This document discusses modeling magnetic fields of microtrap arrays for trapping ultracold atoms. It describes how double-loop microtraps can be used to create arrays of traps and transfer atoms between adjacent traps by varying currents. Adding an additional Ioffe coil can generate a trap with a nonzero minimum field to prevent atom loss due to spin flips. The microtraps use much smaller currents than macroscopic traps and allow controlling trap position and depth through bias fields, which could enable applications in precision measurements, quantum information processing, and more.

1. Modelling Magnetic Fields of MicroTrap
Arrays for Trapping Ultracold Atoms
A. Mouraviev, & W. A. van Wijngaarden
Physics Dept., York University
www.wvanwijngaarden.info.yorku.ca
CUPC 2015

2. Ultracold Atoms
W.A. van Wijngaarden & B. Lu. Phys. in Can. 60, No. 5 (2004)
Why Study Ultracold Atoms?
- Near absolute zero, weird effects such as superfluidity &
superconductivity occur. (D & J Tilley, Superfluidity & Superconductivity,
U. Sussex Press, 1986)
How Do You Get Bose Einstein Condensation (BEC)?
- Bosons condense into lowest state at ultralow temperatures (K.
Stowe, Intro. Stat. Mech. & Thermo., J. Wiley, Toronto, 1984)
- Macroscopic effects of quantum mechanics evident when de
Broglie wavelength λB ~ distance between atoms
h = Planck’s constant
M = atom’s mass
kB = Boltzmann’s constant

3. t = 6 ms t = 12t = 10
t = 14 t = 20t = 18t = 16
t = 8
Measurement of Ultracold Temperature
Observe expansion of atom cloud after trap turned off.

4. How does an Atom Trap Work?
Zeeman Hamiltonian
Zeeman Shift of 87Rb F=2 Hyperfine level
µ = atom’s magnetic moment
B = magnetic field
1
-2
0
mF = 2
-1
B5S1/2 F=2
Atoms in mF = 1, 2 hyperfine levels trapped at minimum magnetic field.

5. Double Loop Microtrap
B. Jian & WvW, JOSA B 30, No. 2, 238 (2013)
Current I2 = -1.23 I1
Radius R2 = 1.4 R1
B0 = I1/R1x
z
-y
R1
R2
I2
zm

6. Microtrap Array
B. Jian & W. A. van Wijngaarden, J. Phys. B 47, 215301 (2014)
Cu Block
Heatsink
2 cm
3 mm

7. Magnetic Field Calculation using Mathematica
-y
S
x
z
I
H
Consider loop in yz plane having radius R & current I. I>0 generates
field in + x direction.

8. i. Atom Transfer between Microtraps
• Atom transferred from double-
loop microtrap A centered at x =
0 to double-loop microtrap B
centered at x = R1/2.
• Current IA (IB) linearly
decreased (increased) from t = 0
to t = 1.
• Trap profile remains virtually
constant throughout transfer.
-3 -2 -1 0 1 2 3
x / R1
2
1.5
1
0.5
0
|B|/B0
t = 0.75
t = 0
t = 1
t = 0.5
t = 0.25
x
z
-y
R1
R2
IA
z

9. ii. Addition of Ioffe Coil
2
1.5
1
0.5
0.0
|B| / B0
IIC
x
z
-y
R1
R2 IIC = 9 I1
-1 -0.5 0 0.5 1
x / R1
z/R1
IIC = 0
z/R1
1.0
0.8
0.6
0.4
0.2
0.0
Generate trap having nonzero minimum field to prevent spin flips using Ioffe Coil
having radius RIC = R1/8 centered at (1.4, 0, 0.15) R1.
-1 -0.5 0 0.5 1
x / R1
1.0
0.8
0.6
0.4
0.2
0.0

10. 1.5
1
0.5
0.0 0 0.5 1 1.5
z / R1
-1 -0.5 0 0.5 1
x / R1
3
2
1
0
Bmin = 0.104 Bo
at (0.48, 0, 0.47) R1
-1 -0.5 0 0.5 1
x / R1
z/R1
1.0
0.8
0.6
0.4
0.2
0.0
2
1.5
1
0.5
0.0
|B|/B0
Trap Potential due to Ioffe Coil
Trap depth = 0.48 B0

11. Bias Field Effect on Trap Position & Depth
x
z
-y
R1
R2
xC
Bzbias
0
0.1
0.2
0.3
0.4
0.5
0
0.2
0.4
0.6
0.8
1
-0.75 -0.5 -0.25 0 0.25 0.5 0.75
TrapDepth/Bo
MicrotrapzPositon/R1
Bzbias / Bo

12. Conclusions
• Microtraps use much smaller currents than macroscopic traps
• Double-loop microtraps useful to create one or two dimensional
arrays of ultracold atoms. Current can be adjusted or bias field
varied to modify trap position & depth – useful for surface studies.
• Modelled how to transfer atoms between two adjacent double loop
microtraps. Trap profile remains constant during transfer
minimizing atom loss.
• Trap having nonzero magnetic field minimum generated by adding
small Ioffe coil, partially embedded in atom chip, useful to prevent
atom loss due to spin flips.
Applications: Precision Measurements, Frequency Standards, Surface
Sensing, Atom Interferometry, Quantum Information
Processing etc.
Additional Information: www.wvanwijngaarden.info.yorku.ca

13. Laser Cooling
H. Metcalf & P. v. d. Straten, Laser Cooling & Trapping (Springer,1999)
Analogous to stopping transport truck on highway (atom) by
bouncing beam of ping pong balls (photons) off it.
10-2
10-6
10-4
100
102
104
Kelvins
Mass M
Velocity v h / λ
Photon Momentum
h = Planck’s constant
# photons to stop thermal 87Rb atom = M v / (h / λ) = 50,000 photons
Stopping Time T = 50,000 x τ excited state lifetime = 1.4 msec
Stopping Distance = v τ / 2 = 20 cm
Laser Power to stop 109 atoms/sec = 109 x 50,000 h ν / T ≈ 10 mW
Doppler cooling limit = h Γ transition linewidth /2 k ≈ 100 µK
Analogous to stopping transport truck on highway (atom) by
bouncing beam of ping pong balls (photons) off it.
10-2
10-6
10-4
100
102
104
Kelvins
10-2
10-6
10-4
100
102
104
Kelvins
Mass M
Velocity v h / λ
Photon Momentum
h = Planck’s constant
# photons to stop thermal 87Rb atom = M v / (h / λ) = 50,000 photons
Stopping Time T = 50,000 x τ excited state lifetime = 1.4 msec
Stopping Distance = v τ / 2 = 20 cm
Laser Power to stop 109 atoms/sec = 109 x 50,000 h ν / T ≈ 10 mW
Doppler cooling limit = h Γ transition linewidth /2 k ≈ 100 µK