BEC in Optical Dipole Trap & Artificial Gauge Potential

Shuai Chen Department of Modern Physics,

University of Science and Technology of China

Lanzhou, August 1st 2011

Outline

• Motivation: Quantum simulation with ultracold atoms

• BEC in Optical dipole trap

– Optical dipole trap

– Experiment process to produce BEC in optical dipole trap

• Artificial gauge potential by Raman coupling

– How to generate gauge field with Raman coupling

– Experiment generation of gauge potential

– Spin Orbit coupling

– Quantum tunneling in Spin-Orbit coupled BEC

• Conclusion

Quantum Simulation

…nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy. - Richard P. Feynman, May 1981

published: Int. J. Theo. Phys. (1982)

Quantum Simulation

• Understand and Design Quantum Materials – One of the biggest challenges of Quantum Physics in the 21st Century

• Technological Relevance – High temperature superconductivity (Power Delivery)

– Magnetism (Storage, Spintronics…)

– Quantum Hall effect (transportation…)

– Quantum Computing

WHAT’S THE PROBLEM? THE CHALLENGE OF QUANTUM MATERIALS

A controllable quantum material is required!

Ultracold Quantum Gas

Open the new era of quantum simulation with ultracold Bose and Fermi Gas!

BEC in 1995 Ultracold Fermi Gas

Superfluid to Mott insulator transition

Greiner et al., Nature 415, 39 (2002)

Mott insulator of Fermions

A. Kastberg et al. PRL 74, 1542 (1995) M. Greiner et al. PRL 87, 160405 (2001)

Brillouin Zones in 2D and the momentum distribution of cold atoms in lattices

Joerdens et al., Nature 455, 204 (2008) U. Schneider et.al., Science 322, 1520 (2008)

Optical lattices

Feshbach resonance

Feshbach resonance of 6Li

BEC-BCS crossover

JILA, MIT, Innsbruck…

K.-K. Ni et.al., Science 322, 231 (2008) K.-K. Ni et.al., Nature 464, 1324 (2010) J. G. Danzl, et.al., Nature Physics 6, 265 (2010)

Formation of ultracold molecule

W. S. Bakr et.al., Nature 462, 75 (2009) J. F. Sherson et.al., Nature 467,69 (2010) Ch. Weitenberg et.al., Nature 471,319 (2011)

Single site resolution and single site addressing

Exchange interaction of spin in super lattice Anderlini et al., Nature 448, 452 (2007) Trotzky et al., Science 319, 295 (2008)

Anderson localization of matter wave Billy et al., Nature 453, 891 (2008) Roati et al., Nature 453, 895 (2008)

Development of quantum simulation

Could ultracold atoms emulate charged particle?

• To simulate Lorenz Force: 𝐹 = 𝑞𝑣 × 𝐵

• To understanding Quantum Hall Effect?

• To form the topological insulator

• Large scale Quantum Computing

Nobel Prize 1985 Quantum Hall Effect

Nobel Prize 1998 Fractional Quantum Hall Effect

What is Bose-Einstein Condensate?

de Broglie wavelength

Phase space density

𝜌𝑝𝑠 > 2.612

In 3D free space

Typical road to BEC

Optical dipole trap

Evaporative cooling

Proper for magnetic trap

Good for optical trap

Rb-87 atom

“Rb-87 D line data”, D. Steck, http://steck.us/alkalidata/

5𝑆1/2

𝐹 = 1

𝐹 = 2

3

2

1 0

2

1

5𝑆1/2

5𝑃3/2

coo

ling

rep

um

pin

g

imag

ing

For ground states:

For laser cooling:

+1 0

+1

−1

+2 −1 −2

0 𝐹 = 2

𝐹 = 1

Optical Dipole Trap

Red: Δ < 0, atoms get attracted to intensity maximum Blue: Δ > 0, atoms get repelled from intensity maximum

𝑈dip ∝ 𝐼/Δ

Γsc ∝ 𝐼/Δ2 Go for large detuning and intensity!

Optical dipole potential:

Photon scattering Rate: ℏ𝜔

Δ

Δ = ℏ𝜔 − ℏ𝜔0

|𝑒

|𝑔

ℏ𝜔0

0

𝐸

Γ

Optical Dipole Trap

Experimental setup

Setup for BEC in Optical trap

Setups for Lasers (cooling, repumping & probe )

Vacuum system

2D MOT chamber

Science Chamber: 3D MOT & Dipole trap

Differential pump stage

Ion pump

Ti: sublimation pump

Ion pump

Science Chamber: 10−11 mbar

2D MOT chamber: 10−9 mbar

MOT loading to Dark Molasses

Dark MOT Atom number: ~ 3 × 109 Density: 𝑛~1 × 1012/cm3 Temperature: 𝑇~200μK

Dark molasses Atom number: ~ 2 × 109 Density: 𝑛~5 × 1011/cm3 Temperature: 𝑇~40μK

Optical Dipole Trap loading

Beam waist: ~80μm Crossing angle: 75° Initial trap depth: ~500μK

Atom number: ~ 1.4 × 107 density: ~1 × 1012/cm3 Temperature: ~100μK

Dipole trap laser: Yb doped fiber laser, 1070nm, 50W

Time of flight: 2ms

Evaporative cooling and Imaging of BEC

Time of Flight Image: CCD pixel cize: 16μm Magnification: 1: 1 N/A: 0.18 Resolution: 16μm

10 ms Time of flight image

~100μK 1.4 × 107 atoms

~20μK 6.0 × 106 atoms

~5μK 2.5 × 106 atoms

Formation of BEC

𝑇 > 𝑇𝑐 Thermal atoms

𝑇 = 𝑇𝑐 BEC appears

𝑇 < 𝑇𝑐 Bi-mode distribution

𝑇 ≪ 𝑇𝑐 Pure BEC

Critical temperature： 𝑇𝑐~100nK Atom number: 𝑁 = 2.5 × 105 Density of atoms: 𝑛 > 1013/cm3 Trapping frequency: *50, 50, 80+Hz Effective temperature: 10nK

Image for BEC: CCD pixel cize: 16μm Magnification: 4: 1 N/A: 0.18 Resolution: ~4.0μm

Could ultracold atoms emulate charged particle?

• To simulate Lorenz Force: 𝐹 = 𝑞𝑣 × 𝐵

• To understanding Quantum Hall Effect?

• To form the topological insulator

• Large scale Quantum Computing

Nobel Prize 1985 Quantum Hall Effect

Nobel Prize 1998 Fractional Quantum Hall Effect

For a particle moving in the potential 𝑉(𝑟), the Hamitonian:

Magnetic field: 𝐵 = 𝛻 × 𝐴

Electric field: 𝐸 = −𝜕𝐴

𝜕𝑡− 𝛻𝜑

Once we could construct such a Hamitonian for the neutron atoms, we can simulate the charged particle with neutron atoms!!

Yes! If we can construct the gauge potential

𝐻 𝑝, 𝑟 =𝑝2

2𝑚+ 𝑉(𝑟)

Vector potential: 𝐴 Scalar potential: 𝜑(𝑟)

For a particle with charge 𝑞, moving in a electromagnetic field,

the form or Hamitonian can be expressed as:

𝐻′(𝑝, 𝑟) =𝑝 − 𝑞𝐴 2

2𝑚+ 𝑉 𝑟 + 𝜑(𝑟)

Possible schemes

• BEC in a rotating trap

– Coriolis force-> Lorentz force

– Only modest effective fields

– limited to rotational symmetric setups and does not allow to study transport phenomena

• Light-induced gauge field (geometric phase)

– Need a spatially varying basis of internal states

BvFLorentz vFcoriolis

· Dark state and Bright state · Spin Hall Effect

G. Juzeliunas, et.al., PRA 73, 025602 (2006) K. J. Guenter, et.al., PRA 79, 011604 (2009) I. B. Spielman, PRA 79, 063613 (2009)

S.-L. Zhu et.al., PRL 97, 240401 (2006)

Laser coupling with spatial gradients

Add an effective electric field

Gauge field generation • based on EIT configuration • retained in the dark state

• Easy to adjust parameters in experiment • Verify the effective gauge field by measuring the momentum distribution

I. B. Spielman, PRA 79, 063613 (2009)

Hamitonian:

0~

min k

Atom detuning with spatial gradients

Some Experiment Progress

Y.-J. Lin, et.al., PRL 102, 130401 (2009)

Y.-J. Lin, et.al., Nature 462, 628 (2009)

Synthetic magnetic fields for BEC: 𝐵 = 𝛻 × 𝐴

Vortices are formed in condensates

the Hamitonian of the dressed state:

Y.-J. Lin, et.al., arXiv: 1008.4864 (2010)

Electric field generation: 𝐸 = −𝜕𝐴/𝜕𝑡

Vector Potential generation:

Spin-Orbit coupling

Y. -J. Lin, et.al., Nature 417, 83 (2011)

Phase transition due to the interaction of atoms for different “spin” in BEC

Proved the SO coupling in 1D

Still many open questions!

How to do it? – Dispersion relations

(𝑝 − 𝑞𝐴)2

2𝑚

𝑝2

2𝑚

𝐸

momentum p

Where 𝐴 is the vector potential In general, “𝐴” could be a number (abel gauge potential) “𝐴” could also be a matrix (non-abelian gauge potential)

𝑞𝐴

Raman coupling: Basic concepts

Resonant Raman Rabi Frequency:

Ω =Ω1Ω2

4Δ Ω1 Ω2

Δ

𝛿

|𝑒

|𝑔1

|𝑔2

ℏ𝜔1 ℏ𝜔2

Δ : Single photon detuning 𝛿 : Raman detuning Ω1, Ω2: Rabi frequency

𝑘2 𝑘1

atoms

𝜃

For 𝜔1 ≈ 𝜔2, 𝑘1 = 𝑘2 = 𝑘

Recoil momentum: ℏ𝑘𝑟 = ℏ𝑘 sin𝜃

2

Recoil Energy: 𝐸𝑟 =ℏ2𝑘𝑟

2

2𝑚

Γsc ∝ Ω/Δ2

Take care: Spontaneous photon scattering

𝑘 𝑥

𝐸

𝛿 = 0 𝛿 ≠ 0

How to realize such a vector potential? Two level atoms with counter propagate Raman coupling

𝐻 = ℏ

ℏ

2𝑚(𝑘 𝑥 + 𝑘𝑟)

2 − 𝛿/2 /2

/2ℏ

2𝑚(𝑘 𝑥 − 𝑘𝑟)

2 + 𝛿/2

ℏ𝑘 𝑥: quasi-momentum in 𝑥 direction ℏ𝑘𝑟: recoil momentum of single laser

|−1 : 𝑘𝑥 = 𝑘 𝑥 + 𝑘𝑟

|0 : 𝑘𝑥 = 𝑘 𝑥 − 𝑘𝑟

for real momentum during probe:

𝑞𝐴

𝑣𝑔 =𝜕𝐸

𝜕𝑘 𝑥= 0

𝛿/2 |−1

𝛿/2

: Raman Rabi frequency 𝛿: Raman detuning

|0

Three level case For Rb87 𝐹 = 1 hyperfine state:

𝐻 =

(𝑘 𝑥 + 2𝑘𝑟)2−𝛿′ Ω/2 0

Ω/2 𝑘 𝑥2− 𝜖 Ω/2

0 Ω/2 (𝑘 𝑥 − 2𝑘𝑟)2+𝛿′

real momentum during TOF detection:

𝛿′

𝛿′

𝜖

𝜖: quadratic Zeeman shift

𝛿′: Raman detuning

|−1

|0

|+1

|−1 : 𝑘𝑥 = 𝑘 𝑥 + 2𝑘𝑟

|0 : 𝑘𝑥 = 𝑘 𝑥

|+1 : 𝑘𝑥 = 𝑘 𝑥 − 2𝑘𝑟

𝑘 𝑥

𝐸

𝑞𝐴

Spin-Orbit Coupling Two level atoms with counter propagate Raman coupling 𝐻 = ℏ

ℏ

2𝑚(𝑘 𝑥 + 𝑘𝑟)

2 − 𝛿/2 /2

/2ℏ

2𝑚(𝑘 𝑥 − 𝑘𝑟)

2 + 𝛿/2

|↑ ′ |↓ ′

𝑘 𝑥

If Ω ≪ 4𝐸𝑟

|↑ ′ = |−1 + ε|0 , |↓ ′ = |0 + ε|−1

𝐻 =ℏ2𝑘2

2𝑚I +

2𝜎𝑥 −

δ

2𝜎𝑧 + 2α𝑘 𝑥𝜎𝑧

Spin-Orbit coupling!

We can mark spin |↑ = |−1 , |↓ = |0

𝐻 =𝑝 − 𝑞𝐴 2

2𝑚

“𝑞𝐴” is not a number, but a matrix! Non-abelian gauge potential!

𝛿/2 |−1

𝛿/2

: Raman Rabi frequency

|0

2-level Raman coupling

3-level Raman coupling

State selection for atoms

With relative large Bias field (large quadratic Zeeman shift) quasi-2-level Raman coupling

+1 0

+1

−1

+2

−1 −2

0 𝐹 = 2

𝐹 = 1

Rb-87 Ground states

First experiment: produce gauge potential

Counter propagate Raman lasers

Raman lasers: Wavelength: 790.07nm Beam waist: 180μm Recoil energy: 𝐸𝑟 = 3.68kHz

Calibration of Raman coupling strength

Raman Rabi Oscillation

Counter propagation

Resonant Raman Rabi Frequency:

Ω =Ω1Ω2

4Δ

0 /2 3/2 2

|1, −1

|2,0

Bias magnetic field: 𝐵 = 3.2 Gauss Quadratic Zeeman shift: 𝜖 = 0.4𝐸𝑟 Coupling Strength: Ω = 4.25𝐸𝑟

𝛿 = −1.5𝐸𝑟 𝛿 = +1.5𝐸𝑟 𝛿 = 0

𝑞𝐴 = 0 𝑞𝐴 > 0 𝑞𝐴 < 0

Formation of the uniform Gauge potential

Spin-Orbit coupling

Stage 1 Stage 2 Stage 3

Increase the strength of counter propagate Raman coupling

Time of Flight Image after Stage 2

|0

kx

|0 |0 |0 |−1 |−1 |−1 |−1

|↑ ′ |↓ ′

|↑ ′ = |−1 + ε|0

|↓ ′ = |0 + ε|−1

Raman laser configuration changed

Raman laser: Wavelength: 803.3nm Beam waist: 180μm Cross angle: 105 Recoil energy: 2.24kHz

𝑘𝜋 𝑘𝜎

2𝑘𝑟

|−1

|0

𝑡 = 0 240ms 60ms 120ms 180ms

Tunneling in momentum space Raman coupling: Ω = 2.2𝐸𝑟 Detuning: 𝛿 = −0.22𝐸𝑟 Bias field: 𝐵 = 7.2 Gauss Quadratic Zeeman shift: 𝜖 = 3.328𝐸𝑟

𝑘 𝑥

|↓ = |0 + ϵ|−1

tunneling

|↑ = |−1 + ϵ|0

|−1 : 𝑘𝑥 = 𝑘 𝑥 + 𝑘𝑟

|0 : 𝑘𝑥 = 𝑘 𝑥 − 𝑘𝑟 Real momentum of atom state during probe:

Josephson effect theory

Make 𝑠 stands for the difference between the population of the two parts, and the 𝜃 stands for the phase difference between the coefficients:

The total wave funtion:

𝑠 = |𝑏|2 − |𝑎|2 𝜃 = 𝜃𝑏 − 𝜃𝑎

𝜓 = 𝑎 𝜓𝑎 + 𝑏|𝜓𝑏

In the Coordinate Space

𝐻 𝑥 = −ℏ2

2𝑚

𝜕2

𝜕𝑥2+ 𝑉(𝑥)

The 1D Hamitonian:

Solve the Schrödinger Equation:

The Schrödinger Equation:

Lead to Josephson oscillation.

|𝜓𝑎 |𝜓𝑏

• In the Momentum Space The 1D Spin-Orbit coupling Hamitonian:

Josephson effect theory

The 𝜌 and the 𝜃 has the samilar definition as in the coordinate space

𝜌 = |𝜉2|2 − |𝜉1|

2 𝜑 = 𝜑2 − 𝜑1

the Schödinger equation:

the similar dynamic process as in the coordinate space

Josephson oscillation in momentum space!

The total wave funtion:

𝜓 = 𝜉1 𝜓1 + 𝜉2|𝜓2

𝜉1 = 𝜉1 𝑒𝑖𝜑1 , 𝜉2 = 𝜉2 𝑒𝑖𝜑2

In harmonic trap,

𝑉𝑒𝑥𝑡 =1

2𝑚𝜔2𝑥2 = −

𝑚ℏ2𝜔2

2

𝜕2

𝜕𝑝2

|𝜓1 |𝜓2

𝐻 = −𝑚ℏ2𝜔2

2

𝜕2

𝜕𝑝2+ 𝐸(𝑝)

Tunneling in momentum space

𝜹 = 0.15𝐸𝑟 𝜹 = 0.5𝐸𝑟 𝜹 = 0.05𝐸𝑟

Raman coupling: Ω = 2.2𝐸𝑟 Bias field: 𝐵 = 7.2 Gauss Quadratic Zeeman shift: 𝜖 = 3.328𝐸𝑟

Conclusion

• Rb-87 BEC in optical dipole trap is produced

• Artificial gauge potential is generated with Raman coupled BEC

• Generation of Spin-Orbit coupled BEC with Raman coupling

• Quantum tunneling and Josephson Effect in momentum space is observed

Future plan

利用BEC和空间多维Raman激光耦合产生非阿贝尔等效规范

场，观测中性超冷原子在非阿贝尔规范场中的行为， 模拟带电

粒子的自旋轨道耦合导致的相变问题。

Focus on Spin-Orbit coupled BEC

Study the phase transition of the Spin-Orbit coupled BEC

Stripe phase of Spin-Orbit coupled BEC

Chunji Wang et.al., PRL 105, 160403 (2010)

Boson pairing and fractional vortices phase of Spin-Orbit coupled BEC

C. -M. Jian & H. Zhai, arXiv: 1105.5700

Acknowledgement • Group Leader: Jian-Wei Pan

• Experiment

– Jinyi Zhang, Zhidong Du, Sicong Ji, Yingzhu and SC

• Theory

– Long Zhang, Ran Wei, YJD

• Cooperation

– Hui Zhai

• Former member

– Bo Yan, Mingfei Han

Supported by: