diff --git a/source/know/concept/bernstein-vazirani-algorithm/bernstein-vazirani-circuit.avif b/source/know/concept/bernstein-vazirani-algorithm/bernstein-vazirani-circuit.avif
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diff --git a/source/know/concept/bernstein-vazirani-algorithm/index.md b/source/know/concept/bernstein-vazirani-algorithm/index.md
index f91c0ba..85017dc 100644
--- a/source/know/concept/bernstein-vazirani-algorithm/index.md
+++ b/source/know/concept/bernstein-vazirani-algorithm/index.md
@@ -36,9 +36,7 @@ However, the Bernstein-Vazirani algorithm
allows a quantum computer to do it with only a single query.
It uses the following circuit:
-
-
-
+{% include image.html file="bernstein-vazirani-circuit.png" width="52%" alt="Bernstein-Vazirani circuit" %}
Where $$U_f$$ is a phase oracle,
whose action is defined as follows,
diff --git a/source/know/concept/bloch-sphere/bloch-small.jpg b/source/know/concept/bloch-sphere/bloch-small.jpg
deleted file mode 100644
index e99c0e1..0000000
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diff --git a/source/know/concept/bloch-sphere/index.md b/source/know/concept/bloch-sphere/index.md
index 2cb7742..0ca6f1b 100644
--- a/source/know/concept/bloch-sphere/index.md
+++ b/source/know/concept/bloch-sphere/index.md
@@ -13,9 +13,7 @@ In quantum mechanics, particularly quantum information,
the **Bloch sphere** is an invaluable tool to visualize qubits.
All pure qubit states are represented by a point on the sphere's surface:
-
-
-
+{% include image.html file="sketch-full.png" width="67%" alt="Bloch sphere" %}
The $$x$$, $$y$$ and $$z$$-axes represent the components of a spin-1/2-alike system,
and their extremes are the eigenstates of the Pauli matrices:
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diff --git a/source/know/concept/deutsch-jozsa-algorithm/index.md b/source/know/concept/deutsch-jozsa-algorithm/index.md
index bbdd58d..5f2f268 100644
--- a/source/know/concept/deutsch-jozsa-algorithm/index.md
+++ b/source/know/concept/deutsch-jozsa-algorithm/index.md
@@ -27,6 +27,7 @@ while classical computers need up to $$2^{N - 1} + 1$$ queries
for an $$N$$-bit $$x$$.
+
## Deutsch algorithm
The Deutsch algorithm handles the simplest case,
@@ -40,9 +41,7 @@ In other words, we only need to determine if $$f(0) = f(1)$$ or $$f(0) \neq f(1)
To do this, we use the following quantum circuit,
where $$U_f$$ is the oracle we query:
-
-
-
+{% include image.html file="deutsch-circuit.png" width="48%" alt="Deutsch circuit" %}
Due to unitarity constraints,
the action of $$U_f$$ is defined to be as follows,
@@ -134,16 +133,15 @@ A classical computer would need to query it twice,
once with input $$x = 0$$, and again with $$x = 1$$.
-## Full Deutsch-Jozsa algorithm
+
+## Deutsch-Jozsa algorithm
The Deutsch-Jozsa algorithm generalizes the above to $$N$$-bit inputs $$x$$.
We are promised that $$f(x)$$ is either constant or balanced;
other possibilities are assumed to be impossible.
This algorithm is then implemented by the following quantum circuit:
-
-
-
+{% include image.html file="deutsch-jozsa-circuit.png" width="52%" alt="Deutsch-Jozsa circuit" %}
There are $$N$$ qubits in initial state $$\Ket{0}$$, and one in $$\Ket{1}$$.
For clarity, the oracle $$U_f$$ works like so:
diff --git a/source/know/concept/dispersive-broadening/index.md b/source/know/concept/dispersive-broadening/index.md
index 4e4cf82..746eb6d 100644
--- a/source/know/concept/dispersive-broadening/index.md
+++ b/source/know/concept/dispersive-broadening/index.md
@@ -59,9 +59,7 @@ This phenomenon is illustrated below for our example of a Gaussian pulse
with parameter values $$T_0 = 1\:\mathrm{ps}$$, $$P_0 = 1\:\mathrm{kW}$$,
$$\beta_2 = -10 \:\mathrm{ps}^2/\mathrm{m}$$ and $$\gamma = 0$$:
-
-
-
+{% include image.html file="simulation-full.png" width="100%" alt="Dispersive broadening simulation results" %}
The **instantaneous frequency** $$\omega_\mathrm{GVD}(z, t)$$,
which describes the dominant angular frequency at a given point in the time domain,
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index 980fa54..6eefc6e 100644
--- a/source/know/concept/fabry-perot-cavity/index.md
+++ b/source/know/concept/fabry-perot-cavity/index.md
@@ -19,9 +19,8 @@ We divide the $$x$$-axis into three domains: left $$L$$, center $$C$$, and right
The cavity $$C$$ has length $$\ell$$ and is centered on $$x = 0$$.
Let $$n_L$$, $$n_C$$ and $$n_R$$ be the respective domains' refractive indices:
-
-
-
+{% include image.html file="sketch-full.png" width="70%" alt="Cavity structure" %}
+
## Microscopic cavity
@@ -129,6 +128,7 @@ $$A_1$$, $$A_2$$, $$A_3$$ or $$A_4$$ freely,
and then the others are determined by $$k_m$$ and the field's continuity.
+
## Macroscopic cavity
Next, consider a "macroscopic" Fabry-Pérot cavity
diff --git a/source/know/concept/fabry-perot-cavity/sketch-full.png b/source/know/concept/fabry-perot-cavity/sketch-full.png
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diff --git a/source/know/concept/feynman-diagram/index.md b/source/know/concept/feynman-diagram/index.md
index c36e7df..ace8dbc 100644
--- a/source/know/concept/feynman-diagram/index.md
+++ b/source/know/concept/feynman-diagram/index.md
@@ -25,6 +25,7 @@ Below, we go through the most notable components of Feynman diagrams
and how to translate them into a mathematical expression.
+
## Real space
The most common component is a **fermion line**, which represents
@@ -37,9 +38,7 @@ Let the subscript $$I$$ refer to the
and $$\mathcal{T}\{\}$$ denote the
[time-ordered product](/know/concept/time-ordered-product/):
-
-
-
+{% include image.html file="fermion-light.png" width="60%" alt="Fermion line diagram" %}
$$\begin{aligned}
= i \hbar G_{s_2 s_1}^0(\vb{r}_2, t_2; \vb{r}_1, t_1)
@@ -59,9 +58,7 @@ Less common is a **heavy fermion line**, representing
a causal Green's function $$G$$ for the entire Hamiltonian $$\hat{H}$$,
where the subscript $$H$$ refers to the [Heisenberg picture](/know/concept/heisenberg-picture/):
-
-
-
+{% include image.html file="fermion-heavy.png" width="60%" alt="Heavy fermion line diagram" %}
$$\begin{aligned}
= i \hbar G_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1)
@@ -75,9 +72,7 @@ which we assume to be instantaneous, i.e. time-independent
hence it starts and ends at the same time,
and no arrow is drawn:
-
-
-
+{% include image.html file="boson.png" width="60%" alt="Boson/interaction line diagram" %}
$$\begin{aligned}
= \frac{1}{i \hbar} W_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1)
@@ -99,9 +94,7 @@ $$\begin{aligned}
One-body (time-dependent) operators $$\hat{V}$$ in $$\hat{H}_1$$
are instead represented by a special vertex:
-
-
-
+{% include image.html file="impurity.png" width="35%" alt="One-body perturbation (e.g. impurity) diagram" %}
$$\begin{aligned}
= \frac{1}{i \hbar} V_s(\vb{r}, t)
@@ -148,6 +141,7 @@ so that a particle with a given spin propagates
from vertex to vertex without getting flipped.
+
## Fourier space
If the system is time-independent and spatially uniform,
@@ -177,9 +171,7 @@ Working in Fourier space allows us to simplify calculations.
Consider the following diagram and the resulting expression,
where $$\tilde{\vb{r}} = (\vb{r}, t)$$, and $$\tilde{\vb{k}} = (\vb{k}, \omega)$$:
-
-
-
+{% include image.html file="example.png" width="40%" alt="Example: fermion-fermion interaction" %}
$$\begin{aligned}
&= (i \hbar)^3 \sum_{s s'} \!\!\iint \dd{\tilde{\vb{r}}} \dd{\tilde{\vb{r}}'}
@@ -274,6 +266,7 @@ then conservation removes all internal variables,
so no integrals would be needed.
+
## Imaginary time
Feynman diagrams are also useful when working with
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diff --git a/source/know/concept/metacentric-height/index.md b/source/know/concept/metacentric-height/index.md
index 3d0c9a5..3d81d44 100644
--- a/source/know/concept/metacentric-height/index.md
+++ b/source/know/concept/metacentric-height/index.md
@@ -26,9 +26,7 @@ does not coincide with the origin in general,
as is illustrated in the following sketch
of our choice of coordinate system:
-
-
-
+{% include image.html file="sketch-full.png" width="75%" alt="Boat's coordinate system" %}
Here, $$B$$ is the **center of buoyancy**, equal to
the center of mass of the volume of water displaced by the boat
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diff --git a/source/know/concept/modulational-instability/index.md b/source/know/concept/modulational-instability/index.md
index a01293c..e29b2d5 100644
--- a/source/know/concept/modulational-instability/index.md
+++ b/source/know/concept/modulational-instability/index.md
@@ -172,9 +172,7 @@ $$\begin{aligned}
= \sqrt{P_0} \sech\!\Big(\frac{t}{T_0}\Big)
\end{aligned}$$
-
-
-
+{% include image.html file="simulation-full.png" width="100%" alt="Modulational instability simulation results" %}
Where $$L_\mathrm{NL} = 1/(\gamma P_0)$$ is the characteristic length of nonlinear effects.
Note that no noise was added to the simulation;
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diff --git a/source/know/concept/optical-wave-breaking/index.md b/source/know/concept/optical-wave-breaking/index.md
index 42064ff..882749f 100644
--- a/source/know/concept/optical-wave-breaking/index.md
+++ b/source/know/concept/optical-wave-breaking/index.md
@@ -34,9 +34,7 @@ Shortly before the slope would become infinite,
small waves start "falling off" the edge of the pulse,
hence the name *wave breaking*:
-
-
-
+{% include image.html file="frequency-full.png" width="100%" alt="Instantaneous frequency profile evolution" %}
Several interesting things happen around this moment.
To demonstrate this, spectrograms of the same simulation
@@ -53,9 +51,7 @@ After OWB, a train of small waves falls off the edges,
which eventually melt together, leading to a trapezoid shape in the $$t$$-domain.
Dispersive broadening then continues normally:
-
-
-
+{% include image.html file="spectrograms-full.png" width="100%" alt="Spectrograms of pulse shape evolution" %}
We call the distance at which the wave breaks $$L_\mathrm{WB}$$,
and would like to analytically predict it.
@@ -183,9 +179,7 @@ $$\begin{aligned}
This prediction for $$L_\mathrm{WB}$$ appears to agree well
with the OWB observed in the simulation:
-
-
-
+{% include image.html file="simulation-full.png" width="100%" alt="Optical wave breaking simulation results" %}
Because all spectral broadening up to $$L_\mathrm{WB}$$ is caused by SPM,
whose frequency behaviour is known, it is in fact possible to draw
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diff --git a/source/know/concept/quantum-fourier-transform/index.md b/source/know/concept/quantum-fourier-transform/index.md
index 113367c..1c68ad0 100644
--- a/source/know/concept/quantum-fourier-transform/index.md
+++ b/source/know/concept/quantum-fourier-transform/index.md
@@ -172,17 +172,13 @@ The quantum circuit to execute the mentioned steps is illustrated below,
excluding the swapping part to get the right order.
Here, $$R_m$$ means $$R_\phi$$ with $$\phi = 2 \pi / 2^m$$:
-
-
-
+{% include image.html file="qft-circuit-noswap.png" width="100%" alt="QFT circuit, without final swap" %}
Again, note how the inputs $$\Ket{x_j}$$ and outputs $$\Ket{k_j}$$ are in the opposite order.
The complete circuit, including the swapping at the end,
therefore looks like this:
-
-
-
+{% include image.html file="qft-circuit-swap.png" width="85%" alt="QFT circuit, including final swap" %}
For each of the $$n$$ qubits, $$\mathcal{O}(n)$$ gates are applied,
so overall the QFT algorithm is $$\mathcal{O}(n^2)$$.
diff --git a/source/know/concept/quantum-fourier-transform/qft-circuit-noswap.avif b/source/know/concept/quantum-fourier-transform/qft-circuit-noswap.avif
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diff --git a/source/know/concept/quantum-gate/index.md b/source/know/concept/quantum-gate/index.md
index 8c251be..e8ff579 100644
--- a/source/know/concept/quantum-gate/index.md
+++ b/source/know/concept/quantum-gate/index.md
@@ -14,6 +14,7 @@ the number of possible quantum gates is uncountably infinite,
so we only consider the most important examples here.
+
## One-qubit gates
As an example, consider the following must general single-qubit state $$\Ket{\psi}$$:
@@ -165,6 +166,7 @@ This is the definition of universality:
any state can be approximated.
+
## Two-qubit gates
As an example, let us consider
@@ -202,9 +204,7 @@ but not always in the basis of $$\Ket{0}_1$$, $$\Ket{1}_1$$, $$\Ket{0}_2$$ and $
With that said, the first two-qubit gate is $$\mathrm{SWAP}$$,
which simply swaps $$\Ket{\psi_1}$$ and $$\Ket{\psi_2}$$:
-
-
-
+{% include image.html file="swap.png" width="22%" alt="SWAP gate diagram" %}
$$\begin{aligned}
\boxed{
@@ -231,9 +231,7 @@ $$\begin{aligned}
Next, there is the **controlled NOT gate** $$\mathrm{CNOT}$$,
which "flips" (applies $$X$$ to) $$\Ket{\psi_2}$$ if $$\Ket{\psi_1}$$ is true:
-
-
-
+{% include image.html file="cnot.png" width="22%" alt="CNOT gate diagram" %}
$$\begin{aligned}
\boxed{
@@ -258,9 +256,7 @@ More generally, from every one-qubit gate $$U$$,
we can define a two-qubit **controlled U gate** $$\mathrm{CU}$$,
which applies $$U$$ to $$\Ket{\psi_2}$$ if $$\Ket{\psi_1}$$ is true:
-
-
-
+{% include image.html file="cu.png" width="22%" alt="CU gate diagram" %}
$$\begin{aligned}
\boxed{
diff --git a/source/know/concept/quantum-gate/swap.avif b/source/know/concept/quantum-gate/swap.avif
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diff --git a/source/know/concept/random-phase-approximation/index.md b/source/know/concept/random-phase-approximation/index.md
index 0f53136..698e1e7 100644
--- a/source/know/concept/random-phase-approximation/index.md
+++ b/source/know/concept/random-phase-approximation/index.md
@@ -71,23 +71,17 @@ leaving only the single most divergent one at each order $$n$$,
i.e. the ones where all $$n$$ interaction lines
carry the same momentum and energy:
-
-
-
+{% include image.html file="self-energy.png" width="92%" alt="RPA self-energy definition" %}
Where we have defined the **screened interaction** $$W^\mathrm{RPA}$$,
denoted by a double wavy line:
-
-
-
+{% include image.html file="interaction.png" width="95%" alt="RPA screened interaction definition" %}
Rearranging the above sequence of diagrams quickly leads to the following
[Dyson equation](/know/concept/dyson-equation/):
-
-
-
+{% include image.html file="dyson.png" width="55%" alt="Dyson equation for screened interaction" %}
In Fourier space, this equation's linear shape
means it is algebraic, so we can write it out:
@@ -104,9 +98,7 @@ with an internal wavevector $$\vb{q}$$, fermionic frequency $$i \omega_m^F$$, an
Abbreviating $$\tilde{\vb{k}} \equiv (\vb{k}, i \omega_n^B)$$
and $$\tilde{\vb{q}} \equiv (\vb{q}, i \omega_n^F)$$:
-
-
-
+{% include image.html file="pairbubble.png" width="45%" alt="Internal variables of pair-bubble diagram" %}
We isolate the Dyson equation for $$W^\mathrm{RPA}$$,
which reveals its physical interpretation as a *screened* interaction:
diff --git a/source/know/concept/random-phase-approximation/interaction.avif b/source/know/concept/random-phase-approximation/interaction.avif
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diff --git a/source/know/concept/repetition-code/index.md b/source/know/concept/repetition-code/index.md
index 99ac630..678211e 100644
--- a/source/know/concept/repetition-code/index.md
+++ b/source/know/concept/repetition-code/index.md
@@ -27,6 +27,7 @@ albeit with some complications,
as discussed below.
+
## Bit flip code
Suppose that we wa
+ {% endif %}
+ 
-
+{% include image.html file="bernstein-vazirani-circuit.png" width="52%" alt="Bernstein-Vazirani circuit" %}
Where $$U_f$$ is a phase oracle,
whose action is defined as follows,
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deleted file mode 100644
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diff --git a/source/know/concept/bloch-sphere/index.md b/source/know/concept/bloch-sphere/index.md
index 2cb7742..0ca6f1b 100644
--- a/source/know/concept/bloch-sphere/index.md
+++ b/source/know/concept/bloch-sphere/index.md
@@ -13,9 +13,7 @@ In quantum mechanics, particularly quantum information,
the **Bloch sphere** is an invaluable tool to visualize qubits.
All pure qubit states are represented by a point on the sphere's surface:
-
-
-
+{% include image.html file="sketch-full.png" width="67%" alt="Bloch sphere" %}
The $$x$$, $$y$$ and $$z$$-axes represent the components of a spin-1/2-alike system,
and their extremes are the eigenstates of the Pauli matrices:
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diff --git a/source/know/concept/deutsch-jozsa-algorithm/index.md b/source/know/concept/deutsch-jozsa-algorithm/index.md
index bbdd58d..5f2f268 100644
--- a/source/know/concept/deutsch-jozsa-algorithm/index.md
+++ b/source/know/concept/deutsch-jozsa-algorithm/index.md
@@ -27,6 +27,7 @@ while classical computers need up to $$2^{N - 1} + 1$$ queries
for an $$N$$-bit $$x$$.
+
## Deutsch algorithm
The Deutsch algorithm handles the simplest case,
@@ -40,9 +41,7 @@ In other words, we only need to determine if $$f(0) = f(1)$$ or $$f(0) \neq f(1)
To do this, we use the following quantum circuit,
where $$U_f$$ is the oracle we query:
-
-
-
+{% include image.html file="deutsch-circuit.png" width="48%" alt="Deutsch circuit" %}
Due to unitarity constraints,
the action of $$U_f$$ is defined to be as follows,
@@ -134,16 +133,15 @@ A classical computer would need to query it twice,
once with input $$x = 0$$, and again with $$x = 1$$.
-## Full Deutsch-Jozsa algorithm
+
+## Deutsch-Jozsa algorithm
The Deutsch-Jozsa algorithm generalizes the above to $$N$$-bit inputs $$x$$.
We are promised that $$f(x)$$ is either constant or balanced;
other possibilities are assumed to be impossible.
This algorithm is then implemented by the following quantum circuit:
-
-
-
+{% include image.html file="deutsch-jozsa-circuit.png" width="52%" alt="Deutsch-Jozsa circuit" %}
There are $$N$$ qubits in initial state $$\Ket{0}$$, and one in $$\Ket{1}$$.
For clarity, the oracle $$U_f$$ works like so:
diff --git a/source/know/concept/dispersive-broadening/index.md b/source/know/concept/dispersive-broadening/index.md
index 4e4cf82..746eb6d 100644
--- a/source/know/concept/dispersive-broadening/index.md
+++ b/source/know/concept/dispersive-broadening/index.md
@@ -59,9 +59,7 @@ This phenomenon is illustrated below for our example of a Gaussian pulse
with parameter values $$T_0 = 1\:\mathrm{ps}$$, $$P_0 = 1\:\mathrm{kW}$$,
$$\beta_2 = -10 \:\mathrm{ps}^2/\mathrm{m}$$ and $$\gamma = 0$$:
-
-
-
+{% include image.html file="simulation-full.png" width="100%" alt="Dispersive broadening simulation results" %}
The **instantaneous frequency** $$\omega_\mathrm{GVD}(z, t)$$,
which describes the dominant angular frequency at a given point in the time domain,
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index 980fa54..6eefc6e 100644
--- a/source/know/concept/fabry-perot-cavity/index.md
+++ b/source/know/concept/fabry-perot-cavity/index.md
@@ -19,9 +19,8 @@ We divide the $$x$$-axis into three domains: left $$L$$, center $$C$$, and right
The cavity $$C$$ has length $$\ell$$ and is centered on $$x = 0$$.
Let $$n_L$$, $$n_C$$ and $$n_R$$ be the respective domains' refractive indices:
-
-
-
+{% include image.html file="sketch-full.png" width="70%" alt="Cavity structure" %}
+
## Microscopic cavity
@@ -129,6 +128,7 @@ $$A_1$$, $$A_2$$, $$A_3$$ or $$A_4$$ freely,
and then the others are determined by $$k_m$$ and the field's continuity.
+
## Macroscopic cavity
Next, consider a "macroscopic" Fabry-Pérot cavity
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diff --git a/source/know/concept/feynman-diagram/index.md b/source/know/concept/feynman-diagram/index.md
index c36e7df..ace8dbc 100644
--- a/source/know/concept/feynman-diagram/index.md
+++ b/source/know/concept/feynman-diagram/index.md
@@ -25,6 +25,7 @@ Below, we go through the most notable components of Feynman diagrams
and how to translate them into a mathematical expression.
+
## Real space
The most common component is a **fermion line**, which represents
@@ -37,9 +38,7 @@ Let the subscript $$I$$ refer to the
and $$\mathcal{T}\{\}$$ denote the
[time-ordered product](/know/concept/time-ordered-product/):
-
-
-
+{% include image.html file="fermion-light.png" width="60%" alt="Fermion line diagram" %}
$$\begin{aligned}
= i \hbar G_{s_2 s_1}^0(\vb{r}_2, t_2; \vb{r}_1, t_1)
@@ -59,9 +58,7 @@ Less common is a **heavy fermion line**, representing
a causal Green's function $$G$$ for the entire Hamiltonian $$\hat{H}$$,
where the subscript $$H$$ refers to the [Heisenberg picture](/know/concept/heisenberg-picture/):
-
-
-
+{% include image.html file="fermion-heavy.png" width="60%" alt="Heavy fermion line diagram" %}
$$\begin{aligned}
= i \hbar G_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1)
@@ -75,9 +72,7 @@ which we assume to be instantaneous, i.e. time-independent
hence it starts and ends at the same time,
and no arrow is drawn:
-
-
-
+{% include image.html file="boson.png" width="60%" alt="Boson/interaction line diagram" %}
$$\begin{aligned}
= \frac{1}{i \hbar} W_{s_2 s_1}(\vb{r}_2, t_2; \vb{r}_1, t_1)
@@ -99,9 +94,7 @@ $$\begin{aligned}
One-body (time-dependent) operators $$\hat{V}$$ in $$\hat{H}_1$$
are instead represented by a special vertex:
-
-
-
+{% include image.html file="impurity.png" width="35%" alt="One-body perturbation (e.g. impurity) diagram" %}
$$\begin{aligned}
= \frac{1}{i \hbar} V_s(\vb{r}, t)
@@ -148,6 +141,7 @@ so that a particle with a given spin propagates
from vertex to vertex without getting flipped.
+
## Fourier space
If the system is time-independent and spatially uniform,
@@ -177,9 +171,7 @@ Working in Fourier space allows us to simplify calculations.
Consider the following diagram and the resulting expression,
where $$\tilde{\vb{r}} = (\vb{r}, t)$$, and $$\tilde{\vb{k}} = (\vb{k}, \omega)$$:
-
-
-
+{% include image.html file="example.png" width="40%" alt="Example: fermion-fermion interaction" %}
$$\begin{aligned}
&= (i \hbar)^3 \sum_{s s'} \!\!\iint \dd{\tilde{\vb{r}}} \dd{\tilde{\vb{r}}'}
@@ -274,6 +266,7 @@ then conservation removes all internal variables,
so no integrals would be needed.
+
## Imaginary time
Feynman diagrams are also useful when working with
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diff --git a/source/know/concept/metacentric-height/index.md b/source/know/concept/metacentric-height/index.md
index 3d0c9a5..3d81d44 100644
--- a/source/know/concept/metacentric-height/index.md
+++ b/source/know/concept/metacentric-height/index.md
@@ -26,9 +26,7 @@ does not coincide with the origin in general,
as is illustrated in the following sketch
of our choice of coordinate system:
-
-
-
+{% include image.html file="sketch-full.png" width="75%" alt="Boat's coordinate system" %}
Here, $$B$$ is the **center of buoyancy**, equal to
the center of mass of the volume of water displaced by the boat
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diff --git a/source/know/concept/modulational-instability/index.md b/source/know/concept/modulational-instability/index.md
index a01293c..e29b2d5 100644
--- a/source/know/concept/modulational-instability/index.md
+++ b/source/know/concept/modulational-instability/index.md
@@ -172,9 +172,7 @@ $$\begin{aligned}
= \sqrt{P_0} \sech\!\Big(\frac{t}{T_0}\Big)
\end{aligned}$$
-
-
-
+{% include image.html file="simulation-full.png" width="100%" alt="Modulational instability simulation results" %}
Where $$L_\mathrm{NL} = 1/(\gamma P_0)$$ is the characteristic length of nonlinear effects.
Note that no noise was added to the simulation;
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diff --git a/source/know/concept/optical-wave-breaking/index.md b/source/know/concept/optical-wave-breaking/index.md
index 42064ff..882749f 100644
--- a/source/know/concept/optical-wave-breaking/index.md
+++ b/source/know/concept/optical-wave-breaking/index.md
@@ -34,9 +34,7 @@ Shortly before the slope would become infinite,
small waves start "falling off" the edge of the pulse,
hence the name *wave breaking*:
-
-
-
+{% include image.html file="frequency-full.png" width="100%" alt="Instantaneous frequency profile evolution" %}
Several interesting things happen around this moment.
To demonstrate this, spectrograms of the same simulation
@@ -53,9 +51,7 @@ After OWB, a train of small waves falls off the edges,
which eventually melt together, leading to a trapezoid shape in the $$t$$-domain.
Dispersive broadening then continues normally:
-
-
-
+{% include image.html file="spectrograms-full.png" width="100%" alt="Spectrograms of pulse shape evolution" %}
We call the distance at which the wave breaks $$L_\mathrm{WB}$$,
and would like to analytically predict it.
@@ -183,9 +179,7 @@ $$\begin{aligned}
This prediction for $$L_\mathrm{WB}$$ appears to agree well
with the OWB observed in the simulation:
-
-
-
+{% include image.html file="simulation-full.png" width="100%" alt="Optical wave breaking simulation results" %}
Because all spectral broadening up to $$L_\mathrm{WB}$$ is caused by SPM,
whose frequency behaviour is known, it is in fact possible to draw
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diff --git a/source/know/concept/quantum-fourier-transform/index.md b/source/know/concept/quantum-fourier-transform/index.md
index 113367c..1c68ad0 100644
--- a/source/know/concept/quantum-fourier-transform/index.md
+++ b/source/know/concept/quantum-fourier-transform/index.md
@@ -172,17 +172,13 @@ The quantum circuit to execute the mentioned steps is illustrated below,
excluding the swapping part to get the right order.
Here, $$R_m$$ means $$R_\phi$$ with $$\phi = 2 \pi / 2^m$$:
-
-
-
+{% include image.html file="qft-circuit-noswap.png" width="100%" alt="QFT circuit, without final swap" %}
Again, note how the inputs $$\Ket{x_j}$$ and outputs $$\Ket{k_j}$$ are in the opposite order.
The complete circuit, including the swapping at the end,
therefore looks like this:
-
-
-
+{% include image.html file="qft-circuit-swap.png" width="85%" alt="QFT circuit, including final swap" %}
For each of the $$n$$ qubits, $$\mathcal{O}(n)$$ gates are applied,
so overall the QFT algorithm is $$\mathcal{O}(n^2)$$.
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diff --git a/source/know/concept/quantum-gate/index.md b/source/know/concept/quantum-gate/index.md
index 8c251be..e8ff579 100644
--- a/source/know/concept/quantum-gate/index.md
+++ b/source/know/concept/quantum-gate/index.md
@@ -14,6 +14,7 @@ the number of possible quantum gates is uncountably infinite,
so we only consider the most important examples here.
+
## One-qubit gates
As an example, consider the following must general single-qubit state $$\Ket{\psi}$$:
@@ -165,6 +166,7 @@ This is the definition of universality:
any state can be approximated.
+
## Two-qubit gates
As an example, let us consider
@@ -202,9 +204,7 @@ but not always in the basis of $$\Ket{0}_1$$, $$\Ket{1}_1$$, $$\Ket{0}_2$$ and $
With that said, the first two-qubit gate is $$\mathrm{SWAP}$$,
which simply swaps $$\Ket{\psi_1}$$ and $$\Ket{\psi_2}$$:
-
-
-
+{% include image.html file="swap.png" width="22%" alt="SWAP gate diagram" %}
$$\begin{aligned}
\boxed{
@@ -231,9 +231,7 @@ $$\begin{aligned}
Next, there is the **controlled NOT gate** $$\mathrm{CNOT}$$,
which "flips" (applies $$X$$ to) $$\Ket{\psi_2}$$ if $$\Ket{\psi_1}$$ is true:
-
-
-
+{% include image.html file="cnot.png" width="22%" alt="CNOT gate diagram" %}
$$\begin{aligned}
\boxed{
@@ -258,9 +256,7 @@ More generally, from every one-qubit gate $$U$$,
we can define a two-qubit **controlled U gate** $$\mathrm{CU}$$,
which applies $$U$$ to $$\Ket{\psi_2}$$ if $$\Ket{\psi_1}$$ is true:
-
-
-
+{% include image.html file="cu.png" width="22%" alt="CU gate diagram" %}
$$\begin{aligned}
\boxed{
diff --git a/source/know/concept/quantum-gate/swap.avif b/source/know/concept/quantum-gate/swap.avif
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diff --git a/source/know/concept/random-phase-approximation/dyson.avif b/source/know/concept/random-phase-approximation/dyson.avif
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index 0000000..dc80856
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diff --git a/source/know/concept/random-phase-approximation/index.md b/source/know/concept/random-phase-approximation/index.md
index 0f53136..698e1e7 100644
--- a/source/know/concept/random-phase-approximation/index.md
+++ b/source/know/concept/random-phase-approximation/index.md
@@ -71,23 +71,17 @@ leaving only the single most divergent one at each order $$n$$,
i.e. the ones where all $$n$$ interaction lines
carry the same momentum and energy:
-
-
-
+{% include image.html file="self-energy.png" width="92%" alt="RPA self-energy definition" %}
Where we have defined the **screened interaction** $$W^\mathrm{RPA}$$,
denoted by a double wavy line:
-
-
-
+{% include image.html file="interaction.png" width="95%" alt="RPA screened interaction definition" %}
Rearranging the above sequence of diagrams quickly leads to the following
[Dyson equation](/know/concept/dyson-equation/):
-
-
-
+{% include image.html file="dyson.png" width="55%" alt="Dyson equation for screened interaction" %}
In Fourier space, this equation's linear shape
means it is algebraic, so we can write it out:
@@ -104,9 +98,7 @@ with an internal wavevector $$\vb{q}$$, fermionic frequency $$i \omega_m^F$$, an
Abbreviating $$\tilde{\vb{k}} \equiv (\vb{k}, i \omega_n^B)$$
and $$\tilde{\vb{q}} \equiv (\vb{q}, i \omega_n^F)$$:
-
-
-
+{% include image.html file="pairbubble.png" width="45%" alt="Internal variables of pair-bubble diagram" %}
We isolate the Dyson equation for $$W^\mathrm{RPA}$$,
which reveals its physical interpretation as a *screened* interaction:
diff --git a/source/know/concept/random-phase-approximation/interaction.avif b/source/know/concept/random-phase-approximation/interaction.avif
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diff --git a/source/know/concept/random-phase-approximation/rpasigma.png b/source/know/concept/random-phase-approximation/rpasigma.png
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diff --git a/source/know/concept/random-phase-approximation/self-energy.avif b/source/know/concept/random-phase-approximation/self-energy.avif
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diff --git a/source/know/concept/random-phase-approximation/self-energy.png b/source/know/concept/random-phase-approximation/self-energy.png
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diff --git a/source/know/concept/repetition-code/bit-flip-detect.avif b/source/know/concept/repetition-code/bit-flip-detect.avif
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diff --git a/source/know/concept/repetition-code/bit-flip-encode.avif b/source/know/concept/repetition-code/bit-flip-encode.avif
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diff --git a/source/know/concept/repetition-code/index.md b/source/know/concept/repetition-code/index.md
index 99ac630..678211e 100644
--- a/source/know/concept/repetition-code/index.md
+++ b/source/know/concept/repetition-code/index.md
@@ -27,6 +27,7 @@ albeit with some complications,
as discussed below.
+
## Bit flip code
Suppose that we wa