Continuous quantum measurements with superconducting circuits:Most likely paths and optimal control

该奖项支持量子力学系统测量和控制的理论研究和教育，使用超导体制造的模型作为人工量子系统。与铜等普通导体相比，超导体表现出不同寻常的特性，比如不耗散导电的能力。量子力学被用来描述像电子和原子这样的微小物体，它们的行为与我们熟悉的宏观物体（如棒球）不同，它们遵循经典力学定律。由许多电子组成的超导状态可以用量子力学描述，其复杂性类似于单个原子。与观察棒球的轨迹不同，对量子力学系统进行测量会改变系统。研究小组将从理论上研究由超导体制造的量子力学系统，以促进对量子力学系统在测量下如何随时间变化的理解。这项工作将与一名实验人员合作进行严格的实验检查，并将理论研究过程中产生的想法和概念付诸实施。精确测量、控制和引导量子系统的能力是量子技术新兴领域的关键要素。这包括量子计算机的操作，其中量子力学状态将被操纵以实现高性能计算。这项研究与利用超导元素作为量子力学比特的量子计算机的建议特别相关。该奖项支持培训参与这项研究工作的研究生，以及PI向高中生教授暑期课程的外展活动。本课程使用最少的数学在定性层面上教授学生量子物理，并侧重于动手演示，实验室参观以及课堂讲座。该奖项支持理论研究和教育，以促进对超导transmon量子位连续量子测量的基本理解。超导量子系统的技术发展取得了长足的进步，使得具有长相干时间的系统得以实现；实验可以连续测量波函数的随机坍缩，并实现连续反馈。连续测量的量子轨迹方法允许在单个测量运行期间对量子态进行瞬时跟踪。研究小组将系统地研究超导传输量子比特中连续量子测量的物理学。PI的小组已经开发了连续量子测量的随机路径积分形式，非常适合研究和表征测量过程的物理特性。将量子系统固定的初始和最终状态视为随机动力学的边界条件，这些边界条件是通过前选择和后选择施加的。在这种情况下，一个重要的主题是量子系统如何在这两种状态之间得到。特别是，通过量子态空间的平均路径或最可能路径以及到达时间的分布为连续量子坍缩过程的物理学提供了深刻的见解。将研究量子态通过其空间的最可能路径，以及在状态被预先选择和后选择的情况下的首次通过时间的分布。这将在量子系统在被测量时发生单一变化时完成。相关的问题，如轨迹的动态稳定性，在量子控制的发展领域是至关重要的。最可能路径的非线性运动方程表明(i)边界条件之间可能存在多条最可能路径的情况，以及（ii）混沌行为的可能性，即对初始条件的指数敏感性。这种行为将被探索和描述。对多个传输子的联合测量也将被研究，以及最有可能从可分离状态开始并以期望的纠缠状态结束的方法。这项工作将在这个快速发展的凝聚态物理领域利用新的研究视野。PI将与同事Irfan Siddiqi的团队（加州大学伯克利分校）合作，测试这些理论，并在他的实验室中实现这里所描述的物理。

NONTECHNICAL SUMMARYThis award supports theoretical research and education on measurement and control of a quantum mechanical system using models fabricated from superconductors as artificial quantum systems. Superconductors exhibit unusual properties as compared to ordinary conductors like copper, such as the ability to conduct electricity without dissipation. Quantum mechanics is required to describe tiny objects like electrons and atoms which behave differently than familiar macroscopic objects like baseballs which obey the laws of classical mechanics. The superconducting state made up of many electrons can be described by quantum mechanics at a level of complexity similar to that of a single atom. Unlike observing the trajectory of a baseball, making measurements on a quantum mechanical system changes the system. The research team will theoretically investigate quantum mechanical systems fabricated from superconductors to advance understanding of how a quantum mechanical system changes in time under measurements. This work will be done in collaboration with an experimentalist to perform rigorous experimental checks, and implement the ideas and concepts developed in the course of the theoretical research. The ability to precisely measure, control and guide a quantum system is a critical element of the emerging field of quantum technology. This includes the operation of a quantum computer in which quantum mechanical states would be manipulated to achieve high performance computing. This research is particularly relevant for proposals of quantum computers that utilize superconducting elements as the quantum mechanical bits.This award supports training graduate students involved in this research effort, as well as an outreach activity in which the PI teaches a summer course to high school students. This class teaches students quantum physics at a qualitative level using minimal mathematics, and focuses on hands-on demonstrations, lab tours, as well as classroom lectures. TECHNICAL SUMMARY This award supports theoretical research and education to advance the fundamental understanding of continuous quantum measurement in superconducting transmon qubits. Technological development of superconducting quantum systems has made great strides leading to the fabrication of systems having long coherence times; experiments can continuously measure the stochastic collapse of the wavefunction as well as implement continuous feedback. The quantum trajectory approach to continuous measurement permits instantaneous tracking of the quantum state during individual measurement runs. The research team will systematically investigate the physics of continuous quantum measurement in superconducting transmon qubits. The PI's group has developed a stochastic path integral formalism of continuous quantum measurements that is well suited to investigate and characterize the physical properties of the measurement process. Fixed initial and final states of the quantum system are considered as boundary conditions on the stochastic dynamics, which are imposed by a pre- and post-selection. In this case, an important topic is how the quantum system gets between these two states. In particular, the average or most-likely path through the quantum state space and the distribution of arrival times provide deep insight into the physics of the continuous quantum collapse process. The most-likely paths the quantum state takes through its space will be investigated as well as the distribution of first passage times in the case where the state is pre- and post-selected. This will be done when the quantum system is unitarily changing while it is being measured. Associated questions, such as the dynamical stability properties of the trajectories, are of fundamental importance in the developing area of quantum control. The nonlinear equations of motion for the most likely path indicate (i) there can be cases when there are multiple most likely paths between the boundary conditions, and (ii) the possibility of chaotic behavior, in the sense of exponential sensitivity to initial conditions. This behavior will be explored and characterized. Joint measurements on multiple transmons will also be investigated, as well as the most likely ways to start in a separable state and end in a desired entangled one. This effort will capitalize on new research horizons in this rapidly developing field of condensed matter physics. The PI will collaborate with colleague Irfan Siddiqi's group (UC Berkeley) to test these theories and implement the physics described here in his laboratory.