Dynamically generated synthetic electric fields for photons

Open Access

Dynamically generated synthetic electric fields for photons

Petr Zapletal, Stefan Walter, and Florian Marquardt

Phys. Rev. A 100, 023804 – Published 6 August 2019

Abstract

Static synthetic magnetic fields give rise to phenomena including the Lorentz force and the quantum Hall effect even for neutral particles, and they have by now been implemented in a variety of physical systems. Moving toward fully dynamical synthetic gauge fields allows, in addition, for backaction of the particles' motion onto the field. If this results in a time-dependent vector potential, conventional electromagnetism predicts the generation of an electric field. Here, we show how synthetic electric fields for photons arise self-consistently due to the nonlinear dynamics in a driven system. Our analysis is based on optomechanical arrays, where dynamical gauge fields arise naturally from phonon-assisted photon tunneling. We study open, one-dimensional arrays, where synthetic magnetic fields are absent. However, we show that synthetic electric fields can be generated dynamically. The generation of these fields depends on the direction of photon propagation, leading to a mechanism for a photon diode, inducing nonlinear unidirectional transport via dynamical synthetic gauge fields.

Received 18 July 2018

DOI:https://doi.org/10.1103/PhysRevA.100.023804

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Nonlinear optics Optomechanics

Coupled oscillators Micromechanical & nanomechanical oscillators

Nonlinear DynamicsAtomic, Molecular & Optical

Authors & Affiliations

Petr Zapletal ^1,2, Stefan Walter^2,3, and Florian Marquardt^2,3

¹Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom
²Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany
³Institute for Theoretical Physics, University Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 100, Iss. 2 — August 2019

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Setup exhibiting dynamically generated synthetic electric fields. (a) In general terms: Limit-cycle oscillator assisting the transition between linear modes. (b) Optomechanical realization: A cavity with a movable membrane (green rectangle) in the middle supporting optical supermodes ${\hat{a}}_{1}$ and ${\hat{a}}_{2}$ (mostly localized left and right, respectively). The mechanical mode $\hat{b}$ undergoes limit-cycle oscillations. Photons tunneling (red arrow) from optical mode ${\hat{a}}_{1}$ to ${\hat{a}}_{2}$ absorb a phonon from the mechanical oscillation, thereby acquiring a phase shift set by the oscillation phase $ϕ$ . Photon transport through the setup can be probed by driving mode ${\hat{a}}_{1}$ or ${\hat{a}}_{2}$ . Optical frequencies are represented by the blue lines. (c) A one-dimensional array, with optical modes ${\hat{a}}_{j}$ of increasing frequency. Mechanical modes ${\hat{b}}_{j}$ assist tunneling between modes ${\hat{a}}_{j}$ and ${\hat{a}}_{j + 1}$ . Some mode, ${\hat{a}}_{d}$ , is driven by a laser (blue arrow), probing photon transport both toward the left and right.
Reuse & Permissions
Figure 2
Dynamically generated synthetic electric fields in the two-site system. (a), (b) Phase evolution on the mechanical limit cycle (green orbit). (a) When the higher-frequency optical mode is laser-driven, the system settles into a stable fixed point, with a phase lag $ϕ - θ = - π / 2$ . (b) When the lower-frequency optical mode is driven, the phase is continuously repelled from an unstable fixed point $ϕ - θ = + π / 2$ , generating a finite synthetic electric field $E = \dot{ϕ} \neq 0$ acting on the photons. (c) The phase diagram. In the white region, $E$ vanishes in the steady state. If the lower-frequency mode, $a_{1}$ , is driven, $E$ bifurcates in the colored region to finite steady-state values. Their absolute values are indicated by the color scale. The blue insets show the effective potential $V (E)$ determining the steady-state value of $E$ . The dashed black line denotes the cut along which $E$ and the optical transmission $T$ are plotted in (d) and (e), respectively. For the higher-frequency mode, $a_{2}$ , being driven, $E$ always vanishes for any values of the system parameters. Consequently, the transmission is never suppressed.
Reuse & Permissions
Figure 3
Light transport in a 1D array with dynamical gauge fields and the generation of a barrier for photon transport induced by synthetic electric fields. (a) The optical amplitudes $| a_{j} |$ as a function of position for different values of the laser amplitude $E$ and $B J / κ = 1$ (shown on a logarithmic scale). The dashed red line denotes the driven site. Transport to the right is strongly suppressed. (b) The ratio $R = | a_{d - 1} / a_{d + 1} |^{2}$ of the optical amplitudes adjacent to the driven site $j = d$ . (Plotted for $n = 81$ sites, site $d = 40$ being driven.)
Reuse & Permissions
Figure 4
The potential for the synthetic electric field. It has a single minimum at $E = 0$ for $k = 2$ (dashed lines) when the higher optical frequency is driven. For $k = 1$ , when the lower optical frequency is driven, the stationary value $E = 0$ becomes unstable with increasing $E J / κ^{2}$ as two minima with a finite frequency shift emerge (solid lines). (The potential is plotted for $B J / κ = 0.5$ .)
Reuse & Permissions
Figure 5
Dynamically generated synthetic electric fields in the two-site system considering the model described by the fundamental Hamiltonian (A1). (a) For the uncoupled optical mode, $a_{L}$ , being driven with a laser at frequency $ν_{1}$ (solid lines), a large synthetic electric field $E$ develops. For driving the uncoupled optical mode, $a_{R}$ , with a laser at frequency $ν_{2}$ (dashed lines), a small synthetic electric field develops; however, it is reduced as the sideband ratio $ω / κ$ increases (see the legend). (b) As a result, the optical transmission $T$ to the right (solid lines) is significantly suppressed in comparison to the transmission to the left (dashed lines). In a one-dimensional array, this transmission ratio gets exponentiated by the length of the array, leading to a very significant suppression of transport in one direction. (Plotted for $B g_{0} / κ = 2, J_{0} / ω = 0.085$ .)
Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information