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

Danieli, Ernesto; Pastawski, Horacio M.; Álvarez, Gonzalo

doi:10.1016/j.cplett.2004.11.056

Cited by 18 publications

(38 citation statements)

References 34 publications

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“…Consider, for example, a spin chain with XY interaction as described in Refs. [19] and [9]. One may assume that the initial condition of the dimer used a swap gate is in the state |1, 0 while the chain that acts as environment is in a mixed state with 1/2 occupation at each site.…”

Section: Discussionmentioning

confidence: 99%

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Dynamical regimes of a quantum SWAP gate beyond the Fermi golden rule

2008

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We discuss how the bath's memory affects the dynamics of a swap gate. We present an exactly solvable model that shows various dynamical transitions when treated beyond the Fermi Golden Rule. By moving continuously a single parameter, the unperturbed Rabi frequency, we sweep through different analytic properties of the density of states: (I) collapsed resonances that split at an exceptional point in (II) two resolved resonances ; (III) out-of-band resonances; (IV) virtual states; and (V) pure point spectrum. We associate them with distinctive dynamical regimes: overdamped, damped oscillations, environment controlled quantum diffusion, anomalous diffusion and localized dynamics respectively. The frequency of the swap gate depends differently on the unperturbed Rabi frequency. In region I there is no oscillation at all, while in the regions III and IV the oscillation frequency is particularly stable because it is determined by the environment's band width. The anomalous diffusion could be used as a signature for the presence of the elusive virtual states.

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Section: Discussionmentioning

confidence: 99%

“…In this work we present a simple and exactly solvable quantum model that, while only describing the coherent part of a spin SWAP gate dynamics [19], presents the dynamical transitions. Our goal is to deepen the understanding of how bath's memory affects the system dynamics beyond the Fermi golden rule.…”

Section: Introductionmentioning

confidence: 99%

Dynamical regimes of a quantum SWAP gate beyond the Fermi golden rule

2008

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“…This "nesting" of two Fermi golden rule rates is formally obtained from a continued fractions evaluation of the self-energies 32,33 involving an infinite order perturbation theory. Another relevant result is that the frequency depends not only on b, but also on τ SE .…”

Section: E Limiting Casesmentioning

confidence: 99%

Environmentally induced quantum dynamical phase transition in the spin swapping operation

Álvarez¹,

Danieli²,

Levstein³

et al. 2006

The Journal of Chemical Physics

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127

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Quantum Information Processing relies on coherent quantum dynamics for a precise control of its basic operations. A swapping gate in a two-spin system exchanges the degenerate states |↑, ↓ and |↓, ↑ . In NMR, this is achieved turning on and off the spin-spin interaction b = ∆E that splits the energy levels and induces an oscillation with a natural frequency ∆E/ℏ. Interaction of strength /τSE, 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/ and τSE 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 ∆EτSE ℏ. Beyond a critical interaction with the environment the swapping freezes and the decoherence rate drops as 1/τ φ ∝ (b/ ) 2 τ SE . The transition between quantum dynamical phases occurs when ω ∝ (b/ℏ) 2 − (k/τ SE ) 2 becomes imaginary, resembling an overdamped classical oscillator. Here, 0 ≤ k 2 ≤ 1 depends only on the anisotropy of the systemenvironment interaction, being 0 for isotropic and 1 for XY interactions. This critical onset of a phase dominated by the Quantum Zeno effect opens up new opportunities for controlling quantum dynamics.

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“…We consider two spin configurations that are representative of physical situations of contrasting topology. One is a spin ladder which is a variant of the linear chains which are exactly solvable [16,23] and are known to have strong mesoscopic echoes [4,5]. The other is a spin star, a cluster with random long range interactions which is a fair representation of many molecular crystals [24].…”

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confidence: 99%

“…Analytical solutions of ensemble dynamics, either from the integration of the Liouville von-Neumann equation [12] or the alternative Keldysh formalism [13,14] are limited to special cases (e.g. [15,16]). Thus, one has to resort to numerical solutions.…”

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Quantum Parallelism as a Tool for Ensemble Spin Dynamics Calculations

Álvarez

Danieli²,

Levstein³

et al. 2008

Phys. Rev. Lett.

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Efficient simulations of quantum evolutions of spin-1/2 systems are relevant for ensemble quantum computation as well as in typical NMR experiments. We propose an efficient method to calculate the dynamics of an observable provided that the initial excitation is "local". It resorts a single entangled pure initial state built as a superposition, with random phases, of the pure elements that compose the mixture. This ensures self-averaging of any observable, drastically reducing the calculation time. The procedure is tested for two representative systems: a spin star (cluster with random long range interactions) and a spin ladder.PACS numbers: 03.67. Lx, 05.30.Ch, 75.40.Gb, 75.10.Pq One of the goals of quantum computers is the simulation of quantum systems. While quantum processing is ideally cast in terms of pure states, experimental realizations are a major challenge [1] met only for very small systems [2]. The alternative use of statistical mixtures of pure states, as the spin ensembles in standard NMR experiments [3,4,5,6], led to the development of the ensemble quantum computation (EQC) [7] that allowed useful algorithm optimizations [8,9]. Recently, EQC was experimentally implemented in a 12-qubits system [10] and much larger quantum registers [11] were prepared to assess its stability against decoherence. Hence, an efficient evaluation of ensemble evolutions is needed. Analytical solutions of ensemble dynamics, either from the integration of the Liouville von-Neumann equation [12] or the alternative Keldysh formalism [13,14] are limited to special cases (e.g. [15,16]). Thus, one has to resort to numerical solutions. One can describe an ensemble of M spins by using the 2 M × 2 M density matrix, but this quickly reaches a storage limit as M increases. Thus, a typical exact diagonalization method in a desktop computer slightly exceeds a dozen of qubits. Alternatively, the use of wave functions combined with the TrotterSuzuki decomposition [17,18] overcomes this limitation because it uses vectors of size 2 M . However, the average over individual evolutions of a large number of components of the ensemble takes a long time. Here, this limitation is overcome by profiting of quantum parallelism [19] to evaluate the ensemble dynamics of any observable evolved from a "local" initial condition. The idea is that when evaluated on single pure states that are a superposition of all the elements of the ensemble, these observables become self-averaging. This reinforces the suggestions that one can avoid ensemble or thermal averages by using a single pure state [20,21]. Ref.[20] considers a subsystem with the reduced density matrix derived from a pure state, where the subsystem is entangled with an environment which has a much bigger size. The resulting reduced density matrix describes the microcanonical ensemble without resorting to the equiprobability postulate of statistical mechanics. Following a similar inspiration, we focus on the non-equilibrium dynamics of any given observable. The key is that the initial non-eq...

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

Cited by 18 publications

References 34 publications

Dynamical regimes of a quantum SWAP gate beyond the Fermi golden rule

Dynamical regimes of a quantum SWAP gate beyond the Fermi golden rule

Environmentally induced quantum dynamical phase transition in the spin swapping operation

Quantum Parallelism as a Tool for Ensemble Spin Dynamics Calculations

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