Decoherence

or why the world behaves classically

Daniel Braun, Walter Strunz, Fritz HaakePRL 86, 2913 (2001), PRA 67, 022101 & 022102 (2003)

1

| 1

| 2

2

1 2

why no interferencefrom superposition

c2 | 1 c2 | 2 ?

modern answer: dissipative influence of environment decoheres superpositions to mixtures

Schrödinger 1935:why no interferences between macroscopically distinct states (“cat” states)?

d

q

λ ?

diss

sys

t

Qt

system reservoir

H Hsys Hres Hint

sys res diss

H int QB

, ,sysQ H 0diss for quantities with meaningful limit

like probabilities or mean values

But: damping brings about vastly different time scales:

for coherences between macroscopically

distinct states

dec

dec diss

0 for d

? 0

0 c1 1 c2 2

sys0 | 0 0 |

c1c2 | 1 2 | c1

c2 | 2 1 |

initial superposition

with interference term

|c1 |2 11t |c2 |2 22t

c1c2 12t c1

c2 21t

dec t diss|c1 |2 | 1 1 | |c2 |2 | 2 2 |

superposition collapsed to mixture

sys0 res0e iHt/ eiHt/ syst Trres

To study collapse, look at how interference term

12t Trres e iHt/ | 1 2 | res0 eiHt/

decays for t dec

/d decreases with dec

dDifferent scenarios by choice of distance :

1. Golden Rule: dec res sys diss

Hint small perturb during dec; all expts thus far

dec3. res sys diss

ineffective during decHsys Hres

12tQt, Hsys, . . .

Hsys

, strong for

ineffective during dec

2. sys diss dec res ,

damping weak for

Scenario 1: Golden Rule, res sys dec diss

not to be (ab)used outside limit of validity! λ/d must not become too small! not applicable to macroscopic superpositions!

res sys dec d

2 diss

long-time limitsin syst

syst sys

wavepackets have width λ and distance d in Q-space

lowest-order perturbation theory w.r.t. toH int QB

current experiments all in GR regime:

• Wineland et al: superpositions of coherent states of translational motion of Be ions in Paul trap, damped through irradiation, dec/ diss 1 1

25

fringes resolved; world record

• Zeilinger et al: multislit diffraction of : /d 10 5 ; all dissipation carefully avoided;

C60 & C70

/1 0

• Haroche et al: superpositions of coherent or Fock states of microwave cavity mode

s 1 sys s diss 6 dec diss 110

,

decrease to 1100

would begin to invalidate GR

current experiments ctd

• Delft, Stony Brook, Orsay, all independent: superpositions of counterpropagating mA super- currents in small loops (SQUIDS); again,

dec/ diss /d2 not very small

λ

d

d 1

Scenario 3: lazy theorist’s favorite: only interaction effective; no free evolution during decoherence; applies to macroscopic superpositions

H H sys H res H int

have width and distance in -spaceQ 1 , 2

QB

q| |q q| |q 12t Trres e iQBt/ | 1 2 | res eiQBt/

q| 12t| q q|Trres e iQBt/ | 1 2 | res eiQBt/ |q

q|Trres e iqBt/ | 1 2 | res eiqBt/ |q

q| 1 2 |q Trres e iqBt/ res eiqBt/

q| 1 2 |q e iq qBt/

requires|q q| d

reservoir mean of exponentiated coupling agent B describes decoh

dec |q q| B2

d B2

many-freedom bath:B i 1N B i, N 1 B 0,

world behaves classically! Universally so! BUT:

e iq qBt/ 2 e q q2B2 t2/ 2 e t/ dec 2

central limit theorem: B Gaussian , ei B e 12 2B2

corrections arise only for t dec, vanish as N

Thus far, superposed packets distinct in Q-space.What if packets far apart in other space (eigen-space of observable not commuting with Q) ?

Same strategy, more technical hokuspokus,

same conclusion:

Scenario 2 with competition of bath correlation decay and decoherence?

Thus far, 1q, 2q taken far apart in Q-space,

i.e. eigenspace of system coupling agent inH int QB

what if 1p, 2p far apart in P-space,P,Q i

?

naïve repetition of previous reasoning gives surprise:

p| 12 |p p|Trrese iQBt/ | 1 2 | reseiQBt/ |p

with p|Q i

p

p|

diffusion w.r.t. c.o.m. momentum, with diffusionconstant independent of both and p p dP

No accelerated decoherence? There is, just work harder!

e12 B2t2

p

p 2

p| 120|p

ei

i Bt

p

p p| 120 |p

a little bit of free motion with

gives

Hsys P2/2M VQ

Q Q Pt/M

dec 8M2 2

B2 dP2

14

p|e i Qt Pt2/2M B e

pBt i

pB2M t2

p|

p| 12t|p e

p p 2B2

8M2 2 t4

p| 120|p

classical world

Scenario 2, at least as interesting: res , dec sys

interaction picture: H t Q tBt QBt

e iqBt/ e iq

0

tdtBt/

e iq qBt/ eiq

0

tdtBt/

e iq

0

tdtBt/

again, Gaussian B by central limit theorem:

eiq

0

tdsBs/

e iq

0

tdsBs/

exp i q q

0

tds

0

sdsqBsBs qBsBs

N12t Trsys 12t 12 t e

d2

2 0

tds

0

sds Bs,Bs

classical world

CONCLUSION

• Collapse of superpositions to mixtures due to interaction with environment

• While all decoherence expts done thus far refer to Golden-Rule regime,

• Classical behavior of the macro-world, caused by extremely rapid decoherence of macroscopic superpositions, understood through simple short-time solution of Schrödinger’s equation