Spontaneous rotational symmetry breaking in a Kramers two-level system

Mário G. Silveirinha

Phys. Rev. B 100, 165146 – Published 30 October 2019

Abstract

Here, I develop a model for a two-level system that respects the time-reversal symmetry of the atom Hamiltonian and the Kramers theorem. The two-level system is formed by two Kramers pairs of excited and ground states. It is shown that due to the spin-orbit interaction it is in general impossible to find a basis of atomic states for which the crossed transition dipole moment vanishes. The parametric electric polarizability of the Kramers two-level system for a definite ground state is generically nonreciprocal. I apply the developed formalism to study Casimir-Polder forces and torques when the two-level system is placed nearby either a reciprocal or a nonreciprocal substrate. In particular, I investigate the stable equilibrium orientation of the two-level system when both the atom and the reciprocal substrate have symmetry of revolution about some axis. Surprisingly, it is found that when chiral-type dipole transitions are dominant the stable ground state is not the one in which the symmetry axes of the atom and substrate are aligned. The reason is that the rotational symmetry may be spontaneously broken by the quantum vacuum fluctuations, so that the ground state has less symmetry than the system itself.

Received 22 July 2019
Revised 11 October 2019

DOI:https://doi.org/10.1103/PhysRevB.100.165146

Physics Subject Headings (PhySH)

Plasmonics Quantum optics Spontaneous symmetry breaking

T-symmetry

Atomic, Molecular & OpticalGeneral PhysicsCondensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Mário G. Silveirinha ^*

University of Lisbon–Instituto Superior Técnico and Instituto de Telecomunicações, Avenida Rovisco Pais, 1, 1049-001 Lisboa, Portugal

^*mario.silveirinha@co.it.pt

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Issue

Vol. 100, Iss. 16 — 15 October 2019

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Images

Figure 1
(a) A two-level system is formed by a minimum of four eigenstates, linked in pairs by the time-reversal operator $T$ (Kramers pairs). The black arrows indicate the nontrivial dipole moment downward transitions ( $γ_{g, e}$ ). The upward transition elements are $γ_{e, g} = γ_{g, e}^{*}$ (not shown in the figure). (b) A two-level system formed by two Kramers pairs is placed at a distance $d$ from a planar material substrate.
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Figure 2
Interaction tensor evaluated for imaginary frequencies ( $ω = i ξ$ ) for $d = 0.01 c / ω_{sp}$ and $Γ_{m} = 0.1 ω_{sp}$ . Solid lines: Exact result [Eq. (C4)]. Dashed lines: Quasistatic approximation [Eq. (14)]. The plot is nearly unchanged for smaller values of the distance $d$ .
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Figure 3
Normalized interaction energy for a two-level system formed by two Kramers pairs placed nearby a gyrotropic magnetized plasma with $ω_{c} = 0.5 ω_{p}$ . (a) $E_{int}^{}$ as a function of the atomic transition frequency for (i) $γ_{d} = γ_{d} (\hat{x} \pm i \hat{y}) / \sqrt{2}$ and an arbitrary ground state (H-pol); (ii) $γ_{d} = γ_{d} (\hat{x} - i \hat{z}) / \sqrt{2}$ (V-pol) and for the states $| g_{1} 〉$ (solid green line), $(| g_{1} 〉 + e^{i δ} | g_{2} 〉) / \sqrt{2}$ (solid black line), and $| g_{2} 〉$ (dashed green line). (b) $E_{int}^{}$ as a function of the direction $φ$ of the atom polarization plane for the stationary states of the interacting system and $ω_{0} = 0.7 ω_{p}$ .
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Figure 4
Normalized Casimir energy as a function of the orientation of the symmetry axis of a Kramers two-level atom with $ω_{0} = 0.1 ω_{sp}$ . The atom symmetry axis orientation is determined by the spherical coordinates $θ$ and $φ$ . In the lower panels, the angle $θ$ is represented as a radial coordinate in the interval $0^{\circ} < θ < 90^{\circ}$ . The Casimir torque acts to push the system state to the valley region. (a) $γ_{c} = 0$ (linear polarization dominates), corresponding to an unbroken rotation symmetry. The atom symmetry axis is vertical with respect to the interface (upper panel). (b) $γ_{c} = 2 γ_{d}$ (circular polarization dominates), corresponding to a spontaneously broken rotation symmetry. The atom symmetry axis is horizontal with respect to the interface (upper panel).
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Figure 5
(a) Normalized Casimir torque acting on a Kramers two-level atom as a function of the orientation $θ$ of the atom symmetry axis with respect to the $z$ direction. The atom transition frequency is $ω_{0} = 0.1 ω_{sp}$ . Solid black line: $γ_{c} = 0$ ; dashed black line: $γ_{c} = γ_{d}$ ; green line: $γ_{c} = \sqrt{2} γ_{d}$ ; dashed blue line: $γ_{c} = \sqrt{3} γ_{d}$ ; solid blue line: $γ_{c} = 2 γ_{d}$ . The stable equilibrium orientation is $θ = 0^{\circ}, 180^{\circ}$ for the black lines (unbroken rotational symmetry) and $θ = 90^{\circ}$ for the blue lines (spontaneously broken symmetry). ( $b i$ ) Normalized Casimir free energy as a function of the orientation of the atom symmetry axis for $γ_{c} = 0$ and $ω_{0} = 0.1 ω_{sp}$ . Dashed black line: zero-temperature limit; black line: $ω_{T} = ω_{0}$ ; blue line: $ω_{T} = 2 ω_{0}$ ; green line: $ω_{T} = 3 ω_{0}$ . The arrow indicates the direction of increasing temperature. ( $b ii$ ) Similar to ( $b i$ ) but for $γ_{c} = 2 γ_{d}$ .
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