1953-36

International Workshop on the Frontiers of Modern Plasma Physics

M. Marklund

14 - 25 July 2008

University of Umea Dept. of PhysicsSweden

Quantum Plasmas – Recent Developments

Mattias MarklundDepartment of PhysicsUmeå UniversitySweden

Quantum Plasmas – Recent Developments

Supported by

International Conference on Plasma Physics, Abdus Salam ICTP, Trieste, 2008

Swedish Research Council

Collaborators

L. Stenflo, G. Brodin, V. Bychkov, B. Eliasson, J. Zamanian (Umeå U.)

P. K. Shukla (Ruhr-U. Bochum, also S. Ali)

G. Manfredi (U. Strasbourg)

C. S. Liu (U. Maryland)

Overview

Why look at quantum plasma effects?

Schrödinger’s description.

Non-relativistic single electron dynamics.

Paramagnetic electrons.

From micro to macro physics.

MHD regime.

Conclusions - what the future might bring.

Quantum plasmas

From G. Manfredi, Fields Inst. Comm 46, 263 (2005)

Non-relativistic QMe at v st c QoWave function dispersion

Electron correlation

Tunneling

Lowest order relativistic QMe e at v st c Qde

Magnetization

Spin–orbit coupling

Relativistic QM

QMMMMMMMMMM

vistic QMativiFermi–Dirac statistics

Zitterbewegung

QFTQ

Pair production

Sourced plasmas

Coherent (computational)

theory?

Manifold applications (Pines, 1961; Kremp et al. 2005)

Condensed matter systems (Gardner, SIAM J. Appl. Math., 1994).

Astrophysical environments (Harding & Lai, Rep. Prog. Phys., 2006).

Ultracold plasmas (Rydberg states) (Li et al., PRL, 2005).

Nanostructured materials (Craighead, Science, 2000).

Laser-plasmas (Glenzer et al., PRL, 2007).

Spintronics.

Interesting fundamental aspects of matter dynamics.

Quantum to classical transition?

Collective quantum systems.

Schrödinger descriptionElectron properties described by complex scalar wavefunction ( probability)

where we have the Hamiltonian operator

and is the external electrostatic potential and being the magnitude of the electron charge.

i�∂ψ

∂t= Hψ

H = −�

2

2me

∇2− eφ

φ

|ψ|2 =ψ

e

Nice approach: allows for easy generalizations, new interactions can be incorporated in Hamiltonian.

Microscopic equations of motion

for some operator , Poisson brackets.

Example:

for previous scalar electron description.

dF

dt=

∂F

∂t+

1

i�[F, H]

F [F, H]

v ≡dx

dt= [x, H] =

p

me

Quantum statistical dynamics

Fluid moments from quantum kinetic equation, or from summing up particle contributions in Madelung picture.

W (t, x,p) =1

(2π�)3

∫d3y exp(ip · y/�)〈ψ∗(t, x + y/2)ψ(t, x − y/2)〉

(∂

∂t+

p

m· ∇

)W +

2e

�φ sin

(�

2←∇ · →

∇p

)W = 0

Quantum pressure (gradient of Bohm-de Broglie potential)

Gives higher order dispersion (spreading of electronic wave function).

mdv

dt∼ �

2

2m∇

(∇2√

n√n

)

Plasmonic devices

Surface plasmon polaritonspropagating on conductor surface.

Classically, surface is sharp.

Broadening of surface layer due to quantum dispersion.

Finite width gives damping!

ω ≈ ω(0)p

(1 + ε(0)d )1/2

⎡⎣1 + (0.6 + 2i)

(�k2

mω(0)p

)1/2⎤⎦

δSP ≈(

λ

100 nm

)4

μm

⇒

λ ∼ 30 nm⇒

δSP ∼ 10 nm

λ ∼ 30 nm

δδδSP ∼ 10 nm10

Spin contributionsElectron properties described by complex spinorwavefunction (spin degrees of freedom)

and the Pauli Hamiltonian operator

Here is the vector potential, is the spin operator, is the Bohr magneton.

ψ =

(ψ1

ψ2

)

H =

1

2me

(1

i�∇ +

e

cA

)2

+ μBB · σ − eφ

A σ

μB = e�/2me

Quantum equations of motion for electrons

where .

v =

1

me

(p +

e

cA

)

me

dv

dt= −e(E + v × B) −

2μB

�∇(B · S)

dS

dt=

2μB

�B × S

S = (�/2)σ

For spin systems: Decompose spinor wave function , unit spinor .

Electron fluid equations (not complete!)

ψα =

√nα exp(iSα/�)ϕα ϕα

me

(∂

∂t+ ve · ∇

)ve = −e (E + ve × B) −

∇pe

ne

−2μB

�Sa∇Ba

(∂

∂t+ v · ∇

)S =

2μB

�B × S + thermal and nonlinear spin terms

Maxwell’s equationsDue to the intrinsic magnetization, given by

,

Ampère’s law is modified according to

Gives dynamic spin contribution to Maxwell’s equations.

M = −2μBne

�S

∇ × B = μ0(j + ∇ × M) +

1

c2

∂E

∂t.

MHD regimeSingle fluid dynamics (for lowest order coherent spin)

Model magnetization using Brillouin function for spin-1/2 particles (for long enough time scales)

where the Zeeman energy gives the degree of alignment through the Brillouin function. For high temp., magnetization 0.

ρdv

dt= −∇

(B2

2μ0

− M · B

)+ B · ∇

(1

μ0

B − M

)− ∇p +

�2ρ

2memi

(∇

2√

ρ√

ρ

)

M = μBne tanhx ˆB

x = μBB/kBTe

∂B

∂t= ∇ ×

{v × B −

[∇ × (B − μ0M)] × B

eμ0ne

− Ma∇Ba

}

For high densities/low temperature:only electrons above Fermi levelcontribute to magnetization; implies reducedmagnetization.

Instabilities and ferrofluidsFerrofluids Nanostructured paramagnetic fluids. Formalism similar to the above applicable.

Normal field instability - saturated by gravity and surface tension (Cowley & Rosensweig 1967).( y g 9 7)

http://mrsec.wisc.edu/Edetc/cineplex/ff/text.html

Spin kinetics

Spin dependent distribution function:

(see also Cowley et al. Phys. Fluids 1986; Kulsrud et al. Nucl. Fusion, 1986)

Quantum equations of motion gives semiclassical dynamics.

See Gert Brodin’s talk.

∂f

∂t+

dx

dt· ∇xf +

dv

dt· ∇vf +

ds

dt· ∇sf = 0

f = f(t,x,v, s)

Formal structure: Wigner matrix from density matrix

Distribution function defined by

Evolution determined by governing equation for density matrix. Lowest order terms gives semiclassical kinetic equation. Generalized spin Wigner equation (J. Zamanian et al., submitted; Arnold & Steinrück, ZAMP, 1989).

Wab(t, x,p) =1

(2π�)3

∫d3y exp(ip · y/�)ρab(t, x + y/2,x − y/2)

f(t, x,p, s) = 12 (1 + s · σ)abWab

Issues in relativistic quantum plasmas

Pair production: “Schwinger mechanism” for fields with spatial and temporal variation.

Temporal compression: increased production rate.

Spatial compression: lower production rate.

Laser fields production rate unknown.

Strong fields + relativistic plasma particles: can computational models be developed?

Quantum field theoretical models (Melrose, 2008)?

t

rate

rate

x

Gigagauss laboratory fields

Currently, gigagauss laboratory fields generated in solid-laser interactions.

Look for quantum plasma effects:

Landau quantization.

Spin effects.

...

ConclusionsNew important effects appear from collective quantum domain.

Wide ranging possibilities for applications.

Nanomaterials.

Astroplasmas.

Ultracold plasmas.

Interesting future possibilities:

Theoretical development: Dense, relativistic plasmas using computationally viable models.

Solitons and plasmonics.