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              <p class="caption"><span class="caption-text">Contents:</span></p>
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<li class="toctree-l1"><a class="reference internal" href="01_overview.html">Krotov Python Package</a></li>
<li class="toctree-l1"><a class="reference internal" href="02_contributing.html">Contributing</a></li>
<li class="toctree-l1"><a class="reference internal" href="03_authors.html">Credits</a></li>
<li class="toctree-l1"><a class="reference internal" href="04_features.html">Features</a></li>
<li class="toctree-l1"><a class="reference internal" href="05_history.html">History</a></li>
<li class="toctree-l1 current"><a class="current reference internal" href="#">Krotov’s Method</a><ul>
<li class="toctree-l2"><a class="reference internal" href="#optimization-functional">Optimization functional</a></li>
<li class="toctree-l2"><a class="reference internal" href="#update-equation">Update equation</a></li>
<li class="toctree-l2"><a class="reference internal" href="#optimization-of-non-linear-problems-or-non-convex-functionals">Optimization of non-linear problems or non-convex functionals</a></li>
<li class="toctree-l2"><a class="reference internal" href="#time-discretization">Time discretization</a></li>
<li class="toctree-l2"><a class="reference internal" href="#choice-of-a">Choice of λₐ</a></li>
<li class="toctree-l2"><a class="reference internal" href="#rotating-wave-approximation">Rotating wave approximation</a></li>
<li class="toctree-l2"><a class="reference internal" href="#optimization-in-liouville-space">Optimization in Liouville space</a></li>
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<li class="toctree-l1"><a class="reference internal" href="07_qutip_usage.html">Using Krotov with QuTiP</a></li>
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<li class="toctree-l1"><a class="reference internal" href="09_howto.html">How-Tos</a></li>
<li class="toctree-l1"><a class="reference internal" href="10_other_methods.html">Other Optimization Methods</a></li>
<li class="toctree-l1"><a class="reference internal" href="99_bibliography.html">References</a></li>
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<div class="section" id="krotovs-method">
<h1>Krotov’s Method<a class="headerlink" href="#krotovs-method" title="Permalink to this headline">¶</a></h1>
<div class="section" id="optimization-functional">
<h2>Optimization functional<a class="headerlink" href="#optimization-functional" title="Permalink to this headline">¶</a></h2>
<p>Quantum optimal control methods formalize the problem of finding
“control fields” that achieve some physical objective, using the time
evolution of a quantum system, including possible constraints. The most
direct example is a state-to-state transition, that is, for a known
quantum state at time zero to evolve to a specific target state at final
time <span class="math notranslate nohighlight">\(T\)</span>, controlling, e.g. a chemical
reaction <a class="bibtex reference internal" href="99_bibliography.html#tannorjcp1985" id="id1">[1]</a>. Another example is the
realization of quantum gates, the building blocks of a quantum computer.
In this case, the states forming a computational basis must transform
according to a unitary transformation <a class="bibtex reference internal" href="99_bibliography.html#nielsenchuang" id="id2">[2]</a>.
The control fields might be the amplitudes of a laser pulse, for the
control of a molecular system, RF fields for nuclear magnetic resonance,
or microwave fields for superconducting circuits. There may be multiple
independent controls involved in the dynamics, such as different color
lasers used in the excitation of a Rydberg atom, or different
polarization components of an electric field.</p>
<p>The quantum control methods build on a rich field of classical control
theory <a class="bibtex reference internal" href="99_bibliography.html#bellmanbook" id="id3">[3]</a><a class="bibtex reference internal" href="99_bibliography.html#pontryaginbook" id="id4">[4]</a>. This includes
Krotov’s method <a class="bibtex reference internal" href="99_bibliography.html#krotov-book" id="id5">[5]</a>, which was originally
formulated to optimize the soft landing of a spacecraft from orbit to
the surface of a planet <a class="bibtex reference internal" href="99_bibliography.html#konnovarc99" id="id6">[6]</a>, before being
applied to quantum mechanical
problems <a class="bibtex reference internal" href="99_bibliography.html#sklarzpra2002" id="id7">[7]</a>. Fundamentally, they rely on
the variational principle, that is, the minimization of a functional
<span class="math notranslate nohighlight">\(J[\{\ket{\phi_k^{(i)}(t)}\}, \{\epsilon_l^{(i)}(t)\}]\)</span> that
includes any required constraints via Lagrange multipliers. The
condition for minimizing <span class="math notranslate nohighlight">\(J\)</span> is then
<span class="math notranslate nohighlight">\(\nabla_{\phi_k, \epsilon_l} J = 0\)</span>. In rare cases, the
variational calculus can be solved in closed form, based on Pontryagin’s
maximum principle <a class="bibtex reference internal" href="99_bibliography.html#pontryaginbook" id="id8">[4]</a>. Numerical methods
are required in any other case. These start from an initial guess
control (or set of guess controls, if there are multiple controls), and
calculate an update to these controls that will decrease the value of
the functional. The updated controls then become the guess for the next
iteration of the algorithm, until the value of the functional is
sufficiently small, or convergence is reached.</p>
<p>Mathematically, Krotov’s method, when applied to quantum
systems <a class="bibtex reference internal" href="99_bibliography.html#tannor92" id="id9">[8]</a><a class="bibtex reference internal" href="99_bibliography.html#reichjcp12" id="id10">[9]</a>, minimizes a functional
of the most general form</p>
<div class="math notranslate nohighlight" id="equation-functional">
<span class="eqno">(1)<a class="headerlink" href="#equation-functional" title="Permalink to this equation">¶</a></span>\[  J[\{\ket{\phi_k^{(i)}(t)}\}, \{\epsilon_l^{(i)}(t)\}]
    = J_T(\{\ket{\phi_k^{(i)}(T)}\})
        + \sum_l \int_0^T g_a(\epsilon_l^{(i)}(t)) \dd t
        + \int_0^T g_b(\{\phi^{(i)}_k(t)\}) \dd t\,,\]</div>
<p>where the <span class="math notranslate nohighlight">\(\{\ket{\phi_k^{(i)}(T)}\}\)</span> are the time-evolved
initial states <span class="math notranslate nohighlight">\(\{\ket{\phi_k}\}\)</span> under the (guess) controls
<span class="math notranslate nohighlight">\(\{\epsilon^{(i)}_l(t)\}\)</span> of the <span class="math notranslate nohighlight">\(i\)</span>’th iteration. In the
simplest case of a single state-to-state transition, the index <span class="math notranslate nohighlight">\(k\)</span>
vanishes. For the example of a two-qubit quantum gate,
<span class="math notranslate nohighlight">\(\{\ket{\phi_k}\}\)</span> would be the logical basis states
<span class="math notranslate nohighlight">\(\ket{00}\)</span>, <span class="math notranslate nohighlight">\(\ket{01}\)</span>, <span class="math notranslate nohighlight">\(\ket{10}\)</span>, and
<span class="math notranslate nohighlight">\(\ket{11}\)</span>. The sum over <span class="math notranslate nohighlight">\(l\)</span> vanishes if there is only a
single control. For open system dynamics, the states
<span class="math notranslate nohighlight">\(\{\ket{\phi_k}\}\)</span> may be density matrices.</p>
<p>The functional consists of three parts:</p>
<ul>
<li><p>A final-time functional <span class="math notranslate nohighlight">\(J_T\)</span>. This is the “main” part of the
functional, and we can usually think of <span class="math notranslate nohighlight">\(J\)</span> as being an
auxiliary functional in the optimization of <span class="math notranslate nohighlight">\(J_T\)</span>.</p></li>
<li><p>A running cost on the control fields, <span class="math notranslate nohighlight">\(g_a\)</span>. The most commonly
used expression (and the only one currently supported by the
<a class="reference internal" href="API/krotov.html#module-krotov" title="krotov"><code class="xref py py-mod docutils literal notranslate"><span class="pre">krotov</span></code></a> package) is <a class="bibtex reference internal" href="99_bibliography.html#palaopra2003" id="id11">[10]</a></p>
<div class="math notranslate nohighlight" id="equation-g-a">
<span class="eqno">(2)<a class="headerlink" href="#equation-g-a" title="Permalink to this equation">¶</a></span>\[g_a(\epsilon_l^{(i+1)}(t))
= \frac{\lambda_{a,l}}{S_l(t)} \Delta\epsilon_l^2(t)\,,
  \quad
\Delta\epsilon_l(t) \equiv \epsilon_l^{(i+1)}(t) - \epsilon_l^{(i)}(t)\,.\]</div>
<p>This introduces two parameters for each control, the (inverse)
Krotov “step width” <span class="math notranslate nohighlight">\(\lambda_{a,l}\)</span> and the update-shape function
<span class="math notranslate nohighlight">\(S_l(t) \in [0, 1]\)</span>. <span class="math notranslate nohighlight">\(\Delta\epsilon_l(t)\)</span> is the update
of the control in a single iteration of the optimization algorithm.
As we will see below, <span class="math notranslate nohighlight">\(\lambda_{a,l}\)</span> determines the overall magnitude
of <span class="math notranslate nohighlight">\(\Delta\epsilon_l(t)\)</span>, and <span class="math notranslate nohighlight">\(S_l(t)\)</span> can be used to ensure
boundary conditions on <span class="math notranslate nohighlight">\(\epsilon^{(i+1)}_l(t)\)</span>.</p>
</li>
<li><p>An optional state-dependent running cost, <span class="math notranslate nohighlight">\(g_b\)</span>, can be
employed, e.g., for penalizing population in a
subspace <a class="bibtex reference internal" href="99_bibliography.html#palaopra2008" id="id12">[11]</a>. This is rarely used, as
there are other methods to achieve the same effect, like using a
non-Hermitian Hamiltonian to remove population from the forbidden
subspace during the time evolution. Currently, the <a class="reference internal" href="API/krotov.html#module-krotov" title="krotov"><code class="xref py py-mod docutils literal notranslate"><span class="pre">krotov</span></code></a> package
only supports <span class="math notranslate nohighlight">\(g_b \equiv 0\)</span>.</p></li>
</ul>
<p>The most commonly used final-time functionals (cf. <a class="reference internal" href="API/krotov.functionals.html#module-krotov.functionals" title="krotov.functionals"><code class="xref py py-mod docutils literal notranslate"><span class="pre">krotov.functionals</span></code></a>)
optimize for a set of initial states <span class="math notranslate nohighlight">\(\{\ket{\phi_k}\}\)</span> to evolve to a
set of target states <span class="math notranslate nohighlight">\(\{\ket{\phi_k^\tgt}\}\)</span>.  The functionals can then
be expressed in terms of the complex overlaps of the target states with the
final-time states under the given control. Thus,</p>
<div class="math notranslate nohighlight" id="equation-tauk">
<span class="eqno">(3)<a class="headerlink" href="#equation-tauk" title="Permalink to this equation">¶</a></span>\[  \tau_k = \Braket{\phi_k^\tgt}{\phi_k(T)}\]</div>
<p>in Hilbert space, or</p>
<div class="math notranslate nohighlight">
\[\tau_k
= \langle\!\langle \Op{\rho}^{\tgt} \vert \Op{\rho}_k(T) \rangle\!\rangle
\equiv \tr\left[\Op{\rho}_k^{\tgt\,\dagger} \Op{\rho}_k(T) \right]\]</div>
<p>in Liouville space.</p>
<p>The following functionals <span class="math notranslate nohighlight">\(J_T\)</span> can be formed from these complex
overlaps, taking into account that any optimization functional <span class="math notranslate nohighlight">\(J_T\)</span> must
be real. They differ by the way they treat the phases <span class="math notranslate nohighlight">\(\varphi_k\)</span> in the
physical optimization goal <span class="math notranslate nohighlight">\(\ket{\phi_k(T)} \overset{!}{=}
e^{i\varphi_k}\ket{\phi_k^{\tgt}}\)</span> <a class="bibtex reference internal" href="99_bibliography.html#palaopra2003" id="id13">[10]</a>:</p>
<ul>
<li><p>Optimize for simultaneous state-to-state transitions, with completely
arbitrary phases <span class="math notranslate nohighlight">\(\varphi_k\)</span>,</p>
<div class="math notranslate nohighlight" id="equation-jtss">
<span class="eqno">(4)<a class="headerlink" href="#equation-jtss" title="Permalink to this equation">¶</a></span>\[J_{T,\text{ss}} = 1- \frac{1}{N} \sum_{k=1}^{N} \Abs{\tau_k}^2\,,\]</div>
<p>cf. <a class="reference internal" href="API/krotov.functionals.html#krotov.functionals.J_T_ss" title="krotov.functionals.J_T_ss"><code class="xref py py-func docutils literal notranslate"><span class="pre">J_T_ss()</span></code></a>.</p>
</li>
<li><p>Optimize for simultaneous state-to-state transitions, with an arbitrary
<em>global</em> phase, i.e., <span class="math notranslate nohighlight">\(\varphi_k = \varphi_{\text{global}}\)</span> for all
<span class="math notranslate nohighlight">\(k\)</span> with arbitrary <span class="math notranslate nohighlight">\(\varphi_{\text{global}}\)</span>,</p>
<div class="math notranslate nohighlight" id="equation-jtsm">
<span class="eqno">(5)<a class="headerlink" href="#equation-jtsm" title="Permalink to this equation">¶</a></span>\[J_{T,\text{sm}} = 1- \frac{1}{N^2} \Abs{\sum_{k=1}^{N} \tau_k}^2
        = 1- \frac{1}{N^2} \sum_{k=1}^{N} \sum_{k'=1}^{N} \tau_{k'}^* \tau_{k}\,,\]</div>
<p>cf. <a class="reference internal" href="API/krotov.functionals.html#krotov.functionals.J_T_sm" title="krotov.functionals.J_T_sm"><code class="xref py py-func docutils literal notranslate"><span class="pre">J_T_sm()</span></code></a>.</p>
</li>
<li><p>Optimize for simultaneous state-to-state transitions, with a global phase of zero, i.e.,
<span class="math notranslate nohighlight">\(\varphi_k = 0\)</span> for all <span class="math notranslate nohighlight">\(k\)</span>,</p>
<div class="math notranslate nohighlight" id="equation-jtre">
<span class="eqno">(6)<a class="headerlink" href="#equation-jtre" title="Permalink to this equation">¶</a></span>\[J_{T,\text{re}} = 1-\frac{1}{N} \Re \left[\, \sum_{k=1}^{N} \tau_k \,\right]\,,\]</div>
<p>cf. <a class="reference internal" href="API/krotov.functionals.html#krotov.functionals.J_T_re" title="krotov.functionals.J_T_re"><code class="xref py py-func docutils literal notranslate"><span class="pre">J_T_re()</span></code></a>.</p>
</li>
</ul>
</div>
<div class="section" id="update-equation">
<h2>Update equation<a class="headerlink" href="#update-equation" title="Permalink to this headline">¶</a></h2>
<p>Krotov’s method is based on a rigorous examination of the conditions for
calculating the updated fields <span class="math notranslate nohighlight">\(\epsilon_l^{(i+1)}(t)\)</span> such that
<span class="math notranslate nohighlight">\(J(\{\ket{\phi_k^{(i+1)}(t)}\}, \{\epsilon_l^{(i+1)}\}) \leq
J(\{\ket{\phi_k^{(i)}(t)}\}, \{\epsilon_l^{(i)}\})\)</span> is true
<em>by construction</em> <a class="bibtex reference internal" href="99_bibliography.html#krotov-book" id="id14">[5]</a><a class="bibtex reference internal" href="99_bibliography.html#konnovarc99" id="id15">[6]</a><a class="bibtex reference internal" href="99_bibliography.html#palaopra2003" id="id16">[10]</a><a class="bibtex reference internal" href="99_bibliography.html#reichjcp12" id="id17">[9]</a><a class="bibtex reference internal" href="99_bibliography.html#sklarzpra2002" id="id18">[7]</a>.
It achieves this by adding a vanishing quantity to the functional that
disentangles the implicit dependence of <span class="math notranslate nohighlight">\(\{\ket{\phi_k}\}\)</span> and
<span class="math notranslate nohighlight">\(\{\epsilon_l(t)\}\)</span> in the variational calculus. Specifically, the
derivation formulates an auxiliary functional <span class="math notranslate nohighlight">\(L[\{\ket{\phi_k^{(i)}(t)}\},
\{\epsilon_l^{(i)}(t)\}, \Phi]\)</span> that is equivalent to
<span class="math notranslate nohighlight">\(J[\{\ket{\phi_k^{(i)}(t)}\}, \{\epsilon_l^{(i)}(t)\}]\)</span>, but includes an
arbitrary scalar potential <span class="math notranslate nohighlight">\(\Phi\)</span>. The freedom in this scalar potential is then
used to formulate a condition to ensure monotonic convergence,</p>
<div class="math notranslate nohighlight" id="equation-krotov-proto-update">
<span class="eqno">(7)<a class="headerlink" href="#equation-krotov-proto-update" title="Permalink to this equation">¶</a></span>\[\begin{split}\left.\frac{\partial g_a}{\partial \epsilon}\right\vert_{\epsilon^{(i+1)}(t)}
= 2 \Im
  \sum_{k=1}^{N}
  \Bigg\langle
    \chi_k^{(i)}(t)
  \Bigg\vert
    \Bigg(
       \left.\frac{\partial \Op{H}}{\partial \epsilon}\right\vert_{{\scriptsize \begin{matrix}\phi^{(i+1)}(t)\\\epsilon^{(i+1)}(t)\end{matrix}}}
    \Bigg)
  \Bigg\vert
    \phi_k^{(i+1)}(t)
  \Bigg\rangle\,.\end{split}\]</div>
<p>For <span class="math notranslate nohighlight">\(g_a\)</span> as in Eq. <a class="reference internal" href="#equation-g-a">(2)</a>, this condition becomes the Krotov update
equation <a class="bibtex reference internal" href="99_bibliography.html#tannor92" id="id19">[8]</a><a class="bibtex reference internal" href="99_bibliography.html#palaopra2003" id="id20">[10]</a><a class="bibtex reference internal" href="99_bibliography.html#sklarzpra2002" id="id21">[7]</a>,</p>
<div class="math notranslate nohighlight" id="equation-krotov-first-order-update">
<span class="eqno">(8)<a class="headerlink" href="#equation-krotov-first-order-update" title="Permalink to this equation">¶</a></span>\[\begin{split}\Delta\epsilon(t)
  =
\frac{S(t)}{\lambda_a} \Im \left[
  \sum_{k=1}^{N}
  \Bigg\langle
    \chi_k^{(i)}(t)
  \Bigg\vert
    \Bigg(
       \left.\frac{\partial \Op{H}}{\partial \epsilon}\right\vert_{{\scriptsize \begin{matrix}\phi^{(i+1)}(t)\\\epsilon^{(i+1)}(t)\end{matrix}}}
   \Bigg)
  \Bigg\vert
    \phi_k^{(i+1)}(t)
  \Bigg\rangle
\right]\,,\end{split}\]</div>
<p>with the equation of motion for the forward propagation of
<span class="math notranslate nohighlight">\(\ket{\phi_k}\)</span> under the optimized controls
<span class="math notranslate nohighlight">\(\epsilon^{(i+1)}(t)\)</span> of the iteration <span class="math notranslate nohighlight">\((i)\)</span>,</p>
<div class="math notranslate nohighlight" id="equation-fw-eqm">
<span class="eqno">(9)<a class="headerlink" href="#equation-fw-eqm" title="Permalink to this equation">¶</a></span>\[\frac{\partial}{\partial t} \Ket{\phi_k^{(i+1)}(t)}
= -\frac{\mathrm{i}}{\hbar} \Op{H}^{(i+1)} \Ket{\phi_k^{(i+1)}(t)}\,.\]</div>
<p>For the moment, we have assumed unitary dynamics; the generalization to
open system dynamics will be discussed later in this section. The
co-states <span class="math notranslate nohighlight">\(\ket{\chi_k^{(i)}(t)}\)</span> are propagated backwards in time
under the guess controls of iteration <span class="math notranslate nohighlight">\((i)\)</span>, i.e., the optimized
controls from the previous iteration, as</p>
<div class="math notranslate nohighlight" id="equation-bw-eqm">
<span class="eqno">(10)<a class="headerlink" href="#equation-bw-eqm" title="Permalink to this equation">¶</a></span>\[\frac{\partial}{\partial t} \Ket{\chi_k^{(i)}(t)}
= -\frac{\mathrm{i}}{\hbar} \Op{H}^{\dagger\,(i)} \Ket{\chi_k^{(i)}(t)}
  + \left.\frac{\partial g_b}{\partial \Bra{\phi_k}}\right\vert_{\phi^{(i)}(t)}\,,\]</div>
<p>with the boundary condition</p>
<div class="math notranslate nohighlight" id="equation-chi-boundary">
<span class="eqno">(11)<a class="headerlink" href="#equation-chi-boundary" title="Permalink to this equation">¶</a></span>\[\Ket{\chi_k^{(i)}(T)}
= - \left.\frac{\partial J_T}{\partial \Bra{\phi_k}}
  \right\vert_{\phi^{(i)}(T)}\,.\]</div>
<p>Here, and in the following, we have dropped the index <span class="math notranslate nohighlight">\(l\)</span> of the
controls and the associated <span class="math notranslate nohighlight">\(\lambda_{a,l}\)</span> and <span class="math notranslate nohighlight">\(S_l(t)\)</span>;
all equations are valid for each individual control.</p>
<p>Frequently, the control field <span class="math notranslate nohighlight">\(\epsilon(t)\)</span> is required to be zero
at <span class="math notranslate nohighlight">\(t=0\)</span> and <span class="math notranslate nohighlight">\(t=T\)</span> in order to smoothly switch on and off.
To ensure that the update maintains this behavior,
<span class="math notranslate nohighlight">\(S(t) \in [0,1]\)</span> is chosen as a function with those same
conditions. A typical example is a <a class="reference internal" href="API/krotov.shapes.html#krotov.shapes.flattop" title="krotov.shapes.flattop"><code class="xref py py-func docutils literal notranslate"><span class="pre">flattop()</span></code></a> function</p>
<div class="math notranslate nohighlight">
\[\begin{split}S(t) = \begin{cases}
  B(t; t_0=0, t_1=2 t_{\text{on}})
    &amp; \text{for} \quad 0 &lt; t &lt; t_{\text{on}} \\
  1 &amp; \text{for} \quad t_{\text{on}} \le t \le T - t_{\text{off}} \\
  B(t; t_0=T-2 t_{\text{off}}, t_1=T)
    &amp; \text{for} \quad T - t_{\text{on}} &lt; t &lt; T\,,
\end{cases}\end{split}\]</div>
<p>with the <a class="reference internal" href="API/krotov.shapes.html#krotov.shapes.blackman" title="krotov.shapes.blackman"><code class="xref py py-func docutils literal notranslate"><span class="pre">blackman()</span></code></a> shape <span class="math notranslate nohighlight">\(B(t; t_0, t_1)\)</span>, which is similar to a
Gaussian, but exactly zero at <span class="math notranslate nohighlight">\(t = t_0, t_1\)</span>.</p>
<p>The scaling factor <span class="math notranslate nohighlight">\(\lambda_a\)</span> controls the overall magnitude of
the pulse update, thereby taking the role of an (inverse) “step size”.
Values that are too large will change <span class="math notranslate nohighlight">\(\epsilon^{(i)}(t)\)</span> by only
a small amount in every iteration, causing slow convergence. Values that
are too small will result in numerical instability, see <a class="reference internal" href="#choiceoflambdaa"><span class="std std-ref">Choice of λₐ</span></a>.</p>
<p>The coupled equations <a class="reference internal" href="#equation-krotov-first-order-update">(8)</a>-<a class="reference internal" href="#equation-chi-boundary">(11)</a> can be
generalized to open system dynamics by replacing Hilbert space states with
density matrices, <span class="math notranslate nohighlight">\(\Op{H}\)</span> with <span class="math notranslate nohighlight">\(i \Liouville\)</span>, and brakets with
Hilbert-Schmidt products, <span class="math notranslate nohighlight">\(\langle  \cdot \vert \cdot \rangle \rightarrow
\langle\!\langle \cdot  \vert \cdot \rangle\!\rangle\)</span>. In full generality,
<span class="math notranslate nohighlight">\(\Op{H}\)</span> in Eq. <a class="reference internal" href="#equation-krotov-first-order-update">(8)</a> is the operator <span class="math notranslate nohighlight">\(H\)</span>
on the right-hand side of whatever the equation of motion for the forward
propagation of the states is, written in the form <span class="math notranslate nohighlight">\(i \hbar \dot\phi = H
\phi\)</span>, cf. Eq. <a class="reference internal" href="#equation-fw-eqm">(9)</a>, see <a class="reference internal" href="API/krotov.mu.html#module-krotov.mu" title="krotov.mu"><code class="xref py py-mod docutils literal notranslate"><span class="pre">krotov.mu</span></code></a>. Note also that the backward
propagation Eq. <a class="reference internal" href="#equation-bw-eqm">(10)</a> uses the adjoint operator, which is relevant both for a
dissipative Liouvillian <a class="bibtex reference internal" href="99_bibliography.html#bartanajcp93" id="id22">[12]</a><a class="bibtex reference internal" href="99_bibliography.html#ohtsukijcp99" id="id23">[13]</a><a class="bibtex reference internal" href="99_bibliography.html#goerznjp2014" id="id24">[14]</a> and a
non-Hermitian Hamiltonian <a class="bibtex reference internal" href="99_bibliography.html#mullerqip11" id="id25">[15]</a><a class="bibtex reference internal" href="99_bibliography.html#goerzqst2018" id="id26">[16]</a>.</p>
</div>
<div class="section" id="optimization-of-non-linear-problems-or-non-convex-functionals">
<span id="secondorderkrotov"></span><h2>Optimization of non-linear problems or non-convex functionals<a class="headerlink" href="#optimization-of-non-linear-problems-or-non-convex-functionals" title="Permalink to this headline">¶</a></h2>
<p>The condition <a class="reference internal" href="#equation-krotov-proto-update">(7)</a> and the update
Eq. <a class="reference internal" href="#equation-krotov-first-order-update">(8)</a> are based on a first-order expansion of the
auxiliary potential <span class="math notranslate nohighlight">\(\Phi\)</span> with respect to the states.
This first order is sufficient if the equation of motion is linear
(<span class="math notranslate nohighlight">\(\Op{H}\)</span> does not depend on the states <span class="math notranslate nohighlight">\(\ket{\phi_k(t)}\)</span>),
the functional <span class="math notranslate nohighlight">\(J_T\)</span> is convex (all the “standard” functionals for
quantum control are convex), and no state-dependent constraints are used
(<span class="math notranslate nohighlight">\(g_b\equiv 0\)</span>). When these conditions are not fulfilled, it is
still possible to derive an optimization algorithm with monotonic
convergence via a second term in Eq. <a class="reference internal" href="#equation-krotov-proto-update">(7)</a>
<a class="bibtex reference internal" href="99_bibliography.html#konnovarc99" id="id27">[6]</a><a class="bibtex reference internal" href="99_bibliography.html#reichjcp12" id="id28">[9]</a>,</p>
<div class="math notranslate nohighlight" id="equation-krotov-proto-update2">
<span class="eqno">(12)<a class="headerlink" href="#equation-krotov-proto-update2" title="Permalink to this equation">¶</a></span>\[\begin{split}\begin{split}
  \left.\frac{\partial g_a}{\partial \epsilon}\right\vert_{\epsilon^{(i+1)}(t)}
  &amp; =
  2 \Im \left[
    \sum_{k=1}^{N}
    \Bigg\langle
      \chi_k^{(i)}(t)
    \Bigg\vert
      \Bigg(
         \left.\frac{\partial \Op{H}}{\partial \epsilon}\right\vert_{{\scriptsize \begin{matrix}\phi^{(i+1)}(t)\\\epsilon^{(i+1)}(t)\end{matrix}}}
      \Bigg)
    \Bigg\vert
      \phi_k^{(i+1)}(t)
    \Bigg\rangle
 \right. \\ &amp; \qquad \left.
    + \frac{1}{2} \sigma(t)
    \Bigg\langle
      \Delta\phi_k(t)
    \Bigg\vert
      \Bigg(
         \left.\frac{\partial \Op{H}}{\partial \epsilon}\right\vert_{{\scriptsize \begin{matrix}\phi^{(i+1)}(t)\\\epsilon^{(i+1)}(t)\end{matrix}}}
     \Bigg)
    \Bigg\vert
      \phi_k^{(i+1)}(t)
    \Bigg\rangle
  \right]\,,
\end{split}\end{split}\]</div>
<p>with</p>
<div class="math notranslate nohighlight">
\[\ket{\Delta \phi_k(t)} \equiv \ket{\phi_k^{(i+1)}(t)} - \ket{\phi_k^{(i)}(t)}\,.\]</div>
<p>This second term is the “non-linear” or “second order” contribution.
The corresponding update quation is, assuming Eq. <a class="reference internal" href="#equation-g-a">(2)</a>,</p>
<div class="math notranslate nohighlight" id="equation-krotov-second-order-update">
<span class="eqno">(13)<a class="headerlink" href="#equation-krotov-second-order-update" title="Permalink to this equation">¶</a></span>\[\begin{split}\begin{split}
\Delta\epsilon(t)
&amp; =
\frac{S(t)}{\lambda_a}  \Im \left[
  \sum_{k=1}^{N}
  \Bigg\langle
    \chi_k^{(i)}(t)
  \Bigg\vert
    \Bigg(
       \left.\frac{\partial \Op{H}}{\partial \epsilon}\right\vert_{{\scriptsize \begin{matrix}\phi^{(i+1)}(t)\\\epsilon^{(i+1)}(t)\end{matrix}}}
    \Bigg)
  \Bigg\vert
    \phi_k^{(i+1)}(t)
  \Bigg\rangle
 \right. \\ &amp; \qquad \qquad \quad \left.
  + \frac{1}{2} \sigma(t)
  \Bigg\langle
    \Delta\phi_k(t)
  \Bigg\vert
    \Bigg(
       \left.\frac{\partial \Op{H}}{\partial \epsilon}\right\vert_{{\scriptsize \begin{matrix}\phi^{(i+1)}(t)\\\epsilon^{(i+1)}(t)\end{matrix}}}
   \Bigg)
  \Bigg\vert
    \phi_k^{(i+1)}(t)
  \Bigg\rangle
\right]\,.
\end{split}\end{split}\]</div>
<p>The prefactor <span class="math notranslate nohighlight">\(\sigma(t)\)</span> to the second order update is a scalar function
that must be chosen appropriately to ensure monotonic convergence.</p>
<p>As shown in Ref. <a class="bibtex reference internal" href="99_bibliography.html#reichjcp12" id="id29">[9]</a>, it is possible to numerically approximate
<span class="math notranslate nohighlight">\(\sigma(t)\)</span>. In Refs <a class="bibtex reference internal" href="99_bibliography.html#wattspra2015" id="id30">[17]</a><a class="bibtex reference internal" href="99_bibliography.html#goerzpra2015" id="id31">[18]</a>, non-convex
final-time functionals that depend higher than quadratically on the states are
considered, for a standard equation of motion given by a linear Schrödinger
equation. In this case,</p>
<div class="math notranslate nohighlight">
\[\sigma(t) \equiv -\max\left(\varepsilon_A,2A+\varepsilon_A\right)\,,
  \label{eq:sigma_A}\]</div>
<p>where <span class="math notranslate nohighlight">\(\varepsilon_A\)</span> is a small non-negative number that can be used to
enforce strict inequality in the second order optimality condition. The optimal
value for <span class="math notranslate nohighlight">\(A\)</span> in each iteration can be approximated numerically
as <a class="bibtex reference internal" href="99_bibliography.html#reichjcp12" id="id32">[9]</a></p>
<div class="math notranslate nohighlight">
\[A  =
  \frac{\sum_{k=1}^{N} 2 \Re\left[
     \langle \chi_k(T) \vert \Delta\phi_k(T) \rangle
  \right]
        + \Delta J_T}
       {\sum_{k=1}^{N} \Abs{\Delta\phi_k(T)}^2}
  \,,\]</div>
<p>cf. <a class="reference internal" href="API/krotov.second_order.html#krotov.second_order.numerical_estimate_A" title="krotov.second_order.numerical_estimate_A"><code class="xref py py-func docutils literal notranslate"><span class="pre">krotov.second_order.numerical_estimate_A()</span></code></a>, with</p>
<div class="math notranslate nohighlight">
\[\Delta J_T \equiv J_T(\{\phi_k^{(i+1)}(T)\}) -J_T(\{\phi_k^{(i)}(T)\})\,.\]</div>
<p>See the <a class="reference internal" href="notebooks/07_example_PE.html"><span class="std std-ref">Optimization towards a Perfect Entangler</span></a> for an example.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Even when the second order update equation is mathematically required to
guarantee monotonic convergence, very often an optimization with the
first-order update equation <a class="reference internal" href="#equation-krotov-first-order-update">(8)</a> will give
converging results. Since the second order update requires
more numerical resources (calculation and storage of the states
<span class="math notranslate nohighlight">\(\ket{\Delta\phi_k(t)}\)</span>), you should always try the optimization with
the first-order update equation first.</p>
</div>
</div>
<div class="section" id="time-discretization">
<h2>Time discretization<a class="headerlink" href="#time-discretization" title="Permalink to this headline">¶</a></h2>
<div class="figure align-default" id="id35">
<span id="figkrotovscheme"></span><a class="reference internal image-reference" href="_images/krotovscheme.svg"><img alt="Sequential update scheme in Krotov’s method on a time grid." src="_images/krotovscheme.svg" width="100%" /></a>
<p class="caption"><span class="caption-number">Fig. 1 </span><span class="caption-text">Sequential update scheme in Krotov’s method on a time grid.</span><a class="headerlink" href="#id35" title="Permalink to this image">¶</a></p>
</div>
<p>The derivation of Krotov’s method assumes time-continuous control fields. Only
in this case, monotonic convergence is mathematically guaranteed. However, for
practical numerical applications, we have to consider controls on a discrete
time grid with <span class="math notranslate nohighlight">\(nt\)</span> points running from <span class="math notranslate nohighlight">\(t=0\)</span> to <span class="math notranslate nohighlight">\(t=T\)</span>, with
a time step <span class="math notranslate nohighlight">\(\dd t\)</span>. The states are defined on the points of the time
grid, while the controls are assumed to be constant on the intervals of the
time grid.  See the notebook <a class="reference external" href="https://nbviewer.jupyter.org/gist/goerz/21e46ea7b45c9514e460007de14419bd/Krotov_time_discretization.ipynb#">Time Discretization in Quantum Optimal Control</a>
for details.  This discretization yields the numerical scheme shown in
<a class="reference internal" href="#figkrotovscheme"><span class="std std-numref">Fig. 1</span></a>.  It proceeds as follows <a class="bibtex reference internal" href="99_bibliography.html#palaopra2003" id="id33">[10]</a>:</p>
<ol class="arabic simple">
<li><p>Construct the states <span class="math notranslate nohighlight">\(\ket{\chi^{(i)}_k(T)}\)</span> according to
Eq. <a class="reference internal" href="#equation-chi-boundary">(11)</a>. These typically
depend on the states <span class="math notranslate nohighlight">\(\{\ket{\phi^{(i)}_k(T)}\}\)</span>
forward-propagated under the optimized pulse from the previous
iteration, that is, the guess pulse in the current iteration.</p></li>
<li><p>Perform a backward-propagation using Eq. <a class="reference internal" href="#equation-bw-eqm">(10)</a> as the equation of
motion over the entire time grid. The resulting state at each point in the
time grid must be stored in memory.</p></li>
<li><p>Starting from the known initial states
<span class="math notranslate nohighlight">\(\ket{\phi_k} = \ket{\phi_k(t=0)}\)</span>, calculate the pulse update
for the first time step according to Eq. <a class="reference internal" href="#equation-krotov-first-order-update">(8)</a>,
with <span class="math notranslate nohighlight">\(t=\dd t/2\)</span> on the left-hand side (representing the first
<em>interval</em> in the time grid, on which the control pulse is defined),
and <span class="math notranslate nohighlight">\(t=0\)</span> on the right-hand side (representing the first
<em>point</em> on the time grid). This approximation of
<span class="math notranslate nohighlight">\(t \approx t + \dd t /2\)</span> is what constitutes the “time
discretization” mathematically. It resolves the seeming contradiction
in the time-continuous Eq. <a class="reference internal" href="#equation-krotov-first-order-update">(8)</a>
that the calculation of <span class="math notranslate nohighlight">\(\epsilon^{(i+1)}(t)\)</span> requires
knowledge of the states <span class="math notranslate nohighlight">\(\ket{\phi_k^{(i+1)}(t)}\)</span> obtained from
a propagation under <span class="math notranslate nohighlight">\(\epsilon^{(i+1)}(t)\)</span>.</p></li>
<li><p>Use the updated field <span class="math notranslate nohighlight">\(\epsilon^{(i+1)}(\dd t/2)\)</span> for the first
interval to propagate <span class="math notranslate nohighlight">\(\ket{\phi_k(t=0)}\)</span> for a single time
step to <span class="math notranslate nohighlight">\(\ket{\phi_k^{(i+1)}(t=\dd t)}\)</span>, with
Eq. <a class="reference internal" href="#equation-fw-eqm">(9)</a> as the equation of motion. The
updates then proceed sequentially, until the final forward-propagated
state <span class="math notranslate nohighlight">\(\ket{\phi^{(i+1)}_k(T)}\)</span> is reached.</p></li>
<li><p>The updated control field becomes the guess control for the next
iteration of the algorithm, starting again at step 1. The
optimization continues until the value of the functional <span class="math notranslate nohighlight">\(J_T\)</span>
falls below some predefined threshold, or convergence is reached,
i.e., <span class="math notranslate nohighlight">\(\Delta J_T\)</span> approaches zero so that no further
significant improvement of <span class="math notranslate nohighlight">\(J_T\)</span> is to be expected.</p></li>
</ol>
<p>For multiple objectives, the scheme can run in parallel, and each objective
contributes a term to the update. Summation of these terms yields the sum
in Eq. <a class="reference internal" href="#equation-krotov-first-order-update">(8)</a>. See <a class="reference internal" href="API/krotov.parallelization.html#module-krotov.parallelization" title="krotov.parallelization"><code class="xref py py-mod docutils literal notranslate"><span class="pre">krotov.parallelization</span></code></a> for
details. For a second-order update, the forward propagated states from step 4,
both for the current iteration and the previous iteration, must be stored in
memory over the entire time grid.</p>
</div>
<div class="section" id="choice-of-a">
<span id="choiceoflambdaa"></span><h2>Choice of λₐ<a class="headerlink" href="#choice-of-a" title="Permalink to this headline">¶</a></h2>
<p>The monotonic convergence of Krotov’s method is only guaranteed in the
continuous limit; a coarse time step must be compensated by larger values of
the inverse step size <span class="math notranslate nohighlight">\(\lambda_a\)</span>, slowing down convergence. Generally,
choosing <span class="math notranslate nohighlight">\(\lambda_a\)</span> too small will lead to numerical instabilities and
unphysical features in the optimized pulse. A lower limit for <span class="math notranslate nohighlight">\(\lambda_a\)</span>
can be determined from the requirement that the change
<span class="math notranslate nohighlight">\(\Delta\epsilon(t)\)</span> should be at most of the same order of magnitude as
the guess pulse <span class="math notranslate nohighlight">\(\epsilon^{(i)}(t)\)</span> for that iteration. The
Cauchy-Schwarz inequality applied to the update equation yields</p>
<div class="math notranslate nohighlight">
\[\Norm{\Delta \epsilon(t)}_{\infty}
  \le
  \frac{\Norm{S(t)}}{\lambda_a}
  \sum_{k} \Norm{\ket{\chi_k (t)}}_{\infty} \Norm{\ket{\phi_k (t)}}_{\infty}
  \Norm{\frac{\partial \Op{H}}{\partial \epsilon}}_{\infty}
  \stackrel{!}{\le}
  \Norm{\epsilon^{(i)}(t)}_{\infty}\,,\]</div>
<p>where <span class="math notranslate nohighlight">\(\norm{\partial \Op{H}/\partial \epsilon}_{\infty}\)</span> denotes the
supremum norm (with respect to time) of the operator norms of the operators
<span class="math notranslate nohighlight">\(\partial \Op{H}/\partial \epsilon\)</span> obtained at time <span class="math notranslate nohighlight">\(t\)</span>.  Since
<span class="math notranslate nohighlight">\(S(t) \in [0,1]\)</span> and <span class="math notranslate nohighlight">\(\ket{\phi_k}\)</span> is normalized, the condition
for <span class="math notranslate nohighlight">\(\lambda_a\)</span> becomes</p>
<div class="math notranslate nohighlight">
\[\lambda_a \ge
  \frac{1}{\Norm{\epsilon^{(i)}(t)}_{\infty}}
  \left[ \sum_{k} \Norm{\ket{\chi_k(t)}}_{\infty} \right]
  \Norm{\frac{\partial \Op{H}}{\partial \epsilon}}_{\infty}\,.\]</div>
<p>From a practical point of view, the best strategy is to start the
optimization with a comparatively large value of <span class="math notranslate nohighlight">\(\lambda_a\)</span>, and
after a few iterations lower <span class="math notranslate nohighlight">\(\lambda_a\)</span> as far as possible
without introducing numerical instabilities. The value of
<span class="math notranslate nohighlight">\(\lambda_a\)</span> may be adjusted dynamically with respect to the rate of
convergence. Generally, the optimal choice of <span class="math notranslate nohighlight">\(\lambda_a\)</span> requires
some trial and error.</p>
</div>
<div class="section" id="rotating-wave-approximation">
<h2>Rotating wave approximation<a class="headerlink" href="#rotating-wave-approximation" title="Permalink to this headline">¶</a></h2>
<p>When using the rotating wave approximation (RWA), it is important to remember
that the target states are usually defined in the lab frame, not in the
rotating frame. This is relevant for the construction of
<span class="math notranslate nohighlight">\(\ket{\chi_k(T)}\)</span>. When doing a simple optimization, such as a
state-to-state or a gate optimization, the  easiest approach is to transform
the target states to the rotating frame before calculating
<span class="math notranslate nohighlight">\(\ket{\chi_k(T)}\)</span>. This is both straightforward and numerically
efficient.</p>
<p>Another solution would be to transform the result of the forward propagation
<span class="math notranslate nohighlight">\(\ket{\phi_k(T)}\)</span> from the rotating frame to the lab frame, then
constructing <span class="math notranslate nohighlight">\(\ket{\chi_k(T)}\)</span>, and finally to transform
<span class="math notranslate nohighlight">\(\ket{\chi_k(T)}\)</span> back to the rotating frame, before starting the
backward propagation.</p>
<p>When the RWA is used the control fields are
complex-valued. In this case the Krotov update equation is valid for
both the real and the imaginary part independently. The most straightforward
implementation of the method is for real controls only, requiring that any
complex control Hamiltonian is rewritten as two independent control
Hamiltonians, one for the real part and one for the imaginary part of the
control field. For example,</p>
<div class="math notranslate nohighlight">
\[\epsilon^*(t) \Op{a} + \epsilon(t) \Op{a}^\dagger
=  \epsilon_{\text{re}}(t) (\Op{a} + \Op{a}^\dagger) + \epsilon_{\text{im}}(t) (i \Op{a}^\dagger - i \Op{a})\]</div>
<p>with two independent control fields <span class="math notranslate nohighlight">\(\epsilon_{\text{re}}(t)= \Re[\epsilon(t)]\)</span> and
<span class="math notranslate nohighlight">\(\epsilon_{\text{im}}(t) = \Im[\epsilon(t)]\)</span>.</p>
<p>See the <a class="reference internal" href="notebooks/02_example_lambda_system_rwa_complex_pulse.html"><span class="std std-ref">Optimization of a State-to-State Transfer in a Lambda System in the RWA</span></a> for an
example.</p>
</div>
<div class="section" id="optimization-in-liouville-space">
<h2>Optimization in Liouville space<a class="headerlink" href="#optimization-in-liouville-space" title="Permalink to this headline">¶</a></h2>
<p>The control equations have been written in the notation of Hilbert space.
However, they are equally valid for a gate optimization in Liouville space, by
replacing Hilbert space states with density matrices, <span class="math notranslate nohighlight">\(\Op{H}\)</span> with
<span class="math notranslate nohighlight">\(i \Liouville\)</span> (cf. <a class="reference internal" href="API/krotov.mu.html#module-krotov.mu" title="krotov.mu"><code class="xref py py-mod docutils literal notranslate"><span class="pre">krotov.mu</span></code></a>), and inner products with
Hilbert-Schmidt products, <span class="math notranslate nohighlight">\(\langle  \cdot \vert \cdot \rangle \rightarrow
\langle\!\langle \cdot  \vert \cdot \rangle\!\rangle\)</span>, cf., e.g.,
Ref. <a class="bibtex reference internal" href="99_bibliography.html#goerznjp2014" id="id34">[14]</a>.</p>
<p>See the <a class="reference internal" href="notebooks/04_example_dissipative_qubit_reset.html"><span class="std std-ref">Optimization of Dissipative Qubit Reset</span></a> for an
example.</p>
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