quantum zeno dynamics induced by temperature b. d. militello 44th symposium on mathematical physics...
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Quantum Zeno Effect Possible transitions from subspace to subspaceTRANSCRIPT
Quantum Zeno dynamics induced by Temperature
B. D. Militello
44th Symposium on Mathematical Physics (Torun)
June 2012
Dipartimento di Fisica
Collaboration: . M. Scala and A. Messina (Palermo) . N. V. Vitanov (Sofia)
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
• Quantum Zeno subspaces
Misra and Sudarshan, J. Math. Phys. 18, 756 (1997)
Facchi and Pascazio, PRL 89, 080401 (2002)
• “Bang-bang” decoupling Facchi, Lidar and Pascazio, PRA 69, 032314 (2004)
• Continuous and Pulsed MeasurementsSchulman, PRA 57, 1509 (1998)
Zeno Phenomena
A decay can effectively be considered as a continuous measurement
=0
=1
=10
=100
t
P(t)The higher the decay rate, the stronger the inhibition of the dynamics:
Zeno Phenomena
The unitary counterpart: a strong coupling can struggle the effects of a weaker one, realizing the partitioning of the Hilbert space:
Zeno Phenomena
Restoring the dynamics through continuous measurements
=0, =0
/=10, =0
/=10, /=1
/=10, /=104
P1(t)
t
Zeno Phenomena
Again, the unitary counterpart:
Implications
• Connected with fundamental aspects of quantum mechanicsHome and Whitaker, Ann. Phys. 258, 237 (1997)
• Plenty of applications in Quantum Information and NanotechnologiesToo many...
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)
• Relevant in Biological processes
QZE and Temperature
• Temperature of the detector affects the Zeno effectRuseckas, PRA 66, 012105 (2002)
• Temperature contributes to determine the QZE-AZE borderManiscalco, Piilo, and Suominen, PRL 97, 130402 (2006)
Bhaktavatsala Rao and Kurizki, PRA 83, 032105 (2011)
• QZE-AZE border vs bath statistics, i.e. fermions vs bosons
Zeno subspaces induced by Temperature
Zeno subspaces induced by Temperature
• Zeno dynamics in Landau-Zener transitions
• Mechanism: a statistical approach
L-Z Transitions
Time
Ener
gy
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)
Transition Probability of dressed states = Survival Probability of bare states
L-Z Transitions
Transitions between time-dependent levels in the presence of a level crossing…
M. Scala, BM, A. Messina, N. Vitanov, PRA 84, 023416 (2011)
Time
Ener
gy
L-Z Transitions
Transitions between time-dependent levels in the presence of a level crossing in the presence of dissipation.
M. Scala, BM, A. Messina, N. Vitanov, PRA 84, 023416 (2011)
Interaction with the environment:
Time
Ener
gy
L-Z Transitions
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)
Asymptotic (after a long time) Survival Probability of the bare non-decaying state as a function of the Temperature:
Time
Ener
gy
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.
|1>
|2>
|3>
We start by considering a finite number of oscillators:
BM, M. Scala, A. Messina, PRA 84, 022106 (2011)
Single Oscillator
Single Harmonic Oscillator (D=1)
In an invariant subspace the dynamics of state |1,n> can be hindered for large n:
The Hamiltonian is divided into invariant blocks:
|1,n>|2,n>|3,n+1>
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
At low Temperature the Zeno dynamics never occurs.
At high Temperature the blocks where the Zeno dynamics occurs are more populated.
0 1 2 3 4 5 6
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>
|2>
|3>
Let us consider many oscillators…
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
An eigenvalue approches the “free energy” 1
The corresponding eigenstate approaches
the “bare” state |A>
|A>|B>|C>
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
Threshold Temperature:
We require that the probability of occupying blocks where the Zeno Dynamics does not occur is small.
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:
Temperature causes Zeno dynamics
Low Temperature:
High Temperature:
More about Temperature
Conditions for Zeno dynamics:
Temperature causes Zeno dynamics
Temperature destroys Zeno dynamics
Low Temperature:
High Temperature:
Thank you!