Driving forbidden transitions in the fluxonium artificial atom
U. Vool, A. Kou, W. C. Smith, N. E. Frattini, K. Serniak, P. Reinhold, I. M. Pop, S. Shankar, L. Frunzio, S. M. Girvin, M. H. Devoret
DDriving forbidden transitions in the fluxonium artificial atom
U. Vool, ∗ A. Kou, W. C. Smith, N. E. Frattini, K. Serniak, P. Reinhold, I. M. Pop, S. Shankar, L. Frunzio, S. M. Girvin, and M. H. Devoret † Department of Applied Physics and Physics, Yale University, New Haven, CT 06520, USA Physikalisches Institut, Karlsruhe Institute of Technology, Karlsruhe 76131, Germany (Dated: June 1, 2018)Atomic systems display a rich variety of quantum dynamics due to the different possible sym-metries obeyed by the atoms. These symmetries result in selection rules that have been essentialfor the quantum control of atomic systems. Superconducting artificial atoms are mainly governedby parity symmetry. Its corresponding selection rule limits the types of quantum systems that canbe built using electromagnetic circuits at their optimal coherence operation points (“sweet spots”).Here, we use third-order nonlinear coupling between the artificial atom and its readout resonatorto engineer the selection rules of our atom, allowing us to drive transitions forbidden by the parityselection rule for linear coupling to microwave radiation. A Λ-type system emerges from these newlyaccessible transitions, implemented here in the fluxonium artificial atom coupled to its “antenna”resonator. We demonstrate coherent manipulation of the fluxonium artificial atom at its sweet spotby stimulated Raman transitions. This type of transition enables the creation of new quantumoperations, such as the control and readout of physically-protected artificial atoms.
I. INTRODUCTION
Atoms exhibit complex level-transition structures,which are governed by the interactions between theircomponents. Superconducting artificial atoms, however,presently have much simpler level-transition structures.As superconducting circuits emerge as a leading platformto investigate quantum information and coherent quan-tum physics [1], there is growing interest in engineeringtheir selection rules [2–7] to implement a larger varietyof quantum dynamics in artificial atoms.All superconducting circuits at their sweet spots - oper-ating points where the circuits are insensitive to certainenvironmental noise mechanisms - obey parity symme-try. This symmetry forbids transitions between states ofthe same parity under a microwave drive [2, 8]. Awayfrom the sweet spots, all transitions become allowed, butat the cost of lower qubit coherence [9]. The ability todrive such transitions can lead to the implementation ofa new class of artificial atoms. Moreover, it is necessaryfor the control and measurement of physically-protectedqubits [10–13] - circuits which implement error correctionat the hardware level, but whose inherent protection re-sults in the quasi-impossibility of manipulating them di-rectly. Is it possible then, in superconducting circuits, tobreak the parity selection rule while still operating at thesweet spot?In this article, we present a method for driving forbid-den transitions in superconducting artificial atoms. Us-ing nonlinear coupling between the atom and an ancillaresonator, we create an atom with engineered selectionrules while maintaining the symmetry, and thus the co-herence properties, of the sweet spot. We implement non-linear coupling using a fluxonium artificial atom [14, 15]inductively coupled to an “antenna” resonator [16, 17].We demonstrate the creation of a Λ-type system in which the states of the fluxonium can be manipulated by a res-onator excitation using the now-allowed transition. Wethen use this structure to cool the fluxonium atom toone of its two lowest energy eigenstates by resonator de-cay through spontaneous Raman scattering. Finally, wepresent coherent manipulation of the fluxonium throughthe Λ-type system by driving Rabi oscillations betweenthe fluxonium ground and excited states, using stimu-lated Raman transitions through a virtual resonator ex-citation. g ,0 g ,1 g ,2 e ,1 e ,0 e ,2 f ,0 f ,1 FIG. 1. Level diagram of a fluxonium atom at Φ f ext / Φ = m + 1 / | g, (cid:105) , | e, (cid:105) and | e, (cid:105) states. a r X i v : . [ qu a n t - ph ] M a y II. FLUXONIUM-RESONATOR SYSTEM
The fluxonium artificial atom is a superconducting cir-cuit made up of a Josephson junction in parallel witha large linear inductance. While the fluxonium can beoperated at any applied external flux through the fluxo-nium loop, Φ f ext , its flux-noise insensitive sweet spots arelocated at Φ f ext / Φ = m, m + 1 / is the mag-netic flux quantum and m ∈ N . Here, we will focus onthe behavior of the fluxonium at its m + 1 / | g (cid:105) and first excited state | e (cid:105) is < φ q across the fluxonium junction. Hence,transitions are only allowed between states of oppositeparity, and transitions such as | g (cid:105) ↔ | f (cid:105) are forbiddenby a parity selection rule [15, 18].Coupling to a resonator breaks the fluxonium par-ity symmetry, but the total parity of the fluxonium-resonator excitations remains conserved and a parity se-lection rule still holds in the coupled system [8, 19]. Thelevel digram for this system, with its allowed and forbid-den transitions, is shown in Fig. 1. The states are labeledas | s, n (cid:105) where s is the state of the fluxonium and n is theresonator photon number. States with even (odd) totalparity are shown in red (blue). Transitions are only al-lowed between states of opposite total parity (solid blacklines).The dashed magenta lines show forbidden transi-tions such as | g, (cid:105) ↔ | e, (cid:105) and | g, (cid:105) ↔ | e, (cid:105) , and theability to drive them gives us access to a Λ-type structureas shown in the figure. Low-frequency transitions suchas | g, (cid:105) ↔ | e, (cid:105) are not forbidden, but are suppresseddue to a small dipole moment and the filtering of thefluxonium environment at low frequencies.The parity selection rule only holds when the externaldrive is coupled to the odd φ q fluxonium operator via lin-ear coupling to the antenna resonator. Nonlinear third-order coupling between the fluxonium and the resonatorwould lead to the drive coupling to operators such as φ q or φ q φ r , where φ r is the resonator flux. These even op-erators can drive transitions of equal parity at the fluxo-nium sweet spot (see Appendix B for more details). Onewould therefore like to implement third-order couplingbetween the fluxonium and its resonator to drive theseforbidden transitions. Note that such coupling would pre-serve the protection offered at the sweet spot with respectto flux noise.Figure 2a shows a diagram of a fluxonium atom non-linearly coupled to its resonator. The fluxonium artificialatom is made up of a small junction (black) shunted bya large linear inductance (dark blue). The resonator iscomposed of a linear inductance and a capacitor (lightblue). The nonlinear coupling is mediated by a nonlin-ear inductance made up of five three-wave-mixing dipoleelements - each named the Superconducting Nonlinear (c) (d) μ m 1 μ m (a) (b) Φ S ext Φ f ext
112 3 43 2 4
FIG. 2. (a) Schematic of the fluxonium artificial atomcoupled to an antenna resonator. The fluxonium is madeup of a small phase-slip junction (black) shunted by a lin-ear inductance (dark blue). It is coupled to the resonator(light blue) by sharing an inductance comprising SNAILS(magenta), which induce the nonlinear coupling. An exter-nal flux Φ f ext is threaded through the fluxonium loop. (b)The SNAIL is composed of three large Josephson junctionsin parallel with a smaller Josephson junction. An externalmagnetic flux Φ S ext is threaded through the SNAIL loop. (c)An SEM image of the device sketched in (a), with colored ar-rows indicating the different circuit elements. The area of thefluxonium loop is A f = 350 ± µ m (d) An SEM image of aSNAIL, where the junctions corresponding to the numberedjunctions in (b) are indicated. The ratio between the largejunction and small junction areas is α = 0 . ± .
02 and thearea of the SNAIL loop is A S = 6 ± . µ m . We can tuneΦ f ext and Φ S ext quasi-independently using a global magneticfield due to the large ratio A f /A S (cid:39) Asymmetric Inductive eLement (SNAIL) [20] (magenta).A circuit diagram of the SNAIL is shown in Fig. 2b, Φ fext / Φ F r equen cy ( G H z ) P ha s e o f s i gna l r e f l e c t ed o ff r e s ona t o r ( deg ) (a) (b) (c) (d) g ,0 e ,1
25 kHz g ,0 e ,1
120 kHz g ,0 e ,1
210 kHz g ,0 e ,1
300 kHz
FIG. 3. (a)-(d) Two-tone spectroscopy of the | g, (cid:105) ↔ | e, (cid:105) transition with a SNAIL-fluxonium device in the vicinity ofΦ f ext / Φ = 0 .
5, 2 .
5, 4 .
5, and 6 .
5, respectively. As the external flux is increased, the third-order coupling strength of the SNAILgrows, and the nominally forbidden transition becomes visible. The values for the | g, (cid:105) ↔ | e, (cid:105) coupling strength are given inthe insets. The change in the sign of the reflected phase near Φ f ext / Φ = 6 .
51 is due to a change in the dispersive coupling χ between the fluxonium and the resonator. which consists of three large Josephson junctions in paral-lel with a smaller Josephson junction. As an external fluxΦ S ext is threaded through the SNAIL, it becomes a non-linear element with third-order nonlinearity. The SNAILdesign appears similar to that of the flux qubit [21, 22],but the devices are operated in very different regimes.In the flux qubit, the area ratio ( α ) between the smalland large junction is chosen to be ≈ .
8, which leadsto a double-well potential at Φ S ext = 0 .
5. The SNAIL isdesigned to have α < . S ext = 0 . − . α = 0 . ± . . f r = 6 .
82 GHz and the fluxonium qubit transition fre-quency at its Φ / f q = 500 MHz. Thissample was housed in a WR-102 waveguide and measuredin reflection through an impedance-matched waveguide-SMA adapter [27]. III. SPECTROSCOPY
In Fig. 3, we show two-tone spectroscopy of the | g, (cid:105) ↔ | e, (cid:105) transition around different fluxonium sweet spots. A continuous-wave tone was swept around thefrequency of the | g, (cid:105) ↔ | e, (cid:105) transition, while anothertone at f r was used to measure the corresponding res-onator response. The measurements were done aroundthe Φ f ext / Φ = m + 1 / m = 0 , , , f ext / Φ = 0 . . T = 6 µs .The frequency of the | g, (cid:105) ↔ | e, (cid:105) transition is low-est at the sweet spots because it is the sum of theflux-dependent | g, (cid:105) ↔ | e, (cid:105) fluxonium transition fre-quency and the approximately constant resonator fre-quency. The resonator frequency decreased slightly athigher external flux, due to the increasing linear induc-tance of the SNAIL. This manifests itself in our measure-ment as the minimum transition frequency being slightlylower for sweet spots of higher external flux. Notice thatin Fig. 3d the sign of the phase response changes. This islikely due to a change in the fluxonium-resonator disper-sive shift, which is also observable in a direct measure-ment of the fluxonium transition [18]. IV. Λ -SYSTEM OPERATIONS Driving parity-forbidden transitions, such as | g, (cid:105) ↔| e, (cid:105) and | e, (cid:105) ↔ | g, (cid:105) , allows us to construct a Λ-typesystem in which the fluxonium qubit states | g, (cid:105) and | e, (cid:105) are the low-energy states, and the excited state is aresonator excitation state such as | g, (cid:105) or | e, (cid:105) . Pre-vious superconducting implementations of Λ-type sys-tems employed flux-tunable qubits away from their sweetspot [28–31], effective driven systems [32, 33], or two-photon transitions [34–36]. Here we present a physicalimplementation of Λ-type system at the fluxonium sweetspot using direct drives.We demonstrated that our circuit can be treated as aΛ-type system by performing both incoherent and co-herent operations using Raman transitions. All mea-surements were performed at the Φ f ext / Φ = 6 . σ width and varying am-plitude was applied at f q . As the amplitude was varied,the qubit population oscillated between thermal equilib-rium and inverted population. From the oscillation am-plitude we inferred that the qubit had 60% probability tobe in the ground state | g (cid:105) , which corresponds to a tem-perature of 62 mK. This experiment was repeated afterapplying a tone resonant with the | e, (cid:105) ↔ | g, (cid:105) tran-sition for 5 µ s. After this duration, the fluxonium had94% probability to be in | g (cid:105) , which corresponds to 9 mK- well below thermal equilibrium. We have thus demon-strated cooling of the fluxonium to its ground state bythe Raman process shown in the inset of Fig. 4a. Wealso initialized the qubit in | e (cid:105) by applying a tone reso-nant with the | g, (cid:105) ↔ | e, (cid:105) transition before performingthe Rabi experiment (blue). This inverted the fluxoniumpopulation and prepared it in | e (cid:105) with 91 .
5% probability.The ground state population was calibrated from qubitmeasurements in conjunction with a theoretical modelfor Raman cooling (see Appendix D).We were also able to coherently control the qubitthrough the Λ-type system. Figure 4b shows Rabi os-cillations of the fluxonium qubit via a virtual transi-tion through the | e, (cid:105) state. Two tones were appliedat the qubit for varying lengths of time after initiallypreparing the fluxonium ground state using Raman cool-ing. One was detuned 150 MHz below the resonator fre-quency (black arrow in the inset of Fig. 4b), and theother was detuned below the | g, (cid:105) ↔ | e, (cid:105) transitionby (150 + ∆) MHz where ∆ was a variable additionaldetuning (magenta arrow in the inset of Fig. 4b). Weobserved a typical Rabi oscillation pattern, which shows g ,0 g ,1 e ,1 e ,0 g ,0 g ,1 e ,1 e ,0 g ,0 Pulse amplitude ∆ (MHz)
150 MHz ∆ T i m e ( µ s ) - π π π -2 π P opu l a t i on i n g P opu l a t i on i n g e ,0 e ,1 preparation (a)(b) Direct g ,0 e ,0 Raman g ,0 e ,0 FIG. 4. (a) Rabi flops of the fluxonium | g, (cid:105) ↔ | e, (cid:105) transi-tion with different initial preparations before applying a 20-ns σ pulse at f q . The black dots correspond to the qubit start-ing in thermal equilibrium, where we measured the qubit tobe 60% in | g (cid:105) , corresponding to 62 mK. The red dots corre-spond to initially cooling the qubit to | g (cid:105) by applying a toneresonant with the | e, (cid:105) ↔ | g, (cid:105) transition (see inset). Thequbit then is in | g (cid:105) with a 94% probability. The blue dotscorrespond to initially preparing the qubit in | e (cid:105) by applyinga tone resonant with the | g, (cid:105) ↔ | e, (cid:105) transition (see inset),which results in a 91 .
5% probability for the qubit to be in | e (cid:105) .The solid lines are sinusoidal fits to the measured Rabi oscil-lations. (b) Rabi oscillations of the fluxonium | g, (cid:105) ↔ | e, (cid:105) transition by a Raman process through the | e, (cid:105) state (see in-set). The | e, (cid:105) ↔ | e, (cid:105) tone is applied 150 MHz detuned fromresonance, and the | g, (cid:105) ↔ | e, (cid:105) is applied at 150 MHz + ∆. that we can control a qubit with transition frequency f q = 500 MHz by only applying tones around 7 GHz.Note that the optimal detuning of the | g, (cid:105) ↔ | e, (cid:105) tran-sition corresponds to ∆ = 60 kHz, due to a Stark shiftof this transition. From ∆, we can extract a drive ampli-tude of g / π = 3 MHz for the | g, (cid:105) ↔ | e, (cid:105) transitiondue to nonlinear coupling (see Appendix E).The methods described above extend the quantum con-trol of atomic physics by Raman transitions to supercon-ducting circuits. In atomic physics, these transitions cou-ple levels whose direct transition is forbidden and thusprotected from environmental noise. However, the cir-cumstances in which this idea can be exploited only occurin a limited number of atoms. By contrast, in supercon-ducting artificial atoms we can engineer circuits to im-plement the transitions needed for this quantum control.Here, as a proof-of-principle, we applied this method tothe fluxonium artificial atom, whose | g (cid:105) ↔ | e (cid:105) transi-tion can be directly driven. This Raman control is ab-solutely essential for more complex physically-protectedqubits [10–13], however, whose inherent protection makesthem impossible to directly control and read out. Notealso that our technique separates qubit control from thequbit transition frequency. Previous two-photon imple-mentations [35, 37] of the | g, (cid:105) ↔ | e, (cid:105) transition reliedon the direct | g, (cid:105) ↔ | e, (cid:105) transition, and thus can-not be used for the control of protected qubits. Withour method, one can perform coherent operations on thequbit while its direct transition remains completely iso-lated. V. CONCLUSIONS
In conclusion, nonlinear coupling between a qubit anda resonator can be used to directly drive transitions for-bidden by parity symmetry at the fluxonium sweet spot.This method implements a Λ-type system within super-conducting circuits. We created a new transition whichcompletes the triangle spanning the | g, (cid:105) , | e, (cid:105) , and | e, (cid:105) states of the fluxonium qubit coupled to an antenna res-onator. We then demonstrated cooling of the fluxoniumqubit by spontaneous Raman scattering, and coherentoscillations between the ground and excited state drivenby a stimulated Raman process.Furthermore, the ability to create new transitions ina superconducting circuit opens the door to applicationsin microwave quantum optics [38] and autonomous errorcorrection [39]. While we have focused here on Λ-systemphysics, our ability to drive a two-excitation transitioncan also be understood as tunable mode coupling, use-ful for the single-drive implementation of protocols forphoton detection [33] and remote entanglement [40–42]. ACKNOWLEDGMENTS
We acknowledge fruitful discussions with Liang Jiang.Facilities use was supported by YINQE, the Yale SEAScleanroom, and NSF MRSEC DMR 1119826. Thisresearch was supported by ARO under Grant No.W911NF-14-1-0011, by MURI-ONR Grant No. N00014-16-1-2270, and by NSF DMR-1609326.
APPENDICESA. Theoretical description of a SNAIL
A sketch of a SNAIL circuit element can be seen inFig. 2b. A detailed derivation and analysis of this circuitis given in Refs. 18 and 20, and in this Appendix weonly give a brief overview. We can express the potential(inductive) energy of the SNAIL as: U SNAIL ( ϕ ) = − αE J cos ϕ − nE J cos (cid:18) Φ Sext /φ − ϕ n (cid:19) , (1)where ϕ is the superconducting phase across the smalljunction of the SNAIL, E J is the Josephson energy of thelarge SNAIL junction, α is the ratio between the smalland large junction, n is the number of large junctions inthe SNAIL loop (in our implementation n = 3), Φ Sext isthe external flux through the SNAIL loop and φ is thereduced magnetic flux quantum. In this description wehave eliminated the dynamics of the modes within the n -junction array, and consider the circuit as a single degreeof freedom with equal phases across the array junctions.To expand Eq. 1 as a non-linear inductor, we must firstfind the minimum ϕ min of the potential. This minimumdepends on Φ Sext and α , and can be numerically obtainedfor each of those. Then we can expand the SNAIL po-tential around the minimum, using the new coordinate (cid:101) ϕ = ϕ − ϕ min . We express the Taylor expansion of thepotential as: U SNAIL ( (cid:101) ϕ ) = c (cid:101) ϕ + c (cid:101) ϕ + c (cid:101) ϕ + ..., (2)where c m is the coefficient of the m -th order in the expan-sion. c is related to the linear inductance of the SNAILas L S = φ c E J . These coefficients also depend on Φ Sext , α , and n , and can be obtained numerically. B. Theoretical description of the device
Fig. 2a shows a sketch of our circuit in which a fluxo-nium artificial atom is coupled to an antenna resonatorby sharing a nonlinear inductance composed of SNAILs.To understand the behavior of the circuit quantitatively,let us simplify it into an effective circuit given in Fig. 5b.The fluxonium is now represented by a small junction(black) with Josephson energy E J and capacitive en-ergy E C , shunted by a linear inductance L q (dark blue).An external flux Φ f ext is threaded through the fluxoniumloop. The resonator (light blue) is represented as an LCoscillator with capacitance C r and unshared inductance L r . The N SNAIL array which couples the two systems(in our implementation N = 5) is reduced to a singleeffective SNAIL (magenta). We represent the SNAIL ashaving only second-order and third-order terms. L r C r φ r φ q (a) (b) Φ f ext L q E C E J L S tot c tot FIG. 5. (a) The fluxonium artificial atom is made up of a small junction (black) shunted by a large inductance, itself made ofan array of larger Josephson junctions (dark blue). Some of this inductance is shared with the resonator (light blue). In thisdesign, the shared elements (magenta) are taken to be SNAILs, which allows us to break selection rules at the fluxonium sweetspot. (b) The equivalent circuit diagram. This circuit has two DOF, and we express the Hamiltonian as a function of ϕ r - theflux across the resonator capacitor and ϕ q , the flux across the fluxonium small junction. We can label the superconducting phase across theSNAIL array as ϕ S , and assume it is divided equallyacross all N SNAILs in the array. Thus, we can calcu-late the coefficients of the total SNAIL array from thoseof the SNAIL. c tot2 ϕ S = N c (cid:16) ϕ S N (cid:17) , (3)where c is the second-order coefficient of a single SNAILin the array, and c tot2 is the second-order coefficient ofthe whole array. Thus, c tot2 = c /N or L totS = N L S . Lin-ear inductances in series are simply added, as expected.A similar calculation shows that c tot3 = c N , and so thethird-order non-linearity is suppressed by an additionalfactor of N . Higher order non-linearities are similarlysuppressed by higher and higher factors, making the low-order non-linearity assumption better.The circuit in Fig. 5b has only two true degrees of free-dom and we choose to use the phase across the Joseph-son junction ϕ q , and the phase across the resonatorcapacitance ϕ r . These are related to the fluxoniumand resonator flux operators given by ϕ q = φ q /φ and ϕ r = φ r /φ , where φ is the reduced magnetic flux quan-tum.We can derive the Hamiltonian for the circuit by fol-lowing the circuit quantization protocol [43]. A very sim- ilar derivation is given in Ref. 44, with a shared linearinductance replacing the SNAIL. The addition of theSNAIL adds a three-wave mixing term to the simplefluxonium-resonator Hamiltonian, of the form: H = c tot3 (cid:18) L r L totS L q ( L r + L totS ) ϕ q + L totS L r + L totS ϕ r (cid:19) , (4)where we have assumed that L q (cid:29) L r , L totS . This three-wave mixing Hamiltonian gives rise to several effectsthrough its different mixing terms, but let us focus ontwo terms of special importance: H | g (cid:105)−| f (cid:105) = 3 c tot3 (cid:18) L totS L q (cid:19) L r L totS ( L r + L totS ) ϕ r ϕ q (5) H | g , (cid:105)−| e , (cid:105) = 3 c tot3 L totS L q L r ( L totS ) ( L r + L totS ) ϕ r ϕ q (6)The term in Eq. 5 is proportional to the term ϕ r ϕ q .With an additional resonator drive, this gives rise to aneven drive term of the form ϕ q which is able to drive thefluxonium | g (cid:105) ↔ | f (cid:105) transition at the fluxonium sweetspot as (cid:104) g | ϕ q | f (cid:105) (cid:54) = 0. To understand this effective cou-pling term, first notice that from circuit quantization ϕ r = ϕ r ZPF ( a r + a † r ) where a r is the resonator decayoperator and ϕ r ZPF are the zero-point fluctuations of theresonator phase operator. We can now add a drive termof the form (cid:15) ( a r + a † r ) in the drive frequency rotatingframe, where (cid:15) is the resonator drive amplitude. By ap-plying the displacement operator, we end up with a driveterm of the form ϕ r ZPF α r ϕ q where α r is the coherent stateamplitude in the resonator.The term in Eq. 6 similarly leads to a drive term ofthe form ϕ q ϕ r . This is another even term, but one thatallows us to drive the forbidden joint transitions such as | g, (cid:105) ↔ | e, (cid:105) and | e, (cid:105) ↔ | g, (cid:105) . Eq. 6 should thus re-mind us of tunable mode coupling, as it gives rise to botha beam-splitter term which enables the | e, (cid:105) ↔ | g, (cid:105) transition, and a two-mode squeezing term which enablesthe | g, (cid:105) ↔ | e, (cid:105) transition. Thus, our selection-rule-breaking drive can also be understood as a tunable cou-pling between modes, such that the parity is preserved.There are two important things to notice in the coef-ficients of Eqs. 5 and 6. First of all, they both dependon L r in the numerator. L r is the unshared resonatorinductance, and thus one would expect that if the an-tenna shares more of its inductance, it is more coupledto the fluxonium and thus the SNAIL is better able todrive the forbidden transitions. This intuition is false, aswhen L r = 0, the SNAIL element is the entire inductanceof the resonator and the phase across it is ϕ r . There isstill coupling between the two modes, mediated by the L q inductor, but the SNAIL does not participate in it andthus there is no three-wave mixing for the qubit mode.Thus, a substantial L r , comparable to L totS , is necessaryto drive forbidden transitions.The second thing to notice is that the coefficient inEq. 6 is larger than Eq. 5 by a factor of L q /L totS , which isexperimentally ≈
50. Thus, this coupling scheme is moresuited to drive the two-mode forbidden transitions suchas | g, (cid:105) ↔ | e, (cid:105) and | e, (cid:105) ↔ | g, (cid:105) . C. Calculating the drive amplitude g The term in Eq. 6 allows us to quantify the effec-tive coupling strength of our nonlinear transitions. Letus take this term, and add a direct drive on the res-onator of the form (cid:15) ( a r + a † r ). As mentioned in Ap-pendix B, we can displace the resonator by the transfor-mation a r → a r + α r , where the displacement is cho-sen to be the coherent steady-state amplitude in the res-onator α r = (cid:15)iκ/ − ∆ r where ∆ r is the drive detuning from The distinction between ϕ q and ϕ q ϕ r and the terms they candrive is not absolute, due to the coupling between the resonatorand the fluxonium. Both of these drive terms can drive thetransitions | g (cid:105) ↔ | f (cid:105) and | g, (cid:105) ↔ | e, (cid:105) to some extent. However,as the coupling between the resonator and the fluxonium is weak(we are in the dispersive regime), we associate each term withthe transitions it couples to more strongly. the resonator resonance frequency and κ is its linewidth.This transformation eliminates the direct drive term andwe end up with an effectively undriven resonator.The term in Eq. 6, however, in this new frame givesrise to the effective coupling term that we require. Thiseffective term is of the form: H eff = 6 ϕ r ZPF α r c tot3 φ L totS L q L r ( L totS ) ( L r + L totS ) ϕ r ϕ q , (7)and drive amplitude term can then be simply obtainedby g = (cid:104) g, | H eff | e, (cid:105) .We can separate the discussion of this term into threecomponents. The first one is the bare coupling term: g bare3 = 6 ϕ r ZPF c tot3 L totS L q L r ( L totS ) ( L r + L totS ) , (8)which includes the specific parameters of the designwhich we have discussed in Appendix B. The flux Φ Sext through the SNAIL strongly influences the value of c tot3 ,and to a lesser extent L totS . Thus g bare3 is responsiblefor the improved ability to drive the forbidden transi-tion with increased flux (see Fig. 3). Fig. 6 shows g bare3 as a function of the flux through the fluxonium loop,Φ f ext . Recall that due to the difference in loop areas,Φ Sext = Φ f ext / Φ fext / Φ g r e ( M H z ) FIG. 6. The bare coupling coefficient g bare3 (see Eq. 8) vs. theexternal flux through the fluxonium loop Φ f ext . The second component in the g term is the matrixelement (cid:104) g, | ϕ r ϕ q | e, (cid:105) . Note that ϕ r and ϕ q are not the field operators of the resonator and fluxoniummodes, but simply convenient bases made up of theirlinear parts. Especially, ϕ q is the field operator of a lin-ear mode very different from the fluxonium qubit. Thestatements (cid:104) g | ϕ q | f (cid:105) = 0 and (cid:104) g | ϕ q | f (cid:105) (cid:54) = 0 are true dueto the selection rules, but calculating the value of thematrix element requires a diagonalization of the fluxo-nium Hamiltonian, and is usually done numerically [44].However, this matrix element is identical for all fluxo-nium sweet spots Φ f ext / Φ = m + 1 / m ∈ N .Thus, is can be treated as a constant for the purposes ofthis paper and from a numerical diagonalization of theHamiltonian we obtain (cid:104) g, | ϕ r ϕ q | e, (cid:105) (cid:39) / (cid:104) g, | ϕ r ϕ q | e, (cid:105) = (cid:104) e, | ϕ r ϕ q | g, (cid:105) and thus our driveterm can excite | e, (cid:105) ↔ | g, (cid:105) and | g, (cid:105) ↔ | e, (cid:105) withequal amplitude, and the selection is made by the tran-sition frequency to which we tune our external drive.The last component is the α r = (cid:15)iκ/ − ∆ r , the coherentpopulation in the cavity during the drive. This term isproportional to (cid:15) and thus shows the increase in transi-tion rate as our drive amplitude increases. This is thecomponent we cannot estimate from system parameters,thus limiting our ability to predict the transition rate ofthe nonlinear transition. However, we have independentcalibrations for the value of g from the spontaneous andstimulated Raman transition measurements, as will bediscussed in Appendix D, and we can use them to esti-mate the population of the resonator.Note, however, that we can directly calculate the ra-tio g /(cid:15) , which is the ratio of the rate at which we driveforbidden transitions such as | g, (cid:105) ↔ | e, (cid:105) and | e, (cid:105) ↔| g, (cid:105) , and the rate at which we directly drive the res-onator transitions such as | g, (cid:105) ↔ | g, (cid:105) or | e, (cid:105) ↔ | e, (cid:105) .From our system parameters at Φ f ext = 6 . , we can es-timate g /(cid:15) = 0 . D. Calibration of the fluxonium ground statepopulation
In Fig. 4 we present the measurements of spontaneousand stimulated Raman transitions in terms of the flux-onium ground state population. This axis is actuallycalibrated by using known parameters and by assuminga Raman cooling model for the results in Fig. 4a, and weexplain this calibration in detail in this Appendix.Recall that we measure the state of the fluxonium viaits effect on the resonator frequency, and there are posi-tions in the I - Q phase-space of the reflected signal whichcorrespond to the fluxonium being in | g (cid:105) and | e (cid:105) . Let usmark half of the distance in phase-space between thesetwo positions as A . Thus, if the initial state of the flux-onium qubit is exactly | g (cid:105) and it performs perfect Rabioscillations, the amplitude for these observed oscillationswould be A . However, as our qubit is in thermal equilib-rium, the actual measured amplitude is: A th = A ( P g th − P e th ) = A (2 P g th − , (9)where P g th ( P e th ) is the probability the fluxonium is | g (cid:105) ( | e (cid:105) ) in thermal equilibrium. Note that A th is the ampli-tude of oscillations measured in the black curve in Fig. 4a. Similarly, we can define the probability in | g (cid:105) after the | e, (cid:105) ↔ | g, (cid:105) (red) Raman cooling sequence as P g red , andthe probability in | e (cid:105) after the | g, (cid:105) ↔ | e, (cid:105) (blue) Ra-man cooling sequence as P e blue . Their corresponding Rabioscillation amplitudes are then: A red = A (2 P g red −
1) (10) A blue = A (2 P e blue −
1) (11)We can also find expressions for P g red and P e blue . Let uslabel the transition rate of the | e, (cid:105) ↔ | g, (cid:105) transitionas g red . The Raman cooling thus involves a coherent ex-citation to the state | g, (cid:105) with a rate g red , followed by anincoherent decay of the resonator to | g, (cid:105) at rate κ . Asour resonator has a large decay rate κ = 2 π × . g red (cid:28) κ . We can thusadiabatically eliminate the higher state. A similar pro-cess can be done for the | g, (cid:105) ↔ | e, (cid:105) transition and itsrate g blue . Note that we can assume g red = g blue = g as their corresponding matrix elements are identical, asdiscussed in Appendix D. Thus, we express the coolingrate for both processes using adiabatic elimination:Γ cool = 4 g κ . (12)The thermal fluxonium population can be described interms of an “up” rate Γ ↑ which is the rate of transition | g (cid:105) → | e (cid:105) , and a “down” rate Γ ↓ which is the rate oftransition | e (cid:105) → | g (cid:105) . Their sum equals the total thermal-ization rate Γ ↓ + Γ ↑ = Γ , and the qubit population isrelated to them via: P g th = Γ ↓ Γ ↓ + Γ ↑ = Γ ↓ Γ (13)from a detailed balance assumption in equilibrium.The Raman cooling tones then enter to aid the differ-ent thermal equilibration rates. The red tone cools thequbit to | g, (cid:105) , and thus the cooling rate Γ cool aids Γ ↓ .Similarly, the blue tone cools the qubit to | e, (cid:105) , and thusthe cooling rate Γ cool aids Γ ↑ . We can thus express thepopulations after cooling as: P g red = Γ cool + Γ ↓ Γ cool + Γ ↓ + Γ ↑ = 4 g + κ Γ ↓ g + κ Γ ↓ + κ Γ ↑ (14) P e blue = Γ cool + Γ ↑ Γ cool + Γ ↓ + Γ ↑ = 4 g + κ Γ ↑ g + κ Γ ↓ + κ Γ ↑ (15)Let us summarize all these relations. Eqs. 9, 10, and11 relate three measured quantities, A th , A red , and A blue ,to expressions with several unknowns. From the follow-ing equations, we see that we have expressed all of theseterms using only three unknowns: A , g , and P g th . Allother unknowns can be expressed using these three, aswell as known quantities such as κ = 2 π × . which is related to the qubit lifetime 1 / Γ = T =5 . µ s. Thus, we can solve a set of three equations withthree unknowns, and extract the thermal population ofour fluxonium qubit.The extracted qubit equilibrium temperature is 62 mK,which corresponds to P g th = 0 .
6. We also obtain the pop-ulation after cooling to | g (cid:105) , P g red = 0 .
94, and the popula-tion after cooling to | e (cid:105) , P e blue = 0 . g =2 π × .
87 MHz. Notice we self-consistently justify ourassumption g (cid:28) κ . We can also compare this mea-surement to the theoretical prediction. In Appendix Dwe discussed the calculation of the rate g (see Eq. 7),where we can independently predict all the coefficientsbesides α r , which is the coherent state population in theresonator which enables this drive. From the measure-ment of g , we can estimate the photon population in theresonator due to this cooling drive as | α r | = 0 . E. Estimation of drive rates from stimulated Ramantransition measurements
Notice that the Rabi-oscillations in Fig. 4b are notquite centered at ∆ = 0, but rather are slightly offsetat ∆ = 60 kHz. This is a result of the Stark shift inthe nonlinear mode, and it is related to the drive ampli-tude and the detuning by ∆
Stark = g ∆ r . Notice that inthis case there is only one nonlinear mode, and only itexperiences a Stark shift.This allows us to estimate g = 2 π × | α r | = 4 .
3. This seems to contradict the transition | g, (cid:105) ↔ | e, (cid:105) as there are more photons in the cavity.But note that while this drive is on we are in a displacedframe, and the states | g, (cid:105) and | e, (cid:105) are defined fromthis displaced value. Also note this value is larger thanthat of the cooling drive by a factor of ≈
10, consistentwith the generator being set 10 dB higher for this mea-surement. This is also the basis of the rate estimate inFig. 2, accounting for the change in the external driveamplitude for the spectroscopy measurement.Fig. 7 shows a cut of Fig. 4b at ∆ = 100 kHz. Toquantify these oscillations, we can compare then to a the-oretical model given by: H Λ / (cid:126) = ω r a † r a r + ω q σ z + χ a † r a r σ z + 2 (cid:15) cos( ω d t )( a r + a † r )+ 2 g cos( ω nl t )( a r σ − + a † r σ + ) , (16)where a r is the resonator annihilation operator andthe fluxonium is modeled as a two-level system in thePauli σ z basis. g is the transition rate of the non-linear drive (same coefficient as discussed in AppendixC), (cid:15) is the coefficient of the direct resonator drive and P opu l a t i on i n g Time ( µ s) FIG. 7. Rabi oscillations of the fluxonium qubit due tostimulated Raman transitions. This is a cut of Fig. 4b at∆ = 100 kHz. The green line shows a fit to Eq. 17 with onlyone fit parameter (cid:15) χ = 2 π × . I - Q response of the fluxonium). ω d = ω r − ∆ r is the drivefrequency of the direct cavity drive where ω r is the res-onator frequency and ∆ r = 2 π ×
150 MHz is the drivedetuning. ω nl = ω r + ω q − ∆ r − ∆ is the drive frequencyfor the nonlinear transition, and it is detuned from the | g, (cid:105) ↔ | e, (cid:105) resonance frequency by ∆ r + ∆ where ∆ isa variable detuning (see Fig. 4b).By moving to the rotating frames U r = e i a † r a r ω d t , U q = e i σ z ωq − ∆2 t and taking the rotating wave approx-imation, we arrive at the time-independent Hamiltonian: H Λ / (cid:126) = ∆ r a † r a r + ∆2 σ z + χ a † r a r σ z + (cid:15) ( a r + a † r ) + g ( a r σ − + a † r σ + ) . (17)Notice that we have independent measurements of ev-ery coefficient in Eq. 17 except (cid:15) . We also know all thedecay constants for the fluxonium and resonator, and theinitial population of the fluxonium (which was cooled to94% in | g, (cid:105) by Raman cooling). Thus, we can simu-late the master equation for our system, and fit it to ourmeasurement in Fig. 7 with only a single fit parameter (cid:15) . This numerical simulation result is shown as the greenline in the figure. Notice we obtain good agreement withthe measurement, and thus we conclude that our Hamil-tonian in Eq. 17 is a good description for the dynamicsof the system.The value we get is (cid:15) = 2 π × . R = g (cid:15) ∆ r = 2 π × g /(cid:15) = 0 . −
25 dB lower0than the nonlinear drive generator in our experiment. ∗ Current Address: Department of Physics, Har-vard University, Cambridge, MA 02138 USA;uri [email protected] † [email protected][1] M. H. Devoret and R. J. Schoelkopf, Science , 1169(2013).[2] Y.-x. Liu, J. Q. You, L. F. Wei, C. P. Sun, and F. Nori,Physical Review Letters , 087001 (2005).[3] F. Deppe, M. Mariantoni, E. P. Menzel, A. Marx,S. Saito, K. Kakuyanagi, H. Tanaka, T. Meno, K. Semba,H. Takayanagi, et al., Nature Physics , 686 (2008).[4] K. Harrabi, F. Yoshihara, A. O. Niskanen, Y. Nakamura,and J. S. Tsai, Physical Review B , 020507 (2009).[5] P. Forn-Daz, G. Romero, C. J. P. M. Harmans, E. Solano,and J. E. Mooij, Scientific Reports , 26720 (2016).[6] J. Goetz, F. Deppe, K. G. Fedorov, P. Eder, M. Fis-cher, S. Pogorzalek, E. Xie, A. Marx, and R. Gross,arXiv:1708.06405 (2017).[7] Y.-H. Lin, L. B. Nguyen, N. Grabon, J. S.Miguel, N. Pankratova, and V. E. Manucharyan,arXiv:1705.07873 (2017).[8] A. Blais, J. Gambetta, A. Wallraff, D. I. Schuster, S. M.Girvin, M. H. Devoret, and R. J. Schoelkopf, PhysicalReview A , 032329 (2007).[9] G. Ithier, E. Collin, P. Joyez, P. J. Meeson, D. Vion,D. Esteve, F. Chiarello, A. Shnirman, Y. Makhlin,J. Schriefl, et al., Physical Review B , 134519 (2005).[10] B. Doucot and L. B. Ioffe, Reports on Progress in Physics , 072001 (2012).[11] P. Brooks, A. Kitaev, and J. Preskill, Phys. Rev. A ,052306 (2013).[12] M. T. Bell, J. Paramanandam, L. B. Ioffe, and M. E.Gershenson, Physical Review Letters , 167001 (2014).[13] A. Kou, W. Smith, U. Vool, R. Brierley, H. Meier,L. Frunzio, S. Girvin, L. Glazman, and D. M. H., Physi-cal Review X , 031037 (2017).[14] V. E. Manucharyan, J. Koch, L. I. Glazman, and M. H.Devoret, Science , 113 (2009).[15] V. E. Manucharyan, Ph.D. thesis, Yale University (2011).[16] I. M. Pop, K. Geerlings, G. Catelani, R. J. Schoelkopf,L. I. Glazman, and M. H. Devoret, Nature , 369(2014).[17] U. Vool, I. M. Pop, K. Sliwa, B. Abdo, C. Wang,T. Brecht, Y. Y. Gao, S. Shankar, M. Hatridge, G. Cate-lani, et al., Phys. Rev. Lett. , 247001 (2014).[18] U. Vool, Ph.D. thesis, Yale University (2017).[19] Y.-x. Liu, L. F. Wei, J. S. Tsai, and F. Nori, PhysicalReview Letters , 067003 (2006).[20] N. E. Frattini, U. Vool, S. Shankar, A. Narla, K. M.Sliwa, and M. H. Devoret, Applied Physics Letters ,222603 (2017).[21] J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian, C. H.v. d. Wal, and S. Lloyd, Science , 1036 (1999).[22] C. H. v. d. Wal, A. C. J. t. Haar, F. K. Wilhelm, R. N.Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd, and J. E. Mooij, Science , 773 (2000).[23] A. B. Zorin, Physical Review Applied , 034006 (2016).[24] A. B. Zorin, M. Khabipov, J. Dietel, and R. Dolata,arXiv:1705.02859 (2017).[25] F. Lecocq, I. M. Pop, Z. Peng, I. Matei, T. Crozes,T. Fournier, Ccile Naud, W. Guichard, and O. Buisson,Nanotechnology , 315302 (2011).[26] I. M. Pop, T. Fournier, T. Crozes, F. Lecocq, I. Matei,B. Pannetier, O. Buisson, and W. Guichard, J. Vac. Sci.Technol. B , 010607 (2012).[27] A. Kou, W. C. Smith, U. Vool, I. M. Pop, K. M.Sliwa, M. H. Hatridge, L. Frunzio, and M. H. Devoret,arXiv:1705.05712 (2017).[28] C.-P. Yang, S.-I. Chu, and S. Han, Physical Review Let-ters , 117902 (2004).[29] K. V. R. M. Murali, Z. Dutton, W. D. Oliver, D. S.Crankshaw, and T. P. Orlando, Physical Review Letters , 087003 (2004).[30] S. O. Valenzuela, W. D. Oliver, D. M. Berns, K. K.Berggren, L. S. Levitov, and T. P. Orlando, Science ,1589 (2006).[31] M. Grajcar, S. H. W. van der Ploeg, A. Izmalkov,E. Ilichev, H.-G. Meyer, A. Fedorov, A. Shnirman, andG. Sch¨on, Nature Physics , 612 (2008).[32] K. Inomata, K. Koshino, Z. R. Lin, W. D. Oliver, J. S.Tsai, Y. Nakamura, and T. Yamamoto, Physical ReviewLetters , 063604 (2014).[33] K. Inomata, Z. Lin, K. Koshino, W. D. Oliver, J.-S. Tsai,T. Yamamoto, and Y. Nakamura, Nature Communica-tions , 12303 (2016).[34] W. R. Kelly, Z. Dutton, J. Schlafer, B. Mookerji, T. A.Ohki, J. S. Kline, and D. P. Pappas, Physical ReviewLetters , 163601 (2010).[35] S. Novikov, T. Sweeney, J. E. Robinson, S. P. Pre-maratne, B. Suri, F. C. Wellstood, and B. S. Palmer,Nature Physics , 75 (2016).[36] N. Earnest, S. Chakram, Y. Lu, N. Irons, R. K. Naik,N. Leung, J. Lawrence, J. Koch, and D. I. Schuster,arXiv:1707.00656 (2017).[37] A. Wallraff, D. I. Schuster, A. Blais, J. M. Gambetta,J. Schreier, L. Frunzio, M. H. Devoret, S. M. Girvin,and R. J. Schoelkopf, Physical Review Letters , 050501(2007).[38] J. Q. You and F. Nori, Nature , 589 (2011).[39] J. Kerckhoff, H. I. Nurdin, D. S. Pavlichin, andH. Mabuchi, Physical Review Letters , 040502 (2010).[40] P. Campagne-Ibarcq, E. Zalys-Geller, A. Narla,S. Shankar, P. Reinhold, L. D. Burkhart, C. J. Ax-line, W. Pfaff, L. Frunzio, R. J. Schoelkopf, et al.,arXiv:1712.05854 (2017).[41] C. Axline, L. Burkhart, W. Pfaff, M. Zhang, K. Chou,P. Campagne-Ibarcq, P. Reinhold, L. Frunzio, S. M.Girvin, L. Jiang, et al., arXiv:1712.05832 (2017).[42] P. Kurpiers, P. Magnard, T. Walter, B. Royer, M. Pechal,J. Heinsoo, Y. Salath, A. Akin, S. Storz, J.-C. Besse,et al., arXiv:1712.08593 (2017).[43] U. Vool and M. Devoret, International Journal of CircuitTheory and Applications , 897 (2017).[44] W. C. Smith, A. Kou, U. Vool, I. M. Pop, L. Frunzio,R. J. Schoelkopf, and M. H. Devoret, Physical Review B94