**Abstract**
We apply a computationally-efficient method to study the superradiant evolution of a set of N two-level systems spontaneously radiating under the effect of phase-breaking mechanisms [1]. We investigate the dynamics generated by non-radiative losses and pure dephasing, and their interplay with spontaneous emission, comparing the exact method to analytical approximations.

The computationally-efficient tool can solve in general the dynamics of large ensembles of N two-level systems evolving under the action of local Lindblad dissipation terms that are permutational invariant [2-5]. The method keeps the computational resources scaling as O(N^2) instead of having them grow exponentially like 2^(2N). I will discuss how this
implementation relates to several other works that are exploiting permutational invariance to reduce the resources of simulation [6-8]. The code is written in Python and readily available for use in other phenomena, such as spin squeezing and phase transitions.

References

[1] N. Shammah, N. Lambert, F. Nori, and S. De Liberato, Superradiance with local phase-breaking effects, arXiv:1704.07066v1 (2017).

[2] B. A. Chase and J. M. Geremia, Collective processes of an ensemble of spin-1 particles, Phys. Rev. A 78, 052101 (2008).

[3] B. Q. Baragiola, B. A. Chase, and J. Geremia, Collective uncertainty in partially polarized and partially decohered spin-1 systems, Phys. Rev. A 81, 032104 (2010).

[4] T. Moroder, P. Hyllus, G. Toth, C. Schwemmer, A. Niggebaum, S. Gaile, O. Guhne, and H. Wein- furter, Permutationally invariant state reconstruction, New Journal of Physics 14, 105001 (2012).

[5] F. Damanet, D. Braun, and J. Martin, Cooperative spontaneous emission from indistinguishable atoms in arbitrary motional quantum states, Phys. Rev. A 94, 033838 (2016).

[6] P. Kirton and J. Keeling, Suppressing and restoring the Dicke superradiance transition by dephasing and decay, Phys. Rev. Lett. 118, 123602 (2017).

[7] Z.-X. Gong, M. Xu, M. Foss-Feig, J. K. Thompson, A. M. Rey, M. Holland, and A. V. Gorshkov, Steady-state superradiance with Rydberg polaritons, arXiv:1611.00797 (2016).

[8] M. Gegg and M. Richter, PsiQuaSP - a library for efficient computation of symmetric open quantum systems, arXiv:1707.01079 (2017).