A continuously monitored quantum bit (qubit) exhibits competition between unitary Hamiltonian dynamics and non-unitary measurement-collapse dynamics, which for diffusive measurements form an enlarged transformation group equivalent to the Lorentz group of spacetime. We leverage this equivalence to develop a four-dimensional generalization of the three-dimensional Bloch ball to visualize the state of a monitored qubit as the four-momentum of an effective classical charge affected by a stochastic electromagnetic force field. Unitary qubit dynamics generated by Hermitian Hamiltonians correspond to elliptic spatial rotations of this effective charge while non-unitary qubit dynamics generated by non-Hermitian Hamiltonians or stochastic measurement collapse correspond to hyperbolic Lorentz boosts. Notably, to faithfully emulate the stochastic qubit dynamics arising from continuous qubit measurement, the stochastic electromagnetic fields must depend on the velocity of the charge they are acting on. Moreover, continuous qubit measurements admit a dynamical delayed choice effect where a future experimental choice can appear to retroactively determine the type of past measurement backaction, so the corresponding point charge dynamics can also exhibit delayed choice Lorentz transformations in which a future experimental choice determines whether stochastic force fields are electric or magnetic in character long after they interact with the particle.
A continuously monitored quantum bit (qubit) exhibits competition between unitary Hamiltonian dynamics and non-unitary measurement-collapse dynamics, which for diffusive measurements form an enlarged transformation group equivalent to the Lorentz group of spacetime. We leverage this equivalence to develop a four-dimensional generalization of the three-dimensional Bloch ball to visualize the state of a monitored qubit as the four-momentum of an effective classical charge affected by a stochastic electromagnetic force field. Unitary qubit dynamics generated by Hermitian Hamiltonians correspond to elliptic spatial rotations of this effective charge while non-unitary qubit dynamics generated by non-Hermitian Hamiltonians or stochastic measurement collapse correspond to hyperbolic Lorentz boosts. Notably, to faithfully emulate the stochastic qubit dynamics arising from continuous qubit measurement, the stochastic electromagnetic fields must depend on the velocity of the charge they are acting on. Moreover, continuous qubit measurements admit a dynamical delayed choice effect where a future experimental choice can appear to retroactively determine the type of past measurement backaction, so the corresponding point charge dynamics can also exhibit delayed choice Lorentz transformations in which a future experimental choice determines whether stochastic force fields are electric or magnetic in character long after they interact with the particle.
Abstract Quantum mechanics admits two distinct evolutions: deterministic unitary dynamics governed by the Schrödinger equation and the probabilistic collapse of the wave function. We show that the continuous collapse of a quantum state under measurement can, on a trajectory-by-trajectory basis, be equivalently described as unitary evolution generated by a time- and state-dependent Hermitian Hamiltonian with stochastic parameters. While the ensemble dynamics remains non-unitary, each individual trajectory thus admits a unitary representation. We derive explicit forms of such Hamiltonians for projective measurements on arbitrary n -level systems and for continuous position measurements of a harmonic oscillator, and we propose experimental schemes to test these predictions. Our framework provides a new approach to modeling and controlling continuously monitored quantum systems using only state-dependent unitary resources.
We consider a stochastic dynamics describing the evolution of a qubit controlled by an external field and subject to continuous-time measurements. Motivated by stabilization techniques recently developed in, e.g., [10], [11], we investigate the support of the corresponding solution, which is a random variable taking values on the space of two-by-two density matrices. By making use of the Strook-Varadhan support theorem and by classical geometric control arguments we compute the support for two possible choices of the measurement and Hamiltonian operators. In one case we show that, in Bloch coordinates, the support is always contained inside an ellipsoid depending on the physical parameters of the system. More precisely, a solution starting from the ellipsoid never exit it, with probability one, and every open subset of the ellipsoid is visited with nonzero probability for some choice of the control function. In the second case the support coincides with (the interior of) the Bloch ball: every open subset of the Bloch ball is visited with nonzero probability up to suitably choosing the control function.
The state of a continuously monitored qubit evolves stochastically, exhibiting competition between coherent Hamiltonian dynamics and diffusive partial collapse dynamics that follow the measurement record. We couple these distinct types of dynamics together by linearly feeding the collected record for dispersive energy measurements directly back into a coherent Rabi drive amplitude. Such feedback turns the competition cooperative and effectively stabilizes the qubit state near a target state. We derive the conditions for obtaining such dispersive state stabilization and verify the stabilization conditions numerically. We include common experimental nonidealities, such as energy decay, environmental dephasing, detector efficiency, and feedback delay, and show that the feedback delay has the most significant negative effect on the feedback protocol. Setting the measurement collapse time scale to be long compared to the feedback delay yields the best stabilization.
In this chapter, we consider a quantum particle wave function with a bound spectrum of velocity c, and obtain the relativistic momentum based on the group velocity of this wave function. With a space-time isometry condition, the Lorentz transformation and the relativistic dynamics were obtained. Considering a field interacting with a quantum particle as a four-vector conjugated to the space-time vector in the time-dependent phase, we obtain the Lorenz force and the Maxwell equations. It is interesting that only the Ampère-Maxwell equation of a magnetic circuit is specific to the electromagnetic field, while the other equations are general for a field interacting with a charged quantum particle. Considering the time-dependent phase of a quantum particle interacting with an electromagnetic field with a space-time homogeneity condition, we obtain Lorentz transformations for this field. For a quantum particle at a non-relativistic velocity, we obtain a wave function with a very rapidly-varying factor, of a frequency proportional to the rest energy of this particle. From the Schrödinger equation of a particle with a relativistic Hamiltonian, we obtain a split of the wave function into four components, describing a proper rotation of this particle with an angular momentum called spin (Dirac’s relativistic electron theory). Moreover, we also calculate electron potential in the magnetic field, and two-electron interaction potential.
In quantum mechanics, standard or strong measurement approaches generally result in the collapse of an ensemble of wavefunctions into a stochastic mixture of eigenstates. On the other hand, continuous or weak measurements have the propensity to dynamically control the evolution of quantum states over time, guiding the trajectory of the state into non-trivial superpositions and maintaining state purity. This kind of measurement-induced state steering is of great theoretical and experimental interest for the harnessing of quantum bits or "qubits", which are the fundamental unit of the emerging quantum computer. We explore continuous measurement-based quantum state stabilization through linear feedback control for a single quantum bit. By applying a time-varying Rabi drive that includes a linear feedback term, we show that the fixed points of the continuous measurement may be relocated. Analytical derivations of Itô Stochastic Master Equations being ideal models, we employ them to derive the projected ensemble average while utilizing numerical simulations to characterize the stability of the set of possible fixed points, as well as their modified collapse time-scales. We include the effects of realistic experimental non-idealities, such as environmental energy relaxation, dephasing, time-delay, and inefficient measurement. Ultimately, we discuss potential experimental implementations from collaborating universities.
We introduce a framework that unifies quantum measurement dynamics, Hamiltonian dynamics, and double-bracket gradient flows. We do so by providing explicit expressions for stochastic Hamiltonians that produce state dynamics identical to those that happen during continuous quantum measurements. When such dynamical processes are integrated over sufficiently long time intervals, they yield the same results and statistics as during wavefunction collapse. That is, wavefunction collapse can be interpreted as coarse-grained (stochastic) Hamiltonian dynamics. Alternatively, wavefunction collapse can be interpreted as double-bracket gradient flows determined by derivatives of (stochastic) potentials defined in terms of observables with direct physical interpretations. The gradient flows minimize the variance of the monitored observable. Our derivations hold for general monitoring described by non-Hermitian jump processes. We show that such reinterpretations of measurement dynamics facilitate the design of feedback processes. In particular, we introduce feedback processes that yield deterministic double-bracket flow equations that prepare ground states of a target Hamiltonian, and state-agnostic feedback processes for state preparation. We apply the latter for entanglement stabilization of a two-qubit system considering a setup with imperfect measurements and feedback delay. We conclude by re-interpreting feedback processes as gradient flows with tilted fixed points.