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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
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