For dielectrics, optical phenomena are typically understood as following: Light drive a polarization field P=\chi E. The coherent coupling between E and P is described by the macroscopic Maxwell equations.

However, the polarizability \chi stems from quantized electric motion in matter. A more accurate description thus replaces the constitutive relation about \chi with equations that describe the interacting electric dipoles. By this step, a prohibitively difficult problem emerges.

In the simplest scenario, let's consider N 2-level atoms interacting with light. For large N, the Hilbert space to describe matter field is huge: 2^N in dimension. Our only hope seems to be that the system could remain in a ``product state'' to be efficiently described. However, when any of the atom is in |e>, its own ``spin-flip'' drives single-photon package of electric dipole radiation outward to interact with all the other atoms. The long-range interaction would force all the atoms to entangle with one another. So, it seems the simplest system become intractable classically!

Fortunately, even in presence of the long-range electric dipole interaction, the ``product state'' description of many-atom system is often still valid. The effective picture is that single atoms interact with the ``mean field'', which is the light composed of the incident beam interfering with those from the dipole radiation by all the other atoms. As such, the many-body problem is reduced to N+1 simple equations: N Shrodinger equations for the driven 2-level atoms, and the Maxwell wave equation for E and average P. The equation set is referred to as Maxwell-Bloch equations.

This project intents to create atom-light interaction scenario where the M-B picture breaks down explicitely. Two key ingredients are identified for a first step toward our goal:

1. Freeze the atomic motion.

The above mentioned many-atom entanglements are real. That they hardly affect our observations is often because that the atoms move rapidly. Atomic motion leads to Doppler dephasing, which are often able to wash out the effects of many-body entanglements generated by the dipole interaction.

Therefore, to preserve effects of many-body entanglements in optics, the first step is to freeze the atomic motion. Fortunately, our laser cooling tools are at hand for freeze the atomic motion. For example, with standard sub-Doppler cooling, alkaline atomic samples are regularly cooled to micro-kelvin temperatures. For rubidium atoms, this means inter-atomic distance hardly change an optical wavelength within a microsecond, which is long enough for interesting quantum optics phenomena to occur.

2. Freeze and restart the light propagation on demand.

To suppress atomic motion is not enough. Optical excitation cannot stay for too long among the atoms, since light propagates at speed of light anyway.

How to generate strong atomic entanglements before the local optical excitation decays away? One might wonder whether long-lived optical transitions, such as those for optical clocks, could help. However, as it turns out interestingly, the lifetime of atomic excitation is directly related to ability of atom to dipolarly interact with one another. Therefore, the choice of transition linewidth does not matter that much.

Another thought is to increase the atomic density to enhance the 1/r^3 near-field interaction strength. This thought is valid. Practically, it turns out simple density enhancement may not help that much either, since an enhanced density typically also leads to strong ``collective interaction'' that enhances the radiation power of the whole ensemble. This ``directional collective emission'' is part of light propagation effects. For example, for continuous wave excitation, the collective emission leads to absorption, phase retardation, group delay, as well as various nonlinear wave mixing effects in nonlinear optics. Quite generally, the enhanced collective emission damps out local optical excitations more rapidly, leaving little time for many-body entanglements to build up either. For true benefit of the density enhancement, the atomic sample may need to be squeezed to a size below the optical wavelength, where the optical phenomena start to look like a lot those in nuclear magnetic resonances with microwaves. However, nanometer-sized ultracold many-atom samples are not only difficult to prepare, but also may not be particularly easy to work with.

Instead of simple density enhancement, Our work intend to control the collective interaction in ultracold atomic gas, so as to steer the many-body entanglement building-up process, or even to transfer the quantum signature into collective emission on demand. We work with Rb87 atoms on the D1/D2 strong transitions. Equipped with a wideband optical pulse shaper (credits to Yizun and his colleagues), we use optical pulses with talored waveforms to modify the spatial profile of D2 atomic excitations - referred to as electric dipole spin wave for obvious reasons -- with geometric phase patterning by cyclically driving the axilary D1 transition.

The technique allow us to redirect, suppress, and recall superradient emission from a cold atomic gas. In other words, the propagation of light in the free-space atomic medium can be freeze and restarted on-demand! Microscopic dynamics of interacting dipoles are built up during the freeze period. Quantum effects can be readout later by a superradient recall. The signals are measured in a back-ground-free fashion.

In our next step, instead of working with free laser-cooled gases, we hope to confine atoms into a regular sub-wavelength array optical lattice. In that scenario, not only the many-body entanglement can be preserved and probed by our spin-wave control technique, but also the interaction can be tailored for generating potentially useful quantum effects into light!

This project is a joint experimental+theoretical endeavor. On the theoretical side, we are inspired by our collaborator, Prof. Darrick Chang team. We also benefit from interactions and supports from the Prof. Lei Zhou group.

We are looking for new team members. If you are interested in deep-immersion experience of experimental quatnum optics with strong theoretical training, you might like our project -- please drop by for a discussion.

电偶极自旋波的量子控制

我们对电介质光学现象的理解常基于如下图像：光场E驱动极化场P=\chi E。E和P的相干耦合由宏观麦克斯韦方程组描述。

然而，物质极化率\chi源于原子的量子化电子运动。因此，更加精确的理解似乎需要由描述电偶极子相互作用的薛定谔方程取代关于\chi的本构关系。这一“更加精确”一步后的量子光学成了一个极端复杂的难题。

让我们考虑N 个二能级原子与光的相互作用这一“最简单场景”。对于大N，描述物质场的希尔伯特空间巨大：其维数为2^N。为了有效地描述系统，我们唯一的希望似乎是系统能够保持“直积态”。然而，当任何一个原子处于| e>时，其 “自旋翻转”形成的电偶极辐射单光子波包会与所有其他原子相互作用。这种长程量子相互作用迫使所有原子相互纠缠。因此，这个最简单体系其动力学在最高级的经典计算机上似乎也无法处理！

幸运的是，在上述长程电偶极相互作用下，多原子系统的“直积态”描述通常仍然有效。其等效图像是单个原子与“平均场”相互作用，而平均场是由入射光束与所有其他原子的偶极辐射干涉而成的光场。因此，该多体问题可以简化为N+1个简单方程：驱动二能级原子的N个Schrödinger方程，以及E和平均P的Maxwell波动方程。该方程组称为Maxwell-Bloch方程组。

本项目旨在构造M-B图像完全失效的光-原子相互作用奇特场景。实现该目标的第一步有两个关键要素：

1、冻结原子运动。

上述多原子纠缠是真实的，但通常这些纠缠并不会影响我们的观察，这是因为原子移动迅速。原子运动导致的多普勒退相干通常能够抑制偶极相互作用产生的多体纠缠对观测的影响。

因此，为了获得光学现象中多体纠缠的可观测效果，第一步是冻结原子的运动。幸运的是，我们的激光冷却手段可用于冻结原子运动。例如，使用标准亚多普勒冷却技术，碱金属原子样品可以冷却至微开尔文级温度。对于铷原子而言，这意味着原子间距离在一微秒内的变化远小于光学波长，足以用于积累有趣的量子光学效应。

2、光传播的可控冻结和重启。

仅仅抑制原子运动是不够的。因为光以光速传播，光激发不能在原子间保存太长时间。

如何在局域光激发衰减之前产生强的原子纠缠？人们可能会觉得，光钟等长寿命光学跃迁可能会有些帮助。然而，有趣的是，原子激发态寿命与激发态原子偶极作用强度直接相关。在多体相互作用光学物理方面，跃迁线宽的选择几乎是无关紧要的。

另一种想法是增加原子密度，以增强1/r^3近场光学相互作用强度。这个想法是对的。然而简单的密度增强可能也没有多大帮助。这是因为通常来说，密度增强会导致强烈的“集体相互作用”，导致整个系综辐射能力增强。事实上，这种“定向集体辐射”是光传播效应的一部分：集体辐射是线性吸收、相位延迟、群速延时，以及非线性光学中的各种混频效应的物理根源。一般来说，集体辐射增强对应于局域光激发的衰减增强，会减少多体纠缠建立的宝贵时间。事实上，通过提高密度来增强光学多体物理效应，很可能需要将原子样品压缩到低于光波长的尺寸。在这种情况下，多体光学现象看起来会类似于微波磁共振的一个“高速版”。然而纳米尺寸的超冷多原子样品不仅难以制备，且可能不特别容易控制。

我们的工作不是简单地增强原子密度，而是通过控制超冷原子气体中的集体相互作用来操控多体纠缠的建立过程，进而根据需求将量子纠缠资源转移致集体辐射，以实现多体作用的远场探测，甚至量子光学应用。我们在D1/D2强跃迁上研究铷87同位素。运用独特发展的高宽带任意波形光脉冲技术，通过波形优化的纳秒脉冲驱动原子系综D2偶极自旋激发 -- 我们称之为电偶极自旋波 – 并通过循环驱动D1辅助跃迁对之进行亚波长分辨的几何相位修饰。

该类新技术使我们能够在时域对冷原子气体的超辐射进行重定向、相干抑制和延时唤起。换句话说，光在自由空间原子介质中的传播可以按需冻结和重启！在“冻结”阶段，电偶极相互作用下的多体关联被建立起来，稍后，其量子纠缠可被重新唤起的超辐射传播致远场，并以无背景的方式精密读取。

不同于自由气体，本项目下一步希望将超冷原子囚禁在规则的亚波长阵列光晶格中。在这种情况下，我们的自旋波控制技术不仅可以保留和探测光学多体纠缠，还可以通过直接调控相互作用在超辐射中产生有用的量子资源！

本项目是一项实验+理论的联合努力。在理论方面，我们受益于合作者Darrick Chang教授团队的启发。我们还受益于周磊教授团队的深入交流和支持。

我们正在寻找新的团队成员。如果你对全身心投入量子光学实验，同时接受“硬核”量子光学理论训练感兴趣，你可能会喜欢我们的项目——请来信来函讨论各种可能。