Perfect Quantum State Revivals: Designing Arbitrary Potentials with Specified Energy Levels

TL;DR

The paper addresses designing quantum potentials that support perfect quantum state revivals for arbitrary bound states. It develops a general inverse-design approach based on the intertwining (Darboux) method to add bound-state levels below the ground state, constructing new potentials with spectra of the form $E_n=a N_n+b$ and revival time $T_{ ext{rev}}=2\pi / a$. Key contributions include Algorithm 1 for iteratively inserting levels, a variety of spectral patterns (bi-periodic, alternating gaps, primes, Fibonacci) and multidimensional extensions validated by spectra, autocorrelations, and quantum carpets, as well as a detailed error analysis showing high spectral accuracy. The work provides a versatile framework for spectral engineering of quantum revivals with potential applications in controlled quantum dynamics and higher-dimensional systems.

Abstract

It is known that there exist a limited number of analytic potentials with the unusual property that any bound quantum state therein will be periodic in time. This is known as a perfect quantum state revival. Examples of such potentials are the infinite well, quantum harmonic oscillator and the Pöschl-Teller potentials; here, we present a general method of designing such potentials. A key requirement is that their energy eigenvalues have integer spacings (up to a prefactor). We first analyze the required conditions which permit quantum state revivals for potentials in general, and then we use techniques of iterated Hamiltonian intertwining to construct potentials exhibiting perfect quantum revivals. Our method can readily be extended to multiple dimensions.

Figures (8)

• Figure 1: Biperiodic potentials. Starting from the harmonic potential $x^2/2$ (the uppermost curve, red), $N=25$ additional energy levels with spacing of 2 at values $-1,-3,\dots,-49$ have been added successively using the scheme in Algorithm 1, creating 25 new potentials. The numerically calculated spectra of each potential are shown towards the left in matching colors; the leftmost column shows the spectrum of the lowermost potential. The fact that the levels of various potentials match is apparent.
• Figure 2: Quantum carpets for the biperiodic potential where 100 levels with spacing of 2 were added to the original harmonic potential $x^2/2$. The absolute value of the wavefunction is indicated by the brightness (larger value corresponds to a darker color), the vertical axis represents time proportional to $T_{\mathrm{rev}}=4\pi$. Two initial wavepackets were used: (a) superposition of two Gaussian states and (b) a shifted cosine oscillating between 0 and 1. It is apparent, especially in (a), that the period of the lower-amplitude Gaussian component is equal to half of the period of the higher-amplitude component.
• Figure 3: The plots show the magnitude of the autocorrelation function, $|A(t)|$, for the following quantum carpets: (a) the carpet from Fig. \ref{['fig:talbot-biperiodic1']}(b), (b) the carpet from Fig. \ref{['fig:talbot-biperiodic2']}(b), (c-d) the carpets from Fig. \ref{['fig:alternating']}(e-f), respectively, (e,f) the carpets from Fig. \ref{['fig:primes-50']}(b,d), respectively.
• Figure 4: Same as Fig. \ref{['fig:biperiodic1']}, but for reverse biperiodic potentials. $N=15$ levels were added successively with spacing of $1/2$ at values $0,-1/2,-1,\dots,-7$. Fewer added levels than in Fig. \ref{['fig:biperiodic1']} were used for the sake of clarity of the plot.
• Figure 5: Quantum carpets for the reverse biperiodic potential where 100 levels with spacing of 1/2 were added to the original harmonic potential $x^2/2$. The initial conditions are similar as in Fig. \ref{['fig:talbot-biperiodic1']}. The biperiodicity is apparent again, this time the period of the lower-amplitude component equals twice the period of the higher-amplitude component. The revival time is $T_{\mathrm{rev}}=4\pi$.