Analog quantum gravity effects in dielectrics. Carlos H. G. Bessa (UFPB-Brazil) ICTP-SAIFR April 2017

Transcription

1 Analog quantum gravity effects in dielectrics Carlos H. G. Bessa (UFPB-Brazil) In collaboration with V. A. De Lorenci (UNIFEI-Brazil), L. H. Ford (TUFTS-USA), C. C. H. Ribeiro (USP-Brazil) and N. F. Svaiter (CBPF-Brazil) ICTP-SAIFR April 2017

2 Light Cone Fluctuation as a Quantum Gravity Effect Gravitons in a squeezed state induce metric fluctuations Metric fluctuations induce light-cone fluctuations Deser, Rev. Mod. Phys. (1957) DeWitt, Phys. Rev. Lett. (1964) Ford, Phys. Rev. D (1995) h μν is an operator < h uv (x)>=0 <h uv (x)h uv (x')> 0 An observer at a distance r from the source detects pulses whose spacing varies by Δt For a pulse which is delayed or advanced by the time Δt Ford, Phys. Rev. D (1995)

3 Analog quantum gravity effects in dielectrics BASIC IDEA: 1) A probe pulse traverses a slab of nonlinear optical material 2) The pulse interacts with the ambient vacuum fluctuations of the quantized electric field 3) These vacuum fluctuations cause variations in flight time of the pulse through the material Probe Pulse Non-linear material <E(x)E(x')> De Lorenci, Ford, Menezes, Svaiter, Ann. Phys. (2013) Bessa, De Lorenci, Ford, Phys. Rev. D (2014) Bessa, De Lorenci, Ford, Svaiter, Ann. Phys. (2015) Bessa, De Lorenci, Ford, Ribeiro, Phys. Rev. D (2016)

4 Non Linear Optics and Light Propagation Using Maxwell's equations in the absence of charges and currents P Polarization ε 0 Permitivity Total electric field: E = E 0 + E 1 E 0 background field E 1 probe field P can be expanded in terms of the components of the susceptibility tensors χ (a)

5 Non Linear Optics and Light Propagation Assumptions: Probe field is small, but rapidly varying, perturbation of the background field We choose E 1 to be polarized in the z-direction and propagating in the x-direction, i. e. E 1 = E 1 (x)z Many materials including non-linear ones have optical parameters which are relatively independent of frequency over a specific broad range In other words χ (a) constant (Dispersion ignored)

6 Non Linear Optics and Light Propagation (1) (2) (3) In this case only χ zz, χ zjz and χ zzjl contribute to the wave equation Working in the linear order in the probe field Wave Eq. For E 1 with Refractive index of the medium measured by the probe pulse when only linear effects take place Where

7 Flight Time BACK TO THE BASIC IDEA Flight time of a pulse propagating in the x-direction for a slab of material with thickness d Without the background field t d = d/v = n p d

8 Analog model for Quantum Gravity Effects So far everything is classical Quantizing the background field E 0 Analog Model QG < h uv (x)h uv (x')> AM < E 0 i (x)e0(x')> i Electric field correlation functions for a nondispersive, isotropic material

9 Analog model for Quantum Gravity Effects The travel time undergoes fluctuations around a mean value < t d > and with a variance of The fractional variance in flight time may be expressed by

10 F. Charra etal, Springer Materials (2000) Results A material of CdSe which has Χ (2) zzz = 1.1 x m/v at a wavelength of 10.6μm and n p = 2.54 at λ p = 1.06μm and n b = 2.43 at λ b = 10.6μm A material of Si which has Χ (3) zzzz = 2.8 x m 2 /V 2 at a wavelength of 11.8μm and n p = at λ p = 1.4μm and n b = at λ b = 11.8μm The model presented here is an illustration of a physical effect of vacuum fluctuations and an analog model for lightcone fluctuations predicted by quantum gravity

11 Thank You Obrigado

12 Non Linear Optics and Light Propagation Maxwell's equations in the absence of charges and currents D Induced electric field B Induced magnectic field P Polarization ε 0 Permitivity μ 0 Permeability

13 Light Cone Fluctuation as a Quantum Gravity Effect Gravitons in a squeezed state induce metric fluctuations Metric fluctuations induce light-cone fluctuations Deser, Rev. Mod. Phys. (1957) DeWitt, Phys. Rev. Lett. (1964) Ford, Phys. Rev. D (1995) h μν is an operator < h uv (x)>=0 <h uv (x)h uv (x')> 0 In the unperturbed spacetime, the square of the geodesic separation of points x and x' In the perturbed spacetime, the square of the geodesic separation of points x and x'

14 Vacuum lightcone fluctuations The travel time undergoes fluctuations around a mean value < t d > and with a variance of

15 Vacuum lightcone fluctuations The integrals of the electric field correlation will be well defined if there is a sampling function which falls smoothly to zero at both ends of the integration range Such function can be provided by the geometry of the slab of nonlinear material Switching function defined by Effect: insert a factor F(x)F(x') in the integrand of the electric field correlation function and to extend the range of integration to all x

16 Vacuum lightcone fluctuations The integrals are The fractional variance in flight time may be expressed by

17 Assumptions Recall that our approximations require (i) E 1 << E 0 dominance of the vacuum field over the probe field, (ii) E 1 /E 1 >> E 0 /E 0, which is equivalent to λ p < λ b, (iii) a range of frequencies in which the material can be assumed free of dispersion (iv) a material which is approximately isotropic, at least for the frequencies which give the primary contribution to the background field

18 Assumptions (i) electric field operator is expanded in a plane wave basis, with a narrow bandwidth of modes with frequencies near ω = ω p (t he probe f ield) excited in a coherent state, and the remaining part of the modes in their ground state. The probe field modes are of higher frequency than the dominant vacuum modes For the sake of an estimate, we ignore the indices of refraction, and write E 2 1 z 2 λ p 4 (Δω/ω)ΔΩ. z 2 mean number of photons per mode Δω/ω fractional bandwidth ΔΩ solid angle subtended by the probe beam In our assumption: E 2 0 a/b3 d. Even though λ p < d, we can still have E 1 E 0 if Δω/ω 1 or rδω 1. The physical reason for vacuum dominance is that many more modes contribute to the vacuum field than to the probe field

19 Assumptions (ii) The rate of decay of the Fourier transform of F bd allows us to test approximations (ii) and (iii). The exponentially decreasing behavior of the Fourier transform of this function suppresses the high energy modes of the background field.

20 Assumptions (ii) and (iii) For the case b 0.01d, at least 90% of the effect will occur in the range 0 z 18π, which means that only wavelengths such that λ b d/9 will significantly contribute. For a slab with d 10μm, the dominant wavelengths of the background field are those with λ b 1.1μm. Furthermore, the larger contribution occurs arises from z 2π, which for b 0.01d, corresponds to a wavelength of λ b 10μm. Thus if the material is relatively free of dispersion when λ b 1.1μm, then our assumption that n b is independent of frequency is justified. We may choose λ p 1μm to satisfy (iv) n b = 2.43 (ordinary ray) and n b = 2.44 (extraordinary ray) at λ = λ b = 10.6 μm The nearly equal values of n b for the ordinary and extraordinary rays justify the isotropy assumption