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Department of physics
Seminar Ia
Quantum Cheshire Cat
Author: Tadej Meˇznarˇsiˇc
Mentor: prof. dr. Anton Ramˇsak
Ljubljana, May 2015
Abstract
This seminar presents quantum phenomenon known as quantum Cheshire Cat. It starts by describing its paradoxical nature in the regime of regular measurement on an example of photon and
its polarization. It continues into detailed description of weak measurement by presenting a double
Stern-Gerlach experiment. Then it shows how to implement the principle of weak measurement
into Cheshire Cat experiment for photons and ends with presentation of another experiment, proving that Cheshire Cat also applies to neutrons and their spin.
Contents
1 Introduction
1
2 Cheshire Cat
2
3 Weak Measurement
3.1 Weak Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Weak Measurement in double Stern-Gerlach Experiment . . . . . . . . . . . . . . .
3.3 Weak Measurement and Quantum Cheshire Cat . . . . . . . . . . . . . . . . . . . .
4
4
6
8
4 Quantum Cheshire Cat in Neutron Interferometry Experiment
9
5 Conclusion
1
10
Introduction
In the world of quantum mechanics we encounter many interesting and unusual phenomena. One
of them is quantum Cheshire cat, named after a cat that vanishes and leaves behind only a grin
from the novel Alice in Wonderland. In quantum mechanics we don’t have cats that grin but
particles with properties (e.g. spin). With carefully assembled experimental setup scientists have
been able to separate the property from the particle just like the cat separates itself from its grin.
This has been done with photons, separating their polarization from them, and more recently
by separating spin from neutrons in a neutron interferometry experiment [1]. In such experiments
it is of utmost importance to carefully choose pre- and post-selected ensemble, meaning we have
to prepare our particles in initial state |Ψi i and when they exit our experimental setup we perform
post-selection so that we get final state |Ψf i. Only if states |Ψi i and |Ψf i are chosen correctly can
we observe quantum Cheshire Cat (from now on QCC).
Figure 1: Picture of quantum Cheshire cat inside an interferometer. Cat travels along the upper
path while its grin travels along the lower one. Source: [1]
1
Let us take a look at an experimental design that can be used for observation of QCC by
separating photons from their polarization that is described in [2].
2
Cheshire Cat
In this experiment we have a photon in two possible locations, |Li and |Ri. Its property, the
cat’s grin, is circular polarization, with two basis states |+i and |−i. In terms of horizontal and
√
vertical linear polarization
states
|Hi
and
|V
i
they
can
be
expressed
as
|+i
=
(|Hi
+
i
|V
i)/
2
√
and |−i = (|Hi − i |V i)/ 2.
Initial state of the photon is
1
|Ψi i = √ (i |Li + |Ri) |Hi ,
2
(1)
horizontally polarized superposition of two positions |Li and |Ri. Such state can be prepared by
sending a horizontally polarized photon into a beam splitter as depicted in figure 2, denoted by
BS1 . The reflected beam |Li gains a phase factor i.
In post-selection we would like to get the state
1
|Ψf i = √ (|Li |Hi + |Ri |V i),
2
(2)
which means we would like to perform a measurement that returns answer ’yes’ when photon is
in state |Ψf i and returns ’no’ if photon is in a state orthogonal to |Ψf i. We will then examine
only cases with an affirmative answer. This measurement can be performed in an optics setup as
depicted in figure 2.
2
Figure 2: Schematic diagram of the experimental setup, which comprises of two beam splitters
(BS1 and BS2 ), half wave plate (HWP), phase shifter (PS), a polarizing beam splitter (PBS) and
three photon detectors (D1 , D2 , D3 ). Source: [2].
Components in this setup perform the following operations: Half-wave plate switches between
polarization states |Hi ↔ |V i, phase
√ shifter adds factor i to the photon, BS2 is chosen such that
if photon in a state (|Li + i |Ri)/ 2 hits it, it will always emerge on the left side, meaning that
the detector D2 will certainly not click. Phase beam splitter transmits state |Hi and reflects |V i.
If components are chosen like this and photon is in state |Ψf i upon entering post-selection process
(i. e. just before HWP) then D1 will click with certainty. And if photon is not in state |Ψf i one
of the other detectors will respond.
Let us now focus only on cases in which detector D1 clicks. Inside the interferometer (between
pre- and post- selection) we can perform measurements to figure out which path the photon took
or what is its polarization.
We can show that with pre-selected state |Ψi i and post-selected state |Ψf i, a photon certainly
followed the left path. We check the location of the photon by inserting non-demolition detectors
along each path. Since detectors are non-demolition they do not absorb the photon or change
its polarization. These detectors measure projection operators ΠL = |Li hL| and ΠR = |Ri hR|.
If we insert such a detector into right path the state of the photon after the measurement
√ will
0
be |Ψ i = |Ri |Hi which is orthogonal to post selected state |Ψf i = (|Li |Hi + |Ri |V i)/ 2 and
detector D1 can never click. This means the photon can not be found along the right path of the
experimental setup and always travels on the left side. The cat is in the left arm, but what about
its grin?
Now we replace position detectors with polarization detectors. We do not expect such a detector
to click if placed in the right arm since photon is always found in the left arm. We define polarization
3
detector as
σz(R) = ΠR σz
(3)
σz = |+i h+| − |−i h−|
(4)
where
(R)
Operator σz has 3 eigenvalues +1, -1 and 0 corresponding to eigenstates |Ri |+i, |Ri |−i. If we
apply operator σz to linear polarization states |Hi and |V i we get
σz |Hi = i |V i
σz |V i = i |Hi
(5)
(6)
If we insert this detector along the right arm we get the intermediate state |Ψ0 i = i |Ri |V i which
is not orthogonal to post selected state |Ψf i and detector D1 clicks. So even though by measuring
its position with ΠR we never find the photon along the right path, we can still find its angular
momentum there. It seems we have found a grin without a cat. But is this true?
We haven’t performed measurements of location and angular momentum at the same time.
(R)
What happens if we insert the detectors ΠR , ΠL and σz simultaneously? (The order of ΠR and
(R)
(R)
σz doesn’t matter since they commute.) We now see that whenever σz yields an non-zero
angular momentum we also get value 1 from ΠR meaning that photon and its polarization both
(R)
travel through the right arm. If σz indicates no angular momentum ΠR yields value 0 indicating
that photon went through the left arm. The paradox vanishes because measurements collapse
quantum states and consequently disturb each other. We might give up at this point stating that
Cheshire cat is nothing more than an illusion, but if a subtler measuring method called weak
measurement is used its existence can still be proven.
3
Weak Measurement
Usually when we perform a measurement of an entangled state such as |Ψi i = α |0i + β |1i the
wavefunction collapses and after the measurement we find the system in one of the eigenstates (|0i
or |1i). The collapse happens because of the interaction between the particle and the measuring
device. After the measurement we cannot restore the wavefunction to its original state. We call
this type of measurement normal or strong measurement.
As opposed to the strong measurement the weak measurement does not significantly disturb
the particle and therefore its wavefunction does not collapse. To achieve this the coupling between
the system and the measuring device must be very weak. To demonstrate what exactly this means
we will take a look at an experiment described by Aharonov, Albert and Vaidman in 1988 [3].
3.1
Weak Value
The Hamiltonian of a standard measuring procedure in quantum mechanics can be written as
H = −g(t)qA,
4
(7)
RT
where g(t) is a normalized function that is zero everywhere but in the measuring device ( 0 g(t)dt =
1, T is time particle spends interacting with the measuring device). q is a canonical variable of the
measuring device with a conjugate momentum p and A is the variable we are measuring, which
has discrete eigenvalues ai . If the initial state is Gaussian in q and p representation it evolves like
this
X
X
R
2
2
2
2
e−i Hdt e−p /4(∆p)
αi |A = ai i =
αi e−(p−ai ) /4(∆p) |A = ai i ,
(8)
i
where
P
i
αi |A = ai i is the initial state of our system [3]. If the width ∆p of the distribution
i
p is small compared to the differences between eigenvalues ai the state of the system after the
interaction will be a superposition of Gaussians located at ai .
In the other limit where ∆p is much bigger than differences between ai the probability distribution
P after2 the measurement will be close to Gaussian with width ∆p and mean value at
hAi = i |αi | ai . But in this limit one measurement is worthless because ∆p hAi. This can
be fixed by repeating the measurement √
for an ensemble of N particles in the same initial state.
Uncertainty is reduced by the factor 1/ N while the mean value hAi remains the same. Both
cases: small ∆p and large ∆p are shown on figure (3).
0.5
0.8
0.4
0.6
0.3
0.4
0.2
0.2
0.1
-3
-2
-1
1
2
3
p
-10
-5
5
10
p
Figure 3: Example figures for observable A with eigenvalues 1 and -1. On the left ∆p = 0.1 is
much smaller than difference between eigenvalues so we get two separated Gaussians around 1 and
-1. On the right ∆p = 2 is large and we get one peak at |Ai = 0. Distributions are not normalized.
The outcome of the measurement can be changed by post-selecting a certain state |Ψf i. We
have a large ensemble of particles in the same initial state |Ψi i and the initial state of the measuring
device is (∆2 (2π))−1/4 exp[−q 2 /(4∆2 )]. After the post-selection we get
R
hΨf |A|Ψi i −q2 /(4∆2 )
−i Hdt
−q 2 /(4∆2 )
hΨf | e
|Ψi i e
≈ hΨf |Ψi i exp iq
e
.
(9)
hΨf |Ψi i
This is true if ∆ is small
∆ max
n
| hΨf |Ψi i |
| hΨf |An |Ψi i |1/n
5
(10)
The state of the measuring device in p representation is
exp[−∆2 (p −
hΨf |A|Ψi i 2
) ],
hΨf |Ψi i
(11)
Which means that the measured value of A will be its weak value [3]:
hΨf |A|Ψi i
,
(12)
hΨf |Ψi i
√
which is decreased by 1/ N and therefore hAiw can be measured to
hAiw =
with uncertainty ∆p =
arbitrary accuracy.
3.2
1
2∆
Weak Measurement in double Stern-Gerlach Experiment
This experiment is designed to measure weak value of z component of spin- 21 particle [3]. Beam
of particles moving in y direction flies trough two consecutive Stern-Gerlach devices (figure 4).
Initially spin of the particle points in ξ direction in xz plane. First Stern-Gerlach device weakly
measures the spin component σz . Because we want the measurement to be weak we only apply
a small gradient of magnetic field ∂Bz /∂z. After the first we have a second Stern-Gerlach device
which splits the beam into two in the x direction. This measurement is strong (∂Bx /∂x is large)
therefore the beam completely splits in the x direction. Somewhere behind both Stern-Gerlach
devices we put a screen that stops the beam with σx = 1 performing the post-selection.
Figure 4: Device for measuring weak value of z component of spin- 12 particle. Source: [3].
Weak value of σz for such experimental setup is
hσz iw =
h↑x |σz | ↑ξ i
α
= tan ,
h↑x | ↑ξ i
2
where α is the angle between x and ξ direction.
6
(13)
To mathematically describe the experiment we write the initial state of the particles with mass
m and magnetic moment µ as
x2
y2
z2
|Ψi i = (2π∆2 )−3/4 e− 4∆2 e− 4∆2 e− 4∆2 e−ip0 y (cos
α
α
|↑x i + sin |↓x i),
2
2
(14)
where p0 is average momentum in y direction [3]. This state is fist affected by the Hamiltonian of
the weak interaction
H1 = −µ
∂Bz
zσz g(y − y1 ),
∂z
(15)
where g(y − y1 ) is only non-zero in weak Stern-Gerlach device, ensuring that H1 only affects
z
the particles while they are inside it. µ ∂B
z is the canonical variable q from equation (7). The
∂z
requirement for weakness of interaction is
∂Bz max[tan(α/2), 1] ∆pz = 1 .
(16)
µ ∂z 2∆
x
After the first Stern-Gerlach device particles enter the second one with Hamiltonian H2 = −µ ∂B
xσz g(y−
∂x
y2 ) which splits the beam into two if the following requirement holds true
∂Bx ∆px = 1 .
(17)
µ
∂x 2∆
The beam with σx = 1 continues its path to the screen at distance l where the wave function just
before the collapse is
"
2 #
p 2 α
lµ
∂B
z
0
tan
exp −∆2
z−
(18)
l
p0 ∂z
2
where
δz =
lµ ∂Bz
α
tan
p0 ∂z
2
(19)
is the displacement of the beam in the z direction which we measure and from this calculate the
weak value hσz iw = tan α2 .
7
Figure 5: Results of the computer simulation based on weak measurement of σz . We see clear separation in x direction due to the strong measurement (post-selection). If we look at the wavepacket
with σx = 1 (small one) we can see slight displacement in the z direction caused by the weak
measurement. Angle α in this picture is close to π. Source: [4].
3.3
Weak Measurement and Quantum Cheshire Cat
Now that we understand what weak measurement is, we need to implement it into the photon
experiment in order to prove that quantum Cheshire Cat really exists. We replace detector D1
with a CCD camera that measures displacement of the beam from the central position [2]. We can
realize measurement of position in the left arm ΠL by putting a thin glass plate into the left arm
perpendicularly to the photon’s path and then slightly tilt it . Direction of the photon passing
through it will change and we will be able to observe a small displacement of the beam on the CCD
detector. Let δ denote this displacement of the beam. For the measurement of angular momentum
we just replace a glass plate with some optical element that changes the direction of the beam
based on its polarization.
If the beam has characteristic width ∆, the degree to which the measurement disturbs the
photon and the precision of the measurement depend on the ratio δ/∆. When δ ∆ the measurement is precise (strong measurement). We can be certain if the beam is displaced or not. On
the other hand δ ∆ marks the regime of the weak measurement. When this condition applies
the photons are not greatly disturbed and a measurement of any single photon cannot reveal if
the beam has been displaced or not.
√ But if the measurement is repeated N times uncertainty can
be reduced to approximately ∆/ N , allowing us to detect beam displacement to desired accuracy
8
by repeating the measurement many times.
If we apply pre- and post-selection the measurement yields weak value of the operator we are
measuring. For example, if we are measuring operator A as described above the average shift of
the beam will be its weak value hAiw as defined by (12).
We can now calculate weak values for the observables measured in our experiment. They are
hΨf |ΠL |Ψi i
= 1,
hΨf |Ψi i
hΨf |ΠR |Ψi i
hΠR iw =
= 0,
hΨf |Ψi i
hΠL iw =
(20)
(21)
(L)
hσz(L) iw =
hΨf |σz |Ψi i
= 0,
hΨf |Ψi i
hσz(R) iw =
hΨf |σz |Ψi i
= 1,
hΨf |Ψi i
(22)
(R)
(L)
(23)
(R)
where σz is defined for the left arm in analogy with σz for the right one. Weak values tell us
that the photon is in the left arm (hΠL iw = 1 and hΠR iw = 0) and its angular momentum is in the
(R)
(L)
right arm (hσz iw = 1 and hσz iw = 0).
These values are obtained via weak measurement that can be applied simultaneously, because
it doesn’t disturb the photon and therefore doesn’t collapse the wave function. That means we
have finally found the Cheshire Cat.
4
Quantum Cheshire Cat in Neutron Interferometry Experiment
Recently the quantum Cheshire Cat has been observed in neutron interferometry experiment done
by T. Denkmayr and his colleagues at Institute Laue-Langevin [1]. Again we must define pre- and
post-selected states |Ψi i and |Ψf i.
1
|Ψi i = √ (|↑x i |Ii + |↓x i |IIi),
2
1
|Ψf i = √ |↓x i (|Ii + |IIi),
2
(24)
(25)
where |Ii and |IIi stand for spatial part of wavefunction along path I or II and |↑x i, |↓x i denote
spin state in x direction. After we pre-select the ensemble we must perform weak measurement of
the population along both paths and a measurement of their spin component in z direction σ
ˆz .
This experiment is very similar to the one with photons. Measurement of position can be
mathematically expressed with projection operator Πj = |ji hj| where j ∈ {I, II}. The actual
measurement of position is performed by inserting an absorber in the desired path and observing
the decline of intensity of the signal. The other parameter we are interested in, the grin, is z
9
component of neutrons’ spin. With an operator it can be expressed as Πj σz . This is measured by
applying additional magnetic field along the path which causes a small spin rotation that can be
measured.
Figure 6: Experimental setup for neutron interferometry experiment. Magnetic birefringent prisms
(P) polarize the neutron beam. Whole experimental device is inside a magnetic field to prevent
depolarization. Neutrons pass trough a spin turner (ST1) that rotates their spin by π/2 into
xy plane. After that they enter a triple interferometer inside which are spin rotators (SRs) that
complete the pre-selection and also perform the measurement of hΠI σz iw and hΠII σz iw . The
absorber (ABS) is inserted when position is determined. The phase shifter (PS) enables tuning of
the phase between the two beams. After the neutron exits the interferometer it goes either into
detector H or into post-selection by the second spin turner and spin analyser. Only the neutrons
that reach O detector are post-selected. Source: [1].
Let’s compare theoretical weak values of the observables with experimentally measured ones
hΠI iw
hΠII iw
hΠI σz iw
hΠII σz iw
theoretical experimental
0
0.14 ± 0.04
1
0.96 ± 0.06
1
1.07 ± 0.25
0
0.02 ± 0.24
Table 1: Source: [1]
We see that the neutron goes trough path II while its spin goes along path I. This once again
proves the existence of QCC.
5
Conclusion
Although only two examples of quantum Cheshire Cat have been presented in this seminar, we
must keep in mind that this is a completely general phenomenon that applies to any quantum
10
particle and any of its properties. For example we could probably also separate electron and its
charge or an atom from its internal energy [2]. This separation could prove useful in measuring
quantities that are overshadowed by another quantity (e.g. spin). We would just have to separate
spin from the particle and then measure the quantity of interest. We have yet to reach this level
but hopes for the future development are high and quantum Cheshire Cat may prove to be one of
the most useful tools for measuring quantum properties of particles.
References
[1] Tobias Denkmayr, Hermann Geppert, Stephan Sponar, Hartmut Lemmel, Alexandre Matzkin,
Jeff Tollaksen, and Yuji Hasegawa. Observation of a quantum cheshire cat in a matter-wave
interferometer experiment. Nat Commun, 5, 07 2014.
[2] Yakir Aharonov, Sandu Popescu, Daniel Rohrlich, and Paul Skrzypczyk. Quantum cheshire
cats. New Journal of Physics, 15(11):113015, 2013.
[3] Yakir Aharonov, David Z. Albert, and Lev Vaidman. How the result of a measurement of a
component of the spin of a spin-1/2 particle can turn out to be 100. Physical Review Letters,
60(14):1351–1354, 1988.
[4] Anton Ramˇsak. Double stern-gerlach computer simulation.
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