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Superconducting qubit in waveguide cavity with coherence time approaching 0.1ms Chad Rigetti,1 Stefano Poletto,1 Jay M. Gambetta,1 Britton Plourde,2 Jerry M. Chow,1 A. D. C´rcoles,1 J. R. Rozen,1 George A. Keefe,1 Mary B. Rothwell,1 Mark B. Ketchen,1 and M. Steﬀen1 o 1 IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA 2 Department of Physics, Syracuse University, Syracuse, New York 13244-1130, USA ∗ We report a superconducting artiﬁcial atom with an observed quantum coherence time of T2 = 95µs and energy relaxation time T1 = 70µs. The system consists of a single Josephson junction transmon qubit embedded in an otherwise empty copper waveguide cavity whose lowest eigenmode is dispersively coupled to the qubit transition. We attribute the factor of four increase in the coherence quality factor relative to previous reports to device modiﬁcations aimed at reducing qubit dephasing from residual cavity photons. This simple device holds great promise as a robust and easily produced artiﬁcial quantum system whose intrinsic coherence properties are suﬃcient to allow tests of quantum error correction. PACS numbers: 03.67.Ac, 42.50.Pq, 85.25.-j Superconducting quantum circuits are one of the lead- fabrication processes, facilities, and measurement setups? ing candidate technologies for large scale quantum com- Second, what is the origin of the dephasing process sup- puting. They have been used to observe a violation of pressing T∗ well below the no-pure-dephasing limit of 2 a Bell inequality [1]; implement a simple two-qubit gate 2T1 ? Is it intrinsic to the junctions or to this qubit ar- favorable for scaling [2]; generate three-qubit entangle- chitecture? The weight and urgency of these questions ment [5]; perform a routine relevant for correction [3]; are increased by implications on scaling potential: if the and very recently to demonstrate a universal set of quan- results are reproducible and decoherence times can be tum gates with ﬁdelities greater than 95% [4]. Most of extended close to the 2T1 limit for observed T1 times, these devices employ small angle-evaporated Josephson this technology becomes an immediate candidate for the junctions as the critical non-linear components that pro- construction of prototype processors for testing QEC, duce an anharmonic spectrum and allow coherent state without signiﬁcant need of longer coherence. It would control within a computational subspace. While existing also suggest that other systems based on small angle- superconduting qubit architectures appear to be consis- evaporated junctions, such as traditional planar inte- tent with basic requirements for quantum error correction grated circuit architectures, may also be able to attain (QEC) and fault tolerance [7], the construction and op- similar performance if present performance limits can be eration of much larger systems capable of meaningfully identiﬁed and overcome. testing such procedures will require individual qubits and In this Letter we report a device in the 3D design junctions with a very high degree of coherence. Current that demonstrates the basic reproducibility of Paik, et estimates for threshold error rates, along with the cu- al. and, further, shows that the decoherence times can be mulative nature of errors originating from control, mea- further extended by taking precautions to protect against surement, and decoherence, makes it likely that coherent qubit dephasing induced by the ﬂuctuations of the cav- lifetimes at least 10−3 times longer than typical gate and ity photon number during qubit operation. With these measurement times are needed [6], corresponding for typ- precautions our system has signiﬁcantly enhanced coher- ∗ ical systems to quantum lifetimes of 20 to 200µs. ence, with with T2 =95µs (Q2 ≈2.5×106 ) and T1 = 70µs 6 Progress towards longer qubit coherence has contin- (Q1 ≈1.8×10 ). This level of performance places this ued for the past decade, spurred largely by clever meth- device well within the regime where microwave control ods of decoupling noise and loss mechanisms from the techniques can be used to perform quantum gates with qubit transition and thereby realizing actual Hamiltoni- ﬁdelities exceeding those required for error correction. ans more closely resembling their idealized versions. Re- In the 3D cQED framework, qubits are manufactured cently, Paik, et al. made a breakthrough advance [9] by with a standard lithographic process while the cavities, embedding a transmon-style qubit [11, 12] in a waveg- simple macroscopic resonant enclosures, are produced in- uide cavity. This system, dubbed three-dimensional cir- dependently and with very diﬀerent techniques, such as cuit QED (3D cQED), led to the observation of signif- precision machining or laser etching. Individual qubits icantly enhanced qubit lifetimes T1 of 25–60µs and T∗ 2 and cavities can be treated as discrete components; their of 10–20µs corresponding to quality factors for dissipa- properties, materials, and designs may be varied indepen- tion and decoherence of Q1 ≈1.8×106 and Q2 ≈7×105 , dently without special eﬀort. The large mode volumes, respectively. structural simplicity, and absence of very large aspect These results lead to two important questions. First, ratio thin ﬁlms make full-device electromagnetic simula- are similar coherence properties observable using other tion highly accessible. These properties in turn facilitate 2 a high degree of practical control and engineerability of a e qubit system and its electromagnetic environment. 50K -1dB -1dB In this work we exploit these and other properties of 3D cQED systems to engineer a device more robust against HEMT -20dB a leading candidate for the dominant dephasing process LNA 2.8K observed in earlier work: qubit dephasing due to the pres- ence and ﬂuctuations of residual photon population in the 4.2mm -20dB 0.08K cavity. We do this by three parallel methods. First, we NbTi/ select qubit and cavity parameters to reduce the expected NbTi qubit dephasing rate per residual photon in the funda- 18.6mm coax -20dB 0.01K mental cavity mode. Second, we engineer the device to 15.5mm limit the spectral proximity of and couplings to higher isolator modes of the cavity. We employ a symmetric cavity b ( x 2) TE101 TE102 TE201 shape such that the next nearest mode after the funda- mental that couples to the qubit is at ≈ 24GHz, or more 12GHz LPF than 20GHz detuned from the qubit transition. This de- 11.84 GHz 18.80 GHz 20.77 GHz sign minimizes the role of higher modes and makes the TE301 TE202 DUT cryoperm standard single-mode cavity approximation more robust. shield Third, we aim to suppress residual cavity population by d following a simple rule: the thermal photon temperature 24.03 GHz 25.19 GHz 28.00 GHz of a resonant mode, be it linear or nonlinear, is upper- c qubit resonator environment bounded by the temperature of the dissipation source limiting its quality factor (Q). Typically, the quality fac- IN tors of cQED resonators are limited by the ohmic envi- OUT ronment external to the resonator to which the device is coupled. But it is notoriously diﬃcult to ensure that the modes of a feedline are thermalized to very low temper- FIG. 1. (color online) Transmon qubit in three-dimensional atures [8]. Rather than solving this problem directly, we copper waveguide cavity with long coherence. (a) Device instead make use of an ideal cold resistor – the interior model (HFSS) showing interior volume of the waveguide en- walls of a bulk oxygen-free high-conductivity (OFHC) closure housing a sapphire chip and transmon qubit, with copper cavity – as the limiting source of dissipation. In two symmetric coaxial connectors for coupling signals in/out. conjunction with the under-coupling of the cavity to the (b) Eigenmodes of the enclosure with sapphire chip (obtained external environment, this internal cavity dissipation is with HFSS eignemode solver) illustrating robustness of single- mode-cavity approximation. The qubit is positioned at a cav- expected to thermalize the cavity photon population to ity symmetry point where the fundamental mode (TE101) is the temperature of the bulk copper, which in turn is eas- maximal. Device dimensions and symmetries imply that next ily anchored to the lowest available temperature. mode interacting with qubit (TE301) is > 20 GHz detuned The three-dimensional circuit QED sytem we report from qubit. (c) Equivalent circuit diagram of device. The in- is described in the two-level, dispersive, and single-mode terior walls of the OFHC copper waveguide cavity provide a cavity approximations by the Hamiltonian [10] readily thermalized cold resistor (light blue) that is expected to sink the residual cavity photon population to the lowest ω01 available temperature. (d) Optical imgage of transmon, con- H/¯ = ωc a† a − h σz − χa† aσz . (1) 2 sisting of two capacitor pads 350x700µm2 each, separated by a 50µm wire interrupted by a shadow evaporated Al/AlOx/Al The term proportional to a† aσz can be interpreted as a Josephson junction. Pads are formed of mesh with 5µm wires cavity photon number dependent shift of the qubit transi- and 25µm2 holes to suppress vortex trapping and motion. (c) tion frequency (ac-Stark shift), or as a qubit state depen- Cryogenic microwave measurement setup. Epxeriments per- dent shift of the cavity frequency (cavity pull). For trans- formed at base temperature of 8mK in a BlueFors cryogen-free dilution refrigerator. mons the cavity pull χ = −g 2 EC /(∆2 − ∆EC ) where g is the bare coupling strength, ∆ = ω01 − ωc is the cavity– qubit detuning and EC = e2 /2CΣ is the transmon charg- surement of the qubit state [10]. It can result from both ing energy and CΣ is the total qubit capacitance [11]. In thermal and coherent cavity photon populations. the strong dispersive coupling regime the cavity pull can We represent thermal driving of the resonator by the be larger than the intrinsic linewidth of the qubit transi- master equation [13] tion. In such systems, ﬂuctuations of the cavity photon number scramble the qubit frequency and place a limit i on coherence. This occurs through the same mechanism ρ = − [H, ρ] + κj (nthj + 1)D[a]ρ + κj (nthj )D[a† ]ρ, ˙ ¯ h that allows the cavity photons to induce a projective mea- (2) 3 ˆ ˆ ˆ ˆ ˆ ˆ ˆ where D[L]ρ = 2LρL† − L† Lρ − ρL† L /2 is the stan- * T2=95µs dard Linblad super-operator for dissipation, κj is the cavity relaxation rate through source j, and nthj = signal [a.u.] ¯ 1/(ehω01 /kTj −1) is the thermal photon number for source j at temperature Tj . Following a similar procedure to Refs. [14, 16] the thermal-induced dephasing rate at times long compared to 1/κtot is 2 κtot 2iχ 8iχ j κj nthj 0 10 20 30 40 50 60 70 80 Γth = Re 1+ + − 1 . time [µs] 2 κtot κ2 tot (3) For large κtot /χ [17], signal [a.u.] 4χ2 j κj nthj Γth = κj nthj + 1 (4) κ2 tot j while it saturates to Γth = j κj nthj for large χ/κtot . T1=70µs These suggest diﬀerent possible strategies to mitigate 0 50 100 150 200 250 300 350 400 cavity photon induced dephasing: suppress ﬂuctuations time [us] of the photon number by using very high Q cavities, ef- fectively pushing photon shot noise to lower and lower FIG. 2. (color online) Quantum state lifetimes. (a) Quantum frequencies; or, suppress the photon number with a cold coherence time. (b) Energy relaxation time. dissipation source internal to the cavity. In the ﬁrst strat- egy, one must ensure that the modes of the feedlines cou- pled to the cavity are cold at all relevant frequencies, Measurement is performed with standard circuit QED as the cavity modes will thermalize to the same tem- dispersive measurement techniques [15] using the low- perature. In the second, the internal cold dissipation is est resonant mode of the enclosure (TE101) which is expected to thermalize all cavity modes. In both cases, found experimentally at 12.1GHz. The cavity is closed one pays a price in eﬀective signal to noise of the mea- with brass screws, wrapped with Eccosorb foam and alu- surement, but for diﬀerent reasons. The cold dissipation minized mylar to protect against stray radiation [18], leads to a loss of information-carrying photons within the placed inside a cryoperm magnetic shield, and thermal- cavity before they can be measured; the high Q approach ized to the mixing chamber stage of a dilution refrigera- requires longer and longer measurement integration and tor. repetition times. Control and measurement signals are supplied from In this Letter we follow the second strategy by em- room temperature electronics to the cavity through stan- ploying the interior surfaces of an enclosure machined dard attenuated wideband coaxial lines. Measurement from bulk OFHC copper as an ideal cold resistor that signals exiting the cavity pass through (respectively) a appears as parallel damping of the eﬀective cavity res- 12GHz low pass ﬁlter (K&L LP12000) thermalized to onant circuit and limits its Q. The cavity, accordingly, 10mK with a copper wire wrap; two double-junction and is under-coupled to the input and output transmission magnetically shielded isolators (Pamtech) thermalized to lines. 10mK via large copper plates with multiple high-pressure Our device is shown in ﬁgure 1. The transmon qubit contact points; a NbTi/NbTi superconducting cable from junction has characteristic energy EJ /h=10.3946 GHz. 10mK to the 2.8K stage and thermalized at each end The qubit capacitance is determined from measurement and at its midpoint (80mK) with wire wrap. At 2.8K to be CΣ =91f F, implying EJ /EC =49, placing the qubit the signal is ampliﬁed by a low-noise wideband HEMT in the transmon regime [10–12]. Qubits are produced in ampliﬁer (Caltech) operating from 6-18GHz with a noise a 3” wafer process on c-plane 330µm thick sapphire prior temperature of 10-15K. The signal is ampliﬁed again at to dicing into 3.2mm×6.7mm chips. The chip is enclosed room temperature before being mixed down to 10MHz in a cavity machined from bulk OFHC copper. The cav- and digitized. In-phase and quadrature components of ity is formed by two halves and has in the assembled the signal are summed to produce a measurement of the state an interior volume 18.6×15.5×4.2mm3 plus sym- qubit energy eigenstate. metric cylindrical perturbations (d=7.7mm; h=4mm) of We performed standard measurements to characterize the ceiling to accommodate commercial bulkhead SMA qubit device properties and performance. We ﬁnd the connectors through which signals are coupled in and out. following parameters: ω01 /2π ≈4.2 GHz; ωc /2π ≈12.1 4 GHz; g/2π =153 MHz; χ=780 kHz; and a cavity with performance, along with the simplicity and discrete na- an internally limited Qc =10,400 [19] implying a Purcell ture of the qubits and cavities, makes this technology limit on qubit energy relaxation time of about 400µS, well an intriguing candidate for the construction of prototype above the measured T1 . Excited state lifetime and Ram- quantum processors with 10-1000 qubits, though more ∗ sey fringes experiments yield T1 = 70µs and T2 =95µs work is called for to determine whether multi-qubit sys- (Figure 2). Our data are consistent with our hypoth- tems can be demonstrated with similar performance. esis that a 3D circuit QED system whose cavity has a We acknowledge support from IARPA under Contract lower internal quality factor can facilitate signiﬁcantly No. W911NF-10-1-0324. We acknowledge discussions improved qubit coherence. and contributions from John Smolin, Seth Merkel, Jack What are the trade-oﬀs with this approach? The ex- Rohrs and Joel Strand. perimenter pays a price in convenience: the increased internal dissipation implies that for every photon exiting the cavity to be ampliﬁed and measured three are dis- sipated in the normal metal walls. This places greater [1] Markus Ansmann, H. Wang, Radoslaw C. Bialczak, Max demands on the performance of the ampliﬁcation chain Hofheinz, Erik Lucero, M. Neeley, A. D. O’Connell, D. to achieve a particular signal-to-noise ratio of the mea- Sank, M. Weides, J. Wenner, A. N. Cleland, John M. surement. Martinis Nature 461, 504-506 (2009) An ideal setup would employ transmission lines whose [2] J. M. Chow, A. D. Corcoles, J. M. Gambetta, C. Rigetti, modes are already thermalized to the lowest available B. R. Johnson, J. A. Smolin, J.R. Rozen, G. A. Keefe, temperature. This is indeed the objective but can be M. B. Rothwell, M. B. Ketchen, and M. Steﬀen, Phys. challenging to realize in practice due to basic materials Rev. Lett. 107, 080502 (2011). [3] M. D. Reed, L. DiCarlo, S. E. Nigg, L. Sun, L. Frunzio, S. properties at mK temperatures and the sensitivity of the M. Girvin, R. J. Schoelkopf, Nature 482 382–385 (2012) system to even small fractions of a photon. In one ex- [4] J. M. Chow,et al.,in preparation. periment known to the authors, great care was taken to [5] L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. reduce this temperature as much as possible and a bound Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. of about 55mK was achieved [8]. Reliably achieving mode Devoret, R. J. Schoelkopf, Nature (London) 467, 574-578 temperatures even that low is a major challenge and can (2010) be diﬃcult to reproduce from one experimental setup to [6] Andrew W. Cross, David P. DiVincenzo, Barbara M. Ter- hal arXiv:0711.1556. another. For this reason we have instead taken the multi- [7] David P. DiVincenzo, arXiv:0905.4839. pronged approach described in this Letter. [8] Lev S. Bishop, J. M. Chow, Jens Koch, A. A. Houck, An auxiliary beneﬁt of the bulk copper cavity is the re- M. H. Devoret, E. Thuneberg, S. M. Girvin, R. J. liability of the thermal link between the qubit substrate Schoelkopf, Nature Physics 5, 105 - 109 (2009) and the coldest temperature stage of the fridge. An inter- [9] Paik, et al, Phys. Rev. Lett. 107, 240501 (2011) esting avenue for future work entails controlled design of [10] J. Gambetta, et al., Phys. Rev. A 74, 042318 (2006). [11] J. Koch, et al, Phys. Rev. A 76, 042319 (2007) the cold dissipation in the cavity such that higher overall [12] J.A. Schreier, et al, Phys. Rev. B 77, 180502(R) (2008) cavity Q’s are obtained while otherwise maintaining the [13] C. W. Gardiner and P. Zoller, Quantum Noise: A properties of the device we’ve described here. This could Handbook of Markovian and Non-Markovian Quantum be done, for example, with bulk copper cavities whose Stochastic Methods with Applications to Quantum Optics interior walls are partially coated with a thin layer of (Springer, New York, 2004). aluminum; we have begun work towards this. [14] M. I. Dykman and M. A. Krivoglaz, Sov. Phys. Solid Conclusion: We have veriﬁed and extended the State, 29, 210–214 (1987) [15] Alexandre Blais, Ren-Shou Huang, Andreas Wallraﬀ, S. breakthrough results of Paik, et al. by constructing a M. Girvin, R. J. Schoelkopf, Physical Review A 69, 3D qubit system based on a single-junction transmon 062320 (2004) ∗ in a copper waveguide cavity with lifetimes T2 =95µs [16] A. A. Clerk and D. Wahyu Utami, Phys. Rev. A, 75, and T1 =70µs. Our results provide evidence that the 042302 (2007). highly coherent properties of qubits based on small [17] P. Bertet, et al., Phys. Rev. Lett. 95, 257002 (2005). shadow evaporated Josephson junctions may be robust [18] A. D. Corcoles, J. M. Chow, J. M. Gambetta, C. Rigetti, to changes fabrication processes, facilities, and measure- J. R. Rozen, G. A. Keefe, M. Beth Rothwell, M. B. Ketchen, and M. Steﬀen, Applied Physics Letters 99, ment setups. By pursuing three parallel approaches to 181906 (2011) improving coherence limits due to cavity photon induced [19] An otherwise identical cavity made of bulk (supercon- dephasing we have attained a factor of four improvement ducting) aluminum was measured to have Q=40,000. in the coherence quality factor Q2 =2.5×106 of a super- conducting qubit relative to previous reports. Our de- vice falls well within the range of coherent performance required for large scale tests of error correction and fault tolerant quantum computing procedures. We believe this

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