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Entanglement entropy of coupled harmonic oscillators: an approach in Fock space

M. Jonker

Published Jan 1, 2016• 0 citations

Abstract

It has been known for decades that black holes behave as thermodynamic objects and, as such, have an entropy, which was acquired in exact form as the Bekenstein-Hawking entropy. However, the origin of this entropy, and in particular its proportionality to the area of the horizon, remained unexplained. Motivated by this mystery, several authors examined the possibility of entanglement entropy as a source of black hole entropy, by considering a scalar ?eld in the ground state and\ntracing out the degrees of freedom inside an imaginary sphere, analogous to the interior of the black hole. The resulting entanglement entropy between the two regions turned out to scale with the area of their mutual boundary. In the simplest model the scalar ?eld is represented by a system of coupled harmonic oscillators. After an introduction to the phenomenon of quantum entanglement and the associated entanglement entropy, we will review the problem of two coupled oscillators in the ground state and derive the entanglement entropy using an approach in Fock space, as opposed to the standard method involving integrals in positon space. This will also\nallow us to look at some excited states of the system. Subsequently we will try to extend this approach to a system of N coupled harmonic oscillators and derive the entanglement entropy between an inner and outer region of oscillators.

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It has been known for decades that black holes behave as thermodynamic objects and, as such, have an entropy, which was acquired in exact form as the Bekenstein-Hawking entropy. However, the origin of this entropy, and in particular its proportionality to the area of the horizon, remained unexplained. Motivated by this mystery, several authors examined the possibility of entanglement entropy as a source of black hole entropy, by considering a scalar ?eld in the ground state and\ntracing out the degrees of freedom inside an imaginary sphere, analogous to the interior of the black hole. The resulting entanglement entropy between the two regions turned out to scale with the area of their mutual boundary. In the simplest model the scalar ?eld is represented by a system of coupled harmonic oscillators. After an introduction to the phenomenon of quantum entanglement and the associated entanglement entropy, we will review the problem of two coupled oscillators in the ground state and derive the entanglement entropy using an approach in Fock space, as opposed to the standard method involving integrals in positon space. This will also\nallow us to look at some excited states of the system. Subsequently we will try to extend this approach to a system of N coupled harmonic oscillators and derive the entanglement entropy between an inner and outer region of oscillators.

The entropy of a black hole scales with its area. This is a puzzling feature in the context of statistical mechanics, where entropy scales with volume. It is argued that entanglement entropy underlays such an area law. This quantity is hard to calculate because a precise understanding of the underlying quantum state is needed. Therefore it is interesting to study a Schwarzschild analogue in a photon condensate which is an exactly known quantum system in the ground-state mimicking the behaviour of an astronomic black hole. A harmonic chain of size

N

with a constant spring constant is studied as a toy model, of which the entanglement entropy is calculated by tracing out

N-l

oscillators. This separation of two regions can be interpret as separating the inner and outer regions of a black hole. The resulting entanglement entropy is considered for the discrete chain as well for

N\\rightarrow\\infty

, which represents the Klein-Gordon field on a flat background and with a UV cut-off. For large

N

, the entropy converge to the limit case. The performed numerical simulations hints at an area law with logarithmic corrections, given by

S=\\frac{0.2776}{3} \\ln{\\bigg{(} \\frac{N}{\\pi }\\sin{\\frac{\\pi l}{N}}\\bigg{)}} + 0.6421

in the discrete case and

S=\\frac{0.3744}{3} \\ln{l} + 0.6421

in the infinite case.

We explore the method of entanglement entropy applied to 2d black holes. We introduce a solvable model of a real scalar field with finite volume and lattice spacing in terms of

N

coupled mechanical oscillators and compute its entanglement entropy in many cases. The large

N

limit of this scheme, with finite lattice spacing, should reproduce the Bekenstein-Hawking formula for black hole entropy.

We reexamine the idea that the origin of black-hole entropy may lie in the entanglement of quantum fields between the inside and outside of the horizon. Motivated by the observation that certain modes of gravitational fluctuations in a black-hole background behave as scalar fields, we compute the entanglement entropy of such a field, by tracing over its degrees of freedom inside a sphere. We show that while this entropy is proportional to the area of the sphere when the field is in its ground state, a correction term proportional to a fractional power of area results when the field is in a superposition of ground and excited states. The area law is thus recovered for large areas. Further, we identify the location of the degrees of freedom that give rise to the above entropy.

The area dependence of entanglement entropy of a free scalar field is often understood in terms of coupled harmonic oscillators. In Schrodinger quantization, the Gaussian nature of ground state wave function for these oscillators is sufficient to provide the exact form of the reduced density matrix and its eigenvalues, thus giving the entanglement entropy. However, in polymer quantization the ground state is not Gaussian and the formalism that can provide the exact analytical form of the reduced density matrix is not yet known. In order to address this issue, here we treat the interaction between two coupled harmonic oscillators in the perturbative approach and evaluate the entanglement entropy in Schrodinger and polymer quantization. Contrary to Schrodinger quantization, we show that in the high frequency regime the entanglement entropy decreases for polymer quantization, keeping the ratio of coupling strength to the square of individual oscillator frequency fixed. Furthermore for the free scalar field we validate the area dependence of entanglement entropy in Fock quantization and demonstrate that polymer quantization produces a similar area law.

It is known that the entanglement entropy of a scalar field, found by tracing over its degrees of freedom inside a sphere of radius [Formula: see text], is proportional to the area of the sphere (and not its volume). This suggests that the origin of black-hole entropy, also proportional to its horizon area, may lie in the entanglement between the degrees of freedom inside and outside the horizon. We examine this proposal carefully by including excited states, to check probable deviations from the area law.PACS Nos.: 04.60.m, 04.62, 04.70.s, 03.65.Ud