The Role of Detector in Which-Way Experiment

doi:10.4236/jqis.2013.31005

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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

Email: [email protected]

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







and







, 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].









  

1111 222

tt Dr

ttCaDr CbDr



 

 



. (1)

In the above, 012

,,DDD are the state vectors of

detector, while



 

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







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

111211

122 112

211 221

221222

Trtttt

CaCaDDrr

CC abDDrr

CCbaD Drr

CbCbD Drr



















, (2)

above

Tr indicates partial trace over the detector de-

grees of freedom. When the state vectors 12

,DD

Z. C. WANG 21

detector are orthogonal to each other, the density matrix

can reduce to

  

31 11222

Carr Cbrr

 

 , (3)

which indicates pure state





has become to a

mixture state. Generally, the pure state





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





. We can evaluate

the above deviation by the difference between 3



and

the density matrix 1



of pure state

 

rarbr







, it is

,nm





. (4)

Considering the density matrix 3



in the above Ex-

pression (2), this deviation can be further written as

22 2

21112

122 1

211 2

2212

CaCaDDa

C CabDDab

C C baDDba

CbCbDDb



 

 



 

. (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

CC , then the deviation will vanish.

In this special case, the state vector of combined parti-

cle-detector system is

  



1112

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

 







and the probability density of particle falling on the screen

  

 

12 21

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

  

  

12122121

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

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

,CC

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

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

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



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

state of particle, which is consistent with the experimen-

tal results of Schully et al.’s and Dürr et al.’s.

[7] M. S. Chapman, T. D. Hammond, A. Lenel, J. Schmied-

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Um ansky, “Dephasing in Electron Interference by a ‘Which-

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4. Acknowledgements [9] W. H. Zurek, “Environment-Induced Superselection Rules,”

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doi:10.1103/PhysRevD.26.1862

This work is supported by the NNSF (Grant No. 10404037

and 11274378).