Free Space QED and Emergent Universal Dynamics with a Single Rydberg SuperatomJan Kumlin1, Sebastian Ho�erberth2, and Hans Peter Büchler1

1Institute for Theoretical Physics III, University of Stuttgart, Germany2Department of Physics, Chemistry and Pharmacy, University of Southern Denmark, Odense, Denmark

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Rydberg blockade

The interaction of a single photon with an individual two-level system is the textbook example of quantum electrodynamics. Achieving strong coupling in this system has so far required con�nement of the light �eld inside resonators of waveguides. Here, we present the theoretical background of the experimental demonstration of strong coherent coupling between a single Rydberg superatom and a propagating light pulse containing only a few photons [1]. We also study the in�uence of the coherent exchange interactions between the atoms inside the superatom mediated by exchange of virtual photons in a simple one-dimensional setup. In contrast to the naive expectation of fast dephasing of the bright state due to exchange interactions, we �nd oscillations and dephasing on a time scale that grows with the number of atoms inside the superatom. Further, we analyze this behaviour analytically using perturbation theory �nding excellentagreement with our numerical simulations.

Setup and Model Master Equation and Dynamical Phase Diagram

coherent exchange interaction (Lamb shift)

Lindblad dissipator (spontaneous decay)

dimensionless parameters of the problem

with

evaluate unequal-time correlation functions within the Born-Markov approximation

time evolution of the density matrix

Quantum Regression Theorem is exact for a single two-level emit-ter [3]

compare result for 2nd order correlation function with exact so-lution using Bethe ansatz [4] for

propagator for density matrix

correlation functions

Rydberg blockade leads to collective states

driven Rydberg atoms

directional emission into forward propagating mode due to collective excitations (averaged over atomic positions)

enhanced decay in forward direction with an opening angle

backscattering suppressed by for a Gaussiandensity distribution with width

de�ne e�ective 1D model with coupling

normal decay rate into free space

bright state:

dark states:

collective enhancement of decay into speci�c mode

derive master equation from e�ective Hamiltonian by integrating outphotonic degrees of freedom for coherent input �eld

open system‘s dynamics of single two-level system

intrinsically damped system due to enhanced decay into forward propagating mode (other decay channelscan be put in phenomenologically)

relate outgoing electric �eld to the dynamics of the atom

imprint correlations onto outgoing �eld

coupling parameter

mean photon number

We study the in�uence of interactions between the atoms inside the superatom mediated by exchange of virtual photons within a simple one-dimensional waveguide model. The atomic positions are assumed to be randomly distributed with the density distribution given by with a characteristic width .

Incoherent DynamicsCoherent Dynamics - Exchange Interaction Hamiltonian

enhanced spontaneous emission rate into forward propagating mode

decay into backward propagating mode (after averaging over atomic positions)

Two Analytical Limits

for smooth density distribution

FT of density distribution

expect fast dephasing out of state by coherent exchange Hamiltonian on time scale

Waveguide Model [5] Emergent Universal Dynamics

integrate out photonic degrees of freedom

Analytical Calculation of the Spectrum

split with

is Poisson distributed

dephasing on time scale

become eigenstates of full

odd

Eigenstates ofSpectrum of and

since only values with small have signi�cant overlap with and have a characteristic energy scale , one can perform degenerate perturbation theory for small

numerical simulations of coherent dynamics show oscillations on time scale which are damped out on time scale

Spectrum in Perturbation Theory

Free Space QED with a Single Rydberg Superatom [1]

In�uence of the Interactions - Emergent Universal Dynamics

Introduction

kumlin@itp3.uni-stuttgart.deGerman-Israeli Minerva School DIP QUANTUM, Jerusalem, October 23- 25, 2017

Quantum Regression Theorem [2]

[1] A. Paris-Mandoki et al., Phys. Rev. X 7, 041010 (2017)[2] M. Lax, Phys. Rev. 129, 2342 (1963)[3] T. Shi et al., Phys. Rev. A 92, 053834 (2015)[4] V. I. Yudson and P. Reineker, Phys. Rev. A 78, 052713 (2008)[5] H. Pichler et al., Phys. Rev. A 91, 042116 (2015)

exact result simulated resultdynamical phase diagram

: waist of Gaussian beam

analytical calculation of

excellent agreement with numerical simulations

saturation at due to equal distribution onto + and - eigenstates

Phys. Rev. X 7, 041010 (2017)

photon-photon correlationsfor damped Rabi oscillations

in general

and �nite and