Steady state engineering of a two-level system by the mixed-state inverse engineering scheme

The mixed-state inverse engineering scheme is a control scheme used for engineering the quantum state of a driven open quantum system from an initial steady state to a final steady state. In this paper, we present an analytical study of this scheme applied to the driven two-level model coupled to a heat reservoir. Typically, when the purity of the quantum state varies, incoherent control techniques are required for mixed-state engineering. However, we show that for both Markovian and non-Markovian dynamics, coherent control protocols can transfer the quantum state into the target state. This simplification comes at a cost, as the evolution of the quantum state must be limited to restricted conditions, resulting in special trajectories in its Hilbert space that connect the initial and target states.


The steady state inverse engineering
In this section, we briefly review the mixed-state inverse engineering scheme 22 , and apply it in the steady state engineering.
Let us consider an open quantum system whose dynamics is governed by the following time-convolutionless master equation in the superoperator description, where Lc (t) is a time-dependent control Liouvillian superoperator and |ρ(t)�� is the density matrix vector.A dynamical invariant Î(t) of such open quantum system is defined as a superoperator which satisfies 25 Î(t) can be formulated in Jordan canonical form with biorthogonal eigenvectors which satisfies 20,26 with i = 0, 1, ..., n α − 1 .|D (i) α (t)�� and ��E (i) α (t)| are the right and left eigenvectors of Î(t) in the Hilbert-Schmidt space with the eigenvalue α .We also assume that the α-th Jordan block is n α -dimensional.Generally, Î(t) can be used to generate an arbitrary solution of the time-convolutionless master equation.The general solution of Eq. (1) can be written as 14 |� α (t)�� is a right vector in the α th Jordan block, which can be written as η α (t) is a complex phase defined as where ��� α (t)| is a left vector of the α-th Jordan block, which satisfies ��� β (t)|� α (t)�� = δ αβ .a α is time-inde- pendent coefficient, which is given by a α = ��� α (0)|ρ(0)��.
where r = (r 1 , r 2 , ..., r N 2 −1 ) is the generalized Bloch vector, and |I�� is the right vector corresponding to an N × N identical operator.
To formulate feasible control protocols, we impose that the control Liouvillians take the form as in which H(t) is the control Hamiltonian, and is the control Lindbladian.The Lindblad operators L α (N α ) are related to some incoherent control parameters N α , such as the decoherence rates and the temperatures of the environments, which will be used in the later dis- cussion on the control Liouvillian.Since the Hamiltonian H c (t) can be expanded in terms of SU (N) Hermitian generators {T k } , i.e., we might choose {c k (t)} as the coherent control parameters.In this way, the control Liouvillians can always present feastible control protocols.Substituting Lc (t) and Î(t) into Eq. ( 2) we can express the control parameters {c k , N α } as a function of the generalized Bloch vector and its derivative {r µ , ∂ t r µ } .On the other hand, we impose that the control Liouvillian is always the same as the reference Liouvillian at the initial and final moment, which leads to boundary conditions for {r µ , ∂ t r µ } .By utilizing the control parameters {c k , N α } and setting proper boundary conditions for {r µ , ∂ t r µ } , the open quantum system can be transferred from the initial steady state into the target steady state along an exact trajectory given by |� 0 (t)�� .Here, we do not require Lc (0), Î(0)]=[ Lc (t f ), Î(t f ) = 0 , because of the difficulties in engineering parameters in the Lindbladian superoperator.

The mixed-state inverse engineering scheme with the incoherent control
Consider a two-level system with a reference Hamiltonian H 0 (t) = ω 0 2 σ z + H c (t), where ω 0 is frequency dif- ference, H c (t) is general control Hamiltonian, and σ z is z component of the Pauli matrix.The two-level system couples to a bosonic heat reservoir at finite temperature T 0 .In the interaction picture respecting to ω 0 2 σ z , the dynamics of the two-level system is governed by a following phenomenological Markovian master equation 28 , in which γ denotes the decoherence strength , and N 0 = [exp(ω 0 /T 0 ) − 1] −1 is the mean excitation number.Throughout this paper, we assume the dimensionless units = k B = 1 .The reference Hamiltonian is expressed as H I c0 (t) = � 0 (t)σ x with the time-dependent control field strength � 0 (t).
Here, we use the "bra-ket" notation for superoperator 29 .On the one hand, the density matrix can be reshaped into 1 × 4 density matrix vector.The density matrix vector can be written as |ρ(t)�� = (ρ 00 (t), ρ 01 (t), ρ 10 (t), ρ 11 (t)) T , where ρ ij (t) = �i|ρ(t)|j� with i = 0, 1 .|0� and |1� are the eigenstates of σ z .On the other hand, the reference Liouvillian superoperator L0 (t) (Eq. ( 9) can be expressed as a 4 × 4 matrix, The steady state of two-level system is given by the condition L0 (t)|ρ 0 (t)�� = 0 , which results in with the normalization factor p = (2N 0 + 1) 2 + 2(� 0 /γ ) 2 .Since the corresponding left eigenstate fulfills L † 0 (t)|ρ ′ 0 (t)�� = 0 , one can obtain ��ρ ′ 0 | = ��I| = (1, 0, 0, 1) , which leads to ��ρ ′ 0 |ρ 0 �� = 1.In what follows, we parameterize the eigenvalue and eigenvector of the invariant Î(t) .For a two-level system, the eigenvector can be parameterized as, (7) where r i is the i-th component of the Bloch vector, and σ i is i-component of the Pauli operators.And we set s as an arbitrary constant with units of energy.Thus Î(t) reads Assume that the control Liouvillian Lc (t) can be written as with H I c (t) = �(t)σ z + �(t)σ x , and and N = [exp(ω 0 /T) − 1] −1 .Here we also assume that the temperature T of the heat bosonic reservoir is tunable, which is the incoherent control parameter in Eq. (8).Following the standard procedure, the control parameters and the boundary conditions of the inverse engineering protocol can be identified.Substituting the dynamical invariant Î(t) and the control Liouvillian Lc (t) into Eq. (1),we have with r = r 2 x + r 2 y + r 2 z .Since our purpose is to accelerate the adiabatic steady state process governed by L0 (t) , we only require that the initial state and the target state are same for the adiabatic engineering and the inverse engineering, but the trajectory from the initial state and the target state can be chosen freely.To make |ρ 0 (t)�� and |̺(t)�� equal at the beginning and the end of the control process, we impose the initial boundary conditions and the final boundary conditions Substituting the above boundary conditions into Eq.( 15), it yields The trajectory of the quantum state is determined by parameterizing the Bloch vector, which can be designed according to control tasks.For the adiabatic trajectory given by the instantaneous steady state Eq. ( 11), the components of the Bloch vector read ( 12) where c is a constant with units of energy.This profile is corresponding to turning down the reference control field from c to 0 smoothly.Since ∂ t � 0 (t)| t=t 0 = 0 and ∂ t � 0 (t)| t=t f = 0 , the boundary conditions Eq. ( 16) are satisfied.Taking Eq. ( 17) into Eq.( 15), we can obtain the control parameters in analytic expression.
As mentioned in the adiabatic theorem, when the control field � 0 (t) is varied slowly enough, the quantum state of the two-level system will track the trajectory of the instantaneous steady state into the target state 27 .Here we denote this trajectory as "an adiabatic trajectory".Since r x (t) = 0 and ∂ t r x (t) = 0 , the detuning �(t) absents for the adiabatic trajectory tracking.Therefore, if we select the adiabatic trajectory, the structure of the control Liouvillian Lc (t) is completely the same as the reference Liouvillian L0 (t).
The fidelities with respect to the instantaneous steady state for both protocols are plotted in Fig. 1a.The numeral results illustrate that the inverse engineering protocol forbids the transition from the steady state ρ 0 (t) and the other part of the Hilbert-Schmidt space.When the adiabatic engineering protocol is considered, the population on the target steady state only occurs in the adiabatic limit.On the other hand, the control parameters in the control Liouvillian L(t) , i.e., the temperature of the bosonic heat reservoir T(t) and the control field �(t) are plotted in Fig. 1b,c.For t f = 10ω −1 0 , the control parameters are very close to the adiabatic case (see blue dash lines in Fig. 1b,c).With the decrease of t f , the control parameters deviate from the adiabatic engineering protocol's, and the target state can still be reached for arbitrary t f ( green solid line1 in Fig. 1a).

Coherent control protocols
The coherent control protocol via r x (t)   In this section, we show that coherent control protocols is enough to drive the open quantum system into the target steady state.What we should do is to set control absents, and there is only a coherent control field with the strength �(t) and the detuning �(t) in the control protocol.
Here, we select a designed r x to satisfy the constraint condition N(t) = N 0 .According to Eq. (15c), r x has to satisfy the following differential equation, with We may consider Eq. ( 18) as a differential equation about r 2 x .Solving this equation is equivalent to select a trajectory with the particular x-component of the Bloch vector.For formally solving the differential equation about r x , we introduce a new variant rx = r x e (2N0+1)γ t/2 , which satisfies At last, we obtain the formal solution about r 2 x , Since r x is one component of the Bloch vector, the following restrictions have to be satisfied: (i) as r x is always real, t 0 dτ �(τ )e (−2N0+1)γ (t−τ ) ≤ 0 has to be ensured; (ii) 0 ≤ r 2 x + r 2 y + r 2 z ≤ 1 .Those restrictions illustrate that the control parameters in Lc (t) must be chosen carefully to make sure that the Bloch vector corresponds to a reasonable quantum state 30 .
By introducing two new variants, ry = r y e (2N0+1)γ t/2 , rz = r z e (2N0+1)γ t , we can transform Eq. ( 18) into the following form, Integrating above equation with time, it yields By partial of the last term in above equation, one finds with Turning back into the Bloch vector, we obtain the relation with r(t) = r 2 x (t) + r 2 y (t) + r 2 z (t).Now let us consider the nonadiabatic control protocol from the initial state |ρ 0 (t f )�� to the target state |ρ 0 (t f )�� .We set the ansatz for i = y, z .And, r x (t) can be obtained by the exact solution Eq. ( 20), or by solving the differential equation Eq. ( 18) numerically.It is worth to notice that under such parameterized protocol the boundary conditions for y and z components of the bloch vector are satisfied automatically, but the boundary condition of r x (t f ) is not.As shown in Eq. ( 21), if r x (t f ) = 0 , it requires (18) ∂ t r 2 x + (2N 0 + 1)γ r 2 x = −�(t), This means that t f can not be chosen arbitrarily.The final x-component of the bloch vector r x (t f ) is related to the decoherence strength γ and the z component of the bloch vector r z (t) .On the one hand, we can get control period t f numerically by optimization methods, such as gradient, conjugate direction and quasi-Newton methods 31 .
On the other hand, since the trajectory of the quantum state can be chosen optionally, one may select a proper components r y (t) and r z (t) to ensure the final boundary condition 32 .
As an example, we select the parameters as follows: the strength of the initial control field � c = 2ω 0 , the deco- herence strength γ = 2ω 0 , the temperature of the reservoir T 0 = 0 , and t f = 10ω −1 0 .The control field �(t) and the detuning �(t) are presented in Fig. 2a.As prediction, the control field and the detuning are turned off at the end of the control process.We also present the fidelity with respect to a reference state ρ r (t) = 1 2 I + r y (t)σ y + r z (t)σ z , where r y (t) and r z (t) are given by Eq. ( 22).As shown in Fig. 2b (the green dash line), the quantum state deviate the trajectory of the reference state at beginning, which is due to the participation of r x (t) in the dynamical evo- lution; two trajectories converge to the target state at the final moment of time, which attributes to r x (t f ) = 0 at the end of the control process.Comparing with the adiabatic engineering protocol (the blue solid line), the final fidelity for the inverse engineering protocol with respect to the target steady state for the inverse engineering is perfect, far better than the adiabatic one.

The coherent control protocol via r y (t)
In this part, we show how to drive the quantum state into the target state by a coherent control protocol without the detuning �(t) .As shown in Eq. ( 15), when r x (t) = 0 and ∂ t r x (t) = 0 , the detuning �(t) equals to zero.Therefore, we consider a modification on r y (t) .As required for the coherent control protocol, the main excitation number of the heat reservoir is constant, i.e., N(t) = N 0 .And we set that r x (t) = 0 and ∂ t r x (t) = 0 at ∀ t .The differential equation about r 2 y (t) can be obtained by considering Eq. (15c) with At this time, the only control parameter used in the control Liouvillian Lc (t) is the coherent control field �(t) , which reads The control task is still that the control field � 0 (t) in the reference Liouvillian L0 (t) decreases from c to 0. The numerical results are shown in Fig. 3, in which we present the trajectory of the quantum state in the Bloch sphere, the control parameters, and the fidelity between the quantum state ρ(t) and the reference state ρ r (t) .As shown in Fig. 3a, when the inverse mixed state engineering scheme is used (red solid line), the quantum system is driven into the target state at the end of the control process, which is also verified by the fidelity in Fig. 3c.In contrast, the quantum state engineered by the adiabatic engineering protocol can not reach the target state.What (23)
Vol:.( 1234567890 www.nature.com/scientificreports/ is more, the detuning for this protocol is always zero (see Fig. 3b), which means the control Liouvillian Lc (t) has the same form with the reference Liouvillian L0 (t) even for the coherent control protocol.

The coherent control protocols for the non-markovian dynamics
The control parameters Now, we are in the position to study the non-Markovian case.Consider a two-level system with frequency ω 0 driven by an external laser of frequency ω L .There is a detuning � = ω 0 − ω L between the two-level system and the external laser.The two-level atom is embedded in a bosonic reservoir at zero temperature.The reservoir has a Lorentzian spectral density, J(ω) = γ 0 2π 2 (ω−ω 0 +δ) 2 + 2 , where δ = ω 0 − ω c is the detuning of ω c to ω 0 , ω c is the center frequency of the cavity, and is the spectral width of the reservoir 28 .The parameter γ 0 is the decoherence strength of the system in the Markovian limit with a flat spectrum.The exact non-Markovian master equation can be written as 33 with the effective Hamiltonian H I 0c (t) = s 0 (t)σ + σ − + � 0 (t)σ x , where s 0 (t) and � 0 (t) are the Lamb shift and the renormalized driving field respectively.The time-dependent decoherence strength Ŵ 0 (t) describes the dissipative non-Markovian dynamics due to the interaction between the system and reservoir.All these time-dependent coefficients can be given explicitly as follows with where k(t) = exp(−( + 2i� − iδ)t/2) and d = ( − iδ) 2 − 2γ 0 .

The coherent control protocol via r x
Our purpose is to find a set of feasible control parameters to ensure that the quantum state can track the trajectory we designed.Here, we consider following trajectory.Firstly, the y and z components of the trajectory are set to be which are the y and z components of the Bloch vector given by the instantaneous steady state |ρ 0 (t)�� .However, we can not select r x (t) as the x-component of the Bloch vector given by |ρ 0 (t)�� , due to r 2 + r z 2 + 2 r z = 0 which results in meaninglessness control parameters (see Eq. ( 33)).Moreover, it is difficult to tune the time-dependent decoherence strength Ŵ(t) properly in the experiment.Therefore, the decoherence strength can not be tuned artificially, i.e., Ŵ(t) = Ŵ 0 (t) .Thus, according to Eq. (33c), it yields a differential equation about r 2 x (t) , which reads Beside with the initial condition r x (0) = 2�� 0 (0)/(� 2 + � 0 (0) 2 ) , we can determine the x-component of the trajectory, i.e.,r x (t).
Consider that the reference renormalized control field � 0 (t) tunes up from 0 to a finite strength c .The The numeral results are presented in Fig. 4. As shown in Fig. 4a, for the inverse engineering scheme (the blue solid line), the quantum state of the open two-level system strictly follows the trajectory we designed.Instead, for the adiabatic engineering protocol (the red dash line), i.e., Ŵ = Ŵ 0 , = 0 , and s = s 0 , there is severe fidelity loss.On the other side, the control field �(t) and the detuning s(t) deviate from � 0 (t) and s 0 (t) respectively, but they coincide with each other at the beginning and the end (see Fig. 4b,c).This attributes to ∂ t � 0 (0) = 0 and ∂ t � 0 (t f ) = 0 .And Ŵ(t) and s(t) become stable at the end, which are illustrated in Fig. 4c, d.If t f is not large enough, the control field �(t f ) and the detuning s(t f ) can not coincide with � 0 (t f ) and s 0 (t f ) , but the fidelity never get loss.As we required, the spectral density can not be tuned, which requires Ŵ(t) = Ŵ 0 (t) all the time.Numerical results in Fig. 4d confirm this requirement, (29) Ŵ(t) = − ∂ t r x r x + ∂ t r y r y + ∂ t r z r z r x 2 + r y 2 + 2 r z 2 + 2 r z . (34) which shows that it is not necessary to engineer the reservoir for obtaining perfect fidelity.At last, s(t) contains two different contributions.One is the Lamb Shift s 0 (t) , and the other comes form the detuning of the control field s ′ (t) .It is easy to obtain the detuning via s ′ (t) = s(t) − s 0 (t) , which is illustrated by the green dot line in Fig. 4c.

The coherent control protocol via r y
As shown in Eq. ( 33), the control parameters are related with r −1 y , which cause singular points of the control parameter if r y = 0 .This can be avoided by picking up a desired trajectory of r y .Here, we consider that r x and r z are tracking the instantaneous steady state, And r y satisfies which corresponds to the constraint condition Ŵ(t) = Ŵ 0 (t).
The control task is to increase the control field � 0 (t) in the reference Liouvillian L0 (t) from 0 to c .The profile of � 0 (t) is chosen as The parameters for the non-Markovian dynamics are assumed to be t f = 1.25/ω 0 , � c = ω 0 , γ 0 = 10ω 0 , � = 0.375γ 0 , = 0.2ω 0 , and δ = 0.1γ 0 .With this setting, the informa- tion will flow back into the system from the reservoir.The numerical results are presented in Fig. 5.As we predict, the x and z components of the Bloch vector track the trajectory of the instantaneous steady state ρ 0 (t) (see Fig. 5a), and the y-component of the Bloch vector does not cross the zero point (see Fig. 5b), which leads to reasonable control parameters as shown in Fig. 5c.
In Fig. 6a, we plot the fidelity between quantum states governed by the control Liouvillian Lc (t) (blue solid line) (the reference Liouvillian L0 (t) (red dash line)) and the quantum state ρ r (t) = 1/2 I + r x (t)σ x + r y (t)σ y + r z (t)σ z with the designed trajectory given by Eqs. ( 36) and (37).The (36) The evolution of the fidelity of the inverse mixed state engineering protocol (the blue solid line) and the adiabatic engineering protocol (the red dash line), (b) the external control field strength, (c) the detuning, and (d) the decoherence strength) as a function of the dimensionless time t/t f .Parameters: γ 0 = ω 0 , = 0.5ω 0 , � = 0.5ω 0 , δ = 0.1ω 0 , � c = ω 0 , t f = 30/ω 0 .We set ω 0 = 1 as the unit of the other parameters.
quantum state engineered by the inverse engineering scheme always tracks the designed trajectory, but the final fidelity approach to 1 only in the long-time limit.Even though the quantum state arrived at the target state at the end of the adiabatic control process in the long-time limit, it still deviate the designed trajectory within the control process due to the oscillation of the deocherence strength.In other words, the adiabatic trajectory tracking can not be achieved for the non-Markovian dynamics.We also present the control parameters �(t) and s(t) with different control periods t f = 10ω −1 0 (Fig. 6b), t f = ω −1 0 (Fig. 6c), and t f = 0.1ω −1 0 (Fig. 6d).As we see in those figures, more fast to drive the quantum state into the target state is, stronger control fields are required.This elucidates the trade-off between speed and energy cost, namely, that instantaneous manipulation is impossible as it requires an infinite cost.
As we have illustrated by examples about the trajectory tracking control of the non-Markovian two-level system, Eq. ( 33) presents a map from the trajectory of the quantum state ρ(r x , r y , r z ) to the control parameters.By using these control parameters, the quantum system will be driven into the desired target state along an exact trajectory (r x , r y , r z ) in the Bloch sphere.There are three issues should be mentioned: firstly, when we consider the trajectory given by the instantaneous steady state, it is not difficult to verify that r 2 + r z 2 + 2 r z = 0 , which leads to all of the control parameters to be meaningless (see Eq. ( 33)).It tells us that the adiabatic trajectory given by the instantaneous steady state ρ 0 (t) can not be tracked by means of the adiabatic engineering protocol.At least, one of components of the Bloch vectors has to deviate from the adiabatic trajectory, which also verifies that the adiabatic evolution for the non-Markovian dynamics does not exist.Secondly, s(t) and �(t) ∝ r y (t) −1 , which require that r y can not be zero for this control protocol.When the dynamics is non-Markonian, r y in the adiabatic trajectory oscillates with time.Due to r y (t) ∝ Ŵ 0 (t) , it can not be avoided that r y (t) crosses the zero point when Ŵ 0 (t) = 0 .However, the inverse mixed state engineering protocol supports the selection of the tra- jectory of the quantum state in the Hilbert space.Therefore, we can select a trajectory whose y-component of the Bloch vector does not cross the zero point.Thirdly, since the differential equation Eq. ( 35) is about r 2 x (t) , the solution of Eq. ( 35) can not be negative.A suitable choice of r y (t) , r z (t) and t f is very essential for this inverse engineering protocol.Therefore, the optional nature of the trajectory choice provides a broader prospect for the optimal control method of STAs.

Conclusion
n this paper, we have demonstrated the effectiveness of the mixed-state inverse engineering scheme in speeding up the adiabatic steady state process of open two-level quantum systems.Using a control Liouvillian based on the same framework as the reference Liouvillian, we can drive the open quantum system into the target state along the adiabatic trajectory with perfect fidelity, although incoherent control is required.We have also shown that a pure coherent control protocol can be designed to drive the quantum state from the initial steady state to the target steady state.This approach can also be used for trajectory tracking control of non-Markovian systems by keeping the reservoir parameters invariant and finding a proper trajectory.
Our results demonstrate the potential of the mixed-state inverse engineering scheme for controlling open quantum systems with coherent control techniques, which is a future development trend in quantum control theory.Extending our ideas to infinite dimensional open quantum systems and open many-body quantum systems is another interesting line of research.Experimentally, our method can be realized in feasible models, such as cavity quantum electrodynamics systems 34 , superconducting circuits systems 35 , nitrogen-vacancy centers systems 36 , and even many-body and spin-chain models 37 .The coherent control schemes in our protocol can be adopted within a reasonable parameter regime used in experiments.If incoherent control is indispensable, reservoir engineering technology can be used to continuously vary the parameters of the environment 38,39 .Therefore, our control protocol can be tested in currently available experimental situations.

Figure 4 .
Figure 4. (a)The evolution of the fidelity of the inverse mixed state engineering protocol (the blue solid line) and the adiabatic engineering protocol (the red dash line), (b) the external control field strength, (c) the detuning, and (d) the decoherence strength) as a function of the dimensionless time t/t f .Parameters: γ 0 = ω 0 , = 0.5ω 0 , � = 0.5ω 0 , δ = 0.1ω 0 , � c = ω 0 , t f = 30/ω 0 .We set ω 0 = 1 as the unit of the other parameters.