Foundations of Physics, Vol . 28, No. 3, 1998
An Operational Analysis of Quantum Eraser
and Delayed Choice 1
Marlan O. Scully 2 and Herbert Walther 2
Received October 10, 1997
In the present paper we expand upon ideas published some time ago in connection
with which path detectors based on the micromaser . Frequently questions arise
concerning the time ordering of detection and eraser events. We here show, by a
detailed and careful analysis of a quantum eraser experimental setup, that the
experimenter can choose to ascertain particle-like which path information or
wavelike interference information even after the atom has hit the screen.
1. INTRODUCTION AND CONCLUSION
Complementarity, e.g., wave± particle duality, lies at the heart of quantum
physics. For example, in Young’s double slit experiment a ``particle’’ displays interference ( wave-like) behavior when it ``goes through’’ both slits.
But as soon as we have which slit ( particle-like) information we lose the
interference pattern.
Just how the acquisition of which way ( Welcher Weg ) information
rubs out the interference fringe is an interesting question. The most common route to incoherence is clearly stated by Feynman who says that it is
impossible to get Welcher Weg information without disturbing the interference pattern, to wit: ( 1 )
``If an apparatus is capable of determining which hole the electron [ or atom
or ...] goes through, it cannot be so delicate that it does not disturb the pattern
in an essential way.’’
1
We dedicate this paper to our friend and colleague Asim Barut who always delighted and
inspired us with his insights and ideas concerning basic physics.
2
Max-Planck-Institut fuÈr Quantenoptik, 85748 Garching, Germany and Department of
Physics, Texas A&M University, College Station, Texas 77843.
399
0015-9018/98/0300-0399$15.00/0
Ñ
1998 Plenum Publishing Corporation
400
Scully and Walther
But, as was pointed out some time ago, ( 2 ) it is not necessarily the
indelicate nature of our probing that rubs out the interference pattern. It
is simply knowing ( or having the ability to know even if we choose not to
look at the Welcher Weg detector) which eliminates the pattern. This has
been verified experimentally.
Hence one is led to ask: what would happen if we put a Welcher Weg
detector in place ( so we lose interference even if we don’t look at the detector)
and then erase the which way information after the particles have passed
through? Would such a ``quantum eraser’’ process restore the interference
fringes? The answer is yes and this has also been verified experimentally.
One particularly vivid, and physically sensible, arrangement demonstrating the quantum eraser concept ( 3 ) is shown in Fig. 1. In this setup the
atoms are detected, one by one, at the screen and an interference pattern
is observed or not, depending on what the experimenter does with his
which path detectors. That is, if the Welcher Weg information is present in
the detectors then there can be no interference fringes observed even after
accumulating a large number of counts. However, if this information is
``erased’’ then fringes can be recovered.
As we have pointed out elsewhere, this erasure ( and fringe retrieval)
can be achieved even after the atoms have hit the screen. This somewhat
Fig. 1. Quantum eraser overview. ( a) Electro-optic shutters separate microwave
photons in two cavities. (b) Detector wall absorbs microwave photons and acts as
a photodetector. Plot is density of particles on the screen depending upon whether
a photocount is observed in the detector wall (``yes’’ ) or not (``no’’), demonstrating
that correlations between the event on the screen and the eraser photocount are
necessary to retrieve the interference pattern.
An Operational Analysis of Quantum Eraser and Delayed Choice
401
subtle point has been the subject of some confusion; for example we are
often asked whether this eraser must be carried out before the atom hits the
screen.
In actual fact, the experimenter’s choice exists as long as the micromaser
Welcher Weg photons are not lost due to finite cavity Q, that is, for times
which are of order 0.1 sec. In other words, the experimenter can indeed
choose to analyze his data in ways that will show fringes ( or not) by
manipulating the which path detectors ( erase or not to erase, experimenter’s choice) even long after the atoms are detected at the screen.
In order to bring the physics into sharper focus we here present a
detailed physical model of an atomic detector array ( screen ) and the
microwave Welcher Weg radiation as it evolves under various quantum
eraser scenarios. Each detector, atomic and photon, will be equipped with
shutters which are opened, for a short time, at the discretion of the
experimenter.
The envisioned setup, as shown in Fig. 2, is a more detailed physical
model of the quantum eraser experiment of Fig. 1, the point being that this
model permits detailed specific calculations yielding answers to physically
realizable experiments.
In particular we consider an array of detectors at the screen at various
®
points r i . Each detector is equipped with a shutter which opens at time
t a for a short time t a . Likewise, the quantum erasure photodetector is
activated by opening the corresponding shutters at time t m for a time t m .
Then, as shown in the following sections and Appendices A and B, ( 4 ) we
may easily calculate the state vector for the combined system.
Fig. 2. Quantum eraser detector array. The photon detector in Fig. 1 is represented
by a two-level system which is excited upon absorption of a photon. The screen
consists of an array of atom detectors.
402
Scully and Walther
For example, suppose that at time t o an atom is sent through the
micromaser-slit arrangement and arrives at the screen at some time
t f t o + d/v, where d is the distance to the screen and v is the mean particle
velocity. Suppose, furthermore, that we consider only detection events
corresponding to times t a substantially later than t o + d/v so that the atom
will surely have arrived at the screen.
Now in order to clarify some of the points raised by interested colleagues, let us consider the state of the i th detector, atomic wave function,
Welcher Weg ( WW) fields and eraser photodetector. In the limit of perfect
screen detectors, in the notation of Fig. 2, for times t> t m and t> t a , the
``erasure’’ state vector
| Y i ( t, t a , t m ) ñ
erase
= {w
( r i , t a ) [ cos gt m | b , s ñ 2
i sin gt m | a, 0 1 , 0 2 ñ ]
®
s
+ w
±
( r i , t a ) | b, s ñ } |e
®
sÅ
i
ñ
( 1a)
which represents the state in which the i th atomic detector is excited to the
state | e i ñ at t a and erasure shutters are opened at time t m , where a g is the
±
photodetector-field coupling strength. The | a, 0 ñ , | b , S ñ , and | b , S ñ are
the state of the excited photodetector± ± vacuum in the cavities and ground
state photodetector± ± and symmetric and antisymmetric radiation field
states ( 8a, b) respectively. The symmetric and antisymmetric probability
®
®
amplitudes w s ( r i , t a ) and w sÅ ( r i , t a ) written in terms of the amplitudes
associated with the i th detector excitation by an atom from hole 1 and 2,
®
®
i.e., w 1 ( r i , t a ) and w 2 ( r i , t a ) are given by
1
®
w
s
( ri , ta ) =
sÅ
( r i , ta ) =
Ï
2
®
(w
1
( r i , ta) + w
(w
1
®
2
( ri , ta) )
( 1b)
and
®
w
Ï
1
2
®
( ri , ta ) + w
®
2
( ri , ta ) )
( 1c)
If we do not open the photodetector shutter then we have ( see following section) the ``Welcher Weg ’’ state vector
|Y i ( t a ) ñ
ww
=
Ï
1
2
[w
1
( r i , t a ) | 1, 0 ñ + w
2
( r i , t a ) | 0, 1 ñ ] | e i ñ | b ñ
( 2)
Suppose we send an atom through the apparatus and find it excites
the i th detector at time t a . If we do not activate the quantum eraser then
An Operational Analysis of Quantum Eraser and Delayed Choice
403
we must use the WW State ( 2b) and we can correlate the frequency ( likeli®
®
hood) of exciting the i th detector, | w 1 ( r i , t a ) | 2/2 and | w 2 ( r i , t a ) | 2 /2 with the
WW information | 1, 0 ñ and | 0, 1 ñ .
But suppose we choose to open the eraser shutter at t m f t a , i.e., after
®
the atom has been detected at r i on the screen, but without making any
WW measurements of the type discussed in the previous paragraph. In this
case we write the joint count probability that the photodetector is excited
( or not) at time t m , and the i th screen detector records a count ( º ``clicks’’ )
at time t a as
P( a, i; t a , t m ) =
á
Y i ( t a , tm ) | { | e i ñ
á
ei | Ä
|añ á a| } | Y i ( t a , t m ) ñ
( 3a)
P( b , i; t a , t m ) =
á
Y i ( t a , tm ) | { | e i ñ
á
ei | Ä
| b ñ á b | } |Y i ( t a , t m ) ñ
( 3b)
We remind the reader that in the present section we assume that t> t m and
t> t a so that the time t is unimportant, i.e., will not appear in the final
results. Thus we could choose to correlate screen events with excited erasure
photodetector counts and ``sort the data’’ so as to record interference fringes
as per ( 3a) , independent of whether t m is earlier or later than t a .
Or we could turn things around and view Eqs. ( 3a, b) as giving the
®
probability that a screen detector event at r i gives us information about the
likelihood of the erasure atom being in state a or b .
Mathematically such conditional probabilities are given by
P( i | a; t a , t m ) =
P( i, a; t a , t m )
P( a; t m )
( 4a)
P( a | i; t a , t m ) =
P( i, a; t a , t m )
P( i; t a )
( 4b)
and
These single and joint count probabilities are easily shown ( see next
sections) to be
®
P( a, i, t a , t m ) = | Y s( r i , t a ) | 2 sin 2 gt m
P( a, t m ) = sin 2 gt m
P( i, t a ) = | w
( 5b)
®
s
( r i , ta) | 2+ |w
( 5a)
®
sÅ
( ri , ta ) | 2
( 5c)
404
Scully and Walther
and the conditional probabilities ( 4a) and ( 4b) are thus given by
P( i | a; t a , t m ) = | w
®
s
( ri , ta) | 2
®
P( a | i; t a , t m ) =
( 6a)
| w s( r i , t a ) | sin gt m
®
®
| w s ( r i , t a ) | 2 + | w sÅ ( r i , t a ) | 2
2
( 6b)
Note that all probabilities are independent of t m .
Thus we can correlate events on the screen given a count in the
erasure detector in order to regain interference fringes. Or we can turn
things around and say that we seek information concerning erasure
photodetector events in view of data present in the screen detectors. Then
( independently of whether t a > t m or t a < t m ) we view ( 6a) as giving the
``betting odds’’ that the i th atomic detector is excited given that the eraser
photodetector is in the a states, and vice versa for ( 6b).
As a case in point, let us consider the symmetric point on the screen such
®
that | r 1 | = | r 2 | = r and, writing the probability amplitudes as w 1 ( r 1 , t a ) =
®
®
w 2 ( r 2 , t a ) º 1/Ï 2 w o ( r, t a ), from Eq. ( 20a) we have
P( i | a; t a , t m ) = | w
o
( r, t a ) | 2
( 7a)
and
P( a | i; t a , t m ) = sin 2 gt m
( 7b)
What could be simpler?
As we said before: ``The choice falls to the experimenter’’: Do we want
particle-like WW information? Then keep the eraser shutters closed and
use Eq. ( 12). Do we want wavelike complementary information? Then
open the eraser shutters and use Eq. ( 1). It matters not in which order the
atomic detection and photodetection occur. The result is the same.
In the following we develop explicitly the states given by Eqs. ( 1, 2)
and discuss the problem in detail.
2. MODEL AND METHODOLOGY
Consider the atomic proximity detector array as depicted in Fig. 2.
®
There we see atom detectors at positions r i which are triggered by the
passage of an atom in the neighborhood of the detector. This is governed
by the atom± detector interaction Hamiltonian V a, d . We envision our
atomic detector as an ionization type detector which, in effect, annihilates
( ionizes) atoms as they pass close by. The detector responds to the emitted
An Operational Analysis of Quantum Eraser and Delayed Choice
405
Fig. 3. (a) Atomic photodetector absorbs a photon and
makes a transition to its excited state. (b) Quantum eraser
photodetector: The b ® a transition is sensitive to a symmetric
combination of photons 1 and 2.
election and is thus promoted from the ground to an excited state; this is
depicted in Fig. 3a.
Then, as shown in Appendix A, the probability for exciting the i th
detector at time t a is given by
| á e i , 0 | Ua, d ( t, t a ) | w
a
, giñ | 2= g |w
®
s
( r i , ta ) | 2
( 8)
where | g i ñ is the detector ground state, and the atomic center of mass state
is given by
|w
a
ñ
=
Ï
1
2
|w
1
ñ
+ |w
2
ñ
]
( 9)
406
Scully and Walther
with |w j ñ being the spherical atomic wave coming from hole j= 1, 2; the
time development operator for the atom± detector system Ua, d ( t, t a ), |0 ñ is
the atomic vacuum and | e i ñ is the excited state of the i ± n detector; g is the
detector efficiency, and the spherical wave w 1( r i , t a ) = á r i | w i ( t a ) ñ with a
similar expression for atoms coming from slit 2.
From Eq. ( 8) the interference cross terms w 1*( r i , t a ) w 2( r i , t a ) are
apparent. Note that it is the atomic wave function at time t a which is
important since the detector shutter is only open for a short time after t a .
Having set the stage by treating the center of mass interference
problem in detail we turn to the main problem, namely the inclusion of
micromaser Welcher Weg detectors and the option of quantum eraser, i.e.,
delayed choice.
The initial state of the atom, Welcher Weg , micromaser fields, quantum
eraser photodetector, and atomic detector array is
| Y ( 0) ñ =
Ï
1
2
[ |w
1
ñ | 1, 0 ñ
+ |w
2
ñ | 0, 1 ñ
] |W d ñ |W m ñ
( 10)
where the atomic states | w j ñ are the same as in the preceding paragraphs;
the Welcher Weg photons are described by the states | 1, 0 ñ and |0, 1 ñ
denoting the photon in cavity 1 or 2; the mitial state of the detector array
| W d ñ it that of all N detectors in their ground ``ready’’ states; and the initial
state of the microwave photodetector atom ( quantum eraser ) | W m ñ is
simply the atomic ground state | b ñ , which upon absorbing a Welcher Weg
photon is excited to | añ .
The microwave± photodetector interaction is turned on by opening a
shutter at time t m which remains open for a time t m and then closes at
t= t n + t m ; see Fig. 3b.
The temporal evolution from the initial state ( 4) is determined by the
usual U matrix expression
U( T ) = T exp 2
i
&
T
0
dt[ V a, d ( t) + V m, p ( t) ]
( 11)
where T is the time ordering operator. We note, however, that V a, d and
V m , p operate in different Hilbert spaces and therefore commute.
Hence, was may write
U( T , t a , t m ) = Ua, d ( T , t a ) Um , p ( T , t m )
( 12)
and it is clear that the time evolution of the atomic detection and
photodetection ( eraser) subsystems are completely independent. The consequences of this are developed in the next section.
An Operational Analysis of Quantum Eraser and Delayed Choice
407
3. ERASURE BEFORE AND AFTER THE ATOM HITS
THE SCREEN: WHAT’S THE DIFFERENCE?
First we note that the erasure photodetector process acts very differently on the symmetric and antisymmetric combinations of Welcher Weg
states
|s ñ =
1
Ï
2
[ | 1, 0 ñ + | 0, 1 ñ ]
( 13a )
[ | 1, 0 ñ 2
( 13b)
and
| s± ñ =
1
Ï
2
| 0, 1 ñ ]
so that
1
( a^ 1 + a^ 2 ) | s ñ =
[ | 0, 0 ñ + | 0, 0 ñ ] =
Ï 2
1
( a^ 1 + a^ 2 ) | s± ñ =
[ | 0, 0 ñ 2
Ï 2
Ï
2 | 0, 0 ñ
( 14a )
| 0, 0 ñ ] = 0
( 14b)
Thus motivated we rewrite Eq. ( 10), making use of the fact that
| s ñ + | s± ñ ]/Ï 2, etc. as
| Y ( 0) ñ = 12 [ | w
1
ñ
+ |w
2
ñ
] | s ñ | W d ñ | W m ñ + 12 [ | w
1
ñ 2 | w 2 ñ | s± ñ
] | W d ñ |W m ñ
( 15)
Hence the state of the system at time t, as found by letting the time
development operator ( 12) operate on ( 15), is
| Y ( t) ñ = Ua, d ( t, t a ) 12 ] | w
ñ
1
+ Ua, d ( t, t a ) 12 ] | w
+ |w
1
2
ñ
] | W d ñ Um , p ( t, t m ) | s ñ | W m ñ
ñ 2 |w 2ñ
] | W d ñ Um, p( t, t m ) | s± ñ | W m ñ
( 16)
where we have included the shutter times t a and t m explicitly in the
U-matrices in order to emphasize the independence of the two observation
times.
408
Scully and Walther
In Appendix B we take the simplest possible photodetector, a twolevel atom with excited state | añ and ground state | b ñ = | W m ñ , and show
that
Um, p( t, t m + t m ) | s, b ñ = cos gt m | s, b ñ 2
±
i sin gt m | 0, 0, añ
±
Um, p( t, t m + t m ) | s , b ñ = | s , b ñ
( 17a )
( 17b)
In view of ( 17a, b) the total state ( 16) reads
1
| Y ( t, t a , t m ) ñ =
Ia, d ( t, t a )
1
Ï
[ |w
Ï
2
3
{ [ cos gt m | s, b ñ 2
+ | b , s ñ h( t m 2
+
1
Ï
2
2
1
ñ
+ |w
2
ñ
] |W d ñ
i sin gt m | 0, 0, añ ] h( t 2
tm)
t) }
Ua, d ( t, t a )
Ï
1
2
[ |w
1
ñ
+ |w
2
ñ
] | W d ñ | s± , b ñ
( 18)
From Eq. ( 18) we easily show that the probability amplitude that the
i th detector clicked at t a and the Welcher Weg detector is excited or not
( erasure complete or not) is given by
á
e i | w ( t, t a , t m ) ñ = w
( r i , t a ){[ cos gt m | s, b ñ 2
®
s
+ | s, b ñ h( t m 2
t) }+ w
sÅ
i sin gt m | 0, 0, añ ] h ( t 2
±
( r i , t a ) |s, b ñ
tm )
( 19)
®
where w s( r i , t a ) and w sÅ ( r i , t a ) are given by Eqs. ( 1b, c). Equation ( 19) is the
key state vector which is Eq. ( 1) of the introduction when t> t m . Further
technical details are given in the appendices.
APPENDIX A. ATOMIC DETECTOR ARRAY AND
INTERFERENCE OF ATOMIC DE BROGLIE WAVES
Consider the atomic proximity detector array as depicted in Fig. 2.
®
There we see atom detectors at positions r i which are triggered by the
passage of an atom in the neighborhood of the i th detector. This is governed
by the atom± detector interaction Hamiltonian, in the interaction picture,
which is given by
®
v g( r i 2
V a, d ( t) = +
i
®
à ie,² ( t) Dà ig ( t) w à ( r, t) + adj
r, t, t a ) D
( A.1)
An Operational Analysis of Quantum Eraser and Delayed Choice
®
409
®
where v g ( r i 2 r , t, t a ) is the coupling strength for excitation of the detector
®
at r i at time t a ( shutter opening time t a < t< t a + t a ), D ie² ( t) and D ig ( t) are
the creation and annihilation operators for the i th atom detector in the
®
excited and ground states, and w à ( r, t) is the corresponding operator which
annihilates an atom at r, t.
The initial state for the atom ( prepared by a ``double-slit’’ assembly)
and the detector array is given by
1
| Y ( 0) ñ =
Ï
[ |w
2
1
ñ
+ |w
2
ñ
] |W d ñ
( A.2)
where | w j ñ represents a spherical wave originating at slit j= 1, 2 and | W d ñ
denotes the ground state of the detector array given by
|W d ñ = *
| gi ñ
( A.3)
i
which represents the state in which all atom detectors are in the ground or
``ready’’ configuration.
The state of the atomic center of the mass± ± detector array at time T
is determined by the usual U matrix, that is,
| Y ( t) ñ = Ua, d ( t) | Y ( 0) ñ
( A.4)
where, to a good approximation,
Ua, d | T
ñ @
12
i
&
t
dt ¢ V a, d ( t¢ )
0
( A.5)
á
Hence the probability amplitude for exciting the i th atomic detector is
given by
e i , O | U ia, d ( T , t a ) | Y ( 0) ñ
@ 2
i
&
t
dt¢ á e i , O | V
0
i
a, d
( t¢ , t a ) | g i , w ( 0) ñ
( A.6)
where | w ( 0) ñ = [ | w 1 ( 0) ñ + | w 2 ( 0) ñ ]/Ï 2. Using a delta function in time to
model the shutter and using ( A.1) we have
á
e i , O | Ua, d | Y ( 0) ñ = 2
3
= 2
i
&
®
dt v e, g( n 2
0 | w à ( r 2 t)[ | w
á
i
ve, g
Ï
2
®
( ri2
à +e ( t) D
à +g ( t) | g ñ
ta ) á e| D
®
r ) d( T 2
2
ñ
+ |w
2
ñ
]/Ï
r) á 0 | w à ( r 1 t a ) | w
2
2
ñ
+ 1«
2
( A.7)
410
Scully and Walther
Writing out the atomic wave function for hole 1 we have
á
à ( r® , t a ) | Y ( 0) ñ =
0| Y
à ( r® ) e {
0 | e iH o ta Y
á
=
n
á
à ( r ) |n ñ
0| Y
á
®
n| e
=
&
dr on á r | n ñ
=
&
dr o G( r , t a ; r o , 0) w
®
á
®
|w
iH o ta
{
1
iH o ta
n | r oñ e{
®
ñ
|w
i en t a
1
á
ñ
ro | w
( r o , 0)
( A.8) )
®
e i 0 | U iad | Y ( 0) ñ = 2
i
Ï
vo
2&
®
ñ
®
1
Inserting ( A.8) into ( A.7) and using the fact that v e, g ( r i 2
®
peaked about r i we have
á
1
®
dr o G( r i , t a ; r o , 0)[ w
®
r ) is sharply
®
1
( r o , 0) + w
®
2
( r o , 0) ]
( A.9)
Hence the probability of exciting the i th detector at time t a is given by
| á e i , 0 | U iad | Y ( 0) ñ | 2 = g | w 1 ( r i , t a ) | 2 º P( r i , t a )
®
®
( A.10)
APPENDIX B. ADDING WELCHER WEG DETECTORS
AND QUANTUM ERASER
Having set the stage by treating the center of mass interference
problem in detail we turn to the main problem, namely the inclusion of
micromaser Welcher Weg ( which way) detectors and the option of quantum
eraser, i.e., delayed choice.
The initial state of the atom, Welcher Weg , micromaser fields, quantum
eraser photodetector and atomic detector array is
| Y ( 0) ñ =
Ï
1
2
[ |w
1
ñ | 1, 0 ñ
+ |w
2
ñ | 0, 1 ñ
] |W d ñ |W m ñ
( B.1 )
where the atomic states | w i ñ and the detector array state vector are the
same as in Appendix A; the Welcher Weg photons are described by the
states | 1, 0 ñ and | 0, 1 ñ denoting one photon in cavity 1 or 2 and the initial
state of the quantum eraser photodetector | w m ñ = | b ñ which upon absorbing
a microwave photon is excited to | añ .
An Operational Analysis of Quantum Eraser and Delayed Choice
411
The total interaction Hamiltonian is now
V ( t) = V a, d ( t) + V m, p ( t)
( B.2 )
where the atom± detector contribution, V a, d ( t), given by Eq. ( A.1) and the
micromaser photon-eraser photodetector term V m , p ( t) may be written as
à ²a( t) R
à b ( t) + adj
V m, p ( t) = g( t m ) ( a^ 1 ( t) + a^ 2 ( t) ) R
( B.3 )
where g is the ( time-dependent) photodetector-radiation coupling constant
which is switched on at time t m , a^ j ( j= 1, 2) is the radiation annihilation
à ²a and R
à b create and annihilate a photooperator for the j th cavity, and R
detector electron in the excited | añ and ground | b ñ states, respectively.
It is convenient to rewrite the initial state ( 11) in terms of the symmetric and antisymmetric micromaser states
|s ñ =
| s± ñ =
1
Ï
2
1
Ï
2
[ | 1, 0 ñ + | 0, 1 ñ ]
( B.4a )
[ | 1, 0 ñ + | 0, 1 ñ ]
( B.4b )
since the interaction Hamiltonian ( B.3) couples only the | s ñ state. That is,
V m, p ( t) | s ñ = g( t o ) R ²a( t) R b ( t) e {
in t
| 0, 0 ñ
±
V m, p ( t) | s ñ = 0
( B.5a )
( B.5b )
where n is the microwave frequency which is the same for both cavities. For
simplicity we shall take the frequency difference between states | añ and | b ñ
to be equal to n, so that the time dependence implicated in Eqs. ( B.5a ) and
( B.3 ) drops out; i.e. the interaction Hamiltonian V m, p is time independent.
The state of the total system at time t is
| Y ( t) ñ = U( t)
Ï
1
2
[ |w
1
ñ | 1, 0 ñ
+ |w
2
ñ | 0, 1 ñ
] |w d ñ | w m ñ
( B.6 )
where the time evaluation operator U( t) is given by
U( t) = Ua, d ( t) U m, p ( t)
( B.7 )
It is important to note that the two parts of ( B.7), corresponding to the
two parts of the interaction Hamiltonian of Eq. ( B.2) , are independent
since V a, d ( t) and V m, p ( t) commute.
412
Scully and Walther
Um ,
The atom-detector U matrix is given by Eq. ( B.7); in order to specify
p we insert ( B.7) into ( B.6) and use ( B.4a, b) to write
| Y ( t) ñ = Ua, d ( t)
1
Ï
2
[ |w
1
+ Ua, d ( t)
Ï
ñ
1
[ |w
2
+ |w
1
2
ñ
] | w d ñ Um, p ( t) | s ñ | w p ñ
] | w d Um , p ( t) | s± ñ | w p ñ
ñ 2 |w 2 ñ
( B.8 )
We proceed by taking our photodetector coupling constant to be
turned on at time t m ( shutters in Fig. 2 open) and off at time t m + t m ( shutters
closed). Then we have
Um, p ( t m + t m ) | s ñ | w d ñ = cos gt m | s ñ | b ñ 2
i sin gt m | 0, 0 ñ | añ
( B.9 )
so that for t= p/2g we have
Um , p | s ñ | w p ñ = | 0, 0 ñ | añ
( B.10a)
whereas, in view of Eq. ( 15b) we have
Um , p | s± ñ | w p ñ = | s± ñ | b ñ
( B.10b )
indicating that half the time the photodetector is not excited.
In view of Eqs. ( B10a, b) we may write ( B.8 ) as
| Y ( t) ñ = Ua, d ( t)
1
Ï
+ Ua, d ( t)
2
[ |w
1
Ï
2
1
ñ
[ |w
+ |w
1
2
ñ
] | w d ñ | 0, 0 ñ | añ
ñ 2 |w 2ñ
] | w d ñ | s± ñ | b ñ
( B.11 )
so that the joint probability amplitude for finding the i th atomic detector
excited at t a and the eraser process carried out at t m ( photodetector in state
| añ ) is given by
á
e i , a, 0 | U( t, t a , t m ) | Y ( 0) ñ =
’
g
(w
2
®
1
( ri , ta) + w
®
2
( ri , ta) )
( B.12 )
which is the same as the result found in Appendix A with an overall factor
of 1/Ï 2. That is, the joint probability for erasing the which way information at t m and finding the i th detector excited at time t a is
®
®
P a( r i , t a , t m ) = 12 P( r i , t a )
where P( r i , t a ) is given by ( 10a) and is independent of t m .
( B.13 )
An Operational Analysis of Quantum Eraser and Delayed Choice
413
Likewise, when the atomic shutters are opened at t a and the erasure
photodetector at t m but no count is registered in the photodetector,
Eq. ( B.13) yields the joint probability amplitude
á
e, b , 0 | U( t, t a , t m ) | Y ( 0) ñ =
’
g
®
( w ( ri , ta) 2
2
w
®
2
( ri , ta ) )
( B.14 )
which again is independent of t m ; but now the interference pattern will be
shifted as in Fig. 3.
REFERENCES
1. R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. III
( Addison Wesley, Reading 1965) , pp. 1± 9.
2. M. O. Scully, R. Shea, and J. McCullen, Phys. Rep. 43, 486 (1978).
3. M. O. Scully, B.-G. Englert, and H. Walther, Nature 351, 111 ( 1991); Sci. Am. 271, 56
( 1994).
4. U. Mohrhoff, Am. J. Phys. 64, 1468 (1996).