# Quantum Zeno Isolation of Quantum Dots

N. Ahmadiniaz  Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400,
01328 Dresden, Germany,  M. Geller  J. König  P. Kratzer  A. Lorke  Fakultät
für Physik and CENIDE, Universität Duisburg-Essen, Lotharstraße 1, 47057
Duisburg, Germany,  G. Schaller  Helmholtz-Zentrum Dresden-Rossendorf,
Bautzner Landstraße 400, 01328 Dresden, Germany,  R. Schützhold  Helmholtz-
Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany,
Institut für Theoretische Physik, Technische Universität Dresden, 01062
Dresden, Germany,

###### Abstract

We investigate whether and how the quantum Zeno effect, i.e., the inhibition
of quantum evolution by frequent measurements, can be employed to isolate a
quantum dot from its surrounding electron reservoir. In contrast to the often
studied case of tunneling between discrete levels, we consider the tunnelling
of an electron from a continuum reservoir to a discrete level in the dot.
Realizing the quantum Zeno effect in this scenario can be much harder because
the measurements should be repeated before the wave packet of the hole left
behind in the reservoir moves away from the vicinity of the dot. Thus, the
required repetition rate could be lowered by having a flat band (with a slow
group velocity) in resonance with the dot or a sufficiently small Fermi
velocity or a strong external magnetic field.

## Introduction

One of the major distinctions between classical and quantum physics is the
role of measurements. As a consequence, it is impossible to directly observe
quantum evolution taking place without actually affecting it. A striking
example is the quantum Zeno effect which describes slowing down or even
stopping quantum evolution by frequent measurements, see, e.g. misra1977a ;
itano1990a ; hackenbroich1998a ; fischer2001a ; streed2006a ; bernu2008a ;
slichter2016a .

Let us discuss the basic picture by means of a simple example, two discrete
states or levels $\left|1\right>$ and $\left|2\right>$ of equal energy which
are tunnel-coupled. Preparing the initial state in one level
$\left|\Psi(t=0)\right>=\left|1\right>$, the time-dependent probability
$P_{2}(t)=|\left<2|\Psi(t)\right>|^{2}$ for the other level reads
$P_{2}(t)=\sin^{2}(\gamma t)$, where the frequency $\gamma>0$ is given by the
tunneling strength. For short times $t\ll 1/\gamma$, we obtain a quadratic
growth $P_{2}(t)\approx\gamma^{2}t^{2}$, because in quantum physics amplitudes
instead of probabilities are added. However, if we measure the level
occupation after such a short time, i.e., within the quadratic-growth regime,
we project the state $\left|\Psi(t)\right>$ back to the initial state
$\left|1\right>$ (assuming a strong, i.e., projective measurement) with high
probability $P_{1}(t)\approx 1-\gamma^{2}t^{2}$ and thus the quantum evolution
has to start again afterwards. In other words, the amplitudes before and after
the measurement do no longer interfere constructively because we have obtained
which-way information via the measurement. Now, repeating this measurement
with a fast rate (much larger than $\gamma$) would effectively keep setting
back the quantum evolution such that the quantum state stays in the initial
level $\left|1\right>$. This inhibition (or slowing down) of quantum evolution
is usually referred to as the quantum Zeno effect misra1977a . Note that this
line of argument crucially relies on the quadratic growth
$P_{2}(t)\approx\gamma^{2}t^{2}$ discussed above. In case of a linear behavior
$P_{2}(t)\sim t$, for example, setting back the evolution by measurements
would not have this retarding effect.

The quantum Zeno effect is a striking example for quantum control facchi2001a
; facchi2002a ; facchi2004a ; wiseman2010 ; alvarez2010a ; paz_silva2012a ;
schaefer2014a ; barontini2015a  and closely related to passive quantum error
correction schemes (similar to the spin-echo method). Thus, this fascinating
phenomenon is of fundamental interest and has already been observed
experimentally in different systems, e.g., for ions in Penning traps
itano1990a , ultra-cold atoms in optical lattices fischer2001a ; zhu2014a ,
Bose-Einstein condensates in magnetic traps streed2006a ; schaefer2014a ,
Rydberg atoms in cavities bernu2008a  or superconducting qubits slichter2016a
.

On much shorter time scales, electrons in quantum dots are also very suitable
systems for studying the quantum Zeno effect: Their energy levels and
tunneling rates can be manipulated (e.g., by the variation of voltages that
are applied to suitably placed gate electrodes) or the size of the dot and
they have quite long coherence times while their states can be read out, i.e.,
measured, quite fast by capacitive charge detectors fujisawa2006a ;
flindt2009a  or optical transitions kurzmann2016a ; kurzmann2019a , for
example.

In most previous works, the quantum Zeno effect has been studied in a regime
where the aforementioned picture based on transitions between discrete levels
can be applied itano1990a ; hackenbroich1998a ; gurvitz2003a ; gurvitz2006a ;
segal2007a ; schaller2018a . In the following, we shall study the more
involved case of transitions between a discrete level and a continuum (a Fermi
gas or liquid), see also e.g. gurvitz1997a ; engelhardt2018a . As a concrete
experimental realization, we consider a quantum dot coupled to a reservoir in
the form of an effectively two-dimensional electron gas (2DEG) ando1982a ;
reimann2002b ; russ2004a ; russ2006a ; geller2019a .

## The Model

We consider the quantum dot-reservoir system in good approximation vdovin2000a
being described by the following many-body Hamiltonian ($\hbar=1$)

$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle\int
d^{2}r\left(\frac{(\mbox{\boldmath$\nabla$}\hat{\psi}_{1}^{\dagger})\cdot(\mbox{\boldmath$\nabla$}\hat{\psi}_{1})}{2m}+V_{1}(x,y)\hat{\psi}_{1}^{\dagger}\hat{\psi}_{1}+\right.$
(1)
$\displaystyle+\left.\frac{(\mbox{\boldmath$\nabla$}\hat{\psi}_{2}^{\dagger})\cdot(\mbox{\boldmath$\nabla$}\hat{\psi}_{2})}{2m}+\left(\gamma\hat{\psi}_{1}^{\dagger}\hat{\psi}_{2}+{\rm
h.c.}\right)\right)\,.$

The first line represents the quantum dot for which we employ the standard
warburton1998a  harmonic potential approximation
$V_{1}(x,y)=V_{0}+m\omega^{2}(x^{2}+y^{2})/2$ where the offset $V_{0}$ can be
tuned by a gate voltage. The second line describes the reservoir and its
tunnel coupling to the dot with the coupling strength $\gamma$. The field
operators $\hat{\psi}_{1}(x,y)$ and $\hat{\psi}_{2}(x,y)$ could be envisaged
as lowest quantum well states in $z$-direction, for example.

If we focus on the ground state of the quantum dot
$\hat{\psi}_{1}(x,y)\to\Phi_{0}(x,y)\hat{a}_{\rm d}$ and Fourier transform the
reservoir modes $\hat{\psi}_{2}(x,y)\to\hat{\psi}_{\mathbf{k}}$, the
Hamiltonian (1) becomes

$\displaystyle\hat{H}=\epsilon_{\rm d}\hat{n}_{\rm d}+\int
d^{2}k\left[\epsilon_{\mathbf{k}}\hat{\psi}_{\mathbf{k}}^{\dagger}\hat{\psi}_{\mathbf{k}}+\left(\gamma_{\mathbf{k}}\hat{a}_{\rm
d}^{\dagger}\hat{\psi}_{\mathbf{k}}+{\rm h.c.}\right)\right]\,,$ (2)

where $\epsilon_{\rm d}$ denotes the energy of the quantum dot level and
$\hat{n}_{\rm d}=\hat{a}_{\rm d}^{\dagger}\hat{a}_{\rm d}$ the corresponding
number operator, while $\epsilon_{\mathbf{k}}=\mathbf{k}^{2}/(2m)$ are the
single-particle energies of the reservoir modes $\mathbf{k}=(k_{x},k_{y})$.
Finally, $\gamma_{\mathbf{k}}$ is determined by the Fourier transform of the
ground-state wave-function inside the quantum dot (see e.g. wibbelhoff2005a ),
i.e., Gaussian

$\displaystyle\gamma_{\mathbf{k}}=\gamma\int\frac{d^{2}r}{2\pi}\Phi_{0}^{*}(x,y)e^{+{\rm
i}\mathbf{k}\cdot\mathbf{r}}=\gamma_{0}\exp\left\\{-\frac{1}{2}\mathbf{k}^{2}\ell^{2}\right\\}\,,$
(3)

where $\ell=\sqrt{\hbar/(m\omega)}$ is the characteristic dot length scale.
Note that the bi-linear Hamiltonian (1) does not explicitly include non-linear
(e.g., Coulomb) interactions between the electrons nor their coupling to other
degrees of freedom (e.g., phonons) whose potential impact will be discussed
below. Therefore, we restrict ourselves to studying the transition between an
empty and a singly-occupied dot, for which a charging energy of the quantum
dot is irrelevant.

## Quantum Zeno Dynamics

Now we are in the position to study the tunneling process of an electron from
the reservoir into the quantum dot. We start with the initial state $\left|\rm
in\right>$ where the dot is empty $\hat{a}_{\rm d}\left|\rm in\right>=0$ while
the reservoir is filled up to the Fermi energy.

Although the above model can be solved exactly using e.g. Greens function
techniques haug2008 , we may already calculate the time-dependent probability
$P(t)$ for the occupation of the quantum dot using standard time-dependent
perturbation theory with respect to $\gamma$,

$\displaystyle P(t)=\langle\hat{n}_{\rm d}(t)\rangle=4\int\limits_{\rm
F}d^{2}k\;|\gamma_{\mathbf{k}}|^{2}\frac{\sin^{2}([\epsilon_{\rm
d}-\epsilon_{\mathbf{k}}]t/2)}{(\epsilon_{\rm
d}-\epsilon_{\mathbf{k}})^{2}}\,,$ (4)

where the subscript F on the integral indicates that the integration runs up
to Fermi energy $\epsilon_{\rm F}$.

## Quantum-Zeno Regime

Even within the region of applicability of perturbation theory (i.e., for
small probabilities $P(t)\ll 1$), we obtain two temporal regimes. For small
times $t$, the probability $P(t)$ grows quadratically

$\displaystyle P(t)=t^{2}\int\limits_{\rm
F}d^{2}k\;|\gamma_{\mathbf{k}}|^{2}+\,{\cal O}(t^{3})\,.$ (5)

For the Gaussian case (3), this simplifies to

$\displaystyle
P(t)=\pi\,\frac{\gamma_{0}^{2}t^{2}}{\ell^{2}}\left(1-\exp\left\\{-k^{2}_{\rm
F}\ell^{2}\right\\}\right)+\,{\cal O}(t^{3})\,,$ (6)

where $k_{\rm F}=\sqrt{2m\epsilon_{\rm F}}$ denotes the Fermi momentum. If the
Fermi energy $\epsilon_{\rm F}$ is large enough $2m\epsilon_{\rm F}\ell^{2}\gg
1$, i.e., $k^{2}_{\rm F}\ell^{2}\gg 1$, we obtain
$P(t)=\pi\gamma_{0}^{2}t^{2}/\ell^{2}$. In the opposite limit $k^{2}_{\rm
F}\ell^{2}\ll 1$, the integration is cut off by $k^{2}_{\rm F}$ and we find
$P(t)=\pi\gamma_{0}^{2}t^{2}k^{2}_{\rm F}$.

## Regime of Fermi’s Golden Rule

For later times (but still in the perturbative regime $P(t)\ll 1$), we can use
the standard approximation of the sinc function in Eq. (4) by a Dirac delta
function and obtain a linear growth in time, consistent with Fermi’s Golden
Rule

$\displaystyle P(t)\approx 2\pi t\int\limits_{\rm
F}d^{2}k\;|\gamma_{\mathbf{k}}|^{2}\delta(\epsilon_{\rm
d}-\epsilon_{\mathbf{k}})\,.$ (7)

Obviously, the above integral vanishes for $\epsilon_{\rm d}>\epsilon_{\rm F}$
and for $\epsilon_{\rm d}<0$, i.e., if there are no filled reservoir modes in
resonance with the dot. Assuming $\epsilon_{\rm F}>\epsilon_{\rm d}>0$, we
find for the Gaussian case (3) with
$\epsilon_{\mathbf{k}}=\mathbf{k}^{2}/(2m)$

$\displaystyle
P(t)\approx(2\pi)^{2}tm\gamma_{0}^{2}\exp\left\\{-2m\epsilon_{\rm
d}\ell^{2}\right\\}\,.$ (8)

The exponential suppression $\exp\left\\{-2m\epsilon_{\rm d}\ell^{2}\right\\}$
for large dot energies $2m\epsilon_{\rm d}\ell^{2}\gg 1$ stems from the wave-
function mismatch between the Gaussian ground-state wave-function of the dot
and the reservoir wave-functions of the corresponding energy $\epsilon_{\rm
d}$ which are rapidly oscillating for large $\epsilon_{\rm d}$. In order to
avoid this suppression, we assume small dot energies $2m\epsilon_{\rm
d}\ell^{2}\ll 1$ in the following.

As a peculiarity of the case of two spatial dimensions, the remaining pre-
factor $P(t)\approx(2\pi)^{2}tm\gamma_{0}^{2}$ is actually independent of
$\epsilon_{\rm d}$ because the energy
$\epsilon_{\mathbf{k}}=\mathbf{k}^{2}/(2m)$ and the volume factor $d^{2}k$
both display the same quadratic scaling in $k$, such that the density of
states per energy is constant.

## Crossover Time

Having found an initial period of quadratic growth (5) for early times
(quantum Zeno regime) followed by a period of linear growth (7) for later
times (regime of Fermi’s Golden Rule), we may estimate the crossover time
$T_{*}$ marking the transition between the two regimes sangeetha  by comparing
Eqs. (5) and (7). For the Gaussian limit we obtain with (6) and (8)

$\displaystyle T_{*}\approx 4\pi m\ell^{2}\frac{\exp\left\\{-2m\epsilon_{\rm
d}\ell^{2}\right\\}}{1-\exp\left\\{-\ell^{2}k^{2}_{\rm F}\right\\}}\,.$ (9)

Assuming small dot excitation energies $2m\epsilon_{\rm d}\ell^{2}\ll 1$ and
considering the limiting cases as discussed after Eq. (6), this simplifies to
$T_{*}\approx 4\pi m\ell^{2}$ for $\ell^{2}k^{2}_{\rm F}\gg 1$ and to
$T_{*}\approx 4\pi m/k^{2}_{\rm F}$ for $\ell^{2}k^{2}_{\rm F}\ll 1$,
respectively.

As explained in the Introduction, only measurements with a repetition time
faster than $T_{*}$ can induce the quantum Zeno effect. This time scale is
typically quite short and can be visualized by the following intuitive
picture.

Let us first consider the case $\ell^{2}k_{\rm F}^{2}\gg 1$. If an electron
tunnels with a small amplitude from the reservoir to the quantum dot, it
leaves behind a hole in the reservoir. The typical size of the initial wave
packet of this hole is determined by the characteristic length $\ell$ of the
ground-state wave function of the dot. Afterwards, this wave packet is
spreading out or moving away with the typical group velocity at that length
scale $v_{\rm group}\propto 1/(\ell m)$. Once the wave packet has spread out
too much or moved away too far, further tunneling amplitudes would not be be
added coherently and probabilities would add up instead. This is precisely the
transition from the quantum Zeno regime to the regime of regime of Fermi’s
Golden Rule occurring at the crossover time $T_{*}\sim\ell/v_{\rm group}$.

In the opposite limit ($k_{\rm F}^{2}\ell^{2}\ll 1$), one should just replace
the length scale $\ell$ of the dot by the Fermi length $\ell_{\rm F}\propto
1/k_{\rm F}$ and the associated group velocity by the Fermi velocity $v_{\rm
F}=k_{\rm F}/m$. Then, the crossover time $T_{*}$ is basically the Fermi time
$T_{\rm F}$ which is related to the Fermi energy $\epsilon_{\rm F}$ by
Heisenberg’s uncertainty relation [$T_{*}=\,{\cal O}\\{\epsilon_{\rm
F}^{-1}\\}$] and determines the minimum response time of hole states in the
reservoir.

## General Band Structure

Let us now discuss possible generalizations of the above results. For
reservoir electrons propagating in a lattice, the single-particle energies
$\epsilon_{\mathbf{k}}$ of the reservoir modes $\mathbf{k}$ should be replaced
by the lattice band structure, which might also modify the (experimentally
accessible zhou2015a ) couplings $\gamma_{\mathbf{k}}$ accordingly. Of course,
the same would happen for non-harmonic quantum dot potentials $V_{1}(x,y)\neq
V_{0}+m\omega^{2}(x^{2}+y^{2})/2$. Nevertheless, one would still expect the
effective Hamiltonian (2) to provide a good approximation. Thus, the results
(4) and thus (5) and (7) remain valid, but with modified
$\epsilon_{\mathbf{k}}$ and $\gamma_{\mathbf{k}}$. This may entail interesting
consequences. For example, if we have a flat band (with
$\epsilon_{\mathbf{k}}=\rm const$) below the Fermi energy in resonance with
the quantum dot (i.e., $\epsilon_{\mathbf{k}}=\rm const=\epsilon_{\rm d}$),
the probability (7) per time would become very large while the initial growth
(5) would remain almost unaffected since it is independent of
$\epsilon_{\mathbf{k}}$. This means that the crossover time $T_{*}$ would also
become very large, which might help observing the quantum Zeno effect in this
scenario. In terms of the intuitive picture sketched above, the group velocity
becomes very small for such a flat band such that the wave packet stays quite
long in the vicinity of the dot.

## Magnetic Field

The above finding regarding flat or nearly flat bands motivates the study of a
strong external magnetic field perpendicular to the reservoir, because this
does also turn the reservoir modes into flat bands in the form the well-known
Landau levels.

As usual, we represent the constant perpendicular magnetic field
$\mathbf{B}=B\mathbf{e}_{z}$ in the symmetric gauge by the vector potential
$\mathbf{A}=\mathbf{B}\times\mathbf{r}/2$. Then, after minimal coupling
$\mbox{\boldmath$\nabla$}\to\mbox{\boldmath$\nabla$}-q\mathbf{A}$ to the
electron charge $q$, the Hamiltonian (1) acquires an additional angular-
momentum contribution
$-2\mathbf{A}\cdot\mbox{\boldmath$\nabla$}=\mathbf{B}\cdot\hat{\mathbf{L}}=B\hat{L}_{z}$
as well as a harmonic confinement potential
$V_{B}=\Omega_{B}^{2}(x^{2}+y^{2})/8$ from the quadratic term
$\mathbf{A}^{2}=B^{2}(x^{2}+y^{2})/4$ where $\Omega_{B}$ is the cyclotron
frequency $\Omega_{B}=qB/m$.

Thus, the reservoir eigen-functions are no longer plane waves as in Eq. (2),
but discrete Landau levels which can be represented by the modes of a two-
dimensional harmonic oscillator in polar coordinates $r$ and $\varphi$
ciftja2020a

$\displaystyle\psi_{n,l}(r,\varphi)=f_{n,l}(r)e^{il\varphi}\,.$ (10)

Here $n\in\mathbb{N}$ and $l\in\mathbb{Z}$ are the quantum numbers
corresponding to energy $\epsilon_{n}=\Omega_{B}(n+1/2)$ and angular momentum
$\hat{L}_{z}$, respectively. The radial mode functions $f_{n,l}(r)$ are given
by polynomials multiplied by a Gaussian $\exp\\{-r^{2}/(2\ell_{B})^{2}\\}$
with the magnetic length $\ell_{B}=1/\sqrt{qB}$. For example, the lowest
Landau levels with $n=0$ have $f_{n=0,l}(r)\propto
r^{l}\exp\\{-r^{2}/(2\ell_{B})^{2}\\}$ with $l\geq 0$, which can be
experimentally mapped russ2004a ; lei2010a .

The ground-state wave function of the quantum dot will also be modified, but
merely by the additional harmonic confinement potential
$V_{B}=\Omega_{B}^{2}(x^{2}+y^{2})/8$ which narrows the Gaussian wave-function
(i.e., decreases $\ell$) and increases the energy $\epsilon_{\rm d}$. Since
this wave function is rotationally symmetric (i.e., has zero angular
momentum), it only tunnel couples to reservoir modes with $l=0$.

If we now assume a strong magnetic field $B$ and/or a small Fermi energy
$\epsilon_{\rm F}<3\Omega_{B}/2$ such that only the lowest Landau levels are
occupied, the effective Hamiltonian (2) can be restricted to the mode $n=l=0$
and reads

$\displaystyle\hat{H}=\epsilon_{\rm d}\hat{n}_{\rm
d}+\frac{\Omega_{B}}{2}\hat{n}_{0,0}+\left(\gamma_{0,0}\hat{a}_{\rm
d}^{\dagger}\hat{a}_{0,0}+\rm h.c.\right)\,,$ (11)

where the effective tunnel coupling $\gamma_{0,0}$ is determined by the
overlap of the two Gaussian wave functions of the dot and
$\psi_{0,0}\propto\exp\\{-r^{2}/(2\ell_{B})^{2}\\}$.

Assuming resonance $\epsilon_{\rm d}=\Omega_{B}/2$, this Hamiltonian (11) is
then formally equivalent to tunneling between two discrete modes (described by
$\hat{a}_{\rm d}$ and $\hat{a}_{0,0}$) such that we are back to the original
picture of the quantum Zeno effect as described in the Introduction.

## Measurement Model

So far, the Hamiltonian (1) does only describe the internal dynamics of the
electrons, but not the actual measurement process – which is causing the
quantum Zeno effect. To incorporate this, let us construct a simple toy model
for the read out. We assume that the dot is strongly illuminated by laser
light (saturation regime) which quickly transfers the electron in the quantum
dot from the ground state level to an excited level $\hat{a}_{\rm
d}\hat{a}_{\rm e}^{\dagger}$ as soon as the dot is occupied. This excited
level then rapidly decays by emitting a photon as described by the bosonic
creation and annihilation operators $\hat{b}^{\dagger}$ and $\hat{b}$, where
we focus on one mode for simplicity. Assuming that these excitation–de-
excitation cycles occur much faster than all the other time scales considered
above (e.g., $T_{*}$), we may use an effective description where the dot emits
photons as soon as it is occupied. Thus, we model the measurement by the
additional Hamiltonian

$\displaystyle\hat{H}_{\rm measure}=\hat{n}_{\rm
d}\left(i\zeta\hat{b}^{\dagger}+{\rm h.c.}\right)$ (12)

with the detector coupling strength $\zeta>0$, which is basically a pointer
Hamiltonian for measuring the observable $\hat{n}_{\rm d}$.

Since $\hat{H}_{\rm measure}$ commutes with the undisturbed Hamiltonian
$\hat{H}_{0}=\epsilon_{\rm d}\hat{n}_{\rm d}+\int
d^{2}k\,\epsilon_{\mathbf{k}}\hat{\psi}_{\mathbf{k}}^{\dagger}\hat{\psi}_{\mathbf{k}}$,
we may incorporate it easily into the time-dependent perturbation theory used
above for calculating $P(t)$. In the interaction picture, the dot annihilation
operator $\hat{a}_{\rm d}$ just acquires an additional operator-valued factor

$\displaystyle\hat{a}_{\rm d}(t)=e^{-i\epsilon_{\rm
d}t}\exp\left\\{(\zeta\hat{b}^{\dagger}-{\rm h.c.})t\right\\}\hat{a}_{\rm
d}(0)\,,$ (13)

which represents the dynamics induced by $H_{0}+H_{\rm measure}$. When acting
on the initial photonic vacuum state $\left|0\right>$ with
$\hat{b}\left|0\right>=0$, this additional factor
$\exp\\{(\zeta\hat{b}^{\dagger}-{\rm h.c.})t\\}$ just generates a coherent
state $\left|\alpha(t)\right>$ of the photon field whose amplitude
$\alpha(t)=\zeta t$ grows linearly with time $t$.

The overlap of these coherent states at different times
$\left<\alpha(t_{1})|\alpha(t_{2})\right>=e^{-\zeta^{2}(t_{1}-t_{2})^{2}/2}$
modifies Eq. (4) via

$\displaystyle P(t)$ $\displaystyle=$
$\displaystyle\int\limits_{0}^{t}dt_{1}\int\limits_{0}^{t}dt_{2}\int\limits_{\rm
F}d^{2}k\;|\gamma_{\mathbf{k}}|^{2}\exp\left\\{-\frac{\zeta^{2}}{2}(t_{1}-t_{2})^{2}\right\\}$
(14) $\displaystyle\times\exp\\{-i(\epsilon_{\rm
d}-\epsilon_{\mathbf{k}})(t_{1}-t_{2})\\}\,.$

For very short times $t\ll T_{*}$ and $t\ll 1/\zeta$, we recover the initial
quadratic growth in Eq. (5). What happens after that initial period depends on
the quantity $\zeta T_{*}$. For small $\zeta T_{*}\ll 1$, we recover Eq. (7).
For large $\zeta T_{*}\gg 1$, on the other hand, the probability is strongly
suppressed

$\displaystyle P(t)\sim t\,\frac{\sqrt{2}}{\zeta}\int\limits_{\rm
F}d^{2}k\;|\gamma_{\mathbf{k}}|^{2}\,,$ (15)

which is a manifestation of the quantum Zeno effect by frequent measurements
with the effective rate $\zeta$.

The detector signal can be very explicitly evaluated with methods of Full
Counting Statistics or quantum polyspectra schaller2018a ; sifft2021a .
However, already the analysis of average values reveals (see supplement) that
– although delay effects occur – a transition between the timescales can be
directly resolved.

## Conclusions

For the example of a quantum dot tunnel coupled to an effectively two-
dimensional electron reservoir, we study whether and how the quantum Zeno
effect can be employed to suppress tunneling between a discrete level and a
continuum. In contrast to tunneling between two discrete levels, we find that
the required measurement repetition rate is set by the time $T_{*}$ it takes
the wave-packet in the reservoir to move away from the vicinity of the quantum
dot. Hence this time scale can be increased (and thus the required repetition
rate decreased) by lowering the Fermi energy or having a flat band (thus
reducing the Fermi velocity) or by applying a strong perpendicular magnetic
field, which effectively localizes the reservoir modes.

In a bigger picture, suppressing tunneling via the quantum Zeno effect and
thereby effectively isolating the quantum dot from its environment is quite
analogous to passive quantum error correcting schemes, such as the spin echo
method. In contrast to active quantum error correcting schemes (typically
involving redundancies), such passive schemes are based on interrupting the
coherence between the quantum dot or qubit and its environment such that the
constructive interference of the error amplitudes is turned into a (at least
partially) destructive interference. As an intuitive picture, these methods
work well as long as the wave-packet associated with the quantum error is
still “lurking” in the vicinity of the quantum dot or qubit. Once they moved
too far away, the impact of measurements or echo sequences is strongly
reduced.

In order to discuss the experimental relevance of our findings, let us insert
typical experimental parameters. For an effective mass of around $7~{}\%$ of
the electron mass fricke1996a  and a 2D electron density between $3$ and
$6\times 10^{11}\,\rm cm^{-2}$, we get Fermi energies between $10$ and
$20~{}\rm meV$ and Fermi velocities between $2$ and $4\times 10^{5}\,\rm m/s$.
Assuming a typical level spacing in the quantum dot of around $50~{}\rm meV$
(fricke1996a, ; warburton1998a, ), the characteristic length scale of the
ground state wave function is $\ell\approx 5~{}\rm nm$. With a Fermi momentum
of order $10^{8}\,\rm m^{-1}$, we have $k_{\rm F}\ell$ of order unity. Thus,
we may estimate the crossover time $T_{*}$ by comparing the Fermi velocity
with the length scale $\ell\approx 5~{}\rm nm$ which yields rather short times
of order $T_{*}=\,{\cal O}(10~{}\rm fs)$. Unfortunately, although measurements
can be performed quite fast (on the nano-second scale kurzmann2019a ), this
$T_{*}=\,{\cal O}(10~{}\rm fs)$ is probably too short to observe the quantum
Zeno effect in this set-up. Increasing this time scale $T_{*}$ by decreasing
the Fermi energy (i.e., reducing the electron density in the reservoir) or by
using materials with a flat band is possible in principle, but experimentally
quite challenging.

Thus, probably the most viable option is to apply a strong perpendicular
magnetic field $B$, say, of around $10~{}\rm T$, such that the 2DEG is in the
Quantum Hall regime 111In the Quantum Hall regime, small residual electric
fields within the 2DEG that originate e.g. from interface roughness or
electrostatic potential fluctuations introduced by the QDs will result in a
drift of the wave packet in crossed $E$ and $B$ fields. The drift velocity
$\mbox{\boldmath$v$}_{\rm
drift}=c(\mbox{\boldmath$E$}\times\mbox{\boldmath$B$})$ is, however, much
lower than the Fermi velocity in the original, field-free 2DEG. Thus, the
problem of a temporally decreasing overlap is diminished. The resulting
magnetic length of $\ell_{B}\approx 8~{}\rm nm$ is roughly of the same order
as the length scale $\ell\approx 5~{}\rm nm$ associated with the quantum dot
and similarly the Landau level spacing of about $15~{}\rm meV$ is not too far
away from the other energy scales. With aligning the energy levels
accordingly, one can arrange resonant tunnel transitions between the discrete
level of the quantum dot and one of the discrete Landau levels in the
reservoir. In this case, the line of arguments sketched in the Introduction
can be applied and thus the measurement repetition rate is set by the
tunneling rate $\gamma$ between these two levels. Since the time associated
scale $1/\gamma$ can be quite long (in the micro- to milli-second regime or
even longer, which would also enable an electric readout fujisawa2006a ;
flindt2009a ), a measurement time in the nano-second regime should be
sufficient to observe the quantum Zeno effect.

In order to enable such an observation, one has to make sure that the
environmental decoherence (which can have basically the same effect as a
measurement; for a review of tunneling in a dissipative environment see e.g.
caldeira1983a ) is weak enough, i.e., the associated coherence time is longer
than the time between measurements. A frequently discussed decoherence
mechanism is the scattering of the electrons or holes in the reservoir at
local impurities. As long as these impurities (which can be characterized
experimentally kerski2021a ) are not too dense and thus far away from the dot,
this mechanism is not relevant here because, as explained above, if the
reservoir wave packet moved away from the dot far enough to see the impurity,
it is already outside the region of applicability of the quantum Zeno effect.
Another potentially important mechanism stems from the Coulomb interaction
between the electrons or holes or their interaction with phonons. In order to
suppress this decoherence mechanism, the temperature should be low enough such
that the available phase space and number of excitations is reduced.

Also in this respect, working in a strong magnetic field can be helpful: While
electron-phonon scattering in a field-free 2DEG happens in two-dimensional
$k$-space, in the Quantum Hall regime electron-phonon scattering is limited to
the one-dimensional (both in real-space and $k$-space) lines along which the
electrons drifts. This reduced dimensionality strongly suppresses electron-
phonon scattering and renders it negligible at milli-Kelvin temperatures. At
these temperatures $T$, electron-electron scattering is the only relevant
mechanism and follows a $T^{2}$ law altshuler1982a  in a 2DEG in a strong
magnetic field. Experimental studies of the universality in the Quantum Hall
regime koch1992a  allow one to estimate the associated time scale.

###### Acknowledgements.

R.S. acknowledges valuable discussions with S. Popescu and W.G. Unruh. Funded
by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –
Project-ID 278162697– SFB 1242.

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##  Appendix A Detector dynamics

To evaluate the detector occupation, we first note that in the interaction
picture, the photon annihilation operator grows linearly in time

$\displaystyle\hat{b}(t)=\hat{b}(0)+\zeta t\hat{n}_{\rm d}(0)\,.$ (16)

The actually emitted photon number of our toy model then becomes to second
order perturbation theory

$\displaystyle\langle\hat{b}^{\dagger}\hat{b}\rangle_{t}$ $\displaystyle=$
$\displaystyle\zeta^{2}\int\limits_{0}^{t}dt_{1}\int\limits_{0}^{t}dt_{2}\int\limits_{\rm
F}d^{2}k\;|\gamma_{\mathbf{k}}|^{2}\exp\left\\{-\frac{\zeta^{2}}{2}(t_{1}-t_{2})^{2}\right\\}$
(17) $\displaystyle\times(t-t_{1})(t-t_{2})\exp\\{-i(\epsilon_{\rm
d}-\epsilon_{\mathbf{k}})(t_{1}-t_{2})\\}\,,$

which also reflects the crossover from a $t^{4}$ scaling for $t\ll T_{*}$ and
$t\zeta\ll 1$ to a reduced – at most $t^{3}$ – scaling for $t\gg T_{*}$. The
initial delayed response can be understood as the detector has to react to the
dynamics of the system first.

The expectation value of a photon annihilation operator (relevant e.g. for
homodyne measurements) would become instead

$\displaystyle\langle\hat{b}\rangle_{t}$ $\displaystyle=$
$\displaystyle\zeta\int\limits_{0}^{t}dt_{1}\int\limits_{0}^{t}dt_{2}\int\limits_{\rm
F}d^{2}k\;|\gamma_{\mathbf{k}}|^{2}\exp\left\\{-\frac{\zeta^{2}}{2}(t_{1}-t_{2})^{2}\right\\}$
(18) $\displaystyle\times(t-t_{2})\exp\\{-i(\epsilon_{\rm
d}-\epsilon_{\mathbf{k}})(t_{1}-t_{2})\\}$

and exhibits a crossover from a cubic to (at most) quadratic scaling.

Considering the effect of a finite detector frequency $\hat{H}_{\rm
measure}\to\hat{H}_{\rm measure}+\omega_{\rm det}\hat{b}^{\dagger}\hat{b}$
implies an oscillatory behaviour of $\hat{b}(t)$ in the interaction picture
and would eventually again lead to a linear long-term scaling in the Fermi
golden rule regime.

##  Appendix B Measuring empty dots

If the detector would be sensitive to the empty state, i.e.,

$\displaystyle\hat{H}_{\rm measure}=\hat{a}_{\rm d}\hat{a}_{\rm
d}^{\dagger}\left({\rm i}\zeta\hat{b}^{\dagger}+{\rm h.c.}\right)$ (19)

instead of (12) in the main text, we would get in the interaction picture

$\displaystyle\hat{a}_{\rm d}(t)$ $\displaystyle=$ $\displaystyle
e^{-i\epsilon_{\rm d}t}\exp\left\\{-(\zeta\hat{b}^{\dagger}-{\rm
h.c.})t\right\\}\hat{a}_{\rm d}(0)\,,$ $\displaystyle\hat{b}(t)$
$\displaystyle=$ $\displaystyle\hat{b}(0)+\zeta t\hat{a}_{\rm d}(0)a_{\rm
d}^{\dagger}(0)\,.$ (20)

Since in the first line, we have only a flip in sign of $\zeta$ in comparison
to (13) in the main text, the arguments raised there for the dot occupation
prevail in this case and we retain (14) in the main text (assuming of course
the same initial conditions).

This is different however for the detector occupation. In this case, due to
${\rm
Tr}\left\\{\hat{b}^{\dagger}(t)\hat{b}(t)\rho_{0}\right\\}=\zeta^{2}t^{2}$, a
detector signal is seen already for $\gamma_{\mathbf{k}}=0$ for an initially
empty dot. Up to second order, we get

$\displaystyle\langle\hat{b}^{\dagger}\hat{b}\rangle_{t}$ $\displaystyle=$
$\displaystyle\zeta^{2}t^{2}$
$\displaystyle+\zeta^{2}\int\limits_{0}^{t}dt_{1}\int\limits_{0}^{t}dt_{2}\int\limits_{\rm
F}d^{2}k\;|\gamma_{\mathbf{k}}|^{2}\exp\left\\{-\frac{\zeta^{2}}{2}(t_{1}-t_{2})^{2}\right\\}$
$\displaystyle\times[t_{1}t_{2}+t|t_{1}-t_{2}|-t^{2}]\exp\\{-i(\epsilon_{\rm
d}-\epsilon_{\mathbf{k}})(t_{1}-t_{2})\\}\,,$

where the temporal crossover of the second order contribution is much harder
to observe as the first order contribution is dominant. Thus, to see Zeno
signatures in such a detector it would be more suitable to consider the
reverse process of an electron tunnelling out of the dot.