quantum fluid description of light-matter...

Introduction DFTs for paramters Bridging DFTs and continuum Conclusion

Quantum �uid description of light-matter

interaction

Michael Ruggenthaler1, T.J.-Y.Derrien1,2,N.Tancogne-Dejean1,H.Appel1, A.Rubio1,3, and N.M.Bulgakova2,4

[1] Max-Planck Institut für Struktur und Dynamik der Materie, Hamburg, Germany

[2] HiLASE Centre, Institute of Physics of the Czech Academy of Sciences, DolníB°eºany, Czech Republic

[3] Nano-Bio Spectroscopy Group and ETSF, UPV, San Sebastian, Spain

[4] Institute of Thermophysics SB RAS, Novosibirsk, Russia

The 1st Annual HiLASE Workshop

October 10-12, 2016 in Stirin, Czech Republic

M.Ruggenthaler FluidQED

Introduction

fzu.cz/sites/default/�les/images/vyzkum/hilase.jpg

Numerical simulation tosupport/guide experiment

⇒ Microscopic equations of matter coupled to laser

Semi-empirical continuummodelling (+Maxwell)

Quantum mechanicaldescription (+photons)

Cong et al., Opt. Express 23, 5357-5367 (2015)M.Ruggenthaler FluidQED 1 / 10

Introduction

Quantum description of large systems is hard

Schrödinger equation

i∂t |Ψ(t)〉 = H(t)|Ψ(t)〉

nist.gov/cnst/epg/atom−manipulation−stm.cfm

M.Ruggenthaler FluidQED 2 / 10

i∂t |Ψ(t)〉 = H(t)|Ψ(t)〉

Single particle

i∂tΨ(rt)=

[− ~2

2m∇2+v(rt)

]︸︷︷︸

=H1(rt)

Ψ(rt)

R3∼5003⇒Ψ(r0)16B∼ 109Bytes

i∂t |Ψ(t)〉 = H(t)|Ψ(t)〉

Single particle

i∂tΨ(rt)=

[− ~2

2m∇2+v(rt)

]︸︷︷︸

=H1(rt)

Ψ(rt)

R3∼5003⇒Ψ(r0)16B∼ 109Bytes

N particles

H(t)=N∑

H1(rkt)+N∑k>l

w(|rk−rl |)︸︷︷︸=W

R3N∼5003N⇒|Ψ(0)〉∼109NBytes

N = 2 leads to one exabyte (google-mail storage capacity of 2012)

i∂t |Ψ(t)〉 = H(t)|Ψ(t)〉

Single particle

i∂tΨ(rt)=

[− ~2

2m∇2+v(rt)

]︸︷︷︸

=H1(rt)

Ψ(rt)

R3∼5003⇒Ψ(r0)16B∼ 109Bytes

N particles

H(t)=N∑

H1(rkt)+N∑k>l

w(|rk−rl |)︸︷︷︸=W

R3N∼5003N⇒|Ψ(0)〉∼109NBytes

N = 2 leads to one exabyte (google-mail storage capacity of 2012)

Quantum-�uid description of matter

Exact reformulation

n(rt)=Ne∫d3(N−1)r|Ψ(rr2...rNt)|2

DensityFunctionalTheories

∂tn(rt) = −∇ · j(rt)

∂t j(rt) = F([j], rt)

R3∼5003⇒n(r, 0)8B∼109Bytes

Kohn-Shamapproach

HKS(t) =N∑

HKS(rkt) + W

R3∼5003⇒|Φ(t)〉16B∼N×109Bytes

N ∼ 104 particles achievable for Kohn-Sham calculations

Exact reformulation

∂tn(rt) = −∇ · j(rt)

∂t j(rt) = F([j], rt)

R3∼5003⇒n(r, 0)8B∼109Bytes

Kohn-Shamapproach

HKS(t) =N∑

HKS(rkt) + W

R3∼5003⇒|Φ(t)〉16B∼N×109Bytes

Exact reformulation

∂tn(rt) = −∇ · j(rt)

∂t j(rt) = F([j], rt)

R3∼5003⇒n(r, 0)8B∼109Bytes

Kohn-Shamapproach

HKS(t) =N∑

HKS(rkt) + W

R3∼5003⇒|Φ(t)〉16B∼N×109Bytes

Exact reformulation

∂tn(rt) = −∇ · j(rt)

∂t j(rt) = F([j], rt)

R3∼5003⇒n(r, 0)8B∼109Bytes

Kohn-Shamapproach

HKS(t) =N∑

HKS(rkt) + W

R3∼5003⇒|Φ(t)〉16B∼N×109Bytes

Exact reformulation

∂tn(rt) = −∇ · j(rt)

∂t j(rt) = F([j], rt)

R3∼5003⇒n(r, 0)8B∼109Bytes

Kohn-Shamapproach

HKS(t) =N∑

HKS(rkt) + W

R3∼5003⇒|Φ(t)〉16B∼N×109Bytes

Quantum-electrodynamical �uid description

∂tn(rt) = −∇ · j(rt)

∂t j(rt) = F′([j], rt)

Coupled to Maxwell

−∇2v(rt) = n(rt)/c

�A(rt) = µ0cj(rt)

Exact reformulation of quantum electrodynamicsa

aMR et al., PRA 90, 012508 (2014)

T=100 (a.u.)Mean-fieldOEPExact

|〈E〉|

T=600 (a.u.)

T=1200 (a.u.)

0.0 2.0 4.0 6.0 8.0 10.0 12.0

T=2200 (a.u.)

x (µm)J.Flick et al., arXiv:1609.03901 (2016), submitted to PNAS

Quantum-electrodynamical �uid description

∂tn(rt) = −∇ · j(rt)

∂t j(rt) = F′([j], rt)

Coupled to Maxwell

−∇2v(rt) = n(rt)/c

�A(rt) = µ0cj(rt)

Exact reformulation of quantum electrodynamicsa

aMR et al., PRA 90, 012508 (2014)

T=100 (a.u.)Mean-fieldOEPExact

|〈E〉|

T=600 (a.u.)

T=1200 (a.u.)

0.0 2.0 4.0 6.0 8.0 10.0 12.0

T=2200 (a.u.)

x (µm)J.Flick et al., arXiv:1609.03901 (2016), submitted to PNAS

Objectives

length scales ∼ Å

time scales ∼ fs

Continuum modelling

∼ µm = 104 Å

∼ ps = 103 fs

Main objective of collaboration

Connect ab-initio quantum-electrodynamical description tosemi-empirical continuum modelling

1 DFTs provide parameters for continuum modelling

2 Averaging DFT equations: towards novel continuum approach

Objectives

time scales ∼ fs

Continuum modelling

∼ µm = 104 Å

∼ ps = 103 fs

Objectives

time scales ∼ fs

Continuum modelling

∼ µm = 104 Å

∼ ps = 103 fs

Objectives

time scales ∼ fs

Continuum modelling

∼ µm = 104 Å

∼ ps = 103 fs

Objectives

time scales ∼ fs

Continuum modelling

∼ µm = 104 Å

∼ ps = 103 fs

Objectives

time scales ∼ fs

Continuum modelling

∼ µm = 104 Å

∼ ps = 103 fs

Objectives

time scales ∼ fs

Continuum modelling

∼ µm = 104 Å

∼ ps = 103 fs

Objectives

time scales ∼ fs

Continuum modelling

∼ µm = 104 Å

∼ ps = 103 fs

Parameters for continuum modelling

T.Derrien et al., in preparation

For instance

∆n(rt) = n(rt)− nGS(r)

slab of silicon irradiated by laser

Number of excited electrons (single color)

For instance

Macroscopic averaging of exact quantum-�uid description

DFT �uid equations (Å, fs)averaging⇒ Continuum approach (µm, ps)

T. J.-Y. Derrien1,2, N. Tancogne-Dejean2, M. Ruggenthaler2, H. Appel2, A. Rubio2,3,4, N. M. Bulgakova1,5

Bridging quantum mechanics and continuum modeling: Macroscopic averaging of exact quantum fluid description

CONCLUSIONS REFERENCES

INTRODUCTION

1HiLASE Centre, Institute of Physics ASCR, v.v.i, Dolní Brežany Czech Republic; 2Max Planck Institute for Structure and Dynamics of Matter, Hamburg, Germany;

3Physics Department, University of Hamburg, and the Hamburg Center for Ultrafast Imaging, Hamburg, Germany; 4Nano-Bio Spectroscopy Group and ETSF, Universidad del Pais Vasco, San Sebastian, Spain;

5Institute of Thermophysics SB RAS, Novosibirsk, Russia.

MACROSCOPIC AVERAGING2,3

Objectives: ♦

Context: ♦

<[email protected]>

♦ Marie Skłodowska Curie Actions (MSCA) Individual Fellowship of the European’s Union’s (EU) Horizon 2020 Programme under grant agreement ”QuantumLaP” No. 657424.♦ ERC Advanced grant "DYNAMO" ♦ State budget of the Czech Republic (project HiLASE: Superlasers for the real world: LO1602).

CAN WE NEGLECT SOME OF THE FLUCTUATION TERMS?

FINANCIAL SUPPORTS:

♦ ♦ ♦ ♦

EXACT QUANTUM FLUID DESCRIPTION

Obtain an averaged form of the exact quantum fluid description of coupled light-matter systems, allowing for accurate and parameter-free large-scale modeling.

Continuum modeling is a powerful and accurate semi-empirical approach for modeling nanostructures and large-scale systems. However, it extensions to new phenomena is not straighfirward and requieres inputs from the experiement or other theoryQuantum mechanics is an exact theory but limited to small-size systems

Idea: Microscopic fluctuations of physical quantities are usually not important from the macroscopic perspective.

Method: Every function can be expressed as the sum of a macroscopic, or averaged part, and a microscopic, or fluctuating part

Importantly, the macroscopic averaging of the product of two functions is given by

which still contains the microscopic fluctuations of the two functions.

Quantum description of matter is usually described by a Schrodinger equation

which can be reformulated in terms of a exact coupled fluid equations of the form1

Pros: These are exact equations, allowing for a parameter-free description of any pertubation. Cons: Cannot be solved in practice for real systems.

,with of the form

MACROSCOPIC QUANTUM FLUID EQUATIONS

Applying the averaging procedure to the continuity equation Eq. (1) gives directly

The macroscopic averaging of Eq. (2) is more problematic as it leads to many terms, some of them including fluctuations.

Using of the fact that the potential is describing the electron-nuclei potential ( ) and the driving laser-field, we obtain the exact equation of motion for the total current

Fluctuation terms

Lorentz force

In order to understand if we can neglect some of the fluctuation terms, we consider the ideal situation of a perfect crystal ( ), under a uniform laser.In this case, the equation of motion of the current reduces to

We obtain that some of the fluctuation terms are negligible, whereas some of them leads to important physical effects.

One obtains also that neglecting the fluctuation terms here is equivalent to assuming a jellium model, i.e., .

Importantly, most of the fluctuation terms can be neglected in some ideal caseswhereas some of them are vital for describing correctly the physics.

Number of electron in the averaging volume

Responsible for HHG in perfect bulk materials4

We derived an exact quantum fluid description of coupled light-matter systemsMacroscopic fluid equations contain microscopic fluctuations.Some fluctuation terms can be neglected, but some of them lead to important physical effects.New framework to analyse approximation in continuum modeling

[1] R. van Leeuwen, Mapping from Densities to Potentials in Time-Dependent Density-Functional Theory. Phys. Rev. Lett. 82, 3863 (1999).[2]W. L. Mochán and R. G. Barrera, Electromagnetic response of systems with spatial fluctuations. I. General formalism.Phys. Rev. B 32, 4984 (1985).[3]R. Del Sole and E. Fiorino, Macroscopic dielectric tensor at crystal surfaces. Phys. Rev. B 29, 4631 (1984).[4]N. Tancogne-Dejean et al., Impact of the electronic band structure in high-harmonic generation spectra of solids. Submitted.

Details see poster of Nicolas