#  Hallmarks of tunneling dynamics with broken reflective symmetry

V.P<EMAIL_ADDRESS>, M.I.
<EMAIL_ADDRESS>and A.J.
<EMAIL_ADDRESS>

$\,{}^{\spadesuit}$ Akhiezer Institute for Theoretical Physics of NSC KIPT
1 Akademicheskaya St., Kharkov 61108 UA
$\,{}^{\vardiamondsuit}$ Department of Physics & Technology, Karazin Kharkov
National University,
4 Svobody Sq., Kharkov 61022 UA
$\,{}^{\varheartsuit}$ Usikov Institute of Radiophysics and Electronics
12 Ak. Proskury St., Kharkov 61085 UA

###### Abstract

We study features of tunneling dynamics in an exactly-solvable model of N=4
supersymmetric quantum mechanics with a multi-well potential and with broken
reflective symmetry. Quantum systems with a phenomenological potential of this
type demonstrate the phenomenon of partial localization of under-barrier
states, possibly resulting in the appearance of the so-called “resonant”
tunneling, or the phenomenon of coherent tunneling destruction, referring to
the complete localization. Taking the partial localization and the coherent
tunneling destruction as basic examples, we indicate main advantages of using
isospectral exactly-solvable Hamiltonians in studies quantum mechanical
systems with two- and three-well potentials. They, in particular, are: having
enough freedom of changing the potential shape in a wide range, that allows
one to choose an exactly-solvable model close to characteristics of the
phenomenological one; ability of changing the number of local minima and
symmetry characteristics of the potential (symmetric or deformed) without
changing the main part of the spectrum; engaging a smart basis of states, that
dramatically decreases the dimensionality of matrices used in the
diagonalization procedure of the corresponding spectral problem.

##  1 Introduction

Tunneling is perhaps the most fascinating, enigmatic and controversial feature
of the quantum world. It came almost a century since this phenomenon was
discovered by Hund, Gamow, Oppenheimer, Condon and Gurney [1, 2, 3, 4, 5], but
we do not have the comprehensive picture of this process yet and the ultimate
conclusions on mechanisms behind. For instance, one of the controversies,
related to the tunneling, is a super-luminal passing speed of a quantum object
through a potential barrier [6, 7, 8, 9, 10] (and refs. therein). Another
intriguing concern is the possibility to tunnel a macroscopic structure. This
issue was risen up in late 80th in different contexts (see, e.g., refs. [11,
12, 13, 14, 15] for reviews), including the macroscopic tunneling in SQUIDs,
magnetic domains relaxation to the ground state, atomic and molecular
processes in chemical physics, biology and so on. The first experimental
evidence of these theoretical ideas comes from pioneering works on the quantum
tunneling of magnetization [16, 17]; the realization of Bose-Einstein
condensates of cold atoms has simplified the observation of the macroscopic
quantum tunneling in all forms [18] (and refs. therein). Another feature of
the tunneling effect worth to mention is that experiments in this branch put
even foundations of Quantum Mechanics [19] on the test, since some of the
tunneling properties in modern artificial materials like graphene [20] are
naturally described within alternative – the Bohmian – interpretation of
Quantum Mechanics. Moreover, in the context of the main subject of the paper,
it is important to note the role of supersymmetry in explaining the prominent
features of carbon nanostructures such as the perfect Klein tunneling and the
no-backscattering effect [21].

Progress in experimental studies of the condensate of atoms, mentioned
insofar, resulted in the foundation and development of a new discipline –
quantum engineering – which currently is considered as the base of developing
Q-bits, quantum transistors, new quantum-type flash-memory modules and other
parts of outcoming quantum computers. We have to emphasize that the quantum
engineering means not just the creation of new type devices, but, what is more
important, of quantum-mechanical objects with predefined properties, together
with the possibility to control these properties with external EM fields
(laser pulses). The tunneling dynamics will certainly have a profound effect
on the design of quantum computer components, and ultimately, on the computing
power of the final gadget. It is also interesting to emphasize the potential
importance of supersymmetry for quantum computation and quantum information,
pointed up in recent work [22].

Evidently, any effects of the tunneling dynamics disclose the specific of the
potential of a quantum-mechanical model in the end. For a symmetric double-
well potential it is worth emphasize the theoretically predicted phenomena of
the coherent tunneling destruction (CTD) [23, 24, 25], and the tunneling
amplification, initiated by dynamical chaos, ref. [26], which were
experimentally discovered and reported in refs. [27, 28]. The coherent
tunneling destruction is the subject of numerous theoretical and experimental
investigations (see, e.g., refs. [29, 30, 31, 32, 33, 34, 35, 36, 37]) and is
one of the aspects in the focus of the present work.

Qualitatively, good enough conclusions on the tunnel dynamics can be reached
with employing phenomenological models of [29, 30, 31, 32, 33, 34, 35, 36, 38]
(with a Higgs-type potential and its multi-well modifications with the number
of wells higher than two). These results are obtained upon the assumption of a
sufficiently high barrier in between wells, or, equivalently, with taking a
few lowest states of the corresponding spectra. That is to say, physics of
these processes is mainly determined by the lowest (under-barrier) states of
Hamiltonians (by the tunnel doublet); nevertheless, the role of more higher
states may be decisive. For example, a control of the tunnel dynamics with
time-dependent fields is achieved in quantum mechanical models with multi-well
potentials. In most cases, a multi-well potential is assigned by an analytical
modification of the Higgs-type potential, that requires, in what follows,
cumbersome numerical computations upon getting the (quasi)energies and the
Floquet eigenstates by the diagonalization of the complete Hamiltonian. The
choice of the appropriate basis of states becomes crucial in the
simplification and optimization of the diagonalization procedure, since the
right choice of the basis essentially reduces the dimensionality of the used
in numerics matrices. From this point of view, the series expansion of states
of phenomenological Hamiltonians over the Harmonic Oscillator basis (see,
e.g., refs. [23, 24, 25, 34]) is not the finest choice: it requires dealing
with hundreds of basic states, that determines the dimensionality of the
matrix to diagonalize.

Another aspect in the focus of the present paper is considering tunneling
processes in quantum systems with broken reflective symmetry (see refs. [39,
40] for early papers in this field). The importance of studies in the
tunneling dynamics of models with asymmetric multi-well potentials can be
illustrated by the analysis of the control processes in a radio-frequency
SQUID phase qubit [41, 42], as well as in widely-discussed prospects of
creation of the THz radiation sources [43, 44]. As we will see, shortly, the
case of asymmetric potentials is more technically complicated, however
encompasses new phenomena.

Specifically, the examination of states, located in different wells of an
asymmetric two-well potential [39], revealed the following non-trivial effect:
the Gaussian wave packet in a metastable state stays at its initial
localization for an infinitely long time (for an arbitrary asymmetry degree of
the potential). Moreover, there is a claim, that, in a system with a symmetric
two-well potential, adding a specific but small perturbation [45], which
breaks the reflective symmetry, results in localization of under-barrier
states. In ref. [46], on the example of a system with the two-well
phenomenological potential, this claim was verified by the direct integration
of the Schrödinger equation. Therefore, the phenomenon of localization of
states in an asymmetric potential becomes an important tool of controlling the
tunneling processes, in which the specific perturbation over the initially
symmetric potential acts as a control parameter.

One more branch of studies of the tunneling dynamics in systems with an
arbitrary asymmetry degree of multi-well potentials is presented in refs. [47,
48, 49, 50, 51, 52, 53, 54, 55]. First works on this subject [47, 48, 49] were
focused on a small deformation of the potential; the value of the tunnel
doublet splitting has been computed in this approximation. Further
developments of refs. [50, 51, 52, 53, 54] made possible to compute the tunnel
doublet levels splitting for an arbitrary deformation of the potential, as
well as to find the under-barrier wave functions within the quasi-classical
approximation. The Gaussian packets evolution in the model with an asymmetric
two-well potential has been considered in ref. [55], where, among others,
arising the resonant tunneling upon the specific values of the deformation
parameters has been examined in details.

Typically, the analysis of quantum systems with the phenomenological-like
potentials is carried out either by numerical computations of the spectra and
wave functions, or by use of an involved WKB computational scheme to determine
the under-barrier wave functions, together with the value of the tunnel
doublet splitting. The usage of multi-well potentials of a polynomial type,
the asymptotic behavior of which is $x^{2n}$ (where $n$ is the number of the
potential minima), faces with essential complications even for $n=2$. With
growing the $n$ number, the computational schemes become more and more hard to
realize. It is worth mentioning, that the aforementioned difficulties with
numeric computations are related to the required high precision of the
results, especially in the low part of the spectrum. Indeed, small errors in
the computed energies and wave functions of the tunnel doublet may result in
large deviations upon the tunneling dynamics, making possible to observe false
effects. Not least at all, roots of the difficulties, one meets with, lie in
the choice of the basis in the wave function series expansion upon performing
the numerics. And here we turn back to what we have pointed out not so far:
the choice of the Harmonic Oscillator basic states is not the optimal one, and
results in growing the rank of matrices under the use.

One of the ways to resolve difficulties, inherent to potentials of a
phenomenological type, is to operate with exactly-solvable models with multi-
well potentials. Examples of these models have been studied and developed in
refs. [56, 57, 58]. The advantages of the exactly-solvable models of [56, 57,
58] consist in: 1) having enough freedom of changing the potential shape in a
wide range, that allows one to choose an exactly-solvable model close to
characteristics of the phenomenological one; 2) the ability of changing the
number of local minima and symmetry characteristics of the potential
(symmetric or deformed) without changing the main part of the spectrum; 3)
engaging a smart basis of states, that dramatically decreases the
dimensionality of matrices, used in the diagonalization procedure of the
corresponding spectral problem. We will illustrate all of these benefits by
different examples in below.

The rest of the paper has the following organization. In Section 2, we briefly
discuss the machinery of construction of exactly-solvable isospectral multi-
well Hamiltonians in a general case; then, we survey the results of employing
this technique to the Harmonic Oscillator (HO) Hamiltonian. In Section 3 and
in the following sections, we focus on the obtained Hamiltonian in different
situations. Specifically, in Section 3, we describe the tunneling dynamics of
states in symmetric and asymmetric two- and three-well potentials, which were
obtained from the HO potential for special values of the deformation
parameters. As a result, we observe a Josephson’s type oscillation of the
Gaussian packet in the two-well potential of a symmetric shape, and detect a
partial localization of the wave packet in the case of asymmetry of the
potential. Here, we also outline the common properties of the tunneling
dynamics of the Gaussian wave packet for three-well symmetric and asymmetric
potentials. In Section 4, we study the response of a system with two- and
three-well potentials on an external disturbance of their shapes. We compute
the corresponding wave functions upon the small disturbance, far from the
local minimum of the undisturbed potential, and show that the most
contribution into rearranging the spectrum and the wave functions comes from
seven first basic states of the exactly-solvable Hamiltonian. Checking the
absence of visible changing in the spectrum and in the coefficients of series
expansion of wave functions with increasing the number of basic states on ten,
we verified, in this way, the efficiency of our approach. In Section 5, we
examine the behavior of a quantum system with a multi-well potential upon the
action of an external time-periodic driving force. We establish the criteria
of the coherent tunneling destruction, and show that ten basic wave functions
of the corresponding time-independent exactly-solvable Hamiltonian is good
enough to describe the tunneling dynamics of the under-barrier states. Our
conclusions are collected in the last section.

Notation and conventions. We work in $\hbar=m=1$ units, recovering, if
necessary, their physical values. Computations were performed for the harmonic
oscillator frequency $\omega_{0}=1$; hence, the natural and the dimensionless
coordinates $x$ and $\xi=\sqrt{\omega_{0}}\,x$ are indistinguishable with this
choice. $\omega$ is reserved for the frequency of an external periodic driving
force.

##  2 Isospectral Hamiltonians with two- and three-well potentials: A brief
survey on previous works

There are several solution-generation methods in mathematical physics, leading
to new analytic solutions of exactly-solvable differential equations upon the
controlled deformation of their structure. The mentioned deformation does not
intact the global structure of the initial equation (viz., its type – elliptic
or hyperbolic), but can significantly change its local form. Hence, the so
specifically obtained equation will describe new dynamics, properties of which
may crucially be distinct of that of the original system.

In the context of supersymmetric quantum mechanics (SQM) one of the frequently
employed solution-generation routines is the method by Crum and Krein [59,
60]. The Crum-Krein method consists of the well-defined procedure of
deformation of an exactly-solvable seed Hamiltonian $H_{0}$ – with the known
spectrum and wave functions – potential of which has the only one well.444Here
and hereafter, $H_{0}$ is not just the free Hamiltonian $p^{2}/2$ (in
$\hbar=m=1$ units); it also includes a potential $V_{0}$. Specifically, this
procedure is realized as the add-on, in step by step manner, new levels of the
energy spectrum below the ground state of $H_{0}$, that results in designing a
new Hamiltonian $\tilde{H}$ with a multi-well potential. The number of wells
in the potential of $\tilde{H}$ is not just determined by the number of sub-
ground levels; it also depends on the particular choice of parameters of the
deformation (see more on that further on). The Crum-Krein method does not
appeal to any particular properties of the Hamiltonians (except of their
solvability), so SUSY is not a necessary prerequisite of the construction.
However, for the supersymmetric quantum mechanical Hamiltonians, the
employment of the Crum-Krein routine becomes equivalent to constructing
Hamiltonians with pre-defined spectral properties within the polynomial
(reducible) SQM [61, 62, 63], the main attribute of which is the sequential
application of the SUSY transformations to the initial Hamiltonian.

Note, that the discussed procedure of generating a new Hamiltonian, considered
in more detail in below, does not restricted to the standard exactly-solvable
models of quantum mechanics and can be extended to more involved cases, when
the considered system obeys a specific symmetry (like, for instance, the
hidden superconformal symmetry [64], or the Klein four-group in superconformal
mechanics [65]) and/or has a specific structure of the spectrum (as, for
example, having a series of “valence bands” with the own fine structure and
gaps in between [66, 67], including the periodic finite gap [68] or zero-gap
[69]). Another interesting generalization of Supersymmetric Quantum Mechanics,
realizing a correspondence of SQM to the inverse scattering method [70], is
related to the description of multi-soliton physics [71] as well as soliton
defects in a crystalline background [72] within (exotic) non-linear
sypersymmetry. However, the tasks aimed in the present paper allow us to stay
within the standard Crum-Krein approach, to the brief consideration of which
in the cases of interest we are now moving on.

Previously, in ref. [58], we gained general expressions, describing a family
of N=4 multi-well SQM Hamiltonians, originated from an exactly-solvable model
with one-well potential, by adding two extra sub-levels of energies
$E_{-2}<E_{-1}<E_{0}$ below the ground state $E_{0}$ of the initial
Hamiltonian $H_{0}$. In accordance with the Crum-Krein recipe, the initial
Hamiltonian gets transformed into

$\tilde{H}=H_{0}-\frac{d^{2}}{dx^{2}}\ln\frac{1}{2}\epsilon^{JI}\mathrm{W}\\{\psi_{-I},\psi_{-J}\\},$
(1)

where the indices $I,J=1,2$ count the extra sub-levels of the energy spectrum
(with energies $E_{-I}\equiv(E_{-1},E_{-2})$); $\epsilon^{IJ}$ is the 2D Levi-
Civita symbol, defined as $\epsilon^{12}=1$. Furthermore, $\psi_{-I}$ are the
wave functions associated with the energies $E_{-I}$ and deformation
parameters $\Lambda_{-I}\in]0,\infty[$. They are constructed out of linearly-
independent non-normalizable solutions – $\varphi^{(A)}_{-I}(x,E_{-I})$,
$A=1,2$ – to the Schrödinger equations with $H_{0}$ at the energies $E_{-I}$
(see ref. [58] for details):

$\psi_{-1}(x,E_{-1},\Lambda_{-1})=\varphi^{(1)}_{-1}+\Lambda_{-1}\,\varphi^{(2)}_{-1},$
(2)
$\psi_{-2}(x,E_{-2},\Lambda_{-2})=\varphi^{(1)}_{-2}-\Lambda_{-2}\,\varphi^{(2)}_{-2}.$
(3)

Finally, $\mathrm{W}\\{\psi,\psi^{\prime}\\}$ stands for the Wronskian of two
independent solutions. The spectral problem for Hamiltonian (1) is solved with
the following set of wave functions [56, 57, 58]:

$\tilde{\Psi}_{0}(x,E_{-2},\Lambda_{-2};E_{-1},\Lambda_{-1})=\Big{[}\tilde{N}_{\Lambda_{-2}}\Big{]}^{-1}\frac{\psi_{-1}(x,E_{-1},\Lambda_{-1})}{\mathrm{W}\\{\psi_{-2},\psi_{-1}\\}},$
(4)
$\tilde{\Psi}_{1}(x,E_{-1},\Lambda_{-1};E_{-2},\Lambda_{-2})=\Big{[}\tilde{N}_{\Lambda_{-1}}\Big{]}^{-1}\frac{\psi_{-2}(x,E_{-2},\Lambda_{-2})}{\mathrm{W}\\{\psi_{-2},\psi_{-1}\\}},$
(5)
$\tilde{\Psi}_{i}(x,E_{i})=\sqrt{\frac{E_{i}-E_{-1}}{E_{i}-E_{-2}}}\,\,\times$
$\times\left(\psi_{i}(x,E_{i})+\frac{E_{-1}-E_{-2}}{E_{i}-E_{-2}}\frac{\mathrm{W}\\{\psi_{i}(x,E_{i}),\psi_{-1}(x,E_{-1},\Lambda_{-1})\\}}{\mathrm{W}\\{\psi_{-2},\psi_{-1}\\}}\psi_{-2}(x,E_{-2},\Lambda_{-2})\right),$
(6)

where $\psi_{i}(x,E_{i})$, $i=0,1,\dots$ are the wave functions of the primary
Hamiltonian $H_{0}$.

Eqs. (1)-(6) determine the general computational scheme for any exactly-
solvable seed Hamiltonian. In what follows we will specify $H_{0}$ to be the
Harmonic Oscillator (HO) Hamiltonian

$H_{0}=\frac{p^{2}}{2}+\frac{\omega^{2}_{0}}{2}\,x^{2}.$ (7)

Then, the non-normalizable solutions at $E_{-I}<E_{0}$ are represented by the
parabolic cylinder functions $D_{\mu}(x)$ (see ref. [58]),

$\varphi^{(1)}_{-1}(\xi,\bar{E}_{-1})=D_{\nu}(\sqrt{2}\xi),\qquad\varphi^{(2)}_{-1}(\xi,\bar{E}_{-1})=D_{\nu}(-\sqrt{2}\xi),$
(8)
$\varphi^{(1)}_{-2}(\xi,\bar{E}_{-2})=D_{\mu}(\sqrt{2}\xi),\qquad\varphi^{(2)}_{-2}(\xi,\bar{E}_{-2})=D_{\mu}(-\sqrt{2}\xi).$
(9)

As customary in SQM computations, we have moved to the dimensionless
“coordinate” $\xi=\sqrt{\omega_{0}}\,x$ and the dimensionless “energy”
$\bar{E}_{-I}=E_{-I}/\omega_{0}$. The orders of the parabolic cylinder
functions are determined by $\nu=-1/2+\bar{E}_{-1}$, $\mu=-1/2+\bar{E}_{-2}$.

The Wronskian of the linear-independent solutions becomes

$\mathrm{W}\,\\{\varphi^{(1)}_{-I},\varphi^{(2)}_{-J}\\}=\frac{2\sqrt{\pi\omega_{0}}}{\Gamma\left(\frac{1}{2}-E_{-I}\right)}\,\delta_{IJ}.$
(10)

Consequently, the normalization constants of the newly added sub-level wave-
functions (4), (5) are [58]

$\left[\tilde{N}_{\Lambda_{-2}}\right]^{-2}=4\Lambda_{-2}\,\frac{\sqrt{\pi\omega_{0}}}{\Gamma(-\mu)}\,(\nu-\mu),\quad\Lambda_{-2}>0,$
(11)
$\left[\tilde{N}_{\Lambda_{-1}}\right]^{-2}=4\Lambda_{-1}\,\frac{\sqrt{\pi\omega_{0}}}{\Gamma(-\nu)}\,(\nu-\mu),\quad\Lambda_{-1}>0.$
(12)

Eqs. (8)-(12) empower recovering the exact expressions for the new Hamiltonian
(1) and the corresponding wave functions (4)-(6) in the case of the HO seed
Hamiltonian (7).

Note that multi-well potential Hamiltonians, constructed this way, inherit the
property of the HO Hamiltonian – $H(p,x)=\omega_{0}H(p_{\xi},\xi)$ – upon
turning to the dimensionless operators of the “coordinate” $\xi$ and its
conjugated “momentum” $p_{\xi}$. Moreover, the specific deformation procedure
does not just lead to a single new Hamiltonian, but to a family of
(isospectral) Hamiltonians, originated from varying the potential shape by
changing the energies $\bar{E}_{-1},\bar{E}_{-2}$ and the positive deformation
parameters $\Lambda_{-1},\Lambda_{-2}$. Turning to the standard $(x,p_{x})$
variables back, another free parameter in hands is the frequency $\omega_{0}$.
For instance, varying $\omega_{0}$ results in changing the positions of local
minima.

Pinning up one of the deformation parameters does not affect the generality of
results, but essentially simplifies the computational scheme. We will assume
hereafter (where it will not be specified apart) the unit value of the
deformation parameter $\Lambda_{-2}$, keeping variable the other parameter
$\Lambda\equiv\Lambda_{-1}$. In addition, in all the computations below, we
will assume $\omega_{0}=1$. According to the scale property of the HO
Hamiltonian, this choice corresponds to $H(p,x)=H(p_{\xi},\xi)$. Therefore, we
will not differentiate, in what follows, the natural and the dimensionless
coordinates.

##  3 Tunneling dynamics of quantum states: symmetric vs asymmetric multi-well
potentials

As we have previously mentioned, the described by (1)-(6) family of
Hamiltonians possesses a remarkable property: in dependence on the choice of
free parameters the deformed potentials get either two or three wells.
Specifically, once the tunnel doublet $\triangle=\bar{E}_{-1}-\bar{E}_{-2}$ is
located far enough from the ground state of the seed Hamiltonian
$\bar{E}_{0}=1/2$, the Hamiltonian potential is shaped by a two-well curve. It
turns out that the set of the parameters, resulting in two-well potentials,
provides a wide “plateau” between the local minima (see Fig.1). This
observation makes possible to employ the two-level approximation upon studying
the tunneling dynamics.

a)

b)

c)

Figure 1: Examples of two-well potentials
$\tilde{U}(\xi,\bar{E}_{-2},\bar{E}_{-1},\Lambda)$ with fixed values
$\nu\equiv-1/2+\bar{E}_{-1}=-3.0$, $\mu\equiv-1/2+\bar{E}_{-2}=-3.02$: a) the
symmetric potential ($\Lambda=1$); b) the asymmetric potential with
$\Lambda=0.5$; c) the asymmetric potential with $\Lambda=0.05$. Solid lines
$\bf{1}$ correspond to the ground state wave functions; dashed lines $\bf{2}$
display the first excited level wave functions; dotted lines $\bf{3}$
demonstrate the behavior of the second excited level wave functions.

In the case of a symmetric potential (Fig.1a) its shape, especially in the
central part of the potential, resembles a quartic, double-well potential (the
Higgs-like potentials), frequently used in the SQM literature. Tuning the
value of the tunnel doublet and its location on the energy scale, one may
match, with a good enough precision, the central part of the Higgs-like
potential to the considered here exactly-solvable potentials. Since the value
of the tunnel doublet and wave functions for exactly-solvable Hamiltonians are
explicitly known (cf. eqs. (4)-(12)), they may be used as a basis in studying
properties of lowest states in models with the phenomenological interaction.
Recall, the lowest states of a spectrum are responsible for the tunneling
dynamics and its features.555Note that the asymptotic behavior of the
phenomenological and exactly-solvable potentials differs. The asymptote of
Hamiltonian (1) is the same as the seed Hamiltonian $H_{0}$, while the
asymptote of the Higgs-like potential is $x^{2n}$, where $n$ is the number of
local minima (wells of the potential). As a result, the tunnel doublet and the
wave functions under the barrier of the phenomenological type potentials are
forced to change (taking into account of more and more highest states of the
spectrum), that leads to essential computational difficulties.

Changing the value of $\Lambda\equiv\Lambda_{-1}$ (remind, we have fixed
$\Lambda_{-2}=1$), we get various potentials of a family of isospectral
Hamiltonians. Decreasing the value of $\Lambda\in\,]0,\infty[$, we alter the
depth of the right well. The location of its minimum shifts to higher values;
the central part of the potential does not undergo essential modifications,
see Fig.1. A deformation of the potential causes the redistribution of the
fraction of a wave function in each of the wells, which is the main effect.
For example, the fraction of the ground state wave function increases in the
left well, while the fraction of the first excited level wave function
decreases therein, but increases in the right well. This effect is mostly
noticeable at $\Lambda\rightarrow 0$ (cf. Fig.1); however, the complete
localization of states does not occur.666Indeed, changing the deformation
parameter does not change the spectrum of the form-invariant Hamiltonians (see
ref. [56] in this respect). Therefore, the value of the tunnel doublet, which
characterizes the tunneling process, does not change as well. To sum up, we
can only get a partial localization of the under-barrier states.

The partial localization of the under-barrier states gets a strong impact on
the tunneling dynamics. To make this point clear, note first the viable
advantage of the considered quantum Hamiltonians for studying dynamical
processes: since the wave functions of the system are explicitly known (cf.
eqs. (4)-(6)), the exact propagator can also be computed analytically [58]. By
use of the exact propagator for the considered here HO, which acts on an
originally located in the right well Gaussian wave packet,

$\Phi(\xi,0)=\left(\frac{e^{2R}}{\pi}\right)^{1/4}\exp\Big{[}{-\frac{(\xi-\xi_{0})^{2}e^{2R}}{2}}\Big{]},$
(13)

and choosing different sets of parameters, indicated in Fig.1, one may observe
the redistribution of the wave function fractions in two wells. Before turning
to the discussion of the tunneling dynamics, caused by such redistribution, we
would like to emphasize that knowing the exact propagator allows us to study
(albeit technically difficult) the tunneling dynamics features of the wave
packet of an arbitrary shape with taking into account all states of
Hamiltonian (1).

The qualitative interpretation of the wave packets localization is essentially
simplified in the case when the under-barrier part of the spectrum, with
$\bar{E}_{n}<U_{\mathrm{loc.max.}}$ ($U_{\mathrm{loc.max.}}$ is the local
maximum of the potential between two wells) together with the corresponding
wave functions $\psi_{n}(\xi)$, are known. The pure tunneling dynamics occurs
for packets of the following form777Formally, the wave functions
$\psi_{n}(\xi)$ and series coefficients $C_{n}$ are supposed to be complex. In
fact, both of them are real for the under-barrier states. The whole wave
function $\Phi(\xi,\tau)$ is complex of course.

$\Phi(\xi,\tau)=\sum_{n}C_{n}e^{-i\bar{E}_{n}\tau}\psi_{n}(\xi),\quad\bar{E}_{n}<U_{\mathrm{loc.max.}},\quad\tau=\omega_{0}t,$
(14)

with

$C_{n}=\int\,dx\,\Phi(\xi,0)\psi^{*}_{n}(\xi).$ (15)

The probability $P^{R}(t)$ (or $P^{R}(\tau)$ with the apparent modification of
the result in below) to find a particle at the time $t$ in the definite local
minimum $R$ is

$P^{R}(t)=\int_{R}\,dx\,|\Phi(x,t)|^{2}=\sum_{m,n}C^{*}_{m}C_{n}e^{i(E_{m}-E_{n})t}\int_{R}\,dx\,\psi^{*}_{m}(x)\psi_{n}(x).$
(16)

Within the two-level approximation (when $n$ runs over $1$ and $2$ and the
wave functions are real; cf. footnote 4),

$P^{R}(t)=P^{R}(0)-4C_{1}C_{2}\sin^{2}{(E_{1}-E_{2})t}\int_{R}\,dx\,\psi_{1}(x)\psi_{2}(x).$
(17)

In the symmetric case (displayed in Fig.1a), the wave packet (13) makes
periodic oscillations between two minima, with the period $T=2\pi/\triangle$;
here $\triangle=\bar{E}_{-1}-\bar{E}_{-2}=\nu-\mu$. We observe that the wave
packet completely transfers from one well to another, so that the “Josephson’s
oscillations” (17) take place. Having a deformation of
$\tilde{U}(\xi,\bar{E}_{-2},\bar{E}_{-1},\Lambda)$ from the symmetric shape,
the picture of the tunneling dynamics is essentially modified (cf. Fig.2). It
is featured the so-called “partial confinement” of the wave packet in the
initial well. The effect is amplified with growing the deformation of the
potential (see Fig.2b). Such a behavior of the tunneling dynamics is
determined by a re-distribution of wave functions between local minima upon
increasing the potential deformation. The wave packet recovery time coincides
with that of the symmetric case because the value of the tunnel doublet does
not change upon the potential deformation. Recall that the just discussed
hallmarks of the tunneling dynamics take place within the two-level
approximation for the potentials in Fig.1, when the main contribution to the
wave packet is given by the wave functions of the tunnel doublet.

a)

b)

Figure 2: $|\Phi(\xi,T)|$ in the case of $\tilde{H}$ with: a) $\nu=-3.0$,
$\mu=-3.02$, $\Lambda=0.5$; b) $\nu=-3.0$, $\mu=-3.02$, $\Lambda=0.05$. $T$
and $\xi$ are changed in the following ranges: $-5\leq\xi\leq 5$, $0\leq T\leq
300$.

To sum up, having the relevant basis in the disposal, it is always possible to
present an arbitrary wave packet in its terms and, by use of the appropriate
approximation, to get a qualitative characteristic of localization in the
definite local minimum. Wave packets dynamics in two-well potentials
$\tilde{U}(\xi,\bar{E}_{-2},\bar{E}_{-1},\Lambda)$ within the two-level
approximation will be characterized by values $P^{R}(\tau)<1$, that means the
absence of full localization in view of a non-trivial tunnel doublet value.
The case of the complete localization of the tunnel doublet states corresponds
to $P^{R}(\tau)\equiv P^{R}(0)$.

Dynamics of wave packets in three-well potentials within the taken here
approach, ref. [58], have a number of differences from the previously
considered case. Hamiltonian (1) contains a set of parameters, in dependence
on which the potential gets two or tree wells. When the first added level (of
energy $\bar{E}_{-1}$ in our notation) is located near the ground state of the
initial Hamiltonian $H_{0}$ and forms, with the $H_{0}$ ground state
$\psi_{0}$ (of energy $\bar{E}_{0}$), the tunnel doublet of
$(\psi_{-1},\psi_{0})$ states, the potential
$\tilde{U}(\xi,\bar{E}_{-2},\bar{E}_{-1},\Lambda)$ possesses three local
minima (see Fig.3).

a)

b)

Figure 3: The potential $\tilde{U}(\xi,\bar{E}_{-2},\bar{E}_{-1},\Lambda)$
with three wells (thin solid lines) and wave functions of first levels. The
parameters are: a) $\nu=-0.02$, $\mu=-1$, $\Lambda=1$; b) $\nu=-0.02$,
$\mu=-1$, $\Lambda=0.05$. Bold solid lines correspond to the ground state wave
functions; dashed lines are that of the first excited level, dotted lines
correspond to the second excited level wave functions. (The dashed-dotted line
is that of the third excited level.)

Figure 3 shows that the ground state wave function of a (deformed) three-well
potential is localized in the central minimum domain independently on values
of the deformation parameter. The excited states are delocalized in between
uttermost minima, and deforming the potential from the pure symmetric shape
results in the redistribution of their wave functions over the uttermost
wells. These facts impact the tunneling process of the wave packet, initially
located in one of the utmost minima (see Fig.4). Both cases in Fig.4 are
characterized by a small value of $|\Phi|$ within the central minimum domain
at any values of the deformation parameter. (We suppose the initial wave
packet is in the right well.)

a)

b)

Figure 4: Two plots of $|\Phi(\xi,\tau)|$ for the parameters of
$\tilde{U}(\xi,\bar{E}_{-2},\bar{E}_{-1},\Lambda)$ in Fig.3: a) $\nu=-0.02$,
$\mu=-1$, $\Lambda=1$; b) $\nu=-0.02$, $\mu=-1$, $\Lambda=0.05$. The ranges of
$\tau\equiv T$ and $\xi$ are chosen to be: $-5\leq\xi\leq 5$, $0\leq T\leq
300$.

##  4 Localization of under-barrier states in asymmetric multi-well potentials

Studies of the wave packets dynamics in an asymmetric double well, ref. [39],
showed the possibility of the complete localization of quantum states in the
false vacuum (the local minimum of the higher energy), in dependence on the
deformation degree. Furthermore, in ref. [45], it was shown that the tunneling
dynamics of symmetric double-well Hamiltonians is exponentially sensitive to a
tiny perturbation of the potential, that results in localization of the under-
barrier states. The disturbance potential is localized in one of the wells,
far from the corresponding minimum; its value is chosen to be
$|V_{1}|\gg\triangle$ ($\triangle$ is the tunnel doublet value). The sign of
the potential $V_{1}$ determines the well, in which the under-barrier states
will be localized. In ref. [46], localization of states was demonstrated by
use of a tricky procedure of the direct integration of the Schrödinger
equation with $U=V_{0}(x^{2}-a^{2})^{2}$ potential and with888For the sake of
generality, we will consider the $(x,t)$ set of the natural variables in
analytical expressions of this section. Numerical computations are performed
in terms of $(\xi,\tau)$.

$V_{1}(x)=\left\\{\begin{array}[]{l}s\cdot\exp\left(\frac{1}{c^{2}}-\frac{1}{c^{2}-(x-b)^{2}}\right)\qquad\mathrm{if}\,\,|x-b|<c\\\
0\qquad\qquad\qquad\qquad\qquad\quad\,\,\mathrm{if}\,\,|x-b|>c\,\,.\end{array}\right.$
(18)

Below we will show the advantages of using the basis (4)-(6) in studying
localization of states of symmetric Hamiltonians with two- and three-well
potentials, on account of the disturbance (18). The spectrum and the wave
functions of the disturbed Hamiltonian $\hat{H}=\tilde{H}+V_{1}$ (recall,
$\tilde{H}$ is Hamiltonian (1) with the HO Hamiltonian $H_{0}$) are recovered
from the diagonalization procedure for
$\hat{H}\Psi^{(n)}(x)=\hat{E}_{n}\Psi^{(n)}(x)$, with the general expansion of
wave functions:

$\Psi^{(n)}(x)=\tilde{C}^{(n)}_{0}\tilde{\Psi}_{0}(x,E_{-2},\Lambda_{-2};E_{-1},\Lambda_{-1})+\tilde{C}^{(n)}_{1}\tilde{\Psi}_{1}(x,E_{-2},\Lambda_{-2};E_{-1},\Lambda_{-1})$
$+\sum^{N}_{i=0}\tilde{C}^{(n)}_{i+2}\tilde{\Psi}_{i+2}(x,E_{i}),\qquad
n=0,1,\dots,N\,,$ (19)

with $\Lambda_{-2}=1=\Lambda_{-1}$. Here $N$ is a number, from which the
spectrum $\hat{E}_{n}$, $n>N$ and the wave functions $\tilde{\Psi}^{(n)}(x)$
are exactly that of the Hamiltonian $\tilde{H}$ (i.e., the wave functions of
eqs. (4)-(6)).999For the parameters $s$, $b$ and $c$ of eq. (18), which are
used in our subsequent calculations, we established, by the trial and error
method, $N=7$.

After diagonalizing the disturbed Hamiltonian $\hat{H}$, we can compute the
probability density of the corresponding wave functions and figure out the
localization degree of the lowest states, forming the tunnel doublet (see
Fig.5). From the shapes of $|\Psi^{(0)}|^{2}$ and $|\Psi^{(1)}|^{2}$ in Fig.5,
it follows that, upon increasing the disturbance parameters, the degree of
localization growths and becomes full at $s=1$. The energy spectrum of first
ten states of the disturbed Hamiltonian is given in Table 1.

a)

b)

Figure 5: a) Localization of the ground state wave function of the disturbed Hamiltonian with the disturbance parameters $b=1.86$, $c=0.5$ (left panel) and $b=1.86$, $c=0.25$ (right panel); b) Localization of the first excited state wave function of the disturbed Hamiltonian with the disturbance parameters $b=1.86$, $c=0.5$ (left panel) and $b=1.86$, $c=0.25$ (right panel). Other parameters of the Hamiltonian are the same as in Fig.1a. Parameters |  $n$=0 |  $n$=1 |  $n$=2 |  $n$=3 |  $n$=4 |  $n$=5 |  $n$=6 |  $n$=7 |  $n$=8 |  $n$=9
---|---|---|---|---|---|---|---|---|---|---
$b$=1.86, $c$=0.25, $s$=0.6 | -2.512 | -2.466 | 0.507 | 1.515 | 2.506 | 3.501 | 4.509 | 5.509 | 6.501 | 7.509
$b$=1.86, $c$=0.5, $s$=0.6 | -2.511 | -2.357 | 0.525 | 1.549 | 2.526 | 3.508 | 4.529 | 5.521 | 6.510 | 7.524
Table 1: The energy spectrum of the distrubed Hamiltonian: first ten levels.

Looking at Table 1, one may notice that the chosen disturbance does not lead
to essential changes in the HO energies $1/2,3/2,5/2,\dots$ upper the tunnel
doublet. On the other hand, the value of the tunnel doublet by itself (recall,
it has been originally chosen to be $\triangle=0.02$) gets essentially
increased: $\triangle=0.046$ for $c=0.25$ and $\triangle=0.154$ for $c=0.5$.
It decisively matters in localization of states of a Hamiltonian with two
symmetric wells, upon incorporating the interaction, which breaks the
reflective symmetry.

Let us present the values of first coefficients in the expansion (19) for the
ground and first excited level wave functions, Table 2. Looking on them, one
may notify the essential contribution into the under-barrier wave function of
the disturbed Hamiltonian comes from the states of the tunnel doublet of
$\tilde{H}$. The contribution of more higher states of $\tilde{H}$ is small,
though their values are comparable to each other. It mainly occurs due to the
choice of parameters of $\tilde{H}$, that provides a good enough fulfillment
of the two-level approximation. Graphically, the choice of the parameters for
the two-level approximation becomes apparent (see Fig.6a and Fig.6b): they do
not deform the initial potential much. However, the Schrödinger equation
$\hat{H}\Psi^{(n)}=\hat{E}_{n}\Psi^{(n)}$ can not only be satisfied due to the
tunnel doublet wave functions. Despite the small fraction of states above the
tunnel doublet in the wave function (19), their contribution to the formation
of the full potential (see the discussion around (20) below) is significant:
the contribution of the under-barrier states within the disturbance region is
commonly small, so the uttermost contribution comes from the high-level
states. In the case of an analysis of the localization problem in a simplest
model of two states [40], the contribution of higher states is
phenomenologically taken into account by the asymmetry degree parameter.

Parameters | Wave functions | $\tilde{C}^{(0)}_{0}$ | $\tilde{C}^{(0)}_{1}$ | $\tilde{C}^{(0)}_{2}$ | $\tilde{C}^{(0)}_{3}$ | $\tilde{C}^{(0)}_{4}$
---|---|---|---|---|---|---
|  | $\tilde{C}^{(1)}_{0}$ | $\tilde{C}^{(1)}_{1}$ | $\tilde{C}^{(1)}_{2}$ | $\tilde{C}^{(1)}_{3}$ | $\tilde{C}^{(1)}_{4}$
$b$=1.86, $c$=0.25, $s$=0.6 | $\Psi^{(0)}$ | 0.8491 | 0.5283 | 0.0013 | 0.0013 | -0.0007
| $\Psi^{(1)}$ | 0.5286 | -0.849 | 0.0059 | 0.006 | 0.003
$b$=1.86, $c$=0.5, $s$=0.6 | $\Psi^{(0)}$ | 0.7558 | 0.6548 | 0.0013 | 0.0012 | 0.0005
| $\Psi^{(1)}$ | 0.6544 | -0.7555 | 0.0205 | 0.0197 | 0.0087
Table 2: The contribution of states of the non-disturbed Hamiltonian
$\tilde{H}$ into the lowest wave functions of $\hat{H}$.

As a criterium to verify the precision of our computations, we used a relation
that links the full potential (i.e., the potential of $\hat{H}$) to the
computed, within the diagonalization procedure, ground state wave functions
and ground state energies. It is known, ref. [73], that these values are
related to each other via

$\hat{U}(x)-\hat{E}_{0}=-\frac{\hbar^{2}}{4m}S(\hat{f}(x)),$ (20)

where $S(f(x))$ denotes the Schwarzian derivative

$S(f(x))=\left(\frac{f^{{}^{\prime\prime}}}{f^{\prime}}\right)-\frac{1}{2}\left(\frac{f^{{}^{\prime\prime}}}{f^{\prime}}\right)^{2},$
(21)

for the function

${\hat{f}(x)=-\int_{x}\,ds\,\frac{1}{\Psi_{(0)}^{2}(s)}}.$ (22)

I.e.,

${S(\hat{f}(x))=-2\left(\frac{d^{2}}{dx^{2}}\ln\Psi^{(0)}(x)+\left(\frac{d}{dx}\ln\Psi^{(0)}(x)\right)^{2}\right).}$
(23)

By use of eq. (20) with the r.h.s. of eq. (23), we find that the results of
Tables 1 and 2 are in agreement to the disturbance potential (18) with the
parameters, indicated inside the Tables.101010Note that a similar to eq. (20)
relation can also be found for excited levels of the complete Hamiltonian
$\hat{H}$, though it will require additional steps to isolate zeros in the
corresponding wave functions. Since the ground state function of the perturbed
Hamiltonian is formed by a finite set of the HO wave functions (see eq. (19)),
we conclude that our choice on the number of the basic wave functions (recall,
$N=7$ in our case) works fine. Another way to justify the chosen cut-off in
the series expansion (19) is to compare the results of the diagonalization for
higher values of the cut-off with that of Tables 1 and 2. One may verify that,
with higher values of $N$ (we checked it up to $N=17$), the energy spectrum
and the series expansion coefficients for the first ten levels are almost
matched with values of Tables 1 and 2; for levels higher than ten, the
spectrum and the wave functions are not visibly changed with adding new
levels.

a)

b)

c)

d)

Figure 6: The disturbance of a two-well symmetric potential in Fig.1a with
$V_{1}(x)$ of eq. (18) with $s=0.6$ and $b=1.86$. Other parameters are choosen
to be: a) $c=0.25$; b) $c=0.5$; c) $c=0.8$; d) $c=2.0$. The upper panel shows
the disturbance, to which the two-level approximation is applicable.

Thus, we have validated that localization of the under-barrier states is
realized on a small number of the basic wave functions upon the relevant
choice of the basis. In the case, when the (small) disturbance of the main
potential is located far from the local minima of the, by now, full potential
(like in Fig.6c), one needs to take into account more basic states in the
series expansion for the wave functions (akin to eq. (19)). Indeed, forming
the spectrum and the wave functions of the disturbed Hamiltonian is strongly
determined by the over-barrier states, hence we need to increase the number of
the basic wave functions. (Our computations shows that the full potential can
be effectively recovered on account of about twenty basic states.)

Localization of states takes place in the vicinity of the quasi-crossing of
energy levels of undisturbed Hamiltonian, and the localization degree depends
on the value of the potential disturbance. In the case of a three-well
potential, the ground state is localized in the central well, whereas the
first and the second excited levels are delocalized (cf. Fig.3). Taking into
account the disturbance (18) in the far right well of a symmetric potential at
a special choice of its parameters, one may reach the complete localization of
states. (See Fig.7.)

a)

b)

Figure 7: Localization of the first and the second excited states of the
symmetric three-well potential ($c=0.25$, $b=3$).

One may notice non-zero values of the wave functions outside of the main
peaks, that is a reflection of having the nodes in the corresponding wave
functions.

We end up this section with emphasizing that localization of initially
delocalized states in symmetric potentials is accompanied with essential
increasing the tunnel doublet value. When the perturbation, which breaks the
reflective symmetry, is turned on, the lion-share of the localized state wave
functions consists of the contribution of the over-barrier states.

##  5 Coherent control with an external driving force

Interesting possibilities in constructing a new type of quantum devices open
upon employing the control by external time-dependent fields on quantum-
mechanical processes with keeping the coherence. Theoretical and experimental
studies of quantum mechanical systems with external driving forces are very
popular and widespread in view of their importance for modeling q-bits, the
quantum telecommunication and transportation, developing new schemes of
encryption and decryption, creation of new quantum interferometers and other
tools for the highly precision measurement. Clearly, tunneling processes are a
significant part of this activity.

From a variety of the observed in the tunneling processes effects, it is worth
mentioning theoretical predictions of the coherent tunneling destruction (CTD)
[23, 25] in symmetric two-well potentials, and experimental observations of
this phenomenon [27, 28]. Importance of the CTD consists in the possibility to
control the tunneling dynamics of wave packets, up to their complete stop, by
changing parameters of an external time-dependent field. We will reveal
further on features of the tunneling in a system with general Hamiltonian (1),
whose potentials are of a various deformation degree, in the presence of a
periodic time-dependent perturbation.

The evolution of quantum systems, characterized by periodic in time
Hamiltonians $H(t)=H(t+T)$, is traditionally described in the Floquet wave
functions basis:

$\Psi_{\alpha}(x,t)=\sum_{\alpha}c_{\alpha}\exp\left(-{i\varepsilon_{\alpha}t}\right)\Phi_{\alpha}(x,t),\quad\Phi_{\alpha}(x,t)=\Phi_{\alpha}(x,t+T),$
(24)

where $c_{\alpha}$ is determined by initial conditions, while the quasi-
energies $\varepsilon_{\alpha}$ and the Floquet wave functions
$\Phi_{\alpha}(x,t)$ follow from the diagonalization procedure of the complete
Hamiltonian.111111Under the diagonalization we mean any of the methods to
determine the quasi-energies and wave functions. They, in particular, include
the Floquet matrix and matrix-continued-fraction methods (see, e.g., [29,
30]). The time periodicity of the Floquet wave functions makes possible to use
the following expansion over the Fourier modes,

$\Phi_{\alpha}(x,t)=\sum_{n=-\infty}^{n=\infty}c^{n}_{\alpha}(x)e^{in\omega
t},$ (25)

with the subsequent expansion of $c^{n}_{\alpha}(x)$ over the chosen complete
orthonormal basis $\\{\varphi_{k}(x)\\}_{k=1}^{\infty}$, so that

$\Phi_{\alpha}(x,t)=\sum_{k=1}^{\infty}\sum_{n=-\infty}^{n=\infty}c^{n}_{\alpha,k}\,\varphi_{k}(x)e^{in\omega
t}.$ (26)

The notable example of time-periodic Hamiltonians is given by a symmetric two-
well potential with an external time-periodic driving force [23, 29].
Following ref. [29], one finds the energy spectrum and the Floquet wave
functions from the diagonalization of the following Hamiltonian

$\hat{H}(\xi,p_{\xi},\tau)=\frac{p^{2}}{2}+\tilde{U}(\xi,\bar{E}_{-2},\bar{E}_{-1},\Lambda)+\xi
S\cos(w\tau+\varphi_{0}),\quad\tau=\omega_{0}t,$ (27)

where $w=\omega/\omega_{0}$ is the dimensionless ratio of the driving force
frequency $\omega$ to the internal frequency of the HO. In what follows, we
choose the phase $\varphi_{0}$ to be equal to zero.

In contrast to the approach of ref. [29], where the basis
$\\{\varphi_{k}(x)\\}_{k=1}^{\infty}$ consists of the wave functions of the
unperturbed harmonic oscillator, we use wave functions of an exactly-solvable
Hamiltonian with symmetric and asymmetric potentials (cf. eqs. (4)-(6)). The
employment of an exactly-solvable model allows us to consider cases with the
predefined deformation of the potential and the computed analytically spectrum
of the under-barrier states, as well as to crucially decrease the dimension of
a diagonalized matrix upon computing the energies and the Floquet wave
functions.121212For instance, the computations of ref. [31] were performed on
the basis set of 200 states of a harmonic oscillator with the optimized
frequency. We use 10 basic states of the unperturbed exactly-solvable
Hamiltonian instead. Recall that the value of the initial tunnel doublet does
not depend on the deformation degree within the considered approach. That, in
particular, means changing the tunnel doublet value is completely determined
by the external time-dependent driving force. States of the Hamiltonian
$H(\xi,p_{\xi})=p^{2}/2+\tilde{U}(\xi,\bar{E}_{-2},\bar{E}_{-1},\Lambda)$ are
suitable for basic functions of the Higgs type phenomenological potential as
soon as the minima locations and the depth of wells of symmetric potentials
will be reconciled by use of the parametric dependence of
$\tilde{U}(\xi,\bar{E}_{-2},\bar{E}_{-1},\Lambda)$. Following this way, one
may control the behavior of wave functions of the under-barrier states, which
mostly impact the tunneling dynamics. Otherwise, within the standard approach,
one needs to modify, in this or those manner, the HO wave functions to fit
them to the phenomenological potentials; it results in the unavoidable
extension of the number of basic states.

To compute the energies and the Floquet wave functions for Hamiltonian (27),
we use states of the exactly-solvable model with the potential
$\tilde{U}(\xi,\bar{E}_{-2},\bar{E}_{-1},\Lambda)$. The parameters of the
driving external force are chosen to be close to that of satisfying the CTD
condition within the two-level approximation (in the symmetric case) [24, 31,
25]:

$J_{0}\left(\frac{2Sr}{w}\right)=0,\qquad r=\langle 1|\xi|2\rangle.$ (28)

Here $J_{0}(x)$ is the zero-order Bessel function; $|1\rangle$ and $|2\rangle$
are the lowest states of the undriven system. Zeros of the Bessel function
correspond to the parameters, upon which the tunneling is completely
suppressed. The resulted Floquet energies for the exactly-solvable Hamiltonian
in an external periodic driving force with symmetric and asymmetric two-well
potentials are collected in Table 3.

Parameters |  $N$=0 |  $N$=1 |  $N$=2 |  $N$=3 |  $N$=4
---|---|---|---|---|---
$\Lambda=1$ |  |  |  |  |
$S$=0.65, $w$=0.9 | 0.39038 | 0.358714 | 0.302197 | -0.291837 | -0.189549
$\Lambda=0.5$ |  |  |  |  |
$S$=0.65, $w$=0.9 | 0.389543 | 0.358666 | 0.30244 | -0.291951 | -0.188954
Parameters |  $N$=5 |  $N$=6 |  $N$=7 |  $N$=8 |  $N$=9
$\Lambda=1$ |  |  |  |  |
$S$=0.65, $w$=0.9 | 0.181939 | 0.181259 | -0.119865 | 0.0367728 | 0.0299879
$\Lambda=0.5$ |  |  |  |  |
$S$=0.65, $w$=0.9 | 0.185352 | 0.177982 | -0.120192 | 0.0377385 | 0.0293748
Table 3: The Floquet energies for the symmetric ($\Lambda=1$) and asymmetric
($\Lambda=0.5$) two-well potentials: first ten levels.

We restricted the basis with 10 basic wave functions $\varphi_{k}(x)$ entering
eq. (26); extending the number of the involved basic states does not
essentially change the results. Note that the level “numbers” $N=0,\dots 9$ do
not sequentially numerate states from the ground to excited, that is common
for the Floquet states (see, for instance, Sec 3.1. of [29]). However, we can
always determine the tunnel doublet following two criteria: the difference
between the quasi-energies of states has to be small; the levels have to be
(quasi)crossed. Inspecting the quasi-energies of different levels on the plot
(see Fig.8 as an example), one may notice that the smallest difference is
between levels $N=5,6$ of Table 3, and they are (quasi)crossed (see Fig.9).
Therefore, just these levels form the tunnel doublet.

Figure 8: The quasi-energies of Table 3 for the asymmetric two-well potential.
The horizontal axis numerates the states; the vertical axis corresponds to the
quasi-energies values in $\omega_{0}$ units
($\bar{\varepsilon}=\varepsilon/\omega_{0}$). The closest quasi-energies
correspond to levels $N=5,6$ and $N=8,9$. However,
$(\bar{\varepsilon}_{5}-\bar{\varepsilon}_{6})<(\bar{\varepsilon}_{8}-\bar{\varepsilon}_{9})$.

a)

b)

Figure 9: The levels (quasi)-crossing diagram inside the fundamental domain
$\\{-w/2,w/2\\}$ (vertical axis; $w=0.9$): a) the symmetric case with
$\Lambda=1$; b) the asymmetric case with $\Lambda=0.5$. In both cases $S$
varies over zero to one (horizontal axis).

Now one may notice the essential change in the tunnel doublets from their
initial value $\triangle=2\cdot 10^{-2}$. In the symmetric case, with the
deformation parameter $\Lambda=1$, we get
$\triangle^{F}_{sym}=0.181939-0.181259\approx 6.8\cdot 10^{-4}$, while in the
asymmetric case, with $\Lambda=0.5$, we have
$\triangle^{F}_{asym}=0.185352-0.177982\approx 7.37\cdot 10^{-3}$. Also, the
parameters of the driving field we used are slightly different from that of
used in eq. (28): according to the CTD criterium, in the two-level
approximation, one exactly needs $w_{sym}\approx 0.802268$ in the symmetric
case and $w_{asym}\approx 0.762012$ in the asymmetric case; $w=0.9$ has been
chosen in our computations with taking into account the contribution of more
high states of the initial Hamiltonian. However, the proximity of the chosen
by us value of $w$ to the values of $w_{sym}$ and $w_{asym}$ justifies the
usage of the two-level approximation, at least in the quantitative analysis of
the tasks under consideration.

Introducing the so-called stroboscopic variables, $t=nT$, where
$T=2\pi/\omega$ is the period of the driving force oscillations, essentially
simplifies studies of quantum states dynamics. By use of such variables we can
compute the dynamics of a wave packet (24), expanded over the basis of the
Floquet states and energies computed before. As the initial state of eq. (24)
we choose the Gaussian wave packet (13) with different values of the
parameters $x_{0}$ (or $\xi_{0}$) and $R$. (Recall, we initially set the
packet in the right well.) The temporal dynamics of the packet and the
parameters of computations can be viewed in Fig.10.

a)

b)

Figure 10: Evolution of the Gaussian wave packet $\Phi(x,t)$ in the external
periodic driving force: a) for the symmetric two-well potential with the
initial tunnel doublet $\triangle=0.02$; $R=0.75$, $x_{0}=1.525$; b) for the
asymmetric potential with $\triangle=0.02$ and $\Lambda=0.5$; $R=0.75$,
$x_{0}=1.607$.

Looking at Fig.10, one may notice that the wave packet fraction in the left
well is negligible in both, symmetric and asymmetric, cases. In other words,
wave packets are mainly localized within the initial local minima. There is a
difference in localization for asymmetric potentials, since it is provided by
a particular localization from the unperturbed case and by the tunneling
suppression due to quasi-crossing of the lowest Floquet levels upon turning on
the periodic driving. In the symmetric case localization is provided by
crossing the Floquet levels of the periodically-driven Hamiltonian [24]. One
may identify a small fracture of the wave packet in the left part of Fig.10a,
related to the contribution of higher states.

##  6 Summary and Conclusions

Studies of the tunneling dynamics by use of exactly-solvable models with
multi-well potentials is one of examples of the so-called quantum-mechanical
engineering. Variations of the reviewed model parameters (value of energies of
added states, deformation parameters etc.) allows one to investigate features
of the tunneling processes in systems with different number of local minima,
and to control the change of their specific parameters (the minima depth, the
barriers width, the deformation degree). Recall that the asymptotic behavior
of the considered here potentials does not change upon altering the amount of
the local minima; models with the phenomenological type potentials do not
share such a hallmark. This fact might be crucial in studies of the Bose-
condensate dynamics of atoms in the Pauli-type traps with several local minima
in their central parts. Moreover, we have shown that changing the values of
energies of added states still does not go beyond two- or/and three-level
approximations, the corresponding parameters of which have been computed
within the approach of N=4 supersymmetric quantum mechanics (SQM).

Treating the models with or without reflection symmetry within the consistent
unified consideration may be regarded as a key advantage of the employed
technique. Viz., we have employed N=4 SQM to examine properties of the
tunneling processes in models with two- or three-well potentials, induced by a
partial localization of levels. The complete localization of levels is
unattainable within the followed approach, because it appears upon the
considerable enlarging the difference in energies of the tunnel doublet. The
latter can not be reached for isospectral Hamiltonians with symmetric or
asymmetric potentials. However, turning on a special, ref. [45, 46], breaking
the reflection symmetry, perturbation provides the complete localization of
the under-barrier states in the considered here examples of multi-well
potentials. As we have highlighted, the tunnel doublet gets significantly
changed for two-well potentials, when its value (on account of the breaking
symmetry term) gets increased in order (in compare to the initial quantity of
the tunnel doublet). We have also noted the specifics in localization of
levels in models with a three-well potential, where the tunnel doublet is
formed by the first and the second excited states. Localization of these
states is occurred in outer wells, while the ground state is placed in the
central well.

The performed studies of the coherent tunneling destruction by a periodically-
driven interaction for models with broken reflective symmetry have shown that
the criterium for this phenomenon to arise is in agreement with that of a
symmetric case. Wave functions and the Floquet spectra are obtained in our
approach upon the complete Hamiltonian diagonalization on account of 30
states, in contrast to the symmetric case of ref. [23], where to diagonalize
the Hamiltonian more than 100 states of the harmonic oscillator have been
used. This observation points out one of the repeatedly mentioned throughout
the paper advantages of the developed approach, viz., engaging a smart basis
of states, that essentially simplifies the computations.

Finally, let us conclude with the following point. Partial or complete
localization of states is an outcome of playing with free parameters of the
model. However, in another language, it may be considered as changing the
geometry of an effective internal manifold of quantum states by changing its
quantum metric [74, 75, 76, 77]. We believe that applying ideas of the quantum
metric approach in exactly-solvable models will give new prospects in studies
various properties of quantum systems, the tunneling dynamics as well.

## Acknowledgements

The authors are thankful to an anonymous reviewer for suggestions in improving
the early version of the paper. We also grateful to S.S. Zub for a technical
support.

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