{
"cells": [
{
"cell_type": "markdown",
"id": "sit-title",
"metadata": {},
"source": [
"# Self-Induced Transparency: Area Theorem and Solitons\n",
"\n",
"The **McCall–Hahn area theorem** governs how the pulse area $\\mathcal{A} = \\int_{-\\infty}^\\infty \\Omega(t)\\,dt$ evolves as a pulse propagates through a resonant absorbing medium:\n",
"\n",
"$$\\frac{d\\mathcal{A}}{dz} = -\\frac{\\alpha}{2}\\sin\\mathcal{A}$$\n",
"\n",
"This has the same form as the equation of motion for a pendulum, and its fixed points tell us everything about what happens to any pulse entering the medium:\n",
"\n",
"| Fixed point | Stability | Outcome |\n",
"|:-----------:|:---------:|--------|\n",
"| 0 | stable | weak pulses absorbed |\n",
"| π | unstable | pulses tip toward 0 or 2π |\n",
"| 2π | stable | **SIT soliton** propagates without loss |\n",
"| 3π | unstable | one 2π soliton emitted |\n",
"| 4π | stable | two 2π solitons |\n",
"| … | | |\n",
"\n",
"Even multiples of π are stable: a pulse with area near $2n\\pi$ reshapes toward exactly $2n\\pi$, splitting into $n$ **solitons** that propagate without attenuation. This is **self-induced transparency** (SIT). Odd multiples are unstable saddle points.\n",
"\n",
"The area theorem is also **shape-independent**: only the integrated area matters, not the pulse envelope. A Gaussian with area $1.8\\pi$ reshapes toward $2\\pi$ just as a sech pulse does.\n",
"\n",
"This notebook demonstrates the full picture, starting from the fundamental SIT soliton and then exploring what the area theorem predicts for other input areas."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "sit-setup",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:02.104177Z",
"iopub.status.busy": "2026-05-05T15:14:02.103979Z",
"iopub.status.idle": "2026-05-05T15:14:02.757305Z",
"shell.execute_reply": "2026-05-05T15:14:02.756707Z"
}
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"t_width=0.001, t_min=-0.0100, t_max=0.0400, interaction_strength=5000.0\n"
]
}
],
"source": [
"import numpy as np\n",
"from maxwellbloch import mb_solve, plot\n",
"\n",
"from maxwellbloch.utility import SECH_FWHM_CONV\n",
"t_width = 0.001\n",
"t_min = -t_width * 10\n",
"t_max = t_width * 40\n",
"interaction_strength = 5 / t_width\n",
"gauss_fwhm = t_width / SECH_FWHM_CONV\n",
"print(f\"t_width={t_width}, t_min={t_min:.4f}, t_max={t_max:.4f}, interaction_strength={interaction_strength:.1f}\")"
]
},
{
"cell_type": "markdown",
"id": "sit-2pi-intro",
"metadata": {},
"source": [
"## The 2π SIT soliton\n",
"\n",
"A sech pulse with area exactly $2\\pi$ is the **central object of self-induced transparency**. It is an exact analytical solution of the Maxwell-Bloch equations: it propagates through the resonant medium without any attenuation, slowed relative to vacuum by a delay proportional to the optical depth. The medium absorbs energy from the leading edge and re-emits it coherently into the trailing edge — the two processes cancel exactly.\n",
"\n",
"This is the stable fixed point at $\\mathcal{A} = 2\\pi$. All other inputs are understood in relation to it."
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "sit-2pi-json",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:02.759107Z",
"iopub.status.busy": "2026-05-05T15:14:02.758948Z",
"iopub.status.idle": "2026-05-05T15:14:02.766397Z",
"shell.execute_reply": "2026-05-05T15:14:02.765964Z"
}
},
"outputs": [],
"source": [
"mbs_2pi = mb_solve.MBSolve().from_json_str(f\"\"\"\n",
"{{\n",
" \"atom\": {{\n",
" \"fields\": [{{\"coupled_levels\": [[0, 1]], \"rabi_freq_t_func\": \"sech\",\n",
" \"rabi_freq_t_args\": {{\"n_pi\": 2.0, \"centre\": 0.0, \"width\": {t_width}}}}}],\n",
" \"num_states\": 2,\n",
" \"decays\": [{{\"channels\": [[0, 1]], \"rate\": 1.0}}]\n",
" }},\n",
" \"t_min\": {t_min}, \"t_max\": {t_max}, \"t_steps\": 500,\n",
" \"z_min\": -0.5, \"z_max\": 1.5, \"z_steps\": 200,\n",
" \"interaction_strengths\": [{interaction_strength}], \"savefile\": \"mbs-sit-2pi\"\n",
"}}\n",
"\"\"\"\n",
")\n",
"mbs_2pi.mbsolve(recalc=False);"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "sit-2pi-spacetime",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:02.767829Z",
"iopub.status.busy": "2026-05-05T15:14:02.767748Z",
"iopub.status.idle": "2026-05-05T15:14:03.388435Z",
"shell.execute_reply": "2026-05-05T15:14:03.387616Z"
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"outputs": [
{
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"text/html": [
" \n",
" \n",
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"output_type": "display_data"
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{
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""
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"source": [
"fig = plot.field_spacetime(mbs_2pi, show_z_bounds=[0, 1])\n",
"fig.update_layout(title=\"2π sech — SIT soliton propagates without loss\")\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "sit-2pi-area",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.392381Z",
"iopub.status.busy": "2026-05-05T15:14:03.392266Z",
"iopub.status.idle": "2026-05-05T15:14:03.411063Z",
"shell.execute_reply": "2026-05-05T15:14:03.410299Z"
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"data": {
"text/html": [
""
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],
"source": [
"fig = plot.pulse_area(mbs_2pi)\n",
"fig.update_layout(title=\"2π sech — area constant throughout\", yaxis_range=[0, 4])\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "markdown",
"id": "sit-1pi-intro",
"metadata": {},
"source": [
"## 1π — absorbed\n",
"\n",
"A sech pulse with area $\\mathcal{A} < \\pi$ lies in the basin of attraction of the zero fixed point: it is simply absorbed. There is no coherent re-emission to balance the absorption, so the pulse decays monotonically as it propagates. The pulse area tracks the area theorem prediction, falling smoothly toward zero."
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "sit-1pi-json",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.412805Z",
"iopub.status.busy": "2026-05-05T15:14:03.412696Z",
"iopub.status.idle": "2026-05-05T15:14:03.420596Z",
"shell.execute_reply": "2026-05-05T15:14:03.419764Z"
}
},
"outputs": [],
"source": [
"mbs_1pi = mb_solve.MBSolve().from_json_str(f\"\"\"\n",
"{{\n",
" \"atom\": {{\n",
" \"fields\": [{{\"coupled_levels\": [[0, 1]], \"rabi_freq_t_func\": \"sech\",\n",
" \"rabi_freq_t_args\": {{\"n_pi\": 1.0, \"centre\": 0.0, \"width\": {t_width}}}}}],\n",
" \"num_states\": 2,\n",
" \"decays\": [{{\"channels\": [[0, 1]], \"rate\": 1.0}}]\n",
" }},\n",
" \"t_min\": {t_min}, \"t_max\": {t_max}, \"t_steps\": 500,\n",
" \"z_min\": -0.5, \"z_max\": 1.5, \"z_steps\": 200,\n",
" \"interaction_strengths\": [{interaction_strength}], \"savefile\": \"mbs-sit-1pi\"\n",
"}}\n",
"\"\"\"\n",
")\n",
"mbs_1pi.mbsolve(recalc=False);"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "sit-1pi-spacetime",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.422394Z",
"iopub.status.busy": "2026-05-05T15:14:03.422271Z",
"iopub.status.idle": "2026-05-05T15:14:03.447904Z",
"shell.execute_reply": "2026-05-05T15:14:03.447368Z"
}
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"outputs": [
{
"data": {
"text/html": [
" \n",
" \n",
" "
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"output_type": "display_data"
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{
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""
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"source": [
"fig = plot.field_spacetime(mbs_1pi, show_z_bounds=[0, 1])\n",
"fig.update_layout(title=\"1π sech — absorbed\")\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "sit-1pi-area",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.451250Z",
"iopub.status.busy": "2026-05-05T15:14:03.451147Z",
"iopub.status.idle": "2026-05-05T15:14:03.468731Z",
"shell.execute_reply": "2026-05-05T15:14:03.468240Z"
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"text/html": [
" \n",
" \n",
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{
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"text/html": [
""
]
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"metadata": {},
"output_type": "display_data"
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],
"source": [
"fig = plot.pulse_area(mbs_1pi)\n",
"fig.update_layout(title=\"1π sech — area decays to 0\", yaxis_range=[0, 4])\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "markdown",
"id": "sit-3pi-intro",
"metadata": {},
"source": [
"## 3π — one 2π soliton emitted\n",
"\n",
"The $3\\pi$ fixed point is **unstable**: any perturbation drives the area away from it. Propagation tips a $3\\pi$ input downward toward the nearest stable point at $2\\pi$. One $2\\pi$ soliton forms and the remaining area ($3\\pi - 2\\pi = \\pi$) falls below the unstable fixed point at $\\pi$ and is absorbed. The area plot shows the transition clearly."
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "sit-3pi-json",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.470850Z",
"iopub.status.busy": "2026-05-05T15:14:03.470717Z",
"iopub.status.idle": "2026-05-05T15:14:03.478048Z",
"shell.execute_reply": "2026-05-05T15:14:03.477587Z"
}
},
"outputs": [],
"source": [
"mbs_3pi = mb_solve.MBSolve().from_json_str(f\"\"\"\n",
"{{\n",
" \"atom\": {{\n",
" \"fields\": [{{\"coupled_levels\": [[0, 1]], \"rabi_freq_t_func\": \"sech\",\n",
" \"rabi_freq_t_args\": {{\"n_pi\": 3.0, \"centre\": 0.0, \"width\": {t_width}}}}}],\n",
" \"num_states\": 2,\n",
" \"decays\": [{{\"channels\": [[0, 1]], \"rate\": 1.0}}]\n",
" }},\n",
" \"t_min\": {t_min}, \"t_max\": {t_max}, \"t_steps\": 500,\n",
" \"z_min\": -0.5, \"z_max\": 1.5, \"z_steps\": 200,\n",
" \"interaction_strengths\": [{interaction_strength}], \"savefile\": \"mbs-sit-3pi\"\n",
"}}\n",
"\"\"\"\n",
")\n",
"mbs_3pi.mbsolve(recalc=False);"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "sit-3pi-spacetime",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.479923Z",
"iopub.status.busy": "2026-05-05T15:14:03.479815Z",
"iopub.status.idle": "2026-05-05T15:14:03.504522Z",
"shell.execute_reply": "2026-05-05T15:14:03.503717Z"
}
},
"outputs": [
{
"data": {
"text/html": [
" \n",
" \n",
" "
]
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"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fig = plot.field_spacetime(mbs_3pi, show_z_bounds=[0, 1])\n",
"fig.update_layout(title=\"3π sech — reshapes to one 2π soliton\")\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "sit-3pi-area",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.507581Z",
"iopub.status.busy": "2026-05-05T15:14:03.507478Z",
"iopub.status.idle": "2026-05-05T15:14:03.524847Z",
"shell.execute_reply": "2026-05-05T15:14:03.524465Z"
}
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"outputs": [
{
"data": {
"text/html": [
" \n",
" \n",
" "
]
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"output_type": "display_data"
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{
"data": {
"text/html": [
""
]
},
"metadata": {},
"output_type": "display_data"
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],
"source": [
"fig = plot.pulse_area(mbs_3pi)\n",
"fig.update_layout(title=\"3π sech — area settles at 2π\", yaxis_range=[0, 4])\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "markdown",
"id": "sit-shape-intro",
"metadata": {},
"source": [
"## Shape independence — Gaussian pulses\n",
"\n",
"The area theorem contains no information about pulse shape — only $\\mathcal{A}$ appears in $d\\mathcal{A}/dz = -(\\alpha/2)\\sin\\mathcal{A}$. A Gaussian, square, or any other envelope with the same area undergoes the same fate. A $1.8\\pi$ Gaussian reshapes toward the $2\\pi$ sech soliton, just as a $1.8\\pi$ sech pulse would — the attractor depends only on the area, not the shape."
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "sit-g18-json",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.526206Z",
"iopub.status.busy": "2026-05-05T15:14:03.526119Z",
"iopub.status.idle": "2026-05-05T15:14:03.533999Z",
"shell.execute_reply": "2026-05-05T15:14:03.533576Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Input area: 1.800π\n"
]
}
],
"source": [
"mbs_g18 = mb_solve.MBSolve().from_json_str(f\"\"\"\n",
"{{\n",
" \"atom\": {{\n",
" \"fields\": [{{\"coupled_levels\": [[0, 1]], \"rabi_freq_t_func\": \"gaussian\",\n",
" \"rabi_freq_t_args\": {{\"n_pi\": 1.8, \"centre\": 0.0, \"fwhm\": {gauss_fwhm}}}}}],\n",
" \"num_states\": 2,\n",
" \"decays\": [{{\"channels\": [[0, 1]], \"rate\": 1.0}}]\n",
" }},\n",
" \"t_min\": {t_min}, \"t_max\": {t_max}, \"t_steps\": 500,\n",
" \"z_min\": -0.5, \"z_max\": 1.5, \"z_steps\": 200,\n",
" \"interaction_strengths\": [{interaction_strength}], \"savefile\": \"mbs-sit-gauss-18pi\"\n",
"}}\n",
"\"\"\"\n",
")\n",
"mbs_g18.mbsolve(recalc=False)\n",
"print(f\"Input area: {np.trapezoid(mbs_g18.Omegas_zt[0, 0, :].real, mbs_g18.tlist) / np.pi:.3f}π\")"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "sit-g18-spacetime",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.535296Z",
"iopub.status.busy": "2026-05-05T15:14:03.535220Z",
"iopub.status.idle": "2026-05-05T15:14:03.559415Z",
"shell.execute_reply": "2026-05-05T15:14:03.558908Z"
}
},
"outputs": [
{
"data": {
"text/html": [
" \n",
" \n",
" "
]
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"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
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],
"source": [
"fig = plot.field_spacetime(mbs_g18, show_z_bounds=[0, 1])\n",
"fig.update_layout(title=\"1.8π Gaussian — reshapes toward 2π soliton\")\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "sit-g18-area",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.563237Z",
"iopub.status.busy": "2026-05-05T15:14:03.563098Z",
"iopub.status.idle": "2026-05-05T15:14:03.579584Z",
"shell.execute_reply": "2026-05-05T15:14:03.579094Z"
}
},
"outputs": [
{
"data": {
"text/html": [
" \n",
" \n",
" "
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"fig = plot.pulse_area(mbs_g18)\n",
"fig.update_layout(title=\"1.8π Gaussian — area asymptotes to 2π\", yaxis_range=[0, 3])\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "markdown",
"id": "sit-g18-analysis",
"metadata": {},
"source": [
"The $1.8\\pi$ Gaussian reshapes toward $2\\pi$ just as a sech pulse does. The attractor is always the same $2\\pi$ sech soliton regardless of input shape — the pulse distorts in transit, shedding the excess as radiation. Having established the area theorem for single-soliton outcomes, we now look at inputs large enough to produce multiple solitons."
]
},
{
"cell_type": "markdown",
"id": "sit-4pi-intro",
"metadata": {},
"source": [
"## 4π — two solitons\n",
"\n",
"A $4\\pi$ input is near a second stable fixed point. It breaks into **two** $2\\pi$ solitons, each carrying half the original area. The two solitons emerge with different widths and therefore different group velocities — the narrower soliton travels faster — so they separate as they propagate through the medium."
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "sit-4pi-json",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.581193Z",
"iopub.status.busy": "2026-05-05T15:14:03.581102Z",
"iopub.status.idle": "2026-05-05T15:14:03.588695Z",
"shell.execute_reply": "2026-05-05T15:14:03.588185Z"
}
},
"outputs": [],
"source": [
"mbs_4pi = mb_solve.MBSolve().from_json_str(f\"\"\"\n",
"{{\n",
" \"atom\": {{\n",
" \"fields\": [{{\"coupled_levels\": [[0, 1]], \"rabi_freq_t_func\": \"sech\",\n",
" \"rabi_freq_t_args\": {{\"n_pi\": 4.0, \"centre\": 0.0, \"width\": {t_width}}}}}],\n",
" \"num_states\": 2,\n",
" \"decays\": [{{\"channels\": [[0, 1]], \"rate\": 1.0}}]\n",
" }},\n",
" \"t_min\": {t_min}, \"t_max\": {t_max}, \"t_steps\": 500,\n",
" \"z_min\": -0.5, \"z_max\": 1.5, \"z_steps\": 200,\n",
" \"interaction_strengths\": [{interaction_strength}], \"savefile\": \"mbs-sit-4pi\"\n",
"}}\n",
"\"\"\"\n",
")\n",
"mbs_4pi.mbsolve(recalc=False);"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "sit-4pi-spacetime",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.590206Z",
"iopub.status.busy": "2026-05-05T15:14:03.590127Z",
"iopub.status.idle": "2026-05-05T15:14:03.614213Z",
"shell.execute_reply": "2026-05-05T15:14:03.613763Z"
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},
"outputs": [
{
"data": {
"text/html": [
" \n",
" \n",
" "
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{
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""
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],
"source": [
"fig = plot.field_spacetime(mbs_4pi, show_z_bounds=[0, 1])\n",
"fig.update_layout(title=\"4π sech — breaks into two 2π solitons\")\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "sit-4pi-area",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.617872Z",
"iopub.status.busy": "2026-05-05T15:14:03.617778Z",
"iopub.status.idle": "2026-05-05T15:14:03.633715Z",
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{
"data": {
"text/html": [
" \n",
" \n",
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{
"data": {
"text/html": [
""
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"metadata": {},
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],
"source": [
"fig = plot.pulse_area(mbs_4pi)\n",
"fig.update_layout(title=\"4π sech — total area conserved\", yaxis_range=[0, 6])\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "markdown",
"id": "sit-6pi-intro",
"metadata": {},
"source": [
"## 6π — three solitons\n",
"\n",
"A $6\\pi$ input breaks into **three** $2\\pi$ solitons. The general rule is clear: a $2n\\pi$ input produces exactly $n$ solitons. The solitons again emerge with different widths and speeds, fanning out as they propagate."
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "sit-6pi-json",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.635164Z",
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"outputs": [],
"source": [
"mbs_6pi = mb_solve.MBSolve().from_json_str(f\"\"\"\n",
"{{\n",
" \"atom\": {{\n",
" \"fields\": [{{\"coupled_levels\": [[0, 1]], \"rabi_freq_t_func\": \"sech\",\n",
" \"rabi_freq_t_args\": {{\"n_pi\": 6.0, \"centre\": 0.0, \"width\": {t_width}}}}}],\n",
" \"num_states\": 2,\n",
" \"decays\": [{{\"channels\": [[0, 1]], \"rate\": 1.0}}]\n",
" }},\n",
" \"t_min\": {t_min}, \"t_max\": {t_max}, \"t_steps\": 500,\n",
" \"z_min\": -0.5, \"z_max\": 1.5, \"z_steps\": 200, \"z_steps_inner\": 2,\n",
" \"interaction_strengths\": [{interaction_strength}], \"savefile\": \"mbs-sit-6pi\"\n",
"}}\n",
"\"\"\"\n",
")\n",
"mbs_6pi.mbsolve(recalc=False);"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "sit-6pi-spacetime",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.643709Z",
"iopub.status.busy": "2026-05-05T15:14:03.643633Z",
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" \n",
" \n",
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""
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"source": [
"fig = plot.field_spacetime(mbs_6pi, show_z_bounds=[0, 1])\n",
"fig.update_layout(title=\"6π sech — breaks into three 2π solitons\")\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "sit-6pi-area",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.672016Z",
"iopub.status.busy": "2026-05-05T15:14:03.671875Z",
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""
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],
"source": [
"fig = plot.pulse_area(mbs_6pi)\n",
"fig.update_layout(title=\"6π sech — total area conserved\", yaxis_range=[0, 8])\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "markdown",
"id": "sit-collision-intro",
"metadata": {},
"source": [
"## Soliton collision\n",
"\n",
"The Maxwell-Bloch equations in the sharp-line limit are integrable, which means SIT solitons collide **elastically**: two solitons passing through each other emerge with their original shapes and speeds intact, shifted only in position. A wide (slow) and a narrow (fast) $2\\pi$ soliton are launched together so the faster one overtakes the slower inside the medium."
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "sit-collision-json",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.689408Z",
"iopub.status.busy": "2026-05-05T15:14:03.689310Z",
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"shell.execute_reply": "2026-05-05T15:14:03.692396Z"
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},
"outputs": [],
"source": [
"t_width_wide = 2.0 * t_width\n",
"t_width_narrow = 1.0 * t_width\n",
"t_max_col = t_width * 80\n",
"\n",
"mbs_col = mb_solve.MBSolve().from_json_str(f\"\"\"\n",
"{{\n",
" \"atom\": {{\"fields\": [{{\"coupled_levels\": [[0, 1]]}}], \"num_states\": 2, \"decays\": [{{\"channels\": [[0, 1]], \"rate\": 1.0}}]}},\n",
" \"t_min\": {t_min}, \"t_max\": {t_max_col}, \"t_steps\": 500,\n",
" \"z_min\": -0.5, \"z_max\": 1.5, \"z_steps\": 200,\n",
" \"interaction_strengths\": [{interaction_strength}], \"savefile\": \"mbs-sit-collision\"\n",
"}}\n",
"\"\"\"\n",
")"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "sit-collision-setup",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.694215Z",
"iopub.status.busy": "2026-05-05T15:14:03.694106Z",
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"shell.execute_reply": "2026-05-05T15:14:03.700508Z"
}
},
"outputs": [],
"source": [
"from maxwellbloch import t_funcs\n",
"\n",
"probe = mbs_col.atom.fields[0]\n",
"probe.rabi_freq_t_func = lambda t, args: t_funcs.sech(1)(t, args) + t_funcs.sech(2)(t, args)\n",
"probe.rabi_freq_t_args = {\n",
" \"n_pi_1\": 2.0, \"centre_1\": 0.0, \"width_1\": t_width_wide,\n",
" \"n_pi_2\": 2.0, \"centre_2\": t_width * 5, \"width_2\": t_width_narrow,\n",
"}\n",
"mbs_col.atom.build_operators()\n",
"mbs_col.init_Omegas_zt()\n",
"mbs_col.mbsolve(recalc=False);"
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "sit-collision-spacetime",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-05T15:14:03.702342Z",
"iopub.status.busy": "2026-05-05T15:14:03.702242Z",
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"source": [
"fig = plot.field_spacetime(mbs_col, show_z_bounds=[0, 1])\n",
"fig.update_layout(title=\"2π soliton collision — elastic scattering\")\n",
"fig.show(renderer='notebook_connected')"
]
},
{
"cell_type": "markdown",
"id": "sit-collision-analysis",
"metadata": {},
"source": [
"The two solitons briefly merge into a high-amplitude transient, then re-emerge unchanged in width and speed. The only lasting effect is a position shift — each soliton arrives slightly earlier or later than it would have without the collision. This is the hallmark of elastic, soliton-preserving scattering."
]
},
{
"cell_type": "markdown",
"id": "sit-summary",
"metadata": {},
"source": [
"## Summary\n",
"\n",
"| Input | Outcome |\n",
"|:-----:|--------|\n",
"| 2π sech | SIT soliton — propagates without loss |\n",
"| 1π sech | absorbed |\n",
"| 3π sech | one 2π soliton emitted, remainder absorbed |\n",
"| 1.8π Gaussian | reshapes toward 2π (shape-independent) |\n",
"| 4π sech | two 2π solitons |\n",
"| 6π sech | three 2π solitons |\n",
"\n",
"The area theorem unifies all of these: $d\\mathcal{A}/dz = -(\\alpha/2)\\sin\\mathcal{A}$ has a pendulum-like fixed-point structure, and the SIT soliton is the $2\\pi$ stable fixed point realised as an exact propagating solution. Soliton collisions are elastic because the underlying equations are integrable.\n",
"\n",
"## References\n",
"\n",
"1. S. L. McCall, E. L. Hahn, *Self-Induced Transparency by Pulsed Coherent Light*, PRL **18**, 908 (1967).\n",
"2. S. L. McCall, E. L. Hahn, *Self-Induced Transparency*, Phys. Rev. **183**, 457 (1969).\n",
"3. L. Allen, J. H. Eberly, *Optical Resonance and Two-Level Atoms* (Dover, 1987), Ch. 4."
]
}
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