Quantum optimal control of the dissipative production of a maximally entangled state
Karl P. Horn Florentin Reiter Yiheng Lin Dietrich Leibfried Christiane P. Koch





Theoretische Physik, Universität Kassel, 
  Heinrich-Plett-Straße 40, D-34132 Kassel,
Germany


Department of Physics, Harvard University, 
  Cambridge, MA 02138, USA


CAS Key Laboratory of Microscale Magnetic Resonance and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China


National Institute of Standards and Technology, 
  Boulder, Colorado 80305, USA


Theoretische Physik, Universität Kassel, 
  Heinrich-Plett-Straße 40, D-34132 Kassel,
Germany
christiane.koch@uni-kassel.de




Entanglement generation can be robust against certain types of noise in approaches that deliberately incorporate dissipation into the system dynamics. The presence of additional dissipation channels may, however, limit 
fidelity and speed of the process.
Here we show how quantum optimal control techniques can be used to both speed up the entanglement generation and increase the fidelity in a realistic setup,
whilst respecting typical experimental
limitations. For the example of entangling two trapped ion qubits [Lin et al., Nature 504, 415 (2013)], we find an improved fidelity by 
simply optimizing the polarization of the laser beams utilized in the
experiment. More significantly, an alternate combination of transitions between internal states of the ions, when combined with optimized polarization,
enables faster entanglement and  decreases
the error by an order of magnitude.


Quantum optimal control of the dissipative production of a maximally entangled state
    Christiane P. Koch
    December 30, 2023
====================================================================================





§ INTRODUCTION

Quantum devices aim to exploit 
the two essential elements of 
quantum physics, quantum coherence 
and entanglement, for practical applications. 
They require the implementation of 
a number of basic tasks such as state 
preparation or generation of entanglement, 
all the while preserving the relevant 
non-classical features at the level of 
device operation. 
The implementation of quantum tasks thus needs to be robust 
with respect to parameter fluctuations and 
external noise that is unavoidable in 
any real physical setup. 


Loss of coherence and noise are commonly 
attributed to the coupling of the quantum 
system with its surrounding 
environment <cit.>. 
One strategy for realizing all necessary
tasks with sufficient accuracy is to 
perform the quantum operations at a 
time scale faster than the time scale 
at which the noise affects the system.
Quantum optimal control theory provides 
a set of tools to derive the corresponding 
protocols <cit.> and  can be used to
identify the quantum speed limit 
<cit.>,
i.e., the shortest possible duration 
within which the operation can be 
carried out with a pre-specified fidelity. 

Nevertheless, there is a fundamental 
limit in that one cannot `beat' the noise,
particlularly, when its time scales are comparable to or
faster than the typical speed limits of 
the target operation.
An alternative is found in approaches that 
deliberately incorporate dissipation into 
the system dynamics, often referred to as 
quantum reservoir engineering <cit.>. 
The basic idea is to implement stochastic 
dynamics whose stationary state is non-classical. 
This is achieved by manipulating the coupling 
to the environment, or reservoir. In its simplest
form, a constant but switchable coupling is
realized by an electromagnetic field that drives
a transition to a state with fast decay <cit.>. 
The dynamics are described by
the quantum optical master equation <cit.>, 
and the system will 
eventually be driven into the fixed point of 
the corresponding Liouvillian 
 <cit.>. 

Applications of this basic idea  are many faceted
—its use has been suggested, for example in generating 
entanglement 
<cit.>, 
implementing universal quantum computing <cit.>,
driving  phase transitions <cit.>
and autonomous quantum error correction
 <cit.>.
Experimentally, the 
generation of non-classical states <cit.>, 
entangled states <cit.>, 
and non-equilibrium quantum phases 
<cit.>
have successfully been demonstrated. 
Engineered dissipation can also be used 
towards a better understanding of open quantum system dynamics, by means of 
quantum simulation <cit.>. 

All of these examples testify to the fact that
dissipation can be a resource <cit.> for quantum technology. 
The ultimate performance bounds that can be 
reached with driven-dissipative dynamics under 
realistic conditions have, however, not yet 
been explored. 
While quantum reservoir engineering has been 
advocated for its robustness, its performance 
in a practical setting is compromised as soon 
as additional noise sources perturb the steady 
state or trap population flowing towards it.
 
This can be illustrated by examining the 
experiment described in Ref. <cit.>.
For a - - - chain
occupying the same linear Paul trap,
the two ions
were entangled via their collective motion using hyperfine 
electronic ground state levels as logical states.
Entanglement was achieved by applying a combination of  
laser and microwave transitions. 
This could be done in an either time-continuous 
manner or by repeating a fixed sequence of steps,
driving the system into a steady state, 
with the majority of population in the targeted, 
maximally entangled singlet state. 
Desired dissipation was brought into play 
by a combination of spin-motion coupling from a 
sideband laser, motion dissipation by sympathetically 
cooling cotrapped ions, and a repump laser 
which addresses the transition to a rapidly
decaying electronically excited state.
The sideband laser beams also lead to 
undesired pumping of spins, so-called
spontaneous emission. 
This resulted in population leakage and was 
the main source of error in that experiment <cit.>. 

The simultaneous presence of both desired 
and undesired dissipation channels is rather generic. 
To harness the full power of dissipative 
entangled state preparation, one would like 
to exploit the former while mitigating the latter. 
Here, we use quantum optimal control theory
 <cit.> to address this problem. 
For the example of preparing two trapped ions in
a maximally entangled state <cit.>, we 
ask whether entanglement can be generated faster
and more accurately when judiciously choosing a 
few key parameters. 
In order to keep in line with the experimental 
setup described in Ref. <cit.>,
we forego the usual assumption of time-dependent
pulses whose shapes are derived by quantum optimal 
control. 
Instead, we employ electromagnetic fields with 
constant amplitude and use tools from non-linear 
optimization to directly determine the best 
field strengths, detunings and polarizations. 
Our approach allows to not only determine the 
optimal values for these parameters, but also,
identify key factors that ultimately limit 
fidelity and speed of entanglement generation. 
Based on this insight, we explore an alternative
set of transitions and show that this scheme can 
outperform the original one both in terms of fidelity and speed. 

The paper is organized as follows. 
<Ref>
recalls the mechanism for entanglement generation in the experiment 
of Ref. <cit.> and 
details the theoretical description of the corresponding
trapped ion system.
Optimization of the transitions used in Ref. <cit.> is discussed in <ref>. 
An alternative set of transitions is introduced in <ref>, 
together with the optimization of the corresponding experimental parameters. 
We conclude in <Ref>. 



§ MODEL



In this section we consider the system described in
Ref. <cit.>,
consisting of a linear Paul trap containing ions and
ions, which interact mutually through their Coulomb
repulsion and with external electric fields.
A unitary idealization of these interactions
is summarized in the Hamiltonian H. The 
mechanism giving rise to dissipation in the state 
preparation process is spontaneous emission
after excitation of internal electronic
states of the ion by the external laser fields. 
The system dynamics is therefore described by 
the quantum optical master equation
in Lindblad form (with ħ = 1),

    ∂_tρ = ℒρ = -i [H,ρ] + ℒ_𝒟ρ .

We refer to ℒ_𝒟 as
the (Lindblad) dissipator, which is
given by

    ℒ_𝒟ρ = ∑_k( 
    		L_kρ L_k^† - 1/2[ 
    		L_k^† L_k, ρ] ),

where the sum over k contains individual contributions 
due to sympathetic cooling, heating
and photon scattering
occurring during stimulated Raman processes and repumping
into an electronically excited state.





 §.§ State space

The model Hamiltonian H accounts for the internal structure 
of two ions as well as two vibrational modes of the trapped ion chain.
The state space of the considered system consists of the 
following tensor product structure,

    (n_qb1)⊗ (n_qb2) ⊗
    	(n_) ⊗ (n_).

In <ref> n_qb1 and n_qb2 designate
hyperfine states of the ions,
specified by the quantum numbers F and their projections
m_F, obtained from coupling the total electronic angular
momentum quantum number J with the nuclear spin quantum number 
I.
<Ref>(a) highlights
the hyperfine states of interest, comprising of 
S_1/2,F=2,m_F=2
and S_1/2,F=1,m_F=1,
the two hyperfine levels to entangle, as well as 
an auxiliary level
aS_1/2,F=2,m_F=1.
The neighbouring levels 
oS_1/2,F=1,m_F=0
and 
tS_1/2,F=2,m_F=0
are also accounted for in the model, since
these are predominantly populated by inadvertent scattering
processes.
In the following, the only electronically excited
state of interest will be eP_1/2,F'=2,m_F'=2.

n_ and n_ are vibrational 
quantum numbers of two of the four shared 
motional modes of the trapped
ionic crystal along its linear axis.
Entanglement generation employs , 
and sideband transitions utilizing 
this mode are essential for the
presented schemes.
Unless specifically required, the mode , which
is not utilized for entanglement
but is included in the model to account for off-resonant coupling, will
be suppressed notationally for the sake of simplicity.
It is assumed that the trap has an
axis of weakest confinement along which the 
four-ion string is aligned and that
the eight radial motional modes can be neglected,
since they are largely decoupled given the 
sideband laser configuration described in Ref.
 <cit.>.
<Ref>(b)
shows three transitions that were driven on a single
ion in Ref. <cit.>. 
These belong to the
coherent part of <ref>, described by H, and one of them results in 
population of the electronically
excited state e with subsequent dissipation
which is modeled by the incoherent part, ℒ_𝒟ρ.
After adiabatic elimination, however, the transition to e
no longer appears in the
coherent part of <ref>,
while the dissipative part
is modified by the result of the adiabatic
elimination to fully account for
the effective decay out of a electronic 
ground state hyperfine level instead <cit.>. 
This is illustrated in  
<ref>(c).



 §.§ Original scheme for entanglement preparation





As represented in <ref>, 
the dissipative entanglement generation of Ref. <cit.> uses three different types of fields to induce population flow in the state space.
The entanglement mechanism can be understood
by qualitatively tracing the flow of
population from state to state as indicated
in <ref>.
Entangling the two ions 
via their joint motion in the trap is made possible by utilizing
sideband transitions driven by Raman lasers. These change the internal
states of the ions whilst simultaneously
exciting or de-exciting the utilized motional mode. In contrast, 
carrier transitions driven by a microwave field change the internal states only. Finally, a repump laser excites population to a short-lived electronically excited state. Specifically, in
Ref. <cit.>, a single
sideband transition between 
 and
,
a carrier transition
between 
a
and 
and a repump transition between a and
e are used.
Figure <ref> indicates
the transitions between the hyperfine levels
of interest for a single ion. 
The above transitions can be driven simultaneously
and time-independently for the duration of the experiment or in a step-wise manner <cit.>. Here, we focus on the continuous case, which resulted in a larger error.
Each ion is affected by the driven transitions
independently and no individual addressing is required.
Starting with both ions in an arbitrary state confined
to the hyperfine subspace { a,,},
in the ideal case, this scheme always leads 
to a steady state in which the the population
is trapped in the singlet entangled state
between  and , S_1/√(2)(  - ).
In the following, all singlet entangled states are
designated by S_ij1/√(2)( i j - j i),
whilst the triplet entangled states are designated
by T_ij1/√(2)( i j + j i), 
∀ i,j ∈{ a,,}.

Let us inspect in more detail the flow of
population from state to state 
in <ref>.
Starting in n_=0, for instance, 
it is possible to reach the target singlet entangled state
S_ by two sideband transitions
leading to n_, followed by a 
carrier transition into a combination of the
a n_ and a n_ states.
Population in the auxiliary state is driven by the repump laser into the electronically excited state from where it 
subsequently decays back into the electronic
ground state hyperfine subspace.
The process of electronic excitation and decay happens
sufficiently fast with respect to the other transitions,
that it can be regarded as `effective decay' directly
out of a, as depicted in 
<ref>(b) and (c).
This decay drives the system into a combination
of n_, the triplet entangled state
T_⊗n_,
and the target state S_⊗n_.
At any stage, sympathetic cooling can counteract the
excitations of the vibrational mode in the trap which are
caused by sideband transitions and heating.
Sympathetic cooling is induced by a different set of sideband lasers driving transitions only between internal states of the ions which share common motional modes with the ions.
The carrier transition between a and
 leads out of the target state
S_ into 
S_a.
This particular transition is highlighted 
specifically in <ref> 
by a dotted black double headed arrow.

By ensuring that the two-photon Rabi frequency
Ω_blue,,^2p
of the stimulated Raman sideband transition
between  and  is
much larger than the carrier Rabi frequency
Ω_car,a,, the latter
transition can effectively be suppressed.
<Ref> also highlights
the state 0 with a 
thick, dotted, black border, since 
the effective decay, proportional to
the square of the repump laser Rabi
frequency Ω_car,a,e
must be made sufficiently weak relative
to Ω_car,a,, in order
to prevent the trapping of population
in 0.
Consequently, a hierarchy of rates is
established in which the maximum
attainable two-photon Rabi frequency
of the stimulated Raman transition 
determines the maximal carrier
Rabi frequency between a and 
, which in turn
determines the maximal repump Rabi
frequency between a and e.



 §.§ Hamiltonian


In the rotating wave approximation
and interaction picture,
the total system Hamiltonian is comprised of
the driven hyperfine transitions

    H = ∑_type,i,f H_type,i,f,

where the sum runs over specific triples
(type, i,f ), designating a
transition of type `red' or `blue' sideband 
or `carrier', between the initial 
and final hyperfine states i
and f.

Transitions of the carrier type 
between the ground state hyperfine levels
are driven by microwave fields
with a Hamiltonian of the form

    H_car,i,f =    Ω_car,i,f( fi⊗1_qb2⊗1_⊗1_
       + 1_qb1⊗fi⊗1_⊗1_)e^-iΔ_car,i,ft + h.c..

Above, Ω_car,i,f denotes the Rabi frequency and Δ_car,i,f a small
detuning between the applied field and the transition energy
between i and f.
Each identity operator 1_j, with
j ∈{qb1, qb2, , }, is labelled
according to the subspace to which it corresponds.
A repump laser is required to drive 
transitions 
between ground and electronically excited 
hyperfine states.
These transitions therefore involve Hamiltonians of the form of <ref>, where
i is a hyperfine ground state
level and f= e is the addressed 
electronically excited hyperfine level.
Since population excited by this repumper decays very rapidly
into the hyperfine ground states, adiabatically eliminating
the excited state is well justified.


Ideally, a blue sideband transition between two hyperfine levels
i and f, utilizing the motional mode
, is represented  by 

    H_blue,i,f =    Ω_blue,i,f( fi⊗1_qb2⊗ b^+⊗1_
       + 1_qb1⊗fi⊗ b^+⊗1_)e^-iΔ_blue,i,ft + h.c.,

where Ω_blue,i,f 
is the sideband Rabi frequency
and Δ_blue,i,f 
a small detuning from the 
energy difference between i and f
plus the energy of one quantum of .
b^+ and b denote the bosonic creation 
and annihilation operators which respectively excite and 
de-excite the harmonic mode .
Analogously, the Hamiltonian of a red sideband 
transition takes the form of 
<ref> but with the 
annihilation operator b
replacing the creation operators b^†
and Δ_blue,i,f replaced by 
Δ_red,i,f.

In the specific case of a stimulated Raman
sideband transition,
Ω_red/blue,i,f in <ref>
becomes Ω_red/blue,i,f^2p, a
two-photon Rabi frequency of a red/blue
sideband transition between i and
f, given by

    Ω_red/blue,i,f^2p = η_μ^2 E_rE_b/4 ∑_kfd·ε_rkkd·ε_bi/Δ_kμ^2.

In the following we assume Lamb-Dicke parameters of
η_=0.180 and η_=0.155
for the utilized () and off-resonant 
() motional modes, respectively <cit.>. 
Above, E_r and E_b are the field strengths of the 
lower (red) and higher (blue) frequency Raman laser beams 
which have 
polarizations ε_r and
ε_b, expressed in spherical components as
ε_r=(r_-,r_0,r_+)
and ε_b=(b_-,b_0,b_+),
respectively.
d is the dipole operator for the ions
(also expressed in the spherical basis)
and the sum runs over all hyperfine levels k
in the electronically excited states P_1/2
and P_3/2.
The laser frequencies are shifted, such that
the ground state to excited state transitions
are detuned by Δ_e
and Δ_e + f_P below the
S_1/2↔ P_1/2
and S_1/2↔ P_3/2 
resonances, respectively.
f_P≈[197.2]GHz is the fine structure splitting between 
P_1/2 and P_3/2.
For the detuning between i and an individual excited state
hyperfine level k, the hyperfine splitting 
is neglected such that

    Δ_k≈Δ_e,     if k∈ P_1/2
    Δ_e+f_P,     if k∈  P_3/2  .

<Ref> utilizes
a characteristic stretched state
transition matrix element,
μP_3/2, F=3, m_F=3 
d_+S_1/2, F=2,m_F=2 
to properly scale 
a given reduced matrix element, 
fd·εi/μ,
with d_+, the right circular
component of the dipole operator.
The Wigner-Eckart theorem
<cit.>
and Breit-Rabi formula 
<cit.>
can then be used to
express an arbitrary transition matrix element
fd·εi
between two hyperfine levels i and f. 

To accurately model the system dynamics, 
it is necessary to account for the undesired 
off-resonant coupling of a given sideband transition
described by <ref> to an additional mode
, given by

    H_blue,i,f^ = 
    	   η_/η_Ω_blue,i,f^2p( fi⊗1_qb2⊗1_⊗ c^+
       + 1_qb1⊗fi⊗1_⊗ c^+) 
           × e^-i(δ -Δ_blue,i,f)t + h.c.,

in the case of a blue sideband transition.
In <ref>, δ is the detuning
between the utilized mode  and 
, which couples off-resonantly.
In the case of a red sideband transition,
the off-resonant coupling takes the form
of <ref> under interchange
of the annihilation and creation operators
of the harmonic oscillator describing
the  motional mode, 
c and c^+, and replacement of
Δ_blue,i,f by 
Δ_red,i,f, respectively.



 §.§ Lindblad operators


Incoherent processes taking place alongside
the driven transitions appear in the dissipative
part ℒ_𝒟 in <ref>, which is comprised
of individual contributions modelled by the Lindblad (jump) 
operators L_k in <ref>. An effective operator formalism <cit.> allows to adiabatically
eliminate the hyperfine excited state addressed by the
repump laser.
It leads to Lindblad operators of the form <cit.>

    L_rep,i,f^(1)   = √(γ_if^eff)fi⊗1_qb2⊗1_⊗1_
    
    	L_rep,i,f^(2)   = √(γ_if^eff)1_qb1⊗fi⊗1_⊗1_,

with effective rates 

    γ_if^eff = γ_ef4Ω_car,i,e^2/γ^2,

where e is the intermediate, rapidly decaying, 
electronically excited state,
Ω_car,i,e the repump 
Rabi
frequency,
γ_ef the decay rate from e into the
hyperfine ground state f and
γ = ∑_f'γ_ef'
the total decay rate out of e into a subspace
of hyperfine ground states.

Similarly to <ref>,
leaking between ground-state hyperfine levels
due to stimulated Raman sideband transition
acts on both beryllium ions according to

    L_sid,i,f^(1)   = √(Γ_if)fi⊗1_qb2⊗1_⊗1_
     
      L_sid,i,f^(2)   = √(Γ_if)1_qb1⊗fi⊗1_⊗1_ .

The scattering rate Γ_if
between an initial hyperfine ground state 
i and a final hyperfine ground state
f, due to a single laser beam is
given by the Kramers-Heisenberg formula

    Γ_i f = Γ_i → f = |E|^2μ^2/4γ| ∑_ka_i f^(k)/Δ_k|^2,

where

    a_i f^(k) = a_i → f^(k) = ∑_q∈{+,0,-}f d_qk/μkd·εi/μ

is the two-photon transition amplitude between i
and f.
As in <ref>, k runs over
all states k belonging to 
the ion P_1/2 and
P_3/2 manifolds.
Again, it sufficies to approximate
the Δ_k of 
k ∈ P_1/2, P_3/2
as Δ_e
and Δ_e+f_P, respectively. 
Rayleigh scattering is modelled by a Pauli σ_z
matrix between pairs of levels. 
Acting at the rate

    ϕ_i f = |E|^2μ^2/4γ| ∑_k( a_i i^(k)/Δ_k 
        -a_f f^(k)/Δ_k) |^2,

Rayleigh scattering is only of concern between
the  and  levels
and in most cases negligibly small.

Sympathetic cooling is achieved by using stimulated Raman
laser cooling and 
can be made to affect either or
both of the considered 
motional modes according to 

    L_cool,   = √(κ_c,)1_qb1⊗1_qb2⊗ b⊗1_
    
    	L_cool,   = √(κ_c,)1_qb1⊗1_qb2⊗1_⊗ c
    	,

where the cooling rates κ_c, 
and κ_c,
are governed by the field strengths of the repump and
stimulated Raman lasers acting on the magnesium ions.

Heating acts on all motional modes. 
It is caused by spontaneous emission occuring during the magnesium 
sideband Raman transitions, as well as photon recoil
from spontaneous emission and also the anomalous heating
of the ion trap.
The total heating can be modelled by

    L_heat,   = √(κ_h,)1_qb1⊗1_qb2⊗ 
        b^†⊗1_
    
      L_heat,   = √(κ_h,)1_qb1⊗1_qb2⊗1_⊗ c^†,

for a set of given heating rates κ_h,
and κ_h,.




§ OPTIMIZING THE ORIGINAL SCHEME



The goal of optimization 
is to maximize 
the 
population in the target state 
S_.
To this end, the final time T is defined as the
time at which the peak population in the
target state is reached and 
all driving fields 
can be turned off.
The target state population at final time is 
defined as the fidelity F
and correspondingly the error 
as
ϵ 1 - F.
The peak population at the final time is an appropriate
quantity to observe, since the stability
of the ionic hyperfine ground states causes
the system to remain in its entangled state
for a long time after all driving fields 
have been turned off.

In the following, the system degrees of freedom 
available for control are introduced and categorised
into two collections
in preparation for the optimization scheme discussed
below.
In contrast to a straightforward parameter optimization
of all degrees of freedom, the specialised
optimization scheme presented here is
less susceptible to running into local minima 
and demonstrates reliable and 
fast convergence.



 §.§ Optimization parameters



As previously discussed, the limitations
of the original scheme <cit.> are fundamentally
linked to the physical process of the stimulated Raman
sideband transition.
The two-photon Rabi frequency 
Ω_blue,,^2p
associated with this transition
should be made as large as possible to drive the
system towards the desired target state whilst ensuring
that the unfavourable transition between S_
and S_a is suppressed.
Consequently, the carrier transition Rabi
rate Ω_car,a, and in
turn the repump transition Rabi rate 
governing the effective decay out of a 
are limited, bottlenecking the flow
of population into the target state.

<Ref>
show that merely increasing
the field strengths of the sideband lasers
has the adverse side effect of also increasing
the chance of photon scattering and therefore
the rates of leaking between hyperfine 
ground states.
As such, a safe way of increasing the
field strength of the sideband lasers
is to compensate by increasing the
detuning Δ_e
from the excited state manifold,
since the two-photon Rabi 
frequency scales inversely with the detuning whilst
the scattering rates between hyperfine
states scale with the square of the inverse detuning.
The field strengths required 
to significantly increase the 
two-photon Rabi frequency
whilst minimising the associated 
scattering rates
are, however, beyond current experimental capabilities
<cit.>.
A third option is given by the polarization
of the two stimulated Raman sideband laser beams
ε_r and ε_b,
which have a great impact on both Ω_blue,,^2p and also {Γ_if}.

The tunable parameters E_r and E_b,
ε_r and ε_b
and Δ_e,
appearing in
<Ref>
constitute a first set of parameters defined as 

    { E_r,E_b,ε_r,
    ε_b, Δ_e}.

These are directly associated with the stimulated 
Raman sideband
transition.
The two-photon Rabi frequency 
Ω_blue,,^2p
scales with the product of field strengths
E_rE_b, whilst the scattering 
rates due to each laser beam
scale with |E_r/b|^2, the magnitude of the field
strength squared.
The polarization is split into its three 
spherical components, 
ε = (ε_-, ε_0, ε_+) 
where ε_i∈[ -1,1 ],∀ i ∈{ -,0,+ } and
with

    | ε_-|^2 + 
    	| ε_0|^2 + 
    	| ε_+|^2 = 1.

Due to the normalisation of the spherical
components, each polarization
possesses two degrees of freedom
which can be represented as
the azimuthal and polar angles on the
unit sphere.

A given configuration
of  fully determines the resulting
two-photon Rabi frequency
Ω_blue,,^2p
and all leakage rates Γ_if
between hyperfine states.
These parameters are deliberately
regarded separately from a
second set of parameters, 

    {Ω_car,a,,
    		Ω_car,a,e,
    		Δ_car,a,,
    		Δ_blue,,,
    		a
    	} ,

consisting of the carrier Rabi
frequencies
and
detunings for both ground state
transitions
and a balance parameter a, which
shall become important during the optimization.
The carrier Rabi frequencies are directly
determined by the applied field strengths and
can be tuned over broad ranges.
The detunings Δ_car,a,
and Δ_blue,, should
be kept small to prevent off-resonant coupling
to additional motional modes.



 §.§ Optimization algorithm



Our optimization algorithm, schematically depicted in <ref>, 
takes the approach of
optimizing the sets
introduced above in a two-step process.
Conceptually, the 
inner optimization over the first set
of parameters  incorporates the
dynamics indirectly and is encapsulated by
an outer optimization over the second
set of parameters , maximizing the
actual fidelity F.
This strategy is motivated by the fact that  
determining
Ω_blue,,^2p = 
Ω_blue,,^2p
(E_r,E_b,ε_r,ε_b,Δ_e)
and {Γ_if = Γ_if
(E_r,E_b,ε_r,ε_b,Δ_e)}
does not require explicit knowledge of the dynamics and is 
therefore computationally inexpensive.

The target functional of the inner step of the
optimization depends
on the field strengths E_r and E_b,
polarizations ε_r and 
ε_b and excited state
detuning Δ_e and is defined as

    J_inner[E_r,E_b,ε_r,ε_b,Δ_e] = ∑_if c_ifΓ_if - αΩ_sid^2p.

Here, α is a balance parameter
which weights up the relative importance of maximizing
Ω_blue,,^2p
versus minimising the sum ∑_if c_ifΓ_if,
for a given set of weights { c_if}.
If the set of weights { c_if} and a are fixed,
the inner optimization can calculate 
Ω_blue,,^2p
and {Γ_if} in terms
of , which are passed back to the
outer part of the optimization, once J_inner
is minimal.



The optimization of the set of parameters  
requires a measurement of the effect
a change in each scattering rate Γ_if has
on F, the overall fidelity of the dynamics.
This runs contrary to the usual practice of 
minimising the total scattering rate ∑_ifΓ_if
between all pairs of ground state hyperfine levels.
Individually weighting each Γ_if 
comes as a consequence of the 
observation that the leaking between each pair
of hyperfine ground states
affects the reached fidelity differently.
Most notably, transitions leading out
of the steady state S_
and transitions leading out of the 
hyperfine subspace { a,,}
into the neighbouring states { o,t }
have the largest negative effect on 
the fidelity.
Taking into account each individual leaking 
rate therefore offers the possibility
of strongly suppressing certain detrimental
Γ_if by carefully tuning the polarization.
We encode the degree to which a certain Γ_if
affects the fidelity by running several simulations
where each individual rate
Γ_if is artificially boosted 
by a factor of 10, whilst keeping
all other rates fixed, resulting in a set of 
fidelities {F_if}. 
Observing the difference F-F_if,
between boosted and unaltered dynamics
leads to a set of weights, 
{ c_if 1000( F-F_if) }.

<Ref> shows two different
sets of weights at the beginning of the optimization and after a few updates. 
As discussed in <cit.>
the biggest scattering error is due to 
the qubit transition between 
and . 
Γ_,
Along with transitions
leading out of the hyperfine subspace
into the neighbouring o and t
levels, recieved the
largest weights c_if for the
duration of the optimization.

A given set of weights can be used 
to optimize , leading to
the best possible
Ω_blue,,^2p
and {Γ_if}
with which to perform the dynamics.

The optimization of the second set 
of parameters, , directly targets 
the fidelity F of the dynamics, 

    J_outer[ Ω_car,a,, Ω_car,a,e,Δ_blue,,, Δ_car,a,,α]  1 - F = ϵ .

For each iteration of the outer optimization,
the inner optimization
over <ref> leading to optimal
Ω_blue,,^2p and 
{Γ_if} is performed
using the set of weights { c_if}
generated during the previous iteration 
(for the first iteration c_if=1, ∀ i,f).
After the inner optimization,
a new set of weights { c_if}
is generated for the next iteration of
the outer optimization, as illustrated in <ref>.

This two-step optimization is 
easily generalised for arbitrary 
combinations of transitions, 
including the possibility for
multiple sideband transitions
between differentground state hyperfine
levels.
Optimization of multiple sideband
transitions follows the rule,
that the j^th
sideband transition
has its own set of polarizations
ε_r^(j) and
ε_b^(j), field
strengths E_r^(j) and E_b^(j),
balance parameter α^(j) and
excited state detuning Δ_e^(j)
but
each contributes towards a set of
total scattering rates 
{Γ_if = ∑_jΓ_if^(j)}.
Furthermore, all transitions except for
the repump transition have a detuning
Δ_type,i,f
and all carrier transitions have a
Rabi frequency Ω_car,i,f to be optimized
directly, along
with the set of balance parameters
{α^(j)}, in the 
outer optimization.



 §.§ Result of optimization




All parameter optimizations have been performed
with the NLopt package <cit.>
using
the Subplex algorithm <cit.>.
While other optimization
methods could also be used 
in the outer and inner optimization loops, we have found these to converge well.
<Ref> compares
the simulated dynamics of the system as described in
Ref. <cit.> 
with the dynamics obtained after optimization.
The peak fidelity 
is increased from F=76% to
F=88%. 
This is due to  a modified steady state, in which the populations
in T_, 
and  each are smaller 
than in the original scheme.
Furthermore, through optimization of
the Γ_if, a significant portion of the
population can be prevented from escaping
the ground state hyperfine subspace 
{ a,,}, which
causes the prominent crest in the
S_ population for the original scheme.
The optimized result was compared to
different realisations of randomly chosen
polarizations ε_r and
ε_b, which leads to
dramatically varying peak fidelities
that can be as low as F=10% but are only
rarely in the vicinity of but never surpass
the peak fidelity reached by optimization.

The optimized values of the various parameters are 
reported in 
<ref>.
After optimization, the two-photon 
sideband Rabi frequency Ω_blue,,^2p
assumes a value of 2π×[7.65]kHz,
which is very close to 
the rate 2π×[7.81]kHz reported in
Ref. <cit.>.
The increase in fidelity can therefore
mainly be attributed to the
adjustments made to the polarization
ε_r, ε_b
and increase in excited state manifold detuning 
Δ_e from
[270]GHz to [662]GHz, which
is feasible, see for example in Ref. <cit.>.
In other words, the outcome of the inner optimization is
a superior set of scattering rates
{Γ_if},
with the parameters of the outer optimization
adjusted to rebalance the system.
Compared to Ref. <cit.>, in
which Ω_car,a,=[495]Hz, 
the carrier Rabi frequency between a and
 drops to [316]Hz after
optimization, thus further suppressing the
unwanted S_↔S_a
transition.
As the optimal fidelity
is approached, the detunings
Δ_car,a, and
Δ_blue,,
become negligibly small, indicating
that for this particular entanglement scheme, 
the shift out of resonance due to the
driven transitions is not much of a
factor.

Nevertheless, the achievable fidelity is inherently
limited in this entanglement scheme.
As demonstrated by 
<ref>, 
even if the field strengths of
the lasers utilized for the stimulated Raman
sideband transition were unconstrained, 
a finite amount of leaking between hyperfine
states would remain present.
Limited field strengths of the sideband lasers
necessitate a trade-off between the error due
to leaking between hyperfine states and the
errors due to population trapping in 
and the unfavourable transition between
S_ and S_a.

As such, the fidelity that can be reached with our optimized parameters 
falls short of the fidelity obtained by switching to the
stepwise scheme presented in Ref. <cit.> which amounts to 
 F=89.2%.
The stepwise scheme negates the error caused
by the unfavourable transition between 
S_ and S_a
by temporally separating the 
ground state hyperfine transitions from the
application of the repumper and also the
sympathetic cooling.
This strategy ensures that population lost out
of S_ into S_a
has nowhere to go and, if precisely timed,
is returned to S_ after
a full Rabi cycle. 
Essentially, the stepwise scheme
lifts the requirement of
balancing the rates at which each
transition can be driven, thereby 
overcoming the limitations associated
with the time-continuous implementation.
In the following we will show that a continuously operated scheme can outperform both variants for entanglement generation of Ref. <cit.> by exploiting a different combination of transitions. 



§ TWO-SIDEBAND SCHEME

 
Alternatively to the original scheme presented in 
<ref>, steady-state entanglement
can be reached using other combinations of continuously
driven carrier and sideband
-hyperfine transitions. 
We consider here a scheme that 
features two sideband transitions:
a blue sideband transition from  to ,
and a second, red sideband transition from  to a.
Note that we assume each sideband transition to be driven by its
own pair of stimulated Raman laser beams. It would also be possible to drive the two sideband transitions using only three beams. This simply requires
 proper choice of the correct
relative detunings.
In addition, and as in the original
scheme, a repump transition between
a and e is driven.
In order for all states in the hyperfine subspace to
be connected to the target state S_,
a carrier transition between 
 and  is included as well.
This choice is similar to the combination of transitions utilized 
for the entanglement of two ^40Ca^+ 
ions in Ref. <cit.>. It  offers 
numerous advantages over the original scheme as detailed below.



 §.§ Entanglement mechanism and optimization parameters



<Ref> illustrates the entanglement
mechanism for this new combination of
transitions.
Crucially, the unfavourable transition
between S_⊗0 and 
S_a ⊗0 due to the carrier
connecting a and 
in the original scheme has been eliminated.
Instead, the red sideband transition from 
to a leads from
S_⊗n_
to S_a⊗n_-1
only when n_ > 0.
Consequently, for this combination of transitions,
in the absence of leakage between
hyperfine states and heating, 
S_⊗n_=0 alone
is the steady state of the dynamics.
In the presence of heating, population in 
S_⊗0 can
only escape due to an excitation of the
utilized vibrational mode 
followed by a sideband transition from
S_⊗n_
to S_a⊗n_-1.
Population in S_a⊗n_-1
can take multiple branching paths,
all of which
eventually lead back to 
S_.
As such, in contrast to the 
original scheme, which relies on
sympathetic cooling, this particular combination of
transitions inherently cools the utilized
mode   of the system
during entanglement generation.

Without the need for sympathetic cooling, the 
ions can be removed. 
This leads not only to
a simplification of the experiment but also reduces the
number of motional modes of the ionic
crystal. It thus effectively eliminates
the error due to off-resonant coupling
to , given by
<ref> in the original scheme. 
As described in <ref>,
in the original scheme
the carrier Rabi frequencies 
Ω_car,a, and 
Ω_car,a,e,
which determine the rate of
effective decay out of a,
are limited by the maximum attainable
Ω_blue,,^2p.
In contrast, in the current scheme
the carrier Rabi frequencies 
Ω_car,, and
Ω_car,a,e can be  
increased significantly, without
causing losses out of the target state
and population trapping in 
.
By driving an additional sideband
transition, the graph of states
in <ref>
is more connected, permitting
population to reach S_
by additional paths.
Comparing the graphs shown in 
<ref>,
the combined effect of additional paths
into the target state and the increase
in Ω_car,a,e which results in larger 
effective decay rates 
{γ_a,f∝Ω_car,a,e^2}
should lead to much faster entanglement preparation.

Optimization of the field strengths
and polarizations for the two-sideband
scheme has been carried out according to the same
principle as described in
<ref>, with the slight
complication of having to address
additional degrees of freedom.
In the specific case of the two-sideband
combination, the corresponding form
of the target functional for the polarization optimization,
<ref>, becomes

    J_inner[   E_b^(1),E_r^(1),ε_b^(1),
    	ε_r^(1),Δ_e^(1), 
        E_b^(2),E_r^(2),ε_b^(2),
    	ε_r^(2),Δ_e^(2)] 
    
    	=    ∑_if c_ifΓ_if - α^(1)Ω_blue,,^2p (1) - α^(2)Ω_red,,a^2p (2).

As in the original scheme
and described in detail in <ref>, 
optimization of the
polarization can be accomplished
without having to simulate the
dynamics in each iteration.
A single inner optimization step determines both
Ω_blue,,^2p (1)
=Ω_blue,,^2p (1)(E_r^(1),E_b^(1),ε_r^(1),
ε_b^(1),Δ_e^(1))
and
Ω_red,,a^2p (2)
=Ω_red,,a^2p (2)(E_r^(2),E_b^(2),ε_r^(2),
ε_b^(2),Δ_e^(2)),
in addition to 
{Γ_if = Γ_if^(1)(E_r^(1),E_b^(1),ε_r^(1),
ε_b^(1),Δ_e^(1)) 
+ Γ_if^(2)(E_r^(2),E_b^(2),ε_r^(2),
ε_b^(2),Δ_e^(2))
},
the set of scattering rates due to each
sideband transition.
As in <ref>, the outer optimization
is performed directly on the fidelity F of the dynamics, 

    J_outer[   Ω_car,,,Ω_car,a,e,Δ_car,,, 
    		
       Δ_blue,,,
    	Δ_red,,a, α^(1), α^(2)]  1 - F = ϵ .

The set 
 now consists of
the carrier Rabi frequencies 
Ω_car,,,
Ω_car,a,e, 
the detunings
Δ_car,,,
Δ_blue,,^(1)
and Δ_red,,a^(2),
of the microwave carrier, first and second sideband
transitions, and 
the weights
α^(1) and 
α^(2). 
Since the scheme now involves a second sideband combination,
an additional weight is required to balance
the maximization of its two-photon sideband 
Rabi frequency against the sideband photon
scattering rates in <ref>.
In order to make sure that
both Rabi frequencies are
maximized without one
dominating the other, however,
the left hand side of 
<ref> can be
modified slightly, such that

    J̃_inner[   E_b^(1),E_r^(1),ε_b^(1),
    		ε_r^(1),Δ_e^(1), 
        E_b^(2),E_r^(2),ε_b^(2),
    	ε_r^(2),Δ_e^(2)]  
        = ∑_if c_ifΓ_if - α( Ω_blue,,^2p (1) 
    		+Ω_red,,a^2p (2)) 
          + β| Ω_blue,,^2p (1) 
    		-Ω_red,,a^2p (2)| ,

where α now balances the
maximization of the sum of
two-photon Rabi frequencies
against the Γ_if, whilst
β 
is a parameter controlling how
strictly the two-photon sideband
Rabi frequencies should be matched.
For simplicity it is assumed that
E_r^(1) = E_r^(2) and
E_b^(1) = E_b^(2) and
that each field strength is
limited to the maximum value 
allowed during the optimization
of the original scheme.



In the absence of sympathetic cooling,
the primary source of heating,
caused by spontaneous emission 
during the stimulated Raman sideband
transition driven on the ions,
is eliminated.
The remaining sources of heating are
photon recoil from the spontaneous
emission out of e after repumping
and electric field noise associated 
with the ion trap <cit.>.
Since the
heating rate
influences the system dynamics
and therefore the obtained fidelity,
the result of the optimization depends
on the specific heating rate assumed,
which can vary, depending on the
motional mode utilized for the sideband
transition.




 §.§ Influence of trap heating rates


<Ref> compares the
reached peak fidelity for different
values of the
heating rate
κ_h 
of 
vibrational mode .
For an assumed heating rate of
κ_h = 2π×[1],
optimization leads to a peak fidelity of F=98.3%, whilst
κ_h = 2π×[10]
is a more realistic heating rate for modern traps
leading to a peak fidelity of F=96.7%.
Finally, when κ_h = 2π×[100],
the peak fidelity is reduced to F=90.3%. 
For each considered heating rate,
the parameters leading to optimal entanglement
are listed in <Ref>.
As the heating rate κ_h
is increased, recovering population
lost from the target
state S_
requires an increase in
the Rabi frequencies of all driven
transitions.
For all reported heating rates,
however, the ratios 
Ω_car,,∝Ω_blue,,^2p (1)≈Ω_red,,a^2p (2)∝Ω_car,a,e^2
remain approximately constant.
Here, the repumper Rabi frequency Ω_car,a,e
enters squared, since the effective decay rates
in <ref>
are proportional to Ω_car,a,e^2.
This observation can be understood, 
since the target state should be reachable as
directly as possible from any given state.
Scaling all transition rates equally
is necessary in order to prevent the flow of
population from being bottlenecked throughout the 
entanglement generation.
The optimized peak fidelities are again
significantly higher than the average fidelity
of F≈ 0.4 (or F≈ 0.5 with the 
fixed scaling of Rabi frequencies mentioned above) 
obtained from simulating the dynamics with
random polarizations 
ε_r^(1), ε_b^(1),
ε_r^(2) and ε_b^(2)
of the sideband laser beams and assuming 
κ_h=2π×[1].
For the two-sideband scheme it is much more difficult
to randomly select a near-optimal polarization, due 
to the increased number of degrees of freedom, which
also causes the peak fidelity to strongly vary depending
on the polarization.
Furthermore, an optimization to minimize the time taken 
to reach a target state population
of F=85% was performed for the heating
rates κ_h∈{2π×[1],2π×[10],2π×[100]},
leading to a preparation time of
t≈[0.3]ms for all 
assumed heating rates.

For each increase in the heating rate, 
the optimization results 
in a different set of polarizations 
ε_r^(1), 
ε_b^(1), 
ε_r^(2) and 
ε_b^(2).
As the heating rate is increased, 
the minimization of leakage rates 
{Γ_if} becomes less important.
A given Γ_if is determined
by the polarization of each stimulated Raman laser beam
and scales with the
squared magnitude of the field strength |E|^2
whilst scaling inversely with the squared detuning
from the excited state Δ_e 
(<ref>) of the considered laser beam.
Instead, the maximization of the
two-photon stimulated Raman
sideband transition rates 
Ω_blue,,^2p (1) and
Ω_red,,a^2p (2) (<ref>)
is prioritised.
The two-photon stimulated Raman sideband 
transition Rabi frequencies depend on
the polarizations ε_r and 
ε_b of both laser beams, 
the product of field strengths E_rE_b
and the detuning of both stimulated Raman laser beams
from the excited state Δ_e.
Larger sideband two-photon Rabi frequencies 
ensure that population can flow back into 
S_⊗0
much faster than the heating can allow it to escape.

Increasing all of the transition rates
has the side effect of speeding up the entanglement
but limits the attainable fidelity, with an increased
error due to population leaking outside of the 
hyperfine subspace { a,,}.
The behaviour of the Δ_e^(1)
and Δ_e^(2) is 
non-monotonic and appears to be strongly dependent on the
particular polarization profile.
As in the original scheme, 
each of the Δ_car,,,
Δ_blue,, and
Δ_red,,a becomes
smaller as the optimal fidelity is reached.

The error due to heating can only be
reduced by increasing the flow
of population into the target state
S_⊗0,
since there is no straightforward
way to compensate for heating.
This comes at the cost of 
increasing 
∑_if c_ifΓ_if and 
thus the error due to leakage between the
hyperfine states, as explained above.
Assuming optimal polarization and
balancing of the driven rates,
the only way to reduce
one error without compounding
the other error
is by increasing the maximum field strengths E_r^(1),
E_b^(1),E_r^(2) and E_b^(2). This explains why the
field strengths take their maximal allowed value in
<Ref>. 



 §.§ Comparison to the original scheme


The two-sideband scheme  represents a promising alternative
to the original scheme even with optimized parameters, as
discussed in <ref>.
In terms of fidelity, the two-sideband 
scheme outperforms the original one, 
regardless of the assumed heating
rate κ_h.
Even in the worst case considered, with 
κ_h = 2π×[100],
the resulting error is under 10% after optimization.
In comparison, the previously best
fidelity, reached by the stepwise scheme 
in Ref. <cit.>, corresponds to an error of about 
11%.
The corresponding errors for the original scheme in <ref>
are slightly larger for the polarization optimized case and
two and a half times as large for the non-optimized case.
In terms of speed, the two-sideband scheme
outperforms the original scheme.
Given traps with sufficiently small heating rates,
entangling speed can be sacrificed in order to maximize fidelity.
The lowest regarded heating rate
κ_h=2π×[1],
can be optimized over [3]ms, attaining a
fidelity of F=98.3%, or optimized over [6]ms,
in order to increase the fidelity to F=98.7%.
In contrast, the original scheme peaks after
approximately [6]ms but at the much lower fidelity F=76%.

To summarize, when considering the experimental modifications 
necessary to go from the protocol 
in Ref. <cit.> to
the two-sideband scheme, the overall complexity
is reduced.
Instead of a four ion setup consisting of two and
two ions, with their respective sympathetic cooling
laser beams, now only the two to be entangled need to be trapped without sympathetic cooling laser beams.
Given sufficient power, the four laser 
beams required for both of the stimulated
Raman sideband transitions can all be derived from the
same [313]nm laser and frequency shifted using
acousto-optic modulators.
The only further complication is the ability to 
independently manipulate the polarization
of each individual stimulated Raman sideband
transition laser beam.
One may wonder of course how sensitive the Bell state fidelity is with respect to small deviations from the optimized polarizations. We have found 
fluctuations in the polarization components of up to 5% to only have a neglible effect on the entanglement error, whilst fluctuations above 10% will noticeably reduce the fidelity.



 §.§ Fundamental performance bound


Given the superior performance of the two-sideband scheme compared to
the original protocol <cit.>, one may wonder
whether there are ultimate limits to the fidelity of a Bell state
realized in this way. 
There are two main sources of error that limit the fidelities in this
dissipative state preparation scheme—anomalous heating and spontaneous
emission. As discussed above, the obtainable fidelity is determined by a
trade-off between utilizing fast enough sideband transitions in order
to beat trap heating, and minimization of the spontaneous
emission rates associated with the sideband transitions.
While anomalous heating can in principle be made
arbitrarily small by improving the ion trap, undesired spontaneous
emission is an inherent and unavoidable loss mechanism accompanying
the desired spontaneous emission at the core of the dissipative state
preparation. In order to explore the 
fundamental performance bound posed by spontaneous emission, we assume
a realistic trap with κ_h=2π×[10], a close to perfect 
trap, with κ_h=2π×[1]
or no heating at all (κ_h=[0]), and investigate how much
laser power is needed to achieve a certain fidelity, or error. 

In the absence of all heating, the optimization will favor slow
sideband transitions that are detuned far below the
P_1/2 and P_3/2 levels with laser
beams polarized such that there is minimal spontaneous 
emission. Identifying the conditions under which it is possible to reach
Bell state fidelities of F=99.9% or even F=99.99%
allows us to benchmark the performance of the current dissipative scheme.
For comparison, Ref. <cit.> 
examines the dependence 
of fidelity on laser power for gate-based 
entanglement creation for various ion species.
Of all observed ion species, 
the gate error of entanglement 
was lowest for a given power P, related to
the laser field strengh E by

    P = π/4 E^2w_0^2cϵ_0  ,

where w_0 is the laser beam waist, 
c the speed of light and  ϵ_0 the vacuum
permittivity <cit.>. We assume here an (idealized)
beam waist of w_0=[20]μ m, to directly compare
to Ref. <cit.>.

During optimization, the highest regarded threshold, F=99.99%
was reached after [0.33]ms  using field strengths of 
E_r/b≈[752]kV/m per beam and detunings
up to [25]THz.
For the sake of comparison with Ref. <cit.>, and
for the case of negligible heating, the timescale in the master equation 
	(<ref>) can be changed, t →τ= t/χ.
In order to match the same entangling speed and duration of [10]μ s as reported in Ref. <cit.>, we require [4.4]MV/m per beam,
corresponding to a total power of 4×[16]W at the same 
detuning.
For this very fast entanglement, the negative effects of heating are limited,
leading to errors of ϵ = 6.5× 10^-5, 
ϵ=1.0× 10 ^-4 and
ϵ=4.47× 10^-4 for heating rates of κ_h ∈{2π×[1],
2π×[10], 2π×[100]}, respectively.

If we fix the available field strength to the value 
E_r/b≈[200]kV, corresponding to a total power 
of 4×[36]mW,
as reported in Ref. <cit.>, the target fidelity is reached after  [4.6]ms.
Again, despite the much lower field strengths, the detuning remains unchanged.
It should be noted here, that an entangling duration of  [4.6]ms is 
still faster than that of the original entanglement scheme <cit.>.
At this lower extreme in field strengths, the effects of 
heating are more noticeable, since heating is allowed to act
for almost 500 times longer relative to the [10]μ s case.


<Ref> shows the Bell state, i.e.,
entanglement error, obtained when rescaling the duration 
to [1]ms, for different field strengths
E_r/b and heating rates
(shown in red, orange, and grey, respectively). 
A fidelity of F=99% is reached for all regarded heating rates, requiring 
field strengths between E_r/b
= [31]kV/m (κ_h=2π×[0]) and 
E_r/b=[38]kV/m (κ_h=2π×[10]).
The next threshold, F=99.9%,
is only crossed for κ_h=0 and 
κ_h = 2π×[1] at field strengths of 
E_r/b=[100]kV/m and E_r/b=[125]kV/m, corresponding 
to total powers of P≈ 4×[8.3]mW 
and P≈4×[13]mW, respectively.
Obtaining this fidelity requires detunings on the order of [6]THz.
The highest threshold,  F=99.99%, is reached for 
κ_h=2π×[0] whilst requiring field strengths 
of the order of E_r/b≈[325]kV/m 
corresponding to a total power of 4×[89]mW.
This is about two and a half times more power than for the gate based approach
in Ref. <cit.>.
For neglible heating, the required field strength per beam can be reduced by using three instead of four beams to drive the two sideband transitions (blue curve in Fig. <ref>). 
This finding illustrates that parameter optimization is prone to trapping in local optima, in particular for a larger number of optimization parameters <cit.>.
We attribute the improvement to the fact that omission of one beam reduces the inadvertent scattering. More specifically, the extra constraint on the beam detunings
appears to aid the optimization algorithm in finding a 
configuration for which the contribution of each beam 
towards the scattering error is distributed in a more favorable
way than in the four beam setup. 




§ CONCLUSIONS



We have addressed the problem of additional noise sources that limit fidelity 
and speed of  dissipative entanglement generation. 
Combining quantum optimal control theory <cit.>
with the effective operator approach <cit.>,
we have shown how to improve both fidelity
and speed for the example of entangling two
hyperfine qubits in a chain of trapped ions
 <cit.>. 
The detrimental noise source in this case 
is undesired spontaneous decay brought
about by the sideband laser beams that
are necessary for coupling the qubits
 <cit.>. 
This decay leads to the irrevocable 
loss of population from the hyperfine 
subspace of interest.
Whilst the undesired spontaneous decay 
cannot be eliminated entirely,
an optimal choice of the experimental parameters increases the
fidelity from 76% to 85%, with minimal changes to the
setup. Key to the improvement is  optimization of the sideband laser
beam polarizations which enter the decay rates of each individual
hyperfine level. Due to their interdependence,  the various parameters
of the experiment need to be retuned when changing the
polarization. 
The two-stage optimization process that we have
developed here can easily resolve this issue, demonstrating the power
of numerical quantum control.  

Further limitations to fidelity and speed can be identified graphically,
by visualizing the connections between states
due to the various field-driven transitions. 
This allows to  qualitatively trace the flow of population and shows that, 
depending on the
relative transition rates, population can get trapped in states other than 
the target state or be transferred out of the target state by an unfavourable 
transition. The latter in particular implies that the target state does not 
fully coincide with the steady state of the evolution. 
In order to overcome this limitation, we suggest to adapt the entanglement scheme
presented in Ref. <cit.> to using two sideband transitions. 
Of course, adding a second sideband makes the
suppression of the error due to detrimental spontaneous emission even more 
important. Our optimization method had no difficulty to cope with this task, 
despite the increase in the number of tunable parameters. 

Analysis aided by the graph of connected states for the two-sideband scenario 
reveals that the limitations of the original scheme of
Ref. <cit.> can indeed be overcome, whilst
providing additional advantageous properties such as higher
entanglement speed and  
inherent cooling. This offers the possibility of reducing the
complexity of the experiment by removing the need for sympathetic
cooling and all sources of error that come with it. The entangled, or, 
Bell state fidelity that we predict for  
the two-sideband scenario strongly depends on the heating rate. It can
be as high as 98% under conditions similar to those of
Ref. <cit.>, in particular in terms of the
available laser field strengths. The maximum attainable fidelity is
primarily limited by the heating rate of the motional mode that is
used to couple the qubits. It dictates the timescale at which the
sideband transitions must take place. Weaker sideband transition rates
in turn enable better suppression of spontaneous emission errors. 
Whilst a fidelity of 98% represents an order of magnitude
improvement over the originally obtained fidelity, execution of most
quantum protocols requires fidelities in excess of 99%. These
could be achieved through experimental refinements, such as ion traps
with weaker anomalous heating, 
more powerful sideband lasers <cit.>, or use of
optical instead of hyperfine
qubits <cit.>. 

Provided that heating rates can be made negligible,
for instance by implementing the two-sideband
scheme on the stretch rather than the center of mass mode of the two ions,
one may wonder whether
spontaneous emission ultimately limits the performance of dissipative
Bell state preparation. Spontaneous emission can be reduced by using larger
detunings which in turn requires more laser power or longer durations. Compared to
gate-based entanglement preparation <cit.>, we find,
for the same laser power of 4× 36mW into 20μm beam
waist as in Ref. <cit.>, the entangling duration to
realize  a Bell 
state fidelity of 99.99% to be increased from 10μs to 4.6ms
in an ideal trap. The advantages of the dissipative approach, in
particular its inherent robustness against noise, might easily outweigh this
time requirement, making dissipative entanglement production a viable
resource for quantum information protocols. Consider, for example,
carrying out primitives such as  gate teleportation. This 
could be driven by an entanglement machine that produces 200 pairs/s in serial
mode (per node) or  output one pair per  10μs when run with 250
nodes in parallel. A further speed up is possible by using more 
laser power.


Our study provides a first example for how to use quantum optimal
control theory to push driven-dissipative protocols to their ultimate
performance limit, despite imperfections in a practical
setting. Performance limits include, in addition to maximal fidelity,
also the highest speed. Here, we have obtained a speed up of about a
factor of four compared to Ref. <cit.>. Speed is of particular concern when scaling up
entanglement generation since some undesired decoherence rates are
known to scale with system size <cit.>. Deriving the
fastest possible protocol is therefore key if dissipative generation
of many-body entanglement <cit.> is to succeed. As
we have shown, optimal control theory is a tool ideally suited to tackle this
task, and targeting a multipartite entangled state is a natural next step. 

The optimal control theory framework for dissipative entanglement generation that we have introduced here is not limited to the specific example of trapped ions. In fact, our technique is applicable to generic multi-level quantum systems in the presence of dissipation for which the time evolution can be obtained within reasonable computation time [Typically of the order of one hundred propagations are necessary to converge the optimization]. This includes also systems with multiple steady states <cit.> which would be interesting for e.g. quantum error correction, or systems with  
non-Markovian dynamics such as solid state devices <cit.>. 
In the latter case, the  generalization requires the combination of the present optimization approach with one of the methods for obtaining non-Markovian dynamics <cit.>, such as partitioning the environment into strongly and weakly coupled parts <cit.>. Non-Markovianity has been shown to assist entanglement generation in coupled dimers subject to dephasing noise <cit.>. Our approach would allow to investigate, for more complex systems  and other types of dissipation,
whether non-Markovianity is beneficial or detrimental to the speed and overall success of entanglement generation.



  We thank Dave Wineland for discussions and Yong Wan and Stephen 
  Erickson for helpful suggestions on the manuscript. 
  Financial support
  from the Deutsche Forschungsgemeinschaft (Grant No. KO 2301/11-1) 
  is gratefully acknowledged.
  Florentin Reiter acknowledges support by a Feodor-Lynen fellowship
  from the Humboldt Foundation.










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@bib@innerbibempty

[Breuer and Petruccione(2002)]breuer2002theory
  author author H.-P. Breuer and author F. Petruccione, @noop  title The theory of open
  quantum systems (publisher Oxford University Press,  year 2002)NoStop
[Glaser et al.(2015)Glaser,
  Boscain, Calarco, Koch,
  Köckenberger, Kosloff, Kuprov, Luy, Schirmer, Schulte-Herbrüggen, Sugny, and Wilhelm]glaser2015training
  author author S. J. Glaser, author U. Boscain,
  author T. Calarco, author C. P. Koch, author
  W. Köckenberger, author
  R. Kosloff, author I. Kuprov, author B. Luy, author S. Schirmer, author T. Schulte-Herbrüggen, author D. Sugny,  and author F. K. Wilhelm, http://dx.doi.org/10.1140/epjd/e2015-60464-1 journal journal Eur. Phys. J. D volume 69, pages 279 (year 2015)NoStop
[Caneva et al.(2009)Caneva,
  Murphy, Calarco, Fazio,
  Montangero, Giovannetti, and Santoro]caneva2009optimal
  author author T. Caneva, author M. Murphy,
  author T. Calarco, author R. Fazio, author
  S. Montangero, author
  V. Giovannetti,  and author
  G. E. Santoro, 10.1103/PhysRevLett.103.240501 journal journal
  Phys. Rev. Lett. volume 103, pages
  240501 (year 2009)NoStop
[Goerz et al.(2011)Goerz,
  Calarco, and Koch]goerz2011quantum
  author author M. H. Goerz, author T. Calarco,   and author C. P. Koch, http://stacks.iop.org/0953-4075/44/i=15/a=154011 journal
  journal J. Phys. B volume 44,  pages 154011 (year 2011)NoStop
[Patsch et al.(2018)Patsch,
  Reich, Raimond, Brune,
  Gleyzes, and Koch]patsch2018fast
  author author S. Patsch, author D. M. Reich,
  author J.-M. Raimond, author M. Brune, author
  S. Gleyzes,  and author
  C. P. Koch, @noop  journal journal Physical Review A volume 97, pages 053418 (year
  2018)NoStop
[Sørensen et al.(2016)Sørensen, Pedersen, Munch, Haikka, Jensen, Planke, Andreasen, Gajdacz, Mølmer,
  Lieberoth, and Sherson]sorensen2016exploring
  author author J. J. W.  Sørensen, author M. K.  Pedersen, author M. Munch, author P. Haikka,
  author J. H. Jensen, author T. Planke, author
  M. G. Andreasen, author
  M. Gajdacz, author K. Mølmer, author A. Lieberoth,  and author J. F. Sherson, https://www.nature.com/articles/nature17620 journal journal Nature volume 532, pages
  210 (year 2016)NoStop
[Poyatos et al.(1996)Poyatos, Cirac, and Zoller]poyatos1996quantum
  author author J. F. Poyatos, author J. I. Cirac,
   and author P. Zoller, https://link.aps.org/doi/10.1103/PhysRevLett.77.4728 journal
  journal Phys. Rev. Lett. volume 77,  pages 4728 (year 1996)NoStop
[Kraus et al.(2008a)Kraus, Büchler,
  Diehl, Kantian, Micheli,  and Zoller]kraus2008preparation
  author author B. Kraus, author H. P. Büchler, author S. Diehl,
  author A. Kantian, author A. Micheli,  and author P. Zoller, 10.1103/PhysRevA.78.042307 journal journal Phys.
  Rev. A volume 78, pages 042307
  (year 2008a)NoStop
[Verstraete et al.(2009)Verstraete, Wolf, and Cirac]verstraete2009quantum
  author author F. Verstraete, author M. M. Wolf,  and author J. I. Cirac, http://dx.doi.org/10.1038/nphys1342 journal
  journal Nat. Phys. volume 5,  pages 633 (year 2009)NoStop
[Plenio et al.(1999)Plenio,
  Huelga, Beige, and Knight]plenio1999cavity
  author author M. B. Plenio, author S. F. Huelga,
  author A. Beige,  and author P. L. Knight, 10.1103/PhysRevA.59.2468 journal journal Phys.
  Rev. A volume 59, pages 2468
  (year 1999)NoStop
[Benatti et al.(2003)Benatti, Floreanini, and Piani]benatti2003environment
  author author F. Benatti, author R. Floreanini,
   and author M. Piani, 10.1103/PhysRevLett.91.070402 journal journal Phys. Rev. Lett. volume 91, pages 070402 (year 2003)NoStop
[Vacanti and Beige(2009)]vacanti2009cooling
  author author G. Vacanti and author A. Beige, http://stacks.iop.org/1367-2630/11/i=8/a=083008
  journal journal New Journal of Physics volume 11, pages 083008 (year
  2009)NoStop
[Wolf et al.(2011)Wolf,
  Chiara, Kajari, Lutz, and  Morigi]wolf2011entangling
  author author A. Wolf, author G. D. Chiara,
  author E. Kajari, author E. Lutz,  and author
  G. Morigi, http://stacks.iop.org/0295-5075/95/i=6/a=60008 journal
  journal Europhys. Lett. volume 95,  pages 60008 (year 2011)NoStop
[Gonzalez-Tudela et al.(2011)Gonzalez-Tudela, Martin-Cano, Moreno,
  Martin-Moreno, Tejedor, and Garcia-Vidal]gonzalez2011entanglement
  author author A. Gonzalez-Tudela, author D. Martin-Cano, author E. Moreno,
  author L. Martin-Moreno, author C. Tejedor,  and author F. J. Garcia-Vidal, 10.1103/PhysRevLett.106.020501 journal journal Phys. Rev. Lett. volume 106, pages 020501 (year 2011)NoStop
[Cho et al.(2011)Cho,
  Bose, and Kim]cho2011optical
  author author J. Cho, author S. Bose,  and  author M. S. Kim, 10.1103/PhysRevLett.106.020504 journal journal Phys. Rev. Lett. volume 106, pages 020504 (year 2011)NoStop
[Kastoryano et al.(2011)Kastoryano, Reiter, and Sørensen]kastoryano2011dissipative
  author author M. J. Kastoryano, author F. Reiter,
   and author A. S. Sørensen, 10.1103/PhysRevLett.106.090502 journal journal Phys. Rev. Lett. volume 106, pages 090502 (year
  2011)NoStop
[Rao Bhaktavatsala and Mølmer(2013)]bhaktavatsala2013dark
  author author D. D. Rao Bhaktavatsala and author K. Mølmer, 10.1103/PhysRevLett.111.033606 journal journal Phys. Rev. Lett. volume 111, pages 033606 (year
  2013)NoStop
[Habibian et al.(2014)Habibian, Zippilli, Illuminati, and  Morigi]habibian2014stationary
  author author H. Habibian, author S. Zippilli,
  author F. Illuminati,  and  author G. Morigi, 10.1103/PhysRevA.89.023832 journal journal Phys. Rev. A volume 89, pages 023832 (year 2014)NoStop
[Taketani et al.(2014)Taketani, Fogarty, Kajari, Busch, and Morigi]fogarty2014quantum
  author author B. G. Taketani, author T. Fogarty,
  author E. Kajari, author T. Busch,  and author
  G. Morigi, 10.1103/PhysRevA.90.012312 journal journal Phys.
  Rev. A volume 90, pages 012312
  (year 2014)NoStop
[Bentley et al.(2014)Bentley, Carvalho, Kielpinski, and  Hope]bentley2014detection
  author author C. D. B.  Bentley, author A. R. R.  Carvalho, author D. Kielpinski,  and author J. J. Hope, 10.1103/PhysRevLett.113.040501 journal journal Phys. Rev. Lett. volume 113, pages 040501 (year
  2014)NoStop
[Morigi et al.(2015)Morigi,
  Eschner, Cormick, Lin,
  Leibfried, and Wineland]morigi2015dissipative
  author author G. Morigi, author J. Eschner,
  author C. Cormick, author Y. Lin, author
  D. Leibfried,  and author
  D. J. Wineland, 10.1103/PhysRevLett.115.200502 journal journal
  Phys. Rev. Lett. volume 115, pages
  200502 (year 2015)NoStop
[Aron et al.(2016)Aron,
  Kulkarni, and Türeci]aron2016photon
  author author C. Aron, author M. Kulkarni,   and author H. E. Türeci,  10.1103/PhysRevX.6.011032 journal journal Phys. Rev. X volume 6, pages
  011032 (year 2016)NoStop
[Reiter et al.(2016)Reiter,
  Reeb, and Sørensen]reiter2016scalable
  author author F. Reiter, author D. Reeb,  and  author A. S. Sørensen,  10.1103/PhysRevLett.117.040501 journal
  journal Phys. Rev. Lett. volume 117,  pages 040501 (year 2016)NoStop
[Cormick and Morigi(2012)]cormick2012structural
  author author C. Cormick and author G. Morigi, 10.1103/PhysRevLett.109.053003 journal journal Phys. Rev. Lett. volume 109, pages 053003 (year
  2012)NoStop
[Diehl et al.(2008)Diehl,
  Micheli, Kantian, Kraus,
  Büchler, and Zoller]diehl2008quantum
  author author S. Diehl, author A. Micheli,
  author A. Kantian, author B. Kraus, author
  H.-P. Büchler,  and author
  P. Zoller, 10.1038/nphys1073 journal journal Nat. Phys.  volume 4, pages 878 (year
  2008)NoStop
[Habibian et al.(2013)Habibian, Winter, Paganelli, Rieger, and Morigi]habibian2013bose
  author author H. Habibian, author A. Winter,
  author S. Paganelli, author H. Rieger,  and author
  G. Morigi, 10.1103/PhysRevLett.110.075304 journal journal
  Phys. Rev. Lett. volume 110, pages
  075304 (year 2013)NoStop
[Pastawski et al.(2011)Pastawski, Clemente, and Cirac]pastawski2011quantum
  author author F. Pastawski, author L. Clemente,
   and author J. I. Cirac,  10.1103/PhysRevA.83.012304 journal journal Phys. Rev. A volume 83, pages 012304 (year 2011)NoStop
[Mirrahimi et al.(2014)Mirrahimi, Leghtas, Albert, Touzard, Schoelkopf, Jiang, and  Devoret]mirrahimi2014dynamically
  author author M. Mirrahimi, author Z. Leghtas,
  author V. V. Albert, author S. Touzard, author
  R. J. Schoelkopf, author
  L. Jiang,  and author
  M. H. Devoret, 10.1088/1367-2630/16/4/045014 journal journal
  New J. Phys. volume 16, pages 045014
  (year 2014)NoStop
[Reiter et al.(2017)Reiter,
  Sørensen, Zoller, and Muschik]reiter2017dissipative
  author author F. Reiter, author A. S. Sørensen, author P. Zoller,
   and author C. Muschik, 10.1038/s41467-017-01895-5 journal journal Nat. Commun. volume 8, pages
  1822 (year 2017)NoStop
[Kienzler et al.(2014)Kienzler, Lo, Keitch, de Clercq, Leupold, Lindenfelser,
  Marinelli, Negnevitsky, and Home]kienzler2014quantum
  author author D. Kienzler, author H.-Y. Lo,
  author B. Keitch, author L. de Clercq, author
  F. Leupold, author F. Lindenfelser, author M. Marinelli, author V. Negnevitsky,  and author J. Home, 10.1126/science.1261033
  journal journal Science , pages
  1261033 (year 2014)NoStop
[Krauter et al.(2011)Krauter, Muschik, Jensen, Wasilewski, Petersen, Cirac, and  Polzik]krauter2011entanglement
  author author H. Krauter, author C. A. Muschik, author K. Jensen,
  author W. Wasilewski, author J. M. Petersen, author
  J. I. Cirac,  and author
  E. S. Polzik, 10.1103/PhysRevLett.107.080503 journal journal
  Phys. Rev. Lett. volume 107, pages
  080503 (year 2011)NoStop
[Lin et al.(2013)Lin,
  Gaebler, Reiter, Tan,
  Bowler, Sørensen, Leibfried, and Wineland]lin2013dissipative
  author author Y. Lin, author J. Gaebler,
  author F. Reiter, author T. Tan, author
  R. Bowler, author A. Sørensen, author D. Leibfried,  and author D. Wineland, https://www.nature.com/articles/nature12801 journal journal Nature volume 504, pages
  415 (year 2013)NoStop
[Shankar et al.(2013)Shankar, Hatridge, Leghtas, Sliwa, Narla, Vool, Girvin,
  Frunzio, Mirrahimi, and Devoret]shankar2013autonomously
  author author S. Shankar, author M. Hatridge,
  author Z. Leghtas, author K. Sliwa, author
  A. Narla, author U. Vool, author S. M.  Girvin, author L. Frunzio, author M. Mirrahimi,
   and author M. H. Devoret,  10.1038/nature12802 journal journal Nature volume 504, pages
  419 (year 2013)NoStop
[Kimchi-Schwartz et al.(2016)Kimchi-Schwartz, Martin, Flurin,
  Aron, Kulkarni, Tureci, and  Siddiqi]kimchi2016stabilizing
  author author M. E. Kimchi-Schwartz, author L. Martin, author E. Flurin,
  author C. Aron, author
  M. Kulkarni, author
  H. E. Tureci,  and author
  I. Siddiqi, 10.1103/PhysRevLett.116.240503 journal journal
  Phys. Rev. Lett. volume 116, pages
  240503 (year 2016)NoStop
[Syassen et al.(2008)Syassen, Bauer, Lettner, Volz, Dietze, García-Ripoll,
  Cirac, Rempe, and Dürr]syassen2008strong
  author author N. Syassen, author D. M. Bauer,
  author M. Lettner, author T. Volz, author
  D. Dietze, author J. J. García-Ripoll, author J. I. Cirac, author G. Rempe,  and author S. Dürr, 10.1126/science.1155309
  journal journal Science volume 320, pages 1329 (year
  2008)NoStop
[Schindler et al.(2013)Schindler, Müller, Nigg, Barreiro, Martinez, Hennrich, Monz, Diehl, Zoller, and Blatt]schindler2013quantum
  author author P. Schindler, author M. Müller, author D. Nigg,
  author J. T. Barreiro, author E. A. Martinez, author
  M. Hennrich, author
  T. Monz, author S. Diehl, author P. Zoller,  and author R. Blatt, http://dx.doi.org/10.1038/nphys2630
  journal journal Nat. Phys. volume 9, pages 361 (year 2013)NoStop
[Barreiro et al.(2011)Barreiro, Müller, Schindler,
  Nigg, Monz, Chwalla,
  Hennrich, Roos, Zoller, and  Blatt]barreiro2011open
  author author J. T. Barreiro, author M. Müller, author P. Schindler, author D. Nigg,
  author T. Monz, author
  M. Chwalla, author M. Hennrich, author C. F.  Roos, author P. Zoller,  and author R. Blatt, https://www.nature.com/articles/nature09801 journal journal Nature volume 470,  pages 486 (year 2011)NoStop
[Sobel'man(1972)]sobel'man1967introduction
  author author I. I. Sobel'man, @noop  title Introduction to the
  theory of atomic spectra (publisher Pergamon Press,  year 1972)NoStop
[Woodgate(1999)]woodgate1970elementary
  author author G. K. Woodgate, @noop  title Elementary atomic
  structure (publisher Oxford University Press, year 1999)NoStop
[Gardiner and Zoller(2004)]gardiner2004quantum
  author author C. W. Gardiner and author P. Zoller, @noop  title Quantum noise: a
  handbook of Markovian and non-Markovian quantum stochastic methods with
  applications to quantum optics, series Springer Series in
  Synergetics, Vol. volume 56 (publisher
  Springer-Verlag, address Berlin, Heidelberg, year
  2004)NoStop
[Reiter and Sørensen(2012)]reiter2012effective
  author author F. Reiter and author A. S. Sørensen, 10.1103/PhysRevA.85.032111 journal journal Phys. Rev. A volume
  85, pages 032111 (year 2012)NoStop
[Ozeri et al.(2007)Ozeri,
  Itano, Blakestad, Britton,
  Chiaverini, Jost, Langer,
  Leibfried, Reichle, Seidelin,
  Wesenberg, and Wineland]ozeri2007errors
  author author R. Ozeri, author W. M. Itano,
  author R. B. Blakestad, author J. Britton, author
  J. Chiaverini, author
  J. D. Jost, author
  C. Langer, author D. Leibfried, author R. Reichle, author S. Seidelin, author J. H.  Wesenberg,  and author D. J.  Wineland, 10.1103/PhysRevA.75.042329
  journal journal Phys. Rev. A volume 75, pages 042329 (year
  2007)NoStop
[Johnson(2014)]johnson2014nlopt
  author author S. G. Johnson, @noop  title The nlopt
  nonlinear-optimization package, howpublished Available at
  http://ab-initio.mit.edu/nlopt (year 2014)NoStop
[Rowan(1990)]rowan1990functional
  author author T. H. Rowan, title Functional stability analysis of
  numerical algorithms, @noop  Ph.D. thesis, school
  University of Texas at Austin (year 1990)NoStop
[Gaebler et al.(2016)Gaebler, Tan, Lin, Wan,
  Bowler, Keith, Glancy,
  Coakley, Knill, Leibfried,  and Wineland]gaebler2016high
  author author J. P. Gaebler, author T. R. Tan,
  author Y. Lin, author
  Y. Wan, author R. Bowler, author A. C.  Keith, author S. Glancy, author K. Coakley,
  author E. Knill, author D. Leibfried,  and author D. J. Wineland, 10.1103/PhysRevLett.117.060505 journal journal
  Phys. Rev. Lett. volume 117, pages
  060505 (year 2016)NoStop
[Goetz et al.(2016)Goetz,
  Merkel, Karamatskou, Santra,  and Koch]GoetzPRA16b
  author author R. E. Goetz, author M. Merkel,
  author A. Karamatskou, author R. Santra,  and author
  C. P. Koch, 10.1103/PhysRevA.94.023420 journal journal Phys.
  Rev. A volume 94, pages 023420
  (year 2016)NoStop
[Monz et al.(2011)Monz,
  Schindler, Barreiro, Chwalla,
  Nigg, Coish, Harlander,
  Hänsel, Hennrich, and Blatt]monz2011fourteen
  author author T. Monz, author P. Schindler,
  author J. T. Barreiro, author M. Chwalla, author
  D. Nigg, author W. A. Coish, author M. Harlander, author W. Hänsel, author M. Hennrich,  and author R. Blatt, 10.1103/PhysRevLett.106.130506
  journal journal Phys. Rev. Lett. volume 106, pages 130506 (year
  2011)NoStop
[Note1()]Note1
  note Typically of the order of one hundred propagations are
  necessary to converge the optimizationNoStop
[Kraus et al.(2008b)Kraus, Büchler,
  Diehl, Kantian, Micheli,  and Zoller]KrausPRA08
  author author B. Kraus, author H. P. Büchler, author S. Diehl,
  author A. Kantian, author A. Micheli,  and author P. Zoller, 10.1103/PhysRevA.78.042307 journal journal Phys.
  Rev. A volume 78, pages 042307
  (year 2008b)NoStop
[Albert(2017)]AlbertPhD
  author author V. V. Albert, title Lindbladians with multiple steady
  states: theory and applications, @noop  Ph.D. thesis, school Yale University (year 2017), note
  arXiv:1802.00010NoStop
[Doucet et al.(2018)Doucet,
  Reiter, Ranzani, and Kamal]Doucet18
  author author E. Doucet, author F. Reiter,
  author L. Ranzani,  and  author A. Kamal, @noop 
  journal journal arXiv:1810.03631  (year 2018)NoStop
[de Vega and Alonso(2017)]deVegaRMP17
  author author I. de Vega and author D. Alonso, 10.1103/RevModPhys.89.015001 journal journal Rev. Mod. Phys. volume
  89, pages 015001 (year 2017)NoStop
[Reich et al.(2015)Reich,
  Katz, and Koch]ReichSciRep15
  author author D. M. Reich, author N. Katz,  and  author C. P. Koch, 10.1038/srep12430 journal journal Sci.
  Rep. volume 5, pages 12430 (year 2015)NoStop
[Huelga et al.(2012)Huelga,
  Rivas, and Plenio]HuelgaPRL12
  author author S. F. Huelga, author A. Rivas,   and author M. B. Plenio,  10.1103/PhysRevLett.108.160402 journal
  journal Phys. Rev. Lett. volume 108,  pages 160402 (year 2012)NoStop



