Model of time-dependent geometric graph for dynamical Casimir effect
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ORIGINAL PAPER
Model of time-dependent geometric graph for dynamical Casimir effect I S Lobanov, D S Nikiforov, I Y Popov*
, A I Trifanov and E S Trifanova
ITMO University, Kronverkskiy pr. 49, Saint Petersburg, Russia 197101 Received: 13 October 2019 / Accepted: 07 May 2020
Abstract: Geometric graph model is suggested for the dynamical Casimir effect. The wave equation is considered at the graph edges and the Kirchhoff condition at the internal vertex. It is assumed that the edge lengths depend on time. Photon generation is considered in the framework of the Dodonov–Klimov model. Keywords: Quantum graph; Spectrum; Time evolution
1. Introduction The Casimir effect is a purely quantum phenomenon which can be explained by the following example. Consider two uncharged conductive plates in a vacuum, placed a few nanometers apart. In a classical description, the lack of an external field means that there is no field between the plates, and no force would be measured between them. In quantum electrodynamics, the vacuum can produce virtual photons and the plates are affected by the virtual photons which constitute the field and generate a net force—either an attraction or a repulsion depending on the specific arrangement of the two plates [1–3]. The dynamical Casimir effect is the production of particles and energy from an accelerated moving mirror. This reaction was predicted by certain numerical solutions to quantum mechanics equations made in the 1970s [4]. In 2011 the dynamical Casimir effect was detected. In the experiment, microwave photons were generated out of the vacuum in a superconducting microwave resonator [5]. As for theoretical models, there are models of box with moving walls or varying boundary conditions [6–9]. It was pointed out in [6] that the problem of 1D box with a moving wall can be mapped onto that of an harmonic oscillator with time-dependent frequency confined inside the static box. We use a model based on metric graphs with edges having time-depending lengths. Using the quantum graph, one can vary the spectrum of the system essentially and can make it close to the spectrum of the real system under consideration. Although the metric graph model is
well developed (see, e.g., [10]), there are only a few works dealing with time-dependent graphs [11–14]. As for photon generation, we use the ideas of [7]. The structure of the present paper is as follows. At first, we describe the model of metric graph with time-depending edges. The last part of the paper is devoted to the photon generation.
2. Stationary problem Let us consider the stationary problem for star-like quantum graph C with N leads. An eigenfunction / satisfies the following equation on j-th edge
d2 / ðyÞ ¼ k2 /j ðyÞ; 0 y lj ; j ¼ 1; 2; . . .; N; dy2 j
ð1Þ
where /j is the value of / on j-th edge. The following conditions take place at the graph vertices: 8 /1 ð0Þ ¼ /2 ð0Þ ¼ ¼ /N ð0Þ ¼ 0; > > > < / ðl Þ ¼ / ðl Þ ¼ ¼ / ðl Þ; 1 1 2 2 N N ð2Þ N P > o/ > j > : oy ðlj Þ ¼ 0: j¼1
Conditions (2) lead to the spectral eq
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