# Evanescent Electron Wave Spin

Ju Gao<EMAIL_ADDRESS>Fang Shen  University of Illinois, Department of
Electrical and Computer Engineering, Urbana, 61801, USA

###### Abstract

Our study shows that an evanescent electron wave exists outside both finite
and infinite quantum wells, by solving exact solutions of the Dirac equation
in a cylindrical quantum well and maintaining wavefunction continuity at the
boundary. Furthermore, we demonstrate that the evanescent wave spins
concurrently with the wave inside the quantum well, by deriving analytical
expressions of the current density in the whole region. Our findings suggest
that it is possible to probe or eavesdrop on quantum spin information through
the evanescent wave spin without destroying the entire spin state. The wave
spin picture interprets spin as a global and deterministic property of the
electron wave that includes both the evanescent and confined wavefunctions.
This suggests that a quantum process or device based on the manipulation and
probing of the electron wave spin is deterministic in nature rather than
probabilistic.

###### pacs:

3.50, 32.80, 42.50

††preprint: APS/123-QED

##  I Electron wave spin

The electron spin has attracted significant attention especially in recent
years due to its potential applications in information sciences and
technologies, such as quantum computing [1],[2] and spintronics [3]. If future
computers rely on the precise manipulation of the electron spin, it is crucial
to develop a deeper understanding of this property beyond a simplistic
interpretation as a superluminal spinning particle.

In recent works [4],[5], it has been argued that the electron spin is a wave
property that can be fully described by current densities calculated using the
Dirac theorem. The authors demonstrate the existence of a stable circulating
current density for an electron confined in an infinite quantum well, which
exhibits a multi-vortex topology at excited states. The electron wave spin
exhibits geometric and topological characteristics when it interacts with an
electromagnetic field, leading to various effects, including fractional spin
and modified Zeeman splitting.

It is noted that the electron wave spin was investigated previously in the
infinite quantum well, where the wave function outside the well is typically
assumed to be zero [6]. Although this model is widely used that has helped
gain much insight into many quantum effects inside the quantum well, it
assumes the wavefunction outside the well to be exactly zero, which causes a
problem in maintaining continuity for both the wavefunction and its derivative
at the boundary in both the Schrödinger and Dirac theorems [7]. In reality,
all potentials are finite, and therefore, there exists a non-negligible
wavefunction outside the well, even when the eigenenergy of the electron is
below a finite or an infinite quantum well potential. The wavefunction outside
the well shall decay rapidly, thus is evanescent in analogy to the evanescent
waves observed in optics.

An intriguing question arises as to whether the evanescent electron wave also
spins. An intuitive answer to this question suggests that an evanescent wave
shall also spin due to the continuation of wavefunctions and the prevailing
algebraic structure of the Dirac equation. The existence of an evanescent wave
spin shall raise intriguing possibilities for probing the spin from outside
the well without collapsing the spin inside, analogous to the evanescent
optical wave sensing [8], [9], [10].

In this paper, we plan to demonstrate the evanescent wave spin by solving the
Dirac equation in a quantum well rigorously and obtaining analytical
expressions for the wavefunctions and current densities both inside and
outside the quantum well. We will demonstrate that the evanescent wave
diminishes at high potentials but remains within the skin depth region even
for the infinite quantum well due to the wavefunction continuity at the
boundary. Finally, we intend to discuss the implications of these findings on
quantum detection and quantum information technologies.

##  II Dirac electron in a finite cylindrical quantum well

We will derive exact solutions of the Dirac equation in a finite cylindrical
quantum well and obtain analytical expressions for the current density. The
eigenenergy values will be solved numerically through the boundary condition
equations.

The Dirac equation in the cylindrical coordinate is

$\frac{1}{c}\frac{\partial}{\partial
t}\psi(\pmb{r},t)=\\{-\pmb{\alpha}\cdot\pmb{\nabla}-i\frac{mc^{2}}{\hbar
c}\gamma^{0}-i\frac{1}{\hbar c}U(\pmb{r})\\}\psi(\pmb{r},t),$ (1)

where $(\rho,\phi,z)$ represent polar, azimuthal angle and z coordinate,
respectively. The operator

$\pmb{\alpha}\cdot\pmb{\nabla}=\alpha_{\rho}\frac{\partial}{\partial\rho}+\alpha_{\phi}\frac{1}{\rho}\frac{\partial}{\partial\phi}+\alpha_{z}\frac{\partial}{\partial
z}$ (2)

contains $\alpha-$matrix in cylindrical coordinate

$\displaystyle\alpha_{\rho}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0&0&0&e^{-i\phi}\\\
0&0&e^{i\phi}&0\\\ 0&e^{-i\phi}&0&0\\\ e^{i\phi}&0&0&0\\\ \end{array}\right);$
(7) $\displaystyle\alpha_{\phi}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0&0&0&-ie^{-i\phi}\\\
0&0&ie^{i\phi}&0\\\ 0&-ie^{-i\phi}&0&0\\\ ie^{i\phi}&0&0&0\\\
\end{array}\right);$ (12) $\displaystyle\alpha_{z}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&0&0&-1\\\ 1&0&0&0\\\
0&-1&0&0\\\ \end{array}\right),$ (17)

with the following properties

$\displaystyle\sigma_{\rho}^{2}=\sigma_{\phi}^{2}=\sigma_{z}^{2}=1,$
$\displaystyle\sigma_{\rho}\sigma_{\phi}=i\alpha_{z},$
$\displaystyle\sigma_{\rho}\sigma_{\phi}+\sigma_{\phi}\sigma_{\rho}=0.$

We now let the potential

$U(\pmb{r})=U(\rho)=\Bigl{\\{}\begin{array}[]{cc}0,&\rho<R\\\ U,&\rho>R\\\
\end{array}$ (19)

represent a finite cylindrical quantum well of potential $U$ and radius $R$.
The wavefunction in the quantum well can be expressed by the separation of
variables

$\psi(\pmb{r},t)=e^{-i\mathcal{E}t/\hbar}e^{iP_{z}z/\hbar}\tilde{\psi}(\phi,\rho),$
(20)

where the momentum along z-direction is set $P_{z}=0$ for our discussion and
$\mathcal{E}$ is the eigenenergy to be determined by the boundary conditions.

Plugging Eq. 20 into Eq. 1 to obtain the equation

$\\{-i\frac{\mathcal{E}}{\hbar
c}+\alpha_{\rho}\frac{\partial}{\partial\rho}+\alpha_{\phi}\frac{1}{\rho}\frac{\partial}{\partial\phi}+i\frac{mc^{2}}{\hbar
c}\gamma^{0}+i\frac{1}{\hbar c}U(\rho)\\}\tilde{\psi}(\phi,\rho)=0.$ (21)

$\tilde{\psi}(\phi,\rho)$ is a four-spinor that can be written as

$\tilde{\psi}(\phi,\rho)=\left(\begin{array}[]{c}\mu_{A}(\phi,\rho)\\\
\mu_{B}(\phi,\rho)\\\ \end{array}\right),$ (22)

where $\mu_{A}(\phi,\rho)$ and $\mu_{B}(\phi,\rho)$ are two component spinor
wavefunctions known as the large and small components of the Dirac
wavefunctions that follow the equations

$\displaystyle-i\frac{\mathcal{E}-U(\rho)-mc^{2}}{\hbar c}\mu_{A}(\phi,\rho)=$
$\displaystyle\left(\begin{array}[]{cc}0&e^{-i\phi}\frac{\partial}{\partial\rho}-ie^{-i\phi}\frac{1}{\rho}\frac{\partial}{\partial\phi}\\\
e^{i\phi}\frac{\partial}{\partial\rho}+ie^{i\phi}\frac{1}{\rho}\frac{\partial}{\partial\phi}&0\\\
\end{array}\right)\mu_{B}(\phi,\rho);$ (25)
$\displaystyle-i\frac{\mathcal{E}-U(\rho)+mc^{2}}{\hbar c}\mu_{B}(\phi,\rho)=$
$\displaystyle\left(\begin{array}[]{cc}0&e^{-i\phi}\frac{\partial}{\partial\rho}-ie^{-i\phi}\frac{1}{\rho}\frac{\partial}{\partial\phi}\\\
e^{i\phi}\frac{\partial}{\partial\rho}+ie^{i\phi}\frac{1}{\rho}\frac{\partial}{\partial\phi}&0\\\
\end{array}\right)\mu_{A}(\phi,\rho).$ (28)

The above equations are combined to give the equation for
$\mu_{A}(\phi,\rho)$,

$\left(\frac{\partial^{2}}{\partial\rho^{2}}+\frac{1}{\rho}\frac{\partial}{\partial\rho}+\frac{1}{\rho^{2}}\frac{\partial^{2}}{\partial\phi^{2}}\right)\mu_{A}(\phi,\rho)=\Bigl{\\{}\begin{array}[]{cc}-\zeta^{2}\mu_{A}(\rho),&\rho<R;\\\
\\\ \xi^{2}\mu_{A}(\rho),&\rho>R,\end{array}$ (30)

where $\zeta$ and $\xi$ are wave numbers inside and outside the quantum well,
respectively,

$\displaystyle\zeta$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\mathcal{E}^{2}-m^{2}c^{4}}{\hbar^{2}c^{2}}},$
$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\sqrt{\frac{m^{2}c^{4}-(\mathcal{E}-U)^{2}}{\hbar^{2}c^{2}}},$
(31)

and are both real numbers, since the quantum well potential falls in the range
of

$\mathcal{E}-mc^{2}<U<mc^{2}.$ (32)

We now separate the variables of $\mu_{A}(\phi,\rho)$ for the spin-up
electron,

$\mu_{A}(\phi,\rho)=e^{il\phi}\mu_{A}(\rho)\left(\begin{array}[]{c}1\\\ 0\\\
\end{array}\right),$ (33)

where $l$ is the azimuthal quantum number for the angular wavefunction
$e^{il\phi}$.

The radial wavefunction $\mu_{A}(\rho)$ follows the equation by plugging Eq.
33 into Eq. 30,

$\left(\frac{\partial^{2}}{\partial\rho^{2}}+\frac{1}{\rho}\frac{\partial}{\partial\rho}-\frac{l^{2}}{\rho^{2}}\right)\mu_{A}(\rho)=\Bigl{\\{}\begin{array}[]{cc}-\zeta^{2}\mu_{A}(\rho),&\rho<R;\\\
\\\ \xi^{2}\mu_{A}(\rho),&\rho>R.\end{array}$ (34)

$\mu_{A}(\rho)$ can be readily solved from Eq. 34 and $\mu_{B}(\phi,\rho)$ can
be subsequently obtained from Eqs. II. Finally, the four component spinor
wavefunction $\tilde{\psi}(\phi,\rho)$ in Eq. 22 for the spin up electron is
obtained

$\tilde{\psi}(\phi,\rho)=\bBigg@{5.5}\\{\begin{array}[]{ll}e^{il\phi}\left(\begin{array}[]{c}J_{l}(\zeta\rho)\\\
0\\\ 0\\\ ie^{i\phi}\frac{\hbar
c}{\mathcal{E}+mc^{2}}\\{\frac{1}{2}\zeta\left[J_{l-1}(\zeta\rho)-J_{l+1}(\zeta\rho)\right]-\frac{l}{\rho}J_{l}(\zeta\rho)\\}\\\
\end{array}\right),&\rho\leq R;\\\ \\\
e^{il\phi}\left(\begin{array}[]{c}\kappa K_{l}(\xi\rho)\\\ 0\\\ 0\\\
ie^{i\phi}\kappa\frac{\hbar
c}{\mathcal{E}-U+mc^{2}}\\{\frac{1}{2}\xi\left[-K_{l-1}(\xi\rho)-K_{l+1}(\xi\rho)\right]-\frac{l}{\rho}K_{l}(\xi\rho)\\}\\\
\end{array}\right),&\rho>R,\\\ \end{array}$ (35)

where $J_{l}$ and $K_{l}$ are the Bessel function and modified Bessel
functions of order $l$, respectively. The constant $\kappa$ measures the
relative magnitude between the wavefunctions inside and outside the quantum
well.

We now apply the boundary condition of wavefunction continuation at $\rho=R$
to obtain

$\displaystyle\kappa K_{l}(\xi R)=J_{l}(\zeta R)$
$\displaystyle\kappa\\{\frac{1}{2}\xi\left[-K_{l-1}(\xi R)-K_{l+1}(\xi
R)\right]-\frac{l}{R}K_{l}(\xi R)\\}=$
$\displaystyle\frac{\mathcal{E}-U+mc^{2}}{\mathcal{E}+mc^{2}}\\{\frac{1}{2}\zeta\left[J_{l-1}(\zeta
R)-J_{l+1}(\zeta R)\right]-\frac{l}{R}J_{l}(\zeta R)\\},$

from which the eigenenergies $\mathcal{E}_{ln}$ and constant $\kappa$ can be
solved numerically. Here $l$ and $n$ denote the azimuthal and radial quantum
numbers, respectively. Numerical calculation can then be carried out for
studying the wave properties in all regions.

##  III Evanescent electron wave

To simplify the discussion on the evanescent electron wave, we choose the
lowest azimuthal quantum number $l=0$ for the wavefunction

$\tilde{\psi}(\phi,\rho)=\bBigg@{5.5}\\{\begin{array}[]{ll}\left(\begin{array}[]{c}J_{0}(\zeta\rho)\\\
0\\\ 0\\\ -ie^{i\phi}\frac{\hbar c}{\mathcal{E}+mc^{2}}\frac{1}{2}\zeta
J_{1}(\zeta\rho)\\\ \end{array}\right),&\rho\leq R;\\\ \\\
\left(\begin{array}[]{c}\kappa K_{0}(\xi\rho)\\\ 0\\\ 0\\\ -i\kappa
e^{i\phi}\frac{\hbar c}{\mathcal{E}-U+mc^{2}}\frac{1}{2}\xi K_{1}(\xi\rho)\\\
\end{array}\right),&\rho>R.\\\ \end{array}$ (37)

The boundary conditions become

$\displaystyle\kappa K_{0}(\xi R)$ $\displaystyle=$ $\displaystyle J_{0}(\zeta
R)$ $\displaystyle\kappa\xi K_{1}(\xi R)$ $\displaystyle=$
$\displaystyle\frac{\mathcal{E}-U+mc^{2}}{\mathcal{E}+mc^{2}}\zeta J_{1}(\zeta
R),$

which can be combined to give the equation

$\xi J_{0}(\zeta R)K_{1}(\xi
R)=\frac{\mathcal{E}-U+mc^{2}}{\mathcal{E}+mc^{2}}\zeta K_{0}(\xi
R)J_{1}(\zeta R),$ (39)

from which the eigenenergies $\mathcal{E}$ and constant $\kappa$ can be solved
numerically.

We now conduct the numerical study by choosing a quantum well of radius
$R=10~{}\text{n}m$ and a series of potentials $U=0.01,0.1,1,10~{}\text{e}V$.
For each potential, multiple eigenenergies can be found by solving the Eq. 39
numerically. As an example, for $U=0.01~{}\text{e}V$, only two eigenenergies
are found within the quantum well, $\mathcal{E}_{01}-mc^{2}=1.53(meV)$ and
$\mathcal{E}_{02}-mc^{2}=7.63(meV)$ that correspond to the ground state and
first excited state, respectively. Fig. 1 shows the two eigenenergy solution
as the inception points of functions $\xi J_{0}(\zeta R)K_{1}(\xi R)$ and
$\frac{\mathcal{E}-U+mc^{2}}{\mathcal{E}+mc^{2}}\zeta K_{0}(\xi R)J_{1}(\zeta
R)$ from Eq. 39.

Figure 1: Numerical solution of the eigenenergies for the electron in a
quantum well of $U=0.01~{}\text{e}V$ and $R=10~{}\text{nm}$.
The blue and orange curves representing $\xi J_{0}(\zeta R)K_{1}(\xi R)$ and
$\frac{\mathcal{E}-U+mc^{2}}{\mathcal{E}+mc^{2}}\zeta K_{0}(\xi R)J_{1}(\zeta
R)$ from Eq. 39 respectively intercept at eigenenergies
$\mathcal{E}_{01}-mc^{2}=1.53(meV)$ and $7.63(meV)$ for the ground and excited
state respectively.

We now calculate the ground state eigenenergies $\mathcal{E}_{01}-mc^{2}$ and
$\kappa_{01}$ for all potentials, followed by the calculation of wave numbers
$\zeta_{01}$ and $\xi_{01}$ with the help of Eqs. II. The results are listed
in Table 1.

Table 1: Eigenenergy $\mathcal{E}_{01}$ and relative constant $\kappa_{01}$ $U$ (eV) | $\mathcal{E}_{01}-mc^{2}(meV)$ | $\kappa_{01}$ | $\zeta_{01}\text{(}m^{-1})$ | $\xi_{01}\text{(}m^{-1})$
---|---|---|---|---
$0.01$ | $1.53$ | $44.1$ | $2.00\times 10^{8}$ | $4.71\times 10^{8}$
$0.10$ | $1.95$ | $2.23\times 10^{6}$ | $2.26\times 10^{8}$ | $1.60\times 10^{9}$
$1.00$ | $2.12$ | $2.32\times 10^{21}$ | $2.36\times 10^{8}$ | $5.12\times 10^{9}$
$10.0$ | $2.18$ | $1.75\times 10^{69}$ | $2.39\times 10^{8}$ | $1.62\times 10^{10}$

The numerical calculation shows that the electron wave tunnels out of the
quantum well and behaves evanescent due to the characteristics of the modified
Bessel functions. Fig. 2 plots the wavefunctions inside and outside the well
for the ground state of $U=0.01~{}\text{e}V$. It is shown that substantial
evanescent wave tunnels out of the quantum well, but satisfies wavefunction
continuity at the boundary.

Figure 2: Wavefunction plot for the Dirac electron in a quantum well of
$U=0.01~{}\text{eV}$ and $R=10~{}\text{nm}$.
The large component $\mu_{A}(\phi,\rho)$ (left) and small component
$\mu_{B}(\phi,\rho)$ (right) wavefunctions demonstrate that substantial
evanescent wave tunnels out of the quantum well, but satisfies wavefunction
continuity at the boundary.

Table 1 shows that at higher potential, for example $U=10~{}\text{e}V$, the
wave number $\zeta_{01}=2.39\times 10^{8}$ already approaches the wave number
for the infinite quantum well $\zeta_{01}^{\inf}=2.40\times 10^{8}$, where
zero wavefunction is assumed outside the well

$J_{0}(\zeta_{01}^{\inf}R)=0.$ (40)

Therefore, the wavefunctions $\mu_{A}(\phi,\rho)$ and $\mu_{B}(\phi,\rho)$
inside the quantum well are nearly the same as the wavefunctions in the
infinite quantum well, as illustrated by Fig. 3. However, contrary to the
conventional assumption, the wavefunctions outside the quantum well are
diminished but not exactly zero, especially at the boundary for
$\mu_{B}(\phi,\rho)$, so that the wavefunction continuity is always satisfied.
The wavefunctions remain significant within a narrow region near the boundary
known as the skin depth, but decay rapidly. The skin depth becomes narrower as
the potential becomes higher, as is the case for an optical field confined in
a waveguide [9].

Figure 3: Wavefunction plot for the Dirac electron in a quantum well of
$U=10~{}\text{eV}$ and $R=10~{}\text{nm}$.
The large component $\mu_{A}(\phi,\rho)$ (left) and small component
$\mu_{B}(\phi,\rho)$ (right) wavefunctions (blue) inside the quantum well are
nearly identical to the wavefunctions (orange) inside an infinite quantum
well. The wavefunctions outside the quantum well are diminished but not
exactly zero, especially at the boundary for $\mu_{B}(\phi,\rho)$, so that the
wavefunction continuity is always satisfied.

The above analysis validates the infinite quantum well model to account for
the quantum behavior of the electron inside the well. However, it also reveals
that the evanescent wave diminishes but never vanishes to mathematical zero,
circumventing the discontinuity problem at the boundary of the commonly-used
infinite quantum well model.

##  IV Evanescent electron wave spin

We now investigate the spin nature of the evanescent electron wave that
resides within the skin depth region outside the quantum well. Analytical
expression of the current density is obtained by using the wavefunctions in
Eq. 37,

$\displaystyle j_{\rho}$ $\displaystyle=$ $\displaystyle-
ec\psi^{\dagger}\alpha_{\rho}\psi=0,\textrm{e}verywhere;$ $\displaystyle
j_{\phi}$ $\displaystyle=$ $\displaystyle-
ec\psi^{\dagger}\alpha_{\phi}\psi=\biggl{\\{}\begin{array}[]{ll}-\frac{e\hbar
c^{2}}{\mathcal{E}+\text{m}c^{2}}\zeta
J_{0}(\zeta\rho)J_{1}(\zeta\rho),&\rho\leq R\\\ \\\ -\kappa^{2}\frac{e\hbar
c^{2}}{\mathcal{E}-U+\text{m}c^{2}}\xi
K_{0}(\xi\rho)K_{1}(\xi\rho),&\rho>R,\\\ \end{array}$ (44)

where $-e=-1.602\times 10^{-19}\text{C}$ to represent the electron charge.

Eq. IV demonstrates that stable circulating current density exists both inside
and outside the quantum well, as evidenced by the non-zero component
$j_{\phi}$ in all regions and zero component $j_{\rho}$ everywhere. The
evanescent electron wave is shown to spin concurrently with the electron wave
inside the quantum well, as illustrated by the vector plot of Fig. 4. The
current density continuity is observed at the boundary as a result of the
wavefunction continuity at the boundary in Eqs. 35 and 37, which also ensure
the charge density continuity.

The above analysis highlights the wave spin picture, in which the spin is a
wave property encoded in the entire Dirac field. In other words, the
evanescent wave is an integral part of the entire electron wave that possesses
the wave spin property.

Figure 4: Vector plot of the current density for a spin-up ground state
electron in a quantum well of $U=0.01~{}\text{e}V$ and $R=10~{}\text{nm}$. The
evanescent wave spins concurrently with the wave inside the quantum well.

##  V Evanescent electron wave spin sensing

The discussion above raises questions about the security of spin quantum
information and the possibility of novel evanescent electron wave spin
sensing, which is analogous to evanescent optical wave sensing [8], [9], [10]
that has already been employed in real applications. The evanescent optical
wave is a part of an eigen electromagnetic wave outside an optic waveguide or
fiber that can interact with matters around the waveguide. The main optic wave
confined inside the waveguide is perturbed but largely maintained. Similarly,
the evanescent electron wave is a part of an eigen electron wave outside a
quantum well that can interact with an electromagnetic wave around the quantum
well. The two schemes share the same fundamental matter-field interaction:
$\pmb{j}(r)\cdot\pmb{A}(r)$, where $\pmb{j}(r)$ and $\pmb{A}(r)$ represent the
electron wave and the electromagnetic wave, respectively. The interaction is
usually small due to the nature of evanescent waves; therefore, the main wave
inside the confinement is only perturbed but not destroyed. Thus, evanescent
wave sensing offers a unique detection scheme as a partial wave interaction
without destroying the main wave inside the confinement.

However, the evanescent electron wave contains the spin property as described
by the current density $\pmb{j}(r)$ in Eq. IV. When it interacts with the
electromagnetic field $\pmb{A}(r)$, only partial spin participates in the
process that could result in the fractional spin effects as discussed in the
previous study [5]. Such partial spin concept conflicts with the particle spin
picture, where a single particle electron possesses a unit spin and can only
manifest the full spin effect during interaction. In the particle electron
spin picture, the electron carrying the full spin tunnels out of the quantum
well with a probability density given by the square of the wavefunction
$\psi^{\dagger}\psi$. If the spin is detected outside the quantum well, the
spin information inside is destroyed since the electron that carries the full
spin no longer exists within the quantum well.

The conflicting pictures are illustrated in Fig. 5, which comprises two
figures for the electron in the same quantum well of $R=10~{}\text{nm}$ and
$U=0.01~{}\text{e}V$.

Figure 5:  Conflicting views on the electron spin and tunnelling.
Upper figure illustrates the wave spin view that partial wave spin tunnels out
of the quantum well with full certainty. The figure shows the three-dimension
distribution of the current density that spins as a whole in all regions for
the spin-up electron in a finite quantum well of $R=10~{}\text{nm}$ and
$U=0.01~{}\text{e}V$.
Lower figure illustrates the particle spin view that full particle spin
tunnels out of the quantum well with partial certainty. The figure shows the
probability density $\psi^{\dagger}\psi$ of the particle electron of spin-up
(represented by the ball and arrow) in the same quantum well.

The upper figure illustrates the wave spin picture by showing the spinning
current density in all regions. The current density resembles the current
density inside an infinite square well discussed previously [4]. Here, partial
wave spin exists outside the quantum well but with full certainty.

The lower figure depicts the particle spin picture by showing the probability
density and particle electron spin. Here, full particle spin tunnels out of
the quantum well but with partial certainty.

The conflicting views reflect different interpretations of quantum mechanics,
but resolving them has significant implications for emerging technology
development, such as quantum computing. Quantum computing is generally
regarded as a probabilistic process at the fundamental level due to the
probabilistic interpretation of the wavefunction that describes the behavior
of subatomic particles. By adopting the particle spin picture, which
interprets spin as a local property pertaining to the particle electron, a
spin-based quantum computer should also be considered as a probabilistic
machine.

However, the wavefunction that is supposed to provide the probability density
$\psi^{\dagger}\psi$ is deterministic itself since it is a vector in the
Hilbert space. Consequently, the charge density $e\psi^{\dagger}\psi$ and
current density $ec\psi^{\dagger}\alpha\psi$ are deterministic observables. We
have shown that the electron spin is a wave property that can be fully
described by the current density. Therefore, the electron wave spin is also a
deterministic property. It is our view that a quantum process or device based
on the manipulation and probing of the electron wave spin is deterministic in
nature rather than probabilistic.

##  VI Conclusions

  1. 1.

Our study shows that an evanescent electron wave exists outside both finite
and infinite quantum wells, by solving exact solutions of the Dirac equation
in a cylindrical quantum well and maintaining wavefunction continuity at the
boundary.

  2. 2.

Furthermore, we demonstrate that the evanescent wave spins concurrently with
the wave inside the quantum well, by deriving analytical expressions of the
current density in the whole region. Our findings suggest that it is possible
to probe or eavesdrop on quantum spin information through the evanescent wave
spin without destroying the entire spin state.

  3. 3.

The wave spin picture interprets spin as a global and deterministic property
of the electron wave that includes both the evanescent and confined
wavefunctions. This suggests that a quantum process or device based on the
manipulation and probing of the electron wave spin is deterministic in nature
rather than probabilistic.

##  VII Acknowledgement

The authors would like to express their gratitude to Sumit Ghosh for the
insightful discussion on the boundary condition issue of an infinite quantum
well.

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# Convergent dynamics of optimal nonlinear damping control

Michael Ruderman<EMAIL_ADDRESS>University of Agder, 4604-Norway

###### Abstract

Following Demidovich’s concept and definition of convergent systems, we
analyze the optimal nonlinear damping control, recently proposed [1] for the
second-order systems. Targeting the problem of output regulation,
correspondingly tracking of $\mathcal{C}^{1}$-trajectories, it is shown that
all solutions of the control system are globally uniformly asymptotically
stable. The existence of the unique limit solution in the origin of the
control error and its time derivative coordinates are shown in the sense of
Demidovich’s convergent dynamics. Explanative numerical examples are also
provided along with analysis.

###### keywords:

Convergent dynamics , optimal nonlinear damping , tracking control ,
convergence analysis , stability analysis

##  1 Introduction

Unlike the linear feedback systems, the nonlinear controllers and plants
require, mostly, some non-universal methods and tools for analysis and design.
Depending on the complexity of underlying nonlinear dynamics, the local or
global approach, and (not least) the control specification, e.g. whether a
set-point or output regulation [2] problem is of interest, the stability and
convergence analysis can require a more specific approach than that based on
the standard Lyapunov stability theory [3, 4]. For analyzing the behavior of
nonlinear control systems subject to external inputs, such as reference or
disturbance signals, one needs to prove the existence and global asymptotic
stability of a solution, along which the output regulation is ensured. The so-
called incremental stability [5] and contraction analysis [6], or more
generally contraction theory, see e.g. [7], provide the means to show that all
system trajectories converge uniformly to a unique solution for which the
output regulation error is zero.

Being motivated by the Demidovich’s [8] concept and definition of the
convergent systems, reviewed in [9], this note analyzes and develops the
output regulation properties of the optimal nonlinear damping control [1].

##  2 Preliminaries

###  2.1 Convergent dynamics

In this section, we briefly recall the main definition and properties of a
convergent system dynamics, according to Demidovich [8], while we will closely
follow the developments and notations provided in [9].

For a large class of $n$-dimensional nonlinear systems

$\dot{x}=f(x,t),$ (1)

where the state vector $x\in\mathbb{R}^{n}$, with $2\leq n<\infty$, is
continuous in time $t$, and $f(\cdot)$ is the vector field which is
differentiable in $x$, the following notion of _convergent systems_ can be
given according to [8].

######  Definition 1.

The system (1) is said to be convergent if for all initial conditions
$t_{0}\in\mathbb{R}$, $\bar{x}_{0}\in\mathbb{R}^{n}$ there exists a solution
$\bar{x}(t)=x(t,t_{0},\bar{x}_{0})$ which satisfies:

  1. (i)

$\bar{x}(t)$ is well-defined and bounded for all $t\in(-\infty,\infty)$;

  2. (ii)

$\bar{x}(t)$ is globally asymptotically stable.

Such solution $\bar{x}(t)$ is called a _limit solution_ , to which all other
solutions of the system (1) converge as $t\rightarrow\infty$. In other words,
all solutions of a convergent system ’forget’ their initial conditions after
some transient time, which is depending on the exogenous values like reference
or disturbance signals, and converge asymptotically to $\bar{x}(t)$.

######  Remark 1.

If a globally asymptotically stable limit solution exists, it may be non-
unique. Yet if $\bar{x}(t)$ is the single solution defined and bounded for all
$t\in(-\infty,\infty)$, then the system (1) is said to be _uniformly
convergent_.

The unform convergence requires the system (1) to have an unique limit
solution $\bar{x}(t)$, like in case of linear systems. Otherwise, a non-
uniformly convergent system might have also another globally asymptotically
stable solutions $\tilde{x}(t)$, bounded for all $t\in(-\infty,\infty)$, so
that for any such pair of solutions it is valid
$\bigl{\|}\bar{x}(t)-\tilde{x}(t)\bigr{\|}\rightarrow 0$ as
$t\rightarrow\infty$.

######  Remark 2.

The system (1) is, moreover, _exponentially convergent_ if it is uniformly
convergent and the limit solution $\bar{x}(t)$ is globally exponentially
stable.

For further details on uniform, asymptotic, and exponential properties of
stability of the system solutions we refer to seminal literature, e.g. [3, 4].
The existence and uniqueness of a limit solution of system (1) has an
essential application to the output regulation problems [2, 10]. Here, for a
given reference signal of the closed-loop system, one can seek for
demonstrating the control system (1) is convergent, i.e. has an asymptotically
stable limit solution along which the regulated output control error is zero.
The property of a system to be convergent follows from the sufficient
condition, given by Demidovich [8], which is formulated in the following
theorem [9]:

######  Theorem 1.

Consider the system (1). Suppose, for some positive definite matrix
$P=P^{T}>0$ the matrix

$J(x,t):=\frac{1}{2}\Bigl{(}P\,\frac{\partial f}{\partial
x}(x,t)+\Bigl{[}\frac{\partial f}{\partial x}(x,t)\Bigr{]}^{T}P\Bigr{)}$ (2)

is negative definite uniformly in $(x,t)\in\mathbb{R}^{n}\times\mathbb{R}$ and
$|f(0,t)|\leq\mathrm{const}<+\infty$ for all $t\in\mathbb{R}$. Then the system
(1) is convergent.

The proof of the Theorem 1 can be found in [9]. Note that a particular case
$f(0,t)\equiv 0$ of the Theorem 1 implies the well celebrated Krasovskii
stability theorem [11]. The uniform negative definiteness of (2) implies a
vanishing difference between any two solutions $x_{i}(t)$ and $x_{ii}(t)$ of
the system (1), while for an exponentially convergent system (cf. Remark 2)
that means

$\bigl{|}x_{i}(t)-x_{ii}(t)\bigr{|}<\alpha\exp\bigl{(}-\beta(t-t_{0})\bigr{)}\bigl{|}x_{i}(t_{0})-x_{ii}(t_{0})\bigr{|}$
(3)

for all $t>t_{0}$, where $\alpha,\beta>0$ have the same values for all pairs
of the solution $\bigl{[}x_{i}(t)$, $x_{ii}(t)\bigr{]}$.

###  2.2 Optimal nonlinear damping control

In the following, we will summarize the optimal nonlinear damping control,
insofar as necessary for the main results presented in Section 3, while for
that recently introduced control and its properties we refer to [1].

Figure 1: Phase portrait of the control system (4), (5).

The second-order closed-loop control system with an optimal nonlinear damping
is written as

$\displaystyle\dot{x}_{1}$ $\displaystyle=$ $\displaystyle x_{2},$ (4)
$\displaystyle\dot{x}_{2}$ $\displaystyle=$ $\displaystyle-
kx_{1}-x_{2}^{2}|x_{1}|^{-1}\mathrm{sign}(x_{2}),$ (5)

where $k>0$ is the single arbitrary design parameter. The system (4), (5),
where $x_{1}$ is the output of interest, is globally asymptotically stable and
converges to the unique equilibrium in the origin: (i) along the attractor

$x_{2}+\sqrt{k}x_{1}=0$ (6)

of trajectories in vicinity of the origin, and (ii) without crossing the
$x_{2}$-axis, see Figure 1. Note that the (ii)-nd property prevents
singularity (owing to $x_{1}=0$) in the solutions of (4), (5), and that for
all initial conditions $x_{1}(t_{0})\neq 0\>\wedge\>x_{2}(t_{0})\in\mathbb{R}$
and all trajectories outside the origin i.e. $\|x_{1},x_{2}\|(t)\neq 0$. The
control system (4), (5) assumes an unperturbed system dynamics, so that the
robustness against external perturbations is the subject of future research.
Other, here relevant remarks follow.

######  Remark 3.

The output convergence of the control system (4), (5) is quadratic on the
logarithmic scale $\log|x_{1}|$. Hence, the control constitutes a fast
alternative to the standard proportional-derivative controller which, with the
same proportional gain factor $k$, converges only linearly on the logarithmic
scale $\log|x_{1}|$, cf. [1, Fig. 5].

It is also worth noting that the nonlinear damping law is completely
independent of the proportional feedback gain $k$. The latter scales solely
the state trajectories in the $(t,x_{2})$-coordinates, see examples depicted
in Figure 2.

Figure 2: State trajectories $x_{1}(t)$ in (a) and $x_{2}(t)$ in (b) for the
varying values of the control gain $k=[10,\,100,\,1000]$ of (4), (5).

######  Remark 4.

The closed-loop control system (4), (5) allows, in addition, for a bounded
control action $|\dot{x}_{2}|<S$, with $S=\mathrm{const}>0$ to be the
controller saturation (see [1]), which is affecting yet neither the stability
nor convergence performance of the state trajectories.

Although a saturated control of (4), (5) enables bypassing singularity in the
solutions of a set-point problem, the output regulation problem requires us to
allow for $x_{1}=0\>\wedge\>x_{2}\neq 0$ at all times $t_{0}\leq
t\rightarrow\infty$. For ensuring non-singularity of the solutions in the
entire state-space $(x_{1},x_{2})\in\mathbb{R}^{2}$, we will next make a
necessary regularization of the nonlinear damping term in (5). With keeping
this in mind, we are now in the position to formulate the main results of this
work.

##  3 Main results

For output tracking of a reference trajectory $r(t)\in\mathcal{C}^{1}$, we
introduce the error state $e_{1}=x_{1}-r$. Its time derivative is
$e_{2}=x_{2}-\dot{r}$, respectively. Note that for output tracking of
$\mathcal{C}^{1}$-trajectories, one can assume $\ddot{r}(t)=0$ for $t>\tau$,
while $t\leq\tau$ characterizes some transient phase where
$\dot{r}\neq\mathrm{const}$. In the sense of a motion control, for instance,
all $t\leq\tau$ will correspond to the transient acceleration or deceleration
phases of a reference trajectory. If a reference trajectory $r(t)$ contains
multiple, but finite in time, transient phases with $\ddot{r}(t)\neq 0$, these
will act as temporary perturbations upon which the convergent dynamics of the
control error, i.e. $\|e_{1},e_{2}\|\rightarrow 0$, must be guaranteed.

With the introduced above states of the control error and the steady-state
reference (i.e. $\ddot{r}=0$), the closed-loop control system (4), (5) can be
rewritten as

$\displaystyle\dot{e}_{1}$ $\displaystyle=$ $\displaystyle e_{2},$ (7)
$\displaystyle\dot{e}_{2}$ $\displaystyle=$ $\displaystyle-
ke_{1}-\frac{|e_{2}|\,e_{2}}{|e_{1}|+\mu}.$ (8)

Note that the newly introduced regularization term $0<\mu\ll k$ does not act
as an additional design parameter, yet it prevents singularity in the
solutions of the system (4), (5), cf. Section 2.2. Evaluating the Jacobian of
$f(x,t)$, with $x=[e_{1},\,e_{2}]^{T}$ cf. (7), (8) and (1), one obtains

$\displaystyle\frac{\partial f}{\partial x}=$ (9)
$\left[\begin{array}[]{cc}0&1\\\\[5.69054pt]
-k+|e_{2}|\,e_{2}\,\mathrm{sign}(e_{1})/\bigl{(}|e_{1}|+\mu\bigr{)}^{2}&-2|e_{2}|/\bigl{(}|e_{1}|+\mu\bigr{)}\\\
\end{array}\right].$

Then, suggesting the positive definite matrix

$P=\frac{1}{2}\,\left[\begin{array}[]{cc}k&0\\\ 0&1\\\ \end{array}\right],$
(10)

one can show that the matrix $J(x,t)$, which is the solution of (2), is
negative definite and, correspondingly, the Theorem 1 holds. For proving it,
we substitute (9) and (10) into (2) and evaluate the matrix definiteness as

$x^{T}J(x,t)\,x=-\frac{3}{4}\,\frac{|e_{2}|\,e_{2}^{2}\,\bigl{(}|e_{1}|+2\mu\bigr{)}}{\bigl{(}e_{1}+\mu\,\mathrm{sign}(e_{1})\bigr{)}^{2}}\,\leq
0\;\;\forall\;x\neq 0.$ (11)

Note that the obtained inequality (11) proves only the negative _semi-
definiteness_ of $J(x,t)$, since it is visible that $x^{T}J(x,t)\,x=0$ for
$e_{2}=0\,\wedge\,e_{1}\neq 0$. Yet it appears possible to show that
$[e_{1},\,e_{2}]=0$ is the unique limit solution by evaluating the
$\dot{e}_{2}$ dynamics at $e_{2}=0$. Substituting $e_{2}=0$ into (8) results
in $\dot{e}_{2}=-ke_{1}$. It implies that $[e_{1}\neq 0,\,e_{2}=0](t)$ cannot
be a limit (correspondingly steady-state) solution, since any trajectory will
be repulsed away from $e_{2}=0$ as long as $e_{1}\neq 0$. Hence, the closed-
loop control system (7), (8) reveals as uniformly convergent. Respectively,
the origin of the control error $[e_{1},e_{2}](t)=0\equiv\bar{x}$ is the
unique limit solution, for all times $t_{0}<\tau<t$ and independent of the
initial conditions $t_{0}$ and $[e_{1},e_{2}](t_{0})$.

######  Remark 5.

When assuming a quadratic Lyapunov function candidate

$V(x)=x^{T}P\,x=\frac{1}{2}ke_{1}^{2}+\frac{1}{2}e_{2}^{2},$ (12)

which represents the entire energy level (i.e. potential energy plus kinetic
energy) of the system (7), (8), its time derivative results in

$\frac{d}{dt}V(x)=-\frac{|e_{2}|\,e_{2}^{2}}{|e_{1}|+\mu}.$ (13)

Thus, the rate at which the control system (7), (8) reduces its energy is
cubic in the error rate, i.e. $\sim|e_{2}|^{3}$, and hyperbolic in the error
size, i.e. $\sim|e_{1}|^{-1}$, cf. Figure 3.

Figure 3: Energy reduction rate $|\dot{V}|$ of the system (7), (8): depending
on $e_{1}$ in (a), depending on $e_{2}$ in (b), and as the overall function,
according to (13), in (c).

It becomes apparent, cf. Figure 3 (a), that the regularization factor $\mu$
prevents an infinite energy rate and, thus, ensure a finite control action as
$|e_{1}|\rightarrow 0$. At the same time, a hyperbolic energy rate allows to
accelerate the convergence as $|e_{1}|\rightarrow 0$. The cubic dependence of
energy rate on the error rate enables the control to react faster to the error
dynamics. It reveals as relevant in case of, for example, non-steady
trajectory phases (i.e. $\ddot{r}(t)\neq 0$) or sudden external perturbations
which can provoke high $|e_{2}|$ values.

### Numerical examples

Following numerical examples are provided for the implemented system (7), (8),
assuming $k=100$ and $\mu=0.0001$ parameters and $r(t)\in\mathcal{C}^{1}$
reference trajectories.

First, the output regulated trajectories are shown for the different initial
values $x_{0}\equiv[x_{1},x_{2}](t_{0})$:

$x_{0}=\bigl{\\{}(0.5,50),\,(0.1,20),\,(1,0),\,(1.5,-30),\,(0.3,-20)\bigr{\\}}.$

The assigned reference trajectory is a linear slope $r(t)=t$. The output
response $x_{1}(t)$ under control is depicted in Figure 4 (a). The
corresponding phase portrait of the error states, i.e. $(e_{1},e_{2})$, is
depicted Figure 4 (b).

Figure 4: Trajectories of the system (7), (8), with $k=100$, $\mu=0.0001$, for
different initial values $x_{0}\equiv[x_{1},x_{2}](t_{0})$: the output
$x_{1}(t)$ versus reference $r(t)$ in (a), phase portrait of the error states
in (b).

Next, we demonstrate the control performance of the output tracking when
$r(t)$ is only piecewise $\mathcal{C}^{1}$ and contains the finite phases
where $\dot{r}(t)\neq\mathrm{const}$. Furthermore, in order to emphasize a
practical control applicability, both $x_{1}(t)$ and $x_{2}(t)$ signals, used
for the feedback control in (8), are subject to a non-correlated bandlimited
white-noise. The output response $x_{1}(t)$ under control is depicted in
Figure 5 (a) over the reference trajectory.

Figure 5: Trajectories of the system (7), (8), with $k=100$, $\mu=0.0001$: the
output $x_{1}(t)$ versus reference $r(t)$ in (a), the $x_{2}(t)$ state in (b)
– compared with case without regularization (i.e. $\mu=0$) and also with
critically damped proportional-derivative linear control.

The $x_{2}(t)$ state, which corresponds to the relative velocity if (for
example) a motion control is intended, is depicted in Figure 5 (b). For the
sake of comparison, the case without regularization, i.e. $\mu=0$, is also
shown. Moreover, the $x_{2}(t)$-response of a critically damped proportional-
derivative linear control is also demonstrated, for the sake of comparison.
The assigned linear controller, with the same $k=100$, features the
$\dot{e}_{2}=-100e_{1}-20e_{2}$ error dynamics, correspondingly.

##  4 Conclusions

The Demidovich’s concept of convergent system dynamics was used for analyzing
the convergence properties of the optimal nonlinear damping control,
introduced in [1]. The existence of unique limit solution in the coordinate
origin of the output tracking error and its time derivative was demonstrated.
A regularization term was introduced, comparing to [1], which prevents
singularity in the solutions when an output zero crossing occurs outside of
origin. The provided analysis and results can be relevant for using the
optimal nonlinear damping control as an alternative to a standard
proportional-derivative controller. The optimal nonlinear damping control
performs as significantly faster converging and is, moreover, robust against
the measurement noise of the feedback states.

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