M. Dube and L. Thompson- Quantum Solitons in Magnetic Systems: Dynamics, Dissipation and Decoherence

8/3/2019 M. Dube and L. Thompson- Quantum Solitons in Magnetic Systems: Dynamics, Dissipation and Decoherence

1/34

PCE STAMP: LECTURE 3

QUANTUM SOLITONS inMAGNETIC SYSTEMS:

DYNAMICS, DISSIPATION & DECOHERENCE

Some o f the w ork he rein w as done w i th

M Dube

L Thompson


2/34

3.1: INTRODUCTION t o QUANTUM VORTICES

Here a qu ick survey of the many di f fe ren t k inds o f quantum s t r ing exc i ta t ion

ex is t , ie ., 1-d imensional t opo log ica l ex c i t a t ions o f some quantum f ie ld . The

low -T laborato ry is o f ten t he best p lace t o s tudy t hese, and many qu i te

in te res t ing ideas about the ro le o f s t r ingy exc i ta t ions in e lementary par t ic le

physics and c osmology have come f rom such s t ud ies.

Fina l ly i t i s no ted how t he fundamenta l quest ion o f how quantum vor t ic es

MOVE (ie ., the i r equat ion o f m ot ion) is s t i l l unansw ered. Th is is desp i t e some40 years o f debate.


3/34

VORTICES in SUPERCONDUCTORS

Superconductors show the Meissner effect (expulsion of flux), but in

Type-II superconductors, flux penetrates through quantized vortices, withflux = h/e. The flux is confined by screening currents to a length scale (penetration depth). In thin films (thickness d ), the screening currents

spread to a much larger length = 2/d.Outside the superconductor, the fluxballoons out into space, and can bephotographed using Aharonov-Bohmphase effects.

In an applied field the vortic es

in a type- II superc onduc tor form alattice, which feels a Lorentz force

in a field, but whic h is p inned by

imp urities, defec ts,

etc..


4/34

VORTICES in superf lu id 4He

In a neutral superfluid

like 4He, the vortices have

a quantized circulation:

= h/m4These vortices cannot be

observed directly, but they

can be decorated with electrons and thenPhotographed (by sucking the electrons

out); and single vortex rings can be

produced by quantum nucleation around

ions which move through the superfluid

at high velocities).


5/34

SUPERFLUID 3He

In superfluid 3He a huge variety of

vortices can be seen- their propeties are

observed using sensitive NMR measurements,

sometimes in rotating cryostats. The coherence length is long (15 nm at

T=0) and many of the core textures are not even singular (and dependingon which phase one is in there will be many different possible vortex

phases). The cores are thus full of bound or resonant quasiparticle states.


6/34

One can even have vortex sheets in3

He superfluid, in the A phase. Alarge variety of phases is supported

under different conditions of rotatio

magnetic field, temperature, etc.,

either as disordered vortex tangles

or in lattice arrays.


7/34

VORTICES inNeutron Stars

These can only be seen

very indirectly, using the

So-called glitch

phenomenon. There aremany gaps in our

understanding, because of

incomplete knowledge of the nuclear

equation of state at these densities


8/34

STRINGS & t he UNIVERSE

The relationship between

cosmic strings (which have

never been observed) andthe strings of string theory

has yet to be clarified. However

theory does indicate a crucial

role for cosmic strings in the

history of the universe


9/34

WHAT is t he EQUATION of MOTION of a QUAN TUM VORTEX?

One can ask s im i lar (and even more d i f f ic u l t ) quest ions about domain w al ls .

How ever , super fl u ids are no t t he on ly sys tems w e c an look a t ..


10/34

3.2: QUANTUM DYNAMICSof a

MAGNETIC QUANTUM VORTEX

Here w e look a t the p rob lem o f a magnet ic vor tex , w here (i ) one can ge t

a muc h be t t e r theore t ic a l hand le on the fundamenta l ques t ion posed in

t he last sect ion, and (i i ) w here exper im ents should be a lo t eas ier . The

answ er tu rns ou t t o be su rpr i sing the magnon envi ronmen t g ives a

non-loca l d iss ipat ion w h ich c auses s t rong long-t im e m emory e f fec t s .Thus the rea l dynamics o f the vor tex is muc h more complex t han w as

prev ious ly t hought .


11/34

Topolog ic a l So l i t ons in MAGNETS

SOLID3

He

SOLITONS in 2D MAGNETS


12/34

Quant um Vort ex in 2D Easy-p lane Ferrom agnet

Lattice Hamiltonian

L. Thompson, PCE Stamp, to be publ ished

L Thompson, MSc thes is (UBC)

Continuum Limit

In Lagrangian: (Berry phase)

VORTEX PROFILE

MAGNON SPECTRUM

Core Radius

Spin Wave velocity


13/34

VORTEX DYNAMICSDensity matrix propagator

Influence Functional

For the Magnetic vortex we find

Effective interaction

Effective coupling


14/34

K INETIC/BERRY PHASE TERMS

Assembly of Vortices:

One immediately recovers gyrotropic Magnus force:

and for vortex assembly a NON-LOCAL MASS:


15/34

In f luenc e Func t ional : Phase t erm sTotal Phase

(1) Longitudinal Phase terms

, etc.


16/34

(2) Mixed memory term:

(3) Transverse Damping term:

It immediately becomes clear that the real dynamics of a vortex, magnetic or

otherwise, has both reactive and dissipative terms that are more complex than

those that have been discussed so far.

We note that there is definitely a transverse dissipative force having thesymmetry of the Iordanskii term! If we have more than one vortex these have

non-local contributions.

There are however other terms which come in, not discussed ever before for

vortices in superfluids or superconductors- these should however exist in these

systems as well.

SHAPE f MOVING VORTEX


17/34

SHAPE of a MOVING VORTEX

The pro f i le of t he vortex s low ly d isto r t s as i t

moves m ore qu ick ly ; t he in-p lane sp ins are

forc ed s l ight ly out o f the p lane, even somed is tance aw ay f rom t he vor tex core . Th is

d is to r t ion is very import an t no t on ly does

i t inc rease the energy o f the vor t ex (leading

to a k ine t ic energy te rm , and de fin ing the

e f fec t ive mass o f the vor tex) , bu t i t a lso

crea t es an ex t ra sc a t t e r ing po ten t ia l fo r the

sp in w aves in the system , con t r ibu t ing to t he

fo rces ac t ing on the vor tex .


18/34

In f luenc e Func t ional : Dam ping/Q Noise t erm sMulti-vortex damping/noise term:

etc.

with propagator:

This gives a quantum noise term on the right hand-side of a Quantum

Langevin equation. However the noise is not only non-Markovian (highly

coloured in fact) but also non-local.

CONCLUSIONS(1) There is no reason whatsoever to exclude transverse dissipative forces. In fact

they are even more complex than previously understood

(2) The equations of motion for an assembly of vortices involve all sorts of forces

(non-local in time and space) that have not previously been studied.


19/34

EXPERIMENTS on VORTICES in M AGNETS?

Ex am ple o f T im e-reso lved Im agingin m agnet ic d isc (c ourt esy M Freem an)


20/34

3.3: QUANTUM DYNAMICS of DOMAIN WALLS

The in te res t ing th ing about a domain w a l l i s tha t the rea l ly th ic k & so f t(ie ., flex ib le ) w a l ls a re the l ightes t ones, and there fo re the ones most

l ik e ly to show large-sc a le quant um behaviour . Nave est im ates t hen

ind ica te t ha t rea lly very la rge w a l ls , con t a in ing ~ 1012 spins, should have

large quantum f luc t uat ions and show , eg., t unnel ing behaviour . Wal ls

t hemselves can have in terest ing quant um num bers, l ik e ch i ra l i t y , and alsohave in terest ing in terna l order ing and m odes and in terna l modes.

How ever one finds tha t the e f fec t o f decoherence can be ra ther d ras t i c .

For exam ple , bo th phonons and t ransverse sp in bath m odes qu ick ly k i l l o f f

ch i ra l i t y f luc t ua t ions . In sp i te of th is , de ta i led theore t ic a l w ork does pred ic t

Tunne l ing o f very la rge domain w a l ls , and some ev idence fo r th is hasappeared in exper iments on m agnet ic w i res (a g rea t dea l more needs t o be

done). Th is tunnel ing is on a large sca le , simi lar t o that found in SQUIDs.

How ever sp in bath e f fec t s on magnet ic tunne l ing tend to be fa r more serious

t han t hat in SQUIDs (cur ious ly, spin bat h ef fec t s on COHERENCE in SQUIDs

are very large, as a l ready pred ic t ed a very long t ime ago).

QUANTUM DYNAMICS of a


21/34

Consider the Hami l t on ian

Where in the f lat w a l l approx im at ion

We then have the w al l pro f i le

w i th (ch i ra l i t y )

and ( charge )

One then has a w a l l th ick ness

(in la t t i ce un i ts )and a w a l l surface energy

Now le t us assume s low w a l l ve loc i t ies , w here

is t he Walk er ve loc i t y .

Then the w a l l behaves as a par t ic le , w i th e f fec t ive Hami l t on ian

The ef fect ive m ass is per un i t area.

One of ten assumes so t hat

The e f fec t ive mass is then per un i t a rea

QUANTUM DYNAMICS of a

BLOCH WALL

- PCE Stamp, PRL 66, 2802 (1991)

- G Tatara H Fukuyama, PRL 72,772 (1994)- M Dube PCE Stamp,

J Low Temp Phys 110, 779 (1998)

RELEVANT TERMS in t he HAMILTONIAN


22/34

Assume t he la t t i ce fo rm

(1) Elec t ron ic Spin term s

Of ten go to t he con t inuum fo rm

w here in f l at w a l l app rox ima t ion:

(2) Phot on-spin int erac t ions

(3) Spin-phonon int erac t ions

(4) St at ic impur i t y /defect p inn ing pot ent ia l

(5) Dynamic impur i t ies (Nuc lear sp ins, paramagnet ic im pur i t ies)

WALL TUNNELING PROBLEM


23/34

WALL TUNNELING PROBLEM

The nave Hami l t on ian is

One eas i ly f inds a c oerc ive f ie ld

Now assume w e are c lose to t he coerc ive f ie ld . Def ine

Then the po ten t ia l i s w i th escape po int

Then the nave tunnel ing ra te is g iven by

w i th exponent :

The exponent c an a lso be w r i t t en as

w i th f requency The c rossover t o t unnel ing occ ursa t a tempera tu re

Th is w a l l con ta ins a to t a l number o f sp ins


24/34

PUTTING IN SOME NUMBERS

Now le t

BASIC FORM of COUPLINGS


25/34

(1) OSCILLATOR BATH:

This g ives a term :

w i th and c oup ling

(2) SPIN BATH:

We have t he s tandard bath Ham i l t on ian

The in te rac t ion tak es the fo rm:

Typ ica l ly w e in t roduce an e f fec t ive po ten t ia l

w h ich f luc t ua tes over a d is t r ibu t ion :

WALL-MAGNON INTERACTIONS


26/34

WALL MAGNON INTERACTIONS

We assume a co l lec t ive coord ina te decoup l ing o f the w a l l p rof i le f rom t he spin

w aves, ie ., a separat ion

The e f fec t ive Lagrang ian t akes the fo rm, in te rms o f Ho ls te in -Pr imak of f bosons:

These bosons are def ined

along the loca l ax es def ined

by the w a l l p rof i le :

w i t h

The key poin t i s tha t quadra t ic in te rac t ionsw i th magnons cause no d iss ipa t ion o r

dec oherenc e; one needs TRIPLETS of

magnons to do th is . The t r ip le ts involve

com binat ions o f w a l l & bu lk magnons, &

g ive a diss ipa t ion ra te a t low ve loc i t y o f

WALL-PHONON INTERACTIONS M Dube, PCE St am p, J Low Tem p Phys 110, 779 (1998)


27/34

p p y ( )

Free phonons w i th

In te rac t ion :

Here

These cont r ibu te to an e f fec t ive ac t ion

One-phonon c oupl ings Tw o-phonon c oupl ings

Mat r i x e lemen ts

We see tha t t h is p rob lem is com plex because w e again have totak e account o f non-l inear te rms in the coup l ing in fac t t heca lc u la t ions show tha t in te rac t ions w i t h pa irs o f phonons (above)and 4-phonon sc at t er ing processes ( le f t ) are c ruc ia l .The phonons a lso com plet e ly suppress CHIRALITY Fluct uat ions( thus t hese are suppressed even i f w e have no sp in bath) . For a l ldet a i ls see reference above.

EFFECT OF NUCLEAR SPINS & PARAMAGNETIC IMPURITIES


28/34

One deals w i th bo t h o f these in t he same w ay. The key here is tha t th is is t he sp in ba th , &

so i t can c ause a g rea t dea l o f decoherence.

Le t s tak e the inte rac t ion w i th nuc lear spins :

As w e saw be fore , w e should sp l i t t h is in to longi tud ina l & t ransverse par ts . Wi th som e

a lgebra w e t rans form t h is to an in te rac t ion be tw een nuc lear spins & t he domain w a l l

coord ina te

whe re

We t hen f ind t ha t one depends on the c h i ra li t y , & the o t her on the t opo log ica l c harge

Longi tud ina l :

Transverse:

The t ransve rse sp in bath f l uc t uat ions couple to t he ch i ral i t y w e a l ready saw how

c ruc ia l th is is in s tudy ing the sp in ne t .

The long i t ud ina l termg ives our e f fec t ive po ten t ia l :

MQT in Magnet s: Theory vs. Ex per im ent .QT in Magnet s: Theory vs. Ex per im ent .


29/34

So, can we get Large Scale Quantum Phenomena with theseSo, can we get Large Scale Quantum Phenomena with these solitonssolitons??

Initial theory on domain walls showed that any QUANTUMInitial theory on domain walls showed that any QUANTUM

SOLITON in a magnet should show largeSOLITON in a magnet should show large--scale quantumscale quantum

PropertiesProperties provided theprovided the decoherencedecoherence could be suppressed.could be suppressed.

Quantitative predictions could be made hereQuantitative predictions could be made here-- indicating thatindicating that

in some situations magnetic domain walls containing up toin some situations magnetic domain walls containing up to

1010

1010

spins should be able to tunnel.spins should be able to tunnel.

Experiments in Ni wires

and in large particles

bore out the theory

THEORY : P.C.E. Stamp, PRL 66, 2802 (1991)M.Dube, P.C.E.Stamp, JLTP 110, 779 (1998)

EXPT :K. Hong,N. Giordano, Europhys. Lett.36, 147 (1996)

W Wernsdorfer et al. PRL 78, 1791 (1997)

Example:

Not ic e the p rob lem there is a very w ide dispers ion,

w h ich i s caused by the in te rac t ion w i th t he spin bath(w h ich t u rns ou t t o be a com bina t ion of paramgnet ic

O impur t ies and nuc lear sp ins).

Macroscopic Quantum Tunneling (MQT) inMacroscopic Quantum Tunneling (MQT) in SQUIDsSQUIDs: Theory: Theory vsvs ExptExpt..


30/34

p Q g ( Q )p Q g ( Q ) QQ yy pp

It is interesting to compare with this previously studied phenomIt is interesting to compare with this previously studied phenomenon. It was shown by Leggett thatenon. It was shown by Leggett that

essentially all previous arguments against largeessentially all previous arguments against large--scale Quantum phenomena were flawed, becausescale Quantum phenomena were flawed, because

(i)(i) matrix elements between macroscopic states can be controlled bymatrix elements between macroscopic states can be controlled by microscopic energiesmicroscopic energies

(ii) Because what really matters is the(ii) Because what really matters is the behaviourbehaviour in time ofin time of

OOABAB(t) =(t) = (transition matrix)(transition matrix)and that in particular,and that in particular, Macroscopic Quantum TunnelingMacroscopic Quantum Tunneling between 2 different Flux states of abetween 2 different Flux states of aSQUID should be possibleSQUID should be possible-- a QUANTATIVE THEORETICAL PREDICTION was given.a QUANTATIVE THEORETICAL PREDICTION was given.

Caldeira & Leggett: Ann. Phys. 149, 374 (1983)

Voss & Webb: PRL 47, 265 (1981)

Clarke et al: Science 239, 992 (88)

Experimental confirmationExperimental confirmation

came very quicklycame very quickly

The compar ison betw een theory and exper iment fo r these MQT exper iments w as very

good. The exper iment ONLY took acc ount o f t he osc i l la to r ba th env i ronment ,

pa ramet r i sed by the e lec t ronic c i r cu i t r y such m odes cause a l l t he dissipat ion .

How ever t h is t heory fa i ls c omplet e ly t o deal w i t h DECOHERENCE in SQUIDS..

SQUID DECOHERENCE RATES


31/34

The basic problem with any theory-experiment comparison here is that most of the 2-level systemare basically just junk (coming from impurities and defects), whose

characteristics are hard to

quantify. Currently ~10 groupshave seen coherent oscillationsin superconducting qubits, andseveral have seen entanglementbetween qubit pairs. It has

become clear in the last yearthat most of the low-Edecoherence is coming from

TLS in the junction.RW Simmonds et al., PRL 93, 077003 (2004)

I Chiorescu et al., Science 299, 1869 (2003)

We c an a lw ays der ive an e f fec t ive Hami l ton ian o f

sp in bath form for a SQUID c oupled to a sp in bath. In

t he w eak decohe rence reg ime w e ge t

p lus no ise t e rms, g iv ing a decoherence ra te

This ra te is FAR h igher than t he d iss ipat ion

ra te . Exper iment s now c onf irm t h is is the m a indecoherence sou rce

- PCE Stam p, PRL 61 , 2905 (1988)- NV Prokof ev, PCE Stam p,

Rep Prog Phys 63, 669 (2000)

Thus w hat i s requ i red is a paramet er -f ree w ay o f re lat ing the t heory t o

exper iment , in w h ich t he dis t r ibu t ion o f coup l ings is ex t rac t ed from ex pt . &

used to pred ic t o ther ex per iment a l p roper t ies . We do not ye t have th is .


32/34

3.4: DOMAIN WALLS in SOLID 3He

There is one domain w a l l tha t has rather

exc ept iona l p roper t ies , at leas t

on paper . Th is is the dom ain w al l in so l id

3He, w hic h has never been exp lored

ex per iment a l ly . In fact so l id 3He has anumber o f ex t raord inary p roper t ies ,

w h ich w e desc r ibe b rie fl y below .

3He SOLID: HAMILTONIAN and

EQUATIONS f MOTION


33/34

EQUATIONS of MOTION

The bas ic s t ruc tu re o f t he under ly ing U 2D2 sta t e has pa i rs o f

FM- coup led p lanes w h ich are an t i f e rromagne t i ca l l y coup led to

each ot her . Th is in terest ing com binat ion o f FM and AFM order a r ises because of the c ompet i t ion

betw een d i f ferent order FM and

AFM exc hange proc esses.

We w i l l be in terest ed in t he equat ions o f m ot ion in t he inhomogeneous c ase, so one

genera l ises the OCF equat ions to ge t c oup led equat ions fo r d(r , t ) and S(r , t ):

The f i rs t equat ion here has d(r , t ) p recess ing in a comb inat ion o f ex te rna l f ie ld and the

f ie ld o f S(r,t).The 2nd equat ion invo lves precess ion of S i t se l f , and a lso a d ipo lar c oupl ingbe tween l(r, t ) and d(r , t ); and gradient t erms.

A tex t u re in d(r , t ) fo r smal l HA tex t u re in d(r , t ) for large H

S

d

S

d


34/34

The remain ing mat er ia l on He-3 w as d iscussed on the b lack board

(For m ore on He-3 sol id, see lec t ures of Osherof f )

M. Dube and L. Thompson- Quantum Solitons in Magnetic Systems: Dynamics, Dissipation and Decoherence

Documents

Transcript of M. Dube and L. Thompson- Quantum Solitons in Magnetic Systems: Dynamics, Dissipation and Decoherence