# Facilitation of controllable excitation in Rydberg atomic ensembles††thanks:
The National Science Foundation of China (Grants No. 12174106, No. 11474094),
and the Science and Technology Commission of Shanghai Municipality (Grant No.
18ZR1412800).

Han Wang  Jing Qian
State Key Laboratory of Precision Spectroscopy, Department of Physics, School
of Physics
and Electronic Science, East China Normal University, Shanghai 200062, China
E-mail<EMAIL_ADDRESS>
###### Abstract

Strongly-interacting Rydberg atomic ensembles have shown intense collective
excitation effects due to the inclusion of single Rydberg excitation shared by
multiple atoms in the ensemble. In this paper we investigate a counter-
intuitive Rydberg excitation facilitation with a strongly-interacting atomic
ensemble in the strong probe-field regime, which is enabled by the role of a
control atom nearby. Differing from the case of a single ensemble, we show
that, the control atom’s excitation adds to a second two-photon transition
onto the doubly-excited Rydberg state, arising an excitation facilitation for
the ensemble atoms. Our numerical studies depending on the method of quantum
Monte Carlo wavefunction, exhibit the observation constraints of this
excitation facilitation effect under practical experimental conditions. The
results obtained can provide a flexible control for the excitation of Rydberg
atomic ensembles and participate further uses in developing mesoscopic Rydberg
gates for multiqubit quantum computation.

Keywords: Excitation facilitation; Rydberg atom; Monte Carlo Wave Function;
many-body system

PACS: 33.80.Rv, 32.80.Qk, 05.10.Ln, 73.20.Mf, 45.50.Jf

##  1 Introduction

Collective excitation enhancement associated with strongly-interacting Rydberg
atoms constitutes the basis for versatile applications in observing e.g. many-
body effects [1, 2, 3, 4], large-scale quantum computation [5, 6, 7, 8],
quantum entanglement [9, 10, 11], generation of single photons [12, 13],
collective emission of photons [14], etc. The manipulation of a strongly-
interacting atomic ensemble can enable the implementation of single ensemble
qubit gate [15], a many-particle GHZ state [16, 17, 18], a photon reflection
phase [19], and the creation of control-ensemble entanglement [20]. So far,
many of these achievements depend on the Rydberg blockade mechanism since it
causes a single atom’s excitation shared by $N$ ensemble atoms within the
range of blockade radius [21, 22, 23, 24]. This fully-blockaded atomic
ensemble (or so-called a superatom) can be described by a reduced three-level
structure using symmetric Dicke states [25], and benefits from a
$\sqrt{N}$-enhancement of the weak probe strength in the collective excitation
environment [26, 27, 28].

Although the superatom model could fundamentally exhibit the mechanism of
Rydberg blockade, it only works with a perfect eletromagnetically induced
transparency condition where the probe strength is much weaker than that of
the coupling one [29]. A stronger probe field will lead to unexpected multi-
photon processes associated with various intermediate and Rydberg states,
which renders the incoherent dissipative decays onto other asymmetric
collective states non-negligible [31, 32]. It is therefore important to
completely capture the feature of collective Rydberg excitation with an
improved superatom model, which is beyond the weak probe regime [33].

Motivated by a pioneering work in [34], Gärttner and co-authors discovered a
novel collective excitation enhancement effect in the weak probe regime
because of the multi-photon transition among collective Dicke states. In the
present work, we extend this mechanism by investigating the collective
excitation enhancement of Rydberg-ensemble atoms with the help of a control
atom nearby. This enhancement effect is found to occur beyond the weak probe
regime where the probe strength is comparable to or larger than the coupling
strength [35]. We show that, due to the use of a co-excited control atom, the
coupled control-ensemble system would obtain an auxiliary excitation channel
towards the doubly-excited Rydberg state. When this state is adjusted to be
resonantly coupled via an antiblockade facilitation, the steady-state Rydberg
population will redistribute between the singly- and doubly-excited Rydberg
states, exhibiting an apparent excitation enhancement effect [36, 37, 38, 39].
By utilizing the quantum Monte Carlo wavefunction method [40], we numerically
verify that, this facilitated excitation can persistently exist with strong
probe drivings. By using realistic parameters from experimental setups, we
show the regimes where the excitation facilitation effect can happen as long
as the temperature of atoms and the control-ensemble distance are
appropriately determined.

##  2 Theoretical Strategy

###  2.1 Atomic ensemble and Dicke states

Figure 1: Controllable excitation facilitation in a coupled control-ensemble
system. (a) Full level scheme of a control atom
$\\{|g_{c}\rangle,|e_{c}\rangle,|r_{c}\rangle\\}$ [see (a2)] interacting with
a strongly-blockaded Rydberg atomic ensemble which is described by five
symmetric collective states
$\\{|G\rangle,|E^{1}R^{0}\rangle,|E^{0}R^{1}\rangle,|E^{2}R^{0}\rangle,|E^{1}R^{1}\rangle\\}$
[see (a3)]. The cascade excitation obeying the transition of
$|g_{c}G\rangle\to|g_{c}E^{1}R^{0}\rangle\to|g_{c}E^{0}R^{1}\rangle\to|e_{c}E^{0}R^{1}\rangle\to|r_{c}E^{0}R^{1}\rangle$(marked
by the blue-shaded and red-shaded areas), is facilitated by an exact
antiblockade condition $\delta=-U(\bm{r}_{j})$ with respect to the doubly-
excited Rydberg state $|r_{c}E^{0}R^{1}\rangle$. Insets: Level schemes and
atom-light couplings with (a1) the $j$th ensemble atom, (a2) the control atom
and (a3) the atomic ensemble.

We consider a strongly-interacting atomic ensemble within the blockade volume
which is composed by $N$ three-level atoms [see Fig.1(a1)]. For the $j$th
ensemble atom, the ground state $|g_{j}\rangle$ is off-resonantly coupled to
an intermediate state $|e_{j}\rangle$ with Rabi frequency $\Omega_{p}$ and
frequency detuning $\Delta$. A second laser resonantly drives the subsequent
transition between $|e_{j}\rangle$ and a Rydberg level $|r_{j}\rangle$ with
Rabi frequency $\Omega_{c}$. The strong intraspecies interactions $U_{rr}$
would lead to a perfect dipole blockade effect that accommodates just one
Rydberg excitation here [41, 42]. Also, the Hamiltonian is invariant under the
exchange of particles since all the ensemble atoms are identical. Therefore
the atomic ensemble can be described by a set of collective symmetric Dicke
states $\\{|G\rangle,|E^{m}R^{0}\rangle,|E^{m-1}R^{1}\rangle\\}$ [43], in
which

$|G\rangle=|g_{1},g_{2},...,g_{N}\rangle$ (1)

stands for all atoms being in the ground state. When the probe field is not so
strong it is reasonable to assume at most one ensemble atom can be pumped into
intermediate state $|e_{j}\rangle$ via a one-photon excitation, arising

$|E^{1}R^{0}\rangle=\frac{1}{\sqrt{N}}(\sum_{j=1}^{N}|e_{j}\rangle\langle
g_{j}|)|G\rangle.$ (2)

However for a stronger probe field with more photons inside the ensemble,
other states like
$|E^{2}R^{0}\rangle,|E^{3}R^{0}\rangle,|E^{4}R^{0}\rangle...$ are also
essential [44]. We have to consider a general form as

$|E^{m}R^{0}\rangle=\frac{1}{\sqrt{N_{e}^{m}}}(\sum_{j=1}^{N}|e_{j}\rangle\langle
g_{j}|)^{m}|G\rangle$ (3)

with $N_{e}^{m}=N!m!/{(N-m)!}$. State $|E^{m}R^{0}\rangle$ indicates $m$ atoms
in the bare state $|e_{j}\rangle$ and $m\in[0,N]$. Additionally, state

$|E^{m-1}R^{1}\rangle=\frac{1}{\sqrt{N_{r}^{m}}}(\sum_{j=1}^{N}|e_{j}\rangle\langle
g_{j}|)^{m-1}(\sum_{j=1}^{N}|r_{j}\rangle\langle g_{j}|)|G\rangle$ (4)

with $N_{r}^{m}=NN!(m-1)!/(N-m+1)!$, means one atom being excited into the
Rydberg state $|r_{j}\rangle$ while others are either in the ground or
intermediate states. If $m=0$, $|E^{m-1}R^{1}\rangle$ is not present. Here we
treat the targeted atomic ensemble represented by symmetric Dicke states that
accommodates only one Rydberg excitation due to the Rydberg blockade, so state
$|E^{m-2}R^{2}\rangle$ is not present.

In addition, note that under the condition of $\Delta\gg\Omega_{p}$ (the probe
field is not so strong), the transitions into states $|E^{>1}R^{1}\rangle$ and
$|E^{>2}R^{0}\rangle$ are also negligible [45]. See Fig.2a(inset), we have
numerically verified that the probability of multi-excitation onto these
states is always below $10^{-5}$. Therefore, the atomic ensemble can be safely
described by five collective states
$\\{|G\rangle,|E^{1}R^{0}\rangle,|E^{0}R^{1}\rangle,|E^{2}R^{0}\rangle,|E^{1}R^{1}\rangle\\}$
[Fig.1(a3)], where the corresponding transition coefficients as shown in the
figure, depend on the number of atoms $N$ in the ensemble. Based on such an
atomic ensemble, a collective excitation enhancement effect has been observed
due to the coherent multi-photon couplings among collective states [34]. While
this finding occurs only in the weak probe regime
$0.15\leq\Omega_{p}/\Omega_{c}\leq 0.4$ where a small $\Delta$ value is
required.

###  2.2 Control-ensemble system

Now we introduce another three-level control atom
$\\{|g_{c}\rangle,|e_{c}\rangle,|r_{c}\rangle\\}$, individually addressed near
the atomic ensemble [Fig.1(a2)]. In this manner, single control atom and a
mesoscopic atomic ensemble can be confined in two separate optical tweezers
[46]. The way to load an optical tweezer with a desired number of atoms can
adopt the probabilistic loading technique [47]. Here we assume that the
ensemble atoms(blue) distribute with a random position $\bm{r}_{j}$ while the
auxiliary control atom(green) is placed at the zero point $O$ with a distance
$\bm{r}_{0}$ apart from the ensemble center. The control atom features a same
three-level structure, and is simultaneously driven by $\Omega_{p}$ and
$\Omega_{c}$ on the transitions $|g_{c}\rangle\to|e_{c}\rangle$ and
$|e_{c}\rangle\to|r_{c}\rangle$. Let us now discuss the concrete realization
of the excitation facilitation mechanism.

Taking account of all atom-light couplings and interatomic interactions, the
full level scheme as illustrated in Fig.1(a), has totally 15 collective
states. By assuming a sufficiently large detuning
$\Delta\gg\Omega_{p},\Omega_{c}$, numerical estimation for the probability
onto these off-resonance energy levels(marked by thin lines) is only
$7.13\times 10^{-4}$ when $N=3$, $\Omega_{p}=2\Omega_{c}$, which can be safely
eliminated. So we pay attention to merely five collective states(marked by
thick lines),

$\\{|g_{c}G\rangle,|g_{c}E^{1}R^{0}\rangle,|g_{c}E^{0}R^{1}\rangle,|e_{c}E^{0}R^{1}\rangle,|r_{c}E^{0}R^{1}\rangle\\}.$
(5)

From Fig.1(a), we know that if the control atom is unexcited, i.e. it is at
$|g_{c}\rangle$, only states $|g_{c}G\rangle$, $|g_{c}E^{1}R^{0}\rangle$,
$|g_{c}E^{0}R^{1}\rangle$ (blue shaded area) can compose a resonant two-photon
transition mediated by a large detuning $\Delta$ to the intermediate
$|g_{c}E^{1}R^{0}\rangle$ state. In this case the lower transition between
$|g_{c}G\rangle$ and $|g_{c}E^{1}R^{0}\rangle$ is enhanced by
$\sqrt{N}\Omega_{p}$ [48]. As a result, the steady-state Rydberg population
$f_{r}$($=P_{g_{c}}(\infty)$, see definition in Eq.(20)) of ensemble atoms,
equivalent to the Rydberg population on state $|g_{c}E^{0}R^{1}\rangle$, can
be simply described by
$f_{r}=N\Omega_{p}^{2}/(N\Omega_{p}^{2}+\Omega_{c}^{2})$, as same as the
prediction by the superatom model [34]. While remarkably, accounting for the
use of a control-ensemble system we find that the excitation of a
simultaneously-driven control atom, adds to a subsequent two-photon transition
which obeys

$|g_{c}E^{0}R^{1}\rangle\to|e_{c}E^{0}R^{1}\rangle\to|r_{c}E^{0}R^{1}\rangle.$
(6)

When the detuning $\delta$ with respect to $|r_{c}E^{0}R^{1}\rangle$ can be
overcome by an appropriate control-ensemble interaction $U(\bm{r}_{j})$, the
steady-state Rydberg population of ensemble atoms described by the total
population on states $|g_{c}E^{0}R^{1}\rangle$ and $|r_{c}E^{0}R^{1}\rangle$
would be enhanced, which is irrelated to the status of the control atom.
Meanwhile the population on the intermediate state $|e_{c}E^{0}R^{1}\rangle$
will be suppressed due to a large detuning $\Delta$.

###  2.3 Imperfect antiblockade condition

Ideally we expect an exact two-photon resonance
$\delta=-U(\bm{r}_{j})$(antiblockade) for each ensemble atom, which makes the
double Rydberg state $|r_{c}E^{0}R^{1}\rangle$ resonantly-coupled [49, 50,
51]. However accounting for the thermal motion of atoms the real control-
ensemble interaction $U(\bm{r}_{j})$ must be space-dependent. The thermal
distribution of ensemble atoms should be characterized by a Gaussian function
[52]

$f(\bm{r}_{j})=e^{-\frac{(\bm{r}_{j}-\bm{r}_{0})^{2}}{2\sigma^{2}}}$ (7)

with its width $\sigma=\sqrt{k_{B}T/m\omega^{2}}$ where $T$, $m$, $\omega$ are
the atomic temperature, atomic mass and trap frequency respectively.
Correspondingly, we obtain a fluctuated dipolar interaction [53]

$U(\bm{r}_{j})\approx U(\bm{r}_{0})+\delta U(\bm{r}_{j}).$ (8)

where $U(\bm{r}_{0})=\frac{C_{6}}{|\bm{r}_{0}|^{6}}$ with $C_{6}$ the
interaction coefficient for state $|r_{c}r_{j}\rangle$ and $\bm{r}_{0}$ the
displacement between the ensemble center and the control atom. The small
fluctuated interaction shift is

$\delta
U(\bm{r}_{j})=-\frac{6C_{6}(|\bm{r}_{j}-\bm{r}_{0}|)}{|\bm{r}_{0}|^{7}}$ (9)

caused by a random displacement $\bm{r}_{j}$ of atoms obtained from the
function $f(\bm{r}_{j})$. As a consequence we set $\delta=-U(\mathbf{r}_{0})$
in the calculation. For a sufficiently low temperature, the exact antiblockade
condition $\delta\approx-U(\bm{r}_{j})$ is approximately satisfied since the
atoms are frozen and $\delta U(\bm{r}_{j})\to 0$. Whereas the fluctuated
component $\delta U(\bm{r}_{j})$($\propto|\bm{r}_{j}-\bm{r}_{0}|$) becomes
larger if the temperature grows leading to the breakdown of antiblockade
facilitation (see more details in Sec.5.2). Our numerical results depend on a
fluctuated interaction $U(\bm{r}_{j})$.

##  3 Hamiltonian and Numerical method

The associated Hamiltonian of the control-ensemble scheme can be written as

$\mathcal{H}=\mathcal{H}_{en}+\mathcal{H}_{c}+\sum_{j=1}^{N}U(\bm{r}_{j})|r_{c}\rangle\langle
r_{c}|\otimes|r_{j}\rangle\langle r_{j}|$ (10)

where the first term

${\mathcal{H}}_{en}=\sum_{j=1}^{N}\mathcal{H}_{0}^{(j)}+\sum_{k>j}^{N}U_{rr}(\bm{r}_{j},\bm{r}_{k})|r_{j}\rangle\langle
r_{j}|\otimes|r_{k}\rangle\langle r_{k}|$ (11)

represents the Hamiltonian of an atomic ensemble. For each ensemble atom $j$,
the single-atom Hamiltonian $\mathcal{H}_{0}^{(j)}$ reads

${\mathcal{H}}_{0}^{(j)}=-\Delta|e_{j}\rangle\langle
e_{j}|+(\Omega_{p}|e_{j}\rangle\langle g_{j}|+\Omega_{c}|r_{j}\rangle\langle
e_{j}|+\text{H.c.}).$ (12)

Besides, we assume the intraspecies interaction
$U_{rr}(\bm{r}_{j},\bm{r}_{k})$ is also strong that leads to a perfect
blockade effect within the ensemble atoms, where $\bm{r}_{j}$ and $\bm{r}_{k}$
are the random atomic positions. The second term $\mathcal{H}_{c}$ describing
the Hamiltonian of single control atom, can be described by

$\mathcal{H}_{c}=-\Delta|e_{c}\rangle\langle e_{c}|+\delta|r_{c}\rangle\langle
r_{c}|+(\Omega_{p}|e_{c}\rangle\langle g_{c}|+\Omega_{c}|r_{c}\rangle\langle
e_{c}|+\text{H.c.})$ (13)

and the third term means the control-ensemble interaction, see Eq.(8).

To our knowledge, exact treatment with a master equation method without mean-
field approximation is rather computationally-demanding [54], especially for a
many-body system containing complex interactions. Below we numerically solve
the time-dependent evolution with the Monte Carlo wave-function method [55].
In the simulation, state $|\Psi\rangle$ of system evolves according to a
stochastic Schr$\ddot{o}$dinger equation

$\partial_{t}|\Psi\rangle=-i(\mathcal{H}-\frac{i}{2}\mathcal{L}^{2})|\Psi\rangle$
(14)

where the Liouvillian dissipative operator serving as a non-Hermitian term,
reads

$\mathcal{L}^{2}=\sum_{j=1}^{N}(\hat{{\mathcal{L}}}_{e}^{j\dagger}\hat{{\mathcal{L}}_{e}^{j}}+\hat{{\mathcal{L}}}_{r}^{j\dagger}\hat{{\mathcal{L}}_{r}^{j}}+\hat{{\mathcal{L}}}_{z1}^{j\dagger}\hat{{\mathcal{L}}_{z1}^{j}}+\hat{{\mathcal{L}}}_{z2}^{j\dagger}\hat{{\mathcal{L}}}_{z2}^{j})+(\hat{{\mathcal{L}}}_{e}^{c\dagger}\hat{{\mathcal{L}}_{e}^{c}}+\hat{{\mathcal{L}}}_{r}^{c\dagger}\hat{{\mathcal{L}}_{r}^{c}}+\hat{{\mathcal{L}}}_{z1}^{c\dagger}\hat{{\mathcal{L}}_{z1}^{c}}+\hat{{\mathcal{L}}}_{z2}^{c\dagger}\hat{{\mathcal{L}}}_{z2}^{c}).$
(15)

Following Eq.(15), the whole dissipation process in the control-ensemble
system contains three parts:

(i) The spontaneous decay from the intermediate state $|e_{j(c)}\rangle$ with
rate $\Gamma_{e}$ is denoted by

$\hat{{\mathcal{L}}}_{e}^{j(c)}=\sqrt{\Gamma_{e}}|g_{j(c)}\rangle\langle
e_{j(c)}|,$ (16)

(ii) the spontaneous decay from the Rydberg state $|r_{j(c)}\rangle$ with rate
$\Gamma_{r}$ reads

$\hat{{\mathcal{L}}}_{r}^{j(c)}=\sqrt{\Gamma_{r}}|g_{j(c)}\rangle\langle
r_{j(c)}|.$ (17)

(iii) the dephasing effect with rates $\gamma_{ge}$ and $\gamma_{er}$ caused
by atomic motions and collisions takes form of

$\displaystyle\hat{{\mathcal{L}}}_{z1}^{j(c)}=\sqrt{\gamma_{ge}}(|e_{j(c)}\rangle\langle
e_{j(c)}|-|g_{j(c}\rangle\langle g_{j(c)}|)$
$\displaystyle\hat{{\mathcal{L}}}_{z2}^{j(c)}=\sqrt{\gamma_{er}}(|r_{j(c)}\rangle\langle
r_{j(c)}|-|e_{j(c)}\rangle\langle e_{j(c)}|).$ (18)

In each quantum trajectory $m$, the evolution of $|\Psi_{m}\rangle$ obeys the
stochastic Schr$\ddot{o}$dinger equation (14) [56]. However it interrupts via
a random quantum jump
$|\Psi_{m}\rangle\to\hat{\mathcal{L}}_{a}^{k}|\Psi_{m}\rangle$ with
$a\in\\{e,r,z1,z2\\}$ and $k\in\\{j,c\\}$, determined by its weight
$dt\langle\Psi_{m}|\hat{\mathcal{L}}_{a}^{k\dagger}\hat{\mathcal{L}}_{a}^{k}|\Psi_{m}\rangle$.
At the same time $|\Psi_{m}\rangle$ should be re-normalized via
$|\bar{\Psi}_{m}(t)\rangle=|\Psi_{m}(t)\rangle/\sqrt{\langle\Psi_{m}(t)|\Psi_{m}(t)\rangle}$
for keeping the preservation of population [57]. Finally the calculated
density matrix $\bar{\rho}(t)$ is obtained by averaging over sufficient
quantum trajectories $M=300$(used), which takes form of

$\bar{\mathcal{\rho}}(t)=\frac{1}{M}\sum^{M}_{m=1}|\bar{\Psi}_{m}(t)\rangle\langle\bar{\Psi}_{m}(t)|.$
(19)

The population dynamics of arbitrary Rydberg-excited states in the control-
ensemble system can be written as

$P_{s}(t)=\text{Tr}[\bar{\rho}\hat{\mathbf{O}}_{s}]=\frac{1}{M}\sum_{m=1}^{M}\langle\bar{\Psi}_{m}(t)|\hat{\mathbf{O}}_{s}|\bar{\Psi}_{m}(t)\rangle$
(20)

where the operator is

$\displaystyle\hat{\mathbf{O}}_{s}$ $\displaystyle=$
$\displaystyle|s\rangle\langle s|\otimes\\{\sum_{j=1}^{N}|r_{j}\rangle\langle
r_{j}|\prod_{k\neq j}^{N}(|g_{k}\rangle\langle g_{k}|)\\}$ (21)

with $s\in(g_{c},r_{c})$ denoting the state of the control atom.
$P_{g_{c}}(t)$ or $P_{r_{c}}(t)$ stand for the time-dependent Rydberg
population of the targeted ensemble when the control atom is unexcited or
excited. Note that we treat $P_{g_{c}}(t)+P_{r_{c}}(t)$ as the Rydberg
excitation probability of ensemble atoms. Replacing the operator
$\hat{\mathbf{O}}_{s}$ by

$\hat{\mathbf{O}}_{g_{c}G}=|g_{c}\rangle\langle
g_{c}|\otimes\prod_{j=1}^{N}(|g_{j}\rangle\langle g_{j}|)$ (22)

in Eq.(20) arises a new expression $P_{g_{c}G}(t)$ which stands for the
population dynamics of the ground state $|g_{c}G\rangle$.

Figure 2: (a) Steady-state Rydberg population $f_{r}$ of the targeted atomic
ensemble vs. the variation of $\Omega_{p}/\Omega_{c}$. The facilitated
excitation range is shaded by gray. Inset: The total steady population of
multi-excitation onto states $|E^{2}R^{1}\rangle$ and $|E^{3}R^{0}\rangle$ for
$N=3$. Results in the presence(absence) of control atom are shown by the
black(red-dashed) line. Rydberg population dynamics under the cases of (b1) no
control atom and (b2) one control atom where $\Omega_{p}/\Omega_{c}=0.4$.
Similar behaviors with $\Omega_{p}/\Omega_{c}=1.0$ and $2.0$ are displayed in
(c1-c2) and (d1-d2). Populations $P_{g_{c}}(t)$, $P_{r_{c}}(t)$,
$P_{g_{c}}(t)+P_{r_{c}}(t)$ are labeled by the black-solid line, the red-
dashed line and the blue-solid line, respectively.

##  4 Numerical Verification

We next employ the above Monte Carlo method to study the realistic population
dynamics in a coupled control-ensemble system. We consider the explicit case
of 87Rb atoms trapped in two optical tweezers separated by $\bm{r}_{0}$, with
energy levels $|g_{c(j)}\rangle=|5S_{1/2}\rangle$,
$|e_{c(j)}\rangle=|5P_{3/2}\rangle$ and $|r_{c(j)}\rangle=|55S_{1/2}\rangle$.
The corresponding interaction coefficient for state $|r_{c(j)}\rangle$ is
$C_{6}/2\pi=50$ GHz$\cdot\mu$m6 and the non-fluctuated distance is
$\bm{r_{0}}=3.062$ $\mu$m, where
$\bm{r_{0}}>R_{b}=(\gamma_{eg}C_{6}/\Omega_{c}^{2})^{1/6}\approx 1.6$ $\mu$m
is assumed for a vdWs interaction between the control-ensemble atom pairs
[58]. At $T\approx 1$ $\mu$K the trap frequency is $\omega/2\pi=100$ kHz,
leading to the standard deviation $\sigma=14.3$ nm. The spontaneous decay
rates are $\Gamma_{e}/2\pi=6.06$ MHz, $\Gamma_{r}/2\pi=2$ kHz, the dephasing
rates are $\gamma_{re(eg)}/2\pi=12.12$ kHz and $\Omega_{c}/2\pi=6.06$ MHz,
$\Delta/2\pi=121.2$ MHz.

In Fig. 2(a), we show the steady-state Rydberg population $f_{r}$ of the
targeted ensemble as a function of $\Omega_{p}/\Omega_{c}$(by varying
$\Omega_{p}$) for $N=3$. When the control atom is unexcited we have
$f_{r}=P_{g_{c}}(\infty)$; otherwise, we have
$f_{r}=P_{g_{c}}(\infty)+P_{r_{c}}(\infty)$. For a weak probe field
($\Omega_{p}/\Omega_{c}<0.5$), $f_{r}$ is almost same. That means the second
two-photon process from $|g_{c}E^{0}R^{1}\rangle$ to $|r_{c}E^{0}R^{1}\rangle$
plays a negligible role and the system evolves samely in both cases. Yet once
$\Omega_{p}/\Omega_{c}>0.5$, the steady Rydberg population can be clearly
enhanced with the help of an excited control atom and quickly approaches
0.9204 at $\Omega_{p}/\Omega_{c}=2$. While this value is only 0.7994 in the
case of no control atom. This enhancement effect comes from the second two-
photon transition onto the double Rydberg state $|r_{c}E^{0}R^{1}\rangle$
enabled by the antiblockade condition which becomes more apparent in the case
of a stronger probe field. Other off-resonance collective states are almost
unpopulated in our calculation. Especially, the population onto multi-
excitation intermediate states keeps below $10^{-5}$, see the inset of
Fig.2(a).

To further verify the above steady-state analysis, we study the time-dependent
dynamics of the Rydberg-state populations in Fig.2(b1-d2), which visibly
presents the population transfer among different collective states. From top
to bottom we choose $\Omega_{p}/\Omega_{c}=(0.4,1.0,2.0)$. It is clear that
with a weak probe driving $\Omega_{p}=0.4\Omega_{c}$, the excitation of the
targeted ensemble mainly occupies state $|g_{c}E^{0}R^{1}\rangle$ with a small
fraction of $P_{g_{c}}(t)$(black-solid) except the ground state(not shown), no
matter whether the control atom is excited or not. By comparing (b1) and (b2)
we find $P_{r_{c}}$(red-dashed) in (b2) is substantially suppressed agreeing
with our theoretical prediction. However for a stronger probe driving, the
control atom’s excitation would result in a subsequently new transition
channel, which leads to an enhanced Rydberg probability redistributed on
states $|g_{c}E^{0}R^{1}\rangle$ and $|r_{c}E^{0}R^{1}\rangle$. In (c2,d2) the
population $P_{r_{c}}$(red-dashed) obtains a dramatic increases for a larger
$\Omega_{p}$, arising the steady-state Rydberg population of ensemble atoms
described by $P_{r_{c}}+P_{g_{c}}$(blue-solid) greatly enhanced. So we verify
that the coupled control-ensemble system can exhibit a clear excitation
facilitation effect for the targeted ensemble atoms when the probe field is
not so weak.

##  5 Scheme Feasibility

###  5.1 $N$ ensemble atoms

Figure 3: (a1-a2) Steady-state Rydberg population $f_{r}$ vs. the change of
atomic number $N$ in the ensemble. Results with(or without) the control atom
are shown by the black line with squares(or the red line with circles). Every
point is obtained by averaging over 300 stochastic realizations at $T=1$
$\mu$K and the error bar is given accordingly. (b1-b4) Time-dependent
population dynamics for $N=2$ under the cases with (b1)
$\Omega_{p}=0.4\Omega_{c}$ and no control atom; (b2)
$\Omega_{p}=0.4\Omega_{c}$ and one control atom; (b3) $\Omega_{p}=2\Omega_{c}$
and no control atom; (b4) $\Omega_{p}=2\Omega_{c}$ and one control atom.
$P_{g_{c}}(t)$, $P_{r_{c}}(t)$, $P_{g_{c}G}(t)$ are denoted by the black-solid
line, the red-solid line and the blue-dashed line, respectively. Other
intermediate-state population is labeled by the green-dotted line. (c1-c4) are
same as (b1-b4) except for $N=5$.

We first discuss the excitation facilitation with different number of ensemble
atoms. The experimental realization of a few-atom ensemble with a desired
number of cold atoms could utilize a microscopic dipole trap [59]. Depending
on the local density of the MOT cloud around the trap, the exact number of
trapped atoms can be desirably changed in experiment [60].

Figure 3(a1-a2) show the steady-state Rydberg population $f_{r}$ of the
targeted atomic ensemble as a function of atomic number $N$. We choose
$\Omega_{p}/\Omega_{c}=(0.4,2.0)$ representing the cases of weak and strong
probe drivings. As $N$ is increased, an intuitive feature is that $f_{r}$
slowly grows and approaches its saturation as shown in (a1) where a weak probe
is applied. Because the probe Rabi frequency can obtain a usual
$\sqrt{N}$-enhancement allowing for an enhanced steady-state Rydberg
population [61]. However, if beyond the weak probe limit [see (a2)] the steady
Rydberg population $f_{r}$ exhibits an anomalous decrease. To understand this
unusual effect we revisit the collective states as shown in Fig.1(a3) for
single atomic ensemble. Under the strong probe limit state
$|E^{2}R^{0}\rangle$ can be ignored due to its far-off-resonance detuning
$2\Delta$. Therefore the Hamiltonian of the atomic ensemble can be rewritten
by using four collective states

$\displaystyle{\mathcal{H}}_{en}^{\prime}$ $\displaystyle=$
$\displaystyle-\Delta(|E^{1}R^{1}\rangle\langle
E^{1}R^{1}|+|E^{1}R^{0}\rangle\langle E^{1}R^{0}|)$ (23) $\displaystyle+$
$\displaystyle(\sqrt{N}\Omega_{p}|G\rangle\langle
E^{1}R^{0}|+\Omega_{c}|E^{1}R^{0}\rangle\langle
E^{0}R^{1}|+\sqrt{N-1}\Omega_{p}|E^{0}R^{1}\rangle\langle
E^{1}R^{1}|+\text{H.c.}).$

When $\Delta\gg\Omega_{p}$ we safely ignore the doubly-excited state
$|E^{1}R^{1}\rangle$ and ${\mathcal{H}}_{en}^{\prime}$ can be reduced into

$\hat{\mathcal{H}}_{en}^{\prime\prime}=-\Delta|E^{1}R^{0}\rangle\langle
E^{1}R^{0}|+\frac{(N-1)\Omega_{p}^{2}}{\Delta}|E^{0}R^{1}\rangle\langle
E^{0}R^{1}|+(\sqrt{N}\Omega_{p}|G\rangle\langle
E^{1}R^{0}|+\Omega_{c}|E^{1}R^{0}\rangle\langle E^{0}R^{1}|+\text{H.c.}).$
(24)

Based on Eq.(24), it is obvious that the energy shift related to state
$|E^{0}R^{1}\rangle$ is proportional to $N$ which means $|E^{0}R^{1}\rangle$
is highly-shifted when the atomic number $N$ is increased leading to a smaller
Rydberg excitation probability. Numerical results(red line with circles in
(a2)) also agree with the above discussions. It is remarkable that, by placing
a control atom nearby, $f_{r}$(black line with squares) can be kept at a high
level with an arbitrary $N$ value see (a2). This effect is mainly contributed
by the second two-photon process between $|g_{c}E^{0}R^{1}\rangle$ and
$|r_{c}E^{0}R^{1}\rangle$, leading to an enhanced Rydberg population
redistributed on these states. The large spontaneous loss from intermediate
state $|g_{c}E^{1}R^{0}\rangle$ can be overcome by this facilitation effect.

Figure 3(b1-b4) and (c1-c4) display the population dynamics on three prominent
states $|g_{c}G\rangle$, $|g_{c}E^{0}R^{1}\rangle$, $|r_{c}E^{0}R^{1}\rangle$
for $N=2$ and $5$. When the probe field is weak i.e.
$\Omega_{p}=0.4\Omega_{c}$ in (b1,b2,c1,c2), a large fraction of population is
observed to stay on the ground state $|g_{c}G\rangle$ (blue-dashed). But with
the help of control atom, the Rydberg population $P_{r_{c}}$(red-solid)
obtains a clear enhancement with the atomic number $N$. Therefore even in the
weak probe regime, the excitation facilitation effect inside a control-
ensemble system is easily observable as long as ensemble atoms are sufficient.
Most importantly, when the probe driving is strong $\Omega_{p}=2\Omega_{c}$,
$f_{r}$ suffers from a strong decrease as $N$ increases. Hence, by comparing
(b3) and (c3) we find $P_{g_{c}}$(black-solid) suffers from a reduction
accompanied by the increase of $P_{g_{c}G}$(blue-dashed) on the ground state.
However in the coupled control-ensemble system, benefiting from an enhanced
Rydberg population which is redistributed on states $|g_{c}E^{0}R^{1}\rangle$
and $|r_{c}E^{0}R^{1}\rangle$, the leakage from intermediate states can be
suppressed. The total Rydberg population $P_{g_{c}}+P_{r_{c}}$ of the atomic
ensemble can be persistently kept at a high level with arbitrary $N$. This
means the controllable excitation facilitation requires a relatively strong
probe strength and should be more observable in a realistic atomic ensemble
with sufficient atoms.

###  5.2 Thermal motion of atoms

Figure 4: Steady-state Rydberg population $f_{r}$ of the targeted ensemble vs.
the atomic temperature $T$ for $N=3$. Results with(or without) the control
atom are shown by the black line with squares(or the red line with circles).
The excitation facilitation regime is shown in blue. Insets (a1-a2): The
corresponding time-dependent population dynamics at $T=50$ $\mu$K. The
linetype and parameters are same as used in Fig.2 except for
$\Omega_{p}=2\Omega_{c}$.

To provide a profound study for this effect we further investigate the
behavior of population dynamics under different temperatures. Intuitively the
increase of atomic temperature would easily break the antiblockade condition,
leading to the fluctuated component $\delta U(\bm{r}_{j})$ non-negligible
[62]. As a consequence the doubly-excited Rydberg state
$|r_{c}E^{0}R^{1}\rangle$ becomes off-resonance accompanied by a big reduction
of the population $P_{r_{c}}$. A comparison for the behaviors of steady
Rydberg populations vs. the temperature $T$, is illustrated in Fig.4. By
increasing $T$ it is apparent that $f_{r}$ almost keeps a constant in the case
with no control atom due to the perfect blockade effect preserved inside the
ensemble atoms. Because even at $T=50$ $\mu$K the distance deviation of
ensemble atoms is only $\sigma\approx 0.101$ $\mu$m, which is smaller than the
blockade radius $R_{b}\approx 1.6$ $\mu$m by one order of magnitude.

As turning to the case with a control atom the steady-state Rydberg population
reveals a clear decrease due to the influence of fluctuated interaction
$\delta U(\bm{r}_{j})$ that increases with $T$. From Eq.(9) a rough estimation
shows $\delta U(\bm{r}_{j})/2\pi\approx-11.98$ MHz if
$|\bm{r}_{j}-\bm{r}_{0}|=0.1$ $\mu$m at $T=50$ $\mu$K. This value leads to a
non-negligible off-resonance detuning to the doubly-excited Rydberg state
$|r_{c}E^{0}R^{1}\rangle$, arising the breakdown of antiblockade facilitation
$\delta\neq-U(\bm{r}_{j})$. Therefore the controllable excitation facilitation
merely exists at a relatively lower temperature. Based on our simulations when
$T\leq 20$ $\mu$K the effect should be observable within a few-atom ensemble
containing three atoms [63]. Since $f_{r}$ decreases with $N$ in a single-
ensemble system [see Fig.3(a2)], a wider regime of temperature for excitation
facilitation can be obtained if more ensemble atoms are involved. So far a
desired number of cold atoms can be prepared in an optical dipole trap [64]
which makes this excitation facilitation effect more observable
experimentally. Realistic population dynamics as shown in Fig.4(a1-a2) agrees
with the above steady-state discussions, in which the population
$P_{r_{c}}$(red-solid, (a2)) has been deeply suppressed due to the off-
resonantly coupled state $|r_{c}E^{0}R^{1}\rangle$. Therefore no facilitation
can be found at $T=50$ $\mu$K because
$f_{r}$(=$P_{g_{c}}(\infty)+P_{r_{c}}(\infty)$ in
(a2))$<f_{r}$(=$P_{g_{c}}(\infty)$ in (a1) ).

###  5.3 Adjustable control-ensemble distances

Figure 5: (a-f) Steady-state Rydberg population $f_{r}$ of the targeted
ensemble vs. the control-ensemble distance $\bm{r_{0}}$ under various $T$(in
unit of $\mu$K), corresponding to the distance deviations $\sigma=(4.5\times
10^{-5},0.014,0.045,0.101,0.202,0.451)$ $\mu$m. Linetypes and parameters are
same as used in Fig.4.

To manipulate the internal states of ensemble atoms, the assisted control atom
which is individually addressed in a dipole trap, should keep an appropriate
distance $\bm{r}_{0}$ from the ensemble trap. This relative distance is
typically adjustable within a range of several $\mu$m via tuning the incidence
angle of the focusing laser beams [65]. Here we study the influence on the
steady-state Rydberg population of ensemble atoms by using a variable distance
$\bm{r}_{0}$, equivalent to a variable $U(\bm{r}_{0})$. It should be noted
that the condition $\delta=-U(\bm{r}_{0})$ set for calculation makes the
detuning $\delta$ changeable with $\bm{r}_{0}$ at the same time. Intuitively,
the manipulation of control atom would vanish as $\bm{r}_{0}\to\infty$ and
both the control and atomic ensemble behave independently then. And when
$\bm{r}_{0}$ is adjusted within an appropriate range, the excitation of
ensemble atoms must be affected.

Figure 5(a-f) show the steady-state Rydberg population as a function of the
control-ensemble distance $\bm{r}_{0}$ under different atomic temperatures. It
is evident that $f_{r}$ of single atomic ensemble is unvaried because of the
preservation of perfect blockade effect, which agrees with the results in
Fig.4. While we note that in the vicinity of a control atom, by varying
$\bm{r}_{0}$ there exists a clear dip $\bm{r}_{0}=\bm{r}_{dip}$ at which the
$f_{r}$ attains its minimum. To explain this point, we revisit the full level
scheme in Fig.1(a). At $\bm{r}_{0}=\bm{r}_{dip}$, another transition channel

$|g_{c}G\rangle\to|e_{c}G\rangle\to|r_{c}G\rangle\to|r_{c}E^{1}R^{0}\rangle\to|r_{c}E^{0}R^{1}\rangle$
(25)

is facilitated, where the effective detuning $\Delta_{r_{c}G}$ with respect to
$|r_{c}G\rangle$ can be estimated as

$\Delta_{r_{c}G}=\frac{\Omega_{c}^{2}-\Omega_{p}^{2}}{\Delta}+\delta_{dip}+\frac{N\Omega_{p}^{2}}{\Delta-\delta_{dip}}.$
(26)

By considering $\Delta\gg|\delta|$ and $\Delta_{r_{c}G}=0$, the steady Rydberg
population eventually stays on state $|r_{c}G\rangle$ and the position
$\bm{r}_{dip}$ can be analytically solved by

$\bm{r}_{dip}=(\frac{C_{6}}{-\delta_{dip}})^{1/6}\approx(\frac{C_{6}\Delta}{\Omega_{c}^{2}+(N-1)\Omega_{p}^{2}})^{1/6}.$
(27)

Notice that at $\bm{r}_{0}=\bm{r}_{dip}$, the doubly-excited Rydberg state
$|r_{c}E^{0}R^{1}\rangle$ is off-resonance due to a non-zero effective
detuning
$\frac{\Omega_{c}^{2}}{\Delta-\delta_{dip}}-\frac{\Omega_{p}^{2}}{\Delta}\neq
0$. Hence the excitation of control atom prevents the ensemble atoms from
excitation at this position. By using $\Omega_{c}=\Gamma_{e}=2\pi\times 6.06$
MHz, $\Delta=20\Gamma_{e}$, $\Omega_{p}/\Omega_{c}=2.0$ and $N=3$, it gives
$\bm{r}_{dip}\approx 5.1$ $\mu$m, agreeing with the numerical results in
Fig.5.

In addition we find that, the excitation facilitation effect is observable
only if $\bm{r}_{0}$ is appropriate. Based on Fig.5, as expected there is no
facilitation when $\bm{r}_{0}\to\infty$, where the ensemble atoms would obtain
an independent excitation just like a single atomic ensemble. On the contrary
if $\bm{r}_{0}\to 0$, a stronger fluctuated interaction $\delta U(\bm{r}_{j})$
would result in a complete breaking of the antiblockade condition by the
control atom. Hence no excitation facilitation occurs when $\bm{r}_{0}$ is
very small [see Fig.5(a-f)]. Only if the control-ensemble distance
$\bm{r}_{0}$ is appropriately adjusted the excitation facilitation effect can
take place. Moreover no facilitation exists when $T>50$ $\mu$K. Because at a
higher temperature, the thermal motion of ensemble atoms strongly breaks the
antiblockade condition, see more details in Sec.5.2.

Finally it is worth stressing that an experimental observation of this
excitation facilitation effect depends on an appropriate adjustment of both
the control-ensemble distance and the atomic temperature, as well as the
inclusion of sufficient ensemble atoms.

##  6 Conclusion

We propose a new scheme to realize excitation facilitation of a strongly-
interacting Rydberg ensemble enabled by an auxiliary control atom. By
utilizing a relatively strong probe driving, a simultaneously-excited control
atom can counterintuitively add to a second two-photon transition onto the
doubly-excited Rydberg state, which arises an excitation facilitation effect
for the steady Rydberg population of ensemble atoms. Differing from the
single-ensemble model where only the singly-excited Rydberg state can be
occupied, this enhanced Rydberg probability would be redistributed on the
singly- and doubly-excited Rydberg states under the antiblockade facilitation
condition. By studying various experimentally-feasible parameters which set
some constraints for this effect, we show the combination of a strong probe
driving and a low temperature is an important prerequisite for observing
excitation facilitation in a realistic Rydberg atomic ensemble.

Our achievements, on one hand, arise a more profound understanding of the
collective excitation feature of Rydberg-atom ensembles (see a review in
[66]); and on the other hand, are able to realize a flexible control for the
facilitated excitation in a single atomic ensemble. The next step towards the
use of this mechanism would be realizing a mesoscopic Rydberg controlled-phase
gate [67]. Because the excitation of control atom could add to a quasi-dark
eigenstate between $|G\rangle$ and $|E^{0}R^{1}\rangle$ mediated by
$|E^{1}R^{0}\rangle$. By varying the amplitudes of probe laser and the two-
photon detuning, the system will experience an adiabatic evolution along this
quasi-dark state which can accumulate an effective dynamic phase through the
nonzero eigenenergy. A careful modulation of the laser profiles can be used
for implementing a mesoscopic Rydberg gate with an arbitrary phase factor [70,
71, 72], which will leave for our future work.

## References

  * [1] Urvoy A, Ripka F, Lesanovsky I, Booth D, Shaffer J P, Pfau T , and Löw R 2015 Phys. Rev. Lett. 114, 203002
  * [2] Gutiérrez R, Simonelli C, Archimi M, Castellucci F, Arimondo E, Ciampini D, Marcuzzi M, Lesanovsky I, and Morsch O, 2017 Phys. Rev. A 96, 041602
  * [3] Lampen J, Duspayev A, Nguyen H, Tamura H, Berman P R , and Kuzmich A, 2019 Phys. Rev. Lett. 123, 203603
  * [4] Stiesdal N, Busche H, Kumlin J, Kleinbeck K, Büchler H P, and Hofferberth S, 2020 Phys. Rev. Research 2, 043339
  * [5] Brion E, Mølmer K, and Saffman M 2007 Phys. Rev. Lett. 99, 260501
  * [6] Zhang Z Y, Ding D S, and Shi B S 2021 Chinese Physics B 30, 020307
  * [7] Wu X, Liang X, Tian Y, Yang F, Chen C, Liu Y C, Tey M K, and You L 2021 Chinese Physics B 30, 020305
  * [8] Li P C and Chu S I 2020 Chinese Phys.B 29 083202
  * [9] Khazali M 2018 Phys. Rev. A 98, 043836
  * [10] Yang C W, Yu Y, Li J, Jing B, Bao X H, and Pan J W, 2022 Nature Photonics 16, 658–661
  * [11] Sun P F, Yu Y, An Z Y, Li J, Yang C W, Bao X H, and Pan J W 2022 Phys. Rev. Lett. 128, 060502
  * [12] Yang C W, Li J, Zhou M T, Jiang X, Bao X H, and Pan J W 2022 Optica 9, 853–858
  * [13] PadrónBrito A, Lowinski J, Farrera P, Theophilo K, and de Riedmatten H 2021 Phys. Rev. Research 3, 033287
  * [14] Petrosyan D and Mølmer K 2021 Phys. Rev. A 103, 023703
  * [15] Guo C Y, Yan L L, Zhang S, Su S L, and Li W 2020 Phys. Rev. A 102, 042607
  * [16] Gujarati T P 2018 Phys. Rev. A 98, 062326
  * [17] Haase T, Alber G, and Stojanović V M 2022 Phys. Rev. Research 4, 033087
  * [18] Saffman M, Walker T G, and Mølmer K 2010 Rev. Mod. Phys. 82, 2313–2363
  * [19] Motzoi F and Mølmer K 2018 New Journal of Physics 20, 053029
  * [20] Müller M, Lesanovsky I, Weimer H, Büchler H P, and Zoller P 2009 Phys. Rev. Lett. 102, 170502
  * [21] Heidemann R, Raitzsch U, Bendkowsky V, Butscher B, Löw R, Santos L, and Pfau T 2007 Phys. Rev. Lett. 99, 163601
  * [22] Petrosyan D, Höning M, and Fleischhauer M 2013 Phys. Rev. A 87, 053414
  * [23] Zeiher J, Schauß P, Hild S, Macrì T, Bloch I, and Gross C 2015 Phys. Rev. X 5, 031015
  * [24] Mei Y, Li Y, Nguyen H, Berman P R, and Kuzmich A 2022 Phys. Rev. Lett. 128, 123601
  * [25] Petrosyan D, Otterbach J, and Fleischhauer M, 2011 Phys. Rev. Lett. 107, 213601
  * [26] Dudin Y O, Li L, Bariani F, and Kuzmich A 2012 Nature Physics 8, 790–794
  * [27] Höning M, Muth D, Petrosyan D, and Fleischhauer M 2013 Phys. Rev. A 87, 023401
  * [28] Tian X D, Liu Y M, Bao Q Q, Wu J H, Artoni M, and La Rocca G C 2018 Phys. Rev. A 97, 043811
  * [29] Pritchard J D, Maxwell D, Gauguet A, Weatherill K J, Jones M P A, and Adams C S 2010 Phys. Rev. Lett. 105, 193603
  * [30] Ding Y, Bai Z, Huang G, and Li W 2023 Phys. Rev. Appl. 19, 014017
  * [31] Qiao C and Zhang W 2021 Journal of Physics B: Atomic, Molecular and Optical Physics 54, 205501
  * [32] Berman P R, Nguyen H, and Rojo A G 2022 Phys. Rev. A 105, 043715
  * [33] Liu Y M , Yan D, Tian X D, Cui C L, and Wu J H 2014 Phys. Rev. A 89, 033839
  * [34] Gärttner M, Whitlock S, Schönleber D W, and Evers J 2014 Phys. Rev. Lett. 113, 233002
  * [35] Bai S Y, Bao Q Q, Tian X D, Liu Y M , and Wu J H 2018 Journal of Physics B: Atomic, Molecular and Optical Physics 51, 075502
  * [36] Ates C, Pohl T, Pattard T, and Rost J M 2007 Phys. Rev. Lett. 98, 023002
  * [37] Amthor T, Giese C, Hofmann C S, and Weidemüller M 2010 Phys. Rev. Lett. 104, 013001
  * [38] Kara D, Bhowmick A, and Mohapatra A K 2018 Scientific Reports 8, 5256
  * [39] Bai S, Tian X, Han X, Jiao Y, Wu J, Zhao J, and Jia S 2020 New Journal of Physics 22, 013004
  * [40] Petrosyan D 2013 Journal of Physics B: Atomic, Molecular and Optical Physics 46, 141001
  * [41] Lukin M D, Fleischhauer M, Cote R, Duan L M, Jaksch D, Cirac J I, and Zoller P 2001 Phys. Rev. Lett. 87, 037901
  * [42] Dudin Y O and Kuzmich A 2012 Science 336, 887–889
  * [43] Carmele A, Vogell B, Stannigel K, and Zoller P 2014 New Journal of Physics 16, 063042
  * [44] Yan D, Cui C L, Liu Y M, Song L J, and Wu J H 2013 Phys. Rev. A 87, 023827
  * [45] Yan D, Liu Y M, Bao Q Q, Fu C B, and Wu J H 2012 Phys. Rev. A 86, 023828
  * [46] Andersen M F 2022 Advances in Physics: X 7, 2064231
  * [47] Yavuz D D, Kulatunga P B, Urban E, Johnson T A, Proite N, Henage T, Walker T G, and Saffman M 2006 Phys. Rev. Lett. 96, 063001
  * [48] Spong N L R , Jiao Y, Hughes O D W ,Weatherill K J, Lesanovsky I, and Adams C S 2021 Phys. Rev. Lett. 127, 063604
  * [49] Ding Z X, Hu C S, Shen L T, Yang Z B, Wu H, and Zheng S B 2019 Laser Physics Letters 16, 045203
  * [50] Zuo Z and Nakagawa K 2010 Phys. Rev. A 82, 062328
  * [51] Li R, Yu D, Su S L, and Qian J 2020 Phys. Rev. A 101, 042328
  * [52] Marcuzzi M, Minář J c v, Barredo D, x de Léséleuc D, Labuhn H, Lahaye T, Browaeys A, Levi E, and Lesanovsky I 2017 Phys. Rev. Lett. 118, 063606
  * [53] Liu F, Yang Z C, Bienias P, Iadecola T, and Gorshkov A V 2022 Phys. Rev. Lett. 128, 013603
  * [54] de Hond J, van Bijnen R, Kokkelmans S J J M F, Spreeuw R J C, van den Heuvell H B v L, and van Druten N J 2018 Phys. Rev. A 98, 062714
  * [55] Petrosyan D, Rao D, and Mølmer K 2015 Phys. Rev. A 91, 043402
  * [56] Lee T E and Cros M C 2012 Phys. Rev. A 85, 063822
  * [57] Lee T E, Häffner H, and Cross M C 2012 Phys. Rev. Lett. 108, 023602
  * [58] Hofmann C S, Günter G, Schempp H, Robert-de Saint-Vincent M, Gärttner M, Evers J, Whitlock S, and Weidemüller M 2013 Phys. Rev. Lett. 110, 203601
  * [59] Kruse J, Gierl C, Schlosser M, and Birkl G 2010 Phys. Rev. A 81, 060308
  * [60] Yoon S, Choi Y, Park S, Kim J, Lee J H, and An K 2006 Applied Physics Letters 88, 211104.
  * [61] Liu Y M, Tian X D, Yan D, Zhang Y, Cui C L, and Wu J H 2015 Phys. Rev. A 91, 043802
  * [62] Ostmann M, Marcuzzi M, Minár J, and Lesanovsky I 2019 Quantum Science and Technology 4, 02LT01
  * [63] Fuhrmanek A, Bourgain R, Sortais Y R P, and Browaeys A 2012 Phys. Rev. A 85, 062708
  * [64] Frese D, Ueberholz B, Kuhr S, Alt W, Schrader D, Gomer V, and Meschede D 2000 Phys. Rev. Lett. 85, 3777–3780
  * [65] Béguin L, Vernier A, Chicireanu R, Lahaye T, and Browaeys A 2013 Phys. Rev. Lett. 110, 263201
  * [66] Browaeys A and Lahaye T 2020 Nature Physics 16, 132–142
  * [67] Yin H D and Shao X Q 2021 Opt. Lett. 46, 2541–2544
  * [68] Xu J and Liu T 2022 Quantum Information Processing 21, 201
  * [69] Saffman M and Mølmer K 2009 Phys. Rev. Lett. 102, 240502
  * [70] Li X, Shao X, and Li W 2022 Phys. Rev. Appl. 18, 044042
  * [71] Sun Y, Xu P, Chen P X, and Liu L 2020 Phys. Rev. Appl. 13, 024059
  * [72] Saffman M, Beterov I I, Dalal A, Páez E J, and Sanders B C 2020 Phys. Rev. A 101, 062309


# Stochastic Quantum Dynamics Stabilization: A Lyapunov-Based Control Approach
with Homodyne-Mediated Filtering

Nahid Binandeh Dehaghani, A. Pedro Aguiar, Rafal Wisniewski N. B. Dehaghani
and A. P. Aguiar are with the Research Center for Systems and Technologies
(SYSTEC), ARISE, and the Electrical and Computer Engineering Department,
Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465
Porto, Portugal<EMAIL_ADDRESS>Wisniewski is with Department
of Electronic Systems, Aalborg University, Fredrik Bajers vej 7c, DK-9220
Aalborg, Denmark<EMAIL_ADDRESS>The authors gratefully acknowledge the
financial support provided by the Foundation for Science and Technology
(FCT/MCTES) within the scope of the PhD grant 2021.07608.BD, the Associated
Laboratory ARISE (LA/P/0112/2020), the R&D Unit SYSTEC through Base
(UIDB/00147/2020) and Programmatic (UIDP/00147/2020) funds and project
RELIABLE (PTDC/EEI-AUT/3522/2020), all supported by national funds through
FCT/MCTES (PIDDAC). The work has been done in honor and memory of Professor
Fernando Lobo Pereira.

###### Abstract

Efficient control of stochastic dynamics in quantum systems is pivotal for
various applications, including quantum information processing and metrology.
This paper introduces a Lyapunov-based control approach with homodyne-mediated
filtering. We employ a modified extended Kalman filtering method to directly
estimate the evolution of the quantum density operator $\rho$, considering
sequential homodyne current measurements. Our method explicitly addresses the
dynamics of a stochastic master equation with correlated noise, ensuring by
construction the quantum properties of the estimated state variable
$\hat{\rho}$. Moreover, our proposed switching based Lyapunov control scheme
that is fed with $\hat{\rho}$, guarantees noise-to-state practically stable in
probability of the desired stationary target set with respect to the
estimation error variance. We demonstrate our approach’s efficacy in
stabilizing a qubit coupled to a leaky cavity under homodyne detection.

##  I Introduction

A particularly fertile domain within quantum control is that of feedback
control, where the control strategy is dynamically adjusted based on real-time
feedback signals emanating from the evolving system [1]. In this regard,
significant advancements have been achieved in the development and enhancement
of the underlying principles of quantum probability and filtering theory [2,
3, 4, 5, 6, 7].

Figure 1: Illustration of the proposed configuration: A qubit linked to a
leaky cavity undergoes constant monitoring through homodyne detection. The
emitted radiation from the qubit combines with a local oscillator laser at a
beamsplitter. The resulting signal $I_{c}(t)$ is transmitted to a filter,
where the quantum state $\rho$ is estimated, and then, is sent to a control
system. This system is responsible to drive the field $\Omega(t)$ through the
control Hamiltonian to the qubit, with the aim of generation and stabilization
into a desired target set.

In this study, see Fig. 1, we spotlight a qubit undergoing continuous homodyne
detection described by a stochastic master equation. Crafting a controller is
pivotal, as it furnishes a control scheme tailored to fulfill specific state
transformation tasks based on the homodyne current measurements. Inherent
stochasticity of the analyzed problem accentuates the complexity of the
control task, as the controller must dynamically adapt to the random evolution
of the quantum state in each trajectory. Additionally, discerning the actual
state of the qubit solely from the instantaneous homodyne current proves
challenging. Thus, a sophisticated filtering mechanism is imperative to
extract pertinent information about the qubit’s state from the time series of
measurement results. Herein, we outline our contributions.
-We develop an observer to directly estimate the evolution of the density operator $\rho$ that completely characterizes the state of a $n$-level quantum system taking into account the sequence in time of homodyne current measurements. It is worth noting that Kalman-based filtering methodologies have been previously implemented in the quantum domain [7] and have found applications in quantum systems [8]. Here, we go beyond the usual Kalman techniques, e.g., in [9], by explicitly addressing the fact that the quantum system exhibits correlated process noise with measurement noise. More importantly, in our design, we guarantee by construction the properties of valid quantum states (norm preservation, Hermiticity and positive semi-definiteness). Note that $\rho$ is not even an element of a Lie Group $G$, but rather as part of a stratified space that accommodates quantum states, and therefore the techniques presented in e.g., [10] cannot be directly applied. In addition, to address the nonlinearity and complexity of the associated stochastic master equation, we utilize the coherence vector $r$ representation, [11], to capture the estimation uncertainty and enjoy the linearity properties in $r$. We emphasize that the presented approach adapts the classical Kalman filtering process to the quantum domain, leveraging the mathematical structure of quantum mechanics to estimate the density matrix $\rho$, which encapsulates the complete information about the quantum state over time.

-We introduce a robust switching-based Lyapunov control strategy tailored explicitly for $n$-level quantum stochastic systems with state uncertainty. It is important to recognize that Lyapunov control approaches have been widely used in the field of quantum control for ensuring the stability of quantum systems or guiding them to a desired state [12, 13, 14]. More recently, a quantum measurement-based feedback control via switching strategies has been proposed in [15] for quantum systems that can be controlled with a series of driving dynamics. Our contribution lies in the design of a controller in a different setup of [15] with the aim of steering the system state $\rho$ towards a pre-defined stationary target set, by directly using the estimate of $\rho$, and explicitly guaranteeing noise-to-state practically stable in probability with respect to the variance of the estimation error.
-We consider a complete framework that combines quantum Kalman-based filtering with switching Lyapunov-based feedback control, ensuring robustness despite uncertainty and noise. It probabilistically guarantees practical convergence of the system’s state to a small neighborhood of the desired invariant set, whose size degradates gracefully with the intensity of the estimation error.

##  II Quantum Stochastic Evolution

Traditionally, processes in quantum mechanics that are irreversible, often
resulting from interaction with an unbounded environment, have been explained
using a master equation applied to a density operator. Consider an open
$n$-level quantum system indicated by a Hamiltonian operator $\hat{H}$,
explaining the unitary aspects of the dynamics, and a Schrödinger picture
system operator describing the dissipative part of the dynamics induced by an
environment through an arbitrary Lindblad operator $\hat{c}$. Then, the
continuous-time evolution for system’s density operator
$\rho\in\mathcal{H}_{\rho}$ is governed by the quantum Markovian master
equation [16, 17]

$d\rho(t)=-idt[\hat{H},\rho(t)]+dt\mathcal{D}[\hat{c}]\rho(t)\equiv\mathcal{L}[\hat{c}]\rho
dt$ (1)

where
$\mathcal{D}[\hat{c}]\rho=\hat{c}\rho\hat{c}^{\dagger}-\frac{1}{2}(\hat{c}^{\dagger}\hat{c}\rho+\rho\hat{c}^{\dagger}\hat{c})$.
Note that $\mathcal{H}_{\rho}$ denotes the set of linear bounded operators on
the finite-dimensional Hilbert space characterized by $n\times n$ complex
matrices that are positive semi-definite, Hermitian, and has trace one.

###### Proposition 1

For any arbitrary $\upsilon\in\mathbb{C}$, the evolution of Lindblad generator
$\mathcal{L}$ is invariant under transformations

$\hat{c}\mapsto\hat{c}+\upsilon
I\qquad\hat{H}\mapsto\hat{H}+\frac{1}{2i}\left({{\upsilon}^{*}}\hat{c}-\upsilon{{{\hat{c}}}^{\dagger}}\right)$
(2)

where $I$ denotes the identity matrix.

###### Proof:

See [16, Chapter III]. ∎

Notably, there has been a focus on “unraveling” these master equations,
transforming density operators into stochastic trajectories Some methods,
incorporating jump processes, initially stemmed from experimental observations
of such phenomena, while others involving diffusion processes have their roots
in quantum measurement theory. Corresponding to the measurement outcomes, we
indicate the jump superoperator as
${\mathcal{J}}\left[{\hat{c}}\right]\rho={\hat{c}}\rho\hat{c}^{\dagger}$. In
the customary approach, the solution of (1) is usually implemented as a
generalized Dyson expansion in $\mathcal{J}$. However, a more direct approach
can be achieved by regarding the stochastic master equation for conditioned
density operator $\rho_{c}$, i.e., the density operator $\rho$ is considered
to be conditioned on the occurrence of jumps in the past [18]. In the sequel,
the classical expectation value is indicated by $\mathbb{E}[X]$, and the
quantum expectation value is denoted by $\left\langle
X\right\rangle=\Tr\left(X\rho\right)$. We now state the evolution of
$\rho_{c}$.

###### Proposition 2

Let $\rho_{c}$ be the conditional density operator governed by the stochastic
master equation (SME), [18],

$d{{\rho}_{c}}\left(t\right)=\left(dN\left(t\right)\mathcal{M}\left[{\hat{c}}\right]+dt\mathcal{N}\left[i\hat{H}+\frac{1}{2}{{{\hat{c}}}^{\dagger}}\hat{c}\right]\right){{\rho}_{c}}\left(t\right)$
(3)

where the superoperators $\mathcal{M}$ and $\mathcal{N}$, which are nonlinear
in $\rho$, are defined on an operator $\hat{r}$ as
$\mathcal{M}\left[{\hat{r}}\right]\rho_{c}=\frac{\mathcal{J}[\hat{r}]\rho_{c}}{\Tr\left(\mathcal{J}[\hat{r}]\rho_{c}\right)}-\rho_{c}$,
$\mathcal{N}\left[\hat{r}\right]\rho_{c}=\hat{r}\rho_{c}+\rho_{c}\hat{r}^{\dagger}-\Tr\left(\hat{r}\rho_{c}+\rho_{c}\hat{r}^{\dagger}\right)\rho_{c}$,
and the stochastic increment $dN(t)$ is a random real variable satisfying

$\mathbb{E}\left[dN(t)\right]=\left\langle{{c}^{\dagger}}c\right\rangle
dt=\Tr\left(\mathcal{J}\left[\hat{c}\right]{{\rho}_{c}}\left(t\right)\right)dt,\quad\\!\\!\\!dN{{\left(t\right)}^{2}}=dN\left(t\right)$

Then, the ensemble average $\rho(t)=\mathbb{E}[\rho_{c}(t)]$ satisfies (1).

###### Proof:

See [18] and the references therein. ∎

###### Remark 1

For every value of displacement $\upsilon$ chosen, one can derive a distinct
unraveling of the quantum master equation (1), each linked to a specific
continuous measurement. This transformation, stated in Proposition 1, is
fundamental and can be achieved by homodyne measurements.

The configuration being illustrated in Fig. 1 pertains to a balanced homodyne
setup. We deal with the local oscillator as a classical coherent field
characterized by
$\beta(t)=\sqrt{\gamma}\beta\exp\left(-i\left(\omega_{d}+\Omega(t)\right)t\right)$,
where ${\omega}_{d}$ represents the drift angular frequency and $\Omega(t)$ is
the modulation frequency of the driving field. In addition, the operator
$\hat{c}=\sqrt{\gamma}\hat{a}$ is considered as the output field from the
cavity, ignoring vacuum fluctuations The transformations of the field
operators through the beam splitter are expressed as
${\hat{C}_{1}}\left(t\right)\\!=\\!\sqrt{\frac{\gamma}{{2}}}\left(\hat{a}\\!-\\!\beta^{\prime}(t)I\right){{e}^{-i{{\omega}_{d}}t}}\\!\\!,{\hat{C}_{2}}\left(t\right)\\!=\\!i\sqrt{\frac{\gamma}{{2}}}\left(\hat{a}\\!+\\!\beta^{\prime}(t)I\right){{e}^{-i{{\omega}_{d}}t}}$
where $\beta^{\prime}(t)=\beta\exp\left(-i\left(\Omega(t)\right)t\right)$. We
consider that the local oscillator has phase $\varphi$ relative to the driving
field.

###### Proposition 3

Consider the setup illustrated in Fig. 1 and the annihilation operators
${\hat{C}_{1}}\left(t\right)$ and ${\hat{C}_{2}}\left(t\right)$. The SME (3),
can be expanded to
$d{{\rho}_{c}}\left(t\right)=\left(\sum\limits_{k=1,2}{d{{N}_{k}}(t)\mathcal{M}\left[{{{\hat{C}}}_{k}}\right]}-dt{\mathcal{N}^{{}^{\prime}}}\left[i\hat{H}+\frac{\gamma}{2}{{{\hat{a}}}^{\dagger}}\hat{a}\right]\right)\rho_{c}(t)$
where
${\mathcal{N}^{{}^{\prime}}}\left[\hat{r}\right]\rho_{c}=(\hat{r}\rho_{c}-\rho_{c}\hat{r}^{\dagger})-\Tr(\hat{r}\rho_{c}-\rho_{c}\hat{r}^{\dagger})$.

###### Proof:

See [18] and the references therein. ∎

In the framework illustrated, the signal current is defined by the
differential photocurrents from two detectors. This outcome, resembling a
quadrature measurement, is termed the homodyne signal. For a sufficiently
small time interval $\Delta t$, where the number of photodetections is large
but the system experiences minimal evolution, the photocounts at each detector
$D_{k}$ is approximated as Gaussian random variables $v_{k}$, related to
independent real Wiener increments $\Delta W_{k}$ as
${{v}_{k}}=\frac{\gamma\left|{{\beta}^{2}}\right|}{2}\left(1+{{\left(-1\right)}^{k}}{{\left\langle\frac{\hat{a}}{\beta}+\frac{{{\hat{a}}^{\dagger}}}{{{\beta}^{*}}}\right\rangle}_{c}}\left(t\right)\right)\Delta
t+\sqrt{\frac{\gamma}{2}}\left|\beta\right|\Delta{{W}_{k}}$, [18].
Transitioning to infinitesimal time $\Delta t\to dt$ and stochastic increments
$\Delta W_{k}\to dW_{k}$, with the detectors achieving unit efficiency, the
homodyne signal current is derived as

$I_{c}(t)=\frac{v_{2}-v_{1}}{dt}=\left|\beta\right|\left(\gamma{{\left\langle{{{\hat{x}}}_{\varphi}}\right\rangle}_{c}}\left(t\right)+\sqrt{\gamma}\xi\left(t\right)\right)$
(4)

Here, the deterministic part correlates with the quantum expectation of the
system quadrature
$\hat{x}_{\varphi}=e^{-i\varphi}\hat{a}+e^{i\varphi}\hat{a}^{\dagger}$ at time
$t$, chosen via homodyne detection. The stochastic term $\xi(t)=dW(t)/dt$,
representing shot noise from the local oscillator, follows Gaussian white
noise properties with
$\mathbb{E}[\xi(t)\xi(t^{\prime})]=\delta(t-t^{\prime})$. The differential
$dW(t)$, derived from $\frac{1}{\sqrt{2}}(dW_{2}(t)-dW_{1}(t))$, satisfies
Itô’s lemma with $\mathbb{E}[dW(t)]=0$, $\mathbb{E}[dW^{2}(t)]=dt$.

###### Proposition 4

Consider the setup in Fig. 1. The density operator
$\rho_{c}(t)\in\mathcal{H}_{\rho}$ satisfies a norm-preserving SME

$d\rho_{c}(t)=\left(\Delta(\rho_{c})+\mathrm{E}(\rho_{c})\Omega(t)\right)dt+\mathrm{Z}(\rho_{c})dW(t)$
(5)

where
$\mathrm{Z}(\rho_{c})=\sqrt{\gamma}\left(\hat{a}{{\rho}_{c}}(t)+{{\rho}_{c}}(t){{{\hat{a}}}^{\dagger}}-{{\left\langle\hat{a}+{{{\hat{a}}}^{\dagger}}\right\rangle}_{c}}{{\rho}_{c}}(t)\right)$,
$\Delta(\rho_{c})=-i[\hat{H}_{d},\rho_{c}(t)]+\gamma\mathcal{D}\left[{\hat{a}}\right]{{\rho}_{c}}(t)$,
$\mathrm{E}(\rho_{c})=-i[\hat{H}_{l},\rho_{c}(t)]$.

###### Proof:

In the continuum limit, we transit from discrete photodetection events to a
continuous photocurrent representation. This analysis starts with considering
the operator $\tilde{\rho}(t)$, evolving according to the norm-nonconserving
SME:
$\tilde{\rho}_{c}\left(t+dt\right)=\rho_{c}+\mathcal{L}[\hat{c}]{{\rho}_{c}(t)}dt+\frac{\sqrt{\gamma}{{I}_{c}}\left(t\right)}{\beta}\left(\hat{a}{\rho}_{c}\left(t\right)+{\rho}_{c}\left(t\right){{{\hat{a}}}^{\dagger}}\right)dt$,
which elegantly highlights how variations in the system hinge on the
stochastic outcome of the measurement, [18]. Here, $\tilde{\rho}$ indicates
the norm-nonconserving density operator. To ensure norm preservation, we
normalize $\tilde{\rho}_{c}(t+dt)$ by its trace, i.e.,
$\Tr\left(\tilde{\rho}_{c}\left(t+dt\right)\right)=1+\sqrt{\gamma}\frac{{{I}_{c}}\left(t\right)}{\beta}{{\left\langle\hat{a}+{{\hat{a}}^{\dagger}}\right\rangle}_{c}}$.
By multiplying out the terms and considering up to the second order in the
infinitesimal time interval $dt$, the norm-nonconserving terms, i.e.,
$\sqrt{\gamma}\frac{{{I}_{c}}\left(t\right)}{\beta}\left(\hat{a}{\rho}_{c}\left(t\right)+{\rho}_{c}\left(t\right){{{\hat{a}}}^{\dagger}}\right)$
cancel out. This process introduces a new term that conserves the norm, i.e.,
$\sqrt{\gamma}\left(\left(\hat{a}{{\rho}_{c}}\left(t\right)+{{\rho}_{c}}\left(t\right){{\hat{a}}^{\dagger}}\right)-{{\left\langle\hat{a}+{{\hat{a}}^{\dagger}}\right\rangle}_{c}}{{\rho}_{c}}\left(t\right)\right)$.
Noting that $\hat{H}(t)=\hat{H}_{d}+\hat{H}_{l}\Omega(t)$, the norm preserving
evolution equation is
${{\rho}_{c}}(t+dt)={{\rho}_{c}}(t)-i[\hat{H}(t),\rho_{c}(t)]dt+\gamma
dt\mathcal{D}\left[{\hat{a}}\right]{{\rho}_{c}}(t)+\sqrt{\gamma}dW\left(t\right)\left(\hat{a}{{\rho}_{c}}(t)+{{\rho}_{c}}(t){{{\hat{a}}}^{\dagger}}-{{\left\langle\hat{a}+{{{\hat{a}}}^{\dagger}}\right\rangle}_{c}}{{\rho}_{c}}(t)\right)$.
By considering $d\rho_{c}(t)=\rho_{c}(t+dt)-\rho_{c}(t)$, one obtains (5). ∎

##  III Quantum Observer

In this section, we introduce an observer to estimate the quantum state
$\rho_{c}(t)$. This estimation leverages the system’s dynamics (5) and the
measurement homodyne current $I_{c}(t)$, whose value is obtained in a sequence
of observations at sampling times $t_{k},\,k=0,1,\ldots$. More precisely, from
(4), we consider the output equation

$y(t)=h_{y}(\rho_{c}(t))+V_{y}(t),\quad t=t_{k}$ (6)

where for simplicity, we set the local oscillator to be in phase with the
driving field, i.e., $\varphi=0$. In (6),
$h_{y}(\rho_{c})=|\beta|\gamma{{\left\langle{{{\hat{x}}}_{0}}\right\rangle}_{c}}(t)$,
${{\left\langle{{{\hat{x}}}_{0}}\right\rangle}_{c}}=\Tr(\hat{x}_{0}\rho_{c})$,
and the output noise $V_{y}=|\beta|\sqrt{\gamma}\xi+V_{I}$ contains a second
noise $V_{I}$ that is considered to be the measurement noise associated with
the spectrum analyzer to obtain the current $I_{c}$. At this point, it is
important to stress that we are in the presence of several challenges: (i) the
considered $n$-level quantum system is nonlinear; (ii) the measurement noise
$V_{y}$ is correlated with process noise in (5) by noting that $dW=\xi dt$;
and (iii) the state to estimate is a density operator, which presents extra
challenges because it is not even an element of a Lie Group, but rather a
positive semi-definite, Hermitian operator of trace one acting on the Hilbert
space associated to the quantum system. The set of density matrices is a
convex subset of the space of Hermitian operators whose extreme points
represent pure states.

To address (i), we apply some of the ideas presented in [11] regarding the
coherence vector formulation and apply a Kalman-based filter strategy (more
specifically, an Extended Kalman Filter (EKF) in an hybrid sense, that is,
with continuous-time states and discrete-time observations). For (ii), we make
use of a de-correlating approach, see [19], but tailored for our case, such
that the new transformed equivalent system satisfies the property of
uncorrelated process and measurement noises, and therefore one can then make
use of the standard EKF equations. To address (iii), we integrate into the
observer equations structural properties by defining the following operator
that acts on $\rho\in\mathcal{H}_{\rho}$ and a real vector
$x\in\mathbb{R}^{n^{2}-1}$ and output a density matrix according to

$\rho\oplus
x=\frac{1}{n}I+\sum\limits_{k=1}^{n^{2}-1}{\text{Proj}\left({{r}_{k}}+{{x}_{k}}\right){\hat{F}_{k}}},\quad
r_{k}=\Tr(\rho\hat{F}_{k})$ (7)

where the operator $\text{Proj}(x)=1/\max\\{1,\|x\|\\}x$ projects $x$ onto the
closed unit ball, the $n\times n$ matrices $\hat{F}_{k}$ form a complete
orthonormal matrix basis set satisfying
$\Tr(\hat{F}_{i}\hat{F}_{j})=\delta_{ij}$, $\Tr(F_{k})=0$, and
$r\in\mathbb{R}^{n^{2}-1}$ denotes the so-called coherence vector of $\rho$.
Operator (7) is pivotal for the a posteriori update equation within our
proposed filtering algorithm, ensuring that the updated state adheres to the
properties of a density operator, maintaining the physical relevance of our
estimated states. With the established setup, we proceed to construct a
quantum observer. In this framework, we treat the noise from the local
oscillator $\xi(t)$ and the output noise $V_{I}(t)$ as white, Gaussian,
uncorrelated random processes with variances $\sigma_{\xi}^{2}(t)$, and
$\sigma^{2}_{I}(t)$, respectively, and compute the correlation $\mathbb{E}[\xi
V_{y}]=|\beta|\sqrt{\gamma}\sigma_{\xi}^{2}$ and variance
$\sigma_{V}^{2}=\mathbb{E}[V_{y}^{2}]=|\beta|^{2}\gamma\sigma_{\xi}^{2}+\sigma_{I}^{2}$.
Now, we establish the following result.

###### Proposition 5

Consider the SDE

$d\bar{\rho}=f(\bar{\rho})dt+Z(\bar{\rho})d\bar{W}(t)$ (8)

where
$f(\bar{\rho})=\Delta(\bar{\rho})-\bar{\sigma}Z(\bar{\rho})h_{y}(\bar{\rho})+\mathrm{E}(\bar{\rho})\Omega(t)+\bar{\sigma}Z(\bar{\rho})y(t)$,
$\bar{\sigma}=\frac{|\beta|\sqrt{\gamma}\sigma_{\xi}^{2}}{|\beta|^{2}\gamma\sigma_{\xi}^{2}+\sigma_{I}^{2}}$,
and $d\bar{W}=dW-\bar{\sigma}dV_{y}$. Then, the process and measurement noises
associated with the system (8) with (6) are uncorrelated, that is,
$\mathbb{E}[\bar{W}V_{y}]=0$. Moreover, (8) is equivalent to (5) in the sense
that if both SDEs start with the same initial condition and all the exogenous
signals have the same trajectory, then $\rho(t)=\bar{\rho}(t)$, for every
$t\geq t_{0}$.

###### Proof:

To show uncorrelated noises, we compute
$\mathbb{E}[\bar{W}V_{y}]=\mathbb{E}[\xi
V_{y}]-\bar{\sigma}\mathbb{E}[V_{y}V_{y}]=0$. To show the equivalence of both
SDEs, we first start to note using (6) that the term
$\bar{\sigma}Z(\rho)\big{(}y(t)-h_{y}(\rho(t))-V_{y}(t)\big{)}$ is zero. If we
now add this null term to the right-hand-side of (5), and replace the variable
$\rho_{c}$ by $\bar{\rho}$, we obtain precisely (8). ∎

Algorithm 1 Quantum Kalman Filtering

1:Choose initial quantum state $\hat{\rho}_{0}$ and $P_{0}$

2:Loop over each time step $t_{k}$ to $t_{k+1}$:

3:for $t_{k}\leq t<t_{k+1}$ do

4: Propagation: Compute $\hat{\rho}^{-}(t)$ and $P^{-}(t)$
$\displaystyle\hat{\rho}^{-}\left(t\right)$
$\displaystyle=\int_{t_{k}}^{{t_{k}}+t}\left(-i[\hat{H}(\tau),\hat{\rho}(\tau)]+\gamma\mathcal{D}\left[{\hat{a}}\right]{{\hat{\rho}}}(\tau)\right.$
$\displaystyle\qquad\qquad\left.+Z(\hat{\rho}(\tau))\bar{\sigma}(y-h_{y}(\hat{\rho}(\tau))\right)d\tau$
$\displaystyle{P}^{-}\left(t\right)$
$\displaystyle=\int_{t_{k}}^{{t_{k}}+t}A\left(\tau\right)P\left(\tau\right)+P\left(\tau\right){{A}^{\dagger}}\left(\tau\right)+G\left(\tau\right)\sigma_{\bar{W}}^{2}\left(\tau\right){{G}^{\dagger}}\left(\tau\right)d\tau$

5:end for

6:Measurement update at $t=t_{k+1}$:

7:Compute Kalman gain
$K(t_{k+1})=P(t_{k+1})C(t_{k+1})^{\dagger}S^{-1}(t_{k+1})$ with
$S(t_{k+1})=C(t_{k+1})P(t_{i+1})C^{\dagger}(t_{k+1})+\sigma_{V}^{2}(t_{k+1})$

8:Compute $\hat{\rho}(t_{k+1})$ and $\hat{P}(t_{k+1})$ using update equations
$\displaystyle\hat{\rho}(t_{k+1})$
$\displaystyle=\hat{\rho}^{-}(t_{k+1})\oplus
K\left(t_{k+1}\right)\left(y\left(t_{k+1}\right)-h_{y}(t_{k+1})\right)$
$\displaystyle P(t_{k+1})$
$\displaystyle=P^{-}(t_{k+1})-K(t_{k+1})C(t_{k+1})P^{-}(t_{k+1})$

9:End Loop

We can now propose our state estimator that continuously updates its state
estimate $\hat{\rho}(t)$ as new measurements are acquired at sampling times
$t_{k},\,k=0,1,\ldots$ through the propagation and update steps for the state
and covariance matrix declared in Algorithm 1. Note that we define the initial
condition’s representation for the system, incorporating uncertainties, as
follows $\hat{\rho}_{0}=\hat{\rho}(t_{0})={\bar{\hat{\rho}}}_{0}\oplus\zeta,$
where ${\bar{\hat{\rho}}}_{0}\in\mathcal{H}_{\rho}$ is a deterministic initial
quantum state, and $\zeta$ signifies the influence of uncertainties on this
state, which is modeled as a perturbation on the coherence vector defined as a
random Gaussian vector with zero mean and covariance matrix $P_{0}$, (whose
size is $(n^{2}-1)\times(n^{2}-1)$). To evaluate the propagation of $P$, we
need to compute the matrices $A(t)$ and $G(t)$. To this end, we use (7) by
noting that for the deterministic part
$\dot{r}_{i}=\Tr(\dot{\bar{\rho}}\hat{F}_{i})=\Tr(f(\bar{\rho})\hat{F}_{i})$.
Thus, replacing $\bar{\rho}$ by $I/n+\sum r_{k}F_{k}$, we obtain
$\dot{r}_{i}=\bar{f}_{i}(r)$, where $\bar{f}_{i}(r)=\Tr(f(I/n+\sum
r_{k}F_{k})F_{i})$. Therefore, each $i$-row of $A(t)$ in Algorithm 1 is given
by $\frac{\partial\bar{f}_{i}(r)}{\partial
r}\big{|}_{r=[\Tr(\hat{\rho}F_{1}),\Tr(\hat{\rho}F_{2})\ldots]}$. The same
procedure applies to compute $G$, but now using the term $Z(\bar{\rho})$. It
turns out that only the term $Z(\rho)$ introduces a nonlinear term (in fact
quadratic) in $r$. All the other ones are linear with $r$, and therefore the
linearization (for this terms) are not approximations. This is not surprising
since the Lindblad equation can be transformed into an inhomogeneous linear
vector equation using the coherence vector $r$ formulation, [11]. In addition,
we compute
$C=|\beta|\gamma\,\big{[}\Tr(\hat{x}_{0}\hat{F}_{1}),\Tr(\hat{x}_{0}\hat{F}_{2}),\ldots,\Tr(\hat{x}_{0}\hat{F}_{n^{2}-1})\big{]}$.

We can now state the following result.

###### Theorem 1

Consider the proposed Kalman-based filter with initial condition
$\hat{\rho}_{0}\in\mathcal{H}_{\rho}$. Then, the state estimate
$\hat{\rho}(t)$ leaves in $\mathcal{H}_{\rho}$ for every $t\geq 0$.

###### Proof:

In the propagation step, the density operator $\hat{\rho}^{-}(t)$ can be seen
as the solution of a norm-preserving stochastic master equation given by (5).
Note that $\Tr(Z(\rho))=0$. In the measurement step, we make use of the
operator (7), which guarantees the properties of a density operator by noting
that 1) the (weighted) sum of Hermitian matrices is Hermitian; 2)
$\|r+x\|^{2}\leq 1$, which implies positive semi-definiteness of the result;
and 3) norm preservation since the $\Tr(I/n+\sum(r_{k}+x_{k})F_{k})=1$ using
$\Tr(F_{k})=0$. ∎

##  IV Lyapunov-based state feedback control

In this section, we design a controller that commands the signal $\Omega(t)$
with the aim of steering the system’s state $\rho(t)$ from an initial
condition $\rho_{0}\in\mathcal{H}_{\rho}$ towards a predetermined stationary
point within the orbit
$O_{f}(\rho_{0}):=\\{\hat{X}\rho_{0}\hat{X}^{\dagger}\mid\hat{X}\in U(n)\\}$,
where any target state $\rho_{f}$ within $O_{f}(\rho_{0})$ is characterized by
its commutation with the drift Hamiltonian $\hat{H}_{d}$, that is,
$[\rho_{f},\hat{H}_{d}]=0$. Here, $U(n)$ denotes the group of $n\times n$
unitary matrices. The aim is to identify states that are unitarily equivalent
to $\rho_{0}$, implying these states share the same spectrum as the initial
state. Prior to delving into the control design, we introduce the following
stability definitions pertinent to the stochastic quantum system problem at
hand, drawing upon the frameworks in [20, 21].

###### Definition 1

(Asymptotically Practically Stable in Probability): Let $\chi$ be a closed and
invariant set under the dynamics governed by the stochastic differential
equation in Itô form $d\rho_{c}=f(\rho_{c})dt+h(\rho_{c})\sigma(t)dW(t)$,
where $\rho_{c}$ denotes the state in $\mathcal{H}_{\rho}$, $\sigma(t)$ is the
intensity of noise at time $t$, and $dW(t)$ denotes the increment of a
1-dimensional standard Wiener process. The set $\chi$ is asymptotically
practically stable in probability if, for every $\varepsilon>0$, there exists
a function $\beta\in\mathcal{K}\mathcal{L}$ and a non-negative real number
$b\geq 0$ such that for every initial state
$\rho_{c}(0)\in\mathcal{H}_{\rho}$, the following condition holds:

$\mathbb{P}\left\\{d(\rho_{c}(t),\chi)<\beta\left(d(\rho_{c}(0),\chi),t\right)+b\right\\}\geq
1-\varepsilon,\quad\forall t\geq 0$

where $d(\rho_{c},\chi)$ denotes a distance from state $\rho_{c}$ to the set
$\chi$, measured in an appropriate metric. If $b=0$, $\chi$ is asymptotically
stable in probability. If $\beta(r,t)=cre^{-\lambda t}$ for some $c>0$ and
$\lambda>0$, $\chi$ is exponentially stable in probability.

###### Definition 2

(Noise-to-State Practically Stable in Probability): Adopting the conditions
from Definition 1, the set $\chi$ is noise-to-state practically stable (NSpS)
in probability if, for every $\varepsilon>0$, there exists a function
$\beta\in\mathcal{K}\mathcal{L}$, a function $\gamma\in\mathcal{K}$, and a
non-negative real number $b\geq 0$, ensuring that for every initial state
$\rho_{c}(0)\in\mathcal{H}_{\rho}$, the following holds with probability at
least $1-\varepsilon$, for every $t\geq 0$

$\displaystyle\mathbb{P}\Big{\\{}d(\rho_{c}(t),\chi)<$
$\displaystyle\beta\left(d(\rho_{c}(0),\chi),t\right)+\gamma\left(\sup_{\tau\in[0,t]}\sigma^{2}(\tau)\right)+b\Big{\\}}\geq
1-\varepsilon$

If $b=0$, $\chi$ is noise-to-state stable (NSS) in probability.

Definition 2 is introduced to measure the resilience of a quantum system’s
stability in the face of noise. It ensures that, probabilistically, the
system’s state stays within a practical vicinity of the invariant set $\chi$,
notwithstanding the influence of quantum noise. The extent of this vicinity is
contingent upon the intensity of the noise, as characterized by the variance
$\sigma^{2}$.

### Design of the Lyapunov function

Drawing inspiration from the concepts presented in [22], we begin by
considering a state $\rho_{f}\in O_{f}(\rho_{0})$, and select a matrix
$\hat{P}=I-\rho_{f}$. This selection leads us to define the Lyapunov function
as

$V(\rho)=\Tr(\hat{P}\rho).$ (9)

It is important to note that $\hat{P}$ is Hermitian, ensuring that it
possesses real eigenvalues. Furthermore, $\hat{P}$ is positive semi-definite,
which guarantees that the Lyapunov function $V(\rho)$ is non-negative for all
states $\rho$, i.e., $V({\rho}(t))\geq 0$. Additionally, a significant
characteristic of $\hat{P}$ is its commutativity with the final state
$\rho_{f}$, that is, $[\rho_{f},\hat{P}]=0$.

###### Remark 2

Stationary points of the Lyapunov function $V(\rho)$ are identified as states
$\bar{\rho}$ that commute with $\rho_{f}$, see [22], i.e.,
$[\bar{\rho},\rho_{f}]=0$. Given the definition of $\hat{P}=I-\rho_{f}$, this
commutation property implies $[\bar{\rho},\hat{P}]=0$.

We initially derive a controller to minimize $V(\rho)$, assuming access to the
actual state $\rho(t)$. Subsequently, we adapt this approach for situations
lacking direct access to $\rho(t)$, employing an estimated state
$\hat{\rho}(t)$.

###### Theorem 2

Consider the quantum system (5) and let
$\chi_{f}=\\{\rho\in\mathcal{H}_{\rho}:[\rho,\hat{P}]=0\\}$ be the desired
stationary set. The feedback control strategy is implemented as

$\Omega(t)=\begin{cases}-i\frac{\Upsilon_{0}}{\Upsilon_{1}}&\text{ for
}t\in[t_{k-1},t_{k}),\ k=1,3,5,\ldots\\\ \quad 0&\text{ otherwise}\end{cases}$
(10)

where
${{\Upsilon}_{0}}=\gamma\Tr\left(\hat{P}\mathcal{D}\left[{\hat{a}}\right]{{\rho}_{c}}\right)+\alpha\Tr\left(\hat{P}{{\rho}_{c}}\right)$,
with $\alpha\in\mathbb{R}^{+}$, and
${{\Upsilon}_{1}}=\Tr\left(\hat{P}\left[{{{\hat{H}}}_{l}},\rho_{c}\right]\right)$.
The control is adjusted at discrete switching times
$\\!\\{t_{0},t_{1},\ldots\\!\\}$,  with $t_{0}$ the initial time and
$t_{k}$ as

$\displaystyle\left\\{\begin{array}[]{ll}t_{k}=\min\limits_{t}\\{t\geq
t_{k-1}+\Delta_{1}:\Upsilon_{1}\leq\varepsilon\\}&\text{for }k\text{ odd},\\\
t_{k}=\min\limits_{t}\\{t\geq
t_{k-1}+\Delta_{2}:\Upsilon_{1}\geq\varepsilon\\}&\text{for }k\text{
even},\end{array}\right.$

for some appropriate dwell times $\Delta_{1},\,\Delta_{2}\in\mathbb{R}^{+}$.
Then, the resulting closed-loop system exhibits the property that the set
$\chi_{f}$ is practically exponentially stable in probability.

###### Proof:

Given that $\rho_{f}$ is selected to commute with $\hat{P}$, it follows that
$\rho_{f}$ becomes a stationary point of the Lyapunov function $V(\rho)$, as
discussed in [22]. Furthermore, according to Remark 2, the set $\chi_{f}$ is
identified as a stationary set. To investigate its stability, we consider the
infinitesimal generator of $\rho(t)$ applied to $V(\rho)$. This analysis
involves the diffusion process described in (5) and leverages the Ito formula,
resulting in

$L(V(\rho))=\Tr(\hat{P}^{T}(\Delta(\rho)+\mathrm{E}(\rho)\Omega(t)))+\frac{1}{2}\Tr\left(Z(\rho)H_{V}Z(\rho)^{T}\right)$

where the Hessian matrix $H_{V}$ of $V(\rho)$ is zero in this context.
Considering the commutation relation $[\rho_{f},\hat{H}_{d}]=0$ and the
definition $\hat{P}=I-\rho_{f}$, we deduce $[\hat{P},\hat{H}_{d}]=0$. By
employing the trace property $\Tr(A[B,C])=\Tr(C[A,B])$, it is evident that
$\Tr(\hat{P}[\hat{H}_{d},\rho])=0$. Thus, we obtain

$L(V(\rho))=-i\Omega(t)\Tr(\hat{P}[\hat{H}_{l},\rho_{c}])+\gamma\Tr(\hat{P}\mathcal{D}[\hat{a}]\rho_{c}).$
(11)

At this point, our objective is to identify a control function $\Omega(t)$
that satisfies $L(V(\rho))\leq-\alpha V(\rho)$, ensuring an exponential
decrease in $V(\rho)$ with a rate $\alpha>0$. Among the potential solutions,
we opt for the one that minimizes the control effort $\Omega^{2}(t)$ pointwise
in time. Hence, we seek $\Omega(t)$ that solves the optimization problem:
$\min\Omega^{2}(t)$ subject to $-i\Omega(t)\Upsilon_{1}+\Upsilon_{0}\leq 0$.
The solution to this optimization problem is precisely the control law
presented in (10). Caution is advised when $\Upsilon_{1}$ approaches zero to
prevent computational inaccuracies and excessive control signal magnitudes. As
a preventive measure, the control strategy switches $\Omega(t)$ to zero once
$\Upsilon_{1}\leq\varepsilon$. To avoid Zeno behavior, characterized by an
infinite number of switches within a finite time, we enforce a minimum dwell
time between switching, $t_{i}-t_{i-1}>\Delta$.

Taking now into account the results and reasoning presented in [23] for
switching systems with stable and unstable modes, and acknowledging that
$L(V)$ remains bounded when $\Omega=0$, the practical stability in probability
of the set $\chi$ is conclusively established by noticing that
$L(V)\leq-\eta(V)+\eta_{0}$, for some class $\mathcal{K}$ function $\eta$ and
constant $\eta_{0}\geq 0$. ∎

###### Theorem 3

Consider the same conditions of Theorem 1, but now replacing (10) by

$\Omega\left(t\right)=-i\displaystyle{\frac{{{\hat{\Upsilon}}_{0}}}{{{\hat{\Upsilon}}_{1}}}},\quad
t\in[t_{k-1},t_{k}),\ k=1,3,5,\ldots$ (12)

where
${\hat{\Upsilon}_{0}}=\gamma\Tr\left(\hat{P}\mathcal{D}\left[{\hat{a}}\right]{\hat{\rho}}\right)+\alpha\Tr\left(\hat{P}{\hat{\rho}}\right)$,
${{\hat{\Upsilon}}_{1}}=\Tr\left(\hat{P}\left[{{{\hat{H}}}_{l}},\hat{\rho}\right]\right)$,
$\alpha\in\mathbb{R}^{+}$, and $\hat{\rho}$ is the output estimate of the
proposed Kalman-based filter. Let $\tilde{\sigma}^{2}(t)$ be the incremental
variance associated to the noisy estimation error
$\Tr(\tilde{\rho})=\Tr(\rho_{c}-\hat{\rho})$, which is considered to be
bounded. Then, the set $\chi_{f}$ is NSpS with respect to
$\tilde{\sigma}^{2}(t)$.

###### Proof:

From (11), it follows that $L(V(\rho))$ satisfies

$L(V(\rho))=-i{{\hat{\Upsilon}}_{1}}\Omega\left(t\right)+{{\hat{\Upsilon}}_{0}}-i{{\tilde{\Upsilon}}_{1}}\Omega\left(t\right)+{{\tilde{\Upsilon}}_{0}}$

where $\tilde{\Upsilon}_{0}={\Upsilon}_{0}-\hat{\Upsilon}_{0}$ and
$\tilde{\Upsilon}_{1}={\Upsilon}_{1}-\hat{\Upsilon}_{1}$. Since $\Omega$ is
bounded, $\tilde{\Upsilon}_{0}$ and $\tilde{\Upsilon}_{1}$ are functions that
can be bounded by class $\mathcal{K}$ functions of $\tilde{\sigma}^{2}$, it
follows that using the same arguments in the previous proof and [23] one can
conclude that for $V(\rho)\geq\eta_{1}(\tilde{\sigma}^{2})$, the generator
$L(V)\leq-\eta(V)+\eta_{0}$, for some class $\mathcal{K}$ function $\eta$,
positive definite function $\eta_{1}$ and constant $\eta_{0}\geq 0$.
Therefore, the results follow. ∎

##  V Application to Qubits in Leaky Cavities

The methodology presented here for the illustrated example is universally
applicable to any two-level atom experiencing classical influences such as
driving, damping, and detuning. Our examination specifically focuses on a
qubit interacting with a leaky cavity in a rotating frame. The drift
Hamiltonian, $\hat{H}_{d}=\frac{\omega_{d}}{2}\hat{\sigma}_{z}$, incorporates
the effective detuning,
$\omega_{\Delta}=\omega_{d}+\Delta\omega_{d}-\omega_{0}$, which represents the
adjusted frequency discrepancy of the qubit. Here, $\omega_{d}$ denotes the
frequency gap between the excited $\left|e\right\rangle$ and ground
$\left|g\right\rangle$ states, $\omega_{0}$ signifies the oscillation
frequency of the driving field, and $\Delta\omega_{d}$ refers to the frequency
shift. The Pauli operator $\hat{\sigma}_{z}=\left|g\right\rangle\left\langle
g\right|-\left|e\right\rangle\left\langle e\right|$ serves as the inversion
operator for the qubit. The control Hamiltonian,
$H_{c}=\frac{\Omega(t)}{2}\hat{\sigma}_{x}$, where $\Omega(t)$ is the Rabi
frequency, is directly proportional to the transition dipole moment and the
amplitude of the driving electromagnetic field. For the annihilation and
creation field operators, we utilize the Pauli lowering and raising operators,
$\hat{a}=\hat{\sigma}_{-}$ and $\hat{a}^{\dagger}=\hat{\sigma}_{+}$,
respectively. The initial density operator is chosen as $\rho_{0}=\small
0.5\begin{pmatrix}1&i\\\ -i&1\end{pmatrix}$. We compare three distinct quantum
dynamics scenarios in Fig. 2. Notably, all stochastic evolution graphs are
plotted across 100 realizations alongside their mean values. The efficacy of
our Lyapunov control method is demonstrated in Fig. 3, where the initial state
evolves towards the target final state $\rho_{f}=\small\begin{pmatrix}0&0\\\
0&1\end{pmatrix}$, achieving a fidelity exceeding 99% to the set $\chi_{f}$.
$\rho_{f}$ underscores that the system conclusively transitions to the state
$|1\rangle$ post-measurement. In this vein, we also plot the expected system
quadrature $\langle\hat{x}_{0}\rangle_{c}(t)$, reflecting the expected value
of $\sigma_{x}$ and the output signal in Fig. 4. Finally, we integrate our
control methodology with the quantum observer to implement the framework
depicted in Fig. 1. As shown in Fig. 5, the estimated state tracks with a
small error the evolution of the true state.

Figure 2:  Comparative evolution of quantum state populations.  Figure 3:
Controlled state evolution and respective fidelity.  Figure 4: Expected value
of $\sigma_{x}$, and the mean value of output. Figure 5: Estimated evolution
of quantum states.

##  VI Conclusion

This study presents a Lyapunov-based control framework, integrated by
homodyne-mediated filtering, tailored for quantum systems. By employing a
Kalman-based filtering, exploiting adequately the properties of a coherence-
vector representation, and combining with input to state concepts, our
approach maintains inherent quantum characteristics, which efficiently handle
stochastic dynamics ensuring practical stability against noise. Demonstrated
through qubit state stabilization in a leaky cavity, our method paves the way
for robust quantum information processing.

## References

  * [1] C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, S. Gleyzes, P. Rouchon, M. Mirrahimi, H. Amini, M. Brune _et al._ , “Real-time quantum feedback prepares and stabilizes photon number states,” _Nature_ , vol. 477, no. 7362, pp. 73–77, 2011.
  * [2] V. Belavkin, “Theory of the control of observable quantum systems,” _Automatica and Remote Control_ , vol. 44, no. 2, pp. 178–188, 1983.
  * [3] V. P. Belavkin, “Quantum stochastic calculus and quantum nonlinear filtering,” _Journal of Multivariate analysis_ , vol. 42, no. 2, pp. 171–201, 1992.
  * [4] L. Bouten, R. Van Handel, and M. R. James, “An introduction to quantum filtering,” _SIAM Journal on Control and Optimization_ , vol. 46, no. 6, pp. 2199–2241, 2007.
  * [5] J. E. Gough, M. R. James, H. I. Nurdin, and J. Combes, “Quantum filtering for systems driven by fields in single-photon states or superposition of coherent states,” _Physical Review A_ , vol. 86, no. 4, p. 043819, 2012.
  * [6] P. Rouchon and J. F. Ralph, “Efficient quantum filtering for quantum feedback control,” _Physical Review A_ , vol. 91, no. 1, p. 012118, 2015.
  * [7] M. F. Emzir, M. J. Woolley, and I. R. Petersen, “A quantum extended kalman filter,” _Journal of Physics A: Mathematical and Theoretical_ , vol. 50, no. 22, p. 225301, 2017.
  * [8] K. Ma, J. Kong, Y. Wang, and X.-M. Lu, “Review of the applications of kalman filtering in quantum systems,” _Symmetry_ , vol. 14, no. 12, p. 2478, 2022.
  * [9] E. Corcione and C. Tarin, “State and parameter estimation for stochastic open two-level quantum systems,” in _2023 American Control Conference (ACC)_. IEEE, 2023, pp. 2270–2275.
  * [10] A. Barrau and S. Bonnabel, “The invariant extended kalman filter as a stable observer,” _IEEE Transactions on Automatic Control_ , vol. 62, no. 4, pp. 1797–1812, 2016.
  * [11] R. Alicki and K. Lendi, “N-level systems and applications to spectroscopy, lect. notes phys.” _Quantum Dynamical Semigroups and Applications_ , vol. 717, pp. 47–107, 2007.
  * [12] M. Mirrahimi and R. Van Handel, “Stabilizing feedback controls for quantum systems,” _SIAM Journal on Control and Optimization_ , vol. 46, no. 2, pp. 445–467, 2007.
  * [13] S.-C. Hou, M. Khan, X. Yi, D. Dong, and I. R. Petersen, “Optimal lyapunov-based quantum control for quantum systems,” _Physical Review A_ , vol. 86, no. 2, p. 022321, 2012.
  * [14] G. Cardona, A. Sarlette, and P. Rouchon, “Exponential stochastic stabilization of a two-level quantum system via strict lyapunov control,” in _2018 IEEE Conference on Decision and Control (CDC)_. IEEE, 2018, pp. 6591–6596.
  * [15] T. Grigoletto and F. Ticozzi, “Stabilization via feedback switching for quantum stochastic dynamics,” _IEEE Control Systems Letters_ , vol. 6, pp. 235–240, 2021.
  * [16] H.-P. Breuer and F. Petruccione, _The theory of open quantum systems_. OUP Oxford, 2002.
  * [17] C. Altafini and F. Ticozzi, “Modeling and control of quantum systems: An introduction,” _IEEE Transactions on Automatic Control_ , vol. 57, no. 8, pp. 1898–1917, 2012.
  * [18] H. M. Wiseman and G. J. Milburn, _Quantum measurement and control_. Cambridge university press, 2009.
  * [19] D. Simon, _Optimal state estimation: Kalman, H infinity, and nonlinear approaches_. John Wiley & Sons, 2006.
  * [20] H. Ito and Y. Nishimura, “Stability of stochastic nonlinear systems in cascade with not necessarily unbounded decay rates,” _Automatica_ , vol. 62, pp. 51–64, 2015.
  * [21] H. Deng, M. Krstic, and R. J. Williams, “Stabilization of stochastic nonlinear systems driven by noise of unknown covariance,” _IEEE Transactions on automatic control_ , vol. 46, no. 8, pp. 1237–1253, 2001.
  * [22] D. d’Alessandro, _Introduction to quantum control and dynamics_. Chapman and hall/CRC, 2021.
  * [23] A. P. Aguiar, J. P. Hespanha, and A. M. Pascoal, “Switched seesaw control for the stabilization of underactuated vehicles,” _Automatica_ , vol. 43, no. 12, pp. 1997–2008, 2007.