Floquet control of quantum dissipation in spin chains
aa r X i v : . [ qu a n t - ph ] M a y Floquet control of quantum dissipation in spin chains
Chong Chen, Jun-Hong An,
1, 2, 3, ∗ Hong-Gang Luo,
1, 2
C. P. Sun, and C. H. Oh Center for Interdisciplinary Studies & Key Laboratory for Magnetism andMagnetic Materials of the MoE, Lanzhou University, Lanzhou 730000, China Beijing Computational Science Research Center, Beijing 100084, China Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore
Controlling the decoherence induced by the interaction of quantum system with its environmentis a fundamental challenge in quantum technology. Utilizing Floquet theory, we explore the con-structive role of temporal periodic driving in suppressing decoherence of a spin-1/2 particle coupledto a spin bath. It is revealed that, accompanying the formation of a Floquet bound state in thequasienergy spectrum of the whole system including the system and its environment, the dissipationof the spin system can be inhibited and the system tends to coherently synchronize with the driving.It can be seen as an analog to the decoherence suppression induced by the structured environmentin spatially periodic photonic crystal setting. Comparing with other decoherence control schemes,our protocol is robust against the fluctuation of control parameters and easy to realize in practice.It suggests a promising perspective of periodic driving in decoherence control.
PACS numbers: 03.65.Yz, 03.67.Pp, 75.10.Jm
I. INTRODUCTION
As a ubiquitous phenomenon in microscopic world, de-coherence is a main obstacle to the realization of any ap-plications of quantum coherence, e.g., quantum informa-tion processing [1], quantum metrology [2], and quantumsimulation [3]. Many methods, such as feedback control[4, 5], decoherence-free subspace encoding [6, 7], and dy-namical decoupling [8–11], have been proposed to beatthis unwanted effect. The dynamical decoupling schemecan be generally described by the so-called spectral filter-ing theory in the first-order Magnus expansion [12–16],which is valid when the control pulses are sufficientlyrapid. Based on the spin echo technique, this schemeis widely exploited to suppress dephasing [8, 10, 11, 13],where the system has no energy exchange with the en-vironment, and classical noises [14–16]. It requires ahigh controllability to the system due to its sensitivity tothe time instants at which the inverse pulses are applied[17]. Furthermore, when dissipation and quantum noisesare involved, it generally cannot perform well. Althoughthe dissipation control was partially touched in the orig-inal form of the spectral filtering theory [12], whetherthe first-Markovian approximation used there can cap-ture well the physics for such time-dependent systems isstill an open question. This is because that new timescales would be introduced to the systems by the time-dependent control field, which might invalidate the ap-plication of the Markovian approximation.As a main inspiration, we notice that the decoherenceof dissipative systems connects tightly with the energy-spectrum characters of the total system consisting of thesystem and its environment [18–20]. If a bound stateresiding in the energy bandgap of the whole system is ∗ [email protected] formed by changing the environmental spectral density,the decoherence of the system can be suppressed. Thusone can artificially engineer the bound state to suppressdecoherence of quantum emitters by introducing spatialperiodic confinement in a photonic crystal setting [21–24]. Here the spatial periodic confinement dramaticallyalters the dispersion relation of the radiation field of thequantum emitter such that a certain bandgap structureis present in the spectral density. If the frequency of thequantum emitter resides in the bandgap region, then abound state is formed and thus the decoherence of theemitter can be suppressed. However, in practical solid-state systems, one generally faces that it is hard to ma-nipulate the spectral density via changing the spatial con-finement once the material of system is fabricated. Thusa more efficient way in engineering the bound state thanchanging the spectral density is desired.Recently, temporal periodic driving has become ahighly controllable and versatile tool in quantum control.Many efforts have been devoted to explore non-trivial ef-fects induced by periodic driving on physical systems. Ithas been proven to play profound role not only in con-trolling single-quantum-state of microscopic systems [25–32] and implementing geometric phase gates in quantumcomputation [33, 34], but also in generating novel statesof matter absent in the original static system [35–42].Different from static systems, periodically driven systemshave no stationary states because the energy is not con-served. Due to Floquet theory, they have well-definedquasi-stationary-state properties described by the Flo-quet eigen-values, which are called quasienergies. Thedistinguished role of periodic driving in these diverse sys-tems is that the versatility of driving schemes can inducemore colorful quasi-stationary-state behaviors than thestatic case by controlling the quasi-energy spectrum.In this paper, we explore the possibility of periodicdriving on engineering the bound state of a spin-1/2system interacting with a XX-type coupled spin bath.Via manipulating the quasi-energy spectrum by peri-odic driving, we find that a Floquet eigenstate withdiscrete quasienergy, which we name a Floquet boundstate (FBS), can be formed within the bandgap of thequasienergy spectrum. We further reveal that the pres-ence of the FBS would dynamically cause the dissipationof the system spin inhibited. The result suggests thatwe can manipulate the periodicity in a temporal domaininstead of the one in a spatial domain to suppress deco-herence, which relaxes greatly the experimental difficultyin fabricating periodic confinement in a photonic crystal.Our paper is organized as follows. In Sec. II, wepresent our model of a periodically driven spin-1 / II. MODEL AND DYNAMICS
We consider a periodically driven spin-1 / L spin-1 / H ( t ) = ˆ H S ( t ) + ˆ H I + ˆ H E withˆ H S ( t ) = 12 [ λ + A ( t )]ˆ σ z , ˆ H I = g σ x ˆ σ x + ˆ σ y ˆ σ y ) , (1)ˆ H E = λ L X j =1 ˆ σ zj + J L − X j =1 (ˆ σ xj ˆ σ xj +1 + ˆ σ yj ˆ σ yj +1 ) , (2)where ˆ σ αj ( α = x, y, z ) are the Pauli matrices with j = 0and 1 , · · · , L , respectively, labeling the system spin andthe spins in the chain; λ denotes the longitudinal mag-netic field exerted homogeneously on all the spins; A ( t )is the always-on periodic driving [43, 44] only on the sys-tem; J and g are, respectively, the coupling strengthsbetween the nearest-neighbour spins of the chain and be-tween the system and the first-site spin of the chain. ˆ H E yields a phase transition at the critical point | λ | = 2 J [45]. This type of system has been widely used to realizequantum state transfer, where the XX-coupling chain isused as a bridge [46, 47], and to analyze decoherencecaused by a spin bath [48]. Diagonalizing ˆ H E in thesingle-excitation subspace, we can obtain its eigenstate | ϕ k i = P Lj =1 e ikjx √ L ˆ σ + j |{↓ j }i , which is a spin wave withwave vector k , and the eigenenergy E k = λ + 2 J cos kx with x being the spatial separation of the two neigh-bor sites. Here |{↓ j }i is the ferromagnetic state of thechain with all its spins pointing to the − ˆ e z direction andˆ σ + j = (ˆ σ xj + i ˆ σ yj ) /
2. Obviously, the spin chain defines anenvironment with finite bandwidth 4 J . We are interested in how the spin chain results in de-coherence to the system spin and how it can be sup-pressed by periodic driving. Since the excitation num-ber ˆ N ≡ P Lj =0 ˆ σ + j ˆ σ − j is conserved, the Hilbert spaceis divided into independent subspaces with definite N .Consider that the spin chain is initially polarized in aferromagnetic state and the system is in an up state | Ψ(0) i = | φ i ⊗ |{↓ j }i with | φ i = | ↑ i , and its evolutioncan be expanded as | Ψ( t ) i = e i Lλt P Lj =0 c j ( t )ˆ σ + j |{↓ j }i ,where c ( t ) satisfies˙ c ′ ( t ) + i [ λ + A ( t )] c ′ ( t ) + Z t f ( t − τ ) c ′ ( τ ) dτ = 0 , (3)with c ′ ( t ) = c ( t ) e − i R t [ λ + A ( τ )] dτ and f ( x ) ≡ ( g /L ) P k e − iE k x and c (0) = 1. Denoting the excited-state probability of the system, | c ( t ) | characterizes theenvironmental decoherence effect on the system. Equa-tion (3) provides us with the exact description to thedecoherence of the system. III. FLOQUET QUASI-ENERGY SPECTRUM
For a static system governed by ˆ H , any time-evolvedstate can be expanded as | Ψ( t ) i = X n C n e iE n t | ϕ n i (4)where C n = h ϕ n | Ψ(0) i , E n and | ϕ n i determined byˆ H | ϕ n i = E n | ϕ n i are called as eigenenergies and station-ary states, respectively.A temporal periodic system governed by ˆ H ( t ) = ˆ H ( t + T ) can be treated by Floquet theory [49], which, as apowerful approach to map a non-equilibrium system un-der driving to a static one, can be seen as the applicationof Bloch theorem in the time domain. According to thistheory, the periodic system has a complete set of basis | u α ( t ) i determined by[ ˆ H ( t ) − i∂ t ] | u α ( t ) i = ǫ α | u α ( t ) i (5)such that any state can be expanded as | Ψ( t ) i = X α C α e − iǫ α t | u α ( t ) i (6)with C α = h u α (0) | Ψ(0) i . The similar time-independenceof C α as C n in Eq. (4) implies that ǫ α and | u α ( t ) i playthe same roles in a periodic system as eigenenergies andstationary states do in static system. Such similarityleads us to call them quasienergies and quasi-stationarystates, respectively. Carrying all the quasi-stationary-state characters, the quasienergy spectrum formed by all ǫ α is a key to study periodic system. Note that ǫ α isperiodic with period 2 π/T because e ilωt | u α ( t ) i with ω =2 π/T is also the eigenstate of Eq. (5) with eigenvalue ǫ α + lω .The Floquet operator acts on an extended Hilbertspace named Sambe space, which is made up of the usualHilbert space and an extra temporal space [50, 51]. Tocalculate the quasienergies, one first expands | u α ( t ) i ina complete set of basis of the temporal space, which isgenerally chosen as { e ikωt | k ∈ Z } . We have | u α ( t ) i = P k | ˜ u α ( k ) i e ikωt , with which Eq. (5) is recast into X k ∈ Z [ ˆ˜ H l − k + kωδ l,k ] | ˜ u α ( k ) i = ǫ α | ˜ u α ( l ) i , (7)with ˆ˜ H l − k ≡ R T ˆ H ( t ) e − i ( l − k ) ωt T dt . Then expanding eachˆ˜ H l in the complete basis of Hilbert subspace with N = 1,we get an infinite matrix equation. The quasienergies areobtained by truncating the basis of the temporal spaceto the rank such that the obtained magnitudes converge. IV. DECOHERENCE INHIBITION BYPERIODIC DRIVINGA. The mechanism of the decoherence inhibition
To reveal the mechanism of decoherence inhibition bythe periodic driving, we consider explicitly that the en-ergy splitting of the system is modulated as [48] A ( t ) = (cid:26) a , nT < t ≤ nT + τa , nT + τ < t ≤ ( n + 1) T . (8)It is realizable by adding a time-dependent longitudinalmagnetic field. Note that although only the driving pe-riodic in this step function is considered, the mechanismrevealed in the following is also applicable to other forms.To Eq. (8), we haveˆ˜ H l = ( ˆ H E + ˆ H I ) δ l, + ( ω l / σ z , (9) ω l = a (1 − e − ilωτ ) − a ( e − i πl − e − ilωτ )2 iπl . (10)We first study the asymmetric driving situation bychoosing a = 0. Figure 1(a) shows the time evolutionof the excited-state probability P t = | c ( t ) | with thechange of the driving amplitude a via numerically solv-ing Eq. (3). When the driving is switched off, i.e., a = 0, P t decays monotonically to zero, which means a completedecoherence exerted by the spin chain to the system spin.When the driving is switched on, it is interesting to seethat, dramatically different from the switch-off case, P t is stabilized repeatedly with the increase of a . To ex-plain this, we plot in Fig. 1(b) the quasienergy spectrumobtained by solving Eq. (7). We can find that an FBSis possible to be formed within the bandgap with the in-crease of a . It is remarkable to see that the regimeswhere the decoherence is inhibited match well with theones where the FBS is present. To understand the deco-herence inhibition induced by the FBS, we, according to FIG. 1. (Color online) (a) Evolution of the excited-state prob-ability P t of the system spin in different driving amplitude a .(b) Floquet quasienergy spectrum of the whole system withthe change of the driving amplitude a in step δa = 0 . J .The parameters T = 0 . πJ − , a = 0, τ = 0 . πJ − , g = 1 . J , λ = 20 . J , and L = 800 are used.FIG. 2. (Color online) Evolution of P t for | φ i = | ↑ i in (a)and F t for | φ i = ( | ↑ i + | ↓ i ) / √ a = 36 . J with the FBS (cyan solid line) and a = 1 . J without theFBS (red dashed line) via numerically solving Eq. (3). Theblue dotdashed lines show the results obtained via analyti-cally evaluating the contribution of the FBS to the asymp-totic state, which match with the numerical ones. The pa-rameters are the same as Fig. 1 except for T = 0 . πJ − and τ = 0 . πJ − . Eq. (6), rewrite | Ψ( t ) i = e i Lλt [ xe − iǫ FBS t | u FBS ( t ) i + X α ∈ Band y α e − iǫ α t | u α ( t ) i ] , (11)where x = h u FBS (0) | Ψ(0) i and y α = h u α (0) | Ψ(0) i . Thenone can get that P t evolves asymptotically to P ∞ ≡ x |h Ψ(0) | u FBS ( t ) i| with all the components in the quasi-energy band vanishing due to the out-of-phase interfer-ence contributed by the continuous phases (see AppendixA), as confirmed in Fig. 2(a). In the absence of the FBS, FIG. 3. (Color online) Floquet quasienergy spectrum of thewhole system in (a) and evolution of P t in (b) with the changeof τ when T = 0 . πJ − . The increase step δτ = 0 . J − is used in (a). Floquet quasienergy spectrum in (c) and timeevolution of P t in (d) with the change of of T when τ =0 . πJ − . The increase step δT = 0 . J − is used in (c).Other parameters are the same as Fig. 1. although it is dramatically interrupted by the driving, P t decays to zero finally. Whenever the FBS is formed, P t would be stabilized to P ∞ , which is periodic withperiod T [see the inset of Fig. 2(a)]. It means thatthe presence of the FBS would cause P t to survive inthe only component of the FBS and thus synchronizewith the driving field [52]. Figure 2(b) plots the perfor-mance of the formed FBS in an arbitrary initial state | φ i = ( | ↑ i + | ↓ i ) / √
2. We can see that the decayof the initial-state-fidelity F t ≡ h φ | Tr E [ | Ψ( t ) ih Ψ( t ) | ] | φ i ,to 50% can be stabilized even as high as the ideal loss-less case (i.e., between 0 and 1) with the formation of theFBS. Characterizing the quantum coherence between thetwo spin states, such stabilized oscillation means that thequantum coherence is preserved. We can check that F t tends to F ∞ = h φ | ρ | φ i , where ρ = (1 − | x | | ↓ ih↓ | + | x | ρ FBS ( t )+ { x ∗ µ ( t )Tr E [ |{↓ j }ih u FBS ( t ) | ] + h.c. } (12)with ρ FBS ( t ) = Tr E [ | u FBS ( t ) ih u FBS ( t ) | ] and µ ( t ) = e i R t λ + A ( t ′ )+2 ǫ FBS2 dt ′ (see Appendix B). We plot this F ∞ with the blue dotdashed line in Fig. 2(b), which matcheswith the asymptotical result from numerically solving Eq.(3).The result reveals that we can manipulate thequasienergy spectrum forming the FBS to suppress de-coherence. A prerequisite for forming the FBS is theexistence of finite quasienergy gap in the spectrum. Weplot in Fig. 3 the Floquet quasienergy spectrum and P t with the change of τ as well as T . We can see that, irre-spective of which driving parameter is changed, the firm correspondence between the formation of the FBS andthe decoherence inhibition can be established. The com-mon character between Fig. 1(b) and Fig. 3(a) is thatthe width of the formed bandgap is kept constant duringthe change of driving parameters, which is not true forFig. 3(c). This can be understood in the following way.Periodic in 2 π/T , the quasienergy has a full width 2 π/T .The energy band of the whole system is 4 J . Therefore,a bandgap with finite width 2 π/T − J can be presentin the quasienergy spectrum only in the high-frequency(i.e. 2 π/T > J ) driving case. This can be tested by Fig.3(c) where the bandgap vanishes whenever 2 π/T < J .It leads to the continuous energy band of the environmentfilling up the Floquet spectrum. Thus there is no roomfor forming the FBS here. Reflecting on P t in Fig. 3(d),although it is greatly slowed, P t approaches zero eventu-ally. Therefore, we conclude that the FBS can be presentonly in the high frequency driving case 2 π/T > J , whichsupplies a necessary condition to stabilize decoherence. Itis a very useful criterion on designing a driving schemefor decoherence control.Our finding in the periodically driven system is an ana-log to the bound-state-induced decoherence suppressionrevealed in a static system [18–20]. For a static two-levelsystem [19, 20] or a harmonic oscillator [53] interactingwith an environment, depending on the parameters inthe spectral density, the total system may possess a sta-tionary state named a bound state [19] localized out ofthe continuous energy band of the environment. As astationary state, the bound state contained as one su-perposition component in the initial state does not loseits quantum coherence during time evolution. Thus thesystem evolves exclusively to the time-invariant compo-nent of the bound state with other components in thecontinuous band vanishing due to their out-of-phase in-terference. This idea was used previously to suppressspontaneous emission of quantum emitters via introduc-ing spatial periodic confinement to the radiation field in aphotonic crystal setting [21–24]. The spatial periodicityintroduces a bandgap structure to the environmental en-ergy spectrum such that an emitter-environment boundstate is formed when the frequency of the emitter fallsin the bandgap. Here we demonstrate that the parallelpicture can be set up by introducing temporal period-icity to the system. The benefit of using the temporalperiodic driving instead of the spatial periodic confine-ment is that its high controllability greatly relaxes theexperimental difficulty in fabricating the spatial periodicconfinement. Thus it is easier to realize in practice. B. Comparisons with the previous methods
There are several methods in the literature to explorethe effects of periodic driving on quantum systems. Forexample, via neglecting the coupling between differenttemporal subspaces of the Floquet eigenequation (7) inthe high-frequency driving condition, it was shown that
FIG. 4. (Color online) The renomalization factor | F | in (a)and the evolution of P t in (b) with the change of a in thesymmetric driving case (i.e. a = − a ). The inset of (a) showsthe Floquet quasienergy spectrum revealing the absence of theFBS. The parameters are T = 0 . πJ − and τ = 0 . πJ − andthe others are the same as Fig. 1. the periodic driving can induce the suppressed tunnel-ing of a quantum particle, a phenomenon called coherentdestruction of tunneling [25–27], and the decoupling be-tween open system and its environment [30, 31]. It wasalso revealed that, via introducing the first-Markovianapproximation to Eq. (3), the dynamics of the open sys-tem under periodic control can be characterized by anoverlap integration of the noise spectrum and the spec-trum of the control and thus one can craft the filter-transfer function of the control field to suppress decoher-ence [12]. It is called spectral filtering theory and hasbeen generalized to give a unified description to a dy-namical decoupling method [12–16]. In the following wecompare our exact treatment with the above approximatemethods.First, our decoherence inhibition mechanism is morerobust to the imperfect fluctuation of the driving param-eters than the decoupling mechanism revealed in Ref.[30, 31], where the decoupling is achieved only in cer-tain single values of the driving parameters. To see this,we resort to the same approximate method as in Refs.[27, 30, 31]. Expanding | u α ( t ) i in a new set of basisof the temporal space as | u α ( t ) i = P k ˆ U t e ikωt | ˜ u α ( k ) ii ,where ˆ U t = exp[ − ( i/ R t ( A ( t ′ ) − ¯ A )ˆ σ z dt ′ ] with ¯ A =(1 /T ) R T A ( t ) dt subtracted to guarantee the periodicityof | u α ( t ) i , we can obtain a similar form as Eq. (7) butˆ˜ H l − k = [ λ + ¯ A σ z + ˆ H E ] δ l,k + g ( F l − k ˆ σ +1 ˆ σ − + h.c.) , (13) F l − k = Z T exp {− i R t [ A ( t ′ ) − ¯ A ] dt ′ } e − i ( l − k ) ωt T dt. (14)Using the approximation in Refs. [30, 31], we neglect theterms F l − k with l = k and keep only F . It reduces to aspin system coupled to an environment with the couplingstrength renormalized by a factor F . In Fig. 4, we plot | F | and P t with the change of a in the symmetric driv-ing situation, i.e. a = − a . It shows that although noFBS is formed, F = 0 is achievable in certain values ofdriving parameters. As expected, it induces the decoher-ence inhibited [see Fig. 4(b)]. However, the decouplingis sensitive to the driving parameters and any small de-viation to the decoupling driving values would cause theasymptotic vanishing of P t . Different from this, it is awide parameter regime in our mechanism which makesdecoherence inhibited (see Figs. 1 and 3), which is morestable to the parameter fluctuation in the practical ex-periments than the decoupling one.Second, we emphasize that, our mechanism is substan-tially different from the spectral filtering theory [12–16].That theory works only in the first-Markovian approxi-mation, with which the convolution in the exact evolutionequation (3) can be removed [12], i.e.,˙ α ( t ) ≈ − α ( t ) Z t dτ ε ∗ ( t ) ε ( τ ) f ( t − τ ) e iω a ( t − τ ) , (15)with α ( t ) = c ′ ( t ) e i R t [ λ + A ( τ )] dτ , ε ( t ) = e i R t [ A ( τ ) − ¯ A ] dτ ,and ω a = λ + ¯ A . Its solution can be obtained readily as | α ( t ) | = | c ( t ) | = exp[ − R ( t ) Q ( t ) / , (16)where R ( t ) ≡ π Z + ∞−∞ G ( ω + ω a ) | ε t ( ω ) | Q ( t ) dω, (17) Q ( t ) = Z t dτ | ε ( τ ) | (18)with the environmental spectral density G ( ω ) relat-ing to its correlation function f ( t − τ ) as f ( t − τ ) = R G ( ω ) e − iω ( t − τ ) dω and ε t ( ω ) = √ π R t ε ( τ ) e iωτ dτ . Thusit is only under the first Markovian approximation that | c ( t ) | can be denoted by such filtered spectrum form. Tocheck the physics missed by this approximation, we plotin Fig. 5 the comparison of our exact result with the oneobtained firmly from the spectral filtering theory. Wecan see from Fig. 5(a) that the spectral filtering theoryshows a complete decoherence to zero because of a dra-matic overlap between the environmental spectrum andthe control spectrum [see Fig. 5(b)]. However, our exactresult in Fig. 5(c) shows a stabilization on decoherencedue to the existence of the FBS in the quasi-energy spec-trum [see Fig. 5(d)]. It means that the spectral filteringtheory totally breaks down in describing the long-timesteady state behavior here. To give more evidence onthe dominate role of the formed FBS in the steady-statebehavior, we plot in Fig. 5(c) the fidelity of the FBSin the time-evolved state, which matches well with P t in the long-time limit. Therefore, it confirms again thatthe formed FBS is the physical reason for decoherenceinhibition in long-time limit of our model. Thus the de-coherence cannot be simply described as an overlap be-tween the noise spectrum and the control field here andthe spectral filtering theory is inapplicable to explain ourresult. FIG. 5. (Color online) The comparison of P t calculated bythe spectral filtering method in (a) and our exact method in(c). (b): The noise spectrum G ( ω + ω ) and the spectrum ofthe control F t ( ω ) ≡ | ε t ( ω ) | /Q ( t ) used in the spectral filter-ing method to determine P t . The contribution of the formedFBS to P t is also plotted in (c). The Floquet quasi-energyspectrum in (d) shows the existence of the FBS. a = 3 . J isfurther used in (a-c) and other parameters are same as Fig.1. As a final remark, the mechanism revealed in ourspin-bath model can also be readily extended to otherexcitation-number-conserving models, e.g. a two-levelsystem in a coupled cavity array [30, 31] and a harmonicoscillator in a bosonic bath model [53].
V. CONCLUSIONS
We have studied the decoherence dynamics of a peri-odically driven spin-1/2 particle interacting with an XXcoupled spin chain. It is found that the decoherence ofthe system can be inhibited by the periodic driving. Wehave revealed that the mechanism of such decoherenceinhibition induced by the periodic driving is the forma-tion of a FBS in the quasienergy spectrum. This can beseen as a close analog of the bound-state induced deco-herence suppression in a photonic crystal system, but itrelaxes greatly the experimental difficulties of a photoniccrystal system in fabricating specific spatial periodicityto engineer a bound state. It opens a door to beat de-coherence by tailoring temporal periodicity. Comparedwith the conventional schemes of decoherence control us-ing periodic driving or pulses, our scheme is robust tothe practical driving parameter fluctuation. Given thefact that periodic driving offers a high controllability toquantum system, our decoherence inhibition mechanismprovides us with a promising and realistic way to practi-cal decoherence control.
ACKNOWLEDGMENTS
This work is supported by the Fundamental Re-search Funds for the Central Universities, by the Spe-cialized Research Fund for the Doctoral Program ofHigher Education, by the Program for NCET, the Na-tional 973-program (Grant No. 2012CB922104 andNo. 2014CB921403), by the NSF of China (Grants No.11175072, No. 11174115, No. 11121403, No. 11325417,and No. 11474139), and by the National Research Foun-dation and Ministry of Education, Singapore (Grant No.WBS: R-710-000-008-271).
Appendix A: The contribution of the formed FBS tothe long-time steady state
For the initial state | Ψ(0) i = | ↑i⊗|{↓ · · · ↓ L }i , | Ψ( t ) i can also be expanded in the Floquet basis as | Ψ( t ) i = e i Lλt [ xe − iǫ FBS t | u FBS ( t ) i + X α ∈ B y α e − iǫ α t | u α ( t ) i ] , (A1)where | u FBS ( t ) i is the formed FBS with quasienergy ǫ FBS , | u α ( t ) i are the Floquet eigenstates in the continu-ous band with quasienergies ǫ α , x = h u FBS (0) | Ψ(0) i , and y α = h u α (0) | Ψ(0) i . Then we can calculate the probabil-ity of the system spin keeping in up state as P t = | x | |h Ψ(0) | u FBS ( t ) i| + X α,β ∈ B y ∗ α y β e − i ( ǫ β − ǫ α ) t h Ψ(0) | u β ( t ) ih u α ( t ) | Ψ(0) i + X α ∈ B [ xy ∗ α e − i ( ǫ FBS − ǫ α ) t h Ψ(0) | u FBS ( t ) ih u α ( t ) | Ψ(0) i +c.c.] (A2)Due to the out-of-phase interference contributed from e − i ( ǫ β − ǫ α ) t with α = β and e − i ( ǫ FBS − ǫ α ) t , P t tends to P ∞ = | x | |h Ψ(0) | u FBS ( ∞ ) i| + X α ∈ B | y α | |h Ψ(0) | u α ( ∞ ) i| = | x | |h Ψ(0) | u FBS ( ∞ ) i| + X α ∈ B | y α | |h u α (0) | u α ( ∞ ) i| (A3)where the orthogonality of Floquet eigenstates has beenused. Noticing the fact that P α ∈ B | y α | = P Lα =1 | y α | ∼ L Floquet eigenstates forming thecontinuous quasienergy band), we can estimate that | y α | ∼ /L . In the thermodynamics limit L ⇒ ∞ , thelast term tends to zero. Thus we have P ∞ = | x | |h Ψ(0) | u FBS ( ∞ ) i| . (A4)From the above analysis, we can see that the preservedexcited-state probability is determined by the weight of | u FBS (0) i in the initial state | Ψ(0) i and the excited-state i P i æ æ æ æ æ æ æ æ æ æ æ FIG. 6. (Color online) The distribution of excited-state pop-ulation of the formed Floquet bound state at time t = T / T = 0 . πJ − , τ = 0 . πJ − , a = 3 . J , and a = 0. probability of the system spin in | u FBS ( ∞ ) i itself. In Fig.6, we plot the distribution of excited-state population ofthe formed FBS at time t = T /
Appendix B: The effect of periodic driving on theinitial superposition state
For the general initial state | Ψ(0) i = | φ i ⊗ |{↓ j =0 }i with | φ i = α | ↑ i + β | ↓ i under | α | + | β | = 1, itsevolved state | Ψ( t ) i can be expanded as | Ψ( t ) i = e i Lλt (cid:16) α L X j =0 c j ( t )ˆ σ + j |{↓ j }i + βe i R t λ + A ( t ′ )2 dt ′ |{↓ j }i (cid:17) , (B1)where c ( t ) satisfies Eq. (3) in the main text. The fidelityof the system in its initial state | φ i can be calculated as F t = h φ | Tr E [ | Ψ( t ) ih Ψ( t ) | ] | φ i = (cid:12)(cid:12)(cid:12) | α | c ( t ) e − i R t λ + A ( t ′ )2 dt ′ + | β | (cid:12)(cid:12)(cid:12) + | αβ | | [1 − | c ( t ) | ] , (B2)Since | φ i is not an eigenstate of the system even in the ab-sence of the environmental influence, F t is a temporallyoscillating function even in the long time limit. To qual-itatively reflect the performance of the periodic driving on suppressing decoherence, we use the maximal value F t to characterize it. This happens at a set of times τ n such that c ( τ n ) e − i R τn λ + A ( t ′ )2 dt ′ = | c ( τ n ) | . Under thiscondition, Eq. (B2) has the form F τ n = 1 − | α | [1 − | c ( τ n ) | ] − | αβ | [1 − | c ( τ n ) | ] ≥ | β | + | α | | c ( τ n ) | . (B3)When the FBS is absent, | c ( ∞ ) | = 0 and thus F τ n = | β | . This corresponds to the complete decoherence (i.e,the system spin decays totally to its low-energy spin downstate). Whenever the FBS is formed, a non-zero | c ( ∞ ) | would be achieved. Then we could have F τ n > | β | inthe steady state. From this analysis, we can see that thepreserved probability for arbitrary initial state is deter-mined by the same long-time behavior of | c ( ∞ ) | as theone for the spin up initial state. This proves well that ourmechanism of dissipation suppression can also be appliedto the initial superposition state.More precisely, we can evaluate the contribution of theformed FBS to the steady state. | Ψ( t ) i can also be ex-panded in the Floquet basis as | Ψ( t ) i = e i Lλt h βe i R t λ + A ( t ′ )2 dt ′ |{↓ j }i + α ( xe − iǫ FBS t × | u FBS ( t ) i + X γ ∈ B y γ e − iǫ γ t | u γ ( t ) i ) i . (B4)Due to the out-of-phase interference, the reduced densitymatrix tends to ρ ( ∞ ) = Tr E [ | Ψ( ∞ ) ih Ψ( ∞ ) | ] = | β | | ↓ ih↓ | + | α | {| x | ρ FBS ( t ) + X γ | y γ | Tr E [ | u γ ( t ) ih u γ ( t ) | ] } + { βα ∗ x ∗ µ ( t )Tr E [ |{↓ j }ih u FBS ( t ) | ] + h.c. } , (B5)where ρ FBS ( t ) = Tr E [ | u FBS ( t ) ih u FBS ( t ) | ] and µ ( t ) = e i R t λ + A ( t ′ )+2 ǫ FBS2 dt ′ . Noticing thefact that Tr E [ | u γ ( t ) ih u γ ( t ) | ] is dominated by | ↓ ih↓ | and P γ | y γ | + | x | = 1, we have P γ | y γ | Tr E [ | u γ ( t ) ih u γ ( t ) | ] ≈ (1 − | x | ) | ↓ ih↓ | .Thus the asymptotic state of the system spin is ρ ( ∞ ) = (1 − | α | | x | ) | ↓ ih↓ | + | α | | x | ρ FBS ( t )+ { βα ∗ x ∗ µ ( t )Tr E [ |{↓ j }ih u FBS ( t ) | ] + h.c. } . (B6)Then the analytical form of the fidelity in the long-timelimit can be calculated by F ∞ = h φ | ρ ( ∞ ) | φ i . It givesthe contribution of the formed FBS to the asymptoticalstate and can be used to check the validity of our FBStheory in explaining the dynamics of the system spin. [1] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information (Cambridge University Press,2010). [2] A. W. Chin, S. F. Huelga, and M. B. Plenio, Phys. Rev.Lett. , 233601 (2012).[3] J. W. Britton, B. C. Sawyer, A. C. Keith, C.-C. Joseph
Wang, J. K. Freericks, H. Uys, M. J. Biercuk, and J. J.Bollinger, Nature , 489 (2012).[4] H. M. Wiseman and G. J. Milburn, Phys. Rev. Lett. ,548 (1993).[5] M. P. A. Branderhorst, P. Londero, P. Wasylczyk, C.Brif, R. L. Kosut, H. Rabitz, and I. A. Walmsley, Science , 638 (2008).[6] D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev.Lett. , 2594 (1998).[7] L. Viola, E. M. Fortunato, M. A. Pravia, E. Knill, R.Laflamme, and D. G. Cory, Science , 2059 (2001).[8] L. Viola and S. Lloyd, Phys. Rev. A , 2733 (1998).[9] L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. ,2417 (1999).[10] J. Du, X. Rong, N. Zhao, Y. Wang, J. Yang, and R. B.Liu, Nature , 1265 (2009);Y. Wang, X. Rong, P. Feng, W. Xu, B. Chong, J.-H. Su,J. Gong, and J. Du, Phys. Rev. Lett. , 040501 (2011).[11] G. de Lange, Z. H. Wang, D. Rist`e, V. V. Dobrovitski,and R. Hanson, Science , 60 (2010).[12] A. G. Kofman and G. Kurizki, Phys. Rev. Lett. ,270405 (2001).[13] G. S. Uhrig, Phys. Rev. Lett. , 100504 (2007).[14] L. Cywi´nski, R. M. Lutchyn, C. P. Nave, and S. DasSarma, Phys. Rev. B , 174509 (2008).[15] M. J. Biercuk, A. C. Doherty, and H. Uys, J. Phys. B:At. Mol. Opt. Phys. , 154002 (2011).[16] T. J. Green, J. Sastrawan, H. Uys, and M. J. Biercuk,New J. Phys. , 095004 (2013).[17] A. M. Souza, G. A. ´Alvarez, and D. Sutter, Phil. Trans.R. Soc. A , 4748 (2012).[18] M. Miyamoto, Phys. Rev. A , 063405 (2005).[19] Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, Phys.Rev. A , 052330 (2010).[20] P. Zhang, B. You, and L.-X. Cen, Opt. Lett. , 3650(2013).[21] M. R. Jorgensen, J. W. Galusha, and M. H. Bartl, Phys.Rev. Lett. , 143902 (2011).[22] M. Fujita, S. Takahashi, Y. Tanaka, T. Asano, and S.Noda, Science , 1296 (2005).[23] D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang,T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuˇckovi`c,Phys. Rev. Lett. , 013904 (2005).[24] M. D. Leistikow, A. P. Mosk, E. Yeganegi, S. R. Huis-man, A. Lagendijk, and W. L. Vos, Phys. Rev. Lett. ,193903 (2011).[25] F. Grossmann, T. Dittrich, P. Jung, and P. H¨anggi, Phys.Rev. Lett. , 516 (1991).[26] M. Grifoni and P. H¨anggi, Phys. Rep. , 229 (1998).[27] X. Luo, L. Li, L. You, and B. Wu, New J. Phys. ,013007 (2014).[28] H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zen-esini, O. Morsch, and E. Arimondo, Phys. Rev. Lett. ,220403 (2007);A. Eckardt, M. Holthaus, H. Lignier, A. Zenesini, D. Ciampini, O. Morsch, and E. Arimondo, Phys. Rev. A , 013611 (2009).[29] Q.-J. Tong, J.-H. An, L. C. Kwek, H.-G. Luo, and C. H.Oh, Phys. Rev. A , 060101(R) (2014).[30] L. Zhou, S. Yang, Y.-x. Liu, C. P. Sun, and F. Nori, Phys.Rev. A , 062109 (2009).[31] L. Zhou and L.-M. Kuang, Phys. Rev. A , 042113(2010).[32] A. Das, Phys. Rev. B , 172402 (2010).[33] A. Bermudez, P. O. Schmidt, M. B. Plenio, and A. Ret-zker, Phys. Rev. A , 040302(R) (2012).[34] A. Lemmer, A. Bermudez, and M. B. Plenio, New J.Phys. , 083001 (2013).[35] D. Vorberg, W. Wustmann, R. Ketzmerick, and A.Eckardt, Phys. Rev. Lett. , 240405 (2013).[36] P. Hauke, O. Tieleman, A. Celi, C. ¨Olschl¨ager, J. Si-monet, J. Struck, M. Weinberg, P. Windpassinger, K.Sengstock, M. Lewenstein, and A. Eckardt, Phys. Rev.Lett. , 145301 (2012).[37] N. H. Lindner, G. Refael, and V. Galitski, Nature Physics7, 490 (2011).[38] M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, S.Nolte, M. Segev, and A. Szameit, Nature 496, 196 (2013).[39] J. Cayssol, B. D´ora, F. Simon, and R. Moessner, Phys.Status Solidi RRL , 101 (2013).[40] M. Nakagawa and N. Kawakami, Phys. Rev. A ,013627 (2014).[41] Q.-J. Tong, J.-H. An, J. Gong, H.-G. Luo, and C. H. Oh,Phys. Rev. B , 201109(R) (2013).[42] L. Guo, M. Marthaler, and G. Sch¨on , Phys. Rev. Lett. , 205303 (2013).[43] J.-M. Cai, B. Naydenov, R. Pfeiffer, L. P. McGuinness,L. D. Jahnke, F. Jelezko, M. B. Plenio, and A. Retzker,New J. Phys. , 113023 (2012).[44] N. C. Jones, T. D. Ladd, and B. H. Fong, New J. Phys. , 093045 (2012).[45] S. Katsura, Phys. Rev. , 1508 (1962).[46] M. Christandl, N. Datta, A. Ekert, and A. J. Landahl,Phys. Rev. Lett. 92, 187902(2004).[47] D. Zueco, F. Galve, S. Kohler, and P. Hanggi, Phys. Rev.A 80, 042303(2009).[48] Z.-M. Wang, L.-A. Wu, J. Jing, B. Shao, and T. Yu,Phys. Rev. A , 032303 (2012).[49] A. Verdeny, A. Mielke, and F. Mintert, Phys. Rev. Lett. , 175301 (2013).[50] J. H. Shirley, Phys. Rev. , B979 (1965).[51] H. Sambe, Phys. Rev. A , 2203 (1973).[52] A. Russomanno, A. Silva, and G. E. Santoro, Phys. Rev.Lett. , 257201 (2012).[53] Y.-Q. L¨u, J.-H. An, X.-M. Chen, H.-G. Luo, and C. H.Oh, Phys. Rev. A88