Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps

Fabrizio Minganti, Adam Miranowicz, Ravindra W. Chhajlany, and Franco Nori

Phys. Rev. A 100, 062131 – Published 26 December 2019

Abstract

Exceptional points (EPs) correspond to degeneracies of open systems. These are attracting much interest in optics, optoelectronics, plasmonics, and condensed matter physics. In the classical and semiclassical approaches, Hamiltonian EPs (HEPs) are usually defined as degeneracies of non-Hermitian Hamiltonians such that at least two eigenfrequencies are identical and the corresponding eigenstates coalesce. HEPs result from continuous, mostly slow, nonunitary evolution without quantum jumps. Clearly, quantum jumps should be included in a fully quantum approach to make it equivalent to, e.g., the Lindblad master equation approach. Thus, we suggest to define EPs via degeneracies of a Liouvillian superoperator (including the full Lindbladian term, LEPs), and we clarify the relations between HEPs and LEPs. We prove two main theorems: Theorem 1 proves that, in the quantum limit, LEPs and HEPs must have essentially different properties. Theorem 2 dictates a condition under which, in the “semiclassical” limit, LEPs and HEPs recover the same properties. In particular, we show the validity of Theorem 1 studying systems which have (1) an LEP but no HEPs and (2) both LEPs and HEPs but for shifted parameters. As for Theorem 2, (3) we show that these two types of EPs become essentially equivalent in the semiclassical limit. We introduce a series of mathematical techniques to unveil analogies and differences between the HEPs and LEPs. We analytically compare LEPs and HEPs for some quantum and semiclassical prototype models with loss and gain.

Received 27 June 2019

DOI:https://doi.org/10.1103/PhysRevA.100.062131

Physics Subject Headings (PhySH)

Open quantum systems Open quantum systems & decoherence Quantum optics

Atomic, Molecular & OpticalStatistical Physics & ThermodynamicsCondensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Fabrizio Minganti ^1,*, Adam Miranowicz ^1,2,†, Ravindra W. Chhajlany ^1,2,‡, and Franco Nori ^1,3,§

¹Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
²Faculty of Physics, Adam Mickiewicz University, PL-61-614 Poznan, Poland
³Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA

^*fabrizio.minganti@riken.jp
^†adam@riken.jp
^‡ravi@amu.edu.pl
^§fnori@riken.jp

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 100, Iss. 6 — December 2019

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Spectral properties of the Liouvillian in Eq. (36) in the case of $γ_{y} = 2 ω$ . This Liouvillian has a LEP, but the associated NHH has no HEP. The thick curves represent $γ_{-} = 0$ , while the thin dashed ones are for $γ_{-} = ω$ . (a) Real and (b) imaginary parts of the Liouvillian eigenvalues in Eq. (38) as a function $γ_{x} / ω$ , i.e., the dissipation in the ${\hat{σ}}_{x}$ direction rescaled by the qubit energy $ω$ . (c) Scalar product between the two eigenmatrices ${\hat{ρ}}_{1}$ and ${\hat{ρ}}_{2}$ in Eq. (39) as a function of $γ_{x} / ω$ .
Reuse & Permissions
Figure 2
Spectral properties of the NHH in Eq. (42) showing an HEP. (a) Imaginary and (b) real parts of the eigenvalues in Eq. (43) as a function of $γ_{-} / ω_{x}$ : the ratio between the spin-flip rate and the drive. (c) Scalar product between $| ϕ_{1} 〉$ and $| ϕ_{2} 〉$ , given in Eq. (44), as a function of $γ_{-} / ω_{x}$ .
Reuse & Permissions
Figure 3
Spectral properties of the Liouvillian in Eq. (41), exhibiting a LEP, but for the same parameters as the NHH studied in Fig. 2. (a) Real part and (b) imaginary parts of the Liouvillian eigenvalues in Eq. (45) as a function of $γ_{-} / ω_{x}$ . Note that according to Eq. (5), one should compare $R e [λ_{i}]$ with $I m [h_{i}]$ in Fig. 2, and $I m [λ_{i}]$ with $R e [h_{i}]$ in Fig. 2. (c) Scalar product between the two eigenmatrices ${\hat{ρ}}_{2}$ and ${\hat{ρ}}_{3}$ , given in Eq. (46), as a function of $γ_{-} / ω_{x}$ .
Reuse & Permissions
Figure 4
Expectation value of $〈 {\hat{σ}}_{z} 〉$ as a function of $γ_{-} / ω_{x}$ . In both panels the dashed black curves represent the mean value of photons in the steady state. (a) NHH eigenvectors [Eq. (44)]. (b) Eigenvectors obtained via the spectral decomposition of ${\hat{ρ}}_{3, 4}$ [cf. Eq. (48)].
Reuse & Permissions
Figure 5
Spectral properties of the NHH in Eq. (51). (a) Imaginary and (b) real parts of the eigenvalues as a function of $g / ω$ , i.e., the coupling between the cavities rescaled by the frequency of each cavity. (c) Scalar product between the eigenvectors associated with the EPs as a function of $g / ω$ . Here, the parameters used are $γ_{a} = ω$ , $γ_{b} = ω / 2$ .
Reuse & Permissions
Figure 6
Spectral properties of the full Liouvillian in Eq. (50). (a) Real and (b) imaginary parts of the eigenvalues as a function of $g / ω$ . Note that according to Eq. (5), one should compare $R e [λ_{i}]$ with $I m [h_{i}]$ in Fig. 5, $I m [λ_{i}]$ with $R e [h_{i}]$ in Fig. 5. (c) Scalar product between the eigenvectors associated with the EP as a function of $g / ω$ . We were not able to extract this quantity for all the values of $g / ω$ due to numerical problems in correctly sorting the eigenvectors. Indeed, the eigenmatrices of $L$ no longer obey the total-excitation conservation law valid for the NHH. Here the parameters used are $γ_{a} = ω$ , $γ_{b} = ω / 2$ .
Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information