Magnetism in BoseEinstein condensates Qiang Gu Department of
Magnetism in Bose-Einstein condensates Qiang Gu Department of Physics University of Science and Technology Beijing (USTB) PKU, Beijing, April 24, 2007
Outline • Ferromagnetism: a brief review • Ferromagnetic transition in spinor Bose gas • Thermodynamics of ferromagnetic Bose gas • Spin dynamics in ferromagnetic condensates • Summary
Ferromagnetism: a brief review
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
Magnetic order in insulators
Itinerant ferromagnetism in Fermi gas • Ideal fermi gas: Pauli paramagnetism where is the density of state at Fermi surface is the Bohr magneton due to the intrinsic magnetic moment of electrons
Itinerant ferromagnetism in Fermi gas • Ideal Fermi gas: Landau diamagnetism 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 • Itinerant ferromagnetism Increase in Band energy when The Stoner criterion:
Itinerant ferromagnetism in Fermi gas • The transition temperature • where
Ferromagnetism in spinor bosons • Prototypical Bose system: 4 He scalar particles does not display magnetism at all • Alkali atoms: 23 Na, 87 Rb, . . . 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, P 125 Progress in Ferromagnetism Research (Nova Science Publishers, New York, 2006)
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 C 2>0 C 2<0 Polar state Ferromagnetic state 23 Na 87 Rb Ho, Phys. Rev. Lett. 81, 742 (1998) Ohmi and Machida, J. Phys. Soc. Jpn 67, 1822 (1998)
Ferromagnetism in spinor bosons • FM phase transition induced by FM couplings TF • BEC: intrinsic phase transition in bosons • Competing of Two energy scale TF > T C TF < T C for large I for small I Is that true? TC TF & T C
Ferromagnetic transition in spinor Bose gas
Phase transitions I • Hamiltonian The first term describes a free Bose gas, so we use The second term describes FM couplings Gu and Klemm, Phys. Rev. A 68, 031604(R) (2003)
Phase transitions I • Mean-field approximation
Phase transitions I • Mean-field Hamiltonian We consider a homogeneous spinor Bose gas by using the grand canonical ensembles.
Phase transitions I • Mean-field equations
Phase transitions I • Mean-field equations Here we suppose only spin-1 bosons can condense.
Phase transitions I • The polylogarithm function Asymptotic behaviors (a<<1): • The value of a
Phase transitions I • Phase diagram Acrobat Document
Phase transitions I • Phase diagram Acrobat Document
Phase transitions I Kis-Szabo et al. , Phys. Rev. A 72, 023617 (2005)
Phase transitions I Wolters, Gelderen, Stoof, 2006, Itinerant ferromagnetism in an ultracold Bose gas
Phase transitions II • Ginzburg-Landau free energy for the BEC Acrobat Document Gu, Bongs and Sengstock, Phys. Rev. A 70, 063609 (2004)
Phase transitions II • Ginzburg-Landau free energy for the FM phase Acrobat Document • The coupling between the two phases
Phase transitions II • Minimizing the total free energy, one gets Acrobat Document
Phase transitions II Acrobat Document • Expansions near I = 0 I (P 0 -P) Walker and Samokhin, Phys. Rev. Lett. 88, 207001 (2002)
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 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
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
Spin-1 condensates The Hamiltonian Spin dynamics : Internal JT
Spin-1 condensates The reduced Hamiltonian with Rabi oscillation at q=0:
Spin-1 condensates Coherent spin mixing in Spin-1 Bose condensate Coherent Spin-mixing M. -S. Chang. Nature Physics, 1, 111(2005)
Spin-1 condensates Coherent Spin-mixing M. -S. Chang. Nature Physics, 1, 111(2005)
Spin-1 condensates with domains The schematic view of domain inside a ferromagnet Gu, Qiu, Bongs and Sengstock, Phys. Rev. Lett 98, xx (2007) in production
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
• Our Hamiltonian: where
Spin-1 condensates with domains • The spin dynamics part of Hamiltonian the overlap factor
Spin-1 condensates with domains • For the unmagnetized state • Equations of motion • Bloch relaxation
Spin-1 condensates with domains Phase Diagram under different overlap factor
Spin-1 condensates with domains Spin dynamics under different relaxation time Gu, Qiu, Bongs and Sengstock, Phys. Rev. Lett 98, xx (2007) in production
Spin-1 condensates with domains For initially magnetized (m≠ 0) case: a=0. 5 a=0. 1
Summary
Summary • Phase diagram: 3 different statistics Weiss Theory 1907 T Stoner Theory 1936 Gu and Klemm, PRA 68, 031604(R) (2003) T TF TF Bose Gase f TF I Ferromagnetic Insulator I 0 Fermi Gas Insulator
Summary • Thermodynamics Specific heat: two transition points different behaviors Susceptibility:
Summary • Coherence dynamics of domain formation
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