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arXiv:2206.12976 · Physical Review Letters

Test of Causal Nonlinear Quantum Mechanics by Ramsey Interferometry with a Trapped Ion

Joseph Broz, Bingran You, Sumanta Khan, Hartmut Häffner, David E. Kaplan, Surjeet Rajendran

Published May 15, 2023• 9 citations
arXivHTMLPDF

Abstract

Quantum mechanics requires the time evolution of the wave function to be linear. While this feature has been associated with the preservation of causality, a consistent causal nonlinear theory was recently developed. Interestingly, this theory is unavoidably sensitive to the full physical spread of the wave function, rendering existing experimental tests for nonlinearities inapplicable. Here, using well-controlled motional superpositions of a trapped ion, we set a stringent limit of 5.4×10^{-12} on the magnitude of the unitless scaling factor ε[over ˜]_{γ} for the predicted causal nonlinear perturbation.

DOI
10.1103/physrevlett.130.200201

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Quantum mechanics requires the time evolution of the wave function to be linear. While this feature has been associated with the preservation of causality, a consistent causal nonlinear theory was recently developed. Interestingly, this theory is unavoidably sensitive to the full physical spread of the wave function, rendering existing experimental tests for nonlinearities inapplicable. Here, using well-controlled motional superpositions of a trapped ion, we set a stringent limit of 5.4×10^{-12} on the magnitude of the unitless scaling factor ε[over ˜]_{γ} for the predicted causal nonlinear perturbation.
Linear time evolution is one of the fundamental postulates of quantum theory. Past theoretical attempts to introduce nonlinearity into quantum evolution have violated causality. However, a recent theory has introduced nonlinear state-dependent terms in quantum field theory, preserving causality [D. E. Kaplan and S. Rajendran, Phys. Rev. D 105, 055002 (2022)PRVDAQ2470-001010.1103/PhysRevD.105.055002]. We report the results of an experiment that searches for such terms. Our approach, inspired by the Everett many-worlds interpretation of quantum theory, correlates a binary macroscopic classical voltage with the outcome of a projective measurement of a quantum bit, prepared in a coherent superposition state. Measurement results are recorded in a bit string, which is used to control a voltage switch. Presence of a nonzero voltage reading in cases of no applied voltage is the experimental signature of a nonlinear state-dependent shift of the electromagnetic field operator. We implement blinded measurement and data analysis with three control bit strings. Control of systematic effects is realized by producing one of the control bit strings with a classical random-bit generator. The other two bit strings are generated by measurements performed on a superconducting qubit in an IBM Quantum processor and on a ^{15}N nuclear spin in a nitrogen-vacancy center in diamond. Our measurements find no evidence for electromagnetic quantum state-dependent nonlinearity. We set a bound on the parameter that quantifies this nonlinearity |ε_{γ}|<4.7×10^{-11}, at 90% confidence level.
We propose an adiabatic Ramsey interferometry technique for detecting weak nonlinearities with trapped ions. The method relies on using the quantum Rabi model as a probe, which is sensitive to nonlinear symmetry-breaking perturbations. We show that the couplings which arise either from anharmonic terms of the trapping potential or due to higher order terms in the Coulomb interaction expansion can be efficiently estimated by measuring the spin state probabilities alone. We show that the spin signal is amplified by the mean-phonon excitations, which results in the estimation precision reaching the super-Heisenberg limit. Notably, achieving such high-precision estimation does not require specific entangled state preparation and can be reached even for initial thermal motion state. Furthermore, we show that the super-Heisenberg scaling can be observed even in the presence of weak spin-dephasing.
We add nonlinear and state-dependent terms to quantum field theory. We show that the resulting low-energy theory, nonlinear quantum mechanics, is causal, preserves probability and permits a consistent description of the process of measurement. We explore the consequences of such terms and show that nonlinear quantum effects can be observed in macroscopic systems even in the presence of decoherence. We find that current experimental bounds on these nonlinearities are weak and propose several experimental methods to significantly probe these effects. The locally exploitable effects of these nonlinearities have enormous technological implications. For example, they would allow large-scale parallelization of computing (in fact, any other effort) and enable quantum sensing beyond the standard quantum limit. We also expose a fundamental vulnerability of any nonlinear modification of quantum mechanics---these modifications are highly sensitive to cosmic history and their locally exploitable effects can dynamically disappear if the observed universe has a tiny overlap with the overall quantum state of the universe, as is predicted in conventional inflationary cosmology. We identify observables that should persist in this case and discuss opportunities to detect them in cosmic ray experiments, tests of strong field general relativity and current probes of the equation of state of the universe. Nonlinear quantum mechanics also enables novel gravitational phenomena and may open new directions to solve the black hole information problem and to uncover the theory underlying quantum field theory and gravitation.
It is proposed to set stringent limits on possible nonlinear corrections to ordinary quantum mechanics by searching for the detuning of resonant transitions. A suggested nonlinear generalization of quantum mechanics is used to show that such detuning would be expected in the rf transition in 9Be+^{9}\mathrm{Be}^{+}9Be+ ions that is used to set frequency standards. Measurements at the National Bureau of Standards already set limits of order ${10}^{\ensuremath{-}21}$ on the fraction of the energy of the 9Be^{9}\mathrm{Be}9Be nucleus that could be due to nonlinear corrections to quantum mechanics, with good prospects of improving this by 2-3 orders of magnitude.
This thesis describes new results on the entanglement of atomic spins in Ramsey interferometry, optical atomic clocks and trapped ions. It is divided into three parts: First, we investigate improvements to conventional Ramsey interferometry with entanglement and adding only rotations of the collective spin to adjust the signal and measurement directions. The geometric degrees of freedom, connected to the rotations, are analytically optimized for a large class of generalized Ramsey protocols to allow efficient optimization of all parameters. Besides a unification of existing approaches, the main result is that there is only one new protocol, where a previously unused double inversion is applied. Studies of the local sensitivity show that this protocol reaches the fundamental quantum Fisher information limit and is yet robust against errors during preparation and measurement. In the second section we investigate the conditions under which optical atomic clocks exhibit increased long-term stability when applying weakly entangled, spin squeezed states. We discuss the common case of an atomic clock with a single ensemble, typical Brownian frequency noise and finite dead time. Theoretical modelling of the servo loop allows quantitative predictions of the optimal stability for given values of dead time and laser noise, in very good agreement with numerical simulations of the closed feedback loop. The main result is that, even with the current most stable lasers, the clock stability can only be improved for ensembles below a critical atom number of about one thousand in optical Sr lattice clocks. Even with a future improvement of the laser performance by one order of magnitude, the critical atom number still remains below 100,000. In contrast, clocks based on smaller, non-scalable ensembles, such as ion clocks, can already benefit from squeezed states with current clock lasers. Thus the last section considers the robust generation of entanglement in ion traps. An error budget including relevant experimental error sources is calculated for state-of-the-art quantum gates, driven by oscillating microwave gradients in surface traps. Amplitude modulation of the driving fields is shown to efficiently counteract the current limitations from motional mode instability. The predicted increase of the gate quality was demonstrated by the group of C. Ospelkaus at PTB Braunschweig, who measured gates with errors as low as ~ 10^(-3). In a similar approach, interactions between spin and motion can also be generated by combining oscillating rf-fields with a static magnetic field gradient. Penning traps designed for precision spectroscopy already feature large magnetic field gradients at the edge of a magnetic bottle configuration. We present parameters and conditions under which laser-free coupling of spin and quantized motion for (anti-)protons is possible at these points, in a step towards quantum logic spectroscopy for (anti-)protons.
We analyze a proposed experiment [Boixo et al., Phys. Rev. Lett. 101, 040403 (2008)] for achieving sensitivity scaling better than 1/N1/N1/N in a nonlinear Ramsey interferometer that uses a two-mode Bose-Einstein condensate (BEC) of NNN atoms. We present numerical simulations that confirm the analytical predictions for the effect of the spreading of the BEC ground-state wave function on the ideal 1/N3/21/{N}^{3/2}1/N3/2 scaling. Numerical integration of the coupled, time-dependent, two-mode Gross-Pitaevskii equations allows us to study the several simplifying assumptions made in the initial analytic study of the proposal and to explore when they can be justified. In particular, we find that the two modes share the same spatial wave function for a length of time that is sufficient to run the metrology scheme.