Designing reservoirs for 1/t decoherence of a qubit
aa r X i v : . [ qu a n t - ph ] N ov Designing reservoirs for /t decoherence of a qubit Filippo Giraldi ∗ and Francesco Petruccione † Quantum Research Group, School of Physics and National Institute for Theoretical Physics,University of KwaZulu-Natal, Durban 4001, South Africa
Anomalous decoherence in the Jaynes-Cummings model emerges for a certain class of bosonicreservoirs, described by spectral densities with a band edge frequency coinciding with the qubittransition frequency. The special reservoirs are piecewise similar to those usually adopted in Quan-tum Optics, i.e., sub-ohmic at low frequencies and inverse power laws at high frequencies. Theexact dynamics of the qubit is described analytically through Fox H -functions. Over estimatedlong time scales, decoherence results in inverse power laws with powers decreasing continuously tounity, according to the particular choice of the special reservoir. The engineering reservoir approachis a new way of strongly delaying the decoherence process with possible applications to QuantumTechnologies, due to the simple form of the designed reservoirs. PACS numbers: 03.65.Yz,03.65.Ta,03.65.-w
I. INTRODUCTION
Decoherence indicates the process that a quantum sys-tem of interest undergoes through the interaction withits external environment. The corresponding time evolu-tion and the destructive effects on quantum coherence aretreated in the Theory of Open Quantum Systems [1, 2].Great attention has been devoted to the dissipative ef-fects of a two-level system (TLS), a qubit in QuantumInformation Theory, interacting with an external envi-ronment modeled by a reservoir of bosons [3–5]. Theapplications of this simple model are most various: fromNanotechnology to Quantum Information and QuantumComputing, from Quantum Optics to circuit QED, toname a few. Still, the central issue and one of the great-est challenges remains the way to control or delay the de-structive effect of the external environment on coherence.For example, the decoherence time [6] in Magnetic Reso-nance has orders of magnitude ranging between nanosec-onds and seconds.Various techniques are adopted in order to give an an-alytical description of the exact dynamics of the qubit.Interesting results emerge from the adoption of the re-solvent operator [7] in rotating wave approximation [5],with a Lorentzian distribution of field modes [8]. Theassumption that the coupling constants vary slowly withfrequency, allows a complete analytical treatment and theexact dynamics results in oscillating behaviors envelopedin exponential decays. For a detailed report we also referto [9].An interesting model for the spontaneous decay of aTLS in a structured reservoir, has been introduced byGarraway [10, 11] and solved exactly for a generic dis-crete or a continuous distribution of field modes, de-scribed by Lorentzian type spectral densities and a spe-cial non-Lorentzian one with two poles in the lower half ∗ Electronic address: [email protected],fi[email protected] † Electronic address: [email protected] plane. The crucial conjecture that the frequency range is( −∞ , + ∞ ), allows the analytical evaluation of the exactdynamics in terms of the pseudomodes [12] and the polesof the spectral density in the lower half plane. The timeevolution results in oscillations enveloped in exponentialrelaxations. Recently [13], the model adopted by Gar-raway has been chosen to compare the qubit dynamics ofan exact master equation in time convolution less formwith the Nakajima-Zwanzig master equation, through aperturbation expansion of the memory kernel.The realization of structured environments providing adiscontinuity in the distribution of the frequency modes,also named photonic band gap (PBG) [14–17], introducesnew phenomena in the atom-cavity interactions. For ex-ample, the spontaneous emission of a two-level atom nearthe edge of a PBG exhibits oscillatory relaxations [18] in-stead of a purely exponential decay. A theoretical modelproviding a PBG structure in a N -period one dimensionallattice has been proposed in Ref. [19] by arranging anappropriate sequence of the unit lattice cells. The den-sity of the frequency modes is analytically evaluated as afunction of the transmission coefficient of each unit cell.In line with the attempt to contain the destructive ef-fect of the external environment on a qubit, decoherence,a special reservoir of bosons is designed in Ref. [20] witha PBG structure, excluding modes with frequencies lowerthan the transition frequency of the qubit. The exact dy-namics is described analytically by a linear combinationof incomplete Gamma functions and decoherence resultsin a 3 / II. THE MODEL
The interaction between the qubit and the dissipativeenvironment is described through the Jaynes-Cummingmodel with a continuous distributions of field modes [1,2, 5, 10, 13]. By choosing ~ = 1, the Hamiltonian of thewhole system is H S + H E + H I , where H S = ω σ + σ − , H E = ∞ X k =1 ω k a † k a k ,H I = ∞ X k =1 (cid:16) g k σ + ⊗ a k + g ∗ k σ − ⊗ a † k (cid:17) . The rising and lowering operators, σ + and σ − , respec-tively, act on the Hilbert space of the qubit, definedthrough the equalities σ + = σ †− = | ih | , while a † k and a k are the creation and annihilation operators, respec-tively, acting on the Hilbert space of the k -th boson,fulfilling the commutation rule h a k , a ′ † k i = δ k,k ′ for ev-ery k, k ′ = 1 , , , . . . . The constants g k represent thecoupling between the transition | i ↔ | i and the k -thmode of the radiation field, while ω is the qubit transi-tion frequency. In the following we refer to the systemof a TLS interacting with a cavity supplying a reservoirof field modes, as studied by Garraway [10] and adoptedin Ref. [13]. Starting from the initial state of the totalsystem | Ψ(0) i = ( c | i + c (0) | i ) ⊗ | i E , (1)where | i E is the vacuum state of the environment, theexact time evolution is described by the form | Ψ( t ) i = c | i ⊗ | i E + c ( t ) | i ⊗ | i E + ∞ X k =1 b k ( t ) | i ⊗ | k i E , | k i E = a † k | i E , k = 0 , , , . . . . The dynamics is easily studied in the interaction picture, | Ψ( t ) i I = e ı ( H S + H E ) t | Ψ( t ) i = c | i ⊗ | i E + C ( t ) | i ⊗ | i E + ∞ X k =1 B k ( t ) | i ⊗ | k i E , where ı is the imaginary unity, C ( t ) = e ıω t c ( t )and B k ( t ) = e ıω k t b k ( t ) for every k = 1 , , . . . . TheSchr¨odinger equation gives the forms:˙ C ( t ) = − ı ∞ X k =1 g k B k ( t ) e − ı ( ω k − ω ) t , ˙ B k ( t ) = − ı g ∗ k C ( t ) e ı ( ω k − ω ) t , leading to the following convoluted structure equation forthe amplitude h | ⊗ E h || Ψ( t ) i I , labeled as C ( t ),˙ C ( t ) = − ( f ∗ C ) ( t ) , (2) where f is the two-point correlation function of the reser-voir of field modes, f ( t − t ′ ) = ∞ X k =1 | g k | e − ı ( ω k − ω ) ( t − t ′ ) . For a continuous distribution of modes described by η ( ω ), the correlation function is expressed through thespectral density function J ( ω ), f ( τ ) = Z ∞ J ( ω ) e − ı ( ω − ω ) τ dω, where J ( ω ) = η ( ω ) | g ( ω ) | and g ( ω ) is the frequencydependent coupling constant.The exact dynamics of the qubit is described by thetime evolution of the reduced density matrix obtainedby tracing over the Hilbert space of the bosons, ρ , ( t ) = 1 − ρ , ( t ) = ρ , (0) | G ( t ) | , (3) ρ , ( t ) = ρ ∗ , ( t ) = ρ , (0) e − ıω t G ( t ) . (4)The function G ( t ), fulfilling the convolution equation˙ G ( t ) = − ( f ∗ G ) ( t ) , G (0) = 1 . (5)The function drives both the dynamics of the levels pop-ulations and the decoherence term. III. THE EXACT DYNAMICS
We study the exact dynamics of the reduced densitymatrix of the qubit, interacting in rotating wave approx-imation with a reservoir of bosons described by the con-tinuous spectral density J α ( ω ) = 2 A ( ω − ω ) α Θ ( ω − ω ) a + ( ω − ω ) , (6) A > , a > , > α > . This simple form exhibits a PBG edge in the qubit tran-sition frequency, has an absolute maximum M α at thefrequency Ω α , M α = J α (Ω α ) = A α α/ a α − (2 − α ) − α/ , Ω α = ω + a α / (2 − α ) / . The above spectral densities are piece-wise similar tothose usually adopted, i.e. sub-ohmic at low frequen-cies, ω ≃ ω , and inverse power laws at high frequencies, ω ≫ ω , similar to the Lorentzian one, though with dif-ferent power, J α ( ω ) ∼ A/a ( ω − ω ) α , ω → ω +0 ,J α ( ω ) ∼ A ω α − , ω → + ∞ . The exact dynamics of a qubit interacting with a reser-voir of bosons described by the spectral density J α ( ω ),is driven by the function G α ( t ), solution of Eq. (5), G α ( t ) = ∞ X n =0 n X k =0 ( − n n ! z kα z n − k t n − αk k !( n − k )! × (cid:16) E n +12 , n − αk +1 (cid:0) − z t (cid:1) − a t E n +12 , n − αk +3 (cid:0) − z t (cid:1) (cid:17) , (7)expressed as a series of Generalized Mittag-Leffler func-tions [21, 22], E γα,β ( z ) = ∞ X n =0 ( γ ) n z n n ! Γ ( αn + β ) ,α, β, γ ∈ C, ℜ { α } > , ℜ { β } > , where ( γ ) = 1 and ( γ ) n = Γ ( γ + n ) / Γ ( γ ). The param-eters involved read z = πAa α − sec ( πα/ − a , z = ıπAa α cos ( πα/ ,z α = − ıπAe − ıπα/ csc ( πα ) . (8)The proof is performed below.The Generalized Mittag-Leffler function, fundamentalin Fractional Calculus [21], is a particular case of theFox H -function, defined through a Mellin-Barnes typeintegral in the complex domain, H m,np,q " z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( a , α ) , . . . , ( a p , α p )( b , β ) , . . . , ( b q , β q ) = 12 πı × Z C Π mj =1 Γ ( b j + β j s ) Π nm =1 Γ (1 − a l − α l s ) z − s Π pl = n +1 Γ ( a l + α l s ) Π qj = m +1 Γ (1 − b j − β j s ) ds. Under the conditions that the poles of the Gamma func-tions in the dominator, do not coincide, also the emptyproducts are interpreted as unity. The natural numbers m, n, p, q fulfill the constraints: 0 ≤ m ≤ q , 0 ≤ n ≤ p ,and α i , β j ∈ (0 , + ∞ ) for every i = 1 , · · · , p and j =1 , · · · , q . For the sake of shortness, we refer to [23] fordetails on the contour path C , the existence and the prop-erties of the Fox H -functions. The relation E γα,β ( − z ) = 1Γ ( γ ) H , , " z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 − γ, , , (1 − β, α ) , leads to a series solution of Fox H functions, G α ( t ) = ∞ X n =0 n X k =0 ( − n z kα z n − k t n − αk k !( n − k )! × H , , " z t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( − n, , , ( αk − n, − a t H , , " z t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( − n, , , ( αk − n − , . (9)The Generalized Hypergeometric, the Wright [24] andthe Meijer G -functions [ ? ] are particular cases of the Fox H -function, thus, G α ( t ) can be expressed as a seriesof each of these Special functions, as well.Particular cases give simplified solutions. For example,the condition A = A ( ⋆ ) , A ( ⋆ ) = a − α π cos ( πα/ , (10)corresponding to z = 0, gives a power series solution, G ( ⋆ ) α ( t ) = ∞ X n =0 n X k =0 ( − n n ! z kα z n − k t n − αk k ! ( n − k )! Γ (3 n − αk + 1) × ( − a Γ (3 n − αk + 1)Γ (3 n − αk + 3) t ) , > α > . (11)If the parameter α takes rational values, p/q , where p and q are distinct prime numbers such that 0 < p < q ,the solution of Eq. (5) can be expressed as a modulationof exponential relaxations [26], G p/q ( t ) = Z ∞ dη Z ∞ dξ Φ p/q ( η, ξ ) e − ξt , (12)where Φ p/q ( η, ξ ) = n X l =1 m l X k =1 b l,k ( ζ l ) π η m l − k × sin (cid:16) η ξ /q sin ( π/q ) (cid:17) e η ( ζ l − cos( π/q ) ξ /q ) . The rational functions b l,k ( z ) read b l,k ( z ) = d k − /dz k − [( z q − a ) ( z q + a ) ( z − ζ l ) m l /Q ( z )]( m l − k )! ( k − , for every l = 1 , · · · , n , and k = 1 , . . . , m l . The complexnumbers ζ , · · · , ζ n are the roots of the polynomial Q ( z ) = z q + z z q + z α z p + z (13)and m l is the multiplicity of ζ l , for every l = 1 , . . . , n ,which means Q ( z ) = Π nl =1 ( z − ζ l ) m l and P nl =1 m l = 3 q .The case α = 1 / α = 3 / A = a / cos (3 π/ /π , the parameter z vanishes and the roots ζ , . . . , ζ l can be evaluated ana-lytically from the solutions of a quartic equation. We donot report the expressions for the sake of shortness. Inthe remaining cases of rational values of α , the roots of Q ( z ) must be evaluated numerically, once the numericalvalues of both A and a are fixed. These details completethe necessary analysis of the function G α ( t ).Finally, the exact time evolution of the qubit is ob-tained from Eqs. (3) and (4), by replacing the function G ( t ) with G α ( t ), analyzed in the present Section. IV. INVERSE POWER LAWS
The theoretical analysis of the exact dynamics, per-formed above, leads to the following concrete result: a time scale τ emerges such that, for t ≫ τ , the function G α ( t ) exhibits inverse power law behavior described bythe asymptotic form G α ( t ) ∼ −D α t − − α , t → + ∞ , > α > , (14)where D α = 2 ı α a − α ) e − ıπα/ csc ( πα ) sec ( πα/ πA Γ (1 − α ) . A simple choice is τ = max ( , (cid:12)(cid:12)(cid:12)(cid:12) z (cid:12)(cid:12)(cid:12)(cid:12) / , (cid:12)(cid:12)(cid:12)(cid:12) z α z (cid:12)(cid:12)(cid:12)(cid:12) /α , (cid:12)(cid:12)(cid:12)(cid:12) z z (cid:12)(cid:12)(cid:12)(cid:12)) , (15)the proof is performed below.Thus, over long timescales, t ≫ τ , the qubit exactdynamics is described by inverse power law relaxations: ρ , ( t ) = 1 − ρ , ( t ) ∼ ρ , (0) |D α | t − − α , (16) ρ , ( t ) = ρ ∗ , ( t ) = ρ , (0) D α e − ıω t t − − α , (17)for every α ∈ (0 , J / ( ω ) spectral densities are com-pared. The exponential type relaxations emerging in theLorentzian case, vanish faster than the inverse powerlaws related to the spectral densities J α ( ω ), for every α ∈ (0 , H -functions, and, over long timescale, t ≫ τ , decoherenceresults in an inverse power law relaxation proportional to t − − α for every α ∈ (0 , V. CONCLUSIONS
Anomalous forms of qubit decoherence emerge fromthe Jaynes-Cummings model for reservoirs of bosons de-scribed by special continuous spectral densities with aPBG edge coinciding with the qubit transition frequency.The designed spectral densities are piece-wise similar tothose usually adopted, i.e., sub-ohmic at low frequenciesand inverse power laws at high frequencies, similar tothe Lorentzian one. Initially, the qubit and the reservoir, versing in the vacuum state, are unentangled. The ex-act dynamics is described analytically through series ofFox H -functions. Over estimated long time scales, qubitdecoherence results in inverse power law relaxations withpowers decreasing continuously to unity , according to thechoice of the special reservoir.An environment supplying the designed reservoir offield modes can in principle be realized with materialsproviding the PBG structure. For example, a N -periodone dimensional lattice can reproduce a band gap by ar-ranging the appropriate sequence of dielectric unit cells.The corresponding density of frequency modes is struc-tured by the transmission coefficients of each unit cell.Also, the advanced technologies concerning diffractivegrating and photonic crystals allows the realization oftunable 1D PBG microcavities [27, 28]. The simple formof the designed reservoir may be accessible experimen-tally. The action of such a structured environment on aqubit could be a way of delaying the decoherence processwith fundamental applications to Quantum Informationprocessing Technologies. VI. PROOF
A detailed demonstration of the solutions driving theexact dynamics follows. The reservoirs of bosons de-scribed by the following class of non negative, non di-vergent and summable spectral densities are considered: Z ∞ J ( ω ) dω < ∞ , J ( ω ) = Θ ( ω − ω ) Λ ( ω − ω ) , (18)in this way, Eq. (5) gives˜ G ( u ) = { u − ı S (Λ) ( − ıu ) } − , (19) ℜ { u } > , | arg {− ıu }| < π, where S is the Stieltjes transform. The constraints: R ∞ Λ ( ω ) dω < ∞ , and ℜ { u } >
0, guarantee the uniformconvergence of integrals involved in the Integral trans-forms, so that the above equality holds true. The classof spectral densities (6) and Eq. (5) lead to the followingLaplace transform:˜ G α ( u ) = u − a u + z u + z α u α + z , (20)the parameters z , z α and z are defined by Eq. (8).The function G α ( t ) is obtained through the convergent term by term Laplace inversion of the series expansion ofEq. (20), u − a u + z u + z α u α + z = ∞ X n =0 n X k =0 ( − n n ! z kα z n − k k !( n − k )! × u αk − n − ( u − a )( u + z ) n +1 , (cid:12)(cid:12)(cid:12)(cid:12) z α u α + z u + z u (cid:12)(cid:12)(cid:12)(cid:12) < . This way, a series solution of Eq. (5) can be built interms of either Generalized Mittag-Leffler functions, Eq.(7), or Fox H functions, Eq. (9), for details we refer to[22, 23].The inverse power law behavior of G α ( t ) over long timescales, is found through the series expansion u − a u + z u + z α u α + z = ∞ X n =0 n X k =0 k X j =0 ( − n n ! j !( n − k )!( n − j )! × z − n − z n − kα z k − j ( u − a ) u α ( n − k )+ k +2 j , (21)holding true under the constraint (cid:12)(cid:12)(cid:12)(cid:12) u + z u + z α u α z (cid:12)(cid:12)(cid:12)(cid:12) < . (22)The formal term by term Laplace inversion of Eq. (21)gives G α ( t ) ∼ ∞ X n =1 n X k =0 k X j =0 ( − n +1 n ! z − n − z n − kα z k − j j ! ( n − k )! ( n − j )! × (cid:18) a Γ ( α ( k − n ) − k − j − α ( k − n ) − k − j ) − t − (cid:19) × t − − α ( n − k ) − k − j Γ ( α ( k − n ) − k − j − , t → + ∞ , (23) leading to the asymptotic solution (14). The time scalefor inverse power law behavior descends from the inequal-ity (22) and the constraint t ≫
1, requested by Eq. (23).A possible choice is obtained by imposing that the abso-lute value of each term of the left side of the inequality(22) is less than 1 /
3. This way, an upper bound for | u | isobtained and the corresponding inverse estimates a timescale for inverse power laws, given by Eq. (15). Acknowledgments
This work is based upon research supported by theSouth African Research Chairs Initiative of the Depart-ment of Science and Technology and National ResearchFoundation. F.G. is deeply grateful to Prof. R. Nigmat-ullin for the useful discussions, to Dr. J. Riccardi for thehints about the numerical check and to Prof. F. Mainardifor the continued suggestions on Integral Transforms andSpecial Functions over the years. [1] H.P. Breuer and F. Petruccione,
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