Quantum-Mechanical Description of Two Coupled Harmonic Oscillators

Estes, Lee E.; Keil, Thomas; Narducci, L. M.

doi:10.1103/physrev.175.286

Cited by 69 publications

(75 citation statements)

References 17 publications

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“…Photons propagating through the coupled waveguide array shown in Fig. 2(a) are modelled assuming nearest neighbour interaction with the Hamiltonian for coupled oscillators [26], setting = 1:…”

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confidence: 99%

Quantum Walks of Correlated Photons

Peruzzo

Lobino

Matthews

et al. 2010

Science

903

893

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Quantum walks of correlated particles offer the possibility to study large-scale quantum interference, simulate biological, chemical and physical systems, and a route to universal quantum computation. Here we demonstrate quantum walks of two identical photons in an array of 21 continuously evanescently-coupled waveguides in a SiOxNy chip. We observe quantum correlations, violating a classical limit by 76 standard deviations, and find that they depend critically on the input state of the quantum walk. These results open the way to a powerful approach to quantum walks using correlated particles to encode information in an exponentially larger state space.With origins dating back to observations by Lucretius in 60BC and Brown in the 1800's, random walks are a powerful tool used in a broad range of fields from genetics to economics [1]. The quantum mechanical analoguequantum walks [2, 3]-corresponds to the tunnelling of quantum particles into several possible sites, generating large coherent superposition states and allowing massive parallelism in exploring multiple trajectories through a given connected graph (eg. Fig. 1). This quantum state evolution is a reversible (unitary) process and so requires low noise (decoherence) systems for observation. In contrast to the diffusive behaviour of (classical) random walks, which tend towards a steady state, the wave function in a quantum walk propagates ballistically (Fig. 2(c)). These features are at the heart of new algorithms for database-search [4], random graph navigation, models for quantum communication using spin chains [5], universal quantum computation [6] and quantum simulation [7].Quantum walks have been demonstrated using nuclear magnetic resonance [8,9], phase [10,11] and position [12] space of trapped ions, the frequency space of an optical resonator [13], single photons in bulk [14] and fibre [15] optics and the scattering of light in coupled waveguide arrays [16]. However, to date, all realisations have been limited to single particle quantum walks, which have an exact mapping to classical wave phenomena [17], and therefore cannot provide any advantage from quantum effects (note that the quantum walk with two trapped ions [11] encodes in the centre of mass mode and is therefore effectively a single particle quantum walk on a line). Indeed single particle quantum walks have been observed using classical light [16,18]. In contrast, for quantum walks of more than one indistinguishable particle, classical theory no longer provides a sufficient description-quantum theory predicts that probability amplitudes interfere leading to distinctly non-classical correlations [19,20]. This quantum behaviour gives rise to a computational advantage in quantum walks of two identical particles, which can be used to solve the graph isomorphism problem for example [21]. The major challenge associated with realising quantum walks of correlated particles is the need for a low decoherence system that preserves their non-classical features.The intrinsically low decoherence properti...

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confidence: 99%

Quantum Walks of Correlated Photons

Peruzzo

Lobino

Matthews

et al. 2010

Science

903

893

View full text Add to dashboard Cite

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“…In particular, a Cooper Pair Box (CPB) has a typical transition EDM of d CPB = 20, 000 atomic units ( 36,000 D) [39,40]. Hence, at the operation frequency in those circuit-QED experiments of ∼7 GHz, the ratio between both EDMs is 30. This implies that the interaction of a photon with a CPB is about 30 times stronger than with a trapped electron.…”

Section: Comparison With Other Single Microwave Photon Detectorsmentioning

confidence: 89%

“…The symbolh is the reduced Planck's constant and Ω represents the coupling strength (with the dimensions of a frequency). The dynamics of the Hamiltonian in Equation (4) has been solved in [30], where incoherent losses of the microwave radiation and spontaneous emission are ignored. The electron's cyclotron degree of freedom is a multi-level system and the dynamics in Equation (4) can be very complex when including interactions with waves carrying more than one quantum of energy.…”

Section: Dynamics Of Two Coupled Quantum Harmonic Oscillatorsmentioning

confidence: 99%

Single Microwave Photon Detection with a Trapped Electron

et al. 2016

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Abstract:We investigate theoretically the use of an electron in a Penning trap as a detector of single microwave photons. At the University of Sussex we are developing a chip Penning trap technology, designed to be integrated within quantum circuits. Microwave photons are guided into the trap and interact with the electron's quantum cyclotron motion. This is an electric dipole transition, where the near field of the microwave radiation induces quantum jumps of the cyclotron harmonic oscillator. The quantum jumps can be monitored using the continuous Stern-Gerlach effect, providing the quantum non demolition signal of the microwave quanta. We calculate the quantum efficiency of photon detection and discuss the main features and technical challenges for the trapped electron as a quantum microwave sensor.

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“…In these conditions the rotating wave approximation can be applied. It consists of keeping in the Hamiltonian [8] only the terms that conserve the energy in first order: (3) The lattice, which is thermally excited, does not appreciably change the population of the modes during the relaxation process. They are given by where p = (kT)-'.…”

Section: The Modelmentioning

confidence: 99%

The interference in the decay of two ½ spins in a molecular medium, studied by the Nakajima–Zwanzig technique

Hardisson

Lefèbvre

Mauricio

et al. 1977

Int J of Quantum Chemistry

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AbstractsThis paper studies the decay, due to the spin-lattice coupling, of two 5 spins with slightly different Zeeman energies when the lattice is thermally excited. The analysis is based on obtaining, by means of the Nakajima-Zwanzig projection operator technique, an equation for the evolution of the reduced density operator of the spin system which manifests the influence of one spin on the relaxation process of the other. The zero-order solutions obtained for the evolution of the expectation values of the spin dynamics operators are essentially equivalent to the Bloch equations; higher order solutions are also obtained.Nous etudions la dtsinttgration, due au couplage spin-rbseau, de deux spins 5 avec des tnergies Zeeman, lkgerement difftrentes quand le rtseau est thermiquement excitt. L'analyse mtne avec les optrateurs de projection de Nakajima-Zwanzig B une equation dtcrivant l'tvolution de I'opkrateur densitt rtduit du systkme des spins, qui montre ]'influence d'un spin sur le processus de relaxation de I'autre. Les solutions d'ordre zero qu'on obtient pour l'bvolution des valeurs moyennes des optrateurs de spin sont essentiellement kquivalentes aux tquations de Bloch. On obtient aussi des solutions d'ordres superieurs.Wir studieren den von der Spin-Gitter-Kopplung herriihrenden Zerfall von zwei Spins 1 mit leicht verschiedenen Zeemanenergien, wenn das Gitter thermisch angeregt ist. Die Analyse fuhrt mittels der Nakajirna-Zwansig'schen Projektionsoperatoren zu einer Gleichung fur die Entwicklung des reduzierten Dichteoperators des Spinsystems, die den Einfluss eines Spins auf den Relaxationsverlauf des anderen zeigt. Die Lijsungen nullter Ordnung, die fur die Entwicklung der Erwartungswerte der Spinoperatoren erhalten werden, sind wesentlich den Bloch-Gleichungen aquivalent. Losungen hoherer Ordnung werden auch erhalten.

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Quantum-Mechanical Description of Two Coupled Harmonic Oscillators

Cited by 69 publications

References 17 publications

Quantum Walks of Correlated Photons

Quantum Walks of Correlated Photons

Single Microwave Photon Detection with a Trapped Electron

The interference in the decay of two ½ spins in a molecular medium, studied by the Nakajima–Zwanzig technique

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