# Quantized Acoustoelectric Floquet Effect in Quantum Nanowires

Christopher Yang  Department of Physics, IQIM, California Institute of
Technology, Pasadena, CA 91125, USA  Will Hunt  Department of Physics,
University of Cambridge, Cambridge, United Kingdom  Department of Physics,
IQIM, California Institute of Technology, Pasadena, CA 91125, USA  Gil Refael
Department of Physics, IQIM, California Institute of Technology, Pasadena, CA
91125, USA  Iliya Esin  Department of Physics, IQIM, California Institute of
Technology, Pasadena, CA 91125, USA

###### Abstract

External coherent fields can drive quantum materials into non-equilibrium
states, revealing exotic properties that are unattainable under equilibrium
conditions—an approach known as “Floquet engineering.” While optical lasers
have commonly been used as the driving fields, recent advancements have
introduced nontraditional sources, such as coherent phonon drives. Building on
this progress, we demonstrate that driving a metallic quantum nanowire with a
coherent wave of terahertz phonons can induce an electronic steady state
characterized by a persistent quantized current along the wire. The
quantization of the current is achieved due to the coupling of electrons to
the nanowire’s vibrational modes, providing the low-temperature heat bath and
energy relaxation mechanisms. Our findings underscore the potential of using
non-optical drives, such as coherent phonon sources, to induce non-equilibrium
phenomena in materials. Furthermore, our approach suggests a new method for
the high-precision detection of coherent phonon oscillations via transport
measurements.

Introduction—Recent advances in the creation of phonon sources have enabled
on-demand access to coherent phonon beams across a broad spectrum of
frequencies Ng  _et al._ (2022); Wallentowitz  _et al._ (1996); Vahala  _et
al._ (2009); Ip  _et al._ (2018); Pettit  _et al._ (2019); Grudinin  _et al._
(2010); Jing  _et al._ (2014); Guo  _et al._ (2023). These phonon excitations
in solids Först  _et al._ (2011); Li  _et al._ (2012); Balandin and Nika
(2012); Maldovan (2013); Balandin (2020); Subedi  _et al._ (2014); Mankowsky
_et al._ (2016); Juraschek  _et al._ (2017) can induce new optical properties
Trovatello  _et al._ (2020); Khalsa  _et al._ (2021), strong correlation
physics Mankowsky  _et al._ (2014); Knap  _et al._ (2016); Babadi  _et al._
(2017); Kennes  _et al._ (2017); Mitrano  _et al._ (2016); Li  _et al._
(2019), tunable magnetic properties Mankowsky  _et al._ (2017); Nova  _et al._
(2019); Ng  _et al._ (2022); Balandin (2020), and acousto-electric effects
Parmenter (1953); Weinreich and White (1957); Kalameitsev  _et al._ (2019);
Reulet  _et al._ (2000); Mahfouzi and Kioussis (2022); Kawada  _et al._
(2021). The unique characteristics of coherent phonons—specifically their
finite momentum, low energy, and coupling to the electrons—pave the way for
Floquet engineering of non-equilibrium spatial-temporal electronic phenomena
Peng (2022); Gao and Niu (2022); Hübener  _et al._ (2018); Chaudhary  _et al._
(2020). Notably, terahertz (THz) frequency phonons Esin  _et al._ (2023);
Beardsley  _et al._ (2010); Cartella  _et al._ (2017); Barajas-Aguilar  _et
al._ (2023) can potentially serve as tools for the dynamic manipulation of
materials with narrow bandwidths, such as moiré systems Katz  _et al._ (2020);
Yang  _et al._ (2023); Esin  _et al._ (2023). Recent proposals have
demonstrated that coherent phonon beams can be used for Floquet engineering of
nontrivial band topology in trivial materials Chaudhary  _et al._ (2020); Peng
(2022).

We demonstrate that a continuous propagating wave of coherent THz phonons can
drive a quantum wire into a non-equilibrium Floquet steady state, resulting in
a persistent charge current [see Fig. 1(a)]. Furthermore, under optimal
conditions of doping and driving strength, this current can achieve a
quantized value, $J=e\omega/\pi$, where $\omega$ represents the phonon driving
angular frequency and $e$ is the electron charge [see Fig. 1(b)]. In the
adiabatic limit, $\omega\to 0$, the quantization of current aligns with the
principles of a topological pump Nakajima  _et al._ (2015a); Hotzen Grinberg
_et al._ (2020); Nakajima  _et al._ (2015b); Bustos-Marún  _et al._ (2013);
Citro and Aidelsburger (2023). Remarkably, we find that quantized transport
can be maintained over a wide range of THz frequencies, even beyond the
adiabatic limit. This suggests a robust mechanism underpinning the
quantization, stabilized by the non-equilibrium electronic steady state, which
is set by the coupling to a bath of low-temperature thermal phonons and
electron-electron interactions Bruch  _et al._ (2018); Esin  _et al._ (2024);
Nakagawa  _et al._ (2018); Walter  _et al._ (2023); Privitera  _et al._
(2018).

Figure 1: (a) Schematic experimental setup. A THz-frequency coherent phonon
wave (with atomic displacements sketched by red arrows) of momentum $q$ and
angular frequency $\omega$ induces a quantized current $J=e\omega/\pi$ in a
CNT. A screening material of dielectric constant $\epsilon$ surrounds the CNT,
and anti-reflective material minimize reflection of the phonon wave. (b)
Current $J$ vs. Floquet gap $\Delta/\hbar\omega$ for different $\epsilon$ (see
inset). Vertical line: gap below which the drive is non-adiabatic
($\Delta<\hbar\omega$). Inset: $\Delta^{*}$ vs. $\epsilon$, where $\Delta^{*}$
is the minimal Floquet gap at which $J=0.96e\omega/\pi$. (c) Band structure of
a $(10,0)$ armchair CNT. Inset: Fermi energy lies near the band bottom of the
lowest positive-energy band, and the electronic density $n_{e}$ is chosen to
be commensurate to $q$. (d) Quasienergy spectrum of the driven system. Blue
shading on the $\alpha=0$ band: optimal filling of the bands resulting in
quantized current. Scattering transitions (black arrows) generated by
incoherent phonons relax electrons into the $\alpha=0$ band.

A device for robust generation of quantized current on demand through coherent
phonon illumination has numerous potential applications in metrology Niu
(1990); Pekola  _et al._ (2013); Kaneko  _et al._ (2016); Stein  _et al._
(2015); Zhao  _et al._ (2017); Giblin  _et al._ (2019), electronics, and
quantum computing Das  _et al._ (2006); You  _et al._ (2022); Bäuerle  _et
al._ (2018); Fève  _et al._ (2007). Furthermore, this device can be used as a
tool for the characterization and detection of coherent phonons through
transport measurement. Traditionally, the detection of coherent phonons has
relied on optical methods such as reflectivity measurements Lanzillotti-Kimura
_et al._ (2011); Eisfeld and Renk (1979); Giorgianni  _et al._ (2022); Yoon
_et al._ (2023a, b). Leveraging current quantization as a probe of the phonon
field offers a more direct and sensitive method of coherent phonon detection.

In this work, we focus on a concrete experimental setup of carbon nanotube
(CNT) Sanders  _et al._ (2009); Fathi (2011); Kane and Mele (1997); Saito  _et
al._ (1992); Mintmire  _et al._ (1992); Hamada  _et al._ (1992) coupled to a
continuous source of coherent phonons. The phonon waves propagate from the
right to the left end of the CNT, where they are absorbed by the absorbing
material, see Fig. 1(a). Anti-reflective material with appropriately chosen
thickness on both ends of the CNT [see blue slabs in Fig. 1(a)] suppress
reflected phonon waves that destructively interfere with the coherent phonon
mode. The electronic steady state in the CNT is formed from the balance
between the interaction with the coherent phonon wave, as well as interactions
with incoherent phonons of the CNT and other electrons. In our model, we
consider a detailed microscopic description of the phononic spectrum of the
CNT that serves as the low-temperature heat bath for the phonons and the
electron-electron interactions.

Phonon-driven carbon nanotube.—To analyze the steady state properties of this
setup, we use an effective model of the CNT. We define
$\hat{\boldsymbol{\psi}}_{k,s}^{\dagger}\equiv\begin{pmatrix}\hat{\psi}_{{k},A,s}^{\dagger}&\hat{\psi}_{{k},B,s}^{\dagger}\end{pmatrix}$
as the creation operator of a Bloch state on the CNT, where
$\hat{\psi}_{{k},j,s}^{\dagger}$ creates an electron of crystal momentum ${k}$
on sublattice $j=A,B$, and ${k}$ is the Bloch momentum in the direction
$\boldsymbol{\hat{k}_{\parallel}}$ along the tube axis. The index
$s=0,\ldots,N-1$ enumerates the electronic momentum $s\boldsymbol{k}_{\perp}$
along the direction $\hat{\boldsymbol{k}}_{\perp}$ around the circumference of
the tube. Here, $\boldsymbol{k}_{\perp}=(2\pi/P)\hat{\boldsymbol{k}}_{\perp}$,
$P$ is the perimeter of the CNT, $N$ is the number of graphene unit cells
contained in a length $|\boldsymbol{T}|$ along the tube axis, and the CNT is
periodic along the tube axis by translations of vector $\boldsymbol{T}$. The
electronic momentum $k\in[-\pi/|\boldsymbol{T}|,\pi/|\boldsymbol{T}|]$ along
$\hat{\boldsymbol{k}}_{\parallel}$ is approximated as continuous for a long
tube. Corresponding eigenenergies of the electronic states are given by
$E_{s\nu}(k)$, where $\nu=+,-$ denote the conduction and valence bands,
respectively, see Fig. 1(c).

We focus on a semiconducting CNT whose Fermi surface in equilibrium lies near
the bottom of the lowest conduction band [see inset of Fig. 1(c)]. The lowest
conduction and highest valence bands are described by the effective
Hamiltonian $\hat{H}_{\text{e}}=\int{dk}/{2\pi}\
\hat{\boldsymbol{\psi}}_{{k}}^{\dagger}H_{\text{e}}(k)\hat{\boldsymbol{\psi}}_{{k}}$,
where $H_{\text{e}}(k)=\hbar v_{F}(\delta
k\hat{\boldsymbol{k}}_{\perp}+k\hat{\boldsymbol{k}}_{\parallel})\cdot\boldsymbol{\sigma}$,
$\delta k$ is a constant dependent on the chirality indices (see the
Supplemental Materials Sup  for details), $v_{F}$ is the Fermi velocity of
graphene, $\boldsymbol{\sigma}=(\sigma_{x},\sigma_{y})$ is a vector of Pauli
matrices acting in the graphene sublattice basis, and the $x$ axis is aligned
along a bond between carbon atoms. In what follows, we omit the index $s$,
assuming only the two bands described by $\nu=+,-$ with eigenergies
$E_{\nu}(k)$ and eigenstates $|\nu k\rangle$.

To describe electron-phonon interactions in the CNT, we consider the
Hamiltonian
$\hat{H}_{\text{ep}}({t})=\sum_{\lambda}\hat{H}_{\text{ep}}^{(\lambda)}({t})$,
where $\hat{H}_{\text{ep}}^{(\lambda)}({t})=\int d^{2}\boldsymbol{r}\
\hat{\boldsymbol{\psi}}_{\boldsymbol{r}}^{\dagger}\hat{V}^{(\lambda)}(\boldsymbol{r},t)\hat{\boldsymbol{\psi}}_{\boldsymbol{r}}$
and $\hat{V}^{(\lambda)}(\boldsymbol{r},t)=\hbar
v_{F}\hat{\boldsymbol{\mathcal{A}}}_{\text{ph}}^{(\lambda)}(\boldsymbol{r},t)\cdot\boldsymbol{\sigma}+\hat{\phi}_{\text{ph}}^{(\lambda)}(\boldsymbol{r},t)$
describe electronic coupling to a phonon mode indicated by $\lambda$,
$\boldsymbol{r}=(x,y)$ is the spatial coordinate along the tube, and
$\hat{\boldsymbol{{\psi}}}_{\boldsymbol{r}}^{\dagger}=P^{-1}\int
d{k}/(2\pi)e^{-i(\delta
k\hat{\boldsymbol{k}}_{\perp}+k\hat{\boldsymbol{k}}_{\parallel})\cdot\boldsymbol{r}}\hat{{\boldsymbol{\psi}}}^{\dagger}_{k}$.
The electrons interact with the phonons through the effective vector potential
$\hat{\boldsymbol{\mathcal{A}}}_{\text{ph}}^{(\lambda)}(\boldsymbol{r},t)=\sqrt{3}\beta/(2a)(\hat{u}^{(\lambda)}_{xx}(\boldsymbol{r},t)-\hat{u}^{(\lambda)}_{yy}(\boldsymbol{r},t),2\hat{u}^{(\lambda)}_{xy}(\boldsymbol{r},t))$
and through the local scalar potential
$\hat{\phi}_{\text{ph}}^{(\lambda)}(\boldsymbol{r},t)=D[\hat{u}^{(\lambda)}_{xx}(\boldsymbol{r},t)+\hat{u}^{(\lambda)}_{yy}(\boldsymbol{r},t)]I$,
where $a=0.246\ \mathrm{nm}$, $\beta\approx 3.14$, and deformation potential
$D=15\ \mathrm{eV}$ Esin  _et al._ (2023); von Oppen  _et al._ (2009). Here,
$\hat{\boldsymbol{u}}^{(\lambda)}(\boldsymbol{r},t)$ is the displacement
operator of the phonon mode, $\hat{u}^{(\lambda)}_{b}(\boldsymbol{r},t)$ is
its $b$-th component, and
$\hat{u}^{(\lambda)}_{bc}(\boldsymbol{r},t)=[\partial_{b}\hat{u}^{(\lambda)}_{c}(\boldsymbol{r},t)+\partial_{c}\hat{u}^{(\lambda)}_{b}(\boldsymbol{r},t)]/2$.

We assume that a source of phonons [see Fig. 1(a)] generates a coherent sound
mode with index $\lambda=\lambda_{0}$ that propagates through the CNT, while
the rest of the phonons are in low-temperature thermal equilibrium. The phonon
mode $\lambda_{0}$ has momentum
$\boldsymbol{q}=q\hat{\boldsymbol{k}}_{\parallel}$, angular frequency
$\omega$, and finite displacement expectation value
$\langle\hat{\boldsymbol{u}}^{(\lambda_{0})}(\boldsymbol{r},t)\rangle=u_{0}\cos(\boldsymbol{q}\cdot\boldsymbol{r}-\omega
t)\hat{\boldsymbol{k}}_{\parallel}$, where $u_{0}$ is the displacement
amplitude Lon .

The electronic dynamics can be divided to coherent components described by the
time- and spatially-periodic Hamiltonian
$\hat{H}_{0}(t)=\hat{H}_{\text{e}}+\hat{H}_{\text{ep}}^{(\lambda_{0})}(t)$ and
incoherent components due to coupling to thermal phonon modes,
$\hat{H}_{\text{b}}(t)=\sum_{\lambda\neq\lambda_{0}}\hat{H}_{\text{ep}}^{(\lambda)}(t)$
Com . The single-particle Hamiltonian $\hat{H}_{0}(t)$ can be diagonalized by
the Floquet-Bloch states,
$|\psi_{k\alpha}(\boldsymbol{r},t)\rangle=e^{-i(\boldsymbol{k}\cdot\boldsymbol{r}+\varepsilon_{k\alpha}t/\hbar)}\sum_{n\in\mathbb{Z}}e^{-in(\boldsymbol{q}\cdot\boldsymbol{r}-\omega
t)}|\phi^{(n)}_{k\alpha}\rangle$ Rudner and Lindner (2020); Peng (2022); Gao
and Niu (2022); Hübener  _et al._ (2018). Here, $\varepsilon_{k\alpha}$ is the
quasienergy satisfying

$(\varepsilon_{k\alpha}+m\hbar\omega)|\phi^{(m)}_{k\alpha}\rangle={H}_{\text{e}}({{k}+m{q}})+\sum_{m^{\prime}\neq
0}V_{m-m^{\prime}}|\phi^{(m^{\prime})}_{k\alpha}\rangle$ (1)

where $\alpha$ enumerates the Floquet bands and $V_{n}$ are the Fourier
harmonics of $\langle\hat{V}^{(\lambda_{0})}(\boldsymbol{r},t)\rangle$, i.e.,
$\langle\hat{V}^{(\lambda_{0})}(\boldsymbol{r},t)\rangle=\sum_{n\neq
0}e^{-in(\boldsymbol{q}\cdot\boldsymbol{r}-\omega t)}V_{n}$. The quasienergy
spectrum $\varepsilon_{k\alpha}$ arises from replicas of the original energy
bands $E_{\nu}(k)$ shifted in energy and momentum by $m\hbar\omega$ and $mq$,
respectively, where $m\in\mathbb{Z}$ [see light grey, dashed curves in Fig.
1(d)]. Rabi-like gaps of size $\Delta\approx
qu_{0}[D+\hat{k}_{\parallel}^{y}\hbar v_{F}\sqrt{3}\beta/2a]$ open at momenta
$k=k^{*}+mq$ where the Floquet replicas cross, resulting in the quasienergy
spectrum sketched with solid curves in Fig. 1(d) (see the Supplemental
Materials Sup  for details). Remarkably, the quasienergy satisfies the
periodicity condition
$\varepsilon_{k\alpha}=\varepsilon_{k+q,\alpha}-\hbar\omega$, which is the
basis for the quantized current presented in this work. Specifically, the
current is given by Esin  _et al._ (2018)

$J=\frac{2e}{\hbar}\sum_{\alpha}\int_{0}^{q}\frac{dk}{2\pi}\
\frac{d\varepsilon_{k\alpha}}{dk}F_{k\alpha}.$ (2)

Here,
$F_{k\alpha}(t)=\langle\hat{f}^{\dagger}_{k\alpha}(t)\hat{f}_{k\alpha}(t)\rangle$
is the occupation of the Floquet state
$|\psi_{k\alpha}(\boldsymbol{r},t)\rangle$ created by operator
$\hat{f}_{k\alpha}^{\dagger}(t)$ Spi . When only the $\alpha=0$ band is fully-
occupied, $J=e\omega/\pi$, resulting in quantized current.

Floquet-Boltzmann Equation.—To model the dynamics under electron-phonon
interactions, we consider the microscopic Hamiltonian $\hat{H}_{\text{b}}(t)$
for electronic coupling to incoherent bath phonons. The incoherent low-energy
longitudinal acoustic phonons of speed $c_{\text{ph}}$, momentum $p$, and
energy $\hbar\omega_{l}(p)=\hbar
c_{\text{ph}}|s\boldsymbol{k}_{\perp}+p\boldsymbol{\hat{k}}_{\parallel}|$
dominate the electron-phonon scattering near the Fermi surface of the CNT.
Additionally, we consider electron-electron coupling:

$\begin{split}\hat{H}_{\text{ee}}=&\int\frac{dk_{1}dk_{2}dp}{(2\pi)^{3}}\mathcal{V}_{{k}_{1},{k}_{2}}(p)\hat{c}_{{k}_{1}+{p},+}^{\dagger}\hat{c}_{{k}_{2}-{p},+}^{\dagger}\hat{c}_{{k}_{2},+}\hat{c}_{{k}_{1},+},\end{split}$
(3)

where
$\mathcal{V}_{{k}_{1},{k}_{2}}(p)=U(p)\mathcal{W}_{{k}_{1},{p}}\mathcal{W}_{{k}_{2},-{p}}/(2\epsilon)$,
$\mathcal{W}_{k,p}\equiv\langle+,k+p|+,k\rangle$ is the form-factor, $U(p)$ is
the Coulomb potential, $\epsilon$ is the dielectric constant of the
surrounding screening medium, and $\hat{c}_{k\nu}$ creates an electron in the
eigenstate $|\nu k\rangle$ of $\hat{H}_{\text{e}}$. We consider scattering
within the $\nu=+$ band only, because the Fermi level [see inset of Fig. 1(c)]
is energetically well-separated from other bands, restricting scattering to
small momentum transfers $p\lesssim q$ near the Fermi surface. For small $p$,
we approximate $U(p)\approx 1\ \mathrm{eV}$, consistent with estimates in
Refs. Zhao and Mazumdar (2004); Malic  _et al._ (2008).

We assume that the CNT incoherent phonons are coupled to an external heat bath
and remain in thermal equilibrium at temperature $T$. Under these conditions,
the electrons form a steady state distribution with occupations
$F_{k\alpha}(t)$ determined by solving the Floquet-Boltzmann equation (FBE)
Seetharam  _et al._ (2019, 2015); Genske and Rosch (2015); Hone  _et al._
(2009),
$\dot{F}_{{k}\alpha}(t)=I^{\text{b}}_{{k}\alpha}[\\{F_{{k}\alpha}(t)\\}]+I^{\text{ee}}_{{k}\alpha}[\\{F_{{k}\alpha}(t)\\}]$,
for $\dot{F}_{{k}\alpha}(t)=0$. The FBE is valid when $\hat{H}_{\text{ee}}$
and $\hat{H}_{\text{b}}$ weakly scatter electrons between single-particle
Floquet eigenstates. Here, we use the Fermi golden rule modified for
transitions between Floquet states Seetharam  _et al._ (2019, 2015) to
calculate the electron-phonon and electron-electron collision integrals
$I^{\text{b}}_{{k}\alpha}[\\{F_{{k}\alpha}(t)\\}]$ and
$I^{\text{ee}}_{{k}\alpha}[\\{F_{{k}\alpha}(t)\\}]$, respectively. (For
details, see the Supplemental Materials Sup .) The coherences
$\langle\hat{f}^{\dagger}_{k\alpha}(t)\hat{f}_{k\alpha^{\prime}}(t)\rangle$
for $\alpha\neq\alpha^{\prime}$ are suppressed when
$1/\tau^{\text{ph}}_{k\alpha\alpha^{\prime}}+1/\tau^{\text{el}}_{k\alpha\alpha^{\prime}}\ll|\varepsilon_{k\alpha}-\varepsilon_{k\alpha^{\prime}}|/\hbar$,
where $1/\tau^{\text{ph}}_{k\alpha\alpha^{\prime}}$ and
$1/\tau^{\text{el}}_{k\alpha\alpha^{\prime}}$ are respectively the electron-
phonon and electron-electron interband scattering rates between bands $\alpha$
and $\alpha^{\prime}\neq\alpha$ Kohn (2001); Seetharam  _et al._ (2015, 2019).
Interband scattering transitions also broaden the electronic spectral function
by an energy scale of roughly
$\delta\varepsilon\approx\hbar/\tau^{\text{tot}}_{k\alpha\alpha^{\prime}}$,
relaxing energy conservation in the FBE.

Figure 2: (a) Floquet bands upon driving by a coherent phonon wave. Dashed
light purple arrows (1-3): dominant electron-phonon intraband and cooling
processes relaxing electrons into the $\alpha=0$ band. Solid red arrows (4-5):
electron-phonon heating processes exciting electrons into the $\alpha=1$ band,
which are kinematically suppressed when $\Delta\gg\delta\varepsilon$. (b)
Pairs of zigzag and squiggly red arrows: electron-electron scattering exciting
electrons into the $\alpha=1$ band. Dashed light purple arrow: electron-phonon
process relaxing excited electrons into $\mathcal{S}_{+}$. (c)-(e) Occupation
$F_{i}$ of the patches $\mathcal{S}_{j}$ for $i=+,-,0$ [see panel (a)] vs
$\Delta$ for various dielectric constants $\epsilon$. Vertical lines:
$\Delta=\Delta^{*}$. Note $F_{+}\to 0,F_{-}\to 1$ as $\Delta\to\Delta^{*}$,
resulting in quantized current.

Phenomenological rate equation.—The current in Eq. (2) attains a quantized
value when $F_{k\alpha}=F^{\text{opt}}_{k\alpha}$, where
$F^{\text{opt}}_{k0}=1$, and $F^{\text{opt}}_{k,\alpha\neq 0}=0$. Our goal in
this section is to find the conditions on the scattering rates that lead to
$F_{k\alpha}\approx F^{\text{opt}}_{k\alpha}$ in the steady state. To this
end, we identify the key scattering processes that contribute to the steady
state, indicated by arrows in Fig. 2(a)-(b). These processes connect three
patches of Floquet states denoted by $\mathcal{S}_{+}$, $\mathcal{S}_{-}$, and
$\mathcal{S}_{0}$, with approximately uniform electronic occupation $F_{+}$,
$F_{-}$, and $F_{0}$ respectively. The patch $\mathcal{S}_{0}$ includes
Floquet states with momentum $k^{*}\leq k<q/2$ in the $\alpha=0$ band, while
$\mathcal{S}_{+}$ and $\mathcal{S}_{-}$ encloses those with momentum $-q/2\leq
k<k^{*}$ in the $\alpha=1$ and $\alpha=0$ bands, respectively [see Fig. 2(a)].

To estimate the electronic occupations of the patches, let us first consider
the limit $\epsilon\to\infty$ and $I^{\text{ee}}_{k\alpha}\to 0$ in which
scattering is mediated by acoustic phonons. Averaging the FBE over the
patches, we obtain rate equations for their occupations,
$\dot{F}_{i}=\sum_{j}[\mathcal{R}_{ji}F_{j}(1-F_{i})-\mathcal{R}_{ij}F_{i}(1-F_{j})]$,
where $i,j=+,-,0$ and $\mathcal{R}_{ij}$ denotes the average scattering rate
of an electron from patch $\mathcal{S}_{i}$ to $\mathcal{S}_{j}$. We begin by
assuming that the system is optimally doped, i.e., $n_{e}=q$, where $n_{e}$ is
the density of the electrons. The optimal steady state distribution
$F^{\text{opt}}_{k\alpha}$, corresponding to $F_{-}=F_{0}=1$, and $F_{+}=0$,
is obtained when the “Floquet-cooling” processes $\mathcal{R}_{+0}$,
$\mathcal{R}_{0-}$ and $\mathcal{R}_{-0}$ [dashed, light purple arrows
numbered 1-3 in Fig. 2(a)] dominate the scattering rates. The rest of the
scattering rates $\mathcal{R}_{ij}$ create excitations in $\mathcal{S}_{+}$,
giving rise to deviations from $F^{\text{opt}}_{k\alpha}$ and therefore are
dubbed “Floquet-heating” rates [solid red arrows numbered 4-5 in Fig. 2(a)].

When the incoherent phonons remain at temperature $T=0$, all “Floquet-heating”
processes mediated by acoustic phonons require a small energy and large
momentum transfer. This kinematically constrains the rates $\mathcal{R}_{+-}$
and $\mathcal{R}_{0+}$ at high drive intensities, i.e., when
$\Delta>\delta\varepsilon$, disabling all “Floquet-heating” processes.
Simultaneously, the processes described by the rates $\mathcal{R}_{0-}$ and
$\mathcal{R}_{-0}$ are kinematically allowed. These processes are of the
Floquet-Umklapp (FU) type and therefore increase with the drive intensity as
$(\Delta/\hbar\omega)^{2}$ yielding $F_{k\alpha}\to F^{\text{opt}}_{k\alpha}$
Rudner and Lindner (2020); Esin  _et al._ (2021); Seetharam  _et al._ (2015).
This is a remarkable result that shows that the coupling to acoustic phonons
can stabilize a quantized current in the non-adiabatic regime
$\delta\varepsilon<\Delta<\hbar\omega$. In contrast, in the low intensity
limit ($\Delta\to 0$) of the drive, the rates $\mathcal{R}_{0-}$ and
$\mathcal{R}_{-0}$ vanish, whereas the rates $\mathcal{R}_{-+}$ and
$\mathcal{R}_{0+}$ become kinematically enabled. This situation leads to
$F_{+}=F_{0}=1$ and $F_{-}=0$, recovering the equilibrium Fermi-Dirac
distribution.

A finite incoherent phonon temperature $k_{B}T<\hbar\omega$ and electron-
electron interactions (finite $\epsilon$) cause deviations of the steady state
from $F^{\text{opt}}_{k\alpha}$ in the regime $\Delta>\delta\varepsilon$.
Absorption of incoherent phonons yields a finite but weak electron-phonon
heating rate $\mathcal{R}_{0+}$, generating small electron and hole densities
in $\mathcal{S}_{+}$ and $\mathcal{S}_{-}$ respectively. Heating processes due
to electron-electron interactions excite electrons into the patch
$\mathcal{S}_{+}$ in the $\alpha=1$ band [pairs of squiggly red arrows in Fig.
2(b)]. Other electron-electron scattering processes [pairs of zigzag arrows in
Fig. 2(b)] excite electrons to states elsewhere in the $\alpha=1$ band, which
are relaxed to $\mathcal{S}_{+}$ by electron-phonon cooling [dashed, light
purple arrow in Fig. 2(b)]. These processes result in a net increase in
$F_{+}$ and reduced current response. The electron-electron heating can be
suppressed by increasing the drive intensity, since the phase space for such
scattering processes is constrained to small energy and momentum transfers
near the Floquet gap [see Fig. 2(b)] and is reduced as $\Delta$ is increased.
In Fig. 2(c)-(e), we show the average occupations of the patches as a function
of $\Delta$ for various dielectric constants $\epsilon$. We define
$\Delta^{*}$ as the minimal Floquet gap at which $J=0.96e\omega/\pi$. The
equilibrium Fermi-Dirac distribution ($F_{+}=F_{0}=1-F_{-}=1$) transitions to
$F^{\text{opt}}_{k\alpha}$ as $\Delta\to\Delta^{*}$.

Figure 3: (a) Steady state occupation of the phonon-driven CNT for a weak
phonon drive ($\Delta<\Delta^{*}$), dielectric constant $\epsilon=80$, and
optimal doping $n_{e}=q$. Inset shows excitations in the $\alpha=1$ band near
the Floquet gap. (b) Same as (a) but for a strong drive amplitude
($\Delta>\Delta^{*}$) where the occupation of the $\alpha=1$ band is
negligible, and the $\alpha=0$ band is fully occupied. (c)-(d) Same as (b),
but with two different electronic densities $n_{e}$ away from optimal doping.
(e) Steady state current vs $n_{e}$ evaluated at
$\Delta=0.8\hbar\omega>\Delta^{*}$ for $\epsilon=80$.

Numerical analysis.—To test our prediction, we solve the FBE numerically in
the steady state. We consider a coherent acoustic phonon mode of angular
frequency $\omega=6\ \mathrm{meV}/\hbar$ with speed of sound
$c_{\text{ph}}=20\ \mathrm{km/s}$, incoherent phonon modes at temperature
$T=7\ \mathrm{K}$, and optimal electronic density $n_{e}=q$. Fig. 3(a)-(b)
compares the steady state distributions for weak
($\delta\varepsilon<\Delta<\Delta^{*}$) and strong
($\Delta^{*}<\Delta<\hbar\omega$) non-adiabatic drives. As predicted using the
phenomenological model, the density of excitations in the $\alpha=1$ band is
suppressed in the strong drive limit, approaching the optimal distribution
$F^{\text{opt}}_{k\alpha}$.

Fig. 1(b) shows the current $J$ as a function of ${\Delta}/\hbar\omega$ for
different dielectric constants $\epsilon$. The vertical line indicates
$\Delta=\hbar\omega$, the boundary between the non-adiabatic and adiabatic
drive regimes. The current approaches quantization in the non-adiabatic regime
$\Delta<\hbar\omega$ when $\epsilon$ is sufficiently large, verifying the
phenomenological model. The inset of Fig. 1(b) shows that the optimal Floquet
gap $\Delta^{*}$ decreases as a function of $\epsilon$.

Finally, we study the current as a function of the doping. The optimal
distribution, $F^{\text{opt}}_{k\alpha}$, is obtained at optimal doping
$n_{e}=q$. Fig. 3(c)-(d) compares the steady state distributions for
electronic densities below ($n_{e}<q$) and above ($n_{e}>q$) optimal doping,
where the steady state deviates significantly from $F^{\text{opt}}_{k\alpha}$.
Fig. 3(e) shows $J$ as a function of $n_{e}$ for $\Delta\approx
0.8\hbar\omega>\Delta^{*}$ and $\epsilon=80$. The quantized current is reached
at optimal doping.

Experimental realization.—Lastly, let us comment on the experimental setup
needed to detect a phonon-induced quantized current. In Fig. 1(a), we show a
CNT surrounded by a dielectric material of relative permittivity $\epsilon$.
The source of coherent phonons is placed on the right end of the CNT, and the
CNT is enclosed on both ends with a conducting anti-reflective material of
speed of sound $c_{\text{ph}}^{r}$ and thickness of $2\pi
c_{\text{ph}}^{r}/(4\omega)$ tuned to minimize reflection of the phonon wave.
The current is measured using gold leads attached to the anti-reflective
material. Lastly, the phonon drive amplitude $u_{0}$ must be much weaker than
that which causes melting, i.e., $u_{0}\ll 0.1q^{-1}$ as predicted by the
Lindemann criterion Khrapak (2020); Lindemann (1910). The inset of Fig. 1(b)
shows that, for experimentally accessible values of $\epsilon$,
$\Delta^{*}/\hbar\omega\sim 0.5$, corresponding to the coherent phonon
amplitude $u^{*}\sim 0.001a\ll 0.1q^{-1}$.

Conclusion.—Phonon-driven systems exhibit exciting non-equilibrium physics.
Our work shows that screened quantum wires can exhibit quantized transport
when driven non-adiabatically by coherent phonons. Low-temperature incoherent
phonons relax electronic occupation to a topological Floquet energy band [see
Fig. 1(c)] that hosts a quantized current response. Because the topological
transport does not rely on adiabaticity, the quantized current is realized for
short-wavelength phonons with weak amplitudes ($\Delta<\hbar\omega$). This
phenomenon differs from the non-interacting and adiabatic drive limit
realizing a Thouless pump Thouless (1983); Nakajima  _et al._ (2015a); Hotzen
Grinberg  _et al._ (2020); Nakajima  _et al._ (2015b); Bustos-Marún  _et al._
(2013), or the quantized acousto-electric effect induced by GHz-frequency
phonons Talyanskii  _et al._ (2001); Leek  _et al._ (2005); Buitelaar  _et
al._ (2006); Ahlers  _et al._ (2004); Ebbecke  _et al._ (2004); Fletcher  _et
al._ (2003). While the adiabatic quantized acousto-electric effect induces
nano-Ampere quantized currents, our proposal may be realized for higher, THz-
frequency phonons which result in much stronger quantized currents on the
order of $0.1\ \mu\mathrm{A}$. It also differs from proposals to realize
quantized transport in systems far from equilibrium via strong disorder Kundu
_et al._ (2020), or one-way transport without guaranteed quantized current in
driven open systems Shu  _et al._ (2023).

The setup can serve as a new detector of coherent phonons relying only on
transport measurements, in contrast to conventional methods utilizing optical
reflectivity measurements Lanzillotti-Kimura  _et al._ (2011); Eisfeld and
Renk (1979); Giorgianni  _et al._ (2022); Yoon  _et al._ (2023a). Carbon
nanotubes [see Fig. 1(a)] could provide a suitable experimental platform to
realize such phonon detection. The detection would be sensitive even to weak
coherent phonon amplitudes [see Fig. 1(b)] because adiabaticity is not
required. Topological transport in higher-dimensional phonon-driven systems is
a subject of future work, in which the direction of the transport could be
sensitively controlled by the direction of the coherent phonon propagation.

###### Acknowledgements.

We thank Yang Peng, Michael Kolodrubetz, and Erez Berg for valuable
discussions. C.Y. gratefully acknowledges support from the DOE NNSA
Stewardship Science Graduate Fellowship program, which is provided under
cooperative agreement number DE-NA0003960. W.H. gratefully acknowledges
support from the Caltech Cambridge Scholars Exchange Program. G.R. and I.E.
are grateful for support from the Simons Foundation and the Institute of
Quantum Information and Matter, as well as support from the NSF DMR grant
number 1839271. This work is supported by ARO MURI Grant No. W911NF-16-1-0361,
and was performed in part at Aspen Center for Physics, which is supported by
National Science Foundation grant PHY-1607611.

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