SU(1,1) Coherent States for the Generalized Two-Mode Time-Dependent Quadratic Hamiltonian System

In this paper we study a two-dimensional [2D] rotationally symmetric harmonic oscillator with time-dependent frictional force. At the classical level, we solve the equations of motion for a particular case of the time-dependent coefficient of friction. At the quantum level, we use the Lewis-Riesenfeld procedure of invariants to construct exact solutions for the corresponding timedependent Schrödinger equations. The eigenfunctions obtained are in terms of the generalized Laguerre polynomials. By mean of the solutions we verify a generalization version of the Heisenberg's uncertainty relation and derive the generators of the su(1, 1) Lie algebra. Based on these generators, we construct the coherent statesà la Barut-Girardello andà la Perelomov and respectively study their properties. *

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“…With respect to the equations, we rewrite the eigenfunction of the invariant operator in equation (38) in the form φ ℓ n (u) = N(ρ, α)…”

Section: The Orthogonality Relation Ismentioning

confidence: 99%

“…Lie algebraic group through different approaches. The Barut-Girardello and the Perelomov coherent states gained lot of applications, for instance in the fields of quantum optics [35,36], quantum computation [37,38] and quantum mechanics [39][40][41].…”

Section: Introductionmentioning

confidence: 99%

Lewis-Riesenfeld quantization and SU(1, 1) coherent states for 2D damped harmonic oscillator

Lawson

Avossevou

Gouba

2018

Journal of Mathematical Physics

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“…As a result, one should consider the code states as catstate-like superpositions of pair-coherent states, so we refer to them as "pair-cat" states (noting that they have previously been studied in quantum optics [93][94][95][96]). Note also the connection to NOON states in the γ 1 limit.…”

Section: B Pair-cat Code Statesmentioning

confidence: 99%

Pair-cat codes: autonomous error-correction with low-order nonlinearity

Albert

Mundhada

Grimm

et al. 2019

Quantum Sci. Technol.

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We introduce a driven-dissipative two-mode bosonic system whose reservoir causes simultaneous loss of two photons in each mode and whose steady states are superpositions of pair-coherent/Barut-Girardello coherent states. We show how quantum information encoded in a steady-state subspace of this system is exponentially immune to phase drifts (cavity dephasing) in both modes. Additionally, it is possible to protect information from arbitrary photon loss in either (but not simultaneously both) of the modes by continuously monitoring the difference between the expected photon numbers of the logical states. Despite employing more resources, the two-mode scheme enjoys two advantages over its one-mode cat-qubit counterpart with regards to implementation using current circuit QED technology. First, monitoring the photon number difference can be done without turning off the currently implementable dissipative stabilizing process. Second, a lower average photon number per mode is required to enjoy a level of protection at least as good as that of the cat-codes. We discuss circuit QED proposals to stabilize the code states, perform gates, and protect against photon loss via either active syndrome measurement or an autonomous procedure. We introduce quasiprobability distributions allowing us to represent two-mode states of fixed photon number difference in a twodimensional complex plane, instead of the full four-dimensional two-mode phase space. The twomode codes are generalized to multiple modes in an extension of the stabilizer formalism to nondiagonalizable stabilizers. The M -mode codes can protect against either arbitrary photon losses in up to M − 1 modes or arbitrary losses and gains in any one mode.

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“…Here, we review the single-mode generalizations and introduce an Mmode generalization of cat-codes, making contact with the Lindbladians necessary to generate these codes. We note that the states we consider have been studied in a quantum optical context for M = 2 [89,181] and M = 3 [17].…”

Section: Collision Gatementioning

confidence: 99%

Lindbladians with multiple steady states: theory and applications

Albert¹

2018

Preprint

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Lindbladians with multiple steady states: theory and applications Victor V. Albert 2017Markovian master equations, often called Liouvillians or Lindbladians, are used to describe decay and decoherence of a quantum system induced by that system's environment. While a natural environment is detrimental to fragile quantum properties, an engineered environment can drive the system toward exotic phases of matter or toward subspaces protected from noise. These cases often require the Lindbladian to have more than one steady state, and such Lindbladians are dissipative analogues of Hamiltonians with multiple ground states. This thesis studies Lindbladian extensions of topics commonplace in degenerate Hamiltonian systems, providing examples and historical context along the way.An important property of Lindbladians is their behavior in the limit of infinite time, and the first part of this work focuses on deriving a formula for the asymptotic projection -the map corresponding to infinite-time Lindbladian evolution. This formula is applied to determine the dependence of a system's steady state on its initial state, to determine the extent to which decay affects a system's linear or adiabatic response, and to determine geometrical structures (holonomy, curvature, and metric) associated with adiabatically deformed steady-state subspaces. Using the asymptotic projection to partition the physical system into a subspace free from nonunitary effects and that subspace's complement (and making a few other minor assumptions), a Dyson series is derived to all orders in an arbitrary perturbation. The terms in the Dyson series up to second order in the perturbation are shown to reproduce quantum Zeno dynamics and the effective operator formalism.

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SU(1,1) Coherent States for the Generalized Two-Mode Time-Dependent Quadratic Hamiltonian System

Cited by 6 publications

References 34 publications

Lewis-Riesenfeld quantization and SU(1, 1) coherent states for 2D damped harmonic oscillator

Lewis-Riesenfeld quantization and SU(1, 1) coherent states for 2D damped harmonic oscillator

Pair-cat codes: autonomous error-correction with low-order nonlinearity

Lindbladians with multiple steady states: theory and applications

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