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Monitoring of dynamical phase-space trajectories with sub-Heisenberg indeterminacy

Jascha Zander, Roman Schnabel

Published Aug 7, 2020• 1 citation
OpenAlex

Abstract

The Heisenberg uncertainty relation describes the fact that the phase-space trajectory of a single quantum system cannot be precisely determined with respect to semi-classical reference values to better than twice the quantum system's ground state uncertainty. Einstein, Podolsky and Rosen pointed out that according to quantum theory, however, there are pairs of quantum systems whose properties with respect to each other can be arbitrarily precisely determined. Here we report the experimental proof that even the dynamics of a quantum system, i.e. its phase-space trajectory, can be precisely determined with respect to another quantum system. We present measurements with a remaining indeterminacy of canonical conjugate variables ten times smaller than the lowest possible for a semi-classical reference. Our result may trigger research on the foundations of quantum mechanics and supports quantum technology for entanglement-enhanced metrology and secure communication.

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The Heisenberg uncertainty relation describes the fact that the phase-space trajectory of a single quantum system cannot be precisely determined with respect to semi-classical reference values to better than twice the quantum system's ground state uncertainty. Einstein, Podolsky and Rosen pointed out that according to quantum theory, however, there are pairs of quantum systems whose properties with respect to each other can be arbitrarily precisely determined. Here we report the experimental proof that even the dynamics of a quantum system, i.e. its phase-space trajectory, can be precisely determined with respect to another quantum system. We present measurements with a remaining indeterminacy of canonical conjugate variables ten times smaller than the lowest possible for a semi-classical reference. Our result may trigger research on the foundations of quantum mechanics and supports quantum technology for entanglement-enhanced metrology and secure communication.
The change of a quantum state can generally only be fully monitored through simultaneous measurements of two non-commuting observables X and Y spanning a phase space. A measurement device that is coupled to the thermal environment provides at a time a pair of values that have a minimal uncertainty product set by the Heisenberg uncertainty relation, which limits the precision of the monitoring. Here we report on an optical measurement setup that is able to monitor the time dependent change of the quantum state's displacement in phase space (< X (t)>; < Y (t)>) with an imprecision 10\,dB below the Heisenberg uncertainty limit. Our setup provides pairs of values (X(t_i); Y(t_i)) from simultaneous measurements at subsequent times t_i. The measurement references are not coupled to the thermal environment but are established by an entangled quantum state. Our achievement of a tenfold reduced quantum imprecision in monitoring arbitrary time-dependent displacements supports the potential of the quantum technology required for entanglement-enhanced metrology and sensing as well as measurement-based quantum computing.
Common knowledge suggests that trajectories of particles in quantum mechanics always have quantum uncertainties. These quantum uncertainties set by the Heisenberg uncertainty principle limit precision of measurements of fields and forces, and ultimately give rise to the standard quantum limit in metrology. With the rapid developments of sensitivity of measurements these limits have been approached in various types of measurements including measurements of fields and acceleration. Here it is shown that a quantum trajectory of one system measured relatively to another “reference system” with an effective negative mass can be quantum uncertainty–free. The method crucially relies on the generation of an Einstein‐Podolsky‐Rosen entangled state of two objects, one of which has an effective negative mass. Such state has been created experimentally and thus, from a practical perspective, these ideas open the way towards force and acceleration measurements at new levels of sensitivity far below the standard quantum limit.
Abstract The change of a quantum state can generally only be fully monitored through simultaneous measurements of two non-commuting observables X^\hat{X}X^ X ̂ and Y^\hat{Y}Y^ Y ̂ spanning a phase space. A measurement device that is coupled to the thermal environment provides at a time a pair of values that have a minimal uncertainty product set by the Heisenberg uncertainty relation, which limits the precision of the monitoring. Here, we report on an optical ensemble measurement setup that is able to monitor the time-dependent change of the quantum state’s displacement in phase space ( ⟨X^(t)⟩;⟨Y^(t)⟩\langle \hat{X}(t)\rangle ;\langle \hat{Y}(t)\rangle⟨X^(t)⟩;⟨Y^(t)⟩ ⟨ X ̂ ( t ) ⟩ ; ⟨ Y ̂ ( t ) ⟩ ) with an imprecision 10 dB below the Heisenberg uncertainty limit. Our setup provides pairs of values (X(ti); Y(ti)) from simultaneous measurements at subsequent times ti. The measurement references are not coupled to the thermal environment but are established by an entangled quantum state. Our achievement of a tenfold reduced quantum imprecision in monitoring arbitrary time-dependent displacements supports the potential of the quantum technology required for entanglement-enhanced metrology and sensing as well as measurement-based quantum computing.
Quantum uncertainty emerged from two strikingly different routes to quantum mechanics. Heisenberg's matrix mechanics replaced the classical description of atomic orbits by relations among observable transition quantities and led, in 1927, to the uncertainty relation and its influential measurement-based interpretation. Schrödinger's wave mechanics grew instead from the Bose--Einstein and de Broglie developments and restored a continuous space-time description through matter waves. The two formalisms were soon shown to be closely related, but neither historical route by itself settles the ontological question of whether particles may possess underlying trajectories. After reviewing these developments, including Einstein's criticism of a simple observability doctrine and Mott's wave-mechanical account of straight cloud-chamber tracks, we distinguish precise quantum uncertainty relations from the stronger claim that quantum mechanics excludes definite microscopic histories. We then compare Feynman paths, Bohmian trajectories, Nelson's stochastic mechanics, and finite-speed persistent Goldstein--Kac dynamics. The Kac process has continuous, piecewise differentiable finite-speed paths; its diffusion limit approaches Wiener kinematics, while analytic continuation of its two-sector dynamics is related to Dirac evolution. These examples show that quantum uncertainty relations constrain observable statistics without, by themselves, uniquely determining an underlying path geometry.
Abstract The Heisenberg limit is acknowledged as the ultimate precision limit for parameter estimation in quantum metrology. Recent studies have demonstrated super-Heisenberg-scaling limits, where the root mean square errors of estimation decrease faster-than-linearly with the number of probes or evolution processes. This casts doubt on what the ultimate precision limit of quantum metrology really is and, in particular, whether there is a deeper connection with Heisenberg's uncertainty principle. By linking the dynamics of the metrological process with properties in parameter space, we construct a Heisenberg's uncertainty relation between the unknown parameter and its canonical momentum. We propose a novel quantum metrological scheme that utilizes a generating process with indefinite time direction to increase the canonical momentum nonlinearly. This approach achieves a nonlinear-scaling precision limit that improves quadratically with the increase in evolution time length or the number of processes, in accordance with Heisenberg's uncertainty relation. Then we experimentally demonstrate in quantum optical systems that this nonlinear-scaling enhancement can be achieved with a fixed amount of energy in the probes. Consequently, our results provide a deeper insight into the Heisenberg limit in quantum metrology, and shed new light on improving the precision of practical metrological and sensing tasks in realistic quantum systems.
One of the most striking deviations of quantum mechanics from classical mechanics is that any measurement of a microscopic system involves the interaction with the measuring instruments. The interaction cannot be neglected and in fact leads to uncontrollable disturbance on the system. In 1927, Werner Karl Heisenberg in his celebrated paper established the quantitative expression for unavoidable momentum disturbance caused by any position measurement. The crucial word here is simultaneously. The uncertainty relation gives the lower bound on the product of the uncertainties of two pairs of variables. But it says nothing about the accuracy with which individual observables can be measured. The physical meaning of energy-time uncertainty is different from that of x-px uncertainty. Position and momentum are observables to which Hermitian operators are assigned and which are measured at a particular time. An important implication of the uncertainty principle is that quantum systems do follow classical trajectories which conflict with our everyday experience.