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Life after charge noise: recent results with transmon qubits

A. A. Houck,1 Jens Koch,2 M. H. Devoret,2 S. M. Girvin,2 and R. J. Schoelkopf2

1Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA2Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA

(Dated: December 10, 2008)

We review the main theoretical and experimental results for the transmon, a superconductingcharge qubit derived from the Cooper pair box. The increased ratio of the Josephson to chargingenergy results in an exponential suppression of the transmon’s sensitivity to 1/f charge noise. Thishas been observed experimentally and yields homogeneous broadening, negligible pure dephasing,and long coherence times of up to 3 µs. Anharmonicity of the energy spectrum is required forqubit operation, and has been proven to be sufficient in transmon devices. Transmons have beenimplemented in a wide array of experiments, demonstrating consistent and reproducible results invery good agreement with theory.

PACS numbers: 03.67.Lx, 85.25.-j, 42.50.-p

I. INTRODUCTION

The idea of harnessing the power of quantum mechan-ics for specific computational tasks, first proposed in theearly 1980s (see e.g. [1] for an early review), has inspiredphysicists, engineers, and computer scientists alike. Itcontinues to act as a prime driving force behind the on-going research on quantum control, measurement, deco-herence, and quantum information. The basic buildingblocks of a universal quantum computation scheme [2]are quantum bits (qubits), which are quantum coherenttwo-level systems. Despite some impressive progress, thelast decades have clearly established the difficulty of im-plementing even just a few qubits.

Nature offers only a few true two-level systems, such asspin-1/2 systems, or massless spin-1 systems (e.g. polar-ization of photons). As an alternative, sufficiently anhar-monic multi-level systems can be used as effective qubits.In principle, they also offer the possibility of multi-levelquantum logic [3]. All such systems bear in commondiscrete energy spectra and can be understood as gener-alized atoms. Superconducting circuits have been estab-lished as promising systems for tunable artificial atoms:they utilize the quantum coherence of the superconduct-ing state to minimize undesired dissipation, and employJosephson junctions as the fundamental nonlinear anddissipationless element to obtain an anharmonic spec-trum [4, 5, 6]. Moreoever, fabrication of superconduct-ing qubits benefits from the existence of well-establishedmicrofabrication techniques, spurring hope that the re-quired scaling towards multi-qubit systems will not posea fundamental obstacle.

Here, we review the characteristics of the transmonqubit [7], a superconducting charge qubit derived fromthe Cooper pair box [8, 9], with minimal sensitivity to1/f noise. The transmon made its debut in an experimentdemonstrating the photon-number dependent qubit fre-quency shift in the strongly dispersive limit [10]. Sincethen it has been successfully employed in a growing num-ber of experiments, and has demonstrated an excellentlevel of agreement with theory. Coherent coupling be-

tween two transmon qubits via virtual microwave pho-tons was reported by J. Majer et al. [11]. A comprehen-sive verification of predicted transmon properties withhigh accuracy has been presented in Ref. [12], with re-laxation and dephasing times in the microsecond range.The coherence times of seven different transmon deviceshave been analyzed and shown to be both reproducibleand predictable over more than an octave in qubit fre-quency [13]. Transmons have further been involved inrecent studies of the

√n anharmonicity of the Jaynes-

Cummings ladder [14, 15], and benchmarks of single-qubit operations [16].

The scope of this paper is to provide a comprehensivereview of the transmon basics, together with a summaryof its fabrication and its overall performance as observedin recent experiments.

II. GENERAL IDEA OF THE TRANSMON

Most naturally, the transmon qubit is understood asa modified version of the prototypical charge qubit, theCooper pair box (CPB) [8, 9]. As depicted in Fig. 1, thetransmon shares with the CPB the same underlying cir-cuit topology. More specifically, its circuit is most closelyrelated to the differential single Cooper pair box [17],which is composed of two superconducting islands andno reservoirs. Because the circuit provides no externalconnection between the two islands, the operator for thecharge difference n between the two islands has a discretespectrum, and the superconducting phase difference ϕ isonly defined modulo 2π, i.e., it is a compact variable forwhich ϕ and ϕ + 2πz are considered as identical for anyinteger z [33]. The corresponding Hamiltonian is givenby

H = 4EC(n − ng)2 − EJ cosϕ, (1)

where the two terms describe the contribution fromcharging effects and Josephson tunneling, respectively.The magnitudes of these terms are set by the single-electron charging energy EC = e2/2CΣ with CΣ =

2

FIG. 1: The transmon qubit. (a) The circuit of the transmonis identical to the circuit of a differential single Cooper pairbox, consisting of two superconducting islands coupled by twoJosephson junctions. The coupling to ground is purely capac-itive. The use of two Josephson junctions allows for tuningof the effective Josephson energy via the external magneticflux penetrating the superconducting loop. (b) and (c) showthe optical and SEM image of a transmon device positionedinside a coplanar waveguide. While the size of the junctionsand the superconducting loop is very similar to CPB devices,the inter-island capacitance Cs is increased drastically dueto the large size of the islands and the interdigitated fingerstructure. This capacitance is matched by comparably largecapacitances Cg and C′

g to the ground plane and centerpin ofthe transmission line resonator.

Cs + (C−1g + C′−1

g )−1, and the Josephson energy EJ ,which is set by the junction’s normal-state conductanceGt and the supeconducting gap ∆ via the Ambegaokar-Baratoff relation EJ = hGt∆/8e2 [18]. The offset chargeis denoted by ng and can be tuned by the external gatevoltage.

While the CPB and transmon share the same Hamil-tonian, they belong to different parameter regimes: theCPB is typically operated with EJ ≈ EC , and the trans-mon with EJ ≫ EC . This transmon regime is reachedprimarily by lowering the charging energy EC . In prac-tice, this is achieved by increasing the island sizes [34],thus adding a large shunt capacitance as shown in Fig.1(b). Shunt capacitances have also been independentlyproposed to improve dephasing times in flux qubits bya factor of 3 [19], and have been implemented in phasequbits to avoid spurious resonances [20].

Both the principle benefits and drawbacks of the trans-mon are evident from examining how the energy spec-trum changes as one increases the ratio of Josephsonenergy and charging energy EJ/EC from the chargingregime to the transmon regime, cf. Fig. 2(a). In thecharge regime, the spectrum of the CPB is dominated bycharge parabolas with avoided crossings at the charge de-generacy points due to Josephson tunneling. If operated

FIG. 2: Level spectrum and charge sensitivity of the Cooperpair box in the charge and transmon regime. (a) Eigenener-gies Em (first three levels, m = 0, 1, 2) of the CPB Hamil-tonian (1) as a function of the effective offset charge ng fordifferent ratios EJ/EC . All energies are given in units ofthe transition energy E01 (evaluated at the degeneracy pointng = 1/2). The zero point of energy is chosen as the bottomof the m = 0 level. The sequence of plots highlights the ex-ponentially increasing flatness of energy levels and the slowloss of anharmonicity as EJ/EC is increased. (b) Order-of-magnitude estimates of dephasing times Tϕ caused by chargenoise, assuming Gaussian fluctuations around a fixed offsetcharge, cf. Eq. (2). Arrows on the right-hand side mark theworst-case estimates for Tϕ, as determined by the total chargedispersion. The results demonstrate the exponential gain inthe dephasing time due to charge noise under an increase ofEJ/EC .

away from a charge degeneracy point, it is readily appar-ent that the qubit transition energy E01 varies rapidlywith gate charge ng, thus resulting in fast dephasing dueto random fluctuations in the local electrostatic potential(1/f charge noise). To some extent, this can be combat-ted by careful biasing of the qubit at a charge degeneracypoint, where the levels are first-order insensitive to chargenoise, a special operating point termed “sweet spot” [21].When biased at such a point, dephasing decreases; how-ever, second-order effects of charge noise can still limitdephasing rates. As confirmed in an experiment by M.Metcalfe et al. [22], second-order charge noise can indeedbe identified as the clear limitation on dephasing timesin the CPB charge regime. As an additional complica-tion, drifts in the offset charge can quickly result in adeparture from the sweet spot.

The main idea of the transmon is to eliminate theseproblems by making the charge-dependence of energylevels negligibly small. This is achieved precisely by anincrease of the EJ/EC ratio, as shown in the sequence

3

of plots in Fig. 2(a). The graphs demonstrates that theenergy levels become increasingly flat, i.e., independentof charge, as EJ/EC is increased. We note, however,that the energy spectrum always remains 2e periodic inthe offset charge, illustrating that the transmon is stilla charge qubit. (No such periodicity exists, e.g., for thephase qubit where ϕ is not a compact variable.) The in-creased flatness of levels at EJ/EC ≫ 1 effectively gen-erates a “sweet spot everywhere” so that sensitivity tocharge noise is suppressed to high order, and device per-formance is not strongly dependent on a particular biaspoint anymore. A detailed analysis [7] shows that thesuppression of charge sensitivity is exponential in the pa-rameter

√

8EJ/EC .The effect of this exponential suppression can be re-

cast into a statement about the dephasing induced by 1/fcharge noise. A useful order-of-magnitude estimate of thedephasing time Tϕ can be obtained by using characteris-tic amplitudes of charge noise and evaluating the fluctu-ations in qubit frequency in terms of a Taylor expansion.Assuming a Gaussian process for the offset charge, theroot-mean-square fluctuations in the qubit frequency aregiven by

δωrms =

[

n2

rms(∂ω01

∂ng)2 +

3n4rms

4(∂2ω01

∂n2g

)2]1/2

. (2)

To obtain a finite variance, the 1/f spectrum has to becut off, and all results depend weakly (logarithmically)on the specific choice of the cutoff. Here, we use a typicalvalue of nrms = 0.5 · 10−3 [23]. The estimate predicts asix orders of magnitude improvement in dephasing timedue to charge noise Tϕ ≈ 1/δωrms by changing EJ/EC

by a factor of 50, cf. Fig. 2(b).A worst-case estimate of the dephasing time can also

be obtained by considering the maximum possible varia-tion of the qubit transition energy, known as the chargedispersion ǫ01. Dephasing cannot occur faster than Tϕ ∼~/ǫ01, independent of the amplitude of 1/f charge noise.This scenario is depicted by arrows in Fig. 2(b). Dueto the exponential suppression of level variation in thetransmon regime, dephasing remains negligible even withthis worst case estimate. Furthermore, a detailed analysis[7] shows that the increase in EJ/EC does not heightensensitivity to any of the other known 1/f noise mecha-nisms of dephasing, such as flux or critical current noise.In fact, as is evident from Table I, by operating in thetransmon instead of the charge regime, one gains a fac-tor of 2 in both the insensitivity with respect to criticalcurrent and flux noise.

The only drawback of the EJ/EC increase is revealedby examining the level spacings in Fig. 2(a). Althoughthe charge dependence has been suppressed exponen-tially, the level spectrum approaches that of a pure har-monic oscillator, which would prevent the system fromacting as a qubit. However, the anharmonicity α, deter-mined by the difference between the fundamental qubittransition and the next higher transition frequency, de-

Transmon CPB

EJ/EC = 100 EJ/EC = 1

Noise source 1/f amplitude, A Tϕ (µs) Tϕ (µs)

Charge 10−4-10−3e [23] 24,600 1.1∗

Flux 10−6-10−5Φ0 [29, 30] 3,600∗ 1,800∗

Crit. current 10−7-10−6I0 [31] 35 17

∗ These values are evaluated at a sweet spot(i.e., second-order noise).

TABLE I: Comparison of dephasing times for the transmonand Cooper pair box qubit with ω01/2π = 7GHz. Contribu-tions to Tϕ are theoretical predictions based on [7]. Entriesin bold face mark the dominant noise channel. For the CPB,second-order charge noise at the sweet spot limits the perfor-mance of the qubit. In contrast, for the transmon dephasingis suppressed to an extent that coherence times are limitedby relaxation (T1) processes only.

creases only slowly as function of the EJ/EC ratio, fol-lowing α/ω01 ∼ (EJ/EC)−1/2 [7]. Thus, increasing theEJ/EC ratio from 1 to 50 can virtually eliminate the ef-fects of charge noise, and still maintain sufficient anhar-monicity to act as an effective two-level system. We illus-trate this statement by considering the concrete exampleof a device with transition frequency ω01/2π = 7 GHzand EJ/EC = 100. For these parameters, one obtains anabsolute anharmonicity of 260 MHz and a dephasing timedue to charge noise of Tϕ ≥ 25 ms (worst-case estimate).

At the large EJ/EC ratios characteristic for the trans-mon regime, readout of the qubit state via charge de-tection or measurement of the quantum capacitance [24]as in the CPB is not possible. In fact, the transmondoes not possess any dc measurable state-dependent pa-rameters (charge, flux, etc.). However, the transmonstill exhibits a strong coupling between charge and acvoltage, rendering it an ideal candidate for an artificialatom in the circuit QED architecture [25]. A full deriva-tion of the coupling strength is given in [7], which canbe intuitively understood from adding an ac componentto the offset charge to describe the quantized resonatorfield, ng → ndc

g + CgVrms(a + a†)/2e. Here, Vrms de-notes the root-mean-square voltage of the resonator atthe transmon position, and a, a† are the annihilationand creation operators for microwave photons in the rel-evant resonator mode. Carrying out the square in Eq.(1) generates the coupling term, from which one obtainsthe coupling strength gij = 2eVrms〈 i |n | j 〉Ceff

g /CΣ with

Ceffg = (C−1

g + C′g−1

)−1. In the transmon regime, thematrix elements are significant only between nearest-neighbor transmon levels, i = j ± 1, and their overallmagnitude is larger by a factor of typically 3 − 5 thanfor the CPB, due to the participation of more than oneCooper pair. In all transmons fabricated so far, the ratioCg/CΣ is also ∼ 3 times larger than in the CPB, leadingto a total increase in coupling strength of more than anorder of magnitude.

4

III. TRANSMON FABRICATION

All transmons have thus far been fabricated in a cir-cuit QED architecture, as the cavity offers a convenientmeans of reading out the qubit state. The cavity con-sists of a 50 Ω niobium coplanar waveguide cavity witha 4.2 µm gap between the center pin and ground planes.Standing waves form between transverse capacitors ateither end of the waveguide, designed with a 3 µm spac-ing and a range of capacitances to vary the cavity Qfrom 100 to 500, 000. Cavities are patterned with pho-tolithography and reactive ion etching of 180 nm filmsof sputtered niobium. Transmon qubits are patternedwith electron beam lithography, and consist of two layers(20 and 80 nm thick) of lifted-off aluminum, depositedwith an angle evaporation process without venting sam-ples to air. Samples have been fabricated on bare andoxidized silicon and sapphire substrates, with a commonQ ∼ 70, 000 limit on relaxation in the devices on sap-phire. Details of fabrication and processing are likely toprove important for future improvements in coherence,especially given that all transmon qubits now achieve acommon limit to T1.

IV. TRANSMON PERFORMANCE

The major transmon features have been confirmed byexperiments, which have been published in Refs. [12] and[13]. Here, we summarize these results and provide anoverview of the most current transmon data.

To establish that the anharmonicity is still sufficient totreat the transmon as an effective two-level system, thelevel spectrum of the lowest two transitions is measuredby spectroscopy, as depicted in Fig. 3(a). It is immedi-ately apparent that the transitions 0 → 1 and 1 → 2 arewell resolved in frequency space. The observed 455 MHzanharmonicity allows for single-qubit manipulation withpulse durations of only a few nanoseconds without occu-pying the third level.

The insensitivity of the transmon to charge noise,which should lead to long coherence, can directly be ver-ified by probing the charge dependence of the transmonlevel spectrum. The suppression of this dependence isquantified by using the notion of charge dispersion, de-fined as the total variation of the qubit transition fre-quency (as a function of gate charge). As demonstratedby spectroscopic data in Fig. 3(b), this exponential sup-pression agrees well with theoretical prediction and re-sults in virtually charge-independent qubit frequenciesat sufficiently large EJ/EC . Instead of a single, nearlysinusoidal curve, the data displayed in Fig. 3(b) showstwo such curves with a relative displacement of half aperiod. This can be explained by the presence of one orseveral quasiparticles, which tunnel between the two is-lands. At low EJ/EC this phenomenon has been termedquasiparticle-poisoning [26], and it leads to complete de-phasing of the device [27]. However, the frequency shift

FIG. 3: (a) Anharmonicity of a transmon qubit. Data pre-sented for a transmon qubit at EJ/EC = 40. The qubit tran-sitions measured are the 01 transition in a single tone spec-troscopic measurement (bottom), and the 12 transition (top,offset) while populating transmon excited state with a seconddrive on the 01 transition. The second excited state of thetransmon is not populated with the 12 transition at normalspectroscopy powers (bottom). The 01 and 12 transitions areseparated by 455 MHz; the transmon can therefore be treatedas a two-level system even during fast control operations. (b)Exponential suppression of charge dispersion. Data presentedfor four different values of EJ , where a EJ/EC = 28.6, b 16.3,c 13.3, and d 10.4. Spectroscopic measurements of qubit fre-quency while changing a gate voltage reveal the expected si-nusoidal frequency bands. The width of the band (chargedispersion) is decreased from 74 to 0.8 MHz. Two sinusoidsare evident as random quasiparticle tunneling events causethe frequency curve to shift by one electron. The measuredcharge dispersion agrees well with the theoretical prediction(right). Reprinted from [12].

due to such a tunneling event is bounded from above bythe charge dispersion, and therefore becomes exponen-tially small in the transmon regime. Hence, these dataconfirm that the transmon is insensitive to fluctuationsin local charge, including the special fluctuations due toquasiparticle poisoning.

Since charge noise was the limiting source of dephasing

5

FIG. 4: Long coherence times in a transmon. Data mea-sured at the flux sweet spot for a transmon qubit withEC = 380 MHz, EJ/EC = 52. (a) Relaxation from excitedstate. Measurements of the occupation probability Pexc of theexcited qubit state while varying the time delay after a π pulseyield an exponential decay with T1 = 1.57±0.04 µs. (c) Ram-sey fringes, measured by varying the time delay between twoπ/2 pulses, show a long dephasing time T ∗

2 = 2.94 ± 0.04 µs(no echo).

in charge qubits, the transmon has lead to dramaticallyimproved dephasing times. Currently, the best T ∗

2 for atransmon is measured at a flux sweet spot where T ∗

2 =2.94 ± 0.04 µs (without any echoing) and T1 = 1.57 ±0.04 µs, see Fig. 4(a) and (b) [32]. Here, T ∗

2 is close toidentical to 2T1, indicating a homogeneously broadenedqubit, and a very long pure dephasing time of Tϕ ≥ 35 µs,consistent with our predictions.

These recent results of long coherence times are to becontrasted with the performance of the first generationof transmons [10, 11], which showed shorter relaxationtimes of ∼ 200 ns. The improved times shown in Fig. 4are the result of a more complete understanding of relax-ation, and by now have been reproduced in a number ofsamples. Over more than an octave in frequency, the cur-rent limit on transmon coherence has been shown to beimposed by spontaneous emission of photons through thecavity, a process known as the Purcell effect [28]. Thisrequires proper modeling of the cavity impedance includ-ing all higher modes of the resonator, which serves as afilter between the qubit and the evironment [13]. In fact,all transmon qubits now reach the same limit on intrinsicT1, as shown in Fig. 5. Towards the lowest frequencieswhere the Purcell effect is least severe, a non-Purcell T1

limitation is observed to set in, which is possibly due todielectric loss with a Q value of 70,000.

A separate issue for qubit control and usability of thequbit spectrum for operation regards the presence ofcoupling to unwanted degrees of freedom, such as spu-rious two-level systems. We have performed system-atic searches of transmon spectra for such spurious res-

FIG. 5: Relaxation times for seven transmon qubits. Predic-tions for qubit lifetime (colored lines) agree well with observedrelaxation times (points). Solid lines represent predictionsfor input side qubits located at the resonator input side (L),while dashed lines correspond to output side (R) qubits. Allsapphire qubits (blue and green) reach the same common in-trinsic limits (black line), with lifetimes limited to a constantQ ∼ 70, 000. Qubit lifetimes are accurately predicted over awide range of frequencies and more than two orders of mag-nitude in time. Reprinted from [13].

onances, which have revealed both very good agreementwith theory predictions with errors as low as one part in104, and have enabled us to estimate the average numberof spurious resonances in current transmon devices to be1 per 5 GHz per qubit [12].

V. SUMMARY

In summary, the transmon is a robust superconductingqubit for use in the circuit QED architecture. Its primaryfeature is an insensitivity to 1/f charge noise, the domi-nant source of dephasing in other charge qubits, withoutany detrimental effects on the sensitivity to other knownnoise channels. Experiments have directly verified thepredicted exponential suppression of the sensitivity tocharge fluctuations by monitoring the transmon energylevels, and have confirmed the resulting gain in dephasingtimes with current devices reaching the T2 = 2T1 limitand dephasing times of up to 3 µs. Due to the limitationof T2 by relaxation, future improvements may becomeeasier as they will focus on removing sources of dissipa-tion. The fact that all recent transmon qubits reach aconsistent T1 limit suggests that relaxation is caused bya single source, giving hope that it can be identified andeliminated in the near future.

6

Acknowledgments

The work reviewed in this paper was performed withthe members of the Yale circuit QED collaboration: L. S.Bishop, A. Blais, J. M. Chow, L. Frunzio, J. Gambetta,B. Johnson, J. Majer, J. A. Schreier, D. I. Schuster, A.

Wallraff, and Terri M. Yu. This work was supported inpart by Yale University via a Quantum Information andMesoscopic Physics Fellowship (AAH, JK), by NSA un-der ARO contract number W911NF-05-1-0365, and theNSF under grants DMR-0653377, and DMR-0603369.

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principle difference between the CPB/transmon systemand the phase qubit.

[34] The island area is increased by a factor of ∼ 1000.