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Physical Review A

Noise-dependent uncertainty relations for the harmonic oscillator

Michael J. W. Hall

Published Jan 1, 1994• 14 citations
Publisher

Abstract

The Gaussian noise model for a harmonic oscillator subjected to random linear fluctuations (e.g., thermal noise) is used to derive generalized Heisenberg and entropic uncertainty relations for the position-momentum and number-phase observables. The lower bounds are partitioned into pure quantum and pure noise terms, and the minimum-uncertainty states are determined. Formally similar results of Abe and Suzuki [Phys. Rev. A 41, 4608 (1990)] for the position-momentum case, derived using the thermofield formalism, are shown to correspond to amplification noise rather than thermal noise.

DOI
10.1103/physreva.49.42

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The Gaussian noise model for a harmonic oscillator subjected to random linear fluctuations (e.g., thermal noise) is used to derive generalized Heisenberg and entropic uncertainty relations for the position-momentum and number-phase observables. The lower bounds are partitioned into pure quantum and pure noise terms, and the minimum-uncertainty states are determined. Formally similar results of Abe and Suzuki [Phys. Rev. A 41, 4608 (1990)] for the position-momentum case, derived using the thermofield formalism, are shown to correspond to amplification noise rather than thermal noise.
We calculated the Shannon entropy of position and momentum for the stationary quantum states of the harmonic oscillator as a function of its energy and determined the corresponding entropic uncertainty relations for them. We found an approximate phenomenological function for the dependence of position and momentum entropies on the large quantum numbers and the corresponding asymptotic entropy - energy relation for the stationary harmonic oscillator. We also studied the time evolution of the position and momentum entropies of the non-stationary harmonic oscillator for the coherent states, squeezed vacuum and Schrödinger cat states.
We analyze the position and momentum uncertainties of the energy eigenstates of the harmonic oscillator in the context of a deformed quantum mechanics, namely, that in which the commutator between the position and momentum operators is given by $[\stackrel{^}{x},\stackrel{^}{p}]=i\ensuremath{\hbar}(1+\ensuremath{\beta}{\stackrel{^}{p}}^{2})$. This deformed commutation relation leads to the minimal length uncertainty relation $\ensuremath{\Delta}x\ensuremath{\ge}(\ensuremath{\hbar}/2)(1/\ensuremath{\Delta}p+\ensuremath{\beta}\ensuremath{\Delta}p)$, which implies that $\ensuremath{\Delta}x\ensuremath{\sim}1/\ensuremath{\Delta}p$ at small $\ensuremath{\Delta}p$ while $\ensuremath{\Delta}x\ensuremath{\sim}\ensuremath{\Delta}p$ at large $\ensuremath{\Delta}p$. We find that the uncertainties of the energy eigenstates of the normal harmonic oscillator (m>0m>0m>0), derived in L. N. Chang, D. Minic, N. Okamura, and T. Takeuchi, Phys. Rev. D 65, 125027 (2002), only populate the $\ensuremath{\Delta}x\ensuremath{\sim}1/\ensuremath{\Delta}p$ branch. The other branch, $\ensuremath{\Delta}x\ensuremath{\sim}\ensuremath{\Delta}p$, is found to be populated by the energy eigenstates of the ``inverted'' harmonic oscillator (m<0m<0m<0). The Hilbert space in the inverted case admits an infinite ladder of positive energy eigenstates provided that $\ensuremath{\Delta}{x}_{\mathrm{min}}=\ensuremath{\hbar}\sqrt{\ensuremath{\beta}}>\sqrt{2}[{\ensuremath{\hbar}}^{2}/k|m|{]}^{1/4}$. Correspondence with the classical limit is also discussed.
The well-known Heisenberg--Robertson uncertainty relation for a pair of noncommuting observables, is expressed in terms of the product of variances and the commutator among the operators, computed for the quantum state of a system. Different modified commutation relations have been considered in the last years with the purpose of taking into account the effect of quantum gravity. Indeed it can be seen that letting [X,P]=iℏ(1+βP2)[X,P] = i\hbar (1+βP^2)[X,P]=iℏ(1+βP2) implies the existence of a minimal length proportional to β\sqrtββ​. The Bialynicki-Birula--Mycielski entropic uncertainty relation in terms of Shannon entropies is also seen to be deformed in the presence of a minimal length, corresponding to a strictly positive deformation parameter βββ. Generalized entropies can be implemented. Indeed, results for the sum of position and (auxiliary) momentum Rényi entropies with conjugated indices have been provided recently for the ground and first excited state. We present numerical findings for conjugated pairs of entropic indices, for the lowest lying levels of the deformed harmonic oscillator system in 1D, taking into account the position distribution for the wavefunction and the actual momentum.
The well-known Heisenberg–Robertson uncertainty relation for a pair of noncommuting observables, is expressed in terms of the product of variances and the commutator among the operators, computed for the quantum state of a system. Different modified commutation relations have been considered in the last years with the purpose of taking into account the effect of quantum gravity. Indeed it can be seen that letting [ X , P ] = i ℏ ( 1 + β P 2 ) implies the existence of a minimal length proportional to β . The Bialynicki-Birula–Mycielski entropic uncertainty relation in terms of Shannon entropies is also seen to be deformed in the presence of a minimal length, corresponding to a strictly positive deformation parameter β . Generalized entropies can be implemented. Indeed, results for the sum of position and (auxiliary) momentum Rényi entropies with conjugated indices have been provided recently for the ground and first excited state. We present numerical findings for conjugated pairs of entropic indices, for the lowest lying levels of the deformed harmonic oscillator system in 1D, taking into account the position distribution for the wavefunction and the actual momentum.
In this paper, we study Renyi and Tsallis information entropies for a Hamiltonian system with position-dependent mass confined in harmonic oscillator potential. Gegenbauer polynomials are used to obtain the position eigenfunction of such a system, and the modified Bessel function of the second kind is used to determine the equivalent momentum eigenfunction. By means of probability densities of both representations, we analytically and numerically evaluate the Heisenberg-like uncertainty of this system. Because the Renyi and Tsallis information entropies in position representation are described by integral functionals of the Gegenbauer polynomials, these quantities are much more difficult to calculate. To get around this problem, we evaluate these information entropies at the system s asymptotical limit, which corresponds to the behavior of an undeformed harmonic oscillator. Nevertheless, no approximation method is used to obtain the Tsallis and Renyi information entropies in momentum representation. In both representations, we find that these information entropies approach the Shannon entropy when the entropic parameter alpha-> 1 and are closed to the results of similar models of the literature. Finally, we evaluate the related entropic uncertainty relations numerically to validate the latter obersevations.
A single closed-form expression is obtained for the probability amplitude ψ(x,t) for a harmonic oscillator whose initial state ψ(x,0) has a minimum value for the uncertainty product of position x and momentum p, Δx Δp=ℏ/2. This form describes both coherent and squeezed states. Oscillations in time for uncertainties in position and momentum, characteristic of squeezed states, are shown to follow directly from the derived form of ψ(x,t) and from Heisenberg equations of motion for operators x̂ and p̂. Expressions for average photon number and its uncertainty are obtained from calculations of average oscillator energy and fluctuations about this energy.