Quantum model for Impulsive Stimulated Raman Scattering (ISRS)

Transcription

1 Quantum model for Impulsive Stimulated Raman Scattering (ISRS) University of Trieste Trieste Junior Quantum Days May 18, 2018

2 Outline Introduction 1 Introduction

3 INCEPT INhomogenieties and fluctuations in quantum CohErent Phases by ultrafast optical Tomography Experiments: Prof. Daniele Fausti (P.I.), Theory: Prof. Fabio Benatti Ultrashort dynamics in complex materials (sub-picosecond time scales) Cross-fertilization between quantum optics (quantum state tomography) and condensed matter physics (pump-probe experiments)

4 Pump-probe experiments Pump pulse excites the material Probe pulse (less intense) test the evolution after a delay time t

5 Raman scattering Raman scattering is a kind of inelastic scattering for light One photon loses energy exciting one phonon in the material (Stokes process)... or One photon increases its energy destroying one phonon (anti-stokes process)

6 Probe-target interaction Initial state (probe + target): α α ϱ t Refraction at the boundary H ref = ( ) η (0) µµ + b + b t η (1) µµ )(a µj r µ j + a µj r µ j j,µ,µ Raman scattering H Ram := [( χ µ,µ a µj a µ j+ Ω µ,µ δ a µj, r µj, b are bosonic operators j ) ( b + j a µj a µ j+ Ω δ ) ] b,

7 Dynamics and observables Evolution operator U(τ) = U bulk (τ) U ref U ref = exp( i H ref ), U bulk (τ) = exp( iτ H Raman ) ( = 1), Average of an observable X phot X phot (τ) t = Tr[U(τ) α α ϱ t U (τ) X phot ] Coherent state of the probe ( α = exp α xj a xj αxja xj ) 0, a xj α = α xj, a yj α = 0, α xj = exp j ( (jδ)2 2σ 2 ) e iϕ

8 Pump-target interaction Same Hamiltonian for the light-matter interaction but different approximation Mean field for photons a λj α P λj Explicit dependence on the polarization angle (with respect to x) α P xj = α P 0j cos(θ P ), α P yj = α P 0j sin(θ P ) Phononic operator shifted b t = Tr(ϱ t b) = Tr(ϱb t ) b b t =U ref U bulk (τ) U free (t) b U free(t) U bulk (τ) U ref e iωt b iτ. j,λλ χ λλ α P λj α P λ j+ Ω δ

9 Assumptions on the interaction (good for Quartz) Zeroth order refraction matrix η (0) = ( ) η (0) xx η (0) xy η (0) xy η (0) xx First order refraction matrix depending on the phonon involved (same for χ) ( ) ( ) A : η (1) η (1) xx 0 = 0 η (1), E L : η (1) η (1) xx 0 = xx 0 η (1), xx E T : η (1) = ( 0 η (1) xy η (1) xy 0 )

10 Geometry Phonons selected by the angle between the polarization of the pump and the x axis (θ P ) A : b t = C A e iω At iπ/2 E L : b t = C E cos(2θ P ) e iω E t iπ/2, E T : b t = C E sin(2θ P ) e iω E t iπ/2, Remember: b U ref U bulk (τ) U free (t) b U free(t) U bulk (τ) U ref e iωt b iτ. j,λλ χ λλ α P λj α P λ j+ Ω δ

11 Mode occupation numbers (y polarization) N yk (τ) t b + b t α xk 2 F y ref ( i b b t α xk α xk+ Ω δ ) α xk Ω F y δ Ram (τ) A : E L : E T : F y ref = 0, F y Ram (τ) = 0, F y ref = 0, F y Ram (τ) = 0, F y ref = 2η (1) xy η (0) xy sin 2 (η (0) xx ), F y Ram (τ) = 0.

12 Pump at 45 : A and E T phonons excited Orthogonal polarization (leading term): E T Refractive

13 Mode occupation numbers (x polarization) N xk (τ) t cos 2 (η (0) xx ) α xk 2 + b + b t α xk 2 Fref x ( ) i b b t α xk α xk+ Ω α δ xk Ω FRam(τ) x δ A : E L : E T : F x ref = η (1) xx sin(2η (0) xx ), F x Ram(τ) = χ xx τ cos 2 (η (0) xx ), F x ref = η (1) xx sin(2η (0) xx ), F x Ram(τ) = χ xx τ cos 2 (η (0) xx ), F x ref = 2η (1) xy η (0) xy cos 2 (η (0) xx ), F x Ram(τ) = 0.

14 Pump at 0 : A and E L phonons excited Parallel polarization: Raman and Refractive effects are both visible

15 Results Experiment 1 (summary) Phase mismatch between Raman and refractive modulation Selection of different phonons according to the polarization of the pump Different behaviour of Raman and refractive modulation depending on the phonon involved and on the polarization selected by the analyzer Good agreement between theory and experiment

16 Quadrature: Homodyne detection + Time-resolved spectroscopy We combine two different experimental techniques to probe the nonequilibrium response of the material

17 Average quadrature Measured quantity: Current difference I I = j ( ) c xj c xj d xj d xj, c j = a xj + axj LO 2, d j = a xj axj LO. 2 Quadrature: X s = 1 ( a xj zj 2 Theoretical prediction: j e iφ j (s) + a xj z j e iφ j (s) ) I X s (τ) = A t cos(ω 0 s + Φ t ) where A t A (1 + η sin(ωt)), Φ t 2χ sin(ωt).

18 Average quadrature A t A (1 + η sin(ωt)), Φ t 2χ sin(ωt).

19 Variance of the quadrature: work in progress For a coherent initial state: variance is time-independent up to second order in the coupling Higher order effects or (more likely) signature of a statistical mixture

20 s and Outlook Results: Fully quantum model for Impulsive Stimulated Raman Scattering (ISRS) Outcomes of two different experiments correctly reproduced Future work: Complete tomography of the state of light (variance of the quadrature) Role of quantum correlations More interesting (complex) dynamics in the sample (e.g. interaction between the vibrational and electronic degrees of freedom)

21 Thank you for your attention! Social dinner: Pizzeria "Al Barattolo" at