# Random Kronig-Penney-type potentials for ultracold atoms using dark states

Mateusz Łącki  Instytut Fizyki Teoretycznej, Wydział Fizyki, Astronomii i
Informatyki Stosowanej, Uniwersytet Jagielloński, ulica Łojasiewicza 11,
PL-30-348 Kraków, Poland<EMAIL_ADDRESS>Jakub Zakrzewski  Instytut
Fizyki Teoretycznej, Wydział Fizyki, Astronomii i Informatyki Stosowanej,
Uniwersytet Jagielloński, ulica Łojasiewicza 11, PL-30-348 Kraków, Poland
Mark Kac Complex Systems Research Center, Uniwersytet Jagielloński, PL-30-348
Kraków, Poland<EMAIL_ADDRESS>
###### Abstract

A construction of a quasi-random potential for cold atoms using dark states
emerging in $\Lambda$ level configuration is proposed. Speckle laser fields
are used as a source of randomness. Anderson localisation in such potentials
is studied and compared with the known results for the speckle potential
itself. It is found out that the localisation length is greatly decreased due
to the non-linear fashion in which dark-state potential is obtained. In
effect, random dark state potentials resemble those occurring in random
Kronig-Penney-type Hamiltonians.

##  I Introduction

A particle moving in the potential consisting of narrow peaks may be described
by Kronig-Penney-type Hamiltonians [1]. When the potential is periodic, the
problem is solved by a simple Bloch approach. The presence of disorder
enriches the physics. Here one can imagine that periodicity is broken either
by different potential amplitudes at periodically distributed sites - the case
sometimes called a compositional disorder [2] or by random position of
scatterers having then structural (or positional) disorder. In both cases one
typically expects Anderson localization [3] at all energies for one-
dimensional (1D) system and uncorrelated disorder. The presence of
correlations leads to mobility edges as predicted and verified experimentally
for a number of models [4, 5, 6, 7, 8, 9, 2, 10, 11, 12, 13].

A standard way to implement potentials for ultracold atoms is to use off-
resonant laser standing waves via an AC Stark effect [14]. Such light-shift
potentials enabled experiments typical for condensed matter systems as
manifested by e.g. the pioneering observation of Mott insulator to superfluid
quantum phase transition [15]. Later research in optical lattice potentials
involved the use of different atomic species that feature strong, long range
interactions [16, 17, 18], creation of topological insulators [19] or studies
of non-equilibrium dynamics [20]. In particular, the 1D experiments with
ultracold atoms in random potentials have been conducted with the far off-
resonant speckle potential [21], bichromatic fields [22] or digital mirror
devices [23].

The AC Stark based approach leads, naturally, to diffraction-limitations that
prohibit creating potentials with features much sharper than half of the laser
wavelength. To remedy that, a construction based on ultracold atoms in many-
levels coupling schemes [24, 25] was proposed. Coherent population of a dark
state in the three-levels $\Lambda$ configuration was used to create a
periodic comb potential consisting of subwavelength peaks [26]. Involving more
than three atomic levels [27, 28] opens possibilities for more complex
potentials [29, 30].

In this work we shall use a similar $\Lambda$ scheme to create random
correlated potentials featuring sharp peaks, even beyond the diffraction
limit. The underlying model and creation of a random dark state potential is
described in Section II. The shapes of the potential peaks and basic
statistical properties of the potential are quantitatively analyzed in Section
III. We focus on two cases. In Subsection III.1 both lasers forming the legs
of the $\Lambda$ system are due to a speckle field. In Subsection III.2 one of
them is due to a running wave, with corresponding potential consisting of
equidistant sharp tall peaks of a (quasi)random height.

In Section IV, we study Anderson localization of a random potential from
Subsection III.1, linking the localization length $L_{\textrm{{loc}}}$ to the
correlations functions of the potential. We discuss the possibility for
approximation of the potential by properly placed Dirac-delta scatterers.

In Section V we analyze Anderson localization in potential defined in III.2.
We also discuss the role of correlations between potential peaks height on
singularity in a dependence of the localization length on energy.

Figure 1: The $\Lambda$ level configuration considered in this work. The
states $|1\rangle,|2\rangle,$ are assumed to be the ground state sublevels
while $|3\rangle$ is the excited state with the spontaneous emission rate
$\Gamma_{e}.$ The Rabi frequencies $\Omega_{i}(x)$ may be due to laser
standing waves or a speckle field and are typically position dependent.

##  II The model

We consider a gas of ultracold atoms of mass $m$ confined to a 1D tube along
the $x$ axis by a tight transverse harmonic confinement in $y,z$ realised by
the potential $V(x,y,z)=m\omega_{\perp}^{2}(y^{2}+z^{2})/2$. The
$\hbar\omega_{\perp}$ is assumed to be sufficiently large for excited
transverse modes to remain unpopulated. We assume no confinement along the $x$
direction, but in real experiment one would use either harmonic confinement or
sheet light implementing hard-wall boundary condition [31].

The atoms are driven by resonant laser light coupling three atomic (sub)levels
in the $\Lambda$ configuration as shown in Fig. 1. We assume the gas is non-
interacting. In [26] this was realised using fermionic 171Yb, where $s$-wave
contact interactions were suppressed. Whether the scheme can be successfully
implemented with bosons is still an open question, due to possible detrimental
effect of collisional losses.

The Hamiltonian of the model takes the form

$\displaystyle H$ $\displaystyle=-$
$\displaystyle\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+H_{a},$ (1)
$\displaystyle H_{a}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2}\left(\begin{array}[]{ccc}0&0&\Omega_{1}^{*}(x)\\\
0&0&\Omega_{2}^{*}(x)\\\
\Omega_{1}(x)&\Omega_{2}(x)&-i\Gamma_{e}\end{array}\right).$ (5)

The Rabi frequencies $\Omega_{i}$, $i=1,2$ describe laser driving of the
corresponding transitions between basis states $|i\rangle$ and $|3\rangle$.
The $\Gamma_{e}$ denotes the spontaneous emission decay rate of the upper
state in the $\Lambda$ scheme. The fields $\Omega_{i}(x)$ can be due to a
laser standing wave or a speckle field as discussed later on.

The “atomic” part of the Hamiltonian, $H_{a}$ for each $x\in\mathbb{R}$ has a
zero eigenvalue with associated “dark state” eigenvector:

$\displaystyle|D(x)\rangle\\!\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\\!\frac{-\Omega_{2}(x)|1\rangle\\!+\\!\Omega_{1}(x)|2\rangle}{\sqrt{|\Omega_{1}(x)|^{2}\\!+\\!|\Omega_{2}(x)|^{2}}}\\!=\\!\cos\alpha_{x}|2\rangle\\!-\\!\sin\alpha_{x}|1\rangle,$
(6)

The remaining eigenvectors $|B_{j}(x)\rangle,j=1,2$ are called bright states
since they have a nonzero contribution from the excited state $|3\rangle$.
When $\Gamma_{e}\neq 0$ the matrix $H_{a}$ is non-Hermitian and the set of
right eigenvectors ${\cal
B}=\\{|D(x)\rangle,|B_{1}(x)\rangle,|B_{2}(x)\rangle\\}$ has the associated
“ket” states $\langle D(x)|,\langle B_{j}(x)|$ that complete the biorthonormal
system. The latter are always meant as proper “left” eigenvectors, and in
general $\langle B_{i}(x)|\neq|B_{i}(x)\rangle^{\dagger}.$

The bright state energies are:

$E_{j}(x)=\frac{\hbar}{4}\left(-i\Gamma+(-1)^{j}\sqrt{-\Gamma_{e}^{2}+4|\Omega_{1}(x)|^{2}+4|\Omega_{2}(x)|^{2}}\right).$
(7)

The gap to bright states is non-zero if both $\Omega_{1}(x),\Omega_{2}(x)$ do
not vanish at some $x$. This can be ensured e.g. when one of $\Omega_{1}(x)$
is position-independent and non-zero.

When expressed in the position-dependent basis ${\cal B}$, defined above, the
Hamiltonian (1) takes the form (see [32, 29, 30]):

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2m}(p-A)^{2}+\sum_{i=1}^{2}E_{j}(x)|B_{j}\rangle\langle
B_{j}|$ (8)

with $A_{ij}=-i\hbar\langle{\cal B}_{i}|\partial_{x}|{\cal B}_{j}\rangle.$ One
can always choose the local phases of basis vectors
$|1\rangle,\ldots,|3\rangle$ such that $\Omega_{i}(x)$ are real. Then, after
projection onto $|D(x)\rangle$ state the Hamiltonian (1) reduces to the form:

$\displaystyle H$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+V_{D}(x),$ (9)

where

$\displaystyle V_{D}(x)$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2m}\langle D(x)|\partial_{xx}|D(x)\rangle,$
(10)

is the dark state potential. Using Eq. (6) one obtains:

$V_{D}(x)=\frac{\hbar^{2}}{2m}\frac{(\Omega_{1}^{\prime}(x)\Omega_{2}(x)-\Omega_{1}(x)\Omega_{2}^{\prime}(x))^{2}}{(\Omega_{1}^{2}(x)+\Omega_{2}^{2}(x))^{2}}=\frac{\hbar^{2}}{2m}(\alpha_{x}^{\prime})^{2}.$
(11)

Under the condition

$\displaystyle|E_{j}(x)|$ $\displaystyle\gg$ $\displaystyle|A_{kl}(x)|,\quad
j\in\\{1,2\\},\ k\neq l,$ (12)

valid for sufficiently large $\Omega_{i}$ [24, 33], the dark state is only
very weakly depopulated.

#### The Rabi frequencies.-

The Rabi frequencies in the Hamiltonian (1) considered in this work are due to
a standing/running laser wave or a speckle field. In the former case they are
of the form:

$\Omega_{i}(x)=\tilde{\Omega}_{i}\sin(k_{i}x+\phi)+\tilde{\Omega}_{i}^{0}.$
(13)

The $k_{i}=2\pi/\lambda_{i}$ and $\lambda_{i}$ is the wavelength of the laser
implementing $\Omega_{i}(x)$. The value of $k_{i}$ in Eq. (13) may be also
(smoothly) controlled if the lasers creating the standing wave propagate at a
finite angle with respect to $\hat{x}$. The intensity of the lasers controls
the amplitude $\tilde{\Omega}_{i}$. Implementation of the term
$\tilde{\Omega}_{i}^{0}$ requires phase coherent projection of a running wave
in the direction perpendicular to the $\hat{x}$ axis (see [34]).

The wave number $k_{1}$ defines the recoil energy:

$E_{r}=\frac{\hbar^{2}k_{1}^{2}}{2m}.$ (14)

in this work we always use the recoil energy defined with respect to $k_{1}$.
Thus $E_{r}$ carries no index "$i$".

The potential $V_{D}(x)$ is randomized by using random Rabi frequency
$\Omega_{i}(x)$. That may be accomplished by driving the corresponding
transition with a quasi-random electric field in the form of the speckle
field. It is created by propagating a laser beam from the direction
perpendicular to the $x$ axis through a diffusive plate, and focusing it with
the lens. The complex amplitude of the electric field along the system, near
the focal point of the lens, is then given by the formula [35]:

$F(x)\sim\frac{e^{i\frac{2\pi f}{\lambda}}}{i\lambda f}e^{i\frac{\pi}{\lambda
f}x^{2}}\int_{-R/2}^{R/2}d\rho\mu(\rho)w(\rho)e^{i\frac{\pi}{\lambda
f}\rho^{2}}e^{-i\frac{2\pi}{\lambda f}x\rho}.$ (15)

The $\lambda$ is the laser wavelength, $f$ – the focal distance of the used
lens and $R$ indicates the radius of the diffusive plate (we assume it to be
identical to the radius of the lens). Here we skip the index $i$. The
$\mu(\cdot)$ are random complex phases imprinted by the diffusive surface.
They are assumed to be completely random phase factors with a homogeneous
probability density over a unit circle. The above formula is valid in the
paraxial approximation, namely $f\gg R$. The ratio $R/f$ determines the degree
to which the laser field is focused. This ratio controls the effective length
scale of $F(x)$ (15). Specifically

$\displaystyle\sigma_{R}$ $\displaystyle=$ $\displaystyle\frac{\lambda f}{\pi
R},$

is the correlation length for the speckle potential, a convenient length unit
for the speckle field. In this work even if several speckle fields are used
simultaneously in some laser configuration, it is assumed, for simplicity,
that they have the same $\sigma_{R}$. We then define

$E_{\sigma_{R}}=\frac{\hbar^{2}}{2m\sigma_{R}^{2}},$ (16)

as a characteristic speckle energy scale. It is interesting to compare the
above expression to the recoil energy for a laser with same wavelength. It is

$E_{r}=(2f/R)^{2}E_{\sigma_{R}},$ (17)

which for the assumed in this work ratio $R/f=1/3$, leads to
$E_{r}=36E_{\sigma_{R}}$. The field $F(x)$ generates a Rabi frequency
${\Omega}_{i}(x)$ which may be for convenience expressed as a product of its
mean value $\tilde{\Omega}_{i}$ and the dimensionless function $S_{i}(x)$:

$\Omega_{i}(x)=\tilde{\Omega}_{i}S_{i}(x),$ (18)

where $\frac{1}{L}\int_{0}^{L}|S(x)|dx\to 1$ as $L\to\infty$. The $\Omega_{i}$
as above is non-zero, but it takes arbitrary small value with a finite
probability. To overcome this problem (recall small $\Omega_{i}$ may be
harmful to our $\Lambda$ scheme properties) one can add a phase coherent laser
field which leads to:

$\Omega_{i}(x)=\tilde{\Omega}_{i}S_{i}(x)+\tilde{\Omega}_{i}^{0},$ (19)

where both $\tilde{\Omega}_{i}^{0},\tilde{\Omega}_{i}$ are independently
controlled by intensity of the respective laser field. Again, without a loss
of generality
$S_{i}(x),\tilde{\Omega}_{i}^{0},\tilde{\Omega}_{i}\in\mathbb{R}$.

#### The Speckle potential.-

The speckle laser field can be used to create an optical speckle potential via
the AC Start shift in the two level system. The speckle laser field with Rabi
frequency $\Omega(x)$, detuned by $\Delta$ from the resonance creates the
optical potential

$V_{sp}(x)=\hbar\frac{\Omega^{2}(x)}{4\Delta}.$ (20)

#### The Window function.-

The formula (15) takes into account the window function $w(\cdot)$ which can
be used to tune the statistical properties of $F(x)$. We consider windows of
the form

$w(\rho)=\Theta(|\rho|-R/2+W)-\Theta(|\rho|-R/2)$ (21)

which form a double-slit system [35]. In the simplest case, $W=R/2$,
$w(\rho)=1$ for $|\rho|\leq R/2.$

##  III The dark state potential

The features of the dark state potential $V_{D}(x)$ depend solely on those of
the dark state $|D(x)\rangle$, compare (10). In contrast to potentials created
by AC-Stark shifts, tuning the laser intensity does not necessarily modify the
amplitude of the potential. Scaling of all $\Omega_{i}$ by a common factor
leaves the dressed states and $V_{D}(x)$ unaffected due to a functional form
of $|D(x)\rangle$ [see (6)].

The amplitude and shape of $V_{D}(x)$ is rather controlled by relative
magnitudes of the two Rabi frequencies $\Omega_{1}(x)$ and $\Omega_{2}(x)$,
which prompts us to define the dimensionless parameter:

$\epsilon_{12}=\frac{\tilde{\Omega}_{1}}{\tilde{\Omega}_{2}}=\epsilon_{21}^{-1},$
(22)

controlling that aspect of the setup. Obviously when in a specific situation
roles of $\Omega_{1}(x)$ and $\Omega_{2}(x)$ are interchangeable, then
configurations for $\epsilon_{12}=\epsilon$ and $\epsilon_{12}=\epsilon^{-1}$
are equivalent. From Eq. (10) one sees that potentials peaks in $V_{D}(x)$
occur where $|D(x)\rangle$ changes substantially over a short distance. This
may occur e.g in those places where $\Omega_{1}(x),\Omega_{2}(x)$ go from
$\Omega_{1}(x)\ll\Omega_{2}(x)$ to $\Omega_{1}(x)\gg\Omega_{2}(x)$ regime or
vice versa.

To get more insight into the genesis and shape of $V_{D}(x)$, we first look in
more detail at two important special cases.

First, when $\Omega_{2}(x)$ is due to a speckle field and $\Omega_{1}(x)$ is
constant or slowly varying on a scale much larger than the wavelength of the
speckle. The potential typically consists of double-peak structures that
appear near minima of $\Omega_{2}(x)$. This is discussed below in III.1
together with basic statistical properties of this potential.

Secondly, we consider the case when $\Omega_{1}(x)$ is due to a running wave,
Eq. (13). Then $V_{D}(x)$ has sharp potential peaks near zeros of
$\Omega_{1}$, where $\Omega_{2}(x)$ may be considered constant locally. To
randomize the heights of $V_{D}(x)$ peaks, the $\Omega_{2}(x)$ may come from a
speckle field or a running wave, Eq. (19) with a wavelength incommensurate
with the $\Omega_{1}(x)$, creating a quasiperiodic pattern.

###  III.1 The $V_{D}(x)$ near finite minima of $\Omega_{2}$ due to a speckle
field

Figure 2: Panel a): shape of double peak structure of $V_{D}(x)$ potential for
the 3-level $\Lambda$ system near quadratic minimum of $\Omega_{2}(x)$ – Eq.
(23). To reach a substantial peak height, one needs $b\ll\epsilon_{12}\ll 1$.
Panel b) shows two exemplary realizations of the speckle shape functions
$S_{i}(x)$. Panels c) and d) show the dark state potential $V_{D}(x)$ when
$\Omega_{2}(x)=\tilde{\Omega}_{2}S_{2}(x).$ Panel c) is for a constant
$\Omega_{1}(x)=\tilde{\Omega}_{1}$ while in panel d)
$\Omega_{1}(x)=\tilde{\Omega}_{1}S_{1}(x)$. The relative strengths of
$\Omega_{1}(x)$ and $\Omega_{2}(x)$ are indicated in the legends.

The $\Omega_{2}(x)$ coming from the speckle field does not feature exact
zeros, but rather it has local minima. Consider a minimum of $\Omega_{2}(x)$
at $x=x_{0}$. For $x\approx x_{0}$ we approximate $\Omega_{2}(x)$ as:

$\Omega_{2}(x)\equiv\tilde{\Omega}_{2}\left[b+\frac{\kappa^{2}}{2}(x-x_{0})^{2}\right].$
(23)

The part in bracket is a quadratic expansion of the function $S_{2}(x)$ around
a particular minimum. We do not include the value of $b$ in
$\tilde{\Omega}_{2}$ as we assume that the $\tilde{\Omega}_{2}$ is defined by
Eq. (19) for a given realization of $\Omega_{2}(x)$.

We consider $\Omega_{1}(x)=\tilde{\Omega}_{1}=\textrm{const.}$ and
$|\Omega_{2}(x_{0})|\ll\tilde{\Omega}_{1}$. Under these assumptions, the dark
state potential $V_{D}(x)$ reveals, locally, a double peak structure (see Fig.
2a). Analytically, we have (see [34]):

$V_{D}(x)=\frac{\hbar^{2}\kappa^{2}}{2m}\frac{\epsilon_{12}^{2}\kappa^{2}(x-x_{0})^{2}}{\left[[b+\frac{\kappa^{2}}{2}(x-x_{0})^{2}]^{2}+\epsilon_{12}^{2}\right]^{2}},$
(24)

where $\epsilon_{12}$ is given by Eq. (22). The value of this parameter
depends only on amplitudes of the Rabi frequencies, and is the same for
different minima of a single realization of $\tilde{\Omega}_{1}(x)$.

For arbitrary $\epsilon_{12},b$ the width of this structure is

$\Delta
x(b,\epsilon_{12},\kappa)=2\kappa^{-1}\sqrt{\frac{2}{3}}\sqrt{\sqrt{4b^{2}+3\epsilon_{12}^{2}}-b}$
(25)

and its height is

$V_{\textrm{max}}(b,\epsilon_{12},\kappa)=\frac{\hbar^{2}\kappa^{2}}{2m}\frac{27\epsilon_{12}^{2}\left(\sqrt{4b^{2}+3\epsilon_{12}^{2}}-b\right)}{8\left(b\left(\sqrt{4b^{2}+3\epsilon_{12}^{2}}+2b\right)+3\epsilon_{12}^{2}\right)^{2}}.$
(26)

For $\epsilon_{12}\gg b$ the width:

$\Delta
x(b,\epsilon_{12},\kappa)\to\frac{\sqrt{8\epsilon_{12}}}{\sqrt[4]{3}}\kappa^{-1},$
(27)

and the height :

$V_{\textrm{max}}(b,\epsilon_{12},\kappa)\to\frac{3\sqrt{3}}{8\epsilon_{12}}\frac{\hbar^{2}\kappa^{2}}{2m}.$
(28)

If additionally $\epsilon_{12}\to 0$ the two potential peaks converge to
$\frac{\pi}{2\sqrt{\epsilon_{12}}}\delta(x-x_{0})$.

For $b\gg\epsilon_{12}$ the width is

$\Delta x(b,\epsilon_{12},\kappa)\to 2k^{-1}\sqrt{\frac{2}{3}b},$ (29)

and the height of both peaks is

$V_{\textrm{max}}(b,\epsilon_{12},\kappa)\to\frac{27\epsilon_{12}^{2}}{128b^{3}}\frac{\hbar^{2}\kappa^{2}}{2m}.$
(30)

The Figure 2c) shows the exemplary dark state potentials obtained for
$\Omega_{2}(x)$ equal to the speckle shown in Fig. 2b) with a black line,
while $\Omega_{1}$ remains position independent. The plot shows two cases
$\epsilon_{12}=1$ and $\epsilon_{12}=0.2$ with relative peak heights following
(28) and (30). For fixed $\epsilon_{12}$ we may ascribe the value of parameter
$b_{i}$ from Eq. (23) to each of the minima of $\Omega_{2}(x)$ at $x_{i}$,
indexed by $i$. If the value of $\epsilon_{12}$ is lowered, potential peaks
for which $\epsilon_{12}\gg b_{i}$ are made higher and narrower, but those
that already passed to the opposite $\epsilon_{12}\ll b_{i}$ regime have their
height further reduced (see Eq. (30)). Decrease of $\epsilon_{12}$ results in
fewer sharp peaks in $V_{D}(x)$ but height of some of those peaks can
increase.

Similar observations may be made in the case when $\Omega_{1}(x)$ is not
constant but is due to a speckle field itself. Figure 2d) shows the
corresponding exemplary potential for same $\Omega_{2}(x)$ as in Fig. 2c) and
$\Omega_{1}(x)$ given by the red curve in Fig. 2b). Most of the potential
peaks occur where one of $\Omega_{1}(x)$, $\Omega_{2}(x)$ has a minimum and
the other may be considered approximately constant. Similarly one can use Eq.
(23) applied to $\Omega_{1}(x)$ or $\Omega_{2}(x)$ and ascribe $b_{i}$’s to
each minimum.

Since now $\Omega_{1}(x)$ is position dependent, in order to characterize
individual peaks near minima of $\Omega_{2}(x)$ via (24) we have to substitute
$\epsilon_{12}\to\epsilon_{12,i}$ where

$\epsilon_{12,i}=\frac{\tilde{\Omega}_{1}S_{1}(x_{i})}{\tilde{\Omega}_{2}},$
(31)

with $\epsilon_{12,i}$ specific for each minimum. In case of the minima of
$\Omega_{1}$, we consider $\epsilon_{21,i}$ defined as above with swapped
$\Omega_{1}$ and $\Omega_{2}$.

Let us consider reducing the amplitude $\tilde{\Omega}_{1}$. As
$\epsilon_{12,i}\sim\tilde{\Omega}_{1}$, the discussion of regimes
$\epsilon_{12,i}\ll b_{i}$ vs $\epsilon_{12,i}\gg b_{i}$ carried out for
constant $\Omega_{1}(x)$ still applies. The potential peaks near the minima of
$\Omega_{1}(x)$ are characterized by
$\epsilon_{21,i}\sim\tilde{\Omega}_{1}^{-1}$. Thus for smaller and smaller
$\tilde{\Omega}_{1}$ height of the latter family of peaks is reduced as well.

Figure 3:  Panel (a) shows the mean height $\bar{V}_{D}$ (red lines) and
standard deviation $\bar{\Delta}V_{D}$ (black lines) for potential $V_{D}(x)$
for $W=R/2$ (solid) and $W=R/5$ (dashed lines). In both panels
$\Omega_{2}(x)=\tilde{\Omega}_{2}S_{2}(x)$. In panel a)
$\Omega_{1}(x)=\tilde{\Omega}_{1}$ and in b)
$\Omega_{1}(x)=\tilde{\Omega}_{1}S_{1}(x)$ is due to a speckle field.

Let us now see how the above observations manifest in statistical properties
of the potential $V_{D}(x)$. Figure 3 presents $\bar{V}_{D}$ – the mean, and
$\bar{\Delta}V_{D}$ – the standard deviation of $V_{D}(x)$ as a function of
$\tilde{\Omega}_{1}/\tilde{\Omega}_{2}$ for the case of constant
$\Omega_{1}(x)=\tilde{\Omega}_{1}$ (panel a) and for the case when
$\Omega_{1}(x)=\tilde{\Omega}_{1}S_{1}(x)$ (panel b). In the latter there is
an obvious symmetry
$(\tilde{\Omega}_{1},\tilde{\Omega}_{2})\to(\tilde{\Omega}_{2},\tilde{\Omega}_{1})$.

In both cases for $\tilde{\Omega}_{1}/\tilde{\Omega}_{2}<1$, as this ratio
decreases, the standard deviation of $V_{D}(x)$ grows, and mean $\bar{V}_{D}$
converges to a constant. This is consistent with increasingly more sparse
minima satisfying $b_{i}\ll\epsilon_{12}$ (or $b_{i}\ll\epsilon_{12,i}$ for
panel b) ).

For large values of $\tilde{\Omega}_{1}/\tilde{\Omega}_{2}$, in case of
constant $\Omega_{1}(x)=\tilde{\Omega}_{1}\gg\Omega_{2}(x)$, we approximately
have:

$|D(x)\rangle\approx-\frac{\Omega_{2}(x)}{\tilde{\Omega}_{1}}|1\rangle+|2\rangle,$
(32)

and:

$V_{D}(x)\approx\frac{\Omega_{2}^{\prime}(x)^{2}}{\tilde{\Omega}_{1}^{2}}=(S_{2}^{\prime}(x))^{2}\epsilon_{12}^{-2}.$
(33)

This means that both the mean height $\bar{V}_{D}$ and the standard deviation
${\bar{\Delta}}V_{D}$ decrease to 0, for incraesing
$\tilde{\Omega}_{1}/\tilde{\Omega}_{2}$. Their ratio
$\bar{\Delta}V_{D}/\bar{V}_{D}\to 1.32\pm 0.02$ as seen in Fig. 3a). This
limit is larger than the $\bar{\Delta}V_{sp}/\bar{V}_{sp}\to 1$ for the far-
detuned AC-Stark optical potential $V_{sp}(x)$ created by laser speckle, in a
standard optical lattice setting where
$V_{sp}(x)\sim\Omega_{1}^{2}(x)/(4\delta)$ (with $\delta$ the detuning from
the resonance).

In the situation when both $\Omega_{1}(x)$ and $\Omega_{2}(x)$ are due to
speckle fields, the standard deviation ${\bar{\Delta}}V_{D}$ decreases towards
a minimum at exactly $\tilde{\Omega}_{1}=\tilde{\Omega}_{2}$. For both
$\tilde{\Omega}_{1}/\tilde{\Omega}_{2}\to 0$ and
$\tilde{\Omega}_{1}/\tilde{\Omega}_{2}\to\infty$ the behaviour of
$\bar{\Delta}V_{D}$ is similar to the case of constant $\Omega_{1}(x)$. The
marked difference is that $\bar{V}_{D}$ is
$\tilde{\Omega}_{1}/\tilde{\Omega}_{2}$-independent. Qualitatively speaking,
this is because change of $\tilde{\Omega}_{1}/\tilde{\Omega}_{2}$ has the
opposite effect on potential peaks near minima of $\Omega_{2}(x)$ and
$\Omega_{1}(x)$ when it comes to their height and width.

In Fig. 3 we mark with the dashed lines results for two cases discussed above
when the obstacle is put onto the diffusive plate. We chose to illustrate this
by setting the parameter $W=R/5$, in Eq. (15) (note that the case $W=R/2$
corresponds to no obstacle). The obstacle suppresses low frequencies from the
Fourier expansion of the $F(x)$ and the resulting potential $V_{D}(x)$ has
higher mean and variance.

###  III.2 The dark state potential near zeros of $\Omega_{i}$’s

Let us now consider the situation when $\Omega_{1}(x)$ posses a zero over the
real axis at $x=x_{i}$, as in, e.g., the case of $\Omega_{1}(x)$ being due to
a standing wave, Eq. (13). We assume $\Omega_{2}(x)$ to be locally constant
$\Omega_{2}(x)\approx\tilde{\Omega}_{2,i}$ near $x_{i}$. This creates the
setting similar to the dark state lattice proposal [24]. We then linearize

Figure 4:  Panel a): shape of peak structure in $V_{D}(x)$ potential for the
3-level $\Lambda$ system near a zero of $\Omega_{1}(x)$ at $x=x_{0}$\- Eq.
(35) for different values of $\tilde{\Omega}_{2}/\tilde{\Omega}_{1}$. Panel b)
shows $V_{D}(x)$ when both $\Omega_{1}(x),\Omega_{2}(x)$ are due to a standing
wave. The $\Omega_{1}(x)=\tilde{\Omega}_{1}\sin(k_{1}x)$ and
$\Omega_{2}(x)=\tilde{\Omega}_{2}\sin(k_{2}x)+\tilde{\Omega}_{2}^{0}$ such
that $\epsilon_{+}=0.15,\epsilon_{-}=0.1$. The ratio $k_{1}/k_{2}$ takes the
examplary value $k_{1}/k_{2}=(1+\sqrt{5})/2$. Panel c) dark state potential
$V_{D}(x)$ for the $\Lambda$ system with
$\Omega_{1}(x)=\tilde{\Omega}_{1}\sin(k_{1}x),$
$\Omega_{2}(x)=\tilde{\Omega}_{2}S_{2}(x)+\tilde{\Omega}_{2}^{0}$ such that
$\epsilon_{+}=0.15,\epsilon_{-}=0.1$. The $k_{1}$ is set such that
$k_{1}\sigma_{R}=1$, and $R/f=1/3$.

$\Omega_{1}(x)\approx\tilde{\Omega}_{1}k_{1}(x-x_{0}),$ (34)

which gives $V_{D}(x)$ of the form:

$V_{D}(x)\approx\frac{\epsilon_{21,i}^{2}E_{r}}{[k_{1}^{2}(x-x_{0})^{2}+\epsilon_{21,i}^{2}]^{2}}$
(35)

It describes a peak of width $\sim\epsilon_{21,i}\lambda_{1}$ and height
$\epsilon_{21,i}^{-2}E_{r}$ with
$\epsilon_{21,i}=\tilde{\Omega}_{2,i}/\tilde{\Omega}_{1}$ (see Fig. 4a). In
the limit $\epsilon_{21,i}\to 0$ each of the potential peaks converges to
$\frac{\pi}{2\epsilon_{21,i}}\delta(x-x_{0})$. If $\Omega_{2}(x)$ were truly
constant, the subsequent peaks would create a lattice of narrow peaks of
identical shape and height, just as in [24]. To randomize them, we use
pseudorandom $\Omega_{2}(x)$. We discuss two possibilities.

One option is to choose $\Omega_{2}(x)$ as in Eq. (13) with
$k_{1}/k_{2}\neq\mathbb{Q}$. In that case for different $x_{i}$ such that
$\Omega_{1}(x_{i})=0$ we have $\epsilon_{21,i}$ that vary between
$\epsilon_{-}=\max(0,[-\tilde{\Omega}_{2}+\tilde{\Omega}_{2}^{0}]/\tilde{\Omega}_{1})$
and
$\epsilon_{+}=[\tilde{\Omega}_{2}+\tilde{\Omega}_{2}^{0}]/\tilde{\Omega}_{1}.$
This translates into pseudo-random height and width of subsequent peaks of
$V_{D}(x)$ determined by subsequent $\epsilon_{21,i}$’s. The resulting
potential consisting of equidistant pseudorandom peaks is shown in Fig. 4b)
for $\epsilon_{+}=0.15$ and $\epsilon_{-}=0.1$. The expressions for
$\epsilon_{-}$, $\epsilon_{+}$ show that one can control the amplitude of the
disorder simply by changing $\tilde{\Omega}_{2}$, $\tilde{\Omega}_{2}^{0}$ and
$\tilde{\Omega}_{1}$

One should note that, in general, there are additional potential peaks near
minima of $\Omega_{2}(x)$ at points designed $x^{\prime}_{i}$. Such peaks are
described by Eq. (35) or Eq. (24) with values of
$\epsilon_{12,i}=\Omega_{1}(x^{\prime}_{i})/\Omega_{2}(x^{\prime}_{i})$ for
$x_{i}^{\prime}$ far from any $x_{j}$ we have $\epsilon_{12,i}\gg$ and for
$x^{\prime}_{i}$ equal to some $x_{j}$ the potential peak is mainly due to
zero of $\Omega_{1}(x)$. These peaks are automatically included in the
numerical treatment of the model that takes exact value of $V_{D}(x)$.

One can make similar construction with $\Omega_{2}(x)$ due to a speckle field,
Eq. (19). In contrast to the sine function case, the $S_{2}$ in Eq. (19) is
strictly limited only from below (by zero). The probability for taking the
value above 2 is nevertheless exponentially suppressed. This means that for
most $\epsilon_{i}$ characterizing individual peaks, we have
$\epsilon_{i}\in[\epsilon_{-},\epsilon_{+}]$ where
$\epsilon_{-}=\max(0,[-2\tilde{\Omega}_{2}+\tilde{\Omega}_{2}^{0}]/\tilde{\Omega}_{1})$
and
$\epsilon_{+}=[2\tilde{\Omega}_{2}+\tilde{\Omega}_{2}^{0}]/\tilde{\Omega}_{1}.$
The resulting potential $V_{D}(x)$ is shown in Fig. (4)c) for
$\epsilon_{+}=0.15$ and $\epsilon_{-}=0.1$. In broad terms it is similar to
the previously considered $\Omega_{2}(x)$ as in Eq. (13), but differs in
statistical properties of peak heights. This is discussed further in Section V
where we calculate tight-binding parameters for movement in this kind of
random potential.

##  IV Localization

In the case when $\Omega_{1}(x)=\textrm{const.}$ and
$\Omega_{2}(x)=S_{2}(x)\tilde{\Omega}_{2}$ the $V_{D}(x)$ consists of
relatively narrow random double peaks. In this settings it is natural to
consider Anderson localization which has been traditionally studied in the
optical potential created by a speckle field via AC-Stark effect. To that end
we first discuss the two-point correlation function of the $V_{D}(x)$
potential in such a case.

###  IV.1 Correlation functions

Let us consider two point correlation function

$C_{2}(\delta x)=\overline{V(x)V(x+\delta x)}.$

For random potentials it is directly related to the so-called Anderson
localization length, $L_{\textrm{loc}}$ of the eigenstates [36, 35].
Generically in one-dimensional systems with a random potential $V(x)$,

$\displaystyle H$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+V(x),$ (36)

one expects [3, 36] that eigenstates $\psi_{i}(x)$ are exponentially
localized:

$\displaystyle|\psi_{i}(x)|$ $\displaystyle\sim$
$\displaystyle\exp[-|x|/L_{\textrm{loc}}(E)].$ (37)

The $E$ in $L_{\textrm{loc}}(E)$ is the energy of $\psi_{i}$. Often
$L_{\textrm{loc}}(E)$ quickly grows with $E$.

The localization length may be related to the correlation function via a
series expansion with respect to increasing powers of
$(\bar{\Delta}V/\sqrt{E_{\sigma_{R}}(E-\bar{V})})^{1/2}$, where
$\bar{V},\bar{\Delta}V$ are the mean and standard deviation of $V(x)$.
Specifically:

$L_{\textrm{loc}}^{-1}(E-\bar{V})=\sum_{n\geq 2}\gamma^{(n)}(E-\bar{V}).$ (38)

The lowest term $\gamma^{(2)}$ is given by

$\gamma^{(2)}(E-\bar{V})=\frac{m}{4\hbar^{2}(E-\bar{V})}\tilde{C}_{2}\left[2\sqrt{\frac{2m(E-\bar{V})}{\hbar^{2}}}\right].$
(39)

Here, $\tilde{C}_{2}$ is a Fourier transform of $C_{2}$. Higher order terms
contain multi-point correlation functions, beyond two-point $C_{2}$. The
expansion holds for a small $\bar{\Delta}V$. The other cases can be handled by
numerical determination of $L_{\textrm{{loc}}}^{-1}$.

For the speckle optical potential, $V_{sp}(x)$, with a constant window
function $W=R/2$, the correlation function is:

$C_{2}(\delta
x)=\overline{V_{sp}}^{2}\left\\{1+\left[\frac{\sin(x/\sigma_{R})}{x/\sigma_{R}}\right]^{2}\right\\},$
(40)

and its Fourier transform (see Fig. 5a):

$\tilde{C}_{2}(k)=\overline{V_{sp}}^{2}\left\\{2\pi\frac{\delta(k)}{\sigma_{R}}+\pi\max\left(0,1-\frac{|k|\sigma_{R}}{2}\right)\right\\}.$
(41)

It is important to note that for $|k|\geq k_{0}=\frac{2}{\sigma_{R}}$
$\tilde{C}_{2}(k)$ vanishes [37, 36]. This implies significantly longer
localization lengths for energies $E$ above $\bar{V}+E_{0},\
E_{0}=\frac{\hbar^{2}k_{0}^{2}}{2m}.$ This is because the value of
$L_{\textrm{loc}}$ is solely due to higher order terms in the expansion (38).

The insertion of the obstacle in the optical system, that amounts to $W\neq
R/2$ in (21), has a profound impact on the correlation function
$\tilde{C}_{2}(k)$. For $W\leq D/3$ the $\tilde{C}_{2}(k)$ vanishes not only
for $|k|\geq k_{0}$ but also for some intermediate values of $|k|$ within the
interval $[0,k_{0}]$ as well. This is illustrated for $W=D/5$ in Fig. 5a).

Figure 5: Panel (a) shows $\tilde{C}_{2}$ for the speckle potential $V_{sp}$
for $W=D/2$ (solid line), $W=D/5$ (dashed line). Panel (b) same as above for
the dark state $V_{D}(x)$ potential with
$\Omega_{2}(x)=\tilde{\Omega}_{2}S_{2}(x)$ – a speckle field) for
$\tilde{\Omega}_{2}/\tilde{\Omega}_{1}=0.1$ and $W=D/2$ (solid line), $W=D/5$
(dashed line). The dashed-dotted line shows data for
$\tilde{\Omega}_{2}/\tilde{\Omega}_{1}=0.3$ and $W=D/2$. Panel (c) shows the
spatial correlation function $C_{2}(x)$ for all of the above potentials with
matching colors. In all of the above the normalization of the plot has been
chosen so that the maximal value of each line is 1.

Consider now the dark state potentials $V_{D}(x)$ as in the preceding section,
for the case when $\Omega_{1}(x)=\tilde{\Omega}_{1}$ and
$\Omega_{2}(x)=\tilde{\Omega}_{2}S_{2}(x).$ For such a configuration the
correlation functions $C_{2}$ and $\tilde{C}_{2}$ are shown the Fig. 5b) and
Fig. 5c). Contrary to the $V_{sp}$ potential case, here $\tilde{C}_{2}(k)$ is
non-zero for large values of momenta, $k$. This corresponds, in the position
space, to the shape of $C_{2}$ shown in Fig. 5c) where the dark-state
$C_{2}(x)$ features a narrow peak. These statements hold for both $W=R/2$ and
$W=R/5$. In the latter case, when the obstacle is put in front of the
diffusive plate, the strong modulation of $\tilde{C}_{2}(x)$ occurs.

Let us track the reason why high Fourier components $\tilde{C}_{2}$ behave
differently for $V_{sp}$ and $V_{D}(x)$. The speckle potential $V_{sp}(x)$ is
proportional to the square of the Rabi frequency $\Omega(x)$, as in Eq. (20).
Taking the square at most doubles the extent of $k$ that index non-zero
Fourier components of $V_{sp}$. This allows $\tilde{C}_{2}(k)=0,\ k\geq
k_{0}$. In the case of the dark state potential, the highly nonlinear
dependence of $V_{D}(x)$ on $\Omega_{i}(x)$’s in Eq. (15) produces arbitrarily
large Fourier components in $V_{D}(x)$ and there is no reason for
$\tilde{C}_{2}(k)$ to vanish for large $k$. This is a manifestation of the
origin of the dark state potential coming from position-dependent dark state
in contrast to the conventional AC-Stark shift.

Another feature worth pointing out is that by changing the ratio
$\tilde{\Omega}_{2}/\tilde{\Omega}_{1}$ one controls the shape of the
potential as proven by manifestly different $\tilde{C}_{2}$ for
$\tilde{\Omega}_{2}/\tilde{\Omega}_{1}$ set to two exemplary values of 0.1 and
0.3. In case of the speckle potential change of $\Omega(x)$ changes the
constant factor in $\tilde{C}_{2}(x)$, but keeps the overall shape of
$\tilde{C}_{2}$ from Fig. 5a).

###  IV.2 Anderson localization in a dark state potential

To quantitatively analyze the physical implications of a particular form of
$\tilde{C}_{2}$, we simulate the Anderson localization of a particle moving in
$V_{D},V_{sp}.$ Specifically, we look for eigenstates of Hamiltonian (36) at
energy $E$ such that $\hbar^{2}k^{2}=2m(E-\bar{V})$. The resulting Schrödinger
equation is solved over an interval $x\in[0,L]$ with the condition $\psi(x)\to
e^{-ikx},x\to 0+$. This is the outgoing amplitude of a particle that has
entered the sample at $x=L$. Near $x=L$ the wavefunction has the incoming and
reflected components $\psi(x\to L)=Ae^{-ikx}+Be^{ikx}$ proportional to $A$ and
$B$ respectively. The values of $A$, $B$ are determined numerically.

Figure 6:  Localization lengths $L_{\textrm{loc}}$ for various considered
potentials. Panel a) compares the localization length in a speckle potential
(dashed lines) and $V_{D}$ potential for the $\Lambda$ system with
$\Omega_{1}(x)=\tilde{\Omega}_{1},\Omega_{2}(x)=\tilde{\Omega}_{2}S_{2}(x)$.
For the speckle field we show two cases
$\bar{V}=\bar{\Delta}V={0.04E_{\sigma_{R}}},0.5E_{\sigma_{R}}$ respectively
with black dashed and blue dashed lines. Dark state potential $V_{D}(x)$ for
matching $\bar{\Delta}V$ is shown with same colors and solid line.
Respectively $\tilde{\Omega}_{1}/\tilde{\Omega}_{2}=\epsilon_{12}=2.357$
(black) and $\epsilon_{12}=0.333$ (blue). Panel b) shows the effect of putting
the obstacle in optical paths. The solid lines show
$\sigma_{R}/L_{\textrm{loc}}$ for $W=R/2$ (lines repeated from a) for easing
the comparison) and $W=R/5$ (dashed lines). The $\epsilon_{12}=2.357$ and
$3.398$ ensure $\bar{\Delta}V=0.04E_{\sigma_{R}}$ for the $V_{D}(x)$ potential
for $W=R/2$ and $W=R/5$ respectively.

We define the localization length $L_{\textrm{loc}}$ by the condition

$\displaystyle\langle\log|A|\rangle$ $\displaystyle\to$ $\displaystyle
L/L_{\textrm{loc}},\quad L\to\infty,$ (42)

where $\langle\cdot\rangle$ denotes averaging over disorder realizations.

Figure 6 shows $L_{\textrm{loc}}$ for large $L=5\times 10^{4}\sigma_{R},$ and
$10^{4}$ disorder realizations. Let us focus on the black dashed curve
corresponding to $\sigma_{R}/L_{\textrm{loc}}$ for $V_{sp}$ with shallow
$\bar{V}_{sp}=\bar{\Delta}V_{sp}=0.04E_{\sigma_{R}}.$ Its dependence on $k$
shows a kink at $k=k_{0}$ such that $k_{0}\sigma_{R}=1$. By Eq. (39) this
corresponds to a transition from $\tilde{C}_{2}\neq 0$ for $k\leq 2k_{0}$ to
$\tilde{C}_{2}=0$ for $k\geq 2k_{0}$. The kink is followed by a sudden
increase of $L_{\textrm{loc}}$ as first observed in [36].

We now show $\sigma_{R}/L_{\textrm{loc}}$ computed numerically for the dark
state potential $V_{D}.$ We focus on the case where
$\Omega_{1}(x)=\tilde{\Omega}_{1}$ and
$\Omega_{2}(x)=\tilde{\Omega}_{2}S_{2}(x)$ and present it in the same Fig.
6a). We chose the value of $\tilde{\Omega}_{1}/\tilde{\Omega}_{2}\approx
2.357$ to ensure that
$\bar{\Delta}V_{D}=\bar{\Delta}V_{sp}=0.04E_{\sigma_{R}}$. One sees that for
low momenta, smaller than the threshold value set by $k_{0}$, the localization
length is similar to that for the speckle potential. At $k_{0}$ both
potentials feature a kink. For such a small potential variance, the main
contribution to the correlation length comes from $\gamma^{(2)}(k_{0})$. It is
proportional to $\tilde{C}_{2}(2k_{0})$. For $k>k_{0}$ $\tilde{C}_{2}(k)=0$
for the the speckle potential while for the dark state potential a notable
kink in $\tilde{C}_{2}$ at $k_{0}$ remains.

The main difference comes for $k\sigma_{R}\geq k_{0}\sigma_{R}$ where the
localization is strongly suppressed in the speckle potential but not in the
dark state potential $V_{D}$, again easily explained by properties of
$\tilde{C}_{2}$. Thus the non-linear dependence of the potential $V_{D}$ on
$\Omega$’s translated directly to an observable much stronger localization for
large particle energies.

For a sufficiently large amplitude of the disordered potential, the
$\gamma^{(2)}$ term is no longer a dominant contribution to the inverse
localization length. This is evident in Fig. 5a) where
$\bar{\Delta}V_{D}=\bar{\Delta}V_{sp}=0.5E_{\sigma_{R}}$. The kink at $k_{0}$
no longer can be observed in the dependence of $\sigma_{R}/L_{\textrm{loc}}$
on $k\sigma_{R}$ for both $V_{sp}$ and $V_{D}$, and the localization length is
strongly decreased. Still for large momenta the localization is much stronger
in the non-linear dark state potential.

When a non-trivial window function is used, the correlation function
$\tilde{C}_{2}$ for the $V_{D}$ potential, for increasing $k\sigma_{R}$ shows
oscillations as in Fig. 5b). These oscillations find their way to the
dependence of $\sigma_{R}/L_{\textrm{loc}}$ on the free momentum of the wave-
function [see Fig. (6)b)].

###  IV.3 Dirac Delta approximation

Figure 7: Inverse localization length $\sigma_{R}/L_{\textrm{loc}}$ for
potentials $V_{sp}$, $V_{D}$ and their Dirac delta approximations
$V_{\delta,sp}$, $V_{\delta,D}$ (see legend) for [panel a)] shallow disorder:
$\bar{\Delta}V_{D}=\bar{\Delta}V_{sp}=0.04E_{\sigma_{R}}$ and [panel b)]
$\bar{\Delta}V_{D}=\bar{\Delta}V_{sp}=0.5E_{\sigma_{R}}$. In all cases the
peak approximation parameter $\delta=0.25$.

In this Section we determine if localization in the dark state potential
$V_{D}$ may be approximately described using a potential consisting of series
of Dirac-delta peaks (Kronig-Penney model), $V_{\delta,D}(x)$:

$H=\frac{p^{2}}{2m}+\underbrace{\sum_{n}V_{n}\delta(x-x_{n})}_{V_{{\delta,D}}(x)}.$
(43)

Specifically, we compare the Anderson localization length for both $V_{D}$ and
$V_{\delta,D}$.

To choose $V_{n}$ and $x_{n}$ for a particular potential realization of
$V_{D}(x)$ and obtain the approximate $V_{\delta,D}(x)$, we define a sequence
of intervals $I_{n}=(a_{n},b_{n})\subset\mathbb{R}$ such that:

  * (A)

$V(x)$ has at least one local maximum in $I_{n},$

  * (B)

$V(a_{n}),V(b_{n})\leq\delta\max_{x\in I_{n}}V(x)$ for $\delta$ being a small
positive real number,

  * (C)

no sub-interval contained in $I_{n}$ satisfies the above.

Intuitively, we want each intervals to contain a large portion of a single
potential peak. The small value of $\delta$ ensures that the $V(x)$ is small
outside of each interval $I_{n}$ with respect to the maximum value. On the
other hand $\delta$ should not be chosen too small as it would lead to too
large $I_{n}$ encompassing more than one peak. We opt to choose $\delta=0.25$.

The above definition does not automatically imply that different intervals are
disjoint. To ensure that, we actually find $I_{n}$ in the following way:

  1. 1.

For numerics we consider a particular realization over a finite interval
$x\in[0,L]$.

  2. 2.

We store all local maxima of $V(x),\ x\in[0,L]$ in the decreasing order with
respect to their value,

  3. 3.

We find the interval $I_{1}$ encompassing largest maximum that satisfies
(A)-(C),

  4. 4.

After first $n$ of intervals $I_{n}$ are determined, the (A)-(C) define a
candidate for the next interval $I_{n+1}^{\prime}$. The set
$I_{n+1}:=I_{n+1}^{\prime}\setminus\bigcup_{i=1}^{n}I_{i}$ is an interval. If
it is empty then it is not added to the ${I_{n}}$ sequence.

Each $I_{n}$ allows us to define an effective peak height

$V_{n}=\int_{I_{n}}V(x)\textrm{d}x,$ (44)

and position

$x_{n}=\frac{1}{V_{n}}\int_{I_{n}}xV(x)\textrm{d}x.$ (45)

Let us note that it is possible that two very close maxima, for which $V(x)$
does not fall below the threshold defined by the $\delta$ will be approximated
by a single Dirac Delta.

#### Localization length calculation

We have performed the transfer-matrix calculation of
$\sigma_{R}/L_{\textrm{loc}}$ for potentials $V_{\delta,sp}$ and
$V_{\delta,D}$ the Dirac-delta approximations of the potentials $V_{sp}$ and
$V_{D}$ respectively. We focus on two cases where the disorder of the
pottential is $0.04E_{\sigma_{R}}$ or $0.5E_{\sigma_{R}}$. When generating
potentials $V_{\delta,sp}$ and $V_{\delta,D}$ we assume that it is the
variance of the potential being approximated that is equal to one of the above
values.

For the case of low variance of the potentials $0.04E_{\sigma_{R}}$ the
inverse localization lengths is shown in Fig. 7a). For small $k\sigma_{R}$ the
inverse localization lengths in all four cases are similar. This is because,
for shallow disorder the series expansion given by (38) holds and
$\sigma_{R}/L_{\textrm{loc}}$ is determined by the variance of the potential
that closely match.

For $k\sigma_{R}$ near 1, we observe "kinks" in the dependence of
$\sigma/L_{\textrm{loc}}$ on $k\sigma_{R}$. In case of the speckle potential
this is followed by a sudden drop of $\sigma_{R}/L_{\textrm{loc}}$. This is in
a stark contrast to the Dirac-delta approximation of the speckle potential
$V_{\delta,{sp}}$ (and $V_{\delta,D}$). This is not surprising as the speckle
potential is smooth and Fourier transform of its correlation function has
finite support. We saw in previous sections that for dark state potentials the
$\tilde{C}_{2}$ contained arbitrarily high nonzero Fourier components
explaining why $\sigma_{R}/L_{\textrm{loc}}$ for $V_{D}$ and $V_{\delta,D}$
are closer than for $V_{sp}$ and $V_{\delta,sp}$. The agreement of
$\sigma_{R}/L_{\textrm{loc}}$ for $V_{\delta,D}$ of $V_{D}$ may be regarded as
at most qualitative for $k\sigma_{R}$. Still the Dirac-delta approximation of
$V_{D}$ reproduces the fine details of the dependence of
$\sigma_{R}/L_{\textrm{loc}}$ on $k\sigma_{R}$ such as the kink at
$k\sigma_{R}=1$.

For the deeper disorder with potential variance of $0.5E_{\sigma_{R}}$, we see
in Fig. 7b) that the $\sigma_{R}/L_{\textrm{loc}}$ for $V_{D}$ and
$V_{\delta,D}$ nearly match. This is because the dark state potential consists
now of well-defined narrow peaks, which are well approximated by discrete
Dirac-deleta peaks of $V_{\delta,D}$. The difference shows up for only very
high momenta, beginning from $k\sigma_{R}\approx 3.5$.

For both shallow and deeper disorder potential, one can reach the conclusion
that the Dirac delta $V_{\delta,D}$ potential is a valid approximation for the
low-energy part of the spectrum of Hamiltonian of a particle moving in the
dark state potential $V_{D}$ (only qualitative for a shallow disorder). This
is in contrast to the localization in a speckle field which cannot be
described by a Kronig-Penney-like model.

##  V Tight-binding description of movement in random comb potential

In this Section we discuss localization in the dark state potenial $V_{D}$ for
the configuration presented in Section III.2 for
$\Omega_{1}(x)=\tilde{\Omega}_{1}\sin(k_{1}x),$
$\Omega_{2}(x)=\tilde{\Omega}_{2}S_{2}(x)+\tilde{\Omega}_{2}^{0}$, when the
potential consists narrow peaks separated by $a=\pi/k_{1}$. The low energy
dynamics in such a potential is captured by a Dirac-delta approximation
$V_{D,\delta}$, Eq. (43) with $V_{n}$ given by (44) and $x_{n}=na$.
Localization in such a lattice has been previously intensively studied [38].
Following that review, we consider the Schrödinger equation in the following
form:

$\displaystyle\left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}\\!\\!+\\!\\!\\!\\!\\!\sum\limits_{n=-\infty}^{\infty}\\!\\!\\!E_{r}(\bar{V}\\!+\\!\delta
V_{n})\delta(k_{1}x\\!-\\!k_{1}x_{n}\\!)\\!\right]\\!\psi(x)\\!=$
$\displaystyle=\\!\frac{\hbar^{2}q^{2}}{2m}\psi(x),$ (46)

where $\bar{V}+\delta V_{n}=V_{n}$, $\langle\delta
V_{n}\rangle_{n\in\mathbb{Z}}=0$, $\sigma^{2}=\langle(\delta
V_{n})^{2}\rangle_{n\in\mathbb{Z}}\ll\bar{V}^{2}$. Under above assumptions the
inverse localization length, $L_{\textrm{loc}}$ is:

Figure 8: The inverse localization length in Kronig-Penney-like potentials
with compositional disorder. Panel (a) random Dirac delta peaks, thick lines:
$\epsilon_{+}=0.15,\epsilon_{-}=0.1$, thin lines:
$\epsilon_{+}=0.126,\epsilon_{-}=0.124$; solid black lines: numerical
calculation of $\sigma_{R}/L_{\textrm{loc}}$ for sample length $L=16\cdot
10^{6}/k_{1}$; red dashed lines show analytical, Eq. (47). The gray areas
denote $q$’s not in applicability interval of this equation. Panel (b) green
line: $\sigma_{R}/L_{\textrm{loc}}$ for $\epsilon_{+}=0.15,\epsilon_{-}=0.1$
for $V_{\delta,D}$, $\delta=0.0005$. Black: random uncorrelated Dirac delta
scatterers $\epsilon_{+}=0.1444,$ $\epsilon_{-}=0.1087$, red: localization
length in the dark state potential $V_{D}$ for
$\Omega_{1}(x)=\tilde{\Omega}_{1}\sin(k_{1}x),\Omega_{2}(x)=\tilde{\Omega}_{2}S_{2}(x)+\tilde{\Omega}_{2}^{0}$,
$\epsilon_{+}=0.15,\epsilon_{-}=0.1$, green: $V_{D,\delta}$ approximation of
$V_{D}(x)$ given by the red curves. The $k_{1}$ is set such that
$k_{1}\sigma_{R}=1$ and $R/f=1/3$.

$\frac{a}{L_{\textrm{loc}}}=\frac{1}{8}\frac{k_{1}^{2}\sin
qa}{q^{2}\sin^{2}ka}\sum\limits_{l=-\infty}^{\infty}\\!\\!\langle\delta
V_{n}\delta V_{n+l}\rangle_{n\in\mathbb{Z}}\cos(2kal).$ (47)

The wavevector $k$ is obtained from

$\cos(ka)=\cos(qa)+\frac{\bar{V}k_{1}}{2E_{r}q}\sin(qa).$ (48)

for those $q$ that correspond to the band in case of $\delta V_{n}=0$. The
equation (47) is valid only for those $q$’s and it cannot be applied in the
forbidden bands. There, in presence of disorder, the density of states is
exponentially suppressed [39, 40, 41], but it is non-zero. The localization
for those energies can be addressed numerically. Additionally, the analytic
expression is not expected to hold near the bottom and the top of the band.
Another limitation follows from the details of derivation of Eq. (47) (see
[38]): the latter does not yield an anomaly in the localization length at the
band centre. It predicts a smooth dependence of $L_{\textrm{loc}}$.

#### Random uncorrelated disorder.-

We first consider (46) with random, uncorrelated $\delta V_{n}$. The exact
values $V_{n}=\frac{\pi}{2\epsilon}$ are based on random value of $\epsilon$
with uniform distribution in $[0.124,0.126]$ (weak disorder case) and in
$[0.1,0.15]$ (strong disorder case). The mean potential heights are
$\bar{V}=12.56$ and $12.73$ respectively. The intervals of applicability of
Eq. (47) are $q/k_{1}\in[0.9096,1]$ and $q/k_{1}\in[0.9085,1]$. There (48) can
be solved for Bloch momentum.

In Figure 8a) we compare the localization length given by Eq. (47) to the
numerically determined $L_{\textrm{loc}}$ as the function of $q/k_{1}$ for $q$
within the regions of validity marked with vertical gray dashed lines 111the
line corresponding to $q/k_{1}=0.9096$ is very close to $q/k_{1}=0.9085$ and
was thus omitted. We find the quantitatively good agreement between the
localization length obtained from Eq. (47) (red lines) and from numerics
(black lines). This is true for both weak disorder (thin lines) and strong
disorder cases (thick lines). The discrepancies appear near band edges where
the analytical expression for the inverse correlation length diverges or
equals to zero The singularity present in the dependence of $L_{\textrm{loc}}$
on $q/k_{1}$ determined numerically [see inset in Fig. 8a)] is absent in the
analytic expression, Eq. (47).

#### Dark state potential.-

We now consider the full dark state potential for
$\Omega_{1}(x)=\tilde{\Omega}_{1}\sin(k_{1}x),$
$\Omega_{2}(x)=\tilde{\Omega}_{2}S_{2}(x)+\tilde{\Omega}_{2}^{0}$. For
comparison, we again consider Hamiltonian Eq. (47) with $\delta V_{n}$ based
on $V_{\delta,D}$ approximation, Eq. (45) and (44). For the parameters
considered in this section the potential $V_{D}$ consists of isolated, well-
defined peaks. This allows us to use $\delta=0.0005$, much smaller than
$\delta=0.25$ used in Section IV.3. We focus on the strong disorder case where
for the dark state potential $\epsilon_{+}=0.15$ and $\epsilon_{-}=0.1$. The
majority of peaks of the dark state potential is between $44.4E_{r}$ and
$100E_{r}$. When the integral (44) is calculated, this gives the
$V_{\delta,D}$ consisting of Dirac deltas with $\bar{V}=12.51$. We can also
find parameters of the Hamiltonian with uncorrelated $\delta V_{n}$’s that
will have the same $\bar{V}$, the amplitude of the disorder is matched by
requiring that the standard deviation of $\delta V_{n}$’s is the same.
Fulfilling those two requirements results in parameters $\epsilon_{+}=0.1444$
and $\epsilon_{-}=0.1087$ for the random $\delta V_{n}$ case.

We now compare the localization length determined numerically for the dark
state potential $V_{D}(x)$ and for the $V_{\delta,D}$ Dirac-delta
approximation. In Fig. 8b) we show $a/L_{\textrm{loc}}$ as the function of
$q/k_{1}$ (respectively red and green lines). In both cases the inverse
localization length shows a dip for the values of $\hbar^{2}q^{2}/{2m}$ that
can be traced back to the conduction band in the case of no disorder. The
visible difference in the location of this region in $q/k_{1}$ is due to a
finite width of potential peaks.

One also can observe that the singularity near the band centre is very
pronounced. It is much larger than in the random uncorrelated disorder case
(see black curve in Fig. 8b)). This occurs for both $V_{D}(x)$ potential and
its Dirac-delta approximation $V_{\delta,D}(x)$. This is in contrast to the
model with random and uncorrelated $\delta V_{n}$’s with $\bar{V}_{n}$ and the
standard deviation of $V_{n}$ matching that of $V_{\delta,D}(x)$. We find that
correlations between $V_{n}$’s in $V_{\delta,D}$ and in the $V_{D}(x)$
potential enhance the amplitude of the band centre anomaly. This is a known
possible effect of disorder correlation [43, 38].

Moreover for the actual dark state potential $V_{D}(x)$ the localization
length $L_{\textrm{loc}}$ does not monotonically increase with $q/k_{1}$. The
maximal $L_{\textrm{loc}}$ is reached near the anomaly band center, not at the
top of the band like in $V_{\delta,D}(x)$.

##  VI Conclusions and outlooks

We have shown the construction of the potential for ultracold atoms using a
three-level atomic system. The potential applies to the ultracold atoms
populating the dark state. The potentials consist of narrow pseudo-random
peaks, with randomness driven by the speckle laser field. We have contrasted
the properties of the dark state potential against the off-resonant optical
lattice potential given by the speckle field.

We have found substantially enhanced localization in the dark state potential,
especially for high kinetic energy of the particle. This is explained by a
slow decay of the two-point correlation function in the Fourier space, a
manifestation of the non-linearity of the dark state potential. This is rooted
in different mechanism of generation of the dark state potential than that of
the speckle potential which is due to far off-resonant AC-Stark process.

Our findings indicate that the potential generation via a dark state of a
three level system enhances the resolution of the speckle potential and
preserves its randomness properties. This can be further extended by replacing
speckle fields generating Rabi frequency $\Omega_{1}$ with a laser standing
wave. It leads to a completely different class of potentials that consist of
tall, pseudorandom potential peaks implementing e.g a Kronig-Penney model
structural disorder akin to [44, 2].

###### Acknowledgements.

M.Ł. and J.Z. acknowledge support from National Science Centre (Poland)
through grants No. 2019/35/B/ST2/00838 and 2019/35/B/ST2/00034, respectively.
The research has been supported by a grant from the Priority Research Area
(DigiWorld) under the Strategic Programme Excellence Initiative at
Jagiellonian University. No part of this work was written by the artificial
intelligence.

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# The Instructor’s Guide to Real Induction

Pete L. Clark  Department of Mathematics
Boyd Graduate Studies Research Center
University of Georgia
Athens, GA 30602-7403
USA<EMAIL_ADDRESS>
The author thanks François Dorais, William G. Dubuque, Joel D. Hamkins, Niles
Johnson and Iraj Kalantari for helpful conversations and/or pointers to the
literature.

##  1\. Real Induction

In this article we discuss real induction, a proof technique similar to
mathematical induction but applicable to proving statements indexed by an
interval in $\mathbb{R}$. This is not a new idea – in fact among certain
researchers in real analysis it is a commonplace – but for such a natural and
useful idea it is strangely poorly known by the mathematical community at
large. We also give a new inductive principle valid in a linearly ordered set
which simultaneously generalizes real induction and transfinite induction (and
thus also ordinary mathematical induction).
This article is written primarily for an audience of teachers of undergraduate
mathematics, especially honors calculus, real analysis and topology, hence the
phrase “instructor’s guide”. However I have for the most part backed away from
explicit _pedagogy_ : that is, while I submit to you that there is interesting
material here for several undergraduate courses, I make few specific
recommendations as to how (or even whether) to teach it. The ideas and
techniques presented here should be of interest (especially?) to those who
already have preferred proofs of these theorems. This material could also
serve as a source of challenges and enrichment for bright undergraduates – and
their instructors! – and I have left some things for the reader to work out
for herself as well as what I believe to be open problems.

###  1.1. “Induction is fundamentally discrete…”

Real induction is inspired by the principle of mathematical induction. In
broadest terms, the idea is to show that a statement holds for all real
numbers in an interval by “pushing from left to right”.

In many circles it seems to be a truism that such a thing is not possible: how
many times have you heard – or said! – that induction is fundamentally
“discrete” and that, alas, inductive methods are not available for “continuous
variables”? I held this idea myself until rather recently.

###  1.2. …is dead wrong!

Remarkably, this “induction is fundamentally discrete” idea has been refuted
repeatedly in the literature, going back at least 93 years! The earliest
instance I know of is a 1919 note of Y.R. Chao [Ch19].

######  Theorem 1.

(Chao) Let $a\in\mathbb{R}$, and let $S\subset\mathbb{R}$. Suppose that:
(CI1) $a\in S$.
(CI2) There is $\Delta>0$ such that $\forall x\in\mathbb{R}$, $x\in
S\implies(x-\Delta,x+\Delta)\cap[a,\infty)\subset S$.
Then $[a,\infty)\subset S$.

Since this is not the formulation I want to discuss, I leave the proof to you,
along with Chao’s remark that the Archimedean nature of the ordering of
$\mathbb{R}$ is being critically used here. This Archimedean feature is absent
in later formulations.
But this is just the first – if it actually is the first – of many similar
formulations of “continuous induction”. A literature search turned up the
following papers, each of which introduces some form of “continuous
induction”, in many cases with no reference to past precedent: [Kh23], [Pe26],
[Kh49], [Du57], [Fo57], [MR68], [Sh72], [Be82], [Le82], [Sa84], [Do03],
[Ka07], [Ha11].

###  1.3. Real Induction

Consider “conventional” mathematical induction. To use it, one thinks in terms
of _predicates_ – i.e., statements $P(n)$ indexed by the natural numbers – but
the cleanest statement is in terms of _subsets_ of $\mathbb{N}$. The same goes
for real induction.

Let $a<b$ be real numbers. We define a subset $S\subset[a,b]$ to be inductive
if:

(RI1) $a\in S$.
(RI2) If $a\leq x<b$, then $x\in S\implies[x,y]\subset S$ for some $y>x$.
(RI3) If $a<x\leq b$ and $[a,x)\subset S$, then $x\in S$.

######  Theorem 2.

(Real Induction) For a subset $S\subset[a,b]$, the following are equivalent:
(i) $S$ is inductive.
(ii) $S=[a,b]$.

###### Proof.

(i) $\implies$ (ii): let $S\subset[a,b]$ be inductive. Seeking a
contradiction, suppose $S^{\prime}=[a,b]\setminus S$ is nonempty, so $\inf
S^{\prime}$ exists and is finite.
Case 1: $\inf S^{\prime}=a$. Then by (RI1), $a\in S$, so by (RI2), there
exists $y>a$ such that $[a,y]\subset S$, and thus $y$ is a greater lower bound
for $S^{\prime}$ then $a=\inf S^{\prime}$: contradiction.
Case 2: $a<\inf S^{\prime}\in S$. If $\inf S^{\prime}=b$, then $S=[a,b]$.
Otherwise, by (RI2) there exists $y>\inf S^{\prime}$ such that $[\inf
S^{\prime},y^{\prime}]\subset S$, contradicting the definition of $\inf
S^{\prime}$.
Case 3: $a<\inf S^{\prime}\in S^{\prime}$. Then $[a,\inf S^{\prime})\subset
S$, so by (RI3) $\inf S^{\prime}\in S$: contradiction!
(ii) $\implies$ (i) is immediate. ∎

Theorem 2 is due to D. Hathaway [Ha11] and, independently, to me. But
mathematically equivalent ideas have been around in the literature for a long
time, some of which are _much closer_ to our formulation than the one of Chao
given above. Especially, I acknowledge my indebtedness to [Ka07].
I hope I have been absolutely clear that the enunciation of an inductive
principle for subintervals of $\mathbb{R}$ and its application to basic
results of analysis has a lot of precedence in the literature. In writing this
article, I am wholeheartedly agreeing with their approach and claiming that
there is even more to be said.

##  2\. A Principle of Linearly Ordered Induction

More generally, let $(X,\leq)$ be a linearly ordered set. $X$ has at most one
top element (resp. at most one bottom element); if such an element exists we
denote it by $\mathbb{T}$ (resp. $\mathbb{B}$). $X$ is Dedekind complete if
every nonempty subset which is bounded above has a least upper bound, or,
equivalently, if every nonempty subset which is bounded below has a greatest
lower bound.

The preceding definitions are standard, but this one is new: a subset $S$ of a
linearly ordered set $(X,\leq)$ is inductive if it satisfies all of the
following:

(IS1) There exists $a\in X$ such that $(-\infty,a]\subset S$.
(IS2) For all $x\in S$, either $x=1$ or there exists $y>x$ such that
$[x,y]\subset S$.
(IS3) For all $x\in S$, if $(-\infty,x)\in S$, then $x\in S$.

######  Theorem 3.

For a linearly ordered set $X$, TFAE:
(i) $X$ is Dedekind complete.
(ii) The only inductive subset of $X$ is $X$ itself.

###### Proof.

(i) $\implies$ (ii): Let $S\subset X$ be inductive. Seeking a contradiction,
we suppose $S^{\prime}=X\setminus S$ is nonempty. Fix $a\in X$ satisfying
(IS1). Then $a$ is a lower bound for $S^{\prime}$, so by hypothesis
$S^{\prime}$ has an infimum, say $y$. Any element less than $y$ is strictly
less than every element of $S^{\prime}$, so $(-\infty,y)\subset S$. By (IS3),
$y\in S$. If $y=\mathbb{T}$, then $S^{\prime}=\\{\mathbb{T}\\}$ or
$S^{\prime}=\varnothing$: both are contradictions. So $y<\mathbb{T}$, and then
by (IS2) there exists $z>y$ such that $[y,z]\subset S$ and thus
$(-\infty,z]\subset S$. Thus $z$ is a lower bound for $S^{\prime}$ which is
strictly larger than $y$, contradiction.
(ii) $\implies$ (i): Let $T\subset X$ be nonempty and bounded below by $a$.
Let $S$ be the set of lower bounds for $T$. Then $(-\infty,a]\subset S$, so
$S$ satisfies (IS1).
Case 1: Suppose $S$ does not satisfy (IS2): there is $x\in S$ with no $y\in X$
such that $[x,y]\subset S$. Since $S$ is downward closed, $x$ is the top
element of $S$ and $x=\inf(T)$.
Case 2: Suppose $S$ does not satisfy (IS3): there is $x\in X$ such that
$(-\infty,x)\in S$ but $x\not\in S$, i.e., there exists $t\in T$ such that
$t<x$. Then also $t\in S$, so $t$ is the least element of $T$: in particular
$t=\inf T$.
Case 3: If $S$ satisfies (IS2) and (IS3), then $S=X$, so $T=\\{\mathbb{T}\\}$
and $\inf T=\mathbb{T}$. ∎

If in Theorem 3 we take $X$ to be a closed subinterval of $\mathbb{R}$, we
recover Real Induction and its variants as discussed above. Taking $X$ to be
any well-ordered set, we recover transfinite induction and, in particular,
“ordinary” mathematical induction.

Since $\mathbb{R}$ is the unique Dedekind complete ordered field, we deduce:

######  Corollary 4.

In an ordered field $F$, the following are equivalent:
(i) $F=\mathbb{R}$.
(ii) For all $a<b\in F$, if $S\subset[a,b]$ satisfies (IS1), (IS2) and (IS3),
then $S=[a,b]$.

######  Problem 1.

Characterize the inductive subsets of $[0,1]\cap\mathbb{Q}$.

##  3\. Real Induction In Calculus

######  Theorem 5.

(Intermediate Value Theorem (IVT)) Let $f:[a,b]\rightarrow\mathbb{R}$ be a
continuous function, and let $L$ be any number in between $f(a)$ and $f(b)$.
Then there exists $c\in[a,b]$ such that $f(c)=L$.

###### Proof.

It is easy to reduce the theorem to the following special case: if
$f:[a,b]\rightarrow\mathbb{R}\setminus\\{0\\}$ is continuous and $f(a)>0$,
then $f(b)>0$. Put

$S=\\{x\in[a,b]\ |\ f(x)>0\\},$

so $f(b)>0$ iff $b\in S$. We’ll show $S=[a,b]$. (RI1) By hypothesis, $f(a)>0$,
so $a\in S$. (RI2) Let $x\in S$, $x<b$, so $f(x)>0$. Since $f$ is continuous
at $x$, there exists $\delta>0$ such that $f$ is positive on $[x,x+\delta]$,
and thus $[x,x+\delta]\subset S$. (RI3) Let $x\in(a,b]$ be such that
$[a,x)\subset S$, i.e., $f$ is positive on $[a,x)$. We claim that $f(x)>0$.
Indeed, since $f(x)\neq 0$, the only other possibility is $f(x)<0$, but if so,
then by continuity there would exist $\delta>0$ such that $f$ is negative on
$[x-\delta,x]$, i.e., $f$ is both positive and negative at each point of
$[x-\delta,x]$: contradiction! ∎

######  Theorem 6.

(Extreme Value Theorem (EVT))
Let $f:[a,b]\rightarrow\mathbb{R}$ be continuous. Then:
a) The function $f$ is bounded.
b) The function $f$ attains its maximum and minimum values.

###### Proof.

a) Let $S=\\{x\in[a,b]\ |\ f:[a,x]\rightarrow\mathbb{R}\text{ is bounded}\\}$.
(RI1): Evidently $a\in S$. (RI2): Suppose $x\in S$, so that $f$ is bounded on
$[a,x]$. But then $f$ is continuous at $x$, so is bounded near $x$: for
instance, there exists $\delta>0$ such that for all $y\in[x-\delta,x+\delta]$,
$|f(y)|\leq|f(x)|+1$. So $f$ is bounded on $[a,x]$ and also on $[x,x+\delta]$
and thus on $[a,x+\delta]$. (RI3): Suppose $x\in(a,b]$ and $[a,x)\subset S$.
Since $f$ is continuous at $x$, there exists $0<\delta<x-a$ such that $f$ is
bounded on $[x-\delta,x]$. Since $a<x-\delta<x$, $f$ is bounded on
$[a,x-\delta]$, so $f$ is bounded on $[a,x]$.
b) Let $m=\inf f([a,b])$ and $M=\sup f([a,b])$. By part a) we have

$-\infty<m\leq M<\infty.$

We want to show that there exist $x_{m},x_{M}\in[a,b]$ such that $f(x_{m})=m$,
$f(x_{M})=M$, i.e., that the infimum and supremum are actually attained as
values of $f$. Suppose that there does not exist $x\in[a,b]$ with $f(x)=m$:
then $f(x)>m$ for all $x\in[a,b]$ and the function
$g_{m}:[a,b]\rightarrow\mathbb{R}$ by $g_{m}(x)=\frac{1}{f(x)-m}$ is defined
and continuous. By the result of part a), $g_{m}$ is bounded, but this is
absurd: by definition of the infimum, $f(x)-m$ takes values less than
$\frac{1}{n}$ for any $n\in\mathbb{Z}+$ and thus $g_{m}$ takes values greater
than $n$ for any $n\in\mathbb{Z}^{+}$ and is accordingly unbounded. So indeed
there must exist $x_{m}\in[a,b]$ such that $f(x_{m})=m$. Applying this
argument to $-f$, we see that $f$ also attains its maximum value. ∎

Let $f:I\rightarrow\mathbb{R}$. For $\epsilon,\delta>0$, let us say that $f$
is $(\epsilon,\delta)$-UC on $I$ if for all $x_{1},x_{2}\in I$,
$|x_{1}-x_{2}|<\delta\implies|f(x_{1})-f(x_{2})|<\epsilon$. By definition,
$f:I\rightarrow\mathbb{R}$ is uniformly continuous if for all $\epsilon>0$,
there is $\delta>0$ such that $f$ is $(\epsilon,\delta)$-UC on $I$.

######  Lemma 7.

(Covering Lemma) Let $a<b<c<d$ be real numbers, and let
$f:[a,d]\rightarrow\mathbb{R}$. Suppose that for real numbers
$\epsilon_{1},\delta_{1},\delta_{2}>0$,
$\bullet$ $f$ is $(\epsilon,\delta_{1})$-UC on $[a,c]$ and
$\bullet$ $f$ is $(\epsilon,\delta_{2})$-UC on $[b,d]$.
Then $f$ is $(\epsilon,\min(\delta_{1},\delta_{2},c-b))$-UC on $[a,b]$.

###### Proof.

Left to the reader. ∎

######  Theorem 8.

(Uniform Continuity Theorem) Let $f:[a,b]\rightarrow\mathbb{R}$ be continuous.
Then $f$ is uniformly continuous on $[a,b]$.

###### Proof.

For $\epsilon>0$, let $S(\epsilon)$ be the set of $x\in[a,b]$ such that there
exists $\delta>0$ such that $f$ is $(\epsilon,\delta)$-UC on $[a,x]$. To show
that $f$ is uniformly continuous on $[a,b]$, it suffices to show that
$S(\epsilon)=[a,b]$ for all $\epsilon>0$. We will show this by Real Induction.
(RI1): Trivially $a\in S(\epsilon)$. (RI2): Suppose $x\in S(\epsilon)$, so
there exists $\delta_{1}>0$ such that $f$ is $(\epsilon,\delta_{1})$-UC on
$[a,x]$. Moreover, since $f$ is continuous at $x$, there is $\delta_{2}>0$
such that for all $c\in[x,x+\delta_{2}]$, $|f(c)-f(x)|<\frac{\epsilon}{2}$.
Thus for all $c_{1},c_{2}\in[x-\delta_{2},x+\delta_{2}]$,

$|f(c_{1})-f(c_{2})|=|f(c_{1})-f(x)+f(x)-f(c_{2})|\leq|f(c_{1})-f(x)|+|f(c_{2})-f(x)|<\epsilon,$

so $f$ is $(\epsilon,\delta_{2})$-UC on $[x-\delta_{2},x+\delta_{2}]$. Apply
the Covering Lemma to $f$ with $a<x-\delta_{2}<x<x+\delta_{2}$: we conclude
$f$ is
$(\epsilon,\min(\delta,\delta_{2},x-(x-\delta_{2})))=(\epsilon,\min(\delta_{1},\delta_{2}))$-UC
on $[a,x+\delta_{2}]$. Thus $[x,x+\delta_{2}]\subset S(\epsilon)$. (RI3):
Suppose $[a,x)\subset S(\epsilon)$. As above, since $f$ is continuous at $x$,
there is $\delta_{1}>0$ such that $f$ is $(\epsilon,\delta_{1})$-UC on
$[x-\delta_{1},x]$. Since $x-\frac{\delta_{1}}{2}<x$, by hypothesis there
exists $\delta_{2}$ such that $f$ is $(\epsilon,\delta_{2})$-UC on
$[a,x-\frac{\delta_{1}}{2}]$. Apply the Covering Lemma to $f$ with
$a<x-\delta_{1}<x-\frac{\delta_{1}}{2}<x$: we conclude $f$ is
$(\epsilon,\min(\delta_{1},\delta_{2},x-\frac{\delta_{1}}{2}-(x-\delta_{1})))=(\epsilon,\min(\frac{\delta_{1}}{2},\delta_{2}))$-UC
on $[a,x]$. Thus $x\in S(\epsilon)$. ∎

######  Theorem 9.

Let $f:[a,b]\rightarrow\mathbb{R}$ be continuous. Then $f$ is Darboux
integrable.

###### Proof.

By Darboux’s Criterion [S, Thm. 13.2], $f$ is Darboux integrable iff for all
$\epsilon>0$, there exists a partition $\mathcal{P}$ of $[a,b]$ such that
$U(f,\mathcal{P})-L(f,\mathcal{P})<\epsilon$. It is convenient to prove
instead the following equivalent statement: for every $\epsilon>0$, there
exists a partion $\mathcal{P}$ of $[a,b]$ such that
$U(f,\mathcal{P})-L(f,\mathcal{P})<(b-a)\epsilon$.
Fix $\epsilon>0$, and let $S(\epsilon)$ be the set of $x\in[a,b]$ such that
there exists a partition $\mathcal{P}_{x}$ of $[a,b]$ with
$U(f,\mathcal{P}_{x})-L(f,\mathcal{P}_{x})<\epsilon$. We want to show $b\in
S(\epsilon)$, so it suffices to show $S(\epsilon)=[a,b]$. In fact it is
necessary and sufficient: observe that if $x\in S(\epsilon)$ and $a\leq y\leq
x$, then also $y\in S(\epsilon)$. We will show $S(\epsilon)=[a,b]$ by Real
Induction.
(RI1) The only partition of $[a,a]$ is $\mathcal{P}_{a}=\\{a\\}$, and for this
partition we have $U(f,\mathcal{P}_{a})=L(f,\mathcal{P}_{a})=f(a)\cdot 0=0$,
so $U(f,\mathcal{P}_{a})-L(f,\mathcal{P}_{a})=0<\epsilon$.
(RI2) Suppose that for $x\in[a,b)$ we have $[a,x]\subset S(\epsilon)$. We must
show that there is $\delta>0$ such that $[a,x+\delta]\subset S(\epsilon)$, and
by the above observation it is enough to find $\delta>0$ such that
$x+\delta\in S(\epsilon)$: we must find a partition $\mathcal{P}_{x+\delta}$
of $[a,x+\delta]$ such that
$U(f,\mathcal{P}_{x+\delta})-L(f,\mathcal{P}_{x+\delta})<(x+\delta-a)\epsilon)$.
Since $x\in S(\epsilon)$, there is a partition $\mathcal{P}_{x}$ of $[a,x]$
with $U(f,\mathcal{P}_{x})-L(f,\mathcal{P}_{x})<(x-a)\epsilon$. Since $f$ is
continuous at $x$, we can make the difference between the supremum and the
infimum of $f$ as small as we want by taking a sufficiently small interval
around $x$: i.e., there is $\delta>0$ such that
$\sup(f,[x,x+\delta])-\inf(f,[x,x+\delta])<\epsilon$. Now take the smallest
partition of $[x,x+\delta]$, namely $\mathcal{P}^{\prime}=\\{x,x+\delta\\}$.
Then
$U(f,\mathcal{P}^{\prime})-L(f,\mathcal{P}^{\prime})=(x+\delta-x)(\sup(f,[x,x+\delta])-\inf(f,[x,x+\delta]))<\delta\epsilon$.
Thus if we put
$\mathcal{P}_{x+\delta}=\mathcal{P}_{x}\cup\mathcal{P}^{\prime}$ and use the
fact that upper / lower sums add when split into subintervals, we have

$U(f,\mathcal{P}_{x+\delta})-L(f,\mathcal{P}_{x+\delta})=U(f,\mathcal{P}_{x})+U(f,\mathcal{P}^{\prime})-L(f,\mathcal{P}_{x})-L(f,\mathcal{P}^{\prime})$
$=U(f,\mathcal{P}_{x})-L(f,\mathcal{P}_{x})+U(f,\mathcal{P}^{\prime})-L(f,\mathcal{P}^{\prime})<(x-a)\epsilon+\delta\epsilon=(x+\delta-a)\epsilon.$

(RI3) Suppose that for $x\in(a,b]$ we have $[a,x)\subset S(\epsilon)$. We must
show that $x\in S(\epsilon)$. The argument for this is the same as for (RI2)
except we use the interval $[x-\delta,x]$ instead of $[x,x+\delta]$. Indeed:
since $f$ is continuous at $x$, there exists $\delta>0$ such that
$\sup(f,[x-\delta,x])-\inf(f,[x-\delta,x])<\epsilon$. Since $x-\delta<x$,
$x-\delta\in S(\epsilon)$ and thus there exists a partition
$\mathcal{P}_{x-\delta}$ of $[a,x-\delta]$ such that
$U(f,\mathcal{P}_{x-\delta})=L(f,\mathcal{P}_{x-\delta})=(x-\delta-a)\epsilon$.
Let $\mathcal{P}^{\prime}=\\{x-\delta,x\\}$ and let
$\mathcal{P}_{x}=\mathcal{P}_{x-\delta}\cup\mathcal{P}^{\prime}$. Then

$U(f,\mathcal{P}_{x})-L(f,\mathcal{P}_{x})=U(f,\mathcal{P}_{x-\delta})+U(f,\mathcal{P}^{\prime})-(L(f,\mathcal{P}_{x-\delta})+L(f,\mathcal{P}^{\prime}))$
$=(U(f,\mathcal{P}_{x-\delta})-L(f,\mathcal{P}_{x-\delta}))+\delta(\sup(f,[x-\delta,x])-\inf(f,[x-\delta,x]))$
$<(x-\delta-a)\epsilon+\delta\epsilon=(x-a)\epsilon.$

∎

Remark: Spivak’s text [S] relegates uniform continuity to an appendix, which
creates a slight quandary when it comes to proving Theorem 9. He gives the
standard proof anyway [S, Thm. 13.3], but also [S, pp. 292-293] an alternative
argument establishing equality of the upper and lower integrals by
differentiation. This method goes back at least to M.J. Norris [No52]. The
Real Induction proof given above seems quite different from both of these.

######  Theorem 10.

(Bolzano-Weierstrass) An infinite subset of $[a,b]$ has a limit point.

###### Proof.

Let $\mathcal{A}\subset[a,b]$, and let $S$ be the set of $x$ in $[a,b]$ such
that _if_ $\mathcal{A}\cap[a,x]$ is infinite, it has a limit point. It
suffices to show $S=[a,b]$, which we will do by Real Induction. (RI1) is
clear. (RI2) Suppose $x\in[a,b)\cap S$. If $\mathcal{A}\cap[a,x]$ is infinite,
then it has a limit point and hence so does $\mathcal{A}\cap[a,b]$: thus
$S=[a,b]$. If for some $\delta>0$, $\mathcal{A}\cap[a,x+\delta]$ is finite,
then $[x,x+\delta]\subset S$. Otherwise $\mathcal{A}\cap[a,x]$ is finite but
$\mathcal{A}\cap[a,x+\delta]$ is infinite for all $\delta>0$, and then $x$ is
a limit point for $\mathcal{A}$ and $S=[a,b]$ as above. (RI3) If $[a,x)\subset
S$, then: either $\mathcal{A}\cap[a,y]$ is infinite for some $y<x$, so $x\in
S$; or $\mathcal{A}\cap[a,x]$ is finite, so $x\in S$; or
$\mathcal{A}\cap[a,y]$ is finite for all $y<x$ and $\mathcal{A}\cap[a,x]$ is
infinite, so $x$ is a limit point of $\mathcal{A}\cap[a,x]$ and $x\in S$. ∎

Remark: The standard proofs of Theorem 10 use monotone subsequences, a
dissection / nested intervals argument, or deduce the result from compactness
of $[a,b]$. The Real Induction proof given above seems to be new.

##  4\. Connectedness and Compactness

The preceding applications of Real Induction were at the honors calculus
level. Next we consider applications to undergraduate real analysis and
topology, specifically to results involving connectedness and compactness of
subsets of a linearly ordered topological space. We assume familiarity with
the following standard results.

######  Proposition 11.

Let $f:X\rightarrow Y$ be a surjective continuous map of topological spaces.
If $X$ is connected, then so is $Y$.

######  Proposition 12.

Let $f:X\rightarrow Y$ be a surjective continuous map of topological spaces.
If $X$ is compact, then so is $Y$.

In real analysis, Theorem 5 is _analyzed_ as the conjunction of Proposition 11
and the deep fact that the interval $[a,b]\subset\mathbb{R}$ is connected.
Similarly, Theorem 6 is analyzed as the conjunction of the easy Proposition 12
and the deep fact that the interval $[a,b]\subset\mathbb{R}$ is compact. Real
Induction is very well-suited to proving connectedness and compactness results
in ordered spaces.

######  Theorem 13.

The interval $[a,b]$ is connected.

###### Proof.

Suppose $[a,b]=A\cup B$, with $A$ and $B$ open and closed, and $A\cap
B=\emptyset$. We assume $a\in A$ and prove by Real Induction that $A=[a,b]$:
(RI1) is immediate, (RI2) holds because $A$ is open, and (RI3) holds because
$A$ is closed. We’re done! ∎

The connectedness of $[a,b]$ follows easily from the Intermediate Value
Theorem: a separation of $[a,b]$ yields a surjective continuous function
$f:[a,b]\rightarrow\\{0,1\\}$. But the proof of Theorem 13 is shorter and
easier than that of Theorem 5.

This circle of ideas can be pushed further using the concepts and results of
$\S 2.3$. Namely, let $(X,\leq)$ be a linearly ordered set. A bounded open
interval in a linearly ordered set $X$ is an interval of the form $(a,b)$ or
$[\mathbb{B},b)$ (if $X$ has a bottom element) or $(a,\mathbb{T}]$ (if $X$ has
a top element). The bounded open intervals form a base for the order topology
on $X$. A linearly ordered set $X$ is dense if for all $a<b\in X$, there
exists $c$ with $a<c<b$.

######  Lemma 14.

Let $S$ be a subset of a Dedekind complete ordered set $(X,\leq)$. If $S$ is
closed in $X$ in the order topology, then $(S,\leq)$ is Dedekind complete.

###### Proof.

This is straightforward and left to the reader. ∎

The converse of Lemma 14 does not hold, even for subsets of $\mathbb{R}$:
indeed, _any_ subinterval of $\mathbb{R}$ is Dedekind complete.

######  Theorem 15.

For a linearly ordered set $X$, TFAE:
(i) $X$ is dense and Dedekind complete.
(ii) $X$ is connected in the order topology.

###### Proof.

(i) $\implies$ (ii): Step 1: We suppose $0\in X$. Since $X$ is dense, a subset
$S\subset X$ which contains $0$ and is both open and closed in the order
topology is inductive. Since $X$ is Dedekind complete, by Theorem 3, $S=X$.
This shows $X$ is connected!
Step 2: We may assume $X\neq\varnothing$ and choose $a\in X$. By Lemma 14,
Step 1 applies to show $[a,\infty)$ connected. A similar downward induction
argument shows $(-\infty,a]$ is connected. Since $X=(-\infty,a]\cup[a,\infty)$
and $(-\infty,a]\cap[a,\infty)\neq\varnothing$, $X$ is connected.
(ii) $\implies$ (i): If the order is not dense, there are $a<b$ in $X$ with
$[a,b]=\\{a,b\\}$, so $A=(-\infty,a],\ B=[b,\infty)$ is a separation of $X$.
Suppose we have $S\subset X$, nonempty, bounded below by $a$ and with no
infimum. Let $L$ be the set of lower bounds for $S$, and put $U=\bigcup_{b\in
L}(-\infty,b)$, so $U$ is open and $U<S$. We have $a\neq\inf(S)$, so $a\in U$,
and thus $U\neq\varnothing$. If $x\not\in U$, then $x\geq L$ and, indeed,
since $L$ has no top element, $x>L$, so there exists $s\in S$ such that $s<x$.
Since the order is dense there is $y$ with $s<y<x$, and then the entire open
set $(y,\infty)$ lies in the complement of $U$. Thus $U$ is also closed. Since
$X$ is connected, $U=X$, contradicting $U<S$. ∎

######  Corollary 16.

Let $(F,+,\cdot,<)$ be an ordered field. The following are equivalent:
(i) $F$ is Dedekind complete – i.e., $F\cong\mathbb{R}$.
(ii) Every closed interval $[a,b]$ of $F$ is connected in the order topology.

Now we turn to compactness.

######  Theorem 17.

(Heine-Borel) The interval $[a,b]$ is compact.

###### Proof.

For an open covering $\mathcal{U}=\\{U_{i}\\}_{i\in I}$ of $[a,b]$, let

$S=\\{x\in[a,b]\ |\ \mathcal{U}\cap[a,x]\text{ has a finite subcovering}\\}.$

We prove $S=[a,b]$ by Real Induction. (RI1) is clear. (RI2): If
$U_{1},\ldots,U_{n}$ covers $[a,x]$, then some $U_{i}$ contains $[x,x+\delta]$
for some $\delta>0$. (RI3): if $[a,x)\subset S$, let $i_{x}\in I$ be such that
$x\in U_{i_{x}}$, and let $\delta>0$ be such that $[x-\delta,x]\in U_{i_{x}}$.
Since $x-\delta\in S$, there is a finite $J\subset I$ with $\bigcup_{i\in
J}U_{i}\supset[a,x-\delta]$, so $\\{U_{i}\\}_{i\in J}\cup U_{i_{x}}$ covers
$[a,x]$. ∎

Remark: The Extreme Value Theorem implies the compactness of $[a,b]$, a claim
we leave to the reader. (But again, we prefer the proof of Theorem 17).

These ideas can be used to characterize compactness in linearly ordered
topological spaces. We say a linearly ordered set $(X,\leq)$ is complete if
every subset has a least upper bound; equivalently, if every subset has an
greatest lower bound. $X$ is complete iff it is Dedekind complete and has top
and bottom elements.

######  Theorem 18.

For a nonempty linearly ordered set $X$, TFAE:
(i) $X$ is complete.
(ii) $X$ is compact in the order topology.

###### Proof.

(i) $\implies$ (ii): Let $\mathcal{U}=\\{U_{i}\\}_{i\in I}$ be an open
covering of $X$. Let $S$ be the set of $x\in X$ such that the covering
$\mathcal{U}\cap[0,x]$ of $[0,x]$ admits a finite subcovering. $0\in S$, so
$S$ satisfies (IS1). Suppose $U_{1}\cap[0,x],\ldots,U_{n}\cap[0,x]$ covers
$[0,x]$. If there exists $y\in X$ such that $[x,y]=\\{x,y\\}$, then adding to
the covering any element $U_{y}$ containing $y$ gives a finite covering of
$[0,y]$. Otherwise some $U_{i}$ contains $x$ and hence also $[x,y]$ for some
$y>x$. So $S$ satisfies (IS2). If we have a finite covering of $[0,x)$, then
we get a finite covering of $[0,x]$ just by adding in any element $U_{x}$
which contains $x$, so $S$ satisfies (IS3). Thus $S$ is an inductive subset of
the Dedekind complete ordered set $X$ and $S=X$. So $1\in S$ and the covering
has a finite subcovering.
(ii) $\implies$ (i): For each $x\in X$ there is a bounded open interval
$I_{x}$ containing $x$. If $X$ is compact, $\\{I_{x}\\}_{x\in X}$ has a finite
subcovering, so $X$ is bounded, i.e., has $\mathbb{B}$ and $\mathbb{T}$. Let
$S\subset X$. Since $\inf\varnothing=\mathbb{T}$, we may assume
$S\neq\varnothing$. Since $S$ has an infimum iff $\overline{S}$ does, we may
assume $S$ is closed and thus compact. Let $L$ be the set of lower bounds for
$S$. For each $(b,s)\in L\times S$, consider the closed interval
$C_{b,s}:=[b,s]$. For any finite subset
$\\{(b_{1},s_{1}),\ldots,(b_{n},s_{n})\\}$ of $L\times S$,
$\bigcap_{i=1}^{N}[b_{i},s_{i}]\supset[\max b_{i},\min s_{i}]\neq\varnothing$.
Since $S$ is compact there is $y\in\bigcap_{L\times S}[b,s]$ and then $y=\inf
S$. ∎

######  Corollary 19.

(Generalized Heine-Borel) a) For a linearly ordered set $X$, TFAE:
(i) $X$ is Dedekind complete.
(ii) A subset $S$ of $X$ is compact in the order topology iff it is closed and
bounded.
b) For an ordered field $F$, the following are equivalent:
(i) Every closed, bounded interval $[a,b]\subset F$ is compact.
(ii) $F=\mathbb{R}$.

###### Proof.

a) (i) $\implies$ (ii): A compact subset of any ordered space is closed and
bounded. Conversely, if $X$ is Dedekind complete and $S\subset X$ is closed
and bounded, then by Lemma 14, $S$ is complete and then by Theorem 18, $S$ is
compact.
(ii) $\implies$ (i): If $S\subset X$ is nonempty and bounded above, let $a\in
S$. Then $S^{\prime}=S\cap[a,\infty)$ is bounded, so $\overline{S^{\prime}}$
is compact and thus $\overline{S^{\prime}}$ is complete by Theorem 18. The
least upper bound of $\overline{S^{\prime}}$ is also the least upper bound of
$S$.
b) This follows immediately. ∎

##  5\. Still More Real Induction

###  5.1. Around the Mean Value Theorem

For an interval $I\subset\mathbb{R}$, let us denote by $I^{\circ}$ the
interior of $I$.

######  Theorem 20.

(Mean Value Theorem) Let $f:I\rightarrow\mathbb{R}$ be continuous and
differentiable on $I^{\circ}$. For any $a<b\in I$, there exists $c\in
I^{\circ}$ such that $\frac{f(b)-f(a)}{b-a}=f^{\prime}(c)$.

Many authors – e.g. [Be67], [Co67], [Tu97], [Ko09] – have advocated replacing
Theorem 20 with an alternative that is (somehow) more appealing / intuitive to
calculus students, and especially with one of the following results.

######  Corollary 21.

(Mean Value Inequality) Let $f:I\rightarrow\mathbb{R}$ be differentiable.
Suppose that there exists $M\in\mathbb{R}$ such that for all $x\in I$,
$f^{\prime}(x)\geq M$. Then for all $x<y\in\mathbb{R}$, $f(y)-f(x)\geq
M(y-x)$.

######  Corollary 22.

(Weakly Increasing Function Theorem) Let $f:I\rightarrow\mathbb{R}$ be
differentiable, and suppose that for all $x\in I$, $f^{\prime}(x)\geq 0$. Then
$f$ is _weakly increasing_ on $I$: for all $x,y\in I$, $x\leq y\implies
f(x)\leq f(y)$.

######  Corollary 23.

(Increasing Function Theorem) Let $f:I\rightarrow\mathbb{R}$ be
differentiable, and suppose that for all $x\in I$, $f^{\prime}(x)>0$. Then $f$
is _increasing_ on $I$: for all $x,y\in I$, $x<y\implies f(x)<f(y)$.

######  Corollary 24.

(Racetrack Principle) Let $f,g:[a,b]\rightarrow\mathbb{R}$ be differentiable.
If $g^{\prime}(x)\leq f^{\prime}(x)$ for all $x\in[a,b]$, then $g(x)-g(a)\leq
f(x)-f(a)$.

######  Corollary 25.

(Zero Velocity Theorem) Let $f:I\rightarrow\mathbb{R}$ be differentiable, and
suppose that for all $x\in I^{\circ}$, $f^{\prime}(x)=0$. Then $f$ is
constant.

Corollaries 21, 22, 23 and 24 are equivalent (in the sense that each
immediately implies all the others), are all implied by Theorem 20, and all
imply Corollary 25.
I have no pedagogical misgivings about Theorem 20. But there may be a
mathematical distinction to be made between Theorem 20 and Corollaries 21, 22,
23, 24: the first cannot – so far as I know! – be proved directly by Real
Induction, while the others can. We will be content to give a Real Induction
proof of Corollary 21.

###### Proof.

Let $a<b\in I$. Fix $\epsilon>0$, and define

$S_{\epsilon}=\\{x\in[a,b]\ |\ f(x)-f(a)\geq(M-\epsilon)(x-a)\\}.$

We will show that $S_{\epsilon}=[a,b]$. Then $b\in S_{\epsilon}$, so
$f(b)-f(a)\geq(M-\epsilon)(b-a)$. Since this holds for all $\epsilon>0$, we
deduce $f(b)-f(a)\geq M(b-a)$. (RI1) is immediate. (RI2) Since
$f^{\prime}(x)\geq M$, there is $\delta>0$ such that
$y\in[x,x+\delta]\implies\frac{f(y)-f(x)}{y-x}\geq M-\epsilon$. Thus
$f(y)-f(a)=(f(y)-f(x))+(f(x)-f(a))\geq(M-\epsilon)(y-x)+(M-\epsilon)(x-a)=(y-a)(M-\epsilon),$
so $[x,x+\delta]\subset S_{\epsilon}$. (RI3) Suppose that for $x\in(a,b]$ we
have $[a,x)\subset S_{\epsilon}$. As above, since $f^{\prime}(x)\geq M$, there
exists $\delta>0$ such that $y\in[x-\delta,x]\implies\frac{f(x)-f(y)}{x-y}\geq
M-\epsilon$. Thus

$f(x)-f(a)=(f(x)-f(y))+(f(y)-f(a))$
$\geq(M-\epsilon)(x-y)+(M-\epsilon)(y-a)=(x-a)(M-\epsilon),$

so $x\in S_{\epsilon}$. ∎

######  Problem 2.

Give either a direct proof of the Mean Value Theorem by Real Induction or a
convincing explanation for why such a proof cannot exist.

###  5.2. Some Real Induction Proofs For the Reader

Here are more results amenable to Real Induction. The proofs are left to you.

######  Theorem 26.

(Cantor Intersection Theorem) Let $\\{F_{n}\\}_{n=1}^{\infty}$ be a decreasing
sequence of closed subsets of $[a,b]$. Put $F=\bigcap_{n}F_{n}$. Then _either_
$F\neq\varnothing$ or there exists $n\in\mathbb{Z}^{+}$ such that
$F_{n}=\varnothing$.

######  Theorem 27.

Any open covering of $[a,b]$ admits a Lebesgue number.

######  Theorem 28.

(Dini’s Lemma) Let $\\{f_{n}\\}_{n=1}^{\infty}$ be a sequence of continuous
real-valued functions on the interval $[a,b]$ which is _pointwise decreasing_
: for all $x\in[a,b]$ and all $n\in\mathbb{Z}^{+}$, $f_{n+1}(x)\leq f_{n}(x)$.
If $f:[a,b]\rightarrow\mathbb{R}$ is continuous and $f_{n}\rightarrow f$
pointwise, then $f_{n}\rightarrow f$ uniformly.

######  Theorem 29.

(Arzelà-Ascoli)
Let $\\{f_{n}\\}_{n=1}^{\infty}$ be a sequence of continuous functions on
$[a,b]$ such that:
(i) There is $M\in\mathbb{R}$ such that for all $n\in\mathbb{Z}^{+}$ and all
$x\in[a,b]$, $|f_{n}(x)|\leq M$, and
(ii) For all $x\in[a,b]$ and all $\epsilon>0$, there exists $\delta>0$ such
that if $|x-y|<\delta$, then for all $n\in\mathbb{Z}^{+}$,
$|f_{n}(x)-f_{n}(y)|<\epsilon$.
Then there is a subsequence $\\{f_{n_{k}}\\}$ which is uniformly convergent on
$[a,b]$.

Now that you are aware of the method, I invite you explore its further uses.

######  Problem 3.

Find other theorems which can be proved via Real Induction.

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