Download A quantum delayed choice experiment

Wave-particle
delayed-choice
experiment
Zheng Shi-Biao
Fuzhou University
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Contents
1. Wheeler’s delayed-choice experiment
2. Quantum delayed-choice experiment
with a controlled beam splitter
3. Quantum delayed-choice experiment
with a Schrödinger cat-like beam splitter
4. Conclusion and discussion
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I. Wheeler’s delayed-choice experiment
A. Wheeler's proposal
Wave-particle duality contains the “only
mystery” of quantum mechanics (Feynman).
In quantum mechanics, the definition
of ‘wave’ or ‘particle’ means ‘ability’ or
‘inability’ to exhibit interference.
Quantum interference can occur either
in ordinary space or in abstract space of
quantum states.
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I. Wheeler’s delayed-choice experiment
According to complementarity, test of these two
complementary phenomena needs experimental
arrangements that are mutually exclusive.
This is well illustrated by the Mach-Zehnder
interferometer
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I. Wheeler’s delayed-choice experiment
Local hidden variable model: the photon
knows in advance the experimental
arrangement.
To exclude this possible causal link, Wheeler
proposed the delayed-choice experiment:
The observer randomly chooses to insert BS2
or not after the photon has passed through BS1.
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I. Wheeler’s delayed-choice experiment
According to quantum mechanics, the
delayed choice makes no difference on the
outcomes of measurement.
In Wheeler’s words: “one decides
whether the photon shall have come by one
route or by both routes after it has already
done its travel’’.
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I. Wheeler’s delayed-choice experiment
B. Experimental realization
Experimental realization of Wheeler’s
delayed-choice Gedanken experiment
Science 315, 966 (2007)
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I. Wheeler’s delayed-choice experiment
In the experiment, the choice of inserting BS2
or not is separated from the entrance of the
photon into the interferometer by a space-like
interval.
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I. Wheeler’s delayed-choice experiment
To avoid conflict with relativity, the
definition ''quantun phenomemon'' should be
clarified. In Wheeler's words: ''elementary
phenomenon is a phenomenon until it is a
registered (observed) phenomenon.''
We get a counter reading but we neither know
nor have the right to say how it came. The
process can be illustrated in the cartoon (Miller
and Wheeler, 1983)
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I. Wheeler’s delayed-choice experiment
The sharp tail and head of the dragon correspond
to initial state input and the result of the observation,
respectively. The body of the dragon, is unknown and
smoky.
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II. Quantum delayed-choice experiment
with a controlled BS
A. Theoretical proposal
[Phys. Rev. Lett. 107,230406 (2011)]
Here the Hadamard transformation H plays the
role of beam splitter:
1
0 
0 +1
2

and
1
1 
 0  1 .
2
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II. Quantum delayed-choice experiment
with a controlled BS
After the first transformation, the state of the
photon:
1
0 
 0 + 1 .
2
Then a phase shifter leads to
1
i
 
0

e
1 .

2
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II. Quantum delayed-choice experiment
with a controlled BS
The second transformation (H2) is controlled
by the ancilla.
If the ancilla is in 0 a ，H2 is absence，and the
photon remains in the state
p
1
i

0

e
1 .

2
Then I0=I1=1/2，revealing the particle
behavior.
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II. Quantum delayed-choice experiment
with a controlled BS
If the ancilla is in 1 a ，H2 is present. This
results in
w
1
  0  1   ei  0  1  
2

 
i /2 
 e  cos 0  i sin 1  .
2
2 

Then I 0  cos
2

2
and I1  sin
2

2
, revealing the
wave nature.
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II. Quantum delayed-choice experiment
with a controlled BS
Suppose the ancilla is initially in
cos  0 a  sin  1 a .
The final state of the whole system is
cos   p
Then
s
0 a  sin   w
s
1 a.
1
2
2
2 
I 0  cos   sin  cos .
2
2
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II. Quantum delayed-choice experiment
with a controlled BS
To distuiguish between the wave-like and
particle components, one should measure the
ancilla, and correlate the measurement data of
the photon and that of the ancilla.
When the ancilla is detected in

a
  0 a  1 a  / 2，
the photon collapses to
cos   p
s
 sin   w s .
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II. Quantum delayed-choice experiment
with a controlled BS
The importance includes:
(1). The complementary phenomena can be
observed with a single experiment setup.
(2). The morphing between particle and wave
can be observed.
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II. Quantum delayed-choice experiment
with a controlled BS
B. Experimental realizations
(1). Realization of quantum Wheeler's delayedchoice experiment [Nature Photonics 6, 600 (2012)]
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Remarks: Here the quantum
nature of the wave-particle
superposition has not been
verified: The shapes of the
interference fringes can not
be used to verify the existence
of the quantum superposition.
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II. Quantum delayed-choice experiment
with a controlled BS
(2). Entanglement-enabled delayed-choice
experiment [Science 338, 637 (2012)]
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The result clearly shows the morphing
between the wave and particle behaviors.
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II. Quantum delayed-choice experiment
with a controlled BS
The experiment also verified the quantitative
wave-particle duality
V  D 1
2
V:
D:
2
fringe visibility;
the available amount of which-path
information.
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II. Quantum delayed-choice experiment
with a controlled BS
The quantum nature of the photon’
behavior is manifest in the nonclassical
correlation between the test photon
and ancilla photon.
This is verified by violation of the
Bell's inequalities.
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II. Quantum delayed-choice experiment
with a controlled BS
Suppose that we measure the polarization of the two
photons along the x and y directions, the correlation
function E  x, y  is
E  x, y   P0,0  x, y   P1,1  x, y   P0,1  x, y   P1,0  x, y  .
The Bell's signal can be defined as
S  a, a , b, b   E  a, b   E  a, b    E  a , b    E  a , b  .
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II. Quantum delayed-choice experiment
with a controlled BS
For a classical system
S (a, a ' , b, b' )  2,
while quantum mechanics permits
In this experiment,
S (a, a ' , b, b ' )  2 2,
S (a, a ' , b, b' )  2.77.
It should be noted that when the experiment
is subject to some loophole, a Bell violation is
allowed by a classical model.
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II. Quantum delayed-choice experiment
with a controlled BS
Locality/''lightcone'' loophole: The correlation
between separated events can result from
unknown subluminal signals.
In the experiment,
the two detection
events have a space
-like separation so
that the locality
loophole is closed.
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II. Quantum delayed-choice experiment
with a controlled BS
Detection loophole: When the detection
efficiency is low enough there is a possibility
that the subensemble of detected events agrees
with quantum mechanics, but the entire ensemble
satisfies Bell's inequalities.
This is the case in most optical experiments.
So far, no loophole-free Bell inequality violation
has been experimentally demonstrated.
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II. Quantum delayed-choice experiment
with a controlled BS
(3). A quantum delayed choice experiment
[Science 338, 634 (2012)]
The experiment was performed on a silica-on-silicon
photonic chip.
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II. Quantum delayed-choice experiment
with a controlled BS
Smax  2.45. Again, the Bell test is subject to
the detection loophole.
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
A. The proposal
Ramsey interferometer
A two-level system (qubit) subjected to two pulses, R1 and R2(θ).
R1: splits the qubit state, producing two paths (|g> and |e>) in the
Hilbert space;
R2(θ): recombines them.
Here the tunable relative phase shift θ is incorporated into the
second pulse R2(θ).
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
R1 is produced by the field stored in a resonator,
R2(θ) produced by a classical field.
When the resonator is filled with a coherent field (  ), R1 is
present, leading to
 g 
g
i e

2.
R2(θ) performs the transformations
g   g  ie  i e
e   e  ie  i g


2，
2，
resulting in the state
 w  1  ei  g  i 1  ei  e  2.
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
There exist two paths leading to the final state e ：
g  g  e ,
R1
R2
g  e  e ,
R1
R2
which are indistinguishable, leading to Ramasey interference
Pe  cos
Therefore, the state  w
2

.
2
describes the wave behavior.
For an empty resonator ( 0 ), Pe  1 2 and no Ramsey
interference occurs.
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
Suppose that the resonator is in the Schrödinger
cat state
 b ,i  N  cos    sin  0  ,
acting as the quantum beam splitter (QBS). After R2
the qubit-QBS system is approximately in the
entangled state
 q b, f  Nt cos   w   sin   p 0 .


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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
The probability for finding the qubit in the excited
state approximates to
2 1
2
2
2  
Pe  N t  sin   cos  cos  .
2
2
The wave and particle behaviors of the qubit can be
investigated with the same experimental arrangement
by preparing the resonator in the cat state.
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
The Ramsey interference pattern corresponds to a
superposition of the wave and particle behaviors.
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
B. Experimental implementation
We realize the quantum delayed-choice experiment
with a circuit QED system, in which two phase qubits
are coupled to a resonator. (experiment performed in
ZJU)
Qubit 1: the test qubit for the quantum delayed-choice experiment;
Qubit 2: ancilla qubit used to prepare and probe the resonator field.
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
This figure displays the
measured probability Pe as a
function of  and  , which
clearly demonstrates the
morphing behavior between
the particle and wave. Here
  2.
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
To distinguish the wave and particle
components, the QBS state is examined.
This is achieved by effectively coupling the
ancilla qubit initially in the state g  to the
resonator.
Under certain condition, the ancilla undergoes
the transition g   e for  , while no
evolution for 0 .
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
The test qubit behavior is postselected by
correlating its data with the outcomes of the
measurements on the ancilla.
This figure shows the measured
probabilities Pe ;e and Pe ; g  , versus 
for detecting the test qubit in the state
e conditional upon the detection of
the ancilla in e and g  , respectively.
The parameter  is  / 4.
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
As expected, Pe ;e exhibits the Ramsey
interference fringes with the contrast reaching
0.83, while Pe ; g  almost remains constant.
The experimental sequence for Ramsey
interference is shown in the following figure.
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
C. Verification of quantum coherence
The interference pattern can be produced for
the classical mixture
cos     sin  0 0 .
2
2
To exclude the classical interpretation, the
quantum coherence between  and 0 should
be verified.
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
The quantum state of the resonator field can be
characterized by the Winger function (WF), which
describes the quasiprobability distribution of the field
in phase space. The WF W    associated with the
density operator  is defined as
W   
2

 1


n 0
n
 nn    ,
   D     D    .
This quantity is non-negative for a classical mixture;
the existence of its negative values is a signature of
nonclassical nature.
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
The WFs displayed for
three cases:
(A) Test qubit state not
read out;
(B) measured in |g>;
(C) measured in |e>.
Upper: Simulated; Lower: measured. The parameters:
  2,   / 2 and    / 4.
As a result, the WF exhibits a strongly nonclassical feature
around   1.
The minimum values of the three measured WFs are
-0.258±0.030, -0.342±0.027 and -0.336±0.028, respectively.
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
We also measured the corresponding WFs for
  /8
and
  3 / 8.
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
One measurement outcome was obtained every time,
so that our experiment is not subject to the detection
loophole.
Neither does the other loophole associated with the
Bell test, i.e., the locality loophole, plague our experiment
as the Wigner tomography does not require any
measurement of nonlocal correlation.
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
D. The transition from a quantum to classical
beam splitter
The qubit-resonator interaction is delayed for
a time T. Then, just before this interaction the
field density operator is given by
b  N 2 [cos 2       sin 2  0 0
1   2 1e  T  /2
 e
sin  2     0  0   ],
2
where     e  T / 2 and  is the decay rate of a
photon in the resonator .
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
With suitable choice of the qubit-resonator
interaction time, we have
 g    w ,
0 g  0 p .
We display the measured
Ramsey interference signals
for    / 4 with different
delays.
The result shows that
the fringe contrast is
insensitive to the field
decay.
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III. Quantum delayed-choice experiment
with a Schrödinger cat-like BS
We also measured the negative-valued minimum value
of the WF as a function of delay T, with the WF measured
after R2 but without reading out the test qubit state.
For γT=1/3, the field amplitude
is reduced by only 15%, but the
absolute value of the minimum
negative value of the WF almost
decays to 0, implying the quick
damping of the quantum coherence.
Theoretically the quantum
coherence is shrunk by a factor
of 0.51 after this delay.
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IV. Conclusion and remark
Delayed-choice experiments play an important role in
understanding fundamental aspects of quantum physics:
a. Wheeler's delayed-choice experiments
challenge a realistic explanation of the wave-particle
duality (hidden variable model).
b. Quantum delayed-choice experiments suggest
a reinterpretation of the complementarity principle:
complementarity of the experimental data, rather
than complementarity of the experimental setups.
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IV. Conclusion and discussion
c. For our system, with the increase of size and
fidelity of the cat state, we can distinguish between
the present and absent states more clearly; explore
the gradual transition from a quantum to a
classical beam splitter.
d. Our proposal can also be realized in microwave
cavity QED and ion-trap setups.
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