Periodic driving can be used to control the properties of a many-body state coherently and to realize phases that are not accessible in static systems. For example, exposing materials to intense laser pulses makes it possible to induce metal–insulator transitions, to control magnetic order and to generate transient superconducting behaviour well above the static transition temperature1,2,3,4,5,6. However, pinning down the mechanisms underlying these phenomena is often difficult because the response of a material to irradiation is governed by complex, many-body dynamics. For static systems, extensive calculations have been performed to explain phenomena such as high-temperature superconductivity7. Theoretical analyses of driven many-body Hamiltonians are more challenging, but approaches have now been developed, motivated by recent observations8,9,10. Here we report an experimental quantum simulation in a periodically modulated hexagonal lattice and show that antiferromagnetic correlations in a fermionic many-body system can be reduced, enhanced or even switched to ferromagnetic correlations (sign reversal). We demonstrate that the description of the many-body system using an effective Floquet–Hamiltonian with a renormalized tunnelling energy remains valid in the high-frequency regime by comparing the results to measurements in an equivalent static lattice. For near-resonant driving, the enhancement and sign reversal of correlations is explained by a microscopic model of the system in which the particle tunnelling and magnetic exchange energies can be controlled independently. In combination with the observed sufficiently long lifetimes of the correlations in this system, periodic driving thus provides an alternative way of investigating unconventional pairing in strongly correlated systems experimentally7,9,10.
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We thank D. Abanin, D. Greif, D. Jaksch, M. Landini, Y. Murakami, N. Tsuji, P. Werner and W. Zwerger for discussions. We acknowledge SNF (Project Number 200020_169320 and NCCR-QSIT), Swiss State Secretary for Education, Research and Innovation Contract No. 15.0019 (QUIC) and ERC advanced grant TransQ (Project Number 742579) for funding.
The authors declare no competing financial interests.
Reviewer Information Nature thanks J. Freericks and D. Huse for their contribution to the peer review of this work.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data figures and tables
Nearest-neighbour spin–spin correlator C for the same lattice configuration as in Fig. 3, as a function of the modulation time after the ramp up of the drive. The data allow us to compare the formation and decay of magnetic correlations for two specific sets of interactions and modulation frequencies with the level of correlations in the static case (black). For a driving strength of K0 = 1.30(3) and with U/h = 3.8(1) kHz and ω/(2π) = 3 kHz (red), antiferromagnetic correlations increase with time and reach a level higher than the static case (black, U/h = 3.8(1) kHz). If the interaction is smaller than the driving frequency (blue, U/h = 4.4(1) kHz, ω/(2π) = 6 kHz), then the correlations switch sign and become ferromagnetic after a few milliseconds. For long times, the correlations in each configuration decrease as a result of heating in the lattice. Solid lines show exponential fits of the full data in the static case (grey) and to modulation times longer than 4 ms in the driven lattice for U > ħω (red). The difference between the data and the dashed component of the fit (red) indicates an initial increase in the correlations. The extracted lifetimes decrease from 82(34) ms without drive to 12(4) ms at K0 = 1.30(3). All measurements are averaged over one modulation cycle. Data points and error bars denote the mean and standard error of 13 individual measurements at different times within one driving period (see Methods).
a, b, Nearest-neighbour spin–spin correlator C for the lattice configuration in Fig. 3 and K0 = 1.30(3), as a function of modulation time after the ramp up of the drive, sampled within one oscillation period. We observe substantial micromotion both for the case of enhanced antiferromagnetic correlations (a; U/h = 3.8(1) kHz and ω/(2π) = 3 kHz) and for ferromagnetic correlations (b; U/h = 4.4(1) kHz and ω/(2π) = 6 kHz). For a different set of parameters in the measurement of the micromotion it should be also possible to switch between antiferromagnetic and ferromagnetic correlations within one driving cycle. The open symbols represent a reference measurement in the static case with all other parameters being equal. Solid lines are sinusoidal fits to the data, which results in a fitted frequency of (a) or (b). Error bars denote the standard error of 10 independent measurements.
a, Starting from the static lattice, the modulation amplitude is ramped up and subsequently kept at a fixed value to allow for a 5 ms equilibration time. The ramp up time depends on the chosen configuration and is 3.333 ms (2 ms) for a modulation frequency of ω/(2π) = 3 kHz (6 kHz). We start the detection of nearest-neighbour spin–spin correlations C in the effective Hamiltonian Heff by quenching the tunnelling to zero as we ramp up the lattice depth in all directions during the modulation within 100 μs. To estimate the adiabaticity of the final state, we perform a second type of measurement in which we revert the driving ramp and subsequently wait an additional 5 ms before the detection in the reverted static Hamiltonian . If the ramp scheme of the modulation is fully adiabiatic, we expect a reversal of the correlations to their static value. b, The nearest-neighbour spin–spin correlator C is plotted against the modulation amplitude in the off-resonant driving regime (U/h = 0.93(2) kHz, ω/(2π) = 6 kHz). The filled green circles are measured in the modulated system (same data as in Fig. 2b) and the open green circles after ramping off the modulation. The correlations no longer reach the level of the static case at K0 = 0 after reverting the ramp. We attribute this to some extent to a reduced lifetime of correlations, which is found to be 14(5) ms at K0 = 1.26(4), compared to 92(16) ms in the static case. c, Spin–spin correlator for different driving strengths K0 in the near-resonant regime for U < ħω (blue; U = 4.4(1) kHz, ω/(2π) = 6 kHz) and in the regime of enhanced antiferromagnetic correlations (red; U/h = 3.8(1) kHz, ω/(2π) = 3 kHz). Filled data points represent the effective states in the modulated system and open data points are measured after ramping off the modulation. Again, correlations do not reach the static value after reverting the driving ramp, owing to the finite lifetime (see also Extended Data Fig. 1). Data points and error bars denote the mean and standard error of 10 individual measurements at different times within one driving period (see Methods).
a, Quasi-energy spectrum for two particles in a double well as a function of the onsite interaction U for off-resonant driving (t/h = 350 Hz, K0 = 1.5, ω/(2π) = 8 kHz). Each of the four Floquet states representing the quasi-energy spectrum is shown in a distinct colour. The grey lines show the energy spectrum without modulation. For , the ground state is the spin singlet |s〉 and the first excited state is the triplet |t〉. To lowest order, the driving renormalizes the tunnelling by a zeroth-order Bessel function . b, Calculated exchange energy Jex,off-res (see Methods), defined as the energy difference between the spin singlet and triplet states (see a), as a function of the driving amplitude K0 for an off-resonant modulation (t/h = 350 Hz, U/h = 2.1 kHz, ω/(2π) = 8 kHz; compare with Fig. 4b). The dashed line is the analytical result derived from a high-frequency expansion of the effective Hamiltonian; the solid line is the result of a numerical calculation. The exchange energy is reduced to small values as the tunnelling is renormalized by the zeroth-order Bessel function J0(K0). For large modulation amplitudes, deviations from the result obtained from an expansion up to order 1/ω can be observed. Here, the exchange already becomes weakly ferromagnetic owing to the finite value of the interaction. c, Floquet spectrum of the double-well system as a function of the interactions U for near-resonant driving (t/h = 640 Hz, K0 = 0.8, ω/(2π) = 8 kHz). The grey lines show the energy spectrum without periodic modulation. The drive couples the singlet state to a state that contains double occupancy, which leads to an avoided crossing at U ≈ ħω. As a result, a gap opens that is to lowest order given by 4J1(K0). d, Dependence of the exchange energy Jex,res on the modulation amplitude in the near-resonant regime for two different detunings with t/h = 640 Hz and ω/(2π) = 8 kHz (blue, U/h = 6.5 kHz; red, U/h = 9.1 kHz; compare with Fig. 4c). The dashed line is the analytical result (see Methods) derived from a high-frequency expansion of the effective Hamiltonian; the solid line is the result of a numerical calculation. For U > ħω the exchange energy is greatly increased, whereas for U < ħω it changes sign to ferromagnetic behaviour. In both driving regimes, the analytical result is in very good agreement with the numerics. Our measurements of the exchange energy in Fig. 4 agree well on a qualitative level with the theoretical expectation.
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Görg, F., Messer, M., Sandholzer, K. et al. Enhancement and sign change of magnetic correlations in a driven quantum many-body system. Nature 553, 481–485 (2018). https://doi.org/10.1038/nature25135
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