Transport properties of a superconducting single-electron transistor coupled to a nanomechanical oscillator

Transcription

1 Transport properties of a superconucting single-electron transistor couple to a nanomechanical oscillator V. Koerting,, T. L. Schmit,, C. B. Doiron, B. Trauzettel, an C. Bruer Department of Physics, University of Basel, CH-456 Basel, Switzerlan Institute for Theoretical Physics an Astrophysics, University of Würzburg, D-9774 Würzburg, Germany Date: February, 9) We investigate a superconucting single-electron transistor capacitively couple to a nanomechanical oscillator an focus on the ouble Josephson quasiparticle resonance. The existence of two coherent Cooper pair tunneling events is shown to lea to pronounce backaction effects. Measuring the current an the shot noise provies a irect way of gaining information on the state of the oscillator. In aition to an analytical iscussion of the linear-response regime, we iscuss an compare results of higher-orer approximation schemes an a fully numerical solution. We fin that cooling of the mechanical resonator is possible, an that there are riven an bistable oscillator states at low couplings. Finally, we also iscuss the frequency epenence of the charge noise an the current noise of the superconucting single electron transistor. PACS numbers: j,73.3.Hk,73.5.T I. INTRODUCTION The cooling of nanomechanical systems by measurement has receive a lot of attention recently. Various proceures like the laser sieban cooling schemes evelope for trappe ions an atoms, have been propose as ways to significantly cool a nanomechanical resonator NR) couple to a Cooper-pair box, 5 a flux qubit, 6,7 quantum ots,, trappe ions, an optical cavities On the experimental sie, optomechanical cooling schemes have been shown to be promising: 3 the NR was coole to ultra-low temperatures via either photothermal forces or raiation pressure by coupling it to a riven cavity. Another important nanoelectromechanical measurement evice which both hols the possibility of very accurate position measurements 35 as well as of cooling of an NR, is a superconucting single-electron transistor SSET). Shortly after the theoretical proposals preicting the potential of the SSET to cool a nanomechanical system, 8,9,67 this effect has been experimentally observe. Using other etectors for NRs such as normal-state single-electron transistors 36 or tunnel junctions, 37,68 it is very ifficult to cool the nanomechanical system or rive it into a non-classical state. These etectors usually act as heat baths with effective temperatures proportional to the transport voltage, which is in practice higher than the bath temperature. The SSET system, on the other han, shows sharp transport resonances. At those the effective temperature is voltageinepenent an can be mae very low. To achieve such challenging goals as groun-state cooling of NRs, or the creation of squeeze oscillator states, a better unerstaning of the transport properties of the couple SSET-NR system is require. This system is schematically shown in Fig.. Depening on external parameters such as the gate voltage V G, the bias voltage V, an also the superconucting gap, the SSET supports ifferent types of resonance conitions. The two most prominent ones are the so-calle Josephson quasiparticle JQP) an the ouble Josephson quasiparticle DJQP) cycle. 38,39 Whereas the former involves the coherent tunneling of a Cooper pair at one of the two junctions followe by a successive tunneling of two quasi-particles at the other junction, the latter consists of four steps illustrate in Fig. below) that involve a Cooper pair tunneling at each of the junctions an a quasi-particle tunneling at each of the junctions. The transport properties of the SSET couple to an NR close to the JQP resonance have been analyze in a recent theoretical work. 4 Here, we focus on the analysis of the same couple quantum system at the DJQP resonance. Since the JQP is a oneimensional resonance in the parameter space spanne by V G an V an the DJQP is a zero-imensional resonance in the same parameter space, all action an backaction effects close to the DJQP resonance are much more pronounce than close to the JQP resonance. This is of crucial importance if one wants to manipulate the state of the NR by measurement of the SSET etector because, in experiments, the typical coupling between the two quantum systems turns out to be rather weak. We analyze how the NR can be coole below the temperature of the external heat bath an how it can be brought into a non-thermal) riven state at the DJQP resonance. Uner certain conitions, we fin signatures of bistable solutions of the couple quantum system of NR an SSET. It is of particular interest an experimental relevance, to know how a successful cooling of the NR or the preparation of a riven state can be observe in transport properties of the SSET such as its current or current noise. We show that there is a one-to-one corresponence between interesting state preparations of the NR an the transport properties of the SSET. This provies a powerful an feasible tool to initialize an manipulate NR quantum states by measurement. The article is organize as follows. In section II,

2 V L x islan V G V R V N The left an right leas are connecte to the central islan by quasiparticle tunneling an Cooper pair tunneling. Denoting by φ α the superconucting phase ifference at the junction α = L, R, we use the following quasiparticle tunneling term H T,qp = e iφα/ T kq c αkσ c Iqσ + h. c., 3) α=l,r k,q,σ FIG. : Schematic setup of the SSET-resonator system: Two superconucting leas at voltages V L an V R are couple by tunnel junctions to a superconucting islan. Its chemical potential can be tune by a gate voltage V G. A nearby nanomechanical oscillator acts as an x-epenent gate. we present the moel for the couple quantum system of NR an SSET, iscuss the ifferent approximation schemes of the analytical solutions as well as the calculation scheme behin the exact numerical solution of the unerlying master equation. Then, in section III, we analyze the oscillator properties by means of the ifferent methos, ientifying interesting quantum states of the NR ue to its coupling to the SSET. Subsequently, in section IV, we iscuss the current of the SSET etector an, in section V, the charge an current noise. It turns out that the combination of the two transport properties is sufficient to clearly ientify a successful cooling or riven-state preparation of the oscillator. Finally, we present our conclusions in section VI. Details of the calculations are containe in the Appenices. II. MODEL The system uner investigation consists of a superconucting single-electron transistor SSET) which is capacitively couple to a nanomechanical resonator NR) as shown schematically in Fig.. The total Hamiltonian of the system reas H = H L + H R + H I + H T + H C + H N + H N,I. ) The first three terms H L,R,I are stanar BCS Hamiltonians an escribe two superconucting leas left an right) an a superconucting islan, H α = k,σ ǫ αkσ c αkσ c αkσ. ) Here, c αkσ are annihilation operators for quasiparticles of momentum k an spin σ in the system α α = L, R, I). The ispersion relation ǫ αkσ accounts for the superconucting gap of with which we assume to be equal for the three systems. The chemical potentials in the left an right leas are etermine by the applie bias voltage V = V L V R, while the islan chemical potential can be tune by applying a gate voltage V G see Fig. ). where T kq are the tunneling amplitues which can be use to calculate 4 the quasiparticle tunneling rates Γ L,R. Cooper pair tunneling is accounte for by the term H T,CP = J α cosφ α, 4) α=l,r where J α are the Josephson energies of the two junctions. Hence, the total tunneling Hamiltonian is given by H T = H T,qp + H T,CP. The final ingreient for the SSET Hamiltonian is the Coulomb energy of the islan. If we enote by n L an n R the number of electrons that have tunnele from the islan to the left an right lea, respectively, then n = n L n R is the excess number of electrons on the islan. The charging term can be written as H C = E C n + n ) + ev n R, 5) where E C is the charging energy an n can be controlle by the gate voltage see Appenix A). In terms of the capacitances of the two junctions C L,R, the gate C G an the resonator C N, the charging energy is given by E C = e /C Σ ), where C Σ = C L + C R + C G + C N is the total capacitance. Next, we focus on the coupling of the SSET to the NR. The latter can be regare as a harmonic oscillator of frequency Ω an mass M an is therefore escribe by the Hamiltonian H N = Ω n osc + ) = p M + MΩ x. 6) The NR is hel on a constant voltage V N an hence acts on the SSET as an aitional gate with an x-epenent capacitance C N x). Therefore, the presence of the NR moifies the charging term H C. Expaning the contribution for small isplacements x an retaining only the lowest orer, one fins that the coupling between SSET an NR is given by H N,I = Anx, 7) where the coupling constant A epens in a non-trivial way on the voltages an capacitances of the system an can be regare as an effective parameter see Appenix A). Note that this expansion is only vali for isplacements x which are small compare to the istance between the SSET an the NR, i.e. x/. Upon continuing the expansion, one encounters terms proportional to n an to x which will be neglecte here.

3 3 V/ V/ V/ V/ V/ V/ V/ V/ FIG. : Illustration of the DJQP cycle: i) Cooper pair tunneling through the left junction, ii) quasiparticle tunneling through the right junction, iii) Cooper pair tunneling through the right junction an iv) quasiparticle tunneling through the left junction. Due to the complex structure of the full Hamiltonian ) one shoul not hope for an exact solution in all regimes. Instea, we will make several assumptions which will enable us to investigate the transport properties of this system at a particular point in the parameter space. First, we will briefly review the transport properties of the bare SSET without coupling to the NR. While the capacitances, Josephson energies an quasiparticle tunneling rates are essentially etermine by the experimental setup, the most important tunable parameters are the bias voltage V an the gate voltage V G. The transport properties of the SSET are then etermine by how these voltages are relate to the superconucting gap an the charging energy E C. For high bias voltages ev > 4, the ifference in chemical potentials allows quasiparticles on both junctions to overcome the superconucting gap an a quasiparticle current can flow. But even for lower bias voltages, one observes a finite current at certain values of the gate voltage. A possible mechanism is the Josephson-quasiparticle JQP) resonance which is a cyclic process that starts with the tunneling of a Cooper pair on one of the junctions followe by two subsequent quasiparticle tunneling events on the other junction. 38,4 This process is possible above a lower bias voltage threshol, ev > + E C. For even lower bias voltages, isolate current resonances can be observe which are ue to the onset of the ouble Josephson quasiparticle DJQP) resonance. A schematic picture of this process is shown in Fig.. It starts with a Cooper pair tunneling across, say, the left junction. Next, a quasiparticle tunnels out through the right junction, followe by a Cooper pair. Finally, after a quasiparticle tunnels through the left junction, the initial system state is reache again. This process is energetically allowe only in a restricte parameter regime: Cooper pair tunneling is only possible if the chemical potentials of the lea an the islan taking into account the Coulomb energy) are on resonance while quasiparticle tunneling requires a ifference in chemical potentials sufficient to overcome the superconucting gap. For the DJQP process, it is easy to show that the resonances occur at bias voltages ev = E C an half-integer islan charges n. The parameter regime which we investigate is therefore characterize by a charging energy E C, a superconucting gap an a bias voltage V which are of the same orer of magnitue. Roughly speaking, these energy scales are very large compare to the quasiparticle tunneling rates Γ L,R, the Josephson energies J L,R an the oscillator energy Ω. A. Derivation of a Liouville equation Due to the small tunneling rates, only sequential tunneling will contribute to the transport whereas higherorer cotunneling) processes are suppresse. This suggests escribing the system by a master equation in the Born-Markov approximation. For this purpose, we treat the BCS Hamiltonians H L + H R + H I as a fermionic bath for the remaining system. Then, system an bath are only couple by the quasiparticle Hamiltonian H T,qp. Using the Born approximation correspons to isregaring cotunneling processes while the Markov approximation is vali as long as there is a separation of time scales between the system an the bath egrees of freeom. Introucing the system an bath Hamiltonians H S = H C + H T,CP + H N + H N,I, 8) H B = H L + H R + H I, 9) an using the Born-Markov approximation leas to the following master equation for the reuce ensity matrix ρt) of the system, ρt) = Lρt) ) = i [H S, ρt)] τ Tr B [H T,qp, [H T,qp τ), ρt) ρ B ]], where ρ B is the bath ensity matrix. The time epenence of the H T,qp operator is governe by the Hamiltonian H S + H B. The ensity matrix ρ contains information only about the charge an the oscillator egrees of freeom an can, for example, be written in the basis n, n R, x of islan charge states n, the amount of charge n R which has tunnele through the right junction, an the oscillator coorinate x. This approach allows the calculation of the transport properties of the system via charge counting. 43,44 In orer to investigate the transport at the DJQP resonance, it is sufficient to consier a finite number of basis states for the islan charge n. As a single DJQP cycle involves four charge states, we can restrict the basis to the states,,, an which significantly reuces the complexity of

4 4 the problem, since it is thus sufficient to stuy a reuce ensity matrix as escribe in section II C. This choice of charge states correspons to n = /. As long as one is only intereste in oscillator properties or the current through the SSET, the n R states can be trace out, an an effective master equation acting on the Hilbert space of islan charge an oscillator position, spanne by the states n, x, can be obtaine. On the other han, for the calculation of the current noise, the n R egree of freeom has to be taken into account explicitly, as will be explaine later on. For now, we procee with the n R -inepenent case. As in the case of the JQP, 4 the Liouvillian obtaine from Eq. ) can be written as a sum of three contributions L = L HS + L qp + L CL, ) where L HS governs the coherent evolution of the system, L qp is a issipative term ue to quasiparticle tunneling an L CL is a Caleira-Leggett type contribution introuce to moel the coupling of the harmonic oscillator to a finite-temperature environment. Explicitly, L HS ρ = i [H S, ρ], ) L qp ρ = Γ Lˆp, ρˆp, Γ L {ˆp,, ρ} 3) + Γ Rˆp, ρˆp, Γ R {ˆp,, ρ}, L CL ρ = D [x, [x, ρ]] iγ extm [x, {v, ρ}], 4) where ˆp kj = j k acts on the charge states of the islan an is utilize here to escribe the quasiparticle tunneling event that changes the charge state from k to j. The energy epenence of the quasiparticle tunneling rates Γ L an Γ R is weak an will therefore be neglecte in the following. The iffusion constant D an external amping rate γ ext are relate via a fluctuation-issipation relation ) Ω Ω D = Mγ ext coth k B T B = Mγ ext Ω n osc + ). 5) For Ω k B T B, the iffusion constant can be approximate by D = Mγ ext k B T B. In the general case, it is neither possible to calculate exactly the steay-state properties nor the transport properties of the couple system using the Liouville superoperator ). Thus, approximation schemes must be employe. In the following two subsections, we escribe in etail the two complementary approximation schemes we use to stuy the couple SSET-oscillator system. B. Mean-fiel approach Physical quantities can be calculate by evaluating the matrix elements of the ensity matrix ρt). It turns out that oscillator properties an the average current can be written in terms of expectation values of the form x n v mˆp kj = Tr osc k, n R x n v m ρt) j, n R + k j, n R 6) where x an v are the position an velocity operators of the oscillator an ˆp kj = j k. The trace over the oscillator egrees of freeom Tr osc will be use in the position basis Tr osc ) = x x x as well as in the phonon number basis where Tr osc ) = n n osc= osc n osc. For the uncouple SSET tune closely to the DJQP resonance, the average current can be calculate straightforwarly, as taking the matrix elements of the master equation ) leas to a close set of equations. 43 If the NR is inclue, however, the coupling terms will lea to equations involving matrix elements of the form xˆp kj. Calculating their time evolution leas to ever higher-orer terms of the form x n v mˆp kj, so that the set of ifferential equations never closes. Hence, a truncation scheme is neee. A stanar route is to truncate the system of equations by assuming a vanishing nth-orer cumulant x nˆp kj. This allows one to rewrite nth-orer expectation values in terms of expectation values of orer n an hence to arrive at a close, albeit non-linear, set of equations. We use an compare these approximations for n = which we call thermal-oscillator approximation) an n = Gaussian approximation). These two levels of approximation are relate to what was calle mean an mean in Ref. [4]. In orer to give an estimate of the physical quality of the truncation scheme, we also compare our results to exact numerical calculations. Whereas the expectation values of the form 6) are sufficient for the calculation of the oscillator properties an the SSET current, the calculation of the noise requires a slightly extene approach. In orer to keep track of the transfere charge, one has to investigate the ynamics of n R -resolve expectation values. It turns out see Appenix E) that the noise can be rewritten in terms of expectation values of the operators ˆp n R nr kj = j k n R n R. 7) Note that only elements which conserve the number of charges, j + n R = k + n R, are finite. An analogous truncation scheme can be applie to expectation values containing these operators. Similar approaches have been use extensively to escribe nanoelectromechanical systems. 8,4,44 48 C. Numerical solution of the Liouville equation To complement the analytical mean-fiel approach escribe in the last subsection, we also use a numerical

5 5 approach to stuy the properties of the NR couple to an SSET near the DJQP resonance. First, we present the approach taken for the calculation of the current an the oscillator properties, where the n R egree of freeom plays no role. Subsequently, we will emonstrate how to exten this approach for the noise calculation, where n R has to be taken into account. To calculate the current an the oscillator properties, we write the ensity matrix in the n, n osc basis, with n osc being the phonon quantum number of the oscillator an n the charge of the SSET. The spectrum of the harmonic oscillator is naturally not finite, so we nee to truncate it an consier only its N max lowest energy eigenstates. To escribe the DJQP cycle, the reuce ensity matrix ρ is of imension 4N max 4N max ). The Liouville superoperator L is a two-sie operator, in the sense that it acts both from the left an the right of the ensity matrix [cf. Eq. )]. It can be transforme to a single-sie operator using a property of the matrix vectorization operation: the vectorize form of a prouct of three 4N max 4N max ) matrices A,B,C can be written as a single prouct of an 6Nmax 6N max) matrix with an 6Nmax ) vector via the relation vecabc) = C T A)vecB), where the superscript T enotes the matrix transposition an a Kronecker prouct. 69 The matrix representation of the Liouville superoperator is therefore of orer 6Nmax 6N max ). To illustrate how the aforementione vector ientity can be use, we apply it to the coherent evolution contribution to the Liouville equation [Eq. )]. In this case, we fin L HS ρ = i [H S, ρ], vecl HS ρ) = i I HS + H T S I) vecρ), 8) where the matrix representation of the ientity matrix I an of the system Hamiltonian H S is of orer 4N max 4N max ). To fin the vectorize form of the stationary ensity matrix ρ stat, efine from Lρ stat =, we calculate the null-space of the Liouville matrix. Using the normalization conition Tr[ρ stat ] =, the stationary ensity matrix ρ stat can be etermine uniquely. The ba scaling ON 4 max) of the Liouvillian size with the truncation point in the oscillator spectrum makes the numerical eigenvalue problem very challenging. Luckily, in this problem the Liouville matrix isplays a high sparsity egree, an sparse eigensolvers can be use. Our implementation uses the shift-invert moe of the ARPACK 49 eigensolver in combination with the PARDISO 5,5 linear solver to compute the first few 5) eigenvalues of L with the lowest magnitue λ ) as well as the associate eigenvectors. The calculate magnitue of the smallest eigenvalue can be use to verify the valiity of the truncation scheme: when enough Fock states are kept we fin λ < 5 which is below the esire precision limit of. To improve the spee of the calculation an, more importantly, to increase numerical accuracy, we o not nee to explicitly solve for those matrix elements of ρ stat which, ue to the consiere Hamiltonian, have to be zero. For example, coherence can only be create between two charge states k an j if k j =, since only these pairs of states are couple by Josephson tunneling. Therefore, all ensity-matrix elements k ρ j where k j is o are zero. Using this argument, the size of the Liouville matrix can be reuce to 8N max 8N max ). The use of sparse solvers also minimizes the require memory for the calculation of the eigenvalues, allowing problems of relatively large size N max 5) to be solve on a esktop computer. Also, we note that, contrary to what was iscusse in Ref. [5], no manual preconitioning was neee to achieve high numerical accuracy. To allow for the numerical approach to be use in the riving regime, where we expect the average energy of the oscillator to be relatively high, we ha to make a supplementary approximation. In this case, we assume that coherence coul evelop only between states of the oscillator that are not too far away in energy from each other, setting n osc ρ n osc = for n osc n osc 6. This allowe for N max to be set as high as 75 on a stanar workstation. Moreover, in the cases where it was possible to compare irectly the results of the calculation with an without this last approximation, we foun that they were ientical within our numerical accuracy). While this approach is viable for the calculation of the oscillator properties an the current, it fails to keep track of the tunnele charge n R an thus cannot be use to calculate the current noise. A straightforwar inclusion of the n R states is numerically impossible as the corresponing Hilbert space is of infinite imension. However, this problem can be circumvente by consiering the n R -resolve ensity matrices ρ nr) n R Z), which are submatrices of the complete ensity matrix ρ, whose entries are efine by k ρ nr) j = k, n R ρ j, n R, 9) where the relation between n R an n R is given by charge conservation. At the DJQP resonance, we have n R = n R for k, j) =, ), n R = n R + for k, j) =, ) are n R = n R otherwise. 43 Note that we i not write out the oscillator egree of freeom explicitly in this matrix. Calculating the time evolution of these matrix elements accoring to Eq. ), one fins t ρnr) = [ L I qp I + CP ] I CP ρ n R) ) + I qp ρ nr ) + I + CP ρnr+) + I CP ρnr ). As expecte, the tunneling leas to a coupling between ensity matrices of ifferent n R. It is prouce by the current superoperators escribing the quasiparticle an

6 6 the Cooper pair tunneling, which are efine as I qp ρ = Γ R ρ, ) I + CP ρ = ij R I CP ρ = ij R [ ρ + ρ ], [ ρ + ρ ]. It is important to realize that by writing the Liouville equation in terms of n R -resolve ensity matrices an current superoperators, we have achieve a escription of the system in terms of only the n, n osc states again. This, however, comes at the price of having to eal with an infinite number of ensity matrices, ρ nr). Still, following the approach of Ref. [53] it will turn out that convenient expressions for the current an the noise can be formulate in terms of these current superoperators. III. OSCILLATOR PROPERTIES As mentione before we treat the NR as a harmonic oscillator an we use the master equation to investigate the time evolution of the mean isplacement x = Tr nr Tr n Tr osc [ρt)x], ) an of the velocity v, corresponingly. Here, Tr n ) = n= n n enotes the trace over the islan charge, while Tr nr ) = n n R= R n R traces over the tunnele charge. Likewise, the master equation will allow us to calculate expectation values of higher orer like x an v which are require for the calculation of the oscillator energy. For a linear coupling of the NR to the SSET as in Eq. 7), we fin the following equations escribing the time evolution of the oscillator couple to the SSET x = v, t 3) t v = Ω x γ ext v + A n, M 4) where n = k k ˆp kk is the expectation value of the islan occupation an γ ext accounts for the external amping. The stationary limit, where t v = t x =, can be regare as the long-time limit when the oscillatory behavior has been ampe by the thermal bath an thus v = an x = A/MΩ ) n. If the coupling A to the SSET is zero, the oscillator stays in its equilibrium position at x =. For finite coupling, ue to the electromagnetic repulsion the NR equilibrates in a position x proportional to the coupling an the charge n on the islan. Note that the influence of the SSET on the NR is of the first orer in the coupling A. This regime has alreay been stuie in some etail 8,9 an it was shown that the SSET acts as an effective thermal bath for the NR. As we will illustrate further on, the signature of the NR in the transport properties of the SSET is of secon orer in the coupling an is clearly visible in the current an the noise properties of the SSET. To stuy the influence of the NR on the SSET we introuce imensionless quantities which are normalize to motional quanta of the oscillator. Using the frequency Ω an the harmonic oscillator length, x = MΩ, 5) as units we efine x = x/x, t = Ωt an ṽ = v/ωx, i.e. we normalize all variables with respect to oscillator quantities. This allows an easier comparison with the experiment where, for example, the bias voltage can be varie at constant coupling. Consequently, the equations of motion can be rewritten as x = ṽ, t 6) t ṽ = x γ ext ṽ + à n. 7) where γ ext = γ ext /Ω an à = x A/ Ω. Not only is the equilibrium position of the resonator shifte by the coupling to the SSET, but also the cumulants of the position an velocity of the NR, i.e. x = x x, are influence by this coupling: t x = { x, ṽ} +, 8) t { x, ṽ} + = ṽ x γ ext { x, ṽ} + + 4à n x n x ), 9) t ṽ = { x, ṽ} + γ ext ṽ + 4 γ ext TB + 4à n ṽ n ṽ ), 3) where T B = k B T B / Ω an {, } + enotes the anticommutator. In the stationary limit this leas to { x, ṽ} + =, 3) ṽ = T B + à n ṽ n ṽ )/ γ ext, 3) x = ṽ + à n x n x ). 33) To lowest linear) orer, which we refer to as the thermal-oscillator approximation, we assume that nv = n v. This is ientical to assuming that the correlations between n an v vanish, i.e. n v =. Consequently, the fluctuations of the harmonic oscillator are not influence by the SSET such that the virial theorem ṽ = x an the equipartition theorem ṽ = T B are fulfille in the high-temperature limit T B. The resonator is thus in a thermal state etermine only by T B an γ ext. In the thermal-oscillator approximation analytic expressions for the current an noise in the SSET can be erive an will be iscusse in the upcoming sections.

7 7 The thermal-oscillator approximation is justifie for weak coupling between the SSET an the NR, but fails for stronger coupling, since the observables of the oscillator become entangle with the charge state of the SSET. As was alreay observe before, 8,9 an increase coupling can rive the oscillator to a non-thermal state characterize by a finite nv n v, where the virial an equipartition theorems no longer hol. In orer to investigate this regime, we have to go to the next orer in our approximation which means taking the fluctuations of nx into account, but assuming all higher-orer cumulants to vanish, e.g. nx =. This will be referre to as the Gaussian approximation since for a Gaussian istribution all cumulants x n for n > are zero an the resonator is fully escribe by the two lowest moments. Uner this assumption, we can express expectation values of the form x ˆp kj as proucts of the lower-orer expectation values xˆp kj, x, x an ˆp kj. While this approach leas to a close set of ifferential equations, the set will now be non-linear an has to be solve numerically. In principle, this approximation scheme can be continue to even higher orers. 4 However, since the Gaussian approximation works well for the low-coupling regime we are intereste in, we o not go beyon it. Ultimately, for even stronger coupling, the linear coupling between the SSET an the NR itself becomes questionable. In orer to investigate the oscillator state in more etail, we stuy the energy E = MΩ x + Mv, in imensionless quantities E = x + ṽ ) = n osc + Ω 4. 34) Previous work 9,46 has focuse on the fluctuations of the number of charges on the islan, n, which in linear response can be escribe by an effective amping an effective temperature. We will iscuss this approach in more etail in the context of the charge noise. For an ientification of the oscillator state, though, we choose a ifferent route an investigate the energy of the NR. In the stationary limit, using Eqs. 3) an 33), we fin for the energy E Ω = T B + à nṽ + γ ext n x γ ext + 4 x. 35) A finite x provies aitional potential energy, but the contribution is small, since x = 4à n. Therefore, it is the correlations of the entangle SSET-NR system containe in the secon term, which have the potential to rive the system out of a thermal state. The results for a calculation of the energy in the Gaussian approximation for a typical, experimentally relevant set of parameters are shown in Figs. 3 an 4 where we isplay the oscillator energy as a function of gate voltage V G an the bias voltage V measure away from the resonance position. As the DJQP cycle contains two Cooper pair tunneling events, there are two resonance conitions ev/ hω ev G / hω FIG. 3: Oscillator energy in units of Ω in the Gaussian approximation as a function of the gate voltage ev G/ Ω an the bias voltage ev/ Ω where,) enotes the resonance. The parameters use are Γ L = Γ R =, J L = J R =, γ ext = 4, TB =.5 an à =.. In the re-etune area V <, V < V G < V ), cooling below the bath temperature is visible blue region). Driving can be observe in the blue-etune case V >, V < V G < V ). The highest energies are obtaine in the yellow region. ev/ hω ev G / hω FIG. 4: Oscillator energy in the Gaussian approximation in units of Ω for the Gaussian approximation as a function of the gate voltage ev G/ Ω an the bias voltage ev/ Ω for increase coupling à =.3. Cooling an riving effects are increase as compare to Fig. 3. In the black area two stable an one unstable solution are foun, i.e. the system becomes bistable. The area grows for stronger coupling. which have to be met an which can be controlle by ajusting the bias an gate voltages. The physical picture can be explaine most clearly if we assume V G =, which correspons to a vertical cut in Fig. 3. If the system is blue-etune from a resonance V > ), the tunneling Cooper pairs transfer a part of their energy to the oscillator in orer to be able to tunnel. This leas to riving of the oscillator. On the contrary, for a re-etune resonance V < ), the Cooper pairs can absorb energy from the oscillator, leaing to cooling. A similar result was alreay foun using a linear-response approach in Ref. [9]. In the regime where both resonances involve in the DJQP cycle are blue-etune V >, V < V G < V ),

8 8 Pnosc).. e-6 V = V = V = V = V =3 V =4 V =5 V =6 V =7 à e-8.5 e- e n osc FIG. 5: Distribution Pn osc) of the oscillator phonon number for ifferent values of the bias voltage ev/ Ω = {,,,..., 7} from left to right along the x-axis. The parameters use for this plot are à =., ΓL = Γ R =, J L = J R =.5, γ ext =. an T B = 3. For negative V an small positive V we fin an exponential ecay corresponing to a thermal state. For larger V > the istribution evelops a peak at n which inicates a riven state. we fin a particularly strong riving of the oscillator. In the white regions of Fig. 4, energies of the orer 3 Ω epening on the system parameters) are reache even for rather small coupling of the orer Ã.. The numerical solution of the Liouville equation reveals moreover that the resulting oscillator state is highly nonthermal, i.e. the istribution function of oscillator states Pn osc ) strongly eviates from a Boltzmann istribution. This is calculate in Fig. 5 using the numerical approach for ifferent values of ev/ Ω. We fin an exponential ecay for V corresponing to the hightemperature limit of the Boltzmann istribution an a tren towars a riven state for V >. In the regime where both resonances are re-etune V <, V < V G < V ), we fin a cooling of the oscillator to temperatures well below the bath temperature. This shows up in Figs. 3 an 4 as the little triangularshape regions below the center, where the oscillator energy rops below the value corresponing to the bath temperature. Due to the non-linearity of the master equation, more than one physical solution may emerge an we fin that this is inee the case in the sector where the NR is strongly riven. An analogous effect was foun previously for the same system at the JQP cycle 4 an for a more general class of systems We fin that generally, the response of the system close to a DJQP resonance is much more pronounce than at the JQP in the sense that quantitatively similar effects may be observe at much smaller values of the coupling. This agrees with the preiction 9 that the backaction effects at the DJQP excee those of the JQP by a factor Γ/J) 4. Therefore the DJQP is favorable from the experimental point of ev/ hω FIG. 6: Oscillator energy in arbitrary units as a function of bias voltage ev/ Ω an coupling à calculate in the Gaussian approximation. For V <, backaction leas to cooling of the oscillator blue). On the contrary, strong riving re/yellow) can be observe for V >. Above a critical coupling ÃV ), the system enters a bistable region black). The parameters are the same as in Fig. 3 an ev G/ Ω =. view since achieving a strong coupling is challenging. A plot of the location of the bistabilities foun in the Gaussian approximation as a function of the bias voltage V an the coupling strength à is shown in Fig. 6. Again, cooling of the oscillator is seen in the blue regions for V <, whereas riving happens for V > as is epicte by re regions. Towars stronger coupling, both effects increase in magnitue. Two stable solutions appear only for a blue-etune SSET an the voltage range where such an effect is visible grows with increase coupling. When increasing the coupling for a given voltage V > which correspons to a vertical cut in Fig. 6), the system will evolve from a thermal state via the bistable state to a single riven state. On the contrary, an increase in voltage corresponing to a horizontal cut) carries the system from a thermal state to a riven state, then into the bistable region. Beyon the bistable region, the system will fall back to the thermal state. Note that the effect of riving is much stronger than the cooling of the NR cf. Figs. 3 an 4). We have confirme the existence of bistability using the numerical approach by explicitly calculating the complete probability istribution Px) [results not shown explicitly]. In a thermal state, this istribution shows a single peak at x =. In a riven state, on the contrary, two symmetric peaks at finite values x appear. In the bistable regime, the system switches between these two states, leaing to a istribution Px) that shows three peaks, one thermal) at the origin an two sie-peaks riven) at x >. From these stuies of the bistable regime using the numerical approach, we notice that the parameter range in which the system exhibits bistability is smaller than the one obtaine via the mean-fiel solution. The structure of the bistable region coul be better preicte using the analytical approach by incluing higher than secon orer cumulants, i.e. extening

9 9 the analysis beyon the Gaussian approximation. In the following sections, we will show that these ifferent states of the NR also manifest themselves in the transport properties of the SSET, i.e. the current an the current noise. IV. CURRENT PROPERTIES We showe in the previous section that the coupling of an SSET to an NR can rive the oscillator into a nonthermal state an effect in cooling, potentially even cooling own close to the groun state. 9, In the following, we will stuy if an how it is possible to measure signatures of the resonator state in the current an current noise characteristics of the SSET close to the DJQP resonance. The number of electrons that have left the islan to the right lea, n R, is proportional to the transporte charge an therefore etermines the current flow. Hence, the expectation value of the current is given by I = e) t ˆn R = e)tr{ ρt) ˆn R }. 36) Without loss of generality, we chose to measure the current across the right junction. In the stationary limit, the total current is conserve such that the currents across the left an right junctions are equal. In each DJQP cycle, two tunneling events take place at the right junction, see Fig. : the transfer of a quasiparticle which takes the islan from charge state to the state. Subsequently, a Cooper pair tunnels to the right lea an leaves the islan in the state. Two processes involving only changes in n L an n, which therefore o not contribute to I, close the cycle in which 3 electrons in total have been transporte through the islan. Equivalently, in the stationary state the expectation value of the current I can be written using the superoperator formalism. From Eq. ), one fins I = e)tr n Tr osc I total ρ stat ) 37) where I total = I qp I + CP + I CP is the superoperator escribing the total current. Equation 37) is use in this form in the numerical routine. For the analytic mean-fiel approximations we split the total current into two terms, I = I ND + I D, corresponing to a contribution from the tunnele Cooper pair the non-issipative current) I ND, an a contribution from the quasiparticle tunneling event the issipative part) I D. For these two contributions we can write own the exact expressions for the issipative I D = e)tr n Tr osc I qp ρ stat ), 38a) = e)γ R ˆp, 38b) an non-issipative part I ND = e)tr n Tr osc [I CP I+ CP ]ρ stat), 39a) = e)ij R ˆp, ˆp, ). 39b) In the thermal-oscillator approximation, these expectation values can be calculate by solving for the corresponing elements of the ensity matrix, as shown in etail in the Appenix B. We fin that the vector of all finite ˆp kj, p, is given by p = i J L M c where M is the evolution matrix of the SSET system containing all the system parameters, Eq. B3), an the constant c is the inhomogeneous part of the master equation ue to the normalization of the ensity matrix k ˆp kk =. Using this result the stationary current for the DJQP cycle can be written as I = 3 [ e)ω + + Γ R Γ L γ L x) + ]. 4) γ R x) The inverse of the tunneling rates for quasiparticles an Cooper pairs, Γ L,R an γ L,R, respectively, can be interprete as effective resistances for these processes. Then, Eq. 4) is reminiscent of the current through a series of resistors, where the largest resistance etermines the behavior. The Cooper pair tunneling rates are given by 38 J γ L x) = Γ L R, 4) Γ R /) + ǫ, x) J γ R x) = Γ R L. 4) Γ L /) + ǫ, x) where ǫ k,j enotes the ifference in energy between the charge states k an j an thus measures the etuning from the DJQP resonance. The renormalize tunneling rates of the SSET are efine by Γ α = Γ α /Ω an J α = J α / Ω. If the Cooper pair tunneling, say, to the right lea is resonant, i.e. ǫ, =, the rate γ R reaches a maximum at the value γ R = 8 J R / Γ L = J R /Γ L)/Ω. It ecays like a Lorentzian away from the resonance. Expressions for the current in less general form are for example erive for ǫ jk = in Ref. [43] an for ǫ, = ǫ, in Ref. [59]. Due to the capacitive coupling of the SSET to the NR, the resonance is shifte compare to the uncouple case in the thermal approximation. We fin in imensionless units erivation given in Appenix B) ǫ, x) = ev G Ω + ev à x, Ω 43) ǫ, x) = ev G Ω ev à x. Ω 44) where ev/ Ω an ev G / Ω are the relative bias an gate voltages measure from the values at the DJQP resonance. The SSET is affecte only if the average position of the NR is finite, i.e. x. This shift in the equilibrium position of the NR effectively correspons, from the point of view of the SSET, to a change in V G an will therefore be referre to as an effective backgate behavior later on. This effect is of secon orer in the coupling A since we observe in the previous section that x is linear in A.

10 I Ã) I ).. ev/ hω IV,Ã=) ev/ hω -.4 Ã=.5 Ã=.75 Ã=. FIG. 7: Difference of the inverse current / I Ã) / I à = ) versus bias voltage ev/ Ω for various coupling strengths Ã. From the intersection with the voltage axis, the shift à x of the oscillator position can be rea out. Lines correspon to the numerical analysis an agree well with the Gaussian approximation points) in this parameter regime. Inset: Average current I through the SSET for the parameters Γ L = Γ R =, JL = J R =, γ ext = 4, an T B = 3. Note that the average isplacement of the NR oscillation in the stationary limit, x = à n = à + ) I Γ R γ R x) 3 e)ω, 45) is etermine by the Cooper pair tunneling rate γ R in aition to the quasiparticle tunneling rate Γ R. This is in contrast to the JQP cycle where it is only the necessarily small / Γ R which etermines the isplacement. Since the rates an the current are implicitly epenent on x via the Cooper pair tunneling rate, Eq. 45) is a selfconsistency equation. Going beyon the thermal-oscillator approximation, we use again the truncate master equation an the numerical approach to calculate the current via the general Eqs. 38) an 39). In orer to assess the quality of the Gaussian approximation, we first compare the results of the two approaches for low coupling strength. The result is shown in Fig. 7, where we plot the ifference in the inverse current between the weakly couple an the uncouple system. The results of the Gaussian approximation an the numerically evaluate lines are in excellent agreement. The Lorentzian lineshape of the current inset of Fig. 7) is preserve in case of the weak coupling. The change in the average current ue to the coupling to the oscillator can be most transparently illustrate by plotting the ifference of the inverse currents in the couple an the uncouple cases, see Fig. 7. As obvious from Eq. 4) the inverse current / I is given by the sum of rates involving the various transport processes. In the thermal oscillator approximation, the function / I Ã) / I à = ) changes sign as a function of ev/ Ω at a position which is proportional to à x as IV) Ã=. Ã= ev/ hω FIG. 8: Average current I versus bias voltage for à =. an à =.3, T B =.5, further parameters as in Fig. 7. For increase coupling the Lorentzian peak becomes istorte an two stable solutions emerge which result in two stable values for the current. is shown in Eq. B5). We expect this sign change to be the most feasible way to experimentally observe the influence of the NR on the SSET current an to investigate quantitatively the coupling strength using only the average current. The bistability of the oscillator states, which was alreay iscusse in the previous section, also manifests itself in the current through the SSET. For increase coupling, we fin that the equation of motion erive within the Gaussian approximation has up to three solutions, of which two correspon to stable currents. The resulting current-voltage characteristic is shown in Fig. 8. In the center, the usual DJQP resonance is clearly visible. While for very low coupling Ã., the current still follows approximately a Lorentzian, strong eviations become visible alreay for à =.3. In an experimental setup, we o not expect two stable currents to be istinguishable. Inee, a current measurement of the SSET-resonator system will yiel a weighte average 64 because ecoherence effects will lea to a switching between the two stable configurations on time scales large compare to the oscillation perio but small compare to the measurement resolution. 6 These switching rates can easily be inferre from a comparison of the measure current to the two stable values an from the current noise, as we shall show in section V. V. NOISE PROPERTIES A. Charge noise In the past, linear-response arguments 9 have been use to support the iea that a generic etector acts on the resonator in the same way as a secon thermal bath, an that the backaction on the resonator cause by charge

11 fluctuations on the islan can be escribe essentially by two parameters, a amping rate γ eff an an effective temperature k B T eff. These are relate to the chargefluctuation spectrum via S n ω) = t e iωt nt)n). 46) γ eff ω) = S nω) S n ω) à ω T eff ω) = S nω) + S n ω) à γ eff ω), 47), 48) in the limit k B T eff ω) ω, where ω is given in units of Ω, Teff = k B T eff / Ω an γ eff = γ eff /Ω. Since these expressions follow from a linear-response calculation, both effective quantities are written in terms of the bare charge-noise, calculate in the absence of coupling with the oscillator. Investigating the retare an avance absorption an emission) contribution of the charge correlation explicitly, we can erive an analytic expressions for S n ω) for the uncouple SSET see Appenix D) S n ω) S n ω) = i n [ ω + M ] K p, 49) ω S n ω) + S n ω) = n M [ ω + M ] K+ n ) p, 5) where M enotes again the evolution matrix of the SSET system, n is efine such that n p = n an K ± enote coupling matrices given explicitly in Eqs. C5) an C6). Note that n = n n = n K + p n an K acts only on the off-iagonal elements of ˆp kj, i.e. the Cooper pair tunneling terms. The self-consistency equation for x, that has to be solve in the Gaussian approximation, can be written as where x = γ extt B + γ SSET T SSET x, x ) γ ext + γ SSET x, x ), 5) γ SSET x, x ) = iã) 5) n γ ext γ ext + M) ) + γ ext M + M ) K p, γ SSET T SSET x, x ) = Ã) 53) n γ ext + M ) + γ ext M + M ) K+ n ) p ; We observe that the mean-fiel equation in secon orer provies the same physics as linear-response theory, i.e. γ eff at ω/ω = is of the same form as γ SSET. Since Eq. 5) is a self-consistency equation for x an not for the effective oscillator energy as in Eqs. 47) an 48) the expressions iffer by a factor of 4. The result in Eq. 5) is more accurate in the sense that the parameters of the ampe oscillator are involve: + γ ext M +M = γ ext /) ) + M + γ ext /) with a renormalize frequency of Ω r = Ω γ ext /) an aitional amping ue to γ ext /. Note that Eq. 5) is a self-consistency equation for x since p = p x, x ) an it has to be solve together with x = à n. Even if it is assume that p = p, T B ), the expression contains a correction ue to the finite quality factor of the NR. Whereas the approach escribing the etector as an effective bath prove very successful in proviing a simple physical explanation of experiments, some of its shortcomings have starte to be ientifie in recent theoretical works. 6,6 For example, it has very recently been propose 6 that the signature of the oscillator in the charge noise spectrum of a generic etector is not the one of a thermal oscillator. In the light of these finings, the calculation of the full frequency-epenent charge noise spectrum of the SSET near the DJQP in the presence of an oscillator becomes relevant, even more so since the charge-noise spectrum is an experimentally accessible quantity. As shown in Appenix D, it is possible to use the master-equation approach to erive formal expressions for S n ω), at least for weak coupling in the thermaloscillator approximation. However, it turns out that these expressions are ifficult to evaluate explicitly for stronger couplings in the Gaussian approximation). On the other han, the fully numerical approach presente in section II C can easily be aapte to allow the calculation of finite-frequency correlation functions of system as oppose to bath) operators, using the quantum regression theorem. 63,64 In the following, we therefore iscuss only the charge-noise spectrum obtaine numerically. Note that we verifie that our algorithm reprouces accurately the known charge an position fluctuation spectra in the uncouple à = ) regime. Figure 9 shows the symmetrize in frequency) chargenoise spectra, Sn symm ω) = [S n ω)+s n ω)]/, obtaine for ifferent values of the bias-voltage etuning from the DJQP resonance. The oscillator state, i.e. thermal or riven, can be etermine from the Fock space probability istribution in Fig. 5. The inset shows that the signature of the oscillator in Sn symm ω) appears prominently aroun the natural frequency of the oscillator. Away from ω Ω, the charge spectrum is only weakly affecte by the oscillator, since the coupling of the islan to the resonator changes the effective biasing conitions of the SSET. The main panel of Fig. 9 shows the evolution of the charge-noise spectra when the system is taken from the cooling region V = ) through the resonance point V = ), to the voltage regime where the state of the oscillator becomes highly non-thermal. Unsurprisingly, the overall signal aroun Ω increases ramatically when the oscillator enters the riven regime, reflecting the overall increase in the magnitue of S x ω Ω) when the oscillators energy is increase. Associate with this increase