Asymptotic Expressions for Charge Matrix Elements of the Fluxonium Circuit
AAsymptotic Expressions for Charge Matrix Elements of the Fluxonium Circuit
Guanyu Zhu and Jens Koch Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA (Dated: October 29, 2018)In charge-coupled circuit QED systems, transition amplitudes and dispersive shifts are governed by the ma-trix elements of the charge operator. For the fluxonium circuit, these matrix elements are not limited to nearest-neighbor energy levels and are conveniently tunable by magnetic flux. Previously, their values were largelyobtained numerically. Here, we present analytical expressions for the fluxonium charge matrix elements. Weshow that new selection rules emerge in the asymptotic limit of large Josephson energy and small inductive en-ergy. We illustrate the usefulness of our expressions for the qualitative understanding of charge matrix elementsin the parameter regime probed by previous experiments.
PACS numbers: 85.25.Cp, 03.67.Lx, 42.50.Pq
I. INTRODUCTION
The fluxonium circuit is one of the most recent additionsto the family of superconducting qubits. It is composed of asmall Josephson junction shunted by a large Josephson junc-tion array which primarily acts as a large kinetic inductance.For quantum state manipulation and readout, the fluxoniumcircuit can be capacitively coupled to a microwave resonatorand thus integrated into the circuit QED architecture . Theamplitudes of photon-induced transitions between differentenergy levels are then determined by the charge matrix ele-ments N ll (cid:48) = (cid:104) l | N | l (cid:48) (cid:105) where l and l (cid:48) denote the circuit’seigenstates, and N is the dimensionless charge operator. Forcircuits like the Cooper pair box (CPB) in both charging andtransmon regime , simple selection rules give a very goodapproximation limiting the one-photon transitions to nearest-neighbor levels ( l → l ± ). For the fluxonium circuit, thisselection rule is absent – leading to interesting and useful fea-tures including the experimentally observed large dispersiveshifts over a wide external flux range despite strong detuningbetween the lowest fluxonium energy splitting and the photonfrequency. .In previous work, we have presented numerical results forthe charge matrix elements of the fluxonium circuit . As il-lustrated in Ref. 9 with results obtained for the experimentallyrealized parameter values of Josephson, charging and induc-tive energy, matrix elements indeed do not obey strict selec-tion rules. Nonetheless, trends of certain matrix elements be-ing up to an order of magnitude larger than others hint at thefact that a new set of selection rules emerges asymptoticallyin the limit of large Josephson energy and small inductive en-ergy. In this limit, and making use of the classification of flux-onium eigenstates into metaplasmon and persistent-currentstates , we derive analytical expressions for the charge ma-trix elements. Based on the asymptotic selection rules, wefinally shed light on the different magnitudes of charge matrixelements realized in the experimental parameter regime.Our paper is structured as follows. In Sec. II, we brieflysummarize the classification of the fluxonium eigenstates intometaplasmon and persistent-current states (previously pre-sented in Ref. 10) and derive analytical expressions for the charge matrix elements. Based on the resulting asymptoticselection rules, we distinguish matrix elements of differentmagnitudes and compare the analytical results with numericalresults for the experimentally realized parameters in Sec. III.Finally, we summarize our findings in Sec. IV. II. ANALYTICAL EXPRESSIONS FOR FLUXONIUMCHARGE MATRIX ELEMENTS
The Hamiltonian describing the fluxonium circuit withinthe superinductance model is given by H f = 4 E C N − E J cos ϕ + 12 E L ( ϕ + 2 π Φ ext / Φ ) . (1)Here, the operator ϕ describes the phase difference across thesmall junction. The conjugate operator N = − i ddϕ is asso-ciated with the charge imbalance across the small junction,in units of the Cooper pair charge (2 e ) . The three coeffi-cients represent the three relevant energy scales in the cir-cuit, namely the charging energy E C = e / (2 C J ) , Joseph-son energy E J of the small junction, and the effective induc-tive energy E L of the “superinductor” made by the Josephsonjunction array. It is instructive to view the Hamiltonian H f as describing a fictitious particle in a sinusoidal potential, de-formed by an overall parabolic envelope. In this point of view, ϕ plays the role of the spatial coordinate. Hence, the Joseph-son and inductive energy terms in H f determine the potentialenergy V ( ϕ ) , while the charging term produces the kineticenergy contribution. The external magnetic flux Φ ext (in unitsof the flux quanta Φ = h/ e ) spatially shifts the parabolicenvelope. Due to the presence of the inductive term, the ap-propriate boundary conditions supplementing the stationarySchr¨odinger equation for H f are derived from normalizabilityof its eigenstates | ψ (cid:105) , i.e., (cid:82) R dϕ |(cid:104) ϕ | ψ (cid:105)| < ∞ , implying ψ ( ϕ ) → when ϕ → ±∞ .In the limit of large Josephson and small inductive energy, E J (cid:29) E C (cid:29) E L , (2)the low-lying eigenstates of fluxonium can be classifiedinto two physically distinct types: metaplasmon states and a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y persistent-current states . For clarity and introduction ofnecessary notation, we briefly review this classification as ob-tained in Ref. 10. To do so, we rewrite H f in a more suitablebasis and start by separating off the inductive energy term, H f = H (cid:48) + H ind , where H (cid:48) = 4 E C N − E J cos ϕ . De-spite the tempting appearance of H (cid:48) , we must refrain fromidentifying it as the ordinary Cooper pair box Hamiltonian:in Eq. (1), the spatial coordinates ϕ and ϕ + 2 π are distinctpositions. Hence, H (cid:48) is not subject to periodic boundary con-ditions as the Cooper pair box, but rather obeys the quasi-periodic boundary conditions familiar from Bloch’s theorem,as appropriate for a particle in an infinitely extended periodicpotential. Accordingly, the eigenstates of H (cid:48) are Bloch states, H (cid:48) | p, s (cid:105) = (cid:15) s ( p ) | p, s (cid:105) , (3)where s ∈ N is the band index, p ∈ [0 , the quasimomentumin the first Brillouin zone, and (cid:15) s ( p ) denotes the band disper-sion for the cosine potential (which coincides with the ordi-nary offset charge dispersion of the Cooper pair box levels ).To rewrite the inductive contribution H ind in the Bloch ba-sis, we re-interpret p as a new spatial coordinate. Since it“lives” on a circle with circumference , the resulting expres-sion ϕ = i d/dp + Ω for the phase operator must generallyinclude an inter-band coupling operator Ω . This inter-bandcoupling can be neglected for low-lying bands and sufficientlylarge E J /E C . In that limit, the Hamiltonian H f , hence, be-comes block-diagonal, splitting into individual Hamiltoniansfor each band s , H f ≈ (cid:80) s H s | s (cid:105)(cid:104) s | , where H s = E L (cid:18) i ddp + 2 π Φ ext Φ (cid:19) + (cid:15) s ( p ) . (4)Now, accompanied by periodic boundary conditions in p , eachHamiltonian H s indeed has the same structure as the Hamil-tonian of a Cooper pair box. The only difference lies in theform of the periodic “potential” (cid:15) s ( p ) , which generally devi-ates from a pure cosine. To make the analogy concrete, notethat the variable πp ∈ [0 , π ) in H s takes on the role of theperiodic phase variable of the Cooper pair box, and the ex-ternal flux Φ ext / Φ that of the Cooper pair box offset charge n g .Next, two different types of low-lying fluxonium states canbe distinguished for each band s . First, eigenstates | ν, s (cid:105) with energies below the maximum of the energy dispersion, E νs (Φ ext ) < max p (cid:15) s ( p ) , are metaplasmon states . They arequasi-bound states in the (cid:15) s ( p ) potential analogous to thelowest states of the Cooper pair box in the transmon regime.The corresponding eigenenergies depend only weakly on theexternal flux Φ ext , just as Cooper pair box levels are offset-charge insensitive in the transmon regime . Second, eigen-states with energies above the maximum of the energy dis-persion, are persistent-current states . Their energies stronglydepend on the external flux Φ ext , closely mimicking the offset-charge dependence of the high-lying transmon levels (forwhich, effectively, the charging regime holds). While quasi-itinerant in the (cid:15) s ( p ) potential, persistent-current states local-ize in the individual minima of the V ( ϕ ) potential [Eq. (1)]. They are conveniently expressed in terms of Wannier states | m, s (cid:105) = (cid:90) − dp e − i πmp | p, s (cid:105) . (5)Expressed in this basis, the Hamiltonian Eq. (4) reads: H s ≈ (2 π ) E L ( m + Φ / Φ ) (6) + 12 ∞ (cid:88) m = −∞ (cid:15) s, (cid:20) | m, s (cid:105)(cid:104) m + 1 , s | + H.c. (cid:21) + (cid:15) s, , where we have approximated the potential (cid:15) s ( p ) by thetruncated Fourier series (cid:15) s ( p ) ≈ (cid:15) s, + (cid:15) s, cos(2 πp ) , andused (cid:80) ∞ m = −∞ | m, s (cid:105)(cid:104) m +1 , s | = e − i πp as well as m = i d/d (2 πp ) . Note that in the transmon regime ( E J (cid:29) E C ), (cid:15) , is just the plasmon energy √ E J E C . Analytical approx-imations for (cid:15) s, and (cid:15) s, in the transmon regime are givenin Ref. 7. Based on the classification into metaplasmon andpersistent-current state, we are ready to derive analytical ex-pressions and asymptotic selection rules for the charge matrixelements. Due to the two types of states involved, there arethree possible types of charge matrix elements, which we dis-cuss one by one in the following. a. Matrix elements between persistent-current states. The Wannier states | m, s (cid:105) provide good approximations forthe persistent-current states (away from degeneracies whichoccur at integer and half-integer Φ ext / Φ ). The charge matrixelements between two persistent-current states, possibly fromdifferent bands s and s (cid:48) , are then given by (cid:104) m, s | N | m (cid:48) , s (cid:48) (cid:105) ≈ − i (cid:18) E J E C (cid:19) (cid:90) ∞−∞ dϕ w ∗ ms ( ϕ ) (7) × (cid:20)(cid:112) s (cid:48) w m (cid:48) s (cid:48) − ( ϕ ) − (cid:112) s (cid:48) + 1 w m (cid:48) s (cid:48) +1 ( ϕ ) (cid:21) . Here, w ms ( ϕ ) ≡ (cid:104) ϕ | m, s (cid:105) is the approximate persistent-current state wavefunction in ϕ -space. Due to the strong lo-calization in the minima of V ( ϕ ) , persistent-current states inadjacent minima are nearly orthogonal. One, hence, obtainsthe approximation (cid:104) m, s | N | m (cid:48) , s (cid:48) (cid:105) (8) ≈ − i (cid:18) E J E C (cid:19) δ m,m (cid:48) (cid:20)(cid:112) s (cid:48) δ s,s (cid:48) − − √ s δ s,s (cid:48) +1 (cid:21) . To obtain nonzero values for the charge matrix elements inthis limit, the two states involved must obey two asymptoticselection rules. The first is the neighboring-band selectionrule, demanding ∆ s = s (cid:48) − s = ± . The second is the same-minimum selection rule, given by ∆ m = m (cid:48) − m = 0 . Ac-cordingly, both states involved must belong to the same localminimum m of the potential V ( ϕ ) . This rule implies that thecirculating persistent current (and the flux it generates) can-not change its magnitude or direction during the transition.Both rules follow intuitively from considering the momentummatrix elements of local harmonic oscillators with negligibleneighbor overlap. b. Matrix elements between metaplasmon states. Thecharge matrix elements involving metaplasmon states only,can be brought into the form (cid:104) ν (cid:48) , s (cid:48) | N | ν, s (cid:105) ≈ i (cid:18) E J E c (cid:19) ( √ sδ s,s (cid:48) +1 − (cid:112) s (cid:48) δ s,s (cid:48) − ) × (cid:90) − dp χ ∗ ν (cid:48) s (cid:48) ( p ) χ νs ( p ) . (9)This expression was previously derived in Ref. 10, exceptfor a misprint in the prefactor (fixed here). By χ νs ( p ) ≡(cid:104) p, s | ν, s (cid:105) , we denote the metaplasmon wavefunctions in theBloch basis. The index ν = 0 , , , . . . labels energy levelswithin a fixed band s . The first asymptotic selection rulemanifest in Eq. (9), is the neighboring-band selection rule ∆ s = ± . The matrix elements still depend on the overlapbetween two metaplasmon states, see again Ref. 10 for ana-lytic approximations and asymptotic selection rules in ν, ν (cid:48) . c. Matrix elements between metaplasmon and persistent-current states. The last type of matrix elements involvesboth a metaplasmon and a persistent-current state. Its asymp-totic expression is given by (cid:104) ν, s | N | m, s (cid:48) (cid:105) (10) ≈ − i ν +1 √ ν ν ! (cid:32) E J E L E C (cid:12)(cid:12) (cid:15) s, (cid:12)(cid:12) (cid:33) ( √ sδ s,s (cid:48) +1 − (cid:112) s (cid:48) δ s,s (cid:48) − ) × exp (cid:104) − πF ms (Φ ext ) (cid:105) H ν (cid:104) √ πF ms (Φ ext ) (cid:105) . Here, H ν ( x ) is the Hermite polynomial of order ν and wehave abbreviated F ms (Φ ext ) = ( m + Φ ext / Φ )( E L / | (cid:15) s, | ) / .The only selection rule present is the one for neighboringbands. The magnitude of the matrix elements depends on bothquantum numbers m and ν , and is conveniently tunable bymagnetic flux Φ ext . III. MATRIX ELEMENTS REALIZED IN EXPERIMENTS
The values of Josephson, charging and inductive energyrealized in recent experiments ( E J /h = 8 . GHz ; E C /h =2 . GHz ; E L /h =0 . GHz in Ref. 1) do not quite reach theasymptotic behavior predicted for E J (cid:29) E C (cid:29) E L . Hence,we cannot expect the asymptotic results from Sec. II to quan-titatively match the exact results. Nonetheless, the asymptoticselection rules can still give valuable intuition and qualitativepredictions for the different magnitudes of matrix elements,which will be of immediate use in the design of future fluxo-nium devices.To apply the results derived in Sec. II, we first need to es-tablish the type of each low-lying fluxonium eigenstate (meta-plasmon versus persistent-current), given the experimental pa-rameters. As shown in Fig. 1(a), the energy dispersion of thelowest band, (cid:15) s =0 ( p ) , turns out to be too shallow to supportany metaplasmon states. As a result, the ground state and firstexcited state are found to be s = 0 persistent-current states, −1 −0.5 0 0.5 10102030 s =0 s =1 s =2 s =3 (a) m = m = m = - −10 0 10 m = - (b) V ( ) FIG. 1: (Color online) (a) Fluxonium energy levels (solid curves)as a function of external flux Φ ext , for the parameters realizedexperimentally . Shaded regions in the background show positionand width of the bands (cid:15) s ( p ) . The three s =0 persistent-current stateswith parabolic flux dependence are labeled by their quantum number m . (b) Fluxonium eigenfunctions for Φ ext / Φ =0 . [vertical dashedline in (a)]. The bold black curve shows the fluxonium potential V ( ϕ ) . Local minima are labeled by the quantum numbers m . Thefluxonium wavefunctions (thin curves) are offset by their eigenener-gies. FIG. 2: (Color online) Comparison of numerical results and asymp-totic predictions for charge matrix elements. Solid curves show nu-merical results for the magnitude of the charge matrix elements, |(cid:104) l | N | l (cid:105)| , |(cid:104) l | N | l (cid:105)| and |(cid:104) l | N | l (cid:105)| . Dashed curves show theasymptotic matrix elements between the ground state and the low-est metaplasmon state, namely |(cid:104) l | N | ν , s (cid:105)| , and between the twolow-lying persistent current states, |(cid:104) m , s | N |− m , s (cid:105)| . lying in the gap between the lowest two bands (cid:15) s =0 ( p ) and (cid:15) s =1 ( p ) . Due to inversion symmetry and periodicity in themagnetic flux, we may restrict our discussion in the followingto the flux range ≤ Φ ext / Φ ≤ . without loss of general-ity. Under these conditions and sufficiently away from integerand half-integer flux, the two lowest persistent-current statesare well approximated by the Wannier states | − m , s (cid:105) and | m , s (cid:105) .In the following, we focus on the example flux point Φ ext / Φ = 0 . . The exact wavefunctions at this point are il-lustrated in Fig. 1(b). Note that the ground state (first excitedstate) is indeed primarily localized in the minimum m = 0 ( m = − ). The second and third excited states are metaplas-mon states with band indices s = 1 and s = 2 , respectively.As expected, they delocalize over multiple minima of the po-tential V ( ϕ ) . At Φ ext / Φ = 0 . , the fourth excited state caneasily be identified as a persistent-current state of the s = 0 band, by noting its quadratic flux dependence expected forthe | m , s (cid:105) state. Accordingly, it is strongly localized in the m = 1 minimum. However, due to the large inter-band cou-pling for high-lying levels, this state is already significantlyinfluenced by the nearby metaplasmon state [see the largeavoided crossing of the third and fourth excited states near Φ ext / Φ =0 . in Fig. 1(a)]. As a result, the wavefunction of thefourth excited state slightly spreads out of the m = 1 well. Foreven higher levels, the inter-band coupling becomes strongerand the classification into metaplasmon and persistent-currentstates ceases to apply.The situation of half-integer and integer Φ ext / Φ is spe-cial because of the additional parity symmetry of the poten-tial V ( ϕ ) . For Φ ext / Φ = 0 . , the state | − m , s (cid:105) becomesdegenerate with | m , s (cid:105) . The ground and first excited statesare hence the symmetric and antisymmetric superposition of |− m , s (cid:105) and | m , s (cid:105) . For zero external flux, the ground stateis | m , s (cid:105) , while the first and second excited states becomethe symmetric and antisymmetric superposition of |− m , s (cid:105) and | m , s (cid:105) .With the classification of states in hand, we now employthe analytic results from Sec. II to describe the qualitative be-havior and magnitudes of the charge matrix elements. Awayfrom the degeneracies at integer and half-integer Φ ext / Φ , thecharge matrix element between the ground and the first ex-cited state is approximated by the matrix element between twodifferent persistent-current states, namely (cid:104) l | N | l (cid:105) ≈ (cid:104) m , s | N |− m , s (cid:105) . (11)Here, l enumerates the fluxonium eigenstates in the order oftheir eigenenergies. The magnitude of this matrix elementis relatively small because of the suppression enforced bythe asymptotic selection rules for two persistent-current states[ ∆ s = ± and ∆ m = 0 ; see Eq. (8)]. Figure 2 shows that thecharge matrix element between ground and first excited state, | N | , is indeed significantly smaller than the other elements(especially compared to | N | ) over most of the flux range.We note this discrepancy in matrix element magnitudescould possibly lead to an interesting potential application: ifcoupling to the environment via charge dominates the qubitrelaxation, then the lowest three fluxonium levels ( l = 0 , , )could form a Λ -system over a wide flux range, with the state | l (cid:105) being a relatively long-lived metastable state, as notedpreviously in Ref. 13. The origin of the Λ -configuration isintuitive from Fig. 1(b): the states | l (cid:105) and | l (cid:105) are persistent-current states localized in different minima with only verysmall wavefunction overlap. The state | l (cid:105) , by contrast, is ametaplasmon state and has a large wavefunction overlap with both persistent-current states, resulting in relatively large ma-trix elements (and hence transition rates) between these states.As a result, the state | l (cid:105) may have a significantly longer lifetime than the state | l (cid:105) .It is instructive to assess the deviation of exact results forthe experimental parameters from the asymptotic prediction.For this comparison, we choose two eigenstates which, in agiven flux range, can be approximately classified as a meta-plasmon state and a persistent-current state, respectively. Theapproximate metaplasmon state we choose is | ν , s (cid:105) . Fig-ure 1(a) shows that, in the flux region < Φ ext / Φ < . , thismetaplasmon state approximates the state | l (cid:105) . In the remain-ing flux region, this metaplasmon state approximates the state | l (cid:105) . The persistent-current state we choose is | m , s (cid:105) , whichapproximates the ground state | l (cid:105) away from Φ ext / Φ = 0 . .We employ Eq. (10) to calculate the asymptotic result for thematrix element |(cid:104) m , s | N | ν , s (cid:105)| , where the input parame-ter (cid:15) , /h = 1 . GHz is acquired from the half-width of the s = 1 CPB band by diagonalizing H (cid:48) .The result obtained from Eq. (10) is valid only sufficientlyaway from Φ ext / Φ =0 . . There, the ground state | l (cid:105) be-comes a hybridization of the two states, | − m , s (cid:105) and | m , s (cid:105) , which are energy degenerate states in the absenceof (cid:15) , . To account for this, we consider the × subspacecontaining both persistent-current states. The Hamiltonian inthis subspace is H sub ≈ (cid:32) π E L ( Φ ext Φ ) + (cid:15) , (cid:15) , (cid:15) , π E L ( Φ ext Φ − + (cid:15) , (cid:33) , a truncated version of Eq. (6). Here, the input parameter (cid:15) , /h = 0 . GHz is obtained from the half-width of the s = 0 CPB band. By diagonalizing this matrix, we obtain animproved approximation for the ground state | l (cid:105) , and for theasymptotic prediction of the matrix element |(cid:104) l | N | ν , s (cid:105)| .The asymptotic prediction (dashed curve in Fig. 2) is to becompared with the corresponding solid curves showing thenumerically exact results – in particular, the results for N and N in the previously mentioned flux ranges. Agreementis qualitative rather than quantitative, as expected. Note that,by accounting for hybridization, the complete suppression of N at Φ ext / Φ =0 . enforced by parity symmetry is correctlypredicted. Similarly, the asymptotic prediction for vanishing N agrees qualitatively with the significantly smaller valuesobtained numerically. IV. CONCLUSION
In summary, we have derived asymptotic expressions forthe charge matrix elements of the fluxonium circuit in the pa-rameter limit E J (cid:29) E C (cid:29) E L , presented in Eqs. (8)–(10).Our derivation is based on the classification of fluxoniumeigenstates into persistent-current and metaplasmon states ,and produces simple selection rules for the band indices s andother quantum numbers which can be intuitively understoodfrom the localization properties of the different types of states.We employ our asymptotic predictions to interpret the nu-merically calculated matrix elements for the intermediate pa-rameter regime realized in experiments . Even though quan-titative agreement cannot be expected in this intermediateregime, we find good qualitative agreement and confirm thatthe asymptotic selection rules provide a useful predictor fordifferent magnitudes of charge matrix elements. Thus, our re-sults can easily guide the choice of experimental parametersin order to reach the desired tunability of charge matrix ele-ments in future fluxonium devices. The relatively large degreeof tunability in fluxonium devices can be harnessed for influ-encing transition rates (possibly providing a route towards a Λ -system ), dispersive shifts , as well as the effective qubit-qubit interaction strength when coupling multiple fluxoniumdevices to a single microwave resonator mode . Acknowledgments
We are indebted to Michael Devoret, Leonid Glazman andVladimir Manucharyan for numerous insightful discussions.Our research was supported by the NSF under Grant No.PHY-1055993. V. E. Manucharyan, J. Koch, L. I. Glazman, and M. H. Devoret,Science , 113 (2009). V. E. Manucharyan, N. A. Masluk, A. Kamal, J. Koch, L. I. Glaz-man, and M. H. Devoret, Phys. Rev. B , 024521 (2012). A. Blais, R. S. Huang, A. Wallraff, S. M. Girvin, and R. J.Schoelkopf, Phys. Rev. A , 062320 (2004). R. J. Schoelkopf and S. M. Girvin, Nature (London) , 664(2008). Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, Nature (London) ,786 (1999). V. Bouchiat, D. Vion, P. Joyez, D. Esteve, and M. H. Devoret,Phys. Scr.
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