Liouvillian exceptional points of any order in dissipative linear bosonic systems: Coherence functions and switching between $\mathcal{PT}$ and anti-$\mathcal{PT}$ symmetries

Liouvillian exceptional points of any order in dissipative linear bosonic systems: Coherence functions and switching between $PT$ and anti- $PT$ symmetries

Ievgen I. Arkhipov, Adam Miranowicz, Fabrizio Minganti, and Franco Nori

Phys. Rev. A 102, 033715 – Published 11 September 2020

Abstract

Usually, when investigating exceptional points (EPs) of an open Markovian bosonic system, one deals with spectral degeneracies of a non-Hermitian Hamiltonian (NHH), which can correctly describe the system dynamics only in the semiclassical regime. A recently proposed quantum Liouvillian framework [Minganti et al., Phys. Rev. A 100, 062131 (2019)] enables the complete determination of the dynamical properties of such systems and their EPs (referred to as Liouvillian EPs, or LEPs) in the quantum regime by taking into account the effects of quantum jumps, which are ignored in the NHH formalism. Moreover, the symmetry and eigenfrequency spectrum of the NHH become a part of much larger Liouvillian eigenspace. As such, the EPs of an NHH form a subspace of the LEPs. Here we show that once an NHH of a dissipative linear bosonic system exhibits an EP of a certain finite order $n$ , it immediately implies that the corresponding LEP can become of any higher order $m \geq n$ defined in the infinite Hilbert space. These higher-order LEPs can be identified by the coherence and spectral functions at the steady state. The coherence functions can offer a convenient tool to probe extreme system sensitivity to external perturbations in the vicinity of higher-order LEPs. As an example, we study a linear bosonic system of a bimodal cavity with incoherent mode coupling to reveal its higher-order LEPs; particularly, of second and third order via first- and second-order coherence functions, respectively. Accordingly, these LEPs can be additionally revealed by squared and cubic Lorentzian spectral lineshapes in the power and intensity-fluctuation spectra. Moreover, we demonstrate that these EPs can also be associated with spontaneous parity-time ( $PT$ ) and anti- $PT$ -symmetry breaking in the system studied. These symmetries can be switched in the output fields (the so-called supermodes) of an additional linear coupler with a properly chosen coupling strength. Thus, we show that the initial loss-loss dynamics for the supermodes can be equivalent to the balanced gain-loss evolution.

Received 5 June 2020
Accepted 25 August 2020

DOI:https://doi.org/10.1103/PhysRevA.102.033715

Physics Subject Headings (PhySH)

Open quantum systems PT-symmetric quantum mechanics Quantum fluctuations & noise Quantum optics

Microcavity & microdisk lasers Optical microcavities Quantum cavities

Atomic, Molecular & Optical

Authors & Affiliations

Ievgen I. Arkhipov ^1,*, Adam Miranowicz ^2,3,†, Fabrizio Minganti ^3,‡, and Franco Nori ^3,4,§

¹Joint Laboratory of Optics of Palacký University and Institute of Physics of CAS, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic
²Faculty of Physics, Adam Mickiewicz University, PL-61-614 Poznań, Poland
³Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
⁴Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA

^*ievgen.arkhipov@upol.cz
^†miran@amu.edu.pl
^‡fabrizio.minganti@riken.jp
^§fnori@riken.jp

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Vol. 102, Iss. 3 — September 2020

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Figure 1
Real (red solid curves) and imaginary (blue dashed curves) parts of the eigenfrequencies $ν_{1}$ and $ν_{2}$ of the effective NHH ${\hat{H}}_{eff}$ versus the incoherent coupling rate $γ_{12}$ . These eigenfrequencies constitute a subset of the eigenfrequencies of the Liouvillian $L$ , of a bimodal cavity according to Eq. (23). The chosen system parameters are the frequencies of the modes $ω_{2} = - ω_{1} = 0.5$ [arbitrary units]; the cavity losses for both modes ${\hat{a}}_{1}$ and ${\hat{a}}_{2}$ are $γ = 3$ [arb. units]. The system experiences a spectral-phase transition at the EP $γ_{12}^{{EP}_{1}}$ , according to Eq. (26), due to the interplay between the mode frequency difference $Δ$ and the incoherent mode coupling $γ_{12}$ .
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Figure 2
First-order coherence function $g_{ss}^{(1)} (τ)$ [panels (a)–(c)] and power spectra $S^{(1)} (ω)$ [panels (d)–(f)] of the supermodes ${\hat{c}}_{1}$ (red solid curves) and ${\hat{c}}_{2}$ (blue dashed curves), according to Eqs. (35) and (36), respectively, of the bimodal cavity for various values of incoherent mode coupling rate $γ_{12}$ : (a), (d) $γ_{12} = 0$ [arb. units]; (b), (e) $γ_{12} = γ_{12}^{{EP}_{1}} = 1$ [arb. units]; and (c), (f) $γ_{12} = 2$ [arb. units]. The reference frame is rotating at the central cavity frequency $\bar{ω}$ . The remaining system parameters are the same as in Fig. 1. The spectra of the supermodes ${\hat{c}}_{1}$ and ${\hat{c}}_{2}$ reveal the squared Lorentzian lineshape at the second-order EP, characterized by a plateau at the top of the curve [see panel (e), blue dashed curve]. The inset in panel (e) shows the best Lorentzian fit, represented by a sum of two Lorentzians (gray solid curve) of the spectrum at the EP (blue dashed curve), which highlights the distinctiveness of the squared Lorentzian. Thus, one can experimentally witness the presence of the EP by studying the power spectra for the model under consideration. Moreover, with increasing values of the incoherent mode-coupling rate $γ_{12} > γ_{12}^{{EP}_{1}}$ , the supermode ${\hat{c}}_{2}$ acquires a volcanic cone shape (blue dashed curve), indicating that its spectrum is represented by the difference of two Lorentzians [panel (f)].
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Figure 3
Second-order coherence function $g_{ss}^{(2)} (τ)$ [panels (a)–(c)] and intensity-fluctuation spectra $S^{(2)} (ω)$ [panels (d)–(f)] of the supermodes ${\hat{c}}_{1}$ (red solid curves) and ${\hat{c}}_{2}$ (blue dashed curves), according to Eqs. (15) and (40), respectively, of the bimodal cavity for various values of incoherent mode coupling rate $γ_{12}$ . The system parameters for each panel are the same as in Fig. 2.
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Figure 4
(a) Real and (b) imaginary parts of the eigenvalues $λ$ of the Liouvillian, determined from the evolution matrix $M$ : $λ_{1, 2}$ (green dashed curves), and $λ_{3, 4}$ (gray dash-dotted lines), according to Eq. (48), versus the incoherent coupling rate $γ_{12}$ . The chosen system parameters are the same as in Fig. 1. For comparison, the Liouvillian eigenvalues defined by the NHH from Fig. 1 are shown as blue solid curves. Thus, for a given subspace of the Liouvillian eigenspace, determined by the NHH and evolution matrix $M$ , the EP $γ^{EP}$ in Eq. (26) becomes a LEP of the second and third orders, respectively.
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Physical Review A

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Liouvillian exceptional points of any order in dissipative linear bosonic systems: Coherence functions and switching between $PT$ and anti- $PT$ symmetries

Ievgen I. Arkhipov, Adam Miranowicz, Fabrizio Minganti, and Franco Nori

Phys. Rev. A 102, 033715 – Published 11 September 2020

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Physical Review A

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Liouvillian exceptional points of any order in dissipative linear bosonic systems: Coherence functions and switching between PT and anti-PT symmetries

Ievgen I. Arkhipov, Adam Miranowicz, Fabrizio Minganti, and Franco Nori

Phys. Rev. A 102, 033715 – Published 11 September 2020

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Liouvillian exceptional points of any order in dissipative linear bosonic systems: Coherence functions and switching between $PT$ and anti- $PT$ symmetries