A Schrodinger Cat Living in Two Boxes

4.8 MB·

by  Chen Wang

A Schro¨dinger Cat Living in Two Boxes
Chen Wang,1, Yvonne Y. Gao,1 Philip Reinhold,1 R. W. Heeres,1 Nissim Ofek,1 Kevin
∗
Chou,1 Christopher Axline,1 Matthew Reagor,1 Jacob Blumoff,1 K. M. Sliwa,1 L. Frunzio,1
S. M. Girvin,1 Liang Jiang,1 M. Mirrahimi,1,2 M. H. Devoret,1 and R. J. Schoelkopf1,
†
1Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06511, USA
2INRIA Paris-Rocquencourt, Domaine de Voluceau, B. P. 105, 78153 Le Chesnay cedex, France
(Dated: January 22, 2016)
Quantum superpositions of distinct coherent states in a single-mode harmonic oscillator, known
as “cat states”, have been an elegant demonstration of Schro¨dinger’s famous cat paradox. Here,
we realize a two-mode cat state of electromagnetic fields in two microwave cavities bridged by a
superconducting artificial atom, which can also be viewed as an entangled pair of single-cavity
cat states. We present full quantum state tomography of this complex cat state over a Hilbert
space exceeding 100 dimensions via quantum non-demolition measurements of the joint photon
6 number parity. The ability to manipulate such multi-cavity quantum states paves the way for
1 logicaloperationsbetweenredundantlyencodedqubitsforfault-tolerantquantumcomputationand
0 communication.
2

n
Rapid progress in controlling individual quantum sys- cies, whose amplitudes are prepared equal for simplicity.
a
J temsoverthepasttwentyyears1,2 hasopenedawiderange Each of the two modes are predominantly localized in one
1 of possibilities of quantum information processing. Poten- of the two cavities that are weakly connected. Despite
2 tial applications ranging from universal quantum compu- a nonzero (but small) spatial overlap of the two modes,
tation to long-distance quantum communication share the we will refer for convenience to the state of each mode as

] central theme of exploiting quantum superpositions within the state of each cavity. Quantum superpositions of the
h
alargeHilbertspace. Furtherstimulatedbycuriosityabout form ψ have been previously realized in the optical do-
p
- the quantum-classical boundary, there has been growing main1|2 ±buit were limited to small and non-orthogonal co-
nt interest in generating superpositions of “macroscopically- herent states (α2 = 0.65). For larger α (i.e. α2 & 2),
| | | | | |
a distinguishable” states that are far apart in phase space. ψ canbeconsideredasinglecatstatelivingintwoboxes
u The canonical example is superpositions of coherent states w| h±oise superposed components are coherent states in a hy-
q ofaharmonicoscillator,i.e. (α + α )with 1/√2 bridizedmodeinvolvingbothAliceandBob. Alternatively,
[ N | i |− i N ≈
at large α, known as “cat states”. The two components inthemorenaturaleigenmodebasis, ψ hasbeenknown
1 correspon|d|todistinctquasi-classicalwave-packets,inanal- as the entangled coherent states in t|he±oiretical studies13,
v ogytoSchr¨odinger’sgedankenexperiment ofanunfortunate and may also be understood as two single-cavity cat states
5 catinsideaclosedboxbeingsimultaneouslydeadandalive. that are entangled with each other.
0
Cat states have so far been realized with single-mode opti- Thetwo-modecatstateisaneigenstateofthejointpho-
5
cal3 or microwave fields1,4 with up to about 100 photons5, ton number parity operator P :
5 J
butareincreasinglysusceptibletodecoherenceatlargesize.
0
. Manipulating a large number of excitations in such har- P =P P =eiπa†aeiπb†b (2)
1 J A B
monic oscillator states is one of two possible approaches to
0
6 expand the information capacity of fully-controlled quan- where a(a†) and b(b†) are the annihilation (creation) oper-
1 tumsystems. Catstates,whichspanaHilbertspacewhose ators of photons in Alice and Bob, and P and P are the
A B
: dimension grows linearly with the number of photons, are photon number parity operators in individual cavities. Re-
v
i an attractive approach for redundantly encoding quantum markably, ψ+ (or ψ ) has definitively an even (or odd)
X information for error correction6–8. The other more tradi- number of|phoitons i|n−thie two cavities combined, while the
r tional way to scale up a quantum system is to build many photon number parity in each cavity separately is maxi-
a modesofexcitations,eachoperatedasatwo-levelqubit,so mally uncertain. Quantum non-demolition measurements
that the Hilbert space dimension increases exponentially of such parity operators not only illustrate the highly non-
with the number of modes9,10. Is it possible to combine classical properties of the state, but also are instrumental
thebenefitsofbothapproachesbycreatingacatstatethat for quantum error correction in general.
lives in more than a single mode or box? The idea of non-
Werealizemeasurementsofthejointphotonnumberpar-
localormulti-modecatdatesbacktotheearlydaysofcav-
ityandsingle-modeparitiesusingthedispersiveinteraction
ity quantum electrodynamics (QED)11, but experimental
withthreeenergylevelsofanartificialatom. Basedonjoint
demonstration has remained a formidable challenge.
paritymeasurements,wefurtherdemonstratefullquantum
Here, we deterministically create a two-mode cat state state tomography of the two-cavity system14. This is ob-
of microwave fields in two superconducting cavities, using
tainedintheformofthejointWignerfunctionW (β ,β ),
J A B
thestrongdispersiveinteractionwithaJosephson-junction-
which is a continuous-variable representation of the quan-
based artificial atom. This state can be expressed as:
tumstatewithβ andβ beingcomplexvariablesinAlice
A B
and Bob respectively. Without correcting for the infidelity
ψ = α α α α (1)
A B A B
| ±i N | i | i ±|− i |− i ofthejointparitymeasurementoperator,weobservequan-
where α and (cid:0) α are coherent states(cid:1)of two mi- tum state fidelity of 81% for a two-mode cat state with
A B
|± i |± i
crowave eigenmodes (Alice and Bob) at different frequen- α=1.92. The high-quality and high-dimensional quantum
2
FIG. 1. Sketch (not to scale) of device ar-
chitecture and experimental protocol. (A)
Athreedimensionalschematicofthedevicecon-
sisting of two coaxial cavities (Alice and Bob), a
Y-shapedtransmonwithasingleJosephsonjunc-
tion(markedby“ ”),andastriplinereadoutres-
×
onator. Allcomponentsarehousedinsideasingle
piece of bulk high-purity aluminum, with artifi-
cialwindowsdrawnforillustrationpurposes. (B)
Atopviewofthesamedevice,showingtherela-
tivepositionofthesapphirechip,centerpostsof
the coaxial cavities, transmon antenna, and the
readout resonator. (C) The microwave control
sequences for generating the two-mode cat state
and performing Wigner tomography. D repre-
β
sentscavitydisplacementbyβ,andasuperscript
g is added if the displacement is conditional on
the ancilla being in g . Rge or Ref represents
| i θ θ
ancillarotationbyangleθ(aroundanaxisinthe
X-Y plane) in the g -e Bloch sphere or e -f
Bloch sphere. R00|iis|ani ancilla g -e ro|taiti|oni
π | i | i
conditionalonthecavitiesbeingin 0 0 . C
A B φ
| i | i
represents cavity phase shift of φ conditional on
the ancilla being in an excited state. By choos-
ing φi +φ0i = π or 2π, we can measure photon
number parity of Alice (P ), Bob (P ), or the
A B
two combined (P ), to perform Wigner tomog-
J
raphy of individual cavities or the joint Wigner
tomography.
control is further manifested by the presence of entangle- (i = A or B) represent the dispersive frequency shifts of
mentexceedingclassicalboundsinaCHSH-styleinequality cavity i associated with the two ancilla transitions. The
for two continuous-variable systems14. Finally, our two- readout resonator and small high-order nonlinearities are
cavity space effectively encodes two coupled logical qubits neglected for simplicity. Using time-dependent external
in the coherent state basis, and we present efficient two- classicaldrivesintheformofmicrowavepulses,wecanper-
qubit tomography in this encoded space. form arbitrary ancilla rotations in both g -e and e -f
| i | i | i | i
Our experimental setup uses a three-dimensional (3D) manifolds,andarbitrarycavitystatedisplacementsinAlice
circuit QED architecture15, where two high-Q 3D cavities (DβA = eβAa†−βA∗a) and Bob (DβB = eβBb†−βB∗b) indepen-
andaquasi-planarreadoutresonatorsimultaneouslycouple dently. More importantly, the state-dependent frequency
to a fixed-frequency transmon-type superconducting qubit shifts (χ’s) allow cavity state manipulations conditioned
(Fig. 1A,B)16. The two cavities that host the cat state of on the ancilla level or vice versa using spectrally-selective
microwavephotonsareimplementationsofthelongest-lived control pulses, thus realizing atom-photon quantum logic
quantummemoryincircuitQEDtodate17. Thetransmon, gates5. Itcanbefurthershownthatwithseparatedriveson
while usually considered a qubit, behaves as an artificial thetwocavitiesandadriveontheancilla,thisHamiltonian
atom with multiple energy levels. We use the transmon as
anancillatomanipulatethemulti-photonstatesinthetwo
cavities, and its lowest three levels, g , e and f , are ac-
| i | i | i
cessed in this experiment. The device is cooled down to 20 ω/2π T1 T2∗
mK in a dilution refrigerator, and microwave transmission Cavities: Alice 4.2196612 GHz 2.2-3.3 ms 0.8-1.1 ms
through the readout resonator is used to projectively mea- Bob 5.4467679 GHz 1.2-1.7 ms 0.6-0.8 ms
sure the ancilla state with a heterodyne detection at room Transmon: e g 4.87805 GHz 65-75 µs 30-45 µs
| i→| i
temperature after multiple stages of amplification. (Ancilla) f e 4.76288 GHz 28-32 µs 12-24 µs
| i→| i
WeconsidertheHamiltonianofthesystemincludingtwo
harmonic cavity modes, a three-level atom, and their dis-
χ/2π Alice Bob
persive interaction (with parameters listed in Table I):
χge 0.71 MHz 1.41 MHz
H/~=ωAa†a+ωBb†b+ωge e e +(ωge+ωef)f f χef 1.54 MHz 0.93 MHz
| ih | | ih |
−χgAea†a|eihe|−(χgAe+χeAf)a†a|fihf| TABLE I. Hamiltonian parameters and coherence times of the
two storage cavities and the transmon ancilla, including tran-
χgeb be e (χge+χef)b bf f (3)
− B † | ih |− B B † | ih | sition frequencies (ω/2π), dispersive shifts between each cavity
and each transmon transition (χ), energy relaxation time (T ),
where ω and ω are the angular frequencies of the two 1
cavities(AAliceanBdBob),ω andω arethe e g and andRamseydecoherencetime(T2∗). Thecavityfrequenciesare
ge ef givenwithaprecisionof 100Hzandarestableoverthecourse
| i→| i
f e transition frequencies of the ancilla, χge and χef of several months. ±
| i→| i i i
3
sition shows stronger interaction with Bob (χge > χge),
B A
while the f e transition shows stronger interaction
| i → | i
with Alice (χef >χef). Manipulating the ancilla in differ-
A B
ent superposition states among the three levels allows us
to concatenate conditional phase gates associated with χge
i
and χef with arbitrary weights16. This additional degree
i
of freedom not only allows for joint parity measurement
P (applying C to both cavities), but also enables parity
J π
measurementofeachcavityP orP individuallywithout
A B
affectingtheother(applyingC andC tothetwocavities
π 2π
respectively).
Basedonsingle-cavityparitymeasurements,wecanmea-
sure the Wigner function of individual cavities, W (β ) =
i i
π2Tr ρDβiPiDβ†i (i=A or B)20,22. The Wigner function is
astandardmethodtofullydeterminethequantumstateof
(cid:2) (cid:3)
a single-continuous-variable system, which represents the
quasi-probability distribution of photons in the quadra-
ture space (Re(β)-Im(β)). Our measured W and W for
A B
a two-mode cat state ψ with α = 1.92 (Fig. 2) illus-
trates that the quantu|m−sitate of either Alice or Bob on
its own is a statistical mixture of two clearly-separated co-
herent states with no coherence between them. In other
FIG. 2. Wigner tomography of individual cavities. words,eachcavitydoesnotcontainaregular(single-mode)
Measured single-cavity scaled Wigner function of (A) Alice cat state, which would contain characteristic interference
(π2WA(βA)) and (B) Bob (π2WB(βB)) respectively for the two- fringesintheWignerfunction5,22 (andcanalsobestraight-
mode cats state |ψ−i, each plotted in the complex plane of forwardly generated in our experiment16). However, for a
Re(β )andIm(β )(i=AorB).Foreithercavity,nointerference
i i state involving inter-cavity entanglement like ψ , single-
fringesareobservedinitsWignerfunction,indicatingastatisti- cavity Wigner functions are insufficient for ch|ar−aicterizing
calmixtureoftwocoherentstates,asopposedtoasingle-cavity
the global quantum state. Such entanglement can be in-
cat state, after tracing out the quantum state of the other cav-
ity. The distortion of the coherent states is due to higher order ferred from measurement of the joint photon number par-
Hamiltonian terms (see supplementary). The photon number ity, PJ = 0.81 0.01, even though each cavity alone
h i − ±
parity within each cavity is close to 0, reflected by the value of shows mean photon number parity of P P 0.
A B
h i ≈ h i ≈
respective Wigner functions near the origin. Additional evidence of the joint parity can be seen in a
spectroscopy measurement16.
A full quantum state tomography of the two-cavity sys-
permits universal quantum control of the entire system18.
tem can be realized by measuring the joint Wigner func-
We generate the two-mode cat state ψ deterministi- tion23:
cally using a series of logic gates as sho|w±niin Fig. 1C19.
In particular, we implement effective displacements (Dg ) 4
of both Alice and Bob conditional on ancilla being2αin WJ(βA,βB)= π2hPJ(βA,βB)i
g 16, which realizes a three-way entangling gate, 1 g + 4
|ei 0 0 g 0 0 + e 2α 2α √.2 T| hien = π2Tr ρDβADβBPJDβ†BDβ†A (4)
A B A B A B (cid:0)
a|ni(cid:1)a|nicil|lairot→atioNn(cid:0)(|Riπ0|0)i c|onidition|ail|onith|e ciav(cid:1)ity state WJ isafunctioninthefour-(cid:2)dimensional(4D)phas(cid:3)espace,
|0iA|0iB disentangles the ancilla, and subsequent cavity whose value at each point (Re(βA), Im(βA), Re(βB),
displacements leave the cavities in a two-mode cat state. Im(β )), after rescaling by π2/4, is equal to the expec-
B
Therotationaxiscontrolsthesign(ormoregenerally,phase tation value of the joint parity after independent displace-
angle) of the cat state superposition. ments in Alice and Bob14. For simplicity, we will therefore
We probe the cat state by quantum non-demolition use the scaled joint Wigner function, or “displaced joint
(QND) measurements of the photon number parity. Parity parity” P (β ,β ) to represent the cavity state. P at
J A B J
measurementofasinglecavityusingadispersivelycoupled any givehn point (β ,iβ ) is directly measured by avheragiing
A B
ancilla qubit has been previously demonstrated20,21, where single-shot readout outcomes and takes values between -1
a conditional cavity phase shift4, Cφ =I g g +eiφa†a and +1. To illustrate the core features in this 4D Wigner
⊗| ih | ⊗
e e, of φ = π allows the cavity states with even or odd function of the state ψ , we show its two-dimensional
|phioht|on numbers to be mapped to g or e of the qubit (2D) cuts along the R| e−(βi )-Re(β ) plane and Im(β )-
A B A
| i | i
for subsequent readout. In our multi-cavity architecture, Im(β ) plane for both the calculated ideal state (Fig. 3A,
B
measuring the joint photon number parity requires C in B, also see Ref. 14) and the measured data (Fig. 3C, D).
π
both Alice and Bob, which is difficult to achieve simulta- The Wigner function contains two positively-valued Gaus-
neously with existing techniques21 unless χge = χge. We sian hyperspheres representing the probability distribution
A B
overcome this challenge by exploiting the f -level of the of the two coherent-state components, and an interference
| i
transmon. By designing the frequency of the ancilla to structure around the origin with strong negativity. Excel-
be between those of the two cavities, the e g tran- lent agreement is achieved between measurement and the-
| i → | i
4
FIG. 3. Joint Wigner tomography.
(A, B) Two-dimensional plane-cut along
(A) axes Re(β )-Re(β ) and (B) axes
A B
Im(β )-Im(β )ofthecalculated4Dscaled
A B
joint Wigner function P (β ,β ) of the
J A B
h i
ideal odd-parity two-mode cat state ψ
| −i
with α = 1.92. The red features in
(A) represent the probability distribu-
tion of the two coherent states compo-
nents. The central blue feature in (A)
and fringes in (B) demonstrate quantum
interference between the two components.
(C,D)ThecorrespondingRe(β )-Re(β )
A B
andIm(β )-Im(β )plane-cutsofthemea-
A B
sured joint Wigner function of ψ , to
| −i
be compared with the ideal results in (A)
and (B) respectively. Data are taken in a
81 81 grid, where every point represents
×
anaverageofabout2000binaryjointpar-
ity measurements. (E) Diagonal line-cuts
of the data shown in (A) and (C), cor-
responding to 1D plots of the calculated
(black) and measured (red) scaled joint
Wigner function along Re(β ) = Re(β )
A B
with Im(β )= Im(β )=0. (F) Diagonal
A B
line-cutsofthedatashownin(B)and(D),
correspondingto1Dplotsofthecalculated
(black) and measured (red) scaled joint
Wigner function along Im(β ) = Im(β )
A B
with Re(β )= Re(β )=0.
A B
ory, with the raw data showing an overall 81% contrast of into itinerant microwave signals and/or optical photons24.
the ideal Wigner function. Comprehensive measurements
Compared with other reported quantum states of two
of P intheentire4Dparameterspacefurtherallowusto
h Ji harmonic oscillators, a striking property of the two-mode
reconstructthedensitymatrixofthequantumstate,which
cat state is that its underlying compositions are highly-
shows a total fidelity of also about 81% against the ideal distinguishable. Two-mode squeezed states25–28 have
ψ state. The actual state fidelity may be significantly
|hig−hier if various errors associated with tomography are re- shown strong entanglement, but are Gaussian states with-
out the Wigner negativity and the phase space separation
moved16. Additional visualization of the Wigner function
as in a cat state. The “N00N” state, an entangled state
data is presented in a Supplementary movie16.
inthediscreteFockstatebasis, typicallyrequiresquantum
operations of N photons one by one and so far has been
Analyzed within the energy eigen-mode basis (Alice and
realized with up to 5 photons29,30. The two components
Bob),thetwo-modecatstateisamanifestationofquantum
of the cat state in Fig. 3 have a phase space separation of
entanglement between two quasi-classical systems. The
α ( α) =√15 in each cavity, giving an action distance
entanglement can be tested against a CHSH-style Bell’s | − − |
of√30inthe4Dphasespace,oracatsize22 of30photons.
inequality constructed from displaced joint parity at 4
points in the phase space14. We measure a Bell signal16 Our technique in principle allows generation of two-mode
cat states with arbitrary size using the same operation. So
of2.17 0.01forthestateinFig.3, exceedingtheclassical
± far we have measured cat sizes of up to 80 photons16, and
bound of 2. Without complete spatial separation and fully
more macroscopic states can be achieved by implement-
independentreadoutofthetwomodes,theviolationshould
ing numerically optimized control pulses31 and engineering
be considered a demonstration of the fidelity of the entan-
more favorable Hamiltonian parameters.
glement and the measurement rather than a true test of
non-locality. Nevertheless, various schemes exist to further Compared with single-cavity quantum states, the addi-
separatethetwomodessuchasconvertingthecavityfields tion of the second cavity mode increases the quantum in-
5
FIG. 4. Encoded two-qubit tomogra-
phy. (A) Red bars show tomography of
two logical qubits encoded in the coherent
state basis of two cavities, with the pre-
paredstatebeinganeven-paritytwo-mode
cat state, ψ with α = 1.92. Gray bars
+
| i
represent the ideal state. Insets show the
Re(β )-Re(β )andIm(β )-Im(β )plane-
A B A B
cuts of the measured scaled joint Wigner
function of the same state. The measured
identity operator differs from 1 as a re-
sult of the parity measurement infidelity
and leakage out of the code space. (B)
Encoded two-qubit tomography of an ap-
proximate product state of single-cavity
cat states, 0 α α α
N | i−|− i A⊗ | i−|−
α also with α = 1.92. Insets show
i B (cid:0) (cid:1) (cid:0)
plane-cuts of the measured scaled joint
(cid:1)
Wigner function of the same state. Given
familiarity with single-cavity cat states,
theseWignerfunctionpatternscanbeun-
derstood by considering W (β ,β ) =
J A B
W (β )W (β ) for separable states. For
A A B B
example, the “checkerboard” patterns in
the Im(β )-Im(β ) plane-cuts can be un-
A B
derstoodbymultiplyingorthogonalfringes
from the two independent cat states.
formation capacity significantly. Despite the modest mean space (β ,β )16. The encoded two-qubit tomography of a
A B
photon numbers, a full tomography of the two-mode cat state ψ withα=1.92isshowninFig.4A,providingadi-
+
state (partly shown in Fig. 3) requires a Hilbert space of rectfi|deliityestimation33 of 1 II + XX YY + ZZ
4 h i h i−h i h i
at least 100 dimensions to be described (capturing 99% = 78% against the ideal Bell state, surpassing the 50%
(cid:0) (cid:1)
of the population), comparable to a 6 or 7 qubit GHZ bound for classical correlation. As a comparison, Fig. 4B
state. Our conservatively estimated quantum state fidelity illustrates a product state of single-mode cat states in Al-
iscomparabletothatreportedforan8-qubitGHZstatein ice and Bob, which is identified as X X in the
A B
|− i |− i
trapped ions9 and the largest GHZ state in superconduct- logical space. For both states illustrated here, the two-
ing circuits10 (5 qubits). In addition, a great advantage of qubit tomography suggests that errors within the encoded
continuous-variable quantum control is illustrated by our space are quite small. The reduced contrast compared to
hardware-efficientquantumstatetomographyprotocolthat the ideal state is mostly due to infidelity of the joint par-
covers an enormous Hilbert space by simply varying two ity measurement and leakage out of the code space (due to
complex variables of cavity displacements. higher-order Hamiltonian terms).
An important motivation for creating multi-cavity cat InthisReport,wehavedemonstratedaSchr¨odinger’scat
statesistoimplementapromisingparadigmtowardsfault- that lives in two cavities. This two-mode cat state is not
tolerantquantumcomputation7,32,whereinformationisre- only a beautiful manifestation of mesoscopic superposition
dundantly encoded in the coherent state basis5. This ap- andentanglementconstructedfromquasi-classicalstates13,
proach has recently led to the first realization of quantum but also a highly-desirable resource for quantum metrol-
error correction of a logical qubit achieving the break-even ogy34, quantum networks and teleportation35. Moreover,
point8. Inthiscontext,ourexperimentrealizesanarchitec- the demonstration of high-fidelity quantum control over
ture of two coupled logical qubits. The two-mode cat state the large two-cavity Hilbert space has important implica-
canbeconsideredatwo-qubitBellstate 1 0 0 1 1 , tions for continuous-variable-based quantum computation.
√2 | i| i±| i| i
where the quasi-orthogonal coherent states α in each The measurement of the joint photon number parity real-
(cid:0)|± i (cid:1)
of the two cavities represent 0 and 1 of a logical qubit. ized here is QND by design, and will play a central role
| i | i in quantum error correction6,8,18 and facilitating concur-
Foranytwo-qubitlogicalstateencodedinthissubspace,
rent remote entanglement36 in a modular architecture of
we can perform efficient tomography without extensive
quantum computation.
measurement of the joint Wigner function. This is carried
out by measuring P at 16 selected points of the phase We thank A. Petrenko, Z. Leghtas, B. Vlastakis,
J
h i
6
W. Pfaff, M. Silveri and R. T. Brierley for helpful dis- the Air Force Office of Scientific Research (FA9550-14-1-
cussions, and M. J. Hatridge and A. Narla for assistance 0052). FacilitiesusewassupportedbytheYaleInstitutefor
on the Josephson parametric amplifier. This research was NanoscienceandQuantumEngineering(YINQE),theYale
supported by the U.S. Army Research Office (W911NF- SEAS cleanroom, and the NSF (MRSECDMR 1119826).
14-1-0011), NSF DMR-1301798 and the Multidisciplinary Y. Y. Gao acknowledges support from an A*STAR NSS
University Research Initiatives program (MURI) through Fellowship.
∗ [email protected] Schoelkopf, arXiv:1508.05882 [cond-mat, physics:quant-ph]
† [email protected] (2015).
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Supplementary Material:
A Schro¨dinger Cat living in two boxes
Chen Wang,1 Yvonne Y. Gao,1 Philip Reinhold,1 R. W. Heeres,1 Nissim Ofek,1 Kevin
Chou,1 Christopher Axline,1 Matthew Reagor,1 Jacob Blumoff,1 K. M. Sliwa,1 L. Frunzio,1
S. M. Girvin,1 Liang Jiang,1 M. Mirrahimi,1,2 M. H. Devoret,1 and R. J. Schoelkopf1
1Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06511, USA
2INRIA Paris-Rocquencourt, Domaine de Voluceau, B. P. 105, 78153 Le Chesnay cedex, France
(Dated: January 22, 2016)
MATERIALS AND METHODS
6 A. Device Architecture
1
0
Our cQED system includes two 3D cavities, a quasi-
2
planarlinearresonator,andaY-shapedtransmon. Asingle

n block of high-purity (5N5) aluminum is machined to form
a
a 3D structure that contains both superconducting cavity
J
resonators and also functions as a package for the sapphire
1
chip with deposited Josephson junction. Each of the two
2
cavities can be considered a 3D version of a λ/4 transmis-

] sion line resonator between a center stub 3.2 mm in di-
h ameter and a cylindrical wall (outer conductor) 9.5 mm in
p diameter1. The heights of the stubs control the resonance
-
t frequency,andareabout12.2mmand16.3mmrespectively
n
forAliceandBob. Atunnel(withamaximumwidthof5.8
a
mm and a maximum height of 3.9 mm) is opened from the
u
q outside towards the middle wall between the two cavities,
[ creating a three way joint between the tunnel and the two

cavities (see photo image, Fig. S1). The whole package is
1
chemically etched by about 80 µm after machining to im-
v
5 prove the surface quality of the cavity resonators2.
0 The superconducting transmon is on a 5.5 mm 27.5
×
5 mm chip, which is diced from a 430 µm-thick c-plane sap-
5 phire wafer after fabrication. The fabrication process uses FIG. S1. Overview of the 3D cQED device. (A) A pho-
0 electron-beam lithography and the standard shadow-mask tograph of the full assembly of the device used for this study.
1. evaporation of Al/AlOx/Al Josephson junction. The sap- Themachinedaluminumpackagecontainstwocoaxialstubcav-
0 phirechipisinsertedintothetunnel,withtheantennapads ity resonators. A sapphire chip hosting the transmon ancilla,
6 clampedbyanaluminumchip-holder,isinsertedthroughatun-
of the transmon slightly intruding into the coaxial cavities
1 neltobecoupledtothetwocavitiesfromtheside. Thesapphire
toprovidemodecoupling. Thechipismechanicallyheldat
: chip contains an extra strip of aluminum, forming a stripline
v one end with an aluminum clamping structure and indium
resonatorwiththetunnelwall. (B)Amicrographimageofthe
Xi seal3. Y-shaped transmon. (C) A schematic effective circuit of the
During the transmon fabrication process, a 100 µm 9.8 cQED system containing three LC oscillators and one anhar-
r ×
a mmstripofaluminumfilmisalsodepositedonthesapphire monic oscillator (artificial atom) coupled together.
chip. This metal strip and the wall of the tunnel form a
planar-3D hybrid λ/2 stripline resonator. This resonator
designhastheadvantagesofbothlithographicdimensional
controlandlowsurface/radiationloss,andissystematically
studied in Ref. 3. Here it is capacitively coupled to the
transmon,andstronglycoupledtoa50Ωtransmissionline
ity compared with the other modes, and is treated explic-
for readout.
itly as a three-level artificial atom. Following Black-box
quantization of superconducting circuits4, the other cav-
ity/resonator modes are modeled as near-harmonic oscilla-
B. Sytem Hamiltonian Model
tors with weak nonlinearity inherited from coupling to the
Josephson junction.
The cQED system has four bosonic modes involved in
thisexperiment: thetwo3Dcavities,thereadoutresonator
and the transmon ancilla. The transmon can be under- The full system Hamiltonian can then be written in the
stood as an LC oscillator with much larger anharmonic- followingformuptothefourthorderinthecouplingofthe
2
resonators to the transmon: C. Measurement Setup and Protocol
1 1 1
H/~=ωA(a†a+ )+ωB(b†b++ )+ωR(r†r+ ) Fig. S2 shows the diagram of our measurement setup.
2 2 2
The device package is installed inside a Cryoperm mag-
+ω e e +(ω +ω )f f
ge ge ef netic shield and thermalized to the mixing chamber of a
| ih | | ih |
χgea ae e (χge+χef)a af f dilution refrigerator with a base temperature of 20 mK.
− A † | ih |− A A † | ih | Low-pass filters and infrared (eccosorb) filters are used to
χgeb be e (χge+χef)b bf f
− B † | ih |− B B † | ih | reducestrayradiationandphotonshotnoise. AJosephson
χger r e e (χge+χef)r r f f parametric converter (JPC) is also mounted to the 20 mK
− R † | ih |− R R † | ih | stage, connected to the output port of the device package
K K K
A B R
a†a†aa b†b†bb r†r†rr via circulators, providing near-quantum-limited amplifica-
− 2 − 2 − 2
tion, with a power gain of 20dB and bandwidth of 5MHz.
K a ab b K a ar r K b br r (S1)
− AB † † − AR † † − BR † † We use a field programmable gate array (FPGA) to op-
erate both the quantum-control pulse sequences and the
Eq.(S1)isanexpandedversionofEq.(3)ofthemaintext,
data acquisition process. Our experiment does not rely
whichnowincludesthereadoutresonator(withsubscriptR
on the real-time feedback capability of the FPGA, but our
andoperatorsrandr )aswellasKerrnonlinearitiesofthe
† in-house programmed FPGA offers an important practi-
cavities. Thefirsttworowsrepresenttheexcitationenergy
cal advantage in its ability to compute rather than store
of the all modes, explicitly including the transmon anhar-
sideband-modulation waveforms. This leads to minimal
monicity of ω ω =115.17 MHz. The next three rows
ge ef usage of waveform memory for a large number of differ-
−
are second order terms ( 1 MHz) representing the dis-
ent cavity displacements, allowing us to measure the joint
∼
persive interactions (χ’s) between the transmon and each
Wignerfunctionoveralargenumberofpointsinthephase
of the three resonators. The last two rows are the fourth
space in a single run.
order terms ( 10 kHz), including the self-Kerr energies
∼ Both cavity drives and transmon drives are generated
(K ,K ,K ) of the resonators and the cross-Kerr inter-
A B R by sideband-modulation of continuous-wave (CW) carrier
actionsbetweenanypairsofresonators(K ,K ,K ).
AB AR BR tones produced by respective microwave generators. The 4
AllHamiltonianparametersofourdevicearelistedinTable
FPGAanaloguechannelsareusedas2IQ-pairsthatcontrol
S1.
the cavity drives to implement arbitrary cavity displace-
The key Hamiltonian terms that enable the cat state
ments. Rotations of the transmon ancilla are controlled by
generationandjointparitymeasurementarethedispersive
anotherpairofIQchannelsprovidedbyanarbitrarywave-
shifts,χge,χef,χge,χef,highlightedinthemaintext. Since form generator (AWG) synchronized to the FPGA via a
A A B B
the readout resonator is always kept in the vacuum state digital marker. This IQ pair controls both g -e and e -
until a final measurement is needed, its Hamiltonian terms f transitions by using different intermedia|tie|friequenc|ieis
do not affect the quantum control and quantum state evo- |(IFi).
lution in this experiment. The fourth-order Hamiltonian
Ancillareadoutisperformedbyheterodynemeasurement
terms of Alice and Bob give a minor contribution to the
of the microwave transmission of a readout pulse through
infidelity of the experiment. The self-Kerr terms (K and
A the two ports of the quasi-planar readout resonator near
K )inducedistortionoftheGaussianprobabilitydistribu-
B its resonance frequency. Using the well-established cQED
tion of the coherent state components of the cat state, and dispersive readout6, the amplitude and phase of the trans-
will cause state collapse and revival at long time scales5.
mitted signal depends on the quantum state of the ancilla.
The cross-Kerr interaction K induces spontaneous ent-
AB This readout pulse is produced by a microwave generator
TrueanglementbetweenAliceandBoboverlongtimescales
(RO) gated by a FPGA digital channel. The transmitted
in ways not included in our simple analysis.
signal, after being amplified by the JPC, is further am-
plified by a high electron mobility transistor (HEMT) at
4K and a regular RF amplifier at room temperature. The
Frequency Nonlinear interactions versus: amplified signal is then mixed down to 50 MHz with the
ω/2π Alice Bob Readout output of a “local oscillator” (LO) microwave generator,
e g 4.87805 GHz 0.71 MHz 1.41 MHz 1.74 MHz and analyzed by the FPGA. A split copy of the readout
|fi→|ei 4.76288 GHz 1.54 MHz 0.93 MHz 1.63 MHz pulse is directly mixed with the LO without entering the
| i→| i
Alice 4.2196612 GHz 0.83 kHz -9 kHz 5 kHz refrigerator to provide a phase reference for the measured
Bob 5.4467677 GHz -9 kHz 5.6 kHz 12 kHz transmission.
Readout 7.6970 GHz 5 kHz 12 kHz 7 kHz
The long lifetimes of the cavities allow preparation of
highly coherent cavity quantum states, but severely limits
TABLE S1. Hamiltonian parameters of all cQED components,
the rate one can repeat the measurement process. (With
includingthetransmonancilla,thetwocavityresonators(Alice
T 3msforAlice,ittakes15-20msforthecavityphoton
andBob)andthereadoutresonator. Themeasuredparameters 1 ≈
number to naturally decay to the order of 0.01.) Since to-
include all transition frequencies (ω/2π), dispersive shifts be-
tweeneachresonatorandeachtransmontransition(χ/2π),the mographic measurement of the two-cavity quantum state
self-Kerr of Alice (K /2π) and Bob (K /2π), and the cross- requires large amounts of measurements, we implement
A B
Kerr interaction between Alice and Bob (K /2π). The Kerr four-wave mixing processes to realize fast reset for both
AB
parameters and χef associated with the readout resonator are cavities7. These processes effectively convert photons in
theoretical estimates based on the other measured parameters. Alice or Bob into photons in the short-lived readout res-
3
FIG.S2. Circuitdiagramofthemeasure-
ment setup. We use a field programmable
gatearray(FPGA)tocontroltheexperiment.
TheFPGAhasatotalof4analoguechannels,
which are used to provide I-Q control of the
classical cavity drives via sideband modula-
tionoftheoutputsofmicrowavegeneratorsla-
beled“Alice”and“Bob”. Thesedrivesrealize
arbitrary cavity displacement operations D
α
on the two high-Q cavities (Alice and Bob).
A digital channel from the FPGA triggers an
arbitrary waveform generator (AWG), whose
two analogue channels provide I-Q control of
theancilladrivesforboth g e and e f
| i−| i | i−| i
rotation, modulating the microwave tone la-
beled “ancilla”. The readout pulse is gener-
ated by a “RO” generator gated by a FPGA
digital pulse, which is transmitted through
the readout resonator for measuring the an-
cillastate. Thetransmittedsignalisamplified
by a Josephson parametric converter (JPC)
at 15 mK, a high electron mobility transis-
tor (HEMT) at 4K, a standard (Mini-circuit)
RF amplifier at room temperature, and then
mixed with a local oscillator (“LO”) to pro-
duce a 50 MHz signal to be digitized and
recordedbytheFPGA.Anothertwoanalogue
channels of the AWG controls an off-resonant
pumponthereadoutresonator(“ROpump”).
This drive, together with off-resonant pumps
onAliceandBob,allowsfastresetofthehigh-
Q cavities through 4-wave mixing processes.
FIG. S3. Data acquisition flow chart with state initialization and reset. Purification of the initial ground state against
equilibrium excitations is implemented by performing two additional readouts of the ancilla state preceding each run of the actual
experiment. The first readout is performed without any ancilla/cavity manipulation, intended to filter out the cases where the
ancilla starts in e . Then two π-pulses of ancilla g -e rotations are applied, with the first conditional on 0 0 , and the
A B
| i | i | i | i | i
second unconditional. The second readout of the ancilla state discriminates the presence or absence of photons in the two cavities,
and allows filtering out the cases where either Alice or Bob starts with 1 or more photons. The pulse sequences for the cat state
generationandtomographyfollowthisinitializationsequence,andweselectdatafromrunswherebothreadoutsreturntheoutcome
of g . We use a four-wave mixing method to reset the cavity state close to vacuum by applying three microwave pump tones at
| i
frequencies detuned from Alice, Bob and ancilla respectively for a duration of 400 µs.
4
onator mode using three parametric pumping tones. We Ramsey interference of 0 and 1 Fock states following a
| i | i
apply this reset operation for 400 µs, and acquire our ex- Selective Number Arbitrary Phase (SNAP) gate9 that pre-
perimental data with a repetition cycle of about 900 µs. pares the Fock state superposition. This method, which is
Ideally, in thermal equilibrium at our base temperature described in Ref. 1, extracts the T without being affected
2∗
(20 mK) all modes (Alice, Bob, or ancilla) of our quantum by high-order nonlinearities (Fig. S4). Together with the
system should be in their ground state. However, in our measurement of the cavity T , we find the cavity pure de-
1
experiment there is a non-negligible probability that any phasing time T = 1/( 1 1 ) = 1.1 0.2 ms for Alice
of the three modes is found in an excited state possibly and 0.9 0.2 mφs for BoTb2∗.−Su2cTh1 pure de±phasing times can
due to insufficient thermalization. These erroneous excited be fully±explained by their dispersive frequency shifts due
state populations, about 8% for the ancilla and 2-3% for to the thermal excitation of the transmon ancilla1. The
Alice and Bob, can reduce the fidelity of the subsequently- rate of the g e transition of the transmon, Γ , can be
prepared quantum state and the parity measurement. To determined|frio→m|itis T and thermal population o↑f the e
1
eliminate these effects, we perform two measurements as state, P : Γ = P /T = 7.5%/(70 µs) = 1/(0.9 ms). |Ini
e e 1
shown in Fig. S3 to “purify” the initial state of the system addition,the↑cavityRamseyexperimentalsoallowsextrac-
before we start each run of the experiment. This is imple- tion of the precise resonance frequency of the cavity. Over
mented by post-selecting the cases where our experiment the course of 8 months while the device was continuously
starts from the ground state, g 0 A 0 B, before any non- operated at 20 mK, we observed no slow drift of cavity
| i| i | i
trivial quantum operation is performed, which amounts to frequency exceeding its linewidth ( 200 Hz).
about 80% of the all the data acquired. It should be noted ≈
The dispersive frequency shifts (χ’s) are measured both
thattheuseofpost-selectionhereispurelyforexperimental
in the frequency domain by spectroscopy techniques and
convenienceanddoesnotcompromisethedeterministicna-
in the time domain by a Ramsey-type qubit state revival
tureofthegenerationofcatstates. Onecaninprincipleuse experiment10) (Fig. S5). These techniques established pre-
real-time feedback to prepare the initial state and achieve
viously for 2-level qubits, can both be extended for the
a slightly higher data rate. No post-selection beyond the
3-level artificial atom considered in this study. The latter
ground state initialization is applied in our analyses of the
measurementisalsoavaluableprocedurefortuningupthe
two-mode cat state.
joint parity measurement.
The self-Kerr effect of Alice or Bob (K or K terms in
A B
the Hamiltonian) can be characterized by the collapse and
D. Device Characterization
revival of a single-cavity coherent state5. In addition, with
two long-lived cavities, we are able to directly measure the
The Hamiltonian parameters and coherence properties inter-cavity cross-Kerr effect (K ) by observing the fre-
AB
of the device are mostly characterized by adapting estab- quency shift of Bob in response to the presence of photons
lished techniques, in particular, various forms of Ramsey in Alice.
interferometry. The measured coherence times of the Al-
ice, Bob and the ancilla are listed in Table I of the main
text, and a more complete list is provided in Table S2. We
brieflycommentonafewcharacterizationmethodsthatare
E. Cat State Generation
noteworthy.
Measurementofthecoherenceofhigherexcitedstatesof
a superconducting transmon is a relatively new topic re- Thetwo-modecatstateisgenerateddeterministicallyus-
cently reported in Ref. 8. In addition to using a similar ing conditional operations between the ancilla and the two
sequence to determine T between f and e , we note cavities. As presented in the main text (Fig. 1C), the gen-
2∗,ef | i | i eration sequence is composed of the following steps11: 1)
thatT (between f and g )ismorerelevanttoourex-
2∗,gf | i | i preparingancillasuperposition(Rge ), 2)displacementsof
pTerimaenndt.TThis,bTu2∗t,gcfatnimbeedceatnenromtinbeedsbimypalyRadmersievyedexfproemr- Alice and Bob conditional on the aπn/2cilla state (Dg ), real-
2∗,ge 2∗,ef 2α
iment starting from a 1 (g + f ) superposition. izingathree-wayentanglinggate,3)conditionalflip(reset)
√2 | i | i of ancilla (R00), disentangling it from the cavity state, 4)
ThecoherencetimesofAliceandBobaremeasuredusing π
unconditional displacements of Alice and Bob (D ) to
α
center the cat state in the phase space (which is a−trivial
step purely for convenience of presentation).
T1 T2∗ T2E Pe The conditional displacement (Dg ) is the key step in
Ancilla e 65-75 µs 30-45 µs 55-65 µs 7.5% 2α
Ancilla |fi 26-32 µs 12-24 µsa 20-30 µsa 0.5% this state generation process. Although this operation can
| i be directly implemented using cavity drives with a band-
Alice 2.2-3.3 ms 0.8-1.1 ms N/A 2-3%
widthsmallerthanthedispersiveinteractionstrength(χge,
Bob 1.2-1.7 ms 0.6-0.8 ms N/A 2-3% i
i=A or B), such a method requires a rather long pulse du-
Readout 260-290 ns N/A N/A <0.2%
ration (and therefore higher infidelity due to decoherence
a T2∗=11-19µsfor|fivs.|gi,T2E=18-26µsfor|fivs.|gi. andKerreffects). Alternatively, foreachcavityweusetwo
unconditional displacements separated by a wait time ∆t
TABLE S2. Energy relaxation time (T ), Ramsey coherence
1 in between to effectively realize Dg .
time (T2∗) , coherence time with Hahn echo (T2E), and thermal 2α
population of the excited state (P ) of all components when During the wait time ∆t, due to the dispersive inter-
e
applicable. action, cavity coherent states in both cavities accumulate