Spin dynamics and the Quantum Zeno Effect

QUEST - Centre for Quantum Engineering and Space-Time Research

Spin dynamics and the Quantum Zeno Effect

Fresco in the Library of El Escorial, Madrid.

Carsten Klempt, Luis Santos, Augusto Smerzi, Wolfgang Ertmer


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Zeno’s paradoxes

The quantum Zeno effect

Spin dynamics and the quantum Zeno effect

Entanglement and the quantum Zeno effect

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Zeno’s paradoxes




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Zeno of Elea

490 v. Chr. - 430 v. Chr.


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The paradoxes of Zeno of Elea

• "not less than forty arguments revealing

contradictions" –Proclus

• Only nine are known

• First examples of reductio ad absurdum

• Paradoxes of motion:o The dichotomy paradoxo Achilles and the tortoiseo The arrow paradox


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𝑠𝑔𝑒𝑠=∑𝑖=0

∞

𝑠𝑖=∑𝑖=0

∞

( 12 )𝑖 ( 𝑠𝑔𝑒𝑠2 )=( 1

1− 12 )(

𝑠𝑔𝑒𝑠2 )=𝑠𝑔𝑒𝑠

The dichotomy paradox

That which is in locomotion must arrive at the half-way stage before it arrives at the goal.

–Aristotle


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Achilles and the tortoise


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𝑡1=𝑠1𝑣𝐴

=𝑣 𝑆

𝑣𝐴𝑡 0

𝑡 2=𝑠2𝑣𝐴

=𝑣𝑆

𝑣𝐴𝑡 1

Achilles and the tortoise

𝑡 0=𝑠0𝑣𝐴

𝑠1=𝑣𝑆𝑡 0

𝑠2=𝑣𝑆𝑡 1

𝑠0


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The arrow paradox

If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.

–Aristotle

𝑣=lim❑

∆𝑠∆ 𝑡


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Zeno’s paradoxes




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Zeno with a quantum arrow

Zeno: The spin cannot rotate in the Bloch sphere


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The quantum Zeno setup

Zeno: divide time in m small intervals and follow the dynamics at each time step.

(total time : t = m τ = π )


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Peres, Am. J. Phys. 48, 931 (1980).

Zeno: check at each time step if the spin really rotated: projective measurements

The projective measurement haseigenvalues “yes”, “no”.The “yes” projects on the subspacewith probability


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Zeno: give a look at the survival probability(the probability that at the final time the spin is still pointing up)

The arrow does not rotate if watched !


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The quantum Zeno effect in a BEC


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Level scheme

F=2

F=1

mF= -2 -1 0 +1 +2

5P3/2

5S1/2

6.8 GHz

780 nm


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Pulsed measurements


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Experimental results


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Zeno’s paradoxes




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BEC spin dynamics

-1 0 1

Idea:• Spin dynamics as slow coherent process• Prevent spin dynamics by Zeno measurement• It is sufficient to measure one ±1 component• The creation of the other is blocked by entanglement


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Level scheme

F=2

F=1

mF= -2 -1 0 +1 +2

5P3/2

5S1/2

6.8 GHz

780 nm

?


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Expected result

without Zenomeasurements

with Zeno measurements


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Level scheme

F=2

F=1

5P3/2

5S1/2

6.8 GHz

780 nm

10 Hz

10 kHz

10-100 kHz

6 MHz


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Zeno’s paradoxes




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Zeno dynamics and entanglement

Complicated, extremely entangled,

fragile state

unwanted state

decoherence

Is the stateintact?


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Entangled states are more difficult to protect!


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Level scheme

F=2

F=1

mF= -2 -1 0 +1 +2

5P3/2

5S1/2

6.8 GHz

780 nm


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Two-mode squeezed vacuum

σ(N-1 – N+1) = 0

σ(Φ-1 – Φ+1) /3 N-1 , Φ-1

N+1, Φ+1

Barnett & Pegg, Phys. Rev. A 42, 6713 (1990).


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Level scheme

F=2

F=1

mF= -2 -1 0 +1 +2

5P3/2

5S1/2

6.8 GHz

780 nm


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Jz/J

-1

+1

0

Rotation angle ↔ Variance

Jz2

‹Jz›=0

Probability distribution


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Distribution after rotation


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Level scheme

F=2

F=1

mF= -2 -1 0 +1 +2

5P3/2

5S1/2

6.8 GHz

780 nm


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Expected result

• Twin Fock state can be protected against rotation

• Zeno measurements must be fast.

• They are faster than for a classical state

Entanglement is difficult to protect by Zeno measurements


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Thank you for your attention


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Spin dynamics and the Quantum Zeno Effect

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