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Trade-off relation between generalized which-way information and fringe visibility A. R. Usha Devi Department of physics Bangalore University Bengaluru-560 056 India IOP, , Bhuban banes eswar ar February 9-18, 2016 Einstein and Bohr

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Trade-off relation between generalized which-way information and fringe visibility

  • A. R. Usha Devi

Department of physics Bangalore University Bengaluru-560 056 India

IOP, , Bhuban banes eswar ar February 9-18, 2016

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Einstein and Bohr debated over quantum theory for years, and never agreed. The debates represent one of the highest points of scientific research in the first half of the twentieth century because it called attention to quirky elements of quantum theory, complementarity, non-locality and entanglement, which are central to the modern quantum information science.

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The most beautiful experiment Sep 1, 2002

The most beautiful experiment in physics, according to a poll of Physics World readers, is the interference of single electrons in a Young's double slit.

  • Which is the most beautiful experiment in physics according to you?

This question was asked to Physics World readers - and more than 200

  • replied. Majority vote was for the classic experiments by Galileo,

Millikan, Newton and Thomas Young. But uniquely among the top 10, Young's double-slit experiment applied to the interference of single electrons remained as one of the most beautiful experiments in physics.

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Wave or particle?

  • First decade of 1800: Young – Double slit interference.
  • 1909: Geoffrey Ingram (G I) Taylor – Interference with

feeblest light (equiuivalent to "a candle burning at a distance slightly exceeding a mile“) leads to interference.

  • --- Dirac’s famous statement “each photon interferes with

itself” 1927: Clinton Davisson and Lester Germer -- Diffraction

  • f electrons from Nickel crystal – wave nature of particles

(electrons) -- 1937 Nobel prize for the "discovery of the interference phenomena arising when crystals are exposed to electronic beams“ along with G. P. Thomson.

Thomas Young's sketch of two-slit interference based on observations

  • f water waves.

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Wave nature of electrons in a double slit interference

  • C. Jönsson , Tübingen,

Germany, 1961 4000 clicks

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  • Not a wave of particles
  • Single particles interfere with themselves !!

Intensity so low that

  • nly one electron at a time

Akira Tonomura and co-workers, Hitachi, 1989

Single Particle at a time

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  • Two-slit wave packet collapsing
  • Eventually builds up pattern
  • Particle interferes with itself !!

Single particle interference

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Single electron interference at Hitachi (captured at different times)

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  • A classical particle would follow some single path
  • Can we say a quantum particle does, too?
  • Can we measure it going through one slit or another?

Which path ?

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  • Einstein proposed different ways to

measure which slit the particle went through, without blocking it

  • Each time, Bohr showed how that

measurement would wash out the wave function.

Movable wall; measure recoil Source No: Movement of slit washes out pattern Niels Bohr Albert Einstein

Which path ?

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  • Short answer: no, we can’t tell
  • Anything that blocks one slit washes out the

interference pattern

Which path ?

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Bohr’s Complementarity principle (1933)

Niels Bohr

  • Wave and particle natures are complementary !!
  • Depending on the experimental setup one obtains

either wave nature or particle nature – not both at a time

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Mach-Zehnder Interferometer -- Open Setup 

Single quanton

D0 D1 1 Only one detector clicks at a time

BS1

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Single photon

D0 D1 1 Trajectory can be assigned

BS1

2 1  

 i

e

Mach-Zehnder Interferometer -- Open Setup

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Single photon

D0 D1 1 Trajectory can be assigned

BS1

1 2 1  

 i

e

Mach-Zehnder Interferometer -- Open Setup

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Single photon

D0 D1 1 Trajectory can be assigned : Particle nature !!

BS1

Mach-Zehnder Interferometer -- Open Setup

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Intensity

Intensities are independent of  i.e., no interference

2 / 1 ) 1 ( ...... 1 2 1 2 / 1 ) ( ...... 2 1       p e p e

i i  

Mach-Zehnder Interferometer -- Open Setup

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Single photon

D0 D1 1 Again only one detector clicks at a time !!

BS1 BS2

Mach-Zehnder Interferometer -- Closed Setup

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Single photon

D0 D1 1 Again only one detector clicks at a time !!

BS1 BS2

2 1

 i

e 

   

   

) 2 / ( sin ) 1 ( ) 2 / ( cos ) ( 2 1 1 2 1 2 1 1

2 2

 

  

         p p e e e

i i i

Mach-Zehnder Interferometer -- Closed Setup

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Mach-Zehnder Interferometer -- Closed Setup

Intensities depend on  : Interference!!

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Intensity

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BS2 removes ‘which path’ information Trajectory can not be assigned : Wave nature !! 

Single photon

D0 D1 1

BS1 BS2

Mach-Zehnder Interferometer -- Closed Setup

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Does quanton know the setup ?

 D0 D1 1 Open Setup Closed Setup  D0 D1 1

BS2 BS1 BS1

Particle behavior Wave behavior

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Two schools of thought

Bohr, Pauli, Dirac, ….

  • Intrinsic wave-particle duality
  • Reality depends on observation
  • Complementarity principle

Einstein, Bohm, ….

  • Apparent wave-particle duality
  • Reality is independent of observation
  • Hidden variable theory

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Bohr's complementarity principle: Every quantum system has mutually incompatible properties which cannot be simultaneously measured.

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Delayed Choice Experiments

An idea introduced by John A Wheeler of the University of Texas at Austin in 1978

  • J. A. Wheeler, Mathematical Foundations of Quantum Mechanics, edited by

A.R. Marlow (Academic, New-York, 1978) pp. 9-48; Quantum Theory and Measurement, J. A. Wheeler, W. H. Zurek, Eds. (Princeton Univ. Press, Princeton, NJ, 1984), pp. 182–213.

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Suppose that the path lengths of a Mach-Zehnder interferometer have been tuned to make the quanton come out of one port of the final beam splitter with probability 1. After the quanton has passed the first beam splitter so that it is fully inside the interferometer, and before it has reached the second beam splitter, you decide to whisk away that second beam splitter, preventing any interference between the quanton’s two paths from taking place. Without interference, the quanton behaves like a particle and emerges with equal probability out of either of the two ports of the apparatus where the second beam splitter used to be.

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B.-G. Englert, Fringe Visibility and Which-Way Information: An Inequality,

  • Phys. Rev. Lett. 77, 2154 (1996).

The trade-off between the amount of which-way information encoded in the detector system and the fringe visibility is captured in terms of a generalized complementarity relation

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Which way information

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Visibility |V|

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Trade-off

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D =1 (particle nature) V=0 D=0 (wave nature) V=1

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General Scenario

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So far….

The trade-off between interference visibility and which-path distinguishability for a quantum particle possessing an internal structure -- such as spin or polarization is useful to erase ‘which- path’ information (by appropriate preparations of states of the internal degree of freedom). One can thus recover interference

  • the internal structure could play a manipulative role in

controlling the information about which path in the interferometer arms is taken by the particle.

  • Generalized fringe visibility and detector state distinguishability

show complementarity (trade-off)

  • What happens if detector state has an internal structure??

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Channel discrimination and which-path information in two-slit interference

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Channel Discrimination Distinguishing two channels with input state :

1 0, 

 

Input state Channel Output state

1

1

Channel discrimination  which path information

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All entangled states are useful for Channel discrimination task

  • M. F. Sacchi, Phys. Rev. A 71, 062340 (2005); 72, 014305 (2005)
  • M. Piani and J. Watrous, Phys. Rev. Lett. 102, 250501 (2009)

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We put forth some instances where distinguishability is 0, yet generalized fringe visibility is not equal to 1. Where is the missing information? Our work: Tracking missing ‘which-path’ information via Generalized distinguishability when detector is assisted by an ancilla.

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