# Emergence and Dynamical Stability of Charge Time-Crystal in a Current-
Carrying Quantum Dot Simulator

Subhajit Sarkar<EMAIL_ADDRESS>Department of Chemistry, Ben-Gurion
University of the Negev, Beer Sheva 84105, Israel,  School of Electrical and
Computer Engineering, Ben-Gurion University of the Negev, Beer Sheva 84105,
Israel.  Yonatan Dubi<EMAIL_ADDRESS>Department of Chemistry, Ben-Gurion
University of the Negev, Beer Sheva 84105, Israel,  Ilse Katz Center for
Nanoscale Science and Technology, Ben-Gurion University of the Negev, Beer
Sheva 84105, Israel.

###### Abstract

Periodically-driven open quantum systems that never thermalize exhibit a
discrete time-crystal behavior, a non-equilibrium quantum phenomena that has
shown promise in quantum information processing applications. Measurements of
time-crystallinity are currently limited to (magneto-) optical experiments in
atom-cavity systems and spin-systems making it an indirect measurement. We
theoretically show that time-crystallinity can be measured directly in the
charge-current from a spin-less Hubbard ladder, which can be simulated on
quantum-dot array. We demonstrate that one can dynamically tune the system out
and then back into the time-crystal phase, proving its robustness against
external forcings. These findings motivate further theoretical and
experimental efforts to simulate the time-crystal phenomena in current-
carrying nano-scale systems.

###### keywords:

Quantum Dots Arrays, Discrete Time-Crystals, Quantum Simulation, Dissipative
Quantum Systems, Dynamical Stability

Schrödinger’s equation, which governs quantum physics, is time-translation
invariant. Despite that, time-translation symmetry breaking has been shown to
be possible in quantum systems under specific circumstances, leading to the
so-called discrete time-crystal (DTC) behavior in numerous systems including,
atom-cavity systems, spin-ensembles, solid-state quantum-dot (QD) arrays, and
quantum processors 1, 2, 3, 4, 5, 6, 2, 7, 8, 9, 10, 11, 12, 13. Periodically
driven (Floquet) closed quantum systems that never reach a thermodynamic
equilibrium can indeed exhibit a DTC behavior 14, 15, 16, 17, 18, 19, 1, 2,
12, 13, manifested in the sub-harmonic oscillations of the order parameter.
They typically rely on disorder and localization to avoid reaching a
stationary state of infinite temperature that opposes time-crystalline order
20, 21, 15, 22. Alternatively, dissipation and fluctuations induced by
controlled coupling to suitable environments stabilize DTC in systems without
disorder and localization 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 31, 32, 33, which was confirmed in the recent observation of
dissipative DTC in the atom-cavity system 38. Observations of DTCs in both
closed and open Floquet quantum systems have been limited to optical detection
via spin-dependent fluorescence from spin-full systems, e.g., non-interacting
spin ensembles, spin-Boson, Ising and XXZ systems 1, 2, 3, 13, 1, 2, 3, 4, 5,
6, 10, 11, 38, 12. To this end, the atom-cavity systems and optical lattice
set-ups have been dominant platforms for achieving DTC due to their high
degree of tunability in terms of dissipation engineering.

Although DTC phenomena is interesting in its own right, applications of DTCs
are only beginning to emerge. Advanced fabrication and control of coupled
quantum dot (QD) arrays 39, 40, 41, 42, 43 led to the theoretical proposal of
observing 9 and subsequent experimental realization of a DTC in nano-scale
system, attempting stable quantum information processing using DTC as a
mechanism 10, 11, 44. Gate defined QD array, a.k.a, Quantum-Dot simulator has
been used as a platform to investigate and realize multitude of phenomena that
include Fermi-Hubbard model and Mott transition 45, Heisenberg models 43, 10,
46 and phonon-induced pairing 47. However, it seems that the possibility of
detecting DTC behavior with charge transport measurements has not been
explored. This is somewhat surprising, since charge transport measurements,
e.g., through quantum dot (QD) arrays, allowed for a large range of
discoveries in quantum physics, from Coulomb blockade to Kondo physics,
Superconductivity, quantum-point-contact universal conductance, and quantum
computing 48, 49, 50, 51, 52, 53. A possible reason for that is that it is
a-priori unknown if DTCs survive or die when there is a charge flow through
the system, and previous studies of DTCs (in spin-systems) suggest that fast
decay will overshadow any signature of the DTC 54, 29.

Here we investigate if a dissipative-DTC can be found in a current-carrying
spin-less system and how it can be observed in a transport experiment. In a
spin-less system, the charge density is the primary observable. Therefore, it
suits best to investigate charge transport in a dissipative-DTC state. We
suggest a route for directly measuring dissipative-DTC behavior in transport
measurements in a periodically-driven two-leg spin-less Hubbard ladder, which
a quantum dot array can simulate 47, 55, 56, 57. Specifically, we discuss
system parameters that are relevant to QD arrays 47, 55, 56, where periodic-
driving can be achieved in an electrically tunable set-up 39, 40, 41, 42, 43.
The main advantage of Quantum-Dot simulators over the optical-lattice ones is
the relative ease of transport measurements in the former.

Our main results are as follows: (i) We theoretically show that a stable and a
meta-stable dissipative DTC 111We use the term discrete time crystal also to
describe discrete-time quasi-crystals 58, where the ratio between the driving
and response frequencies in an irrational number. can be observed via
measurement of the charge-current, and identify the conditions necessary for
this to happen. We show that The DTC manifests itself in a sharp peak of the
discrete Fourier transform (DFT) of the charge-current, which is locked at the
DTC sub-harmonic frequency, even in the meta-stable regime where the
oscillation develops a decaying envelope. To this end we clarify that the
notion of meta-stability we use pertains to the finite lifetime of the DTC.
(ii) We show that a reversible transition from a DTC to a normal phase can be
driven by dynamically tuning the system parameters. Since the system is
current-carrying (and hence is an open quantum system), this dynamical
transition is distinct from the existing notion of dynamical transition
between DTC phases proposed in closed (dissipation-less) quantum systems 59,
60.

Our theory reveals the fundamental origin of the DTC behavior in a spin-less
nanoscale system. We show that the two-leg spin-less Hubbard ladder, that can
be emulated through a QD array 57, exhibits a DTC behavior because of the
hidden SU(2) symmetry, arising from its ladder structure. Under proper
engineering of dissipation the hidden SU(2) symmetry ensures the emergence of
a weak-local Floquet-Dynamical symmetry (FDS) 61 in the system. Indeed, as
detailed in Ref. 24, dissipation (in the form of on-site dephasing) is
necessary for inducing the DTC behavior, because it allows for coupling of
different charge sectors of the Floquet Hamiltonian which, in absence of
dephasing, would otherwise remain decoupled. This weak-local FDS provides both
the mechanism for the existence and the necessary decay for the measurement of
DTC behavior 62. The mechanism of weak-local FDS in our model is fundamentally
different from the existing notion of FDS in spin-full systems which is based
on an effective Zeeman effect 25, 24, 37, 31, 32, 33. Understanding the
underlying mathematical structure would motivate further search of dissipative
DTC in various other nanoscale systems 63, 64, 65.

We consider a periodically driven ladder of spin-less Fermions, see Fig. 1
(schematic set-up is motivated by Ref. 66), comprising of $2N$ inter-connected
sites with a Hamiltonian,
$\mathcal{H}(t)=\mathcal{H}_{0}+\mathcal{H}_{hop}^{||}+\mathcal{H}_{int}+\mathcal{H}_{hop}^{\perp}(t)$
where 25, 45

$\displaystyle\mathcal{H}_{0}$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}j=1,\\\
\alpha=a,b\end{subarray}}^{N-1}\epsilon_{\alpha}n_{j,\alpha},~{}~{}\mathcal{H}_{hop}^{||}=-t_{||}^{hop}\sum_{\begin{subarray}{c}j=1,\\\
\alpha=a,b\end{subarray}}^{N-1}(c_{j,\alpha}^{\dagger}c_{j+1,\alpha}+c_{j+1,\alpha}^{\dagger}c_{j,\alpha})$
$\displaystyle\mathcal{H}_{int}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}\sum_{\alpha=a,b}\frac{U}{2}n_{j,\alpha}n_{j,\alpha}+\sum_{j=1}^{N-1}\sum_{\alpha=a,b}\frac{K}{2}n_{j,\alpha}n_{j+1,\alpha}+\sum_{j=1}^{N-1}\frac{K^{\prime}}{2}(n_{j,a}n_{j+1,b}+n_{j+1,a}n_{j,b})$
$\displaystyle+$ $\displaystyle\sum_{j=1}^{N}U^{\prime}n_{j,a}n_{j,b},$
$\displaystyle\mathcal{H}_{hop}^{\perp}(t)$ $\displaystyle=$
$\displaystyle-\sum_{j=1}^{N}\left[t_{\perp}^{hop}(t)c_{j,b}^{\dagger}c_{j,a}+[t_{\perp}^{hop}(t)]^{*}c_{j,a}^{\dagger}c_{j,b}\right],$
(1)

where $\epsilon_{\alpha}$ is the onsite energy (which is uniform along the
chains but different in each chain), $n_{j,\alpha}$ is the charge density at
each site, and $t_{||}^{hop},~{}U,~{}K$ are the nearest-neighbor hopping,
onsite electron-electron, and the nearest-neighbor electron-electron
interaction strengths at each chain, respectively, $U^{\prime}$ and
$K^{\prime}$ are inter-chain nearest-neighbor and inter-chain next nearest-
neighbor interactions, respectively;
$t_{\perp}^{hop}(t)=t_{\perp}^{hop}e^{-i\omega t}$ is the oscillatory inter-
chain tunneling with $\omega=\frac{2\pi}{T}$, leading to the time-periodicity
of the Hamiltonian, $\mathcal{H}(t)=\mathcal{H}(t+T)$. The oscillatory inter-
chain tunneling couples to the charge of the particles only, and will lead to
the notion of weak local charge-FDS if and only if $U^{\prime}=U$ and
$K^{\prime}=K$. It is in this limit the undriven Hamiltonian (i.e., time
independent $\mathcal{H}_{hop}^{\perp}$) exhibits an SU(2) symmetry due to the
ladder structure, see Supporting Information (SI) Sec. S1 for a proof. Such a
Hamiltonian is directly realizable in a DQ-array setup 45, 42, 41, and also in
optical lattice set-ups 67. A similar set-up has been realized for 4 coupled
QDs 57.

Figure 1: (a) Schematic representation of the two-leg ladder Hamiltonian with
legs $\alpha=a$ and $b$. $t_{||}^{hop}$ is the nearest neighbor hopping
strength in each leg, $K$ is the nearest neighbor interaction, $U$ is the
onsite interaction for each leg; $U^{\prime}$ is the inter-chain nearest-
neighbor interaction, and $K^{\prime}$ is the next nearest-neighbor
interaction. $t_{\perp}^{hop}(t)$ is the time-dependent inter-ladder hopping
which is periodically driven. (b) Schematic picture of a possible
(experimental) set-up of an array of 10 gate controlled QDs: (green)
$\epsilon_{j,\alpha}$’s are the gates that control the onsite energies.
$B_{j}$’s are the tunnel barrier that provide a control on $t_{||}^{hop}$,
$T_{j}$’s control the tunnel barrier which are periodically driven with an AC
of frequency $\omega$ that provide $t_{\perp}^{hop}(t)$. Yellow channels
represent suitably placed quantum-point contacts that provide a possible
dephasing mechanism 68. $\sqrt{\Gamma}$ is the strength of the dephasing,
which is further indicated by a rounded box in (a). $L$ and $R$ represent the
left and right electrodes, respectively where suitable bias can be added to
drive electric current through the system.

In order to evaluate currents and other dynamical observable, we study the
time evolution of the density matrix of the system under Floquet-Lindblad
quantum-master equation 69, 70, 71, 72, 73,

$\displaystyle\frac{d\rho}{dt}=\mathcal{L}_{t}[\rho]=-i[\mathcal{H}(t),\rho]+\sum_{\mu}\left(V_{\mu}\rho
V_{\mu}^{\dagger}-\frac{1}{2}\\{V_{\mu}^{\dagger}V_{\mu},\rho\\}\right),$ (2)

($\hbar=1$) with a periodic Liouvillian $\mathcal{L}_{t}=\mathcal{L}_{t+T}$
guaranteed by the periodic Hamiltonian $\mathcal{H}(t)=\mathcal{H}(t+T)$;
index $\mu$ representing a summation over all Lindblad operators. Periodic
drive can be implemented via time dependent control of tunnel barriers
$T_{j}$’s, see Fig. 1(b) 41. Charge flow is modeled by setting Lindbald
operators $V_{in}=\sqrt{\gamma_{L}}c^{\dagger}_{1,b}$ and
$V_{out}=\sqrt{\gamma_{R}}c_{N,a}$ which push charges into and out of the
ladder, respectively (see Fig. 1). Dephasing is introduced via Zeno-type
dephasing operators
$V_{j}=\sqrt{\Gamma}(c^{\dagger}_{j,a}c_{j,a}+c^{\dagger}_{j,b}c_{j,b})$,
which act simultaneously on the $(j,a)-(j,b)$ ladder pairs (see SI Sec. S1-A
for a detailed discussion). In a QD array, for example, such a dephasing can
be induced via suitable placement of quantum-point contacts, 74, 75, 76, 77,
68 e.g., in a parallel 2DEG layer below the QD simulator 78, 79, 80. If the
dephasing is applied on each QD separately then it does not satisfy the
dynamical symmetry criteria (see SI Sec. S1-A) and would destroy the DTC.

We aim to show that in a two-leg ladder system DTC can be measured via charge-
current. We start by showing that a DTC state indeed exists. In SI Fig. S1, we
plot the eigenvalues of the Floquet map (the one period time evolution super-
operator),
$\hat{\mathcal{U}}_{F}=\hat{\mathcal{U}}(T)=\mathcal{T}\left[\exp\left(\int_{0}^{T}\hat{\mathcal{L}}(t^{\prime})~{}dt^{\prime}\right)\right]$
where $\mathcal{T}$ denotes time-ordering 69, for a 6-dot system ($N=3$). The
eigenvalues that lie on the complex unit circle confirms the existence of a
DTC oscillation irrespective of any initial condition 23, 61, 28, 30. This
means that any initial random density matrix would bring in the DTC
oscillation in the long time limit. Moreover, there can be many DTC states
corresponding to the same DTC eigenvalue, and depending on the initial density
matrix the system reaches a particular DTC state 61. We choose
$t_{||}^{hop}=U=K=20\pi~{}\text{MHz}=0.26~{}\mu eV$, all being electrically
tunable. The frequency of the external drive is $\omega=20\pi$ MHz (therefore,
a period of $T=0.1~{}\mu s$) assuming the typical resolution of nano-seconds
in the time-dependent measurements in experiments with QD array. For
$t_{||}^{hop}=\frac{4}{3}\omega$ we find DTC and $t_{||}^{hop}=\omega$ we find
discrete-time quasi-crystal (DTQC). Alternatively, since $\epsilon_{\alpha}$
can be varied via gate voltage, setting the difference between the onsite
energies, $(\epsilon_{b}-\epsilon_{a})=\Delta\epsilon=\frac{7}{3}\omega$ leads
to DTC, $\Delta\epsilon=0$ would lead to DTQC for
$t_{||}^{hop}=t_{\perp}^{hop}$. All the conclusions we obtain remain true for
DTQC also. The system parameters chosen for the numerical calculations are
well within the reach of present day experiments 51, 81, 45, 41, 82, 83, 47,
55, 56.

Having a DTC state is not enough; one must show that this state has a
signature on a measurable quantity (e.g., currents). To show this, we proceed
in the following way. First, we demonstrate that the DTC behavior is
manifested through the inter-chain tunneling currents, which are not directly
measurable. Then, we proceed to show how the charge-current through the system
relates to the inter-chain current, thus proving that the charge-current
exhibit signatures of the DTC state.

In Fig. 2 (a) we plot the synchronized oscillation of the real part of the
time dependent particle tunneling from the chain $a$ to chain $b$, viz.,
$\langle\tau_{j}^{x}(t)\rangle$ on the $j^{\prime}$th inter-chain bond,
$\tau^{x}_{j}=\text{Re}[c^{\dagger}_{j,b}c_{j,a}]$ in the presence of
electrodes. We shall show later that the operator $\tau_{j}^{x}$ represents
the $x-$component of the effective dipole moment appearing along the inter-
chain bonds. We set $\gamma_{L}=\gamma_{R}=\gamma=10^{-4}\omega$, and choose
$\Gamma=t_{hop}$ for faster quantum-synchronization of oscillating effective
dipoles residing on the $(j,a)-(j,b)$ bonds (the choice of magnitude of
$\Gamma$ only determines the synchronization time scale) 61. In the long time
limit between $85T-100T$, oscillations show clear signature of DTC with period
$3T$. This is further fingerprinted in the Discrete Fourier transform (DFT)
$|\tilde{\tau}_{3}^{x}(f)|$ of the effective-dipole oscillation in the form of
a sharp peak at a frequency $f=2\omega/3$, as seen in Fig. 2 (c). For stronger
system lead couplings the DTC oscillation decays with a characteristic decay
time $\approx 2/\gamma$, see SI Sec. S3 for further details.

Figure 2: Plot of DTC oscillation and oscillating charge-current. (a) For a
system with N=5, plot of $\langle\tau_{j}^{x}(t)\rangle$, and (b) of charge-
current $\langle J_{R}(t)\rangle$ through right electrode as a function of
time in the long time limit. (c) Discrete Fourier transform
$|\tilde{\tau}_{3}^{x}(f)|$ (orange line) of $\langle\tau_{j}^{x}(t)\rangle$
and $|\tilde{J_{R}}(f)|$ (blue line) of $\langle J_{R}(t)\rangle$,
respectively. The peak of $|\tilde{J_{R}}(f)|$ at $5/3$ directly provides the
value of the Floquet coherence. Value of parameters:
$t_{||}^{hop}=U=K=\omega=20\pi
MHz,~{}t_{\perp}^{hop}=\frac{4}{3}\omega,~{}\gamma_{L}=\gamma_{R}=10^{-4}\omega$.

Although the effective dipole $\langle\tau_{j}^{x}(t)\rangle$ shows DTC
oscillations, it is not a directly measurable quantity. However, since the
system is attached to a pair of electrodes, one can detect the oscillating
charge imbalance induced by the effective-dipole oscillation at each inter-
chain bond of the ladder through the (charge) current. Fig. 2 (b) shows the
current $\langle J_{R}(t)\rangle=e\gamma_{R}\text{Tr}[n_{N,a}\rho_{DTC}(t)]$
in picco-Ampere (pA) as a function of time, in the long time limit between
$85T-100T$, where $n_{N,a}$ is the density of spin-less Fermions on the
$(N,a)$ site of the ladder attached to the right-electrode (See supplementary
Sec. S4 for the derivation of charge-current) and $\rho_{DTC}(t)$ is the time-
dependent DTC density matrix which exhibits the DTC time period. DFT of the
charge-current indicates that the oscillation exhibits a frequency
$f_{cc}=\frac{5}{3}\omega$ (subscript ‘$cc$’ denotes charge-current). Notably,
$f=f_{cc}~{}(\text{mod }\omega)=f_{cc}-\omega$, i.e., charge-current, being
the primary observable in transport experiments, provides the signature of
DTC.

Having obtained DTC in the model (Emergence and Dynamical Stability of Charge
Time-Crystal in a Current-Carrying Quantum Dot Simulator) we seek to
understand why our system exhibits time-crystallinity and charge-current
manifests the same, from a mathematical perspective. To this end, we recall
that recently it was shown 24, 25, 37, 36, 33 that in open spinfull systems,
for a DTC to survive a necessary condition is the existence of an FDS, a
global operator $A$ that satisfies the following conditions, (i)
$\mathcal{U}_{F}(A\rho)=e^{-i\lambda T}A\mathcal{U}_{F}(\rho)$, i.e., $A$
becomes an eigen-operator of $\mathcal{U}_{F}$ with a characteristic frequency
$\lambda$ (in general a function of system parameters) corresponding to the
Floquet coherence, and (ii) $[V_{\mu},A]=[V_{\mu}^{\dagger},A]=0$, for all
$\mu$.

We first show that the model (Emergence and Dynamical Stability of Charge
Time-Crystal in a Current-Carrying Quantum Dot Simulator) indeed exhibits a
global FDS operator $\tau^{+}=\sum_{j=1}^{N}\tau_{j}^{+}$ in absence of
system-electrode coupling if and only if $U^{\prime}=U$ and $K^{\prime}=K$
(see supplementary Sec. S1 for a proof), where
$\tau_{j}^{+}=c_{j,b}^{\dagger}c_{j,a}$. Physically, $\tau^{+}$ represents the
total particle tunneling operator from chain $a$ to chain $b$, while
$\tau_{j}^{+}$ represents tunneling at each pair of sites. In the Floquet
basis (rotated frame) the equation of motion for
$\tau^{+}=\sum_{j=1}^{N}\tau_{j}^{+}$ can be shown to be generated by a
Floquet Lindbladian $\mathcal{L}_{F}$ whose coherent part is governed by a
Floquet Hamiltonian
$\mathcal{H}_{F}=\mathcal{H}_{hop}^{||}+\mathcal{H}_{int}+\mathbf{E}\cdot\sum_{j}\boldsymbol{\tau}_{j}$.
The Floquet Hamiltonian $\mathcal{H}_{F}$ signifies that in the spin-less
model an FDS can exist, which we call a charge-FDS, as a consequence of an
effective Stark effect under a DC effective electric field
$\mathbf{E}=(t^{hop}_{\perp},0,\Delta\epsilon-\omega)$ in the rotated frame.
Here, $\boldsymbol{\tau}_{j}=(\tau_{j}^{x},\tau_{j}^{y},\tau_{j}^{z})$
represents an effective electric dipole operator situated on the
$j^{\prime}$th inter-chain (red) bond, and $\tau_{j}^{z}$ represents a
particle imbalance on the $j^{\prime}$th inter-chain bond, see Fig. 1.

When the system is coupled to electrodes, the FDS condition is no longer
valid. Recently, we have shown that even if this is the case, DTC signatures
can persist if there exists a weak-local FDS, namely a local operator
$A_{loc}$ obeying $\mathcal{U}_{F}(A_{loc}\rho)=e^{i\lambda
T[1+i\gamma/\lambda]}A_{loc}\mathcal{U}_{F}(\rho)$, where
$[V_{in(out)},A_{loc}]\propto\gamma$ makes the FDS weak 61. This leads to the
appearance of Floquet coherent states $\rho_{m,n}$. The Floquet coherent
states are built from the Floquet steady state $\rho_{\text{FSS}}$ as,
$\rho_{m,n}=(\tau_{loc,|\mathbf{E}|}^{+})^{m}\rho_{\text{FSS}}(\tau_{loc,|\mathbf{E}|}^{-})^{n}$
(with integer values of $m,n$), satisfying
$\mathcal{U}_{F}(\rho_{m,n})=e^{i(m-n)|\mathbf{E}|T}e^{-(m+n)\gamma
T/2}\rho_{m,n},$ and only a few $m,n$ contribute to the weakly decaying
coherent states with decay time $\gamma^{-1}$. For $\gamma T\ll 1$ we find
stable DTC oscillation in the long time limit, otherwise a meta-stable DTC
emerges 61. The characteristic frequency of Floquet coherence is given by
$\lambda(\Delta\epsilon,\omega,t_{\perp}^{hop})=|\mathbf{E}|~{}\text{mod}\omega$,
where $|\mathbf{E}|=\sqrt{(\Delta\epsilon-\omega)^{2}+(t_{\perp}^{hop})^{2}}$.
When
$\frac{f}{\omega}=\frac{\lambda(\Delta\epsilon,\omega,t^{hop}_{\perp})}{\omega}=\frac{q}{p}$
(where $p$, $q$ are integers), DTC emerges with periodicity $pT$, otherwise a
discrete time quasi-crystal emerges.

The charge (weak local) FDS we obtain here exhibits the same mathematical
structure as that of the conventional FDS obtained in spinfull models 25, 24,
37, 33. It protects the quantum coherence between the different particle
imbalance sectors, viz.,
$\frac{\mathbf{E}\cdot\boldsymbol{\tau}}{|\mathbf{E}|}=\tau_{|\mathbf{E}|}^{z}$,
while quantum coherence within each particle imbalance sector is eliminated
due to the dephasing, which indicates the importance of simultaneous dephasing
of $(j,a)-(j,b)$ pair of sites. In long time limit the system reaches a
density matrix of the form
$\rho_{DTC}(t)=\rho_{FSS}+\sum_{\begin{subarray}{c}m,n\\\ m,n\neq
0\end{subarray}}\left(c_{m,n}e^{i(m-n)\lambda(\epsilon,\omega,E)t}\rho_{m,n}+h.c\right)e^{-(m+n)\gamma
t}$ where $c_{m,n}=|\text{Tr}[\rho_{\text{in}}^{\dagger}\rho_{m,n}]|$
corresponding to integer values of $m,n$, $\rho_{\text{in}}$ being the density
matrix of the initial state, are the real coefficients of the superposition,
$\rho_{FSS}$ being the Floquet steady-states, and only a few $m,n$ contribute
61.

Crucially, for a critical value of system parameter $\Delta\epsilon_{c}$ we
find that
$\frac{\lambda(\Delta\epsilon_{c},\omega,t_{\perp}^{hop})}{\omega}=\frac{q}{p}=r$,
where $r$ is an integer, see SI Fig. S2. This further points out that an
oscillation with periodicity $T$ (corresponding to $p=1$), i.e., a Floquet
response, emerges. A natural question arise, can the DTC state recover even
after dynamically reaching a state exhibiting a Floquet response? The answer,
quite surprisingly, is positive.

Figure 3:  Dynamical cross-over between DTC and normal phases. (a) Dynamical
profile of the onsite energy difference $\Delta\epsilon/\omega$ as a function
of time where $\epsilon_{a}/\omega=0$. For a system with $N=5$, (b) plot of
$\langle\text{Re}[\tau_{j}^{+}](t)\rangle$ , and (c) of charge-current
$\langle J_{R}(t)\rangle$, in $pA$ through right electrode as a function of
time plotted over $180T$. (d), (e), and (f) Discrete Fourier transforms
$|\tilde{\tau_{3}^{x}}(f)|$ (solid line) of $\langle
Re[\tau_{j}^{+}](t)\rangle$ and $|\tilde{J_{R}}(f)|$ (dotted line) of $\langle
J_{R}(t)\rangle$, respectively between $5T-55T$, $65T-115T$, and $125T-180T$,
respectively. Value of parameters:
$t_{||}^{hop}=t_{\perp}^{hop}=U=K=\omega=20\pi\text{MHz},~{}\gamma_{L}=\gamma_{R}=10^{-4}\omega$,
strong dephasing strength $\Gamma_{d}=\omega$ ensures fast synchronization.

To show this, we quench (by ramping) the value of $\Delta\epsilon$ during the
dynamics as shown in Fig. 3 (a). Fig. 3 (b) and (c) plot
$\langle\tau_{j}^{x}(t)\rangle$ and the charge-current $\langle
J_{R}(t)\rangle$ from the right lead, respectively, showing that the system
dynamically enters, after a transient relaxation, into a normal state
exhibiting Floquet response from a DTC state once the system reaches the
required critical value $\Delta\epsilon_{c}$, and again relaxes back to the
DTC state once the system is quenched away from $\Delta\epsilon_{c}$. Fig.3
(d), (e), and (f) plot the DFTs of both $\langle\tau_{j}^{x}(t)\rangle$ and
$\langle J_{R}(t)\rangle$ showing that the initial DTC response exhibiting a
peak at $\frac{|\mathbf{E}|}{\omega}=\frac{5}{3}$ in the DFT of $\langle
J_{R}(t)\rangle$, seen in Fig.3 (d), is obtained back, as seen in Fig.3 (f),
after the system dynamically passes trough states exhibiting Floquet response
seen in Fig.3 (e). In supplementary section S4 we discuss dynamical transition
from DTQC to normal phase.

These results imply that once the system relaxes to a DTC state it can never
reach the FSS although it can reach a coherent state exhibiting a Floquet
response which is degenerate to the FSS. This shows the stability of the DTC
state against continuous change of the system parameters.

We identify a paradigmatic experimental setting of a QD array placed between
leads that can identify and directly measure the appearance of DTC (and DTQC)
through the oscillations in the charge-current. A charge-FDS emerges as a
consequence of an effective ‘Stark effect’ in the rotated frame and protects
DTC in the presence of dissipation. This is complementary to the existing
notion of the FDS that appear, as a consequence of an effective Zeeman effect,
in spin-full Hubbard and XXZ spin models 25, 24, 37, 36. Furthermore, we show
that DTC behavior is surprisingly robust against continuous change of system
parameters and support a reversible dynamical transition from DTC state to a
state with Floquet response.

Out theoretical results are obtained by keeping a QD simulator in mind where
parameters can be controlled via gate voltages 51, 81, 45, 41, 82. We choose
$\gamma$, the tunneling rate between the source (and drain) and QDs, in the
range of $10^{-3}\omega=62~{}kHz$ and $10^{-5}\omega=6.2~{}kHz$, which is a
standard in transport experiments with QD-array set-up 84. Although numerical
results are obtained in a set-up of 10 QDs, our analytical formulation is
valid for any number of QDs. In addition, we have verified numerically that
the minimum number of QDs necessary for the FDS based DTC to appear is four
(i.e., N=2), a system size that is routinely achieved in experiments, e.g.,
Refs. [ 45, 43]. The novelty of our finding is that one can detect
dissipative-DTC in a quantum transport experiment performed on a QD array and
perhaps in other nano-scale systems 63, 64, 65 for over 100 driving periods
via measuring charge current of sizable amplitude. Moreover, our
identification of a transport observable of DTC takes the DTC physics beyond
the existing cold-atom set-ups and connects DTC to quantum transport phenomena
in nano-scale systems. Although we have used realistic parameters from QD-
array set-up 51, 81, 45, 41, 82, ultra-cold quantum gases also provide another
promising route 67, 85, 86, 87.

##  1 Supporting Information

charge-FDS of the system, Spectrum of Floquet propagator, Decay of DTC due to
system lead coupling, Current operator, Dynamical transition from DTQC to
normal state

## References

  * Choi et al. 2017 Choi, S.; Choi, J.; Landig, R.; Kucsko, G.; Zhou, H.; Isoya, J.; Jelezko, F.; Onoda, S.; Sumiya, H.; Khemani, V.; von Keyserlingk, C.; Yao, N. Y.; Demler, E.; Lukin, M. D. Observation of discrete time-crystalline order in a disordered dipolar many-body system. _Nature_ 2017, _543_ , 221–225
  * Zhang et al. 2017 Zhang, J.; Hess, P. W.; Kyprianidis, A.; Becker, P.; Lee, A.; Smith, J.; Pagano, G.; Potirniche, I.-D.; Potter, A. C.; Vishwanath, A.; Yao, N. Y.; Monroe, C. Observation of a discrete time crystal. _Nature_ 2017, _543_ , 217–220
  * Kyprianidis et al. 2021 Kyprianidis, A.; Machado, F.; Morong, W.; Becker, P.; Collins, K. S.; Else, D. V.; Feng, L.; Hess, P. W.; Nayak, C.; Pagano, G.; Yao, N. Y.; Monroe, C. Observation of a prethermal discrete time crystal. _Science_ 2021, _372_ , 1192–1196
  * Pal et al. 2018 Pal, S.; Nishad, N.; Mahesh, T. S.; Sreejith, G. J. Temporal Order in Periodically Driven Spins in Star-Shaped Clusters. _Phys. Rev. Lett._ 2018, _120_ , 180602
  * Träger et al. 2021 Träger, N.; Gruszecki, P.; Lisiecki, F.; Groß, F.; Förster, J.; Weigand, M.; Głowiński, H.; Kuświk, P.; Dubowik, J.; Schütz, G.; Krawczyk, M.; Gräfe, J. Real-Space Observation of Magnon Interaction with Driven Space-Time Crystals. _Phys. Rev. Lett._ 2021, _126_ , 057201
  * Smits et al. 2018 Smits, J.; Liao, L.; Stoof, H. T. C.; van der Straten, P. Observation of a Space-Time Crystal in a Superfluid Quantum Gas. _Phys. Rev. Lett._ 2018, _121_ , 185301
  * Dogra et al. 2019 Dogra, N.; Landini, M.; Kroeger, K.; Hruby, L.; Donner, T.; Esslinger, T. Dissipation-induced structural instability and chiral dynamics in a quantum gas. _Science_ 2019, _366_ , 1496–1499
  * Autti et al. 2021 Autti, S.; Heikkinen, P. J.; Mäkinen, J. T.; Volovik, G. E.; Zavjalov, V. V.; Eltsov, V. B. AC Josephson effect between two superfluid time crystals. _Nature Materials_ 2021, _20_ , 171–174
  * Barnes et al. 2019 Barnes, E.; Nichol, J. M.; Economou, S. E. Stabilization and manipulation of multispin states in quantum-dot time crystals with Heisenberg interactions. _Phys. Rev. B_ 2019, _99_ , 035311
  * Qiao et al. 2021 Qiao, H.; Kandel, Y. P.; Dyke, J. S. V.; Fallahi, S.; Gardner, G. C.; Manfra, M. J.; Barnes, E.; Nichol, J. M. Floquet-enhanced spin swaps. _Nature Communications_ 2021, _12_ , 2142
  * Van Dyke et al. 2021 Van Dyke, J. S.; Kandel, Y. P.; Qiao, H.; Nichol, J. M.; Economou, S. E.; Barnes, E. Protecting quantum information in quantum dot spin chains by driving exchange interactions periodically. _Phys. Rev. B_ 2021, _103_ , 245303
  * Randall et al. 2021 Randall, J.; Bradley, C.; van der Gronden, F.; Galicia, A.; Abobeih, M.; Markham, M.; Twitchen, D.; Machado, F.; Yao, N.; Taminiau, T. Many-body-localized discrete time crystal with a programmable spin-based quantum simulator. _Science_ 2021, eabk0603
  * Mi et al. 2021 Mi, X.; Ippoliti, M.; Quintana, C.; Greene, A.; Chen, Z.; Gross, J.; Arute, F.; Arya, K.; Atalaya, J.; Babbush, R., et al. Time-Crystalline Eigenstate Order on a Quantum Processor. _Nature_ 2021, 1–1
  * Sacha and Zakrzewski 2017 Sacha, K.; Zakrzewski, J. Time crystals: a review. _Reports on Progress in Physics_ 2017, _81_ , 016401
  * Khemani et al. 2016 Khemani, V.; Lazarides, A.; Moessner, R.; Sondhi, S. L. Phase Structure of Driven Quantum Systems. _Phys. Rev. Lett._ 2016, _116_ , 250401
  * Else et al. 2016 Else, D. V.; Bauer, B.; Nayak, C. Floquet Time Crystals. _Phys. Rev. Lett._ 2016, _117_ , 090402
  * Surace et al. 2019 Surace, F. M.; Russomanno, A.; Dalmonte, M.; Silva, A.; Fazio, R.; Iemini, F. Floquet time crystals in clock models. _Phys. Rev. B_ 2019, _99_ , 104303
  * Khemani et al. 2019 Khemani, V.; Moessner, R.; Sondhi, S. L. A Brief History of Time Crystals. 2019
  * Wilczek 2012 Wilczek, F. Quantum Time Crystals. _Phys. Rev. Lett._ 2012, _109_ , 160401
  * Yao and Nayak 2018 Yao, N. Y.; Nayak, C. Time crystals in periodically driven systems. _Physics Today_ 2018, _71_ , 40–47
  * Bukov et al. 2015 Bukov, M.; D’Alessio, L.; Polkovnikov, A. Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering. _Advances in Physics_ 2015, _64_ , 139–226
  * Moessner and Sondhi 2017 Moessner, R.; Sondhi, S. L. Equilibration and order in quantum Floquet matter. _Nature Physics_ 2017, _13_ , 424–428
  * Riera-Campeny et al. 2020 Riera-Campeny, A.; Moreno-Cardoner, M.; Sanpera, A. Time crystallinity in open quantum systems. _Quantum_ 2020, _4_ , 270
  * Buča et al. 2019 Buča, B.; Tindall, J.; Jaksch, D. Non-stationary coherent quantum many-body dynamics through dissipation. _Nature Communications_ 2019, _10_ , 1730
  * Chinzei and Ikeda 2020 Chinzei, K.; Ikeda, T. N. Time Crystals Protected by Floquet Dynamical Symmetry in Hubbard Models. _Phys. Rev. Lett._ 2020, _125_ , 060601
  * Iemini et al. 2018 Iemini, F.; Russomanno, A.; Keeling, J.; Schirò, M.; Dalmonte, M.; Fazio, R. Boundary Time Crystals. _Phys. Rev. Lett._ 2018, _121_ , 035301
  * Carollo and Lesanovsky 2021 Carollo, F.; Lesanovsky, I. Exact solution of a boundary time-crystal phase transition: time-translation symmetry breaking and non-Markovian dynamics of correlations. 2021
  * Gong et al. 2018 Gong, Z.; Hamazaki, R.; Ueda, M. Discrete Time-Crystalline Order in Cavity and Circuit QED Systems. _Phys. Rev. Lett._ 2018, _120_ , 040404
  * Zhu et al. 2019 Zhu, B.; Marino, J.; Yao, N. Y.; Lukin, M. D.; Demler, E. A. Dicke time crystals in driven-dissipative quantum many-body systems. _New Journal of Physics_ 2019, _21_ , 073028
  * Gambetta et al. 2019 Gambetta, F. M.; Carollo, F.; Marcuzzi, M.; Garrahan, J. P.; Lesanovsky, I. Discrete Time Crystals in the Absence of Manifest Symmetries or Disorder in Open Quantum Systems. _Phys. Rev. Lett._ 2019, _122_ , 015701
  * Buča and Jaksch 2019 Buča, B.; Jaksch, D. Dissipation Induced Nonstationarity in a Quantum Gas. _Phys. Rev. Lett._ 2019, _123_ , 260401
  * Medenjak et al. 2020 Medenjak, M.; Prosen, T.; Zadnik, L. Rigorous bounds on dynamical response functions and time-translation symmetry breaking. _SciPost Phys._ 2020, _9_ , 3
  * Tindall et al. 2020 Tindall, J.; Muñoz, C. S.; Buča, B.; Jaksch, D. Quantum synchronisation enabled by dynamical symmetries and dissipation. _New Journal of Physics_ 2020, _22_ , 013026
  * Chinzei and Ikeda 2022 Chinzei, K.; Ikeda, T. N. Criticality and rigidity of dissipative discrete time crystals in solids. _Phys. Rev. Research_ 2022, _4_ , 023025
  * Gunawardana and Buča 2021 Gunawardana, T.; Buča, B. Dynamical l-bits in Stark many-body localization. _arXiv:2110.13135v2 [cond-mat.dis-nn]_ 2021, Wed, 19 Jan 2022
  * Buca et al. 2022 Buca, B.; Booker, C.; Jaksch, D. Algebraic Theory of Quantum Synchronization and Limit Cycles under Dissipation. _SciPost Phys._ 2022, _12_ , 97
  * Medenjak et al. 2020 Medenjak, M.; Buča, B.; Jaksch, D. Isolated Heisenberg magnet as a quantum time crystal. _Phys. Rev. B_ 2020, _102_ , 041117
  * Keßler et al. 2021 Keßler, H.; Kongkhambut, P.; Georges, C.; Mathey, L.; Cosme, J. G.; Hemmerich, A. Observation of a Dissipative Time Crystal. _Phys. Rev. Lett._ 2021, _127_ , 043602
  * Ito et al. 2018 Ito, T.; Otsuka, T.; Nakajima, T.; Delbecq, M. R.; Amaha, S.; Yoneda, J.; Takeda, K.; Noiri, A.; Allison, G.; Ludwig, A.; Wieck, A. D.; Tarucha, S. Four single-spin Rabi oscillations in a quadruple quantum dot. _Applied Physics Letters_ 2018, _113_ , 093102
  * Mukhopadhyay et al. 2018 Mukhopadhyay, U.; Dehollain, J. P.; Reichl, C.; Wegscheider, W.; Vandersypen, L. M. K. A 2$\times$2 quantum dot array with controllable inter-dot tunnel couplings. _Applied Physics Letters_ 2018, _112_ , 183505
  * Mills et al. 2019 Mills, A. R.; Zajac, D. M.; Gullans, M. J.; Schupp, F. J.; Hazard, T. M.; Petta, J. R. Shuttling a single charge across a one-dimensional array of silicon quantum dots. _Nature Communications_ 2019, _10_ , 1063
  * Sigillito et al. 2019 Sigillito, A.; Loy, J.; Zajac, D.; Gullans, M.; Edge, L.; Petta, J. Site-Selective Quantum Control in an Isotopically Enriched ${}^{28}\mathrm{Si}/{\mathrm{Si}}_{0.7}{\mathrm{Ge}}_{0.3}$ Quadruple Quantum Dot. _Phys. Rev. Applied_ 2019, _11_ , 061006
  * Qiao et al. 2020 Qiao, H.; Kandel, Y. P.; Deng, K.; Fallahi, S.; Gardner, G. C.; Manfra, M. J.; Barnes, E.; Nichol, J. M. Coherent Multispin Exchange Coupling in a Quantum-Dot Spin Chain. _Phys. Rev. X_ 2020, _10_ , 031006
  * Estarellas et al. 2020 Estarellas, M. P.; Osada, T.; Bastidas, V. M.; Renoust, B.; Sanaka, K.; Munro, W. J.; Nemoto, K. Simulating complex quantum networks with time crystals. _Science Advances_ 2020, _6_ , 1–8
  * Hensgens et al. 2017 Hensgens, T.; Fujita, T.; Janssen, L.; Li, X.; Van Diepen, C. J.; Reichl, C.; Wegscheider, W.; Das Sarma, S.; Vandersypen, L. M. K. Quantum simulation of a Fermi–Hubbard model using a semiconductor quantum dot array. _Nature_ 2017, _548_ , 70–73
  * van Diepen et al. 2021 van Diepen, C. J.; Hsiao, T.-K.; Mukhopadhyay, U.; Reichl, C.; Wegscheider, W.; Vandersypen, L. M. K. Quantum Simulation of Antiferromagnetic Heisenberg Chain with Gate-Defined Quantum Dots. _Phys. Rev. X_ 2021, _11_ , 041025
  * Bhattacharya et al. 2021 Bhattacharya, U.; Grass, T.; Bachtold, A.; Lewenstein, M.; Pistolesi, F. Phonon-Induced Pairing in Quantum Dot Quantum Simulator. _Nano Letters_ 2021, _21_ , 9661–9667, PMID: 34757742
  * Field et al. 1993 Field, M.; Smith, C. G.; Pepper, M.; Ritchie, D. A.; Frost, J. E. F.; Jones, G. A. C.; Hasko, D. G. Measurements of Coulomb blockade with a noninvasive voltage probe. _Phys. Rev. Lett._ 1993, _70_ , 1311–1314
  * Byrnes et al. 2008 Byrnes, T.; Kim, N. Y.; Kusudo, K.; Yamamoto, Y. Quantum simulation of Fermi-Hubbard models in semiconductor quantum-dot arrays. _Phys. Rev. B_ 2008, _78_ , 075320
  * Yang et al. 2011 Yang, S.; Wang, X.; Das Sarma, S. Generic Hubbard model description of semiconductor quantum-dot spin qubits. _Phys. Rev. B_ 2011, _83_ , 161301
  * Barthelemy and Vandersypen 2013 Barthelemy, P.; Vandersypen, L. M. K. Quantum Dot Systems: a versatile platform for quantum simulations. _Annalen der Physik_ 2013, _525_ , 808–826
  * Deshpande et al. 2010 Deshpande, V. V.; Bockrath, M.; Glazman, L. I.; Yacoby, A. Electron liquids and solids in one dimension. _Nature_ 2010, _464_ , 209–216
  * Loss and DiVincenzo 1998 Loss, D.; DiVincenzo, D. P. Quantum computation with quantum dots. _Phys. Rev. A_ 1998, _57_ , 120–126
  * Lazarides and Moessner 2017 Lazarides, A.; Moessner, R. Fate of a discrete time crystal in an open system. _Phys. Rev. B_ 2017, _95_ , 195135
  * Lawrie et al. 2020 Lawrie, W. I. L.; Hendrickx, N. W.; van Riggelen, F.; Russ, M.; Petit, L.; Sammak, A.; Scappucci, G.; Veldhorst, M. Spin Relaxation Benchmarks and Individual Qubit Addressability for Holes in Quantum Dots. _Nano Letters_ 2020, _20_ , 7237–7242, PMID: 32833455
  * Jang et al. 2021 Jang, W.; Cho, M.-K.; Jang, H.; Kim, J.; Park, J.; Kim, G.; Kang, B.; Jung, H.; Umansky, V.; Kim, D. Single-Shot Readout of a Driven Hybrid Qubit in a GaAs Double Quantum Dot. _Nano Letters_ 2021, _21_ , 4999–5005, PMID: 34109799
  * van Riggelen et al. 2021 van Riggelen, F.; Hendrickx, N. W.; Lawrie, W. I. L.; Russ, M.; Sammak, A.; Scappucci, G.; Veldhorst, M. A two-dimensional array of single-hole quantum dots. _Applied Physics Letters_ 2021, _118_ , 044002
  * Giergiel et al. 2019 Giergiel, K.; Kuroś, A.; Sacha, K. Discrete time quasicrystals. _Phys. Rev. B_ 2019, _99_ , 220303
  * Yang and Cai 2021 Yang, X.; Cai, Z. Dynamical Transitions and Critical Behavior between Discrete Time Crystal Phases. _Phys. Rev. Lett._ 2021, _126_ , 020602
  * Kosior and Sacha 2018 Kosior, A.; Sacha, K. Dynamical quantum phase transitions in discrete time crystals. _Physical Review A_ 2018, _97_ , 053621
  * Sarkar and Dubi 2021 Sarkar, S.; Dubi, Y. Signatures of discrete time-crystallinity in transport through quantum dot arrays. _arXiv:2107.04214v2 [cond-mat.mes-hall]_ 2021, Fri, 9 Jul 2021
  * Volovik 2013 Volovik, G. E. On the broken time translation symmetry in macroscopic systems: Precessing states and off-diagonal long-range order. _JETP Letters_ 2013, _98_ , 491–495
  * Petiziol et al. 2021 Petiziol, F.; Chiesa, A.; Wimberger, S.; Santini, P.; Carretta, S. Counteracting dephasing in Molecular Nanomagnets by optimized qudit encodings. _npj Quantum Information_ 2021, _7_ , 133
  * Hussain et al. 2018 Hussain, R.; Allodi, G.; Chiesa, A.; Garlatti, E.; Mitcov, D.; Konstantatos, A.; Pedersen, K. S.; De Renzi, R.; Piligkos, S.; Carretta, S. Coherent Manipulation of a Molecular Ln-Based Nuclear Qudit Coupled to an Electron Qubit. _Journal of the American Chemical Society_ 2018, _140_ , 9814–9818, PMID: 30040890
  * Chiesa et al. 2020 Chiesa, A.; Macaluso, E.; Petiziol, F.; Wimberger, S.; Santini, P.; Carretta, S. Molecular Nanomagnets as Qubits with Embedded Quantum-Error Correction. _The Journal of Physical Chemistry Letters_ 2020, _11_ , 8610–8615, PMID: 32936660
  * Benenti et al. 2019 Benenti, G.; Casati, G.; Rossini, D.; Strini, G. _Principles of quantum computation and information: a comprehensive textbook_ ; World Scientific, 2019; Chapter 10
  * Gross and Bloch 2017 Gross, C.; Bloch, I. Quantum simulations with ultracold atoms in optical lattices. _Science_ 2017, _357_ , 995–1001
  * Granger et al. 2012 Granger, G.; Taubert, D.; Young, C. E.; Gaudreau, L.; Kam, A.; Studenikin, S. A.; Zawadzki, P.; Harbusch, D.; Schuh, D.; Wegscheider, W.; Wasilewski, Z. R.; Clerk, A. A.; Ludwig, S.; Sachrajda, A. S. Quantum interference and phonon-mediated back-action in lateral quantum-dot circuits. _Nature Physics_ 2012, _8_ , 522–527
  * Hartmann et al. 2017 Hartmann, M.; Poletti, D.; Ivanchenko, M.; Denisov, S.; Hänggi, P. Asymptotic Floquet states of open quantum systems: the role of interaction. _New Journal of Physics_ 2017, _19_ , 083011
  * Lindblad (1976 Lindblad, G. On the generators of quantum dynamical semigroups. _Commun.Math. Phys._ (1976), _48_ , 119–130
  * Gorini et al. 1976 Gorini, V.; Kossakowski, A.; Sudarshan, E. C. G. Completely positive dynamical semigroups of N‐level systems. _Journal of Mathematical Physics_ 1976, _17_ , 821–825
  * Breuer and Petruccione 2004 Breuer, H.-P.; Petruccione, F. _The Theory of Open Quantum Systems_ ; Oxford University Press: Oxford, 2004; Chapter 3, p 117
  * Purkayastha et al. 2016 Purkayastha, A.; Dhar, A.; Kulkarni, M. Out-of-equilibrium open quantum systems: A comparison of approximate quantum master equation approaches with exact results. _Phys. Rev. A_ 2016, _93_ , 062114
  * Contreras-Pulido et al. 2014 Contreras-Pulido, L. D.; Bruderer, M.; Huelga, S. F.; Plenio, M. B. Dephasing-assisted transport in linear triple quantum dots. _New Journal of Physics_ 2014, _16_ , 113061
  * Field et al. 1993 Field, M.; Smith, C.; Pepper, M.; Ritchie, D.; Frost, J.; Jones, G.; Hasko, D. Measurements of Coulomb blockade with a noninvasive voltage probe. _Physical Review Letters_ 1993, _70_ , 1311
  * Young and Clerk 2010 Young, C.; Clerk, A. Inelastic backaction due to quantum point contact charge fluctuations. _Physical Review Letters_ 2010, _104_ , 186803
  * Levinson 1997 Levinson, Y. Dephasing in a quantum dot due to coupling with a quantum point contact. _EPL (Europhysics Letters)_ 1997, _39_ , 299
  * Bounouh et al. 2002 Bounouh, A.; Poirier, W.; Piquemal, F.; Geneves, G.; Andre, J. Quantum resistance standards with double 2DEG. Conference Digest Conference on Precision Electromagnetic Measurements. 2002; pp 240–241
  * Nield et al. 2000 Nield, S. A.; Nicholls, J. T.; Tribe, W. R.; Simmons, M. Y.; Ritchie, D. A. Independent contacts to a double-electron gas using mesoscopic surface gates. _Journal of Applied Physics_ 2000, _87_ , 4036–4038
  * Micolich et al. 2002 Micolich, A.; Taylor, R.; Davies, A.; Fromhold, T.; Newbury, R.; Ehlert, A.; Macks, L.; Tribe, W.; Linke, H.; Linfield, E.; Ritchie, D. The dependence of fractal conductance fluctuations on soft-wall profile in a double-2DEG billiard. _Physica E: Low-dimensional Systems and Nanostructures_ 2002, _12_ , 841–844, Proceedings of the Fourteenth International Conference on the Ele ctronic Properties of Two-Dimensional Systems
  * Kouwenhoven et al. 2001 Kouwenhoven, L. P.; Austing, D. G.; Tarucha, S. Few-electron quantum dots. _Reports on Progress in Physics_ 2001, _64_ , 701–736
  * Zajac et al. 2018 Zajac, D. M.; Sigillito, A. J.; Russ, M.; Borjans, F.; Taylor, J. M.; Burkard, G.; Petta, J. R. Resonantly driven CNOT gate for electron spins. _Science_ 2018, _359_ , 439–442
  * Burkard et al. 2021 Burkard, G.; Ladd, T. D.; Nichol, J. M.; Pan, A.; Petta, J. R. Semiconductor Spin Qubits. _arXiv:2112.08863 [cond-mat.mes-hall]_ 2021, Thu, 16 Dec 2021
  * Gustavsson et al. 2007 Gustavsson, S.; Studer, M.; Leturcq, R.; Ihn, T.; Ensslin, K.; Driscoll, D. C.; Gossard, A. C. Frequency-Selective Single-Photon Detection Using a Double Quantum Dot. _Phys. Rev. Lett._ 2007, _99_ , 206804
  * Brantut et al. 2012 Brantut, J.-P.; Meineke, J.; Stadler, D.; Krinner, S.; Esslinger, T. Conduction of Ultracold Fermions Through a Mesoscopic Channel. _Science_ 2012, _337_ , 1069–1071
  * Chien et al. 2015 Chien, C.-C.; Peotta, S.; Di Ventra, M. Quantum transport in ultracold atoms. _Nature Physics_ 2015, _11_ , 998–1004
  * Esslinger 2010 Esslinger, T. Fermi-Hubbard Physics with Atoms in an Optical Lattice. _Annual Review of Condensed Matter Physics_ 2010, _1_ , 129–152

Figure 4: “For Table of Contents Only”