Autonomous Quantum Heat Engine Using an Electron Shuttle

Autonomous quantum heat engine using an electron shuttle

Behnam Tonekaboni,1, 2, ∗ Brendon W. Lovett,3 and Thomas M. Stace1, 2 1School of Mathematics and Physics, University of Queensland, Brisbane, QLD, 4072, Australia 2ARC Centre for Engineered Quantum Systems, The University of Queensland, Brisbane, QLD 4072, Australia 3SUPA, School of Physics and Astronomy, University of St Andrews, St. Andrews KY16 9SS, United Kingdom We propose an autonomous quantum heat engine based on the thermally driven oscillation of a single electron shuttle. The electronic degree of freedom of this device acts as an internal dynamical controller which switches the interaction of the engine with the thermal baths. We show that in addition to energy flux through the thermal baths, a flux of energy is attributed to the controller and it affects the engine power.

I. INTRODUCTION autonomous system. An autonomous rotor engine was proposed [9] inspired by classical rotor engine where the With the advancement in experimental techniques, we state of the rotor determines the interactions and is con- are able to build and control devices with truly quantum sidered as the internal clock. Neither of these papers degrees of freedom. In these quantum devices, the ener- address the energetic cost of the controller or the clock. getics and thermodynamics of the control field become To fully account for the energetic cost of the controller, important. Thermodynamics of such quantum devices we develop a model of an autonomous engine, by replac- have been studied in the context of quantum heat en- ing the external control field with an internal dynami- gines and fridges [1–3], driven nano-systems [4], effect of cal quantum controller that switches the interactions be- energy transfer on quantum coherence [5,6] and many tween the engine and the baths [10]. Formally we replace other topics. Also a quantum heat engine has been built the control parameters in the interaction Hamiltonian using a single ion [7]. The majority of these studies use an with the orthonormal eigenstates of the internal dynamical controller n and write external time-dependent classical control field to switch | ii the interaction between the working system and the heat H = n n Hhot 1cold+ n n 1hot Hcold. (2) baths. The energetic cost of these control fields is often int | 1ih 1|⊗ int ⊗ | 2ih 2|⊗ ⊗ int ignored. Figure1a shows a schematic of a thermodynamic cycle Formally, the external classical control field is de- using a dynamical controller. When the controller is in scribed by a time-dependent control parameter ni [0, 1] the state n the engine interacts with the hot bath and ∈ | 1i in the interaction Hamiltonian between the engine and when it is in the state n2 the engine interacts with the baths, cold bath. | i A heat engine absorbs heat from a hot bath to increase hot cold hot cold Hint = n1(t)H 1 + n2(t)1 H , (1) the energy of the working system. In quantum thermo- int ⊗ ⊗ int dynamics, this process happens in a thermal operation hot where 1 is the identity in Hilbert space of the hot bath [11–14]. One condition on the thermal operation is that hot and Hint is the interaction between working system and the pure state of any ancilla (catalyst or in our case the cold the hot bath, similar for Hint . controller) attached to the system must be the same at To account for the energetic cost of the control field the beginning and the end of the thermal operation. On we develop an autonomous heat engine with a time- arXiv:1809.04251v1 [quant-ph] 12 Sep 2018 the other hand, in an autonomous heat engine, the con- independent Hamiltonian, in which the controller is in- troller has its own dynamics, and stochastic fluctuations cluded in system dynamics. Autonomous heat engines during each cycle make it difficult to unambiguously de- have received recent attention. Frenzel et al. [8] studied fine discrete phases of thermal operations. a quasi-autonomous engine based on an internal quan- In this paper, we introduce a fully autonomous quantum clock which is stabilised by external measurement. tum heat engine based on the oscillation of a single- Although the interaction Hamiltonian used in [8] is time electron shuttle [15, 16], in which the charge state of the independent, the external measurement result is used in shuttle acts as the control system coupling the engine to a feedback loop to intermittently reset the clock. This the hot bath. In section II we describe a model for this induces a back action on the engine, so it is not a fully system and define the engine and the controller. In order to identify the heat flow from the hot and the cold reservoirs, in section III we briefly describe a simpler system, a charged thermally driven oscillator, where the state of ∗ [email protected] the controller is held constant. In section IV we include 2

this noise into a net force, we impose a nonlinear poten- a) b) Hot Bath Cold Bath tial. For this theoretical model we use a half-harmonic ) d s n1 n1 n2 n2 x ( ! | ih | | ih | ! potential (shown with dashed line in Figure1b) which ⇢(t ) U 2 ⇢(t3) is easy to analyse and provides the required nonlinear- Source Drain ity. The energy spectrum of the half-harmonic oscilla-

n n n2 n2 R tor is equally spaced with an energy gap of 2ω, which | 1ih 1| | ih | V (t)= † ⇢(t ) Dissipation 1 ⇢(t4) is described by a simple Hamiltonian, Hosc = 2ωa a ⇢(t5) where a is the bosonic annihilation operator of the os- † x0 x cillator and satisfies [a, a ] = 1. Although the Hamilto- nian of the half-harmonic oscillator is simillar to the full- Figure 1. a) shows thermodynamic cycle of the engine ρ plus harmonic, the position and momentum operators are not the controller |nihn| between two thermal baths, hot and cold. the same as full-harmonic due to the asymmetry in eigen- The state of the engine changes due to thermal operations: wavefunctions which are illustrated by blue shaded area ρ(t1) → ρ(t2) and ρ(t3) → ρ(t4) while it is in contact with in Figure1b (For details see Appendix B). This asymme- hot and cold bath respectively. When the state of the con- try provides larger ˆx for higher energy. Therefore, when h i troller changes, |n1ihn1| → |n2ihn2|(or reverse), engine’s in- Johnson noise adds energy, the oscillator feels a rectified teraction switches from hot to cold (or cold to hot). This force to the right (For details see Appendix C). switching changes the state of the engine ρ(t2) → ρ(t3)(or Finally, the oscillator is coupled to a cold bosonic en- ρ(t ) → ρ(t )). b) shows the diagram of the single electron 4 5 vironment, which provides a cold bath and a sink for the shuttle heat engine. A single electron shuttle (red circle) is confined in a half-harmonic potential (dashed line), between work output of the engine. two leads (source and drain). The leads have the same chem- In our model we assume the total Hamiltonian of the ical potential but different temperatures, indicated by the shuttle, the leads, Johnson noise and the bosonic envi- Fermi distributions shown above them. Johnson noise from ronment is the thermal lead stochastically drives the charged island and ˆ † the anharmonic potential rectifies it into a net force. The H = ~ωI cˆ cˆ (3a) mechanical mode is coupled dissipatively to a cold bath. X ˆ† ˆ ˆ† ˆ + ~ωskbskbsk + ~ωdkbdkbdk (3b) k X h † i the dynamical controller and let it evolve during the ther- + τskFs(ˆx)ˆbskcˆ + H.c. (3c) mal operation to see its effect on the internal energy of k the engine. Also we analyse the energy flow in an unrav- X h † i + τ F (ˆx)ˆb cˆ + H.c. (3d) elling of the system master equation, to derive a system dk d dk k of stochastic dynamic equations for the engine. In sec- † tion V we numerically solve and analyse dynamics of the + 2~ωaˆ aˆ (3e) engine. At the end, in section VI, we calculate power of eξ(t)ˆxˆc†cˆ (3f) the engine in semiclassical and quantum approaches and − X † ˆ ˆ† X ˆ† ˆ show that quantum correlations has an important role in + g(â dp +a ˆdp) + ~ωpdpdp. (3g) the engine power. p p Terms (3a) to (3d) describe dynamics of the controller. Terms (3a) is the single electron energy where ωI is the II. MODEL electronic energy level of the shuttle. We assume the island charging energy is large so that we retain only a Our proposed engine is a modified version of the elec- single electronic mode with fermionic annihilation oper- tromechanical shuttle studied in [16]. Figure1b illus- atorc ˆ which satisfies c,ˆ cˆ† =1. The Hilbert space of the { } trates the engine, consisting of a single electron island electrical degree of freedom is then 0 , 1 , where ne {| i | i} | i that mechanically oscillates in a potential between two is the state of ne electron on the shuttle. leads, shuttling an electron from the source to the drain The Hamiltonian of the leads is given in (3b), ˆ over each cycle. In [16] the voltage bias between the leads where bs(d),k is fermionic annihilation operator for the generates an electric field that drives the charged shut- source(drain) at mode k with energy ~ωs(d)k. Terms tle, so that it is a microscopic electric motor. In contrast, (3c) and (3d) are electron tunneling terms between the we drive the shuttle with a thermal bias (at zero voltage leads and the shuttle. The tunneling functions Fs(d) in- bias), corresponding to a microscopic thermal engine. We dicate the position dependency of the tunneling terms, set the leads to the same chemical potential while allow- and τs(d),k is the tunnelling constant between the shuttle ing them to be at finite temperatures. Johnson noise and mode k of the source(drain). from the finite temperature leads provides a stochastic Expressions (3e) to (3g) describe the mechanical de- driving force. gree of freedom. Expression (3e) is the Hamiltonian of Johnson noise is approximated as white noise with zero the oscillator in the half-harmonic potential while (3f) mean, so the time averaged force is zero. To rectify is the coupling between thermal (Johnson) noise ξ and 3 the shuttle position ˆx. We assume ξ(t) noise = 0 and jump on and off the shuttle. We set the drain tempera- 0 2 0 h i ξ(t)ξ(t ) noise = Ermsδ(t t ), where Erms = 4kBTR is ture to be small so that we can approximate fd(ωI ) 0. hthe noisei amplitude with− temperature T of the source This ensures electrons do not tunnel from the drain≈ to lead and R is the lead resistance. This term can be writ- the shuttle and therefore the shuttle carries an electron ten as eξˆx 1 1 , which is in the form of equation (2). in each cycle from the source to the drain. The shuttle− couples| ih | to the thermal noise (hot bath) when The leads are positioned so that the separation is com- an electron is on board (i.e. the electronic state is 1 ), parable to the oscillator amplitude, therefore the tun- and is uncoupled when it is unoccupied (i.e. the elec-| i nelling amplitude depends on the shuttle position. We −ηˆx tronic state is 0 ). model this by defining Fs(ˆx)= αse and Fd(ˆx)= | i ηˆx The last line in the Hamiltonian, (3g), describes the αde , where η is the inverse of the tunnelling length and ηx0 −ηx0 interaction of the oscillator with the bosonic environment αs=Ae and αd=Ae , where x0 is the midpoint be- where dˆp is the bosonic annihilation operator of mode p tween the source and the drain, as shown in figure1b. with energy ~ωp, while g is a coupling constant. For the purpose of this paper we study the dynamics of We obtain a master equation for the shuttle by tracing the shuttle assuming the temperature of the cold bosonic out the leads and bosonic reservoir and averaging over bath is low and np is near zero (For more general thermo- the noise. We define the reduceed density operator for dynamic analysis this assumption can be lifted, this will the shuttle, following averaging over Johnson noise as be the subject of the future work). The master equation ρ TrL,B ρT noise, where ρT is the density operator of then becomes the≡ total h system.{ }i In the Born-Markov approximation [17] ρ˙ = i 2ωaˆ†a,ˆ ρ + γ [ˆc†cˆˆx]ρ + κ [â]ρ the master equation is (See appendix D for derivation) − L L 2 † −ηˆx 2 −ηˆx + Γs fs(ωI )α [ˆc e ]ρ + (1 fs(ωI )) α [ˆce ]ρ † † sL − sL ρ˙ = i 2ωaˆ a,ˆ ρ + γ [ˆc cˆˆx]ρ (4) 2 ηˆx − L + Γdα [ˆce ]ρ. (5) † dL + κ (np + 1) [â]ρ + κnp [â ]ρ L L Since the engine is the mechanical degree of freedom + Γ f (ω ) [ˆc†F (ˆx)]ρ + (1 f (ω )) [ˆcF (ˆx)]ρ s s I L s − s I L s of the shuttle, the energy of the working system is the † † + Γd fd(ωI ) [ˆc Fd(ˆx)]ρ + (1 fd(ωI )) [ˆcFd(ˆx)]ρ , energy of the oscillator E= Hosc =2ω aˆ aˆ . The rate of L − L the change of the mechanicalh energyi ofh thei oscillator is where [O]ρ = OρO† 1 [O†Oρ + ρO†O] is the Lind- L − 2 dE blad super-operator [18]. Also we use dimensionless units = Tr[H ρ˙]. (6) dt osc where ~ = 1. The first two lines of equation (4) describe the dynam- In the following we analyse this energy rate using a ics of the engine. The first term gives the evolution in fixed charge (in section III) and full dynamics with elec- the half-harmonic potential while the second term is an tron jumps (in section IV). incoherent driving term due to Johnson noise with rate γ = e2E2 T 2. Here we have assumed that Johnson rms ∝ noise is fast compared to the oscillator dynamics. As a III. CHARGED OSCILLATOR result, the stochastic driving field appears as a dissipative term (See Appendix A and [19])1. The second line of the master equation is the damping due to the cold The charge state of the shuttle determines the Johnson ω/T noise coupling. As a prelude to studying the full dynam- bath with the mean phonon number np = 1/(e 1) and damping rate κ. The last two lines of equation− ics, we analyze the evolution for a fixed charge state of the (4) describe the electron tunnelling between the leads shuttle. By using a fixed controller state (fixed charge) and the shuttle which controls the engine–bath coupling. we focus to the thermal operation and identify heat flow to and from the thermal baths. To model this, we set a The rate of tunnelling depends on the Fermi distribution, † (ωI −µ)/Ts(d) fixed chargec ˆ cˆ = ne on the shuttle, and set αs,d = 0. fs(d)(ωI ) = 1/(e + 1) where Ts(d) is temperature of the source(drain) and µ is the common chemical The master equation for the fixed charge shuttle is then potential. † 2 ρ˙ = i 2ωaˆ a,ˆ ρ + γne [ˆx]ρ + κ [â]ρ. (7) We choose ωI slightly greater than the chemical poten- − L L tial of the leads. Thus at zero temperature electrons are We calculate the rate of change of energy in the os- energetically excluded from the shuttle if we neglect co- cillator, i.e. equation (6), during the thermal operation, tunnelling processes. At finite temperature, electrons can and we find dE ωn2γ = e 2ωκN (8) dt 2 − 1 In the voltage driven shuttle in [16], a time-independent electric ˙ H ˙ C, (9) field acts on the shuttle, thus the driving term is coherent and ≡ Q − Q appears as i[ˆc†cˆˆx, ρ]. This is in contrast to the dissipative term where N = a†a is the expectation value of phonon num- in equation (4). ber. In equationh i (9) we have defined the heat flux from 4

2 ˙ ωneγ where E˙ is the contribution of the controller to the the hot bath (i.e. Johnson noise) H = 2 , and the control Q energy of the engine which can be positive or negative, energy ﬂux to the cold bath ˙ C = 2ωκN. When ne = 1, the oscillator couples to JohnsonQ noise which causes a meaning that the controller can add or take energy from steady increase in the mechanical energy of the system the oscillator. Thus the thermodynamic identity for the whole system must include the control power as well as and when ne = 0 the oscillator is only in contact with the bosonic cold bath. We interpret equation (9) as a the thermal operation. thermodynamic identity of the thermal operation with a static control parameter.

IV. AUTONOMOUS ENGINE

Equation (8) and its thermodynamic interpretation equation (9) consider only the energy flow from and to the thermal baths. They do not include power attributed to the dynamical controller (i.e. tunnelling). To account for this, we calculate energy rate, equation (6), from the full dynamical system, including the controller i.e. equation (5). We find To describe the stochastic evolution of the system, we unravel the master equation into a stochastic mas- dE ωγ c†c 2 ter equation (SME), using a jump process for the charge = h i 2ωκN dt 2 − state of the shuttle. In this unravelling the controller charge state fluctuates stochastically between 0 and 1 . 1 2 2 † † −2ηˆx | i | i + ωΓsα η fs(1 c c ) + (1 fs) c c e In an experimental setup, the SME has an interpreta- 2 s − h i − h i h i 1 tion as a model of continuous measurement, in which + ωΓ α2η2 c†c e2ηˆx (10) one monitors the electron jumps and thus the evolution 2 d d h ih i of the controller [20]. The SME for the conditional state ˙ ˙ + E˙ . (11) ≡ QH − QC control of the engine is

dρ = i 2ωaˆ†a,ˆ ρ dt + γ [ˆc†cˆˆx]ρ dt + κ [â]ρ dt (12a) c − c L c L c 1 1 dt Γ f (ω )α2 [ˆccˆ†e−2ηˆx]ρ + Γ (1 f (ω ))α2 [ˆc†ceˆ −2ηˆx]ρ dt Γ α2 [ˆc†ceˆ 2ηˆx]ρ (12b) − 2 s s I sH c s − s I sH c − 2 d dH c + dN s [ˆc†e−ηˆx]ρ + dN s [ˆce−ηˆx]ρ + dN d [ˆceηˆx]ρ , (12c) +G c −G c −G c where and are nonlinear superoperators defined by 2) Electron jumps which switch the bath interactions. To G H show this we write the change in the mechanical energy OρO† [O]ρ = ρ (13) of the oscillator as G Tr[OρO†] − dE = Tr[H dρ ] [O]ρ = Oρ + ρO† Tr[Oρ + ρO†]ρ. (14) c osc c H − = ( ˙ H ˙ C + E˙ M )dt + dEJ (16) Terms in (12a) are the same as the first line of the Q − Q equation (5) while (12c) describes electron jumps on where ˙ H and ˙ C are as before while E˙ M dt = s Q Q and off the shuttle with stochastic increments dN± and Tr[Hosc expression 12b ] describes the energy transfer d { } dN− which are either 0 or 1. The probabilities of these due to continuous current measurement; and dEJ = stochastic terms depend on the state of the shuttle and Tr[Hosc expression 12c ] is a stochastic term due to the { } have the following mean values: electron jumps on and off the shuttle. dN s = Γ f (ω )α2Tr[ˆccˆ†e−2ηˆxρ ]dt (15a) h +i s s I s c dN s = Γ (1 f (ω ))α2Tr[ˆc†ceˆ −2ηˆxρ ]dt (15b) V. DYNAMICAL SIMULATIONS h −i s − s I s c dN d = Γ α2Tr[ˆc†ceˆ 2ηˆxρ ]dt. (15c) h −i d d c In this section we analyze the behaviour of the engine In the SME (12), the controller has two energetic ef- and the effect of the controller by solving the uncondi- fects: 1) Back action of continuous current measurement, tional master equation (5) and SME (12). 5

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0.4 0.10 0.15 0.20 0.25 0.30 0.35 0.40

0.2

0. Figure 3. Shows the phase space diagram of the oscillator coordinates v = hˆpi/m and hˆxi. Solid blue line shows the 0 π 2 π 3 π 4 π 5 π unconditional coordinates up to t = 30π (i.e. 30 cycles), which is substantially longer than 1/κ = 20 representing the damping time scale for the unconditional dynamics. Red dots Figure 2. Shows the time evolution of the expectation value show 50 cycles of a typical stochastic trajectory from t = 450π of (a) position hˆxi, (b) electron number hc†ci and, (c) phonon to t = 500π, i.e. in the long time limit. This illustrates the fact number ha†ai. The solid blue lines show the unconditional so- that typical trajectories do not dwell near the unconditional lution to the master equation (5). Light purple lines are 500 steady state (green point). trajectories of conditional dynamics, equation (12), indicating the typical spread, and the black dashed line highlights a single trajectory. (d) shows the energy flux through the oscil- the control power. The magnitude of the controller effect lator (blue solid line), flux due to thermodynamic reservoirs (red dotted line), and flux due to the dynamical controller increases while electron number increases (and the shut- E˙ (shaded green) tle is near to the source). This is consistent with equation control (10) where electron number, c†c , has an important role h i in E˙ control. A solution to the SME (12) is a quantum trajectory. We use a truncated oscillator basis and find the time The light purple lines in figures2a-c show 500 trajecto- evolution of the matrix elements of ρ in the eigenbasis ries; one is highlighted in black dashed line. The spread of the oscillator and the electrons. Then we calculate of trajectories indicates the stochastic variation of ˆx,ˆc†cˆ the expectation values of the position ˆx , the phonon anda ˆ†aˆ in figure2a-c. number aˆ†aˆ and the electron number hcî†cˆ . The solid h i h i We plot the oscillator phase space coordinates v = blue curves in figures2a–c show these quantities as func- ˆp /M and ˆx , where M is the mass of the oscillator. tions of time. Figures2a and2b show the shuttling Theh i solid blueh i line in figure 3 shows the unconditional co- behaviour of the engine: when the shuttle is near to ordinates over 50 cycles. Phase fluctuations in the cycles the source lead (xsource = 0), average number of elec- cause the oscillator to damp to an unconditional steady trons, cˆ†cˆ , increases and when ˆx is near to the drain h i h i state. (xdrain = 0.6) the electron number decreases. Figure2c In contrast, a typical trajectory in the stochastic so- † shows aˆ aˆ and consequently mechanical energy of the lution to the SME does not converge to a well defined engine hE =i 2ω aˆ†aˆ increases. h i fixed point. Instead it continues to fluctuate around the Figure2d shows the rate of change in mechanical en- unconditional fixed point, rarely dwelling near it. A rep- ergy of the engine. Peaks in E˙ match with the peaks resentative trajectory is shown as red points in Figure3, in cˆ†cˆ and it confirms that the presence of an electron where we have plotted the last 50 cycles to show that the actsh asi the controller: when the shuttle is charged the unconditional steady state is not representative of the heat flux to the engine is the largest. The net heat flux long time stochastic dynamics; that is, stochastic fluctu- ˙ H ˙ C to the engine is plotted as a dotted red line ations remain large in the long time dynamics. whileQ − theQ green shaded area shows the energetic cost of We expect that the swept phase space area grows as the 6

Figure 5. shows power spectrum of oscillator position for diﬀerent values of γ. The peak at ωSR ≈ 2.0ω indicates the existence of stochastic resonance.

driving strength increases. This is demonstrated in Fig- ure4, which shows a series of stochastic solutions to the SME (12) as the strength of the coupling to the Johnson noise increases. Clearly, the phase space region explored by the shuttle grows as the coupling (and heat flux) increases. Discussion– Nonlinear systems subject to noise exhibit various phenomenon including stochastic limit cycle [21] and stochastic resonance [22]. Stochastic resonance (SR) in a nonlinear system occurs when a time-dependent, periodic force is amplified by noise, and arises from a the emergence of stochastic limit cycle, in which two stable attractors become coupled by the noise. Since the dis- covery of SR, Gang et al. [23] and others [24] showed that nonlinear systems with fixed points subject to noise can exhibit autonomous stochastic resonance (ASR) i.e. without an externally applied periodic force. The system we consider in this paper falls into this category: the quantum shuttle has two fixed points as- sociated to each of the charge states of the shuttle, i.e. † for ne = cˆ cˆ = 0 and ne = 1. The position of these fixed pointsh isi shown in Figure4, as a hollow square for ne = 0 and a filled square for ne = 1. Further, the shuttle is driven by both Johnson noise and the stochastic fluctuations in c†c , so it is possible for an autonomous stochastic resonanceh i to appear. Gang et al. [23] characterise ASR phenomena by the Figure 4. shows phase diagrams of typical trajectories of emergence, and subsequent growth, of a peak in the the oscillator for different values of γ and fixed damping rate power spectrum of the system dynamics as noise power κ = 0.05. The unconditional steady states are shown by green increases. Following this approach, we numerically cal- 2 points. The hollow and the filled squares are fixed points for culate the power spectrum Sx(˜ω) = x(˜ω) of the phase | | uncharged, ne = 0, and charged, ne = 1, oscillator respec- space coordinates, where x(˜ω) is the Fourier transform tively. For γ = 0 a typical trajectory fluctuates randomly of ˆx(t) . h i around√ the steady state within the size of the zero point mo- Figure5 shows Sx(˜ω) for different values of γ. At tion 1/ Mω. As γ increases the the stochastic cycles sweep γ = 0, the system has only one fixed point, so it fluc- a larger area. tuates within the zero point motion. A large value of γ is associate with longer distance between the fixed points and consequently larger oscillation. The power spectrum 7 exhibits a clear peak aroundω ˜ = 2ω, which grows with In either approach, we need to calculate the force on increasing strength of the Johnson noise, γ, and is con- the work sink. We first write Heisenberg equations of sistent with the characterisation of ASR phenomena. We motion of phase space coordinates also calculate the power spectrum of velocity (Appendix dpˆ E) and find it has π/2 phase lag relative to the position, = Mω2ˆx κ †[a]ˆp indicating a cyclic orbit in phase space. dt − − D dˆx ˆp Furthermore, in Appendix E we show phase space his- = κ †[a]ˆx, (20) tograms for different trajectories over 400 cycles. The dt M − D ‘crater-like’ depression in the centre of many phase space where †[O]A = O†AO 1 (O†OA+AO†O) is the adjoint histograms is another evidence for existence of stochastic D − 2 Lindblad super-operator.2 We then compute the force resonance. from Newton’s law as follows: d2ˆx dˆp dx M = κM †[a]( ) VI. WORK AND POWER dt2 dt − D dt = Mω2ˆx 2κ †[a]ˆp + κ2M †[a] †[a]ˆx − − D D D In classical thermodynamics, the energy output of a = Fosc + FCold Bath. (21) heat engine is split into a useful part (work) and an en- tropic part (heat), which depend on the process (or path). Considering that the phase coordinates ˆp/M and ˆxare in In quantum machines with high fluctuation in internal dimensionless units and in the same order of magnitude energy, such a classification of work and heat is not ob- (see Figure4), the last term in equation 20 is negligible vious. A common approach is based on the change of when κ ω and κ < 1. The first term is the oscillation 2 the total energy dE = Tr[Hdρ + ρdH], where the first force due to the half-harmonic potential, Fosc = Mω xˆ. term on the right hand side is heat and the second term The second term is the viscous force due to interaction− is work [1,2]. This approach is not suitable for an au- with the cold reservoir. We then write the working force tonomous engine with a time independent Hamiltonian. as the negative of the bath force, Also it has been discussed in [25] that this definition of F = F = +2κ †[a]ˆp. (22) work does not have an operational meaning. Work − Cold Bath D Here we define work based on its classical counterpart. We use equations (17) and (22) to calculate semiclas- The classical definition of infinitesimal mechanical work sical work for different quantum trajectories of the en- is d = F dx, where F is the force on the work sink. gine. Figure6 shows for different value of γ. We EquivalentlyW instentanous power is = F v where v is WSC P have chosen ti well after all transitions to the dynamical the velocity. In quantum mechanics, force and velocity steady state (i.e. system dynamics is independent from can be correlated i.e. F v = F v . In the following h i 6 h ih i the initial state). Thin lines are SC for single trajecto- we define work and power in two different approaches: ries while thick lines are averagedW over trajectories. The Semiclassical and fully quantum. inset plot shows the force-position diagram for the last 10 In semiclassical approach, we ignore quantum correla- cycles of a single trajectory with γ = 4Mω. The semi- tions between force and position (or velocity). The semiclassical work is the cumulative area enclosed by each classical work output during the interval τ = t t is f − i cycle of this plot. The work done on the cold reservoir increases linearly (on average) which indicates a constant Z tf average power. Red circles in Figure7 show the averaged SC(τ) = F (t) d ˆx(t) , (17) W ti h i h i semiclassical power. As we expect this power increases 2 with larger γ THot. and the equivalent instantaneous power is We also calculate∝ power using equation (19) to include quantum correlations. Orange squares in Figure7 shows SC(t) = F (t) ˆv(t) , (18) (averaged over time and over trajectories) for differ- P h ih i PQ ent values of γ. We see that Q (orange squares) are where ˆv(t) = ˆp /M is the velocity of the oscillator. P h i about 5 times larger than SC (red circles). The dis- For the fully quantum approach we include the correla- crepancy between the calculatedP semiclassical power out- tions between F and ˆv.Since these operators do not com- put, and the fully correlated power output indicates that 1 mute we choose the symmetrised operator, 2 (F ˆv+ ˆvF ). quantum correlations in the shuttle are significant But this choice has a nonzero value for the ground Since the cold bosonic reservoir is at zero temperature, state. To avoid a zero point power, we subtract 0 = we suspect that all the transferred energy into it, is at- 1 P 2 0 F ˆv+ ˆvF 0 from symmetric expectation value where tributable as “work”. To investigate this, we compare 0h |is the gground| i state of the half-harmonic. We then |definei quantum power as

1 2 These equations are in rotating wave approximation and the dis- (t) = F ˆv+ ˆvF . (19) PQ 2h i − P0 sipative terms are in both momentum and position. [26] 8

a classical oscillator and we adopted a zero temperature bosonic reservoir as both the cold bath and the work sink. Discriminating heat from work may be done by coupling the engine to a model of a quantum battery that stores the work output as internal energy. This will be the subject of future work. To conclude, we have developed an autonomous heat engine based on the oscillation of a single electron shuttle. In this system the mechanical degree of freedom is the engine and the electric degree of freedom is the controller. We showed that the controller has an energetic cost to

Figure 6. Semiclassical work calculated during the interval τ for different values of γ. The interaction rate to the cold reservoir is chosen as κ = 0.05 and initial time, ti is chosen well after transient to dynamical stationary state. Thin lines are single trajectories while thick lines are averaged over many Figure 7. Shows time and trajectory averaged of PSC (red trajectories. The inset figure shows plot of hF i vs. hˆxi for circles), P (orange squares) and Q˙ (blue diamonds) for dif- the last 10 cycles of the trajectory with γ = 4Mω. The Q C ferent value of γ with fixed κ = 0.05. Dashed lines connecting semiclassical work for each cycle is the area enclosed in the the points to guide the eyes. All the quantities increase by force-position diagram. increasing γ. Quantum power is about 5 times larger than semiclassical which indicates the importance of the force velocity correlation. Also quantum power is close to the heat flux to the cold bath. Inset plot shows P , P and Q˙ for ˙ SC Q C SC and Q to C (blue diamonds in Figure7) and we γ = 4Mω as functions of time. P P Q found that quantum power is close to ˙ . The small PQ QC difference between Q and ˙ C is approximately constant. This is shown in theP inset plotQ in Figure7 which compares the engine power. We also defined both semiclassical and time dependent SC(t), Q(t) and ˙ C (t) for γ = 4Mω. quantum power, and showed that quantum correlation This difference isP a computationalP Q error due to trunca- between force and velocity is significant. tion in oscillator energy basis which is more significant in the momentum operator of half-harmonic oscillator that ACKNOWLEDGMENT is used to calculate FWork. The power is affected by the controller implicitly. It We would like to thank R. Kosloff, A. Nazir, J. Combes can be understood by considering that ˙ and ˙ Q C C and C. Muller, K. Modi, F. Pollock and G. J. Milburn is proportional to phonon number. SinceP phonon' Q numberQ for useful discussions. This work was supported by the is affected by the controller, is affected too. PQ Australian Research Council Centre of Excellence for En- Our definitions of power were based on the damping of gineered Quantum Systems (Grant No. CE 110001013).

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Appendix A: Johnson Noise and Master Equation

Consider an oscillator carries ne electron and oscillates in an arbitrary potential between two grounded leads with no electric bias. We set one of the leads to be at finite temperature while keeping the other at zero temperature. At finite temperature, the electric potential of the hot lead fluctuates, an effect known as Johnson noise. Consequently an stochastic electric field occurs between the leads. We assume that this stochastic field, which we denote by ξ(τ), is a white noise and has following average properties

ξ(τ) = 0 (A1) h inoise ξ(τ)ξ(τ 0) = E2 δ(τ τ 0) (A2) h inoise rms − where Erms is the root mean square of the electric ﬁeld noise. We assume that the distance between the leads, d, is small enough that Erms is constant between the leads and therefore it is proportional to root mean square of Johnson noise as follows V 2 4k TR E2 = rms = B , (A3) rms d d where kB is Boltzmann’s constant, T is the temperature of the hot lead, R is the resistance that connects the hot lead to the ground. The Hamiltonian of the charged oscillator interacting with Johnson noise is

HÎ = H0 + neeξ(τ)ˆx, (A4) where H0 is the oscillator Hamiltonian. This system has two different time scales. One is the time scale of Johnson noise which we denote by τ and the other is the time scale of the oscillator dynamics due to H0 and we label that by t. We consider that time scale of noise is much shorter than dynamics, i.e. dτ dt. The master equation of the oscillator after making Born and Markov approximations is Z ∞ 2 0 0 0 0 ρ˙I = (nee) dτ ξ(τ)ξ(τ ) [ˆxI (τ), [ˆxI (τ ), ρI (τ )]] (A5) − 0 where subscript I indicates that density matrix ρ is in interaction picture. Since we are interested in the dynamics of H0, we average over the noise time scale, Z ∞ D 2 0 0 0 0 E ρ˙I = e dτ ξ(τ)ξ(τ )[neˆxI (τ), [neˆxI (τ ), ρI (τ )]] − 0 τ Z ∞ 2 0 0 0 0 = e dτ ξ(τ)ξ(τ ) τ [neˆxI (τ), [neˆxI (τ ), ρI (τ )]] − 0 h i Z ∞ 0 0 0 0 = γ dτ δ(τ τ )[neˆxI (τ), [neˆxI (τ ), ρI (τ )]] − 0 − 1 = γ [n ˆx (τ), [n ˆx (τ), ρ (τ)]] (A6) −2 e I e I I 2 2 where γ = e Erms. After expanding the commutator we find [n ˆx , [n ˆx , ρ ]] = 2 [n ˆx ]ρ (A7) e I e I I − L e I I † 1 † † where [O]ρ = OρO 2 [O Oρ + ρO O] is the Lindblad superoperator. Using this result, the master equation in the SchrodingerL picture is−

ρ˙ = i[H , ρ] + γ [n ˆx]ρ. (A8) − 0 L e This result shows that the effect of Johnson noise is a dissipative term in the master equation, independent of the shape of the potential. If we include tunnelling on and off the oscillator, we need to substitute ne with the electron number c†c operator in the Hamiltonian (A4), where c is fermionic annihilation operator of an electron. Using the similar derivation we find

ρ˙ = i[H , ρ] + γ [c†cˆx]ρ. (A9) − 1 L where H1 includes all tunnelling dynamics. 11

Appendix B: Quantum Mechanics of the Half-Harmonic Potential

Consider a particle with mass M in a half harmonic potential which is deﬁned as

1 Mω2x2 x 0 V (x) = 2 (B1) x <≥ 0. ∞ The solution to the time-independent Schr¨odingerequation in position space is similar to the full-harmonic oscillator. The boundary condition at x = 0 enforce the wavefunction to be zero for x 0, therefore only the odd solutions are allowed. The wavefunctions and the energy eigenvalues are ≤

1 4 2 1 1 − x ψn(x) = √2 Hn (x) e 2 (B2) π √2nn! 1 E = (n + )ω for n = 1, 3, 5, ... and x 0, (B3) n 2 ≥ q ~ √ where Hn(x) are Hermite polynomials and position x is in units of xc = Mω . An extra factor of 2 in the wavefunction, in comparison to the full-harmonic oscillator, is due to normalising the wavefunction only in x > 0. We deﬁne m with m = 0, 1, 2, 3, ... as excitations of half-harmonic potential and we write the Hamiltonian eigenequation| i

Hˆ m = E m , (B4) | i m| i where Em is the eigenenergy of the eigenstate m . We write the odd integers of the half-harmonic, n, in terms of m, n = 2m + 1. Using this notation, we write the| wavefunctionsi and the energy eigenvalues as

1 4 2 1 1 − x ψm(x) = p H2m+1 (x) e 2 (B5) π 22m (2m + 1)! · 3 E = (m + )(2ω) for m = 0, 1, 2, 3, ... and x 0. (B6) m 4 ≥

Furthermore, we deﬁne the annihilation,a ˆ, and creation,a ˆ†, operators of the half-harmonic as follows X aˆ = √m + 1 m m + 1 (B7) | ih | m X aˆ† = √m + 1 m + 1 m . (B8) | ih | m

These operators obey the usual bosonic commutation relation [ˆa, aˆ†] = 1. Using these operators and equations (B4) and (B6), we write the Hamiltonian in terms of the creation and annihilation operators,

3 Hˆ = aˆ†aˆ + (2ω). (B9) 4

This Hamiltonian is similar to the Hamiltonian of a full-harmonic oscillator with the frequency 2ω and the ground 3 state energy, E0 = 2 ω. But there are more differences between half-harmonic and full-harmonic. Specifically, the position and the momentum operators in half-Harmonic are not trivial and there is no simple form of these operators in terms ofa ˆ anda ˆ†. Instead, we find the matrix elements of these operators in energy basis. X X ˆx= m m ˆx m0 m0 ˆp= m m ˆp m0 m0 | ih | | ih | | ih | | ih | m,m0 m,m0 X X = m ˆx m0 m m0 = m ˆp m0 m m0 h | | i| ih | h | | i| ih | m,m0 m,m0 X 0 X 0 = x 0 m m = P 0 m m mm | ih | mm | ih | m,m0 m,m0 12

0 20 40 60 80 0 20 40 60 80

0 0 8 12 X iP 6 nm 20 10 nm 20 − 4

8 40 40 2

6 0

60 60 2 4 −

4 − 80 2 80 6 − 0 8 −

Main Digonal elements of matrix X Main and next Digonal elements of matrix iP − main diagonal elements 12 ﬁrst upper diagonal elements ﬁrst upper diagonal elements

10 5

8 0 6

4 5 − 2

0 20 40 60 80 100 0 20 40 60 80 100

Figure 8. shows the matrix elements of the position (a) and momentum (b) operators in the energy eigenbasis. (c) shows the diagonal elements of Xnm which are not zero in contrast to the full harmonic potential. (d) shows the main and the ﬁrst upper and lower diagonal elements of Pnm

0 0 where Xmm0 = m ˆx m and Pmm0 = m ˆp m are matrix elements of ˆxand ˆpand we calculate them, using their known wavefunctionsh | | asi follows h | | i Z ∞ Z ∞ 0 0 0 0 Xmm0 = m ˆx m = m dx x x ˆx m Pmm0 = m ˆp m = m dx x x ˆp m h | | i h | 0 | ih | | i h | | i h | 0 | ih | | i Z ∞ Z ∞ ∂ = m dx x x x m0 = i m dx x x m0 h | 0 | i h | i − h | 0 | i∂xh | i Z ∞ Z ∞ ∂ = dx m x x x m0 = i dx m x x m0 0 h | i h | i − 0 h | i∂xh | i Z ∞ Z ∞ ∗ ∗ ∂ = ψm(x)xψm0 (x)dx, = i ψm(x) ψm0 (x)dx. 0 − 0 ∂x Figure8a and b shows matrix elements of these operators. The diagonal elements of the position operator, Figure 8c, are not zero. This feature is the main reason that half-harmonic potential rectiﬁes Johnson noise. These operators obey the canonical commutation relation [ˆx, pˆ] = i, as they must. Another interesting property is the commutation relation of the number operator, a†a, with position operator † i † [ˆx, a a] = 2 pˆ (This commutator in full-harmonic potential is [ˆx, a a] = ipˆ). On the other hand there is no similar commutation relation between a†a and the momentum operator. 13

Appendix C: Rectiﬁction of Johnson Noise with the Half-Harmonic Potential

In this section we show how a half-harmonic potential rectiﬁes Johnson noise. Consider a charged oscillator with ne electrons, inside a half-harmonic potential under the eﬀect of Johnson noise and subject to damping with Hamiltonian

ˆ † H = 2~ωaˆ aˆ eneξ(t)ˆx − X ˆ† ˆ X † ˆ ˆ† (C1) + ~ωpdpdp + g(ˆa dp +a ˆdp). p p

The ﬁrst term is the Hamiltonian of the half-harmonic oscillator, the second term is due to Johnson noise while the second line shows the hamiltonian of the cold bosonic bath which the oscillator is coupled via the last term. Master equation of the oscillator in Born-Markov approximation is

ρ˙ = i(2ω) aˆ†a,ˆ ρ + γn2 [ˆx]ρ + κ [a]ρ (C2) − eL L where we choose the bosonic bath to be at zero temperature and we used dimensionless units. We write equations of motion for the expectation value of phonon number N = aˆ†aˆ h i

dN n2γ = Tr[ˆa†aˆρ˙] = e κN (C3) dt 4 −

and ﬁnd the steady state of the phonon number Nss by equating the equation (C3) to zero

n2γ N = e (C4) ss 4κ

Equation (C4) shows that the average number of phonons at steady state is greater when γ (Johnson noise) is larger and κ (damping rate) is smaller. This shows that Johnson noise act as an energy source (hot bath) and bosonic reservoir acts as an energy sink (cold bath). Also the average number of phonons is zero when ne = 0. This means that an uncharged shuttle looses all mechanical energy to the cold bath. Figure9a shows Nss as a function of γ for a fixed κ. Numerical calculations of steady states for phonon numbers (blue circles) matches with the analytical result of equation (C4) (green line). To show the rectification of the half-harmonic potential we need to show that Johnson noise affects the steady state of the position. In a full-harmonic potential, the steady state of the position is zero even after being heated by Johnson noise. Due to the complication in the position operator of the half-harmonic potential, writing an equation of motion for the position is impossible. Alternatively, we derive the steady state of the master equation numerically and then calculate the expectation value of the position operator. Figure9b shows the expectation value of position at steady state for different value γ. At γ = 0, the steady state of position is as it is for the ground state. Larger γ causes an increase in x ss. This result shows that Johnson noise applies, on average, a net force on the charged shuttle and we concludeh thati the half-harmonic potential rectifies the Johnson noise. In addition we show that the steady states are stable points (i.e. attractors or sinks). To see this, we solve the master equation (7) with an initial state near to the steady states. Specifically, we chose the steady state and displaced it by a small amount. We found that with this choice of initial state the oscillator damps to the related steady state. Figure9 shows unconditional solutions to the master equation (7) with different initial conditions (blue), each near to a steady state (dashed black line). All the solutions damp to their related steady states. 14

Figure 9. a) Numerical (blue circles) and analytical (green line) of expectation value of phonon numbers at steady state vs γ. b) Numerical calculation of expectation value of position at steady state for diﬀernet γ.

Figure 10. Unconditional solutions to the master equation near steady state for diﬀerent values of γ. Each solution eventually damps to the steady state. 15

Appendix D: Derivation of the Master Equation

We showed the eﬀect of Johnson noise after average over the noise and found that adding Johnson noise to the Hamiltonian will add the term due to Johnson noise (the last term of equation (A9) ) to the master equation. In this section we consider an electron shuttle without Johnson noise and ﬁnd its master equation in the Born-Markov approximation and at the end add the Johnson noise term to the master equation. The Hamiltonian of the shuttle, the leads and the bosonic reservoir (for damping) is

H1 = H0 + HI (D1) † † X ˆ† ˆ X ˆ† ˆ H0 = ~ωaˆ aˆ + ~ωI cˆ cˆ + ωkb`,kb`,k + ~ωpdpdp (D2) `,k p X † X † HI = τ`,kF`(ˆx)ˆb`,kcˆ + gpaˆ dˆp + h.c. (D3) `,k p We ﬁrst move to the interaction frame

iH0t/~ −iH0t/~ HI (t) = e HI e (D4) † † † † P † P † X iωaˆ atˆ −iωaˆ atˆ iωI cˆ ctˆ † −iωI cˆ ctˆ i ωkˆb ˆb`kt −i ωkˆb ˆb`kt = τ`k e F`(ˆx)e e cˆ e e `,k `k ˆb`ke `,k `k `,k † † P ˆ† ˆ P ˆ† ˆ X iωaˆ atˆ † −iωaˆ atˆ i ωpdpdpt −i ωpdpdpt + e aˆ e gpe p dˆpe p + h.c. (D5) p

X † iωI t −iωkt † iωt X −iωpt = τ`kF˜`(ˆx)ˆc e ˆb`ke +a ˆ e gpdˆpe + h.c. (D6) `,k p

iωaˆ†atˆ −iωaˆ†atˆ where the time dependence of HI (t) shows we are in the interaction frame and F˜`(ˆx)= e F`(ˆx)e . Then we decompose the system and bath operator by deﬁning

† iωI t −iωI t C1`(t) F˜`(ˆx)ˆc e C2`(t) F˜`(ˆx)ˆce (D7a) ≡ X ≡ X B (t) τ ˆb e−iωkt B (t) τ ˆb† eiωkt (D7b) 1` ≡ `k `k 2` ≡ `k `k k k

† iωt −iωt A1(t) aˆ e A2(t) aeˆ (D8a) ≡ X ≡ X D (t) g dˆ e−iωpt D (t) g dˆ† eiωpt (D8b) 1 ≡ p p 2 ≡ p p p p and we write the interaction Hamiltonian in terms of these operators X H (t) = C (t) B (t) + A (t) D (t). (D9) I i` ⊗ i` i ⊗ i i,` Now the master equation becomes X X Z ∞ ρ˙(t) = dτ [C (t),C (t τ)ρ(t)] ` (τ) + [ρ(t)C (t τ),C (t)] ` ( τ) − i` j` − Bij j` − i` Bji − `=s,d i,j 0 X Z ∞ dτ ([A (t),A (t τ)ρ(t)] (τ) + [ρ(t)A (t τ),A (t)] ( τ)) (D10) − i j − Dij j − i Dji − i,j 0

` 0 0 0 0 where ij(t t ) = Bi`(t)Bj`(t ) E and ij(t t ) = Di(t)Dj(t ) E are baths correlations and to calculate them we use theB following− averages,h i D − h i

b b 0 = Tr[b b 0 ρ ] = 0 d d 0 = Tr[d d 0 ρ ] = 0 (D11a) h `k lk iE `k lk E h p p iE p p E † † † † † † † † b b 0 = Tr[b b 0 ρ ] = 0 d d 0 = Tr[d d 0 ρ ] = 0 (D11b) h `k lk iE `k lk E h p p iE p p E † † † † b b 0 = Tr[b b 0 ρ ] = δ 0 f (ω ) d d 0 = Tr[d d 0 ρ ] = δ 0 n(ω ) (D11c) h `k lk iE `k lk E kk ` k h p p iE p p E pp p † † † † b b 0 = Tr[b b 0 ρ ] = δ 0 (1 f (ω )) d d 0 = Tr[d d 0 ρ ] = δ 0 (1 + n(ω )) (D11d) h `k lk iE `k lk E kk − ` k h p p iE p p E pp p 16

(ωI −µ)/T` ωp/T where f`(ωI ) = 1/(e + 1) is Fermi distribution of lead ` and n(ωp) = 1/(e 1) is the Bose-Einestien distribution for the cold reservoir. Using these we ﬁnd −

` (t t0) = B (t)B (t0) = B (t t0)B (0) = 0 (D12a) B11 − h 1` 1` iE h 1` − 1` iE ` (t t0) = B (t)B (t0) = B (t t0)B (0) = 0 (D12b) B22 − h 2` 2` iE h 2` − 2` iE ` 0 0 0 12(t t ) = B1`(t)B2`(t ) E = B1`(t t )B2`(0) E B − h i h − i 0 X † −iωk(t−t ) = τ τ 0 ˆb ˆb 0 e `k lk h `k lk iE kk0 X 0 = τ 2 (1 f (ω ))e−iωk(t−t ) `k − ` k k Z ∞ −i(t−t0) = dJ`()(1 f`())e (D12c) 0 − ` 0 0 0 21(t t ) = B2l(t)B1`(t ) E = B2`(t t )B1`(0) E B − h i h − i 0 X † iωk(t−t ) = τ τ 0 ˆb ˆb 0 e `k lk h `k lk iE kk0 0 X 2 iωk(t−t ) = τ`kf`(ωk)e k Z ∞ i(t−t0) = dJ`()f`()e (D12d) 0 where we have deﬁned the spectral density J () P τ 2δ( ω ). Similarly we ﬁnd the bosonic bath correlation ` ≡ k | `k| − k

0 0 0 11(t t ) = D1(t)D1(t ) E = D1(t t )D1 E = 0 (D13a) D − 0 h 0 i h − 0 i 22(t t ) = D2(t)D2(t ) E = D2(t t )D2 E = 0 (D13b) D − 0 h 0 i h − 0 i 12(t t ) = D1(t)D2(t ) E = D1(t t )D2 E D − h i h − i 0 X † −iωp(t−t ) = g g 0 dˆ dˆ 0 e p p h d p iE pp0 0 X 2 −iωp(t−t ) = gp(1 + n(ωp))e p Z ∞ −i(t−t0) = dJb()(1 + n())e (D13c) 0 0 0 0 21(t t ) = D2(t)D1(t ) E = D2(t t )D1 E D − h i h − i 0 X † iωp(t−t ) = g g 0 ˆb ˆb 0 e p p h p p iE pp0 0 X 2 iωp(t−t ) = gpn(ωp)e p Z ∞ i(t−t0) = dωJb()n()e (D13d) 0

P 2 where we deﬁned spectral density of bosonic bath as Jb() k gp δ( ωp). Using these correlation functions, we write the master equation as sum of integrals, ≡ | | −

4 4 X X X ρ˙(t) = ` (D14) − Ii − Ki `=s,d i=1 i=1 17 where `s are integrals for lead ` as follows Ii Z ∞ ` ` 1 = dτ [C1`(t),C2`(t τ)ρ(t)] 12(τ) I 0 − B Z ∞ h i Z ∞ † iωI t −iωI (t−τ) −iτ = dτ F˜`cˆ e , F˜`ceˆ ρ(t) dJ`()(1 f`())e 0 0 − Z ∞ Z ∞ † i(ωI −)τ = [F˜`cˆ , F˜`cρˆ (t)] dτ dJ()(1 f`())e (D15) 0 0 − Γ (ω ) = [F˜ cˆ†, F˜ cρˆ (t)] ` I (1 f (ω )) (D16) ` ` 2 − ` I

Z ∞ ` ` 2 = dτ [ρ(t)C2`(t τ),C1`(t)] 21( τ) I 0 − B − Z ∞ h i Z ∞ −iωI (t−τ) † iωI t −iτ = dτ ρ(t)F˜`ceˆ , F˜`cˆ e dJ`()f`()e 0 0 Z ∞ Z ∞ † i(ωI −)τ = [ρ(t)F˜`c,ˆ F˜`cˆ ] dτ dJ`()f`()e (D17) 0 0 Γ (ω ) = [ρ(t)F˜ c,ˆ F˜ cˆ†] ` I f (ω ) (D18) ` ` 2 ` I

Z ∞ ` ` 3 = dτ [C2`(t),C1`(t τ)ρ(t)] 21(τ) I 0 − B Z ∞ h i Z ∞ −iωI t † iωI (t−τ) iτ = dτ F˜`ceˆ , F˜`cˆ e ρ(t) dJ`()f`()e 0 0 Z ∞ Z ∞ † −i(ωI −)τ = [F˜`c,ˆ F˜`cˆ ρ(t)] dτ dJ`()f`()e (D19) 0 0 Γ (ω ) = [F˜ c,ˆ F˜ cˆ†ρ(t)] ` I f (ω ) (D20) ` ` 2 ` I

Z ∞ ` ` 4 = dτ [ρ(t)C1`(t τ),C2`(t)] 12( τ) I 0 − B − Z ∞ h i Z ∞ † iωI (t−τ) −iωI t iτ = dτ ρ(t)F˜`cˆ e , F˜`ceˆ dJ`()(1 f`())e 0 0 − Z ∞ Z ∞ † −i(ωI −)τ = [ρ(t)F˜`cˆ , F˜`cˆ] dτ dJ`()(1 f`())e (D21) 0 0 − Γ (ω ) = [ρ(t)F˜ cˆ†, F˜ cˆ] ` I (1 f (ω )) (D22) ` ` 2 − ` I and are integrals for bosonic bath, Ki Z ∞ 1 = dτ [A1(t),A2(t τ)ρ(t)] 12(τ) K 0 − D Z ∞ Z ∞ h † iωt −iω(t−τ) i −iτ = dτ aˆ e , aeˆ ρ(t) dJb()(1 + n())e 0 0 Z ∞ Z ∞ † i(ω−)τ = [ˆa , aρˆ (t)] dτ dJb()(1 + n())e (D23) 0 0 κ(ω) = [ˆa†, aρˆ (t)] (1 + n(ω)) (D24) 2 18

Z ∞ 2 = dτ [ρ(t)A2(t τ),A1(t)] 21( τ) K 0 − D − Z ∞ Z ∞ h −iω(t−τ) † iωti −iτ = dτ ρ(t)âe , aˆ e dJb()n()e 0 0 Z ∞ Z ∞ † i(ω−)τ = [ρ(t)â, aˆ ] dτ dJb()n()e (D25) 0 0 κ(ω) = [ρ(t)â, aˆ†] n(ω) (D26) 2

Z ∞ 3 = dτ [A2(t),A1(t τ)ρ(t)] 21(τ) K 0 − D Z ∞ Z ∞ h −iωt † iω(t−τ) i iτ = dτ aeˆ , aˆ e ρ(t) dJb()n()e 0 0 Z ∞ Z ∞ † −i(ω−)τ = [ˆa, aˆ ρ(t)] dτ dJb()n()e (D27) 0 0 κ(ω) = [ˆa, aˆ†ρ(t)] n(ω) (D28) 2

Z ∞ 4 = dτ [ρ(t)A1(t τ),A2(t)] 12( τ) K 0 − D − Z ∞ Z ∞ h † iω(t−τ) −iωti iτ = dτ ρ(t)â e , aeˆ dJb()(1 + n())e 0 0 Z ∞ Z ∞ † −i(ω−)τ = [ρ(t)â , aˆ] dτ dJb()(1 + n())e (D29) 0 0 κ(ω) = [ρ(t)â†, aˆ] (1 + n(ω)) (D30) 2

We used the following relation to calculate the time integrals

Z ∞ P dte±it = πδ() i , (D31) 0 ± where P denotes the Cauchy principal value. We only use the real part of the integral since the imaginary part can be absorbed into H0. Also we deﬁnd the following zero temperature rates

Γ`() = 2πJ`() (D32)

κ() = 2πJb(). (D33) 19

Now we write the master equation by adding the calculated integrals X Γ` Γ` ρ˙(t) = (1 f (ω ))[F˜ cˆ†, F˜ cρˆ (t)] + f (ω )[ρ(t)F˜ c,ˆ F˜ cˆ†] − 2 − ` I ` ` 2 ` I ` ` `=s,d Γ Γ + ` f (ω )[F˜ c,ˆ F˜ cˆ†ρ(t)] + ` (1 f (ω ))[ρ(t)F˜ cˆ†, F˜ cˆ] 2 ` I ` ` 2 − ` I ` ` κ κ (1 + n(ω))[â†, aρˆ (t)] + n(ω)[ρ(t)â, aˆ†] − 2 2 κ κ + n(ω)[â, aˆ†ρ(t)] + (1 + n(ω))[ρ(t)â†, aˆ] 2 2 X Γ` Γ` = (1 f (ω )) [F˜ cˆ†, F˜ cρˆ (t)] + [ρ(t)F˜ cˆ†, F˜ cˆ] + f (ω ) [ρ(t)F˜ c,ˆ F˜ cˆ†] + [F˜ c,ˆ F˜ cˆ†ρ(t)] − 2 − ` I ` ` ` ` 2 ` I ` ` ` ` `=s,d κ κ (1 + n(ω)) [â†, aρˆ (t)] + [ρ(t)â†, aˆ] + n(ω) [ρ(t)â, aˆ†] + [â, aˆ†ρ(t)] − 2 2 X = Γ (1 f (ω )) [F˜ cˆ]ρ + Γ f (ω ) [F˜ cˆ†]ρ + κ(1 + n(ω)) [â]ρ + κn(ω) [â†]ρ . (D34) ` − ` I L ` ` ` I L ` L L `=s,d

And the master equation in its ﬁnal form is

ρ˙(t) = Γ (1 f (ω )) [F˜ cˆ]ρ + Γ f (ω ) [F˜ cˆ†]ρ s − s I L s s s I L s + Γ (1 f (ω )) [F˜ cˆ]ρ + Γ f (ω ) [F˜ cˆ†]ρ d − d I L d ` d I L d + κ(1 + n(ω)) [ˆa]ρ + κn(ω) [ˆa†]ρ (D35) L L 20

Appendix E: Evidence for Stochastic Resonance

In this section we show the evidence for existence of Stochastic Resonance. The following table of histograms shows the momentum distribution of solutions to the conditional master equation. The value of γ is increasing from top to bottom and each histogram at each row is a different trajectory for the same parameters. Gaussian fit shown in blue curve and the residual is shown below each histogram. Gaussian fit matches pretty well and show the oscillator is mostly in thermal equilibrium. In the middle of histograms, some disagreements with the gaussian fit are seen. These are due to non equilibrium state of the oscillator after electron jumps. 21

The following 3D histograms show distribution of the phase space coordinates over 400 cycles. Similar to the previous table of histograms, the value of γ increases from top to bottom and each histogram at each row represents a diﬀerent trajectory for the same parameters. The ‘crater-like’ depression in the centre of these histograms suggests the existence of a stochastic resonance. 22

Following ﬁgures show the power spectrum of the position (Sx(˜ω)) and the velocity (Sv(˜ω)). The peak atω ˜ 2.0ω shows stochastic resonance frequency. ≈