Damped harmonic oscillators in the holomorphic representation

Benatti, Fabio; Floreanini, Roberto

doi:10.1088/0305-4470/33/45/310

Cited by 9 publications

(15 citation statements)

References 37 publications

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“…Finally, we also get that |D| 2 − |C| 2 = 1, so D, C differ from A * , B * by a phase. We choose the phase so that it is easily expressible in terms of β, so that we get Consider the holomorphic quantization of the harmonic oscillator, also called the Bargmann representation, see, e.g., [38,49] for reviews. An arbitrary state vector |f in the Hilbert space of the harmonic oscillator is represented by a holomorphic function f (z) and the inner product is given by…”

Section: Discussionmentioning

confidence: 99%

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Entanglement entropy converges to classical entropy around periodic orbits

Asplund

Berenstein

2016

Annals of Physics

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We consider oscillators evolving subject to a periodic driving force that dynamically entangles them, and argue that this gives the linearized evolution around periodic orbits in a general chaotic Hamiltonian dynamical system. We show that the entanglement entropy, after tracing over half of the oscillators, generically asymptotes to linear growth at a rate given by the sum of the positive Lyapunov exponents of the system. These exponents give a classical entropy growth rate, in the sense of Kolmogorov, Sinai and Pesin. We also calculate the dependence of this entropy on linear mixtures of the oscillator Hilbert space factors, to investigate the dependence of the entanglement entropy on the choice of coarse-graining. We find that for almost all choices the asymptotic growth rate is the same.then the typical pure state in H has very close to the maximal amount of entanglement allowed between H A and H B , and this is in turn maximized if dim(H A ) = dim(H B ). We will call such factorization of H into "observable" and "non-observable" physics a coarsegraining of the system. This suggests that if we evolve a system randomly from an initial configuration with zero entanglement entropy, then it will eventually forget essentially all of the information of the initial state if we only measure observables sensitive to H A . There are many studies of this kind of process in specific systems. The rate at which the entanglement grows towards saturation depends, in general, on the details of these systems, although some general bounds exist [2][3][4][5]. Linear growth in time appears in many systems, for example, studies of decoherence and quantum chaos [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21], and quenches of extended systems [22][23][24][25].When trying to apply these results in the context of black hole physics, we are usually confronted with two problems. First of all, the Hilbert space H is big. In the gauge/gravity duality [26] the dynamics takes place in an infinite-dimensional Hilbert space: it is the Hilbert space of a relativistic quantum field theory on the conformal boundary.A very naive application of the results of [1] would suggest that typical states have infinite entropy when splitting H in two pieces of the same size, since both are infinite dimensional.However, the notion of splitting along a random factorization has no meaning, because once we have factored into infinite dimensional pieces, we can factorize the pieces again: there is no natural notion of splitting in half. Thus, the question of the entanglement entropy for a typical state is ill-defined without additional structure on the Hilbert space.An example of such a structure is two operators algebras, one for H A the other one for H B . It is natural to do the splitting with respect to a choice of algebras with reasonable properties determined by features of the dynamics. Once that splitting is done, instead of computing the entanglement entropy of the typical state, we can compute the rate of growth of the entanglement entrop...

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Section: Discussionmentioning

confidence: 99%

“…To compute the entropy of these density operators we would like to find an exponential form so that we can write down log ρ. Their entropy was computed this way in [38] and we reproduce their computation here, giving more details of the derivation. To this end first consider the parametrization…”

Section: Discussionmentioning

confidence: 99%

Entanglement entropy converges to classical entropy around periodic orbits

Asplund

Berenstein

2016

Annals of Physics

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“…Among all density matrices ρ, the so-called quasi-free or Gaussian states are of particular interest [19][20][21][22][23][24][25][26][27] : as mentioned before, they can be easily produced in experiments in quantum optics. These states are defined by the property that the expectations W (z) = tr W (z) ρ of the Weyl operators W (z) = e za+z * a † are in Gaussian form, i.e.…”

Section: Single Mode Dissipative Dynamicsmentioning

confidence: 99%

“…More specifically, we shall study the behaviour of two independent bosonic oscillators, one evolving with a dissipative Markovian semigroup, while the other with a standard unitary dynamics. We shall limit our considerations to the phenomenologically relevant set of Gaussian states [19][20][21][22][23][24][25][26][27] and to dissipative dynamics that preserve this set, the so-called quasi-free semigroups. [27][28][29] Although very simple, this settings actually corresponds to a specific physical situation in quantum optics, 30-33 that of two coherent states travelling in optical fibers, of which only one is subjected to noise, producing dissipative phenomena, while the other gives rise only to unitary, standard birefringence effects.…”

Section: Introductionmentioning

confidence: 99%

Non-Positive Semigroup Dynamics in Continuous Variable Models

Benatti

Floreanini

2007

Int. J. Quantum Inform.

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Non-positive, Markovian semigroups are sometimes used to describe the time evolution of subsystems immersed in an external environment. A widely adopted prescription to avoid the appearance of negative probabilities is to eliminate from the admissible initial conditions those density matrices that would not remain positive by the action of the semigroup dynamics. Using a continuous variable model, we show that this procedure leads to physical inconsistencies when two subsystems are considered and their initial state is entangled.

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“…‡ Further discussions on the notion of symmetry invariance for quantum dynamical semigroups can be found in [43]. This transformation leaves the equation (3.1) form-invariant, i.e.…”

Section: The Effective Time-evolutionmentioning

confidence: 99%

Irreversibility and dissipation in neutral B-meson decays

2001

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The propagation and decay of neutral B-mesons can be described in terms of quantum dynamical semigroups; they provide generalized timeevolutions that take into account possible non-standard effects leading to loss of phase coherence and dissipation. These effects can be fully parametrized in terms of six phenomenological constants. A detailed analysis of selected B-meson decays shows that present and future dedicated experiments, both at colliders and B-factories, will be able to put stringent bounds on these non-standard parameters.

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Damped harmonic oscillators in the holomorphic representation

Cited by 9 publications

References 37 publications

Entanglement entropy converges to classical entropy around periodic orbits

Entanglement entropy converges to classical entropy around periodic orbits

Non-Positive Semigroup Dynamics in Continuous Variable Models

Irreversibility and dissipation in neutral B-meson decays

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