The Unruh quantum Otto engine

Journal of High Energy Physics, Feb 2018

Enrique Arias, Thiago R. de Oliveira, M. S. Sarandy

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The Unruh quantum Otto engine

HJE The Unruh quantum Otto engine Enrique Arias 0 2 Thiago R. de Oliveira 0 1 M.S. Sarandy 0 1 0 28625-570 Nova Friburgo , Brazil 1 Instituto de F sica, Universidade Federal Fluminense 2 Instituto Politecnico , Universidade do Estado do Rio de Janeiro We introduce a quantum heat engine performing an Otto cycle by using the thermal properties of the quantum vacuum. Since Hawking and Unruh, it has been established that the vacuum space, either near a black hole or for an accelerated observer, behaves as a bath of thermal radiation. In this work, we present a fully quantum Otto cycle, which relies on the Unruh e ect for a single quantum bit (qubit) in contact with quantum vacuum uctuations. By using the notions of quantum thermodynamics and perturbation theory we obtain that the quantum vacuum can exchange heat and produce work on the qubit. Moreover, we obtain the e ciency and derive the conditions to have both a thermodynamic and a kinematic cycle in terms of the initial populations of the excited state, which de ne a range of allowed accelerations for the Unruh engine. Black Holes; Thermal Field Theory - 1 Introduction 2 Quantum Otto engine 3 Quantum Otto engine via Unruh e ect 3.1 3.2 3.3 3.4 3.5 Kinematic cycle of the qubit Adiabatic expansion of the energy gap Contact with the hot quantum vacuum Adiabatic contraction of the energy gap Contact with the cold quantum vacuum 4 Quantum vacuum uctuations e ects 5 E ciency and cyclicity of the Unruh heat engine 6 Conclusions A Dynamics of the qubit-vacuum interaction B Regularization by Lorentzian pro le demons, and advantages of work extraction by global operations over quantum systems [ 3 ]. Even though the rst works in this direction appeared already in the 1950's [ 4 ], it regained interest through the works by Kieu [5, 6], where a quantum heat engine is proposed by using as working substance a single two-level system, i.e, a quantum bit (qubit). There, it has been analyzed the quantum Otto cycle, which is the quantum generalization of the classical prototype model of combustion engines. The Otto cycle is composed of two adiabatic and two isochoric steps operating between a hot and a cold reservoir. In order for the machine to produce a nite amount of work, one needs to ful ll the condition TH > ( 2= 1)TC , with TH (TC ) the temperature of the hot (cold) reservoir and 1 and 2 the energy gaps in the qubit Hamiltonian spectrum before and after the adiabatic expansion, respectively. Therefore, for quantum systems, the classical inequality TH > TC has been shown to be { 1 { insu cient for a heat engine to produce work in individual thermodynamic cycles. Another possibility considered in refs. [5, 6] is to use a gas of bosons as the thermal reservoir and let the interaction with the system to last a nite time, such that the system does not fully equilibrate with the reservoir. In that case, a cycle can be built by suitably adjusting the initial population of the excited state.These works have been generalized to include di erent models for the reservoir and also for more complex working substances, e.g. coupled qubits, where quantum correlations such as entanglement may improve the thermal e ciency [7{9]. On the other hand, the relation between quantum thermodynamics, relativistic quantum mechanics, and black hole physics remains little explored. The standard thermodynamics of black holes has been elucidated a long time ago [10], with striking results such particularly relevant phenomenon is the Unruh e ect and its implications to the domain of quantum thermodynamics. The Unruh e ect predicts the existence of thermal properties to the vacuum state of a quantum eld. More speci cally, it establishes that the quantum vacuum uctuations with respect to an accelerated observer in at space-time exhibit thermal properties with temperature proportional to the observer acceleration [15, 16]. Even though an experimental realization of the Unruh e ect remains as a challenge, it has recently appeared a number of proposals of physical observations, such as in superconducting qubits [17], ion traps [18], quantum metrology [19], oscillating neutrinos [20], and classical electrodynamics [ 21 ] (see also ref. [22] for other previous proposals). Here, our aim is to exploit the thermal properties of the vacuum uctuations predicted by the Unruh e ect to model a thermal reservoir and de ne a relativistic quantum heat engine. We consider as working substance a single qubit linearly coupled to a quantum eld prepared in its vacuum state. This is the model used by Unruh and DeWitt in order to detect the quantum vacuum uctuations e ects [ 16, 23 ]. The Unruh quantum thermodynamical cycle proposed in our work consists of a series of quantum adiabatic and isochoric processes. During the adiabatic processes, we consider that the qubit undergoes an expansion/contraction of its energy gap due to the in uence of some external eld. Since the processes are adiabatic, there is no heat exchange between the qubit and the environment. However, the adiabatic expansion/contraction would cost/produce some amount of work. On the other hand, the isochoric processes are de ned here by coupling the qubit to the quantum vacuum for nite periods of time and considering two di erent constant accelerations for the qubit at these stages. In this manner, if the periods of time are very long or in nite lasting we have that, due to the Unruh e ect, the qubit would feel the vacuum as a thermal reservoir. For our case, the exact thermal nature of the vacuum uctuations cannot be recovered, but the nite-time Unruh e ect of the vacuum uctuations can generate a non-trivial response on the accelerated qubit at these stages [24, 25]. In this way the qubit can absorb some heat from the vacuum uctuations with no work at these stages. Heat and work are provided by the prescriptions of quantum thermodynamics and are nd using a perturbation theory approach in the coupling constant between the qubit and the quantum eld. In order to run as a thermal machine, we must also consider the conditions for a closed trajectory of the qubit in real space, since we are dealing with simultaneously thermodynamic and kine{ 2 { matic cycles. The cycle conditions are then derived in terms of the initial populations of the excited state, which de ne a range of allowed acceleration pairs for the Unruh quantum engine. Throughout the paper, we will adopt natural units, such that the speed of light c, the Planck constant ~, and the Boltzmann constant kB are taken as c = ~ = kB = 1. 2 Quantum Otto engine In this section, we brie y review some notions of quantum thermodynamics and quantum heat engines. The basic de nitions here are those of work and heat in a quantum process. For a quantum system described by a density operator given by (t) evolving under a We see that the rst term on the right hand side of eq. (2.1) is associated with changes in the inner state of the system, which are related to the populations of (t), while the second term has its origin only in changes of the external parameters in H(t). We thus identify the rst as the heat transfer hQi and the second as the work hW i realized on the system. This association provides (2.1) (2.2) (2.3) and hQi = hW i = Z T 0 Z T 0 dt Tr dt Tr d (t) dt H(t) ; where T denotes the total time of evolution and quasi-static processes are adopted. For the processes of interest here, either hQi or hW i will be vanishing. Then, from the rst law of thermodynamics, this will allow to take these quantities as dependent only on the initial nal state of the system. More general cases must be carefully examined (for a review see, e.g., ref. [ 26 ]). It is worth observing here that the notion of heat in the quantum thermodynamics domain is entirely compatible with the notions of Boltzmann entropy S = Tr ( ln ) and that of Boltzmann distribution at equilibrium = e H=T =Z, where Z is a normalization constant and T is the temperature of the system. By using the de nition of heat Q = T dS, it is straightforward to show that Q = Tr(d H), which is equivalent to the de nition of mean quantum heat given above. We now brie y review the Otto quantum engine rst proposed in ref. [5]. The classical Otto cycle is composed of two adiabatic processes together with two isochoric processes, which is common in automobile piston engines. In its quantum version, the working substance is a quantum system instead of an ideal gas. Here, we adopt the simplest quantum system, namely, a qubit. The two adiabatic processes, which in the classical engine are provided by the adiabatic expansion and contraction of a piston, here are adiabatic increase and decrease of the upper energy level of the qubit, respectively. The two isochoric process of the classical case are replaced in the quantum version by keeping the system with constant gaps while in contact with a thermal reservoir, which implies in vanishing work. { 3 { Let us now consider a single qubit with eigenlevels fjei; jgig, which are associated with energies f!; 0g, respectively. Hence, by applying the spectral theorem [ 27 ], the qubit Hamiltonian can be written as H = ! jeihej. In our cycle we consider that, during an adiabatic process, the energy level ! changes. This change could be caused by some external eld or changes in the external conditions as the size of the recipient where the qubit is con ned. Here, we only assume it can be done in a smooth way such that the quantum adiabatic theorem is valid [28{30]. The four stages of a quantum Otto cycle are (i) The qubit begins at the initial state in = pjeihej + (1 p)jgihgj with an energy gap !1 and undergoes an adiabatic expansion to a larger value !2. This expansion is at the cost of some work over the system. On the other hand, since the process is adiabatic, no heat is exchanged with the environment. (ii) The qubit in the state in and with xed energy gap !2 is put in contact with a hot bath at temperature TH . After some interaction time T2, the qubit ends up at the state = (p + pH )jeihej + (1 p pH )jgihgj. Only heat is transfered in this process, with no work performed. (iii) The qubit at the state is isolated from the hot bath and undergoes an adiabatic compression, which reduces its energy gap from !2 to !1. This process is adiabatic, with no change in the populations of the energy levels. Thus, work is performed by the system, with no heat transfered. (iv) The qubit still with the state is put in contact with a cold reservoir at temperature TC . After some interaction time T1, the qubit ends up at the nal state n = (p + pH + pC )jeihej + (1 p pH pC )jgihgj. Heat is transfered between the qubit and the environment, with no work performed. One of the novelties in the quantum regime is that the processes are now probabilistic, i.e., while we expect in the classical case that heat ows from the hot bath to the system for every single cycle at stage (ii), namely, pH > 0, there is in the quantum case a nite probability, for individual cycles, of the opposite to occur, namely, pH < 0. Thus, it is possible to observe a violation of the behavior expected by the second law of thermodynamics for individual quantum cycles. Naturally, conciliation with the second law is recovered on average over a long sequence of cycles [5]. Claims of violation of the second law have been considered, but they all typically use non-thermal baths (see, e.g., ref. [ 31 ]). Note also that the random violation of the second law for individual quantum cycles are due to quantum uctuations rather than to thermal uctuations. The probabilistic character of a quantum engine implies that we need to enforce the cyclicity of the engine through the constraint that pH + pC = 0, which ensures that the state of the system at the end of stage (iv) is equal to the initial state at the beginning of stage (i). By imposing such a constraint, we obtain that the total amount of work performed by the heat engine is W = pH (!2 !1): (2.4) { 4 { So it is clear that in order to have a positive work and therefore a heat engine, instead of a refrigerator, we need to require pH > 0. When the system is let to thermalize with the bath, with T1; T2 ! 1, the state of the qubit at the end of stage (ii) is such that p + pH = Tr [ jeihej ] = 1= [1 + exp( H !2)], where H = 1=TH . Similarly at the end of stage (iv), as the system returns to its original state, we have p = Tr [ jeihej n ] = 1= [1 + exp( C !1)], where C = 1=TC . It then follows that the condition for positive work is, on average, TH > (!2=!1)TC . This is stronger than the classical condition of TH > TC . It can be understood by the fact that the system is expected to receive heat from the hot bath if TH > !2 and to transfer heat to the cold bath if TC < !1. 3 Quantum Otto engine via Unruh e ect We now present the Unruh quantum Otto engine. In this case, the thermal bath comes from the e ect of the vacuum uctuations over the accelerated qubit given by the nite-time Unruh e ect. Thus, the thermal reservoir has now a relativistic quantum origin, which is associated with the vacuum uctuations of a quantum eld. This implies that the Unruh e ect is able to yield useful work in a cyclic process. As a working substance, we take a single qubit, which is designed to transform heat extracted from the vacuum into work. The stages proposed to the Unruh thermodynamic cycle are illustrated in gure 1. 3.1 Kinematic cycle of the qubit Since the observation of the Unruh e ect depends on the state of motion of the qubit as it interacs with the quantum vacuum, a description of the kinematics of the qubit on the whole cycle is required. In this way, in addition to a thermodynamic closed cycle, we also assume a kinematic closed trajectory for the qubit traveling in spacetime. The Unruh e ect over closed oscillatory trajectories have been consider in [32]. However, since in our case the conctact of the qubit with the vacuum uctuations is restricted to some stages of the thermodynamic cycle, we only need to analyze the Unruh e ect for a nite part of the qubit trajectory. In this way, the continuous qubit cyclic motion will impose further constraints { 5 { of contact between the qubit and the quantum vacuum in the interaction time with the quantum vacuum, as it will be shown below. Thus the kinematic cycle we analyze is such that, in the rst and third step of the cycle, we assume the qubit keeps constant speed, while in the second and fourth steps, it moves with constant acceleration. This is explicitly shown in gure 2. Notice that, while traveling at constant acceleration the qubit is brought into contact with the vacuum uctuations. On the other hand, while at constant velocity and going through an adiabatic expansion/contraction of the energy gap, there is no contact between the qubit and the quantum eld. In order to assure that there is no change in the state of the system, caused by the vacuum, during the adiabatic processes of the cycle, we should keep the qubit decoupled of the vacuum at these steps. We are thus assuming that one can control the contact between the system and the vacuum in the very same way that one can control the coupling or decoupling between a system and a thermal reservoir in a classical thermal machine. For simplicity we assume a one-dimensional motion of the qubit. Thus, our closed trajectory is composed by the following steps: Step 1: constant velocity v during a period of time T . Step 2: constant acceleration ates from speed v to v. H during a period T2. At this step, the qubit acceler Step 3: constant velocity v during a period of time T . Step 4: constant acceleration C during a period T1. At this step the qubit accelerates from speed v to v. It is worth noticing that interaction times T1 and T2 are xed by the values of v and H;C . More speci cally, we will shown that there is a relation between H and C in order to ensure that the system returns to its original state after one cycle. For now, we say that it is expected the accelerations (temperatures) satisfy H > C to ensure that at step 2 the vacuum behaves like a hot reservoir while at step 4 it acts like a cold reservoir. The non-trivial motion of the qubit occurs when it is in contact with the vacuum. Indeed, in order for the vacuum to behave as a thermal bath, we have to ensure a motion with constant proper acceleration for the qubit [15]. Therefore, during the stages of contact with the vacuum, we suppose that the qubit moves with constant proper acceleration, as described by the step 2 and step 4 of the kinematics cycle. This would ensure the thermal { 6 { HJEP02(18)6 x partially causally disconnected from the qubit. In dark gray is the region totally causally disconnected from the qubit. The white region is the causally accesible region of the qubit, the border of this region represents an event horizon for the accelerated qubit. behavior of quantum vacuum as predicted by the Unruh e ect if a long interaction time is considered. However, even for nite interaction times one can recover a non-null e ect over the qubit due to vacuum uctuations. In general, if one considers a relativistic motion with constant proper acceleration , one nds that the spacetime coordinates of the particle are given by ( ) = (t; x) where HJEP02(18)6 being the particle proper time. This hyperbolic spacetime trajectory of the qubit with constant acceleration is shown in gure 3. From eqs. (3.1), the velocity of the qubit is given by Then, we see then that if the qubit velocity begins at v at time , it will have velocity v at time , where Then, the time that it takes for the qubit to change from velocity v to v using a constant acceleration is 2 = 2 arctanh(v)= . Hence, we have that, when the qubit has acceleration H , the interaction time with the vacuum is T2 = 2 arctanh(v)= H , whereas when the qubit has acceleration C the interaction time with the vacuum is T1 = 2 arctanh(v)= C . Therefore, these are the precise values of the nite interaction times with the vacuum when it would behave as a hot or cold reservoir, respectively. 3.2 Adiabatic expansion of the energy gap Now we describe the rst stage of our engine. This process corresponds to an adiabatic expansion of the energy gap, where we assume that the energy gap of the qubit increases (3.1) (3.2) (3.3) t = x = 1 1 sinh( cosh( ); ); v = tanh( ): = 1 arctanh(v): { 7 { smoothly in time. Recalling that the qubit Hamiltonian is given by H = !(t)jeihej and that we consider as the initial state of the qubit the density operator in = pjeihej + (1 p)jgihgj, it follows that the state of the qubit all along this adiabatic process is kept constant, so (t) = in. Therefore, we obtain that during the adiabatic expansion there is no heat absorbed by the qubit, i.e. hQ1i = dt Tr d (t) dt H(t) = 0: However the energy gap expansion of the qubit comes at the cost of work on it, which reads We now use that Tr ( injeihej) = p, where p is the probability of nding the system in the excited state jei. Moreover, considering that the excited state increases its energy level in the adiabatic expansion from !(t = 0) = !1 to !(t = T ) = !2, we get a positive work Contact with the hot quantum vacuum (3.4) (3.5) (3.6) (3.7) (3.8) In this part of the cycle we take advantage of the Unruh e ect to simulate a thermal bath to the qubit. We take the qubit as interacting with a quantum massless scalar eld in its ground state of zero particles, i.e. in its vacuum state j0i. Due to the Unruh e ect we know that this quantum vacuum uactuations act in a non-trvial way over the qubit when it moves in an accelerated trajectory, even for nite-time interactions. Therefore the kinematics description of the qubit previously presented is crucial for this part of the cycle. An important remark here is that we suppose that the qubit system is small enough so that it does not a ect the state of the scalar eld and disturb the quantum vacuum state. During the interaction of the qubit with the quantum scalar eld, we consider that the Hamiltonian of the total qubit- eld system is given by H = H0 + Hint; H0 = H + H eld; { 8 { where H0 is the free Hamiltonian and Hint is the qubit- eld interaction Hamiltonian. The Hamiltonian H0 reads where H denotes the qubit free Hamiltonian H = !2jeihej, with !2 the constant gap along this stage, and H eld = R d3x(1=2)f(@t')2 + (r')2g is the Klein-Gordon Hamiltonian and the scalar eld '( ( )) is evaluated on the spacetime point ( ) = (t; x) where the qubit is located. We suppose at this stage that the qubit has a constant acceleration with a spacetime trajectory analog to that given in eq. (3.1). We consider that the scalar eld start its evolution at the vacuum state and the initial state of the qubit- eld is the tensor product %in = in j0ih0j. We adopt here the interaction picture, where we apply a perturbative approach and a Dyson series to obtain the nal state of the qubit up to second order in the small coupling constant g. This is shown in details in appendix A. We then show that the nal state of the qubit after interacting some time with the vacuum is (t) = in + pH (t) 3; where 3 = (jeihej contact is restricted to jgihgj) is the third Pauli matrix and the time t during this vacuum 2 < t < 2, with 2 = arctanh(v)= H . The change in the population of the excited state is given by associated to a free massless scalar eld '. The qubit- eld interaction Hamiltonian is given by the linear coupling [ 23 ] dt Tr d (t) dt H(t) : { 9 { where g is the coupling constant of the interaction, m is the qubit monopole operator, which is given by H , HJEP02(18)6 (3.9) (3.10) (3.11) (3.13) (3.14) pH (t) = g2 Z t 2 d Z t 2 0 and G+H ( ; 0) is the Green correlation function of the scalar eld evaluated at two points of the qubit accelerated trajectory. This vacuum correlation function evaluated on the accelerated trajectory of the qubit would capture the thermal nature of the vacuum uctuations. Therefore, at some time t during the interaction with the quantum vacuum the density operator of the qubit is given by (t) = p + pH (t) 0 0 1 (p + pH (t)) ! : One can see that the increase of the population of the excited state, eq. (3.12), is caused by the interaction with the vacuum and this would give us, in some limit, a thermal like response, as predicted by the Unruh e ect. We also observe that the change in the probability of nding the qubit in its excited state depends on the initial population of this state p, on the energy gap !2 and on the acceleration H , which plays the role of hot temperature in the Unruh quantum machine. We can then evaluate the average heat and work over the qubit during the contact with the hot quantum vacuum. From the state of the qubit given by eq. (3.13), at any instant in the interval of contact from the quantum vacuum 2 < t < 2, we obtain the mean quantum heat absorbed Since at this part of the cycle the free qubit Hamiltonian is H = !2 jeihej, the mean quantum heat extracted from the vacuum is dt Tr dt Tr d (t) dt H ; dt d( pH (t)) dt d( pH (t) 3) !2jeihej ; Tr( 3jeihej); pH ( 2)) ; Again, the change in the energy levels of the qubit Hamiltonian does not a ect the state of the qubit. Then, the state of the qubit remains xed as in eq. (3.18) all over the adiabatic contraction and there is no heat transfer. On the other hand, the qubit realizes work, which is proportional to the energy gap change and to the population of the excited state. In this way we straightforwardly obtain that, during the adiabatic contraction, heat and work read hQ3i = 0; hW3i = (p + pH )(!1 !2); where the energy gap contraction is taken from !2 to the initial lower value !1 considered at stage 1 of the quantum cycle. Note that hQ2i depends on the energy gap !2 and on the change pH of the excited level population due to the hot vacuum uctuations. We will discuss ahead the conditions to ensure that pH and the heat absorbed from the vacuum hQ2i are both positive, so the quantum vacuum here really behaves as a hot reservoir. As at this stage of the cycle, the qubit Hamiltonian is constant H = !2jeihej, which implies that we get no work, i.e. hW2i = dt Tr = 0: 3.4 Adiabatic contraction of the energy gap We suppose next that the energy gap is contracted by means of external elds. The speci c way this contraction occurs is not important, but in order to ensure a closed cycle for the quantum heat engine, we have to get back to the initial energy gap. Then, at this stage we decouple the qubit from the vacuum. Hence, the system is kept in the state = p + pH 0 0 1 (p + pH ) ! : where we have used that, from eq. (3.12), we have pH ( 2) = 0 and we have also de ned the total correction pH = pH ( 2), which is given by the nite-time integral pH = g2 d 0 (1 where we have de ned p0 = p + pH . Then we have that the nal state of the qubit is where the new change in the probability of the excited state is given by = p0 as discussed in appendix A. By considering only terms up to second order in the coupling constant and realizing that the rst correction pH of eq. (3.16) is already of order two, we can show that Here we consider that the qubit is bring into contact with the quantum vacuum again, but now with acceleration C < H . So, at this stage, the vacuum plays the role of a the cold bath reservoir. The qubit has a lower energy gap !1 in this process and its initial state is (3.20) (3.21) (3.22) (3.24) Note that here we have considered that the energy gap of the qubit is !1 and that the acceleration in this stage of the cycle is lower that previously C < H . This expression is very similar to that one we have obtained for the case of a hot quantum vacuum. However, in the same way that we must ensure that, during the rst contact of the qubit with the quantum vacuum, it absorbed heat, i.e. pH > 0, here we must require that, in order for the vacuum to behave as a cold reservoir, the qubit have to transfer heat to the vacuum, i.e., pC < 0. As we will see, these conditions are not independent of each other, since we also need to ensure that during a cycle the qubit comes back to its original state. As before, we can obtain the average quantum heat and the work produced in this part of the cycle, which read hQ4i = !1 pC ; hW4i = 0: After this stage of the cycle, we must impose that the system comes back to its initial state. Then the nal state of the qubit, given by eq. (3.21), requires pC + pH = 0. This cyclicity condition, together with the e ciency of the cycle, are analyzed in the next sections. 4 Quantum vacuum uctuations e ects From our previous results, we have found that the corrections induced by the vacuum uctuations to the qubit excited state population can be written in a general form as a nite-time integral given by is the qubit acceleration, ! is the qubit energy gap, p is the initial population of the excited state, and 2 is the interaction time of the qubit with the vacuum. However, as we need to ensure a kinematically closed cycle for the qubit, we require that the interaction time of the qubit with the vacuum is such that = arctanh(v)= ; (4.2) g2 arctanh(v): this precise value ensures us that the qubit enters the vacuum with velocity v and goes out with velocity v, in such a way that its movement in spacetime is continuous and smooth. The nite-time integral eq. (4.1), is worked out properly in appendix B. In order to avoid spourios e ects caused by a sharp switching of the interactions as shown in ref. [33], we consider a continuous smooth regularizator function that limits the interation to a nite domain. A similar procedure have been adopted in [34], in order to consider a nite interaction time between two Unruh-DeWitt detectors with the vacuum. However our method is independent, since we use a Lorentzian regularizator instead of a Gaussian one, we understand that both precedure would be equivalent in some limit. On the other hand, as it is shown in appendix B, the qubit acceleration and the qubit energy gap appear as a single variable, which motivates the de nition of the ratio a =!, here called as reduced acceleration. In turn, the correction to the excited state population eq. (B.18), is then written as p = p(a; p; v) = g2 (1 2p)J gure 4. In fact, by imposing j pj which leads to j pj We will now numerically analyze the behavior of p given in eq. (4.3) for di erent regimes of the Unruh machine. The behavior of p can be provided as a function of the reduced acceleration a = =!, for di erent values of initial probabilities p and velocities v. The results are illustrated in gure 4. Note that the corrections p always increase with the acceleration, which plays here the role of the quantum vacuum temperature. In particular, these corrections asymptotically tend to a velocity-dependent constant value for su ciently high accelerations. Moreover, notice that perturbation theory requires that p. This condition may be violated for small a, which indicates a breakdown of the perturbative approach. Indeed, as given by eq. (4.2), the interaction time tends to increase as we decrease a for a xed velocity v. Since the integral in eq. (4.1) implies that j pj / g2! p, we have that a large j pj may be yielded in the region of small a. This regime is forbidden by perturbation theory and is roughly indicated by the hatched gray areas in p, for p > 0, we then obtain the condition g2! Let us discuss in more details the special cases for initial probabilities p = 0, 0 < p < 1=2 1, (4.4) 0.06 0.04 δp/g2 (a) Completely empty excited state. (b) Lowly populated excite state. (c) Highly populated excited state. (d) Behavior for xed velocity. and 1=2 < p < 1, which are exhibited in gures (4a, 4b, 4c), respectively. For the case p = 0, we have that the excited state is initially empty and the qubit begins the interaction with the vacuum in its ground state. The breakdown of the perturbative approach appears here in the negative corrections p for small a and this is represented by the gray area. This area, between a = 0 and a = 20, encloses one order of magnitude greater that g2 arctanh(v), for g = 1 and v = 0:99, as dictated by eq. (4.4). For larger values of a, the correction p, gets positive and asymptotically increases for larger velocities v. This result is consistent with the Unruh e ect since, if the system is prepared at the ground state, then the thermal uctuations created by the Unruh e ect would induce an increase in the excited state population. In addition, if v is increased, this means a longer interaction time for a xed acceleration a, which is expected to yield an asymptotically larger correction p. For p = 1=4, as shown in gure (4b), we again see for high accelerations an asymptotically constant value of p that increases as a function v, as expected by the Unruh e ect. For small acceleration values, we also observe the breakdown of the perturbative approach through the crossings of the curves for distinct values of v. The crossings would mean that di erent interaction times would imply the same excitation p, which is a clearly a limitation of the perturbation theory. Moreover, for the point of vanishing acceleration, a = 0, we see that a negative divergence appears. The origin of this behavior can be traced back to the integral in eq. (4.1), which is proportional to the interaction time interval. Since 1=a, then a divergence comes out for small a. This occurs because we would need an in nite time interval to change the velocity from v to v, by moving the qubit with zero acceleration. 0.0 -0.2 -0.6 -0.8 0.4 0.2 -0.2 -0.4 -0.6 a=0.1 p a=10 v=0.1 v=0.3 v=0.7 v=0.9 v=0.9999 v=10-10 v=0.6 v=0.9 v=0.999 v=0.999999 v=0.01 v=0.05 v=0.2 v=0.5 v=0.9 v=10-10 v=0.6 v=0.9 v=0.999 v=0.999999 δp/g2 of its initial population, p, for di erent accelerations and velocities. The lighter gray area indicates corrections that obey 0 < p + p < 1, where g = 1 has been considered. The darker gray area indicates points where the perturbative regime is satis ed, ie. j pj The case p = 3=4 is presented at gure (4c). We observe that, for this population of the excited state, the vacuum uctuations would always cause a decrease of the original highly populated excited state. Then, spontaneous and stimulated decay in the qubit energy state would overcome the excitations caused by the Unruh e ect. As the acceleration a becomes larger, we have that the corrections p increase, but keeping always negative as a function of a. In gure (4d), we show the behavior of p as a function of a for di erent initial populations p of the excited state for a xed velocity. This plot summarizes the previously discussed behaviors for both p < 1=2 and p > 1=2. For the special case p = 1=2, any value of accelerations the Unruh thermal radiation produces a decay in the qubit energy state. However, in the limit of a ! 1, the corrections are vanishing, i.e. p = 0. Then, in this case, high accelerations produce a balance on the original equally populated states and no net e ect can be detected. In conclusion, it follows that, in order to gain heat from the quantum vacuum, we would require an initial state with 0 p < 1=2. Moreover, perturbation theory requires (4.4) as a validation approach. We now illustrate in gure 5 the behavior of p as a function of the initial population p of the excited state. As it can be seen, the correction p is linear with p, as expected from eq. (4.3). In addition, this linear response is such that p decreases as the initial population p increases. This result is consistent with the Unruh e ect over the qubit. For p have an initially lowly populated excited state, so that one expects that the Unruh e ect yields an increase of its population, i.e. p > 0. On the other hand, for p 1, the vacuum uctuations are expected to yield a decay, i.e. p < 0. Therefore, as p goes from 0 to 1, one expects the corrections p to decrease and invert its signal, as shown in gure 5. In all the plots of gure 5, we also show a lighter gray area, which represents the allowed values for the corrections p in order to get nal positive probabilities less than one. Then, this lighter gray area ensures 0 < p + p < 1, for illustration in this area it has been adopted g = 1. In order to evidence the limits of the perturbative approach we see that there exist values of p that are not in the lighter gray area. However, by using a lower value of the coupling constant as, for example, g = 0:1, we get that the lighter gray area expands, making better the perturbative results. Then, the lighter gray area is a necessary condition for the validity of perturbation theory. In gure 5 the darker gray area indicates a stronger constraint over perturbation theory, provided by eq. (4.4). This condition gives us the value of the maximum allowed velocity vmax given a value for the acceleration a. From eq. (4.4), we adopt vmax = tanh(a=g2). In gure (5a), we have analyzed the case for a small reduced acceleration. In this case, we nd that only very small values of the velocity remain in the domain of validity. As we increase acceleration, gure (5b), we note that perturbation theory remains valid for a wider range of velocities and only very high velocities v 1 are capable to break down the perturbative results. The cases of high and ultrahigh accelerations, shown in gures (5c) and (5d), respectively, are rather robust against violations of the perturbative regime. 5 E ciency and cyclicity of the Unruh heat engine We can summarize the Unruh heat engine cycle as shown in gure 6. There, we explicitly exhibit the absorbed heat and produced work at each part of the cycle, as well as the changes in the qubit density operator. As we mentioned before, the rst contact of the qubit with the quantum vacuum occurs as it moves at a high acceleration H . In this part of the cycle the qubit should absorb heat hQ2i > 0 from the quantum vacuum and increase the population of the excited state by pH > 0. After this rst contact, the qubit undergoes a contraction of the energy gap. Then, it is put in contact with the quantum vacuum again. During the second contact of the qubit with the quantum vacuum, it should transfer heat hQ4i < 0 and changes its excited state population by pC < 0. Therefore, in order for the cycle to yield positive net work, we need to assure both that the vacuum behaves like the hot thermal reservoir in the rst contact with the qubit and that it acts as a cold thermal reservoir in the second contact with the qubit. We can explicitly analyze the regimes for these conditions to be obeyed. As we previously mentioned, to obtain a closed cycle, we require that the qubit comes back to its original state as the machine returns to gure illustrates the di erent states of the qubit, the absorbed heat, and the produced work at each stage of the cycle. its rst stage, which means pH + pC = 0: Since we expect that pH > 0, then we should also have pC < 0. The periodic condition given by eq. (5.1) is equivalent to require that the rst law of thermodynamics remains valid on average for our process, as we will see. Consequently, this leads us to the total mean quantum heat absorbed by the qubit from the vacuum hQi = hQ2i + hQ4i; = !2 pH + !1 pC ; = (!2 !1) pH : hW i = hW1i + hW3i; = (!2 = (!1 !1)p + (!1 !2) pH : !2)(p + pH ); On the other hand, work is realized only during the adiabatic expansion or contraction of its energy gap. Hence, the total work produced over the qubit is Then, as one can see, the net work produced over the qubit and the total heat absorbed from the vacuum satis es hQi + hW i = 0, as is expected from the rst law of thermodynamics. (5.1) (5.2) (5.3) |δp|/g2 0.02 0.03 0.01 = hW iext ; hQ2i !1 !2 : 1.0 p of p, for various reduced accelerations a = =! and velocity v = 0:8. Continuous lines contain points representing positive corrections, with the vacuum behaving as a hot thermal bath. Dashed lines show negative corrections, with the vacuum acting as a cold thermal bath. For the magenta region, the vacuum only acts as a hot reservoir. For the cyan region, the vacuum only acts as a cold reservoir. In the yellow region, we show that the cycle can be closed, since the vacuum acts either as hot or cold reservoir for the same initial value of p, with hot or cold depending on the di erent reduced accelerations associated with the trajectory. The black point intersection de nes a triplet (aH , aC , p) for which we can de ne a closed cycle. Since hW i is the work performed over the qubit by external forces, then the work performed by the qubit over the exterior is given by hW iext = h W i. Hence, we de ne the e ciency as which yields The e ciency depends only on the ratio of the minimum and maximum energy gaps and is independent of the accelerations (temperatures) we have considered. Moreover, this is the same e ciency found by using a classical thermal bath [5], which suggestes a kind of universal bound independently of the nature of the thermal reservoir. Let us now, proceed to analyze the kinematic constraints for the cyclic condition of the qubit state, eq. (5.1). By de ning the hot aH and cold aC reduced accelerations for the qubit in its two contacts with the vacuum as aH = H =!2 and aC = C =!1 and using the general result eq. (4.3), then the cyclicity condition given by eq. (5.1) can be written as p(aH ; p; v) + p(aC ; p; v) = 0: We also have to ensure that, during the rst contact with the vacuum, the qubit has a larger energy gap !2 and that the qubit absorb heat from the vacuum uctuations, so that (5.4) (5.5) (5.6) p(aH ; p; v) > 0. This, together with the cyclic condition eq. (5.6), would imply that during the second contact with the vacuum, were the qubit has a lower energy gap !1, the vacuum acts as a thermal sink, p(aC ; p; v) < 0. Then, for the same values of p and v, but di erent values of reduced accelerations aH and aC , we must ensure that the vacuum acts like a hot or cold thermal bath, respectively. In gure 7 we show the behavior of the absolute value of the correction j pj as a function of the initial probability p, for a velocity v = 0:8 and di erent reduced accelerations. In solid lines, we show the points where the qubit absorbs heat from the vacuum ( p > 0) and, in dashed lines, the points where the qubit emits heat to the vacuum ( p < 0). The yellow region represents the possible values of aH , aC and p, where we can nd an intersection between solid lines and dashed lines. Therefore, these regions show points corresponding to pairs of accelerations for which, by choosing the initial probability of the excited state as p, we will have a closed thermodynamic cycle and extract work from the vacuum. As an illustration, the intersection of the red solid line (aH = 40) and the dashed orange line (aC = 15) occurs at the value p = 0:293. This triplet satis es eq. (5.6), with v = 0:8, allowing for a closed cycle in the Unruh heat engine. In a more general analysis, eq. (5.6) can be interpreted as an equation for nding the initial population of the excited state p, in such a way that the vacuum acts as a hot reservoir for the qubit with reduced acceleration aH , while at the same value of p, the vacuum acts as a cold reservoir for the qubit with reduced acceleation aC . In this manner, we de ne the critical probability p0 that solves eq. (5.6) as the value for our initial population such that the cycle is closed and the qubit returns to its initial state. Then, this probability, which represents intersections of solid and dashed lines in gure 7, depends on the values of the hot and cold reduced accelerations (vacuum temperatures), reading p0 = p0(aH ; aC ; v): we can shown that the solution for the initial critical probability p0 is p0 = P 1 + 2P : By using eq. (4.3) into eq. (5.6) and de ning the function P(aH ; aC ; v) = 2aH aC (aH +aC ) arctanh(v) J 1 aH 1 aC ; 2 arctanh(v) +J ; 2 arctanh(v) ; The behavior of the solution given by eq. (5.9) for a di erent pair of reduced accelerations is shown in gure 8a. There, we can see that 0 < p0 < 1=2, as expected from previous discussions. Now, using this solution we can de ne the correction pH such that pH = pH (aH ; aC ; v) = p(aH ; p0; v): The behavior of this function is shown in gure 8b). There, we observe that heat is absorbed from the hot quantum vacuum, pH > 0, and the cycle behaves as a thermal machine instead of a refrigerator, only when aH > aC , which implies in the condition (5.7) (5.8) (5.9) (5.10) (5.11) Figure 8 (a). Critical value of the initial probability p0 as function of aH and aC , for v = 0:8, such that we have a closed Unruh cycle. 20 40 80 100 60 Figure 8 (b). Correction pH for di erent accelerations aH and aC . We get positive heat from the quantum vacuum pH > 0, only for aH > aC . This relation is analog to the result nd in [5], however in our case the role of temperatures is interpreted by the accelerations of the qubit when interacts with the vacuum uctuations. This is in accordance with the thermal nature of the quantum vacuum predicted by Unruh. This condition, eq. (5.11), is stronger that the classical relation H > C , and establish which gradients of accelerations (temperatures) could be used in order to run a quantum heat engine. Note also that we have nd the initial critical probability, given by eq. (5.9), in order to run the quantum thermal machine. Finally, by de ning the change due to the cold reservoir as pC = p(aC ; p0; v), we get that pC = pH , so in the domain de ned by eq. (5.11) we obtain that the vacuum acts like a cold reservoir when the qubit is prepare to have initial critical probability p0 and has a small reduced acceleration aC . All this ensure us that the cycle behaves as a heat engine instead of a refrigerator. In order to conclude our analysis, we will realize a comparison between the works performed by Unruh quantum Otto engine and the usual quantum heat engine relying on a classical thermal bath. For a classical thermal bath, we have that the work provided by the machine is given by Wcl = (!2 !1) pcl, where pcl = 1 1 + e H !2 1 1 + e C!1 ; is the change in the population of the excited state when the qubit pass through the hot thermal bath. Here we suppose that the qubit begins this contact with an equilibrium distribution, which has a temperature given by the cold reservoir and, after this contact, the qubit state has an equilibrium distribution with temperature given by the hot reservoir. We compare our result W = (!2 !1) pH with the classical result when the temperatures TH;C = 1= H;C coincide with the accelerations H;C we used here. Then, in this case, we can rewrite the classical correction in terms of the reduced accelerations pcl(aH ; aC ) = 1 1 + e1=aH 1 1 + e1=aC : 100 80 40 20 0.004 0.002 0 -0.002 HJEP02(18)6 (5.12) (5.13) Classic v=0.9 v=0.7 v=0.4 v=0 HJEP02(18)6 velocities and the usual quantum heat engine with a classical thermal reservoir. Here we have used !2 !1 = 1. Since, our correction pH (aH ; aC ; v) also depends on the velocity of the qubit in the kinematic cycle, we can compare this correction for di erent values of the qubit velocity versus the classical thermal bath correction. This is shown in gure 9. There we can see that, as we increase the qubit velocity v in the quantum Unruh heat engine, the results approach the expected result of a classical thermal bath. This can be understood in terms of the nite interaction time with the vacuum. Since we are dealing with a nite trajectory in space-time, the interaction time of the qubit with the vacuum given by T = 2 arctanh(v)= . So, as we increase the qubit velocity, the interaction time with the vacuum uctuations also increases and we approach the exact thermal nature of the vacuum predicted by Unruh. It is worth noting here that perturbation theory constrains us to small values of T , so very small values of as well as ultrahigh velocities v 6 1 must be avoided. Conclusions We have introduced a relativistic quantum thermal machine based on the Unruh e ect. This is achieved by taking the quantum vaccuum as a thermal reservoir. By using quasistatic processes and perturbation theory, we are then able to establish the conditions over the initial excitation probability p0 and qubit accelerations aH and aC for ensuring closed cycles, both from the thermodynamic and kinematic point of view. More speci cally, we determine sets of triplets (p0; aH ; aC ) such that a closed cycle emerges. Moreover, we have also shown that the classical e ciency of the Otto cycle persists in the relativistic regime, suggesting a universal bound as long as quasi-static processes are adopted. On the other hand, work is nontrivially a ected by kinematics, being dependent on the velocities and accelerations of the qubit throughout its time evolution. This analysis has been analytically provided via perturbation theory, with its limitations delineated. Even though realizations of the Unruh e ect are still challenging, the thermal machine proposed here opens the possibility of posing relativistic quantum themodynamics in an experimental setting for quantum motors and refrigerators as long as alternative approaches [17{22] for probing the Unruh e ect are well-succeeded. As further developments, it remains the extension of our approach to the non-equilibrium regime in open systems, e.g., with the entropy production taken into account [35]. Moreover, it is still promising the investigation of the role of correlations, such as entanglement, for the machine e ciency. These topics are left for future research. Acknowledgments E.A. would like to thank Guillermo Duen~as for many useful discussions while this work has been prepared and to FAPERJ for nancial support. T.R.O. would like to thank CNPq-Brazil for nancial support. M.S.S. acknowledges support from CNPq-Brazil (No.303070/2016-1) and FAPERJ (No 203036/2016). The authors also acknowledge the Brazilian National Institute for Science and Technology of Quantum Information (INCT-IQ). A Dynamics of the qubit-vacuum interaction During the interaction of the qubit with the quantum scalar eld we have that the Hamiltonian of the total qubit- eld system is given by HJEP02(18)6 H = H0 + Hint; eld; Hint = g m'( ( )); i d%(t) dt the free Hamiltonian H0 is the part that does not consider interaction between the qubit and the eld where H eld = R d3x(1=2)f(@t')2 + (r')2g is the Klein-Gordon Hamiltonian associated to a free massless scalar eld ' and H is the qubit free Hamiltonian given by H = ! jeihej. The qubit- eld interaction Hamiltonian is given by where g is a small coupling constant of the interaction, m is the qubit monopole operator given by and '( ( )) is the scalar eld evaluated on the spacetime point ( ) = (t; x), where the qubit is located. These coordinates are given by eq. (3.1), with the qubits moving with constant acceleration . Let us denote % the total density operator of the qubit- eld system. Then this operator satis es the quantum Liouville's equation (A.1) (A.2) (A.3) (A.4) (A.5) The qubit is prepared in the mixed state in and the scalar eld starts at the pure vacuum state j0i. The initial density operator of the total system is taken as the tensor product %in = in j0ih0j. In order to simplify the dynamics, we move to the interaction picture. Then, we de ne the total density operator in the interaction picture %I (t) as a pull-back over %(t), yielding %I (t) = U0 1(t; t0)%(t)U0(t; t0); where U0(t; t0) is the total evolution operator when there is no interaction between the qubit and the eld. Therefore, this operator satis es The dynamics of the density operator in the interaction picture then reads i d where HiInt is the interaction Hamiltonian in the interaction picture. As before, the initial condition is given by %I (t0) = %(t0) = %in = in given in terms of a Dyson series j0ih0j. Hence, the solution to eq. (A.8) is We use now the temporal order product operator T , which acts as T HiInt( ); HiInt( 0); %in ) HiInt( ); HiInt( 0); %in with 0. Then, we can write %I (t) as i d 1 Z t 2 t0 d Z t t0 d 0T HiInt( ); HiInt( 0); %in + : : : (A.11) We consider here that the contact with the vacuum occurs at general times < < , where we have that = arctanh(v)= . From eq. (A.11), we need to express the interaction Hamiltonian, Hint = g m'( ( )), in the interaction picture. Given an operator in the Schrodinger picture OS, its version in the interaction picture is i = H0U0(t; t0): i d%I (t) dt = [HiInt; %I (t)]; Z t t0 d Z t0 = ( + ( ) HiInt( 0); HiInt( ); %in ; (A.10) Therefore the interaction Hamiltonian in the interaction picture is given by OI (t) = U0 1(t; t0)OSU0(t; t0): HiInt( ) = g U0 1( ; t0) m'( ( )) U0( ; t0); where we have de ned U0( ; t0) = e iH0( t0): (A.6) (A.7) (A.8) (A.12) (A.13) (A.14) Since H0 = H + H eld is time independent and H; H eld = 0, it follows where U ( ; t0) = e iH( t0) and U eld( ; t0) = e iH eld( t0). Therefore Since operators in the interaction picture, eq. (A.17), evolve as free operators without taking into account the in uence between the eld and the quibt, then the scalar eld 'I ( ( )) represents the free scalar eld evaluated at the position of the qubit. The vacuum expected values of products of this operator give us the free Green correlation functions of the eld. For instance, the free Wightman function of the free scalar eld is given by G+( ; 0) = h0j'I ( )'I ( 0)j0i: Let us now explicitly work the monopole operator at the interaction picture mI ( ). As the initial time during the contact with the vacuum is t0 = and the Hamiltonian of the qubit H = ! jeihej is time independent, then g2 Z t 2 Z t d d [mI ( )'I ( ( )); %in] d 0 T mI ( )'I ( ( )); [mI ( 0)'I ( ( 0)); %in] : U0( ; t0) = U0( ; = I + e i!2( + ) 1 ~; = m = e i!2( + ) 0 0 The monopole operator is given by m = jeihgj + jgihej. Then its matrix representation is In the interaction picture mI ( ) reads The state of the qubit- eld system at the instant < t < after the interaction have begun is, up to second order in perturbation theory, given by (A.15) (A.16) (A.17) (A.18) (A.19) (A.20) (A.21) (A.22) where we have de ned Then (a)( ; 0) = (b)( ; 0) = (c)( ; 0) = (1 Therefore we can write Since we are interested only in the qubit state independently of the eld, we perform a partial trace over the eld degrees of freedom on the total state of the qubit- eld system %I (t), which yields By taking into account that %in = in j0ih0j, the partial traces produce I (t) = Tr eld %I (t): Tr eld %in = in; Tr eld [mI ( )'I ( ( )); %in] = 0; Tr eld mI ( )'I ( ( )); [mI ( 0)'I ( ( 0)); %in] = mI ( )mI ( 0) inG+( ; 0) where we have evaluated the Wightman functions of the scalar eld at di erent points of the qubit trajectory, denoting them as G+( ; 0) = G+( ( ); ( 0)). In this notation we make explicit the fact that the Whightman function depends only on the di erence of proper times and that the trajectory have a proper acceleration . In this way and by using the properties of the temporal ordered products, we obtain that the nal state of the qubit in the interaction picture, eq. (A.23), is given by I (t) = in + g2 Z t d Z t d 0 ( ; 0)G+( ; 0); ( ; 0) = (a)( ; 0) + (b)( ; 0) + (b)( ; 0); with each term that corrects the state of the qubit given by (A.23) (A.24) (A.25) (A.26) (A.27) (A.28) (A.29) (A.30) (a)( ; 0) = mI ( 0) inmI ( ); (b)( ; 0) = (c)( ; 0) = )mI ( )mI ( 0) in; ) inmI ( )mI ( 0): ( ; 0) = (1 p ei! 3 ; 0 p ei! ! 0 ; 0 (1 p) e i! (1 p) e i! ! ; ! : where 3 is the Z Pauli matrix. Hence, we can write the nal state of the qubit after interacted with the vacuum as I (t) = in + p(t) 3 where we have de ned the change in the population of the excited state as p(t) = g2 Z t Z t p ei! Therefore, after the interaction with the quantum vacuum, the density operator of the nal increase of the population of the excited state p = p( ), eq. (A.32), is caused by the interaction with the vacuum and this would give us a thermallike response, as predicted by Unruh [16]. We also realize that the change in the probability of the excited state depends also on the initial population of this state p, on the energy gap ! of the qubit system and also on the acceleration of the qubit when interacting with the vacuum. B Regularization by Lorentzian pro le In this appendix, we will show more explicitly how the vacuum uctuations transfer heat to the qubit system. The integral that shows the connection between vacuum uctuations and the speci c trajectory of the qubit is given by (A.31) (A.32) (A.33) J ( ; !; T ) = Z T =2 Z T =2 T =2 d T =2 d 0ei! (B.1) Here, the nite-time interval T is de ned in terms of the velocity v and the acceleration . This nite-time integral can be extended to in nite by using a regulator that is compact in the domain we are interested in and vanish outside. This procedure is in the same spirit of the regularization by a cut-o function of an in nite divergent integral. In particular, nite-time integrals related to the Unruh e ect have been previously analyzed, e.g., in refs. [24, 25]. However, since we are dealing with a kinematically closed trajectory there is no need of taking the limit of in nite time. Hence, we will identify eq. (B.1) with the expression where T the kind gauss complex plane 1 d 1 d 0 T ( ) T ( 0)ei! G+( ; 0); (B.2) ( ) is a compact smooth function that is nonvanishing for T =2 < < T =2, being approximately zero outside that domain. We could try to adopt a Gaussian regulator of exp( 2 =T 2). However, as we expand this Gaussian regulator to the ! z 2 C, we see that it does not vanish for any jzj ! 1. Actually, depending on the phase in z = jzjei , this regulator can be divergent. In order to bypass this di culty, we adopt as a regulator T pro le, i.e. ( ) a Lorentzian pro le rather than a Gaussian The expression above for T ( ) yields T ( ) = (T =2)2 2 + (T =2)2 : T ( ) T ( 0) = (T 2 T 4 T12)(T 2 T 2) ; 2 ( ) T ( 0) in the complex plane of T = + 0. (B.3) (B.4) (B.5) (B.6) (B.7) (B.8) T ( ) T one gets where we have de ned T1 = + iT and T2 = + iT . Hence, eq. (B.4) shows ( 0) explicitly as a function of T = + 0. This function posses a pole structure in the complex plane of T , as shown in gure 10. Naturally, the polar structure depends on 0. Now, performing the change of integration variables from ( ; 0) to (T; ) 1 d( ) T ( ) T ( 0)ei! G+( ): J ( ; !; T ) = d( dT T ( ) T ( 0): Then, performing the integral on T rst, using the pole structure in gure 10 and the Cauchy-Riemann residue theorem of complex variables [36] one gets Here, we have made explicit the dependence on the Wightman correlation function of the scalar eld evaluated at an accelerated trajectory, which is given by T ( ) T ( 0), such that G+( ; 0) = G+( ) = 2 1 16 2 sinh2( 2 i ) : Therefore, one can see that the only dependence on T is given by the regulator factors 1 Z 1 2 1 1 ) 1 3 T 2 + T 2 : dT T ( ) T ( 0) = 2 for the terms in eq. (B.11). Putting this result into eq. (B.7) and using the alternative representation of the Wightman function on the accelerated trajectory [15] (B.9) (B.10) (B.11) (B.12) (B.13) we get that Splitting the term in the above sum, it follows that G+( ) = T 16 1 d( ) T 16 1 + X k=0 1 + X k=0 k= 1 ) i + 2 ik= )2( 1 i 2 ik= )2 ei! 2 ik= )2( ei! i )2( 2 + T 2) 2 + T 2) ; : 2 + T 2) : 2 + T 2) All the terms in the above expression have simple poles in iT . The rst term corresponds to k = 0 and possess a pole of second order at i . In the second and third terms, we have neglected the i contribution, since in these terms the double pole is nite at 2 i= , respectively. This pole structure is shown in gure 11. By using the Residue theorem and the de nition of the transcendental Lerch-Hurwitz function we obtain then that (z; s; a) = z k 1 X k=0 (k + a)s ; J ( ; !; T ) = ( T =2)2e j!jT 8 sin2( T =2 i ) + j!jT (!) 4 + 2 2 T e 2 j!j=j j + j! jT 2e 2 j!j=j j 32 2 16 e 2 j!j=j j; 2; 1+ j 2jT e 2 j!j=j j; 1; 1+ j 2jT e 2 j!j=j j; 2; 1 j jT e 2 j!j=j j; 1; 1 j jT ; 2 2 J (x; y) = then we can rewrite (y=2)2 e jxjy 8 sin2 (y=2) y2e 2 jxj 32 2 + jxjy2e 2 jxj 16 1 8 + jxjy 4 (x) e 2 jxj; 2; 1 + e 2 jxj; 1; 1 + y 2 y 2 e 2 jxj; 1; 1 y 2 y 2 J ( ; !; T ) = J ; T : By using this analytical expression, we can derive the general result shown in the appendix A. There, we obtained that the nal general correction to the population of the excited state of the qubit due to the vacuum Then, using the de nition of the integral in eq. (B.1) and also eq. (B.15), we can write the small imaginary part i in the rst term of the equation above can be iterpreted using the Cauchy principal value 1=(x i ) = (P=x) i (x), see [15]. In this way, with > 0, we have that the function J tends to zero in the limit of vanishing interaction time T . This function, eq. (B.13), apparently depends on three independent variables. However, as one can check, the combination between these variables leads us to a two-variable function. Indeed, if we de ne the function In the nal line we have used the property J (x; y) J ( x; y) = xy=4, which follows from eq. (B.14). This concludes the analysis of the regularized perturbative expression for the excitation correction p. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. p = g2 ((1 = g2 (1 p)J ( ; !; 2 ) p J ( ; !; 2 )) ; p J De ning the reduced acceleration of the qubit as the ratio a = =! and taking into account that the interaction time of the qubit with the vacuum have to be such = arctanh(v)= , we obtain p=g2 = (1 ; 2 arctanh v ; 2 arctanh v ; 1 a 1 a = (1 2p)J Thermodynamic Behavior Within Composite Quantum Systems, Lect. Notes Phys. 657, Springer-Verlag, Berlin Heidelberg, Germany (2004). 2 (1959) 262. 39 (2006) 115. [5] T.D. Kieu, The second law, Maxwell's demons, and work derivable from quantum heat engines, Phys. Rev. Lett. 93 (2004) 140403. [6] T.D. Kieu, Quantum heat engines, the second law and Maxwell's daemon, Eur. Phys. J. D [7] T. Zhang, W.-T. Liu, P.-X. Chen and C.-Z. 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Enrique Arias, Thiago R. de Oliveira, M. S. Sarandy. The Unruh quantum Otto engine, Journal of High Energy Physics, 2018, 168, DOI: 10.1007/JHEP02(2018)168