Dynamics of dispersive single qubit read-out in circuit quantum electrodynamics
R. Bianchetti, S. Filipp, M. Baur, J. M. Fink, M. Göppl, P. J. Leek, L. Steffen, A. Blais, A. Wallraff
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Dynamics of dispersive single qubit read-out in circuit quantum electrodynamics
R. Bianchetti, S. Filipp, M. Baur, J. M. Fink, M. G¨oppl, P. J. Leek, L. Steffen, A. Blais, and A. Wallraff Department of Physics, ETH Zurich, CH-8093 Z¨urich, Switzerland D´epartement de Physique, Universit´e de Sherbrooke, J1K 2R1 Sherbrooke, Canada
The quantum state of a superconducting qubit non-resonantly coupled to a transmission lineresonator can be determined by measuring the quadrature amplitudes of an electromagnetic fieldtransmitted through the resonator. We present experiments in which we analyze in detail the dy-namics of the transmitted field as a function of the measurement frequency for both weak continuousand pulsed measurements. We find excellent agreement between our data and calculations basedon a set of Bloch-type differential equations for the cavity field derived from the dispersive Jaynes-Cummings Hamiltonian including dissipation. We show that the measured system response can beused to construct a measurement operator from which the qubit population can be inferred accu-rately. Such a measurement operator can be used in tomographic methods to reconstruct single andmulti qubit states in ensemble averaged measurements.
PACS numbers: 42.50.Ct, 42.50.Pq, 78.20.Bh, 85.25.Am
I. INTRODUCTION
Among several stringent requirements like scalabilityor precise coherent control, high-fidelity read-out of thequbit state is an important aspect of all experimen-tal efforts in quantum information science [1]. For su-perconducting qubits [2] a number of read-out strate-gies [3, 4, 5, 6, 7, 8, 9, 10] specific to various implemen-tations have been pursued. Early charge qubit read-outsimplemented with single electron transistors (SET) [11]were limited by strong current noise back-action from themeasurement device on the qubit. Similar limitationsapplied to flux qubit read-out, measuring the switch-ing current of a nearby superconducting quantum in-terference device (SQUID) [12, 13]. One strategy toachieve high read-out fidelities is to perform a quantumnon-demolition (QND) measurement which preserves theeigenstates of the system Hamiltonian [14]. Repeatedmeasurements yield identical results and consequentlyan improved signal-to-noise ratio. In the circuit quan-tum electrodynamics (QED) architecture, where a su-perconducting qubit is strongly coupled to a transmis-sion line resonator [15, 16], the qubit can both be con-trolled and read-out via the cavity using microwave sig-nals. The read-out can be accomplished by detecting thedispersive qubit-state dependent shift of the resonatorfrequency [17]. In the dispersive limit, where the qubittransition frequency is far detuned from the resonatorfrequency, and for small photon numbers the measure-ment of the transmitted resonator field forms a QNDmeasurement [15, 16, 17, 18, 19]. Similarly, QND mea-surements have been employed to demonstrate explic-itly the repeatability of this type of measurement for aflux qubit dispersively coupled to a nonlinear oscillator[20, 21, 22]. In a related experiment the phase of a tankcircuit coupled to a flux qubit was monitored, demon-strating resonant tunneling [23]. Note also, that in thecircuit QED architecture a QND measurement has beenproposed to generate and detect multi-qubit entangledstates [24, 25, 26]. Here we analyze the time-dependent response of thequadrature amplitudes of an electromagnetic field trans-mitted through a resonator to changes in the qubit stateunder dispersive interaction. We derive and discuss aset of Bloch-type equations describing accurately the dy-namics of the qubit and the resonator for continuous mea-surements. We also extend the analysis to pulsed read-out which avoids measurement-induced dephasing duringqubit manipulation and allows for stronger measurement.We analyze experimental data for different measurementfrequencies and find excellent agreement with theory us-ing a single set of independently measured parameters.This approach has already been successfully used in ex-periments [17, 27, 28, 29] but not yet discussed in litera-ture.Within this framework, we demonstrate our abilityto infer the state of the qubit embedded in the cavityfrom a measurement of both field quadratures transmit-ted through the resonator. The construction of the corre-sponding projective measurement operator based on thestate-dependent resonator response is outlined. For Rabioscillation measurements, we discuss the extraction of thequbit state population and compare to numerical simu-lations.
II. THE PHYSICAL SYSTEM AND ITS MODEL
We consider a transmon-type qubit [30, 31] embeddedin a transmission line resonator, as illustrated schemat-ically in Fig. 1. The qubit is coupled to the resonatorthrough the effective capacitance C g , leading to a qubit-resonator coupling of strength g [16, 32]. The qubit tran-sition frequency ω a is tunable by an externally appliedflux Φ [30]. The resonator with resonance frequency ω r determined by its geometric and dielectric properties [33]is modeled as an LC circuit.When the qubit/resonator detuning ∆ ar = ω a − ω r ismuch larger than the coupling strength g , this system isdescribed by the dispersive approximation of the Jaynes- ADCC g
300 K1.5 K20 mK300 K C in C out ω m ω LO ω s Φ Mixer ω r ω a FIG. 1: (color online) Circuit diagram of the experimen-tal setup. A harmonic oscillator modeled as an LC circuitwith resonance frequency ω r is coupled to a transmon-typequbit [30] through the coupling capacitance C g . The qubittransition frequency ω a is controlled by an externally appliedmagnetic flux Φ. The qubit state is coherently manipulatedby a pulsed microwave source at the frequency ω s . The res-onator is probed by a signal applied to the input capacitor C in at the frequency ω m . The transmitted signal is amplified anddownconverted by mixing with a local oscillator at frequency ω LO and then digitized using an analog to digital converter(ADC). Cummings Hamiltonian [15] H disp = ~ ( ω r + χ ˆ σ z )ˆ a † ˆ a + ~ ω a + χ )ˆ σ z . (1)Here, χ ≈ − g E c ∆ ar (∆ ar − E c ) (2)is the dispersive coupling strength between the resonatorand the transmon qubit approximated as a two-level sys-tem [30]. E c is the charging energy. The dispersivecoupling leads to a qubit state-dependent shift of theresonator frequency, which we use to measure the qubitstate.As illustrated in Fig. 1, the qubit state is controlledby a coherent microwave field of amplitude Ω( t ) and fre-quency ω s applied directly to the qubit while the mea-surement tone with amplitude ǫ m ( t ) and frequency ω m isapplied to the input port of the resonator. These exter-nally applied control fields are modeled by the Hamilto-nian H d = ~ (cid:0) ǫ m ( t )ˆ a † e − iω m t + Ω( t )ˆ σ + e − iω s t + h . c . (cid:1) , (3)where we have taken ǫ m ( t ) and Ω( t ) to be real for sim-plicity of presentation.The dynamics of the system in presence of dissipationand dephasing is described by a Lindblad-type masterequation [34]˙ ρ = − i ~ [ H, ρ ]+ κ D [ a ] ρ + γ D [ˆ σ − ] ρ + γ φ D [ˆ σ z ] ρ ≡ L ρ, (4) where H = H disp + H d and D [ ˆ A ] ρ = ˆ Aρ ˆ A † − ˆ A † ˆ Aρ/ − ρ ˆ A † ˆ A/
2. Here, γ = 1 /T is the qubit decay rate, γ φ thequbit pure dephasing rate and κ the photon decay rate.Since all experiments discussed in this paper are done ata small photon number n ≪ n ncrit = | ∆ | / g and as γ exceeds the Purcell decay rate [35] we neglect higher-order corrections to this dispersive master equation [36,37].To study the dynamics of the coupled qubit/resonatorsystem, we derive Bloch-like equations of motions for theexpectation value of the qubit operators h ˆ σ i i (i = x , y , z)and the resonator field operators h ˆ a i and h ˆ a † ˆ a i . However,the master equation (4) leads to an infinite set of coupledequations for these expectation values. For instance, thedifferential equation for h ˆ a i involves terms proportionalto h ˆ a ˆ σ z i , h ˆ a † ˆ a ˆ a ˆ σ z i and h ˆ a ˆ σ x i , which in turn involve evenhigher order terms. We therefore truncate this infiniteseries by factoring higher order terms h ˆ a † ˆ a ˆ σ i i ≈ h ˆ a † ˆ a ih ˆ σ i i and h ˆ a † ˆ a ˆ a ˆ σ i i ≈ h ˆ a † ˆ a ih ˆ a ˆ σ i i , but keeping the terms h ˆ a ˆ σ i i which ensures that the field contains information aboutthe qubit state. This choice of factorization yields thecorrect average values for coherent and Fock states [38]and leads to a complete set of eight coupled differentialequations d t h ˆ a i = − i ∆ rm h ˆ a i − iχ h ˆ a ˆ σ z i − iǫ m − κ h ˆ a i , (5a) d t h ˆ σ z i = Ω h ˆ σ y i − γ (1 + h ˆ σ z i ) , (5b) d t h ˆ σ x i = − (cid:20) ∆ as + 2 χ (cid:18) h ˆ a † ˆ a i + 12 (cid:19)(cid:21) h ˆ σ y i− (cid:16) γ γ φ (cid:17) h ˆ σ x i , (5c) d t h ˆ σ y i = (cid:20) ∆ as + 2 χ (cid:18) h ˆ a † ˆ a i + 12 (cid:19)(cid:21) h ˆ σ x i− (cid:16) γ γ φ (cid:17) h ˆ σ y i − Ω h ˆ σ z i , (5d) d t h ˆ a ˆ σ z i = − i ∆ rm h ˆ a ˆ σ z i − iχ h ˆ a i + Ω h ˆ a ˆ σ y i− iǫ m h ˆ σ z i − γ h ˆ a i − (cid:16) γ + κ (cid:17) h ˆ a ˆ σ z i , (5e) d t h ˆ a ˆ σ x i = − i ∆ rm h ˆ a ˆ σ x i − (cid:2) ∆ as + 2 χ (cid:0) h ˆ a † ˆ a i + 1 (cid:1)(cid:3) h ˆ a ˆ σ y i− iǫ m h ˆ σ x i − (cid:16) γ γ φ + κ (cid:17) h ˆ a ˆ σ x i , (5f) d t h ˆ a ˆ σ y i = − i ∆ rm h ˆ a ˆ σ y i + (cid:2) ∆ as + 2 χ (cid:0) h ˆ a † ˆ a i + 1 (cid:1)(cid:3) h ˆ a ˆ σ x i− iǫ m h ˆ σ y i − (cid:16) γ γ φ + κ (cid:17) h ˆ a ˆ σ y i− Ω h ˆ a ˆ σ z i , (5g) d t h ˆ a † ˆ a i = − ǫ m Im h ˆ a i − κ h ˆ a † ˆ a i , (5h)which we refer to as Cavity-Bloch equations . Here, wehave defined ∆ as = ω a − ω s and ∆ rm = ω r − ω m asthe detuning of the control and measurement microwavefields from the qubit and cavity frequency, respectively.While these equations are apparently more complex thanEq. (4), they can be analytically solved in some cases andare much faster to solve numerically. Note, that they donot include measurement-induced dephasing caused byphoton shot-noise [15, 39], because only the expectationvalue of ˆ a † ˆ a is taken into account, and higher order mo-ments are omitted. This is of no consequence for the un-derstanding of the experiments presented here becauseof the small photon numbers present during the mea-surement. In experiments where measurement-induceddephasing is important, Eq. (4) has to be solved di-rectly [19], or higher order terms have to be taken intoaccount [36]. III. CONTINUOUS MEASUREMENTRESPONSE
To experimentally determine the state of the qubit,we probe the dynamics of the resonator-qubit systemby measuring the resonator transmission at different fre-quencies ω m . A time resolved, phase sensitive measure-ment of the transmission quadrature amplitudes is re-alized by down converting the measurement signal atfrequency ω m in a mixer to an intermediate frequency∆ mLO = ω m − ω LO = 25 MHz using a local oscilla-tor of frequency ω LO and phase φ , yielding one inde-pendent data point every 40 ns, see Fig 1. This inter-mediate frequency (IF) signal is digitized in an analog-to-digital converter (ADC). The whole experiment isrepeated and averaged 650’000 times to enhance the π ω s ω m Weak measurement t ε m Ω I • • Χ Χ Χ ΧΧ Χ Χ Χ • • Q D e t un i ng m r M H z ææææææææææææææææææææææææææææææææææææææææææææææææææææææææ Q @ m V D ææææææææææææææææææææææææææææææææææææææææææææææææææææææææ I @ m V D @ Μ s D @ Μ s D b) c)a) d) e) FIG. 2: (color online) a) In a continuous measurement, thecavity is driven with the qubit in its ground state, populatingthe resonator with ¯ n = 1 photons on average at the cavityresonance frequency. The qubit is then prepared in the ex-cited state with a π -pulse ending at t = 0. b) and c) averagedmeasurement response Q , I versus time t for a continuousweak measurement at the frequency ω m = ω r − χ . Solid linesshow the predicted response from the Cavity-Bloch equations,Eq. (5). Time resolved data taken at different detunings ∆ mr is shown in d) and e). The arrows indicate the detuning atwhich the data shown in b) and c) is taken. The colormapcodes red for a positive amplitude and white for zero. signal-to-noise-ratio and then digitally down convertedto DC, leading to a measurement time of 26 ms perpoint for a total of 6.5 seconds for 250 points in thetrace. From this we obtain both the in-phase ( I ) andthe quadrature ( Q ) components of the transmitted field A sin(∆ mLO t + φ ) ≡ I sin ∆ mLO t + Q cos ∆ mLO t . Usinginput-output theory [14], these quadratures at the outputof the resonator are related to the cavity Bloch equationsby I ( t ) = p Z ~ ω r κ Re h ˆ a ( t ) i ,Q ( t ) = p Z ~ ω r κ Im h ˆ a ( t ) i , (6)where Z is the characteristic impedance of the transmis-sion line connected to the resonator.When arbitrary qubit rotations can be performed, it issufficient to consider the measurement response for thequbit prepared in either its ground | g i or excited state | e i for a full characterization of the qubit state [40]. Fig-ure 2a) shows the pulse scheme used for the measure-ment. The time dependent quadrature amplitudes I and Q are measured at the cavity resonance frequency withthe qubit in the ground state ( ω m = ω r − χ ). The res-onator is continuously driven at a measurement drive am-plitude of ǫ = κ/
2, populating the resonator with ¯ n ≈ π pulseending at time t = 0 and resonant with the ac-Stark[41] and Lamb shifted [42] qubit transition frequency ω s = [ ω a + 2 χ ( h a † a i + 1 / ω s , res is then applied tothe qubit, see Figs. 2b) and c). Qubit relaxation duringthe π pulse limits the achievable | e i state population to ge a)b) ge ΧΚ •••••••••••••••••••••••••••••••••••••••••••••••••••••••••• œœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ Χ Χ Χ Χ - Χ - Χ - Χ I @ m V D •••••••••••••••••••••••••••••••••••••••••••••••••••••••••• œœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - - - - - D mr @ MHz D Q @ m V D
0 ns180 ns740 ns • œ‰ FIG. 3: (color online) Transmission spectrum of the resonator.The Q quadrature of the field is shown in a) and the I quadra-ture in b). The datapoints represent the instantaneous aver-aged response of the field before (red solid points), 180 ns(blue diamonds) and 740 ns after (green crosses) the π -pulseis applied. The underlying lines show the numerical simu-lations and the dotted line shows the expected response ofthe system for the qubit in the excited state | e i with infinitelifetime.
99% [43], as obtained by solving the Cavity-Bloch equa-tions. This is within the statistical uncertainty of thedetection. Furthermore, thermal excitations of the qubitare expected to be very low and are therefore neglected.The dependence of the quadrature components I and Q on the detuning ∆ mr of the measurement frequencyfrom the bare resonator frequency is plotted in Figs. 2d)and e). For clarity, the quadratures are rotated in the IQ -plane for each measurement frequency ω m such thatthe Q quadrature is maximal in the steady-state (qubitin the ground state), resulting in Q = A and I = 0 for t → ∞ . As a result, before the π -pulse the I quadratureis always zero.The time and frequency dependence of the measure-ment signal is accurately described by the Cavity-Blochequations with a single set of independently measured,non adjustable parameters as indicated by the solid linesin Figs. 2b) and c). The cavity resonance frequency isdetermined as ω r / π = 6 . ± . κ/ π = 1 . ± .
02 MHz. Thequbit transition frequency is determined spectroscopi-cally as ω a / π ≈ . ± .
001 GHz with a chargingenergy of E c /h = 232 . ± . g/ π = 134 ± ω a = ω r [16].The cavity pull χ/ π = − . ± .
02 MHz is de-termined spectroscopically. This is done by measuringthe cavity resonance frequency leaving the qubit in theground state and then measuring its frequency shift ap-plying a continuous coherent tone to the effective qubittransition frequency ω s , res . When the qubit transitionis saturated (Ω ≫ γ ), the resonator is shifted on aver-age by χ . This value is in good agreement with the fulltransmon model taking into account higher levels [30]( χ/ π = − .
71 MHz).In fitting the measurement response in Fig. 2, the qubitdecay rate γ / π = 0 . ± .
01 MHz is used as an ad-justable parameter which is equal to an independent mea-surement of γ within the statistical uncertainty. In prac-tice, the qubit decay rate is determined for one measuredtrace, and then kept fixed for all other traces. Note that,for short π -pulses, the dephasing rate γ φ has no measur-able influence on the solution of the equations. Addi-tionally, a single scaling factor is introduced to relate thequadrature voltages at the output of the resonator to thedigitized voltages after amplification.To interpret the time and frequency dependence of themeasurement signal shown in Fig. 2 it is instructive toplot I and Q as a function of ω m at fixed times t , asshown in Fig. 3. With the qubit in | g i , red points inFig. 3, the resonator transmission exhibits the expectedline shapes for both quadratures. When applying a π -pulse to prepare the qubit in | e i , the cavity resonancefrequency shifts by 2 χ , but the transmitted quadraturesrespond only on a time scale corresponding to the photonlifetime T κ ≡ /κ . The lineshape of the cavity transmis- sion spectrum centered at + χ will only be reached in thelimit of T ≫ T κ , see dotted line in Fig. 3. The interplayof the cavity field rise time and the qubit decay time re-sults in the observed dynamics of the cavity transmissionin Figs. 2 and 3.At time t = 180 ns ∼ . κ ∼ . after the prepa-ration of | e i the shift of the cavity resonance to lower fre-quency towards + χ is clearly visible, see blue diamondsin Fig. 3. At t = 740 ns ∼ . κ ∼ . , when ≈ P e is decayed, themeasured curve is approximately the average between thesteady state | g i and | e i responses, see green crosses inFig. 3.When looking at the time traces in Fig. 2d), the ef-fective shift of the resonance to lower frequency ex-plains the reduction of the signal in the Q quadraturefor measurement detunings ∆ mr > − . ∼ χ . For∆ mr < − . π -pulse because the resonator is driven closer to resonance.Given our choice of the rotation of the traces in the IQ -plane, the I quadrature of Fig. 2e) acts like a phase andalways shows a positive response to the π -pulse.The same considerations explain the features seen inthe single measurement trace in Figs. 2b) and c) taken ata measurement frequency corresponding to ∆ mr = − χ .The change of the I and Q quadratures on a timescale T κ after the π -pulse reflects the relaxation of the fieldin response to the qubit excitation. The time scale ofthe return of the quadratures to their initial values isdetermined by the qubit decay at rate γ . IV. PULSED MEASUREMENT RESPONSE
To avoid measurement-induced dephasing during thequbit manipulation most of the recent circuit QED exper-iments have been performed by probing the qubit statewith pulsed measurements [27, 28, 29, 31, 43, 44]. Incontrast to a continuous measurement, the measurementtone is switched on only after the qubit state prepara-tion is completed, see Fig. 4a) for the pulse scheme. Theabsence of measurement photons during qubit manipula-tion also avoids the unwanted ac-Stark shift of the qubittransition frequency, thus simplifying qubit control.With the qubit in | g i the resonator response reachesits steady state at the rate κ , which is seen in the ex-ponential rise of the Q quadrature, see blue crosses inFig. 4b). Since the resonator is measured on resonanceat its pulled frequency ∆ mr = − χ , the I quadrature isleft unchanged, see blue crosses in Fig. 4c). As in the con-tinuous case, the resonator frequency is pulled to ω r + χ when the qubit is prepared in | e i , see red dots in Figs. 4b)and c). Since the resonator is now effectively driven off-resonantly, the transmitted signal has non vanishing I and Q quadrature components both of which contain in-formation about the qubit state. With the measurementfrequency still at ∆ mr = − χ , ringing occurs at the differ-ence frequency ( ω r + χ ) − ω m = 2 χ . At later times, the π ω s ω m Strong measurement t ε m Ω a)b)c)d)
048 0 0.5 1.5Time @ Μ s D I @ m V D ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰
048 0 0.5 1.5Time @ Μ s D Q @ m V D I quad r a t u r e Q quadrature ‰‰‰‰‰‰‰ ‰ ‰ ‰ ‰ ‰ ‰ ‰‰ ‰ ‰‰ ‰‰‰‰ ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ @ mV D @ m V D ge t FIG. 4: (color online) a) Pulsed measurement scheme. b),c)Measurement response versus time for a pulsed averaged mea-surement taken in the same conditions as Fig. 2 for the Q and I quadrature respectively. Red points show the response forthe qubit being prepared in the excited state | e i . The bluecrosses are the measured response to the qubit prepared in | g i . The trajectory of the field in the IQ -plane is plotted ind), the arrows indicate the direction of the time. average response is approaching again the steady-statevalue as the qubit decays to | g i at the rate γ . As in thecontinuous case, the qubit lifetime in presence of mea-surement photons is obtained from a fit to the Cavity-Bloch equations. Note, that the decay of the quadratureamplitudes shown in Figs. 4b) and c) does not directlycorrespond to the exponential decay of the qubit popu-lation h ˆ σ z i , but rather is determined by the interplay ofresonator and qubit evolution.The dynamics of the I and Q quadrature amplitudescan also be represented in a phase-space plot, see Fig. 4d).The response for the qubit in | g i follows a straight linewhile the response for the qubit in | e i is more complex.The nontrivial shape of this curve reinforces the factthat both field quadratures contain information aboutthe qubit state. It is obvious that a simple rotation inthe IQ -plane cannot map the signal into a single quadra-ture.Data taken at different measurement frequencies areshown in Fig. 5. As in Sec. III, the I and Q componentsare rotated such that Q = A and I = 0 in steady-state. For the theoretical curves (solid lines), the same set ofparameters as for the analysis of the continuous measure-ment are used, leading to very good agreement. Fig. 5a)shows the measured response for the qubit in | g i . The Q quadrature shows the expected exponential rise in thecavity population and for t & . µ s we recover the con-tinuous measurement response. The I quadrature showsthe transient part of the response during the initial pop-ulation of the resonator, having a negative value (bluecrosses) for measurements at a frequency above ω r − χ (blue detuned) and a positive value (red dots) at frequen-cies below ω r − χ (red detuned). Ringing can be observedwhen the measurement is off-resonant from the pulledcavity frequency. Fig. 5b) shows the response with thequbit prepared in | e i . The response is similar to the oneshown in Fig. 2 for the continuous measurement, if oneomits the initial 100 ns where the resonator is populated. V. RECONSTRUCTION OF QUBIT STATE
The detailed understanding of the dynamics of the dis-persively coupled qubit/resonator system can be used toinfer the qubit excited state population p e = ( h ˆ σ z i +1) / s ρ ( t ) and the ground state response s g ( t ), which corresponds to the shaded area indicated inFig. 4b) and c), is directly proportional to p e .To explicitly state this relation, we introduce an ef-fective qubit measurement operator ˆ M i ( t ), an approachthat we have already employed to perform two-qubitstate tomography using a joint dispersive read-out [28].Here, i = I, Q denote the I and Q field quadratures usedto measure the qubit state. In terms of this measurementoperator, the I and Q components of the signal s iρ ( t ) forthe qubit in state ρ before the measurement are given by s iρ ( t ) ≡ h ˆ M i ( t ) i = Tr[ ρ ˆ M i ( t )] , (7)where ˆ M i ( t ) is determined by the solution to the masterequation (4). Analytical solutions can be found in thelimit of vanishing qubit decay [28],ˆ M I ( t ) = ǫ e − κt/ [2 ˆ χ cos ( ˆ χt ) + κ sin ( ˆ χt )] − χ ˆ χ + ( κ/ , (8a)ˆ M Q ( t ) = ǫ e − κt/ [ κ cos ( ˆ χt ) − χ sin ( ˆ χt )] − κ ˆ χ + ( κ/ , (8b)which depend on the operator ˆ χ ≡ ∆ rm + χ ˆ σ z for thequbit state-dependent cavity pull. As a consequenceof performing a quantum non-demolition measurementwith only a few photons populating the resonator, mix-ing transitions between the two qubit states can be ne-glected [15] and ˆ M i ( t ) is diagonal at all times. Thequbit then remains in an eigenstate during the mea-surement [37] and we can write ˆ M i ( t ) = s ig ( t ) | g ih g | + s ie ( t ) | e ih e | . The signals s ig ( t ) = Tr[ | g ih g | ˆ M i ( t )] and s ie ( t ) = Tr[ | e ih e | ˆ M i ( t )] are determined by Eq. (8) for the |g> response |e> response Difference a) b) c) Q I Q I Q I I ‰œ· ‰œ· Q ‰œ· ‰œ· - - - D e t un i ng D m r @ M H z D I ‰œ· ‰œ· Χ Χ Χ ΧΧ - Χ- Χ- Χ- Χ Q ‰œ· ‰œ· I ‰œ· ‰œ· Q ‰œ· ‰œ· ••••••••••••••••••••••••••••••••••••••••••••• œœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ @ Μ s D ••••••••••••••••••••••••••••••••••••••••••••• œœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ @ Μ s D ••••••••••••••••••••••••••••••••••••••••••••• œœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ @ Μ s D ••••••••••••••••••••••••••••••••••••••••••••• œœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ @ Μ s D ••••••••••••••••••••••••••••••••••••••••••••• œœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - @ Μ s D I @ m V D ••••••••••••••••••••••••••••••••••••••••••••• œœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœœ‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰‰ - @ Μ s D Q @ m V D FIG. 5: (color online) I and Q quadratures for pulsed measurements at different detunings from the resonance frequency. Theplots in a) are taken with the qubit in the ground state | g i while b) displays the response of the system with the qubit preparedin the excited state | e i . c) shows the result of the pointwise difference of the data acquired with the qubit in the ground stateand the excited one. The lower panels show time traces taken at different detunings (blue crosses: ∆ mr = 1 . mr = 0 . mr = − . values h ˆ χ i = ∆ rm ± χ corresponding to the qubit in theground or excited state. To account for qubit relaxation,the Cavity-Bloch equations (5) are solved to determine s ig/e ( t ).The qubit excited state population p e ( ρ ) in a givenstate ρ is determined by the normalized area between themeasured signal s iρ and theoretical ground state response s ig , p e ( ρ ) = 1 T X j s iρ ( t j ) − s ig ( t j ) s ie ( t j ) − s ig ( t j ) ∆ t, (9)where ∆ t denotes the discrete time step between data-points. s ig/e ( t j ) are solutions to the Cavity-Bloch equa-tions (5) with independently determined parameters. Re-placing s iρ ( t j ) with the corresponding expressions fromEq. (7), we notice that Eq. (9) simplifies to p e ( ρ ) =Tr[ ρ | e ih e | ], demonstrating that the excited state popu-lation of an arbitrary input state is proportional to thenormalized area between signal and ground state. Thus,the effective measurement operator ˆ M ′ i = | e ih e | definedby this procedure is equivalent to a projective measure-ment of the excited qubit state.The measurement protocol can be summarized as fol-lows: First, the relevant system parameters are deter-mined in separate measurements as discussed in SectionIII. The qubit lifetime T , the single remaining param-eter, is determined by applying a π -pulse to the qubitand analyzing the resulting transmitted signal. Fromthis complete set of parameters, the signals s ig ( t ) and s ie ( t ) are computed. Finally, the excited state popula-tion p e is calculated from the recorded signal s iρ ( t ) of anarbitrary qubit state ρ and the theory lines s ig ( t ) and s ie ( t ), using Eq. (9), which amounts to a measurement ofˆ M ′ i = | e ih e | . For the particular case of the qubit being in | e i after a π -pulse, the point-by-point difference signalis shown in Fig. 5c). Note, that the excited state popula-tion can also be directly inferred from a fit of the Cavity-Bloch equations to s iρ ( t ) with p e as free fit parameter.It is, however, computationally less intensive to calculatethe population with the area method from Eq. (9), thatis, to perform algebraic operations for the data analysisrather than employing a non-linear fit-routine. We havechecked that both techniques provide the same resultswithin the experimental precision.To test our method experimentally, we perform a Rabi-oscillation experiment [17], where a pulse of variablelength τ and amplitude Ω is applied at the effective qubittransition frequency ω s , res . Indeed, the population p e ob-tained with the area method (Fig. 6, points) has an rms @ ns D p e FIG. 6: (color online) Rabi oscillations of the qubit popula-tion p e reconstructed by analyzing with Eq. (9) the pulsed Q response of the resonator (dots). The black line correspondsto the theoretical prediction calculated using Cavity-Blochequations with the following parameters: ω a / π = 4 .
504 GHz, χ/ π = − .
02 MHz, T = 860 ns, Ω / π = 50 MHz. deviation of less than 1% from the population predictedby Eq. (10) (Fig. 6, solid line). The data is also in goodagreement with a simplified expression p e ( t ) ∼ = 12 − e − t (3 γ +2 γ φ ) cos(Ω t/ . (10)predicting the time-dependent population of the qubit inthe limit of large driving fields (Ω ≫ γ , γ φ ) [45]. VI. CONCLUSION
In conclusion, we have presented a simple set of equa-tions describing the dynamics of the average values ofthe quadrature amplitudes of the transmitted microwavefields in dependence on the qubit state in a circuit QEDsetup operated in the dispersive regime. The measuredtime dependent response of the cavity field to a change in the qubit state is in excellent agreement with calcula-tions. The dependence of the measured response on mea-surement frequency is well understood both for continu-ous and pulsed measurements. From the time dependentmeasurement response we reconstruct the qubit excitedstate population that is used in tomographic measure-ments to accurately measure both single and two-qubitdensity matrices [28, 29].
Acknowledgments