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Driving-assisted open quantum transport in qubit networks

Donny Dwiputra,1, ∗ Jusak S. Kosasih,1, 2 Albertus Sulaiman,2, 3 and Freddy P. Zen1, 4, †

1Theoretical Physics Laboratory, Faculty of Mathematics and Natural Sciences,Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia

2Indonesian Center for Theoretical and Mathematical Physics (ICTMP), Indonesia3Badan Pengkajian dan Penerapan Teknologi, BPPT Bld. II (19 th floor),

Jl. M.H. Thamrin 8, Jakarta 10340, Indonesia4Indonesian Center for Theoretical and Mathematical Physics (ICTMP), Bandung 40132, Indonesia

(Dated: January 9, 2020)

We determine the characteristic of dissipative quantum transport in a coupled qubit networkin the presence of on-site and off-diagonal external driving. The work is a generalization of thedephasing-assisted quantum transport where noise is beneficial to the transport efficiency. UsingFloquet-Magnus expansion extended to Markovian open systems, we analytically derive transportefficiency and compare it to exact numerical results. We find that periodic driving may increase theefficiency at frequencies near the coupling rate. On the other hand, at some other frequencies thetransport may be suppressed. We then propose the enhancement mechanism as the ramification ofinterplay between driving frequency, dissipative, and trapping rates.

I. INTRODUCTION

The properties of quantum systems can be engineeredin myriad ways through the application of coherent exter-nal driving. This type of quantum engineering is basedon the Floquet theorem and in the past few years it hasgained much interest in both experiment and theory [1–7]. Particularly, in the field of quantum transport, theefficient transport of optical excitation through a net-work of coupled many-body quantum systems has beenextensively studied in both natural and artificial systems[8–10]. The role of periodic driving is to modify themany-body system of interest so that it presents certaindesired properties. For example, near-resonant periodicdriving of many-body quantum systems can be applied todemonstrate full control of the Floquet state population[11]. Given the potential of such Floquet engineering, ithas now become one of the tools to realize quantum sim-ulators, which paves the way to understand the complexand inaccessible many-body phenomena [12].In the past decade, theoretical approaches have demon-

strated the potentially beneficial role of noise in quantumtransport [13–16], usually in the spirit of delocalized exci-tons in natural light-harvesting complexes [17]. That is,the existence of noise in the warm and wet environmentof photosynthetic complexes can increase the transportefficiency instead of suppressing it, a phenomenon calledenvironment-assisted quantum transport (ENAQT). It isexplained by a model which usually consists of a net-work of coupled two-state systems (qubits) in a thermalbath. The mechanism of ENAQT is initially understoodas a result of the destruction of Anderson localization[13, 14, 18] in a disordered system (having different en-ergies) by dephasing noise. One may think that ENAQT

∗ donny.dwiputra@s.itb.ac.id† fpzen@fi.itb.ac.id

ceases to exist in an ordered system, but it is shown inRef. [16] that the ENAQT is impossible only for end-to-end transport in an ordered linear chain. Thus the afore-mentioned mechanism is not the whole story. Instead, itshould include the interference effects due to the inter-play between dephasing, dissipation, and trapping rates.Furthermore, in some specific scenarios ENAQT can alsobe viewed as a momentum rejuvenation to counter thebroad momentum distribution induced by classical noise[19].

However, the existence of the long-lived electronicquantum coherence, which was initially thought to be re-sponsible in the delocalized excitonic transport [20, 21],is disputed in a recent experiment [22]. Nevertheless,ENAQT is evident in the system and has been studiedin recent experiments of environment engineering in thespirit of quantum simulations [23–27].

In this paper, we extend the qubit network model ofENAQT to contain external driving in the spirit of Flo-quet engineering. It should be noted that our model isnot intended to extend the understanding of biologicalenergy transport in vivo. Instead, our aim is to illus-trate phenomena in driven-dissipative quantum simula-tions, since our model can be implemented in state-of-the-art experiments. We suggest some experimental pro-posals: periodically driven on-site energies can be real-ized in dissipative single-molecule junctions [23] by ap-plying AC bias voltage, or alternatively, in a network oflaser-written waveguides [24] by a time-dependent refrac-tive index using the Pockels effect. Time-dependent hop-ping terms can be implemented by varying the distancebetween the sites (to manipulate the dipolar interactions)in a chain of trapped-ions [25] or Rydberg-dressed-atoms[26]. In the mentioned experiments, the dephasing rateshave been simulated using controllable schemes.

Here we will show that in the presence of on-site andoff-diagonal periodic driving, the efficiency is enhancedwithin a range of parameters including the amplitude,frequency, noise, and trapping rates. We refer to this

2

phenomenon as driving-assisted open quantum transport(DAOQT), while in another regime, the driving plays adetrimental role for the efficiency. To this end, we solvethe problem analytically, by utilizing an extension of theFloquet theorem to open systems in the Lindblad picture,as well as numerically. To find the transport efficiency, wefind the approximate time-independent Markovian gen-erator using Magnus expansion. The analytical resultsare compared to exact numerical results.

II. MODELS

A. On-Site Driving

We consider a network of N coupled qubit sites thatmay support excitations and are subject to an externaldriving. First, we consider on-site periodic driving (seeFig. 1) with a period T = 2π/Ω. In a tight-bindingapproximation, the Hamiltonian is

H(t) =

N∑

k=1

(

ωk +∆k cos(Ωt))

σ+k σ

−k

+∑

k<l

νk,l

(

σ+k σ

−l +H.c.

)

, (1)

where σ+k (σ

−k ) are the raising (lowering) operators for site

k, wk is the site excitation energy, ∆k is the magnitudeof on-site driving, and vk,l is the hopping rate betweenthe sites k and l whose values determine the topologyof the network. Note that we have set ~ = 1. Here weconsider the one-exciton manifold, which is reasonablefor low energy systems such as the light-harvesting com-plexes [20, 22]. In a one-exciton manifold one replacesthe lowering operator σ−

k → |0〉〈k|, which is a projec-tion operator from a localized excitation at site k to thevacuum |0〉. In this manner, the Hamiltonian reads

H(t) =

N∑

k=1

(

ωk +∆k cos(Ωt))

|k〉〈k|

+∑

k<l

νk,l

(

|k〉〈l|+H.c.)

. (2)

Our model without the driving term is comparable tothose in Refs. [13, 16].To study the transport efficiency mediated by the

Hamiltonian, we introduce environmental effects mod-eled by two distinct types of Markovian noise processesof Lindblad type [28, 29]. The first is a dissipative pro-cess at a rate µk that reduces the exciton population,which is described by the super-operator

Ddiss ρ =

N∑

k=1

µk

[

− |k〉〈k|, ρ+ 2|0〉〈k|ρ|k〉〈0|]

, (3)

where ·, · is an anti-commutator. The second ispopulation-conserving dephasing process (phase random-ization due to vibrational modes of phonon bath) at a

1 2 3 N trap

FIG. 1. (Color online) Illustration of the network of qubitsforming a linear chain. All sites are subject to dissipationand dephasing noises. On-site driving is applied to site 1 andoff-diagonal driving is applied to all the couplings. Excitationinitiates at site 2. The N-th site is irreversibly connected toa trap site.

rate γk,

Ddeph ρ =N∑

k=1

γk

[

− |k〉〈k|, ρ+ 2|k〉〈k|ρ|k〉〈k|]

. (4)

In the microscopic derivation, one derives the Born-Markov approximation using a specific spectral density(for instance in Ref. [18]) to get the master equation. Theinformation about the bath temperature is contained inγk. To calculate the transport efficiency, we connect thesite m to a trap site, denoted by |N+1〉, at a rate κ. Theexcitation is transfered irreversibly from site m to N +1,which is also described by a super-operator,

Dtrap ρ = κ[

− |m〉〈m|, ρ+ 2|N + 1〉〈m|ρ|m〉〈N + 1|]

.

(5)The Hilbert space H is extended to contain the trap andthe vacuum states:

H = Hsites +Htrap +H0. (6)

Hence, the complete equation of motion is

ρ(t) = L(t)ρ(t) = −i[H(t), ρ(t)] +∑

i

Diρ(t) (7)

where i = diss, deph, trap. Note that the Liouvillian Lis time-dependent. The transport efficiency η is definedas the long-time population of the trap site,

η = limt→∞

pN+1(t). (8)

Likewise, the loss-probability is the long-time evolutionof vacuum population p0(t). The completeness of H de-

mands that∑N+1

n=0 pn(t) = 1 for all t ≥ 0 and this maybe used to check the consistency of later calculations.

B. Off-Diagonal Driving

The second interesting case is when the coupling ratesvary in time. For example, in a network of trapped ionsthe coupling may be varied by altering the distance be-tween the ions. The Hamiltonian is

H(t) =

N∑

k=1

ωkσ+k σ

−k +

∑

k<l

νk,l

×(

1 + fk,l cos(Ωt))(

σ+k σ

−l +H.c.

)

. (9)

3

The dissipators are the same as in the on-site drivingcase.Our study will focus on how periodic driving affects

the transport efficiency in a linear qubit network. Herewe take a common coupling νk,l = νδk+1,l and set itas our energy scale. To find out the parameter region inwhich the driving increases the transport, we analyticallycalculate the efficiency in the case of a localized initialexcitation. We compare the analytical calculations toexact numerical results in specific parameter values.

III. THE FLOQUET-MAGNUS-MARKOV

EXPANSION

To analytically obtain the efficiency η we need to findthe asymptotic solution ρ(∞) for the corresponding mas-ter equation. In principle, one could do this by calculat-

ing ρ(∞) = limt→∞ T exp(

∫ t

0dτ L(τ)

)

ρ(0) where T is

the time-ordering product, but this approach is compu-tationally cumbersome even for a small system. Instead,we solve the following steady-state equation: [16],

Lǫρ(∞) = limǫ→0

ǫρ(0), (10)

using Gaussian elimination to find η, defined in Eq. (8).Here Lǫρ = Lρ+ ǫpN+1|N +1〉〈N +1|. This is the situa-tion where ρ(0) is being injected at rate ǫ, and the limitis used to avoid trivial solutions. An intricacy arises be-cause in our case the superoperator L depends explicitlyon time, and at t → ∞ the value is indefinite in thepresence of periodic driving. We then consider the stro-boscopic evolution using the Floquet theorem, but in ourcase it should be extended for open systems described bytime-periodic Linblad master equations.We adopt the method in Ref. [7] which we refer to as

Floquet-Magnus-Markov (FMM) expansion. We beginwith a time-dependent Markovian master equation,

ρ(t) = L(t)ρ(t). (11)

We can formally write the solution as

ρ(tf ) = V(tf , ti)ρ(ti) (12)

where the propagator V(tf , ti) satisfies the divisibilitycondition, V(tf , ti) = V(tf , t0)V(t0, ti), and takes the

form of V(tf , ti) = eL(tf )δtf . . . eL(tk)δtk . . . eL(ti)δti . Re-call that if L(t) is time periodic, the propagator is alsoperiodic,

V(tf , ti) = V(tf + T, ti + T ), (13)

and depends only on tf − ti. In the Floquet picture, anyperiodic evolution operator can be decomposed into twoparts: one contains the time-independent effective Flo-quet Liouvillian LF[t] which is a functional of a startingtime t, and another is the periodic micromotion of thedriven system. Thus we can divide the propagator into

three parts containing the Floquet Liouvillian in the mid-dle, and the micromotions to account for the evolutionbefore and after the Floquet part,

V(tf , ti) = V(tf , t0 + nT )V(t0 + nT, t0)V(t0, t1)= V(tf , t0 + nT )enLF[t0]TV(t0, ti)= V(tf , t0 + nT )e−LF[t0]δtf e−LF[t0](tf−ti)

× eLF[t0]δtiV(t0, ti)= K(δtf )e

LF[t0](tf−ti)J (δti), (14)

where K(t) = V(t0 + t, t0)e−LF[t0]t and J (t) =

eLF[t0]tV(t0, t0 + t) are the ”kick” superoperators de-scribing the micromotions, δtf = tf − (t0 + nT ), andδti = ti − t0. A different starting time t0 corresponds toa different V(t0+T, t0). Without loss of generality, we setti = t0 = 0 and tf = t, resulting in K(δtf ) = K(t − nT )and J (δti) = 1. From now on, we simply refer to LF[t0]as LF. Thus the propagator in Eq. (14) becomes

V(t, 0) = K(t− nT )eLFt. (15)

Now we can describe the dynamics of the driven opensystem stroboscopically. However, this method is gener-ally available only at high driving frequency Ω. In thecalculation of transport efficiency η, one is interested inthe asymptotic solution, t → ∞, so that the micromo-tion Eq. (15) is negligible, K(t−nT ) ≈ 1. Hence, for theasymptotic dynamics in Eq. (10) we can replace L withLF.Having the asymptotic dynamics governed by the Flo-

quet Liouvillian LF, now we want to derive its approxi-mate form. To this end, we use Magnus expansion [30]

for LF = L(0)F + L(1)

F + L(2)F + . . . . The first three terms

are

L(0)F =

1

T

∫ T

0

dt L(t),

L(1)F =

1

2T

∫ T

0

dt1

∫ t1

0

dt2 [L(t1),L(t2)],

L(2)F =

1

6T

∫ T

0

dt1

∫ t1

0

dt2

∫ t2

0

dt3

(

[L(t1), [L(t2),L(t3)]]

+ [[L(t1),L(t2)],L(t3)])

. (16)

To apply this method, we cast the system Hamiltonianinto the following form,

H(t) = H0 +H1F (t) (17)

where F (t) is the time-dependent part of H(t). In ourcase F (t) = cos(Ωt). Using the FMM expansion, we workout the first three leading terms of LF,

L(0)F ρ = −i[H0, ρ] +

∑

i

Diρ, (18)

L(1)F ρ = 0, (19)

and, after a tedious calculation, the second order is

4

L(2)F ρ =− i

Ω2[H1,

∑

i

∑

j

DiDjρ] +2i

Ω2

∑

i

Di[H1,∑

j

Djρ]−i

Ω2

∑

i

∑

j

DiDj [H1, ρ] +2

Ω2[H0, [H1,

∑

i

Diρ]]

− 1

Ω2[H1, [H0,

∑

i

Diρ]]−1

4Ω2[H1, [H1,

∑

i

Diρ]]−1

Ω2[H0,

∑

i

Di[H1, ρ]]−1

Ω2[H1,

∑

i

Di[H0, ρ]]

+1

2Ω2[H1,

∑

i

Di[H1, ρ]]−1

Ω2

∑

i

Di[H0, [H1, ρ]] +2

Ω2

∑

i

Di[H1, [H0, ρ]]−1

4Ω2

∑

i

Di[H1, [H1, ρ]], (20)

and LF = L(0)F + L(1)

F + L(2)F + O(1/Ω3). In this paper,

we work in the fast driving regime Ω > νk,l, in which theanalytical FMM expansion is plausible. However, in thenext section we show that maximum efficiency enhance-ment is achieved at Ω slower but near the coupling rateνk,l.It should be noted that Eq. (20) is not in Lindblad

form and thus the FMM expansion is not completely pos-itive (CP). Instead, only the zeroth order is CP, and therest is only approximately CP as one considers finite ex-pansion terms.Numerical results are done by calculation of the ex-

act dynamics using master equation solver in the QuTiP

package [31].

IV. RESULTS AND DISCUSSION

In order to obtain results which are not blurred bythe network complexity, we consider a short linear chainwhere N = 3 and the rates µ, γ are equal on all sites. Itis shown in Ref. [16] that ENAQT persists in an orderedsystem, in contrast to previous studies [13, 14, 18] wherepreviously it was thought that the ENAQT would bepossible only in a disordered system due to interplay be-tween Anderson localization [32] and dephasing noise. Infact, in an ordered system, the only case where ENAQTis impossible is in end-to-end transport. To this end,we consider an ordered system in which we renormalizeωk = 0. The renormalization does not break the validityof the local master equation Eq. (7) as long as ω ≫ ν[33]. The excitation initiates at site i = 2, ρ(0) = |2〉〈2|,and the trap is connected to site m = 3.

A. On-Site Driving

We begin with a brief analysis of the transport wherethe periodic driving is applied equally on all the sites,∆k(t) = ∆. In this case there is no enhancement to ηand, in fact, there is no effect to the transport at all. Onecan check in an ordered system that η is independent ofω; see Eq. (23). Thus a homogenous driving has no effectto η since it keeps the system ordered in a rotating frame.Here we set ∆k nonzero only for site k = 1.In the following we will apply the FMM expansion to

find η via the infinite-time stroboscopic evolution. To thisend, we separate the time-dependence of the Hamiltonian

in the form of Eq. (17), where

H0 = νN=3∑

k<l

(

|k〉〈l|+ |l〉〈k|)

, H1 = ∆|1〉〈1|, (21)

and F (t) = cos(Ωt). Inserting Eq. (21) into Eqs. (18)–(20), one obtains a set of linear differential equationswritten in Appendix A1. Transport efficiency is obtainedby solving Eq. (10). The solution is found using Gaus-sian elimination, resulting in η in the form of a rationalfunction,

η = κν2

5∑

n=0An(ν, γ, µ, κ,Ω)∆

2n

6∑

n=0Bn(ν, γ, µ, κ,Ω)∆2n

, (22)

with the coefficients An and Bn written in the Supple-mentary Material [link will be inserted by the publisher].The solution without the presence of driving, η0, is ob-tained by putting ∆ = 0, or equivalently by taking thelimit Ω → ∞ so that cos(Ωt) averages out to zero,

η0 = κν2α2γ

2 + α1γ + α0

β3γ3 + β2γ2 + β1γ + β0(23)

where

α0 = (κ+ 2µ)(ν2 + 2µ2),

α1 = 2γ[µ(κ+ 4µ) + ν2],

α2 = 4µ,

β0 = [(κ+ 2µ)ν2 + µ2(κ+ µ)]

×[2m(κ+ 2µ)2 + (κ+ 4µ)ν2],

β1 = 2[µ2(κ+ µ)(κ+ 2µ)(κ+ 6µ)

+µ(5κ2 + 20κµ+ 18µ2)ν2 + (κ+ 3µ)ν4],

β2 = 4µ[2κµ(κ+ µ) + 6µ2(κ+m) + 3κν2 + 4µν2],

β3 = 8µ2(κ+ µ). (24)

This expression matches with the results in Refs. [13, 16].ENAQT is defined as the difference between the maxi-mum efficiency (η0max), where the dephasing is optimum,γopt, and the efficiency without driving (η0) [16]. The ex-istence of γopt is possible whenever ∂η0/∂γ = 0, which isdominantly occurring for small µ and κ in the order ofmagnitude up to O(10−1ν).To understand the role of the driving to the transport,

we first analyze η at certain values of ∆ and Ω. We re-fer to the efficiency enhancement η − η0 due to driving

5

10−4

10−2

100

102

104κ/ν

(a)η

Ω/ν=2

(b)η− η0

10−3 10−2 10−1 100 101μ/ν

10−4

10−2

100

102

104

κ/ν

(c)

10−3 10−2 10−1 100 101μ/ν

Ω/ν=5

(d)

0

1

0.000

0.056

0

1

0.000

0.009

no DAOQT here

FIG. 2. (Color online) Analytical characterization of effi-ciency of on-site DAOQT as a function of loss (µ) and trap-ping (κ) rates in a N = 3 system with ∆ = 2ν and γ = 0(no dephasing). The efficiency η is computed using secondorder FMM expansion whose result is in Eq. (22), and η0 isfrom Eq. (23). (a) Efficiency for Ω = 2ν and (b) the cor-responding DAOQT defined as the difference between η andη0. The white area between the two dotted lines indicatesthe region where DAOQT is not possible (see text). (c) Ef-ficiency for Ω = 5ν and (d) the corresponding DAOQT; hereDAOQT occurs dominantly at a low κ regime, in contrast tothe Ω = 2ν case.

as the DAOQT where η0 is the efficiency for ∆ = 0 (orΩ → ∞ when the driving averages out). Figures 2(a)and 2(c) shows η as a function of µ and κ in the N = 3chain with the trap site at one end, the driven site atthe other end, and the initial site in the middle. Asone may expect, in general high efficiency is achieved atsmall dissipation µ/ν. The symmetric triangle-like profilecorresponding to high η in Figs. 2(a) and 2(c) is char-acteristic for ENAQT in ordered systems [16]. Severallimiting cases can immediately be obtained, regardless ofthe number of sites: (1) at κ ≪ µ, the excitation is easilylost before it is efficiently trapped, (2) at κ ≫ µ, the sup-pression of transport is due to the quantum Zeno effect,in which the coherence vanishes due to rapidly measure-ment done by the sink, (3) at ∆ > 0 the triangle center

is shifted away from κ =√2ν. Thus the driving opens

some regimes which were previously not available for anefficient transport.

The DAOQT is apparent in Figs. 2(b) and 2(d) wherethe white area below the dotted line indicates η− η0 < 0(no enhancement). According to the FMM expansion,driving affects the transport as a second order processproportional to Ω−2; see Eq. (20). Thus we expectDAOQT to occur significantly at low Ω/ν. We firsttake γ = 0 to isolate the effect from the enhancementdue to ENAQT. For small dissipation rates, the trans-port regimes depend on the competition between κ and

10−2 10−1 100 101 102Ω/ν

0.54

0.60

0.66

η

Γ(a)

ExactFMM

0 4 8 12νt

0.0

0.2

0.4

0.6

p 4(t)

(b)

Ω/ν = 0Ω/ν = 0.746Ω/ν = 30

10−3 10−1 101μ/ν

−0.005

0.000

0.005

0.010

0.015

0.020

0.025

η−η 0

(c) 00.250.50.75

10−3 10−1 101μ/ν

0.000

0.002

0.004

0.006 (d) 00.250.50.75

FIG. 3. (Color online) (a) Transport efficiency η as a functionof driving frequency for ∆ = 2ν, γ = 0, µ = 0.1ν, and κ =0.8ν [corresponds to high DAOQT region in Fig. 2(a)]. TheFMM approximation is accurate in high driving frequencies.DAOQT is maximized at Ω = 0.746ν with enhancement Γ =8.14% and the global minimum is at Ω = 1.56ν. (b) Exacttime evolution for trap site population with the parametersfrom (a), Ω = 0 is the case of static disorder, Ω = 0.746νis the optimum, and Ω = 30ν is the high frequency limit.Effect of increasing dephasing γ (analytical) is shown in (c)for Ω = 2ν, where it contains negative enhancement for γ = 0,and (d) Ω = 5ν. The thinner line indicates higher value of γ.

∆ν2/Ω2. Positive DAOQT is achieved in κ regimes de-pendent of Ω/ν. In Fig. 2(b), where Ω = 2ν, DAOQT oc-curs dominantly at the quantum Zeno regime with κ ≫ ν.In contrast, in Fig. 2(d) DAOQT occurs dominantly atslow trapping rates, although the enhancement is smallfor sufficiently large Ω/ν.

In Fig. 3(a), we compare the efficiency from exact nu-merical result and second order FMM expansion. Theanalytical calculation is accurate for high Ω/ν and cap-tures the local maximum around Ω = 2ν although notexactly. At small Ω/ν, the FMM does not capture the in-teresting global maximum. The validity of second-order

FMM expansion, according to L(2)F in Eq. (20), relies on

the values of κ2µ/Ω2ν, ∆γµ/Ω2ν, and such coefficients,being small [see Eq. (A1) for on-site driving]. In short-time evolution of ρ(t), the approximation still mimics theexact dynamics stroboscopically, but for long-time evo-lution it may lead to a different steady state—althoughfor the parameters in Fig. 3(a) the difference is relativelysmall for Ω/ν & 2. It is shown that the FMM expansionclearly deviates from the exact result for Ω < ν. Never-theless, it reproduces the local peak at around Ω = 2ν.For frequencies larger than this the FMM method is re-liable. The efficiency contour in Fig. 2 takes the value ofΩ = 2ν, which corresponds to the local maximum thatis adequately reproduced by the FMM expansion, andΩ = 5ν, which falls in the range where the FMM methodis reliable.

6

10−1 100 101 102Δ/ν

0Δ5

0Δ6

0Δ7η

Ω/ν=0Ω/ν=2Ω/ν=5

FIG. 4. (Color online) Exact transport frequency as a func-tion of driving amplitude with paramaters as in Fig. 3(a). Inthe high amplitude limit, η converges to 0.73 and is limitedby the static disordered case (Ω/ν = 0). At nonzero driv-ing frequencies, higher ∆/ν does not always guarantee higherefficiency than the static disordered chain.

Here, we define the maximum DAOQT as

Γ(γ, µ, κ,Ω) = ηmax(γ, µ, κ,Ω)− limΩ→∞

η(γ, µ, κ,Ω). (25)

where the maximizing frequency is Ωopt. The Ω → ∞case is when the driving is averaged out and the efficiencydraws back to η0. On the other hand, η(Ω = 0) corre-sponds to the existence of a static disorder at site 1—though this is not always the case with different regimesof µ and κ. The maximum efficiency ηmax is larger thanη(Ω = 0), indicating that the existence of driving mayenhance the transport further than ENAQT in a disor-dered chain. On the other hand, we can observe that thedriving does see a negative enhancement near Ω = ν.The DAOQT is also interrelated with quantum coher-

ence between the sites. Figure 3(b) shows that in lowfrequencies (Ω < ν) the coherent evolution, indicated bywiggly lines in the population dynamics, in the trap siteis suppressed, while in Ω = 5ν it is retained. In this case,the driving may have a detrimental effect on the coher-ence. Nevertheless, efficient transport due to dephasingnoise (ENAQT) also destroys quantum coherence [13].This is because coherence increases the time needed bythe excitation to wander around the chain, resulting inless efficient transport.In general the behavior of the driven system under a

dephasing noise is similar to ENAQT for a disorderedchain. DAOQT tend to arise for γ > 0 if µ and κ falls inthe ENAQT regime. Onsite driving creates a dynamicaldisorder that is time periodic, combined together with γ,they suppress the Anderson localization and increases thedirectivity of the transport. The subtleties arise from thefact that the driving frequency Ω may not always createa constructive interference when it competes with µ andκ. In Fig. 3(c)and 3(d), we compare the DAOQT asfunction of µ in the presence of dephasing γ. Note thatfor Ω = 2ν and γ = ν the transport is slightly suppressedfor some values of µ. For Ω = 5ν, where the DAOQTis comparatively small, γ > 0 suppresses a transport.

However, in this case the difference is negligible and thedriving already starts to average out.The underlying mechanism of the on-site DAOQT lies

at the formation of periodic dynamic disorder at thedriven site. By periodically altering the energy level ofthat site, one can increase (or decrease) the directivityof the transport by controlling the interference effects atthe sites. If the periodic driving can direct the excita-tion to site 3 (connected to the trap) at a right periodsuch that the excitation remains at site 3 for a long time,there will be a gain in efficiency. The range of Ω thatbrings the benefit is of course depends on the coupling,trapping, and dissipative rates. In general, the maximumenhancement Γ is found to be near Ω = ν. Note that theperiodic driving of site energy may yield a higher effi-ciency enhancement than in the case of static disorder.This is because a static disorder cannot provide a dy-namical control to alter the interplay between coherenceand dissipation, i.e., to make the excitation stays longerat a site connected to the trap—this is the key mecha-nism of DAOQT. We term the elongation of a visit bythe excitation ”population congestion”.Figure 4 shows the effect of driving strength to effi-

ciency. Ω = 0 (dashed line) corresponds to static dis-order, in comparison to the dynamic ones. At ∆ & ν),the driven transport (for Ω = 2ν and Ω = 5ν) is moreeffecient than the static one. This shows that periodicdriving enhances the transport even in the presence ofstatic disorder, while, at large ∆/ν, the periodic driv-ing is not beneficial to the transport compared to staticdisorder, and at ∆ > 100ν the efficiency stops gainingas the site energy becomes too high for the excitation toenter. Note that the global minimum for Ω = 2ν existsbecause in the population congestion mechanism tendsto direct the excitation to the wrong site, i.e., to site 1instead of site 3, which is connected to the trap. Negativeenhancement also occurs in Fig. 3(a) for Ω slightly largerthan ν. We will find again the negative enhancement inoff-diagonal (coupling) driving with different congestiondynamics.

B. Off-Diagonal Driving

We implement a homogeneous driving, fk,l(t) = f , toall the site couplings in N = 3 ordered linear chain. Thetrap site is still connected to site 3, and the excitationalso initiates at site 2. The corresponding Hamiltonianin the form of Eq. (17) is

H0 =H1

f= ν

N=3∑

k<l

(

|k〉〈l|+ |l〉〈k|)

(26)

and F (t) = cos(Ωt). The procedures are the same asin the on-site driving and the corresponding dynamicalequations are written in Appendix A2. In the limits off = 0 or Ω → ∞, we also obtain η0 as in Eq. (23).

7

10−2 10−1 100 101Ω/ν

−0.1

0.0

0.1η−η 0

Γ(a)

ExactFMM

10−2 10−1 100 101Ω/ν

0.0

0.1

0.2 (b)

ExactFMM

0 10 20 30νt

0.0

0.2

0.4

p 4(t)

(c)

Ω/ν = 0Ω/ν = 1.41Ω=Ωopt

0 4 8 12νt

0.0

0.2

0.4

0.6

0.8

1.0(d)

Ω/ν = 0Ω/ν = 2Ω/ν = 30

FIG. 5. (Color online) Characterization of off-diagonalDAOQT. (a) The DAOQT for f = 1, γ = 0, µ = 0.05ν,and κ = 0.1ν (slow trapping). Maximum DAOQT is atΩopt = 2.8ν with Γ = 8.25% while the global minimum isat Ω = 1.41ν and η0 = 0.33. (b) Same parameters as in (a)but with κ = 5ν (fast trapping), here η0 = 0.64. The behavioris totally different because there is no oscillating pattern suchas in (a). The exact time evolution for trap site population isshown in (c) for the slow trapping and (d) fast trapping.

The off-diagonal DAOQT characteristics are shown inFig. 5, where we plot η − η0 for slow and fast trappingin Figs. 5(a) and 5(b), respectively. Fast trapping isthe case for κ in the quantum Zeno regime. Within thisregime, DAOQT ceases to exist and Ω = 0 is the mostefficient case for transport. Again, the FMM expansionis accurate only with large Ω/ν as we have discussed inon-site driving. However, in this case it does not capturethe maxima as it does before. For f = 1, Ω = 0 meansthat the coupling strength is doubled to 2ν. This dou-bling has little or no effect for the slow trapping but isprominent for the fast trapping due to the dominance ofthe incoherent population hopping over coherent oscilla-tion. One can observe an interesting oscillation patternwith sharp peaks and troughs in Fig. 5(a), which indi-cates that the system is sensitive to external driving onlybetween Ω = 10−1ν and 3ν. This is in contrast with theonsite driving in Fig. 3(a) where the driving averagesout smoothly. Here the driving appears to be suddenlyaveraged out after the global maximum in Ω ≈ 3ν. Wewill discuss this pecular behavior when we point out theDAOQT mechanism for off-diagonal driving below.

Figures 5(c) and 5(d) show the time evolutions forthe trap site corresponding to the slow and fast trap-ping, respectively. It appears at first sight that the signof coherent oscillation (wiggly lines) exists even for fasttrapping, whereas incoherent transport is expected dueto the quantum Zeno effect. Instead, this is becausethe sites are periodically uncoupled—observe that, attn = (2n + 1)π/Ω with n ∈ Z, H(tn) = 0. At thesetimes, the transport is governed only by dissipation and

0.0

0.5

1.0 (a) Ω/ν = 0.00Undriven

0.0

0.5

1.0 (b) Ω/ν = 0.573rd maxim m

0.0

0.5

1.0 (c) Ω/ν = 0.942nd maxim m

0.0

0.5

1.0 (d) Ω/ν = 1.41Global minim m

0.0

0.5

1.0 (e) Ω/ν = 2.80Global maxim m

0 5 10 15 20 25νt

0.0

0.5

1.0 (f) Ω/ν = 30.00Averaged o t

FIG. 6. (Color online) Exact population dynamics for initialsite population, p2(t) (thick lines), and site 3 population, p3(t)(thin lines), with the frequencies (a)–(f) correspond to theinteresting values in Fig. 5(a) with the same parameters.

trapping, while the evolution is incoherent. This peri-odic freezing of the system turns out to be the key ofthe DAOQT mechanism for off-diagonal driving whichwe will explain below.The DAOQT mechanism is different with the one in

on-site driving case since in this case the driving does notcreate energy disorder. It is purely the interplay betweenperiodic coupling and noise rates. We start with the un-driven chain, Ω = 0, where the sites coupling strength iseffectively 2ν. The diagonalized Hamiltonian has eigen-frequencies of λ = 0,±2

√2ν, and correspondingly it

can be seen in Fig. 6(a) that the populations p2 and p3oscillates with a period of π/2

√2ν up to an exponen-

tial decay with the rate 2µ for p2 and 2(µ + κ) for p3.When the driving is nearly averaged out, Fig. 6(f), the

frequency sets back to√2ν. These two cases portray

the typical dynamics in ENAQT. Figures 6(b),6(c), and6(e) shows the dynamics when the system takes benefitfrom driving (the values of Ω/ν correspond to maxima ofFig. 5(a)). At these frequencies, it is apparent that theexcitation is congested at site-3 and completely vanishesfrom site-2 (while the other half of the population is atsite-1). The complete suppression of p2 indicates that atthese frequencies the population congestion mechanismshould give a maximum benefit to the transport, i.e., re-sulting in the local maxima. We can observe that, at thefrequencies corresponding to local maxima, the popula-

8

tions evolve with a pattern: for the global maximum thepopulation p2(t) oscillates with one peak, for the secondmaximum with three peaks, and for the n + 1-th maxi-mum with 2n+ 1 peaks. The driven dynamics with onepeak, Fig. 6(e), gives the longest time possible for theexcitation being in site 3, thus letting it be transferredto the trap at most. In contrast, in Fig. 6(d) the drivingis congesting the population at the wrong site, resultingin negative enhancement. The mentioned mechanism isresponsible for the existence of the sharp peaks and theapparently sudden averaging of Fig. 5(a). In fact, shortlyafter the global maximum, Ω & 2.8ν, the driving is farfrom being averaged [compare Figs. 6(e) and 6(f)]. Itappears so because the minimum number of peaks in thepopulation dynamics is already reached.

Numerical studies indicate that µ and γ are not dom-inant in determining the position of the global maxi-mum and the spacing between the peaks in Fig. 6(a)—although the oscillating pattern still terminates shortlyafter the global maximum. This applies for slow trappingrates. Meanwhile µ does not affect the dynamics in thesecond order; see the corresponding dynamical equationsin Appendix A2.

V. CONCLUSIONS

We have discussed the transport characteristics of anexciton in a dissipative qubit network under on-site andoff-diagonal periodic drivings. As a clear example, wehave considered a linear ordered chain with N = 3 con-sisting of an initial site, a driven site and a trap-connectedsite at opposite sides of the chain. We have shown thatexternal periodic driving may increase the transport ef-ficiency at driving frequencies near the coupling rate foron-site and off-diagonal driving. The maximum enhance-ment of on-site DAOQT occurs at frequencies Ω just be-low ν. However, at other dissipative and trapping rates(µ and κ) we find that there is no enhancement sincethe static disorder case has the highest efficiency. On thecontrary, at Ω & ν the efficiency may be suppressed lowerthan the undriven case.

In general, the on-site driving opens the regimes whichare previously not available for efficient transport. Inthe presence of off-diagonal driving, we found that theDAOQT shows an oscillatory pattern at low trappingrates and ceases to exist at high trapping rates due to thequantum Zeno effect. We proposed the DAOQT mech-anism termed population congestion for both cases. Foron-site driving, the efficiency enhancement is due to theformation of periodic dynamic disorder at the driven site.Periodic alteration of the energy level controls the inter-ference effect at the sites, which in turn may increaseor decrease the directivity of the transport depending onat which site the driving is directing the transport. Foroff-diagonal driving, the enhancement is the result of in-terplay between periodic coupling and noise rates.

ACKNOWLEDGMENTS

F.P.Z. thanks Ministry of Higher Education and Re-search of Indonesia for Research Funding Desentralisasi2019. The numerical results was obtained using codewritten in NumPy [34] and QuTiP [31], and the figureswere made using matplotlib [35].

Appendix A: Analytical calculation of transport

efficiency

1. On-Site Driving

The efficiency is calculated by finding the asymptoticsolution using Eq. (10) with L = LF (FMM expansionto the second order) and initial condition ρ(0) = |2〉〈2|.Each of the component LFρij consists of a linear equa-tion. The corresponding equations for on-site driving arewritten below, and is solved via Gaussian elimination.

LFρ11 = −2µρ11 − 2νℑρ12+

fν

2Ω2(∆ℑρ12 + 8γℜρ12 + 4νℑρ13),

LFρ22 = −2µρ22 + 2νℑ(ρ12 − ρ23)

− ∆ν

2Ω2(∆ℑρ12 + 8γℜρ12),

LFρ33 = −2(µ+ κ)ρ33 + 2νℑρ23 −2∆ν2

Ω2ℑρ13,

LFρ44 = 2κρ33,

LFρ12 = −2(γ + µ)ρ12 + iν(ρ11 − ρ22 + ρ13)

+i∆ν

2Ω2

(

ν(4ρ12 + ρ∗23)−8iγ +∆

4(ρ11 − ρ22)

)

,

LFρ13 = −(2(γ + µ) + κ)ρ13 + iν(ρ12 − ρ23)

+i∆ν

4Ω2(∆ρ23 + 4ν(ρ33 − ρ11 + 2ρ13)),

LFρ23 = −(2(γ + µ) + κ)ρ23 − iν(ρ33 − ρ22 + ρ13)

+i∆ν

4Ω2(fρ13 − 4ν(ρ∗12 + 2ρ23)). (A1)

2. Off-Diagonal Driving

The corresponding linear equations from Eq. (10) foroff-diagonal driving are written below. The solution η forthis case is also found using Gaussian elimination (notshown in this paper).

LFρ11 = −2µρ11 − 2νℑρ12 +fν

Ω2

(

− 8γ2ℑρ12

+ν

2(f − 4)[κℜρ13 − 2γ(2ρ11 − 2ρ22 + ℜρ23)]

)

,

LFρ22 = −2µρ22 + 2νℑ(ρ12 − ρ23) +fν

Ω2

(

8γ2ℑρ12−2(2γ + κ)2ℑρ23 + ν(f − 4)[−κℜρ22+2γ(ρ11 − 2ρ22 + ρ33 + ℜρ13)]

)

,

9

LFρ33 = −2(µ+ κ)ρ33 + 2νℑρ23

−fν

Ω2

(

2(−2γ + κ)2ℑρ23 +ν

2(f − 4)[4γρ22

+(κ− 2γ)(2ρ33 + ℜρ13)])

,

LFρ44 = 2κρ33 −fνκ

Ω2

(

κ2ℑρ13

−κ

4(f − 4)(ℜρ23 − 2ℜρ12)

)

,

LFρ12 = −2(γ + µ)ρ12 + iν(ρ11 − ρ22 + ρ13)

+ifν

4Ω2

(

4[κ2ρ13 + 4γ2(ρ11 − ρ22)]

−iν(f − 4)[8γ(2ρ12 − ρ23) + κ(ρ∗23 − 2ρ12)

+8γℜ(ρ23 − 2ρ12)])

,

LFρ13 = −(2(γ + µ) + κ)ρ13 + iν(ρ12 − ρ23)

+ifν

4Ω2

(

κ2ρ12 −iν

4(f − 4)[κ(ρ11 + 2ρ13 + ρ13)

−2γ(ρ11 − 2ρ22 + ρ33)]),

LFρ23 = −(2(γ + µ) + κ)ρ23 − iν(ρ33 − ρ22 + ρ13)

+ifν

4Ω2

(

(2γ + κ)2ρ22 − (−2γ + κ)2ρ33

− iν

4(f − 4)[κρ∗12 + 8iγℑ(2ρ23 − ρ12)]

)

. (A2)

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Supplemental Material: Driving-assisted open quantum transport in qubit networks

I. SOLUTION OF THE FLOQUET-MAGNUS-MARKOV EXPANSION

Here we write the coefficients A1–B6 for the solution of the analytical on-site transport efficiency inEq. (22). The corresponding differential equation is Eq. (A1). The solution is obtained via

Gaussian elimination. Setting ν = 1, the numerator coefficients A0–A5 are

A0 = 128Ω16[

2γ + κ+ 2µ][

κ+ 2γ(2 + (2γ + κ)2) + 4µ+ 2(2γ + κ)(6γ + κ)µ+ 8(3γ + κ)µ2 + 8µ3][

1 + 2µ(γ + µ)]

,

A1 = 64Ω12

− 128γ6− 192γ5

[

κ+ 2µ]

+ 4γ[

κ3(9 + 4µ2) + κ2µ(79 + 24µ2) + 8κ(3 + 28µ2 + 6µ4) + 4µ(15

+55µ2 + 8µ4)]

− 2γ[

4κ+ κ3 + 10µ+ 11κ2µ+ 32κµ2 + 32µ3]

Ω2− 8µ2(1 + 2µ2)(−13 + Ω2)

−2κ3µ(−10 + Ω2)− 16γ4[

− 12 + 6κ2 + 24κµ + 16µ2 + Ω2]

− 8γ3[

2(−22κ+ κ3− 53µ + 6κ2µ− 16µ3)

+(3κ+ 8µ)Ω2]

+ 4γ2[

34 + 50κ2 + 248κµ + 4(83 + 6κ2)µ2 + 96κµ3 + 96µ4− 3− (κ+ 3µ)(3κ+ 8µ)Ω2

]

+κµ(80− 7Ω2 + 4µ2(64− 5Ω2))− κ2(−12 + Ω2 + 2µ2(−58 + 5Ω2))

,

A2 = 8Ω8

768γ4− 1024γ6 + 144κ2 + 1312κµ + 1856µ2

− 768γ4[

κ+ 2µ][

κ+ 4µ]

− 512γ5[

3κ+ 8µ]

+32γ[

63κ+ 8κ3 + 146µ+ 42κ2µ+ 88κµ2 + 72µ3]

− 128γ2[

− 12(2 + κ2) + κ(−41 + κ2)µ+ 6(−7 + κ2)µ2

+12κµ3 + 8µ4]

− 128γ3[

κ3− 30µ+ 12κ2µ+ 32µ3 + κ(−19 + 36µ2)

]

− 16[

38γ2 + 32γ4 + 22γκ+ 56γ3κ

+κ2 + 30γ2κ2 + 5γκ3 + (51γ + 92γ3 + 13κ+ 96γ2κ+ 21γκ2− κ3)µ+ (16 + (38γ − 7κ)(2γ + κ))µ2

+4(γ − 4κ)µ3− 12µ4

]

Ω2 +[

16γ4 + 2κ3µ+ 8γ3(3κ+ 8µ) + 5κµ(3 + 4µ2) + κ2(1 + 10µ2) + 2γ(κ(8 + κ2)

+11(2 + κ2)µ+ 32κµ2 + 32µ3) + 16(µ2 + µ4) + 4γ2[

7 + (κ+ 3µ)(3κ+ 8µ)]

]

Ω4

,

A3 = 16Ω4

− 32γ[

− 7κ+ 7µ+ γ[

− 5 + (2γ + κ+ 2µ)(10γ + κ+ 10µ)]

]

− 4[

48γ4 + 88γ3(κ+ 2µ)

−(κ+ 3µ)(3κ+ 16µ) + 4γ2[

17 + 2(κ+ 2µ)(6κ + 13µ)]

+ 4γ[

2κ3− 2µ+ 13κ2µ+ 20µ3 + 4κ(2 + 7µ2)

]

]

Ω2

+[

− κ2 + κ(−2 + κ2)µ+ (−11 + 7κ2)µ2 + 16κµ3 + 12µ4 + 4γ3[

2κ+ 7µ]

+ γ2[

62 + 6κ2 + 40κµ + 68µ2]

+γ[

κ3 + 41µ + 13κ2µ+ 52µ3 + 6κ(5 + 8µ2)]

]

Ω4−

[

γ + µ][

2γ + κ+ µ]

Ω6

,

A4 = Ω2

− 128γ[

34γ + 14κ+ 35µ]

− 8[

16γ4 + 8γ3(κ+ 4µ) − (κ+ 3µ)(κ+ 10µ) − 2γ(26κ+ 61µ)

+4γ2[

− 27 + 2µ(κ+ 2µ)]

]

Ω2− 16γ

[

6γ + 3κ+ 5µ]

Ω4 +[

γ + µ][

2γ + κ+ µ]

Ω6

,

A5 = −32γ2− 16γ

[

4γ + 2κ+ 5µ]

Ω2 +[

γ + µ][

2γ + κ+ 3µ]

Ω4, (S3)

and the denominator coefficients B0–B6 are

B0 = 128Ω16

κ+ 2γ(2 + (2γ + κ)2) + 4µ+ 2(2γ + κ)(6γ + κ)µ+ 8(3γ + κ)µ2 + 8µ3

8γ3µ2(κ+ µ)

+4γ2µ[

3κ+ 2(2 + κ2)µ+ 8κµ2 + 6µ3]

+[

κ+ 2µ+ κµ2 + µ3][

κ+ 2(2 + κ2)µ+ 8κµ2 + 8µ3]

+2γ[

κ+ (3 + 5κ2)µ+ κ(20 + κ2)µ2 + 9(2 + κ2)µ3 + 20κµ4 + 12µ5]

,

1

2

B1 = 64Ω12

− 512γ7µ[

κ+ µ]

− 128γ6[

κ+ 3µ+ 8κ2µ+ 28κµ2 + 20µ3]

− 64γ5[

12κ3µ+ κ2(3 + 76µ2)

+µ2(12 + 80µ2 + Ω2) + κµ(6 + 144µ2 + Ω2)]

−

[

κ+ 4µ+ 2κ2µ+ 8κµ2 + 8µ3][

12µ2(1 + µ2)(−10 + Ω2)

+2κ3µ(−8 + Ω2) + κµ(9(−10 + Ω2) + 4µ2(−47 + 5Ω2)) + κ2(−12 + Ω2 + 2µ2(−44 + 5Ω2))]

−16γ4[

16κ4µ+ 320µ5 + 2κ3(3 + 80µ2) + 3µ(−12 + Ω2) + 22µ3(−6 + Ω2) + 4κ2µ(−13 + 132µ2 + 2Ω2)

+κ(−12 + 704µ4 +Ω2 + µ2(−212 + 30Ω2))]

− 4γ2[

8κ5µ2 + 2κ4µ(−61 + 36µ2 + 4Ω2) + µ(−102

+128µ6 + 9Ω2 + 16µ4(−150 + 13Ω2) + 8µ2(−197 + 17Ω2)) + 2κ2µ(−300 + 224µ4 + 23Ω2 + 5µ2(−350

+29Ω2)) + κ3(−50 + 256µ4 + 3Ω2 + 2µ2(−538 + 41Ω2))

+κ(−34 + 384µ6 + 3Ω2 + 12µ2(−154 + 13Ω2) + 8µ4(−613 + 53Ω2))]

− 8γ3[

4κ5µ+ 320µ6

+κ4(2 + 68µ2) + 34µ2(−12 + Ω2) + 48µ4(−21 + 2Ω2) + 4κ3µ(−37 + 88µ2 + 3Ω2) + κ2(−44 + 800µ4 + 3Ω2

+µ2(−916 + 80Ω2)) + 2κµ(−161 + 416µ4 + 13Ω2 + µ2(−882 + 82Ω2))]

− 2γ[

2κ5µ(−16 + Ω2)

+2µ2(−230 + 21Ω2 + 16µ4(−78 + 7Ω2) + 4µ2(−316 + 29Ω2)) + κ4(−18 + Ω2 + µ2(−428 + 34Ω2))

+κ3µ(−346 + 27Ω2 + 4µ2(−531 + 46Ω2)) + 2κ2(2(−12 + Ω2) + 4µ4(−634 + 57Ω2) + µ2(−928

+81Ω2)) + κµ(−322 + 29Ω2 + 528µ4(−11 + Ω2) + µ2(−3788 + 346Ω2))]

,

B2 = 8Ω8

− 1024γ6[

κ+ 3µ]

+ 4κ5µ2Ω4 + 4κ4µ(64− 8Ω2 + (1 + 9µ2)Ω4)− 64γ5[

24κ2 + κµ(280 + 16Ω2− Ω4)

+µ2(336 + 16Ω2−Ω4)

]

+ 48µ3(200− 40Ω2 + 3Ω4 + 2µ4Ω4 + 6µ2(50− 10Ω2 + Ω4)) + κ3(144− 16Ω2

+Ω4 + 136µ4Ω4 + 4µ2(980− 184Ω2 + 19Ω4)) + κ2µ(2304 − 400Ω2 + 29Ω4 + 264µ4Ω4 + 4µ2(4020

−808Ω2 + 81Ω4)) + 4κµ2(64µ4Ω4 + 12(200 − 40Ω2 + 3Ω4) + µ2(6560− 1336Ω2 + 133Ω4))− 8γ3[

16κ4

+4κ3µ(148 + 88Ω2− 3Ω4) + 2κµ(−1576 + 484Ω2

− 21Ω4 + µ2(4064 + 1216Ω2− 82Ω4)) + κ2(−304

+112Ω2− 3Ω4 + µ2(3936 + 1632Ω2

− 80Ω4)) + 2µ2(−1808 + 544Ω2− 25Ω4

− 48µ2(−52− 12Ω2

+Ω4))]

− 16γ4[

48κ3 + µ3(2880 + 320Ω2− 22Ω4)− 3µ(48− 32Ω2 + Ω4)− 8κ2µ(−130− 24Ω2 + Ω4)

−κ(48− 32Ω2 +Ω4 + µ2(−3584− 512Ω2 + 30Ω4))]

+ 4γ2[

8κ4µ(−8− 32Ω2 + Ω4) + 3µ(768 − 152Ω2 + 7Ω4)

+16µ5(−192− 112Ω2 + 13Ω4) + 4µ3(4292 − 1004Ω2 + 59Ω4) + κ3(3(128 − 40Ω2 + Ω4) + 2µ2(−512− 864Ω2

+41Ω4)) + κ(768 − 152Ω2 + 7Ω4 + 48µ2(455− 111Ω2 + 6Ω4) + 8µ4(−768− 576Ω2 + 53Ω4)) + 2κ2µ(3488

−888Ω2 + 39Ω4 + µ2(−2048− 2144Ω2 + 145Ω4))]

+ 2γ[

2κ5µΩ2(−32 + Ω2) + κ4(128− 8(5 + 72µ2)Ω2

+(1 + 34µ2)Ω4) + κ3µ(3808 − 8(111 + 256µ2)Ω2 + (43 + 184µ2)Ω4) + 2κ2(4µ4Ω2(−448 + 57Ω2)

+4(126− 22Ω2 + Ω4) + 3µ2(3696− 824Ω2 + 51Ω4)) + 2µ2(5216 − 1000Ω2 + 57Ω4 + 16µ4Ω2(−32 + 7Ω2)

+4µ2(3528 − 736Ω2 + 55Ω4)) + κµ(7696 − 1488Ω2 + 81Ω4 + 48µ4Ω2(−64 + 11Ω2)

+µ2(44896 − 9744Ω2 + 682Ω4))]

,

B3 = 16Ω4

2κ4µΩ4− 64γ5µ

[

κ+ µ]

Ω4− κ3Ω2

[

− 12 + (1− 46µ2)Ω2 + 4µ2Ω4]

− 2µ3[

− 1568 + 384Ω2

−3(31 + 30µ2)Ω4 + (8 + 9µ2)Ω6]

+ κ2µ[

392− 4Ω2 + (15 + 202µ2)Ω4− 2(1 + 10µ2)Ω6

]

− 2κµ2[

− 1568

+384Ω2− (93 + 167µ2)Ω4 + (8 + 17µ2)Ω6

]

− 64γ4[

κ2µΩ4 + 3µ(10 + 3Ω2 + µ2Ω4) + κ(10 + 3Ω2

+4µ2Ω4)]

− 4γ3[

4κ3µΩ4 + κ2(96 + 88Ω2 + (−2 + 36µ2)Ω4) + 4µ2(400 + 196Ω2

+4(−7 + 3µ2)Ω4 + Ω6) + κµ(1248 + 728Ω2 + (−109 + 80µ2)Ω4 + 4Ω6)]

+ γ[

κ4Ω2(−32 + Ω2)

+κ3µΩ2(−736 + 131Ω2− 4Ω4)− 2µ2(−3024 + 8(147 + 160µ2)Ω2

− 9(25 + 48µ2)Ω4 + (11 + 26µ2)Ω6)

−κ2(−224 + 128(1 + 25µ2)Ω2− 2(15 + 389µ2)Ω4 + (1 + 36µ2)Ω6) + κµ(5040 − 8(265 + 624µ2)Ω2

+2(191 + 745µ2)Ω4− (17 + 84µ2)Ω6)

]

− 2γ2[

32µ5Ω4 + κ3(16 + 96Ω2 + (−3 + 8µ2)Ω4) + 3µ(−80 + 136Ω2

−31Ω4 + Ω6) + 2κ2µ(184 + 768Ω2 + 5(−25 + 4µ2)Ω4 + 4Ω6) + µ3(2240 + 2560Ω2− 566Ω4 + 25Ω6)

+κ(−80 + 136Ω2− 31Ω4 + 64µ4Ω4 + Ω6 + µ2(2304 + 3840Ω2

− 794Ω4 + 33Ω6))]

,

3

B4 = Ω2

− 4352γ2[

κ+ 3µ]

− 1792γ[

κ2 + 12κµ+ 13µ2]

− 8[

16γ4[

κ+ 3µ]

+ 8γ3[

κ2 + 25κµ+ 30µ2]

−(κ+ 10µ)(κ2 + 27κµ+ 30µ2)− 2γ[

26κ2 + 521κµ + 580µ2]

+ 4γ2[

− 81µ+ 4κ2µ+ 48µ3

+3κ(−9 + 16µ2)]

]

Ω2− 16

[

24γ3µ(κ+ µ) + 2µ(κ2 + 15κµ+ 15µ2) + 2γ2[

3κ+ 9µ+ 16κ2µ+ 44κµ2 + 28µ3]

+γ[

8κ3µ+ 4µ2(19 + 8µ2) + κ2(3 + 40µ2) + κµ(69 + 64µ2)]

]

Ω4−

[

− 16γ3µ(κ+ µ) − 2γ2[

κ+ 3µ+ 8κ2µ

+33κµ2 + 25µ3]

− 2µ(κ2 + 2κ(9 + κ2)µ+ 2(9 + 5κ2)µ2 + 17κµ3 + 9µ4)

−γ[

κ2 + κ(37 + 4κ2)µ+ 6(7 + 6κ2)µ2 + 84κµ3 + 52µ4]

]

Ω6

,

B5 = 32γ2[

κ+ 3µ]

+ 32γ[

κ2 + 19κµ+ 20µ2 + 2γ(κ+ 3µ)]

Ω2 +[

8γ3µ(κ+ µ)− 2µ(κ2 + 15κµ

+15µ2)− γ[

κ2 + 103κµ+ 108µ2]

+ γ2(−6µ+ 8µ3 + κ(−2 + 8µ2))]

Ω4 + 3µ[

γ + µ][

κ+ µ]

Ω6,

B6 =Ω2

8µ[

κ+ µ][

− 32γ + (γ + µ)Ω2]

. (S4)

The solution converges to η0 in Eq. (23) in absence of driving (setting ∆ = 0 or taking limit Ω → ∞).

This figure "off_diag.png" is available in "png" format from:

http://arxiv.org/ps/1906.03702v2