the superposition state, the measurement has to be repeated many times on an identical qubit state.
2.3.6 Two-Qubit Gates for Atoms in Optical Lattices
Several interactions between neutral atoms have been suggested for entanglement:
Electric dipole-dipole interaction [31, 32], magnetic dipole-dipole interaction [33], and cold collisions [34]. We will review the electric and magnetic dipole interaction schemes in detail here, since we will refer to them as part of the discussion of our 1D and 2D optical lattices in the next chapter.
30 2.3.6.1 Two-Qubit Gates Using the Electric Dipole-Dipole Interaction
Neutral atoms can interact with each other through their induced electric dipole moments. The qubit states are often ground states of a neutral atom that are not coupled to any other states. For symmetry reasons these states do not have an induced electric dipole moment. In all suggested electric dipole-dipole entanglement schemes, transitions to other states have to be driven by a laser to induce an electric dipole. The electric dipole-dipole potential is (in SI units):
1 1 r r r r ˆ ˆ U elec.dip. = 3 (d1 ⋅ d 2 − 3(d1 ⋅ r)(d 2 ⋅ r)), 4πε 0 r
r r where d1 and d 2 are the induced dipole moments of qubits 1 and 2, respectively, r is the distance between them, and we chose the axis rˆ is a unit vector pointing from qubit 1 to
qubit 2. ε 0 is the permittivity of free space.
Thus, to make fast qubit gates, the qubits have to have large dipole moments and/or have to be close together. To achieve large dipole moments, it has been suggested to excite the qubits to higher lying states (for example Rydberg states, i.e. states with a large principal quantum number) during the duration of the gate.
We will discuss here two entanglement schemes suggested by Jaksch et al. [32] in
2000. As shown in Figure 2.8 (a), the qubit basis states are e and g , and a third state
r (for “Rydberg” state) can be excited with a laser. The first entanglement scheme is achieved in the following manner (see Figure 2.8 (b)): In this scheme, the two qubits can be addressed by one and the same laser – no individual addressing of qubits is needed.
But, to prevent interference from neighboring Rydberg atoms, it is necessary to resolve
31 individual pairs of qubits. We assume that the laser intensity is high (hΩ >> U elec.dip. ), and the detuning from the g → r transition is ∆ = 0 . A phase gate is performed by the following sequence of laser pulses: (1) Apply a π pulse to both atoms, (2) wait for a time
hϕ t gate = , (3) apply another π pulse to both atoms. The effect of this sequence on U elec.dip. the qubits is as follows:
ee ee ee ee eg er er − eg ⎯⎯→(1) ⎯⎯→(2) ⎯⎯→(3) . ge re re − ge gg rr e −iϕ rr e −iϕ gg
In step (1) the resonant π pulse transfers all of the population of state g to r .
The parts of the qubit wavefunctions that are initially in state e are unaffected by this pulse, since the laser is far detuned from this transition. The ge and eg components undergo an overall 2π pulse during step (2), flipping their sign. The gg component
U t will acquire a phase of ϕ = elec.dip. gate during step (2), in addition to a sign change due h to the second π pulse.
The minus signs in front of the ge and eg components are single-qubit phases acquired by the qubit components due to the (external) laser field. Unlike the internal phases, these phases are relevant for the computation, and have to be undone to complete the gate action. This is done by applying a 2π pulse to each of the two qubits individually, resulting in the transformation:
32 ee single ee qubit − eg 2π eg ⎯⎯→pulses⎯ − ge ge e −iϕ gg e −iϕ gg
Altogether, we have performed a cPHASE gate:
ee + eg + ge + gg → ee + eg + ge + e−iϕ gg .
The second scheme makes use of the so-called dipole blockade, where the dipole- dipole interaction causes a significant increase in the detuning (see Figure 2.8 (c)) that reduces the phase acquired by one qubit component significantly (compared to all other components), also resulting in a phase gate: In this scheme two lasers address the two qubits individually. We assume a large electric dipole interaction (U elec.dip. >> hΩ j , with j=1, 2 for the laser beams addressing qubit 1 and 2, respectively). Again, the laser detuning from the g → r transition is zero for both lasers. A gate is performed with the following sequence: (1) Apply a π pulse to the first atom, (2) a 2π pulse (i.e. twice the duration of (1)) to the second atom, and (3) another π pulse to the first atom. The state
ee remains unaffected by the pulses. The two components eg and ge both undergo a 2π pulse, which flips their sign (corresponding to an accumulated phase of π). The gg component will undergo the transformation:
iπ i(π −ϕ~) ~ gg ⎯⎯→(1) e 2 rg ⎯⎯→(2) e 2 rg ⎯⎯→(3) ei(π −ϕ ) gg .
The phase ϕ~ acquired in the second step is:
∆E t ϕ~ = g (2) . h
33 Using equation (2.1) for the ground state light shift, and plugging in the time for a
2π 2π pulse (t(2) = ), we get the expression: Ω 2
Ω 2 2π πΩ ϕ~ = 2 = 2 . 4∆ Ω 2 2∆
The detuning from the rg → rr transition is U elec.dip. h , which is large
(U elec.dip. >> hΩ 2 ). We therefore get:
π Ω ϕ~ = h 2 << π . 2U elec.dip.
The dipole blockade prevented a phase accumulation of the rg state during step
(2), so that overall, gg experiences a 2π pulse as well. After correcting for single qubit phases ( − ge ,− eg → ge , eg ), we have achieved a π cPHASE gate:
ee + eg + ge + gg → ee − eg − ge − gg
single qubit operations→ ee + eg + ge − gg .
Though more difficult to implement since addressing of individual qubits is required, this scheme has the advantage that, due to the dipole blockade the population of the second qubit never actually gets transferred to the Rydberg state. This reduces the momentum transferred to the atoms during gate operation, and, hence, the decoherence probability.
In a third scheme suggested by Jaksch et al. a designer laser pulse with Ω(t) and
∆(t) applied to both qubits (no individual addressing necessary) results in a cPHASE
34 gate. This is a more realistic situation since the laser beams do not turn on instantly, like we have been assuming thus far.
2.3.6.2 Two-Qubit Gates Using the Magnetic Dipole-Dipole Interaction
The magnetic dipole-dipole interaction has also been investigated as a possible entanglement scheme [33]. Though the permanent magnetic dipole moment is very small
(magnetic dipole-dipole interaction energies are typically ~10-4 times smaller than electric dipole-dipole interaction energies), the decoherence due to spontaneous emission is greatly suppressed, since magnetic substates of ground states can be used throughout gate operation instead of excited states. Since the magnetic dipole moment is permanent the interaction is always on. This problem has previously been treated in the context of
NMR quantum computing [35]. Two-qubit gates are implemented by undoing some of the evolution with appropriate single qubit pulses. r r For two qubits with magnetic dipole moments µ1 and µ2 , separated by a distance r, the magnetic dipole interaction is (in SI units):
µ 1 U = 0 (µr ⋅ µr − 3(µr ⋅ rˆ)(µr ⋅ rˆ)) magn.dip 4π r 3 1 2 1 2
µ µr ⋅ µr = − 0 1 2 , 2π r 3
where rˆ is a unit vector pointed from qubit 1 to qubit 2, and µ0 is the permeability of free space.
A possible choice of qubit states (see [33]) are the two magnetic substates of the
2 S1/2 level of an alkali atom, 0 ≡ J, M J = −1/ 2 and 1 ≡ J, M J = 1/ 2 with magnetic
35 moments of − µ B and µ B , respectively, where µB is the Bohr magneton. Rather than making the qubits interact for a certain gate time, the interaction between them is always on, and through clever application of single-qubit gates, an effective cNOT gate can be performed [35]. The requirement for being able to perform the gate is that the frequency
2µ µ 2 1 ∆f = 0 B CNOT π r 3 h can be resolved [33]. The corresponding gate time is:
1 tCNOT = . (2.3) 2πf CNOT
The closer together the qubits are, the higher the frequency difference and the shorter the
gate time t CNOT becomes.
However, single qubits have to be addressable. Derevianko et al. [33] suggest to address single qubits by shifting their frequency through a magnetic field gradient. The closer together the qubits are the higher a magnetic field gradient is required to resolve them. The frequency of the 0 → 1 transition is given by:
hf = ∆E 0 → 1
= (µ 0 − µ 1 )B
= 2µ B B ,
where µ 0 , µ 1 are the magnetic dipole moments of states 0 , 1 , and µ B is the Bohr magneton. The difference in transition frequency of neighboring atoms in a magnetic
dB field gradient is: dx
36 2 ∆f = µ ∆B h B
2 dB = µ ∆x . h B dx
In order to resolve the transition frequencies of neighboring atoms within a single
qubit gate time of t gate , we need a frequency difference between neighboring qubits of
∆f ≥ 1 . This requires a magnetic field gradient of at least t gate
dB h = . (2.4) dx 2t gate µ B ∆x
This concludes our review of relevant background material: the main ideas of quantum computing and the requirements for the physical implementation of quantum computing, the optical dipole force and potential that result in trapping of neutral atoms in optical lattices, and the state of the art of quantum computing with neutral atoms in optical lattices.
37
Figure 2.1: Single qubit gates. (a) Visualization of a single qubit state on the Bloch sphere. (b) List of most common single qubit gates. (c) Illustration of a Hadamard gate on the Bloch sphere. (Figures taken from [16].) 38
Figure 2.2: Entanglement sequence for a cPHASE gate. The 11 component acquires an additional phase beyond the single qubit evolution, due to the electric dipole-dipole interaction. For the gate timing shown, the gate is a π cPHASE gate.
39
Figure 2.3: Dressed atom energy diagram (shown for ∆ > 0 ). (a) Without interaction, the energy levels of the laser-atom system are nearly degenerate manifolds (splitting of ∆ ), separated by the energy of one laser photon, hωL . (b) The laser-atom interaction mixes the two states, creating the dressed states, with an energy splitting of Ω' . (c) Level shift in a position dependent laser field (e.g. Gaussian laser intensity profile). The energy splitting Ω'(r ) is now position dependent. (Figure adapted from [24].)
40
Figure 2.4: Energy level structure of alkali atoms. (a) For very large laser detunings
( ∆ >> ∆'FS ) the fine structure is not resolved and the atom can be treated as a two-level
system. (b) For large detunings ( ∆'FS ≥ ∆ >> ∆ HFS ), the fine structure becomes important.
2 2 2 2 The two major transitions are D1 ( S1/ 2 → P1/ 2 ) and D2 ( S1/ 2 → P3 / 2 ). (c) For
sufficiently small laser detuning ( ∆ HFS > ∆ ), the light potential depends on the hyperfine state F of the atom. The hyperfine levels shown are for nuclear spin I = 3/2.
41
Figure 2.5: One-dimensional optical lattice configurations. (a) Laser polarizations for different 1D optical lattice configurations. (b) σ + (blue dash) and σ − (red, solid) components of electric field for lin||lin and lin⊥lin configurations. (For lin||lin the σ + field was shifted up for clarity). (c) Light potential contributions from σ + (blue, long dash) and σ − (red, short dash) light and total potential (solid) for an 87Rb atom in
F = 1,mF = 1 . (d) Light potential for atoms in mF = 1 (black) and mF = −1 (red/gray).
All potentials are normalized to the maximum U0 of the total lin||lin potential.
42
Figure 2.6: 3D optical lattices. (a) Three pairs of counter-propagating laser beams result in a 3D optical lattice. Two pairs are in lin||lin configuration, while the third has an adjustable polarization angle θ (“linθlin” configuration), allowing for adjustment of the
+ − relative positions of the σ and σ lattices. (b) Alternative beam configuration for a 3D
+ − optical lattice. This configuration does not allow for shifting of the σ and σ lattices.
(Adapted from [36].) (c) 3D lattice structure. The cubic array of sites does not permit addressing of atoms on the inside of the lattice. 43
Figure 2.7: Single qubit operations for a neutral alkali atom (F numbers for 87Rb). (a)
Single qubit gates can be accomplished via laser pulses resonant with a Raman transition
(light red/gray and solid, blue arrows), or with an RF pulse resonant with the 0 ↔ 1 transition (dotted arrow). (b) Read-out is accomplished by detecting fluorescence from a cycling transition. If the qubit is in state 0 no fluorescence is detected.
44
Figure 2.8: Level diagrams for entanglement schemes using the electric dipole-dipole interaction. (a) Single qubit level diagram. (b) Sequence of entanglement pulses for first scheme in [32]. This is a cPHASE gate. For ϕ = 2π the entanglement phase is π . (c)
Sequence of entanglement pulses for the second entanglement scheme from [32] (also a
π cPHASE gate). 45
CHAPTER 3
ONE- AND TWO-DIMENSIONAL OPTICAL LATTICES ON A CHIP
In this chapter we discuss a way to make one-dimensional and two-dimensional optical lattices on a chip. Numerous research groups study the physics of optical lattices.
They are deemed a useful tool for neutral atom quantum computing. A major difficulty with a three-dimensional optical lattice as a quantum register is that the qubits (atoms) are arranged in a 3D array (recall Figure 2.6 (c)), and it is difficult to address atoms on the inside of the lattice with laser beams without disturbing other atoms along the path. This lack of addressability limits the possibilities for performing quantum gates in this setup.
2D optical lattices on a chip are a solution to this problem. The qubits in such a lattice are arranged in a flat array and each site can be addressed individually with laser beams from above or below. While it is possible to create two-dimensional optical lattices with two laser beam standing waves in free space, the trapping in the third dimension is, at best, weak, since it simply stems from the Gaussian intensity profile of the laser beams. In addition, the number of lattice sites achievable in such a geometry is much smaller than the number of qubits needed for quantum computation. We have found a method of creating a scalable 2D optical lattice above a chip with tight trapping in all three dimensions. The nearby surface may allow integration of current-carrying wires to
46 construct a hybrid magneto-optical atom chip. The research presented here is an initial investigation of these lattices. We have computationally examined geometries that seem especially promising for use as a quantum register.
To create such a lattice, we make use of the evanescent wave “leaking” out of an optical waveguide. As described below, optical waveguides allow propagation of several confined waves (“modes”) of light. Though most of the light is confined to the waveguide itself, there is some light “leaking” out at the interfaces of the waveguide. The intensity of this light wave drops off exponentially with the distance from the waveguide surface. By interfering two such modes of the waveguide, we can achieve destructive interference such as to create nodes away from the waveguide surface. For blue-detuned light, these nodes can be used to trap atoms, creating a two-dimensional array of qubits.
Our proposal, as far as we know, is the first 2D array of atomic microtraps this close (~100 nm) to a dielectric surface. It is as of yet unknown what effects the nearby surface may have on the coherence of qubits in the 2D optical lattice. The lattice could be used to measure the effects of the surface on the trapped atoms, giving insight into the dominant decoherence mechanisms.
The following is a summary of previous work closely related to ours. Atom trapping with the light potential of an evanescent wave has been demonstrated by Grimm and co-workers [37]. They used a combination of red-detuned and blue-detuned light to create a single atom trap above a prism. They were able to confine 20,000 Cs atoms to a
~0.3 mm pancake-shaped region about 1 µm above the prism. Theoretical proposals [38] have studied the use of evanescent waves above an optical waveguide to create a cold atom waveguide, again by use of both a blue and a red detuned laser. Several waveguide
47 configurations for creating one-dimensional arrays of traps have been suggested by Burke et al. [39]. One of these configurations is a pair of waveguides, one mounted a small distance above the other. Blue-detuned laser beams are coupled into each of the waveguides, and a blue-detuned evanescent wave leaks out of each waveguide. In the center between the two waveguides the electric fields from the two evanescent waves cancel, creating a dark channel that can serve as an atom waveguide. Modulating the intensity of one of the evanescent waves creates a 1D array of microtraps along the waveguide. A 2D array of dipole microtraps was demonstrated by Dumke et al.[40].
They trapped 85Rb atoms at the foci of a laser beam that was passed through a substrate patterned with an array of microlenses. The traps were spaced 125 µm apart and confined up to 1000 atoms each, 625 µm above the substrate surface.
3.1. How to Create a Two-Dimensional Optical Lattice on a Chip
A slab waveguide (see Figure 3.1), i.e. a planar layer of high refractive index
material (refractive index nW ) on a lower index substrate ( nS ), supports certain, distinct optical field distributions. The allowed electric field distributions can be calculated analytically from Maxwell’s equations [41]. There are two kinds of solutions for the electric field: Transverse electric (TE) modes with a single component of electric field
(along x in Figure 3.1), and transverse magnetic (TM) modes that have a single component of magnetic field, but two components of electric field (e.g. along y and z in
Figure 3.1). For the optical lattices described below we use TE modes. Finding a mode configuration that achieves cancellation of only one electric field component to create field nodes is more straightforward than cancellation of two components. 48 Let us briefly review the properties of the TE modes of an optical waveguide.
Each mode in an optical waveguide is characterized by an effective refractive index
neff ,m , with nS < neff ,m < nW . The effective index decreases with increasing mode order, up to the “cutoff”, the highest-order confined mode. The number of confined modes depends on the thickness of the waveguide layer. The thicker the waveguide layer, the more modes are confined. The electric field for a TE mode above an optical waveguide
(the evanescent wave) (see Figure 3.1 (a)) is given by:
−δ m y i(km z−ωt+ϕm ) E x (y, z) = E0m e e .
Here, E 0m is the (real) field amplitude of mode m at the waveguide surface that is
determined by laser intensity. δ m and km are the decay and propagation constants of
mode m, respectively, and ϕm is an additional phase. The latter is adjustable by changing the phase of the laser beam coupled into this mode.
The propagation constant km is related to the propagation constant k in free space, k = 2π (for laser wavelength λ ), through k = n k . The decay constant δ λ m eff ,m m
2 of mode m is δ m = k neff ,m −1 .
How can we combine these modes to create a 2D optical lattice? Superposition of two modes (m=1,2) co-propagating along z (see Figure 3.1 (b)) results in:
−iωt −δ1 y i(k1z+ϕ1 ) −δ 2 y i(k2 z+ϕ2 ) ETotal (y, z) = e (E01e e + E02e e )
−iωt i(k1z+ϕ1 ) −δ1 y −δ 2 y −iϕ (z) = e e (E01e + E02e e ) ,
with ϕ(z) = (k1 − k2 )z + (ϕ1 −ϕ2 ). For values z * of z with
49 (2n +1)π − (ϕ −ϕ ) z* = 1 2 , k1 − k2
ϕ(z*) is an odd multiple of π , which means the two fields are out of phase:
−iωt i(k1z+ϕ1 ) −δ1 y −δ 2 y ETotal (y, z*) = e e (E01e −E 02 e ) .
The distance between successive values of z * is:
2π ∆z = . k1 − k2
As we will see shortly, this is the distance between trapping sites. The distance is
depends on neff ,1 and neff ,2 , which are determined by the choice of waveguide material and thickness. For realistic waveguide materials we have found that this distance can be several microns. This is adequate to enable a tightly focused laser beam to address a single qubit without disturbing its neighbors.
Since the decay constants δ1 and δ 2 are different, there will be at most one value y* of the height above the waveguide, y, where the two fields cancel. This distance depends on the relative intensities of the two modes:
ln(E / E ) y* = 01 02 . δ1 − δ 2
By adjusting the relative intensities of the laser beams coupled into the two
modes, one can adjust the ratio E01 / E02 and, thus, control the height, y*, of the field node above the waveguide. We have thus found a way to create a one-dimensional array of nodal lines along x, away from the waveguide surface, spaced apart by ∆z (see Figure
3.1 (b)). For blue detuning ( ∆ > 0 ), these nodes confine atoms along the y and z directions. There is, however, no trapping force along x. In fact, the Gaussian profile of a
50 real laser beam will lead to a slight repulsive force pushing the atoms out of the light. To achieve confinement in the x direction, we add a second pair of modes perpendicular to the first pair, as shown in Figure 3.1 (c). This results in a 2D array of microtraps with tight confinement in all three dimensions, which, if filled with a single atom per site, can serve as a quantum register. We have calculated the properties of this 2D optical lattice and a 1D optical lattice created in a similar fashion above a ridge waveguide. We next present the results of these computations [42].
3.2 Properties of the Basic Two-Dimensional Optical Lattice
We have, in principle, found a way to make a 2D array of atomic microtraps. But, is it suitable for quantum computing? To answer this question, we calculated the trap properties for realistic waveguide and laser parameters. The most commonly used atoms in cold atom experiments are alkali atoms. We chose to model the 2D optical lattices for
Rb. It is the element of choice for the trapping experiments we perform in our laboratory, mainly because the appropriate laser wavelength can be created with common diode lasers. The D2 transitions in Rb correspond to a wavelength of 780 nm. We calculated the trap parameters for 85Rb. We used a laser detuning of ∆ = 1000Γ ≈ 2π ×6 GHz to the blue of the D2 line. The waveguide consists of a 230 nm thick layer of As2S3 (refractive
index nW = 2.4 ) on a NaF (3nS = 1. 2) substrate, as shown in Figure 3.2 (a). We picked a low index substrate because it results in deeper traps. The index of the waveguide material was chosen to be close to the optimal refractive index for creating deep traps 150 nm above the surface. The waveguide thickness we chose was the thinnest we could make the waveguide and still confine two TE modes (TE0 and TE1). The thinner the 51 waveguide, the deeper the traps. The two waveguide modes used are thus the TE0 and
TE1 modes. The results of our calculations are listed in Table 3.1. We find that the nodes are ∆z = 0.98 µm apart, as can be seen in Figure 3.2 (b). We arbitrarily chose an input power of 1 mW per µm of waveguide width in the stronger mode (TE0). We used 100
µm wide input laser beams. For a beam power of 100 mW in each of the TE0 modes we find that we can create traps 150 nm above the surface with a power of 4.2 mW in each
TE1 mode. We calculated the trap potential for atoms in the mF = 0 magnetic hyperfine substate. For this case the trapping potential from equation (2.2) is: r r 2 hΓ Γ I(r ) U dipole (r ) = . 3 8 ∆ I sat
In the field of cold atom physics it is customary to express energies in units of temperature (therefore the nomenclature “cold” as opposed to “low energy” atoms). The conversion is:
Energy = k BT ,
where k B is the Boltzmann constant, and T is the energy expressed in units of temperature. We will follow this convention.
We found that the trap depth is limited by the trapping in the y dimension. In this direction, the trap depth is 1.5 mK (see Figure 3.2 (c)). The trap depth in the x and z dimensions is approximately 40 mK. Sub-Doppler cooling techniques produce atom samples of approximately 10 µK, allowing loading of these traps from a cold atom cloud.
We calculated the trap frequencies ( f x, y,z ) and the corresponding energy difference
hf x, y,z ( ∆Tx, y,z = ) in units of temperature between the vibrational levels of the trapping k B 52 potential along the three dimensions: f x,z = 1.4 MHz, f y = 2.5 MHz; ∆Tx,z = 68 µK,
∆Ty = 120 µK.
Because of the proximity of the traps to the surface, one may be concerned about the van der Waals force that is attracting the atoms to the surface. As the traps get closer and closer to the surface, the van der Waals force will eventually overcome the light potential and pull the atoms onto the waveguide surface. For the parameters used here, this would happen at approximately 30 nm from the surface. The van der Waals potential
for an atom-surface distance of 150 nm is only U vdW / k B ~ 5 µK, much less than the light potential.
The vibrational levels are very far apart, reducing the motional heating rate during quantum computation. For a sample temperature of T = 10 µK (achievable with evanescent wave Sisyphus cooling in a gravito-optical surface trap [43]), the number of atoms in the first excited (motional) state ( n = 1) compared to the number in the ground state ( n = 0) (assuming a Maxwell-Boltzmann distribution) is
⎛ ∆E ⎞ ⎛ ∆T ⎞ N(n = 1) −⎜ x ⎟ −⎜ x ⎟ = e ⎝ kBT ⎠ = e ⎝ T ⎠ = 0.001. N(n = 0)
In other words, 99.9% of atoms will be in the motional ground state.
For quantum computing it is desirable for the 1/e size of the wavefunction of the atoms in the ground state of the trapping potential (“ground state size”) is much smaller
than the separation between qubits. The ground state sizes for our traps are: β x,z = 9.2
nm, β y = 6.9 nm, approximately 1/100 of the interqubit separation.
53 Another important issue is decoherence through spontaneous scattering of light from the optical lattice. One advantage of blue-detuned traps is that the intensity is low near the trapping sites, since atoms are trapped at intensity nodes. For a two-state atom, the average spontaneous emission rate η for a blue-detuned dipole trap is related to the trap frequencies by
π Γ η = ( f + f + f ) . (3.1) 2 x y z ∆
This result is not quoted in the literature, so we include a derivation in Appendix A. For the laser parameters above, we find a spontaneous emission rate of 8 kHz. While spontaneous emission does not lead to loss from the trap, it will destroy the coherence necessary for quantum computing. In order to perform 104 gate operations, the gate time would have to be ~10-8 s.
Coherence times can be improved tremendously by increasing the detuning. The r trapping potential (equation (2.2)) is proportional to I(r) , whereas the spontaneous ∆ emission rate (equation (3.1)) is proportional to:
f (U )1/ 2 (I(r))1/ 2 x, y,z ∝ light ∝ . ∆ ∆ ∆3/ 2
Increasing both the detuning and the intensity by a factor of ten will leave the potential unaltered while decreasing the scattering rate by a factor of ten. Note the difference from r red-detuned light, for which the average scattering rate scales as I(r ) . ∆2
In conclusion, we have found that these lattices are realistic candidates for good quantum registers: The trap frequencies in all directions are so high that the atoms can be kept in their motional ground state with standard cooling techniques. The ground state 54 size is very small (~10-2) compared to the distance between qubits. The decoherence rate can be made so small that it is possible to perform ~104 gates before decoherence occurs.
Qubits can be addressed individually by laser beams from above or below.
3.3 One-Dimensional Optical Lattices on a Chip
A ridge waveguide is a waveguide that has a small width in both transverse dimensions (i.e. along x and y in see Figure 3.3 (a)). We considered creating 1D arrays of atom traps using ridge waveguides. A 1D array of qubits can especially useful for building a quantum computing architecture. For example, two parallel 1D arrays with a distance of ~1µm between them could be used to line up two qubits next to one another.
An electric dipole-dipole gate could then be selectively implemented by exciting the two atoms with a laser. Other combinations can be envisioned. For example, a 2D quantum register with a 1D quantum bus waveguide attached to it would allow for shuttling around of qubits.
The modes of a ridge waveguide are not purely TE or TM modes anymore. The solutions for the electric field distribution can no longer be determined analytically.
Numerical solutions for the electric field are required. We have calculated the waveguide modes of the waveguide shown in Figure 3.3 (a) by using a finite element analysis. We found that there are pairs of modes that allow the creation of 3D traps (i.e. nodes in the optical field) by interference of the evanescent waves of two modes, similar to the basic
2D optical lattice described above (see Figure 3.3 (b) and (c)). The waveguide geometry leads to electric field modes that are not constant along x (as was the case in a slab waveguide). Thus, interference of two appropriate modes automatically leads to trapping
55 along x, as well (Figure 3.3 (b)). The result is, in principle, a 1D array of individually addressable qubits – a quantum register.
For the geometry shown in Figure 3.3 (a), we found two modes, one with
7 -1 7 -1 k1 = 1.304×10 m and one with k2 = 1.105×10 m , that create traps spaced 3.2 µm apart. With 1 mW of laser power in mode 1 and 0.081 mW in mode 2 traps are created
150 nm above the waveguide surface. We found a trap depth of 770 µK, and trap
frequencies of f x = 1.8 MHz, f y = 1.9 MHz, and f z = 0.65 MHz. The corresponding
energy differences (in units of temperature) of the vibrational modes are ∆Tx = 82 µK,
∆Ty = 86 µK, and ∆Tz = 30 µK. The ground state sizes were calculated to be β x = 8.1
nm, β y = 7.7 nm, and β z = 13.3 nm. The scattering rate was found to be 7 kHz. The results of these calculations are listed in Table 3.2. The same scaling with detuning applies as for the 2D optical lattice. The above results show that these arrays may be suitable for quantum computation.
3.4 Variations of the Basic Two-Dimensional Optical Lattice
In the course of our studies of one-dimensional and two-dimensional optical lattices on a chip, we identified a few variations of the basic 2D optical lattice idea that seemed particularly well-suited for application in a quantum computing architecture.
These variations are described here. By no means do we want to imply that these are the only useful variations of 1D and 2D optical lattices above waveguides.
56 3.4.1 The Counter-Propagating 2D Optical Lattice
It is advantageous to have large distances between qubits in order to be able to address them individually. Fast quantum gates, on the other hand, often require small distances between qubits. In the context of magnetic dipole-dipole interaction as entanglement scheme, it was suggested to apply a magnetic field gradient in order to distinguish individual qubits. The magnetic field gradient shifts the transition frequencies between the two qubit states. The 0 → 1 transition frequency therefore depends on the position of the qubit in the magnetic field gradient. As mentioned above, the transition is driven either by a microwave pulse or a pair of Raman laser beams. If the gradient is high enough to shift the frequency of neighboring atoms more than the linewidth of the laser or microwave pulse, qubits can be individually addressed, even though the spatial extent of the laser beam (or microwave) is larger than the separation between qubits. With a high enough magnetic field gradient, even qubits separated by much less than a laser wavelength can be resolved. Thus, it may be advantageous to create a 2D optical lattice with closely spaced qubits for fast gate operation, at the same time retaining the ability to address individual atoms with the help of a magnetic field gradient. This can be achieved by replacing one of the co-propagating mode pairs in Figure 3.3 (c) by a counter- propagating mode pair as shown in Figure 3.4 (a). Along the counter-propagating direction the spacing between traps is now
∆z = λ , 2neff
57 where neff is the effective refractive index of the mode used. With waveguide parameters
as used for the basic lattice (see Figure 3.2 (a)), this is usually around neff = 2. The spacing of traps thus becomes λ/4, only half of that in a free-space standing wave (λ/2).
The counter-propagating pair alone creates nodal planes (along x and y), which means atoms would hit the surface of the waveguide if one were to attempt to use these nodes as atom traps. The co-propagating pair along x, however, provides tight trapping in both the x and y directions, away from the waveguide surface. We have thus created an array of microtraps with a trap spacing of about λ/4 in the z dimension.
Using counter-propagating TE0 modes of 100 mW along z and a co-propagating pair of TE0 and TE1 along x (with the same parameters as for the basic 2D optical lattice) we create a 2D array of traps 150 nm above the waveguide surface with the following
properties: neff = 2.13, ∆x = 0.98 µm, ∆z = 183 nm, trap depth 710 µK, trap frequencies
f x = 1.4 MHz, f y = 1.6 MHz, f z = 7.6 MHz, energy differences between motional
levels ∆Tx = 68 µK, ∆Ty = 76 µK, ∆Tz = 364 µK, and ground state sizes of β x = 9.2
nm, β y = 8.7 nm, β z = 4.0 nm. The scattering rate was found to be 17 kHz, with the usual scaling. Note that the trap parameters along x are the same as for the basic 2D lattice. The trap depth along y is reduced by a factor of two, because the counter- propagating mode pair does not contribute to the trapping along this dimension. These results are summarized in Table 3.3.
The question remains, can we produce a magnetic field gradient high enough to
resolve two qubits that are 183 nm apart? For a gate time of t gate = 1 ms, we need a
58 frequency shift between neighboring atoms of ∆f ≥ 1 = 1000 Hz, and, using t gate equation (2.4) from section 2.3.6.2, we find that we require a magnetic field gradient of at least 19.5 G/cm. This is a modest magnetic field gradient, easily achieved by common magnetic coils.
3.4.2 The Double-Well Optical Lattice
Another variation of our 2D optical lattice is a 2D optical lattice consisting of double wells, with adjustable spacing between the wells. In every scheme we discussed in chapter 2.3.6 it is advantageous for atoms to be close together during gate operation since it increases the interaction and therefore decreases the gate time. Between two-qubit gates, however, it is important to be able to access individual qubits for single qubit rotations. For the always-on magnetic dipole-dipole interaction, we get less unwanted two-qubit evolution between gates if the qubits are spaced further apart. Therefore, it would be advantageous to bring the qubits closer together during two-qubit gate operation and separate them between two-qubit gates. We investigated one such adjustable 2D optical lattice arrangement, and present the results here.
For certain waveguide thicknesses and materials, three modes whose propagation constants satisfy the condition
k1 − k2 = k2 − k3 = 1/ 2(k1 − k3 ) (3.2) may exist. Coupling these three modes into the waveguide as shown in Figure 3.5 (a) leads to an array of double-wells. The distance between the two traps of each well can be adjusted by changing the intensity of mode 2. One can adiabatically bring the traps close
59 together (e.g. for gate operation) and further apart (between gates). Note that as the traps are brought together the barrier between them will also be reduced and the trap parameters will change accordingly. Too small a barrier leads to tunneling between the two wells, which makes the two qubits indistinguishable. This limits how close one can bring the two qubits. The tunneling time (in this case the limit on the coherence time) is related to the barrier height by (see [33]):
3 / 4 5 / 2 π ⎛ E ⎞ 2 V E h ⎜ R ⎟ 0 R . ttunnel = ⎜ ⎟ e (3.3) 2ER ⎝ V0 ⎠
We have found that three modes satisfying the condition of equation (3.2) exist for an As2S3 waveguide (nW = 2.4) with a thickness of 663 nm on a substrate with nS =
1.2827. To make the k values match exactly as above, we had to adjust the value of nS.
NaF has a refractive index of 1.32. It is conceivable that a suitable material can be developed by varying the components of the substrate. Thus, the refractive indices used are not unrealistic. For this waveguide, the TE0, TE2 and TE3 modes satisfy condition
(3.2):
7 -1 k1 = k(TE0) = 1.891×10 m ,
7 -1 k2 = k(TE2) = 1.526×10 m , and
7 -1 k3 = k(TE3) = 1.161×10 m .
We begin by considering a 2D optical lattice with light coupled into the TE0 and
TE3 modes along z, and light coupled into the TE2 and TE3 modes along x. The lattice spacing will be adjustable along z, and fixed along x. We chose to use the TE2 and TE3 modes along x because they result in the deepest traps at this particular waveguide 60 thickness. We choose the intensity ratios of the mode pairs so that traps form 100 nm above the surface. We use a laser beam power of 100 mW for a 100 µm wide beam coupled into the TE2 mode and 21 mW coupled into the TE3 mode along x. With 100 mW in a 100 µm wide beam coupled into the TE0 mode and 0.9 mW into the TE3 mode along z we get a regular single well pattern (similar to Figure 3.1 (c)) with a trap spacing
of ∆z1 = 860 nm (see Figure 3.5 (b)). The trap depth is 1.8 mK, limited by trapping along
y. The trap frequencies are f x = 2.1 MHz, f y = 2.5 MHz, and f z = 0.85 MHz. The corresponding energy differences between vibrational states (in units of temperature) are
∆Tx = 99 µK, ∆Ty = 120 µK, and ∆Tz = 41 µK. The ground state sizes were evaluated
to be β x = 7.6 nm, β y = 6.9 nm, and β z = 11.9 nm. The scattering rate is 8.5 kHz.
Now, we add TE2 and increase its intensity. The wells of each double-well start moving towards one another along z (see Figure 3.5 (d)). The trap height above the waveguide remains 100 nm, as indicated in Figure 3.5 (e). The distance between double wells is 1.72 µm. The barrier height between the two wells reduces as one adds more and more intensity of the TE2 mode. At a TE2 power of 16 mW, the trap spacing between the
wells is reduced to ∆z2 =136 nm (see Figure 3.5 (c)), even closer than in the counter- propagating optical lattice described in the last section. The barrier height between wells
is now ∆E kB = 10 µK, which we arbitrarily chose as the minimum acceptable barrier
height (limited by the tunneling rate). The trap properties are as follows: f x = 2.1 MHz,
f y = 2.3 MHz, f z = 0.22 MHz; ∆Tx = 99 µK, ∆Ty = 112 µK, ∆Tz = 11 µK; β x = 7.6
nm, β y = 7.1 nm, β z = 23 nm. The scattering rate is 7.2 kHz. All results are summarized
61 in Table 3.4. As expected the trap frequency along z is greatly reduced, and the spread of the ground state wavefunction is increased. Note, however, that since the trap depth is less than the difference between vibrational levels, there will only be one confined motional state. If there is an atom in the trap, it will be in the motional ground state.
We can use this type of 2D optical lattice to implement the first electric dipole- dipole scheme described in chapter 2.3.6.1. The scheme does not require us to resolve the two qubits. However, if all wells are closely spaced neighboring pairs could inadvertently be excited simultaneously, whereas in our lattice the large spacing between double wells allows us to address individual pairs of qubits. The gate time for a cPHASE gate with
ϕ = π is:
π t gate = h . U elec.dipole
With a barrier height of 10 µK, the tunneling time (see equation (3.3)) between the two wells for 85Rb is 7 s. The typical gate time (see equation (2.3)) for a magnetic dipole-dipole gate with a qubit distance of 136 nm is 4 ms. Between gates, we separate the qubits to a distance of 860 nm, reducing the interaction between the two qubits by a factor of
3 (∆z2 ) 3 = 0.004 = 0.4 %. (∆z1 )
This means that the single qubit pulses that correct for the two qubit evolution, which is always on, do not need to be applied as often, or as long as when the qubits are always close together. This speeds up the calculation. Also, the magnetic field gradient needed to resolve two qubits spaced 860 nm apart is easy to create (1 G/cm according to equation (2.4)). 62
In conclusion, we have proposed and theoretically investigated a method of creating 1D and 2D optical lattices on a chip by interfering evanescent waves from different modes of an optical waveguide. Nodes are created away from the waveguide surface at points of destructive interference. For blue-detuned light these nodes constitute
3D atomic microtraps. The 1D and 2D structure of these arrays should allow addressing of qubits from above or below. Addressing of individual qubits is a problem in conventional 3D optical lattices.
We found that the optical lattices we investigated may be suitable for use as quantum registers: The energy difference between motional states is much bigger than temperatures achievable with common cooling techniques, greatly suppressing motional heating during quantum computation. The ground state sizes are much smaller than the inter-qubit separation. The distance between qubits is large enough to use laser beams to address individual qubits. Detuning of the optical lattice lasers allows for lowering the spontaneous emission rates as needed.
We have examined variations of the lattice in which the sites are spaced much more closely than in a free space lattice, increasing the interaction strength (and reducing the gate time). With interference of three modes, we find that it should be possible to create a lattice of adjustable double-wells which allows us to bring qubits close together during gate operation and separate them in between gates. This may be a useful tool for implementing fast two-qubit gates.
63
Trap parameters Results
Trap spacing 0.98 µm
Trap depth 1.5 mK
Trap frequencies f x,z = 1.4 MHz, f y = 2.5 MHz
Energy difference ∆Tx,z = 68 µK, ∆Ty = 120 µK
Ground state sizes β x,z = 9.2 nm, β y = 6.9 nm
Spontaneous emission rate 8 kHz
Table 3.1: Results for trap parameters of the basic 2D optical lattice above a slab waveguide. The input parameters were: laser wavelength of 780 nm, detuning
85 ∆ = 1000 Γ = 2π × 6 GHz to the blue of the Rb D2 line, waveguide material As2 S3
( nW = 2.4), substrate material NaF ( nS = 1.32), and waveguide thickness 230 nm. The traps were created using a co-propagating pair of TE0 and TE1 modes in each dimension
(x and z), with a power of 100 mW (for a beam width of 100 µm) in each TE0 mode and
4.2 mW in each TE1 mode, resulting in a trap height of 150 nm above the waveguide surface.
64
Trap parameters Results
Trap spacing 3.2 µm
Trap depth 770 µK
Trap frequencies f x = 1.8 MHz, f y = 1.9 MHz, f z = 0.65 MHz
Energy difference ∆Tx = 82 µK, ∆Ty = 86 µK, ∆Tz =30 µK
Ground state sizes β x = 8.1 nm, β y = 7.7 nm, β z = 13.3 nm
Spontaneous emission rate 7 kHz
Table 3.2: Results for trap parameters of the 1D optical lattice above a ridge waveguide.
The input parameters were: laser wavelength of 780 nm, detuning ∆ = 1000 Γ = 2π × 6
85 GHz to the blue of the Rb D2 line, waveguide material As2 S3 ( nW = 2.4), substrate
material NaF ( nS = 1.32), and waveguide dimensions of 1µm × 0.45 µm (see Figure 3.3
7 - (a)). The traps were created using a co-propagating pair of two modes ( k1 = 1.304×10 m
1 7 -1 , k2 = 1.105×10 m ) along z, with a power of 1 mW in mode 1 and 0.081 mW in mode
2, resulting in a trap height of 150 nm above the waveguide surface.
65
Trap parameters Results
Trap spacing ∆x = 0.98 µm, ∆z = 183 nm
Trap depth 710 µK
Trap frequencies f x = 1.4 MHz, f y = 1.6 MHz, f z = 7.6 MHz
Energy difference ∆Tx = 68 µK, ∆Ty = 76 µK, ∆Tz =364 µK
Ground state sizes β x = 9.2 nm, β y = 8.7 nm, β z = 4.0 nm
Spontaneous emission rate 17 kHz
Table 3.3: Results for trap parameters of the counter-propagating 2D optical lattice above a slab waveguide. The input parameters were: laser wavelength of 780 nm, detuning
85 ∆ = 1000 Γ = 2π × 6 GHz to the blue of the Rb D2 line, waveguide material As2 S3
( nW = 2.4), substrate material NaF ( nS = 1.32), and waveguide thickness 230 nm. The traps were created using a counter-propagating pair of TE0 modes along the z direction and co-propagating pair of TE0 and TE1 modes in the x direction, with a power of 100 mW (for a beam width of 100 µm) in each TE0 mode and 4.2 mW in the TE1 mode, resulting in a trap height of 150 nm above the waveguide surface.
66
Trap Results parameters
Trap spacing ∆z1 = 860 nm ∆z2 =136 nm Trap depth 1.8 mK 10 µK
Trap f x = 2.1 MHz, f y = 2.5 MHz, f x = 2.1 MHz, f y = 2.5 MHz, frequencies f z = 0.85 MHz f z = 0.22 MHz
Energy ∆Tx = 99 µK, ∆Ty = 120 µK, ∆Tx = 99 µK, ∆Ty = 112 µK, difference ∆Tz = 41 µK ∆Tz = 11 µK
Ground state β x = 7.6 nm, β y = 6.9 nm, β x = 7.6 nm, β y = 7.1 nm, sizes β z = 11.9 nm β z = 23 nm Spontaneous 8.5 kHz 7.2 kHz emission rate
Table 3.4: Results for trap parameters of the 2D double-well optical lattice above a slab waveguide. The input parameters were: laser wavelength of 780 nm, detuning
85 ∆ = 1000 Γ = 2π × 6 GHz to the blue of the Rb D2 line, waveguide material As2 S3
( nW = 2.4), substrate material NaF ( nS = 1.2827), and waveguide thickness 663 nm. The double-wells were created along z using the TE0, TE2 and TE3 modes, which satisfy equation (3.2). Along the x direction, a co-propagating pair of TE2 and TE3 modes is used with a power of 100 mW (for a beam width of 100 µm) in the TE2 mode and 21 mW in the TE3 mode, resulting in a trap height of 100 nm above the waveguide surface.
The second column shows the results for 100 mW (in a 100 µm beam) in the TE0 mode
(along z), 0 mW in the TE2 mode and 0.9 mW in the TE3 mode. The third column shows the results for the closest possible double-well spacing, which we achieved by increasing the power in the TE2 mode up to 16 mW. 67
Figure 3.1: Creation of a 2D optical lattice above a chip. (a) TE mode in a slab waveguide. (b) Interference of two co-propagating modes leads to nodal lines away from the waveguide surface. (c) Two pairs of co-propagating modes create a 2D array of 3D traps – a 2D optical lattice.
68
Figure 3.2: Trapping potential for realistic waveguide parameters. (a) Waveguide and substrate materials and thickness. (b) Trap potential along z. The trap potential along x is identical to that along z due to the geometry of the beam configuration. (c) Electric fields of TE0 (red) and TE1 (blue) modes at z* above the waveguide. The electric fields are out of phase. The relative intensities were chosen so that the fields cancel 150 nm above the waveguide surface, resulting in a 1.5 mK deep trapping potential along y (black).
69
Figure 3.3: 1D optical lattice above a ridge waveguide. (a) Illustration of waveguide materials and size. (b) Resulting electric fields and trap potential along x (at node z*).
The blue (dotted) lines are the y-component of the electric field; the red (dashed) lines are the x-components of the electric field. The solid, black line is the resulting trap potential.
(c) y-components of the electric field at node z* (red, long dash = mode 1; blue, short dash = mode 2) along y. The resulting field (purple, dotted line) for relative intensities as in the text cancels 150 nm above the waveguide surface, resulting in a 1.5 mK deep trapping potential (black, solid line) along y. The trap potential along z is periodic as in the basic 2D optical lattice.
70
Figure 3.4: The counter-propagating 2D optical lattice. (a) Beam configuration of the counter-propagating optical lattice. (b) Trap potential along the counter-propagating direction z. As indicated, the trap spacing is approximately λ/4. The trap potential along x is identical to the potential for the basic 2D optical lattice. (c) Electric fields of the TE0
(red, long dash) and TE1 (blue, short dash) modes, total electric field (purple, dotted) and trap potential (black, solid) along y. The trapping is weaker than in the basic 2D lattice, because the counter-propagating mode pair does not contribute to the trapping along y.
71
Figure 3.5: The double-well lattice. (a) Beam configuration for the double-well lattice.
Three modes are necessary along z. (b) Trap potential for 860 nm spacing between wells
(P(TE2)=0). (c) Trap potential for 136 nm spacing between wells (P(TE2)=16 mW). (d)
The trap spacing varies with the power coupled into the TE2 mode. (e) The trap potential along y stays about the same as the power in the TE2 mode is adjusted from a trap
(double-well) spacing of 860 nm to 136 nm.
72
CHAPTER 4
EXPERIMENTAL SETUP
In order to experimentally realize the proposed 2D optical lattice, an experimental system was designed and constructed to load cold atoms into the 2D optical lattice above a waveguide. The construction of the system largely followed the instructions given in
[44]. In order to be able to load the atoms into the 2D lattice they have to be sufficiently cold (temperature less than the trap depth), so that the atoms will settle into the traps. In order to achieve a long lifetime of the atoms in the traps, the background pressure has to be very low (~10-10 Torr). Any “hot” background gas left in the vacuum chamber will eject cold atoms from the trap if it collides with them. On the other hand, to create a large cold atom sample from a Rb vapor, the background pressure will be at least the Rb vapor pressure (~10-7 Torr). One solution to this problem is to build a system comprised of two separate vacuum chambers – one high vacuum (HV) region for loading Rb into the cold atom sample, and one ultra-high vacuum (UHV) region for performing the experiment, undisturbed by hot Rb background gas.
Several cooling and trapping steps have to be completed successfully to achieve an atom sample temperature low enough to load the 2D optical lattice. Initially the atoms are loaded from a Rb vapor into a magneto-optical trap (MOT). Then they are
73 compressed and loaded into a purely magnetic trap. Suspended by the magnetic field only, the atoms can now be moved to the UHV chamber by translating the magnetic trap coils. Here, the atoms are lowered into the evanescent wave above the waveguide sample.
The atoms will bounce off the evanescent wave, and in the process be cooled further.
Finally, the 2D optical lattice is turned on around the cold atoms, and the atoms will be trapped in our 2D array of microtraps. An animation of this process can be found in [45].
Currently, we are able to trap 2 × 108 atoms in the MOT and are working on transferring atoms into the magnetic trap. Unlike Lewandowski’s system, we have the ability to load a waveguide sample remotely, without breaking vacuum, through a straight-through valve on the end of the UHV chamber. An overview of the vacuum chamber is shown in Figure 4.1.
In this chapter a detailed description of our system is presented, including construction of the vacuum chamber, the lasers used to control the atoms, the experimental setup for both MOT and magnetic trap, and control circuits and codes needed for the transfer of atoms into the magnetic trap. The work is presented at a high level of detail with the intention that it might be helpful for others that want to construct similar systems in the future. A list of parts and equipment required for this experiment can be found in Appendix B. Shop drawings are shown in Appendix C, LabVIEW codes in Appendix D, and circuit diagrams can be found in Appendix E.
74 4.1 Vacuum Chamber
4.1.1 Rubidium Loading Chamber
The Rb loading chamber consists of a 2” diameter round glass tube (see Figure
4.2 and Figure C.1), connected to 1-1/3” stainless steel conflat flanges on either end. Rb dispensers from SAES getters serve as a Rb source for the experiment. These dispensers
2 are small metal containers (~12 mm × 1mm ) holding a mixture of Rb2CrO4 and a reducing agent (St101) consisting of 84% Zr and 16% Al. The dispensers have thin metal terminals on either end that were spot-welded to an electric vacuum feed-through. Rb is loaded into the vacuum chamber by running a current through a dispenser. When the dissipated heat creates a high enough temperature, pure Rb is released. Two of these dispensers are mounted in the vacuum chamber, accessible individually, allowing use of one dispenser at a time. Once one of them is used up, the experiment can continue running with the other without having to break vacuum. As per suggestions found in the literature, an initial bake-out of the dispensers was performed for several hours in order to clean them. The manufacturer, however, does not suggest a bake-out, since a lot of Rb is lost in the process. Rather, a slow turn on procedure is suggested every time the dispenser is used. At the beginning of a run, the current through the dispenser in use is ramped up slowly (2A for 1 min, increase by 0.5A every minute, 3.5A for 5 min), loading enough
Rb for the day. Note that the operating current is much lower than what was specified by the manufacturer. Care should be taken not to run too much current through the dispensers, because all the Rb could be released at once. A cold spot (currently a simple ice pack) is placed on the glass wall right above the Rb dispenser, so that the Rb will tend to condense there instead of the glass walls in the MOT region, where it would impair the
75 optical access by the MOT laser beams. At this point the loading chamber is filled with a
Rb vapor of room temperature vapor pressure (10-7 Torr). The MOT is loaded from this vapor. To keep the pressure low in this part of the chamber, an 8 l/s ion pump is attached to one end of the Rb loading chamber. To reduce the pumping rate of the Rb in the chamber, a copper blank with a 5 mm diameter pinhole was inserted between the flanges connecting the ion pump to the chamber.
4.1.2 The Ultra-High Vacuum Chamber
The UHV chamber that will house the waveguide sample consists of a 3” diameter, 2” long glass cylinder (see Figure 4.3 and Figure C.2) that allows optical access to the waveguide sample from above and below the waveguide without having a focusing effect from the round glass walls. The glass cylinder is connected to the other parts of the vacuum system with bellows on either end, so that very little strain is put on the glass.
This is especially relevant during bake-out, where a mismatch in thermal expansion coefficient could lead to enough strain to break the glass. To ensure several orders of magnitude of pressure difference between the UHV chamber and the Rb loading chamber, copper blanks with 5 mm pinholes were placed into either end of the 1-1/3” diameter, 3” long bellows connecting the two chambers, reducing the chance that a hot
Rb atom will have a straight shot into the UHV chamber. While it is essential for the connection between the chambers to be narrow to reduce background gas flow between the two, it would be helpful to have a larger tube connecting the UHV chamber to the straight-through valve at the opposite end of the glass cell (see Figure 4.3). The waveguide sample will be loaded in-situ through this straight-through valve. We chose a
76 setup that allows for remote loading, because the waveguide sample cannot withstand the high temperatures of the vacuum chamber bake-out procedure. In addition, this setup allows us to switch waveguide samples without breaking vacuum. In our chamber, the diameter of the glass tube connecting to the straight-through valve is only about 1” (outer diameter), restricting the size of the sample that can be loaded. This size limitation arose for purely practical reasons: the cylindrical chamber had to be smaller than the distance between the magnets (leaving some room for optics), and for the glass blower to fuse the glass tube to the cylinder, it had to have a smaller diameter than the height of the cylinder.
A 30 l/s ion pump and a titanium sublimation pump (mounted at a 90° angle with the chamber) have been attached to this part of the chamber (see Figure 4.1 (a)). The Ti- sublimation pump works by coating the surrounding walls with titanium. This coating acts as a getter, absorbing background gas from the chamber. After a while, the coating will lose its getter efficiency, and a fresh coating of titanium needs to be added. So, while the ion pump is always on, the Ti sublimation pump is only turned on as needed.
The waveguide sample is loaded through a straight-through, all-metal valve at the end of the UHV chamber. The valve is all-metal so that it will withstand the high baking temperatures. The valve we purchased has two seals: a resealable copper gasket closing off the vacuum chamber and the so-called bonnet seal. The bonnet seal is located where the manipulator (controlling the opening and closing of the valve) is attached to the valve body. Upon pumping down the vacuum chamber for the first time, a leak in the valve was discovered. The leak occurred at the head of a bolt that was holding the resealable gasket to the valve manipulator. We redesigned this part of the valve. We replaced the resealable
77 gasket with a design (see Figure C.3) in which there is no hole in the gasket, but rather it is one solid copper piece with a tapped hole in it for a set screw to attach it to the manipulator. We also replaced the bonnet gasket with a thicker (1/8“) gasket that would allow retightening of the bonnet seal after every time we close the valve, since closing the valve pried apart the bonnet seal slightly.
The waveguide sample is attached to a Macor rod, which in turn is mounted to a copper bracket that can be screwed to a copper finger mounted on a tiltable and translatable flange at the top of the vacuum chamber. The reason the sample is mounted onto an insulator instead of a more stable metal is that during the experiment the magnetic trap will be turned off rapidly, and induced magnetic fields from eddy currents in the metal may exert unwanted forces on the atoms. The flange holding the sample is attached to the vacuum chamber through a bellows, and mounted to a tilt and translation stage that was designed based on the common adjustable optics mounts (see Figure C.9).
It allows tilting of the sample in two dimensions, as well as translation in two dimensions. However, it cannot be rotated about the axis of the copper finger. Thus, careful alignment of the copper finger with the axis of the chamber is necessary, so that the waveguide will not hit the chamber walls when loaded. This alignment was performed by attaching a long square rod (0.5” × 0.5”) to the copper finger similar to what would be done during the loading of the actual sample. The rod was aligned with the chamber, and then the flange holding it was tightened in place, fixing the rotational degree of freedom of the copper finger (see Figure 4.4). A full picture of the vacuum chamber is shown in Figure 4.1 (a).
78 4.1.3 Chamber Construction and Bake-Out
In order to achieve ultra-high vacuum, all parts have to go through a thorough cleaning procedure before assembly. Then, once the chamber is sealed and pumped down, a thorough bake-out is performed to reduce the outgassing rate. Any impurities with a high vapor pressure will be evaporated and pumped out during this process. The remaining impurities have low vapor pressures that do not have any negative effects on the quality of the vacuum. In particular, water has a high vapor pressure and has to be eliminated completely from the vacuum chamber.
The following cleaning procedure is used: a one hour ultrasonic bath in a mixture of Alconox and distilled water, followed by two thorough rinses in distilled water and one thorough rinse in Methanol. The purpose of the Alconox is to degrease the parts.
Then the parts are baked in air at 400 °C (i.e. as high as is possible) for several hours in a home-built furnace. Exceptions are: viewports were found to crack when baked at such high temperatures. We only baked these parts at 300 °C. The straight-through, all-metal valve has several copper gaskets in it which will oxidize when baked in air. This oxide layer is particularly problematic for the resealable gasket, since an uneven oxide layer will make a good seal impossible. This valve therefore can only be baked out under vacuum. The air bake creates an oxide layer on stainless steel parts that will hold in hydrogen. This oxide layer is responsible for the golden color the parts assume.
Once all the components were cleaned, they were assembled into the chamber shown in Figure 4.1 (a). The chamber is supported by stainless steel posts (visible in
Figure 4.1 (b)). Using stainless steel instead of aluminum ensured poor thermal contact between the chamber and the optics table. This is important during bake-out, since heat 79 loss due to heat conduction to the table may reduce the maximum bake-out temperature.
Because we intended to bake the chamber at very high temperatures, we put a non-grease lubricant, molybdenum disulfide suspended in methanol, on all bolts used to assemble the vacuum chamber. This prevents the bolts from seizing up due to the high heat. When finished with the assembly, the chamber was tested for leaks with a helium leak detector.
All noticeable leaks were repaired. The vacuum chamber was vented up to atmospheric pressure in order to add a few components for use during the final bake-out under vacuum: An additional 60 l/s ion pump was attached to the straight-through valve through a 20” long 2-¾” conflat nipple (and a 2-¾” conflat bellows for strain relief), along with a valve at the other end of the ion pump, so that the chamber could be sealed off at this point. This setup is shown in Figure 4.5. On the other side of the valve, we attached a turbo-molecular pump (“turbopump”), backed by a mechanical roughing pump for the initial pump-down and the early stages of the bake-out.
We prepared the chamber for bake-out as follows. Initially, all ion pumps were left off. The magnets are left on the two ion pumps that are part of the chamber, so that the ion pumps can be turned on during bake-out. Heat-resistant high voltage connectors were constructed. The commercial cables are not rated for the high temperatures of the bake-out. For the magnets themselves, the company quoted a maximum temperature of
400 °C. The chamber is first pumped down with the roughing pump. Once the pressure is low enough, the turbo-molecular pump is started. While the chamber is pumping down, it is prepared for bake-out. All glass pieces (including viewports) are covered in aluminum foil to prevent any grease from burning into the glass, permanently impairing the optical qualities of the glass components. Heater tapes are wrapped around all metal components
80 of the vacuum chamber. There were a total of four heater tapes – one around each of the two ion pumps, one around the chamber near the Ti sublimation pump, and one around the straight-through valve. When wrapping heater tape, care has to be taken not to wrap it on top of itself, since intense heat can develop there and destroy the heater tape. In addition to the heater tapes, an oven around the entire chamber was constructed, consisting of aluminum sheets with insulation attached to them. Three band heaters were mounted to the bottom of the oven – one right below the rubidium loading chamber, one below the UHV chamber, and one under the ion pump and Ti sublimation pump area. The reason for building an oven of this design was to ensure that the chamber would heat up more evenly through thermal exchange with the air, rather than heat conduction through the metal parts, thereby putting less strain on the glass due to uneven thermal expansion of glass and metal parts of the chamber. In addition, the oven reduces radiation and convection heat losses. In order to monitor the temperature of the different components of the system, thermocouple wires were attached to both ion pumps, the bellows between the two glass cells, the valve, and the chamber by the Ti sublimation pump. A thermocouple consists of two different conductors (in our case NiCr and NiAl (i.e. “type
K” thermocouple)) which are tied together at one end. At this junction, a voltage difference that is dependent on the temperature at that spot is produced by the thermoelectric effect.
The turbopump pumped down the chamber during the day, but it was not left on overnight, out of concern about oil from the roughing pump getting into the chamber in case the electricity were to shut off. Thus, the valve to the chamber was closed, with the turbopump still running (pumping only on the line connecting to the valve of the
81 chamber). The chamber was pumped down in this fashion for three days before starting the bake-out. The pressure read by an ion gauge near the turbopump was 4.5 × 10-6 Torr when the bake-out was started. The heaters and heater tapes are turned up slowly to make sure all parts of the vacuum chamber heat up evenly. For about four days, the bake-out ran during the day, and was shut off at night after valving off the turbopump, to ensure that no oil could get in the chamber in case of a power failure. The chamber reached temperatures of 250 °C (air) and 315 °C (metal parts) during this part of the bakeout.
After an initial increase in pressure (maximum: 5 × 10-5 Torr measured by ion gauge), the pressure decreased to about 2 × 10-6 Torr (at ion gauge). At this point the 60 l/s ion pump can be turned on, and the valve to the turbopump and roughing pump can be closed.
Since contamination of our chamber by oil was not a concern after this point, the bake- out was left running continuously. The highest temperatures obtained were 280 °C (for air) and 370 °C (for metal parts). The bake-out was left running at these values for three days. After slowly cooling down to room temperature, the pressure indicated by the 60 l/s ion pump controller was down to 10-8 Torr. At this point, the two ion pumps that are part of our chamber (8 l/s and 30 l/s) are turned on, and the straight-through all-metal valve is closed. After the initial pressure spike that occurs when first turning on an ion pump, both pumps finally indicated pressures of 10-9 Torr, or less. For low pressures, we relied on the ion pump current as a measure for the system pressure. At this point we believed that the indicated pressure readings were due to the leakage current across the insulators of the ion pumps. To determine the actual system pressure, we will examine the lifetime of the cold atoms in the magnetic trap.
82 Throughout the bake-out we monitored and recorded all temperatures and pressures by taking digital meter reading of all thermocouples, and ion gauge readings or ion pump controller voltages (which are proportional to the pressure), depending on the stage of the bake-out. The oven temperature and chamber pressure is also monitored with a LabVIEW code that could shut off the bake-out if a sudden increase in pressure or temperature is detected (see Appendix D). The ion pump and all heaters and heater tapes are powered through a home-built relay box, which could be shutdown by a signal from the computer.
4.2 Lasers
Loading atoms into a 2D optical lattice requires several lasers: two lasers are required for the initial trapping stage – the MOT, and one laser is required to create the
2D optical lattice above the waveguide sample. In order to perform the atom trapping experiment, precise control over the laser frequency is required. We also require the laser frequency to be stable, and the laser linewidth needs to be sufficiently narrow to resolve the excited state hyperfine structure very well. Two parameters influence the frequency at which a free-running laser diode lases – the diode current and temperature. To keep the frequency stable, both temperature and current feedback need to be employed. Also, the lasers need to be carefully tuned to the correct frequency with respect to the transitions in
Rb. To achieve this, we have constructed three compact, tunable, grating-stabilized diode lasers, as previously described in [46]. In addition to these three diode lasers, we use a tapered diode optical amplifier (Toptica TA100) to amplify the light from one of the
MOT lasers in order to create the required amount of power. In this section we will
83 discuss the details of the construction of the lasers, and how we tune and stabilize the lasers at the desired frequencies.
4.2.1 External Cavity Diode Lasers and Grating Feedback
The frequency stability and linewidth of a free-running laser diode can be greatly improved by placing it in an external cavity that provides optical feedback. The simplest form of external cavity is the extended cavity. As illustrated in Figure 4.6 (a), it consists of a laser diode with an anti-reflection coating on one facet that allows coupling of the laser to an external optical system with a wavelength selecting and a retro-reflecting optical element. The highly reflective facet of the diode and the retro-reflecting element constitute the extended cavity. This feedback method is readily implemented with common, commercially available diode laser packages with only one accessible output facet. Other, more complex external cavity setups can be found in [47]. The extended cavity lasers in our experiment utilize a diffraction grating both as wavelength selecting and retro-reflecting optical element. The equation governing the diffraction of a grating is:
d(sinθi + sinϕm ) = mλ ,
where d is the grating spacing, θi is the angle of incidence, ϕm is the angle of diffraction of the order m, and λ is the laser wavelength.
For our grating feedback, we mount the grating in the Littrow configuration (see
Figure 4.6 (b)). In the Littrow configuration, the grating is aligned such that the first order
84 ( m = 1) diffracted beam is coupled straight back into the laser diode, i.e. θi = ϕ1 ≡ θ , such that the grating equation becomes:
λ = 2d sinθ .
Thus, by tuning the grating angle θ , one can choose the wavelength λ coupled back into the laser, and thereby control the frequency of the laser that experiences the most gain, thereby tuning the laser frequency.
Unfortunately, the output beam (diffraction order m = 0 ) will move as the grating is tilted. To minimize this effect, we mount a mirror at a right angle with the grating, creating a corner cube, which will always point the beam in the same direction – anti- parallel to the output beam of the laser diode (see Figure 4.6 (c)). There still is a small effect on the beam position, because as the grating is tilted the beam shifts slightly transverse to the propagation direction.
This external optical feedback stabilizes the laser, narrows its linewidth and allows tuning of the laser frequency. The reduction in linewidth due to the external cavity is given by:
∆f sol ∆f ext = 2 , ⎛ τ ⎞ ⎜1+ ext ⎟ ⎝ τ sol ⎠
where ∆f ext ( ∆f sol ) and τ ext (τ sol ) are the linewidth and roundtrip time of the external
(solitary) laser cavity, respectively. For our cavity we estimate a theoretical reduction in linewidth by a factor of 0.005. For our lasers we find a reduction from ~25 MHz for a free-running diode to < 2 MHz for the external cavity laser.
85 4.2.2 Laser Construction
The external cavity diode lasers for the experiment were built following the design suggested by Ricci et al. [48]. Many of the drawings used were adapted from
Stephen Gensemer’s thesis [49]. Figure 4.7 shows a picture of a fully assembled laser.
Shop drawings for the different components can be found in Appendix C.
The base of each laser (see Figure 4.7 (a)) consists of a water-cooled heatsink, a thermoelectric cooler (TEC), and an aluminum plate with a heater, described in more detail below. On top of it sits an aluminum plate with a vertical flex hinge (see Figure 4.7
(c)), and a thermistor. The thermistor sits right below the laser head (see Figure 4.7 (b)), which holds the laser diode and collimating lens, fixing all degrees of freedom besides the distance between lens and diode. This distance has to be adjusted carefully in order to collimate the laser beam. The grating is mounted to a horizontal flex hinge (see Figure
4.7 (c)), which in turn is attached to the laser plate. An arm with a mirror is attached to the grating mount at a 90° angle, forming a corner cube (see Figure 4.7 (b), (d)), as described above.
Most of the pieces were machined from aluminum. Despite its high thermal expansion coefficient it is convenient as it is easy to machine and highly thermally conductive. Although there were concerns about the thermal stabilization of that mass of aluminum, it was found that the temperature feedback (see below in section 4.2.3.1) can keep the temperature stable enough for the cavity length not to change significantly when running the laser. Most of the parts were manufactured with CNC machining equipment but could be made with basic shop tools. The two flex hinges, however, required machining on an electric discharge machine (EDM).
86 A complete list of part numbers for all commercially available parts used in our lasers can be found in Appendix B. We use Sharp GH0781JA2C laser diodes, which can produce up to 120 mW (continuous wave) of laser power at a free running wavelength of
784 nm. The strongly divergent beam emitted by these laser diodes is collimated by a lens with a working distance (i.e. operating distance from the light emitting point of the laser diode to the back of the lens) of 4.57 mm (part 305-0066-780 by Optima Precision
Inc.). The lens is glued into a custom lens mount machined in our main machine shop
(see Figure C.19). The lens mount screws into the laser head mount in Figure C.16. Note the groove in the lens mount. With an appropriate driver it becomes extremely convenient to collimate the beam by turning the lens mount and, thus, moving the lens towards or away from the diode without changing the lateral position or tilt of the lens. Transverse and tilt adjustment of the lens is unnecessary because the mounts are precision machined so that the lens and diode are centered and aligned exactly. The laser diode is mounted with the help of the laser head bracket illustrated in Figure C.18. There are two standard sizes for laser diode packages. We originally were using the bigger diodes (9 mm diameter), but the Sharp diodes we use now are 5.6 mm in diameter. To fit these diodes into the original laser heads, we precision machined adapters (see Figure C.17).
The external feedback grating is glued onto the flexure hinge of the mount in
Figure C.20, which was produced with an EDM. The glue used for mounting both grating and mirror is DUCO cement. The grating mount is equipped with a pocket for a piezoelectric transducer (PZT) the one end of which is glued to the back of the flexure hinge and which is held in place by an adjustment screw. The grating we used is the
Edmund Industrial Optics model no. NT43-775 (i.e. 12.7 × 12.7 mm2, 1800 grooves/mm,
87 visible wavelengths). The PZT is from Thorlabs, part no. AE0203D08 (10 × 3.5 ×4.5 mm3, 9.1 µm displacement at 150 V).
The mounted pieces are assembled on the base plate (Figure C.14) as shown in
Figure 4.7 (b) and (d) (see also diagram in Figure C.10). Similar to the grating mount, the flexure hinge of the base plate was manufactured with an EDM. Two adjustment screws
(Thorlabs UFS075 and N100L5) can be used to move the two flexure hinges and, thus, adjust the vertical and horizontal angle of the grating so as to maximize the optical feedback from the grating. Then the grating adjustment screw and the PZT can be used to change the horizontal angle of the grating to coarse and fine tune the wavelength, respectively. A mirror (Thorlabs BB05-E02, ½” diameter, 6mm thick dielectric mirror) is mounted at an angle of 90º with respect to the grating so as to form a corner cube (see
Figure 4.7 (d) and Figure C.21).
4.2.3 Frequency Stabilization
4.2.3.1 Temperature and Current Control of the Free Running Diode
In order to keep the lasing frequency stable, any fluctuations in temperature and current need to be monitored and corrected. To this end a laser power supply was designed that corrects for temperature fluctuations through a feedback loop.
The temperature of the laser diode is monitored by a thermistor (10 kΩ at 25 ºC) mounted in a hole in the laser baseplate (see Figure 4.7 (b) and Figure C.15), right below the laser head holding the laser diode. The temperature (i.e. thermistor voltage) is compared to a set temperature (control voltage) by a home-built laser power supply. This power supply contains a feedback circuit that adjusts the temperature as needed by 88 changing the voltage across a heating resistor (50 Ω, ¼ W) mounted in an aluminum plate under the laser baseplate (see Figure 4.7 (a)). Two thermoelectric coolers (TEC,
Melcor PT2-12-30) are mounted next to one another underneath the heater plate and are always cooling the laser, and the voltage supplied to it has to be adjusted such that it keeps the laser near the desired temperature, and allows fine adjustment of the temperature through the heater. A thin layer of heatsink compound on each side of the
TEC ensures good thermal conductivity. The hot side of the TEC attaches to a water- cooled heatsink as shown in Figure 4.7 (a). Note that there is no electrical connection between the heatsink and the laser. This is important since the laser case may be at a voltage other than zero, depending on the laser diode used. For the same reason, the different layers of the heatsink and temperature adjustment units are fastened with Nylon screws. Circuit diagrams of the power supply and temperature feedback are shown in
Appendix E (Figure E.1).
To prevent air movements and ambient temperature changes from affecting the laser, we mount the laser in a Plexiglas box with airtight water and electrical feedthroughs. It would be desirable to have an antireflection coated laser light port as was done by Stephen Gensemer [49], but we simply send the laser beam through the Plexiglas wall. As far as we know this does not impair the beam shape, and has only minimal effect on the beam power.
In order to protect the highly static-sensitive laser diode during operation as well as when not in use, the power supply wires are fed through a shorting box (see circuit diagram, Figure E.2) that has two sets of protection diodes to prevent over-bias or reverse-bias of the laser diode. Also, to prevent current spikes from reaching the diode, a
89 protective inductance of 10 mH is in line with the diode and a 1000 µF capacitor is in parallel with the laser diode. High frequency signals such as sudden current spikes thus bypass the laser diode. A 10 kΩ potentiometer can be used to short the two connections to the diode to prevent the development of any current peaks while the laser is not in use.
The current feedback circuit described in detail below bypasses the inductor so that high- speed current regulation is possible.
4.2.3.2 Grating Feedback
The temperature and current control described thus far is the same as that employed for a free running diode laser. Now we will discuss in detail how grating feedback can be used to tune and control the frequency of the laser with respect to transitions in Rb vapor.
The technique we use to tune our lasers to the appropriate frequencies is the dichroic-atomic-vapor laser lock (DAVLL) as suggested by Corwin et al. [50]. We compare the laser frequency to the hyperfine transitions in Rb by comparing the Doppler- broadened DAVLL signal to a saturated absorption signal. We can tune the exact frequency with respect to the hyperfine peaks to MHz precision by adjusting the DAVLL signal electronically. Approximately 2 mW of laser beam power is split off the main laser beams and fed to the locking setup.
The DAVLL (illustrated in Figure 4.8) works as follows: the laser beam is first sent through a linear polarizer. This linearly polarized beam travels through a Rb-vapor cell, placed at the center of a solenoid. After the cell, the laser beam passes through a quarter-wave plate and a polarizing beam splitter. With no magnetic field in the solenoid 90 the quarter wave plate simply makes the incident light circularly polarized, and the polarizing beamsplitter emits two beams of equal intensity. The two resulting beams are detected by two photodetectors.
The photodetectors used in this setup contain a UDT-555 chip of the Photop
Series (see circuit diagram in Appendix E). This chip combines a photodiode with an operational amplifier. The gain of the detector is determined by the value of the feedback resistor. In our designs we employed a 10 kΩ resistor as indicated in Figure E.3, meaning
1 µA of photodiode current creates an output voltage of 10 mV.
While the magnetic field is off, as the laser frequency is tuned across a Rb absorption line, both of the photodiodes detect the Doppler-broadened absorption line.
The two photodiode signals are identical except, possibly, for their relative intensity and gain. One such absorption trace is shown in Figure 4.9 (a). As the solenoid is turned on, a longitudinal magnetic field is applied to the Rb vapor cell, resulting in Zeeman splitting of the Rb hyperfine states. The linearly polarized laser beam is a superposition of the two
+ − + circular polarizations σ and σ . The σ polarized light drives transitions with
∆mF = 1, e.g. from mF = −1 to mF = 0 . Since the lower mF states are shifted down by
the magnetic field (increasing the energy gap between F,mF = −1 and F',mF = 0 ), the energy at which the transition occurs is now higher than for a magnetic field of zero.
The σ + absorption line is, thus, shifted to higher frequencies. Similarly, the absorption features for σ − light are shifted to lower frequencies. The quarter-wave plate converts the two circularly polarized waves into two orthogonal linear polarizations, which are then split by the polarizing beam splitter and each polarization is detected by one of the
91 photodiodes. Thus, the signals measured by the two photodiodes are now shifted relative to one another, as shown in Figure 4.9 (b). The two signals from the photodiodes are subtracted electronically to yield an error signal, shown in Figure 4.9 (c). The laser frequency will be locked to the frequency of the zero voltage crossing. To lock to the frequency we need for the experiment, the zero voltage crossing can be changed (within limits) by either changing the relative gain of the photodiodes or by applying a DC offset.
The error signal is then fed back to the PZT behind the grating. This is a negative feedback setup: If a non-zero voltage develops (due to a shift in laser frequency) the voltage applied to the PZT will shift the grating angle such that the frequency is brought back to its locking point.
Unfortunately, since the grating feedback requires mechanical movement, it can not compensate for very fast frequency fluctuations. To compensate for these fast fluctuations, we feed the error signal back into the laser diode current, which also changes the frequency of the laser. This purely electronic feedback is fast and therefore corrects frequency shifts on the fast scale, which could not be compensated by the grating. Circuit diagrams for the locking electronics can be found in Appendix E (Figure
E.4).
For our experiment we need to set the locking point precisely (within a few MHz) to the appropriate frequency on or near the hyperfine transitions in Rb. We therefore need a means to resolve the excited hyperfine states in Rb, which are about ~100 MHz apart.
In a Rb vapor cell the absorption features are Doppler-broadened to several hundred
MHz. To resolve the hyperfine peaks of Rb we use a saturated absorption setup. Saturated absorption works as follows. A powerful pump beam and a much weaker (<1%) probe
92 beam from the same laser (i.e. same frequency) are sent through a Rb vapor cell in opposing directions, carefully overlapped (see Figure 4.10). On transition, the powerful beam will saturate the transition (i.e. excite all atoms), and the probe beam will pass through the vapor cell without attenuation. Near a transition, the pump laser will excite all the atoms that are moving at a velocity with respect to the beam such that the Doppler shift makes the pump laser seem in resonance. The probe laser, since it is traveling in the opposite direction will now be resonant with atoms going the same speed as those excited by the pump laser, but moving in the opposite direction. The probe laser thus interacts with a different group of atoms than the pump laser. The probe laser gets absorbed. This method creates Doppler-free, narrow (~ a few MHz) features that can be used to tune the locking point to the correct frequency with respect to the hyperfine transitions in Rb. The probe absorption signal is small compared to the absolute intensity of the probe. To distinguish the small saturated absorption signal we use lock-in detection. This is done by amplitude modulating the pump beam with a chopper (rotating with a frequency that is not a harmonic of 60 Hz, the most common interference frequency due to AC electrical lines). This modulates the saturation of the Rb atoms. When the probe laser interacts with the same group of atoms (i.e. on transition), the probe laser intensity is also modulated.
This modulated signal is detected by a lock-in amplifier that will only measure a signal that is modulated with the chopper frequency. It only measures the change in intensity of the probe beam, not its overall intensity. To accurately adjust the frequency the laser is locked to, the laser frequency is scanned by applying a voltage ramp to the grating with the computer. The traces of the DAVLL error signal and the saturated absorption signal are recorded. We then shift the error signal until the zero voltage crossing is at the correct
93 frequency. The resulting scans for the pump and trap lasers for our MOT are shown in
Figure 4.11.
Note that we have two possible ways to change the locking point of the laser with our circuit. We can change the photodiode gains, or we can shift the error signal up and down by applying a DC offset voltage to the error signal. In general it is advisable to use the photodiode gains to adjust the error signal, because it leaves the locking point insensitive to changes in overall intensity of the laser beam. Both photodiode signals change equally with intensity, which leaves the point where they subtract to zero unchanged. The DC offset, however, remains unchanged when fluctuations in intensity occur, changing the zero voltage crossing of the error signal with intensity.
There are, however, situations where the DC offset is required. We use a timed
DC offset pulse from the computer to change the laser detuning rapidly during the trapping sequence of the 2D optical lattice experiment.
4.3 The Atom Trap
The experiment requires an atom cooling and trapping sequence that cools down the atoms enough for loading of the 2D optical lattice. Following Lewandowski et al.
[44] the atoms are first cooled and collected at the center of a magneto-optical trap
(MOT). The atoms are then transferred to a purely magnetic trap and physically translated to the experimental chamber. Both the MOT and the magnetic trap use a magnetic quadrupole field for cooling and/or trapping of atoms, which, in our experiment, is provided by the same pair of magnetic coils, just very different coil currents. We will first review how magnetic and magneto-optical traps work, then
94 describe how we realize them in our setup, and finish by discussing the trapping sequence that prepares our atom sample. Overviews of the optics table and elements necessary to control the experiment are shown in Figure 4.12 and Figure 4.13.
4.3.1 Magnetic and Magneto-Optical Traps r A magnetic field B interacts with a magnetic dipole moment µr via the magnetic dipole interaction. This is also known as the Zeeman effect. The interaction energy is given by:
r r Emagn.dip. = −µ ⋅ B
r For a magnetic field pointing along the z axis, B = Bz , and
Emagn.dip. = −µ z Bz
= g F µ B mF Bz ,
where µ z is the component of the magnetic dipole moment along the quantization axis z,
and g F is the Landé g-factor (for hyperfine states), µ B is the Bohr magneton, and mF is the projection of F along the quantization axis z (magnetic substate). Depending on the
F,mF state the interaction energy is positive or negative, meaning the atoms will be attracted to low magnetic fields (weak-field seekers), or to high magnetic fields (strong- field seekers). An inhomogeneous magnetic field can be created to trap atoms in a certain
F,mF state. The simplest geometry, and the one employed in this work, is the quadrupole trap. A quadrupole field can be created by two magnetic coils with opposing currents as shown in Figure 4.14 (a). The magnetic field is zero at the center of the trap
95 and its magnitude increases linearly away from the center (see Figure 4.14 (b)). Atoms in a low-field seeking state are therefore trapped at the center of the magnetic quadrupole trap. Magnetic trapping requires that the atoms move through the magnetic field adiabatically, so that the magnetic dipole moment of the atom follows the magnetic field direction (i.e. stays aligned or anti-aligned with the magnetic field). Fast motion (i.e. fast change in the magnetic field as seen by the atom) can lead to spin flips into an untrapped state (“Majorana flips”). The condition
∆B << 1, (4.1) B where ∆B is the change in magnetic field during one Larmor precession of the atomic spin, and B is the magnitude of the magnetic field. The period of a Larmor precession is:
1 h ∆t Larmor = = , ω Larmor µ B B
where µ B is the Bohr magneton. For a quadrupole field with a magnetic field gradient
A , we can relate condition (4.1) to the temperature T of the atom sample and the distance of the atom from the trap center ( z ) as follows:
∆B ∆B ∆z ∆t = Larmor B ∆z ∆t Larmor B
h 1 = A⋅ v ⋅ 2 µ B B
h 1 = A⋅ v ⋅ 2 µ B (A⋅ z)
h k B T 1 = 2 << 1, µ B mRb A z
96 where k B is the Boltzmann constant, mRb is the Rb mass, v is the average velocity of the atoms, and ∆z is the distance traveled by the atom during one Larmor precession. There is thus a region near the center (small z) where this condition is not fulfilled and atoms are likely to be ejected from the magnetic trap due to spin flips (so-called Majorana flips).
The lower the temperature of the atom sample and the higher the magnetic field gradient, the smaller the ejection region near the center. For a magnetic field gradient of 210 G/cm and an atom sample temperature of 1 mK, the size of this region is ~10 µm. The expected magnetic trap cloud size is ~1 mm, about 100 times the size of the ejection region. This shows that it is essential that we pre-cool the atoms to low enough temperatures to be able to transfer them to the magnetic trap. We use a magneto-optical trap (MOT) to pre- cool our atom sample.
The MOT combines the energy shift due to magnetic fields with the detuning dependence of optical forces to cool and trap atoms. We will briefly review the MOT technique here. For a more detailed description, refer to [29]. We will first consider the simple case of an atom with two fine structure states g and e with total angular
momentum of J g = 0 and J e = 1. The excited state e has three magnetic substates with
mJ = 0,±1. In a magnetic field along the z axis, these states will shift in energy due to the magnetic dipole interaction:
Emagn.dip. = g J µ B mJ Bz , (4.2)
where g J is the Landé g-factor, µ B is the Bohr magneton, and Bz is the z-component of
the magnetic field. The shift of the magnetic sublevels is proportional to Bz and mJ .
97 As mentioned above, a magnetic quadrupole field has zero magnetic field at the center and the field changes linearly with the distance from the center (see Figure 4.14
(b)). In such a field, according to equation (4.2), the energy shift of the magnetic
substates is, thus, proportional to the distance z from the trap center and mJ (see Figure
4.15).
How can we make use of this to cool and trap atoms at the center of the quadrupole field? As illustrated in Figure 4.15, we shine two equally intense, counter- propagating laser beams, red-detuned from the (unshifted) atomic transition into the trap.
A σ − polarized beam propagates from the positive z direction, and a σ + beam from the negative z direction. Note that we have chosen the positive z direction as quantization axis. If the magnetic field direction were to be chosen as quantization axis instead, both
− laser beams would be σ polarized with respect to that axis. At a point z1 ( z1 >0), away from the trap center, the laser frequency will be closer to resonance with the
mJ = 0 → m'J = −1 than the mJ = 0 → m'J = 1 transition. The atoms at z1 will thus be more likely to scatter light from the σ − polarized laser beam, which drives transitions
+ with ∆mJ = −1, than from the σ beam. After many absorption-spontaneous emission cycles, the net momentum transferred will then be pointed along the σ − polarized laser, i.e. towards the trap center, since the net momentum acquired through spontaneous emission will average to zero over many cycles. Beyond the trap center, the
mJ = 0 → m'J = −1 transition will be shifted to the blue, since the magnetic field is negative (i.e. is pointed in the opposite direction), and the laser frequency is closer to the
mJ = 0 → m'J = 1 transition. Thus, at a point z2 ( z2 <0), more photons are scattered 98 from the σ + laser beam, which will exert a force on the atoms towards the trap center.
This is how atoms are slowed and driven towards the center of the quadrupole field. If three such counter-propagating beam pairs along the three dimensions of space are directed into the trap, the atoms will be pushed towards the center of the trap, regardless of their position. This is the principle of the MOT.
The atoms commonly used in atom cooling and trapping experiments usually have a more complicated level structure than the example described here. In particular, alkali atoms have a complex hyperfine structure (see Figure 2.4). To ensure that the atoms keep interacting, the trapping transition has to be a cycling transition (see Figure
2.7 (b)). One such cycling transition is F = I + S → F'= F +1, since selection rules
2 dictate that ∆F = 0,±1. However, because the hyperfine states of the P3 / 2 level are very close together in frequency (~100 MHz or less), there is a small probability that a laser tuned slightly to the red of the F = I + S → F'= F +1 transition will excite the
F = I + S → F'= F transition instead, which can decay to the F = I − S hyperfine ground state. Since the hyperfine splitting of the ground state is very large (~10 GHz), atoms that decay to F = I − S will not interact with the lasers anymore. Thus, the cooling and trapping is bound to stop once all atoms are in the F = I − S state. To prevent this, another laser, tuned to F = I − S → F'= I + S , is used to pump the atoms back into F = I + S .
In areas where the atoms (initially evenly distributed to all magnetic substates)
− preferably scatter σ light, the atoms will be optically pumped into the mF = −F state,
which, due to selection rules ( ∆mF = 0,±1 for spontaneous emission) forms a closed
99 system with mF ' = −(F +1) . The atom experiences a force toward the trap center.
Similarly, atoms at the other end of the trap scatter more σ + light and are pumped into
mF = F , which forms a closed system with mF ' = F +1, and, again, the atoms are pushed towards the trap center.
Thus, to realize a MOT we need: a magnetic quadrupole field, three pairs of counter-propagating laser beams tuned slightly to the red of the F = I + S → F'= F +1 transition (“trap laser”), and at least one laser beam tuned to the F = I − S → F'= I + S transition (“pump laser”), as illustrated in Figure 4.16.
4.3.2 The Setup of the MOT
A variety of atom traps based on the MOT principle have been devised. A MOT can be formed by three counter-propagating beam pairs in a magnetic quadrupole field.
More compact designs with only four beams (see [51] and references therein), or even a single beam [52] have been realized by various research groups. In this experiment, though, a standard six beam MOT with a magnetic quadrupole field created by two magnetic coils of opposite polarity (see Figure 4.2) is used. Frequently each pair of beams is created by retro-reflecting the beam back on itself. We chose not to retro-reflect our trap laser beams, because we expect a high optical density of our atom sample, meaning that a significant number of photons from the incident beam is scattered by the atoms. Therefore we split our trap laser beam into six individual beams of equal intensity and direct them into the trap. The beam paths of our MOT are indicated in Figure 4.17.
Both the trap and pump lasers are grating-stabilized diode lasers as described in section
100 4.2. The trap laser frequency is tuned approximately 15 MHz to the red of the
F = 2 → F'= 3 transition in 87Rb.
The more power we have in our trapping beams, the more atoms we can capture.
Therefore, we feed our trap laser beam (~50 mW) into an optical amplifier (Toptica
TA100), which produces about 600 mW of laser power at the same frequency as the seed laser. Although the spatial mode of the output laser beam is claimed by the manufacturer to resemble the shape of the seed beam, our experience has been that the light is spread out into a variety of spatial modes, creating dark fringes in our MOT region. To improve the beam quality, we feed it through a spatial filter with 4.4 mm focal length input and output lenses, and a 10 µm diameter pinhole at the center. We commonly get up to 130 mW of laser power out of the spatial filter. While this is sufficient beam power for capture of a large atom sample (~109) the efficiency is so poor that it almost warrants using the diode laser at its maximum power (~120 mW) directly to supply the MOT beams. In this iteration, however, we are using the amplified and spatially filtered beam.
The pump laser is tuned to resonance with the F = 1 → F'= 2 transition of 87Rb, to re- pump atoms that decayed into the untrapped hyperfine ground state.
There are four sets of hyperfine transitions in Rb: F = 1 → F' and F = 2 → F' for 87Rb and F = 2 → F' and F = 3 → F' for 85Rb. In this work, 87Rb is used. We thus need to identify the two sets of transitions of 87Rb in order to tune the two lasers (trap and pump). This is achieved by comparing the saturated absorption signal to a frequency reference, so that the frequency difference between saturated absorption peaks can be measured and compared to the Rb spectrum. In the experimental setup used in this work, a Fabry-Perot cavity with a free spectral range of 300 MHz provides the frequency 101 reference. For a laser in a single, stable mode, the Fabry-Perot signal shows peaks that are spaced 300 MHz apart. These peaks, when compared to the saturated absorption signal provide a frequency scale that allows measuring the frequency differences of the saturated absorption peaks. Comparison to the Rb spectrum allows identification of the transition. The hyperfine transitions of Rb are depicted in Figure 4.18. Typical saturated absorption signals for the four sets of transitions are shown in Figure 4.19.
The two lasers (trap and pump) are placed on the optics table and generate beams at a height of 4” above the table. All locking setups and diagnostic setups (Fabry-Perot and saturated absorption) are at the same height as the lasers. The trap center is elevated,
13” above the optics table, allowing enough space for MOT beam optics underneath the trap.
To get six beams of ~ 3 cm diameter, we split the ~0.5 cm diameter beam up into six equal parts with polarizing beam splitter cubes. We have placed a half-wave plate in front of every beam splitter to adjust the relative power of the two outgoing branches. We have arranged the beam paths such that there are three half-wave plates that control the relative power of opposing beams. This is important when we align the magneto-optical trap center with the center of our magnetic trap. About 1 ft. from the trap center the beam size of each beam is increased to the desired value with a telescope. The telescopes
consist of a pair of lenses with focal lengths of f1 = −50 mm and f 2 = 100 mm and diameters of 1” and 3”, respectively, spaced 50 mm apart. The vertical beam pair is split off sooner than the others and has a somewhat shorter beam path than the horizontal pairs. Because of this, the beam size in these two branches is about 25% smaller than in the other paths. We therefore require higher magnification of the beams in this 102 dimension. Because of spatial constraints below the bottom magnetic coil, we used a higher magnification telescope with the same distance between lenses as for the transverse beams, 50 mm. The telescope lenses for the vertical beams have focal lengths
of f1 = −25 mm and f 2 = 75 mm and diameters of 1” and 3”, respectively. With the trap laser power available we routinely achieve beam intensities of ~8 mW/cm2. To get the required circular polarization in our beams we have placed quarter-wave plates in each of the six beams. The quarter-wave plates and telescope are mounted on a specially designed track (see Figure C.25) that fixes the lateral translational and tilt degrees of freedom of the lenses and wave plate. This greatly simplifies the alignment of the MOT optics and increase the alignment stability.
The same magnetic coils used for the MOT also provide the quadrupole field for the purely magnetic trap. The magnetic trap requires a high magnetic field gradient (~200
G/cm). Because the magnetic field has to be switched on and off rapidly at several points in the trapping sequence, and we require a low inductance (i.e. small number of turns). A small number of turns means that a much higher current is needed in order to achieve the desired magnetic field gradient. A high current results in a large amount of dissipated heat. Therefore, the magnets are wound from 3/16” (outer diameter) refrigerator tubing, which is water-cooled by running tap water through its core. To insulate the tubing, we covered it in a layer of heat shrink tubing. With water-cooling, at least 420 A (the maximum current we can achieve with a 20 V power supply) can pass through these coils continuously without heating up the coils too much (i.e. feels hot to the touch, but does not burn).
103 The coils are wound on a Teflon spool (see Figure C.32), with Plexiglas walls that keep the tubing in place. The coils were wound on a lathe, which allowed for the creation of smooth, even turns by putting constant tension on the tubing while winding. Each coil consists of six layers with four turns each. To prevent the tubing from higher layers from slipping in between turns of lower layers, we covered every layer with a 1/16” thick
Teflon strip. In this way, we were able to wind two identical coils. We mounted the coils onto a Plexiglas mount (see Figure C.27), opposing one another. The distance between coils (center to center) is 10 cm, and the coil radius is 3.75 cm (7 cm) for the innermost
(outermost) turns. To ensure identical currents in both coils we connected them in series, electrically. The water is flowing through the two coils in parallel to ensure sufficient cooling by the water (see Figure C.27).
The magnetic field gradient along the axial dimension of the quadrupole coils was measured to be approximately 0.6 G/cm per A of current through the coils. For the operation of our MOT we use a current of 10 A through the coils to create a magnetic field gradient of 6 G/cm. We found experimentally that the number of atoms in the MOT cloud is not very sensitive to changes in the gradient within a few G/cm.
4.3.3 MOT Optimization
In order to optimize the trapping sequence several measurements were performed.
For optimal MOT operation the light polarization needs to be adjusted appropriately. All six MOT laser beams have to be σ − polarized with respect to the local magnetic field where the beams enter the trap. We performed tests to verify and optimize the
104 polarizations. Once operating, the properties of the MOT (number of atoms and fill time) were measured and optimized.
4.3.3.1 MOT Beam Polarization
One method of measuring the polarization of the laser light is to observe the shift in transition frequency when a magnetic field is applied. The Zeeman effect lifts the degeneracy between different magnetic substates, and therefore the transition frequency for σ − light is shifted towards lower frequencies. While it is straight-forward to align the quarter-wave plates in each MOT beam to yield some (close to) circular polarization, it is not as easy to determine whether the polarization is σ + or σ − . Not all quarter-wave plates have their fast axis marked. The best way to determine whether the MOT beams are polarized correctly would be to measure the frequency shift occurring in the actual
MOT setup. However, it is very difficult to detect absorption or fluorescence near the point where the laser beam enters the MOT, and not to pick up fluorescence from the opposite side of the MOT, where the magnetic field is pointed in the opposite direction.
We therefore perform the test in a mock setup that has incoming linear polarization and magnetic field polarization identical to that in the actual MOT. We modified one of the
DAVLL setups (see Figure 4.8) to achieve this. The polarization test setup is shown in
Figure 4.20.
The laser beam first passes through a polarizing beamsplitter cube of the same orientation as in the MOT setup. A half-wave plate in front of the beamsplitter allows adjustment of the laser polarization to maximize the beam transmitted by the beamsplitter. The quarter-wave plate from the MOT setup is placed downstream from the 105 beamsplitter, oriented so as to yield (nearly) circular polarization. The laser beam then passes through a Rb vapor cell placed in a solenoid. The polarization of the solenoid field is adjusted to match that in the MOT setup for each beam. The absorption signal is detected by a photodiode. The computer scans the laser frequency across the transition and records the photodiode signal and the saturated absorption signal with both the magnetic field on and off. If the laser polarization is correct (σ − ) the absorption signal is shifted to the red, otherwise it is shifted to the blue, with respect to the saturated absorption signal. We performed this test for all six MOT beams and observed a line shift of 133 MHz to the red for all branches.
After determining that the polarization has the correct handedness, we optimized the circularity of the polarization. The above results are possible with an elliptically polarized laser beam. To make sure the laser beam is truly circular, the following test was performed (see Figure 4.21). A polarizing beam splitter is placed before the quarter-wave plate in the MOT setup. A mirror is placed behind the quarter-wave plate to retro-reflect the laser beam. The beam splitter is aligned with the incoming linear polarization by maximizing the transmitted power. The angle of the beam splitter is aligned so that the reflected beam is exactly perpendicular to the incoming beam. A half-wave plate is used to temporarily create a polarization component that is reflected by the beamsplitter for this part of the setup. The mirror is carefully aligned to reflect the beam back on itself and also yield a reflected beam perpendicular to the incoming beam. If indeed the quarter- wave plate is at an angle that creates circular polarization, the back reflection of the light will be linearly polarized perpendicular to the incoming beam. It is thus reflected by the polarizing beamsplitter, and can be detected with a photodiode. The quarter-wave plate
106 angle is adjusted to maximize the reflected beam. After this measurement all six beam have circular polarization with the correct handedness (σ − with respect to the local magnetic field at the point where the laser enters the trap).
4.3.3.2 MOT Beam Intensity
For optimization of the MOT operation it is important that the intensities of all six beams are balanced. The MOT is not very sensitive to intensity imbalances of approximately 10 %. To get the intensities adjusted to within 10 %, we first adjust the beam shape such that all beams are the same size. After careful adjustment of the spatial filter, the laser beam is round and fairly well collimated. At the input of the MOT telescopes the beam diameter is approximately 0.5 cm. The input beams of the vertical
MOT beams are about 25 % smaller due to the shorter beam path, which is compensated by the higher magnification of the telescopes in these beams. After the telescopes the beam diameter is approximately 3 cm. With all beams having approximately the same beam profile, we can equal out the intensities by adjusting the relative power in the six beams. To adjust the relative power, we turn the half-wave plates in front of each polarizing beamsplitter cube to achieve the correct transmission-reflection (T:R) ratio
(2:1 for PBS 1 and 1:1 for all other polarizing beamsplitters). Once the power is adjusted to within 10 %, we also measure the intensity at the center of the beams directly, by placing a 3 mm diameter pinhole in front of the power meter. The typical beam intensity
(for 130 mW of laser power out of the spatial filter) is 8 mW/cm2.
After successful completion of these steps, the MOT is now functional, and fine adjustments can be made through measurements of the MOT cloud properties.
107
4.3.3.3 Characterization of the MOT
There are two properties of the MOT that help gauge the quality of the MOT: the maximum number of atoms captured and the fill time of the MOT.
The number of atoms in the MOT cloud can be determined from the fluorescence observed in the cloud. The fluorescence signal is imaged onto a photodetector with a lens
(see Figure 4.22).
Each atom in the MOT emits photons at a rate that is determined by the laser intensity and detuning. The scattering rate is (see [29] for derivation):
I 0 Γ
I sat 2 Γscat = 2 , I ⎛ ∆ ⎞ 1+ 0 + 4⎜ ⎟ I sat ⎝ Γ ⎠
where I 0 is the total intensity of all six MOT beams combined (approximately 48
2 2 mW/cm in our setup), I sat is the saturation intensity for Rb (1.64 mW/cm ), ∆ is the
detuning ( 2π × 15 MHz), and Γ is the natural linewidth of Rb ( 2π × 6 MHz). Γscat is the number of photons emitted per second by each atom. Since the emission process is spontaneous emission, the emitted photons are evenly distributed over all angles. The rate at which photons hit the photodetector is thus reduced by the solid angle covered by the collection lens:
2 πrlens 4 Γdet = 2 Γscat (0.96) , 4πltrap
108 where rlens is the radius of the collection lens (1.25 cm), ltrap is the distance of the lens from the trap center (30 cm), and the factor of (0.96) 4 accounts for the 4 % loss of photons at the reflection of each non-AR coated glass surface (in our case four).
The power incident on the photodetector is thus:
Pdet = hfΓdet N , where hf is the energy per photon, and N is the number of atoms in the MOT. The photodetectors used in this setup have a typical photodiode current of 0.65 A per Watt of
incident power. This quantity is known as the responsivity ( Rdet ). The output voltage of
the detector is determined by a gain resistor ( Rgain ) in the circuit. We use a gain resistor setting of 100 kΩ. The full expression for the atom number in the MOT is thus:
P N = det hfΓdet
I = det hfΓdet Rdet
V = det , hfΓdet Rdet Rgain
where I det is the photodiode current, and Vdet is the output voltage of the photodiode.
With the parameters listed above, the atom number in our trap routinely reaches 2 × 108 atoms.
To optimize the atom number of the MOT cloud, a fine alignment procedure is performed for all six laser beams. The smaller the size of the MOT laser beams, the more sensitive to alignment the MOT becomes. Apertures of 0.5” and 0.25” diameter were
109 placed into each beam after the telescope, one beam at a time, and the atom number was maximized by adjusting the beam alignment. These apertures were specially designed to be mounted on the optics tracks used for the telescopes, ensuring that the aperture is automatically at the center of the laser beams and the MOT. While good alignment (large atom number in the MOT) of all six beams was achieved routinely with the 0.5” diameter aperture, alignment of all six beams with the 0.25” aperture was difficult at first. This was due to a slight misalignment of the optics tracks in one of the MOT legs. After adjusting this alignment, the MOT was still operating with the 0.25” diameter aperture in each of the beams (one beam at a time). The numbers quoted for the MOT property measurements were performed when the MOT beams were aligned at the 0.5” beam diameter level.
To measure the fill time of the MOT, the photodiode signal was recorded after opening the trap and pump laser shutters. The LabVIEW control code used for this measurement is shown in Figure D.4. The resulting estimate of the fill time was approximately 10 s. Short fill times (~1 s) are indicative of vacuum problems, so the long fill time observed in this measurement seems to indicate that the vacuum in the MOT chamber is sufficient.
4.3.4 Alignment of MOT and Magnetic Trap
In order to enable transfer of atoms from the MOT to the magnetic trap, the trap center of the MOT cloud has to be aligned with the center of the magnetic trap. The center of the magnetic trap is the magnetic field zero of the quadrupole field. The MOT center is determined by the intensities of opposing laser beams. Intensity imbalance
110 between beams can shift the MOT away from the magnetic field zero of the quadrupole field.
To align the two, the magnetic field zero has to be identified. Three infrared (IR) cameras are pointed at the trap from different angles (shown in Figure 4.22). One camera points at the trap horizontally, and is perpendicular to the chamber axis. This camera observes the height of the trap. The other two cameras are placed along each of the horizontal beam pairs. To prevent them from blocking the MOT beams, they have to be placed at an angle. They provide an accurate measure of the displacement of the MOT cloud along each of the horizontal beam axes. To measure the magnetic trap center the magnetic field gradient is increased until the MOT cloud shrinks to the size of little more than a pixel on the TV screens. This usually happens at a current through the magnets of
100 A, corresponding to an axial magnetic field gradient of 60 G/cm. The point is marked on the TV screen. Then the MOT current is reduced to its usual running current. The
MOT displacement usually is no further than the size of the MOT (a few mm). But, to improve the alignment further, we adjust the half-wave plates in the MOT beam paths to move the MOT center towards the magnetic trap center. We have arranged the beam paths (recall Figure 4.17) so that the power in the two MOT beams along each dimension of space is balanced by one half-wave plate. We are thus able to move the MOT cloud along each dimension individually by adjusting each of the three half-wave plates (one for each MOT branch). We perform this alignment at the beginning of every run, since slight drifts in the beam position require realignment of the spatial filter and have some effect on the beam powers.
111 4.3.5 Loading the Magnetic Trap
The following describes how we efficiently transfer atoms from the MOT cloud to the magnetic trap. The procedure is very similar to that used by Lewandowski et al. [44].
To reduce excess energy acquired during the transfer to the magnetic trap due to the increase in Zeeman energy at points away from the magnetic trap center it is advantageous to reduce the size of the MOT cloud (initially ~5 mm) prior to transfer into the magnetic trap. This is achieved by reducing the radiation pressure in the MOT due to re-radiated photons from the spontaneous emission that is part of the MOT cooling process. We reduce the number of radiated photons by increasing the detuning of both the trap and the pump laser. This reduces the number of photons scattered from the trap laser per unit time, and increases the time atoms spend in the F = 1 ground state, which does not interact with the trap laser. The pump laser detuning ( ∆ / 2π ) is increased from 0 to
50 MHz to the blue of the F = 1 → F'= 2 transition, while the trap laser detuning
( ∆ / 2π ) is changed from 15 MHz to 50 MHz to the red of the F = 2 → F = 3 transition.
This so-called compressed MOT (cMOT) stage lasts for 10 ms. This gives the cloud time to collapse, but does not last long enough to lose a lot of atoms from the trap due to the reduced cooling and trapping power. The final cloud size in the cMOT is about 2 mm in diameter. Experimentally, we implement this stage by applying an appropriate offset voltage to the grating with the computer. The offset voltage is determined beforehand by measuring the position of the zero voltage crossing of the DAVLL error signal as a function of applied offset voltage. This is done separately for both lasers. Since both lasers are detuned at the same time, the voltages are split off from one and the same analog output channel on our data acquisition (DAQ) board (National Instruments
112 NI6014) with two adjustable voltage dividers. This output channel is referenced to the
100 µs precision internal clock of the DAQ board, allowing us to perform tasks with sub- ms timing.
The cMOT stage is followed by an optical pumping stage. During this stage the pump laser is turned off, allowing all atoms to be pumped into the F = 1 state. One might expect the atoms to be evenly distributed among the three magnetic substates, since spontaneous decay into all three of the magnetic substates is possible from any of the
F'= 2 substates. Only atoms in the F = 1,mF = −1 state are trapped by the magnetic field, so a transfer efficiency of approximately 1/3 is expected for transfer of the atoms into the magnetic trap. After optically pumping for 1 ms all of the atoms have been pumped to the F = 1 ground state. This can be verified by monitoring fluorescence from the trap laser only. Any remaining fluorescence in the cloud indicates that some atoms are remaining in the F = 2 state, and the optical pumping time has to be increased. The pump beam is shut off with the help of a home-built shutter (shown in Figure 4.23) that reacts to a trigger sent from one of the counter outputs of the DAQ board, which is also synchronized to the internal clock of the board. The required timing (to within << 1ms) cannot be achieved with the digital outputs.
Now, we can transfer the atoms to the magnetic trap by simultaneously turning off the trap laser and turning up the magnetic field. As per Lewandowski’s suggestion, we switch the magnetic field from 8 G/cm to 100 G/cm and then ramp it up to 250 G/cm over 500 ms. To check how many atoms were trapped, we switch off the magnetic field completely and turn on both the pump and trap laser (with their original detunings of 0 and 15 MHz, respectively) and monitor the fluorescence with both a photodetector and a 113 camera. By varying the time we leave the magnetic trap running before we take a fluorescence measurement, we can measure the magnetic trap lifetime, which in the Rb loading chamber we expect to be limited by the Rb vapor pressure (~10-7 Torr).
The trap beam is shut off with a shutter identical to the pump beam shutter, also triggered with a counter output of our DAQ board. The magnetic field is controlled in the following fashion: A TCR20T500 power supply, capable of supplying up to 20 V and
500 A is connected to the magnetic coils, in series with a power transistor (Eupec
FZ600R12KE3). The transistor is controlled by the computer, which applies a gate voltage that allows 10 A through the circuit for MOT operation. For transfer to the magnetic trap, the gate voltage is switched and ramped to achieve the desired magnetic field ramp sequence. The magnetic trap current control circuit is shown in Appendix E.
The Labview code controlling the MOT and magnetic trapping sequence is displayed in
Appendix D. The loading sequence is summarized in Table 4.1.
After the atoms are successfully and efficiently transferred to the magnetic trap, we are ready to move them to the UHV chamber.
4.3.4 Transfer to the UHV chamber
In order to transfer our 2 mm diameter cloud into the UHV chamber through the two 5 mm diameter pinholes on either end of the 3” bellows connecting the two parts of the vacuum chamber, we need to carefully line up the two parts of the chamber with the center of the magnetic field. Turning up the magnetic field to about 40 G/cm reduces the size of the MOT cloud and brings it closer to the magnetic field center. We can use this point as an indicator of the true magnetic trap center. With the help of an IR camera and a
114 ruler we intend to measure the center of the magnetic trap in the MOT region with respect to the glass walls. We then slightly adjust the height and horizontal distance from the magnetic track of the two pinholes so as to create a straight path into the UHV chamber.
This adjustment is not easy with our chamber design. The first pinhole is adjusted by adjusting height and horizontal distance of the second steel post holding the MOT part of the chamber near the small ion pump (see Figure 4.24 (a)). The position of the second pinhole is adjusted with the steel post supporting the bellows between the two parts of the chamber (Figure 4.24 (b)). This adjustment is somewhat easier because this post does not bear any significant weight. At the time of this writing we had not been able to successfully perform this alignment yet.
Once the two pinholes are aligned, we can transfer the atoms by translating the magnets over to the UHV chamber. The servotrack for our magnet assembly is an L20 linear step motor by Compumotor (see bottom of Figure 4.1 (b)). It is controlled by an
SX driver that receives commands from the computer through a serial port. Once the commands are loaded into the buffer of the driver, the driver awaits a trigger before executing its motion. The servotrack is capable of moving the magnets from the loading chamber to the UHV chamber within ≤0.5 s, well before the magnetic trap decays due to the collisions with background gas (Rb vapor) in the loading chamber.
Once we achieve this transfer, we will be able to get a measure for the background pressure in the UHV chamber which should be 10-10 Torr or less to allow us to perform the experiment. Like for the magnetic trap lifetime measurement in the MOT chamber we perform this measurement by leaving the magnetic trap on for variable amounts of time and then take a fluorescence image (i.e. turn off the field completely and
115 turn on a resonant laser). Such a lifetime measurement is essential because it may reveal pre-existing vacuum problems that may prevent us from performing the 2D optical lattice experiment. Modifications or an additional bake-out may be required. But, more importantly, if the pressure is low enough now, but increases significantly after loading the waveguide sample, this may indicate a fundamental problem. The waveguide materials, though bakeable to above 100 °C, are not bakeable as high as the rest of the chamber. We may not be able to rid the waveguide sample of high vapor pressure impurities. Outgassing from the sample itself may make the experiment impossible.
We have described the details of the experimental setup designed and built to demonstrate the proposed 2D optical lattice on a chip in the lab. To date, we have successfully been able to cool the atoms in a MOT, and have developed the timing and control necessary for the transfer to the magnetic trap.
116
Magnetic trap loading sequence Stage Laser detuning Shutter state Magnetic field Time ( ∆ / 2π ) gradient Pump Trap Pump Trap MOT 0 MHz -15 MHz Open Open 6 G/cm ≥ 10 s cMOT 50 MHz -50 MHz Open Open 6 G/cm 10 ms Optical pumping ⎯ -50 MHz Closed Open 6 G/cm 1 ms Magnetic catch ⎯ ⎯ Closed Closed 100 G/cm 3 ms Magnetic ramp ⎯ ⎯ Closed Closed 100 →210 G/cm 500 ms
Table 4.1: Magnetic trap loading sequence used in our experiment.
117
Figure 4.1: (a) Overview of the experimental system. (b) The Rb loading chamber is connected to the UHV chamber through a bellows with two 5 mm pinholes in it. Atoms are transferred from the loading chamber to the UHV chamber by translating the magnets along the servotrack.
118
Figure 4.2: The Rb loading chamber. The Rb filaments are sticking into the vacuum chamber from an electrical feedthrough (off to the left of the picture). The quadrupole magnets (opposing magnetic coils) are also shown in this view.
119
Figure 4.3: Close-up view of the UHV chamber. The straight-through, all-metal valve allowing for remote loading of our waveguide sample can be seen on the left. The top flange is holding a copper finger to which the sample will be attached. A tilt and translation stage will be holding the flange instead of the stainless steel sleeve shown in this picture.
120
Figure 4.4: Alignment of copper finger with chamber. The boron nitride rod is attached to the copper finger, just as it would be after loading the actual waveguide sample.
121
Figure 4.5: Setup for bake-out. A 60 l/s ion pump will take the place of the turbopump- roughing pump station during the bake-out once the pressure is low enough. The scaffolding outlines the position of the oven walls.
122
Figure 4.6: Grating feedback. (a) The extended cavity consists of a laser diode, a retro- reflecting and a wavelength selecting element. In our setup a grating fulfills both functions. (Adapted from [47].) (b) Littrow configuration. (c) Corner cube. While the lateral position of the outgoing beam shifts slightly as the grating is tilted, the angle of the outgoing beam remains steady.
123
Figure 4.7: Grating-stabilized external feed-back laser.
124
Figure 4.8: Illustration of the dichroic atomic vapor laser lock (top view).
125
Figure 4.9: Signals used for dichroic atomic vapor laser lock. (a) Absorption signal of each photodiode (magnetic field in solenoid is off). The frequency is scanned from higher to lower frequencies by applying a 100 ms voltage ramp to the laser grating. (b) Shifted absorption signals (σ + and σ − ) recorded by the two diodes when the magnetic field is on. (c) Error signal generated by subtracting the two photodiode signals. The signal shown has been shifted to move the zero voltage crossings of the error signal to the appropriate frequency.
126
Figure 4.10: Saturated absorption setup. A powerful pump beam and a weak probe beam travel through a Rb vapor cell in opposite directions. The pump beam is amplitude modulated and the signal due to the saturation of atoms (probe beam strength) is detected with a lock-in amplifier.
127
Figure 4.11: Saturated absorption (red/gray) and DAVLL error signal (blue/black) scans for frequency tuning of the lasers. (a) Trap transition. Note that the zero voltage crossing of the error signal is slightly to the right (i.e. red) of the first peak ( F = 2 → F'= 3 transition). (b) Pump transition.
128
Figure 4.12: Overview of the elements of the experimental setup.
129
Figure 4.13: Overview of control circuit of experiment.
130
Figure 4.14: Illustration of magnetic quadrupole field. (a) Two magnetic coils with opposite currents create a quadrupole field. (b) The quadrupole field magnitude is zero at the center of the magnetic coils pair, and increases linearly in every direction. The field direction switches.
131
Figure 4.15: Principle of the MOT (for the simplified case of an atom with J g = 0 and
J e = 1). The magnetic quadrupole field causes an energy level shift proportional to the distance from the trap center. In all points of the trap the net radiation pressure force is towards the trap center, because of the magnetic field induced detuning of the σ + and
σ − transitions (for quantization axis + z ). (Figure adapted from [29].)
132
Figure 4.16: Pump and trap transitions in 87Rb. The trap transition is a cycling transition.
However, there is a small probability of exciting the F’=2 state, which can decay into the
F=1 state, which does not interact with the trap laser. The pump laser lifts atoms in the
F=1 state to F’=2 which can decay either to F=1 or F=2.
133
Figure 4.17: Beam paths for our MOT. (a) Top view. Optics are arranged on two levels:
4” above the table and 13” above the table (except vertical beam optics). Optics on the lower level are indicated by dashed lines. (b) Side view (along chamber axis) of vertical beam optics (starting from “PBS 1”).
134
Figure 4.18: Rb hyperfine transitions in 85Rb and 87Rb. The Doppler-broadened Rb spectrum at the bottom shows four peaks for the four sets of hyperfine transistions.
135
Figure 4.19: Saturated absorption spectra of the hyperfine structure of each of the four sets of hyperfine transitions in Rb.
136
Figure 4.20: Polarization test setup. The orientation of the polarizing beamsplitter cube, quarter-wave plate and magnetic field polarity are matched to those of the actual MOT setup for each beam.
137
Figure 4.21: Polarization optimization setup. (a) The polarizing beamsplitter cube is lined up with the laser polarization by maximizing the transmitted signal. (b) The angle of the cube is aligned so that the reflected beam is perpendicular to the incoming beam. (c) The mirror is aligned to retro-reflect the beam. The quarter-wave plate is adjusted to maximize the reflected signal.
138
Figure 4.22: Illustration of photodetector and camera positions.
139
Figure 4.23: Mechanical Shutter. Two of these shutters are used to shut off the pump and trap beams during the magnetic trap loading sequence.
140
Figure 4.24: Chamber alignment. (a) The alignment of the first pinhole is very difficult because it requires sub-mm precision alignment of the steel post holding the 8 l/s ion pump. (b) The alignment of the second pinhole is achieved by adjusting this steel post, which does not bear any weight.
141
CHAPTER 5
NEUTRAL ATOM DETECTOR
In the course of previous studies of the interaction of cold atoms with surfaces we have developed a high resolution detector for neutral atoms. This detector has a wide range of potential applications in neutral atom experiments. In this chapter we present the design and characterization of this high resolution, high efficiency surface ionization detector for two-dimensional imaging of neutral atoms [53].
5.1 Introduction
The recent rapid increase in experimental research using ultracold neutral atom samples motivates the development of methods to detect neutral atomic species.
Frequently the design or even the feasibility of an experiment is determined by the detection options that are available. An example of this is the case of certain cold atom experiments that use metastable noble gas species [54, 55]. From a purely technical perspective, it is much easier to make ultracold alkali samples than ultracold metastable noble gas samples. Despite this, some cold atom experiments use the latter, largely because it is straightforward to image metastable samples with both high spatial resolution and high detection efficiency. Because the metastable detection scheme shares
142 much with our own, we outline it here. The metastable detector itself consists of a microchannel plate (MCP) with a phosphor screen at its output. The metastable sample to be imaged is directed towards the MCP; in the case of cold atoms, often the sample is simply dropped onto the MCP. Now usually, when a metastable noble gas atom hits the
MCP surface, it will give up its electronic energy by kicking an electron out of the surface and this electron will initiate an electron avalanche pulse in the MCP. In this way, an image of the metastable sample hitting the MCP is created on the phosphor screen. For quantitative work the phosphor screen may be monitored by a television camera feeding a frame grabber. Metastables are detected with order-unity efficiency and a spatial resolution limited only be the pore spacing of the MCP, ~10-15 µm. We have developed a detector with similar characteristics for neutral alkali atomic and molecular species.
Most ultracold atom experiments use alkali species and for these the most widely used detection options are either a Langmuir-Taylor “hot wire” detector based on the surface ionization process or a laser induced fluorescence arrangement. Some detection considerations were recently discussed by Delhuille et al. [56]. Laser induced fluorescence allows for imaging; however, stray light makes it difficult to detect small numbers of atoms. Also, the detection efficiency decreases rapidly as the velocity of the incident atoms increases, which renders this approach unsuitable for analyzing extremely weak thermal atom beams. Surface ionization detectors were originally developed by
Taylor [57] based on the work of Langmuir and Kingdon [58]. A detector usually consists of a heated thin wire or ribbon of tungsten or some other metal with a high work function. Neutral atoms or molecules that strike the wire can boil off as positive ions and these may be either measured as a current or detected, individually, using an electron
143 multiplier and particle counting electronics. 1D spatial information, for example, a 1D transverse profile of a beam, can be obtained by systematically moving the detector around. But, this is time consuming and may not be practical for experiments that require measuring very small numbers of atoms with high spatial resolution.
The basic idea of our detector is to direct the neutral alkali atom – in this work Rb
– sample that we wish to image towards a heated metal surface and allow surface ionization to create positive ions. These are accelerated a short distance into an MCP to initiate electron avalanches, and, in a fashion identical to the noble gas metastable detector, at the MCP output is a phosphor screen that is monitored by a television camera.
The result is a detector with high spatial resolution and high detection efficiency that can detect alkali atoms or any other atomic or molecular species with a low ionization potential. Specifically, in this chapter we describe a device that detects individual alkali atoms, with spatial resolution ~20 µm, detection efficieny ~50 %, and has a background count rate on the order of 30 Hz over a sensitive area of approximately 20 mm2. It can be used for detection of cold as well as thermal atoms.
There were two major technical obstacles in developing the detector. First, we tried several candidate metal samples for the “heated metal surface” and found that for samples of the highest purity that is commercially available, the trace amounts of alkali atoms in the metal itself would also surface ionize and cause enormous backgrounds (i.e. hundreds of kHz). To make a useful detector, these backgrounds had to be reduced to tens of Hz. Second, to have good spatial resolution it is desirable to get the hot metal surface close to the MCP, in our detector the spacing is ≤ 1 cm. But, at the same time,
144 the heat radiating to the MCP is potentially destructive to that device and, as we discuss below, can cause other problems.
Sheehy et al. [59] describe a detector that is in many ways similar to ours. They located a 70 % transmission, heated nickel grid directly in front of an MCP detector. Like us, they displayed the MCP output on a phosphor screen that was monitored by a television camera. In operation, the alkali atoms that hit the nickel grid usually boiled off as ions. The biasing was such that a fraction of these ions would be accelerated on through the grid and onto the MCP surface where they were detected, ultimately by the television system. Their arrangement allowed them to make 2D spatial measurements of the transverse distributions of atoms in atomic beams. They quote a spatial resolution
~500 µm but make no mention of detecting individual atomic particles.
5.2 Design
Figure 5.1 shows the overall design of the detector. Neutral Rb atoms hit a 20 mm long, 3 mm wide heated, tungsten-coated, rhenium foil ribbon. Atoms usually boil off the surface of the ribbon as (singly charged) ions. The ribbon is typically biased about 4000
V above the MCP input, which is held at ground potential. The ribbon and MCP form an
18° pie-shaped segment of a cylinder. Ions travel to the MCP very nearly along the electric field lines, which, for this geometry, are just circular arcs centered on the cylinder axis. The MCP that we use has 10 µm holes spaced on a 12 µm pattern. A third, “guard” electrode is located as shown in Figure 5.1 and biased so as to reduce fringing fields around the ribbon.
145 Our envisioned applications involve looking for DeBroglie wave interference fringes in cold beams of atoms that have passed through various thin slits or transmission gratings. Here, all atoms travel very nearly in the same direction, the beam axis (z in
Figure 5.1), and the detector gives the distribution of atoms in the plane transverse to that direction. In addition, it is the resolution along the direction perpendicular to the slits and grating lines that matters most to us and we arrange this to be the y-direction in Figure
5.1. The tilt of the hot ribbon relative to the incident beam is such that atoms separated by
∆y in the y-direction land on the hot ribbon and in turn the MCP separated by
∆y / sin18° , and this effectively improves the resolution of the detector in that direction by a factor of 1/ sin18° ≈ 3.2 . When we quote a resolution of 20 µm, it refers to the y- direction only. The spatial resolution for the x-direction or randomly directed atoms is 50-
60 µm.
A chevron configured pair of MCP electron multipliers is employed for the ions to generate pulses of electrons that are in turn accelerated onto a phosphor screen. The gap between the MCP output and the phosphor screen is 0.5 mm and the accelerating voltage between the two is typically 1350 V. We chose a phosphor type P22 for the screen because it has high conversion efficiency and is reasonably fast. The screen is viewed by a charge-coupled device (CCD) television camera whose output is directed either to a monitor or a frame grabber board in a computer. Image processing was done using LabVIEW software. Spots in the image were identified and the center of each spot was found. These correspond to the locations of individual Rb atoms in the sample as they hit the ribbon.
146 The heart of the detector is the hot ribbon. It was cut from a 12.7 µm thick rhenium foil using an electric discharge machine in order to minimize arcing problems due to sharp edges. Preliminary measurements with the bare foil yielded backgrounds of at least hundreds of kilohertz due to trace alkali impurities. These were reduced to tolerable levels as follows. First, a 2 µm thick tungsten coating was applied to the ribbon based on a procedure originally described by Frazer et al. [60]: In a separate vacuum chamber the ribbon was heated to ~1400 K (yellow hot) in a W(CO)6 vapor (W(CO)6 is a high vapor pressure solid). When a W(CO)6 molecule strikes the hot ribbon, the reaction
W(CO)6 →W + 6 CO takes place and tungsten is deposited on the ribbon surface. The CO gas released in the process was pumped off every ten minutes. An hour of coating time gave us a coating thickness of 2 µm. After the coating, a hard bake of 1600 K (yellow- white) for one hour removed residual impurities from the surface. The same coating process was also applied to the molybdenum ribbon holder.
After installing the detector into the vacuum system in which our experiments are performed, a soft bake at a temperature of ~1400 K (yellow hot) for several minutes cleaned the ribbon. A similar bake is necessary every time after exposing the detector to air. During these bakes a molybdenum heat shield attached to a linear motion feed- through is positioned between the ribbon and the MCPs to protect the MCPs from the radiation heating by the ribbon. During the operation of the detector, the ribbon is heated to approximately 1000 K (red hot). It is spring loaded to prevent sagging when heated.
A copper heat shield houses the detector. The heat shield, and in turn the detector itself, is cooled by a copper braid connected to a liquid nitrogen trap. Before we added the cooling, we found that the MCPs were significantly heated by the ribbon. This was 147 indicated by a reduction in the resistance of the MCPs when the ribbon was heated and, we believe, the premature, gradual, irreversible loss of gain in the MCPs. Also, we noticed that the background of the detector increased on a minutes time scale after the ribbon was turned on. This, we believe, was due to alkali atoms boiling off as neutrals from the glass of the MCPs as they heated up and then being detected at the ribbon.
Cooling the detector solved these problems. In addition, because Rb has a relatively high vapor pressure (~10-6 Torr) at room temperature, experiments sometimes have problems caused by backgrounds of atoms reemitted from various room temperature surfaces. The cold surfaces of the detector and its housing provide surfaces to trap those atoms and help reduce the overall system pressure.
5.3 Characterization
In this section we describe measurements that we carried out to characterize the detector. These measurements were not intended to calibrate the device but rather to check that it operates, generally, in the expected fashion.
5.3.1 Efficiency
A theoretical estimate for the quantum efficiency (QE ) of the detector is given by
QE = Pion Ppulse Pdetect .
Here, Pion is the probability that an atom incident on the hot ribbon is boiled off as a
positively charged ion, Ppulse is the probability that an ion will initiate an avalanche in the
MCP, and Pdetect is the probability that the television system and image processing
148 software will identify a given pulse of electrons hitting the phosphor screen as a count.
Assuming Rb atoms hitting the ribbon come into thermodynamic equilibrium with the surface, the ratio of atoms emerging as ions to those coming out as neutrals [61] is
~ e −e(I −φ ) / kBT .
Here, eI = 4.18 eV is the ionization potential of Rb, eφ = 4.55 eV is the work function of
tungsten, and k BT = 0.086 eV for a ribbon temperature of 1000 K. Thus Pion ≈ 1. Straub et al. [62] have recently investigated the absolute detection efficiency of a MCP operated
in a fashion similar to ours. They found that Ppulse ≈ 0.58 for 5.4 keV ions of any singly charged ion species. They note that this is very nearly the fractional open area of the first
MCP, 0.61, and that the difference may be understood as the result of ions that land within a channel but very near the surface of the plate. The secondary electrons for these ions are pulled out of the channel by the fringing field of the electric field accelerating the ions into the MCP and therefore do not initiate an avalanche. Ions in our detector impact the MCPs with an energy of ~4 keV and for the purposes of this analysis, it is adequate
and reasonable to expect that Ppulse is on the order of 1/2. Finally, we found that varying the MCP gain (i.e. voltage) has virtually no impact on the actual number of pulses that are counted in the detector. This means that most avalanche electron pulses at the phosphor screen are much larger than the threshold size needed for detection and
therefore Pdetect must be very nearly unity. Combining these results leads us to predict that the absolute quantum efficiency QE of the detector for Rb atoms is on the order of 1/2.
To check that the efficiency of our detector has the order of magnitude of the above theoretical prediction we carried out the following experiment. A sample of 87Rb
149 atoms was slowed in a Zeeman σ − slowing source [63] and captured in a MOT [64]. The atoms were then further cooled by gradient cooling molasses [65] and finally dropped freely onto the detector located 40 cm below. The temperature of the dropped atom sample was estimated using a time-of-flight technique [66]. This involved measuring the temporal pulse width of a fluorescence signal from the dropped atoms as they passed through a standing wave probe beam 24 cm below the trap. From the measured temperature, 13 µK, we find how much the atom sample spreads out at the detector (it is much broader than the ribbon, in all dimensions). The total number of dropped atoms was estimated from an absolute measurement of the fluorescence from the atom trap. We can estimate the number of atoms hitting the ribbon from these two measurements and compare with the number of atoms that are actually detected.
Figure 5.2 is a sequence of frames captured from the television camera showing the dropped atoms. Adding up all spots on frames (a)-(c) gives a (best estimate) measured absolute quantum efficiency of ~50 %. There are large uncertainties (~ a factor of 2) in determining the total number of dropped atoms based on the atom trap fluorescence.
However, the accuracy suffices to show that the order of magnitude of the quantum efficiency is as expected.
5.3.2 Background and Time Response
A notable feature of our detector is the very low background count rate that we achieve. To measure the intrinsic background count rate of the detector we operated the device in an ion pumped ultrahigh vacuum: P < 10-9 Torr. Even after coating the ribbon, the dominant source of background depends on the ribbon temperature and is likely 150 residual low ionization potential species boiling off as ions. We found a few “hot spots” on the ribbon that gave large backgrounds. However, these only constitute <1 % of the ribbon area and can be ignored in operation. Besides this, there is a trade-off that needs to be made in choosing a ribbon temperature: lower temperatures give lower backgrounds but degrade the time response of the device. We empirically chose to operate at a ribbon temperature of T ≈ 1000 K. At this temperature, the background rate across the whole detector was ~30 Hz, and the time response was adequate. The frames in Figure 5.2 were captured at a rate of 30 Hz and from them we can conclude that for the central ~2/3 of the ribbon, the time response of the detector is no worse than 1/30 s.
The particle counting capability of the detector suggests its utility for time-of- flight measurements or other low count rate experiments where timing information is important. For such applications we can theoretically estimate the time response of the detector. For ribbon temperatures between 1000 K and 1400 K the “ribbon release time” will usually dominate the time response of the detector. Note: for the present detector we restrict operation to below 1400 K because of concerns about damaging the MCP.
Hughes and Levinstein [67] have measured release times for Rb on tungsten and found values that range from ~10 ms at 1000 K down to 10 µs at 1400 K. In contrast, the ion travel time from the ribbon to the MCPs is ~0.1 µs with a time-of-flight variation of ≤
0.1 µs due to the ribbon tilt, the electron transit time through the MCPs and to the phosphor is < 1 ns, and the phosphor decay time is ~10 µs. Given the uncertainty of our estimates of the ribbon temperature, the expected 10 ms release time at 1000 K is consistent with the observations of Figure 5.2. Moreover, in principle, at the expense of
151 larger backgrounds, it should be possible to determine timings for the arrival of individual particles to within a couple of tens of µs with suitable electronics.
5.3.3 Resolution
There are several features of the detector that degrade its ultimate resolution,4 including the pore diameter of the MCPs and the finite pixel size of the CCD camera. It turns out that the dominant resolution-limiting mechanism results from the thermal transverse velocities of the ions boiled off the hot ribbon. These velocities cause a spatial spread of impacts at the MCP. We numerically solved the equations of motion of an ion in the electric field between the ribbon and the MCP, and found that, at the center of the ribbon, 1000 K ions accelerated from a single point would produce a 50 µm diameter spot on the MCP. As discussed above, the tilt of the detector spreads out displacements along the y direction and therefore effectively improves the resolution in that dimension by a factor of 3.2. Thus we expected a resolution of 16 µm. However, we had serious concerns that the resolution of the detector might be additionally very significantly degraded if atoms moved large distances along the hot ribbon before they were ionized.
4 Because of the various resolution degrading mechanisms, the apparent size of a sample, measured from the image made by the detector, will always be blurred to somewhat larger than the actual size of the sample. In this work, we use “resolution” to mean, quantitatively, “the apparent width (FWHM) that would be measured for an infinitesimally narrow atomic beam at the detector input.” Note: the resolution of the detector in the x direction is different than in the y direction: a beam that was infinitesimally narrow in both x and y would appear (e.g. in Figure 5.3) to be 50×20 µm2.
152 To check on this and find the order of magnitude of the spatial resolution we carried out the following measurement. First, to make a (near) point source of atoms, we placed a 200 µm diameter pinhole in front of a Rb oven. Next, we positioned a mask with two 15 µm slits separated by 500 µm in a plane 94 cm beyond the pinhole. The detector was located 15 cm further downstream. Figure 5.3 shows the image of the mask created by the atoms at the detector. A quantitative analysis of this and similar images yields a resolution of 15-20 µm, which is consistent with our expectations and eliminates the concern that the atoms move large distances on the heated surface prior to ionization.
5.3.4 Future Improvements
Incremental improvements over the detector described in this chapter are possible.
By increasing the voltage between the ribbon and the MCP or merely using that part of the ribbon nearer the MCP, it should be possible to improve the spatial resolution of the device by a factor of 2 or 3. Deconihout et al. [68] have recently shown that moderate improvements in detection efficiency may be achieved by putting a grid in front of the
MCP and biasing it so as to direct secondary electrons back into the MCP. A faster television camera could improve the response time of the detector. Finally, there is no fundamental limit to the length and the width of the ribbon.
This concludes the presentation of the design and characterization of the high efficiency, high resolution neutral atom detector that was developed as part of our work on cold neutral atoms.
153
Figure 5.1: Surface ionization detector for 2D imaging of neutral atom samples. The neutral atoms incident on the detector are indicated by the blue, narrow arrow along z.
The ions boiled off the hot ribbon are accelerated onto the MCPs as indicated by the red, curved arrow.
154
Figure 5.2: Detection efficiency measurement. (a) – (c) are three consecutive frames recorded at a frequency of 30 Hz by the frame grabber upon dropping a cold atom sample onto the detector. Most of the atoms are detected within one single frame, (b). The ends of the detector ribbon are cooler, and therefore have a longer response time, spreading the detection of incident atoms over all three frames.
155
Figure 5.3: Resolution measurement. The slits imaged here are 15 µm wide and separated by 500 µm. Analysis of this and similar images yields a resolution estimate of 20 µm along the y direction (see Figure 5.1).
156
CHAPTER 6
CONCLUSIONS
We have presented the results of our work towards realizing 2D optical lattices on a chip. Optical lattices are standing waves of light resulting in a periodic one- or multidimensional pattern of nodes and antinodes of light intensity. These light patterns constitute an array of optical dipole traps for neutral atoms. Atoms are trapped at nodes of light intensity for laser light tuned above the atomic resonance (“blue” detuning), and at antinodes for a laser frequency below the atomic resonance frequency (“red” detuning).
The expression “optical lattice” stems from the fact that, when filled with one atom per site, the optical lattice resembles the crystalline lattices of solid-state matter.
This system – an array of single atoms at sites of an optical lattice – has been investigated for potential use as a quantum register for quantum computing. Each atom encodes one qubit. Of all possible lattice geometries (1D, 2D, or 3D), thus far, only 3D optical lattices could provide the number of qubits and tight confinement in all three dimensions needed for quantum computing. A major drawback intrinsic to this geometry is the difficulty of addressing individual atoms (qubits) on the inside of the optical lattice with laser beams, without disturbing other qubits along the beam path. This problem led us to investigate other possibilities for creating optical lattices.
157 We have developed a unique method of creating 1D and 2D optical lattices on a chip. These lattices provide both the large number of qubits as well as tight confinement in all three dimensions needed for quantum computing. Individual atoms (qubits) can be addressed from above or below, or from the sides in the case of the 1D lattice. The optical lattices are created by interfering the evanescent waves of several modes of an optical waveguide to create nodes above the waveguide surface. For blue-detuned light, atoms can be trapped at these nodes. One pair of modes coupled into a ridge waveguide creates a 1D array of nodes above the waveguide. Two orthogonal pairs of modes coupled into a slab waveguide result in a 2D array of nodes. Computational analysis of our method using realistic waveguide materials and laboratory laser parameters showed that for ~100 mW of laser power nodes can be created ~100 nm above the waveguide surface. The lattice sites constitute atom traps that are ~1 mK deep, and have trap frequencies ~1 MHz with ground state sizes of ~10 nm. The spacing between traps is ~1
µm, enabling individual addressing with a focused laser beam. The construction of an optical lattice on a chip bears the additional advantage that may be possible to integrate micro-fabricated wires into the chip, potentially allowing for the development of a hybrid magneto-optical atom chip.
We have also conducted initial studies of variations of the basic 2D optical lattice for implementation of two-qubit gates. The two examples presented in this work are a counter-propagating 2D optical lattice with very close trap spacing (~λ/4) in one dimension, and a 2D optical lattice with adjustable spacing between pairs of traps in one dimension. This would allow for bringing qubits close together for fast two-qubit gate
158 operation and separating them in between gates for easy individual addressing for single- qubit gates.
The optical atom chip proposed by us is, as far as we know, the first array of microtraps this close to a dielectric surface. Its physical realization would give us the unique opportunity to study 1D or 2D systems of atoms and their interaction with the nearby surface. The decoherence effects of a dielectric surface on a nearby atomic qubit are unknown at this point. The physical realization of the proposed chip would allow for the experimental investigation of these decoherence effects.
We designed and built an experimental setup in order to demonstrate a 2D optical lattice on a chip. This kind of cold atom trapping experiment requires ultra-high vacuum
(UHV) conditions and several cooling and trapping steps to prepare the cold atom sample. We use 87Rb atoms in our setup. The experiment will be performed as follows.
The atom sample is collected and pre-cooled in a magneto-optical trap (MOT). Atoms are collected from a Rb background gas. Because the Rb background would rapidly eject atoms from the optical lattice traps, we have built a two-chamber vacuum system with a
Rb loading chamber and a UHV experimental chamber, separated by a small aperture. To transfer atoms into the UHV chamber (where the optical atom chip is located), we first transfer the atom sample from the MOT to a magnetic trap, and then translate the magnetic quadrupole coils carrying the atom sample over to the UHV area. There, the cold atom sample is lowered onto the atom chip, and atoms are loaded into the 2D optical lattice, where they can be detected with a focused laser beam.
To date, we have successfully collected a large (~2 × 108) atom sample in the
MOT, and have developed the necessary circuitry and timing software for the transfer
159 procedure to the magnetic trap. It is clear that there is more work left to do in order to obtain experimental proof for the existence of the proposed optical lattices. But, if successful, an attempt can be made to load single atoms into each lattice site, in principle creating a scalable, addressable quantum register.
In the course of studies of cold atoms near surfaces leading up to the current experiment, we have developed a novel neutral atom detector. The core of the detector is a 3 mm wide hot ribbon that can surface-ionize an incident neutral atom beam and subsequently image it onto a camera. We get high resolution (~20 µm), 2D images of the atom sample at a rate of 30 Hz. This detector has a wide range of applications in experiments with neutral atoms or molecules.
160
APPENDIX A
DERIVATION OF THE AVERAGE SPONTANEOUS EMISSION RATE
Because expression (3.1) for the average spontaneous emission rate for blue- detuned dipole traps is not shown in the literature, we derive it here. The spontaneous
r r emission rate Γsc (r) at position r = (x, y, z) is related to the local light potential
r U light (r ) as follows [25]:
r Γ r Γsc (r ) = U light (r ) . h∆
The average of the spontaneous emission rate for an atom in the vibrational ground state of the trapping potential is thus:
Γ η ≡ Γsc = U light , h∆ where the average is over the spatial degrees of freedom.
The average light potential is:
∞ U = U (x, y, z)ψ (x, y, z) 2 dxdydz , light ∫ light −∞
161 where ψ (x, y, z) 2 is the probability density of a trapped atom in the vibrational ground state of the trapping potential. We assume that the trapping potential near the trap center can be approximated by a 3D harmonic oscillator potential:
m U = (ω 2 x 2 + ω 2 y 2 + ω 2 z 2 ) . light 2 x y z
The probability density is thus calculated from the ground state wavefunction of a 3D harmonic oscillator:
ψ (x, y, z) = ϕ x (x)ϕ y (y)ϕ z (z) , where
1/ 4 1 m 2 ⎛ mω j ⎞ − ω j j ⎜ ⎟ 2 h ϕ j ( j) = ⎜ ⎟ e ⎝ πh ⎠ are the 1D harmonic oscillator ground state wavefunctions for j = x, y, z . The probability density is thus:
2 2 2 2 ψ (x, y, z) = ϕ x (x)ϕ y (y)ϕ z (z)
1/ 2 m 1/ 2 m 1/ 2 m − ω x2 − ω y2 − ω z 2 ⎛ mω x ⎞ x ⎛ mω y ⎞ y ⎛ mω z ⎞ z = ⎜ ⎟ e h ⎜ ⎟ e h ⎜ ⎟ e h ⎝ πh ⎠ ⎝ πh ⎠ ⎝ πh ⎠
3 / 2 m m m − ω x2 − ω y2 − ω z 2 ⎛ m ⎞ 1/ 2 x y z h h h = ⎜ ⎟ (ω xω yω z ) e e e ⎝ πh ⎠
Plugging this into the expression for U light leads to:
∞ 3/ 2 m m m − ω x2 − ω y2 − ω z2 m 2 2 2 2 2 2 ⎛ m ⎞ 1/ 2 x y z U = (ω x + ω y + ω z ) (ω ω ω ) e h e h e h dxdydz light ∫ x y z ⎜ ⎟ x y z −∞ 2 ⎝ πh ⎠
5 / 2 3 / 2 ∞ m 2 ∞ m 2 ∞ m 2 − ω x − ω y − ω z m ⎛ 1 ⎞ 1/ 2 ⎛ 2 2 x y z = (ω ω ω ) ⎜ω x e h dx e h dy e h dz ⎜ ⎟ x y z ⎜ x ∫ ∫ ∫ 2 ⎝ πh ⎠ ⎝ −∞ −∞ −∞ 162 ∞ m ∞ m ∞ m − ω y2 − ω x2 − ω z 2 2 2 y x z + ω y e h dy e h dx e h dz y ∫ ∫ ∫ −∞ −∞ −∞
∞ m ∞ m ∞ m − ω z 2 − ω x2 − ω y2 2 2 z x y ⎞ + ω z e h dz e h dx e h dy⎟ z ∫ ∫ ∫ ⎟ −∞ −∞ −∞ ⎠
5 / 2 3 / 2 m ⎛ 1 ⎞ 1/ 2 2 ≡ ⎜ ⎟ (ω xω yω z ) (ω x I1 (x)I 2 (y)I 2 (z) 2 ⎝ πh ⎠
2 2 + ω y I1 (y)I 2 (x)I 2 (z) + ωz I1 (z)I 2 (x)I 2 (y)) , where we have introduced the following shorthand for the integral expressions:
∞ m − ω j 2 2 j I ( j) = j e h dj 1 ∫ −∞ and
∞ m 2 − ω j j I ( j) = e h dj , 2 ∫ −∞ for j = x, y, z . The solution to these integrals can be found in integral tables:
∞ m − ω j 2 2 j I ( j) = 2 j e h dj 1 ∫ 0
3/ 2 1/ 2 ⎛ m ⎞ = 2π 4⎜ ω j ⎟ ⎝ h ⎠
3 / 2 1/ 2 h π 1 = 3 / 2 3 / 2 2m ω j and
∞ m 2 − ω j j I ( j) = 2 e h dj 2 ∫ 0
163 1/ 2 1/ 2 ⎛ m ⎞ = 2π 2⎜ ω j ⎟ ⎝ h ⎠
1/ 2 ⎛ hπ ⎞ 1 = ⎜ ⎟ 1/ 2 . ⎝ m ⎠ ω j
Plugging these results into the expression for U light we find:
3 / 2 m5 / 2 ⎛ 1 ⎞ 3 / 2π 1/ 2 ⎛ π ⎞⎛ 1 1 1 U = ⎜ ⎟ (ω ω ω )1/ 2 h ⎜ h ⎟⎜ω 2 light 2 x y z 3 / 2 m ⎜ x 3 / 2 1/ 2 1/ 2 ⎝ πh ⎠ 2m ⎝ ⎠⎝ ω x ω y ω z
1 1 1 1 1 1 ⎞ + ω 2 + ω 2 ⎟ y 3 / 2 1/ 2 1/ 2 z 3 / 2 1/ 2 1/ 2 ⎟ ω y ω x ω z ω z ω x ω y ⎠
= h (ω + ω + ω ) 4 x y z
π = h ( f + f + f ) 2 x y z
We thus find the result for the average spontaneous emission rate:
Γ hπ η = ( fx + f y + fz ) ∆h 2
π Γ = ( f + f + f ) . 2 x y z ∆
164
APPENDIX B
LIST OF PARTS
165
List of parts used in the 2D optical lattice experiment
Item Qty. Company Part number Vacuum chamber Rb loading glass cell 1 homebuilt UHV glass cell 1 homebuilt Rb dispensers 2 SAES getters Rb/NF/3.7/12/FT10+10 Electric vacuum feed-through 1 Ion Pump (8 l/s) 1 Ion Pump (30 l/s) 1 Varian 911-5032 Ion pump control unit 2 Varian 921-0062 copper blank for 1-1/3" flange (modified by us) 3 Duniway Stockroom Corp. 1-1/3" Tee 1 4 5/8"-2-3/4" Tee 1 1-1/3" formed bellows 1 2-3/4" welded bellows 1 Straight-through, all-metal valve 1 Thermionics SMV-150-33 Titanium sublimation pump 1 National Electrostatics Corp. TS10 Tilt and translation stage 1 homebuilt 4-1/2"-2-3/4" Adapter 1 4-5/8"-2-3/4" Adapter 2 2-3/4" cube 1 2-3/4" viewports 3 2-3/4" spacer (3/4" thick) 1 Stainless steel posts (assorted sizes) 7 homebuilt Silverplated bolts (5/16-24, 1/4-28, 8-32) Duniway Stockroom Corp.
Vacuum preparation and Bake-out He leak detector 1 Valve 1 Vacuum hose 1 Turbo-molecular pump 1 Pfeiffer TPU 170 Control unit for turbopump 1 Mechanical pump 1 Welch 1402 Thermocouple gauge 1 Varian 801 Ion gauge 1 Kurt J. Lesker G100TF Ion pump (60 l/s) 1 Varian 911-5034 Ion pump controller 1 Varian 921-0015 20" long, 2-3/4" nipple 1
Continued
Table B.1: List of parts for 2D optical lattice experiment.
166 Table B.1 continued
2-3/4" welded bellows 1 Heat-resistant high voltage connector 1 homebuilt Heater tapes 4 Oven 1 homebuilt Band heaters 3 Thermocouple wires 5 Relay box 1 homebuilt
Lasers Optical amplifier (with Faraday isolator) 1 Toptica TA100 Optical amplifier control unit 1 Toptica DC100 3 Grating-stabilized diode laser parts ea. homebuilt Laser diodes 3 Sharp GH0781JA2C Collimating lenses 3 Optima Precision Inc. 305-0066-780 Gratings 3 Edmund Industrial Optics NT43-775 Piezo-electric tranducers 3 Thorlabs AE0203D08 Corner-cube mirrors 3 Thorlabs BB05-E02 Fine adjustments screws 6 Thorlabs UFS075 Nuts for fine adjustment screws 6 Thorlabs N100L5 Thermoelectric coolers 6 Melcor PT2-12-30 Thermistor (10 kW at 25 °C) 3 Heating resistor (50 W, 1/4 W) 3 Heatsink compound 1 Melcor TG-003 Thermal epoxy 1 Melcor TCE-003 Swagelok fittings (1/16" NPT to 1/4" tubing) 6 Nylon screws (8-32, cap screws) 18 Plexiglas boxes 3 homebuilt Shorting boxes 3 homebuilt Laser power supply 3 homebuilt TEC cooler power supplies (3A) 3
Laser tuning Photodiodes 8 UDT sensors incorporated Photop UDT-555 Cases for photodiodes 8 homebuilt Rb vapor cells 4 Solenoids (for DAVLL) 3 homebuilt Laser locking electronics 3 homebuilt Oscilloscopes 3 Tenma 72-720 Chopper 1
Continued
167 Table B.1 continued
Chopper controller 1 Stanford Research Systems SR 540 Lock-in amplifier 1 Stanford Research Systems SR 530 Fabry-Perot cavity 1 Fabry-Perot cavity controller 1 homebuilt
Optics Faraday isolator 1 Isowave I-80T-4 Linear polarizers 3 Quarter-wave plates (AR coated), 0.5" diameter 1 Quarter-wave plates (AR coated), 1" diameter 8 Half-wave plates (AR coated), 0.5" diameter 2 Half-wave plates (AR coated), 1" diameter 4 Polarizing beamsplitters 3 Optics for Research PE-8-NIR Polarizing beamsplitter cubes (1") 5 Beamsplitters 5 Dielectric mirrors, 1" diameter 18 Dielectric mirrors, 1.5" diameter 1 Dielectric mirrors, 2" diameter 2 First surface gold mirrors, 2" diameter 6 Lenses (4.4 mm focal length) 3 Lens (1", collection) 1 Lenses (1.5" diameter) 3 Lenses (1" diameter, focal length -50 mm) 4 Lenses (3" diameter, focal length 100 mm) 4 Lenses (1" diameter, focal length -25 mm) 2 Edmund Industrial Optics F45-922 Lenses (3" diameter, focal length 75 mm) 2 Edmund Industrial Optics F45-368 Pinhole, 10 µm diameter 1 Thorlabs Pinhole, 3 mm diameter 1 Iris 1 ND filters 3 Optics tracks for MOT optics 6 homebuilt Rotation mounts for wave plates (0.5") 3 Rotation mounts for wave plates (1") 12 Mounts for polarizing beamsplitters 3 Mounts for polarizing beamsplitter cubes 5 homebuilt Beamsplitter mounts 5 homebuilt Mirror mounts (1") 18 homebuilt Mirror mounts (1.5") 1 Mirror mounts (2") 8
Continued
168 Table B.1 continued
Lens mounts (<0.5" diameter) 3 Lens mount (1" diameter) 1 Lens mounts (1.5" diameter) 3 Custom lens mounts for MOT telescope (1") 6 homebuilt Custom lens mounts for MOT telescope (3") 6 homebuilt Optics posts (1/2" diameter) 64 homebuilt Optics postholders 64 homebuilt Bases for postholders 64 homebuilt
Atom trapping and transfer Magnetic quadrupole coils 2 homebuilt Magnet assembly 1 homebuilt Magnet power supply 1 Electronic Measurements Inc. TCR20T500 Magnet current control 1 homebuilt Power IGBT module for magnet current control 1 Eupec FZ600R12KE3 Servotrack 1 Compumotor L20 Servotrack driver 1 Compumotor SX IR cameras 3 CCD camera 1 Watec 902B TV screens 4 Shutters 2 homebuilt MOT-magnetic trap alignment apertures 4 homebuilt
Computer control Experiment control computer 1 Image acquisition computer 1 Data acquisition boards 2 National Instruments NI6014 Breaker boxes for DAQ boards 2 homebuilt IMAQ board 1 National Instruments PCI-IMAQ-14
169
APPENDIX C
SHOP DRAWINGS
170
171
Figure C.1: Glass cell for Rb loading chamber.
172
Figure C.2: Glass cell for UHV chamber.
173
Figure C.3: Resealable copper gasket for straight-through, all-metal valve. The sample is the original gasket (with
clearance hole instead of solid copper cap).
174
Figure C.4: 1/8” copper gasket for bonnet seal of straight-through all-metal valve. Sample is a commercially available 3 3/8”
copper gasket.
Figure C.5: Waveguide mount. The sample (5 mm thick glass) is screwed to the end of the macor rod with alumina screws.
175
Figure C.6: Stainless steel rod for attaching the waveguide sample mount to the copper finger.
176
177
Figure C.7: Copper finger for mounting sample holder to chamber.
178
Figure C.8: Modified rotatable flange to hold copper finger.
(a) 179
Continued
Figure C.9: Tilt mount for sample holder flange (attaches to a translation stage). (a) Front piece. (b) Back piece.
Figure C.9 continued
(b) 180
181
Figure C.10: Overview of laser assembly. Only home-built parts are listed here.
182
Figure C.11: Laser heatsink.
183
Figure C.12: Cover plate for laser heatsink and heater plate.
184
Figure C.13: Laser heater plate.
185
Figure C.14: Laser base plate.
186
Figure C.15: Thermistor holder (mounts into base plate with 4-40 flathead screw).
187
Figure C.16: Laser head (precision machined to line up lens and laser diode).
(a) 188
Continued
Figure C.17: Laser diode adapters. (a) Front adapter. (b) Back adapter.
Figure C.17 continued
(b) 189
190
Figure C.18: Laser diode mounting ring (attaches laser diode to laser head).
191
Figure C.19: Collimating lens casing. The laser beam is collimated by screwing this mount in and out of the laser head.
192
Figure C.20: Grating mount. Laser frequency tuning is achieved by adjusting the adjustment screw or PZT voltage.
193
Figure C.21: Mount for corner cube mirror.
Figure C.22: Mount for 1” diameter MOT telescope lenses. When mounted to the optics track, the tilt and lateral position of the lens are fixed, simplifying and stabilizing the
MOT laser beam alignment.
194
Figure C.23: Mount for 3” diameter MOT telescope lenses. When mounted to the optics track, the tilt and lateral position of the lens are fixed, simplifying and stabilizing the
MOT laser beam alignment.
195
Figure C.24: Spacer for MOT mirrors. This spacer lines up the axis of rotation of the
MOT mirror with the center of the precision optics track.
196
Figure C.25: Precision optics track for MOT optics.
197
Figure C.26: Legs for precision optics tracks. Each track sits on two of these legs. The horizontal position of each track can be fine-adjusted by sliding these legs along rulers fixed to the optics table.
198
199
Figure C.27: Overview of the quadrupole magnet assembly.
200
Figure C.28: Plate for mounting the quadrupole magnet assembly to the servotrack forcer.
(a) 201
Continued
Figure C.29: Magnetic coil post. (a) Original design. (b) Modifications for magnetic coil position adjustment, copper tubing
feed-through and counterweight attachment.
Figure C.29 continued
(b) 202
(a) 203
Continued
Figure C.30: Coil holders. The plates for the two quadrupole plates are identical except for the orientation of the Nylon cap
screw clearance holes for attaching the support brackets. (a) For top coil. (b) For bottom coil.
Figure C.30 continued
(b) 204
205
Figure C.31: Brackets for magnetic coil support.
206
Figure C.32: Overview of magnetic coil.
207
Figure C.33: Teflon spool (magnetic coil center).
(a) 208
Continued
Figure C.34: Magnetic coil cover plates (to hold copper tubing in place). (a) “Outside” cover plate (with feed-through for copper
tubing). (b) “Inside” cover plate (groove cut out to increase space between coils near the trap center).
Figure C.34 continued
(b) 209
210
Figure C.35: Coil cover ring (to keep the tubing from springing outward). The ring is attached to the “inside” cover plate with
6-32 Nylon screws (not shown).
211
Figure C.36: Cooling water feed-through. This piece ensures that the weight of the long water tubes providing the cooling
water for the magnetic coils rests on the magnetic coil post rather than the 3/16” copper tubing.
212
Figure C.37: Electrical terminal block. This block ensures that the weight from the 2/0 cables supplying the current for the
magnetic coils rests on the magnetic coil post rather than the 3/16” copper tubing.
213
Figure C.38: Electrical terminals for connecting the 2/0 cables to the copper tubing of the magnetic coils.
214
Figure C.39: Overview of shutter assembly. A spring pulling up on the shutter arm, the BNC connector for the shutter signal and
a protective cover over the solenoid and high voltage electric connections are not shown here.
215
Figure C.40: Shutter base plate.
216
Figure C.41: Shutter arm (made from PCB).
APPENDIX D
LABVIEW CODES
217
(a) 218
Continued
Figure D.1: Bake-out control program. (a) Front panel. (b) Block diagram.
Figure D.1 continued
(b) 219
(a) 220
Continued
Figure D.2: Computer scan program used for tuning the lasers. (a) Front panel. (b) Block diagram (all cases are indicated; sub-
vi “1” is “Assemblegraphclusters.vi”).
Figure D.2 continued
(b) 221
Continued
Figure D.2 continued
222
Continued
Figure D.2 continued
223
(a) 224
Continued
Figure D.3: Assemblegraphclusters.vi, called by the computer scan program. (a) Front panel. (b) Block diagram.
Figure D.3 continued
(b) 225
(a) 226
Continued
Figure D.4: Trapping experiment control program. (a) Front panel. (b) Block diagram (all cases are indicated, the sub-vis
Sequencer.vi (“trap”) and AI_read.vi (“1”) are called).
Figure D.4 continued
(b) 227
Continued
Figure D.4 continued
228
Continued
Figure D.4 continued
229
Continued
Figure D.4 continued
230
Continued
Figure D.4 continued
231
Continued
Figure D.4 continued
232
Continued
Figure D.3 continued
233
Continued
Figure D.4 continued
234
Continued
Figure D.4 continued
235
Continued
Figure D.4 continued
236
Continued
Figure D.4 continued
237
Continued
Figure D.4 continued
238
Continued
Figure D.4 continued
239
(a) 240
Continued
Figure D.5: Sequencer.vi (sub-vi of trapping experiment control program. (a) Front panel. (b) Block diagram.
Figure D.5 continued
(b) 241
(a)
(b) 242
Figure D.6: AI_Read.vi (sub-vi of trapping experiment control program). (a) Front panel. (b) Block diagram.
(a)
243
Continued
Figure D.7: Frame grabber control program. (a) Front panel. (b) Block diagram (all cases are indicated, the sub-vis “Create
path.vi” and Digital output.vi (“D”) are called).
Figure D.7 continued.
(b) 244
245
Figure D.8: Sub-vis of frame grabber control program. (a) Create Path.vi, front panel (b) Create Path.vi, block diagram. (c)
Digital output.vi, front panel. (d) Digital output.vi, block diagram.
APPENDIX E
CIRCUIT DIAGRAMS
246
(a) 247
Continued
Figure E.1: Laser power supply. (a) Current control unit. (b) Temperature control unit.
Figure E.1 continued
(b) 248
Figure E.2: Shorting box. An inductor in series and a capacitor in parallel with the laser diode protect the diode from sudden voltage spikes. The current lock signal (BNC input) bypasses the protective circuit.
249
Figure E.3: Photodetector circuit (taken from manual). For atom number measurements we used RF=10 kΩ.
250
251
Continued
Figure E.4: Laser locking circuit. This circuit provides both the current and grating feedback, allows ramping of the laser frequency manually or by computer, and allows setting the error signal offset manually or by computer.
Figure E.4 continued
252
Continued
Figure E.4 continued
253
Figure E.5: Shutter supply circuit. The circuit can supply up to four shutters simultaneously.
254
Figure E.6: Magnetic coil circuit. The computer controls the current through the magnetic coils by supplying a voltage to the gate of the IGBT Module (FZ600R12KE3).
255
BIBLIOGRAPHY
1. Feynman, R.P., Simulating physics with computers. International Journal of Theoretical Physics, 1982. 21: p. 467.
2. Shor, P., Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. quant-ph/9508027, 1996.
3. Quantum Computing Roadmap. Quantum Information Science and Technology Roadmapping Project, available at http://qist.lanl.gov/qcomp_map.shtml, 2004.
4. Cory, D.G., A.F. Fahmy, and T.F. Havel, Ensemble quantum computing by NMR spectroscopy. Proc. Natl. Acad. Sci. USA - Computer Sciences, 1997. 94: p. 1634-1639.
5. Abe, E., K.M. Itoh, T.D. Ladd, J.R. Goldman, F. Yamaguchi, and Y. Yamamoto, Solid-state silicon NMR quantum computer. Journal of Superconductivity: Incorporating Novel Magnetism, 2003. 16: p. 175-178.
6. Cirac, J.I. and P. Zoller, A scalable quantum computer with ions in an array of microtraps. Nature, 2000. 404: p. 579-581.
7. Cirac, J.I. and P. Zoller, Quantum Computations with Cold Trapped Ions. Physical Review Letters, 1995. 74(20): p. 4091-4094.
8. Deutsch, I.H., G.K. Brennen, and P.S. Jessen, Quantum Computing with Neutral Atoms in an Optical Lattice. Fortschritte der Physik, 2000. 48(9-11): p. 925-943.
9. Briegel, H.-J., T. Calarco, D. Jaksch, J.I. Cirac, and P. Zoller, Quantum computing with neutral atoms. Journal of Modern Optics, 2000. 47(2/3): p. 415- 451.
10. Pellizzari, T., S.A. Gardiner, J.I. Cirac, and P. Zoller, Decoherence, Continuous Observation, and Quantum Computing: A Cavity QED Model. Physical Review Letters, 1995. 75(21): p. 3788-3791.
11. Knill, E., R. Laflamme, and G.J. Milburn, A scheme for efficient quantum computation with linear optics. Nature, 2001. 409: p. 46-52.
256
12. Loss, D. and D.P. DiVincenzo, Quantum computation with quantum dots. Physical Review A, 1998. 57(1): p. 120-126.
13. Kane, B.E., A silicon-based nuclear spin quantum computer. Nature, 1998. 393: p. 133-137.
14. Makhlin, Y., G. Schön, and A. Shnirman, Josephson-junction qubits with controlled couplings. Nature, 1999. 398: p. 305-307.
15. Mooij, J.E., T.P. Orlando, L. Levitov, L. Tian, C.H.v.d. Wal, and S. Lloyd, Josephson Persistent Current Qubit. Science, 1999. 285: p. 1036-1039.
16. Nielsen, M.A. and I.L. Chuang, Quantum computation and quantum information. 2000, New York: Cambridge University Press.
17. Preskill, J., Lecture notes, available at http://www.theory.caltech.edu/people/preskill/ph229/#lecture. 1997.
18. Grover, L.K., Quantum Mechanics Helps in Searching for a Needle in a Haystack. Physical Review Letters, 1997. 79(2): p. 325-328.
19. Deutsch, D., Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. R. Soc. Lond. A, 1985. 400: p. 97.
20. Deutsch, D. and R. Josza, Rapid solution of problems by quantum computation. Proc. R. Soc. Lond. A, 1992. 439: p. 553.
21. DiVincenzo, D.P., Two-bit gates are universal for quantum computation. Physical Review A, 1995. 51(2): p. 1015-1022.
22. Lloyd, S., Almost Any Quantum Logic Gate is Universal. Physical Review Letters, 1995. 75(2): p. 346-349.
23. DiVincenzo, D.P., The Physical Implementation of Quantum Computation. Fortschritte der Physik, 2000. 48(9-11): p. 771-783.
24. Dalibard, J. and C. Cohen-Tannoudji, Dressed-atom approach to atomic motion in laser light: the dipole force revisited. Journal of the Optical Society of America B, 1985. 2(11): p. 1707-1720.
25. Grimm, R., M. Weidemüller, and Y.B. Ovchinnikov, Optical dipole traps for neutral atoms. Advances in Atomic, Molecular and Optical Physics, 2000. 42: p. 95-170.
26. Jessen, P.S. and I.H. Deutsch, Optical Lattices. Advances in Atomic, Molecular and Optical Physics, 1996. 37: p. 95-138.
257
27. Hamann, S.E., D.L. Haycock, G. Klose, P.H. Pax, I.H. Deutsch, and P.S. Jessen, Resolved-Sideband Raman Cooling to the Ground State of an Optical Lattice. Physical Review Letters, 1998. 80(19): p. 4149-4152.
28. Greiner, M., O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature, 2002. 415: p. 39-44.
29. Metcalf, H.J. and P. van der Straten, Laser Cooling and Trapping. 1999, New York: Springer-Verlag. 323.
30. Friebel, S., C. D'Andrea, J. Walz, M. Weitz, and T.W. Haensch, CO2-laser optical lattice with cold rubidium atoms. Physical Review A, 1998. 57(1): p. R20-R23.
31. Brennen, G.K., C.M. Caves, P.S. Jessen, and I.H. Deutsch, Quantum Logic Gates in Optical Lattices. Physical Review Letters, 1999. 82(5): p. 1060-1063.
32. Jaksch, D., J.I. Cirac, P. Zoller, S.L. Rolston, R. Cote, and M.D. Lukin, Fast quantum gates for neutral atoms. Physical Review Letters, 2000. 85(10): p. 2208- 2211.
33. Derevianko, A. and C.C. Cannon, Quantum computing with magnetically interacting atoms. Physical Review A, 2004. 70: p. 062319.
34. Jaksch, D., H.-J. Briegel, J.I. Cirac, C.W. Gardiner, and P. Zoller, Entanglement of Atoms via Cold Controlled Collisions. Physical Review Letters, 1999. 82(9): p. 1975-1978.
35. Leung, D.W., I.L. Chuang, F. Yamaguchi, and Y. Yamamoto, Efficient implementation of coupled logic gates for quantum computation. Physical Review A, 2000. 61: p. 042310.
36. Petsas, K.I., A.B. Coates, and G. Grynberg, Crystallography of optical lattices. Physical Review A, 1994. 50(6): p. 5173-5189.
37. Hammes, M., D. Rychtarik, B. Engeser, H.C. Nägerl, and R. Grimm, Evanescent- Wave Trapping and Evaporative Cooling of an Atomic Gas at the Crossover to Two Dimensions. Physical Review Letters, 2003. 90(17): p. 173001.
38. Barnett, A.H., S.P. Smith, M. Olshanii, K.S. Johnson, A.W. Adams, and M. Prentiss, Substrate-based atom waveguide using guided two-color evanescent light fields. Physical Review A, 2000. 61: p. 023608.
39. Burke, J.P.J., S.-T. Chu, G.W. Bryant, C.J. Williams, and P.S. Julienne, Designing neutral-atom nanotraps with integrated optical waveguides. Physical Review A, 2002. 65: p. 043411.
258
40. Dumke, R., M. Volk, T. Müther, F.B.J. Buchkremer, G. Birkl, and W. Ertmer, Micro-optical Realization of Arrays of Selectively Addressable Dipole Traps: A Scalable Configuration for Quantum Computation with Atomic Qubits. Physical Review Letters, 2002. 89(9): p. 097903.
41. Hunsperger, R.G., Integrated Optics: Theory and Technology. 2nd ed. 1984, New York: Springer-Verlag. 321.
42. Christandl, K., G.P. Lafyatis, S.-C. Lee, and J.-F. Lee, One- and two-dimensional optical lattices on a chip for quantum computing. Physical Review A, 2004. 70: p. 032302.
43. Hammes, M., D. Rychtarik, H.C. Nägerl, and R. Grimm, Cold-atom gas at very high densitites in an optical surface microtrap. Physical Review A, 2002. 66: p. 051401(R).
44. Lewandowski, H.J., D.M. Harber, D.L. Whitaker, and E.A. Cornell, Simplified System for Creating a Bose-Einstein Condensate. Journal of Low Temperature Physics, 2003. 132(5-6): p. 309-367.
45. Future computer: Atoms packed in an "egg carton" of light? available at http://researchnews.osu.edu/archive/eggcarton.htm, 2004.
46. Christandl, K., A compact, grating-stabilized diode laser for atomic spectroscopy, in Physics. 2000, The Ohio State University: Columbus, Ohio. p. 86.
47. Duarte, F.J., Tunable Lasers Handbook. 1995.
48. Ricci, L., M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, and T.W. Hänsch, A compact grating-stabilized diode laser system for atomic physics. Optics Communications, 1995. 117: p. 541-549.
49. Gensemer, S.D., Characterization and Control of Ultracold Collisions, in Physics. 2000, University of Connecticut. p. 150.
50. Corwin, K.L., Z.-T. Lu, C.F. Hand, R.J. Epstein, and C.E. Wieman, Frequency- stabilized diode laser with the Zeeman shift in atomic vapor. Applied Optics, 1998. 37(15): p. 3295-3298.
51. Clifford, M.A., G.P.T. Lancaster, R.H. Mitchell, F. Akerboom, and K. Dholakia, Realization of a mirror magneto-optical trap. Journal of Modern Optics, 2001. 48(6): p. 1123-1128.
52. Lee, K.I., J.A. Kim, H.R. Noh, and W. Jhe, Single-beam atom trap in a pyramidal and conical hollow mirror. Optics Letters, 1996. 21(15): p. 1177-1179.
259
53. Christandl, K., G.P. Lafyatis, A. Modoran, and T.-H. Shih, Two-dimensional imaging of neutral alkali atom samples using surface ionization. Review of Scientific Instruments, 2002. 73(12): p. 4201-4205.
54. Morinaga, M., M. Yasuda, T. Kishimoto, F. Shimizu, J.-I. Fujita, and S. Matsui, Holographic Manipulation of a Cold Atomic Beam. Physical Review Letters, 1996. 77: p. 802.
55. Shimizu, F., Specular Reflection of Very Slow Metastable Neon Atoms from a Solid Surface. Physical Review Letters, 2001. 86: p. 987.
56. Delhuille, R., Optimization of a Langmuir-Taylor detector for lithium. Review of Scientific Instruments, 2002. 73: p. 2249.
57. Taylor, J.B., The reflection of beams of the alkali metals from crystals. Physical Review, 1930. 35: p. 375.
58. Langmuir, I. and K.H. Kingdon, Thermionic Effects Caused by Vapours of Alkali Metals. Proc. R. Soc. Lond. A, 1925. 107: p. 61.
59. Sheehy, B., S.-Q. Shang, R. Watts, S. Hatamian, and H. Metcalf, Diode-laser deceleration and collimation of a rubidium beam. Journal of the Optical Society of America B, 1989. 6: p. 2165.
60. Frazer, J.W., R.P. Burns, and G.W. Barton, Low-Background Tungsten Filaments for Surface Ionization Mass Spectroscopy. Review of Scientific Instruments, 1959. 30: p. 370.
61. Ramsey, N.F., Molecular Beams. 1985, New York: Oxford University.
62. Straub, H.C., M.A. Mangan, B.G. Lindsay, K.A. Smith, and R.F. Stebbings, Absolute detection efficiency of a microchannel plate detector for kilo-electron volt energy ions. Review of Scientific Instruments, 1999. 70: p. 4238.
63. Barrett, T.E., S.W. Dapore-Schwartz, M.D. Ray, and G.P. Lafyatis, Slowing Atoms with σ- Polarized Light. Physical Review Letters, 1991. 67: p. 3483.
64. Sesko, D.W., T.G. Walker, and C.E. Wieman, Behavior of neutral atoms in a spontaneous force trap. Journal of the Optical Society of America B, 1991. 8: p. 946.
65. Lett, P.D., W.D. Phillips, S.L. Rolston, C.E. Tanner, R.N. Watts, and C.I. Westbrook, Optical molasses. Journal of the Optical Society of America B, 1989. 6: p. 2084.
66. Townsend, C.G., N.H. Edwards, C.J. Cooper, K.P. Zetie, and C.J. Foot, Phase- space density in the magneto-optical trap. Physical Review A, 1995. 52: p. 1423. 260
67. Hughes, F.L. and H. Levinstein, Mean Adsorption Lifetime of Rb on Etched Tungsten Single Crystals: Ions. Physical Review, 1959. 113: p. 1029.
68. Deconihout, B., F. Vurpillot, M. Bouet, and L. Renaud, Improved ion detection efficiency of microchannel plate detectors. Review of Scientific Instruments, 2002. 73: p. 1734.
261