Laser Cooling and Trapping of Atom

Ying-Cheng Chen, 陳應誠

Institute of Atomic and Molecular Science, Academic Sinica,

中研院原分所

Outline

• Basic idea & concept
• Overview of laser cooling and cold atom study
• The light force
• Doppler cooling for a two-level atom
• Sub-Doppler Cooling
• Others cooling scheme
• Practical issues about a Magneto-Optical Trap (MOT)
• Atomic species
• Lasers
• Vacuum
• Magnetic field
• Imaging

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Temperature Landmark

To appreciate something is a good motivation to learn something!

Laser cooling and trapping of atom is a breakthrough to the exploration of the

ultracold world. A 12 orders of magnitude of exploration toward absolute zero temperature from room temperature !!!

What is special in the ultracold world?

• A bizarre zoo where Quantum Mechanics governs
• Wave nature of matter, interference, tunneling, resonance

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• Quantum statistics
• Uncertainty principle, zero-point energy
• System must be in an ordered state
• Quantum phase transition

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~1μm for Na @ 100nk

Cold Atom

Cold Molecule

Cold Plasma &

Rydberg Gas

Dipolar Gas

Many-body Physics

Quantum Computation

Atom Chips…

From Physics

to Chemistry

From ground to

highly-excited states

From isotropic to

anisotropic interaction

From fundamental

to application

From atomic to

condensed-matter

physics

Trends in Ultracold Research

Useful References

• Books,
• H. J. Metcalf & P. van der Straten, “Laser cooling and trapping”
• C. J. Pethick & H. Smith ,“Bose-Einstein condensation in dilute gases”
• P. Meystre, “Atom optics”
• C. Cohen-Tannoudji, J. Dupont-Roc & G. Grynberg “Atom-Photon interaction”
• Review articles
• V. I. Balykin, V. G. Minogin, and V. S. Letokhov, “Electromagnetic trapping of cold atoms” , Rep. Prog. Phys. 63 No 9 (September 2000) 1429-1510.
• V S Letokhov, M A Ol'shanii and Yu B Ovchinnikov�Quantum Semiclass. Opt. 7 No 1 (February 1995) 5-40 “Laser cooling of atoms: a review”
• Journal of Opt. Soc. Am. B, Issue 11,1989, special issue on laser cooling

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The Light Force: Concept

Photon posses energy

and momentum !

An exchange of momentum &

energy between photon and atom !

Force on atom

Net momentum exchange

from the photon to atom

absorption

emission

Energy and Momentum Exchange between Atom and Photon

• Atom absorbs a photon and re-emit another photon.

always positive, recoil heating

If the momentum decrease, and if

then ＜ΔK ＞_avg <0 or ＜ω_i＞ >＜ω_s＞ ,

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where avg stands for averaging over photon scattering events.

Criteria of laser cooling

A laser cooling scheme is thus an arrangement of an atom-photo interaction scheme in which atoms absorb lower energy photon and emit higher energy photon on average!

The Light force : quantum mechanical

• Ehrenfest theorem, the quantum-mechanical analogue of Newton’s second law,

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where V(r,t) is the interaction potential.

• Interaction potential: for an atom interacting with the laser field, , where d is atomic dipole moment operator.

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• Semi-classical treatment of atomic dynamics:
• Atomic motion is described by the averaged velocity
• EM field is treat as a classical field
• Atomic internal state can be described by a density matrix which is determined by the optical Bloch equation

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Discussion on semi-classical treatment

• Momentum width Δp is large compared with photon momentum k.

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• Considering slow atoms only simplify the formalism. (Internal variables are fast components and variation of atomic motion is slow components in density matrix of atom ρ(r,v,t))

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• Two conditions are compatible only if

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• If the above conditions is not fullified, full quantum-mechanical treatment is needed. e.g. Sr narrow-line cooling, Γ=2π×7.5kHz ~ ω_r=²k/2m=2π×4.7kHz

or

an lower bound on v

an upper bound on v

J. Dalibard & C. Cohen-Tannoudhi, J. Phys. B. 18,1661,1985

T.H. Loftus et.al. PRL 93, 073001,2004

Why Density Matrix Not Wavefunction?

• Pure versus Mixed ensemble.
• The system that we are studied are usually not in the same state (described by the same wavefunction) but in a statistical mixture, e.g.

atomic population follows Boltzman distribution both in internal states as well as in external states. Atomic system under preparation (like optical pumping) can be in the same internal state. Bose-Einstein condensate is a system in the same state both in internally and externally .

• When dealing with atom-photon interaction, we usually interest in partial system (e.g. atomic system). Spontaneous emission caused by the coupling of atom with infinite degree of freedom of radiation results in a transition from an initial to a final state and can convert a pure state to a statistical mixture since phase information are lost !
• Density matrix formalism establishes a more direct connection with observables!
• Density matrix is a more powerful method for doing calculations.

Density Matrix

• Probablity density to find particle in state |i> is

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• The complete basis for state vector

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• Diagonal elements are probabilities |c_m|² and off-diagonal terms are coherences c_mc_n* since they are depend on phase difference.

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• Expectation value of operator

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• Considering mixed ensemble instead of just pure ensemble, where P_m is classical statistical weight.

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• If we are only interested in part of the system, the density matrix has to be average over the other part of the system.

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The light force for a two-level atom

ρ_ij (or σ_ij)can be determined by the optical Bloch equation of atomic density matrix.

Where d₁₂=d₂₁ are assumed to be real and we have introduced the Bloch vectors u,v, and w.

Remark: dipole moment contain

in phase and in quadrature

components with incident field.

Note! A general form, can be plane wave,Gaussian beam…

Optical Bloch equation

Incoherent part due to spontaneous

emission or others relaxation processes.

The loss of quantum coherence is a big

Issue in quantum computation.

steady state solution

I_sat ~ 1-10 mW/cm² for alkali atom

Rabi frequency Ω characterize the

magnitude of atom-photon interaction.

Two types of forces

radiation pressure or

spontaneous emission force

a dissipative force

Related to v vector

dipole force or

gradient force

a reactive force

related to u vector

Without loss of generality, choose

At r =0,

Take average over one optical cycle

Origin of optical trapping

Origin of optical cooling

Light force for a Gaussian beam

z

k

F_rp

F_dip

F

Optical Tweezers and Dipole Trap

• Laser is far off-resonance, the dipole force dominates and trapping of small

particles occurs.

• For atom, it is called a optical dipole trap. Usually it has a trap depth around

1~1000 μK.

Spontaneous emission force

Decay rate,

,where R_sp is the flourescence rate.

Its maximum value is .

Max deceleration for Na D₂ line !

From

for steady-state

For a plane wave

Dipole Force in a standing wave

• A standing wave has an amplitude gradient, but not a phase gradient. So only the dipole force exists.

Where s₀ is the saturation parameter for each of the two beams that form the

standing wave.

For δ<0 (red detuning), the force attracts atom toward high intensity regions.

For δ>0 (blue detuning), the force repels atom away from high intensity regions.

Velocity dependent force

Atom with velocity v experiences a Doppler shift k∙v.

The velocity range of the force is significant for atoms with velocity such that their

Doppler detunings keeps them within one linewidth considering the power broadening

factor.

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Doppler Cooling

For δ<0, the force slows down the velocity.

[Γ/k]

Doppler Cooling, Energy Point of View

• Red-detune laser photons are absorbed by atoms, spontaneously emitted photons have average energy on the resonance frequency.
• On average, atoms absorb lower energy photons and emit higher energy photon.
• Photons from laser are coherent, photons spontaneously emitted are quite random. Entropy of atoms are carried away by spontaneously emitted photons.

Atom

Laser

Radiation

Reservior

V_AL,

excite the atom

V_AR, Radiation vacuum de-excite atom, Entropy flow

Finite degree of freedom

infinite degree of freedom

Coherent

photon

Incoherent

photon

Doppler Cooling limit

• Doppler cooling : cooling mechanism; Recoil heating : heating mechanism
• Temperature limit is determined by the relation that cooling rate is equal to heating rate.
• Recoil heating can be treat as a random walk with momentum step size k.

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For low intensity s₀<<1

Minimum temperature

T_D ~ 100-1000 μK for alkali atom

Magneto-optical trap (MOT)

• Cooling, velocity-dependent force: Doppler effect
• Trapping, position-dependent force: Zeeman effect

1-D case

3-D case

Position-dependent Force in a MOT

Considering v=0,

Sub-Doppler cooling

• Many laser cooling schemes allow one to cool atoms below the Doppler limit, or even down to the recoil limit.
• Polarization gradient cooling (Sisyphus cooling)
• Already exist in the MOT
• Raman sideband cooling
• Velocity-selective-coherent-population-trapping (VSCPT) cooling

…

Sisyphus Cooling

• Polarization gradient cause a periodic modulation with period of λ/2 for the ac Stark shift of the ground states.
• Atom climbs up the Stark potential and tends to be optically pumped to excited state and then spontaneously emit to the other ground state. It then repeat the same process
• On average, atoms absorb lower energy photons but emit higher energy photon.

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Polarization Gradient Cooling

• A new friction force mechanism for the low velocity atom (vτ_p~λ/4 where

τ_p is the optical pumping time ).

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• Equiliurium temperature

Cs

Optical Pumping

Angular Momentum of Photon

Raman Sideband Cooling

• Atoms are confined in a tight optical dipole trap and prepared in polarized states.
• Cooling cycle : |3,3;v> →Stimulated Raman transition → |3,1;v-1>

→optical pumping →|3,3;v=0> or |3,3;v>

• |3,3;v=0> is dark both to Stimulated Raman transition and to optical pumping light so population will accumulate here.
• Since atoms are tightly trapped, recoil heating is negligible.

PRL81,5768(1998)

π

σ+

VSCPT Cooling

• Atoms are in the CPT dark states when their velocities are almost zero.
• Atomic velocity distribution are non-thermal (Levy flight).
• Longer atom-photon interaction time cause narrower momentum width.

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PRL 61,826(1988)

Beyond Laser Cooling

• Evaporative cooling
• Sympathetic cooling
• Demagnetization cooling
• Stochastic cooling
• Feedback cooling
• ….???

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Part II: Practical Issues about a magneto-optical trap

Laser cooling : demonstrated species

Atomic species

• Different atomic species has its unique feature !

852.35nm

6 ²P_3/2

5.2MHz

6 ²S_1/2

F=5

4

3

2

4

3

cooling

repumping

¹³³Cs, alkali metal, I=7/2

(5s²)¹S₀

(5s5p)³P₁

4.7kHz

(5s5p)¹P₁

32MHz

460.73nm

Broad-line

cooling

689.26nm

Narrow-line

cooling

⁸⁸Sr, alkali earth, I=0

1 ⁰S₁

2 ³S₁

metastable

~20eV

by discharge

⁴He, nobel gas, I=0

2 ³P₂

1.6MHz

1083nm

Lasers

• Diode lasers are extensive use in laser cooling community due to inexpensive cost and frequency tunability.
• Diode lasers in external cavity configuration are used to reduce the laser linewidth.
• Master oscillator power amplifier (MOPA) configuration is used to increase the available laser power.

ECDL in Littrow configuration

ECDL in Littman-Metcalf configuration

master

Tampered

amplifiier

MOPA

Diode laser

Laser frequency stabilization

• Frequency-modulated saturation spectroscopy is the standard setup to generate the error signal for frequency stabilization.
• Feedback circuits are usually built to lock the laser frequency.

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Background subtracted saturation spectrometer

laser

spectrometer

Error signal

Feedback

circuit

Frequency Modulation Spectroscopy

• Frequency modulation and lock-in detection obtain dispersive error signal for frequency stabilization.

Vacuum

• Two different kinds of vacuum setup are mainly used, one is glass vapor cell, the other is stainless chamber.
• Ion pump and titanium sublimation pump are standard setup to achieve ultrahigh vacuum.

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Vapor-cell MOT

Chamber MOT

Magnetic field

• Anti-Helmholtz coils for the MOT
• Magnetic field reach maximum if the distance between two coils equal to the radius of the coil
• Arial field gradient is twice the radial field gradient.
• Helmholtz coils for earth-compensation
• Magnetic field is most uniform ~ x⁴ when the distance between two coils equal to the radius of the coil
• Earth compensation is critical to get good polarization gradient cooling.
• The magnitude of magnetic field scales ~ Γ for different atomic species.

MOT Alignment

• Counterpropagating lasers are with the same polarizations (handness or helicity) but the configuration is referred as σ+σ- configuration in laser cooling.
• Be careful the specifications from vendors on the quarter might be wrong or inconsistent.
• A thumb rule !

B

laser

E

Fast axis

slow axis

Imaging and Number of Atoms

I₀(x,y)

I_transmitted(x,y)

From experiment

Considering the dark count of CCD

From theory

3* = 0~3, depends on laser polarization and

population distribution around Zeeman sublevels

How to determine the temperature?

MOT laser

Magnetic field

Image beam

t

t=3 ms

t=7ms

t=15 ms

TOF(ms)

Size(mm)

Our Exploration, Cold Molecules

Buffer-gas cooling to

prepare 4K large sample

of molecules.

Stark-guiding and optical

pumping to load molecules

into a microwave trap.

Sympathetic cooling of

molecules to mK in a microwave

trap by ultracold atoms.

1 K

1 mK

1 μK

Evaporative cooling of

molecules to μK in a microwave

trap.

Why Cold Molecules ?

• High-resolution spectroscopy
• Better understanding of molecular structure
• Molecular clock
• Cold molecular collision and reaction
• Precise determination of molecular potential energy
• Controlled reaction by electromagnetic field
• Test of fundamental physics,
• e.g. searching for electron dipole moment
• Study of quantum degenerate dipolar gases
• Dipolar effect on Bose condensate
• Cooper pairing by dipolar interaction
• Quantum computation

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Welcome to join us to explore the ultracold world !

Ying-Cheng Chen, 陳應誠

Institute of Atomic and Molecular Science, Academic Sinica,

Ultracold Atom and Molecule Labortory

中研院原分所 超低溫原子與分子實驗室