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arXiv:0912.1602 · Physical Review A

Observing different phases for the dynamics of entanglement in an ion trap

Cecilia Cormick, Juan Pablo Paz

Published Feb 9, 2010• 15 citations
arXivHTMLPDF

Abstract

The evolution of the entanglement between two oscillators coupled to a common thermal environment is nontrivial. The long time limit has three qualitatively different behaviors (phases) depending on parameters such as the temperature of the bath [J. P. Paz and A. J. Roncaglia, Phys. Rev. Lett. 100, 220401 (2008)]. The phases include cases with nonvanishing long-term entanglement, others with a final disentangled state, and situations displaying an infinite sequence of events of disappearance and revival of entanglement. We describe an experiment to realize these different scenarios in an ion trap. The motional degrees of freedom of two ions are used to simulate the system while the coupling to an extra (central) ion, which is continuously laser cooled, is the gateway to a decohering reservoir. The scheme proposed allows for the observation and control of motional entanglement dynamics and is an example of a class of simulations of quantum open systems in the non-Markovian regime.

DOI
10.1103/physreva.81.022306

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The evolution of the entanglement between two oscillators coupled to a common thermal environment is nontrivial. The long time limit has three qualitatively different behaviors (phases) depending on parameters such as the temperature of the bath [J. P. Paz and A. J. Roncaglia, Phys. Rev. Lett. 100, 220401 (2008)]. The phases include cases with nonvanishing long-term entanglement, others with a final disentangled state, and situations displaying an infinite sequence of events of disappearance and revival of entanglement. We describe an experiment to realize these different scenarios in an ion trap. The motional degrees of freedom of two ions are used to simulate the system while the coupling to an extra (central) ion, which is continuously laser cooled, is the gateway to a decohering reservoir. The scheme proposed allows for the observation and control of motional entanglement dynamics and is an example of a class of simulations of quantum open systems in the non-Markovian regime.
We study the dynamics of the entanglement between two oscillators that are initially prepared in a general two-mode Gaussian state and evolve while coupled to the same environment. In a previous paper, we showed that there are three qualitatively different dynamical phases for the entanglement in the long-time limit: sudden death, sudden death and revival, and no sudden death [Paz and Roncaglia, Phys. Rev. Lett. 100, 220401 (2008)]. Here we generalize and extend those results along several directions: We analyze the fate of entanglement for an environment with a general spectral density providing a complete characterization of the evolution for Ohmic, sub-Ohmic, and super-Ohmic environments. We consider two different models for the interaction between the system and the environment (one where the coupling is through position and another where the coupling is symmetric in position and momentum). Finally, we show that for nonresonant oscillators, the final entanglement is independent of the initial state and may be nonzero at very low temperatures.
We study the long-time dynamics of two entangled ions coupled by a single vibrational mode and two lasers with different frequencies as proposed by M\o{}lmer and S\o{}rensen [Phys. Rev. Lett. 82, 1835 (1999)]. It is shown that the time development involves collapses and revivals well known from the dynamics of a single atom in a cavity or in a trap. The revival time is found to be insensitive to the details of the initial collective vibrational state. Therefore, revival and collapse phenomena are obtained for coherent, thermal as well as other vibrational states.
We provide a complete characterization of the evolution of entanglement between two resonant oscillators coupled to a common environment. We identify three phases with different qualitative long time behavior. There is a phase where entanglement undergoes a sudden death. Another phase (sudden death and revival) is characterized by an infinite sequence of events of sudden death and revival of entanglement. In the third phase (no sudden death) there is no sudden death of entanglement, which persists for a long time. The phase diagram is described and analytic expressions for the boundary between phases are obtained. These results are applicable to a large variety of non-Markovian environments. The case of nonresonant oscillators is also numerically investigated.
We cannot translate quantum behavior arising with superposition states or entanglement efficiently into the classical language of conventional computers (Feynman et al. in Int. J. Theor. Phys. 21:467, 1982 ). A universal quantum computer could describe and help to understand complex quantum systems. But it is envisioned to become functional only within the next decade(s). A shortcut was proposed via simulating the quantum behavior of interest in another quantum system, where all relevant parameters and interactions can be controlled and observables of interest detected sufficiently well. For example simulating quantum spin systems within an architecture of trapped ions (Porras and Cirac in Phys. Rev. Lett. 92:207901, 2004 ). Here we specify how we simulate the spin and all necessary interactions and how we calibrate their amplitudes. For example via a two-ion phase-gate operation on two axial motional modes simultaneously at a fidelity exceeding 95%. We explain the complete mode of operation of a quantum simulator on the basis of our simple model case—the proof of principle experiment of simulating the transition of a quantum magnet from paramagnetic into entangled ferromagnetic order (Friedenauer et al. in Nat. Phys. 4:757, 2008 ) and emphasize some of the similarities and differences with a quantum computer.
We propose experimental methods to engineer reservoirs at arbitrary temperature which are feasible with current technology. Our results generalize to mixed states the possibility of quantum state engineering through controlled decoherence. Finite-temperature engineered reservoirs can lead to the experimental observation of thermal entanglement---the appearance and increase of entanglement with temperature---to the study of the dependence of finite-time disentanglement and revival with temperature, quantum thermodynamical effects, and others, enlarging the comprehension of temperature-dependent entanglement properties. Our proposal is discussed in detail in two model systems, consisting of different modes of a single photon and a trapped-ion system.
Quantum simulation of open quantum systems offers a pathway towards better understanding various non-equilibrium physics that would otherwise be challenging to study. While most open quantum systems studied are modeled as being memory-less (obeying the Markov approximation), real baths generally are influenced by the system-bath interaction, and some systems existing in structured non-Markovian environments can display novel behavior as a result. Here we utilize a trapped ion quantum simulator to simulate a single spin-1/21/21/2 driven-dissipative system with a non-Markovian dissipation channel, and experimentally compare steady-states to those from an analogous Markovian bath. We observe that a non-Markovian dissipative channel can shift the steady-state even for a single qubit, to a regime inaccessible for Markovian dissipation. The techniques used here are compatible with many-body extensions of the model, which can not be simulated efficiently on a classical computer in general. Our work also opens up new possibilities in quantum reservoir engineering beyond the Markovian regime.