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)
modern answer: dissipative influence of environment decoheres superpositions to mixtures
Schrödinger 1935:why no interferences between macroscopically distinct states (“cat” states)?
d
q
λ ?
, ,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
|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