Improved Superconducting Qubit Readout by Qubit-Induced Nonlinearities
aa r X i v : . [ qu a n t - ph ] S e p Improved Superconducting Qubit Readout by Qubit-Induced Nonlinearities
Maxime Boissonneault, J. M. Gambetta, and Alexandre Blais D´epartement de Physique, Universit´e de Sherbrooke, Sherbrooke, Qu´ebec, Canada, J1K 2R1 Institute for Quantum Computing and Department of Physics and Astronomy,University of Waterloo, Waterloo, Ontario N2L 3G1, Canada (Dated: October 25, 2018)In dispersive readout schemes, qubit-induced nonlinearity typically limits the measurement fidelity byreducing the signal-to-noise ratio (SNR) when the measurement power is increased. Contrary to seeingthe nonlinearity as a problem, here we propose to use it to our advantage in a regime where it can increasethe SNR. We show analytically that such a regime exists if the qubit has a many-level structure. We alsoshow how this physics can account for the high-fidelity avalanchelike measurement recently reported byReed et al. [arXiv:1004.4323v1].
PACS numbers: 03.65.Yz, 42.50.Pq, 42.50.Lc, 74.50.+r
Quantum measurements are crucial to all quantum infor-mation protocols. In solid-state systems, readout can beperformed by connecting the qubits to noisy measurementelectronics, such as single-electron transistors [1]. Disper-sive readouts based on coupling qubits to high-Q resonatorsare however much less disruptive as all of the energy is dis-sipated away from the qubit [2]. This type of measurementleads to quantum nondemolition (QND) readout. Unfor-tunately, so far the typical signal-to-noise ratio (SNR) hasbeen relatively low, requiring sometimes up to repeti-tions of the experiment to average out the noise added bythe amplifiers [3, 4]. Increasing the qubit-resonator cou-pling is one approach to improve the SNR as it increasesthe amount of information about the qubit state carried bythe measurement photons. This is however at the cost of anincreased Purcell decay rate [5], which decreases the use-ful integration time and in turn the SNR. Another approachhas been to exploit bifurcation in a nonlinear resonator [6].This has already led to the experimental demonstration [7]of single-shot readout of a transmon-type superconductingqubit [8]. Very recently, Reed et al. have also shown thatsingle-shot measurement of a transmon qubit can also berealized in a linear resonator but working at very large mea-surement power [9].Motivated by these results, we study how the nonlinear-ity induced by the qubit in a linear resonator can lead toimprovement of the measurement. We first explore the lowdriving power regime before considering the high-powerregime studied in Ref. [9]. Using a simple model, we findqualitative agreement with these experimental results.For simplicity, we first focus on a two-level system(TLS), of states labeled {| i , | i} , dispersively coupled toa microwave resonator. This system is well described bythe Jaynes-Cummings Hamiltonian, expressed here in thedispersive basis ( ¯ h = 1 ) [10] H = ω r N + ω a − ∆ h − p λ ( N + Π ) i σ z ≈ ( ω r + ζ ) N + ˜ ω a σ z + χN σ z + ζN σ z , (1) where Π ij = | i i h j | for the TLS, and N = a † a . In thisexpression, ω r is the resonator frequency and ω a ( ˜ ω a ) thebare (Lamb-shifted) TLS transition frequency. The qubit-resonator coupling is characterized by χ = g (1 − λ ) / ∆ the dispersive coupling strength and ζ = − gλ is a Kerr-type nonlinearity, with g the bare qubit-resonator couplingstrength, ∆ = ω a − ω r the qubit-resonator detuning and λ = − g/ ∆ . The second line of Eq. (1) is valid to fourthorder in λ and at photon numbers ¯ n ≪ n crit , where n crit =1 / λ is the critical photon number [2]. The limit ζ → of the second line of Eq. (1) is the standard linear dispersiveHamiltonian [11].In this regime, because of the qubit-dependent pull of theresonator frequency χσ z , a nearly resonant drive on the res-onator will displace the resonator’s initial vacuum state toa qubit-state dependent coherent state | α i =0 , i . Homodynemeasurement of the transmitted or reflected signal can thenresolve these pointer states, and hence the qubit states. InRef. [12], it was shown that the SNR of such a homodynemeasurement for an integration time T = 1 /γ is given by SNR = ηκ | α − α | /γ , where κ is the resonator decayrate and η is the measurement efficiency. The SNR reachesits maximal value (SNR) max = 4 η ¯ nχ/γ , for the optimalchoice κ = 2 χ , where ¯ n is the average number of photons.In this limit, the dispersive model Eq. (1) thus predicts thatthe SNR should increase linearly with the number of mea-surement photons ¯ n .Unfortunately, this linear increase of the SNR is not ob-served experimentally and this can, at least partially, be ex-plained by the fourth order corrections in Eq. (1). To thisorder of approximation, an increase of the photon numberleads to a reduction of the cavity pull gλ [1 − λ ( a † a +1)] σ z and hence to a loss of distinguishability of the pointerstates [10]. Thus, the prospects for improving the SNR inhomodyne measurement of a TLS dispersively coupled toa resonator look rather unpromising.Fortunately, most superconducting qubits are well ap-proximated by many-level systems (MLS), often with onlyweak anharmonicity [8, 13–15], rather than by TLS. As isapparent below, it is possible in this situation for χ and ζ tohave the same sign, which yields an increase of the cavitypull with ¯ n and hence an improved SNR.As a good description of a generic superconductingqubit, we thus consider an M-level system, of states la-beled {| i , ..., | M − i} and with the first two statesacting as logical states. The Hamiltonian of the MLS-resonator system takes the generalized Jaynes-Cummingsform H s = H + P M − i =0 g i ( a † Π i,i +1 + a Π i +1 ,i ) , where H = ω r a † a + P M − i =0 ω i Π ii is the free Hamiltonian, ω i the frequency of level | i i , and g i the coupling strength be-tween the resonator mode a and the i ↔ i + 1 MLS transi-tion.In the dispersive regime, where h a † a i ( g i / ∆ i ) ≪ with ∆ i = ( ω i +1 − ω i ) − ω r , it is convenient to approxi-mately diagonalize H s . Following Ref. [10] where this wasdone for a TLS, we obtain to fourth order in λ i = − g i / ∆ i H D s ≈ ˜ H + M − X i =0 S i Π i,i a † a + M − X i =0 K i Π i,i ( a † a ) , (2)where ˜ H includes the Lamb shifts. In this expression, wehave defined the ac-Stark ( S i ) and Kerr ( K i ) coefficients S i = (cid:2) χ i − (1 − λ i ) − χ i (1 − λ i − ) − χ i − λ i − (cid:3) + (9 χ i − λ i − − χ i − λ i − − χ i λ i +1 + 3 χ i +1 λ i ) − g (2) i λ (2) i − g (2) i − λ (2) i − , (3a) K i = (3 χ i − λ i − − χ i − λ i − + χ i λ i +1 − χ i +1 λ i )+ ( χ i − χ i − )( λ i + λ i − ) + g (2) i λ (2) i − g (2) i − λ (2) i − , (3b)with χ i = g i / ∆ i , g (2) i = λ i λ i +1 (∆ i +1 − ∆ i ) , λ (2) i = − g (2) i / (∆ i +1 + ∆ i ) , and χ i = λ i = 0 for i / ∈ [0 , M − .For M = 2 , S = − χ , S = χ (1 − λ ) and K = − K = χ λ , reproducing Eq. (1). The crucial featureof these expressions is that, contrary to a TLS, K = − K for a MLS. Using this state-dependent nonlinearity, we nowshow how to improve the readout in two ways.Weak driving limit. From the dispersive model Eq. (2),it is possible to evaluate the difference in cavity pull δ forlevels | i and | i which we are interested in distinguishingin a measurement. It takes the form δ = χ ′ + ζ ′ ¯ n , with ¯ n = (cid:10) a † a (cid:11) and where we have defined χ ′ = S − S and ζ ′ = K − K . We note that, although we are focusingon the first two levels, the presence of higher MLS levelsis important. This is apparent in the expressions for S i and K i which involve states up to | i + 2 i .Figure 1 shows χ ′ and ζ ′ obtained from Eq. (3) as a func-tion of ω r for a MLS with (a) M = 2 and (b) M = 6 levels. Figure 1(c) has been obtained from exact diagonal-ization of H s for M = 6 . For M = 2 , sgn ( χ ′ ) =sgn ( ζ ′ ) only in regions where the dispersive approximation breaksdown while for M > this is possible in the dispersiveregime (see the caption of Fig. 1). The numerical resultsillustrate that the analytical expressions are good approx-imations. We note that, while these results apply to any -404 (a) Analytics M =2 -404 l og ( χ ´/1 M H z ) , l og ( ζ ´/1 M H z ) (b) Analytics M =6 -4042000 3000 4000 5000 6000 7000 ω r /2 π [MHz] (c) Numerics M =6 FIG. 1: (color online) Analytical (a,b) and numerical (c) ac-Stark χ ′ (black line) and Kerr shifts ζ ′ [lighter (red) line] for a trans-mon qubit taking into account M = 2 (a) and M = 6 levels(b),(c). The parameters are chosen such that ( ω , ω , g ) / π =(6000 , , MHz. The lines change from dotted to fullwhen χ ′ or ζ ′ pass from negative to positive values. Verticalblue dotted lines indicate transitions ω , ω , and ω . Lighter(green) shading indicates regions of interest, while darker (red)indicates regions where the dispersive model breaks down evenat low-photon number. Values of χ ′ and ζ ′ obtained numericallyare not plotted close to divergences.
234 0 50 δ /2 π [ M H z ] Number of photons (a) S N R κ /2 π =2MHz (b) Number of photons κ /2 π =4MHz (c) κ /2 π =6MHz (d) FIG. 2: (color online) Cavity pull δ (a) and SNR (b)–(d), as afunction of the average number of photons. The parameters arethe same as in Fig 1 with T = 1 µ s. Dashed blue lines cor-respond to the second order approximation, for which the cav-ity pull g / ∆ is constant. Full black lines (dotted red lines)are obtained for ω r / π = 4515(7660) MHz, corresponding to n crit ∼ and sgn( χ ′ ) = ( − )sgn( ζ ′ ) . These frequencies arechosen such that | χ ′ | / π = 2 MHz.
MLS, for concreteness we have chosen here parameterscorresponding to a transmon qubit [8].Figure 2(a) shows the cavity pull δ for a cavity fre-quency chosen in the region where sgn ( χ ′ ) =sgn ( ζ ′ ) (fullblack line) and outside of this optimal area (dotted red line).These results are compared to the result of the second orderapproximation (dashed blue line). As expected, the cavitypull increases with ¯ n under the appropriate choice of pa-rameters. Figures 2(b)-2(d) show the corresponding SNR,using the same color scheme, for κ/ χ ′ = 0 . [2(b)], 1[2(c)] and 1.5 [2(d)], with κ/ χ ′ = 1 being the optimalchoice [12]. Because of the increase of the cavity pull with ¯ n , the full black line is always above the dotted red one. For κ/ χ ′ = 1 , an improvement of nearly is expectedat large photon numbers. Biasing the qubit above the res-onator’s fundamental frequency, as is suggested here, canlead to an increase of the Purcell decay. This can howeverbe strongly reduced by a small change of design [16]. Fi-nally, one could also tune the system to a point where χ ′ and ζ ′ have the same sign at the moment of measurementusing a tunable resonator [17, 18].Strong driving limit. The results obtained so far relied onEq. (2), which is valid only below n crit . However, the non-trivial state dependence of the nonlinearity K i should ex-tend well beyond the dispersive regime. To explore this, wetake advantage of the block diagonal structure of the Hamil-tonian H s . There each block of H s corresponds to a fixednumber n of qubit-resonator excitations and is spanned by E n = {| n, i , ..., | n − M + 1 , M − i} . With the rele-vant M being at most ∼ in practice, we can diagonalizeeach block numerically (or analytically for M ≤ ) for ar-bitrary n . In this way, we obtain the dressed energies ¯ E n,i and states (cid:12)(cid:12) n, i (cid:11) , where ¯ E n,i is the energy of the eigenstateclosest to a Fock state with n photons and MLS state | i i .From these expressions, we find the effective resonator fre-quency ω ri ( n ) = ¯ E n +1 ,i − ¯ E n,i . This frequency dependsin a nonlinear way on the MLS state-dependent averagephoton number n i . In steady state and in the absence ofqubit transitions, n i is given by the measurement drive am-plitude ǫ and frequency detuning relative to the effectiveresonator frequency n i ( ǫ, ω m ) = ǫ [ ω ri ( n i ) − ω m ] + [ κ/ , (4)with ω m the measurement frequency. We solve Eq. (4) iter-atively to find n i and ω ri as a function of ǫ and ω m .In Fig. 3, both ω ri [3(a)–3(c)] and n i [3(d)–3(f)] are plot-ted as a function of measurement power for M = 2 , , re-spectively. This is done for ω m = ω r as in Ref. [9]. For allvalues of M , the effective resonator frequency approachesits bare value ω r at large power. This is expected because,at this point, h N i ≫ h√ N i in Eq. (1) and the cavity re-sponds classically [19]. Since K = − K for M = 2 ,this classical crossover occurs at the same input power forboth qubit states. We note that this crossover happens inan avalanche manner, with each additional photon bringing ω ri ( n i ) closer to ω r and facilitating the addition of morephotons. As can be seen in Fig. 3 (b,c), for M = 3 thisavalanche occurs at a state-dependent power, the behaviorchanging only quantitatively for M > . With ω m = ω r ,we thus expect an abrupt change in the average photonnumber in the resonator at a power that is MLS state de-pendent. This is illustrated in Fig. 3(f) where for M = 6 there is a range of ∼ n and n differ significantly and by as much as ∼ at ω r i /2 π [ G H z ] (a) -2 n i levels (d) (b) ǫ /1 MHz) levels (e) (c)
30 40 506 levels (f)
FIG. 3: Effective resonator frequency ω ri (a)–(c) and mean pho-ton number n i (d)–(f) for i = 0 (full red lines), i = 1 (dotted bluelines), and i = 2 [dashed gray lines, (c),(f)] as a function of themeasurement power. Panels (a,d), (b,e) and (c,f) are for M = 2 ,3, and 6 respectively. In (a)–(c), the dashed green horizontal lineis ω r . The parameters are the same as in Fig 1. For clarity ofpresentation the panels have different horizontal scales. l og ( ǫ /1 M H z ) ω m /2 π [MHz] Ground(a) 7000 7010 ω m /2 π [MHz] M e a n ph o t o n nu m b e r n i Excited(b)
FIG. 4: Mean photon number n i for i = 0 (a) and i = 1 (b)as a function of the measurement frequency and power. The fullwhite lines are ω ri ( n i ) for ω m / π = ω r / π = 7 GHz. Thesolid red and dotted blue vertical lines indicate the measurementfrequency used in Fig 3. The parameters are the same as in Fig 1.The horizontal dotted lines delimitate the regime of measurementpower where n − n is maximum. the optimal driving power. This large separation of the Scurves, much larger than typical amplifier noise, leads tosingle-shot readout of the qubit [9]. As shown by the graydashed line in Fig 3(f), pumping the ↔ transition be-fore readout [7, 9] could also help the measurement workat lower power. Finally, Fig. 4 shows the full power versusmeasurement frequency dependence of n i ( ǫ, ω m ) obtainedfrom Eq. (4). In this plot, the full white lines correspondto ω ri shown in Fig. 3(c). Although Fig. 4 shows qualita-tive agreement with the results of Ref. [9], a quantitativecomparison would require proper modeling of the 4-qubitdevice used in Ref. [9].While the dispersive measurement at low-photon num-ber of Fig. 2 is expected to be QND, this is not the casefor the single-shot high-power measurement. To eval-uate the QND character of this avalanche readout, weestimate how the presence of the measurement photonschanges the relaxation and excitation rates of the qubit,as well as causes leakage outside of the logical subspace {| i , | i} . Figure 5(a) shows the Purcell decay rate γ κ /κ ≈ (cid:12)(cid:12)(cid:10) n, (cid:12)(cid:12) a (cid:12)(cid:12) n, (cid:11)(cid:12)(cid:12) and corresponding leakage rate γ lκ /κ ≈ P i =0 , (cid:12)(cid:12)(cid:10) n − i, i (cid:12)(cid:12) a (cid:12)(cid:12) n, (cid:11)(cid:12)(cid:12) as a function of mea-surement power. At low power, we find the expected result γ κ /κ = λ [5]. For large photon number, n + 1 ≈ n and losing a photon through resonator decay does not sig-nificantly change the qubit states. The Purcell decay ratethus goes down with measurement power and does not af-fect the QND character. To evaluate how the qubit dress-ing changes pure relaxation, Fig 5(b) shows the rates fordressed decay γ d /γ ≈ (cid:12)(cid:12)(cid:10) n, (cid:12)(cid:12) Σ − (cid:12)(cid:12) n, (cid:11)(cid:12)(cid:12) and leakage γ l d /γ ≈ P i =0 , (cid:12)(cid:12)(cid:10) n − i, i (cid:12)(cid:12) Σ − (cid:12)(cid:12) n, (cid:11)(cid:12)(cid:12) , where Σ − = P M − i =0 g i g Π i,i +1 . The participation of the higher transmonstates reduces the decay rate γ d from | i to | i as mea-surement power is increased. However, decay of the barehigher states increases the leakage rate, and the total errorrate γ d + γ l d is larger than γ .Finally, dressed-dephasing γ d due to noise responsiblefor dephasing of the bare qubit states can also cause tran-sitions between the dressed-states [10, 20]. For concrete-ness, we consider dephasing due to charge noise on a trans-mon, but the model can be adapted to any source of dephas-ing. Following Ref. [10], this contribution can be evaluatedas γ d /γ ϕ ≈ (cid:12)(cid:12)(cid:10) n + 1 , (cid:12)(cid:12) Σ z (cid:12)(cid:12) n, (cid:11)(cid:12)(cid:12) S (∆ ) /S (1 Hz) ,where Σ z = P M − i =0 Π i,i ǫ i /ǫ , with ǫ i the charge dis-persion of level i [8] and S (∆ ij ) the spectrum ofcharge noise evaluated at the dressed qubit-resonator de-tuning. In the same way, the leakage rate is γ l d /γ ϕ ≈ P i =0 , (cid:12)(cid:12)(cid:10) n + 1 − i, i (cid:12)(cid:12) Σ z (cid:12)(cid:12) n, (cid:11)(cid:12)(cid:12) S (∆ i ) /S (1 Hz) .We note that, even assuming /f charge noise whichwould be times smaller at 1 GHz than at 1 Hz, dresseddephasing can be important for these large photon numbers.Indeed, for the transmon, the charge dispersion ǫ i — andtherefore the susceptibility to charge noise — increases ex-ponentially with i , reaching ǫ /ǫ ∼ for 6 levels. Al-though a quantitative analysis requires a better understand-ing of the noise spectrum at microwave frequencies, ournumerical analysis with /f noise suggests γ d /γ ϕ rangingfrom to and γ ld /γ ϕ from to depending on theparameters and the number of levels.The loss of the QND aspect in such a high-fidelity read-out is not expected to be an issue in the measurement ofthe final state of a quantum algorithm. However, reduc-tion in the QND character is problematic for tasks such asmeasurement-based state preparation, quantum feedbackcontrol and quantum error correction.In summary, we have shown that for a qubit with M > levels dispersively coupled to a resonator, the qubit-induced nonlinearity of the resonator depends in a nontriv-ial way on the qubit state. This can be exploited to increase γ κ ( l ) / κ λ (a)01 20 40 60 γ ( l ) d / γ ǫ /1 MHz) (b)
FIG. 5: Qubit relaxation (full black lines) and leakage rate(dashed blue lines) due to Purcell effect (a) and bare qubit de-cay (b) as a function of the measurement power. The dotted redline in (a) is the expected value for γ κ at low power. The parame-ters are the same as in Fig 1. the SNR ratio in a QND measurement at low-photon num-ber and captures the essential aspects of the high-fidelitynon-QND measurement recently reported [9].We thank the Yale circuit QED team for discussion ofthe results of Ref. [9] prior to publication. M. B. was sup-ported by NSERC; J. M. G. by CIFAR, MITACS, MRI andNSERC; A. B. by NSERC, the Alfred P. Sloan Foundationand CIFAR.Note added. – Theoretical modeling of the high-fidelityreadout has also been reported by Bishop et al. (followingLetter) [21]. [1] K. Lehnert et al ., Phys. Rev. Lett. , 027002 (2003).[2] A. Blais et al ., Phys. Rev. A , 062320 (2004).[3] R. Bianchetti et al ., Phys. Rev. A , 043840 (2009).[4] L. Dicarlo et al ., Nature , 240 (2009).[5] A. A. Houck et al ., Phys. Rev. Lett. , 080502 (2008).[6] R. Vijay et al ., Rev. Sci. Instrum. , 111101 (2009).[7] F. Mallet et al ., Nat. Phys. , 791 (2009).[8] J. Koch et al ., Phys. Rev. A , 042319 (2007).[9] M. D. Reed et al , arXiv:1004.4323v1 (2010).[10] M. Boissonneault, J. M. Gambetta, and A. Blais, Phys. Rev.A , 060305(R) (2008); , 013819 (2009).[11] S. Haroche and J.-M. Raimond, Exploring the Quantum:Atoms, Cavities, and Photons (Oxford University Press,2006).[12] J. Gambetta et al , Phys. Rev. A , 012112 (2008).[13] S. O. Valenzuela et al , Science , 1589 (2006).[14] M. Neeley et al , Science , 722 (2009).[15] J. Koch et al , Phys. Rev. Lett. , 217004 (2009).[16] M. D. Reed et al , Appl. Phys. Lett. , 203110 (2010).[17] A. Palacios-Laloy et al , J. Low Temp. Phys. , 1034(2008).[18] M. Sandberg et al , Appl. Phys. Lett. , 203501 (2008).[19] J. M. Fink et al , arXiv:1003.1161v1 (2010).[20] C. M. Wilson et al , Phys. Rev. B , 024520 (2010).[21] L. Bishop, E. Ginossar, and S. M. Girvin, following Letter,Phys. Rev. Lett.105