# Tuning the universality class of phase transitions by feedback:
open quantum systems beyond dissipation

D. A. Ivanov  Department of Physics, St. Petersburg State University, St.
Petersburg, Russia  T. Yu. Ivanova  Department of Physics, St. Petersburg
State University, St. Petersburg, Russia  S. F. Caballero-Benitez  Instituto
de Física, LSCSC-LANMAC, Universidad Nacional Autónoma de México, Ciudad de
México 04510, México  I. B. Mekhov<EMAIL_ADDRESS>Department of
Physics, St. Petersburg State University, St. Petersburg, Russia

###### Abstract

We shift the paradigm of feedback control from the control of quantum states
to the control of phase transitions in quantum systems. We show that feedback
allows tuning the universality class of phase transitions via modifying its
critical exponent. We expand our previous treatment of Dicke model and go
beyond the approximation of adiabatically eliminated light field. Both
linearized and nonlinear models of spin ensembles are considered. The
tunability of quantum fluctuations near the critical point by the feedbacks of
nontrivial shapes is explained by considering the fluctuation spectra and the
system behavior at single quantum trajectories.

##  I Introduction

In our letter [1], we have studied feedback-induced phase transitions (FPT)
[1, 2, 3] in open quantum systems, which are not dissipative ones, but are
coupled to a measurement device. The first experiment implementing the
feedback control of Dicke phase transition using a Bose–Einstein condensate
(BEC) trapped in a cavity has been reported in Ref. [4]. The general concept
of quantum phase transitions (QPT) [5] is very important not only in physics
of various systems (e.g. atomic and condensed matter ones), but in other
fields as well, e.g., quantum information and technologies [6], machine
learning [7], and complex networks [8]. In contrast to familiar thermal
transitions, QPT is driven by quantum fluctuations existing even at zero
temperature in closed systems. Since the extensive studies of lasers [9] the
notion of phase transitions has been adopted to non-equilibrium open systems.
The dissipation in open systems provides fluctuations via the system-bath
coupling, and the dissipative phase transition (DPT) takes place between
nontrivial non-equilibrium steady states [10, 11].

In Ref. [1] we extended the consideration of phase transitions in open systems
by including the quantum measurements and feedback control. The quantum
description of a system being continuously measured is more detailed than that
of a dissipative system: the latter is its limiting case, where the
measurement results are completely ignored [12]. We have shown that adding the
measurement-based feedback can induce phase transitions. Moreover, the
feedback allows for controlling quantum properties of the transition by tuning
its critical exponents, and thus their universality class [1, 2]. In the
perspective of such feedback-induced phase transitions the role of quantum
fluctuations becomes especially emphasized. The quantum noise here drives the
phase transition similar to QPT for closed systems, but it becomes now “better
observable” via the stochastic nature of the measurement outcome. Moreover,
the effect of quantum noise is now twofold: the direct disturbance of the
system via the measurement backaction and the feedback loop enhanced
fluctuations of the measured signal.

Such tunable feedback-induced phase transitions maybe be obtained in many-body
problems, e.g., the feedback-induced supersolid-like, antiferromagnetic, and
other states in optical lattices [3, 13], time crystals [1, 14], etc. The
influence of quantum measurements on phase transitions in many-body systems
was considered without feedback, e.g., in Refs. [15, 16, 17, 18, 19, 20, 21].

Feedback is a very general idea of modifying system parameters depending on
the measurement outcomes: the examples spread from the contemporary rock and
classical music (e.g. the Sampo device [22]) to reinforcement learning [23].
The concept of feedback control has been successfully extended to quantum
domain [12, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40] resulting in quantum metrology aiming to stabilize nontrivial quantum
states and squeeze their noise [41, 42, 43]. The measurement backaction
typically defines the limit of control, thus, playing an important but
negative role. In Ref. [1] we have shown that the feedback can do more than
just the quantum state control, but it allows for the phase transition control
as well. In this case the measurement fluctuations and feedback enhanced
fluctuations drive the transition playing an essentially positive role.

In its essence the quantum system coupled to a classical feedback loop is a
hybrid system. Such systems are actively studied in the field of quantum
technologies, where various systems have been already coupled [44]: atomic,
photonic, superconducting, mechanical, etc. The goal is to use advantages of
various components, while avoiding using their disadvantages. In this sense,
we address a hybrid quantum-classical system, where the quantum system can be
a simple one providing the quantum coherence, while all other properties
necessary for tunable phase transition are provided by the classical feedback
loop: nonlinear interaction, non-Markovianity, and fluctuations.

We have shown [1] that FPT leads to effects similar to particle-bath problems
(e.g. spin-boson, Kondo, Caldeira-Leggett, quantum Browninan motion,
dissipative Dicke models) describing very different physical systems from
quantum magnets to cold atoms [45, 46, 47, 48, 49, 50, 51, 52]. In particular
it becomes possible to address important questions about quantum-classical
mapping between Floquet time crystals [53, 54] and long-range interacting spin
chains. Our model is directly applicable to many-body systems such as those of
many-body cavity QED (cf. for reviews [55, 56, 57]). These systems have been
recently marked by experimental demonstrations of superradiant Dicke [58, 4],
lattice supersolid [59, 60], and other phase transitions and phenomena [61,
62, 63, 64], as well as theory proposals [65, 66, 67, 68, 69, 70, 71, 72, 73,
49, 50]. Nevertheless, effects we discuss here require to go beyond the
cavity-induced autonomous feedback [74], where the feedback time response is
limited to an exponentially decaying function.

Here we provide a more detailed theoretical description of the feedback
control of phase transitions focusing on the Dicke model. We go beyond the
approximation of the adiabatically eliminated light field [1] and treat light
as a fully dynamical variable. Interestingly, while the adiabatic model
corresponds well to the experiments [58, 59, 4], the non-adiabatic model
corresponds to the experiments [60, 75, 76]. We consider both linearized and
nonlinear models of spin ensembles. The tunability of quantum fluctuations
near the critical point by the feedbacks of nontrivial shapes is explained by
considering the fluctuation spectra and the system behavior at single quantum
trajectories.

##  II Dicke model with feedback

Figure 1:  A possible realization of the feedback-controlled Dicke model with
a BEC. The atoms are trapped in a 1D periodic potential (red) with the
potential depth controlled by changing the intensity of the lattice laser.
Additionally, the atoms are coupled to two light modes: the transverse pump
and a cavity mode (blue). The cavity light is measured by the homodyne
detector (HD). The signal of the detector is processed with the kernel $h(t)$
and used to feedback control the lattice potential.

In order to illustrate new features of a phase transition offered by the
feedback control we consider the Dicke model with an additional feedback loop
[1]. Let us consider a collection of $N$ spins (two-level atoms, qubits)
equally coupled to a single bosonic (optical cavity) mode. If we denote the
cavity frequency as $\delta$ and the spin frequency as $\omega_{R}$ then the
system Hamiltonian becomes

$H=\delta
a^{\dagger}a+\omega_{R}S_{z}+\frac{2g}{\sqrt{N}}S_{x}\left(a^{\dagger}+a\right).$
(1)

Here $a^{\dagger}$ and $a$ are the creation and annihilation operators of the
bosonic mode and $S_{i}=\sum_{j}\sigma_{i}^{(j)}$ are the components
($i=\\{x,y,z\\}$) of the collective spin operator with $\sigma_{i}^{(j)}$
being the Pauli matrices of $j$-th spin. The coupling strength between the
spins and the bosonic mode is characterized by the coupling constant $g$. The
units with $\hbar=1$ are used throughout the paper.

It is well-known [77] that the open Dicke model possesses the superradiant
phase transition that separates the normal phase with the number of quanta in
the boson mode independent from the number of spins $N$ and the superradiant
phase where the number of quanta scales with $N$. In the absence of feedback
the critical value of the coupling constant reads [77]

$g_{\mathrm{crit}}=\sqrt{\frac{\omega_{R}\left(\kappa^{2}+\delta^{2}\right)}{4\delta}}.$
(2)

The direct observation of the superradiant phase transition for two-level
atoms in a cavity is impractical due to low values of the atom-field coupling.
Nevertheless the transition has been observed with internal atomic states
coupled via Raman transition [78, 79] and with momentum states of a Bose-
Einstein condensate in an optical cavity [58, 60]. In principle, the feedback
control can be implemented with both these models. Moreover, other systems
with the Hamiltonian similar to Eq. (1) can be proposed and equipped with a
feedback loop [1].

The critical exponents for the Dicke model has been found to be different for
open and closed systems. In particular, the critical exponent $\alpha$ that
describes the divergence of the number of photons in a cavity
($\sim|g-g_{crit}|^{-\alpha}$) equals $1/2$ for a closed system and 1 for an
open one. This difference demonstrates the effect of the environment-induced
fluctuations on the properties of the phase transition. It was shown [49, 50,
80] that the appropriate environment engineering can gradually change the
critical exponent. However, experimental implementation of required spectral
functions is usually a difficult task: in a BEC the atomic dissipation is
limited by the Beliaev damping [81]. In contrast, the feedback that we
describe here can easily be implemented. We will demonstrate that the feedback
control of phase transitions can easily modify the critical behavior of the
system.

For concreteness we consider an implementation of the feedback control in the
Dicke model based on the momentum states of a BEC [82, 58, 60]. We assume that
the BEC is trapped in a quasi-1D periodic potential $V(t)\cos^{2}(k_{0}x)$,
where $V(t)$ is the amplitude of this potential that is controlled by the
feedback loop, see Fig. 1, $k_{0}$ is the wavenumber of the laser forming such
a lattice.

In addition to this periodic potential, the BEC is coupled to an optical
cavity. The modes of this cavity are labeled as $a_{l}$, the frequencies of
the modes are $\omega_{l}$. Note that we do not require the high finesse of
this cavity. Contrary to the experiments [58, 60] in the feedback scheme the
cavity can be considered as a quantum meter for the atoms or spins. Thus it
would be even beneficial to have a cavity that produces only little effect on
the atoms. Moreover, having enough sensitivity of light detectors the feedback
scheme can possibly be realized without a cavity [2]. The Hamiltonian of the
atoms and the cavity modes reads

$H=\sum_{l}\omega_{l}a_{l}^{\dagger}a_{l}+\int_{0}^{L}\Psi^{\dagger}(x)H_{\mathrm{a1}}\Psi(x)dx+\sum_{l}\zeta_{l}\left(a_{l}^{\dagger}+a_{l}\right),$
(3)

where $\zeta_{l}$ are the external classical pumps of the light modes. Here
the single-atom Hamiltonian $H_{\mathrm{a1}}$ after adiabatic elimination of
the excited atomic level is written as

$\displaystyle H_{\mathrm{a1}}=\frac{p^{2}}{2m_{a}}+V(t)\cos^{2}(k_{0}x)$
$\displaystyle+\frac{1}{\Delta_{a}}\sum_{l,m}g_{l}u_{l}^{*}(x)a_{l}^{\dagger}g_{m}u_{m}(x)a_{m},$
(4)

where the first term is the atomic kinetic energy with $m_{a}$ being the mass
of the atom, and $\Delta_{a}$ is the detuning between the mode frequency and
the atom transition frequency. The coupling between the atoms and the light
modes are described by the constants $g_{l,m}$, the mode functions are
$u_{l,m}(x)$.

The simplest configuration allowing for the detection of the distribution of
atoms along the lattice axis involves only two light modes, see Fig. 1. One
mode is the transverse pump with the uniform distribution along the lattice
axis. The corresponding mode function can be chose as
$u_{\mathrm{pump}}(x)=1$. The other mode has the spatial dependence
$u_{1}(x)\sim\cos k_{1}x$. Assuming the frequency of the two modes equals to
the double frequency of the lattice laser the scattering of the light from the
pump mode $u_{\mathrm{pump}}$ into the mode $u_{1}$ will be maximal if the
atoms will be tightly localize in the lattice sites. Indeed the localized
atoms will serve as a diffraction grating with the period $d=\lambda_{0}/2$
providing the diffraction maximum condition if $\lambda_{0}/2=\lambda_{1}$.
Thus the uniformity of the BEC distribution determines the amount of light
scattered from $a_{\mathrm{pump}}$ into the mode $a_{1}$. This light can be
used for the feedback control.

Let us decompose the field operator according to the spatial dependence of the
Hamiltonian

$\Psi(x)=\frac{1}{\sqrt{L}}c_{0}+\sqrt{\frac{2}{L}}c_{1}\cos k_{1}x,$ (5)

where $c_{0,1}$ are the annihilation operators of the atomic waves with
momenta $0$ and $k_{1}$, respectively. After the substitution of Eqs. (4) and
(5) in Eq. (3) the Hamiltonian transforms to

$H=\delta
a^{\dagger}a+\omega_{R}S_{z}+\frac{2}{\sqrt{N}}S_{x}\left[g\left(a^{\dagger}+a\right)+GI(t)\right],$
(6)

which is the Hamiltonian of the Dicke model (1) with the feedback control part
proportional to $I(t)$. In Eq. (6) we introduced the notations:
$\delta=\omega_{1}-\omega_{0}+Ng_{1}^{2}/(2\Delta_{a})$ is the detuning of the
pump mode and the measured mode including the dispersive shift,
$\omega_{R}=k_{1}^{2}/2m_{a}$ is the recoil frequency, the effective coupling
rate is $g=\Omega_{p}g_{0}\sqrt{(}N/2)/\Delta_{a}$ with the pump Rabi
frequency $\Omega_{p}=g_{\mathrm{pump}}a_{\mathrm{pump}}$. The feedback is
performed via $GI(t)=\sqrt{N/8}V(t)$. The spin operators are
$S_{x}=(c_{1}^{\dagger}c_{0}+c_{0}^{\dagger}c_{1})/2$,
$S_{y}=(c_{1}^{\dagger}c_{0}-c_{0}^{\dagger}c_{1})/2i$ and
$S_{z}=(c_{1}^{\dagger}c_{1}-c_{0}^{\dagger}c_{0})/2$. The total number of
atoms $N=c_{1}^{\dagger}c_{1}+c_{0}^{\dagger}c_{0}$ is assumed to be fixed and
therefore can be considered as a c-number. The obtained Hamiltonian (6)
describes the Dicke model with the possibility to control the lattice
potential.

If the number of atoms $N$ in a BEC is large, the Hamiltonian in Eq. (6) can
be approximated as [1]

$H=\delta
a^{\dagger}a+\frac{\omega_{R}}{2}c_{1}^{\dagger}c_{1}+\left(c_{1}^{\dagger}+c_{1}\right)\left[g\left(a^{\dagger}+a\right)+GI(t)\right].$
(7)

Let us now specify the feedback control. The purpose of the control is to
steer the atoms towards the phase transition and modify the critical behavior.
Thus the natural algorithm is

$I(t)\sim\int h(t-\tau)\left(a+a^{\dagger}\right)d\tau,$ (8)

which acts on the atoms in a similar way as the cavity field. The new feature
due to the feedback is the time-response kernel $h(t)$ that can be customized
to affect the critical behavior of the system. Thus the signal for the
feedback loop should be obtained from the measurement of the cavity mode
quadrature. Equation (8) is the simplest possibility that illustrates the
purpose of the feedback control. Below we will define a more general feedback
algorithm that will be demonstrated to have certain advantages over Eq. (8).

The practical realization of the feedback loop in the system with a BEC should
not possess technical difficulties. The characteristic frequency of the system
dynamics is $\omega_{R}$ which for the rubidium BEC is $\omega_{R}=2\pi\cdot$
4kHz. The bandwidth of modern electronics approaches GHz range. Such a
bandwidth allows for almost any interesting feedback regimes including the
quasi-instantaneous feedback.

One can think of other implementations of the feedback loop as well. A
possible option would be the control of the coupling constant $g$, for
example, via the pump mode $a_{\mathrm{pump}}$ as has been implemented in the
experiment [4]. This would result in highly nonlinear dynamics.

Another option is, instead of creating the external potential $V$ with
$k_{0}=k_{1}/2$, one can inject the feedback signal directly through the
cavity mirror as the pump $\zeta_{1}(t)$. In this case, an additional laser
with doubled wavelength is not necessary. While the initial Hamiltonian Eq.
(6) will be somewhat different, after the adiabatic elimination of light mode
(as in Ref. [1]), the equation for the spin components will be the same as in
the case we consider here, if one chooses the pump through the mirror as
$\zeta_{1}(t)\sim I(t)$.

Further possibility is to use atoms tightly trapped in optical lattices [59,
60]. We demonstrated [3] the stabilization of many-body states by feedback in
such systems (e.g. the supersolid-like and antiferromagnetic states), which is
an example of the feedback-induced phase transition (FPT) as well, as it
appears above some critical feedback strength. In this case, instead of
expanding $\Psi(x)$ in the momentum space, a more appropriate approach is to
expand it in the coordinate space using the localized Wannier functions [55].
The spin operators will then be represented by sums of on-site atom operators
$S_{x}\sim\sum_{i}A_{i}n_{i}$ [83, 84, 16], or the bond operators representing
the matter-wave interference between neighboring sites
$S_{x}\sim\sum_{i}A_{i}b^{\dagger}_{i}b_{j}$ [85, 86, 65, 87]. For example,
the effective spin can correspond to the atom number difference between odd
and even sites [for $A_{i}=(-1)^{i}$] [15, 88], represent the magnetization
[15] or staggered magnetization [89] of fermions, etc. Such a strong
flexibility in choosing the geometrical combination of many-body variables
combined in the effective spin operator enables defining macroscopic modes of
matter fields [90] and assures the competition between the long-range (but
structured in space with a short period comparable to that of the lattice)
light-induced interactions and short-range atom-atom interactions and
tunneling on a lattice [15]. This will open the opportunities for the
competition between the nontrivial feedback-induced interactions and many-body
atomic interactions in atomic and molecular [91] systems subjected to the
measurement.

##  III Heisenber-Langevin approach

###  III.1 Feedback model

We start the theoretical analysis of the system by deriving the Heisenberg-
Langevin equations with feedback [12]. The dynamics of the spins inside the
leaky cavity can be obtained from the Hamiltonian (1) adding cavity decay with
the rate $\kappa$ and vacuum noise $f_{a}$ entering the system via the
partially transparent mirror. The vacuum noise correlation function is
$\langle f_{a}(t+\tau)f_{a}^{\dagger}(t)\rangle=2\kappa\delta(\tau)$. The
resulting Heisenberg-Langevin equations read

$\displaystyle\dot{a}$ $\displaystyle=$ $\displaystyle-i\delta
a-i\frac{2g}{\sqrt{N}}S_{x}-\kappa a+f_{a},$ $\displaystyle\dot{S}_{x}$
$\displaystyle=$ $\displaystyle-\omega_{R}S_{y},$ $\displaystyle\dot{S}_{y}$
$\displaystyle=$
$\displaystyle\omega_{R}S_{x}-\frac{2}{\sqrt{N}}[g(a^{\dagger}+a)+GI(t)]S_{z},$
$\displaystyle\dot{S}_{z}$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{N}}[g(a^{\dagger}+a)+GI(t)]S_{y}.$ (9)

The added feedback term in Eq. (III.1) drives the collective spin with the
strength proportional to the feedback control signal $I(t)$. In order to write
the feedback contribution in the similar form as the direct coupling term the
feedback gain $G$ is explicitly separated. The feedback control $I(t)$ is an
operator in the Heisenberg picture. The expression for $I(t)$ that is even
more general than that given by Eq. (8) is

$I(t)=\sqrt{2\kappa}\int_{-\infty}^{\infty}h(t-z)\mathcal{F}\left[\xi_{\theta}(z)\right]dz.$
(10)

Here the kernel $h(t)$ encodes the timing of the feedback algorithm and the
function $\mathcal{F}$ describes the instantaneous transformation of the
measured homodyne signal $\xi_{\theta}(z)$ by the feedback loop. The simplest
possibility that will be considered below is the linear feedback with
$\mathcal{F}[\xi_{\theta}(z)]=\xi_{\theta}(z)$.

Since the homodyne measurement is performed on the light outside the cavity
the vacuum noise reflected from the output mirror contributes to the signal.
This effect is taken into account using the input-output formalism [92]

$\xi_{\theta}(t)=\sqrt{2\kappa}x_{\theta}(t)-f_{\theta}/\sqrt{2\kappa},$ (11)

where $x_{\theta}=(ae^{-i\theta}+a^{\dagger}e^{i\theta})/2$ is the light
quadrature and the Langevin force
$f_{\theta}=(f_{a}e^{-i\theta}+f_{a}^{\dagger}e^{i\theta})/2$ is the
quadrature of the vacuum noise $f_{a}$.

The analysis of nonlinear operator equations (III.1) is rather difficult. It
is however possible to estimate the critical condition where the non-zero spin
projection $\langle S_{x}\rangle$ emerges. Taking the averages in Eqs (III.1)
and neglecting quantum correlations one obtains for the critical feedback gain

$G_{\rm crit}H(0)=\frac{\omega_{R}(\kappa^{2}+\delta^{2})-4\delta
g^{2}}{4g\kappa C_{\theta}},$ (12)

where $H(0)=\int_{0}^{\infty}h(t)dt$ is the zero-frequency Fourier coefficient
of the feedback kernel $h(t)$ and
$C_{\theta}=\delta\cos\theta+\kappa\sin\theta$. Note that using the feedback
one can reach the phase transition even for the coupling constant
$g<g_{\mathrm{crit}}$ that is below the critical value given in Eq.(2).

The dependence of critical value of the feedback gain on the cavity decay rate
$\kappa$ is shown in Fig 2. The trend for large $\kappa$ depends on the
measured field quadrature. If $x$-quadrature is measured (dashed curve), that
is $\theta=0$, then the critical value $G_{\mathrm{crit}}$ linearly increases
with $\kappa$. Using this regime for an experimental implementation is not
very convenient. The optical cavity in feedback experiments is an auxiliary
component helping to increase measured signal. However choosing to measure
$x$-quadrature one should limit the value of $\kappa$ (or use strong feedback)
taking a sufficiently high-fines cavity. On the other hand, measuring the
$y$-quadrature there will be no restrictions on the value of $\kappa$: the
larger $\kappa$ the closer $G_{\mathrm{crit}}$ to its limiting value
$\omega_{R}/(4g)$. Thus it is advantageous to use the $y$-quadrature
measurement. This type of measurements will be assumed considering all the
numerical examples in this paper. Interestingly, within the approximation of
the adiabatically eliminated light presented in Ref. [1], changing the
quadrature, which is measured, affects only some coefficients of the final
model. In contrast, here in the full model without adiabatic elimination, the
measurement of different quadratures will lead to rather different equations.

Figure 2: Dependence of the critical value of the feedback gain on the
measurement strength $\kappa$ for the quadratures $x_{0}$ ($x$-quadrature) and
$x_{\pi/2}$ ($y$-quadrature). If $x$-quadrature of the cavity field is
measured, $G_{\mathrm{crit}}$ grows for large $\kappa$. If $y$-quadrature is
measured, then $G_{\mathrm{crit}}$ approaches the constant value
$\omega_{R}/(4g)$. For small $\kappa$ the critical gain becomes negative since
the phase transition here happens even without feedback. Other relevant
parameters are $g=\omega_{R}$, $\delta=\omega_{R}$, $h(0)=s$.

Note that for small values of $\kappa$ the critical gain $G_{\mathrm{crit}}$
becomes negative for both measurement scenarios. This indicates that the phase
transition in these cases takes place even without any feedback. Another
perspective would be to use negative feedback $G<0$ to prevent the transition
to superradiant phase and keep the system in the normal phase even for
$g>g_{crit}$. This may be important for enhancing the quantum properties of
light in the field of cavity QED of quantum gases [93, 55, 56, 57]. Moreover,
this will be useful for obtaining fully quantum optical lattices [65, 66, 87,
94, 95, 96] with a plethora of novel phenomena unobtainable in standard
optical lattices created by prescribed classical laser beams. As we have shown
for such systems [65, 95], the quantum properties of light are the most
important, when they are not masked by strong light scattering associated with
the self-organized phase.

Let us now perform the linearization and focus on the quantum behavior of the
system below the phase transition. Using the Holstein-Primakoff representation
$S_{z}=b^{\dagger}b-N/2$, $S_{-}=\sqrt{N-b^{\dagger}b}b$,
$S_{+}=b^{\dagger}\sqrt{N-b^{\dagger}b}$, and $S_{x}=(S_{+}+S_{-})/2$, the
linear Langevin-Heisenberg equations can be found. In terms of spin
quadratures these equations take the following form

$\displaystyle\dot{X}$ $\displaystyle=$ $\displaystyle\omega_{R}Y,$
$\displaystyle\dot{Y}$ $\displaystyle=$ $\displaystyle-\omega_{R}X-2gx-GI,$
$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle\delta y-\kappa
x+f_{x},$ $\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle-\delta
x-\kappa y-2gX+f_{y}.$ (13)

Here $X=\left(b^{\dagger}+b\right)/2$ and $Y=\left(b-b^{\dagger}\right)/2i$
are the atom-field quadratute operators, while
$x=\left(a^{\dagger}+a\right)/2$ and $y=\left(a-a^{\dagger}\right)/2i$ are
their cavity-field analogues. The quadrature Langevin forces are defined as
$f_{x}=\left(f_{a}^{\dagger}+f_{a}\right)/2$ and
$f_{y}=\left(f_{a}-f_{a}^{\dagger}\right)/2i$. The feedback signal operator in
quadrature notation reads

$I(t)=2\kappa\int_{-\infty}^{t}h(t-z)\left[x_{\theta}(z)-f_{\theta}(z)/(2\kappa)\right]dz.$
(14)

The linear operator equations (III.1) can be solved using Fourier
transformation. The algebraic system of equations for the Fourier amplitudes
takes the form

$\displaystyle\begin{bmatrix}\kappa+i\omega&-\delta&0&0\\\
\delta&\kappa+i\omega&2g&0\\\ 0&0&i\omega&-\omega_{R}\\\ 2g+2\kappa
GH(\omega)\cos\theta&2\kappa
GH(\omega)\sin\theta&\omega_{R}&i\omega\end{bmatrix}\begin{pmatrix}\tilde{x}\\\
\tilde{y}\\\ \tilde{X}\\\ \tilde{Y}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\tilde{f}_{x}\\\ \tilde{f}_{y}\\\ 0\\\
GH(\omega)\left(\cos\theta\tilde{f}_{x}+\sin\theta\tilde{f}_{y}\right).\end{pmatrix}$
(15)

The solution of this system is readily found to be given by

$\tilde{X}(\omega)=\mathrm{D}^{-1}(\omega)\left[\mathrm{M}_{x}(\omega)\tilde{f}_{x}+\mathrm{M}_{y}(\omega)\tilde{f}_{y}\right],$
(16)

where the deterministic dynamics is defined by

$\displaystyle\mathrm{D}(\omega)=\omega_{R}^{2}-\omega^{2}$
$\displaystyle-2g\omega_{R}\frac{2g\delta+2\kappa
GH(\omega)\left[\delta\cos\theta+\left(\kappa+i\omega\right)\sin\theta\right]}{\delta^{2}+\left(\kappa+i\omega\right)^{2}},$
(17)

while the noise contribution is mediated via

$\displaystyle\mathrm{M}_{x}(\omega)=\omega_{R}GH(\omega)\cos\theta$
$\displaystyle+\omega_{R}\frac{-2g(\kappa+i\omega)+2\kappa
GH(\omega)\left[\delta\sin\theta-\left(\kappa+i\omega\right)\cos\theta\right]}{\delta^{2}+\left(\kappa+i\omega\right)^{2}},$
(18)

and

$\displaystyle\mathrm{M}_{y}(\omega)=\omega_{R}GH(\omega)\sin\theta$
$\displaystyle-\omega_{R}\frac{2g\delta+2\kappa
GH(\omega)\left[\delta\cos\theta+\left(\kappa+i\omega\right)\sin\theta\right]}{\delta^{2}+\left(\kappa+i\omega\right)^{2}}.$
(19)

The noise spectral correlation function $S(\omega)$ is defined via $\langle
F(\omega)F(\omega^{\prime})\rangle=S(\omega)\delta(\omega+\omega^{\prime})$
and reads

$S(\omega)=\pi\kappa\left|\mathrm{M}_{x}(\omega)-i\mathrm{M}_{y}(\omega)\right|^{2}.$
(20)

This result is obtained assuming the fact that the feedback kernel $h(t)$ is a
real function with the Fourier transform obeying $H(-\omega)=H^{*}(\omega)$.

In a more explicit form the noise spectral density reads:

$\displaystyle
S(\omega)=\pi\kappa\omega_{R}^{2}\left|\frac{2g\left(\kappa+i(\omega-\delta)\right)}{\delta^{2}+(\kappa+i\omega)^{2}}\right.$
$\displaystyle\left.+GH(\omega)e^{-i\theta}\left(\frac{2\kappa\left(\kappa+i(\omega-\delta)\right)}{\delta^{2}+(\kappa+i\omega)^{2}}-1\right)\right|^{2}.$
(21)

The frequency dependence of $S(\omega)$ is due to both the effect of the
feedback transfer function $H(\omega)$ and the dynamics of the cavity mode. As
a result the noise acting on the atoms is non-Markovian even in the absence of
feedback ($G=0$). In the case of the large cavity decay rate $\kappa$ (large
strength of the cavity measurement) the only possible reason of non-Markovian
dynamics is the feedback-induced noise processing. In the Fourier domain the
adiabatic elimination of the cavity mode is formally done by neglecting
$i\omega$ as compared with the decay rate $\kappa$. The adiabatic elimination
results in the spectral correlation function presented in Ref. [1]:

$S_{a}(\omega)=\frac{\pi\omega_{R}^{2}\kappa}{\kappa^{2}+\delta^{2}}\left|2g+G(\kappa-i\delta)e^{-i\theta}H(\omega)\right|^{2}.$
(22)

The critical value of the feedback gain in linear regime $G_{\mathrm{crit}}$
is determined from the condition $D(0)=0$. If this condition is fulfilled then
the low-frequency (long wavelength) fluctuations $\langle X^{2}\rangle$ will
dominate and grow to infinity at the critical point. This is exactly what is
expected at the critical point. Since the low-frequency range determines the
critical value, the result for $G_{\mathrm{crit}}$ is exactly the same as that
without the adiabatic elimination of the cavity field [1]:

$G_{\mathrm{crit}}H(0)=\frac{\omega_{R}(\kappa^{2}+\delta^{2})-4g^{2}\delta}{4g\kappa
C_{\theta}}.$ (23)

Note that this result exactly coincides with the critical value estimation
(12) for the nonlinear model. Thus the conclusions relating the trend of the
critical gain with $\kappa$ for different quadratures remain valid.

The dynamics of the quadrature $X$ can also be formulated in time domain. The
equation of motion reads

$\ddot{X}(t)+\omega_{R}^{2}X(t)-4\omega_{R}g\int_{0}^{t}E^{\theta}\left(t-z\right)X(z)dz=F^{\theta}(t).$
(24)

Here the kernel $E^{\theta}$ is defined as

$\displaystyle E^{\theta}(t)=ge^{-\kappa t}\sin\delta t$
$\displaystyle+G\kappa\int_{0}^{t}h(t-z)e^{-\kappa z}\sin\left(\delta
z+\theta\right)dz,$ (25)

and the Langevin force operator appearing in the right-hand side of Eq. (24)
is expressed as

$\displaystyle
F^{\theta}(t)=-\omega_{R}\int_{0}^{t}dt^{\prime}f_{a}(t^{\prime})\left[ge^{-(\kappa+i\delta)(t-t^{\prime})}\right.$
$\displaystyle-\frac{G}{2}e^{-i\theta}h(t-t^{\prime})$
$\displaystyle\left.+G\kappa
e^{-i\theta}\int_{0}^{t-t^{\prime}}dt^{\prime\prime}h(t-t^{\prime}-t^{\prime\prime})e^{-(\kappa+i\delta)t^{\prime\prime}}\right]+\mathrm{H.c.}$
(26)

In this form Eq. (24) is similar to the equation describing Brownian motion in
a harmonic potential [23].

This integro-differential equation is similar to the one presented in Ref. [1]
for the adiabatically eliminated light field, though with a very different
kernel function. An important consequence of such a difference of kernels,
consists in the fact that here (in the full model without the adiabatically
eliminated light), the measurement of different quadratures (different
$\theta$) leads to rather different equations. In contrast, in the simplified
adiabatic model [1], the measurement of different quadratures affects only
some coefficients of the final simplified model.

The Heisenberg-Langevin approach allows for the calculation of the stationary
fluctuations below the phase transition. The atomic $X$-quadrature fluctuation
can be found from the correlation function

$\langle X^{2}\rangle=\langle
X(t+\tau)X(t)\rangle|_{\tau=0}=\frac{1}{4\pi^{2}}\int_{-\infty}^{\infty}\frac{S(\omega)}{|D(\omega)|^{2}}d\omega.$
(27)

For the linear model this quantity becomes infinite as the feedback strength
approaches the critical value $G_{\mathrm{crit}}$.

We can now analyze the behavior of the quadrature fluctuation $\langle
X^{2}\rangle$ near the phase transition and evaluate the critical exponent
$\alpha$. The dependence of $\langle X^{2}\rangle$ on the feedback gain $G$
for $G<G_{\mathrm{crit}}$ will be numerically approximated by $\langle
X^{2}\rangle=A/|1-G/G_{\mathrm{crit}}|^{\alpha}+B$, where $A$ and $B$ are
constants determined along with the critical exponent $\alpha$ during the
least-mean-square fitting.

##  IV Critical exponents

In Ref. [1], we have shown that a very useful choice of the feedback transfer
kernel $h(t)$ is the power function of the form

$\displaystyle h(t)=h(0)\left(\frac{t_{0}}{t+t_{0}}\right)^{s+1}.$ (28)

In view of the space–time analogy [1] this time dependence corresponds to the
spatial Ising-type interaction extensively used in different models of
condensed matter physics. The instantaneous feedback ($s\rightarrow\infty$)
with $h(t)\sim\delta(t)$ will lead to the short-range in time $S_{x}^{2}$ term
as in the Lipkin-Meshkov-Glick (LMG) model [97, 98] originating from nuclear
physics.

A more exotic $h(t)$ such as a sequence of amplitude-shaped time delays
$h(t)\sim\sum_{n}\delta(t-nT)/n^{s+1}$ will enable studies of discrete time
crystals [53, 99] and Floquet engineering [54] with long-range interaction
$\sum_{n}S_{x}(t)S_{x}(t-nT)/n^{s+1}$, where the crystal period may be
$T=2\pi/\omega_{R}$. The global interaction is given by constant $h(t)$. The
Dicke model can be restored even in the adiabatic limit by exponentially
decaying and oscillating $h(t)$ mimicking a cavity response.

We believe that our proposal will extend the studies in the field of time
crystals [53, 99, 100, 101, 102, 103, 104, 105] and Floquet engineering [54].
The time crystals is a recently proposed notion, where phenomena studied
previously in space (e.g. spin chains, etc.) are now studied in the time
dimension as well. Typically, the system is subject to the external periodic
modulation of a parameter [e.g. periodic $g(t)$], which is considered as
creating a “lattice in time”. Our approach makes possible introducing the
effective “interaction in time.” This makes the modulation in the system not
prescribed, but dependent on the state of the system (via $S_{x}$). This
resembles a true lattice in space with the interaction between particles. In
other words, our model enables not only creating a “lattice in time,” but
introducing the tunable “interaction in time” to such a lattice (without the
necessity of having the standard particle-particle interaction in space [106,
107]).

Moreover, $h(t)$ can have minima, maxima, and even change its sign, which
creates analogies with the molecular potentials (now in time, not in space)
and rises intriguing questions about the creation of time molecules and time-
molecule crystals.

All such $h(t)$ can be realized separately or simultaneously to observe the
competition between different interaction types. It will be interesting to
extend this approach to a broader range of effective spin interactions [108,
109, 98].

Further results of this paper do not rely on the effective Hamiltonians [66].
This discussion motivates us to use in the simulations $h(t)$ given by Eq.
(28), which is unusual in feedback control. In the numerical examples we will
take $h(0)=s$ to make the critical value $G_{\mathrm{crit}}$ insensitive to
the parameter $s$: this will allow us to focus on the influence of the
feedback exponent $s$ on the quantum fluctuations, rather than on a trivial
shift of the transition point.

Before addressing the results with feedback let us recall the behavior of the
quadrature fluctuations when the system approaches the Dicke phase transition.
The quadrature fluctuation spectra for this case are shown in Fig. 3. Fig.
3(a) corresponds to $\kappa=\omega_{R}$ while Fig. 3(b) shows the results for
$\kappa=0.1\omega_{R}$. Different curves in both subplots represent different
values of the coupling constant $g$. All spectra demonstrate strong peaks near
the resonance frequency $\omega_{R}$ that move towards $\omega=0$ as the
coupling constant approaches its critical value, Eq.(2). This is a known “mode
softening” effect.

In Fig.3(b) the quadrature fluctuations have another weaker peak near the
cavity resonance frequency $\delta=2\omega_{R}$. It is the higher frequency
mode of two coupled oscillators, atoms and the cavity field, that typically
corresponds to the relative motion of the oscillators. This resonance is only
slightly shifted to higher frequency as the coupling grows. Thus the
oscillations at the frequency close to $\delta$ remain in the system even
above the phase transition. The “relative motion” mode is also present in
Fig.3(a), but it is not resolved since the resonances for $\kappa=\omega_{R}$
are quite broad.

Figure 3: The quadrature fluctuation spectra for the system without feedback
for different values of the atom–cavity coupling $g$. All spectra show the
strong low frequency peaks that tend to the zero frequency, when one
approaches the critical coupling $g_{\mathrm{crit}}$ (the so-called mode
softening). (a) and (b) correspond to $\kappa=\omega_{R}$ and
$\kappa=0.1\omega_{R}$, respectively. In the case of lower damping (b), the
quadrature fluctuations have another higher frequency peak that is only
slightly shifted up during the increase of the coupling $g$.
$\delta=2\omega_{R}$.

The quadrature fluctuations with feedback are shown in Fig. 4. Fig. 4(a)
corresponds to the slow feedback $s=0.5$, Fig. 4(b) shows the results for
$s=1$, and Fig. 4(c) corresponds to the fast feedback $s=5$. The systems far
from the critical point are represented by solid curves, the systems close to
the transition are represented by dotted curves, the intermediate case is
shown by dashes.

Figure 4: The quadrature fluctuations with feedback. (a) The slow feedback
$s=0.5$, (b) the results for $s=1$, (c) the fast feedback $s=5$. The systems
far from the critical point are represented by solid curves, the systems close
to the transition are represented by dotted curves, the intermediate cases are
shown by dashed curves. While the fast feedback ($s=5$) leads to a typical
scenario of the mode softening (cf. Fig. 3 with no feedback), the curves for
slower feedbacks with $s=0.5$ and $s=1$ show a different behavior.
$g=0.1\omega_{R}$, $\kappa=\omega_{R}$, $\delta=2\omega_{R}$, $h(0)=s$.

The slow feedback, $s=0.5$, demonstrates the trend that is quite different
from the system without the feedback [compare Fig.4(a) with Fig. 3].
Increasing the gain $G$ the resonance near $\omega_{R}$ does not visibly move
towards the zero frequency. Approaching the phase transition the fluctuations
at zero frequency grow resulting in two pronounced peaks: around zero and
around $\omega_{R}$. The feedback frequency transfer function for the slow
feedback has a narrow peak only at the zero frequency, thus the feedback does
not strongly affect the shift of the resonance at $\omega_{R}$. Therefore, the
description of the system behavior in terms of the mode softening is not
strictly valid for this case.

In Fig. 4(b) the fluctuation spectra correspond to $s=1$. Here the shift of
the resonance to lower frequency is more pronounced than for $s=0.5$, but the
growth of the additional peak at zero frequency dominates.

The results for the fast, $s=5$, feedback are shown in Fig. 4(c). The
transformation of the spectrum approaching the critical point is now quite
similar to the transformation without the feedback (compare with Fig. 3). The
feedback transfer function is now broad enough to affect the system resonance
frequencies. The action of feedback is similar to the increase of the coupling
constant $g$. Thus one can expect that the critical behavior for the fast
feedback (which is close to the instantaneous, Markovian, one) is similar to
an open Dicke model without feedback. Note, that in the model with
adiabatically eliminated light of Ref. [1] such a similarity is even stronger,
because, as we have already mentioned, here the full model is more sensitive
to the choice of the quadrature, which is detected, than the simplified model
[1].

Integrating the quadrature fluctuation spectra, Eq. (16), similar to those
presented in Fig. 4, we obtain the average quadrature fluctuations below the
phase transition. The examples of the fluctuation trends as the system
approaches the critical point for different $s$ are shown in Fig. 5(a). The
unlimited growth of the quadrature fluctuations approaching
$G_{\mathrm{crit}}$ is the manifestation of the phase transition in a linear
model. The growing rates for different $s$ looks different.

The dependencies presented in Fig. 5(a) are approximated by the function
$\langle X^{2}\rangle=A/|1-G/G_{\mathrm{crit}}|^{\alpha}+B$ for
$G<G_{\mathrm{crit}}$ fitting the parameters with the least-mean-square
method. The results for the critical exponent $\alpha$ are calculated for
different values of $s$ and are presented in Fig. 5(b). For small $s$ the
critical exponent tends to zero. Increasing $s$ the critical exponent grows
approaching the limiting value $\alpha=1$. Thus the critical behavior of the
system can be controlled by selecting the feedback time response. Choosing the
value of $s$ one can therefore choose the universality class the system
belongs to. This is very simple compared to the reservoir engineering
approach.

Figure 5: Tuning the universality class of phase transitions. Integrated
quadrature fluctuations $\langle X^{2}\rangle$ (a) for different values of $s$
that are used to calculate the critical exponents and the dependence of
critical exponent $\alpha$ on the feedback exponent $s$ (b). The growth of the
quadrature fluctuations approaching $G_{\mathrm{crit}}$ (shown in logarithmic
scale) is the manifestation of the phase transition in a linear model. The
feedback time response determined by $s$ can continuously change the critical
exponent of the transition in a wide range, which corresponds to tuning the
universality class of the phase transition. $g=0.5\omega_{R}$,
$\kappa=\omega_{R}$, $\delta=2\omega_{R}$, $h(0)=s$.

According to the space–time analogy, varying the parameter $s$ of feedback
response $h(t)$ (28) corresponds to varying the length of effective spin-spin
interaction. For $h(t)$ (28), its spectrum is expressed via the exponential
integral $H(\omega)=h(0)t_{0}e^{-i\omega t_{0}}E_{s+1}(i\omega t_{0})$. At
small frequencies its imaginary part behaves as $\omega^{s}$ for $s<1$,
resembling the spectral function of sub-Ohmic baths. For large $s$, $\alpha$
approaches unity, as $h(t)$ becomes fast and feedback becomes nearly
instantaneous such as interactions in the open LMG and Dicke models, where
$\alpha=1$ [49, 50, 110].

##  V Quantum trajectories

Though the numerical simulations on a classical computer are limited to very
few spins, one can get an interesting insight in the system dynamics
simulating single quantum trajectories. Weak measurements constitute a source
of competition with unitary dynamics [15, 88, 89, 16], which is well seen in
quantum trajectories formalism [11, 111, 112, 113, 114, 115, 116], underlining
the distinction between measurements and dissipation. Thus they can affect
phase transitions, including the many-body ones [15, 16, 17, 18, 19, 20, 21].

The evolution of the quantum state of the continuously observed system is
determined by the stochastic master equation

$d\rho_{c}=-i\left[H,\rho_{c}\right]dt+\mathcal{D}[a]\rho_{c}dt+\mathcal{H}[a]\rho_{c}dW,$
(29)

where the dissipation and noise contributions are defined as

$\displaystyle\mathcal{D}[a]\rho_{c}=2\kappa\left[a\rho_{c}a^{\dagger}-\left(a^{\dagger}a\rho_{c}+\rho_{c}a^{\dagger}a\right)/2\right],$
$\displaystyle\mathcal{H}[a]\rho_{c}=\sqrt{2\kappa}\left[ae^{-i\theta}\rho_{c}+\rho_{c}a^{\dagger}e^{i\theta}\right.$
$\displaystyle\left.-\mathrm{Tr}\left\\{ae^{-i\theta}\rho_{c}+\rho_{c}a^{\dagger}e^{i\theta}\right\\}\rho_{c}\right].$
(30)

The increment of the Wiener process representing the quantum noise of the
continuous weak measurement is denoted by $dW$. The Hamiltonian
$H=H_{s}+H_{fb}$ is now the sum of the system Hamiltonian $H_{s}$, which is
given by Eq. (6) for the nonlinear and by Eq. (7) for the linearized spin
models, and the feedback Hamiltonian is

$H_{fb}=G\left(b^{\dagger}+b\right)I_{c}(t).$ (31)

Here $I_{c}$ is the feedback signal expressed similarly to Eq. (10), but it is
not an operator:

$I_{c}(t)=\sqrt{2\kappa}\int_{-\infty}^{\infty}h(t-z)d\xi_{\theta}(z).$ (32)

The increment of the measured signal $d\xi_{c}$ being the outcome of a
homodyne detector reads

$d\xi_{\theta}(t)=\sqrt{2\kappa}\langle x_{\theta}\rangle_{c}dt+dW.$ (33)

The simulation of the stochastic evolution according to Eq. (29) with the
feedback kernels $h(t)$ as defined in Eq. (28) will be presented for a single
spin nonlinear model and $N$-spins linearized model.

###  V.1 Quantum trajectories for linear model

Although we have the quasi-analytical result for the linearized spin model it
is interesting to look at the behavior of the system at a single trajectory.
The results of numerical simulations of Eq. (29) with the linearized
Hamiltonian (7) are shown in Fig. 6. Here the stochastic trajectories of
collective spin quadrature $\langle X\rangle_{c}$ for $s=0.5$ (left column)
and $s=5$ (right column) are shown for different values of $G$ (different
rows).

Figure 6: Stochastic trajectories of the collective spin quadrature $\langle
X\rangle_{c}$ for $s=0.5$ (left column) and $s=5$ (right column) in the linear
approximation. Different rows correspond to different values of the feedback
gain. For $G=0.01G_{\mathrm{crit}}$ the quadrature has an oscillating non-zero
value on a single trajectory. Crossing the critical point $G_{\mathrm{crit}}$
the oscillatory dynamics changes to growth. The growth rate is grater for
grater $s$. The different temporal behaviors for differend $s$ can be compared
to the different fluctuation spectra for different $s$ in Fig. 4. One can see
that the mode softening picture is not applicable to the slow feedback case:
instead of decreasing frequency, one sees the increasing noise due to the
appearance of the peak at zero frequency in Fig. 4(a). Other parameters are
$g=0.1\omega_{R}$, $\kappa=\omega_{R}$, $\delta=2\omega_{R}$, $h(0)=s$.

If $G=0.01G_{\mathrm{crit}}$ (the first row in Fig 6) the spin quadrature
demonstrates quite regular oscillations. These oscillations are induced by the
measurement backaction noise. They correspond to the resonant peaks in the
fluctuation spectra in Figs. 3 and 4.

Increasing $G$ and approaching its critical value we expect to observe the
increase of the oscillation period reflecting the appearance of the soft mode.
This is seen for the fast feedback with $s=5$ (right column in Fig. 6), but
this is not the case for the slow feedback with $s=0.5$ (left column). For
$G=0.99G_{\mathrm{crit}}$ the quadrature oscillations for $s=0.5$ become
affected by the growing fluctuations at the zero frequency [see discussions of
Fig. 4(a) with spectra], while the spectral peak near the resonance frequency
$\omega_{R}$ is still pronounced. This makes the oscillations noisy but still
visible at the original frequency, which is different from the mode softening
scenario.

Above the critical point the oscillations completely disappear and the
exponential growth of the spin quadrature takes place for both values of $s$,
see the lowest row in Fig. 6. Note that growth rate is quite different for the
slow and fast feedback. The time scale for the slow feedback is larger than
that for the fast one. Nevertheless, the phase transition takes place for both
cases at the same value of $G_{\mathrm{crit}}$ (if we normalize the feedback
kernel as $h(0)=s$).

###  V.2 Quantum trajectories for nonlinear single-spin model

Figure 7: Quantum trajectories for the conditional spin component $\langle
S_{x}\rangle_{c}$ of a single nonlinear spin. The plots on the left-hand side,
labeled (a), contain the trajectories for the slow feedback, $s=0.5$. The
right-hand side plots, labeled (b), show the fast feedback trajectories,
$s=5$. For the weak feedback ($G=0.01G_{\mathrm{crit}}$) the spin starts to
almost regularly oscillate due to the homodyne measurement backaction. There
is no difference between $s=0.5$ (a.1) and $s=5$ (b.1). Just above the
threshold, $G=2G_{\mathrm{crit}}$, the feedback still results in oscillatory
dynamics, which may be quite irregular (a.2). Well above the threshold
$\langle S_{x}\rangle_{c}$ reaches a stationary value [(a.3) and (b.3)]. Other
parameters are $g=0.1\omega_{R}$, $\delta=2\omega_{R}$, $\kappa=\omega_{R}$,
$h(0)=s$.

The examples of stochastic trajectories for a single nonlinear spin are shown
in Fig. 7. It demonstrates the spin component $\langle
S_{x}\rangle_{c}=\mathrm{Tr}\left\\{\rho_{c}S_{x}\right\\}$ averaged over the
conditional state of the observed quantum system. The plots on the left-hand
side, labeled (a), contain the trajectories for slow feedback, $s=0.5$. The
right-hand side plots, labeled (b), show the fast feedback trajectories,
$s=5$. For the weak feedback ($G=0.01G_{\mathrm{crit}}$) the spin starts to
almost regularly oscillate due to the homodyne measurement backaction. The
oscillations are around zero value of $S_{x}$, thus averaged over time such a
behavior results in almost zero value of the spin component. This dynamics
corresponds to the normal phase of the spin system and the cases of slow and
fast feedbacks are very similar to each other.

Just above the threshold, $G=2G_{\mathrm{crit}}$, the feedback still results
in oscillatory dynamics, which may be quite irregular, cf. Fig. 7(a.2). This
is similar to the behavior of the quadrature $X$ in the linearized model, cf.
Fig. 6(a.2). Thus the slow feedback with the narrow band transfer function
supports the low-frequency noise and suppresses oscillations [cf. Fig. 4(a)
for the spectra of the linearized model].

Figure 8: Feedback-induced phase transition for a single nonlinear spin. The
stationary values of $\langle S_{x}\rangle$ averaged over many trajectories
depending on the feedback gain $G$. $g=0.1\omega_{R}$, $\kappa=\omega_{R}$,
$\delta=2\omega_{R}$, $s=1$, $h(0)=s$.

Well above the threshold $\langle S_{x}\rangle_{c}$ rapidly saturates to its
maximal value [Fig. 7(a.3) and (b.3)], but the sign of the spin component is
not predefined and can take both positive and negative values. The dependence
of the absolute value of such steady states averaged over many trajectories on
the feedback parameter $G$ is shown in Fig. 8. Below the critical point
$G<G_{\mathrm{crit}}$ the spin component is nearly zero while it starts to
grow for $G>G_{\mathrm{crit}}$. The stationary value of spin component
saturates in the superradiant phase well above the threshold.

##  VI Conclusions

In summary, we presented a theoretical description of the feedback-induced
quantum phase transition in a collective spin system and tuning its
universality class. This shifts the paradigm of the feedback control from the
control of quantum states (as known in quantum metrology) to the control of
phase transitions in quantum systems. The feedback loop is assumed to be based
on the quadrature measurement of the cavity light interacting with the spins.
Starting with a nonlinear model we determined the critical value of the
feedback gain and demonstrate that the dependence of the critical gain on the
cavity decay rate is different for different measurement scenarios. The
measurement of $x$-quadrature results in the critical gain linearly increasing
with the decay rate, while the $y$-quadrature measurement results in the
limited critical gain.

The analysis of the linearized spin model in the Heisenberg-Langevin approach
allowed for the determination of the feedback-induced drift and the quantum
noise acting on the system. The quantum noise is shown to be colored with the
spectral density dependent on the feedback response shape resulting in the
non-Markovian behavior of spins.

By approximating the trend of the spin-quadrature fluctuations near the
critical gain by a power function, the critical exponents $\alpha$ have been
found. The value of the critical exponent can be controlled in the range
$\alpha\in(0,1)$ by changing the feedback loop timing parameter $s$. This
opens new perspective for quantum simulations of systems belonging to
different universality classes.

The dynamics of a single nonlinear spin and linearized $N$-spin models have
been numerically simulated at single quantum trajectories. For the feedback
gain far below the critical value the main effect of the feedback is the
measurement backaction. The quadrature measurement results in breaking the
time symmetry and the appearance of coherent oscillations at a single quantum
trajectory. If one calculates the expectation value of the spin variables,
such oscillations are lost due to averaging over many trajectories, because of
the random phase of the single-trajectory oscillations. Thus, the mean
solution below the threshold is a trivial zero, while, in contrast, the
coherent oscillations are visible in each single realization. Only for a fast
(nearly instantaneous) feedback, approaching the critical point the period of
the oscillations increases reflecting the softening of the spin mode. For the
slow (long-memory) feedback such a simplified picture is not relevant.

While we considered both linear and nonlinear quantum systems, the feedback
was assumed to be a linear one. A possible extension of the model can be the
consideration of a feedback, nonlinearly dependent on the measured light
variable. This can be useful, e.g., for simulations of the spin-boson and
other impurity models with the nonlinear coupling between the particle and
bath [117]. Such models can describe realistic systems such as the quadratic
coupling of a qubit to its environment [118], as well as experiments with
semiconductor quantum dots [119] and bismuth donors in silicon [120].

Another possible extension of the model is considering several feedback loops
simultaneously. Such a configuration can be useful for modeling spins coupled
to several baths [121], which is important for simulating realistic qubits
[122, 123] and impurities in quantum magnets [124, 125, 122, 123].

###### Acknowledgements.

The financial support is provided by RSF 17-19-01097-P (St. Petersburg State
University, AMO physics approach), and DGAPA-UNAM IN109619 and CONACYT- 364
Mexico A1-S-30934 (Universidad Nacional Autónoma de México, CMT approach).

## References

  * Ivanov  _et al._ [2020a] D. A. Ivanov, T. Y. Ivanova, S. F. Caballero-Benitez, and I. B. Mekhov, Phys. Rev. Lett. 124, 010603 (2020a).
  * Ivanov  _et al._ [2020b] D. A. Ivanov, T. Y. Ivanova, S. F. Caballero-Benitez, and I. B. Mekhov, Scientific Reports 10, 10550 (2020b).
  * Mazzucchi  _et al._ [2016a] G. Mazzucchi, S. F. Caballero-Benitez, D. A. Ivanov, and I. B. Mekhov, Optica 3, 1213 (2016a).
  * Kroeger  _et al._ [2020] K. Kroeger, N. Dogra, R. Rosa-Medina, M. Paluch, F. Ferri, T. Donner, and T. Esslinger, New J. Phys. 22, 033020 (2020).
  * Sachdev [2001] S. Sachdev,  _Quantum Phase Transitions_ (Cambridge University Press, Cambridge, 2001).
  * Osterloh  _et al._ [2002] A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature 416, 608 (2002).
  * van Nieuwenburg  _et al._ [2017] E. P. L. van Nieuwenburg, Y.-H. Liu, and S. D. Huber, Nature Phys. 13, 435 (2017).
  * Halu  _et al._ [2013] A. Halu, S. Garnerone, A. Vezzani, and G. Bianconi, Phys. Rev. E 87, 022104 (2013).
  * Haken [1981] H. Haken,  _Light_ (North-Holland, 1981).
  * Kessler  _et al._ [2012] E. M. Kessler, G. Giedke, A. Imamoglu, S. F. Yelin, M. D. Lukin, and J. I. Cirac, Phys. Rev. A 86, 012116 (2012).
  * Daley [2012] A. J. Daley, Advances in Physics 86, 012116 (2012).
  * Wiseman and Milburn [2010] H. M. Wiseman and G. J. Milburn,  _Quantum Measurement and Control_ (Cambridge University Press, Cambridge, 2010).
  * Young  _et al._ [2021] J. T. Young, A. V. Gorshkov, and I. B. Spielman, Feedback-stabilized dynamical steady states in the bose-hubbard model (2021), arXiv:2106.09744 [cond-mat.quant-gas] .
  * Buonaiuto  _et al._ [2021] G. Buonaiuto, F. Carollo, B. Olmos, and I. Lesanovsky, Dynamical phases and quantum correlations in an emitter-waveguide system with feedback (2021), arXiv:2102.02719 [quant-ph] .
  * Mazzucchi  _et al._ [2016b] G. Mazzucchi, W. Kozlowski, S. F. Caballero-Benitez, T. J. Elliott, and I. B. Mekhov, Phys. Rev. A 93, 023632 (2016b).
  * Kozlowski  _et al._ [2016] W. Kozlowski, S. F. Caballero-Benitez, and I. B. Mekhov, Phys. Rev. A 94, 012123 (2016).
  * Ashida  _et al._ [2016] Y. Ashida, S. Furukawa, and M. Ueda, Phys. Rev. A 94, 053615 (2016).
  * Ashida  _et al._ [2017] Y. Ashida, S. Furukawa, and M. Ueda, Nature Commun. 8, 15791 (2017).
  * Ashida  _et al._ [2020] Y. Ashida, Z. Gong, and M. Ueda, Advances in Physics 69, 249 (2020).
  * Li  _et al._ [2019] Y. Li, X. Chen, and M. P. A. Fisher, Phys. Rev. B 100, 134306 (2019).
  * Ippoliti  _et al._ [2021] M. Ippoliti, M. J. Gullans, S. Gopalakrishnan, D. A. Huse, and V. Khemani, Phys. Rev. X 11, 011030 (2021).
  * [22] Sampo, http://musinfo.fr/en/research/sampo.
  * Schuld  _et al._ [2015] M. Schuld, I. Sinayskiy, and F. Petruccione, Contemporary Physics 56, 172 (2015).
  * Hammerer  _et al._ [2010] K. Hammerer, A. S. Sørensen, and E. S. Polzik, Rev. Mod. Phys. 82, 1041 (2010).
  * Hauke  _et al._ [2013] P. Hauke, R. J. Sewell, M. W. Mitchell, and M. Lewenstein, Phys. Rev. A 87, 021601(R) (2013).
  * Campagne-Ibarcq  _et al._ [2016] P. Campagne-Ibarcq, S. Jezouin, N. Cottet, P. Six, L. Bretheau, F. Mallet, A. Sarlette, P. Rouchon, and B. Huard, Phys. Rev. Lett. 117, 060502 (2016).
  * Hacohen-Gourgy  _et al._ [2016] S. Hacohen-Gourgy, L. S. Martin, E. Flurin, V. V. Ramasesh, K. B. Whaley, and I. Siddiqi, Nature 538, 491 (2016).
  * Haroche and Raimond [2006] S. Haroche and J.-M. Raimond,  _Exploring the Quantum: Atoms, Cavities, and Photons_ (Oxford University Press, Oxford, 2006).
  * Steck  _et al._ [2004] D. A. Steck, K. Jacobs, H. Mabuchi, T. Bhattacharya, and S. Habib, Phys. Rev. Lett. 92, 223004 (2004).
  * Wade  _et al._ [2015] A. C. J. Wade, J. F. Sherson, and K. Mølmer, Phys. Rev. Lett. 115, 060401 (2015).
  * Wade  _et al._ [2016] A. C. J. Wade, J. F. Sherson, and K. Mølmer, Phys. Rev. A 93, 023610 (2016).
  * Hush  _et al._ [2013] M. R. Hush, S. S. Szigeti, A. R. R. Carvalho, and J. J. Hope, New J. Phys. 15, 113060 (2013).
  * Szigeti  _et al._ [2009] S. S. Szigeti, M. R. Hush, A. R. R. Carvalho, and J. J. Hope, Phys. Rev. A 80, 013614 (2009).
  * Szigeti  _et al._ [2010] S. S. Szigeti, M. R. Hush, A. R. R. Carvalho, and J. J. Hope, Phys. Rev. A 82, 043632 (2010).
  * Schemmer  _et al._ [2017] M. Schemmer, A. Johnson, R. Photopoulos, and I. Bouchoule, Phys. Rev. A 95, 043641 (2017).
  * Lammers  _et al._ [2016] J. Lammers, H. Weimer, and K. Hammerer, Phys. Rev. A 94, 052120 (2016).
  * Thomsen  _et al._ [2002] L. K. Thomsen, S. Mancini, and H. M. Wiseman, J. Phys. B 35, 4937 (2002).
  * Botter  _et al._ [2012] T. Botter, D. W. C. Brooks, N. Brahms, S. Schreppler, and D. M. Stamper-Kurn, Phys. Rev. A 85, 013812 (2012).
  * Vuletić  _et al._ [2007] V. Vuletić, J. K. Thompson, A. T. Black, and J. Simon, Phys. Rev. A 75, 051405(R) (2007).
  * Ivanov and Wallentowitz [2004] D. Ivanov and S. Wallentowitz, Phys. Rev. Lett. 93, 260603 (2004).
  * Ivanov and Ivanova [2014] D. A. Ivanov and T. Y. Ivanova, Journal of Physics B: Atomic, Molecular and Optical Physics 47, 135303 (2014).
  * Ivanov and Ivanova [2015] D. A. Ivanov and T. Y. Ivanova, J. Exp. Theor. Phys. 121, 179–185 (2015).
  * Ivanov and Ivanova [2016] D. A. Ivanov and T. Y. Ivanova, Journal of Physics B: Atomic, Molecular and Optical Physics 49, 085001 (2016).
  * Kurizki  _et al._ [2015] G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, PNAS 112, 3866 (2015).
  * Iftikhar  _et al._ [2018] Z. Iftikhar, A. Anthore, A. K. Mitchell, F. D. Parmentier, U. Gennser, A. Ouerghi, A. Cavanna, C. Mora, P. Simon, and F. Pierre, Science 360, 1315 (2018).
  * Breuer and Petruccione [2002] H. P. Breuer and F. Petruccione,  _The Theory of Open Quantum Systems_ (Oxford University Press, Oxford, 2002).
  * Leggett  _et al._ [1987] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1 (1987).
  * Hur [2008] K. L. Hur, Annals of Physics 323, 2208 (2008).
  * Nagy and Domokos [2015] D. Nagy and P. Domokos, Phys. Rev. Lett. 115, 043601 (2015).
  * Nagy and Domokos [2016] D. Nagy and P. Domokos, Phys. Rev. A 94, 063862 (2016).
  * Scarlatella and Schiro [2016] O. Scarlatella and M. Schiro, arXiv preprint arXiv:1611.09378 (2016).
  * Chin  _et al._ [2011] A. W. Chin, J. Prior, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett. 107, 160601 (2011).
  * Sacha and Zakrzewski [2017] K. Sacha and J. Zakrzewski, Rep. Prog. Phys. 81, 016401 (2017).
  * Eckardt [2017] A. Eckardt, Rev. Mod. Phys. 89, 011004 (2017).
  * Mekhov and Ritsch [2012] I. B. Mekhov and H. Ritsch, J. Phys. B 45, 102001 (2012).
  * Ritsch  _et al._ [2013] H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, Rev. Mod. Phys. 85, 553 (2013).
  * Mivehvar  _et al._ [2021] F. Mivehvar, F. Piazza, T. Donner, and H. Ritsch, arXiv preprint arXiv:2102.04473 (2021).
  * Baumann  _et al._ [2010] K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, Nature 464, 1301 (2010).
  * Landig  _et al._ [2016] R. Landig, L. Hruby, N. Dogra, M. Landini, R. Mottl, T. Donner, and T. Esslinger, Nature 532, 476–479 (2016).
  * Klinder  _et al._ [2015] J. Klinder, H. Keßler, M. R. Bakhtiari, M. Thorwart, and A. Hemmerich, Phys. Rev. Lett. 115, 230403 (2015).
  * Kroeze  _et al._ [2018] R. M. Kroeze, Y. Guo, V. D. Vaidya, J. Keeling, and B. L. Lev, Phys. Rev. Lett. 121, 163601 (2018).
  * Schuster  _et al._ [2018] S. C. Schuster, P. Wolf, D. Schmidt, S. Slama, and C. Zimmermann, Phys. Rev. Lett. 121, 223601 (2018).
  * Bux  _et al._ [2013] S. Bux, H. Tomczyk, D. Schmidt, P. W. Courteille, N. Piovella, and C. Zimmermann, Phys. Rev. A 87, 023607 (2013).
  * Roux  _et al._ [2020] K. Roux, H. Konishi, V. Helso, and J. Brantut, Nature Commun. 11, 2974 (2020).
  * Caballero-Benitez and Mekhov [2015a] S. F. Caballero-Benitez and I. B. Mekhov, Phys. Rev. Lett. 115, 243604 (2015a).
  * Caballero-Benitez and Mekhov [2015b] S. F. Caballero-Benitez and I. B. Mekhov, New J. Phys. 17, 123023 (2015b).
  * Rogers  _et al._ [2014] B. Rogers, M. Paternostro, J. F. Sherson, and G. De Chiara, Phys. Rev. A 90, 043618 (2014).
  * Fernández-Vidal  _et al._ [2010] S. Fernández-Vidal, G. De Chiara, J. Larson, and G. Morigi, Phys. Rev. A 81, 043407 (2010).
  * Niedenzu  _et al._ [2013] W. Niedenzu, S. Schütz, H. Habibian, G. Morigi, and H. Ritsch, Phys. Rev. A 88, 033830 (2013).
  * Gopalakrishnan  _et al._ [2009] S. Gopalakrishnan, B. L. Lev, and P. M. Goldbart, Nature Phys. 5, 845 (2009).
  * Sheikhan  _et al._ [2016] A. Sheikhan, F. Brennecke, and C. Kollath, Phys. Rev. A 93, 043609 (2016).
  * Piazza and Ritsch [2015] F. Piazza and H. Ritsch, Phys. Rev. Lett. 115, 163601 (2015).
  * Buchhold  _et al._ [2013] M. Buchhold, P. Strack, S. Sachdev, and S. Diehl, Phys. Rev. A 87, 063622 (2013).
  * Kohler  _et al._ [2017] J. Kohler, N. Spethmann, S. Schreppler, and D. M. Stamper-Kurn, Phys. Rev. Lett. 118, 063604 (2017).
  * Keßler  _et al._ [2014] H. Keßler, J. Klinder, M. Wolke, and A. Hemmerich, Phys. Rev. Lett. 113, 070404 (2014).
  * Wolke  _et al._ [2012] M. Wolke, J. Klinner, H. Kessler, and A. Hemmerich, Science 337, 85 (2012).
  * Kirton  _et al._ [2019] P. Kirton, M. M. Roses, J. Keeling, and E. G. Dalla Torre, Advanced Quantum Technologies 2, 1800043 (2019).
  * Dimer  _et al._ [2007] F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, Phys. Rev. A 75, 013804 (2007).
  * Zhiqiang  _et al._ [2017] Z. Zhiqiang, C. H. Lee, R. Kumar, K. J. Arnold, S. J. Masson, A. S. Parkins, and M. D. Barrett, Optica 4, 424 (2017).
  * Lundgren  _et al._ [2020] R. Lundgren, A. V. Gorshkov, and M. F. Maghrebi, Phys. Rev. A 102, 032218 (2020).
  * Kónya  _et al._ [2018] G. Kónya, D. Nagy, G. Szirmai, and P. Domokos, Phys. Rev. A 98, 063608 (2018).
  * Nagy  _et al._ [2010] D. Nagy, G. Kónya, G. Szirmai, and P. Domokos, Phys. Rev. Lett. 104, 130401 (2010).
  * Mekhov and Ritsch [2010] I. B. Mekhov and H. Ritsch, Laser Phys. 20, 694 (2010).
  * Mekhov and Ritsch [2011] I. B. Mekhov and H. Ritsch, Laser Phys. 21, 1486 (2011).
  * Kozlowski  _et al._ [2015] W. Kozlowski, S. F. Caballero-Benitez, and I. B. Mekhov, Phys. Rev. A 92, 013613 (2015).
  * Kozlowski  _et al._ [2017] W. Kozlowski, S. F. Caballero-Benitez, and I. B. Mekhov, Sci. Rep. 7, 42597 (2017).
  * Caballero-Benitez and Mekhov [2016] S. F. Caballero-Benitez and I. B. Mekhov, New J. Phys. 18, 113010 (2016).
  * Mazzucchi  _et al._ [2016c] G. Mazzucchi, W. Kozlowski, S. F. Caballero-Benitez, and I. B. Mekhov, New J. Phys. 18, 073017 (2016c).
  * Mazzucchi  _et al._ [2016d] G. Mazzucchi, S. F. Caballero-Benitez, and I. B. Mekhov, Sci. Rep. 6, 31196 (2016d).
  * Elliott  _et al._ [2015] T. J. Elliott, W. Kozlowski, S. F. Caballero-Benitez, and I. B. Mekhov, Phys. Rev. Lett. 114, 113604 (2015).
  * Mekhov [2013] I. B. Mekhov, Laser Phys. 23, 015501 (2013).
  * Gardiner and Zoller [1999] C. W. Gardiner and P. Zoller,  _Quantum Noise_ (Springer, Berlin, 1999).
  * Mekhov  _et al._ [2007] I. B. Mekhov, C. Maschler, and H. Ritsch, Nature Phys. 3, 319 (2007).
  * Camacho-Guardian  _et al._ [2017] A. Camacho-Guardian, R. Paredes, and S. F. Caballero-Benítez, Phys. Rev. A 96, 051602(R) (2017).
  * Caballero-Benitez  _et al._ [2016] S. F. Caballero-Benitez, G. Mazzucchi, and I. B. Mekhov, Phys. Rev. A 93, 063632 (2016).
  * Lozano-Méndez  _et al._ [2020] K. Lozano-Méndez, A. H. Cásares, and S. F. Caballero-Benítez, Spin entanglement and order competition via long-range interactions in spinor quantum optical lattices (2020), arXiv:2011.07765 [cond-mat.quant-gas] .
  * Morrison and Parkins [2008] S. Morrison and A. S. Parkins, Phys. Rev. Lett. 100, 040403 (2008).
  * Muniz  _et al._ [2020] J. A. Muniz, D. Barberena, R. J. Lewis-Swan, D. J. Young, J. R. K. Cline, A. M. Rey, and J. K. Thompson, Nature 580, 602 (2020).
  * Gong  _et al._ [2018] Z. Gong, R. Hamazaki, and M. Ueda, Phys. Rev. Lett. 120, 040404 (2018).
  * Zhu  _et al._ [2019] B. Zhu, J. Marino, N. Y. Yao, M. D. Lukin, and E. A. Demler, New J. Phys. 21, 073028 (2019).
  * Buca  _et al._ [2019] B. Buca, J. Tindall, and D. Jaksch, Nature Comm. 10, 1730 (2019).
  * Yu  _et al._ [2019] W. C. Yu, J. Tangpanitanon, A. W. Glaetzle, D. Jaksch, and D. G. Angelakis, Phys. Rev. A 99, 033618 (2019).
  * Cosme  _et al._ [2019] J. G. Cosme, J. Skulte, and L. Mathey, Phys. Rev. A 100, 053615 (2019).
  * Keßler  _et al._ [2020] H. Keßler, J. G. Cosme, C. Georges, L. Mathey, and A. Hemmerich, New Journal of Physics 22, 085002 (2020).
  * Keßler  _et al._ [2020] H. Keßler, P. Kongkhambut, C. Georges, L. Mathey, J. G. Cosme, and A. Hemmerich, Observation of a dissipative time crystal (2020), arXiv:2012.08885 [cond-mat.quant-gas] .
  * Fisher  _et al._ [1972] M. E. Fisher, S.-k. Ma, and B. G. Nickel, Phys. Rev. Lett. 29, 917 (1972).
  * Luijten and Blöte [1997] E. Luijten and H. W. J. Blöte, Phys. Rev. B 56, 8945 (1997).
  * Norcia  _et al._ [2018] M. A. Norcia, R. J. Lewis-Swan, J. R. K. Cline, B. Zhu, A. M. Rey, and J. K. Thompson, Science 361, 259 (2018).
  * Davis  _et al._ [2019] E. J. Davis, G. Bentsen, L. Homeier, T. Li, and M. H. Schleier-Smith, Phys. Rev. Lett. 122, 010405 (2019).
  * Öztop  _et al._ [2012] B. Öztop, M. Bordyuh, Ö. E. Müstecaplıoğlu, and H. E. Türeci, New J. Phys. 14, 085011 (2012).
  * Lee and Ruostekoski [2014] M. D. Lee and J. Ruostekoski, Phys. Rev. A 90, 023628 (2014).
  * Pedersen  _et al._ [2014] M. K. Pedersen, J. J. W. H. Sorensen, M. C. Tichy, and J. F. Sherson, New J. Phys. 16, 113038 (2014).
  * Blattmann and Mølmer [2016] R. Blattmann and K. Mølmer, Phys. Rev. A 93, 052113 (2016).
  * Yang  _et al._ [2018a] D. Yang, C. Laflamme, D. V. Vasilyev, M. A. Baranov, and P. Zoller, Phys. Rev. Lett. 120, 133601 (2018a).
  * Yang  _et al._ [2018b] D. Yang, D. V. Vasilyev, C. Laflamme, M. A. Baranov, and P. Zoller, Phys. Rev. A 98, 023852 (2018b).
  * Sørensen  _et al._ [2018] J. J. W. H. Sørensen, M. Dalgaard, A. H. Kiilerich, K. Mølmer, and J. F. Sherson, Phys. Rev. A 98, 062317 (2018).
  * Zheng  _et al._ [2018] D.-C. Zheng, L. Wan, and N.-H. Tong, Phys. Rev. B 98, 115131 (2018).
  * Bertet  _et al._ [2005] P. Bertet, I. Chiorescu, G. Burkard, K. Semba, C. J. P. M. Harmans, D. P. DiVincenzo, and J. E. Mooij, Phys. Rev. Lett. 95, 257002 (2005).
  * Petersson  _et al._ [2010] K. D. Petersson, J. R. Petta, H. Lu, and A. C. Gossard, Phys. Rev. Lett. 105, 246804 (2010).
  * Wolfowicz  _et al._ [2013] G. Wolfowicz, A. Tyryshkin, R. George, H. Riemann, N. Abrosimov, P. Becker, H. Pohl, M. Thewalt, S. Lyon, and J. Morton, Nature Nanotech 8, 561 (2013).
  * Guo  _et al._ [2012] C. Guo, A. Weichselbaum, J. von Delft, and M. Vojta, Phys. Rev. Lett. 108, 160401 (2012).
  * Castro Neto  _et al._ [2003] A. H. Castro Neto, E. Novais, L. Borda, G. Zaránd, and I. Affleck, Phys. Rev. Lett. 91, 096401 (2003).
  * Novais  _et al._ [2005] E. Novais, A. H. Castro Neto, L. Borda, I. Affleck, and G. Zarand, Phys. Rev. B 72, 014417 (2005).
  * Zhu and Si [2002] L. Zhu and Q. Si, Phys. Rev. B 66, 024426 (2002).
  * Zaránd and Demler [2002] G. Zaránd and E. Demler, Phys. Rev. B 66, 024427 (2002).