A central feature of quantum mechanics is that a measurement is intrinsically probabilistic. As a result, continuously monitoring a quantum system will randomly perturb its natural unitary evolution. The ability to control a quantum system in the presence of these fluctuations is of increasing importance in quantum information processing and finds application in fields ranging from nuclear magnetic resonance 1 to chemical synthesis 2 . A detailed understanding of this stochastic evolution is essential for the development of optimized control methods. Here we reconstruct the individual quantum trajectories 3-5 of a superconducting circuit that evolves in competition between continuous weak measurement and driven unitary evolution. By tracking individual trajectories that evolve between an arbitrary choice of initial and final states we can deduce the most probable path through quantum state space. These pre-and post-selected quantum trajectories also reveal the optimal detector signal in the form of a smooth time-continuous function that connects the desired boundary conditions. Our investigation reveals the rich interplay between measurement dynamics, typically associated with wave function collapse, and unitary evolution of the quantum state as described by the Schrödinger equation. These results and the underlying theory 6 , based on a principle of least action, reveal the optimal route from initial to final states, and may enable new quantum control methods for state steering and information processing.Our experiment focuses on the dynamics of two quantum levels of a superconducting circuit (a qubit), which can be continuously measured and excited by microwave pulses. To access individual quantum trajectories, we make use of the fact that fully projective measurement (or wavefunction collapse) happens over an average timescale τ controlled by the interaction strength between the system and the detector. By recording the measurement signal in time steps much shorter than τ with high fidelity, we realize a continuous sequence of weak measurements and track the qubit state as it evolves in a single experimental iteration. Individual weak measurements have been recently employed in atomic physicsCavity input Transmon qubit experiments that probe wave function collapse 7 and perform state stabilization 8 . In the domain of superconducting circuits, weak measurements 9 have only recently been realized due to the challenge associated with high fidelity detection of near single-photon level microwave signals. Advances in superconducting parametric amplifiers have enabled continuous feedback control 10-12 , the observation of individual quantum trajectories 13,14 , the determination of weak values 15,16 , and entanglement of qubits 17,18 .Our experiment consists of a superconducting transmon circuit 19 dispersively coupled to a waveguide cavity 20 (Fig. 1a). Considering only the two lowest levels of the transmon as a qubit, our system is described by arXiv:1403.4992v1 [quant-ph] 19 Mar 2014 2 the Hamiltonian Hwhere ...

We present a stochastic path integral formalism for continuous quantum measurement that enables the analysis of rare events using action methods. By doubling the quantum state space to a canonical phase space, we can write the joint probability density function of measurement outcomes and quantum state trajectories as a phase space path integral. Extremizing this action produces the most-likely paths with boundary conditions defined by preselected and postselected states as solutions to a set of ordinary differential equations.As an application, we analyze continuous qubit measurement in detail and examine the structure of a quantum jump in the Zeno measurement regime.

We generalize and extend the stochastic path integral formalism and action principle for continuous quantum measurement introduced in [A. Chantasri, J. Dressel and A. N. Jordan, Phys. Rev. A 88, 042110 (2013)], where the optimal dynamics, such as the most likely paths, is obtained by extremizing the action of the stochastic path integral. In this work, we apply exact functional methods as well as develop a perturbative approach to investigate the statistical behaviour of continuous quantum measurement. Examples are given for the qubit case. For qubit measurement with zero qubit Hamiltonian, we find analytic solutions for average trajectories and their variances while conditioning on fixed initial and final states. For qubit measurement with unitary evolution, we use the perturbation method to compute expectation values, variances, and multi-time correlation functions of qubit trajectories in the short-time regime. Moreover, we consider continuous qubit measurement with feedback control, using the action principle to investigate the global dynamics of its most likely paths, and finding that in an ideal case, qubit state stabilization at any desired pure state is possible with linear feedback. We also illustrate the power of the functional method by computing correlation functions for the qubit trajectories with a feedback loop to stabilize the qubit Rabi frequency.

We investigate the statistical arrow of time for a quantum system being monitored by a sequence of measurements. For a continuous qubit measurement example, we demonstrate that time-reversed evolution is always physically possible, provided that the measurement record is also negated. Despite this restoration of dynamical reversibility, a statistical arrow of time emerges, and may be quantified by the log-likelihood difference between forward and backward propagation hypotheses. We then show that such reversibility is a universal feature of nonprojective measurements, with forward or backward Janus measurement sequences that are time-reversed inverses of each other.