Introduction

The coherent dynamics of electrons in crystalline potentials under time-periodic driving fields is at the heart of such major phenomena as photon-assisted transport, Rabi oscillations (ROs), dynamic localization, and super-Bloch oscillations1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18. Strong ac fields can modify the band structures of materials and alter the corresponding internal structure of the electronic wave functions19,20,21,22,23, whereas weak ac fields in resonance with two bands of the crystal can induce interband transitions, e.g., absorption and emission of quanta from the field. Momentum conservation ensures that direct transitions must be vertical in the (k, E) plane during absorption and emission, where k and E are the quasi-momentum and energy of the electron. Under coherent dynamics, periodic electron flopping, i.e., ROs between two Bloch bands, can be observed4,5,8. The characteristic frequency of the Rabi flopping is proportional to the electric-dipole moment of the transition and the field strength. In condensed matter systems, dephasing effects generally prevent the observation of Rabi flopping. For such a reason, ROs have been observed mostly in synthetic lattices, such as in cold atoms and photonic crystals5,14.

Topological properties, transport and phase transitions in non-Hermitian crystals, i.e., described by an effective non-Hermitian Hamiltonian, have attracted a huge interest in the past few years24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54. In such crystals, the energy spectrum is strongly sensitive to perturbations, and largely differs under periodic (PBC) and open (OBC) boundary conditions. In systems with OBC the bulk states can get squeezed toward the lattice edges (non-Hermitian skin effect25,26,27,28,29,44,53), and the bulk-boundary correspondence based on Bloch band topological invariants generally fails to predict topological edge states. To correctly describe energy spectra and topological invariants in crystals with OBC one needs to extend Bloch band theory so as the quasi-momentum becomes complex and varies on a generalized Brillouin zone (GBZ)25,26,28,34,44. Bloch and non-Bloch bands show different energy spectra and can undergo different symmetry breaking phase transitions. As major attention is currently devoted to study the topological properties and related symmetries in several non-Hermitian models, the impact of the skin effect on bulk transport properties in non-Hermitian lattices driven by external fields remains so far largely unexplored.

In this work, we show how non-Hermitian influences can disclose a scenario fully distinct from common Rabi flopping, hosting novel features such as enhancement of the effective dipole moment arising from the non-Hermitian skin effect, non-vertical transitions and anharmonic ROs, hence providing unprecedented freedom in controlling the frequency and anharmonicity of ROs. While non-Hermitian systems with complex eigenenergies have often been considered to be of limited experimental interest, since complex eigenenergies seem to lead to rapid decay or divergences, ROs can be sustained without gain or loss, even if the eigenspectrum is complex. This thus greatly expands the scope by which ROs can be controlled or engineered, achieving Rabi frequencies that are orders of magnitudes higher than allowed by the bare dipole moments. More generally, our study sheds light on how non-Hermiticity further enriches the already vibrant field of Floquet dynamics55,56,57 beyond merely causing gain or attenuation.

Results

ROs in a non-Hermitian lattice

Let us consider a minimal non-Hermitian one-dimensional (1D) system comprising two identical chains (sublattices) HO coupled by an inter-chain coupling term HC. Acting on the system is a weak ac field \(F(t)={F}_{0}\cos (\omega t)\) that staggers the energy of the two sublattices and drives the ROs. The system is thus described by a two-component Hamiltonian in real space

$$H(t)=\left(\begin{array}{cc}{H}_{O}&{H}_{C}\\ {H}_{C}&{H}_{O}\end{array}\right)-{d}_{y}F(t)\left(\begin{array}{cc}{\mathcal{I}}&0\\ 0&-{\mathcal{I}}\end{array}\right),$$
(1)

where 2dy is the spatial separation between the two chains of length N, and \({\mathcal{I}}\) is the N × N identity matrix (Fig. 1a). Diagonalizing H(t) in the zero field limit via a basis transformation H →  U−1HU with \(U=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1&1\\ 1&-1\end{array}\right)\), our system takes the form of two effective but inequivalent chains H± = HO ± HC that are coupled by the oscillatory field F(t) (Fig. 1b). In this new basis built from the symmetric and antisymmetric sectors, a two-component state \(\left|\psi (t)\right\rangle ={(\left|{\bf{A}}\right\rangle ,\left|{\bf{B}}\right\rangle )}^{\mathrm{T}}\) obeys the dynamical evolution equation \(i\frac{{\mathrm{d}}\psi }{{\mathrm{d}}t}=H\psi\) which reads

$$i\frac{\mathrm{d}}{{\mathrm{d}}t}\left(\begin{array}{c}{\bf{A}}\\ {\bf{B}}\end{array}\right)=\left[\left(\begin{array}{cc}{H}_{+}&0\\ 0&{H}_{-}\end{array}\right)-{d}_{y}F(t)\left(\begin{array}{cc}0&{\mathcal{I}}\\ {\mathcal{I}}&0\end{array}\right)\right]\left(\begin{array}{c}{\bf{A}}\\ {\bf{B}}\end{array}\right).$$
(2)

To solve Eq. (2), we expand \(\left|{\bf{A}}\right\rangle\) and \(\left|{\bf{B}}\right\rangle\) in terms of the right eigenvectors \(|{u}_{+,n}^{\mathrm{R}}\rangle\) and \(|{u}_{-,n}^{\mathrm{R}}\rangle\), respectively, defined by \({H}_{\pm }|{u}_{\pm ,n}^{\mathrm{R}}\rangle ={E}_{\pm ,n}|{u}_{\pm ,n}^{\mathrm{R}}\rangle\):

$$\left|{\bf{A}}\right\rangle =\sum_{n}{\alpha }_{n}(t){\mathrm{e}}^{-i{E}_{+,n}t}\left|{u}_{+,n}^{\mathrm{R}}\right\rangle,$$
(3)
$$\left|{\bf{B}}\right\rangle =\sum_{n}{\beta }_{n}(t){\mathrm{e}}^{-i{E}_{-,n}t}\left|{u}_{-,n}^{\mathrm{R}}\right\rangle .$$
(4)

Upon substituting into Eq. (2) and left multiplying by left eigenvectors defined by \({H}_{\pm }^{\dagger }|{u}_{\pm ,n}^{\mathrm{L}}\rangle ={E}_{\pm ,n}^{* }|{u}_{\pm ,n}^{\mathrm{L}}\rangle\) and obeying the biorthogonal normalization \(\langle {u}_{\pm ,n}^{\mathrm{L}}| {u}_{\pm ,l}^{\mathrm{R}}\rangle ={\delta }_{nl}\), we obtain coupled equations describing the evolution of the amplitude probabilities αn(t) and βn(t) of the symmetric/antisymmetric sectors:

$$i\frac{{\mathrm{d{\alpha }}}_{n}(t)}{{\mathrm{d}}t}=-{d}_{y}F(t)\sum_{l}{\Gamma }_{n,l}{\beta }_{l}(t){\mathrm{e}}^{i({E}_{+,n}-{E}_{-,l})t},$$
(5)
$$i\frac{{\mathrm{d{\beta }}_{l}}(t)}{{\mathrm{d}}t}=-{d}_{y}F(t)\sum_{n}{G}_{l,n}{\alpha }_{n}(t){\mathrm{e}}^{i({E}_{-,l}-{E}_{+,n})t},$$
(6)

where \({\Gamma }_{n,l}=\langle {u}_{+,n}^{\mathrm{L}}| {u}_{-,l}^{\mathrm{R}}\rangle\) and \({G}_{l,n}=\langle {u}_{-,l}^{\mathrm{L}}| {u}_{+,n}^{\mathrm{R}}\rangle\). Without any restriction on the forms of H± = HO ± HC, Eqs. (5) and (6) generically describe wildly fluctuating dynamics that is generically aperiodic with complex quasi-energy spectrum. To investigate ROs, we specialize to cases where well-defined oscillations exist between two chosen eigenstates \(|{u}_{+,n}^{\mathrm{R}}\rangle\) and \(|{u}_{-,l}^{\mathrm{R}}\rangle\) having the same growth/decay rate, i.e., vanishing imaginary part of E+,n − E−,l (Fig. 1c), and make the crude rotating wave approximation (RWA), assuming as usual that F(t) is modulated at resonance ω = ωnl ≡ E+,n − E−,l and the Rabi frequency is much smaller than ω. Neglecting all non-resonant and cross-coupling terms, from Eqs. (5) and (6) harmonic oscillator equations for the coupled amplitudes αn and βl are obtained, namely \(\frac{{\mathrm{d}}^{2}{\alpha }_{n}(t)}{{\mathrm{d}}{t}^{2}}+\frac{1}{4}({\mathrm{d}}_{y}^{2}{F}_{0}^{2}{\Gamma }_{n,l}{G}_{l,n}){\alpha }_{n}(t)=0\) (and similarly for βl(t)). The Rabi frequency is thus

$${\Omega }_{\mathrm{R}}={d}_{y}{F}_{0}\sqrt{| {\Gamma }_{n,l}{G}_{l,n}| }={d}_{y}{F}_{0}\sqrt{| {\rm{Tr}}[{P}_{n}^{+}{P}_{l}^{-}]| },$$
(7)

where \({P}_{\mu }^{\pm }=|{u}_{\pm ,\mu }^{\mathrm{R}}\rangle \langle {u}_{\pm ,\mu }^{\mathrm{L}}|\) is the biorthogonal projector onto the μth eigenstate of H± = HO ± HC. (We have \({\Gamma }_{n,l}{G}_{l,n}=\langle {u}_{+,n}^{\mathrm{L}}| {u}_{-,l}^{\mathrm{R}}\rangle \langle {u}_{-,l}^{\mathrm{L}}| {u}_{+,n}^{\mathrm{R}}\rangle ={\rm{Tr}}[{P}_{n}^{+}{P}_{l}^{-}{P}_{n}^{+}]={\rm{Tr}}[{P}_{n}^{+}{P}_{l}^{-}]\) since \({P}_{n}^{+}\) is idempotent.) Note that ROs can occur even in the absence of a real spectrum, as long as E+,n − E−,l is real. Under PBCs, this reality condition simplifies to the requirement that HC has a real spectrum. Equation (7) shows that the Rabi frequency ΩR is proportional to the effective dipole moment \({\mu }_{n,l}^{(\mathrm {eff})}\equiv {d}_{y}\sqrt{{\rm{Tr}}[{P}_{n}^{+}{P}_{l}^{-}]}\), which can be enhanced in a non-Hermitian system. In the Hermitian case, this is not possible as \({\rm{Tr}}[{P}_{n}^{+}{P}_{l}^{-}]=| \langle {u}_{+,n}| {u}_{-,l}\rangle {| }^{2}\le 1\), with overlap integrals bounded above by unity. But in non-Hermitian cases, eigenstates are biorthogonally normalized, and there are two scenarios where \({\rm{Tr}}[{P}_{n}^{+}{P}_{l}^{-}]\) can be very large: (i) near an exceptional point and (ii) in the presence of boundary eigenmode accumulation, also known as the non-Hermitian skin effect. For (i), exceptional points have been known to harbor pronounced sensitivity due to the defective nature of their eigenspaces, and the dipole moment amplification is expected17. But more interesting is (ii), where the effective dipole moment and hence Rabi frequency can be controlled just by changing boundary conditions. Moreover, when considering ROs in the (k, E) plane, in the non-Hermitian case boundary conditions drastically challenge the common wisdom that transitions have to be vertical and harmonic, as discussed below.

Fig. 1: Driven two-band lattices.
figure 1

a Schematic of an ac-driven non-Hermitian lattice formed by two chains with sites displayed by blue and red circles. The dashed thin black bonds denote a non-Hermitian hopping, with amplitudes  ±κ1 depending on the direction (left/right) of the hopping, while solid black bonds describe Hermitian hopping. b Equivalent lattice after the basis transformation. The system is basically equivalent to two Hatano–Nelson chains, shifted in energy by  ±Δ and side-coupled by the oscillating bond  −dyF(t). The skin modes under open boundary conditions are squeezed toward opposite edges in the two chains with the same skin length. c Energy bands under periodic boundary conditions in the undriven case (F = 0). Solid blue and dashed red lines refer to the real and imaginary parts of the energy spectrum, respectively.

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Boundary-driven ultrafast and non-vertical ROs

The eigenvectors \(\left|{u}_{\pm ,n}^{{\mathrm{{R,L}}}}\right\rangle\), and thus the scalar products Γn,l, Glml defining the interband transitions, depend on the boundary conditions, and differ for PBC and OBC. Under PBC, \(\langle x| {u}_{\pm ,n}^{{\mathrm{{R,L}}}}\rangle =\exp (iknx)\) are plane waves with quantized quasi-momentum kn = 2nπ/N, and thus Γn,l, Gn,l vanish for n ≠  l, i.e., the ac field can induce only vertical transitions in (k, E) plane, both for Hermitian and non-Hermitian systems (Fig. 2a). Correspondingly, ROs are always harmonic and there is not any enhancement of the dipole moment (\({\mu }_{n,n}^{(\mathrm {eff})}={d}_{y}\)). Under OBC, the quasi-momentum k is not anymore a good quantum number58,59,60,61; however, one can still diagonalize H± in real space and the bulk energy spectrum (with the exception of isolated states) can be obtained from the GBZ25,28,34, i.e., from the analytic continuation of the Bloch energy bands E±(k) where k becomes complex and spans, for each band, a path in complex plane, as detailed in the “Methods” (see also ref. 44). Here we focus our attention to the most interesting case where H±(k), under PBC, can be derived from the same momentum-space Hermitian Hamiltonian HH(k) via uniform complex momentum deformations k → k − ih±, i.e., H±(k) = Q±HH(k − ih±) with Q± and h± real parameters. This case applies to the typical scenario of ROs in lattices with nearest-neighbor (NN) hopping4, in which the non-Hermitian deformations h± are introduced by synthetic imaginary gauge fields28,62,63,64. As explained in the “Methods”, under OBC H± show an entirely real energy spectrum and share the same eigenmodes with quantized wave number \({k}_{+,n}={k}_{-,n}\equiv {k}_{n}=\frac{n\pi }{N+1}\), n = 1, 2, …, N. In the Hermitian limit h1 = h2 = 0, one has Γnl = Gn,l  =  0 for l ≠ n and ROs occur under resonance driving between eigenmodes in the two bands with the same quasi-momentum kn = kl (Fig. 2a). In other words, ROs in the Hermitian limit are the same for PBC and OBC, and transitions remain vertical in the (k, E) plane. This is not the case for non-Hermitian lattices, where boundary conditions affect the dynamics tremendously as non-vertical transitions are allowed (Fig. 2b).

Fig. 2: Interband transitions and Rabi oscillations.
figure 2

a Field-induced transitions between Bloch bands in a system with periodic boundary conditions (PBC). The allowed transitions (vertical dashed lines) are vertical in the momentum–energy (k, E) plane. The resulting Rabi oscillations are always harmonic. b Under open boundary conditions (OBC), transitions between non-Bloch bands can be oblique in the (k, E) plane. Mixed transitions lead to anharmonic Rabi oscilattions. The inset in b shows the two generalized Brillouin zones \({\beta }_{+}=\exp (-h+ik)\) and \({\beta }_{-}=\exp (h+ik)\) of upper and lower bands, corresponding to two circles (in blue color) of different radius \(\exp (\mp h)\). Oblique transitions that do not conserve k correspond to non-radial jumps between the two generalized Brillouin zones, as shown by the dashed red arrow in the inset connecting the inner and outer circles.

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To illustrate this point, let us consider the NN Hamiltonian \({H}_{H}(k)=2\sqrt{{\kappa }_{2}^{2}-{\kappa }_{1}^{2}}\cos k+\Delta\), with Δ and κ2 > κ1 real and positive parameters, and let us assume Q± = ±1 and h± = ∓ h with \(h=(1/2)\mathrm{log}\,[({\kappa }_{2}+{\kappa }_{1})/({\kappa }_{2}-{\kappa }_{1})]\). The resulting Hamiltonians H± read

$${H}_{\pm }(k)=2i{\kappa }_{1}\sin k\pm (\Delta +2{\kappa }_{2}\cos k)$$
(8)

corresponding to the physical single chain Hamiltonian \({H}_{O}=2i{\kappa }_{1}\sin k\) and inter-chain coupling \({H}_{C}=\Delta +2{\kappa }_{2}\cos k\); see Fig. 1a. After the basis transformation, the two decoupled lattices H± are two Hatano–Nelson chains62,63,64 with asymmetric left/right hopping and shifted in energy by  ±Δ; see Fig. 1b. The skin effect squeezes the bulk modes for the two bands towards opposite boundaries. To have well-spaced bands for ROs, we assume that the two bands are separated by a wide gap, i.e., we assume that Δ ≫ κ2. Under PBCs, ROs occur only for vertical transitions kn = kl [Fig. 2a], with corresponding eigenmodes displaying the same lifetime [Fig. 1c]. But under OBC the spectra of the two non-Bloch bands are entirely real and non-vertical transitions are allowed. The OBC energy spectra are readily obtained from the one of HH and read

$${E}_{\pm\!, n}=\pm \left[2\sqrt{{\kappa }_{2}^{2}-{\kappa }_{1}^{2}}\ \cos {k}_{n}+\Delta \right]$$
(9)

with kn = nπ/(N + 1) and n = 1, …, N. To highlight the appearance of non-vertical transitions, let us compute the biorthogonal eigenbasis of H± under OBC, which are identical upon left/right interchange. We have

$$\langle x| {u}_{\pm ,n}^{\mathrm{R}}\rangle ={\mathrm{e}}^{\pm h(N+1)/2}\sqrt{\frac{2}{N+1}}\sin \left({k}_{n}x\right){\mathrm{e}}^{\mp hx}.$$
(10)

Physically, the non-vanishing value of h indicates that the modes in the two non-Bloch bands are squeezed toward the two opposite ends of the lattice (skin effect). The effective dipole moment between OBC eigenstates \(|{u}_{+,n}^{\mathrm{R}}\rangle\) and \(|{u}_{-,l}^{\mathrm{R}}\rangle\) reads

$${\mu }_{n,l}^{(\mathrm {eff})}={d}_{y}\sinh (2h)\frac{({\mathrm{e}}^{h(1+N)}-{\left(-{\mathrm{e}}^{-h}\right)}^{(1+N)})(\cos {\theta }_{-}-\cos {\theta }_{+})}{2(1+N)(\cosh 2h-\cos {\theta }_{-})(\cosh 2h-\cos {\theta }_{+})}$$
(11)

with θ± = kn ± kl. Note that \({\mu }_{n,l}^{(\mathrm {eff})}\) is non-vanishing even for non-vertical transitions n ≠ l. This means that, under appropriate resonance forcing, ROs can be induced between non-vertical modes, as shown in Fig. 3. Note that, in the large N limit, ΩR ~ ehN/∣n − l∣2, which scales exponentially with hN and to the inverse square of ∣n − l∣. The exponential scaling with system size, arising from the skin effect, provides the enhancement of the Rabi frequency, while the inverse power-law dependence on ∣n − l∣ indicates that non-vertical transitions, with decreasing strengths as ∣n − l∣ increases, are allowed when h ≠ 0. Examples of enhanced ROs in a lattice comprising N = 31 sites for increasing values of the non-Hermitian parameter κ1/κ2 are shown in Fig. 4.

Fig. 3: Rabi oscillations involving non-vertical transitions.
figure 3

The figure shows Rabi oscillations in a non-Hermitian lattice comprising N = 31 unit cells with open boundary conditions for parameter values Δ/κ2 = 3, F0/κ2 = 0.005 and κ1/κ2 = 0.1. The initial state is the bulk eigenstate \(|{{\bf{u}}}_{+,n}^{\mathrm{R}}\rangle\) with quasi-momentum kn = nπ/(N + 1) and n = (N + 1)/2 = 16. The modulation frequency ω is tuned to put in resonance the modes kn and kl (with l = n + 4 = 20) in the upper and lower bands. a Evolution of the real-space occupation probabilities ∣An∣2 and ∣Bn∣2 (upper and lower panels). Different k modes are visible from different fringes in upper and lower panels. b Evolution of the occupation probabilities ∣α16∣2 and ∣β20∣2. Also the probability ∣β16∣2 is shown, which remains almost zero because of chosen ac modulation frequency.

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Fig. 4: Ultrafast and anharmonic Rabi oscillations.
figure 4

The figure illustrates enhanced Rabi oscillation frequency induced by skin effect and anharmonic Rabi oscillations for increasing non-Hermitian parameter, measured by the ratio κ1/κ2 of amplitude probabilities. Temporal evolution of the real-space occupation probabilities ∣An∣2 and ∣Bn∣2 on a pseudocolor map for a κ1/κ2 = 0 (Hermitian limit), b κ1/κ2 = 0.05, c κ1/κ2 = 0.1, and d κ1/κ2 = 0.16. Other parameter values are N = 31 lattice unit cell, Δ/κ2 = 3, ω = 2Δ, and F0/κ2 = 0.005. The initial state is the bulk eigenstate \(|{{\bf{u}}}_{n,+}^{\mathrm{R}}\rangle\) with quasi-momentum kn = nπ/(N + 1) and n = (N + 1)/2 = 16. Temporal evolution of the occupation probabilities ∣α16∣2 (blue curves) and ∣β16∣2 (green curves) for e κ1/κ2 = 0 (Hermitian limit), f κ1/κ2 = 0.05, g κ1/κ2 = 0.1, and h κ1/κ2 = 0.16. In h the evolution of ∣α17∣2 is also depicted (dashed red curve), indicating the emergence of mixed and anharmonic Rabi oscillations.

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More generally, we expect to observe ROs as long as the imaginary parts of the eigenenergies of H± = HO ± HC are identical, even if they are not related by imaginary flux deformations Q±HH(k − ih±) as assumed above. In particular, the GBZs of H± are not fundamentally constrained to be circular. For instance, consider a class of models where \({H}_{C}(k)={h}_{C}\ {\mathbb{I}}\), hC(k) a real constant, and HO possessing a generic GBZ such that HO(k + iρ(k)) gives its OBC spectrum. In this case, the spectra of H± are related by real energy offsets  ±hC under both PBCs and OBCs, and thus support ROs. Indeed, ROs allows access into the dynamical properties of such Hamiltonians with non-circular GBZs, which more often than not possess spectra with "arms” extending into the complex energy plane44.

Stability of quasi-energies and anharmonic ROs

Non-vertical transitions and boundary-induced dipole enhancement can invalidate the crude RWA, with the appearance of a complex quasi-energy spectrum (Fig. 5), indicating that the system enters into an unstable regime and ROs are not anymore observed, as shown in the “Methods”. For the periodically driven lattice, one should calculate the complex quasi-energies using Floquet theory and associated two-dimensional lattices in space-frequency plane, as illustrated in Fig. 6a–c. The onset of complex quasi-energies can be regarded as a kind of parametric instability of the ac-driven system65,66, as indicated by the appearance of unstable resonance tongues in the frequency–amplitude plane, (ω, F0), emanating from some of the transition frequencies ωnl (see Figs. 7 and 8). The instability can be physically explained from the field-induced coupling between the two chains H+ and H− that makes it possible a secular amplification of excitation along closed loops (Fig. 6c). We remark that breakdown of the RWA and the onset of unstable dynamics is a genuine boundary-driven non-Hermitian effect arising from non-vertical transitions, i.e., it is not related to counter-rotating terms like in the ultrastrong coupling regime of light–matter interaction. An example of the stability domain in the (F0, h) plane, for fixed value of lattice site N and modulation frequency ω far from any resonance tongue, is shown in Fig. 5a. To observe oscillatory (Rabi-like) dynamics, parameter should be chosen in the stable domain. The largest enhancement factor of effective dipole moment in RO is thus ultimately limited by the onset of the instability, as shown in Fig. 5b. Interestingly, as the system parameters are varied inside the stability domain to approach the stability boundary, ROs become highly anharmonic (see Fig. 4), a phenomenon which is clearly impossible to be observed in any Hermitian lattice.

Fig. 5: Stability domain of Rabi flopping.
figure 5

a Stability domain of quasi-energies in the plane of normalized forcing amplitude F0/κ2 and non-Hermitian parameter κ1/κ2. Parameter values are Δ/κ2 = 3, ω = 2Δ, and N = 31. In the unstable domain quasi-energies are complex, corresponding to exponentially diverging amplitudes. The circles in the stable region correspond to a normalized forcing amplitude F0/κ2 = 0.005. b Behavior of the effective dipole \({\mu }_{n,n}^{(\mathrm {eff})}/{d}_{y}\) versus the non-Hermitian parameter κ1/κ2 for the same parameter values as in a and for the bulk mode corresponding to kn = π/2, i.e., n = (N + 1)/2 = 16. The solid curve shows the theoretical prediction, while the circles are obtained from the full numerical simulations.

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Fig. 6: Floquet lattices.
figure 6

a Equivalent decoupled two-dimensional Floquet lattices of the ac-driven system. Different shading denote alternating Floquet chains \({H}_{+}^{(l)}\) and \({H}_{-}^{(l)}\) in the lattice coupled by the forcing amplitude F0/2, where l = 0, ±1, ±2, …  is the Floquet (synthetic dimension) order. b Reduced model under wide gap and near-resonant forcing. c Simple physical explanation of the onset of instability. In the absence of forcing, excitations are squeezed toward the two opposite edges of the upper and lower chain, and amplification is prevented by the open boundary conditions. As the ac field is switched on, chain coupling realizes closed loops (schematically depicted by arrows), along which excitation can be secularly amplified, resulting in a complex quasi-energy spectrum.

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Fig. 7: Stability domain and resonance tongues.
figure 7

Resonance tongues (shaded areas), showing the onset of complex quasi-energies, in the forcing–frequency plane (F0, ω) for number N of lattice sites a N = 10, and b N = 11. Other parameter values are κ1/κ2 = 0.1 and Δ/κ2 = 3. The forcing F0 and frequency ω are normalized to the hopping amplitude κ2. The dashed vertical lines in the plots show the position of the various resonance transition frequencies ωn,l ≡ (E+,n −  E−,l) between the open boundary condition energy levels in upper and lower bands. Note that the unstable resonance tongues emerge from some (but not all) the resonance transitions.

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Fig. 8: Stability domain and resonance tongues.
figure 8

Resonance tongues (shaded areas) showing the onset of complex quasi-energies in the forcing–frequency plane (F0, ω) for number N of lattice sites a N = 30 and b N = 31. Other parameter values are κ1/κ2 = 0.1 and Δ/κ2 = 3. The forcing F0 and frequency ω are normalized to the hopping amplitude κ2. The dashed vertical lines in the plots show the position of the various resonance transitions at frequencies ωn,l ≡ (E+,n −  E−,l) between the open boundary condition energy levels in upper and lower bands.

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Discussion

The fundamental processes of absorption and emission of quanta in crystals, as well as coherent Rabi flopping, are deeply modified by edge effects when considering non-Hermitian lattices displaying the skin effect. In particular, boundary-driven non-vertical transitions and anharmonic coherent Rabi flopping can be observed. Such results challenge the common wisdom that bulk coherent processes in crystals are largely independent of boundaries, indicating that skin effects not only question the bulk-boundary correspondence but also coherent bulk phenomena. Owing to the recent experimental progresses in the realization of synthetic non-Hermitian lattices displaying the non-Hermitian skin effect in photonic67, mechanical68, and electrical circuit42,69 platforms, as well as pertinent theoretical advances in cold atom engineering70, we expect that the disclosed distinctive physics of absorption and emission of energy in non-Hermitian crystals could be experimentally accessible in the near future.

Methods

GBZ for systems with open boundary conditions

In the absence of the non-Hermitian skin effect, the OBC and PBC eigenstates can be mapped one another in a simple way, and it is physically expected that PBC transitions remain mostly unchanged upon the introduction of open boundaries in a large system. Indeed, for Hermitian lattices with NN hopping and vanishing asymmetry (h = 0), the OBC eigenstates [Eq. (10)] are simply odd superpositions of PBC eigenstates with Bloch profiles \({e}^{\pm i{k}_{n}x}\), where \({k}_{n}=\frac{n\pi }{N+1}\), n = 1, …, N to give rise to N unique OBC states (defined slightly differently from the PBC kn). The introduction of the non-Hermitian skin effect (h ≠ 0) amounts just to an exponential factor that can be obtained by deforming the quasi-momentum via kn → kn ± ih. In this simple example, it is clear that n is still a well-defined label of the eigenstates, even though they are no longer in Bloch form.

More generally, there exists an almost 1-to-1 correspondence between the OBC and PBC eigenstates through the introduction of a GBZ. Intuitively, most of the eigenvalues of the PBC spectrum will adiabatically “flow” en masse towards the OBC eigenvalues when the system is continuously interpolated between PBCs and OBCs, i.e., by slowly switching off the end-to-end couplings, as detailed in ref. 28. In other words, most of the PBC eigenstates, which will have evolved into OBC bulk eigenstates in the absence of the skin effect, will evolve together as boundary-localized skin eigenstates under the skin effect. The exceptions, if any, are topological eigenstates that evolve separately from the rest28,31.

Recently, an “unraveling” picture for the GBZ was developed44 to explain how the most general set of skin eigenstates \({u}_{n}^{{\rm{{OBC}}}}\) for the system with hopping asymmetry h can be understood in terms of non-analytic complex momentum deformations of PBC eigenstates. This complex momentum lives in the so-called GBZ. First, we introduce the concept of the surrogate Hamiltonian \(\bar{H}\) which does not experience the skin effect, defined via a complex momentum deformation of the physical Hamiltonian H in momentum space:

$${\bar{H}}^{{\rm{PBC}}}(k)={H}^{{\rm{PBC}}}(k-i\rho (k)).$$
(12)

Here ρ(k) is the complex momentum deformation required such that there exist a double degeneracy in the decay lengths of the eigenstates, i.e., ρ(k) is determined by the condition that for each k, there exists another \(k^{\prime}\) such that the energy dispersion obeys \(E(k+i\rho (k))=E(k^{\prime} +i\rho (k^{\prime} ))\) with \(\rho (k)=\rho (k^{\prime} )\)28,44. In Hermitian cases, we always have ρ(k) = 0 because as k varies over a period, the energy dispersion E(k) must always retrace itself before going back to its original value after a period in k.

To intuitively understand why the \(\rho (k)=\rho (k^{\prime} )\) gives rise to a surrogate Hamiltonian that does not experience the skin effect, consider explicitly constructing wave functions that satisfy OBC from superpositions of the PBC momentum eigenstates. If the superposition coefficients were to converge in the thermodynamic limit, we need the presence of bulk-boundary correspondence, i.e., the absence of the skin effect. At a particular energy E, OBC wave functions must satisfy two boundary conditions, namely that they vanish at both ends, i.e., at x = 0 and at x = N + 1. A superposition of at least two nonzero eigenstates is required for the wavefunction to vanish at x = 0. For it to also vanish at x = N + 1 in the thermodynamic limit of arbitrarily large N, another prerequisite is that both eigenstates must decay at the same rate, for otherwise one of them will be infinitesimally small compared to the other, and cannot possible cancel it off at x = N + 1. This requirement that both eigenstates decay at the same rate is just that their imaginary momentum components are equal, i.e., ρ(kn) = ρ(km). This is thus also the condition for \(\bar{H}\) to experience no skin effect.

Under OBC where momentum ceases to be a good quantum number, the complex momentum deformation k → k − iρ(k) can be expressed as a similarity transform S with a complex gauge field, i.e.,

$${\bar{H}}^{{\rm{OBC}}}={S}^{-1}{H}^{{\rm{OBC}}}S.$$
(13)

An important corollary of this is that PBC Hamiltonians related by Eq. (12) possess identical OBC spectra. Hence, to understand the OBC spectrum of a generic non-Hermitian Hamiltonian H, it suffices to determine that of its surrogate Hamiltonian \(\bar{H}\), which is also almost equal to the PBC spectrum of \(\bar{H}\) since the latter obeys the bulk-boundary correspondence. In the thermodynamic limit where almost all states (except for isolated edge states) are skin states, the OBC spectrum EOBC is thus almost exactly indexed by EPBC(kn − iρ(kn)).

From Eq. (13), \({u}^{{\rm{OBC}}}=S{\bar{u}}^{{\rm{OBC}}}\), both with the same eigenenergy. Since \({\bar{u}}^{{\rm{OBC}}}\) is not a skin state, it can be expanded in terms of the PBC eigenstates, i.e., \({\bar{u}}_{n}^{{\rm{OBC}}}={\sum }_{n^{\prime} }{c}_{nn^{\prime} }{\bar{u}}^{{\rm{PBC}}}({k}_{n})\) where \(n,n^{\prime}\) label the eigenstates (the label is given through kn in the PBC case) and \({c}_{nn^{\prime} }\) are coefficients that converge in the thermodynamic limit. Hence

$${u}_{n}^{{\mathrm{{OBC}}}}= \;S\ {\bar{u}}_{n}^{{\rm{OBC}}}\\ = \; S\sum_{n^{\prime} }{c}_{nn^{\prime} }{\bar{u}}^{{\rm{PBC}}}({k}_{{n}^{\prime}})\\ = \; \sum_{n^{\prime} }{c}_{nn^{\prime} }{\bar{u}}^{{\rm{PBC}}}({k}_{{n}^{\prime}}+i\rho ({k}_{{n}^{\prime}})).$$
(14)

For our case of one-band Hamiltonians H±, \({k}_{n}=\frac{n\pi }{N+1}\), \({\bar{u}}^{{\rm{PBC}}}({k}_{{n}^{\prime}})\propto {\mathrm{e}}^{i{k}_{{n}^{\prime}}x}\) and \(\rho ({k}_{{n}^{\prime}})=h\ {\rm{sgn}}(\sin {k}_{n^{\prime} })\). Together, these yield Eq. (10), with biorthogonal normalization imposed.

Vertical transitions in systems with OBCs

In the main text, we discussed vertical and non-vertical transitions between the effective chains governed by Hamiltonians H±. Under PBCs, their eigenstates u±,n are labeled by the integer n through the quantized lattice quasi-momentum kn = 2πn/N, and it is clear that vertical transitions refer to those connecting u+,n and u−,n of the same n and thus quasi-momentum. Under OBCs, however, no well-defined lattice momentum exists, and one should discuss how the notions of vertical/non-vertical transitions can be extended more generally.

The Ansatz H±(k) = Q±HH(k − ih±) used in the main text produces H± with real and identical OBC spectra. Since the ih± complex deformation can be implemented as a similarity transform \(S={\rm{diag}}(1,{e}^{{h}_{\pm }},{e}^{2{h}_{\pm }},...,{e}^{(N-1){h}_{\pm }})\) in real space under OBCs, it is evident that H±(k) must possess identical OBC spectra up to a sign Q± [see also Eq. (13)].

More generally, constant complex momentum deformations k → k + ik0 (k0 a constant) generate an equivalence class of models with identical OBC spectra. In particular, since HH(k +  i0) is Hermitian, the OBC spectrum must be real. This construction holds for multiband models as well, where we can further generalize the definition of H± to H±(k) = Q±U−1HH(k − ih±)U where U is a unitary transformation.

The eigenstates of \({H}_{\pm }^{{\rm{OBC}}}\) can be inferred from Eq. (14) with ρ(k) = −h±, where \(\bar{H}\) is just HH. Importantly, it is clear how the nth eigenstates of H± correspond with each other. Again, specializing to our one-band model, we shall recover the expression of Eq. (10).

Having established the correspondence between the spectra and eigenstates of H±, the notion of vertical (n = m) versus non-vertical (n ≠ m) transitions is well-defined.

Quasi-energy spectrum and stability of ROs

The quasi-energy spectrum of the ac-driven lattice under OBC can be determined by a standard Floquet analysis of Eq. (2), or likewise of Eqs. (5) and (6). Let as assume a sinusoidal driving field \(F(t)={F}_{0}\cos (\omega t)\), and let us search for a solution to Eq. (2) in the form of a Floquet eigenstate, i.e.

$${\bf{A}}(t)=\sum_{l}{{\bf{A}}}^{(l)}\exp (-i\mu t+i\omega lt)\ ,\ \ {\bf{B}}(t)=\sum_{l}{{\bf{B}}}^{(l)}\exp (-i\mu t+i\omega lt),$$
(15)

where μ is the quasi-energy, which is uniquely defined in the range (−ω/2, ω/2). Substitution of the Ansatz (15) into Eq. (2) given in the main text yields the Floquet chains

$$\mu {{\bf{A}}}^{(l)}={H}_{+}^{(l)}{{\bf{A}}}^{(l)}+\frac{{F}_{0}}{2}\left({{\bf{B}}}^{(l+1)}+{{\bf{B}}}^{(l-1)}\right)\ ,\,\,\mu {{\bf{B}}}^{(l)}={H}_{-}^{(l)}{{\bf{B}}}^{(l)}+\frac{{F}_{0}}{2}\left({{\bf{A}}}^{(l+1)}+{{\bf{A}}}^{(l-1)}\right),$$
(16)

where we have set \({H}_{\pm }^{(l)}\equiv {H}_{\pm }+l\omega\). The quasi-energies μ can be thus viewed as the eigenenergies of two decoupled static 2D lattices (stripes), in which one dimension (labeled by the index l, varying from  −∞ to ∞) is a synthetic (frequency) dimension while the other dimension (labeled by the index n, varying from n = 1 to n = N) is the physical space, as schematically shown in Fig. 6a. A special case is the one in which the two non-Bloch bands, defined by the Hamiltonians H+ and H− under OBC, are entirely real and spaced by a wide gap 2Δ, like in the model of Fig. 1. Under near-resonant driving ω ≃ 2Δ and assuming weak forcing, application of standard multiple-scale asymptotic analysis (see, e.g., ref. 71) indicates that the full Floquet chains of Fig. 6a basically decouple into equivalent quasi 1D static lattices, as shown in Fig. 6b. Thus, at leading order the 2N quasi-energies μ of the ac-driven system are obtained as the eigenvalues of the linear system

$$\mu {{\bf{A}}}^{(0)}={H}_{+}{{\bf{A}}}^{(0)}+\frac{{F}_{0}}{2}{{\bf{B}}}^{(1)}\ ,\ \ \ \mu {{\bf{B}}}^{(1)}=({H}_{-}+\omega ){{\bf{B}}}^{(1)}+\frac{{F}_{0}}{2}{{\bf{A}}}^{(0)}.$$
(17)

Let us specialize the general theory to the model discussed in the main text [Fig. 1 and Eq. (8)]. For given values of the band spacing 2Δ and non-Hermitian deformation parameter h, the quasi-energy spectrum μ turns out to be strongly sensitive to the modulation frequency ω, force strength F0, and site number N (in particular N odd or even). Extended numerical simulations, based either on diagonalization of the full Floquet chains (Fig. 6a) or the reduced quasi 1D lattice (Fig. 6b), indicate the existence of multi-branch instability domains in (ω, F0) space made of wide and narrow resonance tongues, where the quasi-energy spectrum becomes complex. In such regions the amplitudes exponentially grow in time and the observation of oscillatory (Rabi-like) dynamics is thus prevented. Figures 7 and 8 show typical behaviors of resonance tongues for N odd and N even. Interestingly, the unstable resonance tongues emerge from some (but not all) of the transition frequencies ωn,l ≡ (E+,n − E−,l) between OBC energy levels in upper and lower bands. Physically, the onset of the instability can be readily explained from the diagram of Fig. 6c. Forward (backward) propagating excitations in the upper chain are attenuated (amplified) by the imaginary gauge field h. The opposite holds for the lower chain. In the undriven case (F = 0), the two chains are decoupled. Owing to the OBC, under steady state the excitations in each chain are squeezed at opposite edges and cannot grow/decay anymore: this means that the energy spectrum remains entirely real despite the non-Hermitian term κ1 in the Hamiltonian. When the ac force is switched on, closed loops between upper and lower chains, with an overall net amplification of excitation per round trip, are allowed by the vertical solid bonds in Fig. 6c, resulting in complex quasi-energies. In other words, the ac force effectively changes boundary conditions allowing periodic circulation of excitations between the two chains.