Decoherence or why the world behaves classically Daniel Braun, Walter Strunz, Fritz Haake PRL 86,...

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Transcript of Decoherence or why the world behaves classically Daniel Braun, Walter Strunz, Fritz Haake PRL 86,...

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