Enhanced coherence in superconducting circuits via band engineering

Luca Chirolli1, 2 and Joel E. Moore1, 3 1Department of Physics, University of California, Berkeley, CA 94720 2Istituto Nanoscienze - CNR, I-56127 Pisa 3Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 In superconducting circuits interrupted by Josephson junctions, the dependence of the energy spectrum on offset charges on different islands is 2e periodic through the Aharonov-Casher effect and resembles a crystal band structure that reflects the symmetries of the Josephson potential. We show that higher-harmonic Josephson elements described by a cos(2ϕ) energy-phase relation provide an increased freedom to tailor the shape of the Josephson potential and design spectra featuring multiplets of flat bands and Dirac points in the charge Brillouin zone. Flat bands provide noise- insensitive quantum states, and band engineering can help improve the coherence of the system. We discuss a modified version of a flux qubit that achieves in principle no decoherence from charge noise and introduce a flux qutrit that shows a spin-one Dirac spectrum and is simultaneously quote robust to both charge and flux noise.

Introduction.— Superconducting circuits are among tive interference quenches the kinetic energy. A short the best candidates for quantum computation applica- wavelength counterpart is represented by lattice models tions [1–3] and represent an ideal platform for the study such as the Lieb or the Kagome lattice, where an atomic and the implementation of artificial quantum matter redundancy at the unit cell level typically produces flat [4, 5]. Among the several varieties, the superconducting bands. Spectra featuring multiplets of flat bands pro- flux qubit (FQ) [6] represents one of the early prototypes vide noise-insensitive quantum states, without paying the [7–9], and has recently received renewed interest [10–12]. price of weak anharmonicity. Its two fundamental current-carrying states correspond The compactness of the superconducting phase rules to the minima of the circuit Josephson potential, and out long wavelength perturbations but does not forbid quantum tunneling (phase slip) generates coherent su- short wavelength modulations. To achieve multiplets of perpositions. Due to the Aharonov-Casher effect [13, 14], flat bands, an ordinary Josephson junction described by the spectrum of the qubit acquires a dependence on gate- a cos(ϕ) energy-phase relation alone is not sufficient to controlled offset charges localized on the islands between tailor at will the details of a structured unit cell. We the junctions. The charge degeneracy of the condensate can, however, employ higher-harmonic effective Joseph- confers a 2e periodicity to the spectrum, which resembles son junctions, such as a E4e cos(2ϕ) energy-phase re- a crystal band structure [6, 15–17]. In a small loop host- lation, which effectively describes− tunneling of pairs of ing n independent superconducting islands, the spectrum Cooper pairs. A variety of π-periodic energy-phase rela- in the n dimensional charge Brillouin zone (BZ) can po- tions have been recently proposed [29, 30]. An effective tentially host Dirac and Weyl points [17] or flat bands, cos(2ϕ) is realized through gatemons [31, 32] based on thus simulating quantum materials. An analogous and semiconducting wires [33, 34], or can be obtained by us- dual approach has been recently put forward, where the ing four nominally equal JJs in a rhombus configuration Andreev spectrum of a multi-terminal Josephson junc- [35–39]. In turn, the recently introduced bifluxon qubit tion can host Weyl points and can be seen as a kind of [40], defined by the parity of the flux quanta, is based topological matter [18, 19]. on an effective cos(ϕ/2) energy-phase relation. These el- Typically the dependence on the offset charges is mini- ements provide us with augmented freedom to engineer mized as one of the main source of decoherence in super- the Josephson potential and increase the coherence of the conducting qubits is represented by charge noise [6, 20– system. 22]. This is the case of the transmon qubit design [23–25] We discuss a modified flux qubit that displays multi- where a large capacitance shunts a small superconducting plets of levels that hardly depend on the offset charges, island, resulting in band flattening as a function of the thus rendering the device basically insensitive to charge arXiv:2012.11884v1 [cond-mat.mes-hall] 22 Dec 2020 offset charges, at the price of weakly anharmonic spectra. noise. In turn, by increasing the sensitivity to the offset The same idea has been employed in the later versions of charges, we discuss an implementation of a three-band the flux qubit, where a shunt capacitance reduces charge model mimicking a Lieb lattice and define a flux qutrit sensitivity [10–12]. On the other hand, drawing from the featuring a single Dirac cone and an almost flat band in analogy with crystal band structures, flattening of the the charge BZ. Despite the increased sensitivity to offset bands can result from destructive interference. This is charges, this device shows a good degree of robustness the case in the flat bands of twisted bilayer graphene [26– to charge noise and, perhaps more interestingly, an in- 28], which have recently attracted enormous interest, or creased insensitivity to flux noise compared to conven- the quantum Hall effect, where long-wavelength destruc- tional flux qubits. 2

Charge Noise Insensitive FQ.— The low energy Hamil- a) e)

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pendence on the offset charges via the Aharonov-Casher 4 4

effect. In an extended picture of the Josephson poten- 2 2

2 0 2 0 tial, the low energy Hamiltonian can be constructed via a tight binding model, where the role of the momentum is -2 -2 -4 -4

played by the offset charges q1 = Q1/2e and q2 = Q2/2e. -6 -6

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This way, after recognizing a deformed honeycomb lat- 1 1 tice structure, the off-diagonal matrix element is written 2πiQ1/2e 2πiQ2/2e as ∆q = ∆ t (e + e ), in terms of − 0 − 1 FIG. 1. a) Schematics of a variation of flux qubit employing hoppings ∆0 and t1. The FQ Hamiltonian then reads cos(2ϕ) JJs. b,c) Josephson potential of b) an ordinary Flux Qubit and c) Eq. (3) for the choice α = 0.8 and β = 0.2. + ∗ − H = (ϕx)σz ∆qσ ∆qσ , (1) The unit cell containing three minima is highlighted in white. − − d) Exact spectrum showing the six lowest energy level as a with (ϕx) an energy imbalance between the two current function of the two offset charges q1 and q2. d) Unnormalized wavefunction of the first two levels at q = q = 0. carrying states, and ϕx = Φx/2π the flux threading the 1 2

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with i = ∂H/∂λi the operator that couples to the O a) qAAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2lM120y7dbOLuRCihP8GLB0W8+ou8+W/ctjlo64OBx3szzMwLEikMuu63s7K6tr6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LRzdRvPXFtRKwecJxwP6IDJULBKFrp/rFX7ZXKbsWdgSwTLydlyFHvlb66/ZilEVfIJDWm47kJ+hnVKJjkk2I3NTyhbEQHvGOpohE3fjY7dUJOrdInYaxtKSQz9fdERiNjxlFgOyOKQ7PoTcX/vE6K4ZWfCZWkyBWbLwpTSTAm079JX2jOUI4toUwLeythQ6opQ5tO0YbgLb68TJrVindeqd5dlGvXeRwFOIYTOAMPLqEGt1CHBjAYwDO8wpsjnRfn3fmYt644+cwR/IHz+QMD2I2e 2 b) variable λi in the Hamiltonian H in Eq. (2). Charge fluctuations couple to the dimensionless charge operator 01

ˆ −1 !

Q = 8E ( i∂ + q ) and flux fluctuations couple /

i C j j

ij )

C − q

2π ( to the current operator I = ∂V/∂ϕ . Singling out Φ0 x 01 ! AAACCXicbVC7TsMwFHXKq5RXgJHFokIqS0kKEowVLIxFog+piSLHdVqrthNsB6mKurLwKywMIMTKH7DxN7hthtJypCsdnXOv7r0nTBhV2nF+rMLK6tr6RnGztLW9s7tn7x+0VJxKTJo4ZrHshEgRRgVpaqoZ6SSSIB4y0g6HNxO//UikorG416OE+Bz1BY0oRtpIgQ29mJM+CjLHHVcyL4zgw/j0bE4M7LJTdaaAy8TNSRnkaAT2t9eLccqJ0Jghpbquk2g/Q1JTzMi45KWKJAgPUZ90DRWIE+Vn00/G8MQoPRjF0pTQcKrOT2SIKzXioenkSA/UojcR//O6qY6u/IyKJNVE4NmiKGVQx3ASC+xRSbBmI0MQltTcCvEASYS1Ca9kQnAXX14mrVrVPa/W7i7K9es8jiI4AsegAlxwCergFjRAE2DwBF7AG3i3nq1X68P6nLUWrHzmEPyB9fULI2CZVg== the two lowest energy states, we define the qubit energy basis 0 and 1 , and a set of Pauli matrices such that | i | i σz 0 = 0 . The relaxation time is typically estimated | i −| i qAAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fj04rGi/YA2lM120y7dbOLuRCihP8GLB0W8+ou8+W/ctjlo64OBx3szzMwLEikMuu63s7K6tr6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LRzdRvPXFtRKwecJxwP6IDJULBKFrp/rHn9Uplt+LOQJaJl5My5Kj3Sl/dfszSiCtkkhrT8dwE/YxqFEzySbGbGp5QNqID3rFU0YgbP5udOiGnVumTMNa2FJKZ+nsio5Ex4yiwnRHFoVn0puJ/XifF8MrPhEpS5IrNF4WpJBiT6d+kLzRnKMeWUKaFvZWwIdWUoU2naEPwFl9eJs1qxTuvVO8uyrXrPI4CHMMJnIEHl1CDW6hDAxgM4Ble4c2Rzovz7nzMW1ecfOYI/sD5/AECVI2d 1 by treating the environment as a perturbation and takes the Fermi golden rule expression FIG. 2. a) Energy difference between the two qubit states as 1 1 X 2 a function of the normalized gate charges. b) Full spectrum = 0 i 1 Si(ω01), (6) T 2 |h |O | i| highlighting the two lowest energy levels as a function of the 1 ~ i applied flux δϕx = ϕx −π/2 and matrix elements Mφ and M1 determining the relaxation and pure decoherence rates. where Si(ω) is the environment power spectrum defined as S (ω) = R dτe−iωτ δλ (t + τ)δλ (t) and ω = (E i h i i i 01 1 − E0)/~. The pure dephasing time appears in the time infinite pure dephasing time T and qubit operations can evolution of the coherent superposition 0 + eiδφ01(t) 1 , φ t | i h i| i be performed at twice the qubit frequency [43, 44]. To with δφ (t) = R dt0ω (t0). After statistical average 01 0 01 estimate the relaxation rate we note that the circulating over the fluctuating field and assuming Gaussian noise we −  1  current in the setup is twice the one typically circulat- have e Γφ(t) eiδφ01(t) = exp δφ2 (t) , so that 2 01 ing in a flux qubit, due to the doubled periodicity of [22, 41, 42] ≡ h i − h i the Josephson potential. This way, the relaxation rate X 2 is nominally four times larger than the typical FQ case. Γφ(t) = 1 i 1 0 i 0 Fi(t), (7) 2 2 |h |O | i − h |O | i| The exact matrix elements Mφ = (Φ0/2π) I00 I11 i 2 2 | − | and M1 = (Φ0/2π) I01 determining the dephasing and 2 | | 1 R ∞ dω sin (ωt/2) relaxation rates are shown in Fig. 2b), together with the where Fi(t) = 2 Si(ω) 2 and ωc is a low- ~ ωc 2π (ω/2) circuit spectrum, versus the applied flux. It follows that frequency cutoff. The decay for short time is Gaussian, away from the sweet spot dephasing due to flux noise can the pure dephasing time T is defined as Γ (T ) = 1 and φ φ φ severely affect the device performances. strongly depends on the functions F (t). In what follows, i Flux qutrit.— We now explain how a qutrit design can we approximate F (t) to one, which is the worst case sce- i retain a high degree of robustness to charge noise, as in nario [42]. Given the form of the relaxation and dephas- the above circuit, while having a reduced sensitivity to ing rate, two strategies are typically employed to reduce flux noise. We consider the system shown in Fig. 3a). them: either reduce the coupling to the environment, or It is composed by a flux qubit where two junctions are reduce the fluctuations in the environment. Here we fol- shunted each by a π-periodic JJ. The two new effective low the first route and assess the matrix elements of the junctions are described by the effective potential charge and current operators in the qubit basis. Pure dephasing can be assessed by directly looking at Veff (ϕ) = E2e cos(ϕ0 ϕ) E4e cos(2ϕ), (8) the dependence of the qubit frequency ω01 on charge and − − −

flux. In Fig. 3 we plot the energy difference as a function where ϕ0 is the flux threading the loop between the two of the normalized charges on the entire charge BZ. At elements. The Hamiltonian has the form of Eq. (2), with the sweet spot qi = 0, 1/2 the derivative is exactly zero. the Josephson potential given by More interesting is the bandwidth of energy variation 2 2 X in Fig. 2a), from which is follows that 4δ1t1/(∆0(δE V (ϕ) = (E2e cos(ϕi) + E4e cos(2ϕi)) 2 −5 −1 −10 2 − − δ1)) 10 . This way, T1 10 ∆0Sq(∆0) and i=1,2 −1 ' −5 ∝ T < 10 ∆0. We can then conclude that the dephas- 0 φ E2e cos(ϕx ϕ1 ϕ2), (9) ing time due to charge fluctuations is on order of hun- − − − dreds of microseconds in the worst cases. Flux qubits are where we assumed the two effective junctions to be equal, 0 clearly very susceptible to flux noise due to the intrinsic the third to have Josephson energy E2e, and we set ϕ0 = dependence of the Josephson potential on the external 0. For the choice ϕx = π the Josephson potential has flux. The latter controls the value of the circulating cur- three minima in the unit cell, two degenerate and one rent and couples directly to one qubit axis in Eq. (1). relative, as shown in Fig. 3b). Their relative energy can −1 0 It immediately follows that the relaxation rate T1 has be tuned by varying α = E2e/E4e and β = E2e/E4e and a maximum at ϕx = π/2, for which (ϕx) = 0 and the for α = β the three minima are degenerate. In order to perturbation is purely off diagonal. At the same time, be good fundamental states their energy difference must the point ϕx = π/2 represents a sweet spot of formally be smaller than the plasma frequency ωP = √4EC E4e. 4

1 0.1 0.12 0.03

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FIG. 4. Back panels: lowest energy spectrum of a flux qutrit FIG. 3. a) Schematics of a variation of flux qubit whose spec- as a function of: a) the flux bias δϕ = ϕ − π/2, and b,c) trum realizes a flux qutrit. b) Josephson potential of Eq. (9) x x the normalized offset charges q = −q = q. Front panels: for the choice α = 0.3 and β = 0.4. The unit cell containing 1 2 associated matrix elements determining the relaxation and three minima is highlighted in white. c) Unnormalized wave- pure decoherence rates: a) M and M versus δϕ , b) M function of the first three levels at q = q = 1/2. d) Exact φ 1 x φ 1 2 and c) M versus q. spectrum showing the three lowest energy level as a function 1 of the two offset charges Q1 and Q2.

numerically diagonalize the Hamiltonian in the charge This requires that the 4e Josephson element dominate, basis and the low energy spectrum is shown in Fig. 3d) with α, β < 1. On the other hand, too small a value of for α = 0.3, β = 0.352, and EC /E2e = 0.1. An approxi- α calls into play a fourth minimum that would trivialize mate spin 1 Dirac spectrum is obtained, with dispersive the analysis. Small values of β ensure that only three conduction and valence bands and an approximately flat fundamental minima determine the low energy physics. band, with deviations due to finite hopping t2 between The three minima define a flux qutrit and form a Lieb degenerate minima. The wave functions at q1 = q2 = 1/2 lattice in an extended picture of the potential. It is in- is shown in Fig. 1d) and symmetric and anti-symmetric structive to construct a minimal tight-binding model to combinations of the two degenerate minima appear. describe the low energy bands. The potential Eq. (9) The flux qutrit so far defined shows interesting coher- breaks ”mirror” symmetry, V ( ϕ1, ϕ2) = V (ϕ1, ϕ2) and ence properties. Analogously to the case of a qubit we can V (ϕ , ϕ ) = V (ϕ , ϕ ), so that− two fundamental6 hop- 1 − 2 6 1 2 define qutrit dephasing and relaxation rates between dif- pings t0 and t1 can be introduced. A third hopping t2 nm P 2 ferent levels, Γφ (t) = i n i n m i m Fi(t) has to be introduced in order to account for tunneling nm 1 P |h 2|O | i − h |O | i| and Γ = 2 n i m Si(ωnm), and associated through the barrier between two degenerate minima. The 1 ~ i matrix elements M nm|h ,|O that| i| generically describe the in- effective Hamiltonian reads 1,φ fluence of the environment. Clearly the sensitivity to   0 T (q1) T˜q charge noise is enhanced, as shown in Fig. 4b,c). The ∗ ∗ Hq =  T (q1) (ϕx) T (q2)  , (10) point q1 = q2 = 0 represents a sweet spot. Dephasing ˜∗ − Tq T (q2) 0 and relaxation rates all increase towards the center of nm the charge BZ. Mφ all show a dip at the Dirac point, −2πiq 2πi(q1−q2) 01 with T (q) = t0 t1e and T˜q = t2(1+e ). due to level coalescence and only one M shows a dip, − − − 1 For t0 = t1 and  = t2 = 0, the spectrum realizes a well signaling a transition decoupling. know spin 1 Dirac Hamiltonian, with three degenerate More interesting is the sensitivity to flux noise. In states at q1 = q2 = 1/2, a flat band and a Dirac cone Fig. 4a) we plot the dependence of the relevant matrix at the center of the charge BZ. Both  = 0 and t0 = t1 elements, together with the lowest energy level of the open a finite gap in the spectrum. For t6 = 0 the model6 2 device, as a function of the applied flux ϕx at q = 0. We always contains a flat band given by the state notice that only the energy of the flatband is sensitive to

 2πiq2  the applied flux in a good flux window. We then notice t0 + t1e 01 12 0 that Γ1 = Γ1 = 0 within numerical precision. This uq =  0  . (11) is due to the fact that the flux in Eq. (10) couples only t t e2πiq1 − 0 − 1 states 0 and 2 . It also follows that, as for the case | i | i For t2 = 0 the flat band acquires a weak dispersion of the FQ previously described, the associated dephasing 6 02 02 and two symmetry protected Dirac points develops at its rate Γφ reaches minimum when Γ1 is a maximum. In 01 12 crossing with one of the other two bands. The parame- turn, Γφ and Γφ are nonzero in the energy window in ters t0, t1 and t2 depend on the potential barriers and the which only the flatband energy varies with the flux and 02 effective capacitances of the circuit. Fine tuning is pos- Γ1 is activated only at the Dirac point. Importantly, sible through flux-dependent Josephson junctions. We we notice that the scale of the squared matrix element is 5 nominally two orders of magnitude smaller than in the [9] I. Chiorescu, Y. Nakamura, C. J. P. M. Harmans, and case of the flux qubit in Fig. 2. J. E. Mooij, Science 299, 1869 (2003). The matrix elements M nm also give information on the [10] J. Q. You, X. Hu, S. Ashhab, and F. Nori, Phys. Rev. 1,φ B 75, 140515 (2007). accessibility of the quantum states by external means for [11] F. Yan, S. Gustavsson, A. Kamal, J. Birenbaum, A. P. quantum computing manipulations. We notice a com- Sears, D. Hover, T. J. Gudmundsen, D. Rosenberg, plementarity between charge and flux dependence and G. Samach, S. Weber, J. L. Yoder, T. P. Orlando, complete access to the low energy subspace is provided J. Clarke, A. J. Kerman, and W. D. Oliver, Nature Com- 02 01 12 by M1 (ϕx), M1 (q), M1 (q). Our results show that the munications 7, 12964 (2016). modification of conventional superconducting circuits to [12] L. V. Abdurakhimov, I. Mahboob, H. Toida, include two types of Josephson junction current-phase K. Kakuyanagi, and S. Saito, Applied Physics Letters 115, 262601 (2019), https://doi.org/10.1063/1.5136262. relations, both of current interest, can allow significant [13] Y. Aharonov and A. Casher, Phys. Rev. Lett. 53, 319 gains in robustness to both charge and flux noise, pro- (1984). viding a path beyond current limitations of flux qubits. [14] W. J. Elion, J. J. Wachters, L. L. Sohn, and J. E. Mooij, Acknowledgments.— The authors are grateful to N. Phys. Rev. Lett. 71, 2311 (1993). Yao for very useful discussions. L.C. acknowledges the [15] L. Chirolli and G. Burkard, Phys. Rev. B 74, 174510 European Commission for funding through the MCSA (2006). Global Fellowship grant TOPOCIRCUS-841894. J.E.M. [16] R. P. Tiwari and D. Stroud, Phys. Rev. B 76, 220505 (2007). was supported by the U.S. Department of Energy, Office [17] R. Leone, L. P. L´evy, and P. Lafarge, Phys. Rev. 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