Microscopic derivation of open quantum walks

Sinayskiy, Ilya; Petruccione, Francesco

doi:10.1103/physreva.92.032105

Cited by 13 publications

(20 citation statements)

References 31 publications

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“…(

false| λ false| \sim γ false(0_{\pm} false) ≪ ω_{0}

). The effect of this driving can be restored by rotating the master equation with the unitary operator

U_{λ} = e^{- ı t (λ | 1 ⟩ ⟨ 0 | + λ^{*} | 0 ⟩ ⟨ 1 |)} .

To this end the master equation can be written as,

where the constants

γ_{p}

γ_{z}

and Δ are defined as,

\begin{matrix} true \\ right γ_{p} & = & left α^{2} (γ false(0_{+} false) + γ false(0_{-} false)), γ_{z} = β^{2} (γ false(0_{+} false) + γ false(0_{-} false)), \\ right Δ & = & left α β false(γ (0_{-}) - γ (0_{+}) false) . \end{matrix}

For the case of OQWs the density matrix of the quantum walker has a diagonal form in the position space

ρ false(t false) = \sum_{n} ρ_{n} \otimes false| n false⟩ false⟨ n false|

. Accordingly, in the OQBM case the density matrix will be given by,

ρ_{s} false(t false) = \int_{- \infty}^{\infty} d x ρ false(t, x false) \otimes false| x false⟩ false⟨

…”

Section: Modelmentioning

confidence: 99%

“…To this end the master equation (7) can be written as, (10) where the constants γ p , γ z and are defined as,…”

Section: Original Papermentioning

confidence: 99%

“…For the case of OQWs the density matrix of the quantum walker has a diagonal form in the position space (t) = n ρ n ⊗ |n n| [7][8][9][10]. Accordingly, in the OQBM case the density matrix will be given by,…”

Section: Original Papermentioning

confidence: 99%

See 2 more Smart Citations

Steady‐State control of open Quantum Brownian Motion

Sinayskiy

Petruccione

2016

Fortschritte der Physik

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Open Quantum Brownian motion describes a Quantum Brownian particle with an additional internal degree of freedom.The quantum master equation for a free Open Quantum Brownian walker with two‐dimensional internal degree of freedom interacting decoherently with the dissipative environment is derived. Using the analytical solution for the moments of the position distribution of the Brownian walker, one finds that the steady state average position of the walker can be controlled by the strength of an external classical driving field.

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“…(

false| λ false| \sim γ false(0_{\pm} false) ≪ ω_{0}

). The effect of this driving can be restored by rotating the master equation with the unitary operator

U_{λ} = e^{- ı t (λ | 1 ⟩ ⟨ 0 | + λ^{*} | 0 ⟩ ⟨ 1 |)} .

To this end the master equation can be written as,

where the constants

γ_{p}

γ_{z}

and Δ are defined as,

\begin{matrix} true \\ right γ_{p} & = & left α^{2} (γ false(0_{+} false) + γ false(0_{-} false)), γ_{z} = β^{2} (γ false(0_{+} false) + γ false(0_{-} false)), \\ right Δ & = & left α β false(γ (0_{-}) - γ (0_{+}) false) . \end{matrix}

For the case of OQWs the density matrix of the quantum walker has a diagonal form in the position space

ρ false(t false) = \sum_{n} ρ_{n} \otimes false| n false⟩ false⟨ n false|

. Accordingly, in the OQBM case the density matrix will be given by,

ρ_{s} false(t false) = \int_{- \infty}^{\infty} d x ρ false(t, x false) \otimes false| x false⟩ false⟨

…”

Section: Modelmentioning

confidence: 99%

“…To this end the master equation (7) can be written as, (10) where the constants γ p , γ z and are defined as,…”

Section: Original Papermentioning

confidence: 99%

See 1 more Smart Citation

Steady‐State control of open Quantum Brownian Motion

Sinayskiy

Petruccione

2016

Fortschritte der Physik

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“…The only dissipative process considered for obtaining an OQW was the spontaneous emission in the system. Although this scheme leads to OQW, the dynamics of the walker is relatively poor in comparison to traditional microscopic approaches [22]. The aim of this paper is to generalize the simple case which was developed earlier by [20] to include external driving of the atom and non-zero temperature of the environment.…”

Section: Introductionmentioning

confidence: 99%

“…Not so long ago, Sinayskiy and Petruccione suggested two possible approaches to implement OQWs: first, they suggested a quantum optics implementation of OQWs by using an effective operator formalism [20], second, they followed the traditional theory of open quantum systems and derived OQWs based on the microscopic system-bath setup [21,22]. In the quantum optical implementation of OQWs, [20] used an example of the two-level system in the cavity in the dispersive regime.…”

Section: Introductionmentioning

confidence: 99%

A Generalized Quantum Optical Scheme for Implementing Open Quantum Walks

Zungu

Sinayskiy

Petruccione

2020

Electron. Proc. Theor. Comput. Sci.

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Open quantum walks (OQWs) are a new type of quantum walks which are entirely driven by the dissipative interaction with external environments and are formulated as completely positive trace-preserving maps on graphs. A generalized quantum optical scheme for implementing OQWs that includes non-zero temperature of the environment is suggested. In the proposed quantum optical scheme, a two-level atom plays the role of the "walker", and the Fock states of the cavity mode correspond to the lattice sites for the "walker". Using the small unitary rotations approach the effective dynamics of the system is shown to be an OQW. For the chosen set of parameters, an increase in the temperature of the environment causes the system to reach the asymptotic distribution much faster compared to the scheme proposed earlier where the temperature of the environment is zero. For this case the asymptotic distribution is given by a steady Gaussian distribution.

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Open Quantum Random Walks on the Half-Line: The Karlin–McGregor Formula, Path Counting and Foster’s Theorem

Jacq

Lardizabal

2017

J Stat Phys

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In this work we consider open quantum random walks on the non-negative integers. By considering orthogonal matrix polynomials we are able to describe transition probability expressions for classes of walks via a matrix version of the Karlin-McGregor formula. We focus on absorbing boundary conditions and, for simpler classes of examples, we consider path counting and the corresponding combinatorial tools. A non-commutative version of the gambler's ruin is studied by obtaining the probability of reaching a certain fortune and the mean time to reach a fortune or ruin in terms of generating functions. In the case of the Hadamard coin, a counting technique for boundary restricted paths in a lattice is also presented.We discuss an open quantum version of Foster's Theorem for the expected return time together with applications.

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Microscopic derivation of open quantum walks

Cited by 13 publications

References 31 publications

Steady‐State control of open Quantum Brownian Motion

Steady‐State control of open Quantum Brownian Motion

A Generalized Quantum Optical Scheme for Implementing Open Quantum Walks

Open Quantum Random Walks on the Half-Line: The Karlin–McGregor Formula, Path Counting and Foster’s Theorem

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