[width=1.10cm]logo Nonlocal Wave Particle Dualityconfqic/youqu17/talks/asad.pdf · 3 Ghost Interference 4 Nonlocal Wave-Particle Duality 5 Ghost quantum eraser 6 Conclusions Mohd

Nonlocal Wave Particle Duality

Mohd Asad Siddiqui

National Institute of TechnologyPatna - 800005

March 1, 2017

Mohd Asad Siddiqui (NIT, Patna) Nonlocal Wave Particle Duality March 1, 2017 1 / 37

Outline

1 Introduction

2 Wave-Particle Duality

3 Ghost Interference

4 Nonlocal Wave-Particle Duality

5 Ghost quantum eraser

6 Conclusions


Introduction

Outline

1 Introduction





6 Conclusions


Introduction

Wave Particle Confusion

Huygens: Light is a wave (1678).Newton: It’s a particle (1704).Young: It’s a wave (1801).Fresnel, Poisson/Arago: It’s a wave (1819).Maxwell: It’s a em wave (1864).Planck: Well, it’s emitted as a particle (1900).Einstein: It’s also absorbed as a particle; insome sense, I guess it is a particle (1905).


Introduction

Trouble of Photon

[Fig Source: Internet]


Introduction

Bohr-Einstein Debates

c©Charles Addams

How can a particle go through both slits at once?If I measure which one it went through, how could interference occur betweenthe two of them?


Introduction

Bohr-Einstein Debates

c©Charles Addams

How can a particle go through both slits at once?If I measure which one it went through, how could interference occur betweenthe two of them?


Introduction

Bohr’s "Complementarity"

Light(or electrons, or anything else) has wave-like aspects and particle-likeaspects, and behave one or the other depending on the observation.-truth encompasses both parts, but you cannot observe both simultaneously.


Introduction

Principle of Complementarity

Bohr elevated a concept to the principle:

In describing the results of quantum mechanical experiments,certain physical concepts are complementary. If two concepts arecomplementary, an experiment that clearly illustrates one conceptwill obscure the other complementary one.

N. Bohr (1928).The Quantum Postulate and the Recent Development of Atomic Theory, April 14, 1928.

In the two-slit experiment: the "which-path" information and existence ofinterference pattern are complementary concepts.


Introduction

Without path informationAbsence of path detector

Let |ψ1〉, |ψ2〉 be the state of the particle through slit 1 and 2 resp.

Source

Double Slit

Amplitude of finding the particle at point x: Ψ(x) = ψ1(x) + ψ2(x)Probability density:

|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2+ ψ∗1(x)ψ2(x) + ψ∗2(x)ψ1(x)

cross-terms are responsible for interference.Let us now introduce the path detector in interference experiment.


Introduction



Source

Double Slit


|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2+ ψ∗1(x)ψ2(x) + ψ∗2(x)ψ1(x)



Introduction



Source

Double Slit


|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2+ ψ∗1(x)ψ2(x) + ψ∗2(x)ψ1(x)



Introduction

Path detector follows the quantum measurement rule.

A quantum measurement follows:1 Process 1: Unitary operation establishes the correlation between system

& detector.Initial states: System:

∑ni=1 ci|ψi〉; Detector: |d0〉

n∑i=1

ci|ψi〉|d0〉Unitary evolution−−−−−−−−−→

Process 1

n∑i=1

ci|ψi〉|di〉.

In general, the combined particle-detector state gets entangled.2 Process 2: Non-unitary selection of a single state |ψk〉 with probability|ck|2.

n∑i=1

ci|ψi〉|di〉 −−−−−→Process 2

|ψk〉|dk〉

“The Measurement Problem".


Introduction

Complete path information

The detector states corresponding to each path is |d1〉, |d2〉.The combined state gets entangled: Ψ(x) =|ψ1〉⊗ |d1〉 +|ψ2〉⊗ |d2〉.Probability (intensity):|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2+ψ∗1(x)ψ2(x) 〈d1|d2〉 +ψ∗2(x)ψ1(x) 〈d2|d1〉

Source

Double Slit

Path detector provide "which-path" information

|d

|d

1

2

Detector |d1〉, |d2〉 must be orthogonal→ 〈d1|d2〉 = 0.Interference vanishes for complete path information.


Introduction

Complete path information

The detector states corresponding to each path is |d1〉, |d2〉.The combined state gets entangled: Ψ(x) =|ψ1〉⊗ |d1〉 +|ψ2〉⊗ |d2〉.Probability (intensity):|Ψ(x)|2 = |ψ1(x)|2 + |ψ2(x)|2+ψ∗1(x)ψ2(x) 〈d1|d2〉 +ψ∗2(x)ψ1(x) 〈d2|d1〉

Source

Double Slit

Path detector provide "which-path" information

|d

|d

1

2

Detector |d1〉, |d2〉 must be orthogonal→ 〈d1|d2〉 = 0.Interference vanishes for complete path information.


Introduction

When we don’t go for complete path information

In there any possibility of simultaneous realization of wave and particlenature?

Conclusion: The more we observe the particle nature , the more the

interference pattern become unclear.


Wave-Particle Duality

Outline

1 Introduction





6 Conclusions



Getting Quantitative Phys. Lett. A 128, 391 (1988).

Volume 128, number 8 PHYSICS LETTERS A 18 April 1988

S I M U L T A N E O U S WAVE AND PARTICLE K N O W L E D G E

IN A N E U T R O N I N T E R F E R O M E T E R

Daniel M. G R E E N B E R G E R and Allaine YASIN City College of the City of New York, New York, NY 10031, USA

Received 15 February 1988; accepted for publication 18 February 1988 Communicated by J.P. Vigier

We give a measure of particle knowledge in a neutron interferometer that reflects one's ability to predict in which beam a neutron is located. We can measure wave knowledge by contrast of the interference pattern. Then one's simultaneous knowledge of both is determined by a single parameter (not an uncertainty relation), running from full particle to full wave knowledge. We extend the discussion to partially coherent beams. Our measure of information is much simpler than the conventional one.

1. Fully coherent beams

Wootters and Zurek [ 1 ] caused some surprise when they analyzed the familiar two-slit diffraction experiment and showed that the results contained a significant amoun t of both particle and wave information, and also that one could simultaneously measure position and m o m e n t u m to an unexpectedly high degree of precision, without violating the uncertainty relation. Specifically, if one performs an ex- per iment to determine which beam the electron is in, it does not destroy all wave knowledge, i.e., there is still a discernable interference pattern.

We would like to point out here that a similar sit- uat ion arises constantly in neut ron interferometry, where it is obvious that a significant amoun t of both particle and wave knowledge is obtained, even if one does not make a direct measurement of which beam a particular particle is in. We shall introduce for this a measure of informat ion that is qualitatively similar to the usual informat ion measure, but much easier to handle analytically. (Mathematically, our situa- t ion is similar to that of the case of two unequal slits in the two-slit diffraction experiment) .

As an example, consider the case of a three-eared interferometer (see fig. 1 ), where the beams ABD (beam I) , and ACD (beam II) , of equal intensity, interfere upon recombining and are counted by a detector located at E.

Fig. 1. Top view of a neutron interferometer. The incident beam enters at A and is split into beam I (ABD) and beam II (ACD), which are then recombined at D and detected at E. For our pur- poses, one can imagine a screen at B which records the interference pattern between the two beams. One can also introduce an absorber at F to partially absorb one of the beams.

If now an absorber is placed at F in beam I, such that it absorbs 99% of the intensity of beam I, there will still be a significant interference pattern at point E. In fact if the intensi ty of beam I is reduced from uni ty to ~2, the contrast of the interference pattern will still be 2~, or about 20% in our case, even though the intensity in beam I is only about 1% of that in beam II. This amazing result [2] testifies to the power of the superposition principle, and it has been clearly demonstrated experimentally.

We can use the contrast of the pattern,

W:-- Imax-Imin (1) Imax+Imi.

as a measure of its wave properties (since it mea- sures the amount of interference), so in this case

0375-9601/88 /$ 03.50 © Elsevier Science Publishers B.V. (Nor th-Hol land Physics Publishing Divis ion )

391

Source

Double Slit

Suppose the wave function at screen be ψ= ae+ikx + beiφe−ikx

Then probability density: |ψ|2= |a|2 + |b|2 + 2|a||b| cos(2kx + φ)

Visiblity, V = Imax−IminImax+Imin

= 2|a||b||a|2+|b|2 ; Predictability, P = |a|2−|b|2

|a|2+|b|2 .

The WPDR between Visiblity of interference and Predictability of the path,is given by P2 + V2 ≤ 1 (Equality for pure state).



Meaning of P

[Fig Source: Internet]

Task is to correctly predict the path.

Suppose the probability of particleto pass through slit 1 is say p1 and

the probability of particle to passthrough slit 2 is p2.

Then the Predictability of the path is given by P = |p1−p2|p1+p2

,

For equal probability (no path guessing), P = 0,

For completely true path guessing either p1 = 1 or p2 = 1,which gives P = 1.Physically it gives the predicted amount of path information.



Getting Quantitative Phys. Rev. Lett. 77, 2154 (1996).

Englert followed Wootters and Zurek suggestion presented in PRD (1979).VOLUME 77, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R S 9 SEPTEMBER1996

Fringe Visibility and Which-Way Information: An Inequality

Berthold-Georg Englert*Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany

(Received 21 May 1996)

An inequality is derived according to which the fringe visibility in a two-way interferometer sets anabsolute upper bound on the amount of which-way information that is potentially stored in a which-waydetector. In some sense, this inequality can be regarded as quantifying the notion of wave-particleduality. The derivation of the inequality does not make use of Heisenberg’s uncertainty relation in anyform. [S0031-9007(96)00950-7]

PACS numbers: 03.65.Bz

Bohr’s principle of complementarity [1] states thatquantum systems (“quantons” [2] for short) possess prop-erties that are equally real but mutually exclusive. Thebest known example is what is colloquially termed wave-particle duality. In a loose manner of speaking it is some-times phrased similarly to the following: Depending onthe experimental situation a quanton behaves either like aparticle or like a wave.

To be more specific, let us consider a two-way inter-ferometer such as Young’s double-slit experiment or aMach-Zehnder setup. The wavelike property is then doc-umented by well-visible interference fringes, whereas theparticlelike property is evident if one can tell along whichway the interferometer has been traversed.

The notions of particle and wave are associated withmental pictures that are borrowed from classical (i.e.,prequantum) physics. These associations are dangerousbecause of their obvious limitations. Therefore, “wave-particle duality” should perhaps be abandoned in favor ofa more neutral term, such as “interferometric duality” orsimply “duality.” The general formulation of this conceptcould read as follows.

Duality.—The observations of an interference patternand the acquisition of which-way information are mutu-ally exclusive.

The extreme situations “perfect fringe visibility and nowhich-way information” and “full which-way informationand no fringes” are familiar from textbook discussions.But intermediate stages deserve further study.

The objective of this Letter is the derivation of aninequality that quantifies duality by stating to which extentpartial fringe visibility and partial which-way knowledgeare compatible. The quantitative measure of the fringevisibility is the usual one, and which-way knowledge willbe turned into a number with the aid of an approach thatis originally due to Wootters and Zurek [3].

At the intermediate stage of the interferometer—between beam splitter and beam merger, see Fig. 1—thetwo ways can be labeled by quantum numbers11 and21, say. Accordingly, we are invited to describe therelevant degree of freedom of the quanton by analogsof Pauli’s spin operatorss ssx , sy , szd. Prior to

entering the interferometer the quanton is in an initialstate that is characterized by the statistical operator

rsidQ

1 1 ssid ? s

2

12

s1 1 ssidx sx 1 ssid

y sy 1 ssidz szd (1)

with an initial Bloch vector ssid trQhsrsidQ j. It is

sufficiently general to represent the action of the beamsplitter and the beam merger by

rQ ! exp

µ2i

p

4sy

∂rQ exp

µi

p

4sy

∂,

whereas the phase shifter at the central stage effects

rQ ! exp

µ2i

f

2sz

∂rQ exp

µi

f

2sz

∂. (2)

Consequently, the interferometer of Fig. 1(a) turnsrsidQ

into the final state

rs fdQ

12

s1 1 ss fd ? sd , (3)

with

ss fd s2ssidx , ssid

y cosf 1 ssidz sinf,

ssidy sinf 2 ssid

z cosfd .

After the quanton has passed the beam merger, theobservablesz is measured and the relative frequencywith which the value21 is found reveals the interferencepattern,

pf trQ

Ω12

s1 2 szdrs fdQ

æ

12

s1 2 ssidy sinf 1 ssid

z cosfd ,

so that

V0 fsssidy d2 1 sssid

z d2g1y2

is the correspondinga priori fringe visibility.

2154 0031-9007y96y77(11)y2154(4)$10.00 © 1996 The American Physical Society

Distinguishability is defined as D ≤√

1− |〈d1|d2〉|2(A measure of the distinguishability as determined by detector)

and Visibility (A measure of observed interference) V = |〈d1|d2〉|Stronger WPDR V2 +D2 ≤ 1

Equality when beams and detector are in pure states.The inequality tells how much partial fringe visibility and partial which-wayknowledge are compatible.



Meaning of D

Englert’s approach:Initial state of path detector: ρD = |d0〉〈d0|,Particle through upper slit→ ρ+

D = U†+ρDU+,Particle through lower slit→ ρ−D = U†−ρDU−.Say, ρ+

D = |d1〉〈d1|, ρ−D = |d2〉〈d2|.

Path distinguishability:

D ≡ 12

TrD(|ρ+

D − ρ−D |)

=√

1− |〈d1|d2〉|2

D is the distance between ρ+D and ρ−D in the trace-class norm.

Trace norm of ρ = Tr√ρρ† (Def.)

Physically it gives the the amount of path information that has becomeavailable.


Ghost Interference

Outline

1 Introduction





6 Conclusions


Ghost Interference

A consequence of entanglement

Experimental setup

A pair of entangled photons aregenerated using SpontaneousParametric Down Conversion (SPDC).Photons move in different directions.

ordinaryray

extraordinary ray

Polarizationfixed


Ghost Interference

Experimental results(a)

S

S

(b)

1

1

2

2

D1

D1 D2

L1

D

L2

D

D D

1

1 2

Coincidence

counter

Double Slit

Double Slit

Figure: Schematicdiagram of the two-slitghost interferenceexperiment.

1 No first order interference is observed for photons 1.2 For photons 2, first order interference is neither expected, nor seen.3 Suprisingly, an interference pattern is observed when photons 2 are

detected in coincidence with a fixed D1.4 Here photons 2 do not pass through any double slit!

Ghost interference!Mohd Asad Siddiqui (NIT, Patna) Nonlocal Wave Particle Duality March 1, 2017 20 / 37

Ghost Interference

Fringe-width of ghost interference(a)

S

S1

1

2

2

D1

D1 D2

L1

D

L2

D

D D

1

1 2

Coincidence

counter

Double Slit

Double Slit

Fringe width follows Young’s double-slit formula

w =λDd

Remarkable, particle 2 doesn’t travel the distance D!


Ghost Interference

Author’s explanation

For photon 1the "blurring out" of the first order interferencefringes is due to the considerably large angularpropagation uncertainty of a single SPDC photon.

In terms of the Copenhagen interpretation one cansay that the interference is due to the uncertaintyin the birth place (A or B in Fig.) of a photon pair.


Ghost Interference

Theoretical analysis: Einstein-Podolsky-Rosen (EPR) state

Einstein, Podolsky and Rosen momentum entangled state 1

Ψ(z1, z2) = C∫ ∞−∞

dp eipz1/~e−ipz2/~,

Problems with the EPR state:1 Difficult to normalize2 Wave-function is unbounded in the space variable (z1 + z2)

3 Most likely, non achievable in practice.

Generalized EPR state2

Ψ(z1, z2) = C∫ ∞−∞

dp e−p2/4~2σ2e−ipz2/~eipz1/~e−

(z1+z2)2

4Ω2 ,

In the limit σ,Ω→∞ the above state reduces to the EPR state.1 A. Einstein, B. Podolsky N. Rosen, Phys. Rev. 47, 777 (1935).2T. Qureshi, Am. J. Phys. 73 (2005) 541.


Ghost Interference

Theoretical analysis continued . . .

Integrating generalized EPR state over p, gives

Ψ(z1, z2) =

√σ

πΩe−(z1−z2)2σ2

e−(z1+z2)2/4Ω2.

Ω and σ quantifies the position spread and momentum spread of the particlesalong direction z.After a time t0 particle 1 reaches the double-slit.

Ψ(z1, z2, t0) =1√

π(Ω + i~t04mΩ

)(1/σ + 4i~t0m/σ )

exp

[−(z1 − z2)

2

1σ2 + 4i~t0

m

− (z1 + z2)2

4Ω2 + i~t0m

],

Possibilities for particle 1:Passes through slit A→ |φA(z1)〉Passes through slit B→ |φB(z1)〉Gets blocked by the slit→ |χ(z1)〉.


Nonlocal Wave-Particle Duality

Outline

1 Introduction





6 Conclusions



S1 2

Double slit

D1 D2

Coincidencecounter

L 1D

L 2

Which-path detector

Figure: A ghostinterference experimentwhen which-waydetector is placed behindthe slits.

These three states are orthogonal: any state of particle 1 can be written interms of these:

|Ψ(z1, z2, t0)〉 = |φA〉〈φA|Ψ〉+ |φB〉〈φ|Ψ〉+ |χ〉〈χ|Ψ〉,The combined particle-detector state after the double-slit is given by:

|Ψ(z1, z2)〉 = |d1〉|φA〉|ψA〉+ |d2〉|φB〉|ψB〉+ |d0〉|χ〉|ψχ〉,Throwing away the blocked part and renormalizing it, we get

|Ψ(z1, z2)〉 =1√

〈ψA|ψA〉+ 〈ψB|ψB〉(|d1〉|φA〉|ψA〉+ |d2〉|φB〉|ψB〉)

Assume: |φA〉 and |φB〉 are gaussian states.Mohd Asad Siddiqui (NIT, Patna) Nonlocal Wave Particle Duality March 1, 2017 26 / 37


The state which emerges from the double slit, has the form

Ψ(z1, z2) = c|d1〉e−(z1−z0)2

ε2 e−(z2−z′0)2

Γ + c|d2〉e−(z1+z0)2

ε2 e−(z2+z′0)2

Γ ,

where c = (1/√πε)(√

ΓR + iΓI√ΓR

)−1/2 and

Γ =ε2 + 1

σ2 + ε2

4Ω2σ2 + 2i~t0m

1 + ε2

Ω2 + i2~t04Ω2m + 1

4Ω2σ2

+2i~t0

m.

The time evolution of entangled state is governed by H =p2

12m +

p22

2m .The probability of coincident click of D1 and D2 is given byP(z1, z2) = |Ψ(z1, z2, t)|2, has the form

|Ψ(0, z2, t)|2 = |Ct|2e−2z2

0ε2+[ λL

πε]2 e

−2(z22+z2

0)

γ2+[ λDπγ

]2 cosh

[4z2z0

γ2 + [λDπγ

]2

]1 + |〈d1|d2〉|cos[

4z2z0λDπγ4π2+λ2D2

]cosh

[4z2z0

γ2+[ λDπγ

]2

] ,

An interference pattern for particle 2 is obtained although it has not passedthrough any slit.



Underlying Physics

1 The entanglement helps to form a virtual slit for particle 2 (whendetected in coincidence).

2 One particle carry the which-path information of the other, as a result ofentanglement.

3 By detecting particle 2, one can tell which slit particle 1 passed through.4 Bohr’s complementarity principle compel to show no interference in

such a situation.5 The real reason of the absence of first order interference behind the

double slit.6 Fixing detector D1 loses the knowledge of the path of particle 1.7 Consequently, one loses information on which virtual slit particle 2

passed through.8 Interference is possible in this situation - ghost interference.



Underlying Physics









Underlying Physics









Underlying Physics









Underlying Physics









Underlying Physics









Underlying Physics









Underlying Physics









Trade-off between path distinguishability and visibility

The fringe visibility for particle 2, using V = Imax−IminImax+Imin

, is given by

V2 =|〈d1|d2〉|

cosh( 4z2z0γ2+[λD

πγ]2

),

cosh(θ) ≥ 1 =⇒ V2 ≤ |〈d1|d2〉|.The which-path distinguishability for particle 1, is given by

D1 =√

1− |〈d1|d2〉|2.

Their trade-off relation is given by D21 + V2

2 ≤ 1 .

It connects path information of one particle to interference visibility of other(remote) particle.


Ghost quantum eraser

Outline

1 Introduction





6 Conclusions



Quantum eraser

Erasing path information recovers the interference pattern.

Source

Quarter-wave plate 1

Source Linear Polarizer




Double Slit

Double Slit

Quarter wave plate provide "which-path" information

Linear polarizer erases "which-path" information

Source

Double Slit

M. O. Scully andK. Drühl,Phys. Rev. A 25,(1982).



Mathematics of Quantum eraser

The photon-detector entangled stateΨ(x) = 1√

2

(ψ1(x)⊗ |d1〉+ ψ2(x)⊗ |d2〉

)The probability density is given by

|Ψ(x)|2 =|ψ1(x)|2

2+|ψ2(x)|2

2+ Re ψ1(x)∗ ψ2(x) |〈d1|d2〉| (1)

The interference pattern is quantified by 〈d1|d2〉, and no interference patternwill be observed for 〈d1|d2〉 = 0

Apply linear polarizer in the path, |q1〉 = (|d1〉+ |d2〉)/√

2

|〈q1|Ψ(x)〉|2 =12(|ψ1(x)|2 + |ψ2(x)|2 + 2Re ψ1(x)∗ ψ2(x)

)(2)

Linear polarizer converts the left and right-circular polarized photons to

linearized ones, and thus erases the path information.Mohd Asad Siddiqui (NIT, Patna) Nonlocal Wave Particle Duality March 1, 2017 32 / 37


Ghost quantum erasera doable experiment

Erasing the path information of one particle recovers the interference patternfor other.

M. A. Siddiquiand T. Qureshi,Quant. Stud.:Math. Found.: 3(2016).



Ghost quantum eraser continued . . .

The left-circular polarizations −→ |d1〉The right-circular polarizations −→ |d2〉The linear polarization −→ |q1〉 = (|d1〉+ |d2〉)/

√2

The state of the photons which reach D1 (after passing through the horizontalplate) and D2 now, is given by 〈q1|Ψ(z1, z2, t)〉The coincident probability of detecting photon 2 is now given by

|〈q1|Ψ(0, z2, t)〉|2 =|Ct|2

2e

−2z20

ε2+[ λLπε

]2 e

−2(z22+z2

0)

γ2+[ λDπγ

]2 cosh

[4z2z0

γ2 + [λDπγ

]2

]1 +cos[

4z2z0λDπγ4π2+λ2D2

]cosh

[4z2z0

γ2+[ λDπγ

]2

] ,

The above represents an interference pattern.


Conclusions

Outline

1 Introduction





6 Conclusions


Conclusions

Conclusions

1 Wave nature and particle nature can be seen at the same time, although toa limited degree.

2 The trade relation for entangled particle in two slit ghost experiment isgiven by V2

2 +D21 ≤ 1.

3 The duality relation in ghost expeiment is non-local, the pathdistinguishability of one is related with the fringe visibilty of other.

4 An eraser experiment can also be realized in ghost set-up.


References

M. A. Siddiqui and T. Qureshi,“A nonlocal wave-particle duality,"Quantum Stud.: Math. Found.: 3, 115-122 (2016).

Tabish Qureshi,CTP, JMI, New Delhi.


Documents

[width=1.10cm]logo Nonlocal Wave Particle Dualityconfqic/youqu17/talks/asad.pdf · 3 Ghost Interference 4 Nonlocal Wave-Particle Duality 5 Ghost quantum eraser 6 Conclusions Mohd