Journal of Quantum Information Science, 2013, 3, 20-22
http://dx.doi.org/10.4236/jqis.2013.31005 Published Online March 2013 (http://www.scirp.org/journal/jqis)
The Role of Detector in Which-Way Experiment
Zhengchuan Wang
Department of Physics, Graduate School of the Chinese Academy of Sciences, Beijing, China
Received January 22, 2013; revised Februa ry 25, 2013; accepted March 7, 2013
ABSTRACT
Following Scully et al.’s study on the mechanism of complementarity, we further investigate the role of detector in
which-way experiment. We will show that the initial quantum pure state of particle will reduce to a mixture state be-
cause of the inevitable interaction b etween particle and detector, then the coherence of wavefunction for the particle fal-
ling on the screen will be destroyed, which leads to the disappearance of interference fringes in which-way experiment.
Keywords: Which-Way Experiment; Decoherence; Detector
1. Introduction
Both Bohr’s principle of complementarity [1] and Feyn-
man’s two-slit experiment [2] manifest the “wave-parti-
cle duality of matter”, in which the loss of interference
fringes constitute a mystery of quantum mechanics. There
exists cont roversial expl anations on the mech anism of co m-
plementarity. In which-way experiment, the disappear-
ance of interference fringes is usually explained by the
use of the Heissenberg’s uncertainty principle [3]. How-
ever, in 1991, Scully et al. performed a quantum optical
tests of complementarity [4], and attributed the disap-
pearance of interference fringes to the correlation be-
tween the measuring apparatus and the system being ob-
served, not to the usual position-momentum uncertainty
principle. Their viewpoints were criticized by Storey et
al. [5], the latter still insisted that the uncertainty princi-
ple may account for the loss of interference fringe. In
1998, Dürr et al. proposed a which-way experiment to
further explore the origin of quantum mechanical com-
plementarity [6] by the use of an atom interferometer and
they concluded that correlation between the which-way
detector and the atomic motion will destroy the interfer-
ence fringes, and Heissenberg’s position-momentum un-
certainty principle can not explain the loss of interference
fringes. There are other experiments, such as the atom
interferometer experiment by Chapman et al. [7], the elec-
tron double-path interferometer experiment by Buks et al.
[8], concerned with this scheme, too. In this paper, we will
further elucidate the disappearance of interference fringes
based on the mechanism proposed by Scully et al.
2. Theoretical Formalism
We now consider a which-way experiment, in which the
wavefunction describing the center-of-mass motion of
particle corresponding to the two slits are


1r
 and


2r
 , respectively. When we make use of an arbitrary
detector to determine the path of a particle through a
fixed double slit, the interaction between particle and
detector occurs, which makes the state vectors of particle
and detector become entangled. The quantum state of the
combined system of particle-detector evolves as follows
[9].




  
00
1111 222
tt Dr
ttCaDr CbDr


 
  
. (1)
In the above, 012 ,,DDD are the state vectors of
detector, while

 
12
rarbr   describes
the quantum state of particle before being detected. As a
result of particle-detector interaction, the correlation be-
tween particle and detector has been established after
time 1, the state vectors of particle and detector have
coupled to each other after time 1
t. Expression (1)
clearly demonstrates the violation of pure state
t


r 
after being detected by a detector to determine the path
of particle. We can show this violation b y its density ma-
trix, too. The reduced density matrix of particle is


 

 

 

 
311
2
22
111211
122 112
211 221
2
22
221222
2
2
D
Trtttt
CaCaDDrr
CC abDDrr
CCbaD Drr
CbCbD Drr














, (2)
above
D
Tr indicates partial trace over the detector de-
grees of freedom. When the state vectors 12 ,DD
of
C
opyright © 2013 SciRes. JQIS

Z. C. WANG 21
detector are orthogonal to each other, the density matrix
can reduce to
  
22
31 11222
Carr Cbrr    , (3)
which indicates pure state

r  has become to a
mixture state. Generally, the pure state

r  of parti-
cle will reduce to a mixture state after being detected by
the detector, in the end th e particle is not in the pure state
but a mixture state when it arrives at the screen. The in-
terference fringes will disappear because of the decoher-
ence of the pure state of particle after being detected.
In the general, there exists deviation between the mix-
ture state of particle after interacting with detector and
the initial quantum pure state

r  . We can evaluate
the above deviation by the difference between 3

and
the density matrix 1

of pure state
 
12
rarbr  
 
, it is
2
31
,nm
nm
nm
 
. (4)
Considering the density matrix 3

in the above Ex-
pression (2), this deviation can be further written as
2
2
22 2
21112
2
122 1
2
211 2
2
2
22
2212
2
2
CaCaDDa
C CabDDab
C C baDDba
CbCbDDb

 
 
 


 
2
. (5)
We can see that the deviation is determined by the
state vectors of detector. If we properly chose the detec-
tor and make the state vector 1 DD2
, and the coef-
ficient 12
1
2
CC , then the deviation will vanish.
In this special case, the state vector of combined parti-
cle-detector system is
  

1112
1
2
ttDarbr
  

 , there is
no correlation between state vectors of particle and de-
tector at all, the quantum state of particle still remain in a
pure state after this special measurement by detector. In
this case, the interference fringes will not disappear.
However, the detector will not distinguish the path of
particle because there are no correlation between particle
and detector.
If there are no correlation between particle and detec-
tor, the wavefunction of particle in the interference re-
gion is
 
12
1
2
rr  

and the probability density of particle falling on the screen
is
  
 
22
12
12 21
1
2
PRr r
rr rr

 







(7)
When we want to determine the path of particle, the
correlation will inevitable occur, we should write the
wavefunction of the combined particle-detector system
as Expression (1). However, the probability density at the
screen can not be written as
  
  
22
12
12122121
1
2
PRr r
rrDDrrDD

 







.
(8)
Because the interference fringes originate from the
particle not the detector, only the particle can fall on the
screen, while the detector can not, so the state vectors
1
D, 2 D of detector can not appear in the expression
of probability density at the screen, we can not merely
judge the disappearance of interference fringes by the
factors 12
DD and 21 DD in Expression (8). In
fact, generally, the particle is in a mixture state after de-
tected by the detector, the more precise description of the
disappearance of interference fringes should be based on
Expression (5), this disappearance is determined not only
by the factors 12 DD , 21 DD, but also by the coef-
ficients . Only in the special case of
01
,CC
12
1
2
CC , the disappearance of interference fringe
is determined by 12
DD , 21 DD.
In Scully et al.’s experiment, the detector are two ma-
ser cavity systems, the state vectors of detector are de-
scribed as 12
10 and 12 01 , where 12 10 denotes the
state in which there is one photon in Cav ity 1 and none in
Cavity 2, the interaction between atom beam and maser
cavity system lead to the correlation between them, and
the initial pure state of atom will reduce to a mixture
state when it arrives at the screen, which causes the dis-
appearance of interference fringes. In Dürr et al.’s ex-
periment, two internal electronic states 2 and 3 of
atom are used as a which-way detector system.
Since the states of detector and the states of center-of-
mass motion belong to the same atom, both of them can
appear on the screen, the state vectors
85 b
R
2 and 3 of
detector can appear in the probability density at the
screen similar to (8), then the loss of interferen ce fringes
is determined by the factors 23 and 32 . However,
as pointed out by Dürr et al., there are additional states of
detector must be considered except the internal electron
states of atom, they are the quantum states

and

of microwave field, where

denotes the initial
r

, (6)

Z. C. WANG
22
REFERENCES
state of microwave field,

is the quantum state after
the absorption of one photon.

and

can not
appear on the screen hence in the expression of probabil-
ity density, the complete states of detector should be
2

and 3

, not merely the internal electronic
states 2 and 3 of atom, so in essence, we also need
discuss the disappearance of interference fringes by use
of Formula (5). In Dürr et al.’s experiment, the entan-
glement of the atom with the microwave field can be
simply neglected because the initial state

is a co-
he rent state with a large mean photon number and a large
spread of the photon number, so we can approximately
discuss the disappearance of interference fringes by Ex-
pression (8), otherwise, we must study this issue by Ex-
pression (5).
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3. Conclusion
In summary, we have further shown the role of detector
in which-way experiment. It is this interaction between
particle and detector that leads to the change of quantum
state of particle from initial pure state into a mixture state,
and the disappearance of interference fringes. We also
describe the disappearance of interference fringes by a
deviation between the initial pure state and the mixture
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doi:10.1103/PhysRevD.26.1862 This work is supported by the NNSF (Grant No. 10404037
and 11274378).