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Quantum Robust Stability of a Small Josephson Junction in a Resonant Cavity Ian R. Petersen 3 1 Abstract—This paper applies recent results on the robust this enables us to choose suitable values for the coupling 0 stabilityof nonlinearquantumsystemstothecaseofaJoseph- parameters in the system. For the parameter values chosen, 2 son junction in a resonant cavity. The Josephson junction is we show that the quantum system is robustly mean square n characterizedbyaHamiltonianoperatorwhichcontainsanon- stable according to the definition of [15]. a quadraticterminvolvingacosinefunction.Thisleadstoasector J bounded nonlinearity which enables the previously developed II. QUANTUMSYSTEMS 9 theory to be applied to this system in order to analyze its stability. Themainresultofthepaper[15]considersopenquantum ] systems defined by parameters (S,L,H) where H = H + 1 h I. INTRODUCTION H ;e.g.,see[10],[14].Thecorrespondinggeneratorforthis p 2 quantum system is given by - In recent years, a number of papers have considered the t n feedback control of systems whose dynamics are governed (X)= i[X,H]+ (X) (1) a by the laws of quantum mechanics rather than classical G − L qu mechanics; e.g., see [1]–[13]. In particular, the papers [10], where L(X) = 21L†[X,L] + 12[L†,X]L. Here, [X,H] = [14] consider a framework of quantum systems defined in XH HX denotes the commutator between two operators [ − terms of a triple (S,L,H) where S is a scattering matrix, and the notation † denotes the adjoint transpose of a vector 1 L is a vector of coupling operators and H is a Hamiltonian ofoperators.Also,H1isaself-adjointoperatorontheunder- v operator. lying Hilbert space referred to as the nominal Hamiltonian 8 4 Thepaper[15]considersthe problemofabsolutestability and H2 is a self-adjoint operator on the underlying Hilbert 7 for a quantum system defined in terms of a triple (S,L,H) space referredto as the perturbationHamiltonian.The triple 1 inwhichthequantumsystemHamiltonianisdecomposedas (S,L,H), along with the corresponding generators define 1. H = H +H where H is a known nominal Hamiltonian the Heisenberg evolution X(t) of an operator X according 1 2 1 0 andH isaperturbationHamiltonian,whichiscontainedina to a quantum stochastic differential equation; e.g., see [14]. 2 3 specifiedsetofHamiltonians .Inparticularthepaper[15] We now definea set of non-quadraticperturbationHamil- 1 considers the case in which thWe nominal Hamiltonian H is tonians denoted . The set is defined in terms of the : 1 W W v a quadratic function of annihilation and creation operators followingpowerseries(whichis assumedto convergein the Xi and the coupling operator vector is a linear function of senseoftheinducedoperatornormontheunderlyingHilbert annihilation and creation operators. This case corresponds space) r a to a nominal linear quantum system; e.g., see [4], [5], [7], ∞ ∞ ∞ ∞ [8], [13]. Also, it is assumed that H2 is contained in a set H2 =f(ζ,ζ∗)= Skℓζk(ζ∗)ℓ = SkℓHkℓ. of non-quadratic perturbation Hamiltonians corresponding k=0ℓ=0 k=0ℓ=0 XX XX (2) to a sector bound on the nonlinearity In this special case, Here S = S , H = ζk(ζ )ℓ, and ζ is a scalar operator [15] obtains a robust stability result in terms of a frequency kℓ ℓ∗k kℓ ∗ on the underlying Hilbert space. Also, denotes the adjoint domain condition. ∗ of a scalar operator. It follows from this definition that In this paper, we apply the result of [15] to a quantum system which consists of a Josephson junction in a resonant ∞ ∞ ∞ ∞ cavity as described in the paper [16]. This enables us to H2∗ = Sk∗ℓζℓ(ζ∗)k = Sℓkζℓ(ζ∗)k =H2 analyze the stability of this quantum system. In particular, kX=0Xℓ=0 Xℓ=0Xk=0 and thus H is a self-adjoint operator. Note that it follows 2 ThisworkwassupportedbytheAustralianResearchCouncil(ARC)and from the use of Wick ordering that the form (2) is the most AirForceOfficeofScientificResearch(AFOSR).Thismaterialisbasedon generalformforaperturbationHamiltoniandefinedinterms researchsponsoredbytheAirForceResearchLaboratory,underagreement number FA2386-09-1-4089. The U.S. Government is authorized to repro- of a single scalar operator ζ. duceanddistributereprintsforGovernmentalpurposesnotwithstandingany Also, we let copyrightnotationthereon.Theviewsandconclusionscontainedhereinare thoseoftheauthorsandshouldnotbeinterpretedasnecessarilyrepresenting ∞ ∞ theofficialpoliciesorendorsements,eitherexpressedorimplied,oftheAir f′(ζ,ζ∗)= kSkℓζk−1(ζ∗)ℓ, (3) ForceResearch LaboratoryortheU.S.Government. k=1ℓ=0 Ian R. Petersen is with the School of Engineering and Infor- XX mation Technology, University of New South Wales at the Aus- ∞ ∞ tralian Defence Force Academy, Canberra ACT 2600, Australia. f′′(ζ,ζ∗)= k(k 1)Skℓζk−2(ζ∗)ℓ (4) [email protected] − k=1ℓ=0 XX and consider the sector bound condition fortherobustmeansquarestabilityofthenonlinearquantum 1 system under consideration when H2 : f (ζ,ζ ) f (ζ,ζ ) ζζ +δ (5) ∈W ′ ∗ ∗ ′ ∗ ≤ γ2 ∗ 1 1) The matrix 1 and the condition F = iJM JN JN is Hurwitz; (13) † − − 2 f′′(ζ,ζ∗)∗f′′(ζ,ζ∗) δ2. (6) 2) ≤ γ Then we define the set as follows: E˜#Σ(sI−F)−1D˜ < 2 (14) = H2 of theWform (2) such that . (7) where (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)∞ W conditions (5) and (6) are satisfied D˜ = JΣE˜T, (cid:26) (cid:27) Reference [15] also considers the case in which the I 0 J = , nominal quantum system corresponds to a linear quantum 0 I (cid:20) − (cid:21) system; e.g., see [4], [5], [7], [8], [13]. In this case, H1 is 0 I Σ = . of the form I 0 (cid:20) (cid:21) 1 a This leads to the following theoremwhich is presentedin H = a aT M (8) 1 2 † a# [15]. (cid:20) (cid:21) (cid:2) (cid:3) Theorem 1: Consider an uncertain open quantum system where M C2n 2n is a Hermitian matrix of the form ∈ × defined by (S,L,H) such that H = H1 +H2 where H1 M = MM#1 MM#2 (9) Fisurotfhetrhmeofroer,mass(u8m),eLthiastothfethsetrifcotrmbou(n1d0e)danredalHc2on∈ditWion. (cid:20) 2 1 (cid:21) (13), (14) is satisfied. Then the uncertainquantumsystem is andM1 =M1†,M2 =M2T.Here,thenotation# denotesthe robustly mean square stable. vector of adjoint operators for a vector of operators. Also, In the next section, we will apply this theorem to analyze # denotes denotes the complex conjugate of a matrix for a thestabilityofanonlinearquantumsystemcorrespondingto complex matrix. In addition, we assume L is of the form a Josephson junction in a resonant cavity. L= N N a (10) III. THEJOSEPHSONJUNCTION IN ARESONANTCAVITY 1 2 (cid:20) a# (cid:21) SYSTEM (cid:2) (cid:3) where N Cm n and N Cm n. Also, we write We consider a quantum system consisting of a small 1 × 2 × ∈ ∈ Josephson junction coupled to an electromagnetic resonant L =N a = N1 N2 a . cavity. This system has been considered in the paper [16] L# a# N# N# a# (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) 2 1 (cid:21)(cid:20) (cid:21) where a Hamiltonian for the system is derived. The paper As in [15], we consider a notion of robust mean square [16]considersthesystemasaclosedquantumsystemdefined stability. purelyintermsofasystemHamiltonian.Wemodifythisde- Definition 1: An uncertain open quantum system defined scription of the system to consider an open quantumsystem by (S,L,H) where H = H +H with H of the form model which interacts with external fields by introducing 1 2 1 (8), H , and L of the form (10) is said to be robustly coupling operators for the system in order to apply the 2 ∈W meansquarestableif foranyH ,thereexistconstants results of [15] given above. The coupling operator which 2 ∈W c >0, c >0 and c 0 such that is introduced is taken as a standard coupling operator for 1 2 3 ≥ a resonant cavity coupled to a single field as well as a a(t) † a(t) corresponding coupling operator for the Josephson junction *(cid:20) a#(t) (cid:21) (cid:20) a#(t) (cid:21)+ coupled to an external heat bath. AJosephsonjunctionconsistsofathininsulatingmaterial ≤ c1e−c2t*(cid:20) aa# (cid:21)†(cid:20) aa# (cid:21)++c3 ∀t≥0. (11) baeJtowseeepnhstwonojsuunpcetrioconnidnuactirnegsolnaaynetrsc.aAvsityin. T[1h6is],iwseillcuosntrsaidteedr in Figure 1. The following Hamiltonian for the Josephson a(t) Here denotes the Heisenberg evolution of the junction system is derived in [16] a#(t) vector(cid:20)of operat(cid:21)ors aa# ; e.g., see [14]. H = 21~U′(n′−n¯′)2− J~′ cosφ′+ 21~(p′2+ω2q′2) We define (cid:20) (cid:21) ω Uωn¯2g2 g pn + (15) ζ = E1a+E2a# − r~ ′ ′ 2U′ = E1 E2 aa# =E˜ aa# (12) wi~h.eHreern¯e′ = UUn¯′, U′ =U +~ωg2, [φ′,n′]=i, and [q′,p′]= (cid:20) (cid:21) (cid:20) (cid:21) where ζ is a sca(cid:2)lar operator.(cid:3)Then, the following following φ′ = φ g ω/~q, n′ =n − strict bounded real condition provides a sufficient condition p′ = p+g√pω~n, q′ =q where q and p are the position and momentumoperatorsfor these canonical commutation relations and neglect further the resonant cavity. Also, n is an operator which represents constant terms. Also, the fact that the operators a and a 1 2 the difference between the number of Cooper pairs on the commuteis used to re-distributetermswithinthe matrix M. two superconducting islands which make up the junction. (Formally a and a are defined on different Hilbert spaces 1 2 Furthermore, φ is an operator which represents the phase butfollowingthestandardconventioninquantummechanics difference across the junction. Note that compared to the wecanextendeachoperatortothetensorproductofthetwo expression for the Hamiltonian given in [16], we have Hilbert spaces and then these two extended operators will normalized the expression (15) by dividing through by a commute with each other.) factor of ~ in order to be consistent with the convention We now modify the model of [16] by assuming that the used in [15]. cavityandJosephsonmodesarecoupledtofieldscorrespond- The quantities U, J , n¯, g, ω are physical constants ing to coupling operators of the form ′ associated with the junction and the resonant cavity. In particular, U is the charging energy of the Josephson junc- L= √κ1a1 . tion, J′ is the Josephson energy of the junction, n¯ is an (cid:20) √κ2a2 (cid:21) experimental parameter relating to the gate voltage applied This modification of the quantum system is necessary in tothesuperconductingislands,gisaparameterrelatedtothe order to obtain a damped quantum system whose stability dimensions of the junction, and ω is an angular frequency can be established using the approach of [15] and is quite related to the detuning of the cavity; see [16]. reasonableforanexperimentalsystemwhichwillexperience By a process of completion of squares, defining new damping in both the electromagnetic cavity and in the variables n =n n¯, p =p g√~ω and neglecting the Josephson junction circuit. ′′ ′ ′′ ′ − − constant terms, we can re-write the Hamiltonian as follows: In order to apply Theorem 1 to analyze the stability of this quantum system, we rewrite (16) as = ′ H ω~2 0 0 0 q′ H =H1+H2 1 0 1 g ω 0 p 2[Jq′′ p′′ n′′ φ′] 00 −g0p~ ω~ − U0p~′ ~ 00  nφ′′′′′  Hw−haJ~me′ricelotosHn(iaa12√n+2aa∗2=n)d. aT21nh(cid:2)oatna-qi†su,adawrTaeti(cid:3)chMapvee(cid:20)rtuaarab#aqtui(cid:21)oadnraaHntidacminHloto2mniina=nal. − ~ cosφ′. Then, we define ζ =a /√2 and 2 Now we define new operators a1 = (ωq′ + ip′′)/√2~ω, J′ and a2 = (φ′ + in′′)/√2 which satisfy the canonical f(ζ,ζ∗) = − ~ cos(ζ +ζ∗) commutation relations [a1,a∗1] = 1 and [a2,a∗2] = 1. Then, J′ the Hamiltonian can be re-written in the form f′(ζ,ζ∗) = ~ sin(ζ+ζ∗) H = 21 a† aT M aa# − J~′ cos(a2√+2a∗2) (16) f′′(ζ,ζ∗) = J~′ cos(ζ+ζ∗). (cid:20) (cid:21) (cid:2) (cid:3) From this it follows that a w(9h).erNeoate=th(cid:20)atain12 o(cid:21)rdaenrdtoMwirsitaeHtheermHiatmianiltmonaitarinxionftthhies ffoorrmm f′(ζ,ζ∗)∗f′(ζ,ζ∗)≤4J~′22ζζ∗, f′′(ζ,ζ∗)∗f′′(ζ,ζ∗)≤ J~′22 with the matrix M satisfying (9), it is necessary to apply and γ = ~ . 2J′ The numerical values of the constants ω, g, U, and J ′ are chosen as in [16] as ω = 100GHz, g = 0.15, U = Insulator 2π 2.2087 10 22, J = 3.6652 1011. Also, we calculate − ′ × × γ/2 = 6.8209 10 13. The parameters κ and κ will − 1 2 Superconductor I Superconductor be chosen using×Theorem 1 in order guarantee the robust mean square stability of the quantum system. Indeed, for various values of κ and κ , we form the transfer function 1 2 Gκ1,κ2(s) = E#Σ(sI −F)−1D˜ and calculate its H∞ norm. The H norm of G (s) was found to be virtually κ1,κ2 independe∞nt of κ and so a physically reasonable value of 1 κ = 1011 was chosen. With this value of κ , a plot of 1 1 G (s) versus κ is shown in Figure 2. k κ1,κ2 k∞ 2 Electromagnetic Resonant Cavity From this plotwe can see that stability can be guaranteed for κ > 2.2 1012. Hence, choosing a value of κ = 2 2 × Fig.1. Schematic diagram ofaJosephsonjunction inaresonantcavity. 2.5 1012, it follows that stability of Josephson junction × 16x 10−13 system parameters, it was shown that the robust stability result can be used to choose coupling parameters for the 14 Josephsonjunctionsysteminordertoguaranteerobustmean ∞ k square stability. ) s12 ( κ2 REFERENCES κ,110 [1] M.YanagisawaandH.Kimura,“Transferfunctionapproachtoquan- G k tum control-part I: Dynamics of quantum feedback systems,” IEEE 8 Transactions on Automatic Control, vol. 48, no. 12, pp. 2107–2120, 2003. γ [2] ——,“Transferfunctionapproachtoquantumcontrol-partII:Control 6 2 conceptsandapplications,” IEEETransactionsonAutomaticControl, vol.48,no.12,pp.2121–2132,2003. 4 [3] N. Yamamoto, “Robust observer for uncertain linear quantum systems,”Phys.Rev.A,vol.74,pp.032107–1–032107–10,2006. 2 [4] M. R. James, H. I. Nurdin, and I. R. 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This system involved a cosine function in the system Hamiltonian which was a suitable non-quadratic Hamiltonian leading to a quantum system with a sector bounded nonlinearity. For specific numerical values of the