Hamilton's principle is mysterious: it says when particle travels, it automatically chooses a path with stable action S. The quantum mechanical origin of the mystery is decoded by Feynman, who suggests that the particle actually samples all possible paths, each weighted by a factor \propto e^{iS/\hbar}. The classical path emerges since it survives the phase average of nearby paths. This phase coherence during the path-averge of probability amplitudes is equivalent to the stability of action.

Sometime the classical path is not unique. In this case, wave nature of particle surfaces even in the classical limit. A famous example is Young's double-slit interference for photons, elecrons, neutrons, and atoms. More generally, by beamsplitting, interferometers create two paths for particle to travel, along which actions S_{1,2} can be accumulated respectively. By measuring the output flux at a second beampslitter, the action difference \Delta S_{1,2} is inferred in unit of Planck constant \hbar. Well, \hbar is small, so matterwave interferometry are powerful for sensing tiny difference of actions between the interfering paths.

Now, unlike photons or even electrons, to construct an atom interferometer is not easy. The first atom interferometer was built in 1990's in MIT by Prof. David Pritchard, using state-of-art nanofibrication techniques at that time. There are two major difficulties agaist building atom interferometers:

1. Coherent atom source

Like optical interferometer, atom interferometer require ``monochromatic'' source to certain degree. Monochromatic beams are easier to ``split'' and put into superposition of paths. Related, the classical action S is a nonlinear function of atomic velocity. Too much spread of velocity (and associated deBroglie wavelength) makes it difficult to control the actions or to quantify differential actions. Unfortunately, simple calculation suggests to obtain moderate coherence length, even as those for a cheap LED for light, would typically require atoms to be cooled to sub-pico-kelvin temperatures! (to reach such temperatures, regular techniques might require space-based micro-gravity supports)

Fortunately, with laser cooling people can at least bring down the temperature to micro-kelvin regime, from which ``achromatic'' matterwave interferometry similar to ``white light interferometry'' for photons can be carefully desgined. 

2. Coherent beam splitters

Similar to the optical counterpart, matterewave splitters needs to be ``coherent'' to be interferometrically useful. The critieron is simple: There is fundamentally no ways for anyone, including the atom itself, to know which path to choose. Then, quantum mechanics enforces that motion of the atom to be described by a coherent superposition of propagating modes along both paths.

For photons, this coherent splitting is often achieved with half-transmission mirrors -- Look deeply, the coherent nature of optical splitting is supported by a phenomenum called ``quantum reflection''. To allow this to function, the stepwise interface should change its index substantially over the optical wavelength. However, for atoms quantum reflections by regular material surface hardly works. De Broglie wavelength of atoms easily approaches a length scale to be comparable to the van der Waals interaction. More importantly, atom-surface collisions ususally excite the electronic structure of atoms, as well as the phonon modes of material. The collisions can stick atom to the surface. More fundementally, inelastic collisions means that which-way-information is leaked to a third party, destroying the quantum coherence.

So, not so surprisingly, beamsplitting in atom interferometry do not rely on quantum reflection. Instead, from David Pritchard's nanogratings to Steven Chu's optical pulses, atom interferometry almost entirely relies on diffractions by certain engineered interactions that are spatially periodic. The diffraction shifts the atomic propagation paths in momentum space, in units of ``recoil momentum'' which is simply h/d, with d (or \lambda) to be the ``grating constant'' (or wavelenght of light pulse). Nowadays, light pulse atom interferometry are well developed with exiquisite precision, with applications ranging from probing new physics to inertial navigation.

However, sometime (or maybe a lot of times) the ``recoil momentum'' is a bit too small for advancing the true strength of atom interferometry. Further, the ``diffraction'' is often not so ``accurate'', limiting the fidelity of single-body matterwave control for quantum enhnaced operations. 

This project is also along the line of atomic state control with wideband, arbitrarily shaped pulses. Equipped with wideband optical waveform generation technique, we exploit the geometric robustness of Raman transition to enhance the control fidelity of matterwave. Here, the Raman control means during the momentum shift, the atomic state also ``flips'' from one metas-stable state to another. The ``spin-dependent kick'' technique is a corner stone to the well-developed Raman light pulse interferometry. However, our project push the limit by increasing the control speed (into nanosecond regime) and complexity (using many ``pi'' pulses), in hope of rapidly manipulating matterwave with unprecedented amount of recoil momenta in a high fidelity manner.

The new control technique, if successive, would be useful for enhancing the sensitivity and bandwidth of traditional Raman atom interferometry. The error-resilient control technique may also enable ion-based quantum information processing in novel ways.

This project is a joint experimental + theoretical effort. On the theoretical side, we learn optimal control theory by consulting with experts in the field (Prof. Haidong Yuan) for developing multi-level control strategies. We benefit from frequent exchange of ideas with theoretical experts (Prof. Yiqiu Ma, Prof. Xiaopeng Li) for applications in fundamental and quantum physics.

We are looking for new team members for this project. If you are interested in quantum optimal control techniques with hand-on experimental experience, or if you are enthusiatic to know more about atom interferometry, please drop by for a discussion.

纳秒旋量物质波光学和干涉技术

哈密顿原理很奇妙：当基本粒子运动时，似乎会“自发”的选择一条作用量S稳定的路径。这一奇妙原理的量子力学解释由费曼给出，他认为粒子实际上经历了所有可能的路径，每条路径上的振幅都正比于相位因子（e^{iS/\hbar}）。经典路径的特殊之处在于振幅在对近邻路径相位平均后不为零，该机率幅平均过程的相干性等价于作用量的稳定性。

有时候经典路径不唯一，此时粒子的波动性即便在经典极限下也得已体现。一个著名的案例是光子、电子、中子或原子的杨氏双缝干涉。更加一般地说，通过分束器，干涉仪中粒子拥有两条经典运动路径，作用量S_{1,2}分别在这两条路径上累积。通过测量输出端分束器的粒子流，可以以普朗克常数 \hbar为单位推测两条路径的作用量之差 \Delta S_{1,2}。因为 \hbar很小，因此物质波干涉对于测量干涉路径之间微小的作用量差是非常有效的。

与光干涉仪甚至电子干涉仪相比，构建一个原子干涉仪更为困难。20世纪90年代，麻省理工学院的 David Pritchard教授利用当时最先进的纳米加工技术制造了第一台原子干涉仪。建造原子干涉仪主要困难有两个:

1、相干原子源

和光学干涉仪一样，原子干涉仪在一定程度上需要“单色源”。因为“单色波”更容易“分束”和传播路径的相干叠加。与此相关，经典作用量S是原子初速度的非线性函数。过宽的速度分布（及相关过短的德布罗意波长）将导致作用量的控制及作用量差的量化都变得更为困难。不幸的是，我们通过简单的计算可知，要获得足够长的物质波相干长度，例如和廉价LED输出光类似的毫米级相干长度，通常需要将原子冷却到亚皮开的温度（通常原子物理技术下，可能需要宇宙空间站等微重力环境才能实现）!

幸运的是，通过激光冷却，人们至少可以将原子温度降至微开尔文状态，由此可以精心设计“消色差”，对原子初速度不敏感的物质波干涉技术（类似于光子的“白光干涉技术”）。

2、相干分束装置

类似于光干涉仪，物质波分束装置需要是“相干的”，才能用于干涉测量。判断分束是否“相干”，其标准很简单：如果从基础原理上说，任何观测者，包括原子本身，都不知道该选择哪条路径，则量子力学将迫使原子的运动被描述为两条路径传播模式的相干叠加。

对于光子来说，这种相干分束通常是通过半透半反镜实现——深入一些的理解是，这种相干分束是一种称为“量子反射”的现象。为实现这一现象，光学界面必须在远小于光波长的特征尺度上跃变其折射率。 然而对于原子来说，这种量子反射通常是不行的。原子的德布罗意波长很容易与近表面范德华相互作用尺度接近。更麻烦的是，原子与表面的碰撞通常会激发原子的电子运动，以及材料的声子模式。这样的非弹性碰撞可能使原子粘在表面上。而更根本的困难在于，非弹性碰撞意味着原子运动的路径信息会泄露给第三方，从而破坏量子相干性。

因此，原子干涉技术中的原子分束难以依赖于量子反射技术。相反，从David Pritchard的纳米光栅分束到Steven Chu的光脉冲分束技术，原子干涉几乎完全依赖于设计空间周期性相互作用以驱动物质波衍射。周期结构衍射在动量空间中以“反冲动量”为单位，即h/d,其中d(或 \lambda)为光栅常数(或光脉冲的波长)来移动物质波。目前，光脉冲原子干涉测量技术发展迅速，精度非常高，应用范围包括从探测新物理到惯性导航的方方面面。

然而有时候 (事实上，大部分情况下)， “光反冲动量”有点太小了，还不足以推进原子干涉技术的真正实力。此外，物质波“衍射”通常也不那么“精确”，这限制了多体量子增强原子干涉技术中单体物质波关键控制的保真度。

和其他项目并行，本项目以独特的高带宽、任意波形脉冲调控原子状态。具体来说，通过合成宽带光学波形，驱动拉曼跃迁态矢量高速转动，充分运用拉曼跃迁的几何鲁棒性来提高物质波调控保真度。这里拉曼控制是指在动量转移的同时，原子的电子结构从一个亚稳态“翻转”到另一个亚稳态。该“自旋相关动量转移”方案是当前广泛应用的拉曼光脉冲干涉技术的基石。而我们的项目独特之处在于通过提高控制速度（进入纳秒尺度）和复杂性（许多“pi”脉冲）来突破传统极限，驱动高保真、高速度的大量反冲动量转移，从而实现物质波的相干操控。

这一全新控制物质波技术的成功发展将有助于突破传统拉曼原子干涉仪的灵敏度和带宽。此外，该类容错控制技术可拓展离子阱类实验平台的量子信息处理能力。

本项目基于实验技术和理论发展的协同努力。在理论方面，我们通过咨询量子调控领域的专家(袁海东教授)，学习最优控制理论并制定多能级原子的控制策略。在基础物理和量子物理应用方面，我们受益于与理论专家（马怡秋教授, 李晓鹏教授）的频繁交流。

本项目正寻找新成员。如果你对量子最优控制及精确实验实现感兴趣，或者有兴趣了解更多有关原子干涉技术的基础和前沿知识，请随时与我们联系讨论。