Orthogonally-Driven Superconducting Qubit in Circuit QED

0.21 MB·English

by  M. J. Storcz

Orthogonally-Driven Superconducting Qubit in Circuit QED
M.J. Storcz∗,1 M. Mariantoni∗,2 H. Christ,3 A. Emmert,2 A. Marx,2
W.D. Oliver,4 R. Gross,2 F.K. Wilhelm,1 and E. Solano1,5
1Physics Department, CeNS and ASC, Ludwig-Maximilians-Universit¨at, Theresienstrasse 37, 80333 Munich, Germany
2Walther-Meissner-Institut, Bayerische Akademie der Wissenschaften,
7 Walther-Meissner-Strasse 8, 85748 Garching, Germany
0 3Max-Planck Institute for Quantum Optics, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany
0 4MIT Lincoln Laboratory, 244 Wood Street, Lexington, Massachussets 02420, USA
2 5Secci´on F´ısica, Departamento de Ciencias, Pontificia Universidad Cat´olica del Peru´, Apartado 1761, Lima, Peru

n (Dated: February 3, 2008)
a
J We consider a superconducting charge qubit coupled to distinct orthogonal electromagnetic field
modesbelonging toa coplanar waveguide resonator and a microstrip transmission line. Thisarchi-
7
tectureallows thesimultaneous implementation ofa Jaynes-Cummingsand anti-Jaynes-Cummings

dynamics,aresonantmethodforgeneratingmesoscopicqubit-fieldsuperpositionsandforfield-state
]
l reconstruction. Furthermore, we utilize this setup to propose a field measurement technique that
l
a is, in principle, robust duetoa fast pre-measurement to qubit dephasing and field relaxation.
h
- PACSnumbers: 74.81.Fa,42.50.Dv,32.80.-t
s
e
m
In recent years, superconducting quantum circuits asuperconductingchargequbitcoupledtothequantized,
.
t have demonstrated key elements required for quantum discrete-mode spectrum of a quasi-1D coplanar waveg-
a
information processing [1], including the possibility to uide (CWG) resonator, here called cavity [5], and to a
m
prepare a desired qubit state, coherently manipulate it, multi-layer microstrip transmission line (MTL), which
-
d readitout,andperformpreliminaryconditionalgateop- will be utilized to tune the qubit-cavity resonance. This
n erations [2, 3, 4]. In analogy to quantum-optical cavity archetypical construction ideally subjects the qubit to
o QED, superconducting chargeand flux qubits have been twoorthogonalelectromagneticfieldsandconstitutesthe
c
coupled to on-chip microwave resonators [5, 6], and uni- systemforourtheoreticalinvestigations(SeeFig.1). We
[
versal two-qubit gates mediated by a single cavity mode will show that strongly driving the MTL with coherent

3 havebeenproposed[7]. Insuchanarchitecture,thescal- field pulses modulates the strength of the qubit-cavity
v
ing of the system would require a homogeneouscoupling interaction and, in the strong-driving limit [11], medi-
6
of many qubits with the same cavity mode and a means ates the emergence of a simultaneous JC and anti-JC
2
2 to address each logical qubit individually. Prototypical dynamics in the system, allowing the generation of non-
2 advancedmanipulationschemeshavealreadybeenimple- classical qubit-cavity states [12]. Furthermore, we pro-
1 mented in quantum-optical systems. In trapped ion ex- pose the measurement of relevant observables of the mi-
6
periments, tuned lasers can be switched between single- crowavecavity field via a protocolthat utilizes the MTL
0
ion carrier excitations and ion-motion JC dynamics [8]. as an independent tool for read-out of the qubit popu-
/
at Also, in 3D microwave cavities, a flying atom can per- lation [13]. Finally, we will demonstrate that this tech-
m form local rotations in Ramsey zones before and after nique is, in principle, robust due to a fast qubit-cavity
entering the cavity, in which a Jaynes-Cummings (JC) pre-measurementto the presenceofqubitdephasingand
-
d interaction takes place [9]. In the emerging field of cir- field decoherence processes.
n
cuit QED[5, 7], a higher level of addressabilityand con- The Hamiltonian describing the interaction between a
o
trolisalsodesirable[10]. Forinstance,itisimportantto single Cooper-pair box (CPB) charge qubit, the second
c
: enablecontrolledintracavity qubitrotations,while keep- harmonicofthe undrivenCWGresonator,andthe MTL
v
ing a switchable coupling to the cavity modes, aiming (see Fig. 1), driven with a propagating coherent state
i
X at intercavity qubit-qubit transfer of information. This acting as an AC gate charge,can be written as [14]
r implies the necessity to generate nonclassical field states
a and the possibility of measuring them via rapid qubit Hˆ¯ = 2E 1 2nDC σˆ¯ EJ(Φx)σˆ¯ +~ω aˆ†aˆ
− C − C z − 2 x C 2 2
operations. These requirements could be simultaneously (cid:0) (cid:1)
metifeachqubitwerecoupledtotwo(ormore)indepen- + ~gQCσˆ¯z aˆ†2+aˆ2 +4ECnAMC(t)σˆ¯z. (1)
(cid:16) (cid:17)
dent modes with orthogonal field polarizations.
Here, E =e2/C represents the charging energy of the
C Σ
CPB (C is the total box capacitance), nDC is the num-
InthisLetter,weproposeanarchitectureconsistingof Σ C
ber of pairs induced by the DC gate voltagethrough the
CWG cavity, E is the qubit Josephson energy, which
J
canbetunedbyanexternalquasi-staticfluxbiasΦ ,ap-
x
∗Theseauthors contributedequallytothiswork. plied through a adequately engineered loop, aˆ†,aˆ are
{ 2 2}
2
conditions [E /(2~)=ω =ω ], and within a standard
J C M
Metal 2 rotating-wave approximation (RWA) in the interaction
Layer 2 picture, Eq. (2) can be rewritten as
HˆI = ~g σˆ aˆ +σˆ aˆ† +~gβ σˆ . (3)
QC + 2 − 2 QM x
Metal 1 (cid:16) (cid:17)
A more advanced application of Eq. (3) is obtained
Layer 1
in the strong-driving limit, gβ g . In this case,
QM ≫ QC
following Ref. [11], we utilize an additional interaction
representation with respect to the second term on the
Metal 0
r.h.s. of Eq. (3). Decomposing σ =(σ iσ )/2 allows
± x y
Layer 0 ±
one to omit the quickly-precessing σ term through a
y
second RWA, yielding the effective Hamiltonian
g
Hˆ = ~ QC σˆ +σˆ aˆ† +aˆ . (4)
eff 2 + − 2 2
(cid:0) (cid:1)(cid:16) (cid:17)
This strong-driving limit results in a circuit-QED re-
alization of a simultaneously resonant JC and anti-JC
dynamics. It is characterized here by a large coupling
g /2, and it enables the generation of mesoscopic su-
QC
perpositionstatesbetweenthequbitandthecavityfield.
Figure1: (Coloronline)Sketchoftheproposedsetup,includ- For example, taking the initial qubit-cavity state to be
ingtheblow-upofdifferentlayers, wheretheCPBiscoupled
g,0 = (+ + ) 0 /√2, where are the σˆ
to orthogonal fields produced bya CWG and a MTL. | Ci | i |−i | Ci |±i x
eigenstates of the qubit with eigenvalues 1, the evo-
±
lution associated with Eq. (4) after an interaction time
t yields the following Schr¨odinger cat state
the bosonic creation and destruction operators relative int
to the second harmonic of the CWG resonator, ω and
C (+ α + α )
gQC are the corresponding angular frequency and qubit- |Ψcati= | i| i √|2−i|− i , (5)
cavity vacuum Rabi coupling, respectively, nAC(t) is the
M
number of pairs induced by the AC gate voltage applied with α = ig t /2. Using realistic experimental pa-
via the MTL, and σˆ¯ ,σˆ¯ are Pauli spin-1/2operators. rameters, −e.g.,QCg int 100 MHz [18] and cavity decay
With 4ECnAMC(t) ={(xCQMz}VQ0M/2e)|β|cos(ωMt+ϑβ), we rate κC ≈ 0.1 MQCHz≈, would allow the generation of a
can define ~gβ (C V0 /2e)β , where g is the Schr¨odinger cat state with amplitude α = nˆ =
QM ≡ QM QM | | QM | | h cati
qubit-MTL coupling strength, C and V0 are the g /2κ 1/3 8 (64 photons), obtained fopr a maxi-
QM QM QC C ≈
associated capacitance and vacuum gate voltage, β = m(cid:0) um inter(cid:1)action time t =1/κeff, where κeff = α2κ .
|β|expiϑβ is the amplitude of the driven coherent field This amplitude comparienst well wCith the besCt valu|e|s oCb-
state |βi, ωM its angular frequency, and ϑβ a phase. tained in 3D microwave cavity QED with circular Ry-
The direct coupling between the CWG resonator and dberg atoms [19] and previous proposals in the reso-
the MTL is suppressed due to the high-degree of isola- nant [20] and dispersive regimes [21].
tionbetweentheirrespectivemodes. TheMTLmagnetic In order to generate successfully the specific state
field distribution [15] is predominantly tangential to the Ψ inside the cavity, it is necessary to decouple the
cat
CPB loop, suppressing unwanted jitter of the σˆ¯x term q|ubiti-cavity system after the desired interaction time
in Eq. (1) due to MTL noise. t . This may be implemented by applying a sec-
int
Working in the eigenbasis g , e of the first two ond strong coherent state to the MTL, with decou-
{| i | i}
terms of Eq. (1), the system Hamiltonian takes the form pling frequency ω and amplitude γ , that AC-
dec | dec|
Stark shifts the CPB by an amount δ. At this point,
Ω
Hˆ = σˆ +~ω aˆ†aˆ + ~g aˆ† +aˆ +4E nAC(t) a single-shot measurement of the qubit state g (e )
2 z C 2 2 QC 2 2 C M | i | i
h (cid:16) (cid:17) i would leave the field state in an even (odd) coherent
×(cosθσˆz −sinθσˆx) , (2) state |Ψeαi = (|αi +|−αi)/ 1+exp(−|α|2) (|Ψoαi =
(α α )/ 1 exp( αp2)) [11]. This qubit mea-
where {σˆx,σˆz} are Pauli matrices in the {|gi,|ei} eigen- s|uriem−e|n−t maiy bpe m−ade, fo−r|in|stance, by driving a prob-
basis, Ω=qEJ2+ 4EC 1−2nDCC 2 is the qubit level ing field via the third harmonic of the CWG resonator,
separation, and θ =(cid:2) arct(cid:0)an E /4E(cid:1)(cid:3) 1 2nDC is the which is strongly detuned from the qubit transition fre-
J C − C
mixing angle. Operating t(cid:2)he qubit(cid:0)at the de(cid:1)g(cid:3)eneracy quency by an amount ω /2 g , for a time t ,
C ≫ QC meas
point, i.e., for nDC = 1/2, under complete resonance and thereby allowing an independent QND read-out of
C
3
the qubit population via the resultant phase shift of the result
probing field [22].
d
The architecture proposed here could be applied Yˆ = P+φ(τ) , (7)
directly to implement conventional cavity field mea- D φE dτ e (cid:12)(cid:12)
(cid:12)τ→0+
surements, consisting of an initial qubit-cavity pre- (cid:12)
(cid:12)
measurement in which the qubit acts as the quantum where Yˆ = aˆ e−iφ aˆ†eiφ /2i is a field quadrature,
probewithwhichthecavityfieldisentangled,followedby φ (cid:16) 2 − 2 (cid:17)
a measurement of the qubit [23]. Typically, one requires conjugate to the quadrature Xˆφ = aˆ2e−iφ+aˆ†2eiφ /2
(cid:16) (cid:17)
an interaction time sufficiently long (order of the inverse obtainedbyreplacingφ (φ π/2). Equation(7)shows
→ −
interaction frequency) to entangle strongly the cavity mathematically that the first derivative of the measured
field with the qubit. Accordingly, one typically desires excited-statequbitpopulationobtainedatinfinitesimally
to operate in the qubit-cavity strong-coupling regime to smallinteractiontimecontainsinformationaboutcavity-
minimize this interaction time and, thereby, reduce the field observables with no influence of decoherence pro-
noisyactionofdecoherence[8,9]. Incontrasttothiscom- cesses. If the qubit is now prepared in the excited state
mon approach, we will show below an alternative means e ,itisalsopossibletodeterminethemeanphotonnum-
| i
to implement a measurement of the cavity field with a ber of the cavity field, via
relatively fast qubit-cavity pre-measurement(small frac-
tion of the inverse interaction time) and minimal action 1 d2
nˆ = aˆ†aˆ = Pe(τ) 1. (8)
of decoherence [13]. h i −2d2τ e (cid:12)(cid:12) −
aftWeremcoenassuidreinrgnothwetqhuebeivteinnctohheegrreonutnstdasttea|tΨeeαgio,batsasineeend Fastpre-measu(cid:10)rem(cid:11)ents,plusthenea(cid:12)(cid:12)(cid:12)rτu→n0i+tvisibilityread-
| i
from Eq. (5). Through the MTL, a π/2-pulse of dura- out of qubit populations reported in Ref. [22], turns the
tion t and resonant with the shifted qubit transition proposedmeasurementschemeintoanalternativemeans
rot
can be applied, yielding ρˆ (t ) = + + ρˆe, with to measure cavity-field observables with reduced noise
S S | φih φ|⊗ α
+ = (g + eiφ e )/√2. Here, φ is a relative phase, disturbance.
ρ|ˆeφ=i Ψe| iΨe , an|dit = t +t +t may be set to Although Eqs. (7) and (8) are exact mathematical
tα= 0|. αAiththαi|s point,Sif thientaforemmentriootned decoupling expressions, they do not represent a realistic theoreti-
S
pulse (frequency ω and amplitude γ ) applied via cal description of a measurement. In this sense, it is
the MTL is turneddeocff for a time t |, dtehce| qubit-cavity useful to consider an estimator for the derivatives in
off
state ρˆ (0) will evolve via a resonant JC dynamics for a Eqs. (7) and (8) over short, but non-zero, measurement
S
dimensionless time τ =g t . times ∆τ through the Taylor expansion: P(τ +∆τ) =
QC off
P(τ) + P′(τ)∆τ + P′′(τ)(∆τ)2/2!+ [(∆τ)3]. Subse-
In the presence of a dispersive bath, producing qubit
O
quently, it can be shown that
dephasing, and a thermal bath at zero temperature, in-
ducing field dissipation, the system dynamical equation
∆P+φ(τ) (κ +γ )
becomes e Yˆ 1+ C φ Yˆ ∆τ, (9)
∆τ (cid:12)(cid:12) ≈D φE−(cid:20) 4gQC D φE(cid:21)
d 1 γ (cid:12)τ=0
ρˆ = Hˆ ,ρˆ i Aˆ†Aˆ ,ρˆ 2Aˆ ρˆ Aˆ† , (cid:12)
dτ S i~h QC Si−Xi 2 (cid:16)n i i So− i S i(cid:17) where the (cid:12)second term on the r.h.s. is the dominant
higher-ordercontributiontoEq.(7),which,evenforideal
ensemble averaging,containsthe field decay rateκ and
where Hˆ = ~g σˆ aˆ +σˆ aˆ† , the braces denote C
QC QC + 2 − 2 the qubit dephasing rate γ . Equation (9) shows that in
(cid:16) (cid:17) φ
anti-commutators,andthesumrunsovertwoindicescor- the strong-coupling regime, κ ,γ < g , or even in
responding to qubit dephasing (Aˆ = σˆ ,γ = γ ) and { C φ} QC
1 z 1 φ the weak-coupling regime, κC,γφ >gQC, it is possible
cavitydecay(Aˆ =aˆ ,γ =κ ),respectively. Above,we to identify a time ∆τ that{is long}enough to allow the
2 2 2 C
have neglected the energy relaxation of the qubit which readout of Yˆ and short enough to suppress the effects
h φi
is assumed to happen on a longer timescale [22]. Calcu- of decoherence during the pre-measurement. This result
latingnowdP (τ)/dτ =d e e /dτ =Tr ρˆ˙ e e ,and illustratesthat,inprinciple,strong-couplingbetweenthe
e h| ih |i h S| ih |i probe and the system is not required to implement the
after some algebra, we obtain
proposed measurement scheme.
d 1 Establishinganaccurateestimatorwiththeprescribed
dτPe(τ)= i~Dh|eihe|,HˆQCiE . (6) accuracy for the ensemble averages in Eqs. (7) and (8)
will, in general, require many repetitions of the pre-
This expression does not contain terms involving qubit scribed measurement. There is a trade-off between the
dephasing or field decay rates, which, ideally, are elim- length of the physically implemented ∆τ, the number of
inated by the expectation value calculated at time τ. measurement repetitions, and the strength of the qubit-
Evaluating Eq. (6) in the limit τ 0+ yields the useful cavity coupling required to achieve a desired degree of
→
4
noise immunity. In consequence, the proposed measure- may be applied to the measurement of the field quadra-
ment technique consists of two clear steps. First, a fast tures using a measurement scheme requiring a relatively
pre-measurement allows the pointer (qubit) to encode short interaction time between the system (cavity) and
the information about the system (cavity field), mini- the probe (qubit). We have also shown that full state
mizingdecoherenceprocesses. Second,thereadoutofthe reconstruction of unknown fields can be obtained. The
pointer,happeningtypicallyoverlongertimes,completes multimode concepts described here should bring greater
the quantum measurement procedure. flexibility to the field of circuit QED, in particular those
Finally, another useful application of our scheme is aimingatthesystematicscaling-upofquantumgatesand
the possibility to perform full state reconstruction of an quantum information devices.
unknown field state ρˆ = Ψ Ψ or, equivalently, of its
correspondingcharacterist|icifuhnc|tionχ(α˜)=Tr[ρˆD(α˜)]. This work was supported by DFG through SFB 631.
Here, D(α˜) is the displacement operator and α˜ the com- TheworkatLincolnLaboratorywassponsoredbytheUS
plex amplitude in phase space of an arbitrary coherent DoD under Air Force Contract No. FA8721-05-C-0002.
state [24]. For the sake of convenience, a pure field state ES acknowledges support of EU EuroSQIP project.
willbeconsideredhere,eventhoughtheseresultsremain
valid for any arbitrary mixed state. Following a simi-
lar protocol to the one outlined above, the qubit can be
initially prepared in the state +θ =(+ +eiθ )/√2,
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C d
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g [8] D.Leibfried,R.Blatt,C.Monroe,andD.Wineland,Rev.
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χ(α)= P ,0 +i P , . (11)
g g
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[14] Constant energy offsets, the cavity vacuum term, and
In Fig. 1, dielectric Layer 0 is the substrate for the en-
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