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TESTING QUANTUM MECHANICS ... Initial aim of program: interpret raw data in terms of QM, test (a) vs (b). ... for each sample separately, and also for total.
Typology: Lecture notes
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Anthony J. Leggett
Department of Physics University of Illinois at Urbana-Champaign USA
M ESO/MACROSCOPIC TESTS OF QM: M OTIVATION
At microlevel: (a) | ↑ 〉 + | ↓ 〉 quantum superposition 3 ≠ (b) | ↑ 〉 OR | ↓ 〉 classical mixture ×
how do we know? Interference
At macrolevel: (a) + quantum superposition
OR (b) OR macrorealism
Δ: Decoherence DOES NOT reduce (a) to (b)!
Can we tell whether (a) or (b) is correct? Yes, if and only if they give different experimental predictions. But if decoherence → no interference, then predictions of (a) and (b) identical. ⇒ must look for QIMDS quantum interference of macroscopically distinct states What is “macroscopically distinct”? (a) “extensive difference” Λ (b) “disconnectivity” D
Initial aim of program: interpret raw data in terms of QM, test (a) vs (b).
~large number of particles behave differently in two branches
The Search for QIMDS
1.Molecular diffraction*
( ) 3 e xp ( )^2 / 2 ( ~ 18 ) f υ = A υ − υ −υ (^) o (^) υ m (^) υ o i υ m
~100 nm
C (^60) z
z
I(z) ↑
*Arndt et al., Nature 401 , 680 (1999); Nairz et al., Am. J. Phys. 71 , 319 (2003).
Note:
The Search for QIMDS (cont.)
`
2 2 2 2 ω res (^) ≅ ω o + M H
A n ω o ~ a − bN ←^ no. of spins, exptly. adjustable Nb: data is on physical ensemble, i.e., only total magnetization measured.
*S. Gider et al., Science 268 , 77 (1995).
Apoferritin sheath (magnetically inert)
↑ ↓ ↑ ↑ ↓ ↑ = ↓ ↑ ↓ ↑ ↑ ↓ ~
.... (~5000 Fe 3+^ spins, mostly AF but slight ferrimagnetic tendency) α| 〉 + β| 〉? (M~200μB )
AF : Δ ~ = ω (^) o exp − N K / J
(isotropic) exchange en.
no. of spins uniaxial anisotropy
Interpretation of idealized expt. of this type: (QM theory ⇒) (^) δ J (^) x 1 δ J (^) y 1 ≥| J (^) z 1 | ~ N
1/ 2 ⇒ | δ J (^) x 1 | > N But,
1 2
xtot ytot
xtot x x
δ δ
δ δ δ
⇒state is either superposition or mixture of |n,–n> but mixture will not give (#) ⇒ state must be of form
n^ |^ , n ∑ c^ n^ − n^ >
Note: (a) QM used essentially in argument
(b) D ~ N 1/2^ not ~N. (prob. generic to this kind of expt.)
value of value of J (^) x1 J (^) x
“Macroscopic variable” is trapped flux Φ [or circulating current I]
The Search for QIMDS (cont.)
bulk superconductor
Josephson junction
RF SQUID
London^ trapped flux penetration depth
Josephson circuits
(10 4 –10 10 )
SYSTEM
~10 19 (10^3 –10^15 )
More possibilities for QIMDS:
(a) BEC’s of ultracold alkali gases:
Bose-Einstein condensates
(Gross-Pitaevskii)
Ordinary GP state:
( ( )^ ( )^ )
N
“Schrödinger-cat” state (favored if interactions attractive):
( ( )^ ) ( ( ))
N L
N
problems:
(a) extremely sensitive to well asymmetry ΔΕ (energy stabilizing arg (a/b) ~tN^ ~ exp – NB/=) so ΔΕ needs to be exp’ly small in N
(b) detection: tomography unviable for N»1, ⇒ need to do time-sequence experiments (as in SQUIDS), but period v. sensitive e.g. to exact value of N
single-particle tunnelling matrix element
Ψ (^) L ( ) r^ Ψ^ R ( ) r
WHAT HAVE WE SEEN SO FAR?
NO (fullerene diffraction: N not large enough, SQUIDS: no displacement of COM between branches)
Would MEMS experiments (if in agreement with QM) exclude GRWP?
alas:
⇒ do not gain by going to larger Δx (and small Δx may not be enough to test GRWP)
, ( )^2 Γ (^) col l ∝ Δ x Γ (^) d ce ∝ Δ x
decoherence rate acc. to QM
collapse rate in GRWP theory
HOW CONFIDENT ARE WE ABOUT (STANDARD QM’l) DECOHERENCE RATE?
Theory:
(a) model environment by oscillator bath (may be
questionable)
(b) Eliminate environment by standard Feynman-Vernon
type calculation (seems foolproof)
Result (for SHO):
Tested (to an extent) in cavity QED: never tested (?) on
MEMS.
Fairly urgent priority!
2
0
dec ~^ B
k T x x
⎛ ⎞^ ⎛^ Δ ⎞ Γ Γ (^) ⎜ ⎟ ⎜ ⎟ ⎝ Ω ⎠ ⎝ ⎠
provided kBT»=Ω
zero-point rms displacement
energy relaxation rate (Ω/Q)