Realization of a Λ system with metastable states of a capacitively-shunted fluxonium

N. Earnest,1, ∗ S. Chakram,1, † Y. Lu,1 N. Irons,2 R. K. Naik,1

N. Leung,1 Jay Lawrence,3 Jens Koch,2 and D. I. Schuster1, ‡

1The James Franck Institute and Department of Physics,University of Chicago, Chicago, Illinois 60637, USA

2Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA3Department of Physics, Dartmouth College, Hanover, New Hampshire 03755, USA

We realize a Λ system in a superconducting circuit, with metastable states exhibiting lifetimes upto 7 ms. We exponentially suppress the tunneling matrix elements involved in spontaneous energyrelaxation by creating a “heavy” fluxonium, realized by adding a capacitive shunt to the originalcircuit design. The device allows for both cavity-assisted and direct fluorescent readout, as well asstate preparation schemes akin to optical pumping. Since direct transitions between the metastablestates are strongly suppressed, we utilize Raman transitions for coherent manipulation of the states.

PACS numbers:

Quantum computation has seen rapid progress overthe past decade, with quantum systems of increasingcomplexity being developed in a number of experimen-tal platforms [1–3]. Superconducting circuit quantumelectrodynamics (cQED) is among the most promisingof these, with fast microwave control, rapidly increas-ing coherence times [4, 5] and relatively simple means ofscaling to larger system sizes. Improving the coherencetimes of individual qubits, however, is still an outstand-ing challenge in performing large-scale quantum compu-tation and simulation. According to Fermi’s golden rule,lifetimes of superconducting qubits are governed by twofactors: (1) transition matrix elements, which are de-termined by the qubit-environment coupling as well asthe form of the qubit wavefunctions; (2) the noise spec-tral density associated with the environmental degrees offreedom. To date, improvements of qubit lifetimes haveprimarily been achieved by modifying the noise spectraldensity – for example, by filtering [6, 7] and by embeddingqubits in 3D cavities [8]. Interestingly, the wavefunctionsof popular superconducting qubits such as transmon [9]and flux qubits operated at sweet spots [10–12], actuallyallow for rather large transition matrix elements. Hence,the reduction of environmental noise is particularly crit-ical to enhance energy relaxation times in these devices.

Most current superconducting qubit architectures arebased on variants of the transmon qubit [9, 13–15].Transmons have large dipole matrix elements, simple se-lection rules, and a small non-linearity, sufficient to re-solve the lowest energy levels of the qubit states. In con-trast to the transmon, flux qubits [12, 16, 17], have alarge nonlinearity, rich level structure, and selection rulesthat can be finely engineered to yield a smooth trade-offbetween decay matrix elements and susceptibility to de-phasing. With the realization of a linear superinductanceand the fluxonium qubit [18], this class of qubits has sinceseen enhanced lifetimes and reduced flux-noise induceddecoherence [19]. Furthermore, we use the potential en-ergy landscape of this class of flux/fluxonium qubits to

realize a Λ system in a superconducting artificial atom.

Traditionally, a Λ system is comprised of a ground andmetastable excited state, coherently coupled through theuse of a third intermediate state. Λ systems such as thosebased on ground hyperfine and optical excited states intrapped atoms and ions [20], are commonly observed inatomic physics. Here, the metastability arises not be-cause of small matrix elements, but due to the largedifference in frequency scales (ν) between optical-dipoleand microwave-hyperfine transitions, with the decay rateγ ∝ ν3 as a result of the 3D density of states [21]. Super-conducting qubits have a much smaller dynamic range offrequency scales and a 1D density of states (γ ∝ ν), mak-ing it more challenging to realize the metastability re-quired to explore the physics associated with Λ systems.Nonetheless, previous work in cQED has utilized the Pur-cell effect [22] to modify the density of states and explorecoherent interactions of three-level systems [23, 24].

In this work, we present a Λ system in a capacitivelyshunted fluxonium circuit: the heavy fluxonium. Thiscircuit has suppressed transition matrix elements for en-ergy relaxation and metastable states with lifetimes ashigh as 7 ms. The suppressed transition matrix elementsalso make controlled population transfer to metastablestates a challenge. We overcome this issue by using Ra-man transitions in the Λ system to perform coherent op-erations on the states.

The circuit for the heavy fluxonium device (Fig. 1a,b)consists of a small-area Josephson junction connected inparallel to a capacitance (Cq) and a large superinduc-tor (LjA), realized as a chain of 100 large-area Joseph-son junctions. To ensure idealized inductive behavior ofthe junction array, the chain length and individual junc-tion size must satisfy several conditions as explained in[18], which sets an upper limit on realizable inductancevalues with currently developed fabrication techniques.Once these conditions are satisfied, the Hamiltonian of

arX

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0065

6v1

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ant-

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3 J

ul 2

017

2

Cin

Cout

CC

LR CR

Cq EJ ···Φext LjA

-2π -2π-4π 4π2π 2π0 0Phase, ϕ Phase, ϕ

|g0〉|g-1〉

|f0〉

|e0〉

|g1〉 |g0〉

|e0〉

|g1〉

|e1〉

|g-1〉 |g2〉

Ene

rgy

a b

c d

FIG. 1: a) Equivalent circuit diagram of the heavy fluxoniumcapacitively coupled to a readout resonator (see Supplemen-tary Information). b) Scanning Electron Microscope image ofthe device, with a magnified view of the 100 Josephson junc-tion array, fabricated using the bridgeless method detailed in[25]. c) Potential energy landscape at Φext = 0.02Φ0 demon-strating localized wavefunctions in three wells. We label thestates by the fluxoid state number of the well of primaryresidence (−1, 0, 1) and by the excitation within each well(g, e, f). d) Potential energy landscape at Φext = 0.51Φ0,where fluxoid states (|g0〉,|g1〉) are nearly degenerate.

the fluxonium is given by:

Hf = −4ECd2

dϕ2−EJ cos(ϕ−2πΦext/Φ0)+

1

2ELϕ

2 (1)

where EC = e2/2Cq is the capacitive charging energy, EJthe Josephson energy of the small junction, and EL =Φ2

0/2LjA the inductive energy of the Josephson junctionarray. In contrast to earlier fluxonium devices [18], theheavy fluxonium intentionally shunts the small junctionwith a large capacitance (43 fF, dashed red squares inFig. 1b). This results in a reduced EC/h = 0.46 GHz, in-creases the effective mass of the phase degree of freedom,and produces quasi-localized states in the different wellsof the potential (see Fig. 1c,d). The other parameters ofthe circuit, EJ/h = 8.11 GHz and EL/h = 0.24 GHz, arecomparable to those in previous fluxonium devices. Welabel states by their fluxoid number −1, 0, 1 (number offlux quanta in the superconducting loop formed by thejunctions), and by the plasmon levels within that well|g〉 , |e〉 , |f〉. The heavy fluxonium allows for two typesof transitions: intra-well plasmons (e.g., |g0〉 ↔ |e0〉),and inter-well fluxons (e.g., |g0〉 ↔ |g1〉).

Inter-well transitions involve states with wavefunctionssuch as ψg0(ϕ) and ψg1(ϕ) which are localized in separate

wells. Accordingly, matrix elements∫dϕψ∗g1(ϕ)O ψg0(ϕ)

with respect to local operators O(d/dϕ, ϕ) will be expo-nentially suppressed with ∼ exp[−π2(EJ/8EC)1/2], ascan be inferred from considering the tails of displacedharmonic-oscillator wavefunctions centered in adjacentwells. As a result, the |g1〉 state of the heavy fluxonium ismuch longer lived as compared to the original fluxonium.However, the suppressed transition matrix elements alsomake coherent operations more challenging. This circuitresembles a recently reported design [26], whose dipolemoment (and thereby the fluxon transition rate) is tun-able through the use of a SQUID in place of a singleJosephson junction. Unlike the fluxonium in [26], ourheavy fluxonium – with a fixed EJ/EC ≈ 18 – is suffi-ciently heavy to disallow coherent direct drives. We solvethis issue by realizing a Λ system between the groundstate |g0〉, the metastable state |g1〉, and the excited state|f0〉 of the center well, and performing coherent Ramantransitions between |g0〉 and |g1〉.

For fast readout, the heavy fluxonium is capacitivelycoupled to a lossy resonator (Q ∼ 500). The Hamiltonianof the combined system is given by [27]:

HS = Hf + hνra†a+

∑j,k

hg |j〉 〈k| 〈j| n |k〉 (a+ a†), (2)

where, νr = 4.95 GHz is the bare frequency of the read-out resonator and g = 76 MHz is the coupling betweenthe readout resonator and the fluxonium (as extractedfrom fits to spectra of the combined resonator-fluxoniumsystem). n is the charge operator of the fluxonium andcontrols the transition rates arising from driving on theinput port (Cin in Fig. 1a).

Single-tone spectroscopy (Fig. 2) immediately revealsboth the resonator photon and the plasmon transitions.The curvature of the plasmon transition arises from flux-induced distortion of the well (See supplementary infor-mation for details), and allows one to easily distinguishbetween wells (blue and magenta lines in Fig. 2). Fur-thermore, the strong hybridization of the plasmon andthe resonator (detuned ≈ 155 MHz above the coupled res-onator) allows for fluorescent readout of the metastablestate, over the entire flux range, through cycling the plas-mon transition of the metastable state many times, sim-ilar to quantum non-demolition measurements of singletrapped ions and atoms [28, 29].

The tunnel splitting between the wells can be directlyobserved in the plasmon spectrum at Φext = Φ0/2. Atthis flux location, there are two identical wells with de-generate ground and first excited states. This results inthe feature shown in the left inset of Fig. 2, in which theinterference of the levels results in a unique rhombus-shaped avoided crossing. The separation of the levelcrossings forming the top and bottom corners of the

3

|g0〉 → |e0〉

|g1〉 → |e1〉

|g0〉 → |e1〉

|g1〉 → |e0〉|g0〉 → |g1〉

|g0,0r〉 → |g0,1r〉

|g1,0r〉 → |g1,1r〉 |g1,0r〉 → |g0,1r〉

|g0,0r〉 → |g1,1r〉

Flux (Φext/Φ0)

Pro

be F

requ

ency

(GH

z)

2te

2tg

FIG. 2: Single-tone spectroscopy of the fluxonium-resonatorsystem in the vicinity of the resonator and primary plasmontransition frequencies. Dashed lines indicate predicted en-ergy levels based on device parameters extracted from fits tosingle and two-tone spectra. Transitions that change rapidlywith flux are inter-well fluxon transitions, while the flattertransitions are intra-well plasmon transitions. Left inset: in-terference due to coupling between the ground and excitedstates of neighbouring wells. Right inset: features associatedwith fluxon transitions crossing with the resonator.

rhombus (black arrows) is a direct measure of twice thetunnel coupling of the excited states in the well (|e0〉 and|e1〉), corresponding to te ≈ 7 MHz. In contrast, the tun-nel splitting between ground states is smaller than thelinewidth of the plasmon and fluxon transitions, and isinferred from the fits to be tg ≈ 0.42 MHz, one thousandtimes smaller than in previous experiments [18]. An-other set of avoided crossings is visible in the resonatortransmission peak (right inset of Fig. 2) at Φext ≈ 0.5Φ0.The outer set of crossings arise from the |g1〉 → |e0〉fluxon transition (also seen in the bottom of the left in-set), while the inner crossings are formed by compos-ite resonator/fluxon transitions: |g1, 0res〉 → |g0, 1res〉and |g0, 0res〉 → |g1, 1res〉. The latter indicate that cou-pling between fluxon transitions is increased when aphoton is present in the resonator (see Supplement forphoton-assisted fluxon transitions). The fluxon transi-tions are approximately linear in flux, with slopes givenby ∂f/∂Φext ≈ ±4π2EL/Φ0 = ±9.59 GHz/Φ0.

Fluxon transitions, not seen in single tone spec-troscopy, can be identified via two-tone spectroscopy inwhich we monitor the transmission of the readout res-onator while sweeping the frequency of a second drivetone. A unique feature of the device under study is thatthe plasmon, fluxon, and resonator photon energies areall designed to be very close together, enabling rich inter-play in the neighborhood of Φext = 0 (|Φext| < 0.03 Φ0,Fig. 3a). The lines of largest slope are the single-photoninter-well fluxon transitions |g0〉 → |g±1〉. The restof the lines are two-photon transitions to the second-excited manifold of the fluxonium-resonator system, with

flat features corresponding to plasmons and sloped res-onances corresponding to fluxons. Of particular impor-tance are the plasmon-like two-photon features locatedat ∼ 4.73 GHz corresponding to |g0〉 → |f0〉 two-photontransition, which will assist in performing coherent oper-ations on the qubit.

The rich energy-level structure of the heavy fluxoniumallows for a variety of state-preparation schemes. We canperform T1 measurements from the highest fluxon tran-sition frequency of 4.65 GHz down to about 3 GHz bydirectly driving the fluxon transition at high powers torealize a classically mixed state (100µs pulse duration).Below this point, we perform T1 measurements using aprocess that is similar to optical pumping [30] in atomicphysics. The bright plasmon features shown in Fig. 2provide a means to ‘pump’ the device into the excitedfluxoid state (|g1〉). Through continuous cycling of the|g0〉 → |e0〉 plasmon, we take advantage of a small proba-bility of decaying from |e0〉 to |g1〉 arising from the finitematrix element between these states. Using this methodwe incoherently pump the system into the |g1〉 state, andsubsequently perform a typical T1 measurement.

Since direct fluxon transitions are forbidden, we needto take full advantage of the rich fluxonium energy struc-ture to perform coherent state preparation. We do soby means of Raman transitions that utilize the excitedlevels of the fluxonium. As tunneling is suppressed ex-ponentially by the depth of the well, it is advantageousto use higher plasmon excited states, especially thosecloser to the rim of the wells. Of particular importanceis the |g0〉 → |f0〉 transition shown in the inset of Fig. 3.Though the direct transition is disallowed by the sym-metry of the wavefunctions, we can access it througha two-photon process mediated by the |e0〉 level. Fur-ther, from the inset in Fig. 3b we can see that the|f0〉 wavefunction has a noticeable amplitude in the rightwell, and |g1〉 → |f0〉 is dipole allowed. This indicatesthat we can use the |g0〉, |f0〉, and |g1〉 states to forma Λ system. We explore Raman transitions in this Λsystem by sweeping the pump and probe tone frequen-cies in the vicinity of these transitions, as shown in Fig.3b. We find a shift in the resonator transmission when2νpump − νprobe = E|g1〉 − E|g0〉, corresponding to theintended transfer of population from |g0〉 → |g1〉. TheRaman transition rate is related to the Rabi rates of thetwo Raman tones, Ωprobe from |g1〉 → |f0〉, and Ωpump

from |g0〉 → |f0〉 according to:

Ωg0g1 =ΩprobeΩ2

pump

∆δ2γ, (3)

where ∆ = 2νpump − E|f0〉 = νprobe − (E|f0〉 − E|g1〉)is the detuning of the pump and probe tone from thetwo-photon resonance, while δ2γ = E|e0〉 − νpump is thedetuning of the two-photon |g0〉 → |f0〉 pump tone fromthe intermediate |e0〉 state that mediates the two-photon

4

Δ

νpump

νprobe

|g0〉

|g1〉|e0〉

|f0〉

Flux (Φext /Φ0) Pump Detuning, δνpump (MHz)

Pro

be F

requ

ency

, νpr

obe (

GH

z)

Pum

p Fr

eque

ncy,

νpu

mp (

GH

z)

2νpump-νprobe = Eg1-Eg0

a b

FIG. 3: a) Two-tone spectroscopy showing direct fluxon transitions (orange and green lines) and two-photon transitions tothe two-excitation manifold (|f0〉, |e−1〉,|e1〉). States are labeled by plasmon levels g, e, f and by the persistent current state(fluxoid) number −1, 0, 1 (state labeling for the transitions is valid for Φext > 0). The |f0〉 level serves as the intermediatestate in a Λ system comprising the ground state |g0〉 and the metastable |g1〉 state, and assists in Raman transitions. b)Pump-probe spectroscopy of Raman transitions between fluxoid states (|g0〉 , |g1〉) as a function of pump (near |g0〉 → |f0〉-2γ transition) and probe frequency (near |g1〉 → |f0〉). The Raman transition between the localized states is seen when2νpump−νprobe = E|g1〉−E|g0〉, indicating a Raman transition between these states. The upper-left inset shows the wavefunctionsof the states involved in the transition. The intermediate |f0〉 state couples to |g0〉 via a two-photon process, and has a smallamplitude in the right well, with a dipole-allowed transition to the excited fluxoid state |g1〉, allowing a Λ system to be formed.

transition. Ωpump and Ωprobe are set by the strengthof the drive and by the charge matrix elements of the|g1〉 → |f0〉 and |g0〉 → |e0〉 transitions, respectively.

Having established the Λ system and the necessarytones required to perform a Raman transition betweenthe otherwise forbidden metastable states, we induceRabi oscillations by simultaneously switching on resonantpump and probe drives. The pump is detuned 30 MHzfrom the two-photon |f0〉 transition, and the probe fre-quency is chosen to be νprobe = 2νpump − ∆Eg1g0 . Ata flux value of 0.078Φ0 we achieve a π pulse rate oftπ ∼ 400ns with 90% contrast (Fig. 4b). While thisfidelity can be further optimized in future devices, itdemonstrates several orders of magnitude improvementfrom the direct drive which takes 100µs to generate aclassically mixed state at this flux value (Supplementaryinformation). The upper limit of the Raman transitionrate arises from off-resonant excitation of the resonatorthrough the two-photon pump drive, which additionallyleads to the driving of the |g0, 1res〉 → |g1, 1res〉 transi-tion.

Using these three different methods, direct driving,plasmon pumping, and a three-photon Raman transi-tion, we measure the T1 of the device over the flux range0 ≤ |Φext| < 0.45Φ0, as shown in Fig. 4a. The T1 im-proves as we move toward 0.5Φ0 following the (inversesquare of the) charge matrix elements. Although ad-ditional frequency-dependent contributions exist, this issuggestive of the T1 being limited by a charge-based loss

mechanism, as was also observed in the recent work on asimilar fluxonium device [26]. Furthermore, we success-fully measure the coherence of the fluxon transition us-ing a standard Ramsey sequence (Fig. 4c) with π

2 pulsesextracted from Rabi oscillations (Fig. 4b). The T ∗2 ismeasured to be 530 ns, likely limited by flux-noise. Aspin-echo experiment using Raman transition-based π/2and π pulses gives a T2,echo of 1.3µs with a single insertedπ pulse.

In summary, we have realized a heavy fluxonium in a2D cQED architecture, with metastable states exhibit-ing lifetimes of several milliseconds over a broad range offlux values, likely limited by a charge-based loss mech-anism. We study the coherence of the device by statepreparation schemes that use the rich energy level struc-ture of the device, including a process analogous to op-tical pumping. We perform coherent operations on thelong-lived metastable states using a three-photon Ramantransition using an excited plasmon level as the inter-mediate state, realizing single-fluxon gates (tπ ∼ 400ns)that are several orders of magnitude faster than directlydriving the fluxon transition with comparable microwavepowers. Additionally, the relative proximity of the plas-mon and readout resonator allow for rich photon andplasmon-mediated transitions, that could be useful forhigh-fidelity fluorescent readout and for photon detectionschemes with cQED (Supplementary information).

In future work, we seek to engineer Raman transi-tions with higher fidelity by placing the resonator above

5

0.0 0.1 0.2 0.3 0.4 0.5Flux ( ext/ 0)

0

1

10T 1

(ms)

10

g 0|n

|g1

2

0 15 30 45Time (ms)

0.0

0.5

Popu

latio

n

0 500 1000Idle time [ns]

0.4

0.5

0.6

0.7

Excit

ed st

ate

popu

latio

n

0 1 2Rabi drive time ( s)

1

0

1

Prob

e de

tuni

ng (M

Hz)

1

a

b c

FIG. 4: a) Energy relaxation time (T1) as a function of fluxmeasured through a combination of direct-drive to a mixedstate, plasmon pumping, and Raman transitions. The red lineindicates the inverse square of the charge matrix element ofthe fluxon transition of interest. The inset shows the T1 decaycurve of the point indicated by the red state, after drivingthe fluxon levels to a mixed state following a long Ramandrive. (b) Rabi chevron obtained by detuning the probe-drivetone away from the Raman transition described in Fig. 4bat Φext = 0.078Φ0. The Raman transition is chosen to be60 MHz from the |f0〉 level and the peak π pulse fidelity is∼ 90%. (c) Ramsey experiment at Φext = 0.078Φ0, obtainedusing π/2 pulses extracted from Rabi drive of the Ramantransition resulting in a T ∗2 of 500-550 ns.

the primary plasmon, and by also increasing the detun-ing between them. Additionally, we will explore statepreparation using RF flux pulses and sideband transi-tions made possible by the 2D cQED realization of theheavy-fluxonium and by the inclusion of a tunable EJ .We also hope to further increase the inductance to re-duce dephasing rates. The large inductance fabricationtechniques developed here will be useful for other typesof protected qubits including the 0−π [31] and Josephsonrhombus chain qubits [32].

We thank L. Ocola and D. Czaplewski at Argonne na-tional laboratory for their substantial help in the fabrica-tion of this device, R. Ma for help with the pulsed mea-surement setup, and B. Baker for useful discussions. Thismaterial is based upon work supported by the Army Re-search Office under (W911NF-15-1-0421), and supportedby the National Science Foundation Graduate ResearchFellowship under Grant No. DGE-1144082. Use of theCenter for Nanoscale Materials, an Office of Science userfacility, was supported by the U. S. Department of En-

ergy, Office of Science, Office of Basic Energy Sciences,under Contract No. DE-AC02-06CH11357. We grate-fully acknowledge support from the David and LucillePackard Foundation.

∗ Electronic address: nearnest@uchicago.edu†

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[32] M. T. Bell, J. Paramanandam, L. B. Ioffe, and M. E.Gershenson, Physical Review Letters 112, 167001 (2014).

Device design and fabrication

The device (shown in Fig. 1 in the main text) wasfabricated on a 430 µm thick C-plane sapphire substrate.The base layer of the device, which includes the majorityof the circuit (excluding the Josephson junctions of thetransmon), consists of 150 nm of niobium deposited viaelectron-beam evaporation at 0.9 nm/s, with features de-fined via optical lithography and reactive ion etch (RIE)at wafer-scale. The wafer was then diced into 7x7 mmchips. The junction mask was defined via electron-beamlithography with a bi-layer resist (MMA-PMMA) in thebridgeless junction pattern [25], with overlap pads fordirect galvanic contact to the optically defined capaci-tors. Before deposition, the overlap regions on the pre-deposited capacitors were milled in-situ with an argonion mill to remove the native oxide. The junctions werethen deposited with a three step electron-beam evapora-tion and oxidation process. First, an initial 30 nm layerof aluminum was deposited at 1 nm/s at an angle of -29 relative to the normal of the substrate. Next, thejunctions were exposed to 50 mBar of high-purity O2 for45 minutes for the first layer to grow a native oxide. Fi-nally, a second 60 nm layer of aluminum was depositedat 1 nm/s at 29 relative to the normal of the substrate.Another oxidation step at 3mbar for 5minutes was doneafter to put a clean oxide layer atop the aluminum. Afterevaporation, the remaining resist was removed via liftoffin N-Methyl-2-pyrrolidone (NMP) at 80C for 6 hours,leaving only the junctions directly connected to the baselayer, as seen in the inset of Figure 1 of the main text.After both the evaporation and liftoff, the device was ex-posed to an ion-producing fan for 15 minutes, in order toavoid electrostatic discharge of the junctions.

The device is mounted and wirebonded to a multilayercopper PCB microwave-launcher board. Additional wire-bonds connect separated portions of the ground plane toeliminate spurious differential modes. The device chiprests in a pocketed OFHC copper fixture that pressesthe chip against the launcher board. Notably, the fixturecontains an additional air pocket below the chip to alter3D cavity modes resulting from the chip and enclosure,shifting their resonance frequencies well above the rele-vant band by reducing the effective dielectric constant ofthe cavity volume.

Equivalent Circuit

The actual circuit of the heavy fluxonium device isshown in Fig. S1(a). In this appendix, we exactly treatthe realistic four-node circuit (assuming that we can re-place the Josephson chain as a pure inductor), and showthat it reduces to an effective three-node circuit shownin the main text and again here [Fig S1(b)]. This circuitis described by the Hamiltonian Hf (Eq. 1 of the main

7

CC

CR LRC1

C2

EJ ···Φext LjA

ϕR

ϕ1

ϕ1 − ϕQ

a

CC

CR LR CQ EJ Φext LjA

ϕR ϕQ

b

FIG. S1: (a) The complete four node circuit model of theheavy fluxonium device shown in Fig. 1. This circuit is equiv-alent to the three node circuit shown in (b), whose effective

capacitances, C, are all defined in terms of those in (a), seeEqn. S6, S7, and S8.

text), with effective three-node capacitances defined interms of their four-node counterparts.

The phase variables for each of the circuits are shownat their respective nodes (small black circles). With onenode as ground, the four-node ciruit has three phase vari-ables, while the three-node has just two. The key point inthis reduction is that the potential energy in both casesdepends on only two variables, and we choose the label-ing of nodes in Fig. S1(a) to make the two potentialenergies equivalent. Thus, we need only show that thekinetic energy of S1(a) reduces in form to that of S1(b).The kinetic energy T is the capacitive energy, since thevoltage at each node is Vj = hϕj/(2e). For the four-nodecircuit, then,

T = 12

(h2e

)2[CRϕ

2R + CC(ϕ1 − ϕR)2

+C1ϕ21 + C2(ϕ1 − ϕQ)2] (S1)

The potential energy of the system,

V =1

2ELRϕ

2R +

1

2ELjAϕ

2Q + EJ(1− cosϕQ), (S2)

is independent of ϕ1, so that the Euler-Lagrange equationof motion, d

dt∂L∂ϕ1

= ∂L∂ϕ1

= 0, where L = T − V gives

∂L/∂ϕ1 = h2e [(C1 + C2 + CC)]ϕ1

−CC ϕQ − C2ϕQ] (S3)

is a constant of the motion, which we may set to zero.This allows us to eliminate ϕ1 from equation S1, which,

after some algebra, results in the desired form,

T =1

2

(h

2e

)2 [CRϕ

2R + CQmϕ

2Q − 2CC ϕRϕQ

], (S4)

with three effective capacitances (corresponding to Fig.7b) defined as follows: With the preliminary definition,

CT = C1 = C2 + CC , (S5)

the effective capacitances are

CR = CR + CC(C1 + C2)/CT , (S6)

CQ = C2(C1 + CC)/CT , (S7)

CC = C2CC/CT . (S8)

Physically, CQ is the capacitance between the two

nodes of the qubit, and CR is the capacitance betweenthe resonator and ground nodes. We note that CC van-ishes as C2 → 0 , and approaches CC as C2 →∞. In oursystem, C2 ≈ C1 + CC , so CC ≈ CC/2.

Direct Drive

As mentioned in the main text, we cannot achieve co-herent operations on the metastable states by drivingdirectly on the fluxon transition. The result of such adrive at Φ = 0.078Φ0 is shown in Fig. S2. The width ofthe fluxon transition is found to saturate at high drivepowers, in Fig.S2a. Directly driving the fluxon at thelocation of largest width of the fluxon line results in thepopulation saturating to a mixed state in ∼ 100 µs.

Origin of Plasmon Dispersion

The simulations of the spectra shown in Fig. 3 of themain text account accurately for the observed plasmondispersion, and also for the smaller dispersion observed inthe resonator line. In this and the next section we providea simplified analysis to interpret and trace the origins ofthese and related phenomena. This section is devotedto estimating the change in the bare qubit levels arisingfrom Φext induced distortions of the potential well shape,with resonator interaction effects put off to the followingsection.

The strategy is to expand the potential about itsshifted minimum to characterize the distortion, and thenapply perturbation theory to obtain the spectra. The re-sulting dispersion is quadratic in Φext, with small quarticcorrections.

When Φext is applied, the central well rides up in-side the inductive parabola, making contact at the phasevalue ϕ = 2πΦext/Φ0, where the cosine term vanishes.However, clearly, this point is not the potential minimum;

8

a

0 100 200 300Rabi Drive Time ( s)

0.0

0.1

0.2

0.3

0.4

0.5

Excit

ed p

opul

atio

n80 70 60 50Power at Resonator (dBm)

4.19

4.20

4.21

4.22

4.23Pr

obe

Freq

uenc

y (G

Hz)

b

FIG. S2: a) Power sweep of the probe tone while directlydriving the fluxon transition at Φ = 0.078Φ0. We draw at-tention to two features: 1) The width of fluxon transition(4.21 GHz) saturates at high powers and 2) The appearanceof other photon-assisted transitions immediately above andbelow the fluxon transition, pointing towards a source of lim-itation for the achievable Rabi rates. b) At a power of -52dB, where the fluxon transition has the largest width, a Rabidrive results in no coherent oscillations and a 50/50 mixturein ∼100 µs.

the actual minimum is Stark-shifted by the inductive cur-rent, ELϕ, so that ϕmin = 2π(Φext/Φ0)(1− EL/EJ), tolowest order in EL/EJ . We define the displacement fromthis minimum as ϕ = ϕ−ϕmin, and rewrite the potentialenergy as a function of ϕ:

V = EJ[1− cos

(ϕ− 2π

ELEJ

Φext

Φ0

)]+

1

2EL(ϕ+ ϕmin)2.

(S9)Expanding and ignoring constant terms,

V = −EJ(

cos ϕ cos 2πEL

EJ

Φext

Φ0+ sin ϕ sin 2πEL

EJ

Φext

Φ0

)+ 1

2ELϕ2 + ELϕminϕ. (S10)

Making small-angle expansions and regrouping,

V = −EJ[1− 2

(πEL

EJ

)2(Φext

Φ0

)2]cos ϕ

+ 12ELϕ

2 + 2πEL(

Φext

Φ0

)(ϕ− sin ϕ). (S11)

This equation shows that the symmetric part of the dis-placed well is identical to the original well, except for areduction in the effective EJ by the small relative amountshown, while the odd part appears and cancels the linearpart of the inductive term, leaving cubic and higher oddpowers.

Both symmetric and antisymmetric parts of V con-tribute quadratic terms (∼ Φ2

ext) to the plasmon disper-sion; the even part at first order in perturbation theory,the odd part at second order. To estimate the former,note that in the harmonic approximation, the plasmonenergy EP increases as

√EJ , so that the fractional re-

duction is half that of EJ , and hence

∆EP (symm) = −(πEL

EJ

)2EP(Φext/Φ0

)2

≈ −45MHz(Φext/Φ0

)2. (S12)

Regarding the antisymmetric contribution, applyingsecond-order perturbation theory to the leading cubicterm in ϕ gives:

∆EP (anti) =(π

3EL)2(Φext

Φ0

)2

·(|〈f0|ϕ3|e0〉|2

E′P− 2|〈g0|ϕ3|e0〉|2

EP

)The first term in brackets is the shift in energy of the|e0〉 state due to virtual occupation of |f0〉, with energydenominator E′P = E|f0〉 − E|e0〉 = 4.39 GHz. Virtualoccupation of |g0〉 accounts for half the second term.The other half is (minus) the shift in energy of |g0〉 dueto virtual occupation of |e0〉, with energy denominatorEP = 5.072 GHz. Evaluating the matrix elements usingϕ = 0.60(a+ a†), we find

∆EP (anti) ≈ 39MHz(Φext/Φ0

)2. (S13)

The above two contributions together predict a totalquadratic dispersion ∆EP = 84 MHz (Φext/Φ0)2. Quan-titative agreement with the observed dispersion is decep-tive - interaction with the resonator reduces the predic-tion by about 12 %. While quantitative agreement is notexpected or sought from such a calculation, it suggeststhat a predominantly quadratic plasmon dispersion arisesfrom Φext induced distortions in the well, with compara-ble contributions arising from reduced effective EJ in thesymmetric part, and induced antisymmetry dominatedby cubic anharmonicity. Quartic corrections arise at thenext contributing order in perturbation theory, secondorder for the symmetric part, and fourth order for the an-tisymmetric part. We have found these to be no greaterthan 1% of the quadratic contributions. [The small pa-rameter governing successively higher contributing or-ders in perturbation theory in both cases is (EL/EP )2.]Larger (but still small) quartic corrections will arise fromthe interaction with the resonator, as shown in the fol-lowing.

Resonator/Qubit hybridization

The most direct indication of the strong interactionbetween the principle plasmon and the resonator is pro-vided by the power sweep: At low power levels one seesboth features, separated by 155 MHz, whereas at higherpowers, one sees only the resonator line, shifted upwardby 33 MHz. Assuming that the high-power value is theupcoupled (bare) resonator line, one can infer the inter-action strength g from these three numbers alone. Thisis found to be within the range of experimental uncer-tainty of the value g = 76 MHz (Fig. S3) determined bya global fit to all the data.

9

120 110 100 90 80 70Drive power at Readout Resonator (dBm)

4.90

4.95

5.00

5.05

5.10Pr

obe

Freq

uenc

y (G

Hz)

r=4.95GHzq=5.067GHzr′=4.915GHz

FIG. S3: Single Tone Spectroscopy power sweep at the in-put of the readout resonator. At high powers (> −90 dB),the heavy fluxonium qubit is saturated, and the bare reso-nance, ωr of the readout resonator is seen. At lower powers(< −100 dB) the qubit plasmon ωq and the dressed resonator,ωr′ are seen.

Here we introduce a simple model that describes theplasmon and resonator lines over the entire flux range,excluding the immediate neighborhood of the pointsΦext = 0 and Φ0/2, where more transitions are involved.Consider the coherent superposition of the two lowest ex-cited states of the noninteracting system, one involvingthe resonator and the other involving the qubit:

|ψ〉 = α|1R, g0〉+ β|0R, e0〉. (S14)

The effective Hamiltonian acting on the subspace of S14is

H =

(εR mm∗ εQ

)(S15)

where m is the interaction matrix element specific to theplasmon excitation, m = g〈e0|n|g0〉 = 0.062i. Diago-nalizing H, we find the two lowest excited eigenstates ofthe system. Near Φext = 0, where the bare detuning isεQ − εR ≈ 89 MHz, we find the resonator-like eigenstatedefined by (α/β)R = 1.94i, and the qubit-like eigenstate(α/β)Q = −i/1.94. In the former, there is a probabilityof |βR|2 = 0.21 that the qubit will be found in the (|e0〉state, and conversely, in the latter, the same probabilitythat the resonator will be found in its |1〉 state.

Since both resonance lines are read from the resonator,their relative intensities are given by the relative proba-bilities of finding the resonator in its |1〉 state,

IR/IQ = |αR/αQ|2 = 3.8, (S16)

and a rough comparison of resonance peak heights timeslinewidths is consistent with this estimate. As Φext is in-creased and the detuning is reduced, the eigenstates ex-hibit further hybridization. At Φext = 1/2, for example,the predicted intensity ratio is reduced to IR/IQ = 2.6.

A more obvious consequence of hybridization, quan-titatively, is the curvature of the resonator line, whichappears to be due entirely to level repulsion from thequbit. To demonstrate, we write the bare plasmon en-ergy as εQ = 5.039 GHz −a(Φext/Φ0)2, where a purelyquadratic dependence on Φext is assumed, following ar-guments of the previous section. Assuming that εR is aconstant, we may write down the eigenvalues of H andevaluate the leading quadratic dependence on Φext withsmall quartic corrections. The Φext dependent parts are

∆ER = −0.21a [Φext/Φ0]2 − 0.3a [Φext/Φ0]4, (S17)

∆EQ = −0.79a [Φext/Φ0]2 + 0.3a [Φext/Φ0]4.(S18)

While the estimate of the previous section may not be ac-curate quantitatively, the ratio of coefficients found hereshould be (of both the quadratic and quartic terms). It isinteresting to note that the ratio of quadratic coefficientsis 3.8; compare that of S16.

The calculation above supports the interpretation thatthe observed dispersion of the resonator line is a directconsequence of hybridization with the plasmon, whosedispersion is driven in turn by Φext induced distortions inthe potential well. And, like the plasmon, the resonatorline has two distinct branches visible near the half fluxquantum point, one generated by hybridization with the|g0〉 → |e0〉 branch, the other with the |g1〉 → |e1〉 branch.

Photon/Plasmon Assisted Transitions

The two-photon features of Fig. 4a of the main textare a result of transitions to the two-excitation subspace(|e1,−1〉 , |f0〉) starting with the combined fluxonium-resonator in |g0〉 ⊗ |0〉 state. The avoided crossing asso-ciated with this two-excitation space are also observedin the resonator-photon and plasmon assisted transi-tions shown in Fig. S4a, b. These transitions start withthe coupled resonator-fluxonium system in |g0, 1res〉 and|e0, 0res〉, respectively. The final states of the transitionare the same as those in Fig. 4a of the main text, butare here reached using a single photon, with the sec-ond photon acquired from the readout resonator and theplasmon, respectively. Due to the low Q of the readoutresonator (Q ∼ 500), and the resulting Purcell limitedlifetimes of the primary plasmon features, these transi-tions cannot be used for coherent transitions between thestates. Increasing the Q of the readout resonator, and it’sdetuning from the primary plasmon will allow the use ofthese transitions for coherent operations between fluxoidstates, as will be explored in future devices.

The resonator assisted transitions are useful for photon‘latching’ in which a single photon from the resonator canbe transferred to an excited metastable state, and subse-quently detected by cycling on the excited plasmon tran-sition. This might be potentially useful in dark photondetection schemes.

10

0.04 0.02 0.00 0.02 0.04Flux ( ext/ 0)

4.4

4.5

4.6

4.7

4.8Pr

obe

Freq

uenc

y (G

Hz)

|g0, 1r |g1, 1r|g0, 1r |g1, 1r|g0, 1r |g1, 1r|g0, 1r |g1, 1r|g0, 1r |g1, 1r

0.04 0.02 0.00 0.02 0.04Flux ( ext/ 0)

4.0

4.1

4.2

4.3

4.4

4.5

4.6

4.7

Prob

e Fr

eque

ncy

(GHz

)

|e0, 0r |g1, 1r|e0, 0r |e1, 0r|e0, 0r |f0, 0r|e0, 0r |g 1, 1r|e0, 0r |e 1, 0r

a

b

FIG. S4: Two-tone spectroscopy as a function of flux (nearzero flux) showing (a) resonator-photon and (b) plasmon as-sisted transitions. In (a) the readout tone is placed at the res-onator frequency and the transitions start with the fluxonium-resonator system in |g0〉 ⊗ |1〉. In (b), the ‘readout’ tone isplaced at the plasmon frequency and the transitions start withthe fluxonium-resonator system in |e0〉 ⊗ |0〉. In each case,the transmission of a readout tone is monitored, while thefrequency of a second drive tone is swept (y-axis). Both setsof transitions are absent when the drive and readout tonesare pulsed and staggered, as a result of the short lifetimes ofboth the resonator and the plasmon. (state labeling for thetransitions is valid for Φext¿0)

Hopping in Single Tone Spectroscopy

During the measurement of the device, there was aperiod of time in which the device was in a frustratedexcited state as shown Fig. S5. We take particular noteof the hopping from the |g1〉 → |e1〉 plasmon branch tothe |g0〉 → |e0〉 branch. While we were unable to repro-duce this state of the device, with inverted single tonespectrum, the hopping from one well branch to anotherseems to have been a common feature of this device. Thishopping can also been seen at Φext ≈ 0.6Φ0 in Fig. 2 inthe main text. A more careful analysis to address the ex-act nature of the metastable well hopping is warranted.Fig. S5, however, is a further demonstration of the largedispersive shifts of the readout resonator, and demon-strates the ability to readout the fluxoid state over the

0.75 0.50 0.25 0.00 0.25 0.50Flux ( ext/ 0)

4.90

4.95

5.00

5.05

5.10

Prob

e Fr

eque

ncy

(GHz

)

|g0 |e0|g1 |e1|g0, 0r|g1, 0r|g1 |e0|g1, 0r |g0, 1r|g0 |e1|g0, 0r |g1, 1r|g0 |g1

FIG. S5: Single tone spectroscopy, with the qubit stuck in ametastable excited state, and demonstrating regular hoppingbetween different fluxoid states. Reasons for this hopping areunclear, and will be part of future investigations.

entire flux range.