Quantum theory of a bandpass Purcell filter for qubit readout

# Quantum theory of a bandpass Purcell filter for qubit readout

## Abstract

The measurement fidelity of superconducting transmon and Xmon qubits is partially limited by the qubit energy relaxation through the resonator into the transmission line, which is also known as the Purcell effect. One way to suppress this energy relaxation is to employ a filter which impedes microwave propagation at the qubit frequency. We present semiclassical and quantum analyses for the bandpass Purcell filter realized by E. Jeffrey et al. [Phys. Rev. Lett. 112, 190504 (2014)]. For typical experimental parameters, the bandpass filter suppresses the qubit relaxation rate by up to two orders of magnitude while maintaining the same measurement rate. We also show that in the presence of a microwave drive the qubit relaxation rate further decreases with increasing drive strength.

###### pacs:
03.67.Lx, 85.25.-j, 03.65.Yz

## I Introduction

The implementation of fault-tolerant quantum information processing N-C-book () requires high-fidelity quantum gates and also needs sufficiently fast and accurate qubit measurement. Superconducting quantum computing technology Barends-14 (); Chow-14 (); Weber-14 (); Sun-14 (); Riste-14 (); Stern-14 (); Mlynek-14 (); Lin-14 (); Kelly-15 () is currently approaching the threshold for quantum error correction. Compared with the recent rapid progress in the increase of single-qubit and two-qubit gate fidelities, qubit measurement shows somewhat slower progress. The development of faster and higher-fidelity qubit readout remains an important task.

In circuit QED (cQED) Blais-04 (); Wallraff-04 (), the qubit state is inferred by measuring the state-dependent frequency shift of the resonator via homodyne detection. This method introduces an unwanted decay channel Esteve-86 () for the qubit due to the energy leakage through the resonator into the transmission line, the process known as the Purcell effect Purcell-46 (); Haroche-book (). The Purcell effect is one of the limiting factors for high fidelity qubit readout.

In principle, the Purcell rate can be suppressed by increasing the qubit-resonator detuning, decreasing the qubit-resonator coupling, or decreasing the resonator bandwidth due to damping. However, these simple methods increase the time needed to measure the qubit. This leads to a trade-off between the qubit relaxation and measurement time, whereas it is desirable to suppress the Purcell rate without compromising qubit measurement. Several proposals have been put forward for this purpose, which include employing a Purcell filter Red10 (); Jeffrey-14 (); Bronn-APS-15 (); Bronn-15 (), engineering a Purcell-protected qubit Gam11 (); Srinivasan-11 (), or using a tunable coupler that decouples the transmission line from the resonator during the qubit-resonator interaction, thereby avoiding the Purcell effect altogether Set13 ().

The general idea of the Purcell filter is to impede the propagation of the photon emitted at the qubit frequency, compared with propagation of the microwave field at the resonator frequency, used for the qubit measurement. A notch (band-rejection) filter detuned by 1.7 GHz from the resonator frequency was realized in Ref. Red10 (). A factor of 50 reduction in the Purcell rate was demonstrated when the qubit frequency was placed in the rejection band of the filter. A bandpass filter with the quality factor (and corresponding bandwidth of 0.22 GHz) centered near the resonator frequency was used in Ref. Jeffrey-14 (). This allowed the qubit measurement within 140 ns with fidelities and for the two qubit states. (The bandpass Purcell filter was also used in Ref. Kelly-15 (); it had a similar design with a few minor changes.) A major advantage of the bandpass Purcell filter in comparison with the notch filter is the possibility to keep strongly reduced Purcell rate for qubits with practically any frequency (except near the filter frequency), thus allowing quantum gates based on tuning the qubit frequency, and also allowing multiplexed readout of several qubits by placing readout resonators with different frequencies within the filter bandwidth.

In this work, we analyze the Purcell filter of Ref. Jeffrey-14 () using both semiclassical and quantum approaches and considering both the weak and the strong drive regimes. Our semiclassical analysis uses somewhat different language compared to the analysis in Ref. Jeffrey-14 (); however, the results are very similar (they show that with the filter the Purcell rate can be suppressed by two orders of magnitude, while maintaining the same measurement time). The results of the quantum analysis in the regime of a weak measurement drive or no drive (considering the single-photon subspace) practically coincide with the semiclassical results. In the presence of strong microwave drive, the Purcell rate is further suppressed with increasing drive strength. We have found that this suppression is stronger than that obtained without a filter Set14 ().

In Sec. II we discuss the general idea of the bandpass Purcell filter and analyze its operation semiclassically. Section III is devoted to the quantum calculation of the Purcell rate in the presence of the bandpass Purcell filter. In Sec. IV we discuss further suppression of the Purcell rate due to an applied microwave drive. Section V is the Conclusion. In the appendix we review the basic theory of a transmon/Xmon qubit measurement, the Purcell decay, and the corresponding measurement error without the Purcell filter.

## Ii Idea of a bandpass Purcell filter and semiclassical analysis

In the standard cQED setup of dispersive measurement (Fig. 1) the qubit interaction with the resonator slightly changes the effective resonator frequency depending on the qubit state, so that it is when the qubit is in the excited state and when the qubit is in the ground state. The dispersive coupling is defined as

 χ≡(ω|e⟩r−ω|g⟩r)/2. (1)

In the two-level approximation for the qubit, , where is the qubit-resonator coupling and and are the bare frequencies of the qubit and the resonator, respectively Blais-04 (). For a transmon or an Xmon qubit, is usually significantly smaller, , where is the qubit anharmonicity (); moreover, as well as the central frequency depend on the number of photons in the resonator (see Koch-07 (); Boisson-10 () and the Appendix for a more detailed discussion). The resonator frequency change (and thus the qubit state) is sensed by applying the microwave field with amplitude , then amplifying the transmitted or reflected signal, and then mixing it with the applied microwave field to measure its phase and amplitude (Fig. 1).

In the process of measurement, the qubit decays with the Purcell rate Blais-04 ()

 Γ≈κg2(ωq−ωr)2, (2)

where is the resonator energy damping rate (mostly due to leakage into the transmission line – see Fig. 1). Note that in this formula we do not distinguish the bare and effective frequencies. In the quantum language this can be interpreted as the leakage with the rate of the qubit “tail” , existing in the form of the resonator photon. However, the Purcell decay also has a simple classical interpretation via the resistive damping Esteve-86 (), essentially being a linear effect, in contrast to the dispersion (1) – see Appendix for more details, including dependence of on Set14 ().

The Purcell decay leads to measurement error; therefore, it is important to reduce the rate . This can be done by decreasing the ratio ; however, this decreases and thus increases the necessary measurement time (see Appendix for more details). Another way to decrease is to use a very small leakage rate ; however, this also increases the measurement time because the ring-up and ring-down processes give a natural limitation , and in many practical cases it is even .

It would be good if the rate which governs the measurement time were different from in Eq. (2): specifically if for the Purcell decay were much smaller than for the measurement. This is exactly what is achieved by using the bandpass filter of Ref. Jeffrey-14 (). There are other ways to explain how this Purcell filter works Jeffrey-14 (), but here we interpret the main idea of the bandpass Purcell filter as producing different effective rates for the measurement and for the Purcell decay (so that the measurement microwave easily passes through the filter, while the propagation of the photon emitted by the qubit is strongly impeded by the filter).

The schematic of the qubit measurement with the bandpass Purcell filter of Ref. Jeffrey-14 () is shown in Fig. 2. Besides the readout resonator with qubit-state-dependent frequency or , there is a second (filter) resonator with frequency , coupled with the readout resonator with the coupling . (The coupling is inductive, but we draw it as capacitive to keep the figure simple.) The filter resonator leaks the microwave into the transmission line with a relatively large damping rate , so that its -factor is , while . The leaked field is then amplified and sent to the mixer (not shown) in the same way as in the standard cQED setup. The readout and filter resonators are in general detuned from each other, but not much, (detuning is needed to multiplex readout of several qubits using the same filter resonator Jeffrey-14 (); Kelly-15 (); for simplicity we consider the measurement of only one qubit). The filter resonator is pumped with the drive frequency (close to ) and amplitude . However, for us it will be easier to first assume instead that the readout resonator is pumped with amplitude (Fig. 2), and then show the correspondence between the drives and .

Let us use the rotating wave approximation Allen-Eberly-book (); RWA-note () with the rotating frame based on the drive frequency . Then the evolution of the classical field amplitudes and in the readout and filter resonators, respectively, is given by the equations

 ˙α=−iΔrdα−iGβ−iεr, (3) ˙β=−iΔfdβ−iG∗α−κf2β, (4)

where and are normalized so that and are the average number of photons in the resonators, is normalized correspondingly, and

 Δrd=ωr−ωd,Δfd=ωf−ωd (5)

(recall that depends on the qubit state). If we are not interested in the details of evolution on the fast time scale , then we can use the quasisteady state for [obtained from Eq. (4) using ],

 β=−iG∗κf/2+iΔfdα, (6)

which can then be inserted into Eq. (3), giving

 ˙α=−i(Δrd+δωr)α−κeff2α−iεr, (7) κeff=4|G|2κf11+(2Δfd/κf)2, (8) δωr=−|G|2Δfd(κf/2)2+Δ2fd=−Δfdκfκeff. (9)

Thus we see that the field evolves in practically the same way as in the standard setup of Fig. 1; however, interaction with the filter resonator shifts the readout resonator frequency by and introduces the effective leakage rate of the readout resonator.

Most importantly, depends on the drive frequency. For measurement we use , so is approximately

 κr≡4|G|2κf11+[2(ωr−ωf)/κf]2. (10)

However, when the qubit tries to leak its excitation through the readout resonator, this can be considered as a drive at the qubit frequency, , and the corresponding is then

 κq≡4|G|2κf11+[2(ωq−ωf)/κf]2, (11)

which is much smaller than if the qubit is detuned away from the filter linewidth, . This difference is exactly what we wished for suppressing the Purcell rate : the measurement is governed by , while the qubit sees a much smaller value . Therefore, we would expect that the Purcell rate is given by Eq. (2) with [see Eq. (32) later], while the “separation” measurement error is given by Eqs. (82)–(84) with (see Appendix). As a result, compared with the standard setup (Fig. 1) with the same physical parameters for measurement, the Purcell rate is suppressed by the factor

 F=κqκr=1+[2(ωr−ωf)/κf]21+[2(ωq−ωf)/κf]2≪1. (12)

This is essentially the main result of this paper, which will be confirmed by the quantum analysis in the next section. (To avoid a possible confusion, we note that is not the qubit decay rate; it is the resonator decay, as seen by the qubit.)

Our result for the Purcell suppression factor was based on the behavior of the field amplitude in the readout resonator. Let us also check that the field propagating in the outgoing transmission line behaves according to the effective model as well. The outgoing field amplitude is (in the normalization for which is the average number of propagating photons per second). Using Eq. (6), we find

 γtl=−iG∗√κfκf/2+iΔfdα=eiφ√κeffα, (13)

so, as expected, the outgoing amplitude behaves as in the standard setup of Fig. 1 with , up to an unimportant phase shift . Note that to show the equivalence between the dynamics (including transients) of the systems in Figs. 1 and 2 we needed the assumption of a sufficiently large in order to use the quasisteady state (6). However, this assumption is not needed if we consider only the steady state (without transients).

So far we assumed that the measurement is performed by driving the readout resonator with the amplitude . Now let us consider the realistic case Jeffrey-14 (); Kelly-15 () when the drive is applied to the filter resonator. The evolution equations (3) and (4) for the classical field amplitudes are then replaced by

 ˙α=−iΔrdα−iGβ, (14) ˙β=−iΔfdβ−iG∗α−κf2β−iεf, (15)

so that the quasisteady state for the filter resonator is

 β=−iG∗κf/2+iΔfdα+−iεfκf/2+iΔfd, (16)

and the field evolution in the readout resonator is

 ˙α=−i(Δrd+δωr)α−κeff2α−Gκf/2+iΔfdεf, (17)

with the same and given by Eqs. (8) and (9). The only difference between the effective evolution equations (7) and (17) is a linear relation,

 εr↔−iεfG/(κf/2+iΔfd), (18)

between the drive amplitudes and producing the same effect. Therefore, our results obtained above remain unchanged for driving the filter resonator, and the Purcell rate suppression factor is still given by Eq. (12).

Note that in the quasisteady state the separation between the filter amplitudes for the two qubit states does not depend on whether the drive is applied to the filter or readout resonator, as long as we use the correspondence (18) between the drive amplitudes. The same is true for the separation between the outgoing fields . Similarly, the separation between the outgoing fields for the two qubit states is the same (up to the phase ) as in the standard setup of Fig. 1 with , , and the resonator frequency adjusted by given by Eq. (9). Therefore, these configurations are equivalent to each other from the point of view of quantum measurement, including interaction between the qubit and readout resonator, extraction of quantum information, back-action, etc.

Nevertheless, driving the filter resonator produces a different outgoing field , which now contains an additional term in comparison with Eq. (13), which comes from the second term in Eq. (16). In particular, instead of the Lorentzian line shape of the transfer function when driving the readout resonator, the transfer function for driving the filter is (in the steady state)

 γ(f)tlεf=√κfκf/2+iΔfd2Δrd/κeff1+2i(Δrd+δωr)κeff, (19)

where can be replaced with . (Note a non-standard normalization of the transfer function because of different normalizations of and .) This line shape for the amplitude shows a dip near (note that at ) and is significantly asymmetric when is comparable to ; this occurs when the detuning between the readout and filter resonators is comparable to – see Eq. (9). In terms of the field in the readout resonator, the outgoing field at steady state is

 γ(f)tl=−√κfΔrdGα (20)

The difference between the outgoing fields and when driving the filter or readout resonator (for the same , i.e., the same measurement conditions) may be important for saturation of the microwave amplifier. The ratio of the corresponding outgoing powers is

 |γ(f)tl|2|γ(r)tl|2=(Δrdκr)241+(2Δfd/κf)2, (21)

where we assumed (so that ). For example, if the drive frequency is chosen as , then ; if in this case , then driving the filter resonator is advantageous because it produces less power to be amplified, while driving the readout resonator is advantageous if . However, when , the situation is more complicated.

Figure 3 shows the transient phase-space evolution of the field (coherent state) in the filter resonator when a step-function drive is applied to the readout resonator (red curves) or to the filter resonator (blue curves), with the drive amplitudes related via Eq. (18), and for real . The solid curves show the evolution when the qubit is in the excited state, while the dashed curves are for the qubit in the ground state. We have used parameters similar to the experimental parameters of Ref. Jeffrey-14 (): GHz, GHz (so that MHz), GHz, (so that ns), and ns (so that MHz). The field evolution is calculated using either Eqs. (3)–(4) or Eqs. (14)–(15); in simulations we have neglected the dependence of and on the average number of photons in the readout resonator (see Appendix). The black dots indicate the time moments every 10 ns between 0 and 100 ns, and then every 50 ns. The circles illustrate the coherent state error circles Scully-book () in the steady state. We see that when the drive is applied to the filter resonator (blue curves), the evolution is initially very fast (governed by ), while after the quasisteady state is reached, the evolution is governed by a much slower , eventually approaching the steady state. When the drive is applied to the readout resonator (red curves), the transient evolution is always governed by the slower decay .

In Fig. 3(a) we choose the drive frequency symmetrically from the point of view of the readout resonator field, so that in the steady state , where and are the average photon numbers for the two qubit states. For that we need with given by Eq. (9); for our parameters MHz, so GHz. Such symmetric choice of the drive frequency provides the largest separation between the two coherent states for a fixed drive amplitude if . While the field in the readout resonator is always symmetric in this case, Fig. 3(a) shows that the field in the filter resonator is symmetric only when driving the readout resonator (red curves, ), while it is strongly asymmetric when the filter resonator is driven (blue curves, , ; is very small because for our parameters and therefore ). The number of photons in the filter is much less than in the readout resonator because .

In Fig. 3(b) we choose so that in the steady state for driving the filter; this is the natural choice for decreasing the microwave power to be amplified. This occurs at , which is close to the expected value , but not equal because of the asymmetry of the line shape (19). We choose the amplitudes to produce (then ). The difference between and leads to different values and when driving the readout resonator, while for driving the filter the field in the filter is symmetric, . Compared to the case of Fig. 3(a), there is 5 times less power to be amplified for the state (when driving the filter); however, the state separation is 1.3 times smaller (in amplitude) for the same . Thus, there is a trade-off between the state separation and amplified power in choosing the drive frequency. Comparing the red and blue curves in Fig. 3(b), we see that in the steady state and are smaller for driving the filter rather than the readout resonator. This is beneficial because there is less power to be amplified; however, the ratio is not very big (as expected for a moderate value ).

Note that our definition of in Eq. (10) is not strictly well-defined because the resonator frequency depends on the qubit state, and the drive frequency can be in between and . However, this frequency difference is much smaller than , and therefore not important for practical purposes in the definition of . In an experiment can be measured either via the field decay Jeffrey-14 () or via the linewidth of the steady-state transfer function showing the dip of near the resonance [Eq. (19)], since near the dip .

Thus far we assumed that all decay in the filter resonator is due to the leakage into the outgoing transmission line. If and the decay is due to leakage into the line delivering the drive or due to another dissipation channel, then the only difference compared to the previous discussion is the extra factor for the outgoing field . This will lead to multiplication of the overall quantum efficiency of the measurement by and will only slightly affect the measurement fidelity. Adding dissipation in the readout resonator with rate increases the effective linewidth to and multiplies the quantum efficiency by . Most importantly, since does not change with frequency, the Purcell suppression factor (12) becomes , so that the Purcell filter performance deteriorates; we will discuss this in a little more detail in Sec. IIIC.

Note that our main result (12) for the Purcell suppression factor is slightly different from the result , which was derived in Ref. Jeffrey-14 () using the circuit theory. The reasons are the following. First, in the derivation of Jeffrey-14 () it was assumed that the two resonators have the same frequency, which makes the numerator in Eq. (12) equal to 1. Second, the term 1 in the denominator in Eq. (12) was essentially neglected in comparison with the larger second term. Finally, the role of the factor is not quite clear. In the derivation of Ref. Jeffrey-14 () keeping this factor was exceeding the accuracy of the derivation, while in our derivation we essentially use the rotating wave approximation, which assumes . Aside from these small differences, our result coincides with the result of Ref. Jeffrey-14 ().

## Iii Quantum analysis in single-excitation subspace

In this section we discuss the quantum derivation of the Purcell rate in the presence of the bandpass filter in the regime when the resonators are not driven or driven sufficiently weakly to neglect dependence of the Purcell rate on the number of photons in the resonator Set14 (). More precisely, we consider the quantum evolution in the single-excitation (and zero-excitation) subspace. We apply two methods: the wavefunction approach, in which we use a non-Hermitian Hamiltonian with a decaying wavefunction, and the more traditional density matrix analysis.

In the absence of the drive and in the rotating wave approximation, the relevant Hamiltonian of the system shown in Fig. 2 (without considering decay ) is ()

 H=ωbqσ+σ−+ωbra†a+ωfb†b+g(a†σ−+aσ+) +Ga†b+G∗ab†, (22)

where is the bare qubit frequency, is the bare frequency of the readout resonator, is the filter resonator frequency, raising/lowering operators and act on the qubit state, and are the creation and annihilation operators for the readout resonator, and are for the filter resonator, is the qubit-readout resonator coupling, and is the resonator-resonator coupling. For simplicity we assume a real positive , but can be complex for generality (for the capacitive or inductive coupling between the resonators, is real if the same generalized coordinates are used for both resonators).

Note that in the case without drive it is sufficient to consider only two levels for the qubit because only the single-excitation (and zero-excitation) subspace is involved in the evolution, and therefore the amount of qubit nonlinearity due to the Josephson junction is irrelevant. However, in the presence of a drive (considered in the next section) it is formally necessary to take into account several levels in the qubit (as done in the Appendix). Nevertheless, to leading order the Purcell rate is insensitive to this, because the Purcell decay is essentially a classical linear effect (see discussion in the Appendix). Also note that the lab-frame Hamiltonian (22) assumes the rotating wave approximation (as in the standard Jaynes-Cummings Hamiltonian), since it neglects the “counter-rotating” terms of the form , , , and . This requires assumption that , , , and are small compared to .

Let us choose the rotating frame with frequency , i.e., ; then the interaction Hamiltonian is

 V=Δrqa†a+Δfqb†b+g(a†σ−+aσ+)+Ga†b+G∗ab†, (23)

where

 Δrq=ωbr−ωbq,Δfq=ωf−ωbq, (24)

and the interaction picture is equivalent to the Schrödinger picture because , which is because the starting Hamiltonian (22) already assumes the rotating-wave approximation. The master equation for the density matrix , which includes the damping of the filter resonator is

 ˙ρ=−i[V,ρ]+κf(bρb†−b†bρ/2−ρb†b/2). (25)

In general, the bare basis is , where represents the qubit states, while and represent the readout and filter resonator Fock states, respectively. However, in this section we consider only the single-excitation (and zero-excitation) subspace, so only four bare states are relevant: , , , and .

Note that the interaction hybridizes the bare states of the qubit and the resonators. (Hybridization of the readout resonator mode is essentially what makes the qubit measurement possible.) Therefore, when discussing the Purcell rate for the qubit energy relaxation, we actually mean decay of the eigenstate, corresponding to the qubit excited state. This makes perfect sense experimentally, since manipulations of the qubit state usually occur in the eigenbasis (adiabatically, compared with the qubit detuning from the resonator).

### iii.1 Method I: Decaying wavefunction

Instead of using the traditional density matrix language for the description of the Purcell effect Haroche-book (), it is easier to use the language of wavefunctions, even in the presence of the decay Set14 (). Physically, the wave functions can still be used because in the single-excitation subspace unraveling of the Lindblad equation corresponds to only one “no relaxation” scenario (see, e.g., Kor-13 ()), and therefore the wavefunction evolution is non-stochastic. Another, more formal way to introduce this language, is to rewrite the master equation (25) as Pie07 (); Car93 () , where is an effective non-Hermitian Hamiltonian. Next, the term can be neglected because in the single-excitation subspace it produces only an “incoming” contribution from higher-excitation subspaces, which are not populated. Therefore, in the single-excitation subspace we can use . Equivalently, , which describes the evolution of the decaying wavefunction . Therefore, the probability amplitudes satisfy the following equations:

 ˙ce=−igcr, (26) ˙cr=−iΔrqcr−igc%e−iGcf, (27) ˙cf=−iΔfqcf−iG∗cr−(κf/2)cf, (28)

while the population of the zero-excitation state evolves as or can be found as . Note that Eqs. (27) and (28) exactly correspond to the classical equations (3) and (4) with replaced with , also replaced with , and replaced with .

From the eigenvalues of the matrix representing Eqs. (26)–(28), one can obtain the eigenfrequencies and the corresponding decay rates . These eigenvalues can be found from the qubic equation

 λ3+λ2(iΔrq+iΔfq+κf/2)+λ(−ΔrqΔfq+|G|2+g2 +iΔrqκf/2)+g2(iΔfq+κf/2)=0. (29)

We are interested in the Purcell rate , which corresponds to the decay of the eigenstate close to . Since is close to zero, in the first approximation we can neglect the term in Eq. (III.1), thus reducing it to the quadratic equation. If more accuracy is needed, the equation can be solved iteratively, replacing with the value found in the previous iteration (the second iteration is usually sufficient).

Besides finding the Purcell rate exactly or approximately from Eq. (III.1), we can find it approximately by using quasisteady solutions of Eqs. (27) and (28), to a large extent following the classical derivation in the previous section. Assuming in Eq. (28), we find . Inserting this quasisteady value into Eq. (27) and assuming , we find . Substituting this quasisteady value into Eq. (26), we obtain

 ˙ce=−g2iΔrq+|G|2/(iΔfq+κf/2)ce=λece. (30)

Finally, we obtain the Purcell rate as ,

 Γ=g2|G|2κfΔ2rq[(Δfq−|G|2/Δrq)2+(κf/2)2] (31) ≈g2|G|2κfΔ2rq[Δ2fq+(κf/2)2]=g2κqΔ2rq, (32)

where is given by Eq. (11) and we assumed to transform Eq. (31) into Eq. (32).

The Purcell rate given by Eq. (32) is exactly what we expected from the classical analysis in Sec. II: in the usual formula (2) we simply need to substitute with the readout resonator decay rate seen by the qubit. Since the measurement is governed by a different decay rate , the effective Purcell rate suppression factor is given by Eq. (12), as was expected. This confirms the results of the classical analysis in Sec. II.

### iii.2 Method II: Density matrix analysis

We can also find the Purcell rate in a more traditional way by writing the master equation (25) explicitly in the single-excitation subspace:

 ˙ρee=ig(ρer−ρre), (33) ˙ρer=−iΔqrρer−ig(ρrr−ρee)+iG∗ρef, (34) ˙ρef=−κf2ρ% ef−iΔqfρef+iGρer−igρrf, (35) ˙ρrr=−ig(ρer−ρre)−iGρfr+iG∗ρrf, (36) ˙ρrf=−κf2ρ% rf+iΔfrρrf−iGρff+iG∗ρrr−igρef, (37) ˙ρff=−κfρff−iG∗ρrf+iGρfr. (38)

Note that and (however, we do not use these two equations for the derivation of the Purcell rate).

Using the quasisteady solutions of Eqs. (34)–(38), i.e. assuming , we can obtain a lengthy equation for , which is proportional to . If we use the first-order expansion of this equation in the coupling and neglect terms (there is no contribution), then

 ρer=igiΔrq+|G|2/(iΔfq+κf/2)ρee, (39)

which has the form similar to Eq. (30). Substituting this into Eq. (33), we obtain the evolution equation with given exactly by Eq. (31). If we do not use the above-mentioned approximation for the quasisteady , then the result for the Purcell rate is slightly different and much lengthier,

 Γ=g2|G|2κf[(ΔrqΔfq−|G|2)2+(Δ2rq+g2)(κf/2)2 +g2(Δ2fq+2ΔfqΔrq−|G|2)+g4]−1. (40)

Thus the derivations based on the wavefunction and density matrix languages using the quasisteady-state approximation both lead to practically the same result for the Purcell rate . The most physically transparent result is given by Eq. (32), which corresponds to the semiclassical analysis in Sec. II and simply replaces in Eq. (2) with seen by the qubit, in contrast to the measurement process, which is governed by .

As an example, let us use the parameters similar to that in Ref. Jeffrey-14 (): GHz, GHz, GHz, (so that ns), MHz, and ns (so that MHz). In this case the resonator decay [Eq. (11)] seen by the qubit is s, the Purcell rate [Eq. (32)] is , and the Purcell rate suppression factor [Eq. (12)] is .

Thus, for typical parameters the bandpass Purcell filter suppresses the Purcell decay by a factor of 50. It is easy to increase this factor to 100 by using GHz in the above example; however, further decrease of the Purcell rate is not needed for practical purposes, while increased resonator-qubit detuning decreases the dispersive shift (in the above example MHz for the qubit anharmonicity of 180 MHz, while for GHz the dispersive shift becomes twice less).

Note that for the parameters in the above example, Eq. (32) overestimates the exact solution for via Eq. (III.1) by 5%, the same 5% for Eq. (31), and Eq. (40) overestimates the Purcell rate by 2%. The solution of Eq. (III.1) as a quadratic equation neglecting gives , which overestimates the exact solution by 22%, while the second iteration is practically exact (). The inaccuracies grow for smaller resonator-qubit detuning (crudely as ), but remain reasonably small in a sufficiently wide range; for example, Eq. (32) overestimates the Purcell rate by for GHz, i.e. detuning of 0.3 GHz.

### iii.3 Nonzero readout resonator damping

In the quantum evolution model (25) we have considered only the damping of the filter resonator with the rate . If there is also an additional energy dissipation in the readout resonator with the rate (e.g., due to coupling with the transmission line delivering the drive in Fig. 2), then the master equation (25) should be replaced with

 ˙ρ= −i[V,ρ]+κf(bρb†−b†bρ/2−ρb†b/2) +κr,d(aρa†−a†aρ/2−ρa†a/2). (41)

In the wavefunction-language derivation this leads to the extra term in Eq. (27) for . This does not change the quasisteady value for but changes the quasisteady value , so that the Purcell rate is

 Γ=2Re[g2iΔrq+|G|2/(iΔfq+κf/2)+κr,d/2] (42) ≈g2(κq+κr,d)Δ2rq (43)

instead of Eq. (32). Practically the same result can be obtained using the derivation via the density matrix evolution (assuming ). As expected, the dissipation simply adds to the rate seen by the qubit. Since is not affected by the filter, it adds in the same way to the bandwidth governing the qubit measurement process and thus deteriorates the Purcell rate suppression (12), replacing it with .

## Iv Purcell rate with microwave drive and bandpass filter

The Purcell rate may decrease when the measurement microwave drive is added Set14 (). A simple physical reason is the ac Stark shift, which in the typical setup increases the absolute value of detuning between the qubit and readout resonator with increasing number of photons in the resonator, thus reducing the Purcell rate. However, this explanation may not necessarily work well quantitatively.

The Purcell rate suppression due to the microwave drive was analyzed in Ref. Set14 () for the case without the Purcell filter and using the two-level approximation for the qubit. It was shown that in this case the suppression factor is approximately instead of the factor expected from the ac Stark shift, where is the mean number of photons in the resonator and . This difference results in the ratio between the corresponding slopes of at small , with the ac Stark shift model underestimating the Purcell rate suppression (see the blue lines in Fig. 4). However, when the third level of the qubit is taken into account, then the ac Stark shift model describes correctly the slope of at small when the qubit anharmonicity is relatively small, (see Appendix). In this case the ac Stark shift model predicts at , where is the value of at ; note that and when .

With the filter resonator, we also expect that the ac Stark shift model for the Purcell rate suppression should work reasonably well, so that from Eq. (32) we expect

 Γ(¯n)≈g2|G|2κf[ωr−ωq,eff(¯n)]2{[ωf−ωq,eff(¯n)]2+(κf/2)2}, (44)

where is the effective qubit frequency, if we neglect dependence of on and the “Lamb shift”. Therefore, in a typical situation when and , we expect the suppression ratio

 Γ(¯n)Γ(0)≈[ωr−ωbqωr−ωq,eff(¯n)]4. (45)