---
abstract: 'We explore the advantages of using an asymmetric pulse modulation on a two-level system (TLS) coupled to one or more thermal baths in the context of quantum thermal machines and thermometry. For a thermal machine constructed by coupling the TLS to two thermal baths, we demonstrate that the asymmetric pulse provides an extra degree of control over the mode of operation of the thermal machine. Further, we also show that an asymmetric pulse may provide superior optimality in a recently proposed protocol for quantum thermometry, where dynamical control has been shown to enhance the precision of measurement.'
author:
- Saikat Mondal
- Sourav Bhattacharjee
- Amit Dutta
title: 'Exploring the affects of asymmetric-pulse modulation in quantum thermal machines and quantum thermometry'
---

Introduction {#sec_intro}
============

The last two decades has seen a spurt in research on the modeling and dynamics of open quantum systems [@breuer02; @rivas12; @rotter15; @breuer16]. These systems, in general, exhibit dissipative phenomena and therefore provide natural test-beds for investigating the quantum origin of dynamical processes such as quantum heat exchanges and quantum heat transport. A crucial ramification of this development has been the revitalization of the age-old quest for integrating quantum mechanics and thermodynamics into a single framework, thus leading to the emergence of the field of quantum thermodynamics [@alickiintro; @kosloff13; @anders16; @esposito17; @campaioli18; @campbell19; @kosloffrev19]. A major chunk of research in this field has focused on the building and analysing toy models of ‘quantum thermal machines’, such as quantum equivalents of heat engines and refrigerators [@levy14; @kurizki15; @rezek17; @mitchison19]; the motivation being that understanding the classical version of these machines had led to the foundation of classical thermodynamics.

As in classical thermal machines, the design and working of the quantum thermal machines are also based on cyclic processes, i.e. the system and the environment should return to their initial configuration. In this regard, a theoretical framework of periodically driven open quantum systems has been formulated [@zanardi06; @alickitut; @szczyg13]to deal with quantum systems which, apart from being coupled with external environment, are described by a periodic time-dependent Hamiltonian. Importantly, a direct application of this framework has been shown in the context of designing continuous quantum thermal machines [@kosloff13; @kurizki15; @kurizki13; @alicki14]. In the simplest of realizations, the working substance of these machines consist of a quantum system (with discrete energy levels) which is perpetually coupled to one or more baths or heat reservoirs in contrast to reciprocating thermal machines, such as those based on Carnot or Otto cycles [@rezek17; @kosloff92; @bender00; @nori07; @tang09; @uzdin15]. In addition, the Hamiltonian of the system is periodically modulated with the consequent dynamics resulting in exchange of quantum ‘heat’ and ‘work’. A plethora of works [@kurizki15; @klimovsky15; @niedenzu15; @weitz15; @seifert17; @mukherjee20] have explored the performance of these quantum machines in terms of work output and efficiency, which have remarkably been found to be consistent with thermodynamic principles. On the other hand, periodic modulation of a ‘qunatum probe’ coupled to a thermal bath has also been shown to significantly enhance the precision [@mukherjee19] in low-temperature quantum thermometry [@correa15; @pasquale16; @brunner17; @wang20] by increasing the relevant quantum Fisher information (QFI) [@caves94; @paris09], which in turn lowers the theoretical minimum error set by the quantum Cramer-Rao bound.

However, all of the works mentioned above are based on a symmetric form of the periodic modulations. In other words, the form of the modulation on either side of the half-cycle are mirror copies of each other, for example – sinusoidal or square-pulse modulations. In this work, we explore the consequences of using an asymmetric-pulse modulation (APM) in the two different but related applications discussed above, i.e. in designing quantum thermal machines and enhancement of precision in low-temperature quantum thermometry. The APM we use is of the form of a rectangular pulse whose ‘up’ and ‘down’ time within a single time-period are not necessarily equal (see Fig. \[fig\_asym\_pulse\]).

The motivation behind using an asymmetric modulation is as follows. The Floquet-Lindblad approach [@zanardi06; @alickitut] to the dynamics of periodically driven quantum system reveals that the action of a Markovian thermal bath acting on a periodically driven system can be considered equivalent to infinite number of ‘sub-baths’ acting simultaneously on the system. The multiple sub-baths are artifacts of the infinite number of ‘Floquet side-bands’ or ‘photon sectors’ created as a consequence of the periodic modulation, each of which independently exchange energies with the ‘physical baths’. It is worthwhile to note that each of the sub-baths, when acting independently on the system, steers the system towards different Gibbs states determined by the spectrum of the corresponding photon sector. Nevertheless, the final steady-state is determined by the simultaneous action of all the sub-baths. Importantly, the contribution of a particular sub-bath depends on its bath spectral response function $G(\omega)$ and a weight factor ${P_{\mathrm{q}}}$. While the former encodes all the information regarding the physical baths and is therefore often pre-determined and difficult to tune, the later depends on the characteristics of the modulation itself. In this work, we illustrate that in contrast to a symmetric modulation, an asymmetric pulse can provide a more flexible control on the final steady state, either through a ‘renormalization’ of the excitation energies that are dynamically generated in the steady state, or by lifting the symmetry of the weight factors ${P_{\mathrm{q}}}$ about $q=0$.

Let us briefly recapitulate some of the well-known properties of continuous quantum thermal machines. A continuous quantum thermal machine usually constructed using a two-level system (TLS) perpetually coupled to two thermal baths is known to be capable of working both as a quantum heat engine and quantum refrigerator [@kosloff13; @kurizki15; @kurizki13]. The mode of operation is determined and controlled by the modulation frequency; at the critical value of the modulation frequency where the system switches the mode of operation, the system attains the Carnot efficiency [@carnot], although all heat currents as well as the power generated vanish. This is similar in characteristic to the classical Otto engine, which also achieves the Carnot efficiency in the limit of vanishing work output. Apart from engine-like and refrigerator-like operations, quantum thermal machines are also known to operate as ‘heat distributors’ where an external work is used to supply heat energy to both the hot and cold baths, and as ‘accelerators’ where the heat transfer from the hot bath to the cold bath is accelerated with the help of an external power source [@buffoni19].

Apart from functioning as toy models of quantum heat engines and refrigerators, simple quantum systems, such as a qubit (TLS) or a harmonic oscillator, when connected to baths, have also found applications as efficient ‘quantum probes’ in the field of quantum metrology [@maccone11; @degen17; @zwick17; @pezze18; @sciarrino20]. The idea is to make indirect measurements on the system which in some cases can be more precise than a direct measurement of the small parameter to be estimated. As for example, it has been shown that performing indirect measurements at the Carnot point, where a thermal machine switches opertion from engine-like to refrigerator-like operations or vice-versa, can significantly enhance precision in quantum thermometric and magnetometric measurements [@brunner17; @dutta20]. Recently, it has been shown that a measurement on the steady state populations of a periodically modulated quantum system in conjunction with a thermal bath of unknown temperature, provides a more precise estimation of the bath temperature [@mukherjee19]. Importantly, the quantum Fisher information (QFI) using this protocol scales as $1/T^2$; consequently, the theoretical lower bound of the relative error, set by the quantum Cramer-Rao bound, remains finite till extremely low temperatures.

The rest of the paper is organized as follows. In Sec. \[sec\_asym\], we briefly outline the Floquet-Lindblad formalism used for analyzing dynamical evolution of open quantum systems modulated periodically with an APM and also derive the general form of the steady state. In Sec. \[sec\_eng\], we explore the operation of the thermal machine constructed by modulating a two-level system with an APM and perpetually coupled with two thermal baths. In Sec. \[sec\_thermo\], we highlight the advantage of using an APM in enhancing precision in thermometric measurement using dynamical control. Concluding statements and scope for future research are presented in Sec. \[sec\_con\]. Necessary calculations to support the derivations in Sec. \[sec\_asym\] is provided in Appendix. \[app\_int\_pic\], while a short derivation of the heat currents found in the steady state of the continuous quantum thermal machine is outlined in Appendix \[app\_curr\].

Asymmetric-pulse modulation of open quantum systems {#sec_asym}
===================================================

Let us consider a two-level system (TLS) coupled to an arbitrary number of thermal baths which do not interact with each other. The Hamiltonian of the composite system (including the baths) is, $$H(t) = H_s(t)+\sum_b H_b+H_I,$$ where $H_s(t)$ is the periodically modulated Hamiltonian of the system, $H_b$ is the Hamiltonian describing the $b$-[th]{} thermal bath and $H_I$ denotes the time-independent interaction between the system and the baths. The system Hamiltonian $H_s(t)$ is of the form (see Fig. \[fig\_asym\_pulse\]), $$\begin{aligned}
\label{eq_hs}
H_s(t) =\frac{1}{2}\omega_s(t)\sigma_z=& \frac{1}{2}\left(\omega_0+\mu\Omega\right)\sigma_z, \hspace{0.75cm}0<t<t_0,{\nonumber}\\
       & \frac{1}{2}\left(\omega_0-\mu\Omega\right)\sigma_z, \hspace{0.75cm}t_0<t<\tau,\end{aligned}$$ where $\Omega=2\pi/\tau$ is the frequency of modulation, $\mu$ determines the modulation strength and $\sigma_z$ is a Pauli matrix. We note that the above modulation corresponds to a symmetric pulse when $t_0=\tau/2$; for $t_0\neq \tau/2$, it corresponds to an APM. We do not specify any explicit form of $H_b$ as we are only interested in the reduced dynamics of the system. As for the interaction $H_I$, the following form is assumed,

![Profile of the asymmetric pulse modulation (APM), with time-period $\tau=2\pi/\Omega$ and modulation strength $\mu$, applied on the Hamiltonian of the TLS (see Eq. ). Within a single time-period, the pulse is ‘up’ for a duration $t_0$ and ‘down’ for a duration $\tau-t_0$. For $t_0=\tau/2$, the pulse is symmetric. []{data-label="fig_asym_pulse"}](asym_pulse.pdf){width="\columnwidth"}

$$\label{eq_int}
H_I=\lambda\sum_{b} \sigma_x\otimes B_b,$$

where $\lambda$ denotes the strength of system-bath coupling, which is assumed to be identical in case of all baths while $\sigma_x$ (Pauli matrix) and $B_b$ are Hermitian operators acting locally on the Hilbert spaces of the system and $b$-[th]{} bath, respectively.

To analyze the dynamical evolution, we first note that the Floquet evolution operator which drives the evolution over one period in the absence of dissipation is given by,

$$U_F=\mathcal{T}e^{-i\int_0^\tau H_s(t)dt}=e^{-iH_F\tau},$$

where the Floquet Hamiltonian $H_F$ is given by, $$\label{eq_flo_ham}
H_F=\frac{1}{2}\left(\omega_0+\mu\Omega\left(\frac{2t_0}{\tau}-1\right)\right)\sigma_z=\frac{1}{2}\bar{\omega}\sigma_z.$$

We crucially note here that for symmetric modulation $t_0=\tau/2$, the Floquet Hamiltonian is simply $H_F=\omega_0\sigma_z/2$, i.e. the Floquet spectrum is insensitive to the strength $\mu$ or the frequency $\Omega$ of modulation. Next, we rewrite Eq.  in the interaction picture (see Appendix. \[app\_int\_pic\] for detail) as,

$$\tilde{H}_I=\lambda\sum_{b}\sigma_x(t)\otimes B_b(t),$$

where, $$\sigma_x(t)=\sum_{q\in\mathbb{Z}}\left(\xi_qe^{-i(\bar{\omega}+q\Omega)t}\sigma^-+\xi_q^*e^{i(\bar{\omega}+q\Omega)t}\sigma^+\right),$$ $$B_b(t)=e^{iH_bt}B_be^{-iH_bt}.$$

For the APM, the coefficients $\zeta_q$ assume the form, $$\label{eq_xiq}
\xi_q=\frac{\mu \left(e^{-2i\mu\Omega t_0(1-\frac{t_0}{\tau})+iq\Omega t_0}-1\right)}{i\pi\left(q-2\mu(1-\frac{t_0}{\tau})\right)\left(q+2\mu\frac{t_0}{\tau}\right)}.$$

Proceeding as in Refs. \[\], the Floquet-Lindblad equation (in the interaction picture) governing the evolution of the reduced density matrix of the TLS is,

$$\label{eq_lindblad}
\frac{d\rho}{dt}=\mathcal{L}\rho=\sum_{b,q}\mathcal{L}^b_q\rho,$$

$$\begin{gathered}
\mathcal{L}^b_q=P_q\Big[G^b(\bar{\omega}+q\Omega)\left(\sigma^-\rho\sigma^+-\frac{1}{2}\{\sigma^+\sigma^-,\rho\}\right)\\+G^b(-\bar{\omega}-q\Omega)\left(\sigma^+\rho\sigma^--\frac{1}{2}\{\sigma^-\sigma^+,\rho\}\right)\Big].\end{gathered}$$

where, $$\label{eq_Pq}
P_q=|\xi_q|^2=\frac{4\mu^2\sin^2\left[2\mu\pi\frac{t_0}{\tau}\left(1-\frac{t_0}{\tau}\right)-q\pi\frac{t_0}{\tau}\right]}{\pi^2\left[q-2\mu\left(1-\frac{t_0}{\tau}\right)\right]^2\left[q+2\mu\frac{t_0}{\tau}\right]^2}$$ and $G^b$ is the bath spectral function of $b$-th bath. The above equation can be interpreted as follows. The periodic modulation results in the generation of multiple copies or harmonics of the system Hamiltonian with spectral width $\bar{\omega}+q\Omega$ and are separated by integer multiples $q$ of the modulation frequency $\Omega$. Each of this harmonics interacts with the bath and the resulting dissipation is captured by the Lindblad operators $\mathcal{L}_q^b$. The asymptotic steady state is now easily obtained by solving the eigen equation $\mathcal{L}\tilde{\rho}_{ss}=0$ as,

\[eq\_steady\] $$\tilde{\rho}_{ss}=\frac{1}{1+r}\begin{pmatrix}
            r & 0\\0 & 1\end{pmatrix},$$

where, $$\label{eq_r_ori}
r=\frac{\sum_{q,b}{P_{\mathrm{q}}}G^b\left(\bar{\omega}+q\Omega\right)e^{-\frac{\bar{\omega}+q\Omega}{T_b}}}{\sum_{q,b}{P_{\mathrm{q}}}G^b\left(\bar{\omega}+q\Omega\right)},$$

and $T_b$ is the temperature of the $b$-[th]{} bath. It can be shown that the time-independent steady state $\tilde{\rho}_{ss}$ in the interaction picture translates to a periodic steady state $\rho_{ss}$ in the Schrodinger picture which satisfies $\rho_{ss}(t+\tau)=\rho_{ss}(t)$. However, in the rest of this work, we will analyze the relevant quantities only in the interaction picture.

APM in continuous quantum thermal machines {#sec_eng}
==========================================

We now consider the case where the number of thermal baths is restricted to two. The ‘hot’ bath has a temperature $T_h$ and the cold one has temperature $T_c$, such that $T_h>T_c$. In the steady state, energy is continuosly exchanged with between the TLS and the hot and cold baths, which are identified as the hot current $J_h$ and cold current $J_c$, respectively. In addition, the continuous pumping of energy in or out of the system through the modulation is identified as the work current or power $J_p$. In analogy with classical machines, the thermal machine is considered to operate as a heat engine when $J_h>0$, $J_c<0$ and $J_p<0$. In the refrigeration regime of operation, the quantities reverse their sign. From the steady state defined in Eq. , the currents are calculated as (see Appendix \[app\_curr\]),

$$\label{eq_heat}
J_{h(c)}=\sum_q\frac{\bar{\omega}+q\Omega}{r+1}{P_{\mathrm{q}}}G^{h(c)}\left(\bar{\omega}+q\Omega\right)\left(e^{-\frac{\bar{\omega}+q\Omega}{T_{h(c)}}}-r\right)$$

where $J_h$ and $J_c$ are the heat currents flowing from the hot bath ($T_h$) and cold bath ($T_c$), respectively, to the TLS. Similarly, the power can be calculated using energy conservation principle as, $$\label{eq_power}
J_p=-(J_h+J_c).$$ We now take a closer look at the coefficients ${P_{\mathrm{q}}}$. In this section, we work within the limit $\mu\ll 1$. Consequently, Eq.  reduces to,

$$P_0=\left[\frac{\sin\left(\mu\Omega t_0(1-\frac{t_0}{\tau})\right)}{2\mu\pi\frac{t_0}{\tau}(1-\frac{t_0}{\tau})}\right]^2,$$

$$P_{q\neq 0}=\left[\frac{2\mu\sin\left(\frac{q\Omega t_0}{2}\right)}{\pi q^2}\right]^2$$

The above equation shows that the value of ${P_{\mathrm{q}}}$ diminishes as $|q|$ increases. Therefore, we keep ourselves restricted to only the leading order coefficients $P_0$ and $P_{\pm1}$. Secondly, we also introduce the following cutoffs for the bath spectral functions, $$G^h(\omega)=0\quad \forall~\omega \leq \bar{\omega},\quad G^c(\omega)=0\quad \forall~\omega\geq \bar{\omega}$$

Using Eqs.  and  , the heat currents and the power are now found to be,

\[eq\_all\_curr\] $$J_h=\mathcal{K}\left(\bar{\omega}+\Omega\right)\left(e^{-\frac{\bar{\omega}+\Omega}{T_h}}-e^{-\frac{\bar{\omega}-\Omega}{T_c}}\right),$$ $$J_c=\mathcal{K}\left(\bar{\omega}-\Omega\right)\left(e^{-\frac{\bar{\omega}-\Omega}{T_c}}-e^{-\frac{\bar{\omega}+\Omega}{T_h}}\right),$$ $$J_p=-2\mathcal{K}\Omega\left(e^{-\frac{\bar{\omega}+\Omega}{T_h}}-e^{-\frac{\bar{\omega}-\Omega}{T_c}}\right),$$ where $\mathcal{K}$ is a positive constant given by,

$$\mathcal{K}=\frac{4\mu^2\sin^2\left(\frac{\Omega t_0}{2}\right)G^h(\bar{\omega}+\Omega)G^c(\bar{\omega}-\Omega)}{\pi^2\left[G^h(\bar{\omega}+\Omega)\left(1+e^{-\frac{\bar{\omega}+\Omega}{T_h}}\right)+G^c(\bar{\omega}-\Omega)\left(1+e^{-\frac{\bar{\omega}-\Omega}{T_c}}\right)\right]}$$

We now analyze the different modes of operation as follows. The thermal machine works as a heat engine if $J_h>0$, $J_c<0$ and $J_p<0$. Similarly, refrigeration occurs when $J_h<0$, $J_c>0$ and $J_p>0$. Additionally, the thermal machine is also capable of working as a ‘heater’ when $J_h<0$, $J_c<0$ and $J_p>0$. Let us first assume that $\bar{\omega}>\Omega$. Examining Eq. , it is clear that engine like operation is achieved when $(\bar{\omega}-\Omega)/T_c>(\bar{\omega}+\Omega)/(T_h)$ while refrigerator like operation is achieved for $(\bar{\omega}-\Omega)/T_c<(\bar{\omega}+\Omega)/T_h$. Therefore, there exists a critical point, namely the Carnot point, where the heat currents as well as the power vanish and the thermal machine switches operation from engine like to refrigerator like and vice-versa. The Carnot point is identified by the relation, $$\label{eq_carnot}
\left(\frac{\Omega}{\bar{\omega}}\right)_{cr}=\frac{T_h-T_c}{T_h+T_c}.$$

Note that for a symmetric pulse $t_0=\tau/2$, the above equation reduces to $\Omega_{cr,sym}=\omega_0(T_h-T_c)/(T_h+T_c)$, which is identical for the case of sinusoidal modulation in Ref. \[\]. In this case, the mode of operation of the thermal machine can only be switched by tuning the modulation frequency $\Omega$, which provides the only degree of control over the mode of operation. On the contrary, in the case of the APM, one can rearrange Eq.  as, $$\label{eq_carnot_asym}
\left(\frac{t_0}{\tau}\right)_{cr}=\frac{1}{2}\left[1+\frac{\omega_0}{\mu}\left(\frac{1}{\Omega_{cr,sym}}-\frac{1}{\Omega}\right)\right],$$ which implies that for a fixed modulation frequency $\Omega$, the transition can also be induced by tuning $t_0$, as shown in Fig. \[fig\_ref\_eng\]. In other words, the degree of deviation from a symmetric profile of modulation provides an extra degree of control over the mode of operation of the thermal machine. However, we note that since $0< t_0/\tau< 1$, a critical $(t_0/\tau)_{cr}$ exists only if the following condition is satisfied, $$\left|\frac{\omega_0}{\mu}\left(\frac{1}{\Omega_{cr,sym}}-\frac{1}{\Omega}\right)\right|\leq 1.$$

Next, let us further consider the situation $\Omega=\omega_0$. We therefore have,

$$\bar{\omega}+\Omega=2\omega_0\left(1+\mu\left(\frac{t_0}{\tau}-\frac{1}{2}\right)\right),$$

$$\bar{\omega}-\Omega=2\mu\omega_0\left(\frac{t_0}{\tau}-\frac{1}{2}\right).$$

If we now choose $t_0/\tau<1/2$, we have $\bar{\omega}+\Omega>0$ as $\mu\ll 1$, while $\bar{\omega}-\Omega<0$. Consequently, $e^{-(\bar{\omega}-\Omega)/T_c}>e^{-(\bar{\omega}+\Omega)/T_h}$. With this choice of parameters, it can be easily seen from Eq.  that the heat currents satisfy $J_h, J_c <0$ and the power $J_p>0$. As already mentioned, this corresponds to the thermal machine working as a heater where work is done to supply heat to both the baths. On the other hand, for $t_0/\tau>1/2$, we have $\bar{\omega}-\Omega>0$ but, the condition $e^{-(\bar{\omega}-\Omega)/T_c}>e^{-(\bar{\omega}+\Omega)/T_h}$ still holds as $\bar{\omega}+\Omega\gg\bar{\omega}-\Omega\approx 0$. Hence, only the sign of $J_c$ is reversed, which corresponds to a refrigerator like operation. An example of this transition is illustrated in Fig. \[fig\_ref\_heat\]. Thus, we have demonstrated the possibility of tuning $t_0$ to drive a refrigerator-heater transition if the modulation frequency is in resonance with the un-modulated TLS, i.e. $\Omega=\omega_0$.

APM in quantum thermometry {#sec_thermo}
==========================

We now illustrate how a TLS modulated with an APM can be used to measure the temperature of a thermal bath with a precision higher than that possible with a symmetric pulse. The advantage of using the APM is manifested in the form of a higher QFI and consequently a lower minimum bound on relative error. We briefly outline the measurement protocol below.

.

Let us consider a thermal bath whose temperature $T$ is to be measured. The quantum probe we choose is a TLS which is coupled to the bath and its energy levels are periodically modulated using an APM as discussed in Sec. \[sec\_asym\], with the number of baths now restricted to one. The TLS therefore reaches a thermal steady state $\tilde{\rho}_{ss}$ (see Eq. ). This steady state can be characterized through the parameter $r$ given by, $$\label{eq_r_thermo}
r=e^{-\frac{{\omega_{\mathrm{eff}}}}{T}}=\frac{\sum_{q}{P_{\mathrm{q}}}G\left(\omega_q\right)e^{-\frac{\omega_q}{T}}}{\sum_{q}{P_{\mathrm{q}}}G\left(\omega_q\right)}.$$ where $\omega_q=\bar{\omega}+q\Omega$. First, we note that for given values of $\mu$ and $t_0/\tau$, we can always set the modulation frequency $\Omega$, such that $\omega_{-n}=\bar{\omega}-n\Omega\to0^+$. Further, if $T\sim\omega_{-n}$, we have $e^{-\omega_{-(n-1)}/T}, e^{-\omega_{-(n-2)}/T},\dots\ll e^{-\omega_{-n}/T}$. In addition, the thermal baths satisfy the Kubo-Martin-Schwinger (KMS) condition, $G(-\omega)=G(\omega)e^{-\omega/T}$. Consequently, Eq.  simplifies to, $$\label{eq_rn}
r=\frac{P_{-n} G\left(\omega_{-n}\right)e^{-\frac{\omega_{-n}}{T}}}{\sum_{q=-n}^\infty{P_{\mathrm{q}}}G\left(\omega_q\right)}=\nu e^{-\frac{\omega_{-n}}{T}},$$ where $P_q$ is obtained from Eq. . The steady state therefore assumes the form (see Eq. ). $$\tilde{\rho}_{ss}=\begin{pmatrix}
\varrho_{1} & 0\\
0 & \varrho_{2}
\end{pmatrix}       
= \frac{1}{1+\nu e^{-\frac{\omega_{-n}}{T}}}\begin{pmatrix}
\nu e^{-\frac{\omega_{-n}}{T}} & 0\\
0 & 1
\end{pmatrix}.$$ Measuring the steady state populations, one can hence determine the temperature $T$ of the bath.

However, it is known the minimum error in such a measurement is theoretically lower bounded by the quantum Cramer-Rao bound which states, $$\label{eq_bound}
\frac{\Delta T}{T}\geq\varepsilon=\frac{1}{T\sqrt{\mathcal{M}\mathcal{H}}},$$ where $\mathcal{M}$ is the number of repeated measurements performed and $\mathcal{H}$ is the quantum Fisher information (QFI) [@caves94; @paris09]. For our case, the QFI is easily calculated as, $$\mathcal{H}=\sum_{i=1}^2\frac{1}{\varrho_i}\left|\frac{\partial\varrho_i}{\partial T}\right|^2=\frac{\nu e^{-\frac{\omega_{-n}}{T}}\omega_{-n}^2}{\left(1+\nu e^{-\frac{\omega_{-n}}{T}}\right)^2T^4},$$ where $T\sim\omega_{-n}$.

It is easy to check that the maximum QFI is obtained for $\nu=e^{\omega_{-n}/T}$. At $T\approx\omega_{-n}$, this corresponds to $\nu=2.72$. However, by definition, $\nu\leq1$ and hence optimality is achieved when the following condition is satisfied. $$\label{eq_cond_therm}
P_{-n}G(\omega_{-n})\gg P_{-q}G(\omega_{-q}),~~q\neq n.$$ To demonstrate how this can be easily achieved using the APM, let us consider the simple case of a nearly flat bath spectrum, $$G(\omega\geq\omega_{min}>0)=G_0;~~G(\omega\to 0)=0,$$ Recalling Eq. , one can check that $P_{-q}$ achieves maximum when $2\mu t_0/\tau=q$, for $q>0$. As a simple example, consider $\omega_{-n}=\omega_{-1}\geq\omega_{min}$. $P_{-1}$ is therefore maximum when $\mu=\tau/(2t_0)$.

To illustrate the advantage of using an APM, let us first consider the case of a symmetric pulse ($t_0=\tau/2$). In this case, the maximum value of $P_{-1}$ is achieved for $\mu=1$; however $P_{-1}=0.25\leq P_0, P_1$ as is seen in Fig. \[fig\_pq\_sym\]. This reflects the fact that for a symmetric pulse ${P_{\mathrm{q}}}={P_{\mathrm{-q}}}$, $\forall q$. Consequently, the only way to satisfy the condition given in Eq.  in the case of the symmetric pulse is through careful manipulation of the bath spectral function, for example, by setting a low upper cutoff for $G(\omega)$. We note that the above results obtained for the symmetric pulse are similar to that obtained for a sinusoidal modulation analysed in Ref. . Let us now analyze how the scenario changes in the case of APM. For $t_0\neq \tau/2$, one can immediately see that ${P_{\mathrm{q}}}\neq{P_{\mathrm{-q}}}$. This creates the possibility of manipulating the parameters $\mu$ and $t_0$ such that $P_{-n}>P_q$ for $q\neq -n$. We illustrate this with an example where we choose $\mu=2$ and $t_0/\tau=0.25$ so that the equality $\mu=\tau/(2t_0)$ is satisfied. As shown in Fig. \[fig\_pq\_asym\], for these choice of parameters, we have $P_{-1}\approx(1-t_0/\tau)^2=0.563\gg P_{n\neq-1}$.

To quantify the advantage, let us compare the maximum of the QFI achieved for the APM to that of the symmetric case. In the latter case, the dominant contributions are from $P_0,~P_{\pm1}, P_{\pm2}$ as can be seen from Fig. \[fig\_pq\_sym\]. However, $G(\omega_{-2})$ is exponentially suppressed (as dictated by the KMS condition). Therefore, the parameter $\nu$ evaluates to, $$\nu_{sym}=\frac{P_{-1}G(\omega_{-1})}{\sum_{q=-1}^2P_{q}G(\omega_{q})}\approx0.263,$$ which yields a QFI of $$\mathcal{H}_{sym}=\frac{0.08}{T^2},$$ in the limit $\frac{\omega_{-1}}{T}\approx 1$. On the other hand, for the APM, the dominant contributions arise from $P_{\pm2}, ~P_{\pm1}, ~P_0, ~P_3,~P_4$ and $P_5$. Proceeding as before, we obtain $$\nu_{asym}\approx0.592,$$ and the corresponding QFI value as, $$\mathcal{H}_{asym}=\frac{0.15}{T^2}.$$ Substituting the QFIs calculated above in Eq. , we finally obtain, $$\frac{\varepsilon_{asym}}{\varepsilon_{sym}}\approx0.73,$$ thereby clearly demonstrating that an APM lowers the minimum error bound as compared to the case of symmetric pulse.

Conclusion {#sec_con}
==========

In summary, we have highlighted the advantages of using an asymmetric pulse modulation in the context of two different applications of periodically modulated two-level systems coupled to thermal baths. In the first case, we have explored the consequences of using an APM on a TLS coupled to two different thermal baths. For a symmetric pulse, such a system is known to work both as a quantum heat engine and quantum refrigerator, depending upon the frequency of modulation. We have shown that tuning the length of up time (or down time) of an asymmetric pulse also allows switching of the mode of operation of the thermal machine between heat engine, refrigerator and heater regimes. Thus, an asymmetric modulation provides an extra degree of control over the mode of operation, which may be experimentally useful, particularly in cases when the frequency of the modulation is not easily tunable. Secondly, we have demonstrated the advantage of the APM over a symmetric pulse in quantum thermometry, where the periodically modulated TLS is coupled to a single thermal bath and its steady state populations are measured to read off the bath temperature. In particular, we have shown that the optimality achieved in terms of maximizing the QFI using the APM is superior to that of the symmetric case. In future, it might be worth exploring other forms of asymmetric modulations such as in sinusoidal modulation. The consequences of using APM in the case of strong system-environment couplings and non-Markovian dynamics may also lead to exciting results and applications.

We are grateful to A. Ghosh and V. Mukherjee for their helpful comments and suggestions. We also acknowledge S. Bandyopadhyay and S. Maity for useful discussions. SB acknowledges CSIR, India for financial support. A.D. acknowledges financial support from SPARC program, MHRD, India.

The interaction Hamiltonian $H_I$ in interaction picture {#app_int_pic}
========================================================

In this appendix, we outline the steps to rewrite the interaction Hamiltonian $H_I$ of the main text in the interaction picture. Noting that $[H_s(t),H_b]=[H_a,H_b]=0$ $\forall~a,b$ where $a$, $b$ are bath indices, $H_I$ in the interaction picture is obtained as,

$$\begin{aligned}
\label{app_eq_int}
\tilde{H}_I&=\lambda\sum_{b=1}^k\sigma_x(t)\otimes B_b(t){\nonumber}\\
&=\lambda\sum_{b=1}^k \left(U_s^\dagger(t)\sigma_xU_s(t)\right)\otimes \left(U_b^\dagger(t)B_bU_b(t)\right),\end{aligned}$$

where $$U_s(t)=\mathcal{T}e^{-i\int_{0}^{t}H_s(t')dt'},$$ $$U_b=e^{-iH_bt}.$$

To evaluate Eq. , we first write $$U_s(t)=P(t)e^{-iH_Ft}$$ where $P(t)=\mathcal{T}e^{-i\int_{0}^{t}(H_s(t')-H_F)dt'}$ satisfies $P(t+\tau)=P(t)$. The periodicity of $P(t)$ subsequently allows a Fourier decomposition of $\sigma_x(t)$ as, $$\sigma_x(t)=\sum_{q\in\mathbb{Z}}\left(\xi_qe^{-i(\bar{\omega}+q\Omega)t}\sigma^-+\xi_q^*e^{i(\bar{\omega}+q\Omega)t}\sigma^+\right),$$ where $\bar{\omega}$ is defined in Eq.  of the main text. The coefficients $\xi_q$ are given by, $$\xi_q=\frac{1}{\tau}\int_0^\tau e^{-i\int_0^t(\omega_s(t')-\bar{\omega})dt'}e^{-iq\Omega t}dt.$$ Substituting $\omega_s(t)$ from Eq. \[eq\_hs\] of main text, we explicitly evaluate $\xi_q$ as, $$\begin{gathered}
\xi_q=\frac{1}{\tau}\Big(\int_0^{t_0}e^{-i\int_0^t(\omega_0+\mu\Omega-\bar{\omega})dt'}e^{-iq\Omega t}dt\\+\int_{t_0}^{\tau}e^{\left[-i\int_0^{t_0}(\omega_0+\mu\Omega-\bar{\omega})dt'+\int_{t_0}^t(\omega_0-\mu\Omega-\bar{\omega})dt'\right]}e^{-iq\Omega t}dt\Big)\\
=\frac{\mu \left(e^{-2i\mu\Omega t_0(1-\frac{t_0}{\tau})+iq\Omega t_0}-1\right)}{i\pi\left(q-2\mu(1-\frac{t_0}{\tau})\right)\left(q+2\mu\frac{t_0}{\tau}\right)}.\end{gathered}$$

Heat currents in continuous thermal machines {#app_curr}
============================================

To determine the heat currents, we first note that each sub-bath in principle can individually take the system to a Gibbs-like steady state determined by the eigenvalue equation $\mathcal{L}_{q}^b\rho^{ss}_{b,q}=0$ where $b=h,c$. These steady states are of the form, $$\rho_{b,m}^{ss}=\frac{1}{\mathcal{Z}}\exp{\left(\frac{\bar{\omega}+q\Omega}{\bar{\omega}}\beta_b H_{F}\right)},$$ where $\beta_b=1/T_b$, $\mathcal{Z}={{\rm Tr}}\left(\exp{\left(\frac{\bar{\omega}+q\Omega}{\bar{\omega}}\beta_b H_{F}\right)}\right)$ and $H_F$ is given by Eq.  of main text. Following [@alickitut; @kosloff], we calculate the rate of change of von-Neumann entropy $S(t)=-{{\rm Tr}}\left(\rho(t)\ln\rho(t)\right)$, $$\frac{d S(t)}{dt}=-{{\rm Tr}}\left(\dot{\rho}(t)\ln\rho(t)\right)=-\sum_{b,q}{{\rm Tr}}\left(\mathcal{L}_q^b\rho(t)\ln\rho(t)\right),$$ where we have substituted Eq.  to obtain the second equality. Next, we use Spohn’s inequality, ${{\rm Tr}}\left(\mathcal{L}_q^b\rho(\ln\rho-\ln\rho^{ss}_{q,b})\right)\leq 0$ to arrive at, $$\frac{d S(t)}{dt}\geq-\sum_{b,q}{{\rm Tr}}\left(\mathcal{L}_q^b\rho(t)\ln\rho^{ss}_{q,b}\right)=\sum_j\frac{J_b(t)}{T_b},$$ where the heat currents $J_b$ in the steady state are obtained as, $$J_b=\sum_q\left(\frac{\bar{\omega}+q\Omega}{\bar{\omega}}\right){{\rm Tr}}\Big(\mathcal{L}_q^b\rho^{ss}H_F\Big).$$ Finally, substituting $\rho^{ss}$ from Eqs.  and , the heat currents assume the form, $$J_{h(c)}=\sum_q\frac{\bar{\omega}+q\Omega}{r+1}{P_{\mathrm{q}}}G^{h(c)}\left(\bar{\omega}+q\Omega\right)\left(e^{-\frac{\bar{\omega}+q\Omega}{T_{h(c)}}}-r\right).$$

[breuer02]{} H. P. Breuer, F. Petruccione, *Theory of open quantum systems*, Oxford University Press, Oxford, (2002).

[rivas12]{} A. Rivas and S. Huelga, Open Quantum Systems. An Introduction, Springer, New York (2012).

[rotter15]{} I. Rotter and J. P. Bird, Rep. Prog. Phys. [**78**]{} 114001 (2015).

[breuer16]{} H. P. Breuer, E. M. Laine, J. Piilo, and B. Vacchini, Rev. Mod. Phys. [**88**]{}, 021002 (2016).

[alickiintro]{} R. Alicki, R. Kosloff, arXiv:1801.08314.

[kosloff13]{} R. Kosloff, Entropy [**15**]{}, 2100 (2013).

[anders16]{} S. Vinjanampathy, J. Anders, Contemporary Physics [**57**]{}, 545 (2016).

[esposito17]{} J. Anders and M. Esposito, New J. Phys. [**19**]{} 010201 (2017).

[campaioli18]{} F. Campaioli, F. A. Pollock, and S. Vinjanampathy, *Thermodynamics in the Quantum Regime*, Springer, (2018)

[campbell19]{} S. Deffner and S. Campbell, *Quantum Thermodynamics, An introduction to the thermodynamics of quantum information*, IOP Publishing, 2019.

[kosloffrev19]{} R. Kosloff, J. Chem. Phys. [**150**]{}, 204105 (2019).

[levy14]{} R. Kosloff and A. Levy, Annu. Rev. Phys. Chem. [**65**]{}, 365 (2014).

[kurizki15]{} D. Gelbwaser-Klimovsky, W. Niedenzu, and G. Kurizki, Adv. At. Mol. Opt. Phys. [**64**]{}, 329 (2015).

[rezek17]{} R. Kosloff, Y.Rezek, Entropy, [**19**]{}, 136 (2017).

[mitchison19]{} M. T. Mitchison , Contemporary Physics, [**60**]{}, 164 (2019).

[zanardi06]{} R. Alicki, D. A. Lidar, and P. Zanardi, Phys. Rev. A [**73**]{}, 052311 (2006).

[alickitut]{} R. Alicki, D. Gelbwaser-Klimovsky, and G. Kurizki, arXiv:1205.4552.

[szczyg13]{} K. Szczygielski, D. Gelbwaser-Klimovsky, R. Alicki , Phys. Rev. E [**87**]{}, 012120 (2013).

[kurizki13]{} D. Gelbwaser-Klimovsky, R. Alicki, G. Kurizki Phys. Rev. E [**87**]{}, 012140 (2013).

[alicki14]{} R. Alicki, Open Syst. Inf. Dyn. [**21**]{}, 1440002 (2014)

[kosloff92]{} E. Geva, R. Kosloff, J. Chem. Phys. [**96**]{}, 3054 (1992).

[bender00]{} C. M. Bender, D. C. Brody, B. K. Meister, J. Phys. A: Math. Gen. [**33**]{}, 4427 (2000).

[nori07]{} H.T. Quan, Y.X. Liu, C. P. Sun, F. Nori, Phys. Rev. E [**76**]{}, 031105 (2007).

[tang09]{} L. He, X. He, W. Tang, Sci. China Ser. G-Phys. Mech. Astron. [**52**]{}, 1317 (2009).

[uzdin15]{} R. Uzdin, A. Levy, and R. Kosloff, Phys. Rev. X [**5**]{}, 031044 (2015).

[klimovsky15]{} D. Gelbwaser-Klimovsky, W. Niedenzu, P. Brumer, G. Kurizki, Sci Rep [**5**]{}, 14413 (2015).

[niedenzu15]{} W. Niedenzu, D. Gelbwaser-Klimovsky, and G. Kurizki Phys. Rev. E [**92**]{}, 042123 (2015).

[weitz15]{} D. Gelbwaser-Klimovsky, K. Szczygielski, U. Vogl, A. SaB, R. Alicki, G. Kurizki, and M. Weitz, Phys. Rev. A [**91**]{}, 023431 (2015).

[seifert17]{} K. Brandner, M. Bauer, and U. Seifert Phys. Rev. Lett. [**119**]{}, 170602 (2017).

[mukherjee20]{} V. Mukherjee, A.G. Kofman, and G. Kurizki, Commun. Phys. [**3**]{}, 8 (2020).

[mukherjee19]{} V. Mukherjee, A. Zwick, A. Ghosh, X. Chen, G. Kurizki Communications Physics [**2**]{}, 162 (2019).

[correa15]{} L. A. Correa, M. Mehboudi, G. Adesso, A. Sanpera, Phys. Rev. Lett. [**114**]{}, 220405 (2015).

[pasquale16]{} A. D. Pasquale, D. Rossini, R. Fazio, V. Giovannetti, Nat. Comm. [**7**]{}, 12782 (2016).

[brunner17]{} P. P. Hofer, J. B. Brask, M. Perarnau-Llobet, and N. Brunner, Phys. Rev. Lett. [**119**]{}, 090603 (2017).

[wang20]{} A. Levy, G. Moritz, B. Deng, K. Singer, E. Torrontegui, D. Wang, arXiv:2005.06858.

[caves94]{} S. L. Braunstein, C. M. Caves, Phys. Rev. Lett. [**72**]{}, 3439 (1994).

[paris09]{} M. G. A. Paris, Int. J. Quantum Inf. 07, 125 (2009).

[carnot]{} S. Carnot, Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance (Bachelier, Paris, 1824).

[buffoni19]{} L. Buffoni, A. Solfanelli, P. Verrucchi, A. Cuccoli, and M. Campisi, Phys. Rev. Lett. [**122**]{}, 070603 (2019).

[maccone11]{} V. Giovannetti, S. Lloyd, and L. Maccone, Nature Photonics [**5**]{}, 222 (2011).

[degen17]{} C. L. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod. Phys. [**89**]{}, 035002 (2017).

[zwick17]{} G. Kurizki, G. A. Alvarez, and A. Zwick, Technologies [**5**]{}, 1 (2017).

[pezze18]{} L. Pezze, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Rev. Mod. Phys. [**90**]{}, 035005 (2018).

[sciarrino20]{} E. Polino, M. Valeri, N. Spagnolo, F. Sciarrino, arXiv:2003.05821 \[quant-ph\].

[dutta20]{} S. Bhattacharjee, U. Bhattacharya, W. Niedenzu, V. Mukherjee, A. Dutta, New J. Phys. [**22**]{}, 013024 (2020).
