Tunnel splitting of the spectrum and bilocalization of eigenfunctions in an asymmetric double well

Vybornyi, E. V.

doi:10.1007/s11232-014-0132-7

Cited by 10 publications

(14 citation statements)

References 14 publications

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“…The lower figure is the equivalent analytical model: a particle of variable mass moving in a one-dimensional continuous potential well. This 1D model is seen to exhibit the same qualitative features as the original spin ensemble (upper plot), except for an infinite mass singularity at z = 0 coincident with the hidden second order phase transition (white dashed curve in lower plot), see eqn (19)…”

mentioning

confidence: 75%

“…Recall that β is a ratio of frequencies, and not a Here is illustrated the case that the left well is outside the Rayleigh boundary: {ξ1, β1} → {3.0, 1.5} (white circle marker). Then one may employ WKB-like methods; the gap can become so small on resonance to be dictated by the potential shape and structure away from the parabolic extrema 19 . In such a case the piecewise-parabolic model loses its generality; errors or simplifications in the description of the potential have greater magnitude than the spectral gap calculated via this potential.…”

Section: Appendix A: Details Of the Piecewise-parabolic Potentialmentioning

confidence: 99%

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Quantum speedup at zero temperature via coherent catalysis

Durkin

2019

Phys. Rev. A

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It is known that secondary non-stoquastic drivers may offer speed-ups or catalysis in some models of adiabatic quantum computation accompanying the more typical transverse field driver. Their combined intent is to raze potential barriers to zero during adiabatic evolution from a false vacuum to a true minimum; first order phase transitions are softened into second order transitions. We move beyond mean-field analysis to a fully quantum model of a spin ensemble undergoing adiabatic evolution in which the spins are mapped to a variable mass particle in a continuous one-dimensional potential. We demonstrate the necessary criteria for enhanced mobility or 'speed-up' across potential barriers is actually a quantum form of the Rayleigh criterion. Quantum catalysis is exhibited in models where previously thought not possible, when barriers cannot be eliminated. For the 3-spin model with secondary anti-ferromagnetic driver, catalysed time complexity scales between linear and quadratically with the number of qubits. As a corollary, we identify a useful resonance criterion for quantum phase transition that differs from the classical one, but converges on it, in the thermodynamic limit.

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confidence: 75%

Section: Appendix A: Details Of the Piecewise-parabolic Potentialmentioning

confidence: 99%

Quantum speedup at zero temperature via coherent catalysis

Durkin

2019

Phys. Rev. A

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“…Similarly to this, the probability of finding the particle in the right well is given by (33) The numerical analysis shows that when R > R*, there is always a value, V 0 , of the shell potential such that μ 2 = P R /P L Ӎ 1. Then, the values E L and E R are close to one another with an exponential accuracy (with respect to the tunneling action) [35,36] and the effect of the quasidegeneracy of the lowest ion level occurs. In this case, the following asymptotic formulas for the two lowest levels of the double well Schrod inger operator are valid (34) where the characteristic scale of the exponential smallness of tunnel effects in the double well potential δ is determined by the expression [35,36] ( 35) and ω L, R are the frequencies of the classical periodic motion of a particle with a half of the unit mass in the left and right potential wells.…”

Section: The Tunnel Splitting Of the Lowest Level Of An Ion In A Shellmentioning

confidence: 92%

“…a C < < a 2 , we introduce the following sampling functions (31) The f L, R φ functions are the eigen functions of H L, R with an exponential accuracy [35], where φ is the nor malized wave function of a double well Schrodinger operator that corresponds to E 0 . The probability, P L , of finding the particle in the left well is given by (32) since the integral in the range a 1 < a < a 2 is exponen tially small.…”

Section: The Tunnel Splitting Of the Lowest Level Of An Ion In A Shellmentioning

confidence: 99%

“…If μ Ӷ 1 or μ ӷ 1, the effect of quasidegeneracy is absent [35], whereas the φ 1, 2 values to a first approxi mation are given by φ L and φ R . It is worth noting that when the depths of the left and right potential wells are equal, the quasidegeneracy of the lowest term may absent due to nonequivalence of the potential wells in the radial problem.…”

Section: The Tunnel Splitting Of the Lowest Level Of An Ion In A Shellmentioning

confidence: 99%

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H 2 + in a lattice of cavities: Ammonia-like splitting of the lowest level

Sveshnikov

Tolokonnikov

2015

Moscow Univ. Phys.

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It is shown that reconstruction of the low lying energy levels of the ion under conditions of confinement in a given spatial volume is substantially more significant as compared to the confinement by simple potential barrier. Depending on the cavity parameters, the ground state binding energy of could be significantly greater than that of the free ion, while the behavior of the lowest molecular term of the ion turns out to be quite different when considered as a function of the internuclear distance. In particular, two minima might occur in , while the relationship between them could be sufficiently different depend ing on the cavity parameters, which is shown on the phase diagram that was obtained in the work. We studied the case of a "mexican hat" structure of the effective ion potential in detail. As a result, the lowest electronic level of ion splits into the ground state level and the first excited one, with difference between them being as small as ~10 -4 eV. As in the NH 3 molecule, in the last case the lowest level gives rise to an effective two level system that is separated from the vibrational and rotational modes by a wide energy gap. More concretely, cal culation using the Neumann conditions that actually reproduce the confinement on a lattice of similar cavi ties shows that the splitting of ~10 -4 eV occurs for the linear sizes of the confinement area of the order of some a B and a shell potential magnitude of ~10 eV.

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Quantum confinement in an asymmetric double‐well potential through energy analysis and information entropic measure

Mukherjee¹,

Roy

2015

Annalen der Physik

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Localization of a particle in the wells of an asymmetric double-well (DW) potential is investigated here. Information entropy-based uncertainty measures, such as Shannon entropy, Fisher information, Onicescu energy, etc., and phasespace area, are utilized to explain the contrasting effect of localization-delocalization and role of asymmetric term in such two-well potentials. In asymmetric situation, two wells behaves like two different potentials. A general rule has been proposed for arrangement of quasi-degenerate pairs, in terms of asymmetry parameter. Further, it enables to describe the distribution of particle in either of the deeper or shallow wells in various energy states. One finds that, all states eventually get localized to the deeper well, provided the asymmetry parameter attains certain threshold value. This generalization produces symmetric DW as a natural consequence of asymmetric DW. Eigenfunctions, eigenvalues are obtained by means of a simple, accurate variationinduced exact diagonalization method. In brief, information measures and phase-space analysis can provide valuable insight toward the understanding of such potentials.

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Tunnel splitting of the spectrum and bilocalization of eigenfunctions in an asymmetric double well

Cited by 10 publications

References 14 publications

Quantum speedup at zero temperature via coherent catalysis

Quantum speedup at zero temperature via coherent catalysis

H 2 + in a lattice of cavities: Ammonia-like splitting of the lowest level

Quantum confinement in an asymmetric double‐well potential through energy analysis and information entropic measure

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