Dipartimento di Fisica B D Militello Quantum Zeno

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 44 th 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=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 P 1(t) D=0, G=0 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 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” d 1 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. 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

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