On the relationship between complex potentials and strings of projection operators

Halliwell, J. J.; Yearsley, James M.

doi:10.1088/1751-8113/43/44/445303

Cited by 19 publications

(32 citation statements)

References 42 publications

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“…That is P e − i Hǫ · · · P e − i Hǫ P ≈ P exp − i Hτ − V 0 P , (1.9) where the relationship between V 0 and ǫ was derived in Ref. [17] and found to be…”

Section: Introductionmentioning

confidence: 97%

“…However, the analysis is greatly facilitated by the results of Refs. [16,17], which show that a string of N projections P onto [−L/2, L/2] interspersed with unitary evolution is wellapproximated by evolution in the presence of a complex potential outside the region, V 0P , whereP = 1 − P . That is P e − i Hǫ · · · P e − i Hǫ P ≈ P exp − i Hτ − V 0 P , (1.9) where the relationship between V 0 and ǫ was derived in Ref.…”

Section: Introductionmentioning

confidence: 99%

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Classical limit of the quantum Zeno effect by environmental decoherence

Bedingham

Halliwell

2014

Phys. Rev. A

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We consider a point particle in one dimension initially confined to a finite spatial region whose state is frequently monitored by projection operators onto that region. In the limit of infinitely frequent monitoring, the state never escapes from the region—this is the Zeno effect. In the corresponding classical problem, by contrast, the state diffuses out of the region with the frequent monitoring simply removing probability. The aim of this paper is to show how the Zeno effect disappears in the classical limit in this and similar examples. We give a general argument showing that the Zeno effect is suppressed in the presence of a decoherence mechanism which suppresses interference between histories. We show how this works explicitly in two examples involving projections onto a one-dimensional subspace and identify the key time scales for the process. We extend this understanding to our main problem of interest, the case of a particle in a spatial region, by coupling it to a decohering environment. Smoothed projectors are required to give the problem proper definition and this implies the existence of a momentum cutoff and minimum length scale. We show that the escape rate from the region approaches the classically expected result, and hence the Zeno effect is suppressed, as long as the environmentally induced fluctuations in momentum are sufficiently large. We establish the time scale on which an arbitrary initial state develops sufficiently large fluctuations to satisfy this condition. We link our results to earlier work on the → 0 limit of the Zeno effect. We illustrate our results by plotting the probability flux lines for the density matrix (which are equivalent to Bohm trajectories in the pure-state case). These illustrate both the Zeno and anti-Zeno effects very clearly, and their suppression. Our results are closely related to our earlier paper [ Phys.Rev.A 88 , 022128 ( 2013 )], demonstrating the suppression of quantum-mechanical reflection by decoherence

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“…That is P e − i Hǫ · · · P e − i Hǫ P ≈ P exp − i Hτ − V 0 P , (1.9) where the relationship between V 0 and ǫ was derived in Ref. [17] and found to be…”

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Classical limit of the quantum Zeno effect by environmental decoherence

Bedingham

Halliwell

2014

Phys. Rev. A

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“…The problem of the effect of repeated * shrabantidhar@gmail.com measurements, made at finite time intervals, on the evolution of a quantum system has been studied in various contexts both theoretically and experimentally [28][29][30][31][32][33]. Because of the probabilistic nature of the quantum detection process, we expect the time of detection to be a stochastic variable.…”

Section: Introductionmentioning

confidence: 99%

Detection of a quantum particle on a lattice under repeated projective measurements

et al. 2015

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We consider a quantum particle, moving on a lattice with a tight-binding Hamiltonian, which is subjected to measurements to detect it's arrival at a particular chosen set of sites. The projective measurements are made at regular time intervals τ , and we consider the evolution of the wave function till the time a detection occurs. We study the probabilities of its first detection at some time and conversely the probability of it not being detected (i.e., surviving) up to that time. We propose a general perturbative approach for understanding the dynamics which maps the evolution operator, consisting of unitary transformations followed by projections, to one described by a nonHermitian Hamiltonian. For some examples, of a particle moving on one and two-dimensional lattices with one or more detection sites, we use this approach to find exact expressions for the survival probability and find excellent agreement with direct numerical results. A mean field model with hopping between all pairs of sites and detection at one site is solved exactly. For the one-and two-dimensional systems, the survival probability is shown to have a power-law decay with time, where the power depends on the initial position of the particle. Finally, we show an interesting and non-trivial connection between the dynamics of the particle in our model and the evolution of a particle under a non-Hermitian Hamiltonian with a large absorbing potential at some sites.

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“…This connection is valid as long as ǫ ≪ 1/(∆H) and ǫ and V 0 are related by V 0 ǫ ≈ 4/3 [38]. It strongly suggests that the class operator C nc , normally defined by a string of projection operators, is justifiably defined instead using a complex potential.…”

Section: The Arrival Time Problem In Non-relativistic Quantum Theorymentioning

confidence: 99%

Decoherent histories analysis of minisuperspace quantum cosmology

Halliwell

2011

J. Phys.: Conf. Ser.

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Abstract. Recent results on the decoherent histories quantization of simple cosmological models (minisuperspace models) are described. The most important issue is the construction, from the wave function, of a probability distribution answering various questions of physical interest, such as the probability of the system entering a given region of configuration space at any stage in its entire history. A standard but heuristic procedure is to use the flux of (components of) the wave function in a WKB approximation as the probability. This gives sensible semiclassical results but lacks an underlying operator formalism. Here, we supply the underlying formalism by deriving probability distributions linked to the Wheeler-DeWitt equation using the decoherent histories approach to quantum theory, building on the generalized quantum mechanics formalism developed by Hartle. The key step is the construction of class operators characterizing questions of physical interest. Taking advantage of a recent decoherent histories analysis of the arrival time problem in non-relativistic quantum mechanics, we show that the appropriate class operators in quantum cosmology are readily constructed using a complex potential. The class operator for not entering a region of configuration space is given by the S-matrix for scattering off a complex potential localized in that region. We thus derive the class operators for entering one or more regions in configuration space. The class operators commute with the Hamiltonian, have a sensible classical limit and are closely related to an intersection number operator. The corresponding probabilities coincide, in a semiclassical approximation, with standard heuristic procedures.

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On the relationship between complex potentials and strings of projection operators

Cited by 19 publications

References 42 publications

Classical limit of the quantum Zeno effect by environmental decoherence

Classical limit of the quantum Zeno effect by environmental decoherence

Detection of a quantum particle on a lattice under repeated projective measurements

Decoherent histories analysis of minisuperspace quantum cosmology

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