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Chinese Physics B
Vol. 29, Issue 8, (2020)

Simulation of anyons by cold atoms with induced electric dipole moment

Jian Jing^1,†, Yao-Yao Ma¹, Qiu-Yue Zhang¹, Qing Wang², and Shi-Hai Dong³

Author Affiliations

¹College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 00029, China

²College of Physics and Technology, Xinjiang University, Urumqi 830046, China

³Laboratorio de Información Cuántica, CIDETEC, Instituto Politécnico Nacional, UPALM, CDMX 07700, Mexico

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DOI: 10.1088/1674-1056/ab9737 Cite this Article

Jian Jing, Yao-Yao Ma, Qiu-Yue Zhang, Qing Wang, Shi-Hai Dong. Simulation of anyons by cold atoms with induced electric dipole moment[J]. Chinese Physics B, 2020, 29(8): Copy Citation Text

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Abstract

We show that it is possible to simulate an anyon by a trapped atom which possesses an induced electric dipole moment in the background of electric and magnetic fields in a specific configuration. The electric and magnetic fields we applied contain a magnetic and two electric fields. We find that when the atom is cooled down to the limit of the negligibly small kinetic energy, the atom behaves like an anyon because its angular momentum takes fractional values. The fractional part of the angular momentum is determined by both the magnetic and one of the electric fields. Roles electric and magnetic fields played are analyzed.

Keywords

anyons cold atoms induced electric dipole moment

\begin{array}{r} {\boldsymbol E}^{(1)} = \frac{k}{r} {\boldsymbol e}_{r}, {\boldsymbol E}^{(2)} = \frac{ρ}{2} r {\boldsymbol e}_{r}, \end{array}

(1)

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\begin{array}{r} \boldsymbol B = B {\boldsymbol e}_{z}, \end{array}

(2)

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\begin{array}{r} \boldsymbol d = α (\boldsymbol E + \frac{\boldsymbol v}{c} \times \boldsymbol B), \end{array}

(3)

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\begin{array}{r} L = \frac{1}{2} m {\boldsymbol v}^{2} + \frac{1}{2} \boldsymbol d \cdot (\boldsymbol E + \frac{\boldsymbol v}{c} \times \boldsymbol B) - \frac{1}{2} K {\boldsymbol r}^{2}, \end{array}

(4)

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\begin{array}{r} L = \frac{1}{2} M {\dot{x}}_{i}^{2} - \frac{α B}{c} ε_{i j} {\dot{x}}_{i} E_{j} + \frac{1}{2} α E_{i}^{2} - \frac{1}{2} K x_{i}^{2}, \end{array}

(5)

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\begin{array}{r} p_{i} = \frac{\partial L}{\partial {\dot{x}}_{i}} = M {\dot{x}}_{i} - \frac{α B}{c} ε_{i j} E_{j} . \end{array}

(6)

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\begin{array}{r} {x_{i}, x_{j}} = {p_{i}, p_{j}} = 0, {x_{i}, p_{j}} = δ_{i j} . \end{array}

(7)

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\begin{array}{r} H = \frac{1}{2 M} {(p_{i} + \frac{α B}{c} ε_{i j} E_{j})}^{2} - \frac{1}{2} α E_{i}^{2} + \frac{1}{2} K x_{i}^{2} . \end{array}

(8)

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\begin{array}{r} x_{i} \to x_{i}, p_{i} \to - i ℏ \frac{\partial}{\partial x_{i}}, {} \to \frac{1}{i ℏ} [] \end{array}

()

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\begin{array}{r} J = ε_{i j} x_{i} p_{j}, \end{array}

(9)

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\begin{array}{r} J_{n} = n ℏ, n = 0, \pm 1, \pm 2, \dots . \end{array}

(10)

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\begin{array}{r} L_{r} = - \frac{α B}{c} ε_{i j} {\dot{x}}_{i} E_{j} + \frac{1}{2} α E_{i}^{2} - \frac{1}{2} K x_{i}^{2}, \end{array}

(11)

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\begin{array}{r} p_{i} = \frac{\partial L_{r}}{\partial {\dot{x}}_{i}} = - \frac{α B}{c} ε_{i j} E_{j} . \end{array}

(12)

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\begin{array}{r} H_{r} = \frac{K}{2} x_{i}^{2} - \frac{1}{2} α E_{i}^{2} . \end{array}

(13)

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\begin{array}{r} ϕ_{i}^{(0)} = p_{i} + \frac{α B}{c} ε_{i j} E_{j} \approx 0, \end{array}

(14)

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\begin{array}{r} {ϕ_{i}^{(0)}, ϕ_{j}^{(0)}} = \frac{α ρ B}{c} ε_{i j} . \end{array}

(15)

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\begin{array}{r} J_{r} = ε_{i j} x_{i} p_{j} = \frac{α B}{c} x_{i} E_{i} . \end{array}

(16)

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\begin{array}{r} J_{r} = \frac{α B}{c} x_{i} (E_{i}^{(1)} + E_{i}^{(2)}) = \frac{α B}{c} (k + \frac{ρ}{2} x_{i}^{2}) . \end{array}

(17)

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\begin{array}{r} {x_{i}, x_{j}}_{D} = {x_{i}, x_{j}} - {x_{i}, ϕ_{k}^{(0)}} {ϕ_{k}^{(0)}, ϕ_{l}^{(0)}}^{- 1} {ϕ_{l}^{(0)}, x_{j}} . \end{array}

(18)

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\begin{array}{r} {x_{i}, x_{j}}_{D} = - \frac{c ε_{i j}}{α ρ B} . \end{array}

(19)

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\begin{array}{r} [x_{i}, x_{j}] = - \frac{i ℏ c ε_{i j}}{α ρ B} . \end{array}

(20)

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\begin{array}{r} [J_{r}, H_{r}] = 0, [J_{r}, x_{i}] = i ℏ ε_{i j} x_{j}, \end{array}

(21)

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\begin{array}{r} J_{r n} = \frac{α B k}{c} + (n + \frac{1}{2}) ℏ . \end{array}

(22)

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\begin{array}{r} \bar{L} = \frac{1}{2} M {\dot{x}}_{i}^{2} - \frac{α B}{c} ε_{i j} {\dot{x}}_{i} E_{j}^{(2)} + \frac{1}{2} α {(E_{i}^{(2)})}^{2} - \frac{1}{2} K x_{i}^{2} . \end{array}

(23)

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\begin{array}{r} p_{i} = \frac{\partial \bar{L}}{\partial {\dot{x}}_{i}} = M {\dot{x}}_{i} - \frac{α B}{c} ε_{i j} E_{j}^{(2)} . \end{array}

(24)

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\begin{array}{r} {\bar{L}}_{r} = - \frac{α B}{c} ε_{i j} {\dot{x}}_{i} E_{j}^{(2)} + \frac{1}{2} α {(E_{i}^{(2)})}^{2} - \frac{1}{2} K x_{i}^{2} . \end{array}

(25)

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\begin{array}{r} {\bar{ϕ}}_{i}^{(0)} = p_{i} + \frac{α B}{c} ε_{i j} E_{j}^{(2)} \approx 0. \end{array}

(26)

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\begin{array}{r} {{\bar{ϕ}}_{i}^{(0)}, {\bar{ϕ}}_{j}^{(0)}} = \frac{α B}{c} ρ ε_{i j}, \end{array}

(27)

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\begin{array}{r} \bar{J} = \frac{α B}{c} x_{i} E_{i}^{(2)} = \frac{α ρ B}{2 c} x_{i}^{2} \end{array}

(28)

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\begin{array}{r} \tilde{L} = \frac{1}{2} M {\dot{x}}_{i}^{2} - \frac{α B}{c} ε_{i j} {\dot{x}}_{i} E_{j}^{(1)} + \frac{1}{2} α {(E_{i}^{(1)})}^{2} - \frac{1}{2} K x_{i}^{2} . \end{array}

(29)

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\begin{array}{r} {\tilde{L}}_{r} = - \frac{α B}{c} ε_{i j} {\dot{x}}_{i} E_{j}^{(1)} + \frac{1}{2} α {(E_{i}^{(1)})}^{2} - \frac{1}{2} K x_{i}^{2} . \end{array}

(30)

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\begin{array}{r} {\tilde{H}}_{r} = - \frac{1}{2} α {(E_{i}^{(1)})}^{2} + \frac{1}{2} K x_{i}^{2} . \end{array}

(31)

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\begin{array}{r} p_{i} = \frac{\partial {\tilde{L}}_{r}}{\partial {\dot{x}}_{i}} = - \frac{α B}{c} ε_{i j} E_{j}^{(1)} . \end{array}

(32)

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\begin{array}{r} {\tilde{ϕ}}_{i}^{(0)} = p_{i} + \frac{α B}{c} ε_{i j} E_{j}^{(1)} \approx 0. \end{array}

(33)

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\begin{array}{r} {\tilde{ϕ}}_{i}^{(1)} = {{\tilde{ϕ}}_{i}^{(0)}, H} = - \frac{α k}{r^{2}} E_{i}^{(1)} - K x_{i} \approx 0. \end{array}

(34)

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\begin{array}{r} L = \frac{1}{2} M {\dot{x}}_{i}^{2} - \frac{α B}{c} A_{i}^{e f f} {\dot{x}}_{i} + \frac{1}{2} α E_{i}^{2} - \frac{1}{2} K x_{i}^{2} . \end{array}

()

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Jian Jing, Yao-Yao Ma, Qiu-Yue Zhang, Qing Wang, Shi-Hai Dong. Simulation of anyons by cold atoms with induced electric dipole moment[J]. Chinese Physics B, 2020, 29(8):

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