Quantum dot potentials: Symanzik scaling, resurgent expansions, and quantum dynamics

This article is concerned with a special class of the “double-well-like” potentials that occur naturally in the analysis of finite quantum systems. Special attention is paid, in particular, to the so-called Fokker-Planck potential, which has a particular property: the perturbation series for the ground-state energy vanishes to all orders in the coupling parameter, but the actual ground-state energy is positive and dominated by instanton configurations of the form $\exp (-a∕g)$, where $a$ is the instanton action. The instanton effects are most naturally taken into account within the modified Bohr-Sommerfeld quantization conditions whose expansion leads to the generalized perturbative expansions (so-called resurgent expansions) for the energy eigenvalues of the Fokker-Planck potential. Until now, these resurgent expansions have been mainly applied for small values of coupling parameter $g$, while much less attention has been paid to the strong-coupling regime. In this contribution, we compare the energy values, obtained by directly resumming generalized Bohr-Sommerfeld quantization conditions, to the strong-coupling expansion, for which we determine the first few expansion coefficients in powers of $g^{-2∕3}$. Detailed calculations are performed for a wide range of coupling parameters $g$ and indicate a considerable overlap between the regions of validity of the weak-coupling resurgent series and of the strong-coupling expansion. Apart from the analysis of the energy spectrum of the Fokker-Planck Hamiltonian, we also briefly discuss the computation of its eigenfunctions. These eigenfunctions may be utilized for the numerical integration of the (single-particle) time-dependent Schrödinger equation and, hence, for studying the dynamical evolution of the wave packets in the double-well-like potentials.

Figure 1

(Color online) Eigenvalues $E_{\mathrm{FP}}^{(M=0)}$ (left panel) and $E_{\mathrm{FP}}^{(M=1,2)}$ (right panel) of the Fokker-Planck Hamiltonian as a function of the coupling parameter $g$. Results have been computed by the diagonalization of the Fokker-Planck Hamiltonian (1) in the basis of the harmonic oscillator wave functions.

Figure 2

(Color online) Adiabatic following of the lowest four eigenvalues of the interpolating potential (7) from $\eta =0$ (double-well) to $\eta =1$ (Fokker-Planck potential). The value of the coupling parameter is held constant at $g=0.007$.

Figure 3

(Color online) Energy dependence of (the real part of) the function $Q(E,g)$. Calculations have been performed for fixed $g=0.03$.

Figure 4

(Color online) Energy dependence of (the real part of) the function $Q(E,g)$. As in Fig. 3, the results of linear regression analysis of the function $Q(E,g)$ are depicted by the solid line. Calculations have been performed around the “true” energies of the ground state and for the different values of the coupling parameter: $g=0.03$ (left panel), $g=0.1$ (middle panel), and $g=0.3$ (right panel). The energy eigenvalues obtained using the displayed graphs determine the corresponding entries in the right column of Table .

Figure 5

(Color online) Energy dependence of (the real part of) the function $Q(E,g)$. The results for the zeroes of $Q(E,g)$ obtained by a quadratic regression analysis are depicted by solid lines. Calculations have been performed around the “true” energies of the first excited state with $K=1$, $\epsilon +1$ $(M=1)$ and for three different values of the coupling parameter: $g=0.01$ (left panel), $g=0.03$ (middle panel), and $g=0.05$ (right panel). The energy eigenvalues obtained using the displayed graphs determine the corresponding entries in the right column of Table .

Figure 6

(Color online) Eigenvalues $E_{\mathrm{FP}}^{(M=0)}$ (left panel) and $E_{\mathrm{FP}}^{(M=1)}\equiv E_{\mathrm{FP}}^{(K=1,\epsilon =+1)}$ (right panel) of the Fokker-Planck Hamiltonian as a functions of the coupling parameter $g$. Results have been computed by the diagonalization of the Fokker-Planck Hamiltonian in the basis of the harmonic oscillator wave functions (solid lines) and by solving Eq. (19), where the latter correspond to the data points with the error bars.

Figure 7

(Color online) Relative error (22) of calculations of the eigenvalues as a function of coupling constant $g$. Results are presented for the ground $(M=0)$ and the first excited $(M=1)$ states of the Fokker-Planck potential.

Figure 8

(Color online) Exact values (solid line) for the ground state and the two lowest excited energy levels for the Fokker-Planck potential as a function of $g$, together with the leading asymptotics (dotted line) and the partial sum of the strong-coupling expansion (dashed line) as defined by the first six nonvanishing terms in powers of $g^{-2k∕3}$ [see Eq. (26)], which are listed in Table . Note that the leading strong-coupling asymptotics alone cannot satisfactorily describe the true energy eigenvalues for moderate and small coupling. By contrast, the partial sum of the first six nonvanishing terms of the strong-coupling asymptotics yields numerical values which are indistinguishable from the true eigenvalues on the level of the line thickness used in the plots, down to rather small values of the coupling (dashed vs solid lines).

Figure 9

(Color online) The wave functions ${\Phi }_{\mathrm{FP}}^{\left(M\right)}\left(q\right)$ of the ground $M=0$ and the first excited $M=1,2$ states of the Fokker-Planck Hamiltonian calculated at coupling parameter $g=0.05$. The base lines for the plots of the wave functions correspond to their energies $E_{\mathrm{FP}}^{\left(M\right)}\left(g\right)$.

Figure 10

(Color online) Time propagation of the (initially) Gaussian wave packet in the time-dependent potential (7) with $\eta =\sin (t∕60)$ which oscillates between the double-well and Fokker-Planck potentials. The base line for the plot of the wave function corresponds to its instantaneous energy.