Qiang Gu Ferromagnetism in Bose Systems Department of Physics Workshop on Cold Atom Physics and Quantum Information for Young Researchers Ferromagnetism in Bose Systems Qiang Gu Department of Physics University of Science and Technology Beijing (USTB) Taiyuan, July 4, 2007

Outline Historical introduction to ferromagnetism Curie point of ferromagnetic Bose gas Thermodynamics of ferromagnetic Bose gas Spin dynamics in ferromagnetic condensates Brief summary

Historical introduction to ferromagnetism

Historical introduction 625-564b.c. Earliest known reference to magnetism Thales (台利斯)--Greek philosopher & mathematician In China: magnetite (cishi) mentioned in Guanzi by Guan Zhong (管仲) before 200b.c. 1086 First known application Shen Kuo(沈括) Meng Chhi Pi Than lodestone used as compass by Chinese since 7th-8th) In Europe: compass first mentioned by Neckam in 1187 P. Mohn, Magnetism in the Solid State: An Introduction (Springer-Verlag, Berlin, 2003)

Historical introduction Late 19th ~ early 20th Properties of magnets 1895 P. Curie Linear T-dependence of inverse susceptibility 1905 P. Langevin Theoretical explanation of Curie law & Langevin diamagnetism 1907 P. Weiss Molecular field theory of ferromagnetism 1928 Heisenberg Model 1932-36 Neel Generalied Weiss theory for antiferromagnets Curie law Curie-Weiss law (1907)

Magnetic order in insulators Antiferromagnetic order Ferromagnetic order Heisenberg model I > 0 Antiferromagnetic I < 0 Ferromagnetic

Magnetic order in insulators Weiss Molecular-field (Mean-field) theory (1907)

Magnetic order in insulators The ferromagnetic phase transition

Itinerant ferromagnetism in Fermi gas Ideal fermi gas: Pauli paramagnetism 1927 where is the density of state at Fermi surface is the Bohr magneton due to the intrinsic magnetic moment of electrons

Altogether, free electron gas is paramagnetic Itinerant ferromagnetism in Fermi gas Ideal Fermi gas: Landau diamagnetism 1930 due to the quantization of orbital motions of charged particles Altogether, free electron gas is paramagnetic

Itinerant ferromagnetism in Fermi gas Itinerant ferromagnetism (Stoner Mean-field theory, 1936) Ferromagnetic molecular field energy where is the exchange interaction is the magnetization

Itinerant ferromagnetism in Fermi gas Increase in Band energy when The Stoner criterion:

Itinerant ferromagnetism in Fermi gas The transition temperature where Note: is very difficult to obtained theoretically.

Ferromagnetism in spinor bosons Prototypical Bose system: 4He scalar particles does not display magnetism at all Alkali atoms: 23Na, 87Rb,... Atomic bosons have (hyperfine) spin degree of freedom Atomic bosons now can be confined in purely optical traps Q. Gu, Ferromagnetic phase transition in spinor Bose gases, Chapter 6, Progress in Ferromagnetism Research (Nova Science Publishers, New York, 2006)

Spinor-1 Bose condensate Ferromagnetism in spinor bosons Optical trapping: Focused laser BEC or cold fermions All spin states are trapped, releasing the hyperfine spin degrees of freedom Spinor-1 Bose condensate Stamper-Kurn et al., Phys. Rev. Lett. 80, 2027 (1998); Stenger et al., Nature 396, 345 (1998).

Ferromagnetism in spinor bosons Ground state of spinor Bose gases Effective interactions between F=1 atoms C2>0 Polar state 23Na C2<0 Ferromagnetic state 87Rb Ho, Phys. Rev. Lett. 81, 742 (1998) Ohmi and Machida, J. Phys. Soc. Jpn 67, 1822 (1998)

Ferromagnetism in spinor bosons Mechanism for generating ferromagnetic couplings Spin-flip scattering Super-exchange process Magnetic dipolar interaction Burke and Boh., Phys. Rev. A 59, 1303 (1999) Yang and Li, Inter. J. Mod. Phys, B 17, 1027 (2003) Gu, Phys. Rev. A 68, 025601 (2003)

Ferromagnetism in spinor bosons FM phase transition induced by FM couplings TF BEC: intrinsic phase transition in bosons TC Competing of Two energy scale TF & TC TF > TC for large I TF < TC for small I Is that true?

Curie point of ferromagnetic Bose gas Gu and Klemm, Phys. Rev. A 68, 031604(R) (2003) Gu, Bongs and Sengstock, Phys. Rev. A 70, 063609 (2004)

Phase transitions Hamiltonian The first term describes a free Bose gas, where The second term describes FM couplings Mean-field approximation

Phase transitions Mean-field Hamiltonian We consider a homogeneous spinor Bose gas by using the grand canonical ensembles.

Phase transitions Mean-field equations

Phase transitions Mean-field equations Here we suppose only spin-1 bosons can condense.

Phase transitions The polylogarithm function Asymptotic behaviors (a<<1): The value of a

Phase transitions Phase diagram Acrobat Document

Phase transitions Phase diagram Acrobat Document

Phase transitions Kis-Szabo et al., Phys. Rev. A 72, 023617 (2005)

Phase transitions Wolters, Gelderen, Stoof, 2006, Itinerant ferromagnetism in an ultracold Bose gas

Other theories P. Soltan-Panahi , A. Pelster, and H. Kleinert, 2006, unpublished Isoshima, Ohmi, and machida, J. Phys. Soc. Jpn. 69, 3864 (2000)

Other theories Spin conservation! W. zhang, S. Yi, and L. You, Phys. Rev. A 70, 043611 (2004) Spin conservation!

Experiment Sadler et al., Nature 443, 312 (2006)

Thermodynamics in FM spinor Bose gas Tao, Wang, Qin and Gu, unpublished

Background P. Soltan-Panahi , A. Pelster, and H. Kleinert, 2006, unpublished

Basic formula The free energy The internal energy

Basic formula The specific heat The magnetic susceptibility

Free energy The free energy

Free energy t=0.5

Specific heat The specific heat

Specific heat

Susceptibility The magnetic susceptibility

Susceptibility

Spin dynamics of ferromagnetic condensates Gu and Qiu, PRL 98, 200401 (2007)

Spin-1 condensates The Hamiltonian Spin dynamics : Internal JT H. Schmaljohann et al., Appl. Phys. B 79, 1001 (2004); Romano and de Passos, PR A 70, 043614 (2004)

Spin-1 condensates Single mode approximation The reduced Hamiltonian with

Spin-1 condensates Contour plot of the energy H. Schmaljohann et al., Appl. Phys. B 79, 1001 (2004)

Coherent spin mixing in Spin-1 Bose condensate Spin-1 condensates Coherent spin mixing in Spin-1 Bose condensate M.-S.Chang et al.. Nature Physics,1,111(2005)

Spin-1 condensates Romano and de Passos, PRA 70, 043614 (2004)

Spin-1 condensates with domains The schematic view of domain inside a ferromagnet

Spin-1 condensates with domains Phase separation without dissipation 1 -1 - the overlap constant

Spin-1 condensates with domains The field annihilation operator replaced by its expectation value the normalized distribution

Spin-1 condensates with domains Our Hamiltonian: where

The spin dynamics part of Hamiltonian Spin-1 condensates with domains The spin dynamics part of Hamiltonian Integrate out the spatial degree of freedom

The reduced Hamiltonian Spin-1 condensates with domains the overlap factor The reduced Hamiltonian

Spin-1 condensates with domains For the unmagnetized state

Spin-1 condensates with domains Equations of motion Bloch relaxation

Phase Diagram under different overlap factor Spin-1 condensates with domains Phase Diagram under different overlap factor

Spin dynamics under different relaxation time Spin-1 condensates with domains Spin dynamics under different relaxation time Gu and Qiu, PRL 98, 200401 (2007)

Spin-1 condensates with domains For initially magnetized (m≠0) case: a=0.5 a=0.1 Wu, Qiu, Gu, unpublished

Brief summary

Summary Phase diagram: 3 different statistics T T Bose Gase TF TFf I Weiss Theory 1907 Stoner Theory 1936 Gu and Klemm, PRA68, 031604(R) (2003) T T Bose Gase TF TFf TF I I0 I Fermi Gas Ferromagnetic Insulator Fermi Gas Insulator

Summary Thermodynamics Specific heat: two transition points different behaviors Susceptibility: Tao, Wang, Qin and Gu, unpublished

Summary Coherence dynamics of domain formation Gu and Qiu, PRL 98, 200401 (2007)

谢谢！ Many thanks!