TIME REFRACTION and THE QUANTUM PROPERTIES OF VACUUM
J. T. MendonçaCFP and CFIF, Instituto Superior Técnico
CollaboratorsR. Bingham (RAL), G. Brodin (U. Umea), A. Guerreiro (U. Porto), M. Marklund (U. Umea), E. Ribeiro (IST), P. Shukla (U. Bochum), Ch. Wang (U. Aberdeen)
Outline
1. Time refraction (“flash” ionization);2. Classical: Temporal Fresnel formulae;3. Quantum: photon pair creation;4. Temporal beam splitter;5. Arbitrary time varying media;6. Dynamical Casimir effect;7. Euler-Heisenberg vacuum;8. Contracting plasma bubble;9. Conclusions.
n x
1
1
n2
y
n x
1
1
n2
ct
(Space) refraction Time refraction
Photons cannot travel back in the past Reflection occurs in both cases
Electric field for a given frequency mode
(j = 0, 1)
Temporal Snell’s law:
Sudden change in the medium: n0 --> n1 at t = 0.
Momentum conservation implies a frequency jump (flash ionization)
Field continuity conditions
Temporal Fresnel formulae
Transmission and reflection coefficients
Time refraction leads to (space) reflection!
Quantum theory of time-refraction
Bogoliubov transform. (relating new and old field operators)
Squeezing transf.
Time dependentRefractive index
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n(t) = n1H(−t) + n2H(t)
Creation of photon pairs from vacuum
Relation between the new and the old quantum states
Time refraction for guided propagation
Total electric field
Axial field amplitude
Dispersion relation
Changes in the medium
Frequency shift
Forward propagation
Backward propagation
(For propagation in free space:
Field envelopes for Gaussian pulses
Temporal beam splitter
Two successive jumps in the medium:- n0 for t < 0, and t > - n1 for 0 < t <
Transmitted and reflected intensities
|n1- n0| =0.1
Field operators for the temporal beam splitter
Probability for the emission of m photon pairs
(m=1)
p(m) ~ p(1)m
Temporal beam splitter in guided propagation
Final amplitude of the transmitted pulse
Final amplitude of the reflected pulse
t0
n, kcPerturbation with a finite duration
Pump laser pulse
Optical Fiber
n nn’
Time refraction experiment in guided propagation
Initial Gaussian pulse (t = 0)
Numerical illustration
Formation of a counter-propagating pulse
Secondary pulse resulting from time refraction
Arbitrary time-varying medium
Classical field
Instantaneous frequency
Evolution equations
Approximate solutions for |E| >> |E’|
Transmitted field
Reflected field
Formally identical to reflection in a non-homogeneous medium R (t) --> R (x)
Field operators
Time-dependent Bogoliubov transformations
Manifestations of quantum vacuum
1. Hawking radiation2. Hunruh-Davies effect (accelerated frame)3. Dynamical Casimir effect4. Time refraction5. Superluminal boundary
Time refraction v. Dynamical Casimir
Number of photons created from vacuum
Time refraction stays valid in free space
Squeezing parameter
Photon creation in aperturbed cavity
Superluminal fronts
Reduces to time refraction by a Lorentz transformation
Number of photons produced from vacuum
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< Nk (t − x /u) >= sinh2[α .r(t − x /u)]
Vacuum resonances!
Mendonça and Guerreiro, PRA (2005)
How to create a superluminal front
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Dynamical cavity in vacuum
Dispersion relation of the cavity modes
Geometric factor (
Laser intensity: I (r, t)
Time refraction in a contracting plasma bubble
Possible explanation for sonoluminiscence!
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ω(t / t0) = k 2c 2 +ωp0
2
(1− t / t0)
⎡
⎣ ⎢
⎤
⎦ ⎥
1/ 2
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ω(t)ω0
t/t0
Conclusions
• Time refraction (TR) is a basic first order effect (such as refraction).• TR implies space reflection and photon frequency shifts (temporal Snell’s law).• Temporal interference can be observed and a temporal beam splitter can be built up.• TR of short pulses in optical fibers can used for demonstration experiments.• TR implies photon pair creation in vacuum.•TR is related to the dynamical Casimir effect. It can also be used to study vacuum nonlinearities.• TR can be applied to an expanding or contracting plasma bubble. •TR can explain sono-luminiscence in a simple way (applications to astrophysics?).• Long life to TR...