Random-matrix theory of amplifying and absorbing resonators with $\mathcal {PT}$ or $\mathcal {PTT}^{\prime }$ symmetry

Birchall, Christopher; Schomerus, Henning

doi:10.1088/1751-8113/45/44/444006

Cited by 12 publications

(14 citation statements)

References 47 publications

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“…Longhi (2010) discovered that a PT -symmetric cavity that acts as a laser must behave simultaneously as a coherent perfect absorber; see also . A random-matrix theory for PT -symmetric cavities has been developed by Birchall and Schomerus (2012). An experimental realization of a PT -symmetric cavity in the microwave and in the optical regime has been reported by (Bittner et al, 2012c) and (Chang et al, 2014;Peng et al, 2014), respectively.…”

Section: B Exceptional Pointsmentioning

confidence: 99%

Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics

2015

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Section: B Exceptional Pointsmentioning

confidence: 99%

Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics

2015

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“…Here we restrict the discussion to purely non-unitary evolution L H = 0, leaving the effect of the Hamiltonian for future studies and thus henceforth we use L = L D . An alternative approach has been considered by studying ergodic many-body Hamiltonians perturbed by a nonhermitian term [26][27][28][29].…”

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confidence: 99%

Hierarchy of Relaxation Timescales in Local Random Liouvillians

2020

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To characterize the generic behavior of open quantum systems, we consider random, purely dissipative Liouvillians with a notion of locality. We find that the positivity of the map implies a sharp separation of the relaxation timescales according to the locality of observables. Specifically, we analyze a spin-1/2 system of size with up to n-body Lindblad operators, which are n-local in the complexity-theory sense. Without locality (n = ), the complex Liouvillian spectrum densely covers a "lemon"-shaped support, in agreement with recent findings [Phys. Rev. Lett. 123, 140403; arXiv:1905.02155]. However, for local Liouvillians (n < ), we find that the spectrum is composed of several dense clusters with random matrix spacing statistics, each featuring a lemon-shaped support wherein all eigenvectors correspond to n-body decay modes. This implies a hierarchy of relaxation timescales of n-body observables, which we verify to be robust in the thermodynamic limit.Introduction. For unitary quantum many-body dynamics, the characterization of generic features common to the vast majority of systems is well developed, in the form of an effective random matrix theory [1-6]. It is for example a crucial ingredient to our understanding of thermalization in unitary quantum systems, manifest in the eigenstate thermalization hypothesis (ETH) [7][8][9][10][11][12][13].For open quantum many-body systems analogous organizing principles are yet missing. Only very recently the first developments in this direction appeared with the investigation of spectral features of a purely random Liouvillian [14-18], describing generic properties of trace preserving positive quantum maps. The purely random Liouvillians considered in [14,15,17] constitute the least structured models aiming at describing generic features of open quantum many-body systems. A main result of these recent studies is that the spectrum of purely random Liouvillians is densely covering a lemon-shaped support whose form is universal, differentiating random Liouvillians from completely random Ginibre matrices with circular spectrum [1,[19][20][21].How close is a completely random Liouvillian to a physical system? In this Letter, we make a step towards answering this question by adding the minimal amount of structure to a purely random Liouvillian, namely a notion of locality. We consider a dissipative spin system of size with Lindblad operators given by Pauli strings classified by their length n ≤ , the latter being a measure of their n-locality in a complexity-theory sense [22].In this language, purely random Liouvillians are completely nonlocal and structureless, since they include the full operator basis including Pauli strings of all lengths 1 ≤ n ≤ .Here, we restrict the maximal number of non-identity operators in the Lindblad operators to n max and examine

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“…The transition can then be investigated by combining the effective models set up in the previous section with random-matrix theory [20,23,40], thus, ensembles of Hamiltonians H (usually composed with random Gaussian matrix elements) or timeevolution operators F (distributed according to a Haar measure) which are only constrained by the symmetries of the problem. Closer inspection identifies two natural scenarios [26,28], described in the following two subsections, and a semiclassical source of corrections to random-matrix theory, discussed thereafter [27]. The interplay of the various time and energy scales is illustrated in figure 5.…”

Section: Mesoscopic Energy and Times Scalesmentioning

confidence: 95%

“…Keeping µ/E T in this limit fixed and finite, the real phase is thus completely destroyed, and the focus turns to the typical decay and growth rates encoded in the imaginary parts Im ω n of the complex eigenvalues. Numerical sampling of the random-matrix ensembles suggests that the distribution P (Im ω n /µ; µ/E T ) attains a stationary limit for M ≫ 1 [26][27][28]. At µ E T , one then finds a finite fraction of strongly amplified states N, N T ≫ 1), where M = /τ ∆ is the number of internal modes mixed by multiple scattering with scattering time τ and N is the number of channels coupling the resonators, with transparency T .…”

Section: Growth and Decay Rates In The Semiclassical Limitmentioning

confidence: 99%

From scattering theory to complex wave dynamics in non-Hermitian PT -symmetric resonators

Schomerus

2013

Phil. Trans. R. Soc. A.

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One contribution of 17 to a Theme Issue 'PT quantum mechanics' . Department of Physics, Lancaster University, Lancaster LA1 4YB, UK I review how methods from mesoscopic physics can be applied to describe the multiple wave scattering and complex wave dynamics in non-Hermitian PTsymmetric resonators, where an absorbing region is coupled symmetrically to an amplifying region. Scattering theory serves as a convenient tool to classify the symmetries beyond the single-channel case and leads to effective descriptions that can be formulated in the energy domain (via Hamiltonians) and in the time domain (via time evolution operators). These models can then be used to identify the mesoscopic time and energy scales that govern the spectral transition from real to complex eigenvalues. The possible presence of magneto-optical effects (a finite vector potential) in multi-channel systems leads to a variant (termed PT T symmetry) that imposes the same spectral constraints as PT symmetry. I also provide multi-channel versions of generalized fluxconservation laws.

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Random-matrix theory of amplifying and absorbing resonators with $\mathcal {PT}$ or $\mathcal {PTT}^{\prime }$ symmetry

Cited by 12 publications

References 47 publications

Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics

Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics

Hierarchy of Relaxation Timescales in Local Random Liouvillians

From scattering theory to complex wave dynamics in non-Hermitian PT -symmetric resonators

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