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**Dynamics of Spin-1 Bose-Einstein Condensates**

Ming-Shien Chang Institute of Atomic and Molecular Sciences Academia Sinica

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Outline Introduction to spinor condensates Dynamics of spin-1 condensates Temporal dynamics: coherent spin mixing Spatial dynamics: miscibility and spin domain formation Progress report: BEC experiments at the IAMS Summary

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**Quantum Gases Tests of mean-field theories ground state properties**

Exquisitely clean experimental system Widely variable parameters: Different atomic species Bosons, fermions Internal d.o.f. Spin systems Tunable interactions Feshbach resonances Molecular quantum gases Lattice systems Benefits from 80+ yrs of theoretical many-body research Stimulating much new research Tests of mean-field theories ground state properties Interactions: repulsive, attractive, ideal gas Excitations Free expansion, vortices, surface modes Multi component mixtures Beyond mean field theories Strongly correlated systems Mott-insulator states, BCS Entanglement and squeezing

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**BEC Physics BEC Order parameter χ(r) ~ N1/2 ψs(r) Coherent Matter Wave**

JILA, 1995 Order parameter χ(r) ~ N1/2 ψs(r) Coherent Matter Wave Mean-field theory works

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Phase space density Phase space density De Broglie wavelength BEC occurs when: interparticle spacing, n01/3 ~ de Broglie wavelength Phase space density Ambient conditions Laser cooling Nobel Prize, 1997 BEC Nobel Prize, 2001

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**Bose-Einstein Condensate:**

Mean-field theory Gross-Pitaevskii equation (1961, nonlinear Schrodinger eqn) s-wave scattering length

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**Quest for BEC BEC, 1995 Nobel Prize, 2001 All-optical approach**

A. Cornell C. Wieman Nobel Prize, 2001 All-optical approach Standard recipe Hess, Kleppner, Greytak et al. (1986/7), Pritchard et al. (1989), Ketterle et al. (1993/4), Cornell et al. (1994) slow (60 s)—requires exceptional vacuum not everything can be magnetically trapped magnetic fields difficult to generate M. Chapman (2001) W. Ketterle (1995)

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**All-Optical BEC Gallery**

Cross trap 1-D lattice Single focus Common features: 87Rb CO2 trapping laser Simple MOT < 2 s evaporation time cigar disk ~ spherical 30,000 atoms 30,000 atoms 300,000 atoms

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**F=1 Spinor BEC F = 1 mF = 1 mF = 0 mF = -1**

Stern-Gerlach absorption image of a BEC created in an optical trap (GaTech, 2001)

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**Studies of F=1 Spinor BEC in an optical trap**

A multi-component (magnetic) quantum gas

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**Spinor Condensates A multi-component magnetic quantum gas**

Spinor system Spin mixing Spin domains, spin tunneling (Anti-) Ferromagnetism Rotating spinors Spin textures Skyrmion vortices Quantum Magnetism Spin squeezing, entanglement Spinors in an optical lattice Spin chains QPT, quantum quench

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**Interacting Spin-1 BEC c2 << c0 Intuitive picture: F = 0, 1, 2**

Atomic Parameters a0 (Bohr) a2 c0 (x10-12 Hz·cm3) c2 87Rb 101.8 100.4 7.793 23Na 50.0 55.0 15.587 0.4871 anti-ferromagnetic ferromagnetic Ho, 98 c2 << c0

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**Hamiltonian for Spin-1 BEC**

Ho, PRL (98) Machida, JPS (98) 2nd Quantized Form Spin changing collisions

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**Coupled Gross-Pitaevskii Eqn. for Spin-1 Condensates**

Condensate wave function Cross-phase modulation Modulational instability, domain formation Coherent spin (4-wave) mixing ……. Bigelow, 98-00 Meystre, 98-99

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**When c2 = 0… 3 Zeeman components are decoupled. First BEC in 1995**

Condensate wave function First BEC in 1995 Nobel Prize in 2001 3 Zeeman components are decoupled.

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**Spinors In B fields 72 Hz/G2**

m= m= m=-1 When linear Zeeman effects are canceled, quadratic Zeeman effect favors m0. m= m= m=-1 72 Hz/G2 One can study spinor condensates in mG ~ G regime.

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**Single mode approximation (SMA)**

Simplification on spinor dynamics if all spin components have same spatial wave function (SMA): Hamiltonian reduces to just two variables to describe internal spin : Condensate magnetization Spin-dependent interaction strength Quadratic Zeeman energy Population of ±1 components follows:

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**Spinor energy contours—zero field**

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**Spinor energy contours—finite field**

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**Spin Mixing in spin-1 condensates**

t = 0 s For no interactions, m0 is lowest energy (2nd order Zeeman shift) mF = 2 sec mF =

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**Ferromagnetic behavior**

Anti-ferromagnetic spinor Ferromagnetic spinor You, 03 Chapman, 04 Sengstock, 04

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**Deterministically initiate spin mixing**

At t=0: (ρ1, ρ0, ρ-1) = (0, 0.75, 0.25)

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**Coherent Spin Mixing Chapman, 05**

Josephson dynamics driven only by spin-dependent interactions

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**Coherent Spin Mixing Oscillation Frequency: Bigelow, 99**

Direct measurement of c (c2)

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**Direct measurement of c2 (or aF=2 - aF=0)**

aF=2 - aF=0 = -1.4(3) aB (this work) aF=2 - aF=0 = -1.40(22) aB (spect. + theory) from oscillation frequency from condensate expansion

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**Spin mixing is a nonlinear internal AC Josephson effect**

You, 05 de Passos, 04

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**AC Josephson Oscillations**

For high fields where d >> c, the system exhibits small oscillations analogous to AC-Josephson oscillations: Compare with weakly linked superconductors:

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**Controlling spinor dynamics**

Pulse on a magnetic field Quadratic Zeeman energy when θ (rad)

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**Controlling spinor dynamics**

Change trajectories by applying phase shifts via the quadratic zeeman effect Ferromagnetic ground state θ (rad)

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**Search for ferromagnetic spinor ground state**

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**Coherence of the ferromagnetic ground state**

Restarting the coherent spin mixing by phase-shifting out of the ground state at a later time Spin coherence time = condensate lifetime

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**Beyond the Single-Mode Approx. (SMA)**

Formation of spin domains Miscibilities of spin components Formation of spin waves Atomic four-wave mixing

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Healing length shortest distance ξ over which the wavefunction can change Using Healing length: smoothes the boundary layer and determines the size of vortices.

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**Beyond SMA: formation of spin domains**

weak B gradient during TOF z Single-Mode Approx. (SMA): Spin healing length: Condensate size: (2rc,2zc) ~ (7, 70) m condensate is unstable along the z direction.

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**Miscibility of spin-1 (3-component) superfluid**

Goal: minimize the total mean-field energy 1-fluid M-F 2-fluid M-F 3-fluid M-F MIT, 98-99

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**Miscibility of two-component superfluids**

Total Energy of two-component superfluid If they are spatially overlapped with equal mixture: If they are phase separated: The condensates will phase –separated if

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**Miscibility of spin-1 (3-component) superfluid**

2-fluid M-F 3-fluid M-F 1-fluid M-F

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**Miscibility of two-component superfluid**

Stern-Gerlach Exp. During TOF <1 miscible >1 immiscible 23Na 87Rb Ferromagnetic:

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**Invalidity of the Single-Mode Approx.**

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**Spin waves induced by coherent spin mixing**

(r1, r0, r-1) = (0, 0.75, 0.25) mF 1 -1 - Validate coupled GP eqn. - Theoretical explanation of spin waves. - Atomic 4-wave mixing - Evidence of dynamical instability

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**Domain formation induced by dynamical instability**

(r1, r0, r-1) = (0, 0.5, 0.5) mF 1 -1 total (r1, r0, r-1) = (0, 0.83, 0.17)

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**Miscibility of ferromagnetic spin-1 superfluid**

3 components in the ferromagnetic ground state appear to be miscible Energy for spin waves (external) is derived from internal spinor energy mF 1 -1

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Return to the SMA mF = -1 mF = 0 mF = 1 Single focus trap Cross trap

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**Validity of the SMA Spin healing length:**

Condensate should be physically smaller than spin healing length Spin healing length: Cross trap 1-D lattice Single focus cigar disk ~ spherical (2rc,2zc) ~ (7, 70) m (2rc,2zc) ~ (7, 7) m (2rc,2zc) ~ (1, 10) m Condensate size

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Improving the SMA Single-focus trap result

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Improving the SMA Cross trap result

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Improving the SMA

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**SMA vs. spin waves (domains)**

Single-focused trap Rz = 70 μm ξs = 15 μm mF 1 -1 Cross trap Rz = 7 μm ξs = 11 μm

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**Research projects with ultracold atoms**

at the IAMS Optical dipole trap (ODT) for cold-atom experiments Optical lattice for quantum simulation / quantum information experiment ODT for Single atom trapping All-optical BEC of Potassium / Rubidium Spinor condensates studies of Potassium / Rubidium Determination of the spin nature of potassium complex ground state, SSS spin mixing of only two atoms (entangled pair after mixing) Mixture of bosonic and fermionic spinors Rydberg atom quantum information

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**Quest for all-Optical BEC at the IAMS**

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**Optical Trap - + Far off-resonant lasers work as static field**

Focused laser beam form a 3D trap: gaussian beam: radial focus: longitudinal Importance of optical trap State-Independent Potential Trapping of Multiple Spin States Evaporative Cooling of Fermions

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**All-Optical BEC Gallery**

Cross trap 1-D lattice Single focus cigar disk ~ spherical 30,000 atoms 30,000 atoms 300,000 atoms

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**III. BEC in a Single-Focused Trap**

Initial loading: Scaling for Optical Trap Effective Trap Volume weak focus large trap volume low density Trap frequency Scaling for adiabatic compression Compression and evaporation: Density Elastic collision rate tight focus small trap volume high density

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**Dynamical Trap Compression**

P = 70 w w0 70 30 μm 2.5 mm Time 0.6 s

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**Gallery of optical lattices**

In-situ imaging CO2 lattice constant: 5.3 μm Weiss(07) Bookjen, PhD thesis Time-of-flight(TOF) imaging CO2 lattice constant: 0.43 μm I. Bloch (01) Greiner (09)

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**CO2 laser vs. Nd:YAG laser**

(μm) 10.6 1.06 P (W) 1000 30 (single frequency) Scattering rate (same trap parameters) 1 2200 Rayleigh range (same beam waist) 10 Optics ZnSe / Ge Usual glass Spatial mode Usually better Lattice constant of an optical lattice(μm) 5.3 (easier to resolve) 0.53 (lattice physics)

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**Aspect ratio: CO2 vs. Nd:YAG**

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**Trap frequency: CO2 vs. Nd:YAG**

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**Trap Loading: single-focus beam**

Vapor cell MOT 𝑁=2× 10 7 𝑇=30 𝜇K Dipole trap 𝜆=1.06 μ𝑚 𝑃=8 W (each) 𝑤 0 =30 μ𝑚 𝑇 𝐷 =800 𝜇K ω=2𝜋×(2300, 2300, 19) Hz

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**Trap Loading: cross beams**

Hold time (1 sec) 𝑃=8 W (each) x-angle = 30 ∘ 𝑤 0 =30 𝜇m 𝑇 𝐷 =1.6 mK 𝜔 =2𝜋×2200 Hz

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Free evaporation kBT hot atoms escape

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**Free Evaporation 𝟖 𝑾→𝟖 𝑾 𝒊𝒏 𝟏 𝒔𝒆𝒄 # of atoms N 1.0x105 trap frequency**

2,120 Hz ω 13,300 rad/s temperature T 50 μK peak density n0 E+13 1/c.c. phase space density Λ 8.5x10-4

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**Force evaporation 𝟖 𝑾→𝟎.𝟓 𝑾 𝒊𝒏 𝟏.𝟕 𝒔𝒆𝒄 # of atoms N 3000**

trap frequency f 530 Hz ω 3300 rad/s temperature T 6 μK peak density n0 3.4x1011 1/c.c. phase space density Λ 2.3x10-4

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**Spinor condensates with potassium atoms**

Spinor condensates of potassium in an optical trap Spin mixing Determine nature of the spinors Determine spin-dependent scattering lengths Spinor condensates in an optical lattice Simulation of quantum magnets Mixture of Bosonic and Fermionic spinors

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**Zeeman slower for potassium experiment**

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**Zeeman slower for potassium experiment**

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**Zeeman slower for potassium experiment**

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**Summary Formation of spinor condensates in all-optical traps**

Observation of coherent spinor dynamics Observation of spatial-temporal spinor dynamics Current progress of the BEC experiments at the IAMS Preliminary data of Rb force evaporation Zeeman slowing of K

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Acknowledgement 吳耿碩 陳俊嘉 黃智遠 廖冠博 鄭毓璿 彭有宏

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