spin dynamics and the quantum zeno effect
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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
Content
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Zeno’s paradoxes
The quantum Zeno effect
Spin dynamics and the quantum Zeno effect
Entanglement and the quantum Zeno effect
Content
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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
The quantum Zeno effect
Spin dynamics and the quantum Zeno effect
Entanglement and the quantum Zeno effect
Content
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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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The quantum Zeno effect
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
The quantum Zeno effect
Spin dynamics and the quantum Zeno effect
Entanglement and the quantum Zeno effect
Content
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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
The quantum Zeno effect
Spin dynamics and the quantum Zeno effect
Entanglement and the quantum Zeno effect
Content
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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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