Quantum trajectory approach to circuit QED: Quantum jumps and the Zeno effect

I Introduction

Continuous-in-time measurement theory carmichael:1993a; gardiner:2004b, or quantum trajectory theory, describes how an observer’s state of knowledge of a quantum system (known as the conditional state) evolves given a measurement record. Even in the absence of classical noise, such a trajectory follows a stochastic path in time, with the randomness being due to quantum uncertainty. These stochastic trajectories are either diffusive or jump-like in nature. Diffusive trajectories usually arise when the observable being measured is only weakly coupled to the detector wiseman:2001a, whereas jump-like behavior occurs when there is a large sudden change in the observers knowledge of the system state, a typical example of the latter being detection of a photon with a photomultiplier carmichael:1993a; molmer:1993a. The evolution equation for this trajectory is called a stochastic master equation (SME) carmichael:1993a; gardiner:2004b; wiseman:2001a; molmer:1993a; gambetta:2005a.

In this paper, we consider measurement in circuit quantum electrodynamics (QED) blais:2004a; wallraff:2004a; wallraff:2005a; schuster:2005a; gambetta:2006a. This system consists of a cooper pair box, playing the role of an artificial atom, dispersively coupled to a 1-D transmission line resonator and is the circuit equivalent of cavity QED. It has the advantage that the qubit can be fixed at an antinode of the resonator which, due to its 1D configuration, has a very large vacuum electric field. This leads to very strong Jaynes-Cummings type coupling between the qubit and the resonator. This allows to probe a new regime of parameter space, where the resonator and qubit are strongly coupled via a dispersive interaction and no energy is exchanged between them. In this regime, the qubit causes a large state-dependent shift of the resonator frequency and thus by monitoring the signal transmitted through the resonator we can infer the qubit’s state. An important feature of the quasi-one-dimensional circuit resonator is that it permits dispersive couplings $\sim 10^{6}$ times larger than for ordinary three-dimensional cavities. Coupling of superconducting charge qubits to 3D microwave cavities has also been investigated theoretically you:2003a.

To go beyond previous theoretical works on this system, we employ quantum trajectory theory carmichael:1993a; gardiner:2004b; wiseman:2001a; molmer:1993a; gambetta:2005a and a polaron transformation mahan:2000a to derive an effective SME for the qubit. It is well know in quantum optics that if the voltage of a cavity is weakly monitored, then the evolution of the conditional state of the cavity is given by the homodyne SME carmichael:1993a; wiseman:1993a. Using this SME as a starting point, we show that if the rate at which information is coming out of the resonator is much larger then the rate at which information is being lost into unmonitored baths, then we can use a polaron transformation to eliminate the resonator and obtain a SME for the qubit only.

The effective SME which is derived corresponds to a weak dispersive measurement of the qubit observable $\sigma_{z}$. This is akin to SMEs derived in Refs. korotkov:1999b; korotkov:2001a; korotkov:2001b; wiseman:2001b; goan:2001a; korotkov:2003a; oxtoby:2005a; oxtoby:2007a for a quantum dot monitored by a quantum point contact. In Refs. korotkov:1999b; korotkov:2001a; korotkov:2001b; korotkov:2003a, the SMEs are referred to as the quantum Bayesian equations. They can be seen as the Stratonovich version of the the Itô SMEs presented here and in Refs. goan:2001a; oxtoby:2007a. We note that one can derive a similar equation by adiabatically eliminating the resonator from the qubit-resonator master equation using a similar technique as presented in doherty:1999a. However, by using the present method, we are able to derive corrections to the various system rates. These rates are shown to agree very well with the solution of the total resonator-qubit conditional state found by numerically solving the homodyne SME.

II Cavity QED with superconducting circuits

We consider a Cooper pair box capacitively coupled to a transmission line resonator acting as a simple harmonic oscillator. This system, illustrated schematically in Fig. 1, was first introduced in Ref. blais:2004a and experimentally studied in Refs. wallraff:2004a; schuster:2005a; wallraff:2005a; schuster:2007a; houck:2007a; majer:2007a. Measurement-induced dephasing was theoretically studied in detail in Ref. gambetta:2006a and the applications to quantum information processing investigated in Ref. blais:2007a.

As described in the above references, the system’s Hamiltonian in the presence of a microwave drive of amplitude $\varepsilon_{d}(t)$ and frequency $\omega_{d}$ can be written as blais:2004a

In this expression, $\omega_{r}$ is the resonator frequency, $\omega_{a}$ the qubit transition frequency and $g$ the resonator–qubit coupling strength. Depending on its frequency, the drive can correspond either to measurement of the qubit or can be used to coherently control its state. In the dispersive regime, when $|\Delta|=|\omega_{a}-\omega_{r}|\gg|g|$, the effective Hamiltonian Eq. (1) can be approximated by blais:2004a

It is important to note that the dispersive approximation breaks down as the number of photons in the resonator approaches the critical photon number $n_{\rm crit}=\Delta^{2}/4g^{2}$ blais:2004a. In the present paper, we will work at moderately low photon number and assume the dispersive approximation to hold. That is, we will assume that Eq. (2) is a valid description of the system. The results obtained will therefore also apply to the circuit QED implementation that uses the so-called transmon qubit koch:2007a. A future publication will explore the break down of the dispersive approximation boissonneault:2007a at small detuning and large photon number.

III Master Equation

In the Born-Markov approximation, the master equation describing circuit QED takes the usual Lindblad form lindblad:1976a; carmichael:1993a

where $\varrho(t)$ is the state matrix for both the qubit and the resonator and ${\cal D}[A]$ is the damping superoperator defined by the mapping

The three damping channels are photon loss through the resonator ($\kappa$), qubit decay ($\gamma_{1}$) and dephasing of the qubit ($\gamma_{\phi}$) In the dispersive regime, the operators describing damping, dephasing and measurement should by transformed in the same way as the Hamiltonian. This leads to corrections at order ($g/\Delta$) and can be ignored here. These effects where briefly described in blais:2004a and are investigated in detail in boissonneault:2007a..

III.2 Master equation for the qubit

In Ref. gambetta:2006a the solution of the above master equation, neglecting energy loss due to $\gamma_{1}$ (but keeping its effect on dephasing), was found to be

where the indices $g$ and $e$ label the qubit ground and excited states, respectively. In this expression, the coefficients $c_{i,j}(t)$ are given by

and $|{\alpha_{e(g)}(t)}\rangle$ are coherent states of the resonator with amplitudes determined by

In the expression for $c_{e,g}(t)$, we see that in addition to decay due to dephasing $\gamma_{2}=\gamma_{1}/2+\gamma_{\phi}$, the off-diagonal element of the qubit density matrix decays at a rate that depends on the field amplitudes $\alpha_{g}$ and $\alpha_{e}$. Since the coherent states with these amplitudes act as pointer states in the measurement of the qubit, this decay can be interpreted as measurement-induced dephasing and will depend on how distinguishable the states $|{\alpha_{e}(t)}\rangle$ and $|{\alpha_{g}(t)}\rangle$ are gambetta:2006a. Following Ref. gambetta:2006a, the effect of qubit relaxation in this model has been described by Bonzom et al. bonzom:2007a.

For example, for the case of Fig. 2C), this angle is $\theta_{\beta}=\pi$, which corresponds to the quadrature containing the most information about the state of the qubit. Indeed, for this particular case, all of the information is in the in-phase component $I$, with no information stored in the quadrature $Q$ component.

Our goal in the remainder of this section is to obtain an effective master equation for the qubit by eliminating the resonator degree of freedom from the full master equation, Eq. (3). To achieve this, we first go to a frame defined by the transformation

with $D[\alpha]$ the displacement operator of the resonator,

and $\Pi_{j}=|{j}\rangle\langle{j}|$ projectors on the ground and excited states of the qubit. This is similar to the polaron transformation which has been used extensively in various systems mahan:2000a; leggett:1987a. For example, this was used in Ref. irish:2005a to study a charge qubit coupled to a mechanical oscillator beyond the rotating wave approximation. However, here we use the transformation Eq. (10) on the dispersive Hamiltonian, not on the full Jaynes-Cummings Hamiltonian.

As is shown in Appendix A, applying this transformation on the master equation Eq. (3) and tracing over the resonator state yields the laboratory frame reduced qubit master equation

where $\rho(t)={\rm Tr}_{\rm res}[\varrho(t)]$. This expression is the main result of this section. It is important to note that, within the dispersive approximation, this result is exact.

In addition to dephasing, the qubit transition frequency is also modified by the photon population of the resonator. The shifted qubit frequency is given by

This last term gives rise to the ac-stark shift experimentally measured in Ref. schuster:2005a. In the situation where $\chi\gg\kappa$, the above leads to the number splitting predicted in Ref. dykman:1987a; gambetta:2006a and experimentally observed in Ref. schuster:2007a.

Although the above results are analytically exact, we have compared the numerical solution of the qubit’s dynamics obtained from the full master equation Eq. (3) to that obtained from the reduced model Eq. (12). This is shown in Fig. 3 where the elements of the Bloch vector are plotted as a function of time. It is useful to note that, in terms of the Bloch vector

and is found to be zero at all integration times, up to numerical round off (truncation) errors. This was checked for a wide range of measurement amplitudes $\varepsilon_{d}$ and the trace distance was also found to be zero for all verified amplitudes. Also in this figure is shown the purity (full blue line)

In this master equation description, the purity tends to 1/2, corresponding to a completely mixed state, due to dephasing. On top of this dynamics, relaxation is taking the purity to 1 (a pure state) at a rate $\gamma_{1}$. For the parameters chosen in Fig. 3 relaxation is much weaker than (measurement-induced) dephasing, such that only the effect of the latter is seen.

To summarise this section, we have obtained an exact master equation description of the qubit dynamics that only involves classical solution of the resonator field. In addition to giving direct insights in the qubits dynamics (e.g. measurement induced dephasing and ac-Stark shift) this also provides a tool to significantly reduce the complexity of numerical calculations. Indeed, only a 2-dimensional Hilbert space is now required. We note that this effective model was used to analyze and reproduce, with exceptional agreement, the experimental results reported in Ref. houck:2007a.

IV Stochastic Master equation

In this section, we review homodyne measurement of the field emitted from the resonator. Using these results, we will in the next subsection use the transformation of Eq. (10) to obtain an effective SME for the qubit only.

In a given quantum system, if all the decoherence and decay channels can be monitored continuously, it is possible to describe the system conditioned on the results of this monitoring result $J(t)$ by a pure state $|{\Psi_{J}}\rangle$ rather than by the average state $\varrho$. This pure states is called a conditional state and can be viewed as our state of knowledge of the system. However, in systems where information about only some of the decay channels can be obtained, there is missing information, and it is no longer possible to have a pure state description. In this case, we can assign a conditional state matrix, $\varrho_{J}$ to represent the state of the system under continuous observation of the particular decay channels. The evolution equation of this conditional state is referred to as a stochastic master equation (SME) carmichael:1993a; molmer:1993a; wiseman:1993a; gardiner:2004b; wiseman:2001a; korotkov:2001a; korotkov:2003a; gambetta:2005a. It has the property that its average state $\varrho(t)$,

Although, direct detection of the transmitted microwave photons is possible houck:2007a, here we will consider homodyne processing. That is, we will assume that the signal coming from the output port of the resonator is mixed with a strong local oscillator of phase $\phi$ tuned to the signal frequency. Given the homodyne measurement result $J(t)$, we can assign to the qubit–resonator system the conditional state $\varrho_{J}(t)$ whose evolution is governed by the SME In Ref. wiseman:1993a only Lindblad superoperators with one decoherence term are considered and as such a pure state unraveling is possible, here however we have three decoherence terms and only unravel one of them. That is, we have a partial measurement and must use a SMEs unless we use the numerical techniques of Ref. gambetta:2005a.

with ${\cal L}_{\rm tot}$ given by Eq. (3), $\eta$ is the measurement efficiency which we define below, ${\cal M}[{c}]$ is the measurement superoperator defined as

For homodyne detection, the measurement record observed in an experiment can be expressed as

where $\xi(t)$ is Gaussian white noise and represents the photon shot noise. It is formally defined as gardiner:1985a

with ${\rm E}$ denoting an ensemble average over realizations of the noise $\xi(t)$.

The measurement term in Eq. (21), the one including the superoperator ${\cal M}[{2I_{\phi}}]$, comprises two parts. We will refer to the first one as the homodyne gain and the second as the innovation. The homodyne gain is $\sqrt{\kappa\eta}{\cal M}[{2I_{\phi}}]\varrho_{J}(t)$ and collapses the state towards a $I_{\phi}$ state (eigenstate of $I_{\phi}$). The innovation is $J-\sqrt{\kappa\eta}\langle 2I_{\phi}\rangle_{t}$, it pushes the conditional state to a higher or lower $I_{\phi}$ state depending on whether the current result $J(t)$ is greater or smaller then the average $\sqrt{\kappa\eta}\langle 2I_{\phi}\rangle_{t}$. The last term in Eq. (21), the noisy Hamiltonian term is extra non-Heisenberg backaction (it does not come with any information gain) caused by the measurement. It is this term that stops the conditional state from being driven to a $I_{\phi}$ state as it supplies random $Q_{\phi}$ kicks to the conditional state causing delocalization in $I_{\phi}$.

Finally, we note that the SME Eq. (21) is known to lead to a diffusive-like evolution for the system wiseman:2001a; wiseman:1993a; wiseman:1993b. We will however see in section VI how there can be a cross-over from diffusive to jump-like evolution as the measurement strength is increased.

IV.2 Stochastic master equation for the qubit

In this section, we use the transformation Eq. (10) to obtain an effective SME for the qubit only. As described in Appendix B, this can be done in the limit where

We note that this limit does not imply that the information gain about the state of the qubit is small. For the system to be in this limit, we simply require that the photons come out of the resonator faster than the qubit decay rate. Otherwise, the full SME of Eq. (21) must be used. However, as will be seen from numerical investigations, the validity of the model can extend well beyond this limit in practice.

As described in Appendix B, in the limit where Eq. (26) is valid, the effective SME for the qubit obtained using the transformation of Eq. (10) and tracing over the resonator states is

with the angle $\theta_{\beta}$ being defined in Eq. (9).

In Eq. (27), ${\bar{J}}(t)$ is the processed record coming from the resonator and is given by

which can be related to the homodyne current by

with $\mu(t)=\alpha_{e}(t)+\alpha_{g}(t)$ and $\theta_{\mu}=\arg(\mu)$.

When the resonator dynamics have reached steady-state, it can be shown that

Unlike the average evolution, as we monitor the system we become more certain of its quantum state. This is represented in Fig. 5A) by the $z$ component being stochastically pushed towards $z=-1$. Note that we are showing but one possible trajectory, other realizations will tend to localise the $z$ component to +1. The fact that our state of knowledge about the quantum state is increased by monitoring is also made apparent by the purity which reaches unity [full blue line in panel B)]. Under the average evolution, see Fig. 3b), we end up with a completely mixed state. This is simply due to the fact that the average description does not take into account the information gain due to the measurement.

Also shown in this panel B) is the trace distance between the conditional state found using the homodyne SME Eq. (21) and the effective SME Eq. (27), in both cases using the same parameters as in panel A). The agreement between the two results is excellent, which shows that when the limit Eq. (26) is valid, the effective model derived here is indeed a good description. To see the break down of the validity of the model, we plot in panel C) one minus the trace distance minimized over the simulation time, $300$ ns, as a function of measurement amplitude and for two values of $T_{1}$. This is calculated and averaged over 70 trajectories. For $T_{1}=7\>\mu$s (full red circles), the agreement is very good at all simulated amplitudes. However, for $T_{1}=100$ ns (empty blue squares), the condition Eq. (26) is much less valid. As the measurement amplitude is increased, the approximation that we can neglect elements of Eqs. (100) and (101) in Eq. (99) breaks down and the discrepancy between the effective and full models is apparent. In any cases, it is clear that the effective model is a very good approximation for long $T_{1}$ and can give a relatively accurate description of the system even at low $T_{1}$, provided that the measurement amplitude is not too large. We note that a $T_{1}$ of 7 $\mu$s was reported in Ref. wallraff:2005a. The model obtained here is therefore useful in realistic experimental settings.

Moreover, we find that for the small $T_{1}$ of 100 ns, while the discrepancy reported in panel C) for an average over many trajectories can be large, when inspecting single trajectories we find no notable new features as the measurement amplitude is increased. The only change can be attributed to renormalization of the various system parameters under the measurement. This suggests that the effective model presented here may extend well beyond the limit (26), with the only change being that the measurement rate must be obtained numerically (or experimentally measured).

IV.3 Signal-to-noise ratio

Using the above results, we can define the signal-to-noise ratio (SNR) as

Thus, to extract the most information about the state of the qubit, we need to tune the measurement drive to bare resonator frequency $\omega_{r}$ and choose the system parameters such that $\chi=\kappa/2$. Note that $\kappa$ is chosen at fabrication time (and could be tuned in-situ by a change of resonator design) and that $\chi=g^{2}/(\omega_{a}-\omega_{r})$ can be tuned in-situ by tuning $\omega_{a}$. At a drive strength such that $\bar{n}=n_{\rm crit}$ blais:2004a, the maximum SNR is $\eta\Delta/\gamma_{1}$, which for current experiments is approximately $50-100$. Using the results of Ref. gambetta:2007a, this implies that a measurement fidelity of at least $95\%$ could be achievable in practice. To increase further this value, improvements have to be made to $T_{1}$, to minimise the amplifier noise and/or tune the system to be still further in the dispersive regime.

V Measurement time

This is the information gain/dephasing ‘uncertainty relation’ already discussed by several authors devoret:2000a; averin:2002a; clerk:2003a; makhlin:2001a. The equality is saturated for $\eta=1$ and $\phi=\theta_{\beta}$ and corresponds to the quantum limit discussed in Ref. clerk:2003a.

In the next sections, we define the measurement time with respect to three different measures. As we will see however, there is no unique definition of the measurement time.

For simplicity, let us first discuss the situation where $\gamma_{1}=0$ and with the qubit initially in either $z=\pm 1$. In this situation, the measurement time is very well defined as the qubit state simply remains in its initial state and the task is to estimate that state in the shortest possible time.

We define the measurement signal for an integration time $t$ as

This signal has mean and standard deviation

A problem with this approach is that the condition which defines distinguishability, and thus the measurement time, is somewhat arbitrary. Indeed, separation by one standard deviation is not the only possible choice for distinguishability. Lastly the above discussion only answers the question of how much time it takes to distinguish between two predetermine initial states of the qubit (under a measurement with Gaussian noise). A more natural question is how long it takes the conditional state to collapse to either $z=\pm 1$. This can be characterized by the conditional variance.

V.2 Conditional variance

How fast the ensemble of possible trajectories converge to one of the possible values $z=\pm 1$ is characterized by the conditional variance oxtoby:2006a. This measure is defined as

It ranges from 0 to 1, with $V_{J}=0$ meaning that we are certain that the current value of $z_{J}(t)$ is either $\pm 1$ and $V_{J}=1$ corresponding to complete uncertainty.

For $\gamma_{1}=0$, Eq. (32) can be solved analytically using Itô calculus. Doing this yields the solution

where the signal $s(t)$ is defined in Eq. (39). For example, in the situation where $z(0)=0$ and arbitrary $x(0)$ and $y(0)$ (for example a completely mixed state or an $x$-eigenstate) the conditional variance becomes

Taking the ensemble average of Eq. (45) gives

However, when $\gamma_{1}$ is non-zero, the conditional variance always eventually goes to zero which signifies that we are eventually certain that the system is in its ground state. This is of course not a useful way to define the measurement time. To take into account relaxation, in Ref. gambetta:2007a we studied this question by considering an effective classical model for a dispersive quantum non-demolition measurement which turns out to be equivalent to the model presented here when the initial state of the qubit is either $z(0)=\pm 1$ or a mixture of both (i.e., $x(0)=y(0)=0$). That is, Eq. (32) is the same as the Kushner-Stratonovich equation that describes a measurement of a two state system with Gaussian noise. For completeness and since this is a good approach in the situation where relaxation is non-zero, we will therefore briefly review the relevant results of Ref. gambetta:2007a.

V.3 Initial state fidelity

In Ref. gambetta:2007a, to quantify how good a measurement is at revealing the initial state of the qubit, we imagine preparing the qubit in either the $z(0)=\pm 1$ state and then generate fictitious records ${\bar{J}}(t)$ from that input. Given these fictitious measurement records, but now assuming ignorance of the initial state, we then ask what was the initial state of the qubit. To quantify the efficiency of the measurement at revealing the correct input state, we use the following fidelity measure $F$: the number of correct assignments minus the number of incorrect assignments of the initial state, normalized by the total number of assignments. This measure will range from $1$, indicating that the correct initial state was correctly found for each record, to $-1$, indicating that a wrong assigned was realized for each record. A fidelity of $0$ implies that the assignments are completely random.

where $P(z_{0}=\pm 1|{\bar{J}},t)$ is the probability at time $t$ that the initial state was $\pm 1$ given the record ${\bar{J}}(t)$. This conditional probability is given by Bayes theorem

where $P(z_{0})$ is the initial state probability which we take to be 1/2 for both $i=-1$ and $1$. By introducing a fictitious unravelling of $\gamma_{1}$ (see Refs. gambetta:2005a; gambetta:2007a), the estimate can be rewritten as

and $s(t)$ is the integrated signal defined in Eq. (39).

and using the above definition of the fidelity

we obtain the following compact expression

VI Emergence of quantum jumps

Experimentally, one does not have direct access to $z_{J}(t)$ but reconstructs this quantity from the measured current $J(t)$. As a result, a more striking demonstration of quantum jumps would be to see these jumps directly in $J(t)$. However, due to white noise fluctuations $\xi(t)$, without first averaging the signal over some finite integration time the jumps are not visible in most trajectories. As a result, we consider the quantity

Finally, an interesting situation occurs if the $\gamma_{1}$ channel can be directed (corresponding to the fluorescence of the “atom”). When we register that the qubit has jumped, we know that the energy has been dissipated into the $\gamma_{1}$ channel. If this fluorescence is directed to an additional low-Q resonator, this can be used as a single microwave photon source with a well-defined emission time. Inversely, if the qubit is prepared in the ground state this second resonator can be used as a single photon detector. When a jumped of the qubit is registered, we know that a photon was present in the second cavity. The efficiency of this detector will be limited by the Zeno effect A. Blais, J. Gambetta, C. Cheung, R. Schoelkopf, and S. M. Girvin, manuscript in preparation.

VII Weak Driving of the Qubit – Zeno Effect

In the previous sections, we considered the dynamics of the qubit under measurement. In this section, we will include the possibility of having a control drive. As explained in Sec. II, this is done in practice by adding a drive on the input port of the resonator at the qubit transition frequency. Our goal is to use the transformation Eq. (10) to again obtain an effective SME for the qubit only. However, as shown in appendix C, in the transformed frame, the drive on the qubit induces transitions between the different (transformed) states of the resonator. As a result, it is not possible to integrate out exactly the resonator from the effective description of the qubit.

However, in the small $\beta$ limit, we find we find that qubit evolution is well described by the SME

To see this more clearly, the above SME can be written in terms of the components of the Bloch vectors as

To illustrate this, we have numerically integrated the full homodyne SME (21), with the addition of a qubit drive, for 2 different measurement drives $\varepsilon_{d}/2\pi=0.9$ MHz and 20 MHz, and a Rabi frequency of $\Omega_{R}/2\pi=2.5$ MHz. The results are shown in Fig. 11. From this figure, we see that for the low measurement amplitude (pannel A), measurement only causes small amplitude noise on the Rabi oscillations. However, for the larger amplitude (panel B) the conditional dynamics is jump-like. We note that the rate of the jumps between the ground and excited state is, as expected from the above discussion, slower then the rate $\Omega_{R}$ at which the Rabi drive would coherently drive the system. Note that the x-axis scale is not the same in pannel A and B(C).

These jumps can be observed directly in the signal $J(t)$. This is illustrated in Fig. 11 C) for the case of the larger measurement amplitude. As in the previous section, we are showing here the integrated signal, defined in Eq. (55), and with a integration time $\delta t=32$ ns. The integrated signal clearly reveals the jumps.

We note that the numerical results presented here were obtained using the full homodyne SME, and not the simplified SME (56). This is because while this simplified equation contains the essential features, it is not strictly valid in the limit of large mean photon number. However, by inspecting individual trajectories, we find that trajectories obtained from the full model display the same qualitative behavior as those obtained from the effective model. The difference between the two types of trajectories appear to be only small changes in renormalized system parameters.

VIII Conclusion

We have investigated the measurement of a superconducting qubit using a dispersively coupled resonator. From the fully quantum mechanical model for the resonator and qubit dispersive interaction, we have found an effective master equation for the qubit (only) under the presence of a measurement drive on the resonator. This was done by moving to a frame which takes into account the entanglement between the resonator and the qubit. With respect to the bare qubit-resonator master equation, the effective qubit master equation contains an additional, time-dependent decay channel which has the form of a dephasing process and which depends on the number of photons populating the resonator. This is referred to as measurement-induced dephasing devoret:2000a; averin:2002a; clerk:2003a; makhlin:2001a; gambetta:2006a. In addition to the extra dephasing channel, the photon population of the resonator is also responsible for an ac-Stark shift of the qubit transition frequency. As was shown in Refs. dykman:1987a; gambetta:2006a, and later reported experimentally in Ref. schuster:2007a, these effects lead to number splitting of the qubit spectrum in the limit where $\chi\gg\kappa$. We thus have obtained a very simple to solve (2 dimensional Hilbert space) effective model for the qubit dynamics that contains the essential physics. We believe that this is an interesting analytical, and numerical, tool to further explore the dispersive regime.

We then focused on the situation where one is continuously monitoring the voltage at the output port of the resonator. In that situation, the conditional state of the combined system evolves according to the homodyne SME wiseman:1993a. Again moving to a frame that takes into account the qubit-resonator entanglement, we then obtained an effective SME for the qubit that is valid in the limit $\gamma_{1}\ll\kappa$. The SME corresponds to a weak measurement of the qubit operator $\sigma_{z}$. As a result, the measurement stochastically projects the conditional state into eigenstates of $\sigma_{z}$. This is similar to the result that was obtained in Refs. korotkov:1999b; korotkov:2001a; korotkov:2001b; korotkov:2003a; wiseman:2001b; goan:2001a; oxtoby:2005a; oxtoby:2007a for a double quantum dot qubit monitored continuously by a quantum point contact (although the results are presented in very different forms in all of these references).

From these results, we then showed how, in the limit that the measurement becomes strong, there is a cross-over in the qualitative behavior of the trajectories from diffusive to jump-like. This process relies on the $T_{1}$ decay channel for the qubit to jump from the excited to the ground state. As a result, we expect to see quantum jumps, with Poisson statistics, in the measured homodyne current.

Overall, the model obtained here can be used to obtain qualitative information about the qubit dynamics in the presence of dissipation, measurement and control. We also gave a numerical prescription to go beyond this simple model.

Appendix A Derivation of the effective qubit master equation

In this appendix we show how to obtain the effective qubit master equation Eq. (12) from the full dispersive qubit-resonator master equation Eq. (3). For this purpose, we introduce the the transformation

where $\Pi_{e,g}$ are the projectors on the qubit excited and ground state and $D(\alpha)$ is the field displacement operator walls:1994a.

In this frame, the state of the combined qubit-resonator system $\varrho^{\mathbf{P}}=\mathbf{P}^{\dagger}\varrho\mathbf{P}$ can be written in the energy basis as

Using this expression for the system’s density matrix in the transformed frame, it is simple to express the qubit’s reduced density matrix in laboratory frame as

and $d_{p,q}$ is the matrix element of the displacement operator in the photon number basis

In the remainder of this appendix, we show how by using the transformed master equation for $\varrho^{\mathbf{P}}$ it is possible to obtain the coefficients entering in the expression Eq. (61) for $\rho$.

In the displaced frame, the state matrix $\varrho^{\mathbf{P}}$ obeys the master equation

with $O^{\mathbf{P}}=\mathbf{P}^{\dagger}O\mathbf{P}$ the transformed operators and where the effective Hamiltonian $H_{\rm eff}$ is given by Eq. (2). Using the standard results

Using these results, we then have for the transformed Hamiltonian

We now turn to the damping superoperators. For the field damping, we find

The last terms to take into account are those in the last line of Eq. (64). For these terms, we use the fact that

Choosing $\dot{\alpha}_{e}$ and $\dot{\alpha}_{g}$ in the above expression as in Eq. (7), we finally obtain the transformed master equation

where we have defined the time-dependent shifted qubit transition frequency as

In this transformed frame, the last two terms of Eq. (77) appear because the resonator acts as a non-Markovian bath for the qubit imamoglu:1994a; gambetta:2002a. Moreover, we note that the term proportional to $\gamma_{1}$ in Eq. (77) can be expressed as

Upon taking the trace over the resonator space, this expression will simply take the form of regular Markovian $\gamma_{1}$ damping on the qubit. This is one of the features of the transformed master equation that will allow us below to obtain a simple effective master equation for the qubit only.

A.2 Moving back to the laboratory frame

Using Eq. (60) and Eq. (77), we get the following set of coupled differential equations for the coefficients of the laboratory frame reduced qubit density matrix Eq. (61)

where (82) was obtained by differentiating (62), with respect to time and

To reconstruct the effective master equation from these amplitudes, we note that the time-derivative of the $|{e}\rangle\langle{e}|$ component of Eq. (61) can be written as

where in the last equality we have used the identity

The off diagonal terms are the hardest to deal with, but because in the transformed frame, the photon population is initially zero and there is no mechanism to populate $\lambda_{n,m,p,q}$ terms with $\{n,m,p,q\}>0$ we have

The above expressions correspond to the following effective master equation for the qubit

Appendix B Derivation of the effective qubit stochastic master equation

To derive the effective qubit stochastic master equation we start by using the linear form gambetta:2005a; wiseman:1996a; goetsch:1994a of the full SME Eq. (21). This is

where the bar here is used to signify that the state is not normalized and the linear measurement superoperator is

Moving to the frame defined by Eq. (10) gives

In the same way as Eqs. (80) – (83) were obtained, we can write equations of motion for the coefficients ${\bar{\varrho}^{\mathbf{P}}}_{n,m,e,e}$, ${\bar{\varrho}^{\mathbf{P}}}_{n,m,g,g}$ and $\bar{\lambda}_{n,m,p,q}$ of the decomposition of the conditional density matrix ${\bar{\varrho}^{\mathbf{P}}}_{J}$ in the energy basis. The first term of Eq. (93) yields the same terms as in Eqs. (80) – (83). As a result, we get

In the above expressions, the equation numbers refer to the RHS of the corresponding expressions. They are the contribution of the Lindblad term ${\cal L}_{\rm tot}{\bar{\varrho}^{\mathbf{P}}}_{J}$.

By noting that these matrix elements are not coupled in the qubit basis, we can use the same arguments that were used to obtain Eq. (89). Doing this, we find for the stochastic parts

The amplitude $\bar{\varrho}^{\mathbf{P}}_{1,l}$ obeys the equations of motion

and we find a similar equation for $d_{t}\bar{\varrho}^{\mathbf{P}}_{1,r}$. From this result, we find that provided that

the effect of $\bar{\varrho}^{\mathbf{P}}_{1,l}$ and $\bar{\varrho}^{\mathbf{P}}_{1,r}$ on $\bar{\rho}^{\mathbf{P}}_{g,g}$ will be very small. In this situation, it is possible to construct a laboratory frame linear SME for the qubit only:

Using Eq. (31) and normalizing the above SME gives Eq. (27) with measurement record statistics given by Eq. (30).

Appendix C Including coherent control of the qubit

A drive on the input port of the resonator at a frequency $\omega_{c}$ close to the qubit transition frequency can be used to coherently control the state of the qubit. As shown in Refs. blais:2004a; blais:2007a, this can be represented by the term

in the qubit–resonator Hamiltonian Eq. (2). In this expression, $\Omega_{R}$ is the Rabi amplitude which depends on the drive amplitude and its detuning to the resonator frequency.

To see how this changes the effective qubit master equations obtained in the previous appendices, we apply the transformation of Eq. (10). In a frame rotating at the frequency frequency $\omega_{c}$, we find

In this transformed frame, it is apparent that when the qubit flips, it must ‘drag’ the photon field populating the resonator. In this situation, the equation of motion for the coefficients of Eq. (61) are modified in the following way

where the equation numbers refer to the RHS of the corresponding expressions. These additional terms mix the various coefficients of the decomposition of $\rho$ and obtaining an effective master equation for the qubit only is no longer possible exactly. This can nevertheless be done in the small measurement amplitude limit.

The case where a control and a measurement drive are acting on the system simultaneously is most interesting in the case where the measurement drive is of small amplitude. For example, in Ref. wallraff:2005a, a weak continuous measurement drive corresponding to about one photon was used to monitor Rabi oscillations. When the measurement drive amplitude is increased, the qubit is dephased faster and coherent control is realized with less fidelity.

In the weak measurement limit $\beta\rightarrow 0$, the matrix element of the displacement operator $d_{n,m}$ in the photon number basis can be approximated to

where $L_{n}^{m}(x)$ is an associated Laguerre polynomial. To lowest order in $\beta$, this reduces $d_{p,q}\sim\delta_{p,q}$. Making this replacement in the above expressions for the amplitudes, we find in this limit the following effective qubit master equation

Unsurprisingly, in the small $\beta$ limit, the only effect of a control drive is to flip the qubit at the Rabi frequency $\Omega_{R}$. If we were to include the next order in $d_{p,q}$ we would not be able to obtain an equation for just the qubit, effects such as sidebands would be observed.

References