Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems

Journal name:
Nature Physics
Volume:
7,
Pages:
399–405
Year published:
DOI:
doi:10.1038/nphys1919
Received
Accepted
Published online

Abstract

Quantum simulators are controllable quantum systems that can reproduce the dynamics of the system of interest in situations that are not amenable to classical computers. Recent developments in quantum technology enable the precise control of individual quantum particles as required for studying complex quantum systems. In particular, quantum simulators capable of simulating frustrated Heisenberg spin systems provide platforms for understanding exotic matter such as high-temperature superconductors. Here we report the analogue quantum simulation of the ground-state wavefunction to probe arbitrary Heisenberg-type interactions among four spin-1/2 particles. Depending on the interaction strength, frustration within the system emerges such that the ground state evolves from a localized to a resonating-valence-bond state. This spin-1/2 tetramer is created using the polarization states of four photons. The single-particle addressability and tunable measurement-induced interactions provide us with insights into entanglement dynamics among individual particles. We directly extract ground-state energies and pairwise quantum correlations to observe the monogamy of entanglement.

At a glance

Figures

  1. Mimicking an adiabatic quantum evolution with an analogue quantum simulator.
    Figure 1: Mimicking an adiabatic quantum evolution with an analogue quantum simulator.

    a, Adiabatic quantum evolution. The system is prepared in an initial ground state |ψ(t0,B0,J0,)right fence. Then the gradual change of the system parameters (t, B, J and so on) causes an adiabatic evolution of the system to the final ground state of interest |ψ(t,B,J,)right fence. b, Analogue quantum simulation. The adiabatic evolution of the system to be simulated is mapped onto a controllable evolution of a quantum system. A set of tunable gates gives access to the change of parameters. c, Model used to study the valence-bond states. The nearest-neighbour Heisenberg-type interactions of strengths J1 and J2 among four spin-1/2 particles are drawn as connecting bonds and form a spin-1/2 tetramer. All the properties of the tetramer depend only on the coupling ratio κ=J2/J1. d, Quantum simulation of a spin-1/2 tetramer using a photonic analogue quantum simulator. The initial ground state, |Ψ(θ0)right fence, is prepared by generating the photon pairs 1 and 2 and 3 and 4 in two singlet states. Then the analogue quantum simulation is carried out using the measurement-induced interaction, consisting of quantum interference and the detection of a four-photon coincidence after superimposing photons 1 and 3 on a tunable beam splitter. Mapping the coupling ratio κ on the beam splitter’s splitting ratio tan2θ leads to the ground state of interest, |Ψ(θ)right fence.

  2. Experimental set-up.
    Figure 2: Experimental set-up.

    a, Femtosecond laser pulses (140fs, 76MHz, 404nm) penetrate two β-barium borate (BBO) crystals generating two pairs of photons in the spatial modes 1 and 2 and 3 and 4 (twofold coincident count rate per pair 20kHz). The walk-off effects are compensated with a half-wave plate followed by a BBO crystal in each mode. The photon’s spectral and spatial distinguishability is erased with interference filters (IFs, FWHM=3nm) and single-mode fibres. The polarization of each photon is analysed by a combination of a quarter-wave plate, a half-wave plate and a polarizing beam splitter (PBS). Single photons are detected by single-photon counting modules (SPCMs). b, Schematic diagram of the fibre-based tunable directional coupler (TDC). The view from the top of the TDC illustrates the dependence of the coupling of the evanescent light on the separation of the fibres. The coupling between these two fibres is controlled by adjusting the horizontal position of the D fibre. c, Experimental calibration of the TDC’s transmissivity (red circles) and reflectivity (black circles) with respect to the position of the D fibre(s) is carried out by using weak laser beams and an SPCM. The separations of the fibres for 0%, 50% and 100% transmissivity are shown in the insets. The experimental imperfections originate mostly from detector dark counts, and the error bars, smaller than 0.5% of the mean values, are based on a Poissonian distribution.

  3. Ground-state energy of the spin-1/2 tetramer.
    Figure 3: Ground-state energy of the spin-1/2 tetramer.

    By tuning θ, where represents the splitting ratio of the tunable directional coupler, we gradually change the ground state of the spin-1/2 tetramer. The full range of the coupling ratio is covered by tuning θ from to π/2. We measure the ground-state energy for seven different configurations. Of particular interest are the quantum states |Φparallelright fence+|Φ×right fence, |Φparallelright fence, |Φparallelright fence+|Φ=right fence and |Φ=right fence, shown explicitly. The black circles represent the experimental data and the solid line is the parameter-free theoretical prediction. The error bars follow Poissonian statistics and are smaller than the symbols.

  4. Density matrices of various spin-1/2 tetramer configurations in the computational basis (|H[rang]/|V [rang]).
    Figure 4: Density matrices of various spin-1/2 tetramer configurations in the computational basis (|Hright fence/|V right fence).

    ad, The real parts of the density matrices for the cases of equal superposition of dimer-covering states (a), dimer-covering states (b,d) and resonating valence-bond state (c). The imaginary parts are small and are shown in Supplementary Information. The wire grids indicate the expected values for the ideal case. The density matrices are reconstructed from the experimental four-photon tomography data for the settings of θ=0.197π (a), θ=0.25π (b), θ=0.304π (c) and θ=0.468π (d). The fidelities, F, of the measured density matrix with the ideal state are F=0.745(4) (a), F=0.712(4) (b), F=0.746(6) (c) and F=0.888(2) (d). The uncertainties in fidelities extracted from these density matrices are calculated using a Monte Carlo routine and assumed Poissonian errors.

  5. Experimentally extracted pairwise Heisenberg energies.
    Figure 5: Experimentally extracted pairwise Heisenberg energies.

    a, Experimental observation of quantum monogamy when comparing the pairwise normalized Heisenberg energy, eij; it acts as a two-particle entanglement witness, for the spin pairs 1 and 2 (black square), 1 and 3 (red circle) and 1 and 4 (blue triangle). The highlighted area corresponds to the full range of the coupling ratio . For the case of κ=0 (θ=π/4), the ground state of this spin-1/2 tetramer is |Φparallelright fence=|ψright fence13|ψright fence24 and the amount of entanglement of the pair 1 and 3 reaches its maximum whereas the pairs 1 and 2 and 1 and 4 are not entangled. Similarly, for the case of (θ=π/2), the ground state is reduced to |Φ=right fence=|ψright fence12|ψright fence34, where pair 1 and 2 is now maximally entangled, and pairs 1 and 3 and 1 and 4 are disentangled. In the case of the resonating valence-bond state, entanglement distributions are equal between the pairs 1 and 2 and 1 and 3 (that is, e12=e13). In other cases, entanglement is distributed according to the monogamy relation. b, Experimental demonstration of the complementarity relation in a spin-1/2 tetramer. For each valence-bond configuration we measured pairwise Heisenberg energies, eij, which are normalized by the maximal value, eijmax. The sum of these renormalized energy values is in good agreement with the theoretical prediction (shown as a line in the plot). The uncertainties represent standard deviations deduced from propagated Poissonian statistics.

  6. Directly observed pairwise correlation functions of various valence-bond states.
    Figure 6: Directly observed pairwise correlation functions of various valence-bond states.

    ad, The correlation tensors T12 (photons 1 and 2), T13 (photons 1 and 3) and T14 (photons 1 and 4) are obtained from correlation measurements directly in the bases X=σx, Y =σy and Z=σz. For a convenient graphical representation, the negative values of the correlation tensors are shown. The structures of the superposition state (a) and the resonating valence-bond state (c) show that the quantum correlations are equally distributed among two competing pairs. b,d, Dimer-covering states, in which only one pair is maximally correlated in a singlet state.

References

  1. Giovannetti, V., Lloyd, S. & Maccone, L. Quantum-enhanced measurements: Beating the standard quantum limit. Science 306, 13301336 (2004).
  2. Zoller, P. et al. Quantum information processing and communication. Eur. Phys. J. D 36, 203228 (2005).
  3. Deutsch, D. & Jozsa, R. Rapid solutions of problems by quantum computation. Proc. R. Soc. Lond. A 439, 553558 (1992).
  4. Shor, P. W. in Algorithms for Quantum Computation: Discrete Logarithms and Factoring (ed. Goldwasser, S.) 124134 (Proc. 35th Annu. Symp. Foundations of Computer Science, 1994).
  5. Grover, L. K. Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325328 (1997).
  6. Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21, 467488 (1982).
  7. Feynman, R. P. Quantum mechanical computers. Found. Phys. 16, 507531 (1986).
  8. Lloyd, S. Universal quantum simulators. Science 273, 10731078 (1996).
  9. Farhi, E., Goldstone, J., Gutmann, S. & Sipser, M. Quantum computation by adiabatic evolution. Preprint at http://arxiv.org/abs/quant-ph/0001106v1 (2000).
  10. Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 17041707 (2005).
  11. Trebst, S., Schollwöck, U., Troyer, M. & Zoller, P. d-wave resonating valence bond states of fermionic atoms in optical lattices. Phys. Rev. Lett. 96, 250402 (2006).
  12. Buluta, I. & Nori, F. Quantum simulators. Science 326, 108111 (2009).
  13. Biamonte, J., Bergholm, V., Whitfield, J., Fitzsimons, J. & Aspuru-Guzik, A. Adiabatic quantum simulators. Preprint at http://arxiv.org/abs/1002.0368v1 (2010).
  14. Greiner, M., Mandel, O., Esslinger, T., Hansch, T. W. & Bloch, I. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415, 3944 (2002).
  15. Lewenstein, M. et al. Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond. Adv. Phys. 56, 243379 (2007).
  16. Trotzky, S. et al. Suppression of the critical temperature for superfluidity near the Mott transition. Nature Phys. 6, 9981004 (2010).
  17. Leibfried, D. et al. Trapped-ion quantum simulator: Experimental application to nonlinear interferometers. Phys. Rev. Lett. 89, 247901 (2002).
  18. Friedenauer, A., Schmitz, H., Glueckert, J. T., Porras, D. & Schaetz, T. Simulating a quantum magnet with trapped ions. Nature Phys. 4, 757761 (2008).
  19. Gerritsma, R. et al. Quantum simulation of the Dirac equation. Nature 463, 6871 (2010).
  20. Kim, K. et al. Quantum simulation of frustrated Ising spins with trapped ions. Nature 465, 590593 (2010).
  21. Lu, C-Y. et al. Demonstrating anyonic fractional statistics with a six-qubit quantum simulator. Phys. Rev. Lett. 102, 030502 (2009).
  22. Pachos, J. K. et al. Revealing anyonic features in a toric code quantum simulation. New J. Phys. 11, 083010 (2009).
  23. Lanyon, B. P. et al. Towards quantum chemistry on a quantum computer. Nature Chem. 2, 106111 (2010).
  24. Kaltenbaek, R., Lavoie, J., Zeng, B., Bartlett, S. D. & Resch, K. J. Optical one-way quantum computing with a simulated valence-bond solid. Nature Phys. 6, 850854 (2010).
  25. Somaroo, S., Tseng, C. H., Havel, T. F., Laflamme, R. & Cory, D. G. Quantum simulations on a quantum computer. Phys. Rev. Lett. 82, 53815384 (1999).
  26. Du, J. et al. NMR implementation of a molecular hydrogen quantum simulation with adiabatic state preparation. Phys. Rev. Lett. 104, 030502 (2010).
  27. Verstraete, F., Cirac, J. I. & Latorre, J. I. Quantum circuits for strongly correlated quantum systems. Phys. Rev. A 79, 032316 (2009).
  28. Bakr, W. S., Gillen, J. I., Peng, A., Folling, S. & Greiner, M. A quantum gas microscope for detecting single atoms in a Hubbard-regime optical lattice. Nature 462, 7477 (2009).
  29. Bakr, W. S. et al. Probing the superfluid–to–Mott insulator transition at the single-atom level. Science 329, 547550 (2010).
  30. Sherson, J. F. et al. Single-atom resolved fluorescence imaging of an atomic Mott insulator. Nature 467, 6872 (2010).
  31. Coffman, V., Kundu, J. & Wootters, W. K. Distributed entanglement. Phys. Rev. A 61, 052306 (2000).
  32. Osborne, T. J. & Verstraete, F. General monogamy inequality for bipartite qubit entanglement. Phys. Rev. Lett. 96, 220503 (2006).
  33. Bethe, H. Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Z. Phys. A 71, 205226 (1931).
  34. Born, M. & Fock, V. Beweis des Adiabatensatzes. Z. Phys. A 51, 165180 (1928).
  35. Marshall, W. Antiferromagnetism. Proc. R. Soc. A 232, 4868 (1955).
  36. Lieb, E. & Mattis, D. Ordering energy levels of interacting spin systems. J. Math. Phys. 3, 749751 (1962).
  37. Affleck, I., Kennedy, T., Lieb, E. H. & Tasaki, H. Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett. 59, 799802 (1987).
  38. Balents, L. Spin liquids in frustrated magnets. Nature 464, 199208 (2010).
  39. Mambrini, M., Läuchli, A., Poilblanc, D. & Mila, F. Plaquette valence-bond crystal in the frustrated Heisenberg quantum antiferromagnet on the square lattice. Phys. Rev. B 74, 144422 (2006).
  40. Anderson, P. W. The resonating valence bond state in La2CuO4 and superconductivity. Science 235, 11961198 (1987).
  41. White, A. G., James, D. F. V., Eberhard, P. H. & Kwiat, P. G. Nonmaximally entangled states: Production, characterization, and utilization. Phys. Rev. Lett. 83, 31033107 (1999).
  42. James, D. F. V., Kwiat, P. G., Munro, W. J. & White, A. G. Measurement of qubits. Phys. Rev. A 64, 052312 (2001).
  43. Brukner, Č. & Vedral, V. Macroscopic thermodynamical witnesses of quantum entanglement. Preprint at http://arxiv.org/abs/quant-ph/0406040 (2004).
  44. Amico, L., Fazio, R., Osterloh, A. & Vedral, V. Entanglement in many-body systems. Rev. Mod. Phys. 80, 517576 (2008).
  45. Wootters, W. K. Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 22452248 (1998).
  46. Englert, B-G. Fringe visibility and which-way information: An inequality. Phys. Rev. Lett. 77, 21542157 (1996).
  47. Brukner, Č., Aspelmeyer, M. & Zeilinger, A. Complementarity and information in delayed-choice for entanglement swapping. Found. Phys. 35, 19091919 (2005).
  48. Dürr, S., Nonn, T. & Rempe, G. Fringe visibility and which-way information in an atom interferometer. Phys. Rev. Lett. 81, 57055709 (1998).
  49. Kassal, I., Jordan, S. P., Love, P. J., Mohseni, M. & Aspuru-Guzik, A. Polynomial-time quantum algorithm for the simulation of chemical dynamics. Proc. Natl Acad. Sci. USA 105, 1868118686 (2008).
  50. Knill, E., Laflamme, R. & Milburn, G. A scheme for efficient quantum computation with linear optics. Nature 409, 4652 (2001).

Download references

Author information

  1. These authors contributed equally to this work

    • Xiao-song Ma &
    • Borivoje Dakic

Affiliations

  1. Austrian Academy of Sciences, Institute for Quantum Optics and Quantum Information (IQOQI), Boltzmanngasse 3, A-1090 Vienna, Austria

    • Xiao-song Ma,
    • William Naylor,
    • Anton Zeilinger &
    • Philip Walther
  2. University of Vienna, Faculty of Physics, Boltzmanngasse 5, A-1090 Vienna, Austria

    • Xiao-song Ma,
    • Borivoje Dakic,
    • William Naylor,
    • Anton Zeilinger &
    • Philip Walther
  3. Vienna Centre for Quantum Science and Technology, Boltzmanngasse 3, A-1090 Vienna, Austria

    • Anton Zeilinger

Contributions

X-s.M. and W.N. designed and carried out experiments, analysed data and wrote the manuscript. B.D. provided the theoretical analysis, analysed data and wrote the manuscript. A.Z. supervised the project and edited the manuscript. P.W. designed experiments, analysed data, wrote the manuscript and supervised the project.

Competing financial interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to:

Author details

Supplementary information

PDF files

  1. Supplementary Information (950k)

    Supplementary Information

Additional data