DC Current Generation and Power Feature in Strongly Driven Floquet-Bloch Systems

Introduction.— Periodically driven systems or Floquet systems have long been an intensively studied area in physics, since they provide controllable platforms for realizing many exotic physical phenomena such as the Floquet topological insulator [1, 2] and the (space-)time crystal [3, 4, 5, 6]. The external driving field has a wide range of possibilities: mechanical [7, 8], optical [9, 10], or even acoustic [6] drives, among which the optically driven systems have drawn the most attention due to their applications in industry like solar cell [11]. However, unlike the weakly driven system that can be dealt with using the perturbation theory [10], for periodically driven systems with strong external field strength, Floquet analysis is necessary [12, 13]. Recently, Floquet systems connected to heat sources characterizing more realistic open environments have also attracted many attentions [14, 15, 16, 17, 18, 19]. Being essentially energy non-conserved, Floquet systems constantly involve energy conversions between different forms [6, 19], whose power features are extremely important for energy conversion applications [20].

In this work, we consider a system with periodicity in both space and time, known as the Floquet-Bloch system, connected to a heat bath, and investigate the DC current generations and their power features. We first characterize the heat bath through a relaxation time approximation [12, 21], and obtain the density matrix for a non-interacting Floquet-Bloch system with damping, which is related to two equilibria: an instantaneous Bloch thermal equilibrium and a Floquet quasi-equilibrium whose density matrix is diagonal under the Floquet-Bloch eigenbasis. In the weakly damping but strongly driven regime, we find that the current responses consists of an intrinsic part and an extrinsic correction due to the damping, which are nonzero for systems with broken time-reversal and inversion symmetries. The power features are the following: the intrinsic part is dissipationless and related to the Floquet quasi-equilibrium, while the extrinsic part needs a sustained energy input and originates from a shift between two equilibrium ensembles. Thus the generation of the intrinsic current enhances the overall power efficiency. We also investigate the role of the driving field strength in the DC current generations revealing that the currents become larger under a relatively stronger field but drops dramatically at a certain point where a Floquet gap transition happens.

Floquet Analysis.— We start from a non-driven Bloch system characterized by: $\hat{H}_{B}(\boldsymbol{r})|\varphi^{\alpha}_{\boldsymbol{k}}(\boldsymbol{r})\rangle=E_{\alpha}(\boldsymbol{k})|\varphi^{\alpha}_{\boldsymbol{k}}(\boldsymbol{r})\rangle$, where $\hat{H}_{B}(\boldsymbol{r})=\hat{H}_{B}(\boldsymbol{r}+\boldsymbol{R})$ is the static Hamiltonian with $N_{B}$ multiple bands, $\alpha\in\{1,2,\cdots,N_{B}\}$ stands for the Bloch band index, and $\boldsymbol{k}$ is the lattice momentum. According to Bloch’s Theorem, the wavefunction can be expressed as a plane wave multiplied by a spatially periodic function: $|\varphi^{\alpha}_{\boldsymbol{k}}(\boldsymbol{r})\rangle=e^{i\boldsymbol{k}\cdot\boldsymbol{r}}|u^{\alpha}_{\boldsymbol{k}}(\boldsymbol{r})\rangle$, where $|u^{\alpha}_{\boldsymbol{k}}(\boldsymbol{r}+\boldsymbol{R})\rangle=|u^{\alpha}_{\boldsymbol{k}}(\boldsymbol{r})\rangle$. Then, we apply the time-periodic driving field $\boldsymbol{A}(t)$ to the Bloch system which makes $\hat{H}_{B}\to\hat{H}(\boldsymbol{r},t)$ with periodicity in time $\hat{H}(\boldsymbol{r},t)=\hat{H}(\boldsymbol{r},t+T)$, and the Schrödinger equation reads

To solve Eq.(1), we introduce an extended Floquet-Bloch basis $\{|\phi^{\alpha}_{n,\boldsymbol{k}}(\boldsymbol{r},t)\rangle\equiv e^{-in\Omega t}|\varphi^{\alpha}_{\boldsymbol{k}}(\boldsymbol{r})\rangle\}$ which is the original Bloch wavefunction shifted in energy by integer multiples of $\Omega\equiv 2\pi/T$ [6, 22]. Here $n$ is the Floquet multiplicity. Under the extended Floquet-Bloch basis, we can expand the wavefunction as

where $\omega_{\mu}$ is the quasi-energy with band index $\mu$. We then denote $\sum_{n,\beta}f^{\mu,\beta}_{n,\boldsymbol{k}}|\phi^{\beta}_{n,\boldsymbol{k}}(\boldsymbol{r},t)\rangle\equiv e^{i\boldsymbol{k}\cdot\boldsymbol{r}}|\tilde{u}_{\boldsymbol{k}}^{\mu}(\boldsymbol{r},t)\rangle$ as the corresponding Floquet-Bloch eigenstate with $|\tilde{u}_{\boldsymbol{k}}^{\mu}(\boldsymbol{r},t)\rangle$ referring to the Floquet-Bloch periodic function that has the same periodicity as the Hamiltonian [23]. The expansion coefficient $f^{\mu,\beta}_{n,\boldsymbol{k}}$ satisfies the following eigen-equation: $\sum_{n,\alpha}\mathcal{H}_{m,n;\beta,\alpha}f^{\mu,\alpha}_{n,\boldsymbol{k}}=\hbar\omega_{\mu}f^{\mu,\beta}_{m,\boldsymbol{k}}$ with the kernel matrix

where $\langle\langle\cdot\rangle\rangle\equiv 1/T\int_{0}^{T}\langle\cdot\rangle$ is the spacetime inner product. The quasi-energy $\omega_{\mu}$ inherits the degrees of freedom of the Bloch system, so in principle, we will have $N_{B}$ different quasi-energy bands $\{\omega_{1},\omega_{2},\cdots,\omega_{N_{B}}\}$ within a Floquet Brillouin zone. More details about the matrix $\mathcal{H}_{m,n;\alpha,\beta}$ are given in the Supplementary Material [24].

Approach to Floquet Quasi-Equilibrium.— We now consider a Floquet-Bloch system connected to a heat bath which introduces damping. Such damping system is governed by the following quantum Liouville equation ($\hbar=1$) [12, 21]:

where $\hat{\rho}(\boldsymbol{k},t)$ is the density operator of the system at fixed $\boldsymbol{k}$ and $\hat{H}(\boldsymbol{k},t)\equiv e^{-i\boldsymbol{k}\cdot\boldsymbol{r}}\hat{H}(\boldsymbol{r},t)e^{i\boldsymbol{k}\cdot\boldsymbol{r}}$ is the Hamiltonian of the corresponding non-damping system. In this work, we will restrict our discussion in $\boldsymbol{k}$-conserved systems. The density matrix operator can be expanded using either the static Bloch eigenbasis $\{|u^{\alpha}_{\boldsymbol{k}}\rangle\}$ or the Floquet-Bloch eigenbasis $\{|\tilde{u}^{\mu}_{\boldsymbol{k}}(t)\rangle\}$ (index $\boldsymbol{r}$ is omitted):

where $\rho^{B}_{\alpha,\beta}(\boldsymbol{k},t)$ and $\rho^{F}_{\mu,\nu}(\boldsymbol{k},t)$ are the density matrix elements under two eigenbases, respectively. The operator $\hat{D}$ characterizes the damping due to the heat transfer and the electron scattering. The role of damping is to relax the system to a thermal equilibrium with the heat bath, while the Floquet drive is to strike the system out of equilibrium. Under the relaxation time approximation [21, 25], we can write

where the first line characterizes the thermal contact with damping rate $\Gamma$, and the second line is the pure dephasing caused by the electron scattering [12] with dephasing rate $\Gamma^{\prime}_{\alpha,\beta}$. The $\hat{\rho}^{B,eq}(\boldsymbol{k},t)$ is an instantaneous Bloch equilibrium, which is not diagonal under the static Bloch eigenbasis: $\hat{\rho}^{B,eq}(\boldsymbol{k},t)=\sum_{\alpha,\beta}\rho^{B,eq}_{\alpha,\beta}(\boldsymbol{k},t)|u^{\alpha}_{\boldsymbol{k}}\rangle\langle u^{\beta}_{\boldsymbol{k}}|$ [24]. This equilibrium indicates a field dragging effect (see Eq. (16) for a special case).

Under the Floquet-Bloch eigenbasis, we can rewrite the quantum Liouville equation as [24]:

where $\bar{\rho}^{eq}_{\mu,\nu}(\boldsymbol{k},t)\equiv\sum_{\alpha,\beta}\rho^{B,eq}_{\alpha,\beta}(\boldsymbol{k},t)\langle\tilde{u}^{\mu}_{\boldsymbol{k}}(t)|u^{\alpha}_{\boldsymbol{k}}\rangle\langle u^{\beta}_{\boldsymbol{k}}|\tilde{u}^{\nu}_{\boldsymbol{k}}(t)\rangle$ is still the instantaneous Bloch equilibrium but expanded in the Floquet eigenbasis, and $\mathcal{S}_{\mu,\nu}(\boldsymbol{k},t)\equiv\sum_{\alpha\neq\beta}\Gamma^{\prime}_{\alpha,\beta}(\boldsymbol{k})\langle\tilde{u}^{\mu}_{\boldsymbol{k}}(t)|\rho^{B}_{\alpha,\beta}(\boldsymbol{k})|u^{\alpha}_{\boldsymbol{k}}\rangle\langle u^{\beta}_{\boldsymbol{k}}|\tilde{u}^{\nu}_{\boldsymbol{k}}(t)\rangle$ is the scattering contribution. It is hard to solve the differential equation above, especially when the scattering matrix $\mathcal{S}_{\mu,\nu}$ is unknown. However, the system will evolve into a steady state similar to a forced damping oscillator whose changing frequency synchronizes with the external driving frequency. Therefore, we can do the Fourier transformations: $\rho^{F}_{\mu,\nu}(\boldsymbol{k},t)=\sum_{l}\rho^{F,l}_{\mu,\nu}(\boldsymbol{k})e^{-il\Omega t}$, $\bar{\rho}^{eq}_{\mu,\nu}(\boldsymbol{k},t)=\sum_{l}\bar{\rho}^{eq,l}_{\mu,\nu}(\boldsymbol{k})e^{-il\Omega t}$, and $\mathcal{S}_{\mu,\nu}(\boldsymbol{k},t)=\sum_{l}\mathcal{S}^{l}_{\mu,\nu}(\boldsymbol{k})e^{-il\Omega t}$. By inserting those Fourier expansions into Eq. (27), we obtain a solution for the density matrix:

To further simplify our considerations, we introduce two extremes: one is the pure thermal contact, $\Delta_{\boldsymbol{A}},\Gamma\gg\Gamma^{\prime}_{\alpha,\beta}$, where the electron-electron interaction is surpassed, for example, by large dielectric constant of the material; the other is the isolated system, $\Delta_{\boldsymbol{A}},\Gamma^{\prime}_{\alpha,\beta}\gg\Gamma$, where the system is isolated from any heat sources. The $\Delta_{\boldsymbol{A}}=\min_{\mu\neq\nu,l}\{|\omega_{\mu}(\boldsymbol{k})-\omega_{\nu}(\boldsymbol{k})-l\Omega|\}$ is the minimum direct gap of the Floquet-Bloch system. Our main interest in this work is the first extreme where we can approximately let $\Gamma^{\prime}_{\alpha,\beta}=0$, corresponding to a non-interacting Floquet-Bloch system. Thus, the density matrix becomes:

which is in general non-diagonal and time-dependent. Here $\rho^{B,eq,m}_{\alpha,\beta}(\boldsymbol{k})$ is the Fourier component of $\rho^{B,eq}_{\alpha,\beta}(\boldsymbol{k},t)=\sum_{m}\rho^{B,eq,m}_{\alpha,\beta}(\boldsymbol{k})e^{-im\Omega t}$. One can check that this density matrix is Hermitian and more importantly gauge invariant under the gauge transformation $|u^{\alpha}_{\boldsymbol{k}}\rangle\to e^{-i\theta^{\alpha}_{\boldsymbol{k}}}|u^{\alpha}_{\boldsymbol{k}}\rangle$.

The density matrix in Eq. (9) indicates two special $\Gamma$-independent ensembles. Either, through a very large damping ($\Gamma\to\infty$), the system is quickly relaxed to an instantaneous Bloch equilibrium with the heat bath, in which the density matrix is reduced to $\hat{\rho}^{B,eq}(\boldsymbol{k},t)$. Or, by turning off the thermal contact ($\Gamma\rightarrow 0$), we obtain a diagonal density matrix:

where we have used the fact that $\lim_{\Gamma\rightarrow 0}\frac{-i\Gamma}{\omega_{\mu}-\omega_{\nu}-l\Omega-i\Gamma}=\delta_{\mu,\nu}\delta_{l,0}$. One can also show that this is the instantaneous Bloch equilibrium projected onto the Floquet-Bloch eigenbasis with off-diagonal terms erased by damping: $\rho^{F,eq}_{\mu,\mu}=\langle\langle\tilde{u}^{\mu}_{\boldsymbol{k}}(t)|\hat{\rho}^{B,eq}(\boldsymbol{k},t)|\tilde{u}^{\mu}_{\boldsymbol{k}}(t)\rangle\rangle$, which indicates that with no heat contact (i.e. isolated system), it is a so-called diagonal ensemble [26, 27]:

In the remainder, we refer to this diagonal ensemble as the Floquet quasi-equilibrium since its density matrix element is diagonal and time-independent under the Floquet-Bloch eigenbasis.

Floquet-Liouville Equation: DC Current Response and Power Input.— The essence of a Floquet system is that it evolves periodically in time, which superficially renders the fact that for an operator $\hat{\mathcal{O}}$ also periodic in time, we have $\int_{0}^{T}\frac{d}{dt}\text{tr}(\hat{\rho}\hat{\mathcal{O}})dt=\text{tr}(\hat{\rho}\hat{\mathcal{O}})|_{t=T}-\text{tr}(\hat{\rho}\hat{\mathcal{O}})|_{t=0}=0$. One can combine such fact with the quantum Liouville equation to obtain the following Floquet-Liouville equation under the Floquet-Bloch eigenbasis [24]:

where $\text{Tr}^{F}[\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{0.5}{$\SavedStyle\bullet$}}}}}]\equiv\sum_{\mu}\langle\langle\tilde{u}^{\mu}_{\boldsymbol{k}}(t)|\mathbin{\ThisStyle{\vbox{\hbox{\scalebox{0.5}{$\SavedStyle\bullet$}}}}}|\tilde{u}^{\mu}_{\boldsymbol{k}}(t)\rangle\rangle$ is the Floquet spacetime trace and $\omega$ is the quasi-energy of corresponding Floquet eigenstate: $\omega|\tilde{u}^{\mu}_{\boldsymbol{k}}(t)\rangle\rightarrow\omega_{\mu}(\boldsymbol{k})|\tilde{u}^{\mu}_{\boldsymbol{k}}(t)\rangle$. This Floquet-Liouville equation allows us to study many DC responses in periodically driven systems in a unified manner. In this section, we focus on the DC current response and the driven power. The discussion will be based on the main result in Eq. (9) and the assumptions for obtaining it, where we ignored the electron-electron interactions. In such setting, the damping term in Eq. (22) can be reduced to $[\hat{D},\hat{\rho}]=-\Gamma(\hat{\rho}-\hat{\rho}^{B,eq})$.

The DC charge current can be calculated by (ignoring $-e$ from now on): $\boldsymbol{j}_{DC}=\int\text{d}\boldsymbol{k}\text{Tr}^{F}[\hat{\rho}\hat{\boldsymbol{v}}]$ with $\hat{\boldsymbol{v}}\equiv\partial_{\boldsymbol{k}}\hat{H}(\boldsymbol{k},t)$ the velocity operator. In favor of Eq. (38), we choose $\hat{\mathcal{O}}=i\partial_{\boldsymbol{k}}$ and end up with the following result [24]:

where $\Lambda=O(\frac{\Gamma^{2}}{\Delta_{\boldsymbol{A}}-i\Gamma})$ is a correction that contains higher orders of $\Gamma$ and is negligible when $\Gamma\ll\Delta_{\boldsymbol{A}}$. One can treat $\text{Tr}^{F}[\hat{\rho}^{F(B),eq}i\partial_{\boldsymbol{k}}]$, the ensemble average of the operator $i\partial_{\boldsymbol{k}}$, as the polarization projected onto the Floquet (Bloch) eigenspace. In this work, we focus on the weakly damping but strongly driven regime where $\Lambda\to 0$.

We want to make the following remarks for the result in the Eq. (41): (i) it is gauge invariant [24], allowing us to choose any gauge for convenience; (ii) the linear order correction in $\Gamma$ is geometrical and related to a shift in polarization between two (quasi-)equilibrium ensembles; (iii) via a gauge transformation: $|\tilde{u}^{\mu}_{\boldsymbol{k}}(t)\rangle\to e^{-i\theta^{\mu}_{\boldsymbol{k}}}|\tilde{u}^{\mu}_{\boldsymbol{k}}(t)\rangle$ with $\theta^{\mu}_{\boldsymbol{k}}=\int_{\boldsymbol{k}_{0}}^{\boldsymbol{k}}d\boldsymbol{k}^{\prime}\langle\langle\tilde{u}^{\mu}_{\boldsymbol{k}^{\prime}}(t)|\hat{\rho}^{B,eq}i\partial_{\boldsymbol{k}^{\prime}}|\tilde{u}^{\mu}_{\boldsymbol{k}^{\prime}}(t)\rangle\rangle/\rho^{F,eq}_{\mu,\mu}(\boldsymbol{k}^{\prime})$, we can rewrite Eq. (41) as $\text{Tr}^{F}[\hat{\rho}\hat{\boldsymbol{v}}]=\text{Tr}^{F}[\hat{\rho}^{F,eq}\tilde{\hat{\boldsymbol{v}}}]$ with $\tilde{\hat{\boldsymbol{v}}}\equiv\hat{\boldsymbol{v}}+\Gamma i\partial_{\boldsymbol{k}}$ defined as the shifted velocity operator. Then the current reads:

where $\boldsymbol{\mathcal{A}}^{F}_{\mu}\equiv\langle\langle\tilde{u}^{\mu}_{\boldsymbol{k}}(t)|i\partial_{\boldsymbol{k}}|\tilde{u}^{\mu}_{\boldsymbol{k}}(t)\rangle\rangle$ is the Floquet Berry connection after choosing the above gauge [28]. The current is now fully diagonal in the Floquet band index $\mu$ and contains: $\boldsymbol{j}_{DC}^{\text{In}}$, $\Gamma$-independent and thus an intrinsic current; $\boldsymbol{j}_{DC}^{\text{Ex}}$, linear in $\Gamma$ and an extrinsic contribution that reflects a shift in polarization between two ensembles. It is worth noting that the current response written in Eq. (14) manifests its possible topological natures as the Berry connection comes into play [29].

There are requirements for having nonzero current responses. The first requirement is about symmetry. The extrinsic current $\boldsymbol{j}_{DC}^{\text{Ex}}$ is related to a shift in the polarization of electrons, which is nonzero if the inversion symmetry (parity) is broken. However, for the intrinsic current $\boldsymbol{j}_{DC}^{\text{In}}$, we also need to break the time-reversal symmetry (TRS), since $\partial_{\boldsymbol{k}}\omega_{\mu}$ is odd under TRS. Another requirement is special for $\boldsymbol{j}_{DC}^{\text{In}}$ that the quasi-equilibrium distribution $\rho^{F,eq}_{\mu,\mu}$ should not be a function of quasi-energy $\omega_{\mu}$, otherwise, the intrinsic current will always be zero due to the periodicity of $\omega_{\mu}$ in the Brillouin zone.

To generate the currents, we need to know how much energy should the Floquet drive provide to sustain the DC currents, i.e. the power features. We now specify the Floquet drive as coherent light where the Hamiltonian takes the form $\hat{H}(\boldsymbol{k},t)=\hat{H}_{B}(\boldsymbol{k}+\boldsymbol{A}(t))$ with $\boldsymbol{A}(t)$ the vector potential. Then, the net power provided by the light over one Floquet period is $\bar{P}_{\text{net}}=\int d\boldsymbol{k}\bar{P}(\boldsymbol{k})=\int d\boldsymbol{k}\text{Tr}^{F}[\boldsymbol{E}(t)\cdot\hat{\rho}\hat{\boldsymbol{v}}]$ with $\boldsymbol{E}(t)=-\partial_{t}\boldsymbol{A}(t)$ the AC electric field. By choosing $\hat{\mathcal{O}}=\hat{H}(\boldsymbol{k},t)$ in Eq. (38), we get the input power at specific $\boldsymbol{k}$ [24]:

indicating that the intrinsic current is dissipationless, while the extrinsic current needs a sustained power input. Such result is understandable since for intrinsic current, we ignored the interactions among electrons giving no internal damping channels; but for extrinsic current, damping is involved and the power is related to a energy shift between two ensembles.

It’s important to see the power feature in energy non-conserved systems since it tells us the direction of energy flow which in our case is from Floquet drive to the heat bath. More remarkable is that intrinsic current is dissipationless which can be favored in energy conversion applications. Combined with the symmetry argument, breaking TRS would help to obtain such desirable current. As for the extrinsic current, it becomes dominant in TRS preserved systems and more importantly its power efficiency defined as $\boldsymbol{j}_{DC}^{\text{Ex}}/\bar{P}_{\text{net}}$ reflects some band properties [24]. The field dependence of the DC currents and power feature is also important and will be discussed in the next section.

Case Study: A Two-band System Connected to A Heat Bath.— Consider a two-band model in which we explicitly break the TRS and parity to see the DC current generations and power features under strong driving field. The setup is a two-band Bloch system $H_{B}(\boldsymbol{k})$ with $E_{1}(\boldsymbol{k})$, $E_{2}(\boldsymbol{k})$ and $|u^{1}_{\boldsymbol{k}}\rangle$, $|u^{2}_{\boldsymbol{k}}\rangle$ as its two gaped energy bands and corresponding eigenstates. Then a coherent linearly polarized light, $\boldsymbol{A}(t)=\boldsymbol{A}e^{i\Omega t}+\boldsymbol{A}^{*}e^{-i\Omega t}$, is applied to the Bloch system as a Floquet drive. The Hamiltonian then becomes $H(\boldsymbol{k},t)=H_{B}(\boldsymbol{k}+\boldsymbol{A}(t))$. The system is then connected to a heat bath at temperature $\mathcal{T}$, which, at thermal equilibrium, obeys the Fermi-Dirac distribution: $\rho^{B,eq,s}_{\alpha,\alpha}(\boldsymbol{k})=\frac{1}{e^{[E_{\alpha}(\boldsymbol{k})-\mu_{c}]/k_{B}\mathcal{T}}+1}$, with $\mu_{c}$ the chemical potential sitting in the gap. The instantaneous Bloch thermal equilibrium can be approximated as [24]

where $\boldsymbol{\mathcal{A}}^{B}_{\alpha,\beta}\equiv\langle u^{\alpha}_{\boldsymbol{k}}|i\partial_{\boldsymbol{k}}|u^{\beta}_{\boldsymbol{k}}\rangle$ is the Bloch berry connection, and the second term on the right hand side represents a field dragging effect. The two Floquet bands denoted as $\omega_{\pm}(\boldsymbol{k})$ can be found by diagonalizing the corresponding kernel matrix in Eq. (18). One last thing is that the damping rate $\Gamma$ can in general depend on field and momentum, which makes the current and the total power hard to be evaluated. Here for simplicity, we assume a constant $\Gamma$.

In Fig. 1 and Fig. 2, we show some numerical results. The Bloch model is a 1D diatomic chain with Hamiltonian [30]

and other set-ups are the same as mentioned before, except for restricting in 1D. First, we plot in Fig. 1(a) the static Bloch thermal equilibrium distribution and in Fig. 1(b) the corresponding Floquet quasi-equilibrium distribution. Note that in the Bloch thermal equilibrium, the electron distribution is a function of the energy, while in the Floquet quasi-equilibrium, the distribution is not a function of the quasi-energy, as one can see from Fig. 1(b) that $k$ points with same quasi-energies have different distributions. The reason is that the Floquet quasi-equilibrium is indeed a non-equilibrium state.

Notice that the 1D model with the parameters we choose largely breaks the TRS and parity, which induces nonzero current responses. In Fig. 2(a,b), we plot the intrinsic and extrinsic currents as functions of the field strength $|\boldsymbol{A}|$ for two different driven frequencies. Meanwhile, we also explicitly illustrate the influence of the dragging effect which basically gives numerical corrections. An important observation is that at relatively small $|\boldsymbol{A}|$ (but still $\Delta_{\boldsymbol{A}}\gg\Gamma$), we have approximately $\boldsymbol{j}_{DC}^{\text{In}}\propto|\boldsymbol{A}|$ and $\boldsymbol{j}_{DC}^{\text{Ex}}\propto\Gamma|\boldsymbol{A}|$, latter of which is consistent with the prediction made in Ref. [29]. It is interesting to see that the net power input depicted in Fig. 2(c) shows a strong linear dependence in field strength in a larger field regime. Such linearity is persistent even after a large correction due to the dragging effect. However, when $|\boldsymbol{A}|$ is getting larger, the complexity appears that the currents and power input start to decrease significantly at some turning points. This feature then tells us that there exists an optimized field strength for current generation, namely the DC currents should be generated with strong but not too strong field. Here we give a phenomenological explanation that the decreases in those quantities are consequences of large band shifting under strong driving. To confirm this, we plot the Floquet direct band gaps in Fig. 2(d). When $|\boldsymbol{A}|$ is fairly small, the gap opens at the resonant points where $E_{1}+\Omega-E_{2}=0$ (see Fig. 1(b)), and grows linearly in $|\boldsymbol{A}|$. However, as $|\boldsymbol{A}|$ gets larger, we have to consider the band shifting, namely the self-energy correction to the electron, denoted as $\delta E_{1,2}(k)$. The gap moves to points where $E_{1}+\delta E_{1}+\Omega-E_{2}-\delta E_{2}=0$ and is compressed by the band shifting. Thus we see that the gaps are shrinking at larger $|\boldsymbol{A}|$. As the field gets even larger such that the resonant points disappear, i.e. $|E_{1}+\delta E_{1}-E_{2}-\delta E_{2}|<\Omega$ for all $k$, the gap becomes trivial, which is simply a separation between two shifted Bloch bands. We observe and mark the turning points in Fig. 2(d) which indicates that the gap transfers from a resonant split to a trivial separation. This then explains the decreases in the extrinsic current and the power feature, confirmed by the correspondence of the turning points marked in Fig. 2(b-d). We notice that the intrinsic current decreases a little earlier than the extrinsic, indicating extra possible effects such as more balanced population under stronger field. We emphasize that this strong field effect works in effectively two-band systems and may get complicated in multiband systems.

Discussions and Remarks.— In this work, we discussed DC currents and their power features in a non-interacting Floquet-Bloch system with weak damping and strong drive. The results show the behaviors of the Bloch system under intensive drive. We want to briefly discuss what may happen when considering more aspects.

First is about the weak field limit where the field strength is small enough and $\Gamma\gg\Delta_{\boldsymbol{A}}$. Starting from the density matrix in Eq. (9), we can reproduce the well-known non-linear shift current as previously obtained using perturbation theory or Floquet Green’s function method [24, 29, 31]. Another important aspect is the electron-electron interactions that has been ignored in this work. We want to note that, technically, the interaction normally leads to heating of the system, which tends to erase notable features like the current responses [32, 33, 34]. Lastly, the oversimplification of the constant damping rate $\Gamma$ may fail at large field strength. Therefore, the theory needs to further take into account possible field or even momentum dependence of the damping, for example the treatments in Refs. [14, 15]. However, we believe that in some highly controllable platforms, like the cold atom system, the parameters can be tuned to suit the approximations we used in this work. We also claim that despite various damping mechanisms, the formula in Eq. (38) is general and the intrinsic current response is present under broken TRS, which is important for efficiently generating DC currents.

Q. G. acknowledges financial support of the Provost’s Graduate Excellence Fellowship from The University of Texas at Austin.