Near invariance of quasi-energy spectrum of Floquet Hamiltonians
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| Format: | Preprint |
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2023
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| _version_ | 1866914166186442752 |
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| author | Sagiv, Amir Weinstein, Michael I. |
| author_facet | Sagiv, Amir Weinstein, Michael I. |
| contents | The spectral analysis of the unitary monodromy operator, associated with a time-periodically (paramatrically) forced Schrodinger equation, is a question of longstanding interest. Here, we consider this question for Hamiltonians of the form
$$H^{\varepsilon}(t)=H^0 + \varepsilon^a W(\varepsilon^a t, -i\nabla)\, ,$$ where $H^0$ is an unperturbed autonomous Hamiltonian, $a\geq 1$, and $W(T,\cdot)$ has a period of $T_{\rm per} >0$. In particular, in the small $\varepsilon>0$ regime, we seek a comparison between the spectral properties of the monodromy operator, the one-period flow map associated with the $H^\varepsilon(t)$ dynamics, and that of the autonomous (unforced) flow, $\exp[-iH^0 T_{\rm per} \varepsilon ^{-a}]$. We consider $H^0$ which is spatially periodic on $\mathbb{R} ^n$ with respect to a lattice. Using the decomposition of $H^0$ and $H^\varepsilon(t)$ into their actions on spaces (Floquet-Bloch fibers) of pseudo-periodic functions, we establish a near spectral-invariance property for the monodromy operator, when acting data which are $\varepsilon$-localized in energy and quasi-momentum. Our analysis requires the following steps: (i) spectrally-localized data are approximated by {\it band-limited (Floquet-Bloch) wavepackets}; (ii) the envelope dynamics of such wavepackets is well approximated by an effective (homogenized) PDE, and (iii) an exact invariance property for band-limited Floquet-Bloch wavepackets, which follows from the effective dynamics. We apply our general results to a number of periodic Hamiltonians, $H^0$, of interest in the study of photonic and quantum materials. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_10685 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Near invariance of quasi-energy spectrum of Floquet Hamiltonians Sagiv, Amir Weinstein, Michael I. Analysis of PDEs Mathematical Physics Spectral Theory The spectral analysis of the unitary monodromy operator, associated with a time-periodically (paramatrically) forced Schrodinger equation, is a question of longstanding interest. Here, we consider this question for Hamiltonians of the form $$H^{\varepsilon}(t)=H^0 + \varepsilon^a W(\varepsilon^a t, -i\nabla)\, ,$$ where $H^0$ is an unperturbed autonomous Hamiltonian, $a\geq 1$, and $W(T,\cdot)$ has a period of $T_{\rm per} >0$. In particular, in the small $\varepsilon>0$ regime, we seek a comparison between the spectral properties of the monodromy operator, the one-period flow map associated with the $H^\varepsilon(t)$ dynamics, and that of the autonomous (unforced) flow, $\exp[-iH^0 T_{\rm per} \varepsilon ^{-a}]$. We consider $H^0$ which is spatially periodic on $\mathbb{R} ^n$ with respect to a lattice. Using the decomposition of $H^0$ and $H^\varepsilon(t)$ into their actions on spaces (Floquet-Bloch fibers) of pseudo-periodic functions, we establish a near spectral-invariance property for the monodromy operator, when acting data which are $\varepsilon$-localized in energy and quasi-momentum. Our analysis requires the following steps: (i) spectrally-localized data are approximated by {\it band-limited (Floquet-Bloch) wavepackets}; (ii) the envelope dynamics of such wavepackets is well approximated by an effective (homogenized) PDE, and (iii) an exact invariance property for band-limited Floquet-Bloch wavepackets, which follows from the effective dynamics. We apply our general results to a number of periodic Hamiltonians, $H^0$, of interest in the study of photonic and quantum materials. |
| title | Near invariance of quasi-energy spectrum of Floquet Hamiltonians |
| topic | Analysis of PDEs Mathematical Physics Spectral Theory |
| url | https://arxiv.org/abs/2304.10685 |