Quantum simulation is a rapidly advancing tool for gaining insight into complex quantum states and their dynamics. Trapped-ion systems have pioneered deterministic state preparation and comprehensive state characterization, operating on localized and thus distinguishable particles1. With ultracold atom experiments, one can prepare large samples of delocalized particles, but the same level of characterization has not yet been achieved2. Here, we present a method to measure the positions and momenta of individual particles to obtain correlations and coherences. We demonstrate this with deterministically prepared samples of two interacting ultracold fermions in a coupled double well3. As a first application, we use our technique to certify and quantify different types of entanglement4,5,6.
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The data that support the plots within this paper and other findings of this study are available from the corresponding author on reasonable request.
Journal peer review information: Nature Physics thanks Luca Pezzè and the other anonymous reviewer(s) for their contribution to the peer review of this work.
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Blatt, R. & Roos, C. F. Quantum simulations with trapped ions. Nat. Phys. 8, 277–284 (2012).
Gross, C. & Bloch, I. Quantum simulations with ultracold atoms in optical lattices. Science 357, 995–1001 (2017).
Murmann, S. et al. Two fermions in a double well: exploring a fundamental building block of the Hubbard model. Phys. Rev. Lett. 114, 080402 (2015).
Ghirardi, G., Marinatto, L. & Weber, T. Entanglement and properties of composite quantum systems: a conceptual and mathematical analysis. J. Stat. Phys. 108, 49–122 (2002).
Dowling, M. R., Doherty, A. C. & Wiseman, H. M. Entanglement of indistinguishable particles in condensed-matter physics. Phys. Rev. A 73, 052323 (2006).
Bonneau, M., Munro, W. J., Nemoto, K. & Schmiedmayer, J. Characterizing twin-particle entanglement in double-well potentials. Phys. Rev. A 98, 033608 (2018).
Ott, H. Single atom detection in ultracold quantum gases: a review of current progress. Rep. Prog. Phys. 79, 054401 (2016).
Fölling, S. et al. Spatial quantum noise interferometry in expanding ultracold atom clouds. Nature 434, 481–484 (2005).
Yannouleas, C., Brandt, B. B. & Landman, U. Interference, spectral momentum correlations, entanglement and Bell inequality for a trapped interacting ultracold atomic dimer: analogies with biphoton interferometry. Phys. Rev. A 99, 013616 (2019).
Desbuquois, R. et al. Controlling the Floquet state population and observing micromotion in a periodically driven two-body quantum system. Phys. Rev. A 96, 053602 (2017).
Tichy, M. C., Mintert, F. & Buchleitner, A. Essential entanglement for atomic and molecular physics. J. Phys. B 44, 192001 (2011).
Amico, L., Osterloh, A. & Vedral, V. Entanglement in many-body systems. Rev. Mod. Phys. 80, 517 (2008).
Horodecki, R., Horodecki, P., Horodecki, M. & Horodecki, K. Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009).
Pezzè, L., Smerzi, A., Oberthaler, M. K., Schmied, R. & Treutlein, P. Quantum metrology with nonclassical states of atomic ensembles. Rev. Mod. Phys. 90, 035005 (2018).
Dai, H.-N. et al. Generation and detection of atomic spin entanglement in optical lattices. Nat. Phys. 12, 783–787 (2016).
Kaufman, A. M. et al. Entangling two transportable neutral atoms via local spin exchange. Nature 527, 208–211 (2015).
Lester, B. J. et al. Measurement-based entanglement of noninteracting bosonic atoms. Phys. Rev. Lett. 120, 193602 (2018).
Cramer, M. et al. Spatial entanglement of bosons in optical lattices. Nat. Commun. 4, 2161 (2013).
Islam, R. et al. Measuring entanglement entropy in a quantum many-body system. Nature 528, 77–83 (2015).
Zanardi, P. Quantum entanglement in fermionic lattices. Phys. Rev. A 65, 042101 (2002).
Wiseman, H. M. & Vaccaro, J. A. Entanglement of indistinguishable particles shared between two parties. Phys. Rev. Lett. 91, 097902 (2003).
Fukuhara, T. et al. Spatially resolved detection of a spin-entanglement wave in a Bose–Hubbard chain. Phys. Rev. Lett. 115, 035302 (2015).
Mazza, L., Rossini, D., Fazio, R. & Endres, M. Detecting two-site spin-entanglement in many-body systems with local particle-number fluctuations. New J. Phys. 17, 013015 (2015).
Taguchi, G. et al. Measurement and control of spatial qubits generated by passing photons through double slits. Phys. Rev. A 78, 012307 (2008).
Bergschneider, A. et al. Spin-resolved single-atom imaging of 6Li in free space. Phys. Rev. A 97, 063613 (2018).
Wootters, W. K. Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998).
Jafarpour, M. & Sabour, A. A useful strong lower bound on two-qubit concurrence. Quantum Inf. Process. 11, 1389–1402 (2012).
Mintert, F. & Buchleitner, A. Observable entanglement measure for mixed quantum states. Phys. Rev. Lett. 98, 140505 (2007).
Blume-Kohout, R. Optimal, reliable estimation of quantum states. New J. Phys. 12, 043034 (2010).
Kitagawa, T., Aspect, A., Greiner, M. & Demler, E. Phase-sensitive measurements of order parameters for ultracold atoms through two-particle interferometry. Phys. Rev. Lett. 106, 115302 (2011).
The authors acknowledge insightful discussions with A. Daley, N. Defenu, A. Elben, M. Gärttner, P. Hauke and M. Piani. This work has been supported by ERC consolidator grant 725636, DFG grant JO970/1-1, the Heidelberg Center for Quantum Dynamics and is part of the DFG Collaborative Research Centre SFB 1225 (ISOQUANT). A.B. acknowledges funding from the International Max-Planck Research School (IMPRS-QD). P.M.P. acknowledges funding from the European Union’s Horizon 2020 programme under Marie Sklodowska-Curie grant agreement no. 706487 and from the Daimler and Benz Foundation.
Electronic supplementary material
Supplementary text, Supplementary Figures 1–5 and Supplementary references.