Creating and manipulating non-Abelian anyons in cold atom systems using auxiliary bosons

Yuhe Zhang, G. J. Sreejith, and J. K. Jain

Phys. Rev. B 92, 075116 – Published 10 August 2015

Abstract

The possibility of realizing bosonic fractional quantum Hall effect in ultracold atomic systems suggests a new route to producing and manipulating anyons, by introducing auxiliary bosons of a different species that capture quasiholes and thus inherit their nontrivial braiding properties. States with localized quasiholes at any desired locations can be obtained by annihilating the auxiliary bosons at those locations. We explore how this method can be used to generate non-Abelian quasiholes of the Moore-Read Pfaffian state for bosons at filling factor $ν = 1$ . We show that a Hamiltonian with an appropriate three-body interaction can produce two-quasihole states in two distinct fusion channels of the topological “qubit.” Characteristics of these states that are related to the non-Abelian nature can be probed and verified by a measurement of the effective relative angular momentum of the auxiliary bosons, which is directly related to their pair distribution function. Moore-Read states of more than two quasiholes can also be produced in a similar fashion. We investigate some issues related to the experimental feasibility of this approach, in particular, how large the systems should be for a realization of this physics and to what extent this physics carries over to systems with the more standard two-body contact interaction.

Received 22 May 2015

DOI:https://doi.org/10.1103/PhysRevB.92.075116

Authors & Affiliations

Yuhe Zhang¹, G. J. Sreejith², and J. K. Jain¹

¹Department of Physics, 104 Davey Lab, Pennsylvania State University, University Park, Pennsylvania 16802, USA
²Max Planck Institute for Physics of Complex Systems, Nöthnitzer St 38, 01187 Dresden, Germany

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Vol. 92, Iss. 7 — 15 August 2015

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Images

Figure 1
(a) The modulus of the overlap between the four-quasihole qubit states $Ψ^{(1)}$ and $Ψ^{(ψ)}$ as a function of the minimum distance between two quasiholes in each quasihole-position configuration. The blue squares represent cases when the minimum distance is either $| η_{1} - η_{2} |$ or $| η_{3} - η_{4} |$ , and red squares are other configurations. (b) The modulus of the overlap between the two-quasihole qubit states $Ψ_{(1) (2)}$ and $Ψ_{(1) (2)} - 2 Ψ_{(12) ()}$ as a function of the maximum quasihole position $| η_{i} | / R_{N}$ where $R_{N} = \sqrt{2 N / ν}$ is the radius of the quantum Hall droplet. In both calculations, the quasihole positions are chosen randomly, and the system size is $N = 60$ .
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Figure 2
(a) Black square represents the ratio of the normalization factor of the two different two-quasihole wave functions as shown in Eq. (13) and red circle represents the normalized overlap between them as shown in Eq. (14) as a function of system size $N$ . (b) The natural log of the corresponding quantities as a function of $N$ . In panels (a) and (b), we use $| | ψ_{i} {| |}^{2}$ to denote the normalization constant $〈 ψ_{i} | ψ_{i} 〉$ . (c) The normalized overlap between the two-quasihole qubit states $〈 Ψ_{(1) (2)} | | Ψ_{(1) (2)} - 2 Ψ_{(12) ()} 〉$ as a function of $N$ . In all panels $η_{i}$ 's have been taken as dynamical coordinates in Monte Carlo calculation.
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Figure 3
(a) Energy spectrum obtained by diagonalizing the interaction $V = V_{z z z} + V_{z z η}$ within a Hilbert space formed by $N = 8 z$ bosons and two $η$ bosons with $L = 32$ and $l_{z}^{(max)} = 12$ . There are two zero-energy states in the spectrum. (b) Energy spectrum obtained by adding the $0.01 V_{z η η}$ term to the interaction in (a) and diagonalizing the interaction within the same basis.
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Figure 4
The pair correlation function $g (r, 0)$ of the auxiliary bosons with one boson fixed at the center (a) without any background $z$ particles and (b) in the background of a $ν = 1$ bosonic Pfaffian state of $N = 20 z$ particles. The solid and dashed lines in (b) show the pair correlation function for the different two-quasihole states $Ψ_{(1) (2)} P_{M} (η_{1}, η_{2})$ and $(Ψ_{(1) (2)} - 2 Ψ_{(12) ()}) P_{M} (η_{1}, η_{2})$ , respectively. The pair correlation functions here and in the following are all quoted in units of $ρ_{0} = {(2 π ℓ^{2})}^{- 1}$ , where $ℓ$ is the magnetic length.
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Figure 5
The peak positions of the pair correlation function as a function of the relative angular momentum $M$ for $N = 6 - 60$ , for (a) $Ψ_{(1) (2)}$ state and (b) $Ψ_{(1) (2)} - 2 Ψ_{(12) ()}$ state. The black dashed line plots the predicted behavior in Eq. (32) and the red dashed line plots Eq. (33). (c) The quantity $\frac{1}{4} (R^{2} / l^{2} - 4 M)$ as a function of $M$ for $N = 40$ and 50 for $Ψ_{(1) (2)}$ (black square), $Ψ_{(12) ()}$ (blue circle), and $Ψ_{(1) (2)} - 2 Ψ_{(12) ()}$ (red diamond). The black dashed lines plot the expected $γ$ value: $- \frac{1}{8}$ and $\frac{5}{8}$ . (d) The differences of the quantities $\frac{1}{4} (R^{2} / l^{2} - 4 M)$ between $Ψ_{(1) (2)}$ and $Ψ_{(12) ()}$ are shown in the black square, and between $Ψ_{(1) (2)}$ and $Ψ_{(1) (2)} - 2 Ψ_{(12) ()}$ are shown with the red circle.
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Figure 6
The pair correlation functions $g (r, 0)$ of the $η$ bosons of the ground state obtained from diagonalizing the two-body Hamiltonian with parameters shown in the figure are plotted for different system sizes $N = 6$ (blue square), $N = 8$ (green circle), $N = 10$ (red diamond). As comparison, the pair correlation functions calculated from the model wave functions $Ψ_{(1) (2)} P_{M} (η_{1}, η_{2})$ are also shown with solid lines. The plots for $N = 8$ and 10 have been shifted up by $0.05$ and $0.1$ units, respectively, for ease of depiction. (a), (b) For $M = 2$ , an isotropic contact interaction fails to produce a similar pair correlation function as that of the model wave function, but it is possible to find a universal set of interaction parameters $λ_{1, 2}$ for different system sizes that gives agreement. (c) For $M = 3$ , a slightly different confinement potential for $z$ and $η$ bosons is necessary to produce a desired ground state. However, the proper value of parameter $δ ω$ sensitively depends on the system size.
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