Time-resolved qubit readout via nonlinear Josephson inductance

We propose a generalization of dispersive qubit readout that provides the time evolution of a flux qubit observable. Our proposal relies on the nonlinear coupling of the qubit to a harmonic oscillator with high frequency, representing a dc superconducting quantum interference device. Information about the qubit dynamics is obtained by recording the oscillator response to resonant driving and subsequent lock-in detection. The measurement process is simulated for the example of coherent qubit oscillations. This corroborates the underlying measurement relation and also reveals that the measurement scheme possesses low backaction and high fidelity.


Introduction
The question of how to gain information about the state of a quantum system has intrigued researchers since the early days of quantum mechanics.With the advent of quantum computation, this fundamental question became also of practical interest, mainly because the final stage of a quantum algorithm necessarily is qubit readout.This task only requires distinguishing between two particular qubit states and, thus, can be achieved by projective measurements.Nevertheless, going beyond readout is of interest as well, since one also desires direct experimental evidence for coherent superpositions emerging, e.g., from tunnelling oscillations.
In order to obtain a quantum mechanical description of a measurement process, one usually models the measurement apparatus as a macroscopic quantum environment, i. e., as a heat bath, where the pointer of the apparatus corresponds to an effective bath coordinate.
When interacting with the central quantum system, the bath acquires information about the system state.Owing to the macroscopic nature of the bath, one may assume that already a fraction of the bath possesses the full information about the effective pointer coordinate [1].Therefore one can obtain knowledge of the pointer position without violating fundamental laws of quantum mechanics.
Recently, superconducting quantum circuits have provided a new arena to test fundamental questions of quantum mechanics in the laboratory.Prominent examples are the demonstration of coherent time evolution in charge qubits [2] and of Berry phases [3], as well as testing Bell inequalities [4].Above all, different protocols for quantum measurement were successfully implemented in circuit quantum electrodynamics [5][6][7][8].For a superconducting solid-state qubit, the practical measurement of one of its coordinates is performed by coupling it to a macroscopic environment, as given by external circuitry, via a quantum point contact [9,10] or a harmonic oscillator.Depending on the setup, the oscillator is realised by a dc superconducting quantum interference device (SQUID) [11] or a superconducting resonator [12].In both cases, the resonance frequency of the oscillator depends on the qubit state.Consequently, the response of the oscillator to a close-to-resonant ac-excitation possesses a phase shift which can be measured, and from which one can infer the qubit state.First experiments in this direction worked with an oscillator whose frequency was much lower than the qubit splitting [5,13,14].More recent experiments [6,15] operated in the so-called dispersive regime, where the oscillator frequency and the qubit splitting are of the same order, while their detuning is still larger than their mutual coupling strength.A crucial detail is that the oscillator frequency and bandwidth naturally limit the time resolution in such qubit measurements.Thus, using the said schemes with slow oscillators, it is only possible to extract time-averaged information about the qubit state in general, but there is no possibility to resolve its dynamics in time.
Recently, a first step towards a time-resolved measurement of qubit dynamics has been proposed [16,17]: When a weak high-frequency field acts directly on the qubit, the reflected signal acquires a time-dependent phase shift by harmonic mixing.Lock-in amplification of the reflected signal then allows obtaining information about the qubit dynamics.In this work, we combine both approaches and extend the scheme of Refs.[16,17] to a qubit coupled to a driven high-frequency oscillator.A measurement protocol for such a setup is particularly appealing because an oscillation mode is part of most recent superconducting qubit designs.Moreover, the oscillator serves as filter for quantum noise and, thus, reduces qubit decoherence.Here, we focus on a flux qubit embraced by a dc-SQUID, whose fundamental frequency may even be tunable to some extent [18,19].As a particular feature of this realisation, the qubit-oscillator coupling is non-linear in the oscillator coordinate, that is, the coupling possesses both a significant linear and quadratic contribution.It will turn out that for realistic parameters, the measurement scheme relies on the quadratic part of the coupling.The paper is structured as follows.In section 2, we introduce our model and discuss dispersive qubit readout in generalised terms.The central relation upon which our measurement scheme relies is derived in section 3. Section 4 is devoted to numerical studies in which we test our measurement relation and work out quantitatively measurement fidelity and backaction.Furthermore, we provide an estimate of the signal-to-noise ratio and discuss realistic parameters for a possible experimental implementation.The appendix contains details about the derivation of the measurement relation, the input-output formalism [20] and the Bloch-Redfield master equation which we use for obtaining numerical results.

System-bath model
We consider a superconducting flux qubit coupled to a SQUID [7] as sketched in figure 1.The SQUID is modelled as a harmonic oscillator, which gives rise to the Hamiltonian [6,7,18,21] The first term represents the qubit with energy splitting hω qb = h(ε 2 + δ 2 ) 1/2 and the mixing angle θ = arctan(δ /ε) which depends on the controllable qubit bias energy ε and the qubit gap energy δ , while σ x,z denote the Pauli matrices.The second term describes the oscillator with frequency Ω and the bosonic creation and annihilation operators a † and a, respectively.The qubit couples to the oscillator in two ways.First, via dipole interaction with strength g 1 , which is linear in the oscillator coordinate a + a † .Up to order g 2 1 , this causes a frequency shift for both the oscillator and the qubit.The second coupling term proportional to g 2 , by contrast, is quadratic in the oscillator coordinate.Its physical origin is a non-linear Josephson inductance which depends on the magnetic flux, by which the SQUID is penetrated [7,18].This term provides a frequency shift already in first order of g 2 .The interaction coefficients g 1 and g 2 can be controlled to some extent, as an expansion of the qubit-SQUID interaction to second order in the oscillator coordinate demonstrates [18].In detail, for a small SQUID bias current I b , the coupling coefficient g 1 depends linearly on I b , whereas g 2 is independent of the latter, as we discuss in section 4.4.For I b = 0, one even obtains g 1 = 0, such that the qubit couples only to the square of the oscillator position [7,18].
Regarding a time-resolved measurement of the qubit dynamics via the oscillator, it will turn out that for realistic parameters of flux qubits, this quadratic coupling is crucial, while the linear coupling turns out to be typically too weak.For common circuit-QED setups using charge and flux qubits coupled to a transmission line resonator [11,12,22,23], not only g 1 but also g 2 is too small.Thus, we henceforth focus on setups of flux qubits coupled to SQUIDs possessing a sizable quadratic coupling, as described above.
The qubit-SQUID system is further coupled to external circuitry, which acts as a dissipative environment and is modelled by the system-bath Hamiltonian [24][25][26][27] Here, Q = a † + a is the oscillator coordinate, such that the interaction term represents the inductive coupling between the qubit and the flux degree of freedom of the SQUID.
The system-bath interaction can be fully characterised by the spectral density which is proportional to the real part of the effective impedance of the environment [28].Here we assume an ohmic spectral density, J(ω) = αω, for which the dimensionless damping strength α can be interpreted as effective resistance [29][30][31].

Qubit-oscillator interaction in the dispersive limit
We are interested in the dispersive limit which is characterised by a detuning ∆ = Ω − ω qb larger than the qubit-oscillator couplings, It is then convenient to go to the dispersive picture via the unitary transformation (A.2).As detailed in Appendix A, this yields the effective Hamiltonian [32,33] where the transformed bosonic operators ā and ā † are defined in equation (A.10).The qubitoscillator coupling has been removed formally by shifting it to the operator-valued oscillator frequency where the overbar denotes the dispersive picture, while the qubit operator determines the coupling.The interpretation of equations ( 5) and ( 6) is that the oscillator frequency depends on the qubit state.This allows dispersive qubit readout by measuring the associated phase shift of the oscillator response upon resonant driving, like in the case of a classical oscillator that is driven by an external force.In particular, assuming cos θ = 0 and g 2 = 0, equation (5) predicts the frequency shift Ω = Ω+σ z g 2 1 [1/∆−1/(Ω+ω qb )] −1 .The last contribution in Ω stems from counter-rotating terms in the qubit-oscillator interaction.These must be accounted for the case of large detuning ∆ where a rotating-wave approximation produces inaccurate results [34].Depending on the qubit expectation value σ z , the oscillator is red or blue detuned.Thus, we obtain in this limit the well-known qubit-dependent phase shift corroborated in various experimental realisations [5][6][7][8]22].There, however, the oscillator frequency was smaller than the qubit splitting, Ω ≪ ω qb .As a consequence, it was only possible to obtain time-averaged information about the qubit state.Now the goal of this paper is a generalisation of dispersive qubit readout such that time-resolved information about the qubit state can be obtained as well.This obviously requires oscillator frequencies and bandwidths larger than the qubit transition frequency, that is, Ω ≫ ω qb .We emphasise that equations ( 5) and ( 6) are nevertheless valid as long as the coupling constants are small enough to fulfil condition (3); for details see Appendix A and Ref. [34].If the qubit dynamics is much slower than the oscillator, the qubit can be treated within an adiabatic approximation.This means that the qubit dynamics is assumed to be constant during one oscillator period.In turn, the time evolution of the oscillator depends on the instantaneous qubit state.Then the Schrödinger-picture operators σ x,z in equation ( 6) can be replaced by their time-dependent expectation values, and the operator-valued quantity ω is substituted by Equation ( 7) implies that information about the time-dependent qubit state is encoded in the effective oscillator frequency Ω ≡ Ω(t).This gets expressed as a slow parametric modulation in time, like for a parametric oscillator.In detail, the instantaneous qubit state enters via the qubit expectation values σ x,z t ≡ Tr qb {σ x,z ρ 0 (t)}, where Tr qb denotes the partial trace over the qubit degrees of freedom.The time dependence, indicated by the subscript . . .t , stems from the evolution of the total qubit-oscillator state ρ 0 (t) under the effective system-bath Hamiltonian (A.15).
As an important intermediate result, the found modulation of Ω in time can be traced back to the qubit dynamics in the absence of the driving.This enables measuring the qubit's time evolution via the oscillator response to resonant driving.

Time-resolved measurement of the qubit dynamics
The qubit-oscillator Hamiltonian in the dispersive picture, equation ( 4), together with the effective, modulated frequency (5) already indicates that the oscillator detuning may contain information about the qubit dynamics.As in the case of the traditional dispersive readout, we consider the response of the system to an ac-field that is resonant with the oscillator.Physically, the situation corresponds to that of a classical mechanical oscillator driven by an external periodic force.Owing to the only weak dissipation, the response is manifest in the phase of the reflected ac driving.In the following, we establish a relation between this phase and a time-dependent qubit expectation value.This relation will form the basis of our measurement protocol.

Response of the qubit-oscillator compound to resonant driving
In the theory of optical cavities, the response to an external ac excitation is conveniently calculated with the input-output formalism [20,35].This formalism has also been applied to quantum circuits [5,13,14,16].Its cornerstone is the relation ξ formulated in the Heisenberg picture and derived in Appendix B. It relates the incoming and the outgoing fluctuations of the transmission line, ξ in/out (t), to the time-derivative of the system-bath coupling operator, which in our case is Q = a + a † .The dimensionless dissipation strength α of the ohmic spectral density quantifies the coupling between the oscillator and the electric environment and, thus, appears as prefactor.An ac-driving corresponds to a coherently excited incoming mode, such that the fluctuations can be separated into quantum fluctuations ξ qm in (t) and a deterministic component A cos(Ω ac t).Here, the deterministic part is an ac-field in resonance with the bare oscillator, which implies the expectation value ξ in (t) = A cos(Ωt).Then the input-output relation (8) becomes ξ out (t) = ξ qm in (t) + A cos(Ωt) + 2α Q and, thus, the corresponding expectation value of the outgoing signal reads Also here, it is convenient to work in the dispersive picture obtained by the unitary transformation (A.2).While this leaves the environment operators unchanged, the coordinate by which the oscillator couples to the environment changes as 14).The time-derivative d Q/dt can be obtained from the commutator of Q with the Hamiltonian (4) augmented by a term that describes the driving.This yields terms of the order Ω and terms with prefactors ω qb and g 1 /∆.For a fast oscillator, the latter terms can be neglected, and we obtain the equation of motion This linear, inhomogeneous equation is readily solved with the help of the Green's function for the dissipative harmonic oscillator.Inserting the resulting d Q/dt into the input-output relation (10) and neglecting transient terms yields for the expectation value of the outgoing signal the expression Owing to the weak dissipation, the system energy is almost preserved, such that the amplitude of the incoming and the outgoing signal are practically the same.The phase difference stems from the coupling to the qubit which detunes the oscillator, while the slow time evolution of the qubit renders the phase time-dependent.The approximation is valid if the qubit-oscillator couplings are smaller than the oscillator damping rate, i.e., for g 1 , g 2 ≪ αΩ.
In other words, the first term in the denominator is negligible, since the qubit-induced frequency shift Ω − Ω is of the order g 1,2 .This also ensures ϕ ≪ 1 and, thus, ϕ ≈ tan ϕ.
Next, we insert the effective frequency (5) together with equation ( 7) and obtain to second order in g 1 and first order in g 2 the phase shift This central relation forms the basis for our non-invasive qubit measurement via a resonantly driven harmonic oscillator.It identifies a set of qubit observables, which generate the lowfrequency system dynamics, as the cause of a small phase shift between the ingoing and outgoing signal.In other words, equation ( 14) enables one to monitor the qubit dynamics by continuously measuring the phase shift ϕ(t) with suitable experimental techniques.By evaluating the prefactor for specific setups, we will see below that our measurement scheme is particularly feasible for flux qubits.In this case, the last term of the phase shift ( 14) dominates, and one measures the qubit variable σ z cos θ − σ x sin θ , i.e., the flux degree of freedom by which the qubit couples to the SQUID; cf. the model Hamiltonian (1).

Static versus dynamical phase shift
The terms entering the phase shift ϕ(t) may be static as well as dynamical.In the first instance, this depends on whether or not the related qubit observables undergo any time evolution.At this point, further insight is obtained by a closer look to the Heisenberg equations of motion for the qubit operators σ x and σ z .In the dispersive picture, valid under conditions (3), they are derived from the effective Hamiltonian H0 , given by equation ( 4), and read Thus, in the dispersive qubit-oscillator coupling limit (see section 2.2), the observable σ z is a constant of motion.As a consequence, those contributions to ϕ(t) that depend on σ z t ≡ σ z const are time-independent.This corresponds again to the established scheme for non-invasive qubit state readout.On the contrary, the observable σ x possesses a non-trivial time dependence generated by H0 .Thus, σ x t renders the phase shift ϕ(t) dynamical.This, in turn, enables a time-resolved single-run measurement of the unitary qubit evolution by means of the qubit observable σ x .According to our measurement relation (14), the dynamical phase signal has the amplitude Interestingly, ϕ x max is reciprocal to the damping strength and the oscillator half-bandwidth αΩ.Thus, a large oscillator frequency together with strong damping lead to reduced angular visibility.On the other hand, the adiabatic treatment of the qubit underlying relation (14) becomes invalid if either Ω or α are too small.Moreover, the input-output relation (10) crucially relies on finite damping.Thus, appropriate choices for Ω and α need to be based upon a compromise between good phase resolution and the validity of our approximations.We go into further detail about this issue when discussing the measurement quality in section 4.2.
It is important to note that the amplitude ϕ x max possesses contributions from both the linear and the quadratic qubit-oscillator interaction of Hamiltonian (1).The first term on the r.h.s. of equation ( 17) stems from the linear interaction characterised by the coupling coefficient g 1 .Like the effective Hamiltonian (A.13), this contribution is of second order in the dispersive parameter g 1 /∆.Due to the minus sign inside the round brackets, it is additionally minimised, given that ∆/(Ω + ω qb ) ≈ 1 for large detuning ∆.Thus, for a Cooper- pair box or a flux qubit coupled to a high-frequency transmission line resonator, where the qubit-oscillator coupling merely depends on the circuit characteristics and therefore is purely linear, i. e. g 2 = 0, the maximum amplitude ϕ x max drops below any useful level.On the contrary, a finite quadratic qubit-oscillator interaction g 2 > 0 ensures a noticeable phase signal, independent of the detuning ∆, and even if g 2 ≪ g 1 .Formally, this feature arises from Hamiltonian (1), where the quadratic coupling term already generates a frequency shift in zeroth order perturbation theory.If the qubit-oscillator interaction is transverse, that is, ε = 0, the phase resolution is maximised, whereas it vanishes for purely longitudinal coupling, i.e., for δ = 0. Hence the presence of a non-linear qubit-oscillator interaction, as provided by a non-linear SQUID Josephson inductance, turns out to be a crucial ingredient.

Measurement quality
Still it remains to corroborate the central measurement relations ( 14) and (17), respectively, via comparing them with the phase shift obtained by simulating the actual measurement process.In doing so, we restrict ourselves to the fundamental example of coherent qubit oscillations.For the numerical treatment of the qubit-oscillator state, we employ the quantum master equation (C.1) derived from the full dissipative qubit-oscillator-bath Hamiltonian (2).For a realistic evaluation, we use parameters similar to those of the experiment reported in references [18,19].Furthermore, we employ N = 10 oscillator states, which turns out to be sufficient to reach numerical convergence.

Time-resolved measurement of unitary qubit evolution
If the qubit is only weakly coupled to the oscillator, and if the latter is driven only weakly, the qubit's time evolution is rather coherent (see section 4.3 on qubit decoherence).For this scenario, figure 2(a) depicts the time-dependent phase ϕ(t) computed with the measurement relation (14), while the inset confirms its proportionality to the qubit expectation value σ x t .For a comparison, we wish to recover this phase information directly by analysing the outgoing signal ξ out (t) , as given by equation (10).In an experiment, this can be achieved by lock-in techniques which we mimic in the following way [36]: First, we focus on the associated spectrum ξ out (ω) depicted in figure 2(b).It reflects the qubit dynamics in terms of two sidebands around the central peak related to the oscillator frequency, here chosen as Ω = 10 ω qb .The dissipative influence of the environment, modelled by a transmission line (see figure 1), is reflected in a broadening of this peak.The corresponding oscillator bandwidth is given as 2αΩ, where α denotes the dimensionless damping strength; see Appendix C. Here, we recall that the oscillator is driven resonantly by the external driving signal A cos(Ω ac t), that is, Ω = Ω ac .In the time domain, the sidebands correspond to the phase-shifted signal ξ out (t) = A cos(Ωt −ϕ exp (t)) with slowly time-dependent phase ϕ exp (t).In order to obtain this phase ϕ exp (t), we select the spectral data from a frequency window of size 2∆Ω centred at the oscillator frequency Ω, which means that ξ out (ω) is multiplied with a Gaussian window function exp(−(ω − Ω) 2 /∆Ω 2 ).We choose for the window size the resonator bandwidth, ∆Ω = αΩ, which turns out to suppress disturbing contributions from the low-frequency qubit dynamics.Finally, we centre the clipped spectrum at zero frequency and perform an inverse Fourier transform to the time domain.If the phase shift φ ext was constant, one could use a much smaller measurement bandwidth.Then the outcome of the measurement procedure would correspond to homodyne detection [35] of a quadrature defined by the phase shift and yield a value ∝ cos φ exp .
Figure 2(a) reveals the good agreement of the resulting ϕ exp (t) with the prediction of our measurement relation, ϕ(t) ∝ σ x t , at angular resolutions of 1-2 • .Good agreement is already obtained for a oscillator frequency Ω = 10 ω qb , which obviously represents a good compromise between the validity of the adiabatic approximation (see section 2.2) and a sufficiently strong signal.There is even some room for obtaining a stronger phase signal since the dissipation strength α still can be reduced without violating the validity range of our theory as long as ω qb ∆Ω.The dimensionless oscillator dissipation strength is α = 0.12.The resonator bandwidth is given by 2αΩ = 2.4ω qb .(a) Lock- in amplified phase ϕ exp (t) (dashed green lines), compared to the estimated phase ϕ(t) (solid red line) of the outgoing signal ξ out (t) .Here, ϕ(t) ∝ σ x t [cf.equation ( 14)], which is corroborated by the inset showing that σ x t performs oscillations with (angular) frequency ω qb .(b) Power spectrum ξ out (ω) for the resonantly driven oscillator (blue solid line).The sidebands stemming from the qubit dynamics are visible at frequencies Ω ± ω qb .In order to extract the phase information, we apply a Gaussian window function with respect to the frequency window of half-width ∆Ω = 1.2ω qb , which turns out to be the optimal value for the measurement bandwidth.
Setting either g 1 or g 2 to zero (not shown) reveals that the non-linear coupling g 2 is responsible for the good agreement of the phase shifts in figure 2(a).Thus, the whole protocol is mainly applicable to flux qubits coupled to SQUIDs.For charge qubits, by contrast, the typical values for g 2 are too small.Furthermore, we have verified that the visible constant delay between both phases ϕ(t) and ϕ exp (t) does not depend on the selected parameters, while its detailed origin remains unexplained.

Measurement characterisation: fidelity and backaction
The validity of relation (14) for the phase ϕ(t) is naturally limited to specific parameter ranges due to the various underlying approximations made.The main crucial assumptions to justify the adiabatic treatment of the qubit are a large qubit-oscillator detuning ∆/Ω ≃ 1 and weak mutual interaction, g 2 ≪ g 1 ≪ ∆.Furthermore, the oscillator damping α is assumed to stay within the limits g 1,2 /Ω ≪ α ≪ 1.
In an experiment, the oscillator frequency and the coupling strength are finite, though.Consequently, the actual phase ϕ exp (t), which we extract numerically and which can be measured by lock-in amplification, generally differs from the predicted phase ϕ(t).Thus, the mutual agreement of both phases needs to be tested quantitatively for realistic scenarios.To this end, we employ the measurement fidelity F with the scaled overlap defined as The ideal value of F = 1 is assumed if ϕ(t) ∝ ϕ exp (t).
In figure 3(a) we depict the fidelity defect δ F = 1 − F between ϕ exp (t) and ϕ(t) as a function of the oscillator frequency Ω = Ω ac for different quadratic coupling coefficients g 2 .As expected, the overall fidelity is rather insufficient for small oscillator frequency Ω < 10ω qb , for which the adiabatic approximation of section 2.2 is not valid and, moreover, if the oscillator bandwidth is too small to resolve the qubit dynamics, i.e., if ω qb < αΩ.Along with increasing Ω, the fidelity defect δ F drops to values of 0.05-0.5,independently of the parameter g 2 .Taking into account that the fidelity is arbitrarily lowered by the constant delay between ϕ(t) and ϕ exp (t) visible in figure 2, this still corroborates that Ω = 10ω qb is a good choice.In the limit of large oscillator frequencies, we again observe an increase of the fidelity defect, which occurs the sooner the smaller g 2 .This latter effect, which is only visible for the smallest value of g 2 in figure 3(a), is directly explained by a reduced maximum angular visibility of the phase ϕ(t) ∝ g 2 /Ω.Thus, figure 3(a) provides a pertinent indication for the validity frame of our central relation (14).
Moreover, it is necessary to take into account the backaction upon the qubit that stems from the non-linear qubit-oscillator interaction.An appropriate measure for how much the qubit dynamics is perturbed by the oscillator is given by the time-average D of the trace distance D(t) = 1  2 Tr|ρ qb (t) − ρ qb,0 (t)| between the qubit dynamics with and without the coupling to the driven oscillator.To be specific, we compare the qubit state ρ qb (t) evolving under the full system-bath Hamiltonian (2) to an unperturbed reference state ρ qb,0 (t) that evolves unitarily under the bare qubit Hamiltonian H qb = (hω qb /2)σ z .Thus, the trace distance essentially quantifies the invasiveness of the measurement based upon the secondorder qubit-oscillator interaction.In the absence of perturbations to the qubit, D vanishes by definition, while D = 1 if the density operator of the measured qubit is completely unrelated to that of the reference.
Figure 3(b) shows that the predicted phase ϕ(t) faithfully describes the unperturbed qubit dynamics as long as the coefficient g 2 stays sufficiently small.A reliable operating range appears to be g 2 /ω qb 0.01.For Ω = 10 ω qb , this is consistent with our above reasoning regarding the fidelity.For even weaker quadratic interactions, we first find D ∝ Ω −3 , which implies that the dispersive first-order coupling in terms of g 1 governs the qubit-oscillator interaction when Ω is small.This cubic dependence is due to relation (14) and to the fact that the first-order perturbation acting on the qubit has an inverse quadratic dependence on the detuning ∆ ∝ Ω.Beyond a critical detuning, which individually depends on g 2 , the quadratic interaction prevails again, as is reflected by the saturation of D with increasing oscillator frequency Ω.For g 2 /ω qb 0.05, the effect of the linear coupling withers.Thus, at g 2 = 0.01ω qb , the value of g 1 is rather irrelevant for our measurement protocol.

Signal-to-noise ratio
Generally, a signal can be resolved only if its spectral density exceeds the level of background noise at the measurement frequency.In the present scheme, the desired information is contained in the sidepeaks of the spectrum at Ω ± ω qb ; see figure 2(b).For the corresponding phase-modulated oscillation A cos(Ωt − ϕ max sin(ω qb t)), these sidepeaks correspond in the time domain to the oscillation Aϕ max sin(ω qb t).If the signal is integrated for a time t, its spectral weight becomes (Aϕ max ) 2 t.The phase amplitude ϕ max is given by equation ( 17), but for the present purpose, it is sufficient to consider the dominating contribution which is the one proportional to g 2 .Thus, here ϕ max = 4g 2 /αΩ, while we restrict ourselves to the case θ = π/2.
Since the measured signal corresponds to the state of a highly excited environmental mode, the relevant noise level is determined by the fluctuations of the effective bath coordinate ξ .If the temperature is sufficiently low, such that thermal excitations do not play any role, its spectral density equals the bath spectral density: S ξ ξ (ω) = J(ω) = αω.Thus, the signal is at least as big as the noise background if J(Ω) ≤ (Aϕ max ) 2 t.In other words, the time t during which the output is recorded must fulfil For the parameters used in figure 2, t meas ≈ 4 • 10 3 /ω qb .Since the measurement is performed via a coupling to external degrees of freedom, the qubit experiences unavoidable decoherence, which means that coherent qubit oscillations fade away with a decoherence rate γ φ .The time during which meaningful information can be obtained is therefore limited by the inequality t ≤ 1/γ φ .This condition together with condition (19) can be fulfilled only if γ φ t meas ≥ 1.
The decoherence rate can be estimated upon noticing that our qubit Hamiltonian represents a generalised spin-boson model [24][25][26] with the bath coupling 1  2 σ x η, where η = 2g 2 Q 2 is the effective bath coordinate.For weak dissipation, γ φ is given by the autocorrelation function of the latter evaluated at the qubit splitting, i.e., γ φ = C ηη (ω qb ) [37].This still holds true in the presence of ac driving provided that the driving-induced renormalization of the qubit splitting is negligible [38].We separate the qubit coordinate into the responses to the driving and to the incoming fluctuations, Q = Q + δ Q; cf.equation (9).Then we proceed along the lines of Ref. [39]: The relevant terms are those of second order in δ Q, such that C ηη (t) ≈ 4g 2  2 (A/αΩ) 2 cos(Ωt) δ Q(t)δ Q(0) .By Fourier transformation, we obtain to lowest order in ω qb /Ω the decoherence rate which is the inverse of the required measurement time t meas .A comparison with the numerically computed decay of the qubit coherence (not shown) confirms this value.The obtained relation γ φ t meas = 1 marks the quantum limit of a measurement [39] and allows one to marginally fulfil both conditions on the measurement time t.

A possible experimental implementation
Specific parameters can be obtained for the setup of reference [19] for which the qubitoscillator coupling parameters are determined by the flux bias current I b and read Here, M denotes the mutual SQUID-qubit inductance, I p is the qubit persistent-current, I C the critical current of the SQUID Josephson junctions, and L J = φ 0 [4πI C cos(πφ SQ /φ 0 )] −1 the SQUID Josephson inductance.The flux φ SQ that penetrates the SQUID loop corresponds to the phase ϕ = 2πφ SQ /φ 0 , with φ 0 = h/2e being the flux quantum.For small bias currents I b while neglecting the inductance of the wire that leads to the shunting capacitance C, the oscillator frequency is approximately given by the SQUID plasma frequency Ω = |L J C| −1/2 .For a typical qubit transition frequency of ω qb /2π = 5GHz, resolving the qubit dynamics requires an oscillator frequency of Ω/2π = 10ω qb /2π = 50GHz.The necessary lock-in measurements at a carrier frequency Ω/2π = 50GHz are particularly challenging at low temperatures, but feasible [40].They require rather expensive amplifying technology such as cryogenic amplifiers developed by Low Noise Factory (Sweden).Recently developed suitable devices can be operated at up to 36GHz and possess reasonably low noise temperatures.
Alternatively, a Josephson parametric amplifier [41,42] enables the detection of oscillator frequencies as high as Ω/2π = 20-25GHz.Thus, as a compromise, we restrict ourselves to an oscillator frequency of Ω/2π = 24GHz, which is suitable to detect the dynamics of a qubit with ω qb /2π = 2.5GHz, a value still large enough to avoid thermal excitations at working temperatures of 20mK.The above value for Ω can be realised using the parameters I C = 4.25µA, ϕ = 2.3π, and C = 1pF, which are similar to those of references [18,19].Along with M = 17.5pH,I p = 300nA and I b = 0.4µA, we obtain the qubit-oscillator coupling coefficients g 1 /2π = 45MHz and g 2 = 24MHz.Thus, the relevant dimensionless coupling assumes the value g 2 /ω qb ≃ 0.01 used in our numerical studies.An adequate oscillator halfbandwidth is ∆Ω = 2.9GHz, which implies a low external quality factor of Q ≃ 4-5.
Our two-state model for the qubit does not consider possible excitations to non-qubit states caused by the coupling to the oscillator.Nevertheless our modelling is appropriate, because such leakage has far less relevance than for a Cooper-pair box, owing to the fact that the higher states couple only weakly to the SQUID [43].Apart from this, it is possible to design or tune the oscillator such that its frequency is far from any qubit resonance.The required oscillator frequency of the order 10 GHz is still significantly smaller than the gap energy of aluminium, such that quasi particle excitation should not play a major role.This issue is even less critical for niobium.

Conclusions
We have generalised dispersive qubit readout to time-resolved observation of the qubit dynamics.Concerning the setup, the main difference to dispersive readout is that in the present proposal, the oscillator frequency needs to exceed the qubit splitting by roughly one order of magnitude, and the oscillator bandwidth should be at least twice the qubit frequency.Also here, the oscillator frequency becomes dynamically red or blue detuned, depending on the state of the qubit.When driving the SQUID oscillator at its bare frequency Ω, this detuning turns into a phase shift visible in the reflected signal via lock-in techniques.For such qubit measurement using the oscillator phase, the oscillator frequency represents the sampling rate, which explains the need for high frequencies.
The constituting measurement relation has been derived from the input-output formalism under time-scale separation of the bare qubit dynamics from the oscillator.A numerical solution of the Bloch-Redfield master equation for the full qubit-oscillator dynamics allowed us to compute the phase of reflected signal also directly.Its good agreement with the phase predicted by our measurement relation confirms the validity of the latter even when the oscillator frequency is just moderately large.Thus, there is no need for driving the qubit with extremely high frequencies, which would be quite challenging in an experiment.The found agreement is also reflected by the measurement fidelity, which already for moderate frequencies is rather good.Furthermore, the numerical analysis has demonstrated that the external ac-driving does not significantly modify the qubit dynamics, which means that the backaction of the measurement process is weak.However, it must be emphasised that the whole scheme relies on the coupling of the qubit via the oscillator to a dissipative environment, which causes qubit decoherence already when the external driving is not active.In the limit of far qubit-oscillator detuning, this qubit decoherence gets drastically reduced though.
Evaluating the measurement relation for parameters of recent experiments with flux qubits predicts phase shifts up to 2 • , which can be measured.Moreover, it reveals that the signal mainly stems from the coupling of the qubit to the square of the oscillator coordinate.The linear coupling to the coordinate, by contrast, leads to a rather small phase shift.This means that the measured quantity is essentially the qubit's flux degree of freedom.Likewise, the linear coupling of a superconducting charge qubit to a waveguide resonator is also too weak.Since for this system the non-linear coupling practically vanishes, the measured signal remains tiny.In conclusion, with present technologies, our measurement protocol should be feasible best with flux qubits coupled to SQUIDs that possess a significant non-linear Josephson inductance.All in all, our proposal may initiate further progress on the way towards single-shot experiments that demonstrate quantum coherence in solid-state devices.with p n (t 0 ) = qn (t 0 ).Inserting this solution into the Heisenberg equation of motion for Q yields which only depends on the environmental operators at initial time and, thus, is independent of the central quantum system.We replace the sum ∑ n |c n | 2 by an integral over the spectral density J(ω)/π, which for the ohmic J(ω) = αω becomes the derivative of the delta function δ (t − t ′ ), such that the time integral can be evaluated.In doing so, we arrive at the quantum Langevin equation where we have discarded an initial slip term and a constant potential renormalisation which both are not relevant in the present context and beyond transient behaviour.Notice that dissipation enters via a friction term, while the incoming fluctuations act as stochastic driving force.
The quantum Langevin equation (B.5) can also be expressed in terms of the outgoing fluctuations by solving the equations of motion (B.1) for q n with boundary condition at a later time time t 1 > t, i.e., by backward propagation.Then one obtains q n (t) = q n (t In contrast to ξ qm in (t), this noise operator depends on the time evolution of the system at earlier times t < t 1 .The resulting Langevin equation for the oscillator coordinate Q, is characterised by negative damping and the outgoing noise.The difference of both Langevin equations links the noise terms via twice the dissipative term by means of the input-output relation, which in the stationary limit t 0 → −∞ and t 1 → ∞ reads [20] ξ qm out (t) − ξ qm in (t) = 2α Q .(B.9) Even though we have written this relation for a harmonic oscillator, the derivation does not rely on particular properties of this system.Thus, equation (B.9) is valid as well for non-linear quantum systems coupled to an environment.If a bath mode is coherently excited by an external driving field, the incoming fluctuations are augmented by a deterministic contribution, ξ qm in → ξ qm in + x drive (t).Then the input-output relation allows one to compute both the averaged outgoing signal as well as its fluctuations and noise spectra.

Figure 1 .
Figure1.Sketch of the flux qubit (blue) coupled to a dc-SQUID.The interaction is characterised by the linear coupling g 1 , which depends linearly on the SQUID bias current I b , and the quadratic coupling g 2 .The SQUID with Josephson inductance L J is shunted by a capacitance C. The frequency shift of the resulting harmonic oscillator (green) can be probed by external resonant ac-excitation A cos(Ω ac t) via the transmission line (black), in which the quantum fluctuations ξ qm in (t) are also present.

Figure 2 .
Figure2.Time-resolved measurement of coherent qubit oscillations at the degeneracy point ε = 0.The full qubit-oscillator state was simulated with the quantum master equation (C.1) with N = 10 oscillator states and the parameters Ω = Ω ac = 10 ω qb , g 1 = 0.1 ω qb , g 2 = 0.01 ω qb , A = 1.0 ω qb .The dimensionless oscillator dissipation strength is α = 0.12.The resonator bandwidth is given by 2αΩ = 2.4ω qb .(a) Lock- in amplified phase ϕ exp (t) (dashed green lines), compared to the estimated phase ϕ(t) (solid red line) of the outgoing signal ξ out (t) .Here, ϕ(t) ∝ σ x t [cf.equation (14)], which is corroborated by the inset showing that σ x t performs oscillations with (angular) frequency ω qb .(b) Power spectrum ξ out (ω) for the resonantly driven oscillator (blue solid line).The sidebands stemming from the qubit dynamics are visible at frequencies Ω ± ω qb .In order to extract the phase information, we apply a Gaussian window function with respect to the frequency window of half-width ∆Ω = 1.2ω qb , which turns out to be the optimal value for the measurement bandwidth.

2 =Figure 3 .
Figure 3. (a) Fidelity defect δ F = 1 − F for the phases ϕ(t) and ϕ exp (t) and (b) time-averaged trace distance D between the density operators of a qubit with finite coupling to the oscillator and a reference qubit without oscillator.Both quantities are depicted for various coupling strengths g 2 in dependence of the oscillator frequency Ω.All other parameters are as in figure 2.