Revised index.md (#952)

* Revised index.md

* Added links

* Forgot to add images

docs/index.md

@@ -1,7 +1,7 @@
 <div align="center">
 <picture>
-  <img id="logo_light_mode" src="assets/logo.png" alt="Bloqade Logo">
-  <img id="logo_dark_mode" src="assets/logo-dark.png" alt="Bloqade Logo">
+  <img id="logo_light_mode" src="assets/logo.png" style="width: 70%" alt="Bloqade Logo">
+  <img id="logo_dark_mode" src="assets/logo-dark.png" style="width: 70%" alt="Bloqade Logo">
 </picture>
 </div>
 
@@ -11,34 +11,296 @@
 ![CI](https://github.com/QuEraComputing/bloqade-python/actions/workflows/ci.yml/badge.svg)
 
 # **Welcome to Bloqade**: QuEra's Neutral Atom SDK
+
 ## What is Bloqade?
 
-Bloqade is an SDK designed to make writing and analyzing the results of analog quantum programs on QuEra's neutral atom quantum computers as seamless and flexible as possible. Currently, QuEra's hardware is on Amazon Braket, the primary method of accessing QuEra's quantum hardware. Over the alpha phase, we plan to expand the emulator capabilities to include a performance Python emulator but also a direct integration with Julia via [Bloqade.jl](https://queracomputing.github.io/Bloqade.jl/dev/).
+Bloqade is a Python SDK for [QuEra's](https://www.quera.com/) neutral atom quantum computer *Aquila* (check out our [paper](https://arxiv.org/abs/2306.11727)!). It's designed to make writing and analyzing the results of [analog quantum programs]() on *Aquila* as easy as possible. It features [custom atom geometries](index.md#customizable-atom-geometries) and [flexible waveform definitions](index.md#flexible-pulse-sequence-construction) in both [emulation and real hardware](index.md#hardware-and-emulation-backends). Bloqade interfaces with the [AWS Braket](https://aws.amazon.com/braket/) cloud service where *Aquila*  is hosted, enabling you to submit programs as well as retrieve and analyze real hardware results all-in-one.
 
-## Why A Python Version of Bloqade?
+## Installation
 
-Those coming from Bloqade.jl will most likely have this as their first question in mind. For newcomers the following information is still relevant.
+You can install the package with [`pip`](https://pip.pypa.io/en/stable/) in your Python environment of choice via:
 
-Bloqade.jl was designed as an ***emulation-first*** SDK enabling users to investigate novel physical phenomena and develop complex algorithms with bleeding-edge performance. The result of this focus is that not everything made in Bloqade.jl is compatible with quantum hardware. While the ability to submit to Braket is available in Bloqade.jl it becomes cumbersome to analyze results and keep track of parameter sweeps.
+```sh
+pip install bloqade
+```
 
-In our mission to make neutral atom quantum computing more accessible to a broader community and the valuable feedback we've received in users of our quantum hardware, we took the opportunity to create a ***hardware-first*** SDK. One that is perfectly positioned to make it even easier to reap the benefits of neutral atoms while minimizing the pain points of initial program creation and post-processing results.
+## A Glimpse of Bloqade
 
-## What does Bloqade do?
+Let's try a simple example where we drive a [Rabi oscillation][rabi-oscillation-wiki] on a single [neutral atom](). Don't worry if you're unfamiliar with [neutral atom]() physics, (you can check out our Background for more information!) the goal here is to just give you a taste of what Bloqade can do.
 
-Bloqade makes writing analog quantum programs for neutral atom quantum computers a cinch. Its interface is designed to make designing a program, analyzing its results, and building more complex algorithms as seamless as possible.
+We start by defining where our atoms go, otherwise known as the *atom geometry*. In this particular example we will use a small Honeycomb lattice:
 
-Our interface is designed to guide our users through the process of defining an analog quantum program as well as different methods to run the program, whether it be real quantum hardware or an emulator. Bloqade also provides a simple interface for analyzing the results of the program, whether it is a single run, a batch of runs, or even some types of hybrid quantum-classical algorithms.
+```python
+from bloqade.atom_arrangement import Honeycomb
 
-## Getting Started
+geometry = Honeycomb(2, lattice_spacing = 10.0)
+```
 
-### Installation
+We can verify what the atom geometry looks like by `.show()`'ing it: 
 
-You can install the package with `pip` in your Python environment of choice via:
+```python
+geometry.show()
+```
 
-```sh
-pip install bloqade
+<div align="center">
+<picture>
+  <img src="assets/index/geometry-visualization.png" style="width: 50%" alt="Geometry Visualization">
+</picture>
+</div>
+
+We now define what the [time evolution]() looks like using a *pulse sequence*. The pulse sequence here is the time profile of the Rabi Drive targeting the ground-Rydberg two level transition, which causes the [Rabi oscillations][rabi-oscillation-wiki]. We choose a constant waveform with a value of $\frac{\pi}{2} \text{rad}/\text{us}$ and a duration of $1.0 \,\text{us}$.
+This produces a $\frac{\pi}{2}$ rotation on the [Bloch sphere](https://en.wikipedia.org/wiki/Bloch_sphere) meaning our final measurements should be split 50/50 between the ground and Rydberg state.
+
+```python
+from math import pi
+rabi_program = (
+  geometry
+  .rydberg.rabi.amplitude.uniform
+  .constant(value=pi/2, duration=1.0)
+)
+```
+<!--explain what uniform is-->
+Here `rabi.amplitude` means exactly what it is, the Rabi amplitude term of the [Hamiltonian](). `uniform` refers to applying the waveform uniformly across all the atom locations. 
+
+We can visualize what our program looks like again with `.show()`:
+
+<div align="center">
+<picture>
+  <img src="assets/index/program-visualization.png" alt="Program Visualization">
+</picture>
+</div>
+
+We can now run the program through Bloqade's built-in emulator to get some results. We designate that we want the program to be run and measurements performed 100 times:
+
+```python
+emulation_results = rabi_program.bloqade.python().run(100)
+```
+
+With the results we can generate a report object that contains a number of methods for analyzing our data, including the number of counts per unique bitstring:
+
+```python
+bitstring_counts = emulation_results.report().counts()
 ```
 
-### Developing your Bloqade Program
+Which gives us:
+
+```
+[OrderedDict([('0', 55), ('1', 45)])]
+```
+
+If we want to submit our program to hardware we'll need to adjust the waveform as there is a constraint the Rabi amplitude waveform must start and end at zero. 
+This is easy to do as we can build off the atom geometry we saved previously but apply a piecewise linear waveform:
+
+```python
+hardware_rabi_program = (
+  geometry
+  .rydberg.rabi.amplitude.uniform
+  .piecewise_linear(values = [0, pi/2, pi/2, 0], durations = [0.06, 1.0, 0.06])
+)
+
+hardware_rabi_program.show()
+```
+<div align="center">
+<picture>
+  <img src="assets/index/hardware-program-visualization.png" alt="Hardware Program Visualization">
+</picture>
+</div>
+
+Now instead of using the built-in Bloqade emulator we submit the program to *Aquila*. You will need to use the [AWS CLI](https://aws.amazon.com/cli/) to obtain credentials from your AWS account 
+or set the proper environment variables before hand. 
+
+```python
+hardware_results = hardware_rabi_program.braket.aquila.run_async(100)
+```
+
+`.run_async` is a non-blocking version of the standard `.run` method, allowing you to continue work while waiting for results from *Aquila*. `.run_async` immediately returns an object you can query for the status of your tasks in the queue as well.
+
+You can do the exact same analysis you do on emulation results with hardware results too:
+
+```python
+hardware_bitstring_counts = hardware_results.report().counts()
+```
+
+If you want to try the above at once, we collected the above steps into the snippet below:
+
+```python
+from math import pi
+from bloqade.atom_arrangement import Honeycomb
+
+geometry = Honeycomb(2, lattice_spacing = 10.0)
+rabi_program = (
+  geometry
+  .rydberg.rabi.amplitude.uniform
+  .constant(value=pi/2, duration=1.0)
+)
+emulation_results = rabi_program.bloqade.python().run(100) 
+bitstring_counts = emulation_results.report().counts()
+
+hardware_rabi_program = (
+  geometry
+  .rydberg.rabi.amplitude.uniform
+  .piecewise_linear(values = [0, pi/2, pi/2, 0], durations = [0.06, 1.0, 0.06])
+)
+hardware_results = hardware_rabi_program.braket.aquila.run_async(100)
+hardware_bitstring_counts = hardware_results.report().counts()
+```
+
+<br>
+<br>
+
+## Features
+
+
+### Customizable Atom Geometries 
+
+<!--show that you can add_position on predefined geometries-->
+
+You can easily explore a number of common geometric lattices with Bloqade's `atom_arrangement`'s:
+
+```python
+from bloqade.atom_arrangement import Lieb, Square, Chain, Kagome
+
+geometry_1 = Lieb(3)
+geometry_2 = Square(2)
+geometry_3 = Chain(5)
+geometry_4 = Kagome(3)
+```
+
+
+If you're not satisfied with the [Bravais lattices](https://en.wikipedia.org/wiki/Bravais_lattice) we also allow you to modify existing Bravais lattices as follows:
+
+```python
+geometry_5 = Kagome(3).add_position((10,11))
+```
+
+You can also build your geometry completely from scratch:
+
+```python
+from bloqade import start
+
+geometry = start.add_positions([(0,0), (6,0), (12,0)])
+```
+
+### Flexible Pulse Sequence Construction
+
+Define waveforms for pulse sequences any way you like by either building (and chaining!) them immediately as part of your program:
+
+```python
+from bloqade.atom_arrangement import Square
+
+geometry = Square(2)
+target_rabi_amplitude = geometry.rydberg.rabi.amplitude.uniform
+custom_rabi_amp_waveform = (
+  target_rabi_amplitude
+  .piecewise_linear(values=[0, 10, 10, 0], durations=[0.1, 3.5, 0.1])
+  .piecewise_linear(values=[0, 5, 3, 0], durations=[0.2, 2.0, 0.2])
+)
+```
+
+Or building them separately and applying them later:
+
+```python
+from bloqade.atom_arrangement import Square, Chain
+
+geometry_1 = Square(3)
+geometry_2 = Chain(5)
+
+target_rabi_amplitude = start.rydberg.rabi.amplitude.uniform
+pulse_sequence = target_rabi_amplitude.uniform.constant(value=2.0, duration=1.5).parse_sequence()
+
+program_1 = geometry_1.apply(pulse_sequence)
+program_2 = geometry_2.apply(pulse_sequence)
+```
+
+### Hardware and Emulation Backends
+
+Go from a fast and powerful emulator:
+
+```python
+from bloqade.atom_arrangement import Square
+from math import pi
+
+geometry = Square(3, lattice_spacing = 6.5)
+target_rabi_amplitude = geometry.rydberg.rabi.amplitude.uniform
+program = (
+  target_rabi_amplitude
+  .piecewise_linear(values = [0, pi/2, pi/2, 0], durations = [0.06, 1.0, 0.06])
+)
+emulation_results = program.bloqade.python().run(100)
+```
+
+To real quantum hardware in a snap:
+
+```python
+hardware_results = program.braket.aquila().run_async(100)
+```
+
+### Simple Parameter Sweeps
+
+Use variables to make parameter sweeps easy on both emulation and hardware:
+
+```python
+from bloqade import start
+import numpy as np
+
+geometry = start.add_position((0,0))
+target_rabi_amplitude = geometry.rydberg.rabi.amplitude.uniform
+rabi_oscillation_program = (
+  target_rabi_amplitude
+  .piecewise_linear(durations = [0.06, "run_time", 0.06], values = [0, 15, 15, 0])
+)
+rabi_oscillation_job = rabi_oscillation_program.batch_assign(run_time=np.linspace(0, 3, 101))
+
+emulation_results = rabi_oscillation_job.bloqade.python().run(100)
+hardware_results = rabi_oscillation_job.braket.aquila().run(100)
+```
+
+```
+emulation_results.report().rydberg_densities()
+                0
+task_number      
+0            0.16
+1            0.35
+2            0.59
+3            0.78
+4            0.96
+...           ...
+96           0.01
+97           0.09
+98           0.24
+99           0.49
+100          0.68
+
+[101 rows x 1 columns]
+```
+
+### Quick Results Analysis
+
+Want to just see some plots of your results? `.show()` will show you the way!
+
+```python
+from bloqade.atom_arrangement import Square
+
+rabi_amplitude_values = [0.0, 15.8, 15.8, 0.0]
+rabi_detuning_values = [-16.33, -16.33, 42.66, 42.66]
+durations = [0.8, 2.4, 0.8]
+
+geometry = Square(3, lattice_spacing=5.9)
+rabi_amplitude_waveform = (
+  geometry
+  .rydberg.rabi.amplitude.uniform.piecewise_linear(durations, rabi_amplitude_values)
+)
+program = (
+  rabi_amplitude_waveform
+  .detuning.uniform.piecewise_linear(durations, rabi_detuning_values)
+)
+emulation_results = program.bloqade.python().run(100)
+emulation_results.report().show()
+```
+![](/assets/index/report-visualization.png)
+
+## Contributing to Bloqade
+
+Bloqade is released under the [Apache License, Version 2.0](https://github.com/QuEraComputing/bloqade-python/blob/main/LICENSE). If you'd like the chance to shape the future of [neutral atom]() quantum computation, see our [Contributing Guide](contributing/index.md) for more info!
 
-If you are unfamiliar with Neutral Atom Analog Hamiltonian Simulation (AHS), we recommend you read our [Bloqade 101](home/bloqade_101.md) tutorial where we will give you a in depth introduction to AHS. If you are familiar with AHS, you can skip to the [Getting Started](home/getting_started.md) tutorial, and finally for more advanced usage of Bloqade you can read our [Advanced Usage](home/advanced_usage.md) tutorial.
+[rabi-oscillation-wiki]: https://en.wikipedia.org/wiki/Rabi_cycle