Dispersive regime of circuit QED: photon-dependent qubit dephasing and relaxation rates

I Introduction

Cavity quantum electrodynamics (QED) is a unique tool to study the interaction between light and matter at its most fundamental level raimond:2001a ; mabuchi:2002a . Most interesting is the regime of strong coupling where the frequency associated with light-matter interaction is greater than all relaxation rates thompson:1992a ; boca:2004a . For example, in this regime, experiments with very high-Q cavities have been able to resolve quantum jumps and time-resolved collapse of the cavity field gleyzes:2007a ; guerlin:2007a .

With the strong coupling regime easily accessible, superconducting electrical circuits offer distinctive advantages to the study of light-matter interaction buisson:2001a ; marquardt:2001a ; al-saidi:2001a ; plastina:2003a ; blais:2003d ; you:2003a ; yang:2003a ; blais:2004a , something which has been realized experimentally with charge wallraff:2004a , flux chiorescu:2004a ; johansson:2006 , and phase sillanpaa:2007a superconducting qubits. Although the present work applies to all physical realizations of cavity or circuit QED, here we will focus on superconducting charge qubits coupled to a transmission line resonator blais:2004a ; wallraff:2004a . Because the qubit can be very strongly coupled to the transmission line in this system, it opens the possibility to study new regimes of cavity QED. For example, the strong dispersive regime was theoretically studied in Ref. gambetta:2006a and experimentally investigated in Ref. schuster:2007a .

Here, this is done by pushing the dispersive approximation used in Ref. blais:2004a ; gambetta:2008a to higher order. These results not only apply to circuit QED but also to cavity QED and more generally to any physical situation where a two-level system is dressed by an oscillator. Examples are quatronium siddiqi:2006a ; boulant:2007a or flux qubits lupasu:2006a ; lupascu:2007a ; picot:2008a coupled to bifurcating oscillators for readout purposes. In particular, the authors of Ref. picot:2008a find that the qubit relaxation rate is strongly enhanced when the non-linear oscillator is in its high-amplitude state compared to its low-amplitude state, results which are at least qualitatively consistent with those presented here.

In Sec. VII a quantum trajectory equation describing the evolution of the qubit and resonator under homodyne measurement is obtained, as well as a reduced qubit quantum trajectory equation. This is used to investigate the measurement, where we show that the achievable SNR is decreased substantially by the non-linear effects.

II Circuit QED

In circuit QED, a superconducting charge qubit is fabricated inside a transmission line resonator. This system is illustrated in Fig. 1. Focussing on a single mode of the resonator, the system Hamiltonian describing this circuit takes the Jaynes-Cummings form blais:2004a

In this expression, ${\omega_{r}}$ is the frequency of the mode of interest of the resonator, ${\omega_{a}}$ the qubit transition frequency and $g$ the qubit-resonator coupling. The operators $a^{({\dagger})}$ and $\sigma_{\pm}$ are the creation and annihilation operators for the photon field and the qubit.

Logical operations and readout of the qubit can be achieved by applying a microwave signal on the input port of the resonator. Choosing a frequency that is close to the resonator frequency ${\omega_{r}}$ corresponds to a readout of the qubit’s state, while frequencies that are close to ${\omega_{a}}$ can be used to control the qubit blais:2004a ; gambetta:2006a ; blais:2007a . This can be modeled by the Hamiltonian

The effect of coupling to environmental degrees of freedom can be described by the master equation gardiner:2004b

where $H$ is the total Hamiltonian of the system including drives

and ${{\mathcal{D}}}[L]{\varrho}=\left(2L{\varrho}L^{\dagger}-L^{\dagger}L{\varrho}-{\varrho}L^{\dagger}L\right)/2$. In the above expression, $\kappa$ is the resonator rate of photon loss, $\gamma_{1}$ the qubit energy decay rate and $\gamma_{\varphi}$ is the qubit rate of pure dephasing.

This master equation is obtained in the Markov approximation which assumes that the spectral density of the environment is frequency-independent. For high-quality factor systems, like high-Q transmission line resonators or (most) superconducting qubits, this approximation is accurate as the system is probing the environment in a very small frequency bandwidth. As we will see in Sec. IV, when going to the dispersive approximation, it can be important to take into account the frequency dependence of the environment.

III Dispersive Effects on the Hamiltonian

In the limit that detunning between the cavity and the qubit is large, no energy is exchange. In this situation, the interaction is said to be dispersive. In analyzing this interaction, it is convenient to diagonalize the Jaynes-Cummings Hamiltonian Eq. (1) by using a unitary transformation. In this transformed frame, the new effective qubit and photon operators are combinations of the bare qubit and photon operators. In this sense, the qubit acquires “a photon part” and vice versa. This leads to, for example, the Purcell effect where a qubit can decay through the photon decay channel houck:2008a ; purcell:1946a .

In the limit where $|\Delta|\equiv|{\omega_{a}}-{\omega_{r}}|\gg g$, the Jaynes-Cummings Hamiltonian (1) can be approximately diagonalized using the unitary transformation

with $\lambda=g/\Delta$ a small parameter. Using the relation

to second order in $\lambda$, it is simple to obtain the effective Hamiltonian describing the dispersive regime

The qubit transition frequency is shifted by a quantity proportional to the photon population $2g\lambda\left\langle{{a^{\dagger}}a}\right\rangle$. Alternatively, this shift can be seen as a qubit dependent pull of the resonator frequency $\omega_{r}\rightarrow\omega_{r}\pm g\lambda$.

As a result, shinning microwaves at the input port of the resonator at a frequency close to $\omega_{r}$ and measuring the transmitted signal using standard homodyne techniques serves as measurement of the qubit blais:2004a ; gambetta:2006a ; schuster:2005a ; wallraff:2005a ; gambetta:2008a . In this approximation, this corresponds to a quantum non-demolition (QND) measurement of the qubit gambetta:2006a ; gambetta:2008a .

To see the breakdown of the linear approximation, we have numerically calculated the time-dependent evolution of the system master equation under measurement (see Fig. 2 for details) both in the linear dispersive approximation, using the approach described in Ref. gambetta:2008a , and with the full non-dispersive Jaynes-Cummings model. To compare these results we plot the trace distance

where $\rho_{1}$ is the reduced qubit density matrix found using the linear dispersive model presented in Ref gambetta:2008a , and ${\rm Tr_{r}}\left({\varrho}_{2}\right)$ is the trace over the resonator of the total density matrix of the system found by simulation of the complete master equation (5). This trace distance is the geometrical distance between two Bloch vectors, and ranges from to $2$, with when the two qubit states are the same and $2$ when they are opposite on the Bloch sphere.

We plot, in figure 2 a), the trace distance for a measurement amplitude which is slowly turned on to reach an amplitude corresponding to $\bar{n}\approx 0.08n_{\rm crit.}$ photons in the resonator. Even for this small number of measurement photons (compared to $n_{\rm crit.}$) the trace distance is non-negligible which implies the breakdown of the dispersive approximation. Moreover, we plot in figure 2b) the maximum of the trace distance (over the simulation time), as a function of the maximum measurement amplitude ${\epsilon_{m}}$. Clearly, the trace distance gets worse as the amplitude is increased. The importance of this effect depends on the various parameters entering in the simulation but the results shown here are typical. It is clear from these numerical results, it is important to take into account higher order terms in the dispersive approximation.

III.2 Dispersive Jaynes-Cummings Hamiltonian: Exact transformation

Following the derivation presented in Appendix A, which is similar in spirit to the approach used in Refs carbonaro:1979a ; lo:1998a , we find the unitary transformation that diagonalizes the Jaynes-Cummings Hamiltonian $H_{s}$

$N_{q}$ is an operator representing the total number of excitations, and $\Pi_{e}$ is the projector on the qubit excited state. Applying this transformation to $H_{s}$ yields

As it should, the eigenenergies of this Hamiltonian are the same as those presented in Ref. blais:2004a if $n$ is taken as the eigenvalues of $N_{q}$ and each eigenenergy is shifted by a constant $\hbar{\omega_{r}}$.

In this basis, the qubit is dressed by the field. As a result, qubit operators acquire photon part and similarly for field operators. For example, under the transformation ${\mathbf{D}}$, $\sigma_{z}$ and ${{a^{\dagger}}a}$ become

Both these operators now involve the off-diagonal operator ${I_{+}}$.

Expanding Eq. (14) to third order in $\lambda=g/\Delta$ [one order up from Eq. (9)] we find

is the modified value of Lamb and Stark shift per photon. In addition to a correction to these values, the third order expansion yields a squeezing term $({{a^{\dagger}}a})^{2}$ of amplitude $\zeta=-g^{4}/\Delta^{3}$.

III.3 The drive Hamiltonian under the exact transformation

It is important not only to transform $H_{s}$ but also the drive Hamiltonian $H_{d}$. To do so, we first consider how the qubit and field ladder operators are transformed under ${\mathbf{D}}$. Contrary to $\sigma_{z}$ and $N$, the transformation does not lead to a compact result. To order $\mathcal{O}(\lambda^{3})$ for $a$ and $\mathcal{O}(\lambda^{2})$ for $\sigma_{-}$, we find

such that the drive Hamiltonian Eq. (4) becomes

With $\omega_{k}\sim\omega_{r}$, the first line of the above equation is responsible for measurement of the qubit. Due to the $\lambda^{2}$ term, the effective measurement drive strength is affected by the state of the qubit. It will be slightly larger or smaller depending on the qubit being in its excited or ground state. As will be shown later, this lead to small corrections to the ac-Stark shifted qubit transition frequency and measurement-induced dephasing rate. Moreover, choosing $\omega_{k}\sim\omega_{a}$, one could take advantage of the second line of Eq. (21) to coherently control the qubit. Again due to a $\lambda^{2}$ term, the effective strength of this control drive will be modulated by the number of photons in the cavity.

IV Dispersive Effect on the Master Equation

To obtain a complete description of the system in the dispersive regime, we also need to apply the dispersive transformation to the bath-system coupling. In principle, this can be done by transforming the operators entering the dissipative terms of the Lindblad master equation (5). Once transformed, these terms will typically involve both qubit and field operators which correspond to probing the environment at different frequencies than the untransformed dissipative terms. Since the master equation is obtained in the Markov approximation, this frequency information is lost.

Here, we go beyond this approximation by rederiving the qubit-resonator master equation. We first apply the dispersive transformation on the system-bath Hamiltonian and then trace out the bath degrees of freedom to finally obtain a master equation in the dispersive frame.

Energy damping of the resonator ($\kappa$) and of the qubit ($\gamma$) can be modelled by coupling to baths of harmonic oscillators with free Hamiltonians gardiner:2004b

where $b^{\dagger}_{\kappa,\gamma}(\omega)$ and $b_{\kappa,\gamma}(\omega)$ respectively create and annihilate an excitation of frequency $\omega$ in the resonator or qubit bath. Coupling to these baths is described by gardiner:2004b

where $d_{i}(\omega)$ is the density of modes of bath $i$ and $f_{i}(\omega)$ represents the coupling strength of the mode of frequency $\omega$ to the resonator or the qubit.

Dephasing in the bare basis occurs due to slow fluctuations of the qubit transition frequency. For example, in a superconducting charge qubit this is primarily caused by charge noise astafiev:2004a ; astafiev:2006a . Dephasing can be modeled by adding the Hamiltonian

In this expression, $f_{\varphi}(t)$ is a random function of time with zero mean and $\nu$ is characteristic of the magnitude of the coupling of the qubit to the fluctuations. Defining $f_{\varphi}(t)=\int_{-\infty}^{\infty}f_{\varphi}(\omega)e^{i\omega t}d\omega$, $H_{\varphi}$ can be written in frequency space as

IV.2 Dispersive master equation

As shown in appendix B, applying the dispersive transformation on the above system-bath Hamiltonians and integrating out the bath degrees of freedom leads to the master equation

where we have defined the rates $\kappa=\kappa_{r}$, $\gamma_{\kappa}=\lambda^{2}\kappa_{a}$, $\gamma=\gamma_{a}$, $\kappa_{\gamma}=\lambda^{2}\gamma_{r}$ with

In obtaining these result, we have taken into account the fact that the spectral weight of the environment can be non-white. As a result, although we obtain a Markovian master equation, the rates depend explicitly on the qubit and resonator environments at different frequencies. As is explained in appendix B, in obtaining these results it was assumed that noise at the various relevant frequencies are independent. This assumption is valid if the noise is relatively weak and the various frequencies entering the expression for the rates are well separated one from another. For superconducting charge qubits which experimentally show long coherence times (up to $2~{}\mu$s Schreier:2008a ) and in the dispersive regime (where $\Delta$, and thus the frequency separation, is $10$ GHz), the above model is accurate.

V Effective qubit master equation: Eliminiation of the Cavity

In this section, we eliminate the resonator degree of freedom from Eq. (26) to obtain a master equation for the reduced qubit density matrix in the dispersive frame. Building on Ref. gambetta:2008a , this is done by first moving to a rotating frame for the cavity, and then using a polaron-type transformation to displace the cavity field back to the vacuum. From this frame it is possible to consider only the two classical fields $\alpha_{e}$ and $\alpha_{g}$. These fields correspond to the average value $\left\langle a\right\rangle$ of the cavity field if the qubit is in the excited or ground state.

The resulting master equation is valid as long as the resonator state does not deviate too much from a superposition of coherent state. This can be formalized by two requirements. The first is

where $n_{\kappa}$ is the ratio of the rate $\zeta$ at which the non-linearity is squeezing the resonator state and the rate $\kappa$ at which these deviations are taken back to coherents state by damping. The second requirement is

where ${\gamma_{\downarrow}}$ and ${\gamma_{\uparrow}}$ are given in Eqs (34) and (35) and are the rates at which the superposition of the coherent states $\alpha_{e}$ and $\alpha_{g}$ are getting mixed. This condition implies that the rate at which the superposition of the coherent states gets mixed is much slower than the rate of photon loss.

As derived in appendix C, in a frame rotating at $\omega_{m}$ for the resonator, the effective qubit master equation is

where ${\rho}={\rm Tr_{r}}\left({\varrho}\right)$ is the reduced density matrix of the qubit. The parameters introduced in this master equation are

where the classical parts of the field $\alpha_{g}$ and $\alpha_{e}$ (pointer states) satisfy $(i\in\{e,g\})$

with the top sign for $j=e$ and the bottom sign for $j=g$, and with $\Delta_{rm}^{\prime}=\Delta_{rm}+\zeta$, where $\Delta_{rm}=\omega_{r}-\omega_{m}$. In this expression, $\delta_{j,e}$ is the Kronecker delta. Following the notation of Ref. gambetta:2008a , we have used

Moreover, $n_{e,g}=\left|\alpha_{e,g}\right|^{2}$ is the number of photons in the cavity when the qubit is in the ground or excited state. With $\lambda^{2}=0$, the results of Ref gambetta:2008a are correctly recovered.

Eq. (34) represents the effective qubit decay rate. It’s main contribution is proportional to $\gamma_{1}$ and, again, is reduced by dressing (but can never be negative). The second term $\gamma_{\kappa}\sim\lambda^{2}\kappa$ is the Purcell effect which corresponds to qubit decay through the photon loss channel houck:2008a . The last term of ${\gamma_{\downarrow}}$ and ${\gamma_{\uparrow}}$ are particularly interesting. They describe, respectively, additional relaxation and excitation of the qubit due to the photons populating the resonator.

In summary, the rates ${\gamma_{\uparrow}}$ and ${\gamma_{\downarrow}}$ are due to dressing, by the resonator field, of the qubit operator $\sigma_{z}{}$ causing dephasing in the bare basis. We will therefore refer to this as dressed dephasing.

To verify the validity of the previous results, we have done extensive numerical calculations in the limit $n\ll n_{\kappa}$ to compare results obtained from the reduced master equation (30) to those obtained from the qubit-resonator master equation (5). The results obtained from Eq. (30) are also compared to those obtained from the linear approximation of Ref. gambetta:2008a . From this latter comparison, it will be apparent that the non-linear model obtained here is much more accurate, while adding essentially no additional complexity in numerical simulation.

Figure 4a) presents a typical time evolution of the qubit as obtained by the numerical integration of the full master equation (5) [full black line], the reduced model Eq. (30) [dashed blue line] and the linear model of Ref. gambetta:2008a [dotted red line]. The time evolution of $\left\langle\sigma_{x}\right\rangle$ and $\left\langle\sigma_{y}\right\rangle$ obtained from these three models are indistinguishable. However, because the linear model does not capture dressed dephasing, only the non-linear model reproduces the correct equilibrium value of $\left\langle\sigma_{z}\right\rangle$

We note that the numerical results obtained using the full master equation have been time-averaged to get rid of small amplitude fast oscillations. These oscillations are not contained in the effective models because of the various rotating-wave approximations that have been performed analytically. Experimentally, this averaging is effectively performed due to the finite bandwidth of measurement apparatus. Moreover, for simplicity, for the numerical results, a white noise spectrum was assumed. We have therefore taken $\gamma_{\kappa}=\lambda^{2}\kappa$, $\kappa_{\gamma}=\lambda^{2}{\gamma_{1}}$ and $\gamma_{\pm\Delta}=2\lambda^{2}{\gamma_{\varphi}}$ throughout this section.

Figure 4b) shows the maximum of the trace distance (over time) as a function of measurement power for three values of the pure dephasing rate $\gamma_{\varphi}$. The red curves with square dots are the trace distances between the full and the linear reduced model, while the green curves with triangle dots are the trace distances between the full and the non-linear reduced model. Unsurprisingly, as the measurement power is increased, the trace distance between the reduced models and the exact solution increases. As shown the three different curves for both models, the distance also increases as the dephasing rate is increased. However, the non-linear model obtained here is clearly much more accurate than the linear one, it captures the physics of dressed-dephasing. The non-linear model also shows much less variation in the trace distance with dephasing rate. It is worth pointing out that the maximum measurement power used in Figure 4b) corresponds to a very conservative photon population of the resonator $n\approx 0.4n_{\rm crit}$, much lower than the critical photon number where non-linear effects were thought to become important blais:2004a ; gambetta:2008a .

The effective model developed in this section is both accurate and much less demanding numerically than the full numerical integration of the qubit-resonator Hamiltonian. It should therefore be a useful tool to study the dispersive regime of circuit and cavity QED.

VI Qubit population and effective damping rate

In this section, we focus on the dependance of the qubit mixing rate on photon population and dephasing rate, and on the steady-state value of $\left\langle\sigma_{z}\right\rangle$. These two quantities could be measured experimentally as a test of the present model.

A remarkable feature of the non-linear model is that the qubit up, down and dephasing rates depend on the photon population. In particular, the effective qubit mixing rate is given by

where $n_{is}=\left|\alpha_{is}\right|^{2}$ are understood as the steady-state solutions of Eq. (36). Interestingly, in the situation where $n_{es}\approx n_{gs}$ and for white noise, such that $\gamma_{\Delta}=\gamma_{-\Delta}=2\lambda^{2}{\gamma_{\varphi}}$, if $\gamma_{\varphi}<\gamma_{1}/2$, then increasing photon population leads to a decrease of the effective mixing rate. On the other hand, if $\gamma_{\varphi}>\gamma_{1}/2$, increasing photon population leads to an increase of the mixing rate. This is again a consequence of dressing of the qubit by the photon field.

VI.2 Measurement-induced heat bath

From the reduced qubit master equation (30), the steady-state value of $\left\langle\sigma_{z}\right\rangle$ can be expressed as

While the linear model would predict $\left\langle\sigma_{z}\right\rangle_{s}=-1$, the second order term $\gamma_{-\Delta}$ causes a deviation of $\left\langle\sigma_{z}\right\rangle_{s}$ from this value which increases with $n_{gs}$. This deviation is indicative of the breakdown of the QND aspect of the qubit measurement.

When comparing expectation values, it is of course important to compare expressions computed in the same basis. As a result, it is useful to transform (39) to the bare basis When computing the expectation value of an operator in a transformed basis, the transformation that was applied to the state vector must also be applied to the operator in order to get the expectation value in the non-transformed basis: $\langle A\rangle={\rm Tr}\left(A\rho\right)={\rm Tr}\left({\mathbf{D}}A{\mathbf{D}}^{\dagger}{\mathbf{D}}\rho{\mathbf{D}}^{\dagger}\right)$.. This is done by applying the dispersive transformation to $\sigma_{z}$, see Eq. (15), from which we obtain

It is interestingly to note that, because of the asymmetry of the expression for $\left\langle\sigma_{z}\right\rangle_{s}$ with respect to $n_{e}$ and $n_{g}$, the steady-state value of $\sigma_{z}$ depends on the measurement frequency. Indeed, for a measurements of the phase where ${\omega_{m}}={\omega_{r}}$, in the steady-state $n_{es}\approx n_{gs}\approx\bar{n}_{s}$ and the effect will depend on the average number of photons in the resonator. On the other hand, for amplitude measurements with ${\omega_{m}}={\omega_{r}}+\chi$, $n_{es}\gg n_{gs}$ and the departure from -1 should be less important.

VI.3 Comparison with exact numerics

To compare the results of the analytical expressions Eq. (40) and (38) to numerical integration of the full resonator-qubit master equation (5), we initialize the qubit in its excited state and the resonator in the corresponding steady-state with a continuous measurement drive of amplitude ${\epsilon_{m}}$ and frequency ${\omega_{r}}$. In the absence of coherent driving at the qubit frequency, the qubit then simply decays to reach a steady-state value of $\sigma_{z}$. By fitting the time evolution of $\left\langle\sigma_{z}\right\rangle$ as obtained from numerical integration of Eq. (5) to

we extract the exact effective decay rate and steady-state mean value of $\sigma_{z}$.

These results are shown in Fig. 5, the analytical expressions (40) and (38) (lines) in addition to the values extracted from numerical solution of the full master equation (symbols) are plotted. With the parameters used here (see caption), the critical number of photons is $n_{\rm crit}=400$ such that the figure shows results for $n/n_{\rm crit.}\lesssim 0.4$.

Figure 5a) shows the steady-state value of $\left\langle\sigma_{z}\right\rangle$ as a function of the measurement amplitude for increasing values of the pure dephasing rate ${\gamma_{\varphi}}=\gamma_{\pm\Delta}/2$. The bottom line (black stars) corresponds to ${\gamma_{\varphi}}=0$, which in turns corresponds to the effective heat bath being at zero temperature. As a result, in the dispersive basis, $\left\langle\sigma_{z}\right\rangle_{s}=-1$ and the deviation from $-1$ is only caused by the change from dispersive to bare basis. The lines lying above this result correspond to ${\gamma_{\varphi}}/2\pi=0.05,0.2,0.5$ MHz (dashed grey, dotted red, dashed-dotted blue). Clearly, even for a relatively low number of photons compared to $n_{\rm crit.}$, mixing of the qubit excited and ground states by the effective heat bath can be significant if the pure dephasing rate is large.

Figure 5b) shows the effective decay rate $\gamma_{\rm eff}$ as a function of measurement power for the same dephasing rates ${\gamma_{\varphi}}$ as in pannel a). The lines correspond to Eq. (38) while symbols are extracted numerically. As expected from the discussion surrounding Eq. (38), if the dephasing rate is negligible (black stars), the qubit effective decay rate falls below the bare decay rate ${\gamma_{1}}/2\pi=0.1$ MHz as the measurement power is increased. However, for $2{\gamma_{\varphi}}>{\gamma_{1}}$ (red circles and blue triangles) the effective decay rate increases, again as expected from the model. In this latter case, the photon number and dephasing dependent qubit mixing simply overwhelms the aforementioned decrease in $\gamma_{\rm eff}$ which is no longer visible. For intermediate dephasing, $2{\gamma_{\varphi}}={\gamma_{1}}$ (gray squares), these two processes cancel each other and the effective decay rate is almost independent of the measurement amplitude.

VII Dispersive effects on the quantum trajectory equation

The master equation description of the dynamics does not take into account the result of the measurement. To include this information, we use quantum trajectory theory gardiner:2004b ; carmichael:1993a ; wiseman:1993a ; gambetta:2005a and derive the evolution equation for the conditional state, or quantum trajectory equation (QTE). This was already study in Ref. gambetta:2008a for the linear dispersive model and is extended here to incorporate the non-linear effects.

When monitoring the resonator bath, characterized by the rate $\kappa$, an observer would in principle see two signals: one at the qubit frequency and one at the cavity frequency When monitoring the qubit baths there would also be two signals for both the $\gamma_{1}$ and $\gamma_{\phi}$ baths. To derive the QTE, it is assumed as above that the relevant bath frequencies for the resonator bath are well separated such that they can be treated as two separate Markovian baths with relevant frequencies $\omega_{r}$ and $\omega_{a}$. That is, for an infinitesimal interval $dt$ the unitary operator describing the resonator bath is given by Eq. (77a).

In a homodyne measurement, with a local oscillator set to $\sim\omega_{r}$, the bath is projected in an eigenstate of the operator $dB_{\kappa,r}+dB^{{}^{\dagger}}_{\kappa,r}$ (where $dB_{\kappa,r}$ is defined in appendix B), with measurement result $J$ gardiner:2004b . For many such measurements, each separated by a time $dt$ and with result $J_{k}$, the state conditioned on the complete record $\mathbf{J}(t)=\{J_{1},...,J_{k}\}$ can be expressed as gambetta:2005a

where $\mathbf{L}=\{\sqrt{\kappa_{a}}L_{2},\sqrt{\gamma_{a}}L_{3},\sqrt{\gamma_{r}}L_{4}\}$ and $\Upsilon_{J,\mathbf{F}}$ is the gaussian probability measure

For continuous monitoring, the time step $dt$ between measurements tend towards 0. In this limit, Eq. (44) leads to the QTE whose ensemble average is the unconditional master equation. The corresponding QTE, in Itô form, for the measurement operator Eq. (45) is

where ${{\mathcal{L}}}^{{{\mathbf{D}}}}$ is given by Eq. (26). In this expression, we have defined the $\phi$-dependent field quadratures $2I_{\phi}=ae^{-i\phi}+a^{\dagger}e^{i\phi}$ and $2Q_{\phi}=-iae^{-i\phi}+i{a^{\dagger}}e^{i\phi}$. Moreover, the superoperator ${\cal M}[c]$ is defined as

where $\left\langle c\right\rangle_{t}={\rm Tr}\left(c{\varrho}_{J}^{{{\mathbf{D}}}}(t)\right)$ and the measurement outcome, $J$ can be expressed as

where $\xi(t)$ is Gaussian white noise and $\eta$ is a detection efficiency parameter included for completeness gambetta:2008a .

While only the signal at the cavity frequency was taken into account here, it is interesting to point out that the signal at the qubit frequency could also be measured. This was done experimentally in Ref. houck:2007a to perform qubit state tomography. However, the signal at that frequency is in general much weaker than the signal at the cavity frequency. As a result, while by itself the former signal would lead to a very inefficient QTE (which would have a similar to that of a direct homodyne measurement of the qubit) this additional information could be included in the present treatment to realize even more efficient qubit measurements.

Using the polaron transformation Eq. (90), it is possible to obtain a reduced QTE for the qubit. As show in appendix D, this reduced QTE takes the form

This quantity is linked to the homodyne current by

To demonstrate the different features of the reduced QTE, Fig. 6 presents

a typical trajectory for 3 different measurement powers using the same parameters as in Fig. 2. As in the linear case, increasing the measurement power localizes the qubit state on one of its basis states. As a result, although the QTE is based on homodyne measurement, we do not expect diffusive but rather jump-like trajectories. Moreover, because of the effective upward rate ${\gamma_{\uparrow}}$ which increases with measurement power, the trajectories show telegraph noise rather than a single jump to the ground state. These predictions can be experimentally tested once single shot measurement is achievable. From these results, the waiting time between jumps can be compared to ${\gamma_{\uparrow}}$ and ${\gamma_{\downarrow}}$.

VIII Conclusion

We have investigated circuit QED in the dispersive regime. To take into account large photon population of the resonator, useful for qubit readout, we have shown that it is necessary to push the dispersive treatment to a higher order. We have done this while taking into account the effect of dissipation and external microwave driving. In particular, we have obtained a Markovian model for the effect of dissipation that takes into account frequency dependence of the environment.

Finally, using the quantum trajectory approach, we have obtained an effective stochastic master equation for the qubit. In the single-shot limit, this equation predicts that the measurement-induced heat bath should lead to telegraph-like jumps in the measurement response. Moreover, the non-linearity have been shown to lead to a reduction of the expected signal-to-noise ratio, a result qualitatively consistent with experimental observations boissonneault:2008a .

There are various ways to test experimentally these predictions. First, the qubit effective decay rate ${\gamma_{\uparrow}}(n)+{\gamma_{\downarrow}}(n)$ and the steady-state value of $\left\langle\sigma_{z}\right\rangle$ can be measured experimentally and compared to the results obtained here. Second, when single shot measurements become possible in circuit QED, the effect of the upward transitions caused by the measurement induced heat bath should be observed. The waiting times between the upward and downward transitions can then be related to the rates ${\gamma_{\uparrow}}(n)$ and ${\gamma_{\downarrow}}(n)$ obtained here.

Appendix A Exact diagonalisation of the Jaynes-Cummings Hamiltonian by unitary transformation

In this appendix, the Jaynes-Cummings Hamiltonian is diagonalized exactly using a unitary transformation. In order to simplify the notation, we introduce the commutation linear application

In term of this superoperator, Hausdorff’s relation can be written as

In the same way as for the linear approximation discussed in section III.1, the antihermitian operator ${I_{-}}$ will be key in this diagonalization. Since it commutes with both $H_{s}$ and ${I_{-}}$, another important operator is the total number of quanta $N_{q}$ defined in Eq. (13). Moreover, with a unitary operator of the form

where $\Lambda$ is a function to be defined, $N_{q}$ can be considered as a scalar when applied on $H_{s}$.

Before transforming $H_{s}$ using ${\mathbf{D}}$, it is useful to introduce some important commutators. First, it is simple to show that

Using this result, transformation of $H_{s}$ by ${\mathbf{D}}$ yields

such as to eliminate the off-diagonal term proportional to ${I_{+}}$. Using this result, we finally obtain the exact diagonal form

Using this result, we define the Lamb and ac-Stark shift operators as [we use $H_{s}^{{{\mathbf{D}}}}({{a^{\dagger}}a},\sigma_{z})$]

Developing these expressions in powers of $\lambda$, we obtain

with $\chi=g\lambda(1-\lambda^{2})$ and $\zeta=-g^{4}/\Delta^{3}$. These approximate results are used in Eqs. (9) and (17).

Appendix B Obtaining the dispersive master equation

In this appendix, we find the effect of the dispersive transformation on the non-unitary part of the master equation. Applying the dispersive transformation on the Hamiltonians Eqs. (23), moving to the interaction frame defined by the transformation $\exp[-i(H_{s}+H_{B\kappa}+H_{B\gamma})t/\hbar]$ and performing a rotating-wave approximation (RWA) yields

with $L_{1}=a(1+\lambda^{2}\sigma_{z}/2)$, $L_{2}=\lambda\sigma_{-}$, $L_{3}=\sigma_{-}[1-\lambda^{2}({{a^{\dagger}}a}+1/2)]$ and $L_{4}=\lambda a\sigma_{z}$. The bath operators $z_{i}(t,\omega_{p})$ are given by

We have kept in Eq. (69) only terms that will contribute up to order $\lambda^{2}$ in the master equation. In obtaining this expression, we have taken the (dispersive) system Hamiltonian as $H_{s}\approx\hbar\omega_{r}{{a^{\dagger}}a}+\hbar\omega_{a}\sigma_{z}/2$, where $\omega_{a}$ should be understood as the Lamb and ac-Stark shifted qubit transition frequency and $\omega_{r}$ should be understood as the cavity frequency shifted by the non-linearity (i.e. $\omega_{r}+\zeta$). Moreover, to perform the RWA, we have made the standard assumption that the system-bath interaction is limited to a small band of frequency $B_{i,p}$ around the frequency $\omega_{p}$ of the corresponding system operator $L_{i}$, $B_{i,p}\ll\omega_{p}$ gardiner:2004b .

We now assume that, within the bandwidths $B_{i,p}$, the coupling constants $f_{i}(\omega_{p})$ and the density of modes $d_{i}(\omega)$ do not vary significantly. In this situation, the above system-bath Hamiltonians can be rewritten as

where the decay rates are given by Eq. (27) and the bath temporal modes are defined as

Since $\left[b_{i}(\omega),b^{\dagger}_{j}(\omega^{\prime})\right]=\delta_{i,j}\delta(\omega-\omega^{\prime})$, the commutator of two temporal modes is (after a change of integration variable)

If we now take $|\omega_{p}-\omega_{q}|\gg B_{i,p},B_{i,q}$ for $p\neq q$ then the above becomes

In other words, we assume the bath operators to be independent. In the dispersive regime, this is a reasonable assumption since $|\omega_{p}-\omega_{q}|\sim\Delta$, where the detuning $|\Delta|\gg g$ is large.

We finally make the standard and reasonable assumption that dissipation is not too strong, such that the time scales set by the decay rates $\kappa_{p}$ and $\gamma_{p}$ are much longer than the cutoff time $1/B_{i,p}$. In this situation we can effectively take the limit $B_{i,p}\rightarrow\infty$. This corresponds to the standard Markov approximation gardiner:2004b , which was already successfully applied to describe circuit QED experiments wallraff:2004a ; schuster:2005a ; wallraff:2005a ; schuster:2007a ; houck:2007a ; majer:2007a . In this situation, the above commutation relation reduces to

In this Markov, or white noise, approximation the evolution operators can be written in Itô form as

where $dB_{i,p}=b_{i,p}dt$ is a quantum Wiener increment gardiner:2004b .

We now take the bath to be in the vacuum state and uncorrelated to the system at time $t=0$. By tracing over the bath and keeping terms of order $\mathcal{O}(dt)$ using Itô calculus, we obtain a Lindblad form master equation for the resonator-qubit system. In this master equation, the photon bath $\kappa$ leads to the damping superoperators gardiner:2004b

These terms are the second and third lines of Eq. (26).

B.2 Qubit dephasing

For dephasing, we start with the Hamiltonian (25). Moving to the dispersive basis, it becomes

where we have used the second order expansion of Eq. (15). Moving to a frame rotating at the qubit and resonator frequencies we find

The main contribution to dephasing comes from a small frequency band $B_{0}$ centered around the frequency $\omega_{0}$. In this situation, the integration boundaries in $f_{\omega_{0}}(t)$ can be reduced to

For this rotating-wave approximation to be valid, it is required that $B_{0}\ll\omega_{0}$ gardiner:2004b .

The Wiener-Khinchin theorem can be used to relate $f_{\varphi}(\omega)$ to its noise spectrum $S(\omega)$ gardiner:2004c

where $E[\cdot]$ is an ensemble average. This allows us to write the $\omega$ component of the noise as

with $\xi(\omega)$ white noise obeying $E[\xi(\omega)]=0$ and $E[\xi(\omega)\xi(-\omega^{\prime})]=\delta(\omega-\omega^{\prime})$.

Using these results, we now make similar assumptions as in the last section and take the noise spectrum $S(\omega)$ to be constant within the small frequency band $B_{0}$ around $\omega_{0}$. After a change of integration variable, $f_{\omega_{0}}(t)$ can be written as

In this Markov approximation, we will again assume the noise spectrum to be relatively weak which implies that the time scale corresponding to dissipation is much slower than $1/B_{0}$ gardiner:2004b . In this situation, we take $B_{0}\rightarrow\infty$ which allows us to write

such that the transformed dephasing Hamiltonian becomes

The three $\xi_{\omega_{0}}(t)$ white noise terms in the above expression now correspond to independent noises, centered around three different frequencies.

The above Hamiltonian leads to the following superoperators in the resonator-qubit master equation

with the rates given by Eq. (27). These terms correspond to the fourth and fifth lines of Eq. (26).

Appendix C The polaron transformation

Following the approach developed in Ref gambetta:2008a , a reduced master equation for the qubit is obtained in this appendix. To do so, we start from the dispersive master equation (26) and go to the rotating frame defined by ${\mathbf{R}}=\exp[i{\omega_{m}}{{a^{\dagger}}a}t]$. We then go to a frame defined by the polaron-type transformation

where $D(\alpha)$ is the displacement operator and $\alpha_{g(e)}$ satisfy Eq. (36). In the polaron frame, the field $a$ is described by a classical part given by the complex variables $\alpha_{g}$ and $\alpha_{e}$, and a small quantum part corresponding to quantum noise.

The action of ${\mathbf{P}}$ on various system operators is given by

where we have defined the projection operator

with $\left|\Pi_{\alpha}\right|^{n}=\left|\alpha_{g}\right|^{n}\Pi_{g}+\left|\alpha_{e}\right|^{n}\Pi_{e}$. Using these results, we apply the transformation ${\mathbf{P}}$ to the Hamiltonian $H^{{{\mathbf{D}}}}=H_{s}^{{{\mathbf{D}}}}+H_{d}^{{{\mathbf{D}}}}$ to obtain

with $\Delta_{rm}^{\prime}=\Delta_{rm}+\zeta$. Taking into account the time-dependence of ${\mathbf{P}}$, the transformed Hamiltonian reads

We also apply this transformation to the dissipative terms of the dispersive master equation (26). For the first term of second line ($\kappa$ term), keeping up to order $\lambda^{2}$, we get

We then get for the $\gamma$ and $\kappa_{\gamma}$ terms

Finally, for $\gamma_{\pm\Delta}$, we have

The last line of Eqs (95), (96) and (98) act like a drive Hamiltonian. We will be able to cancel them with $\alpha_{e}$ and $\alpha_{g}$ given by Eq (36). In the above expressions, the quantities $n_{g}=\left|\alpha_{g}\right|^{2}$ and $n_{e}=\left|\alpha_{e}\right|^{2}$ are the number of photons when the qubit in the ground or excited state, and we have ${{\mathcal{D}}}[\sigma_{-}^{{{\mathbf{D}}}}]={{\mathcal{D}}}[D(\beta)\sigma_{-}]$ and ${{\mathcal{D}}}[\sigma_{+}^{{{\mathbf{D}}}}]={{\mathcal{D}}}[D^{\dagger}(\beta)\sigma_{+}]$.

If we put all the results of this section together, we can write the polaron-frame master equation, which is given by applying the polaron transform on Eq.(26). The result is given by combining the results from Eqs (94-98)

In this section, we trace the transformed master equation Eq. (99) over the resonator states to obtain an effective master equation for the qubit only. This is done by first expressing the total density matrix in the polaron-transformed frame as

Since our goal is to obtain the effective equation in the original non-polaron transformed frame, we write the reduced qubit density matrix in this frame as

with $\{s,s^{\prime}\}\in\{g,e\}$, $\lambda^{{{\mathbf{D}}}{{\mathbf{P}}}}_{n,m,p,q}={\varrho}^{{{\mathbf{D}}}{{\mathbf{P}}}}_{n,m,e,g}d_{p,q}e^{-i{\rm Im}[\alpha_{g}\alpha_{e}^{*}]}$, and $d_{p,q}=\left\langle p\left|D[\beta]\right|q\right\rangle$ is the matrix element of the displacement operator in the number basis.

To obtain the master equation for ${\rho}^{{{\mathbf{D}}}}$, we simply find the equation of motion for the matrix elements of ${\varrho}^{{{\mathbf{D}}}{{\mathbf{P}}}}$. More precisely, we will look at the equation of motion for ${\varrho}^{{{\mathbf{D}}}{{\mathbf{P}}}}_{i,j,s}$.

From this equation, we see that the only way the element ${\varrho}^{{{\mathbf{D}}}{{\mathbf{P}}}}_{0,0,e}$ depends on the other elements is through the two last lines. Moreover, the only way the elements $n,m\neq 0$ can be populated from an element $i<n,j<m$ is through the second, third and last lines. The rates at which these mechanisms act are of the order $\zeta\left|\alpha_{e}\right|^{2}=\zeta n_{e}$ and ${\gamma_{\uparrow}}\equiv\gamma_{-\Delta}n_{g}$. On the other side, these elements decay more quickly than the $0,0$ element because of the $\kappa$ term, which we assume is dominant compared to $\kappa_{\gamma}$, $\gamma_{\Delta}$ and $\gamma\lambda^{2}$. If the conditions $n_{e}\ll n_{\kappa}\equiv\kappa/\zeta$ and ${\gamma_{\uparrow}}\ll\kappa$ are satisfied, we can assume there is no significant population of the $n,m\neq 0$ matrix elements. We will have a similar equation for $\dot{\varrho}^{{{\mathbf{D}}}{{\mathbf{P}}}}_{n,m,g}$, with the conditions being $n_{g}\ll n_{\kappa}$ and ${\gamma_{\downarrow}}\ll\kappa$, where ${\gamma_{\downarrow}}$ is defined at Eq. (34). If these conditions are fulfilled, we can reduce the above equation and that for the $g$ component to

On the other hand, the off-diagonal elements of the reduced qubit density matrix involve off-diagonal elements of the resonator density matrix and we must consider the equation of motion for all the terms $\lambda^{{{\mathbf{D}}}{{\mathbf{P}}}}_{n,m,p,q}$

If we do this and compute $\dot{\varrho}^{{{\mathbf{D}}}{{\mathbf{P}}}}_{n,m,e,g}$ according to (99), we get an equation that can be reduced only to the element ${\varrho}^{{{\mathbf{D}}}{{\mathbf{P}}}}_{0,0,e,g}$ in the conditions stated above $(n_{e},n_{g}\ll n_{\kappa})$. Considering that only the $0,0$ element is ever populated significantly, the equation of motion is then

Using the expression (36) for $\alpha_{g(e)}$, we can combine the equations of motion for the reduced qubit density matrix ${\rho}^{{{\mathbf{D}}}}$ to find the reduced qubit master equation (30) with the frequency and rates given by Eqs. (31), (32), (34) and (35).

Appendix D The effective qubit quantum trajectory equation

The QTE is derived using linear quantum measurement theory gambetta:2005a ; gambetta:2008a . The linear form of Eq. (47) is

where the bar means that the state is not normalized and the linear measurement superoperator is

Moving to the frame defined by Eq. (90) yields

As it should, for $\lambda^{2}=0$ the three equations above are of the same form as those obtained in Ref. gambetta:2008a .

As before we now find the equations of motion for the coefficients $\bar{\varrho}^{{{\mathbf{D}}}{{\mathbf{P}}}}_{n,m,e}$, $\bar{\varrho}^{{{\mathbf{D}}}{{\mathbf{P}}}}_{n,m,g}$, and $\bar{\lambda}^{{{\mathbf{D}}}{{\mathbf{P}}}}_{n,m,p,q}$. For the $\bar{\varrho}^{{{\mathbf{D}}}{{\mathbf{P}}}}_{n,m,e}$ element, we find

with a similar equation for the $\bar{\varrho}^{{{\mathbf{D}}}{{\mathbf{P}}}}_{n,m,g}$ component. For the $\bar{\lambda}^{{{\mathbf{D}}}{{\mathbf{P}}}}_{n,m,p,q}$ component, we find

In these expressions, which are the contribution of the Linblad term ${{\mathcal{L}}}^{{{\mathbf{D}}}{{\mathbf{P}}}}\bar{\varrho}_{J}^{{{\mathbf{D}}}{{\mathbf{P}}}}$ of Eq. (111), the equation numbers refer to the RHS of the corresponding expressions.

The added measurement and back-action operators in the evolution equations does not change the approximation used in the previous section. Therefore, in the same limits, we can consider that the only relevant components are $\bar{\varrho}^{{{\mathbf{D}}}{{\mathbf{P}}}}_{0,0,e}$, $\bar{\varrho}^{{{\mathbf{D}}}{{\mathbf{P}}}}_{0,0,g}$ and $\bar{\lambda}^{{{\mathbf{D}}}{{\mathbf{P}}}}_{0,0,0,0}$. We can then write

and it is possible to construct a reduced linear QTE for the qubit in the dispersive frame

Using Eq. (52) and normalizing, the above QTE gives Eq. (50) with measurement record given by Eq. (51).

References