Apparent heating due to imperfect calorimetric measurements

Measurements unavoidably come with noise. In quantum mechanics the effect of a noisy measurement is double, as it does not only influence the outcome but also the state of the system after the measurement.

Perfect continuous measurements lead to the quantum Zeno effect, where the system gets stuck in a state with a small probability to jump away [1]. For imperfect continuous measurements the dynamics are much richer. In case measurement times are sufficiently short and the distribution of measurement outcomes is much broader than the state vector, the dynamics of the measured system are described by a non-linear stochastic Liouville equation [2], see also [3] for a pedagogic introduction. Intermediate regimes have been successfully studied with path-integral methods [4, 5]. The dynamics of a system weakly coupled to an environment under continuous perfect measurements can be described by quantum jump equations [6–10], see also [11, 12]. If the perfect measurement is disturbed by the presence of another bath and one considers a coarse grained time scale on which many jumps happen, the system dynamics undergo quantum state diffusion [11, 12].

In quantum calorimetric measurement schemes one continuously measures the energy, or temperature, of the environment in contact with a system of interest. These measurements allow to indirectly extract information about the system. Quantum calorimetry has been used for example to detect cosmic x-rays [13] and in quantum circuit measurements [14–16]. Recent experiments have shown that quantum calorimeters form a promising tool for single microwave photon detection in quantum circuits [17].

Earlier studies of experimental setups such as [14] have modelled the calorimeter–system dynamics as coupled jump processes of the energy [18] or temperature of calorimeter [19–21] and of the state of the system. The approaches of [18–21] differ from the usual quantum jump schemes [6–10] in that they explicitly update the state of the environment with the measured value of the energy or temperature. Concretely, this means that the jump rates depend on the measured state of the calorimeter. The previous coupled jump equations [18–21] all require that the calorimeter was under continuous perfect energy or temperature measurements. In the current work, we study quantum calorimetric measurement set-ups such as [14] in the case of imperfect detection. By imperfect we do not mean that the calorimeter itself is in contact with another environment but that additional noise acts directly and solely on the detector. Examples include black body radiation, electromechanical noise and elementary particles interfering with the measurement device. The main noise source for quantum circuit calorimeters such as [14] is already well understood [22–24] and can be experimentally characterised by performing the calorimetric measurement itself, see e.g. for a recent experiment [25]. Note that open quantum system approaches keeping track of environmental degrees of freedom similar to [18–21] can improve on predictions by standard Lindblad equations [26, 27].

The authors of [28] developed quantum trajectories for realistic photon detection. They take into account noise on the measurement outcome and delay in obtaining it. Recently these ideas were applied to model single photon measurements in quantum circuit calorimetric measurements [29]. Our approach differs from [28, 29] in the sense that we do not consider noise or delay on the outcome, i.e. imperfect information, but noise on the actual projective operator applied to the calorimeter during the measurement.

In the current work we develop a minimal description for generic noise acting directly on the measurement device in quantum calorimetric measurements. The aim of this scheme is to provide a simple tool to account for additional noise sources when experimental results cannot be accounted for by physical parameters for the experiment. Our scheme relies on introducing an additional noise bath (companion noise) to the modelling of the experiment that does not interact with calorimeter or system. The only influence of this companion noise bath is that its energy is measured simultaneously with the calorimeter energy, instead of just the calorimeter energy. This is similar to the setup where quantum state diffusion applies [11, 12], although we are not interested in coarse graining time.

From a technical point of view, the introduction of the companion bath is a mathematical trick to introduce more complex projection operators in the Nakajima–Zwanwig scheme. The Nakajima–Zwanzig scheme allows to consistently perform the weak coupling approximation for open quantum systems. The projection operator considered in the scheme is directly related to the Born approximation, which in the usual derivation of the Lindblad equation means assuming that the system–environment state is a product state at all times. In [26, 27, 30, 31] more general projection operators and therefore more general Born approximations were considered. The non-interacting companion noise bath greatly extends the system–reservoir states that can result from a Born approximations, while still leading to a consistent weak coupling perturbative expansion.

Under the presence of the companion bath we derive a hybrid master equation and corresponding coupled jump process for the measured energy and system state. Comparing our results to the dynamics described in [18–21], we find that the coupled jump equations have similar structures but the jump rates are modified due to the presence of this additional bath. Note that our approach is not limited to energy measurements but applies to any macroscopic property of the finite size environment such as magnetism.

The paper is structured as follows: in section 2 we introduce our model, a qubit interacting with the calorimeter and additional noise bath, and how we model the imperfect measurements. In section 3 we derive the hybrid master equation for the measured calorimeter–noise bath energy and the qubit state. We also give the corresponding energy-qubit state jump process. In section 4 we discuss the relation between the companion noise bath, the Nakajima–Zwanzig operator and the Born approximation. We consider the specific example of a calorimeter consisting of resonant bosons in section 5. We study the effect of the noise bath on the rates and on the power flowing to the calorimeter. Finally, in section 6 we discuss the results and provide an outlook.

The calorimetric measurement of [14] consists of a quantum circuit with a resistor in contact with a Josephson qubit. The electrons in the resistor form the calorimeter and their temperature is probed by measuring the conductance of an normal metal–insulator–superconductor (NIS) junction on the resistor [32]. The main noise source on the electrons, which we do not study in the present work, are phonons in the resistor themselves in contact with substrate phonons [22–24] which leads the calorimeter to equilibrate with the substrate. In the relevant experimental regime, the electrons can be modelled as a finite reservoir of free electrons in a solid [23, 24].

Motivated by the experiment of [14] we propose a model for imperfect calorimetric measurements, where a companion noise source acts directly and solely on the measurement device. The purpose of the model is to provide a simple, generic scheme to take into account noise sources such as black body radiation, electromechanical noise and elementary particles interfering with the measurement device. We consider a system interacting with a calorimeter (see figure 1) modelled as a finite size environment. A detector permanently measures the combined energy of this calorimeter and the additional noise bath which, however, is not directly coupled to system and calorimeter. Here we introduce the model and in section 3 we derive a joint master equation for the state of the system and the energy actually monitored by the detector. Note that in the current work we always talk about energy instead of temperature measurements. For a concrete description of the calorimeter one can transform the energy E into a temperature T by solving $\mathrm{t}\mathrm{r}(H\,{\mathrm{e}}^{-H/{k}_{\mathrm{B}}T})/\mathrm{t}\mathrm{r}({\mathrm{e}}^{-H/{k}_{\mathrm{B}}T})=E$ where H is the calorimeter Hamiltonian, see e.g. [20].

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While the framework developed below can be applied to any finite dimensional system interacting with arbitrary reservoirs, for the sake of simplicity and being explicit, we here consider a paradigmatic set-up: a qubit linearly coupled a bosonic calorimeter. The total Hamiltonian then consist of

$$\begin{equation}{H}_{\mathrm{Q}}=\omega {\sigma }_{+}{\sigma }_{-}\end{equation} \tag{ 1 }$$

for the system (qubit),

$$\begin{equation}{H}_{\mathrm{B}}=\sum\limits _{k}{\,\omega }_{k}{b}_{k}^{{\dagger}}{b}_{k}\end{equation} \tag{ 2 }$$

for the boson reservoir, i.e. the calorimeter, and

$$\begin{equation}{H}_{\mathrm{I}}=\sum\limits _{k}{\,g}_{k}({\sigma }_{+}{b}_{k}+{\sigma }_{-}{b}_{k}^{{\dagger}})\end{equation} \tag{ 3 }$$

for the interaction between system and calorimeter. Finally, we write the noise bath Hamiltonian as

$$\begin{equation}{H}_{\mathrm{N}}=\sum\limits _{E}\,E{P}_{E}^{\mathrm{N}},\end{equation} \tag{ 4 }$$

where orthogonal projectors ${P}_{E}^{\mathrm{N}}$ project on states of energy E of the noise bath. This way, together with the corresponding operator for the boson reservoir P_E , we can now introduce the projector ${\mathcal{P}}_{E}$ on the joint reservoir and noise bath energy E (detector energy measurement outcome), i.e.,

$$\begin{equation}{\mathcal{P}}_{E}=\sum\limits _{{E}^{\prime }}{\,P}_{{E}^{\prime }}\otimes {P}_{E-{E}^{\prime }}^{\mathrm{N}}.\end{equation} \tag{ 5 }$$

Note that, as per usual, we introduce the calorimeter and noise bath Hamiltonian to be a sum over discrete modes and take the formal continuum limit later on, in the derivation of the hybrid qubit-state measured-energy master equation.

In our derivation of the master equation below, we follow the usual weak coupling treatment and assume that the qubit–calorimeter–noise bath initial state is of the form

$$\begin{equation}{\rho }_{0}=\sum\limits _{E}\,\rho (E,0)\otimes \frac{{\mathcal{P}}_{E}}{{\mathrm{t}\mathrm{r}}_{\mathrm{N}+\mathrm{C}}\,{\mathcal{P}}_{E}}\end{equation} \tag{ 6 }$$

with the partially reduced density ρ(E, 0) of the qubit at fixed energy E of the total environment. Accordingly, taking the trace $\mathrm{t}\mathrm{r}({\mathcal{P}}_{E}{\rho }_{0})={\mathrm{t}\mathrm{r}}_{\mathrm{Q}}\,\rho (E,0)$, we observe that the trace of the unnormalised qubit state ρ(E, 0) is the probability to measure the environmental energy E at the initial time. Eliminating the noise reservoir, leads us to the qubit–calorimeter state

$$\begin{equation}{\mathrm{t}\mathrm{r}}_{\mathrm{N}}\,{\rho }_{0}=\sum\limits _{E}\,\rho (E,0)\otimes \sum\limits _{{E}^{\prime }}{\,P}_{{E}^{\prime }}\frac{{\mathrm{t}\mathrm{r}}_{\mathrm{N}\,{P}_{E-{E}^{\prime }}^{\mathrm{N}}}}{{\mathrm{t}\mathrm{r}}_{\mathrm{N}+\mathrm{C}}\,{\mathcal{P}}_{E}}.\end{equation} \tag{ 7 }$$

The latter distribution captures the energy partition between calorimeter and the noise bath which for the sake of simplicity is taken to be of the form

$$\begin{equation}{\mathrm{t}\mathrm{r}}_{\mathrm{N}}\left\{{P}_{x}^{\mathrm{N}}\right\}\propto {\mathrm{e}}^{-{x}^{2}/k}\end{equation} \tag{ 8 }$$

for all $x\in \mathbb{R}$ with a tunable parameter k. However, any other model originating, for example, from a more microscopic description for a specific experimental set-up is applicable as well.

We are now in a position to derive a qubit-energy hybrid master equation for the partially reduced density ρ(E, t) by generalizing the usual steps of a weak coupling Born–Markov treatment [11]. An alternative route in terms of the Nakajima–Zwanzig projection operator technique [33, 34] is presented in appendix A, see also [26, 27, 30, 31] for similar applications and the recent work [35]. We will not perform the Davies weak coupling limit prescription here explicitly, as it is the equivalent to the Born–Markov treatment we present below.

We start with the interaction-picture dynamics of the qubit–calorimeter–noise bath as described by the Liouville–von Neumann equation

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}t}\tilde{\rho }(t)=-\mathrm{i}[{\tilde{H}}_{\mathrm{I}}(t),\tilde{\rho }(t)]\end{equation} \tag{ 9 }$$

with the interaction picture Hamiltonian defined as

$$\begin{align}\hfill {\tilde{H}}_{\mathrm{I}}(t)=& {\,\text{e}}^{-\text{i}({H}_{\mathrm{Q}}+{H}_{\mathrm{B}})t}\,{H}_{\mathrm{I}}\,{\text{e}}^{\text{i}({H}_{\mathrm{Q}}+{H}_{\mathrm{B}})t}\hfill \\ \hfill =& \underset{k}{\,\sum }{g}_{k}\left[{\text{e}}^{\text{i}(\omega -{\omega }_{k})t}{\sigma }_{+}{b}_{k}+{\text{e}}^{-\text{i}(\omega -{\omega }_{k})t}{\sigma }_{-}{b}_{k}^{{\dagger}}\right]\hfill \end{align} \tag{ 10 }$$

and $\tilde{\rho }$ the corresponding interaction picture state operator.

Born–Markov approximation. Integrating equation (9) and plugging it back into itself gives

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}t}\tilde{\rho }(t)=-\mathrm{i}[{\tilde{H}}_{\mathrm{I}}(t),{\rho }_{0}]-{\int }_{0}^{t}\mathrm{d}s[{\tilde{H}}_{\mathrm{I}}(t),[{\tilde{H}}_{\mathrm{I}}(s),\tilde{\rho }(s)]].\end{equation} \tag{ 11 }$$

To trace out the calorimeter and noise bath degrees of freedom, we proceed by applying a modified version of the Born approximation. Since the calorimeter–noise bath energy is continuously measured, we assume that at any time t the qubit–reservoir–noise bath state $\tilde{\rho }(t)$ has the same structure as the initial state (6), i.e.

$$\begin{equation}\tilde{\rho }(t)\approx \sum\limits _{E}\tilde{\rho }\,(E,t)\otimes {\mathcal{P}}_{E}.\end{equation} \tag{ 12 }$$

Additionally, we perform the Markov approximation by replacing $\tilde{\rho \,}(s)$ in the integral in equation (11) by $\tilde{\rho \,}(t)$. Taking the partial trace ${\mathrm{t}\mathrm{r}}_{\mathrm{C}+\mathrm{N}}({\mathcal{P}}_{E})$ on both sides of (11) gives

$$\begin{align}\hfill \frac{\mathrm{d}}{\mathrm{d}t}\tilde{\rho }(E,t)& =-\mathrm{i}\,{\mathrm{t}\mathrm{r}}_{\mathrm{C}+\mathrm{N}}({\mathcal{P}}_{E}[{\tilde{H}}_{\mathrm{I}}(t),{\rho }_{0}])\hfill \\ \hfill & \quad -{\int }_{0}^{t}\mathrm{d}s\,{\mathrm{t}\mathrm{r}}_{\mathrm{C}+\mathrm{N}}\left(\right.\,{\mathcal{P}}_{E}\left[{\tilde{H}}_{\mathrm{I}}(t),\left[{\tilde{H}}_{\mathrm{I}}(s),\sum\limits _{{E}^{\prime }}\tilde{\rho }({E}^{\prime },t)\otimes {\mathcal{P}}_{{E}^{\prime }}\right]\right].\hfill \end{align} \tag{ 13 }$$

We observe that traces $\mathrm{t}\mathrm{r}({\mathcal{P}}_{E}{b}_{k}{\mathcal{P}}_{{E}^{\prime }})=\mathrm{t}\mathrm{r}({\mathcal{P}}_{E}{b}_{k}^{{\dagger}}{\mathcal{P}}_{{E}^{\prime }})=0$, which allows us to conclude that the first term on the right-hand side of the first line of (13) is zero. Next, the change of variables s → t − s is performed in the remaining integral and it is assumed that the calorimeter–noise correlation functions decay over a time scale τ_B much smaller than the qubit relaxation time τ_R. Under said assumption we are permitted to let the upper limit of the integral in (13) go to infinity to arrive at the master equation

$$\begin{align}\hfill & \frac{\mathrm{d}}{\mathrm{d}t}\tilde{\rho }\,(E,t)=-{\int }_{0}^{\infty }\mathrm{d}s\,{\mathrm{t}\mathrm{r}}_{\mathrm{C}+\mathrm{N}}\left(\right.\,{\mathcal{P}}_{E}{\tilde{H}}_{\mathrm{I}}(t),\left[{\tilde{H}}_{\mathrm{I}}(t-s),\sum\limits _{{E}^{\prime }}\tilde{\rho }({E}^{\prime },t)\otimes {\mathcal{P}}_{{E}^{\prime }}\right].\hfill \end{align} \tag{ 14 }$$

Master equation. From equation (14) we proceed similar to the usual derivation of the Lindblad equation [11]. Evaluating the time integral, reordering terms and dropping the Lamb shift, we arrive in the Schrödinger picture at the hybrid master equation

$$\begin{align}\hfill \frac{\mathrm{d}}{\mathrm{d}t}\rho \,(E,t)& =-\mathrm{i}[{H}_{\mathrm{Q}},\rho \,(t)]+{{\Gamma}}_{{\uparrow}}(E+\omega ){\sigma }_{+}\rho \,(E+\omega ,t){\sigma }_{-}-\frac{{{\Gamma}}_{{\uparrow}}(E)}{2}\left\{{\sigma }_{-}{\sigma }_{+},\rho \,(E,t)\right\}\hfill \\ \hfill & \quad +{{\Gamma}}_{{\downarrow}}(E-\omega ){\sigma }_{-}\rho \,(E-\omega ,t){\sigma }_{+}-\frac{{{\Gamma}}_{{\downarrow}}(E)}{2}\left\{{\sigma }_{+}{\sigma }_{-},\rho \,(E,t)\right\}.\hfill \end{align} \tag{ 15 }$$

The transition rates at fixed environmental energy E are given by

$$\begin{equation}{{\Gamma}}_{{\uparrow}}(E)=\kappa (\omega )\frac{{\mathrm{t}\mathrm{r}}_{\mathrm{C}+\mathrm{N}}\left\{L(\omega ){\mathcal{P}}_{E}\right\}}{{\mathrm{t}\mathrm{r}}_{\mathrm{C}+\mathrm{N}}\left\{{\mathcal{P}}_{E}\right\}}\end{equation} \tag{ 16a }$$

$$\begin{equation}{{\Gamma}}_{{\downarrow}}(E)=\kappa (\omega )\frac{{\mathrm{t}\mathrm{r}}_{\mathrm{C}+\mathrm{N}}\left\{K(\omega ){\mathcal{P}}_{E}\right\}}{{\mathrm{t}\mathrm{r}}_{\mathrm{C}+\mathrm{N}}\left\{{\mathcal{P}}_{E}\right\}},\end{equation} \tag{ 16b }$$

with

$$\begin{equation}L(\omega )=\sum\limits _{\begin{subarray}{c}k\\ {\omega }_{k}=\omega \end{subarray}}{b}_{k}^{{\dagger}}{b}_{k},\quad K(\omega )=\sum\limits _{\begin{subarray}{c}k\\ {\omega }_{k}=\omega \end{subarray}}{b}_{k}{b}_{k}^{{\dagger}}\end{equation} \tag{ 17 }$$

and with coupling strength $\kappa (\omega )\propto {g}_{\omega }^{2}$ [see (3)]. Note that in the above expressions for the rates ${\mathcal{P}}_{E}$ appears in both the numerators and the denominator, therefore we only have to define the noise bath traces (8) up to a constant. Exploiting our assumption on the noise bath traces (8), we obtain the more explicit expressions

$$\begin{equation}{{\Gamma}}_{{\uparrow}}(E)=\kappa (\omega )\frac{{\sum }_{{E}^{\prime }}{\mathrm{t}\mathrm{r}}_{\mathrm{C}}\left\{L(\omega ){P}_{{E}^{\prime }}\right\}{\mathrm{e}}^{-{(E-{E}^{\prime })}^{2}/k}}{{\sum }_{{E}^{\prime }}{\mathrm{t}\mathrm{r}}_{\mathrm{C}}\left\{{P}_{{E}^{\prime }}\right\}{\mathrm{e}}^{-{(E-{E}^{\prime })}^{2}/k}}\end{equation} \tag{ 18a }$$

$$\begin{equation}{{\Gamma}}_{{\downarrow}}(E)=\kappa (\omega )\frac{{\sum }_{{E}^{\prime }}{\mathrm{t}\mathrm{r}}_{\mathrm{C}}\left\{K(\omega ){P}_{{E}^{\prime }}\right\}{\mathrm{e}}^{-{(E-{E}^{\prime })}^{2}/k}}{{\sum }_{{E}^{\prime }}{\mathrm{t}\mathrm{r}}_{\mathrm{C}}\left\{{P}_{{E}^{\prime }}\right\}{\mathrm{e}}^{-{(E-{E}^{\prime })}^{2}/k}}.\end{equation} \tag{ 18b }$$

Note that the sums written here can also be understood as integrals over energies by taking the continuum limit. These results reduce to their perfect measurement counterparts by considering k → 0 so that in (18) ${\mathcal{P}}_{E}$ is replaced by the calorimeter energy projector P_E and in the sums over energies only the contribution with E = E' survives [18]. Furthermore, for sufficiently large reservoirs we expect the size of the calorimeter energy eigenspaces tr_C{P_E } to increase with the energy E. This implies ${\mathrm{t}\mathrm{r}}_{\mathrm{C}}\left\{{b}_{\omega }^{{\dagger}}{b}_{\omega }{P}_{{E}^{\prime }}\right\}$ and ${\mathrm{t}\mathrm{r}}_{\mathrm{C}}\left\{{b}_{\omega }{b}_{\omega }^{{\dagger}}{P}_{{E}^{\prime }}\right\}$ grow with E' and thus, as k increases, the rates primarily feel increased contributions from higher energies. This in turn may appear as an effective heating up of the environment as, by way of example, will be qualitatively confirmed in section 5.

The master equation (15) generates completely positive dynamics. This is ensured by a general result for hybrid classical–quantum master equations by [36], see also appendix C1.

Finally, we remark that the master equation (15) does not preserve the trace P(E, t) = tr(ρ(E, T)), i.e. the probability for the calorimeter to have energy E can change. However, taking the trace and summing over all energies in the master equation (15) gives zero, such that the total probability ∑_E P(E, t) is conserved.

Stochastic evolution. The hybrid master equation (15) can be mapped onto an equivalent stochastic dynamics for a the state vector of the qubit ψ(t) and the energy of the calorimeter E(t)

$$\begin{equation}\rho (E,t)=\mathsf{E}(\vert \psi (t)\rangle \langle \psi (t)\vert \delta (E-E(t))),\end{equation} \tag{ 19 }$$

where $\mathsf{E}(\cdot )$ denotes a proper average over noise realizations. Accordingly, one finds the following set of coupled stochastic differential equations for the state of the qubit ψ and the measured energy E of the calorimeter–noise bath

$$\begin{equation}\begin{cases}\mathrm{d}\psi (t)=\quad \hfill & -\mathrm{i}G(\psi (t))\,\mathrm{d}t+\left(\frac{{\sigma }_{-}\psi (t)}{{\Vert}{\sigma }_{-}\psi (t){\Vert}}-\psi (t)\right)\mathrm{d}{N}_{{\downarrow}}\hfill \\ \quad \hfill & +\left(\frac{{\sigma }_{+}\psi (t)}{{\Vert}{\sigma }_{+}\psi (t){\Vert}}-\psi (t)\right)\mathrm{d}{N}_{{\uparrow}}\hfill \\ \mathrm{d}E(t)=\quad \hfill & \omega (\mathrm{d}{N}_{{\downarrow}}-\mathrm{d}{N}_{{\uparrow}}).\hfill \end{cases}\end{equation} \tag{ 20 }$$

Here, the continuous evolution of the state vector is given by

$$\begin{align*}\hfill -\mathrm{i}G(\psi )=& -\mathrm{i}\omega {\sigma }_{+}{\sigma }_{-}-\frac{1}{2}{{\Gamma}}_{{\downarrow}}(E(t))({\sigma }_{+}{\sigma }_{-}-{\Vert}{\sigma }_{-}\psi {{\Vert}}^{2})\psi \hfill \\ \hfill & \quad -\frac{1}{2}{{\Gamma}}_{{\uparrow}}(E(t))({\sigma }_{-}{\sigma }_{+}-{\Vert}{\sigma }_{+}\psi {{\Vert}}^{2})\psi \hfill \end{align*}$$

and N_↑, N_↓ are Poisson processes with increments obeying

$$\begin{align*}\hfill \mathsf{E}(\mathrm{d}{N}_{{\downarrow}}\vert \psi ,E)=& {{\Gamma}}_{{\downarrow}}(E){\Vert}{\sigma }_{-}\psi {{\Vert}}^{2}\,\mathrm{d}t\hfill \\ \hfill \mathsf{E}(\mathrm{d}{N}_{{\uparrow}}\vert \psi ,E)=& {{\Gamma}}_{{\uparrow}}(E){\Vert}{\sigma }_{+}\psi {{\Vert}}^{2}\,\mathrm{d}t.\hfill \end{align*}$$

In appendix C2, we show that the coupled stochastic differential equation (20) and the average (19) indeed reproduce the master equation (15).

In appendix A we derive the master equation using the Nakajima–Zwanzig scheme which ensures that the weak coupling approximation is consistently performed. In the usual derivation of the Lindblad equation, the projector K used in the Nakajima–Zwanzig scheme acts on a system–reservoir state ρ as

$$\begin{equation}K(\rho )={\mathrm{t}\mathrm{r}}_{\mathrm{R}}(\rho )\otimes {\rho }_{\mathrm{R}},\end{equation} \tag{ 21 }$$

where tr_R is the trace over the reservoir degrees of freedom and ρ_R some reservoir state. The projector K translates into the Born approximation

$$\begin{equation}\rho \approx {\mathrm{t}\mathrm{r}}_{\mathrm{R}}(\rho )\otimes {\rho }_{\mathrm{R}}.\end{equation} \tag{ 22 }$$

The works [26, 27, 30, 31] considered more general projectors T acting on a system reservoir state as

$$\begin{equation}T(\rho )=\sum\limits _{k}{\,\mathrm{t}\mathrm{r}}_{\mathrm{R}}({P}_{k}\rho )\otimes \frac{{P}_{k}}{{\mathrm{t}\mathrm{r}}_{\mathrm{R}}\,{P}_{k}}\end{equation} \tag{ 23 }$$

here the P_K are a set of orthogonal projectors on the reservoir Hilbert space that satisfy ${\sum }_{k}{P}_{k}={\mathbb{I}}_{\mathrm{R}}$, i.e. they sum to the identity operator of the reservoir Hilbert space. Under weak coupling assumptions, the above projector corresponds to the Born approximation

$$\begin{equation}\rho \approx \sum\limits _{k}{\,\mathrm{t}\mathrm{r}}_{\mathrm{R}}({P}_{k}\rho )\otimes \frac{{P}_{k}}{{\mathrm{t}\mathrm{r}}_{\mathrm{R}}\,{P}_{k}}.\end{equation} \tag{ 24 }$$

In appendix A we use a projector P similar to those in [26, 27, 30, 31], which acts on a system–reservoir–noise bath state ρ as

$$\begin{equation}P\rho =\sum\limits _{E}{\,\mathrm{t}\mathrm{r}}_{\mathrm{C}+\mathrm{N}}({\mathcal{P}}_{E}\rho )\otimes \frac{{\mathcal{P}}_{E}}{{\mathrm{t}\mathrm{r}}_{\mathrm{C}+\mathrm{N}}({\mathcal{P}}_{E})}\end{equation} \tag{ 25 }$$

and leads to the Born approximation made in (12). By taking trace over the noise bath degrees of freedom on both sides of the above equation we arrive at

$$\begin{equation}{\mathrm{t}\mathrm{r}}_{\mathrm{N}}P\rho =\sum\limits _{E}{\,\mathrm{t}\mathrm{r}}_{\mathrm{C}+\mathrm{N}}({\mathcal{P}}_{E}\rho )\otimes \sum\limits _{{E}^{\prime }}{\,P}_{{E}^{\prime }}\frac{{\mathrm{t}\mathrm{r}}_{\mathrm{N}}\left\{{P}_{E-{E}^{\prime }}^{\mathrm{N}}\right\}}{{\mathrm{t}\mathrm{r}}_{\mathrm{C}+\mathrm{N}}({\mathcal{P}}_{E})},\end{equation} \tag{ 26 }$$

a much more general system–calorimeter state than (24).

Imagine now that we want to avoid using the noise bath and implement the right-hand side of the above equation directly as a projector $\tilde{P}$ on a system plus calorimeter state $\tilde{\rho }$. The projector would act as

$$\begin{equation}\tilde{P}\tilde{\rho }=\sum\limits _{E}{\,\mathrm{t}\mathrm{r}}_{\mathrm{C}}({{\Lambda}}_{E}[\tilde{\rho }])\otimes \sum\limits _{{E}^{\prime }}\,p\,(E,{E}^{\prime }){P}_{{E}^{\prime }},\end{equation} \tag{ 27 }$$

where the Λ_E are positive operators acting on calorimeter states and are normalised by ${\sum }_{E}{{\Lambda}}_{E}^{\ast }[{\mathbb{I}}_{\mathrm{C}}]={\mathbb{I}}_{\mathrm{C}}$, with ${{\Lambda}}_{E}^{\ast }$ the adjoint operator and ${\mathbb{I}}_{\mathrm{C}}$ the identity operator for the calorimeter Hilbert space. In appendix B we show that the requirement for $\tilde{P}$ to be a projector leads to a complex equation for Λ_E that is not generally expected to be solvable. This shows the advantage of explicitly introducing the non-interacting companion noise bath which allows to effectively cover a broad class of projection operators.

The central ingredients for the hybrid master equation (15) are the rates (18). By way of example and drawing on [18], we study a system of M resonant harmonic oscillators coupled to a qubit. In this case we are able to explicitly compute all traces required to compute the rates (18). The number of micro-states of the calorimeter at a given energy E = nω, i.e. the size of the calorimeter energy eigenspaces, is

$$\begin{equation}{\mathrm{t}\mathrm{r}}_{\mathrm{C}}\left\{{P}_{n\omega }\right\}=\left(\genfrac{}{}{0pt}{}{n+M-1}{M-1}\right)\end{equation} \tag{ 28 }$$

so that

$$\begin{equation}{\mathrm{t}\mathrm{r}}_{\mathrm{C}}\left\{L(\omega ){P}_{n\omega }\right\}=n\left(\genfrac{}{}{0pt}{}{n+M-1}{M-1}\right)\end{equation} \tag{ 29a }$$

$$\begin{equation}{\mathrm{t}\mathrm{r}}_{\mathrm{C}}\left\{K(\omega ){P}_{n\omega }\right\}=(n+M)\left(\genfrac{}{}{0pt}{}{n+M-1}{M-1}\right),\end{equation} \tag{ 29b }$$

where K and L are defined in (17) and since all reservoir operators are resonant K(ω) − L(ω) = M. Note that the above traces indeed grow with the value of n. For the noise bath, we take

$$\begin{equation}{H}_{\mathrm{N}}=\sum\limits _{n=-{M}_{\mathrm{C}}}^{{M}_{\mathrm{C}}}{P}_{n\omega }^{\mathrm{N}},\qquad {\mathrm{t}\mathrm{r}}_{\mathrm{N}}\left\{{P}_{n\omega }^{\mathrm{N}}\right\}\propto {\mathrm{e}}^{-{n}^{2}{\omega }^{2}/k}.\end{equation} \tag{ 30 }$$

The noise bath thus has two free parameters: k and the cut-off M_C. Note that as the rates (18) naturally select energies resonant to the reservoir, we could have equivalently defined the noise bath Hamiltonian as ${H}_{\mathrm{N}}={\int }_{-({M}_{\mathrm{C}}-1/2)\omega }^{({M}_{\mathrm{C}}+1/2)\omega }{P}_{{\epsilon}}^{\mathrm{N}}\,\mathrm{d}{\epsilon}$. Figure 2 shows the dependence of the rates on these parameters. As expected, we observe that the decay rate increases with k and eventually plateau at a value which depends on M_C. The behaviour for the rates is similar at E = 0 (full, dotted and dashed lines) and E ≠ 0 (dash–dotted line). The inset displays the energy dependence of the rates in the case of perfect measurements (i.e. k → 0) which reveals that increasing k has a similar effect as increasing E. Hence, in an imperfect detection considered here, the energy of the calorimeter appears to be elevated compared to the measured value of the energy.

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Energy flow to calorimeter. In a next step, we explore how an imperfect measurement influences the detection of the energy flow towards the calorimeter when the qubit is driven by an external periodic signal, a generic situation for quantum thermodynamical set-ups. For this purpose, a driving term is added to the qubit Hamiltonian (1), i.e.

$$\begin{equation}{H}_{\mathrm{Q}}(t)=\omega {\sigma }_{+}{\sigma }_{-}+\lambda ({\text{e}}^{\text{i}\,\omega t}{\sigma }_{+}{\text{e}}^{-\text{i}\,\omega t}{\sigma }_{-}).\end{equation} \tag{ 31 }$$

The coupled jump equation (20) are simulated (time step dt = 0.03) and the energy flow to the calorimeter is extracted as a function of the energy uncertainty parameter k. Figure 3 reports the average change in the measured energy ΔE = E_f − E_i over five periods while the inset figure depicts the same quantity in the perfect measurement case as a function of the initial energy E_i of the calorimeter. Obviously, in the latter case for increasing initial energy the average energy flowing to the calorimeter decreases. Imperfect measurements exhibit a similar behaviour for increasing k which implies that effectively a less perfect measurement shows the same behavior as a hotter calorimeter.

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Power at steady state. Finally, we address the power flowing through the qubit in steady state which in calorimetric measurements is a viable tool to retrieve qubit properties without 'touching' the qubit directly [14, 16]. To study the steady state properties, we add the loss term −ω dN_loss to the energy jump process (20), where N_loss is a Poisson process with rate

$$\begin{equation}\mathsf{E}(\mathrm{d}{N}_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}})=\gamma \langle E\rangle \mathrm{d}t\end{equation} \tag{ 32 }$$

and ⟨E⟩ is the expected energy of the calorimeter when the energy E was measured

$$\begin{equation}\langle E\rangle =\frac{{\sum }_{{E}^{\prime }}{\mathrm{t}\mathrm{r}}_{\mathrm{C}}\,{E}^{\prime }\,{\mathrm{t}\mathrm{r}}_{\mathrm{C}}({P}_{{E}^{\prime }}){\mathrm{e}}^{-{(E-{E}^{\prime })}^{2}/k}}{{\sum }_{{E}^{\prime }}{\mathrm{t}\mathrm{r}}_{\mathrm{C}}({P}_{{E}^{\prime }}){\mathrm{e}}^{-{(E-{E}^{\prime })}^{2}/k}}.\end{equation} \tag{ 33 }$$

The stochastic loss term corresponds to the extra terms

$$\begin{equation}\gamma \langle E+\omega \rangle \rho \,(E+\omega )-\gamma \langle E\rangle \rho \,(E)\end{equation} \tag{ 34 }$$

in the master equation (15).

When the power from the qubit to the calorimeter and the loss term balance each other out, the calorimeter–noise bath reaches a steady state. We call the average energy at steady state E_s and approximate the power flowing through the calorimeter by

$$\begin{equation}{P}_{\mathrm{s}}=\gamma \langle {E}_{\mathrm{s}}\rangle .\end{equation} \tag{ 35 }$$

Figure 4 shows the power at steady state P_s as a function of the measured average energy E_s for different values of k. Note that we estimated the power at steady state to be proportional to ⟨E_s⟩ and not E_s. The inset shows the measured steady state energies E_s as a function of k. As a comparison the (blue) crosses display the power at steady state when not taking the measurement error into account. In this case our estimate is then ${P}_{\mathrm{s}}^{\prime }=\gamma {E}_{\mathrm{s}}$. We observe that even though the measured steady state energy E_s(k) varies as a function of k, the power at steady state is nearly constant. Thus the expected energy of the calorimeter ⟨E_s(k)⟩ is nearly constant as a function of k. This is not surprising as the qubit driving remains constant when varying k. However, when not taking the measurement errors into account, a significant underestimation of the power can occur.

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In this work we studied the role of noise in quantum calorimetric measurements from the detector point of view, referred to as imperfect measurements. While noise sources disturbing the calorimeter can be directly characterised by performing the calorimetric measurement [25], noise acting on the measurement of the calorimeter itself is more challenging to study. We developed a simple model for imperfect measurements that serves as a tool to analyse unidentified noise sources in quantum calorimetric experiments, for example when measured data cannot be explained by physical parameters.

For the calorimetric measurement we considered the exemplar model of a qubit interacting with a finite reservoir of bosons, which serves as the calorimeter. To model the imperfect measurement we introduced an additional a noise bath which does not interact with the qubit nor calorimeter. The role of the noise bath is that its energy is measured simultaneously with the calorimeter energy and as such disturbs the measurement.

Under weak coupling assumptions we derived a hybrid master equation for the system state and the measured energy (15) and the corresponding stochastic evolution (20). Remarkably, the only change compared to the perfect measurement case is the values of the jump rates. The rates average contributions from different energies which leads to an apparent heating of the calorimeter. We study a simple example of a driven qubit in contact with a reservoir of resonant bosons to study the qualitative behaviour of our model for imperfect measurements. As the measurement imperfection increases, the observed energy transfer from the qubit to the calorimeter decreases.

To study the effect of noise in steady state measurements such as in [14] to extract the power flowing from the qubit to the calorimeter, we added a loss term to the calorimeter energy process. We observe that for growing imperfection of the measurement, not taking the imperfection into account leads to a significant underestimation of the power.

Our approach can be applied to general systems and reservoirs. Further studies could include more complex systems, e.g. multiple qubits, and fermion reservoirs. Another potential avenue is to derive the noise bath from a microscopic description of the noise in an experimental setup. Changes in the noise bath distribution (8) could have a significant impact on the measurement outcomes.

We thank Paolo Muratore-Ginanneschi, Bayan Karimi, Jukka Pekola and Kalle Koskinen for useful discussions. BD acknowledges support from the AtMath collaboration at the University of Helsinki. This work has been supported by IQST, the Zeiss Foundation, and the German Science Foundation (DFG) under AN336/12-1 (For2724).

The data that support the findings of this study are available upon reasonable request from the authors.

We define a projector P, which acts on a qubit–calorimeter–noise bath state ρ as

$$\begin{equation}P\rho =\sum\limits _{E}{\,\mathrm{t}\mathrm{r}}_{\mathrm{C}+\mathrm{N}}({\mathcal{P}}_{E}\rho )\otimes \frac{{\mathcal{P}}_{E}}{{\mathrm{t}\mathrm{r}}_{\mathrm{C}+\mathrm{N}}({\mathcal{P}}_{E})}.\end{equation} \tag{ A1 }$$

It is straightforward to check that P is indeed a projector, i.e. P² ρ = Pρ.

We introduce the orthogonal projector Q = 1 − P, such that QP = PQ = 0 and define the hybrid state

$$\begin{equation}\rho (E,t)\equiv \mathrm{t}\mathrm{r}\,({\mathcal{P}}_{E}P\rho )={\mathrm{t}\mathrm{r}}_{\mathrm{C}+\mathrm{N}}({\mathcal{P}}_{E}\rho ).\end{equation} \tag{ A2 }$$

In order to apply the Nakajima–Zwanzig projection operator technique, see for example [37], we require three hypotheses to be satisfied.

Hypothesis 1. The initial state (6) satisfies Pρ₀ = ρ₀.

Hypothesis 2. For any qubit–calorimeter–noise bath state ρ we have that

$$\begin{equation}[{H}_{\mathrm{C}}+{H}_{\mathrm{N}},P\rho ]=0.\end{equation} \tag{ A3 }$$

Hypothesis 3. For any qubit–calorimeter–noise bath state ρ we have that

$$\begin{equation}{\mathrm{t}\mathrm{r}}_{\mathrm{C}+\mathrm{N}}([{H}_{\mathrm{I}},P\rho ])=0.\end{equation} \tag{ A4 }$$

A direct computation shows that the above three hypotheses are satisfied for our model and choice of P. Going through the Nakajima–Zwanzig projection operator technique, we find that $\rho (E,t)=\mathrm{t}\mathrm{r}({\mathcal{P}}_{E}P\rho )$ satisfies a closed differential equation

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}t}\rho \;(E,t)=-{\int }_{0}^{t}\mathrm{d}s\;\mathrm{t}\mathrm{r}({\mathcal{P}}_{E}[{H}_{\mathrm{I}}(t),G(t,s)[{H}_{\mathrm{I}}(s),P\rho ]]),\end{equation} \tag{ A5 }$$

where G satisfies the differential equation

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}t}G(t,s)=-\mathrm{i}Q[{H}_{\mathrm{I}}(t),G(t,s)].\end{equation} \tag{ A6 }$$

Up to lowest order in the coupling strength G(t, s) ≈ 1, such that (A5) equals (13) up to second order in the coupling.

Imagine one wants to define a projector $\tilde{P}$ on the system plus calorimeter which acts on a state as

$$\begin{equation}\tilde{P}\rho =\sum\limits _{E}{\;\mathrm{t}\mathrm{r}}_{\mathrm{C}}({{\Lambda}}_{E}(\rho ))\otimes \sum\limits _{{E}^{\mathrm{\prime }}}\;p\;(E,{E}^{\mathrm{\prime }}){P}_{{E}^{\mathrm{\prime }}},\end{equation} \tag{ B1 }$$

where the Λ_E are channels acting on the calorimeter state. Since $\tilde{P}$ is a projector, we should have that for all ρ

$$\begin{equation}{\tilde{P}}^{2}\rho =\tilde{P}\rho .\end{equation} \tag{ B2 }$$

Computing both sides leads to

$$\begin{align*}\hfill & \sum\limits _{E}{\,\mathrm{t}\mathrm{r}}_{\mathrm{C}}({{\Lambda}}_{E}(\rho ))\otimes \sum\limits _{x}\sum\limits _{{x}^{\prime }}\sum\limits _{{E}^{\prime }}\,p(E,{E}^{\prime }){\mathrm{t}\mathrm{r}}_{\mathrm{C}}({{\Lambda}}_{x}({P}_{{E}^{\prime }}))p\,(x,{x}^{\prime })\hfill \\ \hfill & \qquad =\sum\limits _{E}{\,\mathrm{t}\mathrm{r}}_{\mathrm{C}}({{\Lambda}}_{E}(\rho ))\otimes \sum\limits _{{E}^{\prime }}\,p\,(E,{E}^{\prime }){P}_{{E}^{\prime }}.\hfill \end{align*}$$

The above equality is satisfied if

$$\begin{equation}\sum\limits _{x}\sum\limits _{{E}^{\prime }}\,p\,(E,{E}^{\prime }){\mathrm{t}\mathrm{r}}_{\mathrm{C}}({{\Lambda}}_{x}({P}_{{E}^{\prime }}))p\,(x,{x}^{\prime })=p\,(E,{x}^{\prime })\quad \forall E,{x}^{\prime }.\end{equation} \tag{ B3 }$$

For a given function p(E, E') this requires solving for a set of positive operators Λ_E , satisfying ${\sum }_{E}{{\Lambda}}_{E}^{\ast }[{\mathbb{I}}_{\mathrm{C}}]={\mathbb{I}}_{\mathrm{C}}$.

C1. Complete positivity

The complete positivity of the hybrid master equation follows from a result by [36]. The authors of [36] consider classical–quantum master equations and lift the classical variables to a quantum description. Concretely, let the classical energy variable E can take d values. We define the d-dimensional Hilbert space ${\mathcal{H}}_{\mathrm{C}\mathrm{l}}$ with basis vectors ${\left\{{v}_{i}\right\}}_{i=1,\dots ,d}$ and we define the projectors v_ij which project from the basis vector j to i. We then define the state

$$\begin{equation}\rho (t)=\sum\limits _{E}{\,v}_{E,E}\otimes \rho \,(t,E)\end{equation} \tag{ C1 }$$

such that tr_Cl(v_E,E ρ(t)) = ρ(t, E) is the hybrid qubit-energy state we are considering in the manuscript. Let us then define the operator

$$\begin{align*}\hfill {\Phi}(a)& =\sum\limits _{E}\left({{\Gamma}}_{{\downarrow}}(E)({v}_{E,E+\omega }\otimes {\sigma }_{-})a({v}_{E+\omega ,E}\otimes {\sigma }_{+})\right.\hfill \\ \hfill & \quad \left.+{{\Gamma}}_{{\uparrow}}(E)({v}_{E,E-\omega }\otimes {\sigma }_{+})a({v}_{E-\omega ,E}\otimes {\sigma }_{-})\right).\hfill \end{align*}$$

Following the results of [36], Φ is completely positive such that

$$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}t}\rho \,(t)=-\mathrm{i}[H(t),\rho \,(t)]+{{\Phi}}^{\ast }(\rho \,(t))-\frac{1}{2}\left\{{{\Phi}}^{\ast }(\mathbb{I}),\rho (t)\right\}\end{equation} \tag{ C2 }$$

generates a completely positive evolution. Projecting both sides of the above equation on v_E,E gives (15).

C2. Unravelling

Let us define the function

$$\begin{equation}f(\psi ,{\psi }^{{\dagger}},\xi )=\vert \psi \rangle \langle \psi \vert \delta (E-\xi ).\end{equation} \tag{ C3 }$$

Such that taking the average gives the hybrid quantum state

$$\begin{equation}\mathsf{E}(\vert \psi \rangle \langle \psi \vert \delta (E-\xi ))=\rho \,(t,E).\end{equation} \tag{ C4 }$$

We then compute the differential

$$\begin{align*}\hfill \mathrm{d}f(\psi ,{\psi }^{{\dagger}},\xi )& =f(\psi +\mathrm{d}\psi ,{\psi }^{{\dagger}}+\mathrm{d}{\psi }^{{\dagger}},\xi +\mathrm{d}\xi )-f(\psi ,{\psi }^{{\dagger}},\xi )\hfill \\ \hfill & =\sum\limits _{\begin{subarray}{c}p=1\\ {p}_{1}+{p}_{2}+{p}_{3}=p\end{subarray}}^{\infty }\frac{{(\mathrm{d}\xi )}^{{p}_{1}}{(\mathrm{d}\psi )}^{{p}_{2}}{(\mathrm{d}{\psi }^{{\dagger}})}^{{p}_{3}}}{{p}_{1}!{p}_{2}!{p}_{3}!}{\partial }_{\xi }^{{p}_{1}}{\partial }_{\psi }^{{p}_{2}}{\partial }_{{\psi }^{{\dagger}}}^{{p}_{3}}f(\psi ,{\psi }^{{\dagger}},\xi ).\hfill \end{align*}$$

According to the stochastic rules of calculus, see e.g. [3], that dN_i  dN_j = δ_i,j  dN_j and equation (20), we find that

$$\begin{align}\hfill \mathrm{d}f(\psi ,{\psi }^{{\dagger}},\xi )& =-\mathrm{i}(G(\psi ){\partial }_{\psi }-G{(\psi )}^{{\dagger}}{\partial }_{{\psi }^{{\dagger}}})f(\psi ,{\psi }^{{\dagger}},\xi )\hfill \\ \hfill & \quad +\left[f({\phi }_{+},{\phi }_{+}^{{\dagger}},\xi -\omega )-f(\psi ,{\psi }^{{\dagger}},\xi )\right]\mathrm{d}{N}_{{\uparrow}}\hfill \\ \hfill & \quad +\left[f({\phi }_{-},{\phi }_{-}^{{\dagger}},\xi +\omega )-f(\psi ,{\psi }^{{\dagger}},\xi )\right]\mathrm{d}{N}_{{\downarrow}}\hfill \end{align} \tag{ C5 }$$

with ϕ_± being the eigenstates of σ_z , i.e. σ_z ϕ_± = ±ϕ_±. Taking the expectation value of the right-hand side of the first line, we get

$$\begin{equation}-\mathrm{i}[H(t),\rho \,(E,t)]-\frac{1}{2}({{\Gamma}}_{{\downarrow}}(E)\left\{{\sigma }_{+}{\sigma }_{-},\rho \,(t)\right\}+{{\Gamma}}_{{\uparrow}}(E)\left\{{\sigma }_{-}{\sigma }_{+},\rho \,(t)\right\})\mathrm{d}t\end{equation} \tag{ C6 }$$

$$\begin{equation}+\frac{1}{2}\mathsf{E}[({{\Gamma}}_{{\downarrow}}(E){\Vert}{\sigma }_{-}\psi {{\Vert}}^{2}+{{\Gamma}}_{{\uparrow}}(E){\Vert}{\sigma }_{+}\psi {{\Vert}}^{2})\vert \psi \rangle \langle \psi \vert ]\mathrm{d}t.\end{equation} \tag{ C7 }$$

The average value of the last two lines of (C5) gives

$$\begin{align}\hfill & {{\Gamma}}_{{\downarrow}}(E-\omega ){\sigma }_{-}\rho \,(E-\omega ,t){\sigma }_{+}\,\mathrm{d}t+{{\Gamma}}_{{\uparrow}}(E+\omega ){\sigma }_{+}\rho \,(E+\omega ,t){\sigma }_{-}\,\mathrm{d}t\hfill \\ \hfill & \qquad -\frac{1}{2}\mathsf{E}[({{\Gamma}}_{{\downarrow}}(E){\Vert}{\sigma }_{-}\psi {{\Vert}}^{2}+{{\Gamma}}_{{\uparrow}}(E){\Vert}{\sigma }_{+}\psi {{\Vert}}^{2})\vert \psi \rangle \langle \psi \vert ]\mathrm{d}t.\hfill \end{align} \tag{ C8 }$$

Summing the above two equations, we recover (15).