Abstract

We propose an alternative experimental scheme to detect the non-Abelian statistics of the Majorana fermions (MF) with a trapped atomic bosonic gas subject to a spin-orbit coupling in 1D optical lattices. The spin-orbit coupling leads to a faithful simulation of an arbitrary tunable XY spin model, where the anisotropy of the effective spin coupling can be controlled by adjusting the optical potentials, Feshbach resonance, and spin-orbit coupling. MF are created at the ends of the optical lattice segment in topological phase. By controlling the different addressing laser beams, one can move, braid, and exchange them. We also show that the non-Abelian statistics of MF can be demonstrated through distinguishing the states before and after braiding by detecting the parity operator of collective spin states.

© 2014 Optical Society of America

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2013

Y. Zhang and C. Zhang, “Bose–Einstein condensates in spin-orbit coupled optical lattices: flat bands and superfluidity,” Phys. Rev. A 87, 023611 (2013).

S. Liu and Z. Y. Xue, “Probing non-Abelian anyonic statistics with cold atoms in an optical lattice,” J. Opt. Soc. Am. B 30, 1720–1724 (2013).
[CrossRef]

2012

D. W. Zhang, L. B. Fu, Z. D. Wang, and S. L. Zhu, “Josephson dynamics of a spin-orbit-coupled Bose–Einstein condensate in a double-well potential,” Phys. Rev. A 85, 043609 (2012).
[CrossRef]

D. W. Zhang, Z. Y. Xue, H. Yan, Z. D. Wang, and S. L. Zhu, “Macroscopic Klein tunneling in spin-orbit-coupled Bose–Einstein condensates,” Phys. Rev. A 85, 013628 (2012).
[CrossRef]

X. J. Liu and H. Hu, “Topological superfluid in one-dimensional spin-orbit-coupled atomic Fermi gases,” Phys. Rev. A 85, 033622 (2012).
[CrossRef]

2011

J. Dalibard, F. Gerbier, G. Juzeliunas, and P. Ohberg, “Colloquium: a gauge potentials for neutral atoms,” Rev. Mod. Phys. 83, 1523–1543 (2011).
[CrossRef]

S. Sinha, R. Nath, and L. Santos, “Trapped two-dimensional condensates with synthetic spin-orbit coupling,” Phys. Rev. Lett. 107, 270401 (2011).
[CrossRef]

Z. Y. Xue, “Detecting non-Abelian statistics of Majorana fermions in quantum nanowire networks,” J. Exp. Theor. Phys. Lett. 94, 213–216 (2011).
[CrossRef]

J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, “Non-Abelian statistics and topological quantum information processing in 1D wire networks,” Nat. Phys. 7, 412–417 (2011).
[CrossRef]

M. Mariantoni, H. Wang, T. Yamamoto, M. Neeley, R. C. Bialczak, Y. Chen, M. Lenander, E. Lucero, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, Y. Yin, J. Zhao, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Implementing the quantum von Neumann architecture with superconducting circuits,” Science 334, 61–65 (2011).
[CrossRef]

A. Fedorov, L. Steffen, M. Baur, M. P. da Silva, and A. Wallraff, “Implementation of a Toffoli gate with superconducting circuits,” Nature 481, 170–172 (2011).
[CrossRef]

M. Aidelsburger, M. Atala, S. Nascimbene, S. Trotzky, Y. A. Chen, and I. Bloch, “Experimental realization of strong effective magnetic fields in an optical lattice,” Phys. Rev. Lett. 107, 255301 (2011).
[CrossRef]

S. L. Zhu, L. B. Shao, Z. D. Wang, and L. M. Duan, “Probing non-Abelian statistics of Majorana fermions in ultracold atomic superfluid,” Phys. Rev. Lett. 106, 100404 (2011).
[CrossRef]

L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, “Majorana fermions in equilibrium and in driven cold-atom quantum wires,” Phys. Rev. Lett. 106, 220402 (2011).
[CrossRef]

S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, “Topology by dissipation in atomic quantum wires,” Nat. Phys. 7, 971–977 (2011).
[CrossRef]

2010

R. M. Lutchyn, J. D. Sau, and S. Das Sarma, “Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures,” Phys. Rev. Lett. 105, 077001 (2010).
[CrossRef]

Y. Oreg, G. Refael, and F. von Oppen, “Helical liquids and Majorana bound states in quantum wires,” Phys. Rev. Lett. 105, 177002 (2010).
[CrossRef]

X. L. Qi and S. C. Zhang, “The quantum spin Hall effect and topological insulators,” Physics Today 63(1), 33–38 (2010).
[CrossRef]

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[CrossRef]

J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, “Generic new platform for topological quantum computation using semiconductor heterostructures,” Phys. Rev. Lett. 104, 040502 (2010).
[CrossRef]

J. Alicea, “Majorana fermions in a tunable semiconductor device,” Phys. Rev. B 81, 125318 (2010).
[CrossRef]

2009

Z.-Y. Xue, S.-L. Zhu, J. Q. You, and Z. D. Wang, “Implementing topological quantum manipulation with superconducting circuits,” Phys. Rev. A 79, 040303(R) (2009).
[CrossRef]

Y. J. Lin, R. L. Compton, K. J. Garcia, J. V. Porto, and I. B. Spielman, “Synthetic magnetic fields for ultracold neutral atoms,” Nature 462, 628–632 (2009).
[CrossRef]

S. L. Zhu, D. W. Zhang, and Z. D. Wang, “Delocalization of relativistic Dirac particles in disordered one-dimensional systems and its implementation with cold atoms,” Phys. Rev. Lett. 102, 210403 (2009).
[CrossRef]

M. Sato, Y. Takahashi, and S. Fujimoto, “Non-Abelian topological order in s-wave superfluids of ultracold fermionic atoms,” Phys. Rev. Lett. 103, 020401 (2009).
[CrossRef]

2008

I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold gases,” Rev. Mod. Phys. 80, 885–964 (2008).
[CrossRef]

A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, “Coherent quantum optical control with subwavelength resolution,” Phys. Rev. Lett. 100, 093005 (2008).
[CrossRef]

C. Zhang, S. Tewari, R. M. Lutchyn, and S. Das Sarma, “px+ipy superfluid from s-wave interactions of fermionic cold atoms,” Phys. Rev. Lett. 101, 160401 (2008).
[CrossRef]

L. Fu and C. L. Kane, “Superconducting proximity effect and Majorana fermions at the surface of a topological insulator,” Phys. Rev. Lett. 100, 096407 (2008).
[CrossRef]

L. Jiang, G. K. Brennen, A. V. Gorshkov, K. Hammerer, M. Hafezi, E. Demler, M. D. Lukin, and P. Zoller, “Anyonic interferometry and protected memories in atomic spin lattices,” Nat. Phys. 4, 482–488 (2008).
[CrossRef]

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[CrossRef]

2007

J. Cho, “Addressing individual atoms in optical lattices with standing-wave driving fields,” Phys. Rev. Lett. 99, 020502 (2007).
[CrossRef]

2006

S. L. Zhu, H. Fu, C. J. Wu, S. C. Zhang, and L. M. Duan, “Spin Hall effects for cold atoms in a light-induced gauge potential,” Phys. Rev. Lett. 97, 240401 (2006).
[CrossRef]

S. L. Zhu, C. Monroe, and L.-M. Duan, “Trapped ion quantum computation with transverse phonon modes,” Phys. Rev. Lett. 97, 050505 (2006).
[CrossRef]

S. L. Zhu, C. Monroe, and L.-M. Duan, “Arbitrary-speed quantum gates within large ion crystals through minimum control of laser beams,” Europhys. Lett. 73, 485–491 (2006).
[CrossRef]

2003

L. M. Duan, E. Demler, and M. D. Lukin, “Controlling spin exchange interactions of ultracold atoms in optical lattices,” Phys. Rev. Lett. 91, 090402 (2003).
[CrossRef]

A. Kitaev, “Fault-tolerant quantum computation by anyons,” Ann. Phys. 303, 2–30 (2003).
[CrossRef]

2001

A. Kitaev, “Unpaired Majorana fermions in quantum wires,” Physics-Uspekhi 44, 131–136 (2001).
[CrossRef]

D. A. Ivanov, “Non-Abelian statistics of half-quantum vortices in p-wave superconductors,” Phys. Rev. Lett. 86, 268–271 (2001).
[CrossRef]

1991

G. Moore and N. Read, “Nonabelions in the fractional quantum hall effect,” Nucl. Phys. B 360, 362–396 (1991).
[CrossRef]

1983

R. B. Laughlin, “Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations,” Phys. Rev. Lett. 50, 1395–1398 (1983).
[CrossRef]

Aidelsburger, M.

M. Aidelsburger, M. Atala, S. Nascimbene, S. Trotzky, Y. A. Chen, and I. Bloch, “Experimental realization of strong effective magnetic fields in an optical lattice,” Phys. Rev. Lett. 107, 255301 (2011).
[CrossRef]

Akhmerov, A. R.

L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, “Majorana fermions in equilibrium and in driven cold-atom quantum wires,” Phys. Rev. Lett. 106, 220402 (2011).
[CrossRef]

Alicea, J.

L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, “Majorana fermions in equilibrium and in driven cold-atom quantum wires,” Phys. Rev. Lett. 106, 220402 (2011).
[CrossRef]

J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, “Non-Abelian statistics and topological quantum information processing in 1D wire networks,” Nat. Phys. 7, 412–417 (2011).
[CrossRef]

J. Alicea, “Majorana fermions in a tunable semiconductor device,” Phys. Rev. B 81, 125318 (2010).
[CrossRef]

Atala, M.

M. Aidelsburger, M. Atala, S. Nascimbene, S. Trotzky, Y. A. Chen, and I. Bloch, “Experimental realization of strong effective magnetic fields in an optical lattice,” Phys. Rev. Lett. 107, 255301 (2011).
[CrossRef]

Baranov, M. A.

S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, “Topology by dissipation in atomic quantum wires,” Nat. Phys. 7, 971–977 (2011).
[CrossRef]

Baur, M.

A. Fedorov, L. Steffen, M. Baur, M. P. da Silva, and A. Wallraff, “Implementation of a Toffoli gate with superconducting circuits,” Nature 481, 170–172 (2011).
[CrossRef]

Bialczak, R. C.

M. Mariantoni, H. Wang, T. Yamamoto, M. Neeley, R. C. Bialczak, Y. Chen, M. Lenander, E. Lucero, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, Y. Yin, J. Zhao, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Implementing the quantum von Neumann architecture with superconducting circuits,” Science 334, 61–65 (2011).
[CrossRef]

Bloch, I.

M. Aidelsburger, M. Atala, S. Nascimbene, S. Trotzky, Y. A. Chen, and I. Bloch, “Experimental realization of strong effective magnetic fields in an optical lattice,” Phys. Rev. Lett. 107, 255301 (2011).
[CrossRef]

I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold gases,” Rev. Mod. Phys. 80, 885–964 (2008).
[CrossRef]

Brennen, G. K.

L. Jiang, G. K. Brennen, A. V. Gorshkov, K. Hammerer, M. Hafezi, E. Demler, M. D. Lukin, and P. Zoller, “Anyonic interferometry and protected memories in atomic spin lattices,” Nat. Phys. 4, 482–488 (2008).
[CrossRef]

Casanova, J.

A. Mezzacapo, J. Casanova, L. Lamata, and E. Solano, “Topological qubits with Majorana fermions in trapped ions,” arXiv:1111.5603 (2011).

Chen, Y.

M. Mariantoni, H. Wang, T. Yamamoto, M. Neeley, R. C. Bialczak, Y. Chen, M. Lenander, E. Lucero, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, Y. Yin, J. Zhao, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Implementing the quantum von Neumann architecture with superconducting circuits,” Science 334, 61–65 (2011).
[CrossRef]

Chen, Y. A.

M. Aidelsburger, M. Atala, S. Nascimbene, S. Trotzky, Y. A. Chen, and I. Bloch, “Experimental realization of strong effective magnetic fields in an optical lattice,” Phys. Rev. Lett. 107, 255301 (2011).
[CrossRef]

Cho, J.

J. Cho, “Addressing individual atoms in optical lattices with standing-wave driving fields,” Phys. Rev. Lett. 99, 020502 (2007).
[CrossRef]

Cirac, J. I.

L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, “Majorana fermions in equilibrium and in driven cold-atom quantum wires,” Phys. Rev. Lett. 106, 220402 (2011).
[CrossRef]

Cleland, A. N.

M. Mariantoni, H. Wang, T. Yamamoto, M. Neeley, R. C. Bialczak, Y. Chen, M. Lenander, E. Lucero, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, Y. Yin, J. Zhao, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Implementing the quantum von Neumann architecture with superconducting circuits,” Science 334, 61–65 (2011).
[CrossRef]

Compton, R. L.

Y. J. Lin, R. L. Compton, K. J. Garcia, J. V. Porto, and I. B. Spielman, “Synthetic magnetic fields for ultracold neutral atoms,” Nature 462, 628–632 (2009).
[CrossRef]

da Silva, M. P.

A. Fedorov, L. Steffen, M. Baur, M. P. da Silva, and A. Wallraff, “Implementation of a Toffoli gate with superconducting circuits,” Nature 481, 170–172 (2011).
[CrossRef]

Dalibard, J.

J. Dalibard, F. Gerbier, G. Juzeliunas, and P. Ohberg, “Colloquium: a gauge potentials for neutral atoms,” Rev. Mod. Phys. 83, 1523–1543 (2011).
[CrossRef]

I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold gases,” Rev. Mod. Phys. 80, 885–964 (2008).
[CrossRef]

Das Sarma, S.

R. M. Lutchyn, J. D. Sau, and S. Das Sarma, “Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures,” Phys. Rev. Lett. 105, 077001 (2010).
[CrossRef]

J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, “Generic new platform for topological quantum computation using semiconductor heterostructures,” Phys. Rev. Lett. 104, 040502 (2010).
[CrossRef]

C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. 80, 1083–1159 (2008).
[CrossRef]

C. Zhang, S. Tewari, R. M. Lutchyn, and S. Das Sarma, “px+ipy superfluid from s-wave interactions of fermionic cold atoms,” Phys. Rev. Lett. 101, 160401 (2008).
[CrossRef]

Demler, E.

L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, “Majorana fermions in equilibrium and in driven cold-atom quantum wires,” Phys. Rev. Lett. 106, 220402 (2011).
[CrossRef]

L. Jiang, G. K. Brennen, A. V. Gorshkov, K. Hammerer, M. Hafezi, E. Demler, M. D. Lukin, and P. Zoller, “Anyonic interferometry and protected memories in atomic spin lattices,” Nat. Phys. 4, 482–488 (2008).
[CrossRef]

L. M. Duan, E. Demler, and M. D. Lukin, “Controlling spin exchange interactions of ultracold atoms in optical lattices,” Phys. Rev. Lett. 91, 090402 (2003).
[CrossRef]

Diehl, S.

S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, “Topology by dissipation in atomic quantum wires,” Nat. Phys. 7, 971–977 (2011).
[CrossRef]

Duan, L. M.

S. L. Zhu, L. B. Shao, Z. D. Wang, and L. M. Duan, “Probing non-Abelian statistics of Majorana fermions in ultracold atomic superfluid,” Phys. Rev. Lett. 106, 100404 (2011).
[CrossRef]

S. L. Zhu, H. Fu, C. J. Wu, S. C. Zhang, and L. M. Duan, “Spin Hall effects for cold atoms in a light-induced gauge potential,” Phys. Rev. Lett. 97, 240401 (2006).
[CrossRef]

L. M. Duan, E. Demler, and M. D. Lukin, “Controlling spin exchange interactions of ultracold atoms in optical lattices,” Phys. Rev. Lett. 91, 090402 (2003).
[CrossRef]

Duan, L.-M.

S. L. Zhu, C. Monroe, and L.-M. Duan, “Trapped ion quantum computation with transverse phonon modes,” Phys. Rev. Lett. 97, 050505 (2006).
[CrossRef]

S. L. Zhu, C. Monroe, and L.-M. Duan, “Arbitrary-speed quantum gates within large ion crystals through minimum control of laser beams,” Europhys. Lett. 73, 485–491 (2006).
[CrossRef]

Fedorov, A.

A. Fedorov, L. Steffen, M. Baur, M. P. da Silva, and A. Wallraff, “Implementation of a Toffoli gate with superconducting circuits,” Nature 481, 170–172 (2011).
[CrossRef]

Fisher, M. P. A.

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Figures (2)

Fig. 1.
Fig. 1.

Braiding of two unpaired MF. (a) The whole optical lattices is in a topological nontrival phase where an unpaired MF occurs at each end. (b) Adiabatically tuning the first laser beam to nonzero in the segment of the optical lattice, i.e., turn the segment to be topological trivial, drives the left-end Majorana mode to the middle part. (c) Braiding of MF can be implemented by introducing an additional optical lattice; MF can be moved to the end of another topological nontrivial parts of the other optical lattice. (d) Using the same methods, adiabatically controlling the laser beams finally drive the original left-end MF to the right end and thus implement the exchange of MF.

Fig. 2.
Fig. 2.

(a) 1D optical lattice inside an optical cavity for cavity assisted interaction. (b) Level structure of atoms and the coupled transitions.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

H=dx(σ,σ=,ψσ[22m2+mωx22x2+Vso(x)]ψσ+Uψ(x)ψ(x)ψ(x)ψ(x)+Uψ(x)ψ(x)ψ(x)ψ(x)+Uψ(x)ψ(x)ψ(x)ψ(x)),
H=ijσ(tσaiσajσ+tsoaiσajσ¯+H.c.)+12i,σUσniσ(niσ1)+Uinini.
tσ=dxw(xxi)[px22m+Vσ(x)]w(xxj),U,,=gs,,dx|w(xxi)|4,tso=αdxw(xxi)xw(xxj).
H12(0)=12i,σUσniσ(niσ1)+Uinini.
H=i(λzSiz+λxSix)+i,j[tzzSizSjz2ttU(Si+Sj+Sj+Si)2tso2U(Si+Sj++SjSi)],
tzz=t2+t22tso2Ut2tso2Ut2tso2U,λz=14(t2Ut2U),
λx=tso(t+tUtUtU).
H=i,j[2ttU(Si+Sj+Sj+Si)+2tso2U(Si+Sj++SjSi)].
H=ti,j[(Si+Sj+Sj+Si)+(Si+Sj++SjSi)]λiSiz.
H=n=1N1t(cncn)(cn+1+cn+1)n=1Nλ(2cncn1),
H=in=1N1tγnBγn+1Ain=1NλγnAγnB,
H=k(γkAγkB)(0iD*(k)iD(k)0)(γkAγkB),
HQND=χaalσlz,
U=exp[iHτ]={Ifornc=0(i)Nlσlzfornc=1,

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