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Dipartimento
di Fisica
B. D. Militello
Quantum Zeno dynamics induced by
Temperature
Collaboration:
. M. Scala and A. Messina (Palermo)
. N. V. Vitanov (Sofia)
June 2012
44th Symposium on Mathematical Physics (Torun)
Quantum Zeno Effect
Very frequent projective measurements
freeze the dynamics and force the system
to stay in its initial state.
Misra and Sudarshan, J. Math. Phys. 18, 756 (1997)
Quantum Zeno Effect
Possible transitions from
subspace to subspace
Quantum Zeno Effect
Possible transitions from
subspace to subspace
Competition between evolution
and measurements: Formation
of invariant subspaces
Zeno Phenomena
• Quantum Zeno effect induced by (pulsed) measurements
Misra and Sudarshan, J. Math. Phys. 18, 756 (1997)
• Continuous and Pulsed Measurements
Schulman, PRA 57, 1509 (1998)
• Quantum Zeno subspaces
Facchi and Pascazio, PRL 89, 080401 (2002)
• “Bang-bang” decoupling
Facchi, Lidar and Pascazio, PRA 69, 032314 (2004)
Zeno Phenomena
A decay can effectively be considered as a continuous measurement
W
G
The higher the decay rate,
the stronger the inhibition
of the dynamics:
P(t)
G/W=0
G/W=1
G/W=10
G/W=100
Wt
Zeno Phenomena
The unitary counterpart: a strong coupling can struggle the effects of a weaker
one, realizing the partitioning of the Hilbert space:
W
L
Zeno Phenomena
Restoring the dynamics through continuous measurements
W
L
G
P1(t)
D=0, G=0
D/W=10, G=0
D/W=10, G/W=1
D/W=10, G/W=104
t
Zeno Phenomena
Again, the unitary counterpart:
W
L
Q
Implications
• Connected with fundamental aspects of quantum mechanics
Home and Whitaker, Ann. Phys. 258, 237 (1997)
• Plenty of applications in Quantum Information and Nanotechnologies
Too many...
• Relevant in Biological processes
Rebentrost, Mohseni, Kassal, Lloyd and Aspuru-Guzik, New J. Phys 11, 033003 (2009);
Caruso, Chin, Datta, Huelga and Plenio, J. Chem. Phys. 131, 105106 (2009)
QZE and Temperature
• Temperature of the detector affects the Zeno effect
Ruseckas, PRA 66, 012105 (2002)
• Temperature contributes to determine the QZE-AZE border
Maniscalco, Piilo, and Suominen, PRL 97, 130402 (2006)
• QZE-AZE border vs bath statistics, i.e. fermions vs bosons
Bhaktavatsala Rao and Kurizki, PRA 83, 032105 (2011)
Zeno subspaces induced by
Temperature
Zeno subspaces induced by
Temperature
• Zeno dynamics in Landau-Zener transitions
• Mechanism: a statistical approach
Transitions between time-dependent
levels in the presence of a level
crossing…
L. D. Landau, Phys. Z. Sowjetunion 2, 46 (1932); C.
Zener, Proc. R. Soc. London A 137, 696 (1932); E. C. G.
Stuckelberg, Helv. Phys. Acta 5, 369 (1932); E. Majorana,
Nuovo Cimento 9, 43 (1932)
Energy
L-Z Transitions
Time
Transition Probability of dressed states = Survival Probability of bare states
Transitions between time-dependent
levels in the presence of a level
crossing…
Energy
L-Z Transitions
Time
M. Scala, BM, A. Messina, N. Vitanov, PRA 84, 023416 (2011)
Transitions between time-dependent
levels in the presence of a level
crossing in the presence of dissipation.
Energy
L-Z Transitions
Time
Interaction with the environment:
M. Scala, BM, A. Messina, N. Vitanov, PRA 84, 023416 (2011)
Asymptotic (after a long time) Survival
Probability of the bare non-decaying
state as a function of the Temperature:
Energy
L-Z Transitions
Time
Temperature is responsible for an
increase of the transition probability
(Zeno effect, in terms of the bare
basis)
M. Scala, BM, A. Messina, N. Vitanov, PRA 84, 023416 (2011)
Mechanism:
a statistical approach
BM, M. Scala, A. Messina, PRA 84, 022106 (2011)
BM, PRA 85, 064102 (2012)
Hamiltonian Model
Three levels, two coupled by an
external field, two coupled by the
bath.
We start by considering a finite
number of oscillators:
|1>
W
|2>
|3>
BM, M. Scala, A. Messina, PRA 84, 022106 (2011)
Single Oscillator
Single Harmonic Oscillator (D=1)
The Hamiltonian is divided into invariant blocks:
|1,n>
|2,n>
|3,n+1>
In an invariant subspace the dynamics of state |1,n> can be hindered for large n:
Hindered dynamics, in the subspace
Single Oscillator
The single Harmonic Oscillator is prepared in a thermal state
Single Oscillator
The single Harmonic Oscillator is prepared in a thermal state
0
1
2
3
4
5
6
At low Temperature the Zeno dynamics
never occurs.
At high Temperature the blocks where the
Zeno dynamics occurs are more populated.
Single Oscillator
Single Oscillator
The higher is the temperature of the oscillator, the more the three-level system is frozen in
the initial state:
The population of the state |1>
approaches unity at every
time, when the temperature
becomes higher and higher.
Many Oscillators
|1>
Let us consider many oscillators…
W
|2>
|3>
Many Oscillators
Many Harmonic Oscillators: an invariant block of the Hamiltonian
Many Oscillators
|A>
|B>
|C>
Many Oscillators
|A>
|B>
|C>
Characteristic equation:
Many Oscillators
|A>
|B>
|C>
An eigenvalue approches the “free energy” d1
The corresponding
eigenstate approaches the
“bare” state |A>
Many Oscillators
For large excitation numbers the dynamics in the block is hindered:
Many Oscillators
We require that the probability of occupying blocks
where the Zeno Dynamics does not occur is small.
Many Oscillators
We require that the probability of occupying blocks
where the Zeno Dynamics does not occur is small.
Many Oscillators
We require that the probability of occupying blocks
where the Zeno Dynamics does not occur is small.
Threshold Temperature:
Many Oscillators
The previous results have been obtained in the following physical scenarios:
The band of the oscillators is
supposed not to intersect the
transition 1-3.
More about Temperature
Two oscillators drive transitions 1-2 and 2-3:
BM, PRA 85, 064102 (2012)
More about Temperature
Two oscillators drive transitions 1-2 and 2-3:
Because of the conservation of the total number of excitations, one has:
BM, PRA 85, 064102 (2012)
More about Temperature
Conditions for Zeno dynamics:
Low Temperature:
High Temperature:
More about Temperature
Conditions for Zeno dynamics:
Low Temperature:
High Temperature:
Temperature causes Zeno dynamics
More about Temperature
Conditions for Zeno dynamics:
Low Temperature:
High Temperature:
Temperature causes Zeno dynamics
Temperature destroys Zeno dynamics
Thank you!