Accelerated quantum control using superadiabatic dynamics in a solid-state lambda system

Journal name:
Nature Physics
Year published:
Published online

Adiabatic processes are useful for quantum technologies1, 2, 3 but, despite their robustness to experimental imperfections, they remain susceptible to decoherence due to their long evolution time. A general strategy termed shortcuts to adiabaticity4, 5, 6, 7, 8, 9 (STA) aims to remedy this vulnerability by designing fast dynamics to reproduce the results of a slow, adiabatic evolution. Here, we implement an STA technique known as superadiabatic transitionless driving10 (SATD) to speed up stimulated Raman adiabatic passage1, 11, 12, 13, 14 in a solid-state lambda system. Using the optical transitions to a dissipative excited state in the nitrogen-vacancy centre in diamond, we demonstrate the accelerated performance of different shortcut trajectories for population transfer and for the initialization and transfer of coherent superpositions. We reveal that SATD protocols exhibit robustness to dissipation and experimental uncertainty, and can be optimized when these effects are present. These results suggest that STA could be effective for controlling a variety of solid-state open quantum systems11, 12, 13, 14, 15, 16.

At a glance


  1. Concept and implementation of three-level superadiabatic transitionless driving.
    Figure 1: Concept and implementation of three-level superadiabatic transitionless driving.

    a, State transfer in a NV centre Λ system by STIRAP. The |+1right fence/ |−1right fence ground-state spin levels are coupled by resonant Stokes ΩS(t) and pump ΩP(t) optical fields to the |A2right fence excited state, which acts as the intermediate state for STIRAP. b, Schematic of possible dynamics. An adiabatic protocol transfers the initial state |ψIright fence to the final state |ψFright fence along the dashed, red trajectory, which is followed exactly only in the infinite time limit. For finite-time realizations, |ψIright fence may be transferred to a different state |φright fence due to non-adiabatic transitions (blue). Our superadiabatic shortcut (solid red) implements modified driving pulses to reproduce the same final transfer of the adiabatic protocol, but for arbitrary evolution time and along a different path determined by the choice of dressed basis. Dissipation leads to errors for all evolutions. c, Example of the modified ΩS(t) pulses for SATD and MOD-SATD, corresponding to two different basis choices. The shape parameter Ashape specifies the appropriate driving pulse under unitary evolution for a particular experimental coupling strength Ω and pulse duration L. The modified ΩP(t) pulses (not shown) mirror the ΩS(t) pulses about the midpoint of the protocol. d, Experimental set-up utilizing EOMs to shape the Stokes and pump pulses from a single laser on sub-nanosecond timescales. AWG, arbitrary waveform generator; IQ, quadrature modulation; SG, signal generator; P/AEOM, phase/amplitude electro-optic modulator; DC, dichroic mirror; APD, avalanche photodiode. e, Optically driven Rabi oscillations between the |−1right fence and |A2right fence levels. The oscillations damp due to excited state dissipation (lifetime and dephasing) and spectral diffusion. The solid line is an example of a fit to a master equation model using the rates given in the main text.

  2. Performance and robustness of superadiabatic pulses.
    Figure 2: Performance and robustness of superadiabatic pulses.

    a, STIRAP transfer efficiency of MOD-SATD, SATD, and adiabatic (Vitanov) pulses as a function of the maximum optical Rabi strength Ω. The superadiabatic protocols utilize pulses prescribed for unitary evolution: the shape parameter Ashape is equal to the experimental adiabaticity A, determined by Ω and the constant pulse duration L = 16.8ns. The right y-axis indicates the absolute population in |+1right fence at the end of the protocol. The left y-axis estimates a transfer efficiency that accounts for imperfect initialization by using direct microwave transfer from |0right fence into |+1right fence to establish a reference transfer efficiency of 1. b, Robustness of the transfer efficiency as a function of the pulse shape Ashape for Ω = 2π × 115MHz. The maximum transfer efficiency for the superadiabatic protocols occurs for a shape parameter Ashapeopt < A, reflecting the presence of dissipation and spectral diffusion. Typical errorbars in a and b correspond to 95% confidence. c, False colour plot of the experimental (left) and simulation (right) transfer efficiency for SATD as a function of A and Ashape. The dashed black lines represent Ashape = A. The data points and fitted cyan line on the experimental plot delineate the extracted Ashapeopt, while the interval corresponds to ±1% in transfer efficiency. The deviation Ashapeopt < A is consistent with the dissipative model (cyan trace denotes Ashapeopt in model results). d, Photoluminescence (PL) (left y-axis) and converted |A2right fence population (right y-axis) measured during the adiabatic, SATD, and MOD-SATD pulses for Ω = 2π × 113MHz, highlighting the designed occupation of |A2right fence (less for MOD-SATD) by the superadiabatic pulses.

  3. Speed-up of superadiabatic protocols.
    Figure 3: Speed-up of superadiabatic protocols.

    STIRAP transfer efficiency for the optimal MOD-SATD and SATD pulses versus the adiabatic pulse as a function of the pulse duration LΩmin−1 for a constant Rabi strength Ω = 2π × 122MHz. The vertical grey bar at 5.8ns represents the quantum speed limit for state transfer via an intermediate state for this coupling strength Ω. The solid grey lines represent interpolating functions used to invert the plot and estimate the pulse length LA (LSA) of the adiabatic (superadiabatic) protocol needed to attain a given transfer efficiency. The inset displays the speed-up factor, given by the ratio LA/LSA, as a function of the desired transfer efficiency. Dashed lines in the inset represent extrapolations outside the range of experimentally attained transfer efficiencies.

  4. Accelerating the transfer and initialization of superposition states.
    Figure 4: Accelerating the transfer and initialization of superposition states.

    a, Bloch sphere schematic for phase-coherent STIRAP processes. (Top) Transfer of superpositions: the phase relation within an initial superposition |ψIright fence of the |0right fence/ |−1right fence states is transferred by STIRAP to a target superposition |ψFright fence of the |0right fence/ |+1right fence states. (Bottom) Initialization of superpositions: fractional STIRAP enables the creation of arbitrary superpositions of the |−1right fence/ |+1right fence states by maintaining a particular phase and amplitude relation between ΩS(t) and ΩP(t) as both fields are simultaneously ramped to zero. b, Visualization of the phase of the transferred superposition |ψFright fence on a polar plot for MOD-SATD, SATD, and adiabatic protocols as the phase of |ψIright fence is incremented. X and Y are the components of the projections of |ψFright fence onto and , respectively, that vary with the initialized phase. The phase visibility can be compared to the square root of the population transfer efficiency (delineated by the solid arcs) to gauge the coherent fraction of the population transfer for each protocol. c, State tomography and fidelity F for the initialization of two different final superposition states by fractional STIRAP via a shortcut SATD protocol (top) and an adiabatic protocol (bottom).


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Author information


  1. Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA

    • Brian B. Zhou,
    • Christopher G. Yale,
    • F. Joseph Heremans,
    • Paul C. Jerger &
    • David D. Awschalom
  2. Department of Physics, McGill University, Montreal, Quebec H3A 2T8, Canada

    • Alexandre Baksic,
    • Hugo Ribeiro &
    • Aashish A. Clerk
  3. Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA

    • F. Joseph Heremans &
    • David D. Awschalom
  4. Department of Physics, University of Konstanz, D-78457 Konstanz, Germany

    • Adrian Auer &
    • Guido Burkard


H.R. and B.B.Z. engaged in preliminary discussions. A.B., H.R. and A.A.C. developed the superadiabatic theory. B.B.Z., C.G.Y., F.J.H. and P.C.J. performed the experiments. A.B., A.A., H.R. and G.B. completed the master equation modelling. D.D.A. advised all efforts. All authors contributed to the data analysis and writing of the manuscript.

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