Spin–motion entanglement and state diagnosis with squeezed oscillator wavepackets


Mesoscopic superpositions of distinguishable coherent states provide an analogue of the ‘Schrödinger’s cat’ thought experiment1,2. For mechanical oscillators these have primarily been realized using coherent wavepackets, for which the distinguishability arises as a result of the spatial separation of the superposed states3,4,5. Here we demonstrate superpositions composed of squeezed wavepackets, which we generate by applying an internal-state-dependent force to a single trapped ion initialized in a squeezed vacuum state with nine decibel reduction in the quadrature variance. This allows us to characterize the initial squeezed wavepacket by monitoring the onset of spin–motion entanglement, and to verify the evolution of the number states of the oscillator as a function of the duration of the force. In both cases we observe clear differences between displacements aligned with the squeezed and anti-squeezed axes. We observe coherent revivals when inverting the state-dependent force after separating the wavepackets by more than 19 times the ground-state root mean squared extent, which corresponds to 56 times the root mean squared extent of the squeezed wavepacket along the displacement direction. Aside from their fundamental nature, these states may be useful for quantum metrology6 or quantum information processing with continuous variables7,8,9.

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Figure 1: Spin population evolution due to spin–motion entanglement.
Figure 2: Revival of the spin coherence.
Figure 3: Evolution of displaced-squeezed-state mixtures.
Figure 4: Mandel Q parameter for the displaced-squeezed states.


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We thank J. Alonso and F. Leupold for comments on the manuscript, and F. Leupold, F. Lindenfelser, J. Alonso, M. Sepiol, K. Fisher and C. Flühmann for contributions to the experimental apparatus. We acknowledge support from the Swiss National Science Foundation under grant number 200021 134776, and through the National Centre of Competence in Research for Quantum Science and Technology (QSIT).

Author information

Experimental data were taken by H.-Y.L., D.K. and L.d.C., using an apparatus primarily built by D.K., H.-Y.L. and B.C.K., and with significant contributions from L.d.C., V.N. and M.M. Data analysis was performed by H.-Y.L. and J.P.H. The paper was written by J.P.H. and H.-Y.L., with input from all authors. The study was conceived by J.P.H.

Correspondence to Hsiang-Yu Lo or Jonathan P. Home.

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Extended data figures and tables

Extended Data Figure 1 Quasi-probability distributions for displaced-squeezed states in phase space using LDA and non-LDA.

a, c, e, The simulation results using LDA with different SDF durations. b, d, f, The results simulated using the full Hamiltonian.

Extended Data Figure 2 Coherence of cat states with fixed magnetic field noise.

The magnetic-field-induced energy-level shift of 1.5 kHz is used in this simulation. a, The duration of both SDF pulses is 60 μs. b, The duration of both SDF pulses is 120 μs. Dashed red and dash–dot green curves show the SDF aligned along the squeezed and anti-squeezed quadratures. The blue trace is for the SDF applied to a ground-state cooled ion.

Extended Data Figure 3 Coherence of cat states with a magnetic field fluctuation distribution.

With the assumption that the magnetic field exhibits a 50 Hz sinusoidal pattern with an amplitude of 2.2 mG, this plot shows the simulation results by taking an average over 100 samples on the field distribution. a, The duration of both SDF pulses is 60 μs. b, The duration of both SDF pulses is 120 μs. Definitions of the curve specification are the same as in Extended Data Fig. 2.

Extended Data Figure 4 Possible application of using SWESs for interferometry.

a, Use of squeezed-state wavepackets. b, Use of ground-state wavepackets. The first SDF pulse is used to create a spin–motion-entangled state. In the middle, a small phase shift Δθ is induced by shot-to-shot fluctuation in the oscillator frequency before the application of the second SDF pulse, which recombines the two distinct oscillator wavepackets.

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Lo, H., Kienzler, D., de Clercq, L. et al. Spin–motion entanglement and state diagnosis with squeezed oscillator wavepackets. Nature 521, 336–339 (2015). https://doi.org/10.1038/nature14458

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