Fermionic formalism for driven-dissipative multilevel systems

Yulia Shchadilova, Mor M. Roses, Emanuele G. Dalla Torre, Mikhail D. Lukin, and Eugene Demler

Phys. Rev. A 101, 013817 – Published 16 January 2020

Abstract

We present a fermionic description of nonequilibrium multilevel systems. Our approach uses the Keldysh path-integral formalism and allows us to take into account periodic drives, as well as dissipative channels. The technique is based on the Majorana fermion representation of spin-1/2 models which follows earlier applications in the context of spin and Kondo systems. We apply this formalism to problems of increasing complexity: a dissipative two-level system, a driven-dissipative multilevel atom, and a generalized Dicke model describing many multilevel atoms coupled to a single cavity. We compare our theoretical predictions with recent QED experiments and point out the features of a counterlasing transition. Our technique provides a convenient and powerful framework for analyzing driven-dissipative quantum systems, complementary to other approaches based on the solution of Lindblad master equations.

Received 6 August 2019

DOI:https://doi.org/10.1103/PhysRevA.101.013817

Physics Subject Headings (PhySH)

Nonequilibrium statistical mechanics Quantum optics Superradiance & subradiance

Dicke model Large-N expansion in field theory Nonequilibrium Green's function Quantum master equation

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Yulia Shchadilova¹, Mor M. Roses², Emanuele G. Dalla Torre ², Mikhail D. Lukin¹, and Eugene Demler^1,*

¹Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
²Department of Physics, Bar Ilan University, Ramat Gan 5290002, Israel

^*demler@physics.harvard.edu

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Vol. 101, Iss. 1 — January 2020

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Images

Figure 1
Sketch of the open quantum systems described in this paper. (a) The two-level system examined in Sec. 2, coupled by dissipative channels (green wiggly arrows). (b) The four-level atom (W scheme) considered in Sec. 3a, including pumping fields (red arrows) and dissipative processes. (c) The Dicke model considered in Sec. 4, describing the coupling between many driven-dissipative atoms and the quantized field of an optical cavity.
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Figure 2
Self-energies for the $f$ and $η$ fermions, $Σ_{f}$ and $Σ_{η}$ . Solid lines correspond to the Dirac fermion Green's function and dashed lines to the Majorana fermion Green's functions; wiggly lines represent the Green's functions of the bosonic bath.
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Figure 3
Self-energy contributions to the Green's functions of the cavity photons, $Π_{a}$ , the Majorana fermions, $Σ_{η, n}$ , and the Dirac fermions, $Σ_{f, n}$ .
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Figure 4
Critical line of the generalized Dicke model with dissipation. Different colors represent different parameters of the system. We considered the cavity mode with frequency $ω_{c} = 100 kHz$ and dissipation $κ = 100 kHz$ , with the two-level splitting $ω_{z} = 77.2 kHz$ . The solid blue line corresponds to the case $Γ = 0$ , $s_{z} = - 0.5$ . The dashed red line is $Γ = 0$ , $s_{z} = - 0.25$ and the dotted dashed green line $Γ = 50 kHz$ , $s_{z} = - 0.25$ .
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Figure 5
Left panel: stability diagram of the generalized Dicke model with dissipation and cavity losses. Here we use the same parameters as in Fig. 6. Colors decode different phases which we characterize by the position of the poles of the cavity photon's Green's function $D_{a}^{R} (ω)$ (53). Phase (0) is white in the main figure. Positions of the poles corresponding to different phases are depicted in the right panel (see also Table 1 for details). Phases (0 and 1) are stable; phases (2–6) are unstable. The Dicke-type transition obtained from Eq. (55) is shown with a solid line. The “counterlasing” instability calculated from Eq. (57) is shown with an arrow.
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Figure 6
Comparison between (a) experimental and (b),(c) theoretical phase transition diagrams for the generalized Dicke system. N, SR, and U denotes normal, superradiant, and unstable phases correspondingly. Parameters used for theoretical calculation correspond to the experimental data: cavity mode frequency $ω_{c} = 100 kHz$ , dissipation $κ = 107 kHz$ , and energy splitting $ω_{z} = 77.2 kHz$ . In theoretical calculations, we consider that the atomic interaction with the incoherent bath partially polarizes the system, $s_{z} = - 0.25$ , and the strength of dissipation is $Γ = 30 kHz$ (b) and $Γ = 0 kHz$ (c).
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