Monitoring quantum Otto engines

For the sake of concreteness we consider an Otto engine with a quantum mechanical working substance, which later will be specified as a two-level system. The subsequent four strokes constituting a full cycle of the engine consist in compressing, heating, expanding, and cooling the working substance, see Fig. 1. The compression and expansion strokes are caused by changing a control parameter of the working substance in thermal isolation. Therefore, the energy change of the working substance during these strokes can be referred to as work. The parameter change in the compression stroke is designed such that the energy level distances of the working substance Hamiltonian increase. Hence, the time evolution of the density matrix caused by the compression is determined by a unitary operator $U$ with $\rho_{2}=U\rho_{1}U^{\dagger}$, with $\rho_{k}=\rho(t_{k})$, $k=1,2,\cdots$, denoting the density matrices at the times indicated in Fig. 1, whereby times $t_{k}$ with $k=1,\cdots,4$ specify instants within the first cycle, with $k=5,\cdots,8$ within the second cycle, etc.. For the sake of simplicity, the expansion is assumed to be the time-reversed protocol of the compression. Therefore, the unitary operator giving the corresponding transformation can be written as $\tilde{U}=K^{*}U^{\dagger}K$ with $K^{*}$ being the adjoint of the antiunitary time-reversal operator $K$ ($K^{*}K=KK^{*}=\mathbb{1}$), [46], such that $\rho_{4}=\tilde{U}\rho_{3}\tilde{U}^{\dagger}$. The relation between $\tilde{U}$ and $U^{\dagger}$ holds as long as $K^{*}H(t)K=H(t)$ for all time $t$ during the work strokes with $H(t)$ being the time-dependent engine Hamiltonian. During the second and the fourth stroke the working substance is brought into contact with a hot and a cold heat bath, respectively, while its Hamiltonian remains unchanged. These two heat strokes are characterized by trace preserving, positive, linear maps $\Phi_{h}$ and $\Phi_{c}$ relating the respective density matrices of the working substance at the beginnings and the ends of these strokes according to $\rho_{3}=\Phi_{h}(\rho_{2})$ and $\rho_{5}=\Phi_{c}(\rho_{4})$. Later, we shall give explicit expressions specifying the unitary operator $U$, $\tilde{U}$ and the maps $\Phi_{h}$ and $\Phi_{c}$ for the case of a two-level system as working substance.

The central quantities characterizing the performance of an engine are the total work $W$ performed on and the total heat $Q$ supplied by the hot bath to the working substance during $N$ cycles. Both quantities follow from the sequence of energies $\mathbf{e}=(e_{1},e_{2},\cdots,e_{4N})$ of the working substance taken at the beginning and the end of each work stroke when the working substance is decoupled from either heat bath, see Fig. 1 yielding

	$\displaystyle W$	$\displaystyle=$	$\displaystyle\sum_{k=1}^{4N}(-1)^{k}e_{k},$		(1)
	$\displaystyle Q$	$\displaystyle=$	$\displaystyle\sum_{k=1}^{4N}\mu_{k}e_{k},$		(2)

where $\mu_{1+4l}=\mu_{4+4l}=0$, $\mu_{2+4l}=-1$, $\mu_{3+4l}=1$ with $l=0,\cdots,N-1$.

In the repeated measurement scheme the energies $e_{k}$ are measured individually by bringing the working substance for a short time into contact with the pointer of a measurement apparatus [47]. This contact causes a unitary transformation of the joint state of the pointer and the working substance given by

$$V_{k}=e^{-i\kappa H_{k}\otimes P_{k}},$$

(3)

where $H_{k}$ is the working substance Hamiltonian at the start of the $k$th stroke, i.e. $H_{1}=H_{4}=H_{5}=\cdots=H_{c}$ and $H_{2}=H_{3}=\cdots=H_{h}$, see also Fig. 1; furthermore, $P_{k}$ denotes the momentum operator conjugate to the pointer coordinate of the $k$th measurement apparatus. The parameter $\kappa$ results as the product of the strength and the duration of the contact which is assumed to be so short that the free motion of the working substance appears as frozen. Above, and throughout this work we will set $\hbar=k_{B}=1$. Before the $k$th interaction, the working substance and the respective pointer are uncorrelated. The latter resides in a Gaussian pure state $\sigma_{k}$ with vanishing mean value and variance $\Sigma^{2}$. Its position matrix element $\sigma(x,y)=\langle x|\sigma_{k}|y\rangle$ is independent of $k$ and is given by

$$\sigma(x,y)=\frac{1}{\sqrt{2\pi\Sigma^{2}}}\exp\left[-\frac{x^{2}+y^{2}}{4\Sigma^{2}}\right].$$

(4)

If, after the contact has taken place, the pointer position is measured projectively at a position $x$, the non-normalized density matrix of the working substance becomes

	$\displaystyle\phi_{x}(\rho)$	$\displaystyle=$	$\displaystyle\langle x|V_{k}\rho\otimes\sigma_{k}V^{\dagger}_{k}|x\rangle$		(5)
		$\displaystyle=$	$\displaystyle\sum_{m,m^{\prime}}\mathcal{P}^{m}_{k}\rho\mathcal{P}^{m^{\prime}}_{k}\sigma_{k}\left(x-e^{m}_{k},x-e^{m^{\prime}}_{k}\right),$

where $e^{m}_{k}$ and $\mathcal{P}^{m}_{k}$ denote the eigenvalues and eigen-projectors of the Hamiltonian $H_{k}=\sum_{m}e^{m}_{k}\mathcal{P}^{m}_{k}$ and $\rho$ denotes the density matrix of the working substance before the contact. For the sake of simplicity and without restriction of generality, the pointer position is measured in units of energy such that $\kappa=1$. The subsequent measurements of the energies $e_{k}$ during $N$ cycles, giving rise to the pointer positions $\vec{x}_{N}=(x_{1},\cdots,x_{4N})$, lead to a mapping of the initial working substance density matrix $\rho$ onto the non-normalized density matrix $\phi^{\mathrm{RM}}_{\vec{x}_{N}}(\rho)$ given by

$$\phi^{\mathrm{RM}}_{\vec{x}_{N}}(\rho)=\sum_{\vec{m}_{N},\vec{m}^{\prime}_{N}}D^{\vec{m}_{N},\vec{m}^{\prime}_{N}}(\rho)\prod_{k=1}^{4N}\sigma\left(x_{k}-e_{k}^{m_{k}},x_{k}-e_{k}^{m_{k}^{\prime}}\right),$$

(6)

where the sum runs over all possible sequences $\vec{m}^{(\prime)}_{N}=(m^{(\prime)}_{1},m^{(\prime)}_{2},\cdots,m^{(\prime)}_{4N})$ of quantum numbers $m^{(\prime)}_{k}$ labelling the eigenstates of the Hamiltonian $H_{k}$. The operator $D^{\vec{m}_{N},\vec{m}^{\prime}_{N}}(\rho)$ describes the action of the alternating time-evolution operators characterizing the subsequent strokes and the interrupting projections due to the energy measurements on the initial density matrix $\rho$. They are recursively given by

$$D^{\vec{m}_{N+1},\vec{m}^{\prime}_{N+1}}(\rho)=S^{\mathbf{m},\mathbf{m}^{\prime}}\left(D^{\vec{m}_{N},\vec{m}^{\prime}_{N}}(\rho)\right),$$

(7)

where the sequences $\vec{m}^{(\prime)}_{N+1}=\vec{m}^{(\prime)}_{N}\oplus\mathbf{m}^{(\prime)}$ denote the concatenation of $\vec{m}^{(\prime)}_{N}$ and $\mathbf{m}^{(\prime)}=(m_{4N+1}^{(\prime)},\cdots,m_{4(N+1)}^{(\prime)})$. Furthermore, the map $S^{\mathbf{m},\mathbf{m}^{\prime}}$ acts as

$\displaystyle S^{\mathbf{m},\mathbf{m}^{\prime}}(\rho)$

$\displaystyle=$

$\displaystyle\Phi_{c}\left(\mathcal{P}^{m_{4(N+1)}}_{1}\tilde{U}\mathcal{P}^{m_{4N+3}}_{2}\Phi_{h}\left(\mathcal{P}^{m_{4N+2}}_{2}U\mathcal{P}^{m_{4N+1}}_{1}\rho\mathcal{P}^{m_{4N+1}^{\prime}}_{1}U^{\dagger}\mathcal{P}^{m_{4N+2}^{\prime}}_{2}\right)\mathcal{P}^{m_{4N+3}^{\prime}}_{2}\tilde{U}^{\dagger}\mathcal{P}^{m_{4(N+1)}^{\prime}}_{1}\right).$

(8)

Taking the trace of $\phi^{\mathrm{RM}}_{\vec{x}}(\rho)$ one obtains for the joint PDF $p(\vec{x})$to find the pointers at the values given by the components of the vector $\vec{x}$ the formal expression

$$p(\vec{x})=\mathrm{Tr}\left[\phi^{\mathrm{RM}}_{\vec{x}}(\rho)\right].$$

(9)

For Gaussian pointers as specified in Eq. (4) this PDF can be written as

$$p(\vec{x})=\sum_{\vec{m},\vec{m}^{\prime}}D^{\vec{m},\vec{m}^{\prime}}e^{-\frac{1}{8\Sigma^{2}}\sum_{k=1}^{4N}\left(e^{m_{k}}_{k}-e^{m_{k}^{\prime}}_{k}\right)^{2}}\prod_{k=1}^{4N}g_{\Sigma^{2}}\left(x_{k}-\frac{1}{2}\left(e^{m_{k}}_{k}+e^{m_{k}^{\prime}}_{k}\right)\right),$$

(10)

where $g_{\Sigma^{2}}(x)=(2\pi\Sigma^{2})^{-1/2}\exp[-x^{2}/(2\Sigma^{2})]$ denotes a Gaussian PDF with vanishing mean value and variance $\Sigma^{2}$. The coefficient $D^{\vec{m},\vec{m}^{\prime}}$ is defined as

$$D^{\vec{m},\vec{m}^{\prime}}=\mathrm{Tr}[D^{\vec{m},\vec{m}^{\prime}}(\rho)].$$

(11)

In order not to overburden the notation we omitted the index $N$ in the vectors $\vec{m}^{(\prime)}$ and $\vec{x}$.

Identifying the components $x_{k}$ as the measured energies, one obtains for the joint probability of work and heat, $P^{\mathrm{RM}}(W,Q)$, the expression

	$\displaystyle P^{\mathrm{RM}}(W,Q)$	$\displaystyle=$	$\displaystyle\int d^{4N}x\,\delta\left(W-\sum_{k=1}^{4N}(-1)^{k}x_{k}\right)$		(12)
			$\displaystyle\times\delta\left(Q-\sum_{k=1}^{4N}\mu_{k}x_{k}\right)p(\vec{x}).$

Going to the characteristic function $G^{\mathrm{RM}}(u,v)=\int dWdQe^{iuW}e^{ivQ}P^{\mathrm{RM}}(W,Q)$ one can perform the $4N$-fold $x$-integration to find

	$\displaystyle G^{\mathrm{RM}}(u,v)$	$\displaystyle=\sum_{\vec{m},\vec{m}^{\prime}}D^{\vec{m},\vec{\tilde{m}}^{\prime}}e^{-\frac{1}{8\Sigma^{2}}\sum_{k=1}^{4N}\left(e^{m_{k}}_{k}-e^{m_{k}^{\prime}}_{k}\right)^{2}}$
		$\displaystyle\quad\times e^{iu\mathcal{W}^{\vec{m},\vec{m}^{\prime}}}e^{iv\mathcal{Q}^{\vec{m},\vec{m}^{\prime}}}e^{-N\Sigma^{2}(2u^{2}-2uv+v^{2})},$

where

	$\displaystyle\mathcal{W}^{\vec{m},\vec{m}^{\prime}}$	$\displaystyle=\frac{1}{2}\sum_{k=1}^{4N}(-1)^{k}\left(e^{m_{k}}_{k}+e^{m_{k}^{\prime}}_{k}\right),$			(14)
	$\displaystyle\mathcal{Q}^{\vec{m},\vec{m}^{\prime}}$	$\displaystyle=\frac{1}{2}\sum_{k=1}^{4N}\mu_{k}\left(e^{m_{k}}_{k}+e^{m_{k}^{\prime}}_{k}\right),$			(15)

with $\mu_{k}$ as defined below Eq. (2). This characteristic function corresponds to a linear superposition of Gaussian PDFs $g_{\mathcal{M}}(W-\mathcal{W}^{\vec{m},\vec{m}^{\prime}},Q-\mathcal{Q}^{\vec{m},\vec{m}^{\prime}})$ with mean values $\langle W\rangle=\mathcal{W}^{\vec{m},\vec{m}^{\prime}}$ and $\langle Q\rangle=\mathcal{Q}^{\vec{m},\vec{m}^{\prime}}$ and with the covariance matrix $\mathcal{M}$ given by

$$\mathcal{M}=\left(\begin{array}[]{cc}\langle(\delta W)^{2}\rangle&\langle\delta W\delta Q\rangle\\ \langle\delta W\delta Q\rangle&\langle(\delta Q)^{2}\rangle\end{array}\right)=2N\Sigma^{2}\left(\begin{array}[]{cc}2&-1\\ -1&1\end{array}\right),$$

(16)

with $\delta X=X-\langle X\rangle$ ($X=W,Q$). Hence, the PDF is given by

	$\displaystyle P^{\mathrm{RM}}(W,Q)$	$\displaystyle=\sum_{\vec{m},\vec{m}^{\prime}}D^{\vec{m},\vec{m}^{\prime}}e^{-\frac{1}{8\Sigma^{2}}{\sum_{k=1}^{4N}}(e^{m_{k}}_{k}-e^{m_{k}\prime}_{k})^{2}}$			(17)
		$\displaystyle\quad\times g_{\mathcal{M}}(W-\mathcal{W}^{\vec{m},\vec{m}^{\prime}},Q-\mathcal{Q}^{\vec{m},\vec{m}^{\prime}}).$

All diagonal coefficients with $\vec{m}=\vec{m}^{\prime}$ are non-negative whereas nondiagonal contributions with $\vec{m}\neq\vec{m}^{\prime}$ may assume complex values. Yet, by construction, the total sum is guaranteed to be always non-negative. Moreover, the nondiagonal contributions are suppressed by exponential factors $\exp[-\sum_{k=1}^{4N}\left(e^{m_{k}}_{k}-e^{m_{k}^{\prime}}_{k}\right)^{2}/(8\Sigma^{2})]$. When the pointer state variance $\Sigma^{2}$ is sufficiently small compared to the minimum squared energy level distance of either Hamiltonian $H_{c}$ or $H_{h}$, all non-diagonal elements are negligibly small and hence only Gaussians having a maximum at $\mathcal{W}^{\vec{m},\vec{m}},\mathcal{Q}^{\vec{m},\vec{m}}$ contribute. Due to the $N$-dependence of the covariance matrix the widths of each of the contributions grow proportionally to the square root of the number of cycles. The marginal work and heat PDFs result as

	$\displaystyle P^{\mathrm{RM}}(W)$	$\displaystyle=\sum_{\vec{m},\vec{m}^{\prime}}D^{\vec{m},\vec{m}^{\prime}}e^{-\frac{1}{8\Sigma^{2}}{\sum_{k=1}^{4N}}(e^{m_{k}}_{k}-e^{m_{k}\prime}_{k})^{2}}g_{4N\Sigma^{2}}(W-\mathcal{W}^{\vec{m},\vec{m}^{\prime}}),$			(18)
	$\displaystyle P^{\mathrm{RM}}(Q)$	$\displaystyle=\sum_{\vec{m},\vec{m}^{\prime}}D^{\vec{m},\vec{m}^{\prime}}e^{-\frac{1}{8\Sigma^{2}}{\sum_{k=1}^{4N}}(e^{m_{k}}_{k}-e^{m_{k}\prime}_{k})^{2}}g_{2N\Sigma^{2}}(Q-\mathcal{Q}^{\vec{m},\vec{m}^{\prime}}).$			(19)

The more pronounced broadening of the Gaussian contributions in the marginal work PDF compared to the marginal heat PDF is caused by the twofold number of energy measurements required for the work compared to the heat.

Instead of registering each of the $4N$ energy values in a separate pointer state one may follow the idea of Ref. [48] and subsequently transfer the actual energy value at the beginning of each stroke with the appropriate sign according to the Eqs. (1) and (2) to two pointer states, one for work and the other one for heat. The transfer of information from the working substance to the pointer states is mediated in a similar way as for repeated measurements by a sufficiently short interaction causing a unitary transformation $V_{k}$ of the form

$$V_{k}=e^{i\kappa H_{k}\otimes((-1)^{k}P_{W}+\mu_{k}P_{Q})},$$

(20)

acting on the joint Hilbert space of the working substance and the two pointer states. Here $P_{W}$ and $P_{Q}$ denote the momentum operators that are canonically conjugate to the position operators of the work and heat pointers, respectively. In contrast to the repeated measurement scheme, the pointers are only read out by a projective position measurements after the completion of $N$ cycles. Again, the pointers are assumed to be gauged in units of energy. Between two subsequent contacts the working substance does not interact with the two pointers but their states remain correlated as a result of the previous contacts. To simplify the further analysis we assume that the pointers have no own dynamics, i.e. they change their states only due to the interactions as specified by the unitary operators defined in Eq. (20). The subsequent strokes and contacts, resulting after the completion of $N$ cycles in pointer state positions at specific values $W$ and $Q$, lead to a non-normalised reduced density matrix of the working substance, which is given by

	$\displaystyle\phi^{\mathrm{RC}}_{W,Q}(\rho)$	$\displaystyle=$	$\displaystyle\sum_{\vec{m},\vec{m}^{\prime}}D^{\vec{m},\vec{m}^{\prime}}(\rho)\sigma(W-\mathcal{W}^{\vec{m}},W-\mathcal{W}^{\vec{m}^{\prime}})$		(21)
			$\displaystyle\times\sigma(Q-\mathcal{Q}^{\vec{m}},Q-\mathcal{Q}^{\vec{m}^{\prime}}),$

where $\rho$ denotes the initial density matrix of the working substance and individual work and heat outcomes are denoted as

	$\displaystyle\mathcal{W}^{\vec{m}}$	$\displaystyle=$	$\displaystyle\sum_{k=1}^{4N}(-1)^{k}e^{m_{k}}_{k},$		(22)
	$\displaystyle\mathcal{Q}^{\vec{m}}$	$\displaystyle=$	$\displaystyle\sum_{k=1}^{4N}\mu_{k}e^{m_{k}}_{k},$		(23)

with $\mu_{k}$ as defined below Eq. (2). The joint PDF $P^{\mathrm{RC}}(W,Q)$ of finding these work and heat values is given by the trace of $\phi^{\mathrm{RC}}_{W,Q}(\rho)$ yielding

$$P^{\mathrm{RC}}(W,Q)=\sum_{\vec{m},\vec{m}^{\prime}}D^{\vec{m},\vec{m}^{\prime}}e^{-\frac{1}{8\Sigma^{2}}(\mathcal{W}^{\vec{m}}-\mathcal{W}^{\vec{m}\prime})^{2}}e^{-\frac{1}{8\Sigma^{2}}(\mathcal{Q}^{\vec{m}}-\mathcal{Q}^{\vec{m}\prime})^{2}}g_{\Sigma^{2}}(W-\mathcal{W}^{\vec{m},\vec{m}^{\prime}})g_{\Sigma^{2}}(Q-\mathcal{Q}^{\vec{m},\vec{m}^{\prime}}).$$

(24)

As for repeated measurements it is a linear superposition of Gaussian PDFs with weights $D^{\vec{m},\vec{m}^{\prime}}$, however, with the following differences:

1. 1.

   for repeated contacts the contributing Gaussians for work and heat factorize while for repeated measurements they are correlated;

2. 2.

   the widths of the contributing Gaussians are independent of $N$ given by $\Sigma$ whereas those for repeated measurements are proportional to $\sqrt{N}\Sigma$;

3. 3.

   for repeated contacts nondiagonal contributions are exponentially suppressed by rates $[(\mathcal{W}^{\vec{m}}-\mathcal{W}^{\vec{m}^{\prime}})^{2}+(\mathcal{Q}^{\vec{m}}-\mathcal{Q}^{\vec{m}^{\prime}})^{2}]/(8\Sigma^{2})$ while for repeated measurements the suppression is determined by rates of the form $\sum_{k=1}^{4N}(e^{m_{k}}_{k}-e^{m_{k}^{\prime}}_{k})^{2}/(8\Sigma^{2})$.

Therefore, for repeated contacts only contributions with different sequences $\vec{m}$ and $\vec{m}^{\prime}$ leading to different values of work and heat are penalized while in the repeated measurement approach deviations of individual components lead to a suppression of coherences due to the monitoring.

For repeated contacts with a non-selective heat measurement the marginal PDF $P^{\mathrm{RC,2P}}(W)$ defined as

$$P^{\mathrm{RC,2P}}(W)=\int dQP^{\mathrm{RC}}(W,Q)=\sum_{\vec{m},\vec{m}^{\prime}}D^{\vec{m},\vec{m}^{\prime}}e^{-\frac{1}{8\Sigma^{2}}(\mathcal{W}^{\vec{m}}-\mathcal{W}^{\vec{m}\prime})^{2}}e^{-\frac{1}{8\Sigma^{2}}(\mathcal{Q}^{\vec{m}}-\mathcal{Q}^{\vec{m}\prime})^{2}}g_{\Sigma^{2}}(W-\mathcal{W}^{\vec{m},\vec{m}^{\prime}})$$

(25)

differs by the extra factor of $\exp[-(\mathcal{Q}^{\vec{m}}-\mathcal{Q}^{\vec{m}\prime})^{2}/(8\Sigma^{2})]$ from the PDF resulting from a mere work measurement performed with a single pointer state. In the latter case one obtains

	$\displaystyle P^{\mathrm{RC,1P}}(W)$	$\displaystyle=$	$\displaystyle\sum_{\vec{m},\vec{m}^{\prime}}D^{\vec{m},\vec{m}^{\prime}}e^{-\frac{1}{8\Sigma^{2}}(\mathcal{W}^{\vec{m}}-\mathcal{W}^{\vec{m}\prime})^{2}}$		(26)
			$\displaystyle\times g_{\Sigma^{2}}(W-\mathcal{W}^{\vec{m},\vec{m}^{\prime}}).$

For repeated contacts the same type of contextuality, i.e. the dependence of the observed results on the presence of a non-selective measurement, also holds for the marginal heat PDF. In both cases it is more pronounced for rather imprecise registration, i.e. for pointer states with relatively large variances $\Sigma^{2}$. For repeated measurements the joint work and heat PDF [see Eq. (17)] results according to Eq. (12) in a classical way from the joint PDF $p(\vec{x})$ of all individual energies. Therefore repeated measurements do not present this particular contextuality feature.

Once the work and heat PDFs are known for a specified diagnostic method, the performance of the engine can be characterized by its efficiency $\eta$ and reliability $R$ which are defined as

	$\displaystyle\eta$	$\displaystyle=$	$\displaystyle-\frac{\langle W\rangle}{\langle Q\rangle},$		(27)
	$\displaystyle R$	$\displaystyle=$	$\displaystyle-\frac{\langle W\rangle}{\sqrt{\langle W^{2}\rangle-\langle W\rangle^{2}}},$		(28)

where $\langle\cdot\rangle$ denotes the average of the indicated quantity with respect to the PDF of the considered diagnostic method. Higher order efficiencies as introduced in Ref. [49] can be obtained from higher order moments of work and heat. Also a fluctuating efficiency, $W/Q$, can in principle be determined [50], but will not be considered here.

The general formulation outlined in Sec. II can be applied to any working substance undergoing adiabatic or non-adiabatic protocols. For a numerically feasible but yet representative example, we will use in this and subsequent sections a two-level system as the working substance and allow for non-adiabatic work strokes, i.e. strokes comprising shifts of the instantaneous energy eigenstates and transitions between them during the unitary evolution. Without specifying the time-dependent Hamiltonian in detail, the system Hamiltonians $H_{c}$ and $H_{h}$ before and after the first work stroke, respectively, are written as

	$\displaystyle H_{c}$	$\displaystyle=$	$\displaystyle\epsilon_{c}\left(|+_{c}\rangle\langle+_{c}|-|-_{c}\rangle\langle-_{c}|\right),$
	$\displaystyle H_{h}$	$\displaystyle=$	$\displaystyle\epsilon_{h}\left(|+_{h}\rangle\langle+_{h}|-|-_{h}\rangle\langle-_{h}|\right),$		(29)

where $|-_{u}\rangle$ and $|+_{u}\rangle$ denote the ground and the excited states, respectively, of the Hamiltonian $H_{u}$ with $u=h,c$. The most general form of the unitary evolution, during the work stroke, then reads

	$\displaystyle U=$	$\displaystyle\sqrt{1-\alpha}\left(e^{-i\varphi}|+_{h}\rangle\langle+_{c}|+e^{i\varphi}|-_{h}\rangle\langle-_{c}|\right)$			(30)
		$\displaystyle-\sqrt{\alpha}\Big{(}|+_{h}\rangle\langle-_{c}|-|-_{h}\rangle\langle+_{c}|\Big{)},$
	$\displaystyle\tilde{U}=$	$\displaystyle\sqrt{1-\alpha}\left(e^{-i\varphi}|+_{c}\rangle\langle+_{h}|+e^{i\varphi}|-_{c}\rangle\langle-_{h}|\right)$			(31)
		$\displaystyle+\sqrt{\alpha}\Big{(}|+_{c}\rangle\langle-_{h}|-|-_{c}\rangle\langle+_{h}|\Big{)}.$

Above $\tilde{U}\equiv C^{*}U^{\dagger}C$ is the evolution operator for the time-reversed protocol with the antiunitary operator $K$ chosen to be the complex conjugation operator $C$. Here $\alpha\in[0,1]$ is the transition probability between the ground and the excited states and $\varphi$ is the corresponding phase (both parameters depend on the stroke protocol and in particular on its duration). The limiting case $\alpha=0$ refers to an adiabatic process while $\alpha=1$ corresponds to a swap. For example, a Landau-Zener protocol is described by the time-dependent Hamiltonian

$$H(t)=-\frac{vt}{2}\sigma_{z}-\epsilon_{c}\sigma_{x}\>,$$

(32)

where the velocity of the drive, $v=4\sqrt{\epsilon_{h}^{2}-\epsilon_{c}^{2}}/T_{1}$, is chosen such that the eigenvalues of the Hamiltonian change from $\pm\epsilon_{c}$ at $t=0$ to $\pm\epsilon_{h}$ at $t=T_{1}/2$, the time corresponding to the duration of a work stroke. The parameters specifying the unitary operator $U$ are given in Ref. [51] by

	$\displaystyle\alpha$	$\displaystyle=$	$\displaystyle e^{-2\pi\delta},\quad\delta=\frac{\epsilon_{c}T_{1}}{4\sqrt{(\frac{\epsilon_{h}}{\epsilon_{c}})^{2}-1}},$
	$\displaystyle\varphi$	$\displaystyle=$	$\displaystyle\frac{\pi}{4}-\delta(\log{\delta}-1)-\mathrm{arg}\Gamma(1-i\delta).$		(33)

where $\Gamma(z)$ denotes the Gamma function. For other protocols characterized by a single parameter see Ref. [52] and for more general cases Ref. [53, 54].

In the first instance we shall not refer to a particular work protocol and use the general form of a unitary time evolution $U$ as specified by Eq. (30). Before we do so, we consider the heating and cooling strokes of the two-level working substance that are achieved by a coupling to the heat reservoirs at temperatures $T_{h}$ and $T_{c}$, respectively. Depending on the duration and the strength of the respective contacts perfect or imperfect thermalization of the two-level system can be reached.

Provided that the contact between the working substance and the heat bath is sufficiently long, the finally reached reduced state of the two-level system is always the same, dependent on the temperature of the heat bath and the strength of the interaction but independent of its state $\rho$ at the beginning of the contact. Hence, the operation $\Phi_{u}$, $u=h,c$ describing a perfect thermalization stroke is given by the projection onto the final state $\rho_{\beta}$ and can be written as

$$\Phi_{u}(\rho)=\rho_{u}\mathrm{Tr}[\rho].$$

(34)

Only for weak coupling $\rho_{u}$, $u=h,c$ coincides with the Gibbs state $\rho_{u}\propto\exp[-\beta_{u}H_{u}]$ defined by the Hamiltonian of the two-level system at the inverse temperature $\beta_{u}$. While these states lack coherence the latter may be found in generalized Gibbs states resulting from the partial trace of the total system’s Gibbs state with respect to the bath [40, 42]. For a perturbative treatment in the coupling strength up to second order see Append. A.

In the asymptotic limit of an initially sharp pointer state with $\Sigma=0$ the joint work and heat PDFs characterizing a single cycle are identical for repeated measurements and repeated contacts (two-pointer scheme), i.e., $P^{\mathrm{RM}}_{1}(W,Q)=P^{\mathrm{RC}}_{1}(W,Q)$. The first two moments can be expressed as

	$\displaystyle\langle W\rangle$	$\displaystyle=$	$\displaystyle\mathcal{A}_{c,h}\,\epsilon_{h}+\mathcal{A}_{h,c}\,\epsilon_{c},\quad\quad\langle W^{2}\rangle=2\mathcal{B}_{c,h}\left(\epsilon_{c}^{2}+\epsilon_{h}^{2}\right)+\left[\frac{1-\mathcal{B}_{c,h}}{1-2\alpha}-(1-2\alpha)\left(1+\mathcal{B}_{c,h}\right)\right]\epsilon_{c}\epsilon_{h},$		(35)
	$\displaystyle\langle Q\rangle$	$\displaystyle=$	$\displaystyle-\mathcal{A}_{c,h}\,\epsilon_{h},\quad\quad\quad\quad\quad~{}~{}\langle Q^{2}\rangle=2\mathcal{B}_{c,h}\,\epsilon_{h}^{2},$		(36)
	$\displaystyle\langle WQ\rangle$	$\displaystyle=$	$\displaystyle-2\mathcal{B}_{c,h}\,\epsilon_{h}^{2}-\frac{1}{2}\left[\frac{1-\mathcal{B}_{c,h}}{1-2\alpha}-(1-2\alpha)\left(1+\mathcal{B}_{c,h}\right)\right]\epsilon_{c}\epsilon_{h},$		(37)

with $\mathcal{A}_{c,h}=2\left(\alpha+d_{c}-d_{h}-2\alpha d_{c}\right)$ and $\mathcal{B}_{c,h}=1-(1-2\alpha)(1-2d_{c})(1-2d_{h})$. Here, $d_{c}$ and $d_{h}$ denote the populations of the exited states of the cold and the hot density matrices, respectively, see the Eqs. (53) and (LABEL:d) in Append. A. Even when the Gaussian pointer has a finite width $\Sigma\neq 0$, the differences between repeated measurements and contacts in the average values of work and heat are of the order $\exp[-\epsilon^{2}_{c(h)}/(2\Sigma^{2})]$ which could be extremely small for a precise measurement apparatus, i.e. if $\epsilon_{c(h)}<\Sigma$. On the other hand, the second moments differ by $3\Sigma^{2}+\mathcal{O}(\exp[-\epsilon_{c(h)}^{2}/(2\Sigma^{2})])$ which is dominated by the $3\Sigma^{2}$ term arising due to the difference in the number of measurements. Accordingly, also the reliability [see Eq. (28)] differs for repeated contacts and repeated measurements. Overall, however, even though the two measurement schemes are quite different they do not show major significant differences for a perfectly thermalizing quantum Otto engine.

Perfect thermalization requires long times during which the working substance is coupled to one of the two heat baths, leading to long cycle periods and correspondingly small power output of the engine. The dynamics of the relaxation process in general is very complicated. A substantial simplification occurs only at week coupling in the Markovian limit in which the dynamics is governed by a Lindblad equation

$$\dot{\rho_{u}}(t)=-i[H_{u},\rho_{u}]+\Gamma_{u}^{+}\mathcal{D}_{u}^{+}[\rho_{u}]+\Gamma_{u}^{-}\mathcal{D}_{u}^{-}[\rho_{u}],$$

(38)

with the subscript $u=c,h$, the dissipators $\mathcal{D}_{u}^{\pm}[\rho_{u}]=2L^{\mp}_{u}\rho_{u}L^{\pm}_{u}-\{L^{\pm}_{u}L^{\mp}_{u},\rho_{u}\}$, the rates $\Gamma_{u}^{\pm}=\gamma\epsilon_{u}/(1+\exp[\mp 2\beta_{u}\epsilon_{u}])$, and the Lindblad jump operators $L^{\pm}_{u}=|\pm_{u}\rangle\langle\mp_{u}|$. The Hamiltonians $H_{u}$ are given by Eq. (29). Even though the stationary solutions of the master equations for the hot and the cold heat baths are given by the Gibbs states, corresponding to the Hamiltonians $H_{u}$ at the respective temperatures $\beta_{u}$ and hence do not possess any coherence, the solutions of the master equations at any finite time in general do contain coherences. This leads to differences between repeated measurements and repeated contacts even in the limit of infinite precision ($\Sigma\rightarrow 0$). For a single cycle, starting at some initial state $\rho=d|+_{\mathrm{c}}\rangle\langle+_{\mathrm{c}}|+(1-d)|-_{\mathrm{c}}\rangle\langle-_{\mathrm{c}}|+[q|+_{\mathrm{c}}\rangle\langle-_{\mathrm{c}}|+\mathrm{h.c.}]$ one finds the following expressions for the first two moments of work and heat for a small pointer state variance including leading order $\mathcal{O}(\Sigma^{2})$ corrections:

	$\displaystyle\langle W\rangle^{\mathrm{RM}}$	$\displaystyle=$	$\displaystyle 2\alpha(1-\alpha)\left[\epsilon_{h}\tanh\left(\beta_{h}\epsilon_{h}\right)(1-e^{-2\gamma\theta})+\epsilon_{c}(1-2d)(1+e^{-2\gamma\theta})\right]$		(39)
			$\displaystyle+\sum_{j=0,1}\left[(-1)^{j}\epsilon_{h}+\epsilon_{c}\right]\left[1+(-1)^{j}\left(2\alpha-1\right)\right]^{2}\left(\frac{e^{(-1)^{j}\beta_{h}\epsilon_{h}}}{e^{\beta_{h}\epsilon_{h}}+e^{-\beta_{h}\epsilon_{h}}}-d\right)\left(1-e^{-2\gamma\theta}\right),$
	$\displaystyle\langle W\rangle^{\mathrm{RC}}$	$\displaystyle=$	$\displaystyle\langle W\rangle^{\mathrm{RM}}-4\epsilon_{c}\alpha(1-\alpha)(1-2d)e^{-\gamma\theta}\cos\left(2(\theta+\varphi)\right),$		(40)
	$\displaystyle\langle W^{2}\rangle^{\mathrm{RM}}$	$\displaystyle=$	$\displaystyle 4\left(\epsilon_{h}^{2}+\epsilon_{c}^{2}\right)\bigg{\{}(\alpha+d-2\alpha d)+\left[(1-2d)(1-2\alpha)-2\alpha^{2}d\right]\frac{e^{-\beta_{h}\epsilon_{h}}}{e^{\beta_{h}\epsilon_{h}}+e^{-\beta_{h}\epsilon_{h}}}\bigg{\}}\left(1-e^{-2\gamma\theta}\right)$		(41)
			$\displaystyle+8\epsilon_{h}\epsilon_{c}\bigg{\{}\alpha^{2}(1-2d)-(1-2\alpha)d-\left[(1-2d)(1-2\alpha)-2\alpha^{2}d\right]\frac{e^{-\beta_{h}\epsilon_{h}}}{e^{\beta_{h}\epsilon_{h}}+e^{-\beta_{h}\epsilon_{h}}}\bigg{\}}\left(1-e^{-2\gamma\theta}\right)$
			$\displaystyle+4\epsilon_{c}^{2}\alpha(1-\alpha)e^{-2\gamma\theta}+4\Sigma^{2},$
	$\displaystyle\langle W^{2}\rangle^{\mathrm{RC}}$	$\displaystyle=$	$\displaystyle\langle W^{2}\rangle^{\mathrm{RM}}-8\epsilon_{c}^{2}\alpha(1-\alpha)e^{-\gamma\theta}\cos\left(2(\theta+\varphi)\right)-3\Sigma^{2},$		(42)
	$\displaystyle\langle WQ\rangle^{\mathrm{RC}}$	$\displaystyle=$	$\displaystyle-4\epsilon_{h}^{2}\bigg{\{}(\alpha+d-2\alpha d)+\left[(1-2d)(1-2\alpha)-2\alpha^{2}d\right]\frac{e^{-\beta_{h}\epsilon_{h}}}{e^{\beta_{h}\epsilon_{h}}+e^{-\beta_{h}\epsilon_{h}}}\bigg{\}}\left(1-e^{-2\gamma\theta}\right)$		(43)
			$\displaystyle-4\epsilon_{h}\epsilon_{c}\bigg{\{}\alpha^{2}(1-2d)-(1-2\alpha)d-\left[(1-2d)(1-2\alpha)-2\alpha^{2}d\right]\frac{e^{-\beta_{h}\epsilon_{h}}}{e^{\beta_{h}\epsilon_{h}}+e^{-\beta_{h}\epsilon_{h}}}\bigg{\}}\left(1-e^{-2\gamma\theta}\right),$
	$\displaystyle\langle WQ\rangle^{\mathrm{RM}}$	$\displaystyle=$	$\displaystyle\langle WQ\rangle^{\mathrm{RC}}-2\Sigma^{2},$		(44)

where $\theta=\epsilon_{h}\tau_{h}$ with $\tau_{h}$ being the duration of the hot bath stroke. A dependence of the moments on the initial coherence parameter $q$ is only seen in higher order $\Sigma^{2}$ corrections. From the expressions of these moments the according variances and the covariance of work and heat can be obtained. Instead of these bulky expressions we present the variances and covariances for the two monitoring schemes in Fig. 2a, b as functions of $\theta$. The difference between repeated measurements and repeated contacts is governed by damped oscillations stemming from the $e^{-\gamma\theta}\cos\left(2(\theta+\varphi)\right)$-term in the deviation of the first two work moments, see Eqs. (40), (42) and Fig. 2c, d. Yet, the maximal amplitude of these oscillations is rather small rendering the (co)variances for repeated measurements and repeated contacts almost identical.

The moments of heat can be obtained by setting $\epsilon_{c}=0$ and multiplying appropriate global signs (negative for odd order moments and positive for even ones) in the expressions for the moments of work. As seen from the above equations, this also implies that the average heat for two-pointer repeated contacts and repeated measurements are the same, i.e., $\langle Q\rangle=\langle Q\rangle^{\mathrm{RM}}=\langle Q\rangle^{\mathrm{RC}}$. The variances of the heat are not displayed. They behave in a similar way as the those of the work as presented in Fig. 2 with solely a constant difference of $3\Sigma^{2}$ between repeated measurements and repeated contacts and therefore are not extra displayed. Note that the engine may starts at an arbitrary initial state specified by the parameters $d$ and $q$. Here and in the following examples, however, we will initiate the monitored sequence of $N$ cycles with the invariant state $\rho^{*}$ of the map $\mathcal{F}$, i.e.

$$\rho^{*}=\mathcal{F}(\rho^{*})$$

(45)

where

$$\mathcal{F}(\rho)=\Phi_{c}(\tilde{U}\Phi_{h}(U\rho U^{\dagger})\tilde{U}^{\dagger})$$

(46)

describes the dynamics of a complete cycle in the absence of any monitoring.

Furthermore we note that the master equation (38), which governs the thermalization strokes of the engine, leads to decoupled equations of motion for the populations and coherences. Hence, the thermalization maps $\Phi_{u}$ resulting as the solutions of the master equation (38) obey the conditions

$$\mathrm{Tr}[L^{\pm}_{u}\Phi_{u}(\mathcal{P}^{k}_{u})]=\mathrm{Tr}[\mathcal{P}^{k}_{u}\Phi_{u}(L^{\pm}_{u})]=\mathrm{Tr}[L^{\pm}_{u}\Phi_{u}(L^{\mp}_{u})]=0,$$

(47)

with swap-operators $L^{\pm}_{u}=|\pm_{u}\rangle\langle\mp_{u}|$, as defined below Eq. (38), and $\mathcal{P}^{k}_{u}=|k_{u}\rangle\langle k_{u}|$ with $k=\pm$, denoting the projection operators onto the eigenstates of the Hamiltonian $H_{u}$. Due to this decoupling, the work PDFs for one- and two-pointer repeated contacts become identical and we denote the average value as $\langle W\rangle^{\mathrm{RC}}$ (see Append. B for more details). In the limit $\theta\rightarrow\infty$, $\Sigma=0$, and $\rho=\rho_{c}$, which leads to perfect thermalization, Eqs. (39)–(44) match Eqs. (35)–(37). For a highly non-adiabatic work stroke ($\alpha$ large) the cosine terms in the expressions for the first two moments play a significant role. Then the sign of $\cos(2(\theta+\varphi))$ determines which one of the two monitoring schemes leads to a larger work on average for a single cycle. This affects also the efficiency of the engine because the average heat is the same for either scheme.

Figure 3 presents the joint and marginal work and heat PDFs of an engine with perfect thermalization for a single cycle ($N=1$). Panel a displays the work PDFs for repeated measurements and repeated contacts for adiabatic work strokes, i.e., for $\alpha=0$. In this case, there is no difference between the marginal two-pointer work PDF given by Eq. (25) and the one-pointer work PDF (26) as explained in Append. D. In accordance with the Eq. (26) the PDF consists of three peaks located at $W=0$ and $W=\pm 2(\epsilon_{c}-\epsilon_{h})$ because transitions between different energy levels are impossible for adiabatic work strokes. These peaks are broader for repeated measurements compared to repeated contacts. This feature prevails for non-adiabatic work strokes. In panel b we present the results for moderately and strongly non-adiabatic work strokes. In order not to clutter the plot we present only the work PDFs resulting from repeated contacts. With decreasing adiabaticity (increasing $\alpha$) six further peaks appear in accordance with Eq. (26) accompanied by a decrease of the weights of the three adiabatic peaks. At the same time the weights of those peaks at positive work values may increase to such an extent that the average work becomes positive. That means that the engine only consumes energy rather than to perform work. As for classical engines any non-adiabaticity is detrimental with respect to the efficiency but must be accepted to achieve finite power. The same features as described for repeated contacts also hold for the repeated measurement PDFs with the only difference that their peaks are broader.

Figures 3c and d present the joint work and heat PDFs for repeated measurements and contacts, respectively. In the case of repeated measurements, each individual Gaussian contribution contains a negative covariance indicating an anti-correlation of the according work and heat contributions. In contrast, for repeated contacts the individual Gaussian contributions are isotropic. In both cases the main contribution to the covariance of work and heat results from the linear superposition of the individual Gaussian distributions.

Numerical results for adiabatic and non-adiabatic work strokes and imperfect thermalization are presented in Fig. 4. The panels a and b display the work PDFs for repeated measurements and repeated contacts for two different numbers of cycles $N$. For the smaller number of cycles, $N=5$, still several peaks are visible in the repeated measurements work PDF. Because their widths grow as $\sqrt{N}$ this multiply peaked structure disappears with increasing number of cycles to finally approach a Gaussian shape, which in the present case is virtually indistinguishable from the exact PDF for $N=30$. In contrast, the fine-structure of the repeated contact work PDF remains visible also for large numbers of cycles due to the fact that the widths of the individual Gaussian contributions are solely determined by the initial pointer state and hence are $N$ independent.