The double slit experiment provides a classic example of both interference and the effect of observation in quantum physics. When particles are sent individually through a pair of slits, a wave-like interference pattern develops, but no such interference is found when one observes which “path” the particles take. We present a model of interference, dephasing, and measurement-induced decoherence in a one-dimensional version of the double-slit experiment. Using this model, we demonstrate how the loss of interference in the system is correlated with the information gain by the measuring apparatus/observer. In doing so, we give a modern account of measurement in this paradigmatic example of quantum physics that is accessible to students taking quantum mechanics at the graduate or senior undergraduate levels.

1.
T.
Young
, “
The Bakerian lecture: On the theory of light and colours
,”
Philos. Trans. R. Soc. London, Ser. A
92
,
12
48
(
1802
).
2.
C.
Jönsson
, “
Elektroneninterferenzen an mehreren knstlich hergestellten Feinspalten
,”
Z. Phys.
161
,
454
474
(
1961
).
3.
A.
Tonomura
 et al, “
Demonstration of single electron buildup of an interference pattern
,”
Am. J. Phys.
57
,
117
120
(
1989
).
4.
M.
Arndt
 et al, “
Wave-particle duality of C60 molecules
,”
Nature
401
,
680
682
(
1999
).
5.
S.
Eibenberger
 et al, “
Matter-wave interference of particles selected from a molecular library with masses exceeding 10000 amu
,”
Phys. Chem. Chem. Phys.
15
,
14696
14700
(
2013
).
6.
M.
Arndt
and
K.
Hornberger
, “
Testing the limits of quantum mechanical superpositions
,”
Nat. Phys.
10
,
271
277
(
2014
).
7.
J.
Beugnon
 et al, “
Quantum interference between two single photons emitted by independently trapped atoms
,”
Nature
440
,
779
782
(
2006
).
8.
P.
Maunz
 et al, “
Quantum interference of photon pairs from two remote trapped atomic ions
,”
Nat. Phys.
3
,
538
541
(
2007
).
9.
M. R.
Andrews
 et al, “
Observation of interference between two Bose condensates
,”
Science
275
,
637
641
(
1997
).
10.
J.
Dziarmaga
,
W. H.
Zurek
, and
M.
Zwolak
, “
Non-local quantum superpositions of topological defects
,”
Nat. Phys.
8
,
49
53
(
2012
).
11.
R. P.
Feynman
,
R. B.
Leighton
, and
M.
Sands
,
The Feynman Lectures on Physics
(
Addison-Wesley
,
Boston
,
1964
), Vol.
3
.
12.
W. K.
Wootters
and
W. H.
Zurek
, “
Complementarity in the double-slit experiment: Quantum nonseparability and a quantitative statement of Bohr's principle
,”
Phys. Rev. D
19
,
473
484
(
1979
).
13.
M. G.
Raymer
and
S.
Yang
, “
Information and complementarity in a proposed which-path experiment using photons
,”
J. Mod. Opt.
39
,
1221
1231
(
1992
).
14.
See Chapter 2.6.2 of Ref. 31.
15.
H.
Ollivier
,
D.
Poulin
, and
W. H.
Zurek
, “
Objective properties from subjective quantum states: Environment as a witness
,”
Phys. Rev. Lett.
93
,
220401
(
2004
).
16.
H.
Ollivier
,
D.
Poulin
, and
W. H.
Zurek
, “
Environment as a witness: Selective proliferation of information and emergence of objectivity in a quantum universe
,”
Phys. Rev. A
72
,
042113
(
2005
).
17.
R.
Blume-Kohout
and
W. H.
Zurek
, “
Quantum Darwinism: Entanglement, branches, and the emergent classicality of redundantly stored quantum information
,”
Phys. Rev. A
73
,
062310
(
2006
).
18.
W. H.
Zurek
, “
Quantum Darwinism
,”
Nat. Phys.
5
,
181
188
(
2009
).
19.
M.
Zwolak
and
W. H.
Zurek
, “
Complementarity of quantum discord and classically accessible information
,”
Sci. Rep.
3
,
1729
1737
(
2013
).
20.
M.
Zwolak
,
C. J.
Riedel
, and
W. H.
Zurek
, “
Amplification, redundancy, and quantum Chernoff information
,”
Phys. Rev. Lett.
112
,
140406
(
2014
).
21.
W. H.
Zurek
, “
Quantum Darwinism, classical reality, and the randomness of quantum jumps
,”
Phys. Today
67
(
10
),
44
50
(
2014
).
22.
W. H.
Zurek
, “
Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?
,”
Phys. Rev. D
24
,
1516
1525
(
1981
).
23.
W. H.
Zurek
, “
Environment-induced superselection rules
,”
Phys. Rev. D
26
,
1862
1880
(
1982
).
24.
W. H.
Zurek
, “
Decoherence and the transition from quantum to classical
,”
Phys. Today
44
(
10
),
36
44
(
1991
).
25.
A.
Albrecht
, “
Investigating decoherence in a simple system
,”
Phys. Rev. D
46
,
5504
5520
(
1992
).
26.
M.
Tegmark
and
H. S.
Shapiro
, “
Decoherence produces coherent states: An explicit proof for harmonic chains
,”
Phys. Rev. E
50
,
2538
2547
(
1994
).
27.
B.
Schumacher
,
M.
Westmoreland
, and
W. K.
Wootters
, “
Limitation on the amount of accessible information in a quantum channel
,”
Phys. Rev. Lett.
76
,
3452
3455
(
1996
).
28.
W. H.
Zurek
, “
Decoherence, einselection, and the quantum origins of the classical
,”
Rev. Mod. Phys.
75
,
715
775
(
2003
).
29.
E.
Joos
 et al,
Decoherence and the Appearance of a Classical World in Quantum Theory
(
Springer-Verlag
,
Berlin
,
2003
).
30.
M. A.
Nielsen
and
I. L.
Chuang
,
Quantum Computation and Quantum Information
(
Cambridge U.P.
,
Cambridge
,
2010
).
31.
M.
Schlosshauer
,
Decoherence and the Quantum-to-Classical Transition
(
Springer-Verlag
,
Berlin
,
2010
).
32.
See Chapter 2.4 of Ref. 30.
33.
R.
Shankar
,
Principles of Quantum Mechanics
(
Plenum Press
,
New York
,
1994
), pp
659
660
.
34.
See Page 105 of Ref. 30.
35.
See Chapter 11 of Ref. 30.
36.
The exact wavefunction emerging from the slits in an experimental set-up will be complicated, depending on the exact slit shape geometry, the incoming state, particle type, etc. A superposition of Gaussian wavepackets, which we consider, leads to a clean presentation of interference and measurement. A square wavepacket, for ψ=Θ(x±L+Δ)Θ(x±LΔ), with Θ being the Heaviside step-function at each slit, which matches the ideal geometry of the slit, is also tractable, and can give a suitable extension of this work for class projects. The actual wavefunction, however, will be more complicated. Indeed, a reasonable approximate form is ψ=exp{1/(α[(Δ/2)2(xL)2])} (and zero for x>±L+Δ/2 and x<±LΔ/2) at each slit of width Δ. This function, while only approximate, is already non-analytic, but can both match the geometry and have smooth boundaries.
37.
J.
von Neumann
,
Mathematische Grundlagen der Quantenmechanik
(
Verlag von Julius Springer
,
Berlin
,
1932
).
38.
Recall that the partial trace is defined by
trA(ρSA)=kAk|ρSA|kA,
where {|siA} is any basis for the apparatus Hilbert space, resulting in an operator that acts only on the system. For example, one has
trA(|MσLΨ,LSAMσRΨ,R|)=0,
as the left hand side is equal to MσL|ΨSΨ|MσRtr(|LAR|). Note that the states |LA and |RA in Eq. (11) do not constitute a complete basis for our apparatus state, but they do give the only nonzero terms in the trace here.
39.
C. M.
Caves
and
G. J.
Milburn
, “
Quantum-mechanical model for continuous position measurements
,”
Phys. Rev. A
36
,
5543
5555
(
1987
).
40.
H. F.
Dowker
and
J. J.
Halliwell
, “
Quantum mechanics of history: The decoherence functional in quantum mechanics
,”
Phys. Rev. D
46
,
1580
1609
(
1992
).
41.
The first-order approximation about x=L is given in Eq. (17) with the replacement LL. The linear coefficient can easily be seen to be bounded graphically, but in this case the denominator includes a factor of mσ(L), which cannot be bounded below by a positive constant. A numerical calculation provides a bound of approximately 0.379/L on the actual coefficient, and there are many functions that can be used to find bounds analytically. For example, inserting the bound
Erfc(x/2)>2/π(x/(x2+1))ex2/2
into the linear coefficient and maximizing yields a bound slightly lower than the 0.4/L used in the main text.
42.
See supplementary material at http://dx.doi.org/10.1119/1.4943585 for a calculation providing dynamic visualization of the distribution for arbitrary parameters.
43.
See Chapter 2.2.6 of Ref. 30.
44.
We note that this statement assumes the system state is initially pure.
45.
E.
Schmidt
, “
Zur theorie der linearen und nichtlinearen integralgleichungen
,”
Math. Ann.
65
,
370
399
(
1908
).
46.
A.
Ekert
and
P. L.
Knight
, “
Entangled quantum systems and the Schmidt decomposition
,”
Am. J. Phys.
63
,
415
423
(
1995
).
47.
See Chapter 2.5 of Ref. 30.
48.
Students reading this might find it enlightening to perform the computation showing that the entropies remain unchanged by independent evolution of the system and apparatus.
49.
The estimate for σ is found by observing that small values of σ correspond to large values in the argument of Erfc. Hence, the small-σ case can be analyzed using an asymptotic expansion, leading to
βσ2σ/L2/πexp(L2/4σ2).
Further expanding this expression as a Taylor series about σ=L/2 and locating the x-intercept then gives
βσ(2/π)1/4e1[1+(5/L)(σ(L/2))]=0,
so we find
σ3L/10,
which is the value stated in the main text.

Supplementary Material

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