Abstract
We theoretically study measurement induced-dephasing of a superconducting qubit in the circuit QED architecture and compare the results to those obtained experimentally by Schuster et al., [Phys. Rev. Lett. 94, 123602 (2005)]. Strong coupling of the qubit to the resonator leads to a significant ac-Stark shift of the qubit transition frequency. As a result, quantum fluctuations in the photon number populating the resonator cause dephasing of the qubit. We find good agreement between the predicted line shape of the qubit spectrum and the experimental results. Furthermore, in the strong dispersive limit, where the Stark shift per photon is large compared to the cavity decay rate and the qubit linewidth, we predict that the qubit spectrum will be split into multiple peaks, with each peak corresponding to a different number of photons in the cavity.
I. INTRODUCTION
Superconducting qubits are promising building blocks for the realization of a quantum computer [1] . Several experiments have shown coherent control of a single qubit [2, 3, 4, 5, 6] and two-qubit experiments have been realized [7, 8, 9, 10, 11] . Recently, it was suggested that superconducting qubits can be strongly coupled to distributed or discrete LC circuits in a way that opens the possibility to study quantum optics related phenomena in solid-state devices [12, 13, 14, 15, 16] . This concept has been successfully demonstrated experimentally [2, 17, 18, 19, 20] and effects associated with the quantum nature of the microwave electromagnetic field have now been seen in the form of vacuum Rabi splitting [17] and measurement induced dephasing via photon shot noise [19] . In this paper we present a detailed analysis of the quantum fluctuations of the photon number in the cavity and its effect on the qubit spectrum. We also show that access to the extreme limit of strong dispersive coupling should allow direct observation of the photon number distribution in the cavity.
An advantage of some of these circuit QED analogs of cavity QED is that the cavity presents a well defined electromagnetic environment to the qubit which can lead to enhanced coherence times of the qubit [13] . This well defined environment makes quantitative predictions for superconducting qubits more tractable. This was shown in Ref. [2] where we have studied Rabi oscillations in a superconducting qubit strongly coupled to a superconducting transmission line resonator. Due to the detailed understanding of the measurement process, we were able to make quantitative predictions about the measured populations in the Rabi oscillations and observe high visibility [2] fringes. Moreover, as we showed in Ref. [19] , pop-ulating the strongly coupled resonator with a coherent microwave field can lead to a significant ac-Stark shift of the qubit, even in the situation where detuning between the cavity and qubit frequencies is large. Due to the shot noise in the number of photons populating the resonator, this ac-Stark shift leads to measurement-induced dephasing of the qubit. This is similar in spirit to the experiment on Rydberg atoms in a 3D cavity reported in Refs. [21, 22, 23, 24] . In those time-domain experiments, the visibility of Ramsey fringes was shown to decay with an increase of the strength of dispersive coupling to the cavity.
In this paper, we will expand on the theoretical model presented in our experimental paper [19] (hereafter referred to as the Letter) where we observed the AC stark shift and measurement induced dephasing in a circuit QED device. We will start in Sec. II with a brief review of the important features of circuit QED. In Sec. III the experimental results reported in the Letter will be reviewed. We then present in Sec. V two theoretical models describing measurement-induced dephasing. We first start with a simple model which assumes Gaussian fluctuations of the qubit's phase. This is the model that was briefly presented in the Letter to explain the experimental results. We then present a more general approach based on the positive P-representation [25] which does not require the Gaussian assumption. For the experimental parameters reported in the Letter, these two approaches give identical results. However, in the limit of strong coupling and very high Q resonators, the second approach shows that qubit spectrum will exhibit structure at several distinct frequencies due to the underlying discrete energy levels of the cavity. That is, we predict that the qubit spectrum will split into multiple peaks, with each peak corresponding to a different number of photons in the cavity. We will refer to this as number splitting of the qubit spectrum.
Experimental observation of this effect would be a direct demonstration of number quantization in the dispersive regime. We also discuss how, by using irradiation which is off-resonant from both the cavity and the qubit, one can obtain substantial ac-Stark shifts without significant dephasing and how this could be used as the basis of a phase gate for quantum computation.
II. CAVITY QED WITH SUPERCONDUCTING CIRCUITS
A. Jaynes-Cummings interaction
In this section, we briefly review the circuit QED architecture first introduced in Ref. [13] and experimentally studied in Refs. [2, 17, 19] . As shown in Fig. 1 , the system consists of a superconducting charge qubit [1, 12, 26] strongly coupled to a transmission line resonator [27] . Near its resonance frequency ω r , the transmission line resonator can be modeled as a simple harmonic oscillator composed of the parallel combination of an inductor L and a capacitor C. Introducing the annihilation (creation) operatorâ ( †) , the resonator can be described by the Hamiltonian
with ω r = 1/ √ LC. Using this simple model, one finds that the voltage across the LC circuit (or, equivalently, on the center conductor of the resonator) is V LC = V 0 rms (â † +â), where V 0 rms = hω r /2C is the rms value of the voltage in the ground state. An important advantage of this architecture is the extremely small separation b ∼ 5 µm between the center conductor of the resonator and its ground planes. This leads to a large rms value of the electric field E 0 rms = V 0 rms /b ∼ 0.2 V/m for typical realizations [2, 17, 19] . As illustrated in Fig. 1 , by placing the qubit at an antinode of the voltage, it will strongly interact with the resonator through the large electric field E 0 rms . In the two-state approximation, the Hamiltonian of the qubit takes the form
where E el = 4E C (1 − 2n g ) is the electrostatic energy and E J = E J,max cos(πΦ/Φ 0 ) the Josephson energy.
Here, E C = e 2 /2C Σ is the charging energy with C Σ the total box capacitance and n g = C g V g /2e the dimensionless gate charge with C g the gate capacitance and V g the gate voltage. E J,max is the maximum Josephson energy and Φ the externally applied flux, with Φ 0 the flux quantum. Due to capacitive coupling with the center conductor, the gate voltage V g = V dc g + V LC has a dc contribution V dc g (coming from a dc bias applied to the input port of the resonator) and a quantum part V LC . When working at the charge degeneracy point n dc g = 1/2 where dephasing is minimized [4] and neglecting fast oscillating terms, the resonator plus qubit Hamiltonian takes the Jaynes-Cummings form [13] 
where ω a = E J /h is the qubit transition frequency and g = e(C g /C Σ )V 0 rms /h is the coupling strength. As shown in Ref. [13] , the qubit can be measured and coherently controlled by applying microwaves, of frequency ω rf and ω s respectively, to the input port of the resonator. This can be described by the additional Hamiltonian
where ε j (t) is the amplitude of the external drives at rf and spectroscopy frequencies.
B. Dispersive regime
In the situation where the qubit is strongly detuned from the cavity, |∆| = |ω r − ω a | ≫ g, the total Hamiltonian H JC + H D can be approximately diagonalized to second order in g/∆ to yield the following quantized version of the dynamical Stark shift Hamiltonian [13] 
(2.5)
Hereω a = ω a + χ is the Lamb shifted qubit frequency and we have defined χ = g 2 /∆. The term proportional toâ †âσ z can be interpreted as a shift of the qubit transition frequency depending on the photon number in the resonator (ac-Stark shift) or as a pull on the resonator frequency by the qubit. As will be shown later, quantum noise in the photon numberâ †â leads to dephasing of the qubit.
Dephasing due to coupling to the cavity field was also studied using Rydberg atoms coupled to a 3D microwave cavity [21, 22, 23] . In this experiment, atoms were sent one at a time through the cavity and interacted for a finite time with the field. The state of the atoms was finally read out by ionization [28] . Visibility of the Ramsey fringes was measured as a function of the detuning from the cavity, hence as a function of χ. Dephasing was shown to increase with the strength of dispersive coupling χ to the cavity [22] . In this paper, we will instead consider dephasing of the qubit due to the resonator field by looking at the qubit spectrum as measured by transmission of the cavity field.
We note that, in practice, ω s is chosen to be close tõ ω a and the last term of Eq. (2.5) with ω s causes Rabi flopping of the qubit. Moreover, as further discussed below, we choose ω rf = ω r − ∆ r to measure the state of the qubit, where ∆ r is the detuning of the measurement probe from the bare cavity frequency. In this situation, the last term of Eq. (2.5) with ω rf is largely detuned from the qubit and does not lead to qubit transitions. As first noted in the original proposal by Brune et al. [29, 30] this measurement Hamiltonian is therefore highly quantum non-demolition [31] with respect to measurement of the qubit state. Conversely, if the dispersive coupling term is dominant in the Hamiltonian, then QND measurement of photon number is also possible. Physical implementation of QND readout for superconducting qubits has been achieved in the microwave regime [2, 17, 18, 19] , but values of the coupling χ large enough to allow QND readout in the dispersive regime for the photon number have not yet been achieved in any system. However remarkable experiments on Rydberg atoms have achieved photon number readout in a non-dispersive (i.e. degenerate) regime [21, 32] .
C. Damping
Coupling to additional uncontrolled bath degrees of freedom leads to energy relaxation and dephasing in the system. Integrating out these degrees of freedom leaves the qubit plus cavity system in a mixed state ρ(t) whose evolution can be described by the master equation [31] 
where D[L]ρ = 2LρL † −L †L ρ − ρL †L /2 describes the effect of the baths on the system in the Markov approximation. The last three terms of Eq. (2.6) correspond to loss of photons at rate κ, energy relaxation in the qubit at rate γ 1 and pure dephasing of the qubit at rate γ φ .
In the dispersive regime, the operators describing energy relaxation and dephasing should be transformed in the same way as was done in Eq. (2.5) . This leads to small corrections, of order (g/∆) 2 , to the master equation that are omitted here. In this section, we briefly review some of the experimental results already presented in the Letter. Only the results that are directly discussed in the present paper will be presented and the details of the experiment can be found in Refs. [2, 17, 19, 27] . In the Letter, we reported spectroscopic measurements of the qubit as a function of measurement power. The qubit spectroscopic line is shown in Fig. 2 for two average photon numbersn in the resonator, corresponding to two input measurement powers. The relevant experimental parameters are ∆/2π = 105 MHz, g/2π = 5.8 MHz, ∆ r /2π = 0, and κ/2π = 0.57 MHz and a dephasing time (in the absence of power broadening) of T 2 > 200 ns. These values correspond to a relatively small cavity pull of χ/2π ≈ 0.32 MHz, or χ/κ ≈ 0.56 in units of the cavity line width. It is important to note that at these relatively low detunings ∆, there can be a qubit contribution to the cavity line width.
III. EXPERIMENTAL RESULTS
As discussed in the Letter, at low measurement power, the line shape of the qubit spectrum is Lorentzian but as the measurement power increases, the line shape approaches a Gaussian. As shown in Fig. 3 , this is also seen in the dependence of the half-width at half-maximum δν HWHM of the qubit line shape onn which goes from ∝n to ∝ √n as measurement power increases. This figure also shows theoretical results that will be discussed below. In the Letter, we have already provided a theoretical explanation for this behavior. In section V, we review and expand on this model. We then explain how in the limit of very large cavity pull, the results can be significantly different.
IV. AC-STARK SHIFT
In the lowest order dispersive approximation [Eq. (2.5)] the predicted ac-Stark shift ω ac = 2χn will be a linear function of the mean photon number,n. However, this approximation only holds at low photon numbers and breaks down on a scale given by the critical photon number n crit = ∆ 2 /4g 2 [13] . This is illustrated in Fig. 4a ) where the ac-Stark shift, calculated from the lowest order dispersive approximation (red solid line) and the exact eigenvalues (blue dashed line) of the Jaynes-Cummings model [13] are plotted for the experimental parameters. Also shown in this figure is n crit (vertical blue line) which for the experimental parameters is about 82 photons.
Here we see that asn increases, the exact Stark shift begins to fall below the lowest order dispersive approximation even beforen reaches n crit .
In Fig. (4b) the experimental results (solid blue points) are plotted as a function of probe power, P RF (extending up to powers larger than those presented in the Letter). To convert betweenn and P RF , we assume that P RF = λhω RF p where p is the photon flux at the res-onator and λ is a scaling factor that takes into account the large attenuation that is placed between the probe generator and the resonator (to eliminate black body radiation). From the lowest order dispersive approximation, the average photon number when driving the cavity at ∆ r = 0 isn = pκ/2 (κ/2) 2 + χ 2 .
(4.1)
By using the lowest order dispersive approximation for the ac-Stark shift and the line of best fit to the experimental points (in the linear regime at low power) λ can be determined. Doing this gives the red solid line in Figs. 4b) and d). The calibration shows that n crit occurs at ≈ 110µW. The experimental results clearly show the breakdown of the lowest order dispersive approximation. The data points fall below the linear prediction but not nearly as much as the blue dashed curve in (a) predicts. In fact, the data points follow fairly closely the linear inn dependence of the lowest order dispersive approximation in Eq. (2.5) for larger powers than expected (up to and well above n crit ). It is possible to understand why the experiment agrees with the simple dispersive approximation for larger probe powers than expected by considering the following simple model. We assume that, at these large powers, the ac-Stark shift is still given by the dispersive approximation 2χn, but we now take into account the non-linear cavity pull (which is χ at lown). From the eigenvalues of the Jaynes-Cummings Hamiltonian the cavity pull can be calculated as a function ofn. This is shown in Fig. 4c ). From this figure, we see that the cavity pull reduces as the number of photons in the resonator is increased. We thus replace χ by χ(n). The second aspect of our simple model is the non-linear dependence of the average photon numbern with input power P due to the power-dependence of the cavity frequency. To account for this we simply replace χ in Eq. (4.1) with χ(n) andn becomes a non-linear function of input power. This nonlinear dependence of the photon number on the input power is illustrated in Fig. 4d ) as the green dotted line. This is a precursor to bistability in this system [31] . Using these two expressions, we have for our simple model of the non-linear ac-Stark shift 2χ(P )n(P ). This expression is plotted in Fig. 4b ) (green dotted line) with a new scaling factor λ ′ ≈ 0.905λ calculated by the best fit for the experimental data (here we use the complete data set). This simple model produces a result that is linear for a larger range of powers and is closely consistent with the experimental results. It happens that for the particular experimental parameters, the two non-linear effects almost cancel each other out and result in the green dotted line being more linear than expected.
We emphasize that in the Letter, only the low power (below n crit ) part of the ac-Stark shift was studied and was fit only with the linear dispersive model. Comparison with the results of the non-linear model shown here in Fig. 4d ) shows that the calibration of the cavity photon The conversion factor from photon number to external microwave drive power was determined by fitting the red curve to the linear portion of the data at low power. Green, dotted curve: Same as red but taking into account the non-linear reduction in the cavity pull (see part c) and the non-linear increase in the average photon number (see part d) with microwave drive power. For the particular experimental parameters these two effects almost cancel each other out and result in the green dotted line being nearly linear out to much greater input powers than expected. (c) Cavity pull as a function of average photon number n. The red solid line is the result of the dispersive approximation (±χ) while the dashed blue curve is obtained from the exact eigenvalues of the Jaynes-Cummings model. number in terms of the drive power is low by approximately 50% at the highest power shown in Fig. 5 of the Letter. We also emphasize that our treatment here of the non-linearities is only approximate.
V. MEASUREMENT-INDUCED DEPHASING
By monitoring the transmission of the cavity using heterodyne detection, one has access to the average of the cavity field â(t) . As shown in Ref. [13] , the phase φ(t) = arg{ â(t) } is directly related to the population of the qubit σ z (t) . As a result, by recording the phase φ(t) as a function of the excitation frequency ω s , one has access to the absorption spectrum of the qubit [25] 
where the subscript s implies that the expectation value is taken in the steady state. The dephasing rate can be determined through the half width at half maximum of S(ω) [33] .
Using the quantum regression formula [25] , the corre-lation function σ − (t)σ + (0) s can be evaluated as
where ρ s is the steady-state density matrix. This allows us to rewrite the spectrum as
We start by calculating this spectrum by assuming gaussian statistics for the qubit's phase noise. This approximation is sufficient for the experimental parameters considered above but breaks down in the situation where the cavity pull is large χ ≫ κ. Moreover, this simple approach has the advantage of presenting the essential physics in a transparent way. We then show how to go beyond the gaussian approximation by using the positive-P function approach [25] .
A. Gaussian approximation for the phase As mentioned above, quantum fluctuations δn in the photon number around its average valuen will lead to accumulation of a random relative phase between the amplitudes of the two basis states of the qubit and hence to dephasing. We first consider the situation where the qubit is prepared in a superposition |ψ(0) = (|g + |e ) / √ 2 of its basis states and that the cavity is populated by a coherent state with average photon numbern. As it evolves under the dispersive Hamiltonian Eq. (2.5), the qubit superposition picks up a relative phase factor
whereω a = ω a + χ is the Lamb shifted qubit transition frequency. It is convenient to express the second term as its mean value plus fluctuations about the mean
withn the average photon number in the cavity leading to the ac-Stark shift (see Fig. 4 ) and δn(t) the random excursions about this mean. In a frame rotating at the Lamb and ac-Stark shifted qubit transition frequency, we obtain for the correlation
where γ 2 = γ 1 /2+γ ϕ . This is the off-diagonal component of the reduced qubit density matrix. Assuming gaussian statistics for the phase δϕ(t), the cumulant expansion is exact and we obtain [34] 
This expression involves the photon-photon time correlator which for a two-sided symmetrically damped driven cavity takes the form [13] 
where θ 0 = tan −1 2χ/κ ≈ 2χ/κ is the magnitude of the accumulated phase shift for the transmitted photons due to the coupling with the qubit in the small pull approximation (χ ≪ κ) and at ω rf = ω r . We now consider two simple limits of the above result. First, in the situation where the mean cavity photon numbern is small, fluctuations of the photon number will only weakly contribute to dephasing. In this situation phase decay occurs on a long time scale with respect to 1/κ. In this limit, exp(−κ|t|/2) ≈ 0 and Eq. (5.9) reduces to
In this situation, the measurement induced-dephasing only adds to the intrinsic dephasing rate γ 2 . This is because, in this long time limit, the phase undergoes a random walk process leading to an exponential decay of the coherence. The Fourier transform of this expression leads to a Lorentzian spectrum with half-width at half maximum γ 2 +Γ m , whereΓ m = 2κnθ 2 0 is the measurement induced dephasing rate in the small pull limit (see next section).
On the other hand, in the largen limit, phase decay can occur on a time scale much shorter than the cavity lifetime, t ≪ κ −1 . Indeed, sincen is large, a small fraction ofn leaking out of the cavity in a time t ≪ κ −1 conveys enough information to infer the state of the qubit and hence to dephase it completely [35] . In this situation, expanding exp(−κ|t|/2) in Eq. (5.9), we obtain
(5.11)
The largen limit does not lead to an exponential decay and the spectrum will be a convolution of a Lorentzian and a Gaussian. This corresponds to inhomogeneous broadening of the qubit due to the Poisson statistics of the coherent state populating the cavity. In this situation, the half width at half maximum therefore scales as √n .
Using the full expression Eq. (5.9) and moving back to the lab frame, we obtain for the spectrum of the qubit in the Gaussian approximation for the phasẽ
whereΓ j = 2(γ 2 +Γ m )+jκ. The spectroscopic line shape is given by a sum of Lorentzians, all centered on the ac-Stark shifted qubit transition but of different widths and weights.
As expected from the above discussion, we see from Eq. (5.12) that if the measurement rateΓ m is much smaller than the cavity decay rate κ/2, then only a few terms in the sum contribute and the spectrum is Lorentzian. On the other hand, when the measurement rate is fast compared to the cavity damping, the spectrum will be a sum of many Lorentzians, resulting in a gaussian profile. In this situation, dephasing occurs before the cavity has had time to significantly change its state, leading to inhomogeneous broadening as discussed above.
The expression Eq. (5.12) for the spectrum can be summed analytically but yields an unsightly result which is not reproduced here. To compare with the experimental results, we evaluate numerically the half-width at half maximum from S(ω). The results are plotted as a function of probe power in Fig. 3 (full red line) . The agreement with the experimental results (symbols) is good, especially given that there are no adjustable parameters apart from γ 2 which only sets the value of the dephasing atn = 0. In this figure we have included more experimental points, for higher powers, than presented in Fig.  5 of the Letter. The experimental points presented here are obtained by fitting a Lorentzian (blue squares) and a Gaussian (red triangles) to the experimentally measured spectroscopic line (see Fig. 2 ). We then keep the fit which has the smallest variance or both points if the variances are approximately the same (purple squares and triangles). The error bars are the standard errors on the halfwidth half-max obtained from the fit. This approach to extracting the error bar is different from what was presented in Fig. 5 of the Letter. In that case, the error bars represented the systematic difference between the Lorentzian and Gaussian fits and the points were the average value. From Fig. 3 (this paper) we see that the first 7 points fit best to a Lorentzian while the later points fit best to a Gaussian, except for the higher powers where the error in the fit is approximately the same. The predicted crossover from Lorentzian to Gaussian is clearly seen.
There are several potential sources of discrepancy between the experimental and the theoretical results. One of them is the breakdown of the lowest order dispersive approximation. Using the same simple non-linear model as in section IV, we plot in Fig. 3 the half-width at half maximum as a function of input power (green dashed line). The effect of this correction is to reduce the width of the spectroscopy peaks.
The breakdown of the dispersive approximation can be seen by the dispersive result (red full line) overestimating the width at high powers. However, while the simple non-linear model used here does correctly show saturation of the width at high powers, it is not a full treatment of the Jaynes-Cummings Hamiltonian and should not be considered too seriously. A complete investigation of the behavior of the system at very large photon numbers will require numerical investigation which is beyond the scope of this paper.
A further possible source of discrepancy comes from the fact that a constant spectroscopy power was used. Since the effective coupling strength χ is not constant with probe power, the effect of the spectroscopy power on the qubit will change with measurement probe power. As a result, spectroscopic power broadening [19] will also depend on measurement probe power. This effect has been taken into account in the green dashed curve of Fig. 3 but only to the accuracy of our simple model. Finally, environmental noise due to two-level systems activated at large photon number could be an additional cause of discrepancy.
B. Beyond the gaussian approximation
To go beyond the gaussian approximation made in the last section, we solve the master equation Eq. (2.6) using the positive P -function method [31] . Following Ref. [36] , we first write the qubit-cavity density matrix as ρ =ρ ee |e e| +ρ gg |g g| +ρ eg |e g| +ρ ge |g e|, (5.13) whereρ ij acts only in the cavity Hilbert space. As shown in appendix A, this leads to four coupled differential equations for the operatorsρ ij . In the absence of qubit mixing (T 1 processes), solving these coupled equations yields the time-evolved full density matrix
where c eg (t) = a eg (t) α − (t)|α + (t) (5.15) describes the decay of the qubit phase coherence. In the above expression, we have
and a ge (t) = a * eg (t) with α
In these expressions, α s ± represent the steady-state value of the field â given that the qubit is either in its ground (−) or excited (+) state. Recall that ∆ r is the detuning of the measurement beam from the bare cavity frequency.
In the limit κt ≫ 1 discussed previously, the decay of a eg (t) is given by
is the generalized measurement-induced dephasing rate. In this expression,
is the stationary average number of photons in the cavity when the qubit is in the excited (+) or ground (−) state. At small pulls, as is the case for the experimental parameters quoted in section III, the measurementinduced dephasing rate is largest at ∆ r = 0 (see Fig. 5 , blue dashed line). At large pulls, cavity transmission decreases at ∆ r = 0 and, as illustrated in Fig. 5 (red solid line), the maximum dephasing rate then occurs at ∆ r = ± χ 2 − κ 2 /4. As we increase χ, the information about the state of the qubit is conveyed more by the amplitude than the phase of the transmitted beam.
From the above, we see explicitly that by introducing a probe (ε rf ) we cause the coherence terms a eg (t) to exponentially decay at a rate Γ m thereby leaving the system in a mixed state with perfect correlation between the eigenstates ofσ z and the pointer states |α + and |α − . As a result, if the pointer states are well separated in phase space, we can regard the cavity as a meter which performs a von Neumann projective measurement of the qubit observableσ z .
A measure of the distinguishability of the cavity states is D = |α + − α − | 2 [31] . If this is large such that | α − |α + | = exp[−D] is small, then the two pointer states are well separated and easily distinguishable. In the steady-state, D is given by
which is related to Γ m in the following way
(5.23) That is, as the measurement becomes more projective in theσ z basis, the qubit dephases faster. This is a clear example of measurement induced dephasing and of the fundamental limit which exists between acquiring information about a quantum system and dephasing of that system [35] . Note that, as shown appendix B, D s can also be related to the measurement time. The present system is a factor of 4 away from the quantum limit. One factor of 2 comes from the fact that we are using a symmetric resonator and only looking at the transmission. Half of the information is lost in the unmeasured reflected signal [13] . The other factor of two comes from the use of heterodyne rather than homodyne detection. As a result and as shown in this appendix, the quantum limit can be reached by using asymmetric resonators and homodyne detection of the transmitted field.
The qubit absorption spectrum
To evaluate the qubit's spectrum, we first need to calculate the correlation function σ − (t)σ + (0) s . Note that (as discussed in appendix A) in calculating this particular correlation function we do not need to assume that T 1 is infinite. This is because it only depends on the off-diagonal coherences and these are not mixed by a T 1 process. Using Eq. (5.2) and the above results, the correlator can be shown to be (t > 0) σ − (t)σ + (0) s = a eg (t), (5.24) with the initial condition α + (0) = α − (0) = α s − . Using Eq. (5.16) with the above initial condition yields (5.25) where
From the above expression, we see that the time dependence of the correlation function is given by three terms. The first one involves Γ m and is simply the Lorentzian part of the measurement-induced dephasing spectrum we have seen before. The second is a frequency shift B which contains a negative term −χD s which gives rise to negative frequency contributions in the spectrum. The last relevant term goes as A exp[{κ/2 + i(χ + ∆ r )}t] and gives rise to non-Lorentizian spectra.
From the expression for the correlation function, it is simple to obtain the spectrum:
, (5.28) where Γ j = 2(γ 2 + Γ m )+ jκ and ω j =ω a + B + j(χ+ ∆ r ). The spectrum, as in the Gaussian approximation, can be written as a sum over different photon numbers j.
In the limit of (∆ r + χ) much different from κ/2, the spectrum is a sum of Lorentzians with decay rate Γ j /2. However unlike Eq. (5.12) where each Lorentzian is centered at the ac-Stark shifted frequency, here each peak has its own frequency shift ω j . As a result, the full theory predicts that the spectrum need not be symmetric whereas in the Gaussian theory only symmetric spectra are possible. Furthermore, in the limit that χ is much larger than κ, A → D s and the spectral weights become Poisson distributed with mean D s . The peaks are separated by (χ + ∆ r ) and the first one is at the frequencỹ ω a + B. That is, the average frequency, which is the ac-Stark shift, occurs at
Taking this limit further with χ much larger than the widths Γ j , the individual peaks will become distinguishable. This is discussed further below. It is interesting to point out that in the limit of largē n − (orn + ) the results obtained here and those obtained in the Gaussian approximation agree. This can be seen by expanding the exponent A exp[{κ/2 + i(χ + ∆ r )}t] in Eq. (5.25) to order t 2 :
(5.30)
For ∆ r = 0,n − =n + ≡n and we recover Eq. (5.11) in the lab frame. As a result, in the largen limit, both the gaussian approximation and the above theory converge to give the same gaussian spectrum. It is only whenn is small that the theories have different predictions. This will be discussed further in section V B 3. The half-width at half maximum of the spectrum obtained from the full expression Eq. (5.28) is plotted at ∆ r = 0 as a function ofn in Fig. 3 using the experimental parameters given in section III. Since the experiment was done in the limit χ < κ/2, the results obtained from Eq. (5.28) cannot be distinguished from those obtained from the Gaussian approximation Eq. (5.12) . This is because in the small cavity pull limit Γ m →Γ m and therefore A → 2Γ m /κ, B → 2nχ. That is, to see the break down of the gaussian approximation at smalln requires a larger cavity pull.
Phase-gate
As discussed above and in appendix B, measurement causes dephasing of the qubit. This is a clear illustration of the Heisenberg type relation between rate of information gain and dephasing Γ m [35] . However, irradiation at the rf frequency does not have to induce dephasing of the qubit. Indeed, the qubit pulls the resonator frequency up or down causing a state dependent phase shift for photons near the cavity frequency. But, in the low χ limit, photons off resonant from the resonator have phase shifts nearly independent of the qubit state. These photons do not become entangled with the qubit, and hence do not cause dephasing.
This can be understood more quantitatively in Fig. 5 , where it can be seen that the dephasing rate is significant only on a frequency range κ around the pulled resonator frequency. In this Figure, we have fixed the input power and scanned ∆ r . In Fig. 6 , we rather keep the number of photons in the cavity fixed (n − = 2) and scan ∆ r . We see that at large detunings, the dephasing rate scales as ∆ −2 r . As a result, off-resonant irradiation can produce large ac-Stark shifts (ω ac ) with minimal dephasing of the qubit. This can be used as a single-bit phase gate for quantum computation.
The observed asymmetry in the large χ case (red solid line of Fig. 6 ) is a result of the fact Γ m depends onn − andn + . By writingn + as, 31) we see that at fixedn − ,n + can be large for negative ∆ r . Thus the overall measurement induced dephasing will be large in that region.
The quality factor for this single qubit gate, Q, can be defined as the coherent phase rotation that can be realized in the total dephasing time (
in the large pull limit and in the ideal situation where dephasing is limited by photon shot noise. Moreover, similarly to Rabi oscillations that have been demonstrated experimentally [2] , this phase gate could be realized on a time scale which is much faster than 1/κ since for offresonant irradiation, the cavity is only virtually populated.
To show that the dephasing is minimal during the phase-gate, we have calculated using Eq. (5.28) the linewidth of the qubit spectrum as a function of input power for the experimental parameters and a large positive detuning ∆ r /2π = 32 MHz (well away from the peak shown in Fig. 6 ). This is shown in Fig. 3 as a blue dashed line. Here we see that in the dispersive model, there is no additional dephasing due to the off-resonant irradiation. That is, the predicted linewidth stays constant at γ 2 for the input powers plotted. At the critical photon number, n crit = ∆ 2 /4g 2 the quality factor for the above experimental parameters is 17.3. This quality factor can however be easily increased by optimizing the system parameters [37] . For example at g/2π = 100 MHz and ∆/2π = 1000 MHz a quality factor of 157 can be reached. (This value is entirely limited by the current value of T 2 ∼ 500ns and not by the direct infidelity of the phase gate.) An advantage of this rf approach over a dc pulse of the flux or gate charge is that the logical operation can be realized while biased at the sweet spot. As explained in the previous sections, the gaussian model and the P-function approach agree in the small pull limit χ ≪ κ/2. In the large pull case, the predicted behavior is however substantially different. Indeed, when the Lorentzians in Eq. (5.28) are separated in frequency by more than their width, the spectrum S(ω) will be split into many peaks with each peak corresponding to a different photon number in the cavity. Number splitting was also predicted by Dykman and Krivoglaz for a different situation, namely an undriven cavity coupled to a thermal bath [38] . The number splitting for our case (driven cavity at zero temperature) is illustrated in Figures 7 and 8 . In Fig. 7 , the spectrum is shown as a function of frequency and of the detuning ∆ r for a fixed χ/κ = 100. In Fig. 7a ) (∆ r > 0),n − is fixed and equal to 2 whereas in Fig. 7b ) (∆ r < 0),n + = 2. The choice of fixing either n + or n − was made so that there is always a small number of photons in the cavity independent of the state of the qubit. For example, for n − fixed, the number of photons in the cavity when the qubit is excited, n + , would be small for ∆ r > 0, equal to n − at ∆ r = 0 and very large at ∆ r = −χ. The inset shows a cross-section at ∆ r /κ = 100 (∆ r = χ). For this value of ∆ r /κ, the peaks are very well separated and the integrated area under each peak obeys Poisson statistics. It is interesting to stress that only at ∆ r = χ does the spectrum have a simple Poisson distribution corresponding to a coherent state with average photon numbern − = 2. At ∆ r < χ there are more peaks than expected for a coherent state of this amplitude. Furthermore, at ∆ r = −χ the spectrum is single peaked.
In Fig. 8 , the spectrum is shown as a function of the cavity pull χ/κ for a detuning of ∆ r = χ, such that the cavity is always driven at the pulled frequency ω r − χ corresponding to the qubit in the ground state. In these plots, we have taken γ 2 = 7.6κ. Assuming an experimentally realistic value of κ/2π ∼ 100 KHz [27] , this corresponds to a conservative T 2 = 200 ns [2] . As seen on Fig 8, for these parameter values, the peaks should be resolvable experimentally starting around χ/κ ∼ 20 (insert in Fig. 8 ). Achieving χ/κ ∼ 20 in the dispersive limit (g/∆ < ∼ 0.1) requires g/2π ∼ 20 MHz. This value of g was already realized experimentally [2] and therefore the experimental observation of number splitting seems feasible.
The behaviour described above can be understood simply as ringing of a high-Q resonator when its resonance frequency is suddenly changed when the qubit changes state. Equivalently we can think of this as a Raman process in which drive photons in the cavity at the time of the transition are lifted up to the final cavity frequency.
As discussed previously, the calculation of the spectrum assumes that the qubit is initially in the ground state with the measurement beam turned on at a frequency ω d detuned by ∆ r from the bare resonator frequency ω r . In the calculation of the correlation function σ − (t)σ + (0) s , the qubit is flipped to the excited state at time t = 0 and the overlap with the ground state is calculated at time t. When the qubit is flipped, the dressed resonator frequency is suddenly changed from ω r − χ to ω r + χ. Depending on the frequency of the measurement drive and the quality factor of the cavity, this sudden change will cause ringing in the cavity. This is illustrated in Fig. 9 where the distance D(t) = |α + (t) − α − (t)| 2 is plotted as a function of time for two values of χ/κ. For the moderate value of χ/κ = 5 (red, full line), the distance is seen to undergo large oscillations before settling to the steady-state value D s . For the low χ/κ ratio of 0.1 (blue, dashed line) there is no ringing due to the abrupt change of dressed cavity frequency and the distance simply rises to D s .
The ringing is also shown in the inset of Fig. 9 where the real and imaginary part of the cavity field α + (t) are plotted as a function of time, again for χ/κ = 0.1 and 5. In the low Q case, as the qubit flips, the cavity field settles to its new steady state value without large excursions in the field amplitude, and therefore large changes in photon number. Only a few different photon numbers contribute and the corresponding spectrum is single peaked as expected. In the high Q case, the field amplitudes performs many cycles before settling to the steadystate value. The cavity therefore probes a large range of photon numbers and the spectrum shows multiple peaks. In the time domain we can see that the qubit correlator, Eq. (5.25), has period recurrences provided χ ≫ κ. It is these recurrences in time which give peaks in the spectrum. These time domain recurrences were observed in the resonant regime (∆ = ∆ r = 0) with Rydberg atoms in Ref. [24] .
In the case where ∆ r = −χ (ω d = ω r +χ), the cavity is driven at the dressed cavity frequency corresponding to the qubit in the excited state. In this situation, flipping the qubit does not produce any inelastic Raman scattering (ringing) since the photons are already at the final cavity frequency. This is seen in Fig. 7 at ∆ r /κ = −100 where the spectrum is single peaked.
Remarkably when the detuning is such that ∆ r < χ, and as can be seen in Fig. 7 , the spectrum has peaks at negative frequencies (i.e. belowω a ) as well as more peaks than expected. In particular, in the limit that χ ≫ κ and at ∆ r = 0, the frequencies start atω a − 2n − χ with peak separation χ and the spectral weights have a poisson distribution with mean 4n and notn. To understand the presence of these peaks, we move the dispersive Hamiltonian Eq. (2.5) to the frame defined by the unitary operator
(5.33)
HereD[α] is the displacement operator for the cavity Here we have considered the situation where ∆ r = 0 and neglected the last term of Eq. (2.5) which only leads to a small shift of the qubit transition frequency in the present situation. This Hamiltonian is pictorially represented in Fig. 10 . We immediately see that, in this frame, the qubit transition frequency is reduced by 2χn from the lamb shifted frequencyω a . Moreover, when the qubit is in the ground state, the Hamiltonian corresponds to a shifted and inverted harmonic oscillator (LHS of Fig. 10 ). On the other hand, when the qubit is in the excited state, the harmonic oscillator is shifted but not inverted (RHS of Fig. 10 ). Starting with the qubit initially in the ground state and the field in a coherent state of amplitude α − corresponds, in the frame defined by Eq. (5.33), to a qubit in the ground state and the vacuum state of the oscillator. In this frame, flipping the qubit at time t = 0 corresponds to applying the operator (for ∆ r = 0)
The result is to both flip the qubit and to displace the oscillator to a coherent state of mean photon number 4n. This is exactly the observed structure in the qubit power spectrum and each of the observed peaks corresponds to one of the possible transitions between these two oscillators. We note that the above is similar to the When calculating the correlator σ−(t)σ+(0) , the qubit is flipped at time t = 0. In the displaced frame, this corresponds to both flipping the qubit and displacing the oscillator from the vacuum state to a coherent state with mean photon number 4n. This distribution is shown on top of the harmonic oscillator corresponding to the qubit in its excited state. This simple picture explains the observed peaks in the qubit power spectrum and the presence of negative shifts.
Mollow triplet, where transitions both above and below the atomic transition frequency are possible due to dressing of the atomic levels by the presence of a strong pump drive [39] . Observation of number splitting would constitute a simple test of number quantization of the field inside the resonator in the dispersive regime. In the resonant regime (∆ = ∆ r = 0), number quantization has been verified in cavity QED using Rydberg atoms [21] . This was done by looking at the Fourier components in the probability to find the atom in the excited state as a function of time. In the dispersive regime (∆ > g, κ) the recurrence time for the cavity field was too long to be observable [22] . In the circuit QED system it should be possible to reach the strong dispersive limit χ > κ, 1/T 2 where it is possible to probe number quantization. In the dispersive regime the qubit spectrum acts as a probe of the cavity field. When the rate at which information about the cavity state is passed to the qubit faster than the rate at which the cavity state changes significantly due to damping and qubit dephasing, it is possible to learn about the statistics of the field from the qubit spectrum. Note the above predictions are only valid in the limit where χ 2 /∆ < κ. When this in not the case the higher order effects in the Jaynes-Cummings Hamiltonian [Eq. (2.3)] will become important and will lead to non-Poissonnian statistics of the resonator field. This will be discussed further elsewhere. Finally, we note an interesting proposal by Brune et al. [29, 30] and a recent experiment [32] to prepare a Fock state of the cavity field (number squeezing) containing a single photon by monitoring the state of a continuous beam of atoms sent through the cavity.
VI. CONCLUSION
We have found that due to the ac-Stark shift, quantum noise in the photon number populating the resonator leads to well characterized measurement-induced dephasing. A simple model based on a gaussian approximation for the phase noise was presented, as well as a more general model based on the positive P-function. For the experimental parameters given in the Letter, both models yield the same quantitative results which are in very good agreement with the experiment. We emphasize that the only adjustable parameter in the theory is the intrinsic qubit dephasing rate whose only effect is to give a constant offset to the predicted linewidth.
In the strong dispersive regime, where the cavity pull χ is much bigger than the cavity field decay rate κ/2, the Pfunction approach predicts a splitting of the qubit spectrum due to the discrete quantum nature of the field populating the cavity. Observation of this prediction would be a confirmation of the quantized nature of the resonator field. This strong dispersive coupling regime should be readily achievable with realistic circuit QED parameters.
To obtain these expressions, we have assumed that P ij (∞, ∞) = 0 as is usual [25] .
These equations can be solved simply by making the ansatzen P ee =δ (2) [α − α + (t)]δ (2) [β − α * + (t)], P gg =δ (2) [α − α − (t)]δ (2) [β − α * − (t)], P eg =a eg (t)δ (2) [α − α + (t)]δ (2) [β − α * − (t)], P ge =a ge (t)δ (2) [α − α − (t)]δ (2) 
and substitute these into each equation. This results iṅ
Solving these simple differential equations completely solves the master equation Eq. (2.6) in the absence of mixing due to T 1 effects. The solution is given in Eqs. (5.14) -(5.18).
APPENDIX B: MEASUREMENT TIME
In this appendix we show how one can calculate the measurement time for this system and show how it relates to the quantum limit [35] . To do this we need to describe how we are measuring the pointer states (i.e. how the information is processed). In this experiment, this is done by using heterodyne detection of the signal that is transmitted from the cavity. In other words, the full quantum trajectory for the system is [40, 41] dρ J (t) = dtLρ J (t) + dtH[(J * (t) − κη â † )â]ρ J (t), (B1) where H[(J * (t)−κη â † )â]ρ J (t) is the superoperator representing the non-linear effects of the continuous monitoring and is defined by
The measurement record (heterodyne signal) is given by
where ζ(t) is a complex gaussian white noise term, which is formally defined as
where E denotes an ensemble average and η = 1/[2(N + 1)] is the inefficiency of the measurement. Here N is the dark noise and the extra factor of 1/2 is due to the fact that information leaks out of the cavity in both directions and we only monitor transmission [13] .
From this quantum trajectory the rate at which information is obtained about â is κη. To convert this to a rate of information gain about σ z we define the measurement observable for a time τ as
where φ determines the quadrature in which the information about the qubit is encoded. This can be determined by
For example, if ∆ r = 0 and φ = 0, the information about the qubit is only encoded into the real part of â . From this observable, the mean and variance is
That is, if we were to measure the system for a time τ many times we are confident that to one standard deviation the value ofĪ is κηRe[ â e −iφ ] ± κη/2τ . If τ is much shorter than 1/γ then we can approximatē I(τ ) withĪ ± (τ ) = κηRe[α s ± e −iφ ], where the ± subscript refers to the state of the qubit. To be able to distinguish between σ z = ±1, we require ∆ σ z ≤ 1 and thus
The equality defines the measurement time t meas . Using the above, this can be rewritten as
That is, even for perfect detection efficiency η = 1, this approach is a factor of two away from the quantum limit t meas Γ m = 1/2 [35] . This is because even though we are selecting the correct quadrature in which the information about the qubit is stored, [using the classical processing defined in Eq. (B7)] we are still measuring the other quadrature as we are performing heterodyne detection. It is well known that heterodyne detection measures both the φ and φ + π/2 quadrature with 1/2 efficiency [42, 43] . Thus if we change the detection scheme to homodyne detection of the φ quadrature we can reach the quantum limit. That is, to reach the quantum limit we require η = 1 which means we need asymmetric cavities and no dark noise as well as a detection scheme which extracts only information aboutσ z . Note if we did not perform any classical processing on the heterodyne signal J(t) then we would be a factor of four away from the quantum limit.

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