Synthesizing and multiplexing autonomous quantum coherences

Quantum coherence is a crucial prerequisite for quantum technologies. Therefore, the robust generation, as autonomous as possible, of quantum coherence remains the essential problem for developing this ﬁeld. We consider a method of synthesizing and multiplexing quantum coherence from a spin systems without any direct drives only coupled to a bosonic baths. The previous studies in this ﬁeld have demonstrated that a back-action of the bath to the spin subsystem is important to generate it, however, it simultaneously gives a signiﬁcant limits to the generated coherence. We propose a viable approach with the bosonic bath that allows overcoming these limits by avoiding the destructive eﬀect of the back-action processes. Using this approach, we suggest advanced synthesis of the quantum coherence non-perturbatively in the spin-boson coupling parameters of multiple bosonic baths to increase and multiplex it for upcoming proof-of-principle experiments.


Introduction
Quantum coherence [1] is a significant and diverse subject of modern quantum physics, phase estimation and thermodynamics [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] and a crucial resource of contemporary quantum technology, specifically, quantum metrology [18,19], quantum communication [20,21], quantum simulators [22,23,24], energy harvesting [25,26], quantum thermodynamics [27,28,29], and quantum computing [30,31].A classical, external and strong coherent drive typically generates such quantum coherence as a superposition of energy states.Recently, it has been proposed that there might be a more autonomous alternative, the quantum coherence, from the coupling between a basic system, like a two-level system, and a thermal bath [32].It used a composite interaction between the two-level system and thermal bath; in one direction, the incoherent energy of the system pushed the bath coherently, but simultaneously it could receive that coherence of the bath.Both interactions must be present to obtain quantum coherence in a single two-level system without any external drive, just from a coherent interaction with a bath.It is therefore conceptually different from a coherence for a pair of two-level systems from thermal baths [33].It triggered further analysis [34,35,36,37,38]; however, it is still not a fully explored phenomenon, without a direct experimental test.For more extensive feasibility, other system-bath topologies generating and detecting more autonomous coherence have to be found.
Here, we present two crucial steps toward such experimental verifications, considering many separate two-level systems to push the bath coherently, many to receive quantum coherence in parallel and also more baths assisting the process in parallel.Advantageously, we split the single two-level system used in Ref. [32] to two separate (drive and output) ones.Using these allowed topologies providing autonomous quantum coherences without a back-action, we propose and study an autonomous synthetisation of coherence from many systems, multiplexing it to many systems and employing jointly different baths to generate the coherences.From the detailed analysis of these cases, we proved a significant result: many systems and baths could be used in parallel to obtain and broadcast autonomous quantum coherences in the experiments.
The paper is organized as follows.In Sec. 2, we propose a general method of the calculation of spin coherences in systems which contain a bosonic bath interacting with many spins and verify this method with the previously obtained results for single spin in Refs.[36,37,38].In Sec. 3, we apply the new method to the case of two spins, input and output, interacting separately with the bosonic bath.In Sec. 4, we extend the previous system to the M input spins and analyze the expression for coherence of single output spin as a function of M .Then, in Sec. 5 we examine the case of M input and two output spins and consider the correlation effects between the output spins.In Sec. 6, we explore the case of two bosonic baths coupled separately to M and N input spins, while the output spin is coupled to both baths.We develop a systematic scheme and calculate the coherence of the output spin in a general form, and then discuss the generalization of this problem to larger number of baths and output spins.Finally, in Sec. 7, we further generalize our method of calculation by substituting the output spin by an oscillator.In Sec. 8, we summarize our results, discuss the advantages and limits of the proposed mechanisms of the generation of the cohherence in the spin-bath systems.Technical details are presented in 4 Appendices.

Multi-spin interaction with thermal bath
First, we modify the model considered in the Refs.[36,37,38] to open more possibilities for the synthesization and multiplexing by separating a single two-level system into driving and receiving two-level systems.
We consider the Hamiltonian of the system H = H B + H S + H SB = H 0 + H SB .Here is the Hamiltonian of bosonic excitations with the spectrum Ω k > 0. The operators b k , b † k satisfy the canonical commutation relations [b k , b † q ] = δ kq , where δ kq is the Kronecker symbol.Furthermore, with ω j > 0 is the Hamiltonian of M + 1 spins, M driving the bath to get coherence there and the (M + 1)th receiving the coherence.Finally, describes the interaction of the spins with the bosonic system.Here, we have introduced the notation σ • n ≡ σ x n x + σ y n y + σ z n z , with the Pauli matrices σ x , σ y , σ z and vector of the coupling strength parameters n = (n x , n y , n z ).
We are interested in the reduced density matrix of the spin system, which can be obtained from the full canonical density matrix ρ ≡ e −βH /Z by tracing out over the bosonic degrees of freedom where Z is the partition function of the full system We evaluate the reduced density matrix using the following method.First we present the operator exponent in the form The operator U (τ ) satisfies the differential equation in the domain τ ∈ [0, β] with the initial condition where H SB (τ ) ≡ e τ H 0 H SB e −τ H 0 .The solution of the equation can be presented in the form of the chronologically ordered (in the imaginary Matsubara time τ ) exponent Using this result we rewrite the reduced density matrix as Here the averaging procedures over the spin (S) and bosonic (B) degrees of freedom read with being the partition functions of the free spin (Z S ) and boson (Z B ) subsystems, respectively.Note that the numerator of Eq. ( 9) can be presented as e −βH ef f , the denominator then transforms into Tr S [e −βH ef f ], see details in Ref. [39].However, in the current study we use another way by evaluating perturbatively the T τ -ordered exponent.The reduced density operator ρ S depends on the M + 1 spin degrees of freedom.Our idea is based on the observation that the bosonic bath effectively couples the spin degrees Self-induced spin coherence method from Refs.[38,37,36].The single spin "polarizes" the bosonic subsystem with spectral density function I(ξ) at temperature T via the f 1 σ z 1 term.The polarized bosonic bath generates indirectly the coherence in the same spin ⟨σ x 1 ⟩ via the f 2 σ x 1 term, i.e., as the back reaction to the spin system from the bosonic bath.(b) Externally induced spin coherence method.The first (input) spin polarizes the bosonic bath via the f 1 σ z 1 term.The polarized bosonic bath generates the coherence in the output spin ⟨σ x 2 ⟩ via the f 2 σ x 2 term, i.e., as the transfer of spin coherence through the bath to the second spin system. of freedom.Part of these coupling terms is responsible for the generation of a non-zero value of the σ x M +1 -operator (coherence), which can be calculated as The aforementioned spin-spin interaction terms correspond to the leading terms of the perturbation series for the reduced density matrix ρ S in the spin-boson coupling parameters and can be deduced from the following expression in the weak-coupling regime (see derivation in Appendix A) Here, we have introduced the bosonic spectral density function the function and τ -dependent multi-spin operator For a comparison, we first reconsider the case with the following single-spin Hamiltonian [38,37,36] It corresponds to the situation of one spin coupled to the bosonic thermal bath simultaneously via the σ z and σ x spin operators, i.e., n = (f 2 , 0, f 1 ).The corresponding system is depicted in Fig. 1 a).Using the result of Appendix A, we can write the leading spin-spin correlation term of the reduced spin density operator in this case In the first line of Eq. ( 19) we introduced the τ -dependent operators with j = x, y, z.Using the algebra of Pauli matrices σ j σ k = δ jk I + iε jkl σ l , where I is a 2 × 2 unit matrix and ε jkl is the Levi-Civita symbol, we evaluated σ x (τ ) = cosh(ωτ )σ x + i sinh(ωτ )σ y , σ y (τ ) = cosh(ωτ )σ y − i sinh(ωτ )σ x , σ z (τ ) = σ z , and then calculated the coherence ⟨σ x 1 ⟩ in the leading order of the perturbation theory This answer coincides with the previously obtained one [36,37,38].Note that the complex structure of the integrand is a result of the dynamical back reaction of the bosonic bath onto the spin system.This is a consequence of the special coupling of the spin system to the bosonic bath, containing both the σ x and σ z coupling terms.It allows self-induced coherence through the bath by the spin itself but also creates dynamical terms that limit the amount of coherence.In order to eliminate these dynamical terms we introduce two groups of spin systems, where the spins of the first (second) group interact with the bosonic bath only via the σ z (σ x ) coupling term.In this configuration the first group of spins influences the bosonic system as driving spins and then the affected bosonic bath generates the nonzero coherence in the second group of output spins.The simplest case of such systems is considered in the next section.
3 New two-spin method (M = 1) For further development and comparison, we propose the basic case n 1 = (0, 0, f 1 ), n 2 = (f 2 , 0, 0), with f 1 , f 2 ∈ R. The corresponding system is depicted in Fig. 1 b).For this case the spin-spin correlation term reads Using this result we obtain where we have introduced the quantity Ω ≡ ∞ 0 dξ I(ξ)/ξ, with meaning of the reorganization energy of the bosonic bath.Note that this result coincides with the mean field (static) result for the original case in the low-temperature limit [38].
Let us compare the results: the original self-induced method [38,36], where a single spin was both the driving and output system, and the new method splitting these roles to separate spins.To do it we first put ω 2 = ω 1 = ω in the new method to discuss a resonant case and rewrite both expressions in the following form Taking into account the positivity of I(ξ) > 0 and the fact that F 2 (β, ω, ξ) > F 1 (β, ω, ξ) > 0 for any value of β and ω we obtain that |⟨σ x 2 ⟩| > |⟨σ x 1 ⟩| for the resonant case.It demonstrates that the new method of generation of the coherence using two spins with distributed roles is more effective than the previously proposed method [36], where a single spin had to play double role, both to coherently displace the bath by its thermal population and receive that coherence back.
In order to know the effectiveness of the generation of the coherence with the new method for a couple of spins we consider a general case of different frequencies ω 1 ̸ = ω 2 with the canonical spectral density function The dimensionless parameter λ describes the strength of the spectral density function, while ω c represents the energy cut-off [32,40,41].Note that for this spectral density the coherence ⟨σ x 2 ⟩ (23) can be calculated analytically where Γ(s) is the Gamma function.The normalized coherence ⟨σ x 2 ⟩/(−4f 1 f 2 λ) for different values of the parameter s = 0.5; 1; 2 as a function of the dimensionless parameters βω 1 and ω 2 /ω 1 is presented in Fig. 2. The coherence linearly growths with the strength λ of the spectral density function and its cut-off energy ω c .As a function of the temperature T = 1/β the coherence is maximized in the T → 0 limit Finally, the generated coherence for the fixed temperature T = 1/β can be increased by increasing the frequency of the first spin ω 1 and decreasing the frequency of the second spin ω 2 .However, the limit ω 2 → 0 can't be taken since it will violate the conditions of the perturbation theory.Namely, the aforementioned perturbation analysis is applicable as long as the condition ω 2 ≫ 4f 1 f 2 Ω is satisfied, see details in Appendix D. The generalized non-perturbative analysis of the case with arbitrary ω 2 is presented later in Sec. 6.Note that ⟨σ x 1 ⟩ and ⟨σ x 2 ⟩ are non-monotonic functions of the parametr s.Therefore it is not obvious for which parameters ω 2 , β and s the coherence ⟨σ x 2 ⟩ is larger than the  coherence ⟨σ x 1 ⟩.In order to clarify this issue we calculate the ratio ⟨σ x 2 ⟩/⟨σ x 1 ⟩ for the dimensionless parameters ω 2 /ω 1 and βω 1 and for three values of the parameter s = 0.5; 1; 2. The corresponding plots for the case ω 1 /ω c = 0.1 are presented in Fig. 3.The plots demonstrate that the new method of generation becomes effective for small ratio ω 2 /ω 1 .It can be understood from analysis of the expression for the coherence of the second spin Eq. ( 23) As one can see the larger ω 1 corresponds to the larger value of 0 < tanh(βω 1 /2) < 1.From other side the multiplier tanh(βω 2 /2)/ω 2 as a function of ω 2 is a decreasing function.Therefore, the larger value of this multiplier can be reached at small values of ω 2 .

Synthesizing coherence from M spins through a single bath
Due to the new method we consider we can now address the first general problem, if the coherence can be synthesized from many driving spins, in general, with specific different couplings and frequencies.
Let us consider the case n j = (0, 0, f 1 ), with j = 1, 2, . . .M , n M +1 = (f 2 , 0, 0).Then we have Using this result we calculate Comparing this result with the previous case (23) one concludes that each of M spins contributes cumulatively to the (M + 1)th spin coherence ⟨σ x M +1 ⟩ ∝ M j=1 ⟨σ z j ⟩ S .It means, the bath can equally accumulate the coherence from the same driving spins and then uses it to make the output spin equivalently coherent even if the coupling f 1 f 2 in Eq. ( 32) decreases M times.
This result can be generalized for the case of M spins with different couplings to the bath system, i.e., for the case with different n j = (0, 0, f (j) 1 ), for j = 1, 2, . . .M and n M +1 = (f 2 , 0, 0).Repeating the previous calculations for this case we obtain It is convenient to introduce the density function of the coupling parameters and rewrite the expression for the coherence in the form This expression easily demonstrates that the main contribution to the generated coherence comes from the domain ω ≫ 2/β, where tanh(βω/2) ≈ 1. Therefore one can use the simplified formula for the coherence Advantageously, even a broad distribution of coupling parameters f 1 (ω) in the frequency domain can be sufficient to induce nearly M times higher coherence if the function f 1 (ω) is localized in the region well above 2/β.Note that the calculation of the average value of σ z M +1 is more complicated, since it needs to take into account the denominator terms in the reduced density operator ρ S from Eq. ( 9).Such a derivation is presented in Appendix B.

Coherence multiplexing to two spins (M + 2 case)
Now we are in position to analyse another problem concerning detection of autonomous coherences, when more spins can be used as output systems.If they all independently receive the same coherence from a single bath, we can observe it more easily without need to repeat the experiment in time.For analysis of coherence multiplexing, we extend the previous study to the case of (M + 2) spins coupled to the bosonic bath, see Fig. 4. The Hamiltonians of the spin system and spin-boson coupling term are and respectively.We again consider the special case n j = (0, 0, f 1 ), with j = 1, 2, . . .M , n M +1 = n M +2 = (f 2 , 0, 0).Repeating the averaging procedure over the bosonic degrees of freedom, see Appendix A, we obtain the following spin-spin correlation term Consequently, After the substituting this result into the expression for the coherence of the (M + p)th spin one can observe that it has the same structure as the coherence, obtained in Sec. 4 for the (M + 1)th spin.Therefore, the level of the coherence of the (M + 1)th spin doesn't decrease the level of the coherence of (M + 2)th spin and vice versa.However, the pair of output spins can be still correlated which will reduce their application as independent resources.In order to check the correlation between these spins we calculate the correlation parameter In the leading order in coupling parameters f 1 , f 2 we get the following expression (see intermediate results of the calculation in Appendix C) where One can observe that the integrand G(ξ, x, y) is a positive regular function on ξ ∈ [0, ∞) domain.It is a symmetric function of the parameters x, y and its values belong to the domain G(ξ, x, y) ∈ [G min (x, y), G max (x, y)] with Therefore we have the following bands on the correlation value In the high temperature limit Therefore in this case we have the following answer Note that the result has the universal character, i.e., it doesn't depend on the parameters of the spin systems ω M +1 , ω M +2 .In the low-temperature limit βω M +1 , βω M +2 ≫ 1 the σ 2 has a non-vanishing value Note that the non-zero correlation between both spins appears in the leading order of the perturbation theory and it can be interpreted as the reaction of the second spin to the first one (and vice versa) using the states of the bosonic bath as intermediate virtual states.Indeed, the leading term of the correlation value contains only the coupling parameter f 2 , but not the coupling parameters f 1 , so the non-zero correlation value will be also at f 1 = 0.
In order to understand the effectiveness of such a scheme, we estimate the signal-tonoise ratio |⟨σ x M +1 ⟩|/σ.We consider the case of the input and output spins with the same frequencies, namely, ω j = ω 1 for j = 1, 2, . . .M and ω M +1 = ω M +2 = ω, as a typical example Using the special case of the power-law (generalized Ohmic) spectral density function (26) we obtain the following expression Here we have introduced the dimensionless parameter η = 4f 1 M √ λ, which doesn't depend on the input and output spin frequencies ω 1 , ω and inverse temperature β, and short notations w = ω/ω 1 , b = βω 1 for brevity.
Using the obtained formula we plot in Fig. 5 the normalized signal-to-noise ratio ⟨σ x M +1 ⟩/ησ as a function of the dimensionless parameters βω 1 and ω/ω 1 for different values of s = 0.5; 1; 2.
6 Synthesizing coherence from two independent baths (M +N +1 case) Now, the last intriguing problem remains for our new method: do we really need a single bath that is coherently manipulated by M driving spins?Cannot we use two different and independent baths coherently pushed by M and N spins if they both transfer coherence through these baths to the same output spin?
We consider an extended version of the previous system, see Fig. 6.We introduce two reservoirs of bosons, with spectra Ω k , Υ k > 0 and creation (annihilation) opera- , respectively.First (second) reservoir is coupled to M (N ) spins via f 1 σ z j (g 1 σ z j ) term.Both reservoirs are coupled to the (M + N + 1)th spin via f 2 σ x and g 2 σ x terms.The Hamiltonian of the system consists of the spin Hamiltonian

the bosnic Hamiltonian
and the spin-bath interaction term The first line of the H SB describes the interaction of the M input spins with the left bath, the second line describes the interaction of the additional N input spins with the right bath, and finally the last term in the H SB describes the coupling of the remaining (M + N + 1)th spin with both baths.The first and second baths are characterized by spectral density functions the reorganization energies We are interested in the evaluation of the average where The polarized baths simultaneously generate the coherence ⟨σ x ⟩ of the (M + N + 1)th output spin via f 2 σ x and g 2 σ x terms.The coherence of this spin in the leading order in the coupling parameters f 1 , f 2 , g 1 , g 2 is (M + N ) times larger than the coherence ⟨σ x 2 ⟩ (colored in red) generated in the direct two-spins scheme, see Fig. 1(b) and Eq. ( 23).
the Appendix D) Here we introduced the partition function of the bosonic system Z B , and the notations R mn = (ω 2 +ω 2 mn ) 1/2 , with ω mn = 4f 1 f 2 Ω(M −2m)+4g 1 g 2 Υ(N −2n), sin θ mn = ω mn /R mn and cos θ mn = ω/R mn .Furthermore, Σ x (τ ) = cosh(R mn τ )[cos θ mn σ x + i cos θ mn σ y ] − sin θ mn σ z , and G(ξ, τ − τ ′ ) is defined in Eq. (81).Finally, the partition function Z of the whole system is Note that the perturbative regime corresponds to the situation ω ≫ [ω mn ] max = 4f 1 f 2 ΩM + 4g 1 g 2 ΥN .In this limit R mn ≈ ω, sin Let's check this formula for the previously considered case of M + 1 spins, where 4f 1 f 2 ΩM ≪ ω, g 2 = 0.For this case the expression for the statistical sum takes the form where we introduced We keep the leading term in T τ exponent, i.e., replace it by 1. Then we use the approximation R m ≈ ω, and remove the small terms ∝ f 2 1 Ω, g 2 1 Υ in the expression for the exponent.Then we get which is nothing but the statistical sum of the non-interacting spin and boson subsystems.
Using the same approximation and sin we obtain the following expression for the coherence in the system which coincides with the previously obtained result (32).Considering the general case with g 2 ̸ = 0 in the same approximation we get Therefore, the impact of several baths on the coherence of the σ's spin has an additive character.It means that independent baths can constructively join to generate coherence of the output spins.Note the several advantages of the results (60), (61), and (115): i) it can be easily generalized for the arbitrary number of baths; ii) it is non-perturbative in the coupling parameters f , f 2 , g 1 , g 2 ; iii) it allows to obtain the answers beyond the weak coupling limit However, the evaluation of the average of the coherence beyond the weak coupling limit requires either more sophisticated analytical methods or direct numerical calculations.

Generalization for the oscillator coherences
To extend the experimental possibilities we finally replace the output spin system of the previous case by an oscillator.The simplest model Hamiltonian of the corresponding system analogously to Sec. 2 is H = H A + H S + H B + H int , where is the oscillator Hamiltonian, the Hamiltoinan of the bosonic excitations of the bath, is the spin Hamiltonian and finally the term describes an interaction of the spin and the oscillator with the bosonic bath.
Applying the general formula for this case we obtain where we have introduced The reduced density matrix of the oscillator can be calculated as a power series in the parameters f, g.We calculate the leading order of the T τ -exponent in the numerator of Eq. ( 70) The denominator of the reduced density matrix then reads Let us use the obtained result for ρ A and calculate the average values for the dimensionless coordinate operator x ≡ (a Note that the non-zero value of x operator exists if both values g and f are non-zero.This result can be understood in terms of a two-step model.First, the spin polarizes the bosonic bath via f σ z term and produces the non-zero value ⟨b k + b † k ⟩ ∝ f in the leading order of the perturbation theory.Then the polarized bath generates the non-zero coordinate shift The logic of the method advantageously follows the previous case of the output spin.The square of the coordinate operator takes the form As a result for the variance σ 2 ≡ ⟨(x − ⟨x⟩) 2 ⟩ we have σ 2 ≈ ⟨x 2 ⟩ in the leading order.Note that the spin subsystem doesn't have an impact onto σ 2 in the leading order of the perturbation theory in coupling parameters f, g.
Using the latter results one can estimate the signal-to-noise ratio |⟨x⟩|/σ, which, in particular, has a simple form in the low-temperature limit Hence, the signal-to-noise ratio in such a system can be increased by increaing the ratio of the reorganization energy of the bosonic bath Ω to the oscillator's energy E.

Summary
To conclude, using a new, more efficient method then in previous case, we proved that autonomous coherences could be synthesised from many independent driving spins through separate low-temperature baths and multiplxed to many output spins.It is a crucial step to allow their experiment at investigation for a much more diverse class of experimental platforms and final verification of such intriguing phenomena.The driving coupling is present as pure dephasing, for example, in quantum dots.Our analysis shows that another output spin system embedded in the same environment can exhibit such autonomous coherence in principle, although it cannot generate it.Moreover, more driving spins will be advantageous, and the same bath can supply many spins with the same autonomous coherence.Additionally, it is not required that driving spins have to generate coherence through the same bath; many baths coupled to the output spin will be equally good.
Our approach qualitatively overcomes the first theoretical proposal suggesting the existence of these new autonomous spin coherences in Refs.[32,36,38].The groundbreaking idea there is limited by a double role of spin in this method and, therefore, unavoidable disruptive back-action effects.We removed that limitation and opened a road towards further much broader investigations.This new approach and extension are critical to observing autonomous quantum coherences experimentally and exploiting them in diverse quantum technology applications.

A Evaluation of the T τ -exponent
To evaluate the reduced density operator in Eq. ( 9) we need to perform the averaging over bosonic degrees of freedom of the T τ ordered exponent (8).To do it we present the spin-boson interaction term (3) as and use the commutativity of the operators under the T τ ordering operation we present Eq. ( 8) in the form Then we the averaging over bosonic degrees of freedom of Eq. ( 8) takes the form The sum P represents the summation over all the transpositions P (1, . . ., n) = j 1 , . . ., j n .Replacing the integration indices τ j 1 , . . ., τ jn → τ 1 , . . ., τ n for each separate term (the total number of such terms is n!) in the expression we obtain Here we introduced a thermal boson Green's function Introducing the spectral density function j=1 σ j (τ ) • n j , see Eq. ( 17) for details.Therefore the leading term of the aforementioned expression is where we introduced the notation Remembering that the term F (τ ) is a linear combination of the spin operators, we conclude that the leading term of the decomposition F (τ )F (τ ′ ) describes the spin-spin correlation between the spin degrees of freedom in spin subsystem induced by its interaction with bosonic bath.
B Average value of σ z M +1 operator for the case of M + 1 spins The expression for the average value of the σ z M +1 operator up to the second order of the perturbation theory in parameters f 1 , f 2 has the following form where ϕ(ξ, τ − τ ′ ) is defined by Eq. (84) and F (τ ) = M +1 j=1 σ j (τ ) • n j , see Eq. (17).Evaluating the integrands and substituting them into the Eq. ( 85) we get where ⟨σ z M +1 ⟩ S = − tanh(βω M +1 /2) is the average value of the σ z M +1 spin operator in the absence of the spin-bath coupling.Therefore the value δ⟨σ z M +1 ⟩ ≡ ⟨σ z M +1 ⟩ − ⟨σ z M +1 ⟩ S , describes the correction to the z component of the (M +1)th spin induced by the spin-bath interaction.The final result reads C Details of the calculation of σ 2 , Eq. (42) In the leading order of the perturbation theory the Eq. ( 42) reads where ϕ(ξ, τ − τ ′ ) is defined by Eq. ( 84), where we introduced Therefore U † σ x U = σ x and the transformed Hamiltonian H = U † HU where are the new spin Hamiltonians, corresponding to the group of spins coupled to the first (H S1 ) and the second (H S2 ) bosonic baths.Here we introduced the reorganization energies of the first and the second bath, respectively.The Hamiltonian of the corresponding baths is Then, is the Hamiltonian of the remaining (output) spin coupled to both bathes by term Finally, the term describes effective coupling between M + N input spins σ z j and the output spin σ x .Using the structure the spin Hamiltonians H S1 and H S2 one can calculate their eigenstates and eigenvalues.Namely, the eigenstates can be written as a product of all possible combinations of the spin-up | ↑⟩ and spin-down | ↓⟩ states.For example, for the Hamiltonian where s j =↑, ↓ for each j = 1, 2, . . .M .The eigenvalues of the Hamiltonian don't depend on the position of the spin-up and spin-down states in the eigenstate in |s 1 , s 2 , . . .s M ⟩, but only on the total number of spin-down and spin-up in the corresponding eigenstate.Therefore all the eigenstates with m spin-down and M −m spin-up states are characterized by the eigenvalue ! times degenerated energy level.The analogous consideration can be performed for the Hamiltonian H S2 .Note that the Hamiltonians H S1 and H S2 commute with the rest of the terms in the H and therefore, we can take a trace over the input spin states where ω mn ≡ 4f 1 f 2 Ω(M − 2m) + 4g 1 g 2 Υ(N − 2n).We transform the last exponent in the expression to the form using the notations Here with θ mn ≡ arccos(ω/R mn ), R mn = ω 2 + ω 2 mn .Now we trace out the bosonic degrees of freedom from the reduced density operator.To do it we introduce the average where Z B = Tr B e −βH B .Then, the σ-dependent reduced density matrix is where Z is the partition function of the full system.After the averaging procedure we obtain Then we calculate the average of the T τ -exponents using the results of Appendix A with the Green's function (81) and and spectral density functions (57).In order to reduce this expression for the most convenient form for further calculations we use the identity and we rewrite ρ S in the form Note that this is the exact result which takes into account non-perturbatively the coupling constants f 1 , f 2 , g 1 , g 2 of the Hamiltonian.Moreover, the perturbative calculation of T τ exponents in this expression requires f 2 , g 2 ≪ 1, which is much better assumption than one used in Secs. 4 and 5, namely f 2 ≪ 1/M with the large number M .In order to complete the new scheme of the derivation of ⟨σ x ⟩, developed in this section, we provide the algorithm of calculation of the T τ -exponent in Eqs.(120) and (121) perturbatively in the small coupling parameter κ(= f 2 or g 2 ) (122) Here I(ξ) represents the bosonic spectral density function, which in our case is I 1 (ξ)/I 2 (ξ) for the left/right bath.Using the previously obtained result for the product of operators Σ x (τ )Σ x (τ ′ ) one can derive the leading correction term of the T τ -exponent, which is needed for the calculation of the coherence ⟨σ x ⟩ in higher orders in the coupling parameters f 2 , g 2

Figure 1 :
Figure1: Generation of the spin coherence in two different schemes of the spin-bath interaction.(a) Self-induced spin coherence method from Refs.[38,37,36].The single spin "polarizes" the bosonic subsystem with spectral density function I(ξ) at temperature T via the f 1 σ z 1 term.The polarized bosonic bath generates indirectly the coherence in the same spin ⟨σ x 1 ⟩ via the f 2 σ x 1 term, i.e., as the back reaction to the spin system from the bosonic bath.(b) Externally induced spin coherence method.The first (input) spin polarizes the bosonic bath via the f 1 σ z 1 term.The polarized bosonic bath generates the coherence in the output spin ⟨σ x 2 ⟩ via the f 2 σ x 2 term, i.e., as the transfer of spin coherence through the bath to the second spin system.

Figure 4 :
Figure 4: Synthesizing and multiplexing of autonomous coherence.The first group of M driving spins polarizes the bosonic bath via the M j=1 f 1 σ z j term.The polarized bosonic bath synthetically generates the coherence in the second group of the output spins via the f 2 σ x M +p terms, where index p = 1, 2, . . .marks the target spins which get the coherence from the bath.For the coherence multiplexing, the figure represents the case of two output spins.The generated coherences of these spins ⟨σ x M +1 ⟩ = ⟨σ x M +2 ⟩ are M times larger than the coherence ⟨σ x 2 ⟩ (colored in red) generated in the direct two-spins scheme, see Fig. 1(b) and Eq.(23).
Z ≡ Tr[e −βH ].The coherence ⟨σ x ⟩ reads (details of the calculation are presented in

Figure 6 :
Figure 6: Autonomous quantum coherence from several independent baths.The group of first M driving spins polarizes the first bosonic bath via M j=1 f 1 σ z j term, the group of another N driving spins polarizes the second bosonic bath via