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arXiv:2008.03228

Full monitoring of phase-space trajectories with 10dB-sub-Heisenberg imprecision

Jascha Zander, Roman Schnabel

Published Aug 7, 2020• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2008.03228

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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.
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.
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.
Heisenberg uncertainty relation in quantum mechanics sets the limit on the measurement precision of non-commuting observables, which prevents us from measuring them accurately at the same time. In some applications, however, the information are embedded in two or more non-commuting observables. On the other hand, quantum entanglement allows us to infer through Einstein-Podolsky-Rosen correlations two conjugate observables with precision better than what is allowed by Heisenberg uncertainty relation. With the help of the newly developed SU(1,1) interferometer, we implement a scheme to measure jointly information encoded in multiple non-commuting observables of an optical field with a signal-to-noise ratio improvement of about 20 % over the standard quantum limit on all measured quantities simultaneously. This scheme can be generalized to the joint measurement of information in arbitrary number of non-commuting observables.
In two articles, the authors claim that the Heisenberg uncertainty principle limits the precision of simultaneous measurements of the position and velocity of a particle and refer to experimental evidence that supports their claim. It is true that ever since the inception of quantum mechanics, the uncertainty relation that corresponds to a pair of observables represented by non-commuting operators is interpreted by many scientists and engineers, including Heisenberg himself, as a limitation on the accuracy with which observables can be measured. However, such a limitation cannot be deduced from the postulates and theorems of quantum thermodynamics.
Determining a beam's full trajectory requires tracking both its position and momentum (angular) information. However, the product of position and momentum uncertainty in a simultaneous measurement of the two parameters is bound by the Heisenberg uncertainty limit (HUL). In this work, we present a proof-of-principle demonstration of a quantum-enhanced beam tracking technique, leveraging the inherent position and momentum entanglement between photons produced via spontaneous parametric down-conversion (SPDC). We show that quantum entanglement can be exploited to achieve a beam tracking accuracy beyond the HUL in a simultaneous measurement. Moreover, with existing detection technologies, it is already possible to achieve near real-time beam tracking capabilities at the single-photon level. The technique also exhibits high resilience to background influences, with negligible reduction in tracking accuracy even when subjected to a disruptive beam that is significantly brighter than SPDC.
Optical phase measurement through its application in quantum metrology has pushed the precision limit with which some physical quantities can be measured accurately. At the very fundamental level, the laws of quantum mechanics dictate that the uncertainty in phase estimations scales as 1/N, where N is the number of quantum resources employed in the protocol [1]. This is the well known Heisenberg limit (HL) which is quadratically better than the traditional precision limit known as the standard quantum limit (SQL) with uncertainty asymptotically scaling as 1/sqrt{N} [1]. Several experiments have demonstrated that the SQL can be beaten by using an entangled state as the probe and a specific measurement scheme for ab initio estimation of unknown phases [2,3]. It has also been shown experimentally that even in the absence of the entanglement one can measure an unknown phase with imprecision scaling at the HL [4]. In this work we first present a new protocol able to estimate an optical phase at the Heisenberg limit, and then experimentally explore fundamental and practical issues in generating high-quality novel entangled states, for use in this protocol and beyond. Our aim in this study is to measure an unknown phase in the interval [0,2pi) with uncertainty attaining the exact HL. There is a condition that should be met to address this objective: preparation of an optimal state [5]. This would cover part of the presentation through which we explain how to experimentally realise such an optimal state with the current technological limitations and the feasibility of the scheme. In particular, we generate an entangled 3-photon (2-photon) state of specific superposition of GHZ (Bell) states. Our numerical simulation of the phase measurement gate together with the experimental outcomes show that the created state should have a high fidelity and purity to be able to have the phase uncertainty achieving the exact HL. Therefore, we briefly explain the modelling for experimental imperfections and finally present the results of experimental phase measurements.