Table Of ContentQuantum 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
(
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2
1
0
1010 1011 1012 1013 1014
Freq rad/s
Fig.3. Magnitude BodeplotofGκ1,κ2(s).
IV. CONCLUSIONS
We have applied a previous result on the robust stability
of nonlinear quantum systems to a quantum system arising
from a Josephson junction coupled to an electromagnetic
resonant cavity. 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
