Department of physics

Seminar Ia

Quantum Cheshire Cat

Author: Tadej Meznarsic

Mentor: prof. dr. Anton Ramsak

Ljubljana, May 2015

AbstractThis seminar presents quantum phenomenon known as quantum Cheshire Cat. It starts by de-scribing its paradoxical nature in the regime of regular measurement on an example of photon andits polarization. It continues into detailed description of weak measurement by presenting a doubleStern-Gerlach experiment. Then it shows how to implement the principle of weak measurementinto Cheshire Cat experiment for photons and ends with presentation of another experiment, prov-ing that Cheshire Cat also applies to neutrons and their spin.

Contents

1 Introduction 1

2 Cheshire Cat 2

3 Weak Measurement 43.1 Weak Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Weak Measurement in double Stern-Gerlach Experiment . . . . . . . . . . . . . . . 63.3 Weak Measurement and Quantum Cheshire Cat . . . . . . . . . . . . . . . . . . . . 8

4 Quantum Cheshire Cat in Neutron Interferometry Experiment 9

5 Conclusion 10

1 Introduction

In the world of quantum mechanics we encounter many interesting and unusual phenomena. Oneof them is quantum Cheshire cat, named after a cat that vanishes and leaves behind only a grinfrom the novel Alice in Wonderland. In quantum mechanics we don’t have cats that grin butparticles with properties (e.g. spin). With carefully assembled experimental setup scientists havebeen 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 recentlyby separating spin from neutrons in a neutron interferometry experiment [1]. In such experimentsit is of utmost importance to carefully choose pre- and post-selected ensemble, meaning we haveto prepare our particles in initial state |Ψi〉 and when they exit our experimental setup we performpost-selection so that we get final state |Ψf〉. Only if states |Ψi〉 and |Ψf〉 are chosen correctly canwe observe quantum Cheshire Cat (from now on QCC).

Figure 1: Picture of quantum Cheshire cat inside an interferometer. Cat travels along the upperpath while its grin travels along the lower one. Source: [1]

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Let us take a look at an experimental design that can be used for observation of QCC byseparating photons from their polarization that is described in [2].

2 Cheshire Cat

In this experiment we have a photon in two possible locations, |L〉 and |R〉. Its property, thecat’s grin, is circular polarization, with two basis states |+〉 and |−〉. In terms of horizontal andvertical linear polarization states |H〉 and |V 〉 they can be expressed as |+〉 = (|H〉 + i |V 〉)/

√2

and |−〉 = (|H〉 − i |V 〉)/√

2.Initial state of the photon is

|Ψi〉 =1√2

(i |L〉+ |R〉) |H〉 , (1)

horizontally polarized superposition of two positions |L〉 and |R〉. Such state can be prepared bysending a horizontally polarized photon into a beam splitter as depicted in figure 2, denoted byBS1. The reflected beam |L〉 gains a phase factor i.

In post-selection we would like to get the state

|Ψf〉 =1√2

(|L〉 |H〉+ |R〉 |V 〉), (2)

which means we would like to perform a measurement that returns answer ’yes’ when photon isin state |Ψf〉 and returns ’no’ if photon is in a state orthogonal to |Ψf〉. We will then examineonly cases with an affirmative answer. This measurement can be performed in an optics setup asdepicted in figure 2.

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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) andthree photon detectors (D1, D2, D3). Source: [2].

Components in this setup perform the following operations: Half-wave plate switches betweenpolarization states |H〉 ↔ |V 〉, phase shifter adds factor i to the photon, BS2 is chosen such thatif photon in a state (|L〉 + i |R〉)/

√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 |H〉 and reflects |V 〉.If components are chosen like this and photon is in state |Ψf〉 upon entering post-selection process(i. e. just before HWP) then D1 will click with certainty. And if photon is not in state |Ψf〉 oneof the other detectors will respond.

Let us now focus only on cases in which detector D1 clicks. Inside the interferometer (betweenpre- and post- selection) we can perform measurements to figure out which path the photon tookor what is its polarization.

We can show that with pre-selected state |Ψi〉 and post-selected state |Ψf〉, a photon certainlyfollowed the left path. We check the location of the photon by inserting non-demolition detectorsalong each path. Since detectors are non-demolition they do not absorb the photon or changeits polarization. These detectors measure projection operators ΠL = |L〉 〈L| and ΠR = |R〉 〈R|.If we insert such a detector into right path the state of the photon after the measurement willbe |Ψ′〉 = |R〉 |H〉 which is orthogonal to post selected state |Ψf〉 = (|L〉 |H〉 + |R〉 |V 〉)/

√2 and

detector D1 can never click. This means the photon can not be found along the right path of theexperimental setup and always travels on the left side. The cat is in the left arm, but what aboutits grin?

Now we replace position detectors with polarization detectors. We do not expect such a detectorto click if placed in the right arm since photon is always found in the left arm. We define polarization

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detector as

σ(R)z = ΠRσz (3)

where

σz = |+〉 〈+| − |−〉 〈−| (4)

Operator σ(R)z has 3 eigenvalues +1, -1 and 0 corresponding to eigenstates |R〉 |+〉, |R〉 |−〉. If we

apply operator σz to linear polarization states |H〉 and |V 〉 we get

σz |H〉 = i |V 〉 (5)

σz |V 〉 = i |H〉 (6)

If we insert this detector along the right arm we get the intermediate state |Ψ′〉 = i |R〉 |V 〉 whichis not orthogonal to post selected state |Ψf〉 and detector D1 clicks. So even though by measuringits position with ΠR we never find the photon along the right path, we can still find its angularmomentum 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.What happens if we insert the detectors ΠR, ΠL and σ

(R)z simultaneously? (The order of ΠR and

σ(R)z doesn’t matter since they commute.) We now see that whenever σ

(R)z yields an non-zero

angular momentum we also get value 1 from ΠR meaning that photon and its polarization bothtravel through the right arm. If σ

(R)z indicates no angular momentum ΠR yields value 0 indicating

that photon went through the left arm. The paradox vanishes because measurements collapsequantum states and consequently disturb each other. We might give up at this point stating thatCheshire cat is nothing more than an illusion, but if a subtler measuring method called weakmeasurement is used its existence can still be proven.

3 Weak Measurement

Usually when we perform a measurement of an entangled state such as |Ψi〉 = α |0〉 + β |1〉 thewavefunction collapses and after the measurement we find the system in one of the eigenstates (|0〉or |1〉). The collapse happens because of the interaction between the particle and the measuringdevice. After the measurement we cannot restore the wavefunction to its original state. We callthis type of measurement normal or strong measurement.

As opposed to the strong measurement the weak measurement does not significantly disturbthe particle and therefore its wavefunction does not collapse. To achieve this the coupling betweenthe system and the measuring device must be very weak. To demonstrate what exactly this meanswe 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, (7)

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where g(t) is a normalized function that is zero everywhere but in the measuring device (∫ T

0g(t)dt =

1, T is time particle spends interacting with the measuring device). q is a canonical variable of themeasuring device with a conjugate momentum p and A is the variable we are measuring, whichhas discrete eigenvalues ai. If the initial state is Gaussian in q and p representation it evolves likethis

e−i∫Hdte−p

2/4(∆p)2∑i

αi |A = ai〉 =∑i

αie−(p−ai)2/4(∆p)2 |A = ai〉 , (8)

where∑i

αi |A = ai〉 is the initial state of our system [3]. If the width ∆p of the distribution

p is small compared to the differences between eigenvalues ai the state of the system after theinteraction will be a superposition of Gaussians located at ai.

In the other limit where ∆p is much bigger than differences between ai the probability dis-tribution after the measurement will be close to Gaussian with width ∆p and mean value at〈A〉 =

∑i |αi|2ai. But in this limit one measurement is worthless because ∆p � 〈A〉. 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 〈A〉 remains the same. Both

cases: small ∆p and large ∆p are shown on figure (3).

-3 -2 -1 1 2 3p

0.1

0.2

0.3

0.4

0.5

-10 -5 5 10p

0.2

0.4

0.6

0.8

Figure 3: Example figures for observable A with eigenvalues 1 and -1. On the left ∆p = 0.1 ismuch 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 |A〉 = 0. Distributions are not normalized.

The outcome of the measurement can be changed by post-selecting a certain state |Ψf〉. Wehave a large ensemble of particles in the same initial state |Ψi〉 and the initial state of the measuringdevice is (∆2(2π))−1/4 exp[−q2/(4∆2)]. After the post-selection we get

〈Ψf | e−i∫Hdt |Ψi〉 e−q

2/(4∆2) ≈ 〈Ψf |Ψi〉 exp

(iq〈Ψf |A|Ψi〉〈Ψf |Ψi〉

)e−q

2/(4∆2). (9)

This is true if ∆ is small

∆� maxn

| 〈Ψf |Ψi〉 || 〈Ψf |An|Ψi〉 |1/n

(10)

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The state of the measuring device in p representation is

exp[−∆2(p− 〈Ψf |A|Ψi〉〈Ψf |Ψi〉

)2], (11)

Which means that the measured value of A will be its weak value [3]:

〈A〉w =〈Ψf |A|Ψi〉〈Ψf |Ψi〉

, (12)

with uncertainty ∆p = 12∆

which is decreased by 1/√N and therefore 〈A〉w can be measured to

arbitrary accuracy.

3.2 Weak Measurement in double Stern-Gerlach Experiment

This experiment is designed to measure weak value of z component of spin-12

particle [3]. Beamof 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 weaklymeasures the spin component σz. Because we want the measurement to be weak we only applya small gradient of magnetic field ∂Bz/∂z. After the first we have a second Stern-Gerlach devicewhich 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-Gerlachdevices 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

〈σz〉w =〈↑x |σz| ↑ξ〉〈↑x | ↑ξ〉

= tanα

2, (13)

where α is the angle between x and ξ direction.

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To mathematically describe the experiment we write the initial state of the particles with massm and magnetic moment µ as

|Ψi〉 = (2π∆2)−3/4e−x2

4∆2 e−y2

4∆2 e−z2

4∆2 e−ip0y(cosα

2|↑x〉+ sin

α

2|↓x〉), (14)

where p0 is average momentum in y direction [3]. This state is fist affected by the Hamiltonian ofthe weak interaction

H1 = −µ∂Bz

∂zzσzg(y − y1), (15)

where g(y − y1) is only non-zero in weak Stern-Gerlach device, ensuring that H1 only affectsthe particles while they are inside it. µ∂Bz

∂zz is the canonical variable q from equation (7). The

requirement for weakness of interaction is

µ

∣∣∣∣∂Bz

∂z

∣∣∣∣max[tan(α/2), 1]� ∆pz =1

2∆. (16)

After the first Stern-Gerlach device particles enter the second one with HamiltonianH2 = −µ∂Bx

∂xxσzg(y−

y2) which splits the beam into two if the following requirement holds true

µ

∣∣∣∣∂Bx

∂x

∣∣∣∣� ∆px =1

2∆. (17)

The beam with σx = 1 continues its path to the screen at distance l where the wave function justbefore the collapse is

exp

[−∆2

(p0

l

)2(z − lµ

p0

∂Bz

∂ztan

α

2

)2]

(18)

where

δz =lµ

p0

∂Bz

∂ztan

α

2(19)

is the displacement of the beam in the z direction which we measure and from this calculate theweak value 〈σz〉w = tan α

2.

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Figure 5: Results of the computer simulation based on weak measurement of σz. We see clear sepa-ration in x direction due to the strong measurement (post-selection). If we look at the wavepacketwith σx = 1 (small one) we can see slight displacement in the z direction caused by the weakmeasurement. 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 photonexperiment 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 canrealize measurement of position in the left arm ΠL by putting a thin glass plate into the left armperpendicularly to the photon’s path and then slightly tilt it . Direction of the photon passingthrough it will change and we will be able to observe a small displacement of the beam on the CCDdetector. Let δ denote this displacement of the beam. For the measurement of angular momentumwe just replace a glass plate with some optical element that changes the direction of the beambased on its polarization.

If the beam has characteristic width ∆, the degree to which the measurement disturbs thephoton and the precision of the measurement depend on the ratio δ/∆. When δ � ∆ the mea-surement is precise (strong measurement). We can be certain if the beam is displaced or not. Onthe other hand δ � ∆ marks the regime of the weak measurement. When this condition appliesthe photons are not greatly disturbed and a measurement of any single photon cannot reveal ifthe beam has been displaced or not. But if the measurement is repeated N times uncertainty canbe reduced to approximately ∆/

√N , allowing us to detect beam displacement to desired accuracy

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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 ofthe beam will be its weak value 〈A〉w as defined by (12).

We can now calculate weak values for the observables measured in our experiment. They are

〈ΠL〉w =〈Ψf |ΠL|Ψi〉〈Ψf |Ψi〉

= 1, (20)

〈ΠR〉w =〈Ψf |ΠR|Ψi〉〈Ψf |Ψi〉

= 0, (21)

〈σ(L)z 〉w =

〈Ψf |σ(L)z |Ψi〉

〈Ψf |Ψi〉= 0, (22)

〈σ(R)z 〉w =

〈Ψf |σ(R)z |Ψi〉

〈Ψf |Ψi〉= 1, (23)

where σ(L)z is defined for the left arm in analogy with σ

(R)z for the right one. Weak values tell us

that the photon is in the left arm (〈ΠL〉w = 1 and 〈ΠR〉w = 0) and its angular momentum is in the

right arm (〈σ(R)z 〉w = 1 and 〈σ(L)

z 〉w = 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 wehave finally found the Cheshire Cat.

4 Quantum Cheshire Cat in Neutron Interferometry Ex-

periment

Recently the quantum Cheshire Cat has been observed in neutron interferometry experiment doneby T. Denkmayr and his colleagues at Institute Laue-Langevin [1]. Again we must define pre- andpost-selected states |Ψi〉 and |Ψf〉.

|Ψi〉 =1√2

(|↑x〉 |I〉+ |↓x〉 |II〉), (24)

|Ψf〉 =1√2|↓x〉 (|I〉+ |II〉), (25)

where |I〉 and |II〉 stand for spatial part of wavefunction along path I or II and |↑x〉, |↓x〉 denotespin state in x direction. After we pre-select the ensemble we must perform weak measurement ofthe population along both paths and a measurement of their spin component in z direction σz.

This experiment is very similar to the one with photons. Measurement of position can bemathematically expressed with projection operator Πj = |j〉 〈j| where j ∈ {I, II}. The actualmeasurement of position is performed by inserting an absorber in the desired path and observingthe decline of intensity of the signal. The other parameter we are interested in, the grin, is z

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component of neutrons’ spin. With an operator it can be expressed as Πjσz. This is measured byapplying additional magnetic field along the path which causes a small spin rotation that can bemeasured.

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 preventdepolarization. Neutrons pass trough a spin turner (ST1) that rotates their spin by π/2 intoxy plane. After that they enter a triple interferometer inside which are spin rotators (SRs) thatcomplete the pre-selection and also perform the measurement of 〈ΠIσz〉w and 〈ΠIIσz〉w. Theabsorber (ABS) is inserted when position is determined. The phase shifter (PS) enables tuning ofthe phase between the two beams. After the neutron exits the interferometer it goes either intodetector H or into post-selection by the second spin turner and spin analyser. Only the neutronsthat reach O detector are post-selected. Source: [1].

Let’s compare theoretical weak values of the observables with experimentally measured ones

theoretical experimental〈ΠI〉w 0 0.14± 0.04〈ΠII〉w 1 0.96± 0.06〈ΠIσz〉w 1 1.07± 0.25〈ΠIIσz〉w 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 againproves the existence of QCC.

5 Conclusion

Although only two examples of quantum Cheshire Cat have been presented in this seminar, wemust keep in mind that this is a completely general phenomenon that applies to any quantum

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particle and any of its properties. For example we could probably also separate electron and itscharge or an atom from its internal energy [2]. This separation could prove useful in measuringquantities that are overshadowed by another quantity (e.g. spin). We would just have to separatespin from the particle and then measure the quantity of interest. We have yet to reach this levelbut hopes for the future development are high and quantum Cheshire Cat may prove to be one ofthe 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-waveinterferometer experiment. Nat Commun, 5, 07 2014.

[2] Yakir Aharonov, Sandu Popescu, Daniel Rohrlich, and Paul Skrzypczyk. Quantum cheshirecats. New Journal of Physics, 15(11):113015, 2013.

[3] Yakir Aharonov, David Z. Albert, and Lev Vaidman. How the result of a measurement of acomponent 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 Ramsak. Double stern-gerlach computer simulation.

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