probing and manipulating majorana fermions in so coupled atomic fermi gases xia-ji liu caous,...
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Probing and Manipulating Majorana Fermions in SO Coupled Atomic Fermi Gases
Xia-Ji Liu
CAOUS, Swinburne University
Hawthorn, July.
Outline
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Majorana Fermion
Probing Majorana Fermion in SO Coupled Fermi Gases
Manipulating Majorana Fermion in SO Coupled Fermi Gases
Majorana Fermion
Majorana Fermion: particle is its own antiparticle
022
jj cc jkkj cc },{Fermion:
12 jkkj 2},{
Quantum statistics of Majorana Fermion: anyon
122121 || M
Majorana Fermion
•Quasiparticles in fractional Quantum Hall effect at n=5/2 Moore Read 1991
Potential System Hosts for Majorana Fermion Bound States
•Unconventional superconductors -Sr2RuO4 Das Sarma, Nayak, Tewari 2006
•Proximity Effect Devices using ordinary s wave superconductors
-Topological Insulator devices Fu, Kene 2008 -Semiconductor/Magnet devices Sau, Lutchyn, Tewari, /das Sarma
2009
•Neutrinos Majorana 1937
•Supersymmetry: photino; neutral gauginos; Higgsinos
Condensed Matter System: quasiparticles
Current Status: Not Observed??
Majorana Fermion
Ultracold Fermionic Atoms near Feshbach Resonance
Das SarmaT. Mizushima, K. Machida, M. Sato, Y. Takahashi, S. Fujimoto, C. Zhang, , et al..
Full Microscopic Calculation ?
Signature for Majorana Fermion ?
Question:
Suggestion:
2D trapped ultracold Fermi gas with SO coupling
Hamiltonian
Single- Particle Hamiltonian
Interaction Hamiltonian
Renormalization
Rashba SO Coupling xyyx kk Dresselhaus SO Coupling xyyx kk
2D trapped ultracold Fermi gas with SO coupling
MF BdG Equantion
Bogoliubov Transformation
BdG Hamiltonian
Gap Function 0Ur
Single VortexSingle Vortex
Quansiparticle wave function
Liu, Hu and Drummond, PRA75, 023614(2007)
2D trapped ultracold Fermi gas with SO coupling
E
E
E
E
EE
0E 00 EE
“trival” “topological”
Majorana Fermion: Particle = Antiparticle
Particle-hole transformation
2D trapped ultracold Fermi gas with SO couplingTopological Invariants
Topological variants
2D trapped ultracold Fermi gas with SO coupling
Question: Does the Majorana fermion bound state exist?
There is not zero mode when the Zeeman field h=0
EE
n
n
n
n
u
v
v
uS-wave interaction without SO coupling 0 nn vuso
Mixed singlet and triplet pairings
2D trapped ultracold Fermi gas with SO coupling
kE
kE
helicity basis k k
chh only Ek (-) is occuped
cvcucvcu0
cvcucvcu ****0
22 ch
p-wave symmetry zero mode bound state
* vu
Two solutions
01 11
* vu 02 i 22
210 i
Majorana fermion: particle is its own antiparticleFermion operator
Majorana fermion is a half of ordinary fermion
2D trapped ultracold Fermi gas with SO couplingPhase diagram
Non-topological state phase separation state topological state
rrrhc22 22
2
1rmr
Fa EE 2.0 1/ FF Ek 0/ FTTparameters
topological state 022 h
2D trapped ultracold Fermi gas with SO coupling
Low-Lying Quasiparticle Spectrum
Three branches with small energy spacing appear:
“Outer edge” state “Inner edge” state
Vortex core GdGM state (garoli-de Gennes-Matricon)
Phase separation phase: “outer edge” and “inner edge”
Topological state: “outer edge” and “GdGM state”
h
2D trapped ultracold Fermi gas with SO couplingWave Function
a bond and anti-bond hybridization and
* vu *
vu
quasiparticle tunneling energy splitting
PS phase: tunneling between two edge states
T state: tunneling between outer edge state and “GdGM state”
Exponentially small energy zeroatomN
Probing Majorana Fermion
Experimental Signature
spin-up and spin-down densities at the trap center
)(0rnrn
)(0rnrn FEh 6.0
Probing Majorana Fermion
Experimental Signature
EEvEEuEr 22 ||||2
1,local density of
state 22 ||||0, rvrur
rf-spectroscopy
40K, B=225.3G, Ea/EF=0.2 N=2000, kHzz 802 Hz1252 Kohl’s group PRL 2011
Application:
Topological quantum computation using Majorana Fermions as qubits
1D trapped ultracold Fermi gas with SO coupling
Hamiltonian
Rashba SO Coupling xyyx kk Dresselhaus SO Coupling xyyx kk
1D trapped ultracold Fermi gas with SO coupling
Topological superfluid xxxhc22
1D trapped ultracold Fermi gas with SO coupling
Phase diagram
1D trapped ultracold Fermi gas with SO coupling
h/EF=0.5
Manipulating Majorana Fermions
Transport Majorana Fermions: Zeeman field
h/EF=0.5h/EF=0.6
Manipulating Majorana Fermions
Manipulating Majorana Fermions
Create Majorana Fermions: Magnetic impurity potential xVimp
h/EF=0.5
Manipulating Majorana Fermions
Magnetic impurity potential
Manipulating Majorana Fermions
40K
SO Coupled ultracold atomic gases
Spielman’s group Nature Vol 471, 83 (2011)
Open Question:
Topological quantum computation
SO coupled bosonic system
Topological superfluidity