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Size-controlled quantum dots reveal the impact of intraband transitions on high-order harmonic generation in solids


Since the discovery of high-order harmonic generation (HHG) in solids1,2,3, much effort has been devoted to understand its generation mechanism and both inter- and intraband transitions are known to be essential1,2,3,4,5,6,7,8,9,10. However, intraband transitions are affected by the electronic structure of a solid, and how they contribute to nonlinear carrier generation and HHG remains an open question. Here we use mid-infrared laser pulses to study HHG in CdSe and CdS quantum dots, where quantum confinement can be used to control the intraband transitions. We find that both HHG intensity per excited volume and generated carrier density increase when the average quantum dot size is increased from about 2 to 3 nm. We show that the reduction in sub-bandgap energy in larger quantum dots enhances intraband transitions, and this—in turn—increases the rate of photocarrier injection by coupling with interband transitions, resulting in enhanced HHG.

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Fig. 1: HHG in CdSe and CdS QD films.
Fig. 2: QD size dependence of HHG.
Fig. 3: TA measurements.
Fig. 4: Calculation results.

Data availability

Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.


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Y.K. acknowledges support from the Japan Society for the Promotion of Science (JSPS KAKENHI grant no. JP19H05465).

Author information

Authors and Affiliations



K.N. and H.H. carried out the experiments. K.N., H.H., S.A.S., H.T., F.S., G.Y. and Y.K. analysed the data. S.A.S. performed the simulations. M.S., R.S. and T.T. synthesized the QDs. H.H. and Y.K. conceived and supervised the project. All the authors discussed the results and contributed to the writing of the paper.

Corresponding authors

Correspondence to Hideki Hirori or Yoshihiko Kanemitsu.

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The authors declare no competing interests.

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Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Additional calculation results.

a, Bandgap energy Eg (blue squares) and subband gap, Δsub (red circles), as a function of the QD diameter (chain length). b, Diameter dependence of I7 obtained by assuming a size-independent Eg (green squares) and that obtained by the full model with a size-dependent Eg (red circles). c, Dependence of I7 on Δsub for different reduced masses.

Source data

Extended Data Fig. 2 Schematics of multiple excitation paths.

In addition to the contribution of the pure interband transition terms (left), the efficient intraband transition in larger QDs (or bulk) opens multiple excitation paths due to the nonlinear coupling between the intra- and interband transitions (right). These additional excitation channels due to the coupling promote nonlinear carrier injection and enhance HHG in larger QDs.

Extended Data Fig. 3 Yield ratio.

Diameter dependence of the yield ratio of the 7th order for CdSe, I7/nd. The data is normalized to the value at d = 6.4 nm. Vertical and horizontal error bars represent the standard deviation of yield ratio and that of diameter. The solid curve is a guide to the eye.

Source data

Supplementary information

Supplementary Information

Supplementary Sections I–VI and Figs. 1–11.

Source data

Source Data Fig. 1

Source data for Fig. 1b.

Source Data Fig. 2

Source data for Fig. 2.

Source Data Fig. 3

Source data for Fig. 3b,c.

Source Data Fig. 4

Source data for Figs. 4a–c.

Source Data Extended Data Fig. 1

Source data for Extended Data Fig. 1.

Source Data Extended Data Fig. 3

Source data for Extended Data Fig. 3.

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Nakagawa, K., Hirori, H., Sato, S.A. et al. Size-controlled quantum dots reveal the impact of intraband transitions on high-order harmonic generation in solids. Nat. Phys. 18, 874–878 (2022).

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