Finite-time performance of a single-ion quantum Otto engine

We consider a single trapped ion Our Paper- Single ion . In the Lamb-Dicke limit, the ion is confined to its two lowest-lying electronic states $|-\rangle$ and $|+\rangle$ and the lowest-lying vibrational states $|0\rangle$ and $|1\rangle$ (see Fig. 1). Both the electronic and the vibrational degrees of freedom can therefore be modeled as two-level systems. The following Hamiltonian describes the dynamics of the ion (in units of Planck’s constant $\hbar=1$):

$$H_{1}(t)=H_{S}(t)+H_{\rm ph}+H_{\rm int}\;,$$

(1)

where

	$\displaystyle H_{S}(t)=g\sigma_{x}+B(t)\sigma_{z},\;\;\;H_{\rm ph}=\omega a^{\dagger}a,$		(2)
	$\displaystyle H_{\rm int}=k\left(a^{\dagger}\sigma_{-}+\sigma_{+}a\right).$		(3)

Here, $H_{S}$ is the Hamiltonian for the electronic degree of freedom of the ion, $H_{\rm ph}$ represents the energy of the vibrational mode with frequency $\omega$, and $H_{\rm int}$ defines the interaction between these two modes with the coupling constant $k$. We consider that the electronic states are driven by a local electric field with Rabi frequency $2g$. A time-dependent magnetic field of strength $B(t)$ is applied along the quantization axis. The operators $\sigma_{z}$ and $\sigma_{x}$ are usual Pauli spin operators in the basis $(|-\rangle,|+\rangle)$, while the operator $a$ is the annihilation operator operating on the vibrational states $(|0\rangle,|1\rangle)$. We consider the electronic degree of freedom of the ion as the working substance $S$ of the engine, while the vibrational mode plays the role of the effective cold bath. The thermal environment at an ambient temperature $T_{H}$ is considered as the relevant hot bath.

Traditionally, a bath is a system with many degrees of freedom. Here we consider a finite-dimensional system that interacts with the system (the electronic states) as equivalent to a bath. Any interaction between the two subsystems, in general entangles them and a partial trace of one of these subsystems leads to decoherence in the other. This basic notion of decoherence is used here, where a finite-dimensional system (the two-level vibrational states) is traced partially. That a finite-dimensional system can act as a bath, leading to decoherence in a spin has been shown explicitly in Ref. Madam decoherence paper . Note that for an average excitation of this mode (vibrational states) $\bar{n}$, one can associate with it an effective temperature using the canonical probability distribution function, as given by $T_{L}=\hbar\omega/\ln(1/\bar{n}+1)k_{B}$, where $k_{B}$ is the Boltzmann constant. For $\bar{n}=0.02$, we can have $T_{L}=1$ mK (see, for example, Ion trap paper C. Monroe ), which is a temperature low enough to approximate the vibrational mode as a two-level system, as often routinely done for quantum computing using trapped ions.

We focus on the operation of a quantum Otto cycle in the following. This cycle consists of four stages: two isochoric and two adiabatic stages. Here we show how to use the system $S$ and the two thermal baths as identified above to implement these stages.

During this stage ($1\rightarrow 2$, Fig. 2), the system $S$ interacts with the hot bath at a temperature $T_{H}$. The magnetic field is kept constant at a value $B=B_{H}$, which means that the Hamiltonian $H_{S}$ of the system is kept fixed during this stage. If we allow the system-bath interaction for a sufficiently long time, much longer than the thermal relaxation time $t_{\rm{relax}}$ of the system, the system reaches thermal equilibrium with the hot bath. Due to full thermalization, only the diagonal elements of the system density matrix remain non-zero, when expressed in the eigenstates basis of $H_{S}$. This means that the system retains no coherence (the off-diagonal elements of the density matrix). On the other hand, the situation is quite different if we consider a partial thermalization, where the time-scale for the system-bath interaction is allowed to be less than or comparable to the thermal relaxation time of the system. In such a case, the coherence in the system does not decay completely. This coherence influences the dynamics of the engine in the next stage of the cycle. We assume here that the system is initially prepared in its ground state $\left|-\right\rangle$ [the ground state of the system, i.e., the lowest-energy eigenstate of $H_{S}(g=0)=\sigma_{z}$], which means that the system does not exhibit any initial coherence.

The dynamics of the ion under the action of the Hamiltonian $H_{1}(t)$ and a heat reservoir can be described by the Liouville-von Neumann equation The Theory of Open Quantum Systems-Book as

	$\displaystyle\frac{d\rho_{1}}{dt}$	$\displaystyle=$	$\displaystyle-\frac{i}{\hbar}\left[H_{1}(t),\rho_{1}(t)\right]$		(4)
			$\displaystyle+\left(\bar{n}_{\text{th}}+1\right)\frac{\Gamma}{2}(2\sigma_{-}\rho_{1}\sigma_{+}-\sigma_{+}\sigma_{-}\rho_{1}-\rho_{1}\sigma_{+}\sigma_{-})$
			$\displaystyle+\left(\bar{n}_{\text{th}}\right)\frac{\Gamma}{2}\left(2\sigma_{+}\rho_{1}\sigma_{-}-\sigma_{-}\sigma_{+}\rho_{1}-\rho_{1}\sigma_{-}\sigma_{+}\right),$

where $\Gamma$ is the vacuum decay rate of the electronic states of the ion, $\bar{n}_{\text{th}}$ is average photon numbers of the hot thermal bath at a temperature $T_{H}$, and $\sigma_{-}=|-\rangle\langle+|$ and $\sigma_{+}=|+\rangle\langle-|$ are the usual annihilation and creation operators in the $(|-\rangle,|+\rangle)$ basis, respectively. In Eq. (4) the assumption is the Born-Markov approximation only. The Born approximation is about the weak coupling between the system and reservoir. The Markov approximation is valid when the correlation among the reservoir (hot bath in this case) decreases much faster compared to the relaxation time of the system. Here, of course, we model the reservoir as a collection of infinitely many harmonic oscillators. Also, Eq. (4) does not address the issue of pure dephasing, as we have not considered any information backflow from the hot bath to the system. In this model, the hot bath is coupled to both of the electronic state and vibrational state. However, the proper choice of the thermalization ensures that there is no anomalous heat flow from a hot bath to a cold bath (vibrational state). Now in Eq. (4) $\rho_{1}$ is the joint density matrix of the electronic state and the vibrational mode of the ion and can be written in terms of the joint basis $\left\{\left|-,0\right\rangle,\left|-,1\right\rangle\left|+,0\right\rangle\left|+,1\right\rangle\right\}$. We find the evolution of $\rho_{1}$ and evaluate the density matrix $\rho_{S}$ of the system $S$ by taking the partial trace over the vibrational state Our Paper- Single ion . We are interested in the heat exchange by the system only. During this stage from $t=t_{0}$ to $t=t_{1}$, the heat $Q_{H}$ absorbed by the system can be expressed in terms of the change in average internal energy of the system as

$$Q_{H}={\rm Tr}\left[\rho_{S}\left(t_{1}\right)H_{S}\left(B_{H}\right)\right]-{\rm Tr}\left[\rho_{S}\left(t_{0}\right)H_{S}\left(B_{H}\right)\right],$$

(5)

where the initial density matrix $\rho_{S}(t=t_{0})=|-\rangle\langle-|$ evolves into $\rho_{S}(t=t_{1})$ during this stage. Note that the system does not do any work, as the magnetic field is maintained at a constant value.

During this stage ($2\rightarrow 3$, Fig. 2), the magnetic field is changed from $B_{H}$ to $B_{L}$ $\left(B_{L}<B_{H}\right)$ through a duration $\tau$. Usually it is assumed that the heat bath is disconnected from the system. This is true if the evolution takes place within the timescale $\tau\lesssim 1/\gamma$ (where $\gamma$ is the decay rate of the system to the heat bath). This ensures that the heat energy exchanged between the system and environment can be neglected and the driven dynamics becomes a unitarity. In addition, to maintain quantum adiabaticity, one needs to vary the magnetic field slow enough such that $\tau\gg\frac{g}{8}\left|\frac{1}{B_{L}^{2}}-\frac{1}{B_{H}^{2}}\right|$ messaiah_book ; Our Paper- Single ion . Clearly for a faster variation, one may attain a nonadiabatic evolution. Here we emphasize that, at the timescale mentioned above, we effectively disconnect the thermal environment from the system, ensuring thermal isolation. This guarantees the thermodynamic adiabaticity. However, during the timescale, it may be possible that internal excitation in the instantaneous energy eigenstates occurs, which indicates that the stroke may be nonadiabatic in the quantum mechanical sense Nonadiabatic single-qubit quantum Otto engine .

We solve the full master equation (4), where the Hamiltonian $H_{1}$ is given in Eq. (1) and the magnetic field varies linearly with time as given by Our Paper- Single ion ; altintas2014quantum

$$B(t)=B_{H}+\frac{B_{L}-B_{H}}{\tau}t,$$

(6)

where $\tau$ is the finite timescale of the change of the magnetic field from $B_{H}$ to $B_{L}$ or vice versa. The output of stage 1, the joint state $\rho_{1}\left(t_{1}\right)$, is considered as the initial condition for solving the master equation. This state evolves into $\rho_{1}\left(t_{2}\right)$ a time $\tau$ later (such that $t_{2}=t_{1}+\tau$). As described in Sec. II.2, we can obtain the reduced density matrix $\rho_{S}\left(t_{2}\right)$ of the system $S$, by taking a suitable partial trace. The change in the internal energy of the system during this stage can now be obtained as

$$W_{1}={\rm Tr}\left[\rho_{S}\left(t_{2}\right)H_{S}\left(B_{L}\right)\right]-{\rm Tr}\left[\rho_{S}\left(t_{1}\right)H_{S}\left(B_{H}\right)\right].$$

(7)

The above could be considered as the work done by the system if there would be no heat exchange with the bath Achieve higher efficiency at maximum power with finite-time quantum Otto cycle . This holds true for the case of a perfect reversible adiabatic process. The time taken to complete the process should ideally be infinite $\left(\tau\rightarrow\infty\right)$, ensuring that the system will remain in the instantaneous eigenstate of the system Hamiltonian. The initial thermal state $\rho_{S}^{\text{th}}(t_{1})$ (when $B=B_{H}$) remains thermal as $\rho_{S}^{\text{th}}(t_{2})$ at the end of the process, when $B=B_{L}$ explain . However, for the finite-time nature of the evolution and non-commutativity of the internal and the external part of the Hamiltonian (2), the adiabatic theorem is not expected to hold. As a result, internal friction arises rezek2010reflections ; kosloff2013quantum ; feldmann2000performance ; Irreversibility in a unitary finite-rate protocol: the concept of internal friction ; ccakmak2017irreversible ; plastina2014irreversible , which has a significant impact on the efficiency of the engine. In this case, the final state of the system deviates from its equilibrium thermal state due to coherence generation in the energy eigenbasis. One can define irreversible work as the extra amount of energy that needs to be done on the system by the driving agent against this friction when the process does not remain reversible. Precisely speaking, if the final state after finite time $\tau$ becomes $\rho_{S}(t_{2})$, instead of the thermal equilibrium state $\rho_{S}^{\text{th}}(t_{2})$, the irreversible work ccakmak2017irreversible can be written as

	$\displaystyle W_{1}^{\rm ir}$	$\displaystyle=$	$\displaystyle W_{1}^{\tau}-W_{1}^{\tau\rightarrow\infty}$
		$\displaystyle=$	$\displaystyle{\rm Tr}\left[\rho_{S}\left(t_{2}\right)H_{S}\left(B_{L}\right)\right]-{\rm Tr}\left[\rho_{S}^{{\rm th}}\left(t_{2}\right)H_{S}\left(B_{L}\right)\right].$

It is possible to connect this quantity $W_{1}^{\text{ir}}$ to the quantum relative entropy plastina2014irreversible . The quantum relative entropy of two density operators $\rho$ and $\sigma$ is defined as, $S\left(\rho||\sigma\right)=k_{B}\left[\rm Tr\left(\rho\ln\rho\right)-\rm Tr\left(\rho\ln\sigma\right)\right]$, which measures the distance between two quantum states. Based on Klein’s inequality, the non-negativity of relative entropy ensures that $W_{1}^{\text{ir}}$ is always a non-negative quantity and is expressed as (in units of $k_{B}=1$)

	$\displaystyle{W^{\text{ir}}_{1}}$	$\displaystyle=$	$T_{H}S\left[\rho_{S}\left(t_{2}\right)||\rho_{S}^{{\rm th}}\left(t_{2}\right)\right]$
		$\displaystyle=$	$T_{H}\left({\rm Tr}\left\{\rho_{S}\left(t_{2}\right){\rm ln}\left[\rho_{S}\left(t_{2}\right)\right]\right\}-{\rm Tr}\left\{\rho_{S}\left(t_{2}\right){\rm ln}\left[\rho_{S}^{{\rm th}}\left(t_{2}\right)\right]\right\}\right).$

Due to the finite-time Hamiltonian driving, the final system state is out of equilibrium due to the inter-level transitions. Here we examine the quantum relative entropy to investigate how close the nonequilibrium system state is to the corresponding equilibrium state. The quantum relative entropy has a significant thermodynamic interpretation plastina2014irreversible . It is related to the inner friction arising due to the finite time adiabatic strokes and the heat exchanges during the cycle’s relaxation, which is defined as the difference between the work done on an actual finite-time process and the work done for the quasistatic transformation. Due to the non-negativity of $S\left(\rho\left\|\sigma\right.\right)$, the heat exchange during the relaxation process is always positive. This can be interpreted as the additional energy that the driving agent should supply to the system to compensate for the same change due to the fast driving. This means an extra amount of energy $W^{\rm ir}$ will be stored in the system, making the system state deviate from the equilibrium state and be prepared in a nonequilibrium state. This extra amount of stored energy will be released to the heat bath during the cycle’s relaxation process at the beginning of the next isochoric stage Quantum Carnot cycle with inner friction ; Generalized Clausius Inequality for Nonequilibrium Quantum Processes . Therefore, the net positive work output is decreased [see Eq. (15)].

During this stage ($3\rightarrow 4$, Fig. 2), the system releases $Q_{L}$ heat to the cold bath and the system Hamiltonian remains fixed at $H_{S}(B_{L})$. Here we employ a projective measurement of the electronic state of the system that post-selects the ground state $|-\rangle$. This essentially is equivalent to a probabilistic cooling of the system Our Paper- Single ion and to heat release out of the system. This increases the effective temperature of the vibrational mode, which therefore can be considered as a heat sink. Note that as the probability that the ion is in the ground state $|-\rangle$ is much higher than that for the excited state, the probability of cooling of the system is quite high. Though at the onset of this stage, the system is in an entangled state $\rho_{1}(t_{2})$ in the electronic-vibrational joint basis, the measurement process disentangles them. After the instantaneous projective measurement, the final state of the system will be

$$\rho_{S}\left(t_{3}\right)=|-\rangle\langle-|.$$

(10)

So, the heat released from the system to the vibrational mode can be calculated as

$$Q_{L}={\rm Tr}\left[\rho_{S}\left(t_{3}\right)H_{S}\left(B_{L}\right)\right]-{\rm Tr}\left[\rho_{S}\left(t_{2}\right)H_{S}\left(B_{L}\right)\right].$$

(11)

The cooling process, as described above, is evidently probabilistic and depends upon the measurement outcome, i.e., post-selection. It may be noted that an alternative way of cooling the system was proposed in Ref. Kurizki alternative way of cooling , which is based on a sequence of nonselective quantum nondemolition measurement of the state of the system $S$, in a continuous-cycle scheme. This leads to decoupling of $S$ from the phonon bath, i.e., $\rho_{\rm S,ph}\rightarrow\rho_{S}\otimes\rho_{\rm ph}$, if the measurement outcomes or, alternatively, the states of the measuring device are not read or averaged out.

The magnetic field strength is changed from $B_{L}$ back to $B_{H}$, during this stroke ($4\rightarrow 1$, Fig. 2). For infinitely slow variation of the magnetic field, the evolution of the system would be adiabatic. However, as discussed in Sec. II.3, for finite-time evolution, certain irreversible work $W_{2}^{ir}$ is done, while the system maintains its interaction with both the hot bath and the vibrational mode. To calculate this quantity, we use the initial state of the system plus vibrational mode as the state obtained during stage 3, which can be expressed as $\rho_{1}(t_{3})=\left|-\right\rangle\left\langle-\right|\otimes\rho^{\text{ph}}$, where $\rho^{\text{ph}}$ is the state of the vibrational mode after the projective measurement. The state $\rho_{1}(t_{3})$ evolves into $\rho_{S}(t_{4})$ during this stage, which can be obtained as a solution of Eq. (4). The work done on the system during this stage now becomes:

$$W_{2}={\rm Tr}\left[\rho_{S}\left(t_{4}\right)H_{S}\left(B_{H}\right)\right]-{\rm Tr}\left[\rho_{S}\left(t_{3}\right)H_{S}\left(B_{L}\right)\right],$$

(12)

which includes the contribution from the irreversible work, as given by

	$\displaystyle{W^{\text{ir}}_{2}}$	$\displaystyle=$	$T_{H}S\left[\rho_{S}\left(t_{4}\right)||\rho_{S}^{{\rm th}}\left(t_{4}\right)\right]$
		$\displaystyle=$	$T_{H}\left({\rm Tr}\left\{\rho_{S}\left(t_{4}\right){\rm ln}\left[\rho_{S}\left(t_{4}\right)\right]\right\}-{\rm Tr}\left\{\rho_{S}\left(t_{4}\right){\rm ln}\left[\rho_{S}^{{\rm th}}\left(t_{4}\right)\right]\right\}\right).$

Note that the above protocol of the heat engine is valid for a single run of the engine cycle. However, as in the case of a non-selective measurement Talkner non-selective measurement heating , this would lead to heating of the system when averaged over many cycles. To circumvent this issue, a feedback control Sagawa feedback control , namely a local unitary transformation in the system based on the measurement outcome, may be used. Precisely speaking, a $\pi$ pulse (as commonly used in a trapped-ion system Quantum dynamics of single trapped ions ) needs to be applied to the system, only if it is measured in the state $\left|+\right\rangle$ to return it to the state $\left|-\right\rangle$ Elouard QHE Maxweels Demon . Otherwise, when averaged over many cycles, this would affect the amount of cooling, and thereby reduce the efficiency of the engine, considering the probabilistic nature of projective measurement. However, we have found that the probability that the system will be projected in the excited state is really much lower compared to that for the ground state. This means that for most of the cycles, the system will get cooled.

Here we specifically focus on the effect of partial thermalization camati2019coherence on the engine efficiency [see Fig. 3(a) for a detailed protocol]. As discussed in Sec. II.2, the interaction time between the system and hot bath is sufficiently short such that the thermalization is incomplete. However, the adiabatic stages (stage 2 and 4) are considered to run for a suitably long time scale, such that the quantum adiabatic condition is satisf–ied. This means that there is no irreversible work that otherwise would get wasted from the total work output. From Fig. 4, it is clear that if the thermalization time $t_{1}$ increases during the first stage, the heat exchange $Q_{H}$ with the hot bath increases as well, and $Q_{H}$ saturates asymptotically for a long thermalization time. When $Q_{H}$ saturates, it signifies full or complete thermalization, i.e., the system has reached thermal equilibrium with the thermal environment. There is no coherence in the energy eigenbasis in this thermal equilibrium state. However, if we terminate the stroke before saturation, certain coherence is sustained in the system. This coherence can enhance the performance of the engine.

From Fig. 5 it can observed that for shorter thermalization time $t_{1}$ during stage 1, the work done by the system $W_{1}$ increases. This is specifically due to residual coherence in the system during the partial thermalization stage. In such a case, the efficiency of the heat engine, $\eta=\left({\rm work\,output}\right)/\left({\rm heat\,input}\right)=\left(W_{1}+W_{2}\right)/Q_{H}\,$, reaches its maximum value (see Fig. 5). Such an enhancement of efficiency can be attributed to the coherence in the system. However, there is a specific value of $t_{1}$ (here $t_{1,{\rm min}}$=14) below which the engine does not operate, as it does not get sufficient heat energy to produce a positive workoutput. When the value of $t_{1}$ is sufficiently large, the engine efficiency $\eta$ reaches the minimum, which is equal to the maximum efficiency of a perfect single-ion Otto engine Our Paper- Single ion ,

$$\eta_{O}=1-\frac{B_{L}}{B_{H}}=0.5\,.$$

(14)

Now we investigate the effect of the non-adiabaticity on the thermal efficiency of the engine [see Fig. 3(b) for a detailed protocol]. When the time $\tau$ for the change in the magnetic field is not long, internal excitations between the energy eigenstates of the system can occur, which act as a source of entropy generation in the engine cycle. In this case, we allow the system to thermalize completely with the hot bath during the first stage, while there will be no coherence in the system. However, stages 2 and 4 are run for a duration, when it does not satisfy the quantum adiabatic condition, such that irreversible work is generated. That is the primary sources of irreversibility in the engine cycle. This is how quantum coherence contributes to irreversibility. Here, we redefine the engine efficiency by considering the effect of irreversible work as

$$\eta^{\text{ir}}=\frac{(W_{1}+W_{2})-(W_{1}^{\text{ir}}+W_{2}^{\text{ir}})}{Q_{H}}\,.$$

(15)

In Figure 6, displays the variation of the thermal irreversible efficiency $\eta^{\text{ir}}$ as a function of $\tau$. It is clear that the engine efficiency has strong dependence on $\tau$. For small values of $\tau$, the entropy production in the adiabatic branches greater, which is reflected in the value of total irreversible work $(W^{\text{ir}}=W_{1}^{\text{ir}}+W_{2}^{\text{ir}})$ (see the inset in Fig. 6). This shows the presence of internal friction in the system. Also, the total workdone ($W=W_{1}+W_{2}$) by the engine is decreased significantly for the small value of $\tau$. This occurs because, due to the short adiabatic time, the system is unable to follow the evolution of the instantaneous eigenstate of the system Hamiltonian and the occupation probability does not remain the same in this basis. Further a critical observation in Fig. 6 reveals that the duration $\tau$ of the expansion and compression stages has a minimum value, (here, $\tau_{{\rm min}}$=8), below which the engine cannot run Turkpence2019 . This is because, due to smaller $\tau$, the entropy generation is so great that the engine will not be able to produce useful work. When the adiabatic strokes are performed quasistatically (i.e. $\tau\rightarrow\infty$) such that the adiabaticity is maintained, the engine extracts the maximum possible amount of work. This is because the quantum adiabatic theorem holds for large $\tau$ and the occupation probability remains the same in the instantaneous eigenstate. As a result, entropy production is minimized, and internal friction is also minimized. The engine efficiency reaches the efficiency of a perfect single-ion Otto engine Our Paper- Single ion .