### Signatures of a quantum dynamical phase transition in a three-spin system in presence of a spin environment 10.1016/j.physb.2007.04.083 : Physica B: Condensed Matter | ScienceDirect.com

Signatures of a quantum dynamical phase transition in a three-spin system in presence of a spin environment

Gonzalo A. Álvarez, Patricia R. Levstein, Horacio M. Pastawski

Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina

Available online 5 May 2007.

## Abstract

We have observed an environmentally induced quantum dynamical phase transition in the dynamics of a two-spin experimental swapping gate [G.A. Álvarez, E.P. Danieli, P.R. Levstein, H.M. Pastawski, J. Chem. Phys. 124 (2006) 194507]. There, the exchange of the coupled states |↑,↓〉 and |↓,↑〉 gives an oscillation with a Rabi frequency *b*/ℏ (the spin-spin coupling). The interaction, ℏ/*τ*_{SE} with a spin-bath degrades the oscillation with a characteristic decoherence time. We showed that the swapping regime is restricted only to *b**τ*_{SE}≳ℏ. However, beyond a critical interaction with the environment the swapping freezes and the system enters to a Quantum Zeno dynamical phase where relaxation decreases as coupling with the environment increases. Here, we solve the quantum dynamics of a two-spin system coupled to a spin-bath within a Liouville–von Neumann quantum master equation and we compare the results with our previous work within the Keldysh formalism. Then, we extend the model to a three interacting spin system where only one is coupled to the environment. Beyond a critical interaction the two spins not coupled to the environment oscillate with the bare Rabi frequency and relax more slowly. This effect is more pronounced when the anisotropy of the system-environment (SE) interaction goes from a purely XY to an Ising interaction form.

### Quantum dynamical phase transition in a system with many-body interactions | Solid State Communications

Quantum dynamical phase transition in a system with many-body interactions

E.P. Danieli, G.A. Álvarez, P.R. Levstein, H.M. Pastawski

Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina

Received 26 October 2006. Accepted 1 November 2006. Available online 16 November 2006. by R. Merlin.

## Abstract

Recent experiments, [G.A. Álvarez, E.P. Danieli, P.R. Levstein, H.M. Pastawski, J. Chem. Phys. 124 (2006) 194507], have reported the observation of a quantum dynamical phase transition in the dynamics of a spin swapping gate. In order to explain this result from a microscopic perspective, we introduce a Hamiltonian model of a two level system with many-body interactions with an environment whose excitation dynamics is fully solved within the Keldysh formalism. If a particle starts in one of the states of the isolated system, the return probability oscillates with the Rabi frequency *ω*_{0}. For weak interactions with the environment , we find a slower oscillation whose amplitude decays with a rate . However, beyond a finite critical interaction with the environment, , the decay rate becomes . The oscillation period diverges showing a quantum dynamical phase transition to a Quantum Zeno phase consistent with the experimental observations.

### Environmentally induced quantum dynamical phase transition in the spin swapping operation | Journal of Chemical Physics

Environmentally induced quantum dynamical phase transition in the spin swapping operation

Gonzalo A. Álvarez, Ernesto P. Danieli, Patricia R. Levstein, and Horacio M. Pastawski

Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000, Córdoba, Argentina

*b*= Δ

*E*that splits the energy levels and induces an oscillation with a natural frequency Δ

*E*/

*ℏ*. Interaction of strength

*ℏ*/

*τ*

_{SE}, with an environment of neighboring spins, degrades this oscillation within a decoherence time scale

*τ*

_{ϕ}. While the experimental frequency

*ω*and decoherence time

*τ*

_{ϕ}were expected to be roughly proportional to

*b*/

*ℏ*and

*τ*

_{SE}, respectively, we present here experiments that show drastic deviations in both

*ω*and

*τ*

_{ϕ}. By solving the many spin dynamics, we prove that the swapping regime is restricted to Δ

*E*

*τ*

_{SE}≳

*ℏ*. Beyond a critical interaction with the environment the swapping freezes and the decoherence rate drops as 1/

*τ*

_{ϕ}∝(

*b*/

*ℏ*)

^{2}

*τ*

_{SE}. The transition between quantum dynamical phases occurs when

*ω*∝ becomes imaginary, resembling an overdamped classical oscillator. Here, 0 ⩽

*k*

^{2}⩽ 1 depends only on the anisotropy of the system-environment interaction, being 0 for isotropic and 1 for

*X*

*Y*interactions. This critical onset of a phase dominated by the quantum Zeno effect opens up new opportunities for controlling quantum dynamics.

© 2006 *American Institute of Physics*

### Quantum dynamics under coherent and incoherent effects of a spin bath in the Keldysh formalism: application to a spin swapping operation | Chemical Physics Letters

Quantum dynamics under coherent and incoherent effects of a spin bath in the Keldysh formalism: application to a spin swapping operation

Ernesto P. Danieli,

Horacio M. PastawskiCorresponding author contact information, E-mail the corresponding author,

Gonzalo A. Álvarez

Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina

Received 30 March 2004. Revised 20 October 2004. Available online 22 December 2004.

## Abstract

We develop the Keldysh formalism for the polarization dynamics of an open spin system. We apply it to the swapping between two qubit states in a model describing an NMR cross-polarization experiment. The environment is a set of interacting spins. For fast fluctuations in the environment, the analytical solution shows effects missed by the secular approximation of the quantum master equation for the density matrix: a frequency decrease depending on the system-environment escape rate and the quantum quadratic short time behavior. Considering full memory of the bath correlations yields a progressive change of the swapping frequency.

### Many-spin quantum dynamics during cross polarization in 8CB | Journal of Chemical Physics

Many-spin quantum dynamics during cross polarization in 8CB

Ana K. Chattah1, Gonzalo A. Álvarez1, Patricia R. Levstein1, Fernando M. Cucchietti1, Horacio M. Pastawski1, Jésus Raya2, and Jérôme Hirschinger2

via Many-spin quantum dynamics during cross polarization in 8CB | Browse – Journal of Chemical Physics.

1Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina

2Laboratoire de RMN de la Matière Condensée, Institut Le bel, Université Louis Pasteur, 67000, Strasbourg, France

(Received 12 May 2003; accepted 23 July 2003)

^{13}C coupled to two protons

^{1}H while the coupling to a spin bath is treated in the fast fluctuation approximation. This model is applied to the methylene and biphenyl groups of the smectic and nematic phases of the liquid crystal 4-

*n*-octyl-4′-cyanobiphenyl (8CB). Experimental data from standard CP, combined with our theoretical results, allow us to separate the homonuclear

^{1}H–

^{1}H and heteronuclear

^{1}H–

^{13}C residual dipolar couplings. These values are in good agreement with those obtained by using a combination of CP under Lee–Goldburg conditions and standard CP data. A well differentiated relaxation behavior among the two phases seems to indicate that while the extreme narrowing approximation is appropriate for the nematic phase, the description of the smectic phase requires consideration of the slow-motion limit. ©

*2003 American Institute of Physics.*

© 2003 *American Institute of Physics*

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