quantum zeno dynamics induced by temperature b. d. militello 44th symposium on mathematical physics...

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.


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)


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

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)


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)


Conditions for Zeno dynamics:

Low Temperature:

High Temperature:



Temperature causes Zeno dynamics

Low Temperature:

High Temperature:



Temperature causes Zeno dynamics

Temperature destroys Zeno dynamics

Low Temperature:

High Temperature:

Thank you!

quantum zeno dynamics induced by temperature b. d. militello 44th symposium on mathematical physics...

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