# Adiabatic Quantum Zeno Dynamics of Bosonic Atom Pairs with Large Inelastic
Losses

Manel Bosch Aguilera1, Alexis Ghermaoui1, Rémy Vatré1, Raphaël Bouganne1,
Jérôme Beugnon1, Fabrice Gerbier1<EMAIL_ADDRESS>1Laboratoire
Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne
Université, 11 Place Marcelin Berthelot, 75005 Paris, France

###### Abstract

We report on experiments exploring the non-Hermitian dynamics of pairs of two-
level atoms tightly confined in an optical lattice and driven by a near-
resonant laser. Although spontaneous emission is negligible for the long-lived
excited state, two-body dissipation arises from strong inelastic collisions
between two atoms in the excited state. We demonstrate quasi-adiabatic control
of the internal state of the pairs in the quantum Zeno regime where inelastic
losses dominate. Preparing each atom pair in the longest-lived eigenstate of
the non-Hermitian Hamiltonian describing the dissipative dynamics, we measure
a lifetime enhanced by more than two orders of magnitude in comparison to the
bare two-body lifetime. The measured enhancement factor is in quantitative
agreement with the expected lifetime of the prepared “non-Hermitian dressed
state”.

##  I Introduction

In recent years, the theory of non-Hermitian systems has emerged as a common
framework to describe classical waves undergoing energy or information loss.
Experiments in optical, mechanical, biological or electrical systems [1] have
demonstrated many phenomena unique to non-Hermitian systems, such as
unidirectional transport, enhanced sensing capability or characteristic
topological properties in the vicinity of the so-called exceptional points
where two (or more) of the eigenmodes coalesce [2, 3, 4].

Non-Hermitian Hamiltonians [5, 6] have also been used for a long time to model
dissipative quantum systems, starting with early studies of radioactive decay
[7, 8]. Dissipation in quantum systems often originates from their weak
coupling to a larger environment, described by a Lindblad master equation [9],

$\displaystyle\frac{d\hat{\rho}}{dt}$
$\displaystyle=\frac{1}{\mathrm{i}\hbar}\Big{[}\hat{H}_{\mathrm{eff}},\hat{\rho}\Big{]}+\sum_{\alpha}\,\Gamma_{\alpha}\,\hat{L}_{\alpha}\,\hat{\rho}\,\hat{L}_{\alpha}^{\dagger}.$
(1)

Here $\hat{\rho}$ is the density matrix, $\hat{L}_{\alpha}$ is the so-called
jump operator for the dissipative channel $\alpha$, and
$\hat{H}_{\mathrm{eff}}=\hat{H}-\mathrm{i}\sum_{\alpha}\frac{\hbar\Gamma_{\alpha}}{2}\hat{L}_{\alpha}^{\dagger}\hat{L}_{\alpha}$
is the effective Hamiltonian, with the Hermitian part $\hat{H}$ describing the
isolated system. The jump operator $\hat{L}_{\alpha}$ describes environment-
induced transitions in the dissipative channel $\alpha$ at an average rate
$\Gamma_{\alpha}$. The quantum trajectory approach [10, 11, 12] interprets the
non-Hermitian operator $\hat{H}_{\mathrm{eff}}$ as an effective Hamiltonian
generating a continuous dissipative evolution interrupted by stochastic
quantum jumps described by the $\hat{L}_{\alpha}$’s. Thus, in general, the
non-Hermitian Hamiltonian captures only part of the dissipative dynamics.

Master equations such as Eq. (1) have a long history and many applications in
low-energy physics, in particular in atomic physics and quantum optics [9, 13,
12]. More recently, master equations and non-Hermitian Hamiltonians have been
used to study many-body systems out of equilibrium. These studies have been
largely driven by experimental progress, for instance with ultracold atoms
subject to strong one-, two- or three-body losses [14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29]. The dissipative part of the Lindblad
master equation can be interpreted as describing generalized measurements
performed on the system [30, 31, 32]. For strong losses, one enters a regime
where the quantum Zeno effect [33, 34, 35] – the freezing of the time
evolution of a quantum system subject to frequent measurements –becomes
relevant. This effect prevents transitions to the highly dissipative states,
thereby effectively suppressing dissipation and dramatically slowing down the
decay [14, 15, 16, 17, 18, 19]. Atomic systems with strong two-body losses are
plentiful, for instance two-electron atoms (as in our work), but also atoms in
highly excited Rydberg states, atoms close to a Feshbach resonance, or spinful
dipolar atoms … These systems also have interesting properties that have led
to many original proposals to engineer correlated quantum states or more
generally study quantum many-body physics [45, 46, 47, 48, 49, 50].

In this article, we investigate the dissipative properties of pairs of two-
level atoms trapped at isolated nodes of an optical lattice and driven by a
near-resonant laser [36, 37]. The two atoms interact elastically in a state-
dependent manner, but also undergo strong inelastic losses when both atoms are
excited. In a quantum trajectory description, quantum jumps project the
quantum state at each lattice site to the vacuum state, which does not
contribute to any observable. As a result, expectation values are completely
determined by the evolution under the non-Hermitian Hamiltonian. We
demonstrate quasi-adiabatic [38, 39, 40, 41, 42, 43, 44] control of the
quantum state of the pairs in the quantum Zeno regime of strong dissipation.
We prepare each atom pair in the longest lived “dressed” eigenstate of the
effective non-Hermitian Hamiltonian, and observe a lifetime longer than the
bare two-body lifetime by at least two orders of magnitude. Preparing
(quasi-)adiabatically many-body systems in loss-protected states is of
paramount importance to realize experimentally the proposals discussed above.
The experiments presented here are a first step towards this goal.

##  II Dynamics of laser-driven, interacting atom pairs and non-Hermitian
Hamiltonian

Figure 1: (a): Sketch of the energy levels of a pair of bosonic atoms with
internal states $g,e$. The parameters $p=(U_{ee}-U_{gg})/2$ and
$q=U_{eg}-(U_{gg}+U_{ee})/2$ characterize the shifts of the transition
frequencies due to state-dependent interactions $U_{\alpha\beta}$
($\alpha,\beta=\\{g,e\\}$), $\delta$ is the laser detuning from the atomic
resonance, and $\Gamma_{ee}$ is a two-body loss rate for two atoms in the
excited state $e$. (b-c): Real and imaginary parts of the eigenvalues of the
effective Hamiltonian in the regime of weak dissipation with
$\Omega/\Gamma_{ee}=1$. (d-e): Same in the quantum Zeno regime
$\Omega/\Gamma_{ee}=0.1$. The thin dotted lines in (b,d) indicate the
uncoupled energy levels $gg,eg,ee$. The vertical dashed lines in (b-e) mark
the position of the one-photon resonances. The dash-dotted lines in (e)
indicate the decay rates calculated from the effective Hamiltonian in the
$\\{gg,eg\\}$ subspace [Eq. (11)]. We used the experimental values from Table
1.

###  II.1 The non-Hermitian Hamiltonian

We consider a pair of bosons with two internal levels confined to the motional
ground state of a tight trap. A near-resonant laser field of frequency
$\omega_{\mathrm{L}}$ couples the two internal states $|g\rangle$ and
$|e\rangle$. Neglecting motional excitations, the internal state is the only
dynamical degree of freedom. The two-particle Hilbert space has only three
exchange-symmetric states, that we label $|gg\rangle,|eg\rangle$ and
$|ee\rangle$ to highlight the occupation numbers of the internal states.

The dynamics is captured by an effective non-Hermitian Hamiltonian (see
Supplemental Material [51]),

$\displaystyle\hat{H}_{\mathrm{eff}}$
$\displaystyle=\hat{H}_{0}\,+\,\hat{W}\,-\mathrm{i}\,\frac{\hbar\Gamma_{ee}}{2}\,|ee\rangle\langle
ee|.$ (2)

The first two terms in the right hand side (rhs) describe coherent internal
dynamics, with

$\displaystyle\hat{H}_{0}$
$\displaystyle=\big{(}p-\hbar\delta)\hat{S}_{z}-q\hat{S}_{z}^{2},\hskip
8.5359pt\hat{W}=\hbar\Omega\hat{S}_{x}.$ (3)

Here the coupling strength is given by the Rabi frequency $\Omega$, the
detuning from the one-atom transition frequency $\omega_{0}$ is
$\delta=\omega_{\mathrm{L}}-\omega_{0}$, and

$\displaystyle\hat{S}_{z}=\begin{pmatrix}-1&0&0\\\ 0&0&0\\\
0&0&1\end{pmatrix},\hskip
8.5359pt\hat{S}_{x}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&1&0\\\ 1&0&1\\\
0&1&0\end{pmatrix},$ (4)

are the standard spin-1 matrices. The uncoupled Hamiltonian $\hat{H}_{0}$
includes the internal energies and the interaction energies of the pair for
each spin configuration $|ij\rangle$,

$\displaystyle U_{ij}$
$\displaystyle=\frac{4\pi\hbar^{2}a_{ij}}{M}\int|w(\mathbf{r})|^{4}\,d^{3}\mathbf{r},$
(5)

with $M$ the mass of an atom, $a_{ij}$ the scattering length describing the
particular interaction process, and $w$ the orbital wavefunction. The relevant
shifts of the transition frequencies are parametrized by $p=(U_{ee}-U_{gg})/2$
and $q=U_{eg}-(U_{gg}+U_{ee})/2$. Relevant values of the various parameters
for our experiment are indicated in Table 1.

The last term in the rhs of Eq. (2) proportional to the projector
$|ee\rangle\langle ee|$ describes inelastic losses. We neglect two-body loss
processes for the $gg$ and $eg$ states, in accordance with experimental
measurements [36, 37]. Physically, the losses originate from “principal
quantum number changing” collisions [52, 53] (also called “energy-pooling”
collisions), concisely summarized by the reaction $e+e\to g+e^{\prime}$ with
$e^{\prime}$ another electronic excited state which eventually radiatively
decays to $g$. The internal energy released in this strongly inelastic process
makes both atoms escape from the trap. The two-body loss rate

$\displaystyle\Gamma_{ee}$
$\displaystyle=\beta_{ee}\,\int|w(\mathbf{r})|^{4}\,d^{3}\mathbf{r},$ (6)

with $\beta_{ee}$ an atom-dependent rate constant, is determined by the same
overlap integral as the elastic interaction in Eq. (5). As a result, the ratio
$\hbar\Gamma_{ee}/U_{gg}=M\beta_{ee}/(4\pi\hbar a_{gg})$ is independent of the
lattice details and of order unity (see Table 1).

###  II.2 Properties of the non-Hermitian Hamitlonian

We consider how the complex eigenspectrum
$\lambda_{n}=\varepsilon_{n}-\mathrm{i}\hbar\gamma_{n}/2$ (with
$\varepsilon_{n}$ and $\gamma_{n}$ real numbers) $\hat{H}_{\mathrm{eff}}$
changes as a function of the detuning $\delta$. We call the eigenstates
$|\lambda_{n}\rangle$ “non-Hermitian dressed states” in analogy with their
Hermitian counterparts [51]. Note that the imaginary part (“decay rate”) of
the non-Hermitian spectrum is related to the population of the bare state
$ee$,

$\displaystyle\gamma_{n}=-\frac{2}{\hbar}\mathrm{Im}(\lambda_{n})=\Gamma_{ee}\Big{|}\Big{|}\langle
ee|\lambda_{n}\rangle\Big{|}\Big{|}^{2}.$ (7)

To set the stage, we first discuss the behaviour of the coherent part
$\hat{H}_{0}\,+\,\hat{W}$. The spectrum of the Hamiltonian $\hat{H}_{0}$ (thin
dotted lines in Fig. 1b,d) gives rise to three level crossings. Two of these
(marked by the thin vertical dashed lines in Fig. 1b-e) correspond to one-
photon transitions and occur when $\delta=(p+q)/\hbar$ ($gg-eg$ crossing) and
when $\delta=(p-q)/\hbar$ ($eg-ee$ crossing). The third crossing corresponds
to a two-photon transition and occurs when $\delta=p/\hbar$ ($gg-ee$
crossing). A non-zero laser coupling $\hat{W}$ turns the level crossings to
avoided crossings.

Turning to the spectrum of the complete non-Hermitian
$\hat{H}_{\mathrm{eff}}$, we find that two regimes emerge either at strong
($\Omega\gtrsim\Gamma_{ee}$) or weak ($\Omega\ll\Gamma_{ee}$) coherent
driving, respectively. For strong driving, the eigenstates are qualitatively
similar to the situation without losses. The real parts $\epsilon_{i}$ display
avoided crossings as for the non-dissipative version of the problem (Fig. 1b).
Near resonance ($|\delta|\lesssim\Omega$), the dressed states are
superpositions of all three bare states with comparable weigths, and all decay
rates $\gamma_{i}$ are comparable to the bare loss rate $\Gamma_{ee}$ (Fig.
1c).

For weak driving (or, equivalently, strong losses), a qualitatively different
situation emerges. Only the $gg-eg$ avoided crossing survives. The other two
disappear and the real parts $\epsilon_{i}$ cross as in the absence of laser
coupling (Fig. 1d). Moreover, the non-Hermitian dressed states no longer mix
near resonance as for strong driving (Fig. 1e). Instead, one notices that the
states $\\{\lambda_{1},\lambda_{2}\\}$ form a “lossless” subspace with a decay
rate orders of magnitude smaller than the bare loss rate $\Gamma_{ee}$. As the
decay rate is varied, this subspace remains well isolated from the lossy
$\lambda_{3}$ state with a decay rate $\gamma_{3}\approx\Gamma_{ee}$ for all
detunings.

###  II.3 Effective Hamiltonian in the lossless subspace

In the quantum Zeno regime, the lossy state remains isolated in the sense that
the complex eigenvalue $\lambda_{3}$ never approaches $\lambda_{1/2}$ because
of the vastly different imaginary parts. This is reminiscent of a common
situation in quantum mechanics, where a low-energy subspace remains well
isolated in energy from the rest of the Hilbert space. The dynamics in the
low-energy subspace can be captured by an effective Hamiltonian given by the
expansion [9],

$\displaystyle\hat{P}\,\hat{H}_{\mathrm{eff}}^{\prime}\,\hat{P}-\hat{P}\,\hat{H}\,\hat{P}$
$\displaystyle=\hat{P}\hat{H}\hat{Q}\,\frac{1}{\hat{H}}\,\hat{Q}\hat{H}\hat{P}+\cdots,$
(8)

where $\hat{Q}$ and $\hat{P}=\mathbf{\hat{1}}-\hat{Q}$ are projectors on the
high- and low-energy subspaces.

The formalism of the effective Hamiltonian is easily adapted to the non-
Hermitian case, with $\hat{Q}=|ee\rangle\langle ee|$ and
$\hat{P}=\mathbf{\hat{1}}-\hat{Q}$ the projectors on the lossy and lossless
subspaces, respectively, and with
$\hat{P}\hat{W}\hat{Q}=(\hbar\Omega/\sqrt{2})\times|eg\rangle\langle ee|$. The
effective Hamiltonian in the lossless subspace spanned by
$\\{|gg\rangle,|eg\rangle\\}$ is given by

$\displaystyle\hat{P}\,\hat{H}_{\mathrm{eff}}^{\prime}\,\hat{P}$
$\displaystyle=\frac{\hbar}{2}\begin{pmatrix}\delta^{\prime}&\sqrt{2}\Omega\\\
\sqrt{2}\Omega&-\delta^{\prime}-\mathrm{i}\Gamma_{\rm eff}\end{pmatrix}.$ (9)

with $\delta^{\prime}=\delta-(p+q-\Delta q)/\hbar$. The $eg$ level acquires an
additional level shift $\Delta q$ and an effective decay rate
$\Gamma_{\mathrm{eff}}$,

$\displaystyle\Delta q+\mathrm{i}\frac{\Gamma_{\mathrm{eff}}}{2}$
$\displaystyle=\frac{\Omega^{2}}{\Gamma_{ee}}\,\frac{x+\mathrm{i}}{1+x^{2}},$
(10)

with $x=2(p-q-\hbar\delta)/(\hbar\Gamma_{ee})$. The small parameter of the
expansion is $\Omega/\Gamma_{ee}\ll 1$ in the quantum Zeno regime, so that
$\Gamma_{\mathrm{eff}},\,|q^{\prime}-q|/\hbar\ll\Omega$. For the experimental
parameters $\Omega/(2\pi)\approx 150\,$Hz and $\Gamma_{ee}/(2\pi)\approx
1.5\,$kHz, we have for instance $\Gamma_{\mathrm{eff}}/(2\pi)\approx 60\,$s-1
on resonance ($x=0$). For $\Omega\ll\Gamma_{ee}$, the eigenvalues of the
$2\times 2$ effective Hamiltonian are approximately

$\displaystyle\lambda_{1/2}\approx\pm\frac{\hbar\Omega_{2\times
2}}{2}-\mathrm{i}\frac{\hbar\Gamma_{\mathrm{eff}}}{4}\left(1\mp\frac{\delta^{\prime}}{\Omega_{2\times
2}}\right).$ (11)

with $\Omega_{2\times 2}=\sqrt{\delta^{\prime 2}+2\Omega^{2}}$. The imaginary
parts of the two eigenvalues are shown in Fig. 1e.

Far off resonance ($|x|\approx 2|\delta|/\Gamma_{ee}\gg 1$), the effective
decay rate $\Gamma_{\mathrm{eff}}$ behaves as
$\Gamma_{\mathrm{eff}}\propto\Gamma_{ee}(\Omega/\delta)^{2}$, which is simply
the bare decay rate $\Gamma_{ee}$ times the occupation probability
$\propto(\Omega/\delta)^{2}$ of the $ee$ state in the large detuning limit.
The decay of the slowest decaying state is even smaller and scales as
$\propto(\Omega/\delta)^{4}$. The scaling reflects that for large detunings,
the slowest decaying state couples to $ee$ by a two-photon transition.

As the detuning is brought closer to the $\delta^{\prime}=0$ resonance, the
parameter $|x|$ eventually becomes small compared to one, and the decay rate
approaches
$\Gamma_{\mathrm{eff}}/2\propto\Omega^{2}/\Gamma_{ee}\ll\Gamma_{ee}$ for both
eigenstates. Moreover, the larger $\Gamma_{ee}$ the smaller the saturation
value, a behavior typical of the quantum Zeno effect. Note that a large bare
decay rate is essential. Indeed, in the opposite limiting case where $|x|\gg
1$ near resonance (or equivalently, $2|q|\gg\hbar\Gamma_{ee}$), one finds
instead a perturbative scaling law $\Gamma_{\mathrm{eff}}\propto\Gamma_{ee}$.

$U_{gg}/h$ | $U_{eg}/U_{gg}$ | $U_{ee}/U_{gg}$ | $\hbar\Gamma_{ee}/U_{gg}$ | $p/U_{gg}$ | $q/U_{gg}$
---|---|---|---|---|---
$1.40\,$kHz | 0.905 | 1.21 | 1.02 | 0.21 | 0.20
|  |  |  |  |
Table 1: Summary of collisional properties. We calibrate experimentally the
interaction strength $U_{gg}$ for ground state atoms as in [36], while
$U_{eg}$, $U_{ee}$ and $\Gamma_{ee}$ are deduced from it using the measured
scattering lengths [37] and inelastic rate constant [36]. The quantities
$p=U_{ee}-U_{gg}$ and $q=(U_{ee}+U_{gg})/2-U_{eg}$ determine the shifts of the
transition frequencies (see Fig. 1).

##  III Description of the experiments

###  III.1 Experimental system

The experiment is performed with ultracold 174Yb atoms trapped at isolated
sites of a three-dimensional optical lattice. The optical lattice operates at
the magic wavelength $\lambda_{\mathrm{m}}\approx 759.4\,$nm, where the
lattice potential is independent of the internal state $g$ or $e$. The lattice
depth is chosen deep in the Mott insulator regime with depth
$V_{x,y,z}=25,25,27\,E_{r}$, where $E_{r}=h\times 1.98\,$kHz is the lattice
recoil energy. We assume that band occupation beyond the fundamental Bloch
band (including interband transitions induced by the coupling laser) and
tunneling within that band are negligible on experimental time scales. In this
regime, the lattice sites can be treated as an ensemble of tight traps
isolated from each other. Due to an auxiliary harmonic trap [36], the system
forms a core of doubly-occupied sites near the trap center, surrounded by a
shell of singly-occupied (or empty) sites (see sketch in Fig. 2). Our focus is
on the dynamics of atom pairs populating the central “Mott core”, as
considered in the previous Section II.

Figure 2: Lifetime measurement for a final detuning
$\delta_{\mathrm{f}}/(2\pi)=650\,$Hz and nominal Rabi frequency
$\Omega/(2\pi)=150\,$Hz. The figure shows the total atom number (irrespective
of the internal state) versus hold time. The decay is attributed to two-body
losses in doubly-occupied sites initially occupying the center of the trap.
The presence of an outer shell of singly-occupied sites explains the non-zero
asymptote. The inset shows a sketch of the temporal profile of the ramp.

###  III.2 Landau-Zener ramps

The atoms initially occupy the electronic ground state
$g\equiv$${}^{1}\textrm{S}_{0}$, coupled to the excited state
$e\equiv$${}^{3}\textrm{P}_{0}$ by a narrow-linewidth “clock” laser with a
wavelength around 578 nm. We prepare the system to the desired final state
using “Landau-Zener ramps” where the detuning changes linearly with time
(inset of Fig. 2). In details, the laser is first turned on with a negligible
intensity at an initial detuning $\delta_{\mathrm{i}}$ far from any resonance.
A first intensity ramp of duration $t_{\Omega}$ brings the Rabi frequency to
its nominal value $\Omega=2\pi\times 150\,$Hz, keeping
$\delta=\delta_{\mathrm{i}}$ fixed. A second ramp of duration $t_{\delta}$
brings the detuning to the final desired value
$\delta_{\mathrm{i}}\to\delta_{\mathrm{f}}$, keeping $\Omega$ fixed at the
nominal value. For all reported experiments, we perform the frequency ramp at
a constant speed $\dot{\delta}\approx 2\pi\times 11.1\,$Hz/ms, so that
$t_{\delta}=(\delta_{\mathrm{f}}-\delta_{\mathrm{i}})/\dot{\delta}$, and chose
$t_{\Omega}=T_{R}/10$, for a total ramp time $T_{R}=t_{\Omega}+t_{\delta}$.

###  III.3 Detection

We measure either the total atom number or the populations in the bare state
$g$ or $e$ using absorption imaging after a time of flight [51]. The $e$
population is measured up to a global repumping efficiency $\eta_{\rm
rp}\approx 0.8$ [36]. The result of a given measurement can be expressed in
terms of the occupation probability $P_{\alpha}$ of the bare atomic states,
weighted by the initial proportions of singly- or doubly-occupied sites. For
instance, the population of state $g$ is given by

$\displaystyle N_{g}$
$\displaystyle=n_{1}\,P_{g}^{\overline{n}=1}\,+\,n_{2}\,\left(P_{eg}+2P_{gg}\right),$
(12)

Here $n_{1/2}$ denote the number of singly/doubly occupied sites determined by
the initial density profile, respectively. A similar equation holds for
$N_{e}$, and the experiment records $\eta_{\rm rp}N_{e}$.

The outer shell of singly-occupied sites thus appears as a “parasitic”
background signal on top of the atom pair signal of interest. This signal is
however easy to correct for, since it merely corresponds to an ensemble of
$n_{1}$ indenpendent two-level systems undergoing a Landau-Zener process.
Provided one is able to determine $n_{1/2}$, the spurious contribution of
singly-occupied sites can be substracted off the measurements to obtain the
contribution of atom pairs.

To measure $n_{1/2}$, we hold the atoms for a variable time $t_{\rm hold}$ in
presence of laser light after the preparation ramp, and record the remaining
$g$ population as a function of $t_{\rm hold}$. An example of such measurement
is shown in Fig. 2. We fit such curves by an exponentially decaying function
$f_{1}+f_{2}e^{-\gamma t_{\mathrm{hold}}}$ where $f_{1/2}=n_{1/2}$ in Eq.
(12), and where $\gamma$ gives the decay rate of the prepared non-Hermitian
dressed state.

##  IV Strongly enhanced lifetime in the quantum Zeno regime

Figure 3: Measurement of decay rates of the non-Hermitian dressed eigenstates.
Solid: numerically calculated decay rates. Dashed: lifetime expected when
treating the non-Hermitian part as a perturbation.

We first discuss the observed lifetime of the state obtained at the end of the
Landau-Zener ramps. We postpone the discussion of (quasi-)adiabatic following
for a non-Hermitian system to the following Section V. We perform the
preparation sequence described above (a) starting from $\delta_{\rm
i}/(2\pi)=-1.5\,$kHz and increasing the laser detuning, or (b) starting from
$\delta_{\rm i}/(2\pi)=+1.5\,$kHz and decreasing the laser detuning. In case
(a), the initial state almost coincide with the dressed state $\lambda_{1}$
and the experimental ramp aims at following “adiabatically” this state. Case
(b) is the same but following $\lambda_{2}$.

After the preparation ramp, we hold the atoms for a variable time $t_{\rm
hold}$ in presence of laser light, and finally record the remaining $e$
population. For both ascending (a) and descending (b) ramps, monitoring the
atom number as a function of hold time $t_{\rm hold}$ allows us to extract the
lifetime of the prepared dressed state as discussed in Section III.3.

We report in Fig. 3 the result of all lifetime measurements using the
ascending and descending frequency ramps (a) and (b) for several final laser
detunings. We compare the measured decay rates to the calculated decay rates
of the eigenstates $\lambda_{1}$ and $\lambda_{2}$. The agreement between the
predictions and the measurements is excellent. This validates a posteriori the
description by a non-Hermitian Hamiltonian and supports the claim of adiabatic
following (see also Section V below).

To highlight the role of the quantum Zeno effect, we also show as dashed lines
the predictions of a naive perturbation theory, where the non-Hermitian
dressed eigenstates are replaced by the corresponding eigenstates of
$\hat{H}_{0}\,+\,\hat{W}$ (i.e. setting $\Gamma_{ee}=0$ in the non-Hermitian
Hamiltonian), and Eq. (7) is then used to compute the decay rates. This
comparison shows that the lifetime is enhanced by two orders of magnitude near
resonance by the quantum Zeno effect.

##  V Quasi-adiabatic ramps

In this Section, we examine the notion of quasi-adiabaticity and relate it to
our experiments. We first remind how the concept of adiabatic following in
Hermitian quantum mechanics can be generalized for non-Hermitian systems [38,
39, 40, 41, 42, 43, 44] , and thereby define the notion of (quasi-)adiabatic
following in a dissipative system. In a second part, we explore in more
details the quasi-adiabaticity of the experimental ramps discussed in the
previous Section.

###  V.1 Quasi-adiabaticity for non-Hermitian systems

We now consider a time-dependent Hamiltonian
$\hat{H}_{\mathrm{eff}}(t)=\hat{H}_{\mathrm{eff}}[\delta(t)]$. The time-
dependence originates from the detuning $\delta(t)$ taken to be a linear
function of time with slope $\dot{\delta}$. We note $|\lambda_{n}\rangle$ and
$\langle\bar{\lambda}_{n}|$ the right and left eigenstates of
$\hat{H}_{\mathrm{eff}}$ associated with the eigenvalue
$\lambda_{n}=\varepsilon_{n}-i\hbar\gamma_{n}/2$ ($\gamma_{n}\geq 0$). We take
the normalization conventions
$\langle\bar{\lambda}_{m}|\lambda_{n}\rangle=\delta_{mn}$,
$\langle\lambda_{n}|\lambda_{n}\rangle=1$ 111For non-Hermitian systems, there
are two arbitrary normalization factors that can be chosen freely [6, 44].
Assuming that the system is initially prepared in one particular eigenstate
$\alpha$ of $\hat{H}_{\mathrm{eff}}(0)$, the non-Hermitian generalization of
adiabatic mapping is [38, 39, 40, 41, 42, 43, 44]

$\displaystyle|\Psi(0)\rangle=\left|\lambda_{\alpha}[\delta_{\mathrm{i}}]\right\rangle\to|\Psi(T_{R})\rangle=\mathrm{e}^{\mathrm{i}\phi_{\alpha}-\frac{1}{2}\kappa_{\alpha}}\,|\lambda_{\alpha}[\delta_{\mathrm{f}}]\rangle,$
(13)

with $T_{R}$ the ramp duration. The system follows the state
$\left|\lambda_{\alpha}\right\rangle$ up to a phase, as in the Hermitian case,
but also up to an attenuation factor $\mathrm{e}^{-\kappa_{\alpha}}$. One may
speak of “quasi-adiabaticity” if the time evolution is well described by Eq.
(13). By this, we mean that the evolution is as close as possible to a truly
adiabatic one, given the dissipative nature of the system at hand.

It is convenient to parametrize the instantaneous detuning as
$\delta(x)=(1-x)\delta_{\mathrm{i}}+x\delta_{\mathrm{f}}$, or equivalently
$x=\dot{\delta}t/(\delta_{\mathrm{f}}-\delta_{\mathrm{i}})\in[0,1]$. The
quasi-adiabatic phase and attenuation exponent can then be written as

$\displaystyle\phi_{\alpha}$
$\displaystyle=\int_{0}^{1}\left(-\frac{\varepsilon_{\alpha}[x]T_{R}}{\hbar}+\mathrm{Re}\mathcal{B}_{\alpha\alpha}[x]\right)dx,$
(14) $\displaystyle\kappa_{\alpha}$
$\displaystyle=\int_{0}^{1}\Big{(}\gamma_{\alpha}[x]T_{R}+2\mathrm{Im}\mathcal{B}_{\alpha\alpha}[x]\Big{)}dx,$
(15)

with
$\mathcal{B}_{\alpha\beta}[x]=\mathrm{i}\Big{\langle}\bar{\lambda}_{\beta}\Big{|}\frac{d\lambda_{\alpha}}{dx}\Big{\rangle}_{x}$
a Berry connection associated with “transport” of the eigenstates in the
$\Omega-\delta$ space. Note that the explicit form of $\phi_{\alpha}$ and
$\kappa_{\alpha}$ depend on the choice of the normalization (see Ref. [44] for
a detailed discussion).

Since the norm of the wavefunction is not conserved by non-Hermitian
evolution, the non-Hermitian dressed eigenstate is transported with a
_survival probability_ less than one,

$\displaystyle P_{\mathrm{s},\alpha}=\mathrm{e}^{-\kappa_{\alpha}}.$ (16)

Eq. (16) provides a lower bound for the actual survival probability, which is
realized only for quasi-adiabatic evolutions following the least dissipative
eigenstate.

A generic validity criterion for the quasi-adiabatic mapping (13) is given by
the inequality [40]

$\displaystyle\frac{1}{\hbar}\left|\left(\frac{d\hat{H}_{\mathrm{eff}}}{dt}\right)_{\beta\alpha}\right|=\frac{|\dot{\delta}|}{2}\left|\langle\bar{\lambda}_{\beta}|\hat{S}_{z}|\lambda_{\alpha}\rangle\right|\ll\frac{|\lambda_{\alpha}-\lambda_{\beta}|^{2}}{\hbar^{2}}\frac{P_{\mathrm{s},\alpha}}{P_{\mathrm{s},\beta}}\,\,\,$
(17)

$\forall\beta\neq\alpha$. There are two main differences from the analogous
Hermitian adiabatic condition. First, the equality of the _real parts_
$\epsilon_{i}$ is only a necessary condition to follow the initial dressed
state. Second, the right-hand-side of Eq. (17) has an extra factor
$P_{\mathrm{s},\alpha}/P_{\mathrm{s},\beta}$. If $\alpha$ is the least
dissipative eigenstate, this factor is positive and exponentially large for
slow ramps: quasi-adiabaticity is then reinforced by the non-Hermitian
dynamics [38].

###  V.2 Experimental ramps

Figure 4: (a): Total atom number and $g$ population following the
$\lambda_{2}$ dressed state (descending frequency ramps at constant speed
$\dot{\delta}$). Note that the frequency axis is reversed, so that the ramp
time increases from left to right. The insets show a sketch of the
“trajectories” in the $\Omega-\delta$ plane. (b): Total populations $N_{2}$
and $N_{1}$ of the central core and of the outer shell, respectively. The
former can be interpreted as a measure of the survival probability of doubly-
occupied sites. We observe an excellent agreement with the expected population
of the $|\lambda_{2}\rangle$ dressed eigenstate (solid lines), which provides
evidence for quasi-adiabatic following. The population $N_{1}$ of singly-
occupied sites remains approximately constant. For such sites, the frequency
ramp amounts to an adiabatic passage from $g$ to $e$ for large final
detunings.

Fig. 4a shows the experimentally measured populations in $g$ and the total
atom number for each final detuning $\delta_{f}$ of a _descending_ frequency
ramp starting from a large and positive initial detuning $\delta_{\rm
i}/(2\pi)=+1.5\,$kHz and lowering the detuning at constant speed
$\dot{\delta}$. This ramp aims at following the $|\lambda_{2}\rangle$
eigenstate of the non-Hermitian Hamiltonian, which connects to $|gg\rangle$
when $\delta\to+\infty$. We focus on descending ramps in this Section for
concreteness, but we obtain equivalent results for ascending ramps following
$|\lambda_{1}\rangle$. The measured populations closely follow the curves
expected from solving numerically the time-dependent Schroedinger equation for
the experimental ramp.

The survival probability of doubly-occupied sites introduced previously
provides a more direct experimental evidence for quasi-adiabatic following.
From the decay measurements discussed in Section III.3, we obtain the total
populations $N_{2}=2n_{2}$ and $N_{1}=n_{1}$ of the central core and of the
outer shell, respectively, for each value of the final detuning
$\delta_{\mathrm{f}}$. The resulting curves are shown in Fig. 4b. The former
quantity can be interpreted as a measure of the survival probability of
doubly-occupied sites times their initial total population. We observe that
the measured $N_{2}$ agrees well with the expected population of the
$|\lambda_{2}\rangle$ dressed eigenstate, _i.e._
$N_{2}(\delta_{\mathrm{f}})=N_{2}(\delta_{\mathrm{i}})P_{s,\lambda_{2}}$. As
discussed before, this supports the interpretation of the experiment in terms
of quasi-adiabatic following of the dressed eigenstate $|\lambda_{2}\rangle$.
Note also that the population $N_{1}$ remains almost constant, as expected.
The small drop in the calculated curve after crossing the resonance is due to
the slightly lower detection efficiency for atoms in $e$, which is is included
in the calculations.

Finally, we briefly mention that there are small deviations from adiabatic
behavior in Fig. 4a, namely large fluctuations of the observables near the
resonance, as well as a small residual $g$ population for large and negative
detunings $\delta_{\mathrm{f}}$. In the Supplementary Material [51], we argue
that these deviations are caused by frequency fluctuations of the driving
laser, and unrelated to the non-Hermitian dynamics of doubly-occupied sites
that is the focus of this paper.

##  VI Conclusion and discussion

In conclusion, we have studied the non-Hermitian dynamics of atoms pairs
tightly confined in an optical lattice. With strong two-body losses, we
observe lifetimes much longer than the natural lifetime set by the inverse
two-body loss rate $\Gamma_{ee}$ by at least two orders of magnitude. We
discussed how this feature can be understood from the quantum Zeno effect, and
demonstrated quasi-adiabatic preparation of the longest-lived eigenstate of
the non-Hermitian Hamiltonian.

The non-Hermitian dressed state prepared after a quasi-adiabatic ramp is of
the form

$\displaystyle|\Psi\rangle$ $\displaystyle=\alpha|gg\rangle+\beta|ge\rangle.$
(18)

Paskauskas and You [55] have discussed criterions to decide whether a two-
particle bosonic state shows quantum correlations, or, equivalently, is not
separable. The state (18) exhibits quantum correlations in the sense of
Paskauskas and You unless either $\alpha$ or $\beta$ vanishes. It is
instructive to consider how the initially “polarized” (and separable) state
$|gg\rangle$ can be mapped to the state (18) using Landau-Zener ramps. Let us
for a moment neglect interactions and dissipation. The three-state system
Hamiltonian $\hat{H}_{0}+\hat{W}$ then reduces to a spin-1 Hamiltonian
involving only generators of SU$(2)$. Consequently, the associated evolution
operator $\exp(-\mathrm{i}\int_{0}^{t}\hat{H}(t^{\prime})dt^{\prime}/\hbar)$
is a rotation $\mathcal{R}$ transforming each spin$-1/2$ particle as
$|g\rangle\to|\mathcal{R}g\rangle$. The final two-particle state retains a
product form $|\mathcal{R}g\rangle\otimes|\mathcal{R}g\rangle$, and there are
no quantum correlations [55]. The same conclusion holds for an interacting
system with $U_{ee}+U_{gg}=2U_{eg}$, _i.e._ $q=0$ with our notations.

For non-symmetric interactions leading to $q\neq 0$, the evolution operator is
no longer a mere rotation and can create correlations. For instance, if
$U_{ee}\gg U_{eg},U_{gg}$, an adiabatic frequency ramp that stops well after
crossing the $gg-eg$ resonance but well before crossing any other would
generate a state of the form (18). Entanglement results in this situation from
an interaction blockade phenomenon.

It is remarkable that a _purely dissipative_ evolution can accomplish the same
thing. One may speak of “dissipative blockade” to highlight the analogy with
the unitary evolution described above. The ideal case would correspond to an
interaction-symmetric situation with $q=0$ but $\Gamma_{ee}\neq 0$, a
situation close to the experimental one where $\hbar\Gamma_{ee}\gg|q|$. In
that sense, one can consider the experiments reported here as a minimal
instance demonstrating dissipation-assisted engineering of non-trivial quantum
states, a topic of farther-reaching significance when applied to more complex
many-body systems [13, 12, 56, 57]. A related theoretical study on a related
system of lossy fermions predicts, for instance, the possibility to create
Dicke-like states [58]. An extension of the present work to a Hubbard-regime
quantum gas, with non-negligible tunneling between neighboring wells, would
bring further progress in this direction.

To conclude this discussion, let us stress that the final state is actually an
incoherent mixture of the two-particle entangled state (18) and of the vacuum
state (as in [58]). In the present work, we only perform ensemble
measurements, and therefore only probe statistical mixtures. The entanglement
could however be revealed in quantum gas microscope experiments [59], where
the spin-resolved population of each site is accessible, and where post-
selecting the doubly-occupied sites could make the entangled state
experimentally detectable.

###### Acknowledgements.

We thank Stefan Rotter and Leonardo Mazza for insightful discussions. LKB is a
member of the network “Science and Engineering for Quantum Technologies in the
Ile-de-France Region”.

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# Behaviour of hot electrons under the dc field in chiral carbon nanotubes

M. Amekpewu  K. A. Dompreh  S. Y. Mensah  N. G. Mensah  S. S. Abukari  R.
Musah  A. Twum  R. Edziah  Department of Applied Physics, University for
Development Studies, Navrongo Campus, Navrongo , Ghana  Department of Physics,
College of Agriculture and Natural Sciences, U.C.C, Ghana  Department of
Mathematics, College of Agriculture and Natural Sciences, U.C.C, Ghana

###### Abstract

Behaviour of hot electrons under the influence of dc field in carbon nanotubes
is theoretically considered. The study was done semi-classically by solving
Boltzmann transport equation with the presence of the hot electrons source to
derive the current densities. Plots of the normalized axial current density
versus electric field strength of the chiral CNTs reveal a negative
differential conductivity (NDC). Unlike achiral CNTs, the NDC occurs at a low
field about $\mathrm{6\ kV/cm}$ for chiral CNT. We further observed that the
switch from NDC to PDC occurs at lower dc field in chiral CNTs than achiral
counterparts. Hence the suppression of the unwanted domain instability usually
associated with NDC and a potential generation of terahertz radiations occurs
at low electric field for chiral CNTs.
Keywords: Hot electrons, Nanotubes, Negative Differential Conductivity

## Introduction

Carbon nanotubes (CNTs) which are unique tubular structures of nanometric
diameter with an extremely high length-to-diameter aspect ratio [1] have a
wide variety of possible applications [2, 3, 4]. The primary symmetry
classification of carbon nanotubes is as either being achiral (symmorphic) or
chiral (non-symmorphic) [5]. Each type of CNT can be either metallic or
semiconducting depending on their diameter and rolling helicity [6]. Much
progress has been made recently showing that carbon nanotubes (CNTs) are
advanced quasi-$1$D materials for future high performance electronics [4, 7,
13]. The behaviour of hot electrons in electronic devices has been observed
since the arrival of the transistor in $1947$ [14]. A number of devices have
been proposed whose very principle is based on effects of hot electrons [15].
In this paper, we present theoretical framework investigations of behaviour of
hot electrons under the influence of dc field in chiral carbon nanotubes using
the semiclassical Boltzmann transport equation. We probe the behaviour of the
electric current density mainly due to the presence of hot electrons in chiral
CNTs as a function of the applied dc field along the axis of the tube.

##  1 Theory

Suppose hot electrons is injected axially in a chiral carbon nanotube which is
considered as an infinitely long chains of carbon atoms wrapped along a base
helix under the influence of dc field $E$ . The current densities in axial and
circumferential directions are calculated by adopting semiclassical
approximation approach. In the presence of hot electrons source, the motion of
quasiparticles in dc field is described by Boltzmann transport equation in the
form [16, 17]:

$\frac{\partial f(p)}{\partial t}+v_{p}\frac{\partial f(p)}{\partial
r}eE(t)\frac{\partial f(p)}{\partial p}=-v[f(p)-f_{0}(p)]+S(p)$ (1)

where $v=1/\tau,v$ is the scattering frequency of the electrons, $\tau$ is
relaxation time of electrons, $S(p)$ is the hot electrons source function,
$f(p)$ is distribution function, $f_{0}(p)$ is equilibrium distribution
function, $\nu_{p}$ is the electron velocity, $r$ is the electron position,
$p$ is the electron dynamical momentum, $e$ is the electronic charge and t is
time elapsed. The energy $\varepsilon(p)$ of the electrons, calculated using
the tight binding approximation is given as expressed in [18] for a chiral
carbon nanotubes:

$\varepsilon(p)=\varepsilon_{0}-\triangle_{s}\cos(\frac{p_{s}d_{s}}{\hbar})-\triangle_{z}\cos(\frac{p_{z}d_{z}}{\hbar})$
(2)

where $\varepsilon_{0}$ is the energy of an outer-shell electron in an
isolated carbon atom, $\triangle_{z}$ and $\triangle_{s}$ are the real
overlapping integrals for jumps along the tubular axis and the base helix
respectively, $p_{s}$ and $p_{z}$ are the components of carrier momentum
tangential to the base helix and along the tubular axis respectively.
$\hbar=h/2\pi$ and $h$ is Planck’s constant. $d_{s}$ and $d_{z}$ are the
distances between the atomic sites $n1$ and $n1+1$, $n_{s}$ and $n_{s}+N_{0}$
respectively along the base helix and the tubular axis where, $(N_{0}>1)$. The
components $v_{s}$ and $v_{z}$ of the electron velocity $v_{p}$ in equation
($1$) are respectively calculated from the energy dispersion relation equation
($2$) as

$v_{s}(p_{s})={\partial\varepsilon(p)\over\partial
p_{s}}={\triangle_{s}d_{s}\over\hbar}\sin({p_{s}d_{s}\over\hbar})$ (3)
$v_{z}(p_{z})={\partial\varepsilon(p)\over\partial
p_{s}}={\triangle_{z}d_{z}\over\hbar}\sin({p_{z}d_{z}\over\hbar})$ (4)

In accordance with [18, 19, 20, 21], we find that the distribution function is
periodic in the quasi-momentum and can be written in Fourier series as:

$\displaystyle f_{0}(p_{s},p_{z})={n_{0}d_{s}d_{z}\over
2I_{0}({\triangle_{s}\over k_{B}T})I_{0}({\triangle_{z}\over
k_{B}T})}\sum^{\infty}_{\alpha=-\infty}I_{\alpha}({\triangle_{s}\over
k_{B}T})\sum^{\infty}_{\beta=-\infty}I_{\beta}({\triangle_{z}\over
k_{B}T})\times$ $\displaystyle\exp\\{i\alpha p_{s}d_{s}/\hbar+\beta
p_{zd_{z}}/\hbar\\}$ (5) $\displaystyle
f_{0}(p_{s},p_{z},t)={n_{0}d_{s}d_{z}\over 2I_{0}({\triangle_{s}\over
k_{B}T})I_{0}({\triangle_{z}\over
k_{B}T})}\sum^{\infty}_{\alpha=-\infty}I_{\alpha}({\triangle_{s}\over
k_{B}T})\sum^{\infty}_{\beta=-\infty}I_{\beta}({\triangle_{z}\over
k_{B}T})\times$ $\displaystyle\exp\\{i\alpha p_{s}d_{s}/\hbar+\beta
p_{zd_{z}}/\hbar\\}\emptyset_{\alpha}(t)$ (6)

where $f(p_{s},p_{z},t)$ is the distribution function, and
$f_{0}(p_{s},p_{z})$ is the equilibrium distribution function,
$\emptyset_{\alpha}(t)$ is the factor by which the Fourier transform of the
non-equilibrium distribution function differs from its equilibrium
distribution counterpart, $I_{\alpha,\beta}$ is the modified Bessel function
of order the ${\alpha,\beta}$, where $\alpha,\beta=0,1,...$, $n_{0}$ is
equilibrium particle density and $k_{B}$ is Boltzmann constant. We consider a
hot electron source $S(p)$ of the simplest form given by the expression [17]

$S(p)=Q\delta(p-p\prime)-{Q\over n_{0}}f_{s}(p)$ (7)

where $f_{s}(p)$ is the stationary solution of Eqn.($1$), $Q$ is the injection
rate of hot electrons and $p\prime$ and $p$ are their momentum. In the case of
constant electric field, the solution to Eqn. ($1$) becomes

$\displaystyle\emptyset_{\alpha}(p_{s},p_{z})={n_{0}d_{s}d_{z}\over
2I_{0}({\triangle_{s}\over k_{B}T})I_{0}({\triangle_{z}\over
k_{B}T})}\sum^{\infty}_{\alpha=-\infty}I_{\alpha}({\triangle_{s}\over
k_{B}T})\sum^{\infty}_{\beta=-\infty}I_{\beta}({\triangle_{z}\over
k_{B}T})\times$ $\displaystyle{v\over(v+ie\alpha dE)}\exp\\{i\alpha
p_{s}d_{s}/\hbar+\beta p_{zd_{z}}/\hbar\\}$ (8)

Therefore Eqn.($6$) becomes

$\displaystyle f_{E}(p_{s},p_{z})={n_{0}d_{s}d_{z}\over
2I_{0}({\triangle_{s}\over k_{B}T})I_{0}({\triangle_{z}\over
k_{B}T})}\sum^{\infty}_{\alpha=-\infty}I_{\alpha}({\triangle_{s}\over
k_{B}T})\sum^{\infty}_{\beta=-\infty}I_{\beta}({\triangle_{z}\over
k_{B}T}){v\over(v+ie\alpha dE)}$ $\displaystyle\times\exp\\{i\alpha
p_{sd_{s}}/\hbar+\beta p_{zd_{z}}/\hbar\\}$ (9)

The stationary homogeneous distribution function $f_{s}(p)$ in the presence of
hot electron source Eqn.($7$) is given by

$f_{s}(p)=f_{E}(p)+f^{\prime}(p)$ (10)

Substituting Eqn.($10$) into Eqn.($1$) we get

${\partial f^{\prime}(\varphi)\over\partial\varphi}+({v\over\Omega}+{Q\over
n_{0}\Omega})f^{\prime}(\varphi)={d_{s}Q\over\Omega}\delta(\varphi-\varphi^{\prime})-{Q\over\Omega
n_{0}}f(\varphi)$ (11)

$\Omega=\Omega_{s,z}={ed_{s,z}E\over\hbar}$ and $d_{s,z}$ is bandwidth,
$\varphi$ and $\varphi\prime$ are the dimensionless momenta of electrons and
hot electrons respectively which are expressed as
$\varphi=\varphi_{s,z}=d_{s,z}p/\hbar$ and
$\varphi^{\prime}=\varphi^{\prime}_{s,z}=d_{s,z}p^{\prime}/\hbar$ for chiral
CNTs. Solving the homogeneous differential equation corresponding to equation
($11$), we obtain

$f^{\prime}(\varphi)=C(\varphi)\exp\\{-[{v\over\Omega}+{Q\over
n_{0}\Omega}]\varphi\\}$ (12)

Then by differentiating Eqn.($12$), we have

$\displaystyle{\partial f^{\prime}(\varphi)\over\partial\varphi}=[{\partial
C(\varphi)\over\partial\varphi}-({v\over\Omega}+{Q\over
n_{0}\Omega})C(\varphi)]\times$ $\displaystyle\exp\\{-[{v\over\Omega}+{Q\over
n_{0}\Omega}]\varphi\\}$ (13)

Substituting for $f^{\prime}(\varphi)$ and ${\partial
f^{\prime}(\varphi)\over\partial\varphi}$ in Eqn.($11$), we obtain

${\partial C(\varphi)\over\partial\varphi}\exp\\{-[{v\over\Omega}+{Q\over
n_{0}\Omega}]\varphi\\}={d_{s}Q\over\Omega}\delta(\varphi-\varphi^{\prime})-{Q\over\Omega
n_{0}}f_{E}(\varphi)$ $\Rightarrow
C(\varphi)=\int\\{{d_{s}p\over\Omega}\delta(\varphi-\varphi^{\prime})-{Q\over\Omega
n_{0}}f_{E}(\varphi)\\}\exp\\{[{v\over\Omega}+{Q\over
n_{0}\Omega}]\varphi\\}d\varphi$

Introducing Dirac-delta transformation

$\delta(\varphi-\varphi^{\prime})={1\over
2\pi}\sum_{\alpha,\beta}\exp\\{i(\alpha\varphi-\beta\varphi^{\prime})\\}$
$C(\varphi)={Q\over 2\pi\Omega}.{n_{0}\over
n_{0}}\int\\{\sum_{\alpha,\beta}\exp\\{i(\alpha\varphi-\beta\varphi^{\prime})\\}-{Q\over\Omega
n_{0}}f(\varphi)\\}\exp\\{[{v\over\Omega}+{Q\over\Omega
n_{0}}]\varphi\\}d\varphi$ (14)

Recall that

$\displaystyle f(\varphi)={n_{0}d_{s}d_{z}\over 2I_{0}({\triangle_{s}\over
k_{B}T})I_{0}({\triangle_{z}\over
k_{B}T})}\sum^{\infty}_{\alpha=-\infty}I_{\alpha}({\triangle_{s}\over
k_{B}T})\sum^{\infty}_{\beta=-\infty}I_{\beta}({\triangle_{z}\over
k_{B}T}){v\over(v+i\exp\\{\beta d_{s,z}E_{s,z})\\}}$
$\displaystyle\times\exp\\{i\alpha d_{s}d_{s}/\hbar+i\beta
p_{z}d_{z}/\hbar\\}$ (15) $\displaystyle C(\varphi)={Q\over
2\pi\Omega}.{n_{0}\over n_{0}}\sum_{\alpha}{\Omega
d_{s}d_{z}\over(i\alpha\Omega+v+{Q\over
n_{0}})}\\{\exp\\{-i\alpha\varphi\\}-{I_{\alpha,\beta}({\triangle_{s}\over
kT},{\triangle_{z}\over kT})\over I_{0}({\triangle_{z}\over
k_{B}T})I_{0}({\triangle_{s}\over k_{B}T})({v\over v+i\alpha\Omega})}\\}$
$\displaystyle\times\exp\\{[{v\over\Omega}+{Q\over
n_{0}\Omega}+i\alpha]\varphi\\}$ (16)

Substituting Eqn.($14$) into Eqn. ($12$) we get

$\displaystyle f^{\prime}(\varphi)={\eta{n_{0}d_{s}d_{z}}\over
2\pi\hbar}\sum_{\alpha}{\Omega\exp\\{i\alpha\varphi_{s}\\}\over(i\alpha\Omega+v+\eta\Omega)}\\{\exp\\{-i\alpha\varphi^{\prime}_{s}\\}-{I_{\alpha,\beta}({\triangle_{s}\over
kT},{\triangle_{z}\over kT})\over I_{0}({\triangle_{z}\over
kT})I_{0}({\triangle_{s}\over kT})}{v\over(v+\alpha l\Omega)}\\}$ (17)

Where the nonequilibrium parameter $\eta={Q\over\Omega n_{0}}$ and
$\eta\Omega={Q\Omega\over\Omega n_{0}}$, we obtain the general expression for
the current density along the tubular axis (z-axis) and the base helix (s) as

$\displaystyle j_{z,s}={-\eta n_{0}\triangle_{s,z}d^{2}_{s,z}\over
I_{0}({\triangle_{s,z}\over
kT})}\sum_{\alpha}{\Omega\over(i\alpha\Omega+v+\eta\Omega)}\\{\exp\\{-i\alpha\varphi^{\prime}_{s}\\}-{I_{\alpha,\beta}({\triangle_{s}\over
kT},{\triangle_{z}\over kT})\over I_{0}({\triangle_{z}\over
k_{B}T})I_{0}({\triangle_{s}\over k_{B}T})}{v\over(v+i\alpha\Omega)}\\}$ (18)
$\displaystyle j_{z}={-\eta n_{0}\triangle_{z}d^{2}_{z}\over
I_{0}({\triangle_{z}\over
k_{B}T})}\sum_{\beta}{\Omega\over(i\beta\Omega+v+\eta\Omega)}\\{\exp\\{-i\beta\varphi^{\prime}_{s}\\}-{I_{\beta}({\triangle_{z}\over
kT})\over I_{0}({\triangle_{z}\over k_{B}T})}{v\over(v+i\alpha\Omega)}\\}$
(19) $\displaystyle j_{s}=-{\eta n_{0}\triangle_{s}d^{2}_{s}\over
I_{0}({\triangle_{s}\over
k_{B}T})}\sum_{\alpha}{\Omega\over(i\alpha\Omega+v+\eta\Omega)}\\{\exp{-i\alpha\varphi^{\prime}_{s}}-{I_{\alpha}({\triangle_{s}\over
kT})\over I_{0}({\triangle_{s}\over k_{B}T})}{v\over(v+i\alpha\Omega)}\\}$
(20)

The axial $j_{a}$ and circumferential $j_{c}$ components of the current
density are given by [18, 22]

$\displaystyle j_{a}=j_{z}+j_{s}\sin\theta_{h}$ (21) $\displaystyle
j_{c}=j_{s}\cos\theta_{h}$ (22)

where $\theta_{h}$ is geometric chiral angle. Substituting Eqn.($17$) and
($18$) into Eqn.($19$) and ($20$) yields

$\displaystyle j_{a}=-{\eta n_{0}\triangle_{z}d^{2}_{z}\over
I_{0}({\triangle_{z}\over
k_{B}T})}\sum_{\alpha}{\Omega\over(i\alpha\Omega+v+\eta\Omega)}\\{\exp\\{-i\alpha\varphi_{s}\prime\\}-{I_{\alpha}({\triangle_{z}\over
kT})\over I_{0}({\triangle_{z}\over kT})}{v\over(v+i\alpha\Omega)}\\}-{\eta
n_{0}\triangle_{s}d^{2}_{s}\over I_{0}({\triangle_{s}\over k_{B}T})}$
$\displaystyle\sum_{r}{\Omega\over(ir\Omega+v+\eta\Omega)}\\{\exp\\{-i\alpha\varphi_{s}\prime\\}-{I_{M}({\triangle_{s}\over
kT})\over I_{0}({\triangle_{s}\over
k_{T}})}{v\over(v+i\alpha\Omega)}\\}\sin\theta_{h}$ (23)

and

$\displaystyle j_{c}=-{\eta n_{0}\triangle_{s}d^{2}_{s}\over
I_{0}({\triangle_{s}\over
k_{B}T})}\sum_{r}{\Omega\over(i\alpha\Omega+v+\eta\Omega)}\\{\exp\\{-i\alpha\varphi_{s}\prime\\}-{I_{\alpha}({\triangle_{s}\over
kT})\over I_{0}({\triangle_{s}\over
kT})}{v\over(v+i\alpha\Omega)}\\}\cos\theta_{h}$ (24)

But from [23],

$\displaystyle\sin\theta_{h}={\sqrt{3}m\over 2\sqrt{n^{2}+m^{2}+nm}}$ (25)

and

$\displaystyle\cos\theta_{h}={2n+m\over 2\sqrt{n^{2}+m^{2}+nm}}$ (26)

where $n$ and $m$ are integers which denote the number of unit vectors along
two directions in the hexagonal lattice of graphene. From Eqn. ($21$), ($22$),
($23$) and ($24$), we finally obtained

$\displaystyle j_{a}=-{\eta n_{0}\triangle_{z}d^{2}_{z}\over
I_{0}({\triangle_{z}\over
k_{B}T})}\sum_{\alpha}{\Omega\over(i\alpha\Omega+v+\eta\Omega)}\\{\exp\\{-i\alpha\varphi^{\prime}_{s}\\}-{I_{\alpha}({\triangle_{z}\over
kT})\over I_{0}({\triangle_{z}\over kT})}{v\over(v+i\alpha\Omega)}\\}$
$\displaystyle-{\eta n_{0}\triangle_{s}d^{2}_{s}\over
I_{0}({\triangle_{s}\over
k_{B}T})}\sum_{\alpha}{\Omega\over(i\alpha\Omega+v+\eta\Omega)}\\{\exp\\{-i\alpha\varphi^{\prime}_{s}\\}-$
$\displaystyle{I_{\beta}({\triangle_{s}\over kT})\over
I_{0}({\triangle_{s}\over k_{T}})}{v\over(v+i\alpha\Omega)}\\}{\sqrt{3}m\over
2\sqrt{n^{2}+m^{2}+nm}}$ (27) $\displaystyle j_{c}=-{\eta
n_{0}\triangle_{s}d^{2}_{s}\over I_{0}({\triangle_{s}\over
k_{B}T})}\sum_{\alpha}{\Omega\over(i\alpha\Omega+v+\eta\Omega)}\left\\{\exp\\{-i\alpha\varphi^{\prime}_{s}\\}-{I_{\beta}({\triangle_{s}\over
kT})\over I_{0}({\triangle_{s}\over kT})}{v\over(v+i\alpha\Omega)}\right\\}$
$\displaystyle\times{2n+m\over 2\sqrt{n^{2}+m^{2}+nm}}$ (28)

##  2 Results and Discussion

In figure $1$, we display the behaviour of the normalized axial as well as
circumferential current density as a function of the electric field for ($4$,
$1$) chiral CNT injected axially with the hot electrons, represented by the
none-quilibrium parameter $\eta=1.7\times 10^{-8}$

Figure 1: A plot of normalized current density versus electric field for a
($4,1$) chiral CNT, $\eta=1.7\times 10^{-8}$, $v=1\mathrm{THz}$ and $T=287.5\
\mathrm{K}$

The normalized axial current density of ($4$, $1$) chiral CNT exhibits a
linear monotonic dependence on the applied electric field at lower field (i.e.
the region of ohmic conductivity). As the electric field increases, the axial
current density increases and reaches a maximum at about $3.5\
\mathrm{kV/cm}$, and drops off, experiencing a negative differential
conductivity (NDC) up to about $25\ \mathrm{kV/cm}$ as shown in figure 1. The
NDC is due to the increase in the collision rate of the energetic electrons
with the lattice that induces large amplitude oscillation in the lattice,
which in-turn increases the scattering rate that leads to the decrease in the
current [24]. When the electric field exceeds about $25\ \mathrm{kV/cm}$, we
once again observe an increase in normalized axial current density of chiral
carbon nanotube. Hence there is switch from NDC to PDC when electric field
value of about $25\ \mathrm{kV/cm}$ is exceeded. This phenomena also occurs
when there is an impact ionization. This has been studied in superlattices
[28] The physical mechanism behind the switch from NDC to PDC has been
explained in [24, 25] The normalized circumferential current density of ($4$,
$1$) chiral CNT exhibits a linear monotonic dependence on the applied electric
field as up to $60\ \mathrm{kV/cm}$ as shown in figure $1$. Especially from
$0$ to about $10\ \mathrm{kV/cm}$, conductivity is mainly axially and hence
circumferential conductivity is negligible or can be ignored. This is due to
an extremely high length-to-diameter aspect ratio of CNTs up to $132000000:1$
which is significantly larger than any other material [26]. Hence CNTs are
quasi-$1$D carbon materials. In figure $2$, we only display the behaviour of
the normalized axial current density as a function of the electric field for
($4$, $1$) chiral CNT stimulated axially with the hot electrons, represented
by the non-equilibrium parameter $\eta$. In figure $2$, there is a switch from
NDC to PDC in a ($4$, $1$) chiral CNT when $\eta=0.9\times 10^{-8}$ near $35\
\mathrm{kV/cm}$. As we increase the rate of hot electrons injection
represented by non-equilibrium parameter $\eta$ from $0.9\times 10^{-8}$ to
$3.0\times 10^{-8}$ chiral CNT, the behviour of hot electrons change leading
to an increase in differential conductivity $|{\partial J\over\partial E}|$ as
well as the peak normalized axial current density as shown in figure $2$

Figure 2: Aplot for Normalized axial current density versus Electric field
$\mathrm{(kV/cm)}$ as non-equilibrium parameter $\eta$ increases,
$\mathrm{v=1THz}$ and $\mathrm{T=287.5K}$

To put the above observations in perspective, we display in figure $3$ a
3-dimensional behaviour of the normalized axial current density $(Ja)$ as a
function of the electric field $(E)$ and non-equilibrium parameter $(\eta)$.
The differential conductivity and the peak of the current density are at the
lowest values when the non-equilibruim parameter $\eta=0$. As non-equilibrium
parameter $\eta$ increases from 0 to $30\times 10^{-9}$, both differential
conductivity and peak current density increase. Therefore, the behaviour of
hot electrons under the influence of dc field in chiral carbon nanotube leads
to a switch from NDC to PDC, differential conductivity and the peak normalized
current density increase.

Figure 3:  A 3D plot of normalized axial current density $(Ja)$ versus
electric field $(E)$ and non-equilibrium parameter $(\eta)$

In figure $4$, we display the behaviour of the normalized axial current
density as a function of the electric field for the achiral CNTs stimulated
axially with the hot electrons, represented by the non-equilibrium parameter
$\eta=1.7\times 10^{-8}$ as in [24]

Figure 4: A plot of normalized current density versus electric field for (a)
($3$, $0$) zigzag CNT and (b) ($3$, $3$) armchair both stimulated axially with
the hot electrons, represented by the nonequilibrium parameter $\eta=1.7\times
10^{-8}$ as in [24] Figure 5: A plot of the normalized axial current density
versus electric field for ($3$, $0$) zigzag CNT, ($3$, $3$) CNT [24] and
($4$,$1$) CNT stimulated axially with the hot electrons, represented by the
non-equilibrium parameter $\eta=1.7\times 10^{-8}$

We generally observed in figure $5$ that with axial injection of hot electrons
in ($4$, $1$) chiral CNT, ($3$, $0$) zigzag CNT and ($3$, $3$) armchair CNT
under the influence of dc field, there is the desirable effect of a switch
from NDC to PDC characteristics. Thus, the most important tough problem for
NDC region which is the space charge instabilities can be suppressed due to
the switch from the NDC behaviour to the PDC behaviour predicting a potential
generation of terahertz radiations [25] which have enormous promising
applications in very different areas of science and technology [27]. However,
there are some differences such as differential conductivity, peak normalized
current density, critical dc field beyond which NDC characteristics occurs and
finally the value of dc field above which there is a desirable switch of NDC
to PDC. First and foremost, the differential conductivity $|{\partial
J\over\partial E}|$ of ($4$, $1$) chiral CNT is the highest, ($3$, $3$)
armchair CNT is the least while that of ($3$, $0$) zigzag CNT is between the
two. Also in figures $4$ and $5$, the peak normalized current density for
($4$, $1$) chiral CNT, ($3$, $3$) zigzag CNT and ($3$, $3$) CNT are about
$14.20$, $2.70$ and $0.46$ respectively. Hence ($4$, $1$) chiral CNT is having
the highest peak normalized current density follow by ($3$, $0$) zigzag CNT
whereas ($3$, $3$) CNT is the least. Furthermore, the critical dc fields that
should be exceeded before NDC are observed in ($4$, $1$) chiral CNT, ($3$,
$0$) zigzag CNT and ($3$, $3$) armchair CNT are about $6\ \mathrm{kV/cm}$,
$12\ \mathrm{kV/cm}$ and $20\ \mathrm{kV/cm}$ respectively in figures $4$ and
$5$. Finally, if the rate of axial injection of hot electrons represented by
non-equilibrium parameter $\eta=1.7\times 10^{-8}$, there is a switch from NDC
to PDC near $35\ \mathrm{kV/cm}$, $50\ \mathrm{kV/cm}$, $75\ \mathrm{kV/cm}$
for ($4$, $1$) chiral CNT, ($3$, $0$) zigzag CNT and ($3$, $3$) armchair CNT
respectively.

##  3 Conclusion

In summary, behaviour of hot electrons under the influence of dc field in
chiral carbon nanotubes have been demonstrated theoretically by adopting
semiclassically approximation approach in solving Boltzmann transport
equation. It leads to changes in the nature of the dc differential
conductivity by switching from an NDC characteristics to PDC characteristics
due to the hot electrons injection. Thus, domain instability can be
suppressed, suggesting a potential generation of terahertz radiations with
enormous promising applications in very different areas of technology,
industry and research. Unlike achiral CNTs, a potential generation of
terahertz radiations in chiral counterpart can take place at lower dc field
which stems from our research findings that the switch from NDC to PDC occurs
at relatively low dc field.

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