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Quantum turnstile regime of nanoelectromechanical systems

Quantum turnstile regime of nanoelectromechanical systems R. Dragomir and V. Moldoveanu National Institute of Materials Physics, Atomistilor 405A, Magurele 077125, Romania S. Stanciu Faculty of Physics, University of Bucharest, Atomistilor 405, Magurele 077125, Romania and National Institute of Materials Physics, Atomistilor 405A, Magurele 077125, Romania B. Tanatar Department of Physics, Bilkent University, Bilkent, 06800 Ankara, Turkey The effects of a turnstile operation on the current-induced vibron dynamics in nanoelectrome- chanical systems (NEMS) are analyzed in the framework of the generalized master equation. In our simulations each turnstile cycle allows the pumping of up to two interacting electrons across a biased mesoscopic subsystem which is electrostatically coupled to the vibrational mode of a nanores- onator. The time-dependent mean vibron number is very sensitive to the turnstile driving, rapidly increasing/decreasing along the charging/discharging sequences. This sequence of heating and cool- ing cycles experienced by the nanoresonator is due to specific vibron-assisted sequential tunneling processes along a turnstile period. At the end of each charging/discharging cycle the nanoresonator is described by a linear combination of vibron-dressed states s associated to an electronic configu- ration ν. If the turnstile operation leads to complete electronic depletion the nanoresonator returns to its equilibrium position, i.e. its displacement vanishes. It turns out that a suitable bias applied on the NEMS leads to a slow but complete cooling at the end of the turnstile cycle. Our calculations show that the quantum turnstile regime switches the dynamics of the NEMS between vibron-dressed subspaces with different electronic occupation numbers. We predict that the turnstile control of the electron-vibron interaction induces measurable changes on the input and output transient currents. I. INTRODUCTION In this theoretical study we focus on the time- dependent control of the entangled electron-vibron dy- namics of a NEMS in the quantum turnstile regime. The nanoelectromechanical systems are hybrid struc- More precisely, we show that the pumping of an inte- tures in which the electrostatic interaction between vibra- ger number of electrons along a turnstile period activates tional modes and open mesoscopic systems is expected to 1 the coupling to the vibrational mode during the charging play a role down to the quantum level . To support this cycle and then renders it ineffective on the discharging idea, the sensing properties of nanoresonators (NR) in cycle when the system is fully depleted. We recall that in the presence of electronic transport have been investi- 11–13 the turnstile setup , electrons are first injected from gated in various experimental settings. the source (left) particle reservoir while the contact to For instance, singly clamped cantilevers or AFM tips the drain (right) reservoir is closed. After this charging were shown to record single-electron tunneling from back- half-period, the left/right contact closes/opens simulta- gate contacts to the excited states of quantum dots de- neously (see the sketch in Fig. 1). posited on a substrate . In another class of experiments, In most experimental investigations on NEMS, a bias a suspended carbon nanotube (CNT) with an embedded voltage continuously supplies the charge flow through the quantum dot is actuated by microwave signals and the mesoscopic system which in turn interacts with the vibra- dips of its resonance frequency are associated to single- tional mode. Then the hybrid structure evolves under electron tunneling . Besides flexural modes, the CNTs the electron-vibron coupling until a stationary transport also develop longitudinal modes with higher frequencies regime is reached. At the theoretical level, the latter is (up to few GHz). Similarly, the vibration energy ~ω of 14,15 recovered by solving rate equations or hierarchical single-molecule junctions is around few meVs . For these quantum master equations (HQME) . Also, the single- systems, refined cooling techniques were used to reach level Anderson-Holstein model provides a sound descrip- the regime ~ω ≫ k T for which the vibrations of the 5,6 tion of the essential spectral properties of NEMS via the nanoresonator must be quantized . Lang-Firsov polaron transformation. On the other hand, the implementation of nanoelec- tromechanical systems as successful devices in quantum Let us stress that recent observation of real-time vibra- 7 8 20,21 22 sensing , molecular spintronics or nano-optomechanics tions in CNTs and pump-and-probe measurements requires an accurate tuning of the underlying electron- provide a strong motivation to scrutinize the time- vibron coupling. For example, the electron-vibron cou- dependent vibron-assisted transport. Few theoretical de- pling can be switched on and off by controlling the loca- scriptions of vibron-assisted transport properties in the tion of a quantum dot (QD) along the suspended CNT presence of pumping potentials acting on the electronic 9,10 in which it is formed . system can be mentioned. The effect of a cosine-shaped arXiv:2005.10466v1 [cond-mat.mes-hall] 21 May 2020 2 where H accounts for the two components of the S,0 NEMS, i.e. the QW accommodating several interacting electrons and the vibrational mode with frequency ω as- sociated to a molecule or a nanoresonator: X X X † † † H = ε c c + V c c c c S,0 iσ iσ ijkl ′ lσ kσ iσ iσ jσ i,σ σ,σ i,j,k,l + ~ωa a. (2) Here c creates an electron with spin σ on the single- iσ particle state ψ of the electronic system with the corre- iσ FIG. 1. Schematic view of the NEMS in the turnstile regime. sponding energy ε , the second term is the two-body iσ Source (L) and drain (R) particle reservoirs with chemical Coulomb interaction within the electronic sample and potentials μ are connected to an electronic structure (e.g. L,R a is the creation operator for vibrons. The eigenstates a quantum wire - QW). The contact regions are modulated by |ν, Ni of H are products of electronic many-body con- switching functions χ - the turnstile operation corresponds S,0 L,R to periodic out-of-phase oscillations of χ . A vibrational figurations |νi with energies E of the electronic system L,R ν mode of frequency ω interacts with the electrons, d being its and N-vibron Fock states |Ni, such that H |ν, Ni = S,0 displacement w.r.t the equilibrium position. (E + N~ω)|ν, Ni. The electron-vibron coupling V ν el−vb reads as V = λ c c (a + a), (3) el−vb i iσ driving of the contact regions has been considered within iσ 23,24 i,σ the Floquet Green’s function formalism . In a very recent paper the HQME method was adapted for a time- where λ is the electron-vibron coupling strength. 25 26 dependent setting . Avriller et al. calculated the tran- We denote by E and |ϕ i the eigenvalues and eigen- ν,s ν,s sient vibron dynamics induced by a step-like coupling of functions of the hybrid system such that molecular junctions to source and drain particle reser- voirs. H |ϕ i = E |ϕ i. (4) S ν,s ν,s ν,s In the present work we rely on the generalized master Since V conserves the electronic occupation and the el−vb equation (GME) method which was previously used to 27 spin, the fully interacting states |ϕ i can still be labeled ν,s study the turnstile regime of single-molecule magnets by a many-body configuration ν and written as: and recently extended for hybrid systems such as NEMS ( ) or cavity-QD systems . The model Hamiltonian em- (ν) bodies both the electron-electron interaction within the |ϕ i = |νi ⊗ A |Ni := |νi ⊗ |s i. (5) ν,s ν sN electronic subsystem and the spin degree of freedom. We also consider turnstile operations where more than one The ν-dependent vibrational overlap |s i contains differ- electron is transferred across the system. The reduced (ν) ent states |Ni, A being the weight of the N-vibron density operator of the hybrid system is calculated nu- sN state. If |ϕ i are obtained by numerical diagonalization merically with respect to vibron-dressed basis. As we ν,s one should truncate the indices N and s at a convenient are interested in the response of the NR to the turn- (ν) upper bound N . In this case, the coefficients A define stile pumping we also calculate its associated displace- 0 sN a finite dimensional unitary matrix which approximates ment which can be, in principle, measured. Note that the exact Lang-Firsov transformation defined by the op- this quantity is mostly derived for the classical regime of erator S = (λ /~ω)c c (a − a) (see e.g., Ref. 30). nanoresonators via the Langevin equation . i iσ iσ i,σ −S The rest of the paper is organized as follows. In Sec- The exact eigenfunctions are then |ϕ i = e |ν, Ni. ν,s tion II we introduce the model and briefly recall the main Let us stress that the electron-vibron coupling con- ingredients of the GME approach. The results are pre- stants λ depend on the single-particle wavefunctions sented in Section III, Section IV being left to conclusions. ψ of the electronic subsystem. In a recent work iσ we took this dependence into account and showed that it leads to different sensing efficiencies when a singly- II. FORMALISM clamped tip is placed above the quantum wire and swept along it. In this work the position of the NR is fixed A typical NEMS setup is sketched in Fig. 1 where a and the transport involves, for simplicity, only the low- quantum wire (QW) is capacitively coupled to a nearby est spin-degenerate single-particle state whose associated nanoresonator (NR) and tunnel-coupled to source and electron-vibron coupling strength will be denoted by λ . drain leads. The closed nanoelectromechanical system It is useful to introduce the Franck-Condon factors (FC): (i.e. not connected to particle reservoirs) is described by (ν) (ν ) ′ ′ ′ the following general Hamiltonian F := hs |s ′i = A A , |n − n | = 1, νν ;ss ν ′ ν ν sN s N N=0 H = H + V , (1) (6) S S,0 el−vb 3 where n is the number of electrons corresponding to the The GME is solved numerically with respect to the many-body configuration ν. We shall see below that for vibron-dressed basis {ϕ } of the hybrid system. Let ν,s a given pair of electronic configurations {ν, ν } one gets a us stress that choosing the fully interacting basis over ′ ′ series of vibron-assisted transitions controlled by F . the ‘free’ one {|ν, Ni} allows us to calculate the matrix νν ;ss i i H t − H t S S In view of vibron-assisted transport the electronic com- ~ ~ elements of e c e which appear in the dissi- iσ ponent of NEMS is also coupled to source (L) and drain pative kernel of the leads (see Eq. (10)). Note also that (R) particle reservoirs characterized by chemical poten- in this representation the Lang-Firsov transformation of tials μ , as shown in Fig. 1. The total Hamiltonian the tunneling Hamiltonian is not needed such that H L,R therefore becomes: does not acquire an additional operator-valued exponen- X tial. By doing so one carefully takes into account the FC H(t) = H + H + H (t), (7) S l T factors which can have both positive and negative signs, l=L,R as pointed out in Ref. . The full information on the system dynamics is em- where H is the Hamiltonian of the lead l and the tunnel- bodied in the populations of various states ing Hamiltonian reads as (h.c. denotes Hermitian conju- P (t) = hϕ |ρ(t)|ϕ i. (12) gate): ν,s ν,s ν,s X X The time-dependent currents in each lead are identified (lσ) † H (t) = dqχ (t) T c c + h.c. . (8) T l iσ from the continuity equation of the charge occupation qi qlσ l=L,R i,σ Q of the system: d d The functions χ (t) simulate the turnstile modulation of Q (t) = eTr N ρ(t) = J (t) − J (t), (13) S ϕ S L R dt dt the contact barriers between the leads and the system (lσ) and T is the coupling strength associated to a pair qi ˆ where N = c c is the particle number op- S iσ iσ i,σ of single-particle states from the lead l and the central erator, Tr stands for the trace with respect to the sample. For simplicity we assume that the tunneling pro- basis {ϕ } of the hybrid system and e is the elec- ν,s (lσ) cesses are spin conserving and that T does not de- qi tron charge. The left and right transient currents pend on σ. We describe the leads as one-dimensional J are then calculated by collecting all diagonal ele- L,R semi-infinite discrete chains which feed both spin up and ments hϕ |ρ˙(t)N |ϕ i which contain the Fermi func- ν,s S ν,s down electrons to the central system. Their spectrum is tion f . The latter appears when performing the l=L,R ε = 2t cosq , where q is electronic momentum in the q L l l partial trace of the integral kernel K(t, t − s; ρ(s)) such n o lead l and t denotes the hopping energy on the leads. ′ ′ that Tr ρ c c = δ δ δ(q − q )f (ε ) Also, L L ′ ′ ′ qlσ ll σσ l q The reduced density operator (RDO) ρ of the hybrid l q l σ system obeys a generalized master equation (GME) (for a note that from the cyclic property of the trace one ˆ ˆ derivation via the Nakajima-Zwanzig projection method has Tr {[H , ρ(t)]N } = Tr {ρ(t)[N , H ]} = 0 and ϕ S S ϕ S S see e.g., Ref. 28): Tr {L [a]ρ(t)N } = 0. ϕ κ S Other relevant observables are the average vibron num- ∂ρ(t) i ber N = Tr {ρ(t)a a} and the nanoresonator displace- = − [H , ρ(t)] − (n + 1)L [a]ρ(t) v ϕ S B κ ∂t ~ ment − n L [a ]ρ(t) − dsTr {K(t, t − s; ρ(s))} ,(9) B κ L d = l Tr {(a + a)ρ(t)}, (14) 2 0 ϕ where l = is the oscillator length and M is the where Tr is the partial trace with respect to the leads’ 0 L 2Mω degrees of freedom and we introduced the non-Markovian mass of the nanoresonator. dissipative kernel due to the reservoirs: h i III. NUMERICAL RESULTS AND DISCUSSION K(t, t − s; ρ(s)) := H (t), U [H (s), ρ(s)ρ ]U . T t−s T L t−s (10) The nanoelectromechanical system considered in our The right hand side of Eq. (9) also contains Lindblad- calculations is made of a two-dimensional quantum type operators which capture the effect of a thermal bath nanowire connected to source and drain reservoirs and described by the Bose-Einstein distribution n and by 32 a vibrational mode. The latter describes either a nearby the temperature T (κ is the loss parameter) : suspended CNT which supports longitudinal stretching modes or a vibrating molecule deposited on a substrate. † † † L [a]ρ(t) = a aρ + ρa a − 2aρa . (11) The length and width of the nanowire are L = 75 nm 2 x and L = 15 nm, while for the mass of the nanoresonator − (HS+HL+HR)t −15 In Eq. (10) U = e is the unitary evo- we set M = 2.5 × 10 kg. The turnstile operation is lution of the disconnected systems (i.e NEMS+leads). switched-on at instant t = 0. The bias applied on the Also, ρ is the equilibrium density operator of the leads. system is given by eV = μ − μ . L R 4 A. Vibron-dressed states and tunneling system and the NR increases with the particle number. Moreover, the diagonal matrix elements of the displace- ment operator are found as: In the following we express the lowest two single- particle energies of the conducting system with respect 2λ to the equilibrium chemical potential of the leads μ . 0 d := hϕ |a + a|ϕ i = n , (16) ν ν,s ν,s ν ~ω Specifically, ε = 0.875 meV and ε = 3.875 meV. 1σ 2σ We choose t = 2 meV and the vibron energy ~ω = and therefore depends only on n . 0.329 meV which is in the range of the observed longi- In view of transport calculations let us denote by tudinal stretching modes of CNTs . The value of the Δ (s, s ) = E − E ′ ′ the energy required to add N,N+1 ν,s ν ,s electron-vibron coupling parameter is λ = 0.096 meV. one electron from the leads such that the hybrid system The temperature of the particle reservoirs equals that of evolves from an N-electron state |ϕ ′ ′i to the (N + 1)- ν ,s the thermal bath. We chose k T = 4.3 μeV which corre- electron state |ϕ i. We calculate these energies for all ν,s sponds to a temperature of 50 mK. pairs of configuration {ν, ν } with a non-vanishing tun- The Hamiltonian H of the hybrid system is diag- S (lσ) neling coefficient T = hϕ |c |ϕ ′ ′if (E −E ′ ′) ′ ′ ν,s ν ,s l ν,s ν ,s νν ;ss σ onalized within a truncated subspace containing ‘free’ which describes the tunneling-in processes from the l-th states |ν, Ni obtained from the lowest-energy 16 elec- (lσ) lead. The tunneling coefficient T appears naturally ′ ′ tronic configurations and up to N = 15 vibronic states. νν ;ss in the Lindblad version of the generalized master equa- In the presence of electron-vibron coupling one gets an tion (see for example Ref. ) and controls the transport N -dimensional vibronic manifold {ϕ } associ- 0 ν,s s=0,...N processes in the quasistationary regime, that is when the ated to each electronic configuration |νi. charge occupation and mean vibron number do not de- For simplicity we set the chemical potentials of the pend on time. The argument of the Fermi function re- leads μ < ε such that the tunneling processes in- L,R 2σ veals the fact that in the quasistationary regime the en- volve only the lowest energy one- and two-particle con- ergy ε of the electron entering the sample matches the figurations. Then only four electronic configurations will l ′ ′ difference E − E between two configurations of the ν,s ν ,s contribute to the transport, namely the empty state |0i, (lσ) two spin-degenerate single-particle states | ↑ i, ↓ i and latter. Note that the tunneling amplitudes T are ′ ′ 1 1 νν ;ss the two-electron ground state | ↑ ↓ i. Henceforth we ′ 1 1 controlled by the FC factors F (see Eq. (6)). The νν ss shall drop the level index and use ↑, ↓ instead of ↑ , ↓ . same energy differences are relevant for tunneling-out 1 1 The transport through the hybrid system is then due ′ ′ processes |ϕ i → |ϕ i which are controlled by the ν,s ν ,s to the states |ϕ i, |ϕ i, |ϕ i and |ϕ i. Clearly, 0,s ↑,s ↓,s ↑↓,s f (x) = 1 − f (x). |ϕ i = |0, si such that s is simply the vibron number of 0,s Now, let us discuss the energy differences ′ ′ a Fock state, because the electron-vibron coupling does Δ (s, s ) in terms of the difference δ = s − s . N,N+1 not change the ‘empty’ states. For a mixed vibrational For tunneling-in processes one has δ > 0 if electrons state |ϕ i, s is related to the integer part of its cor- ν6=0,s have enough energy to excite more vibrons while for responding vibron number w . Indeed, using the Lang- ν,s δ < 0 the vibrations of the hybrid system are absorbed Firsov transformation one obtains the analytical result and allow tunneling of electrons from the leads at lower energies. The role of these transitions changes in the † S † −S w = hϕ |a a|ϕ i = hν, s|e a ae |ν, si ν,s ν,s ν,s case of tunneling-out processes: the system is ‘heated’ λ for δ < 0 and ‘cooled’ down if δ > 0. On the other = s + n , (15) hand, from Eq. (15) one notices that if λ /~ω ≪ 1 the ~ω 0 average vibron number are only slightly changed by the S −S λ ‘diagonal’ processes s = s . where we used the identities e ae = a + N , ~ω ˆ Figure 2 displays the tunneling energies as a function N |ν, si = n |ν, si and the fact that hν, s|a +a|ν, si = 0. S ν of δ and helps us to identify which transitions contribute On the other hand, the numerical diagonalization pro- N (ν) to the current for a symmetric bias window set by μ = 0 2 L,R vides w = N|A | which fits well to Eq. (15), ν,s N=0 sN E −E ′ ±p~ω/2, where p is an odd positive integer. For ν ν at least for the lowest vibronic components. example, the four dashed lines in Fig. (2) correspond to The vibrationally ‘excited’ states correspond to s > 0, μ = Δ ± ~ω/2 and μ = Δ ± ~ω/2. L,R 0,1 L,R 1,2 but it should be mentioned that even the lowest-energy In agreement with the analytical results obtained via states |ϕ i have a non-vanishing vibron number w ν,s=0 ν,0 the Lang-Firsov transformation, the eigenvalues corre- as they are not entirely made of a ‘free’ state |ν, N = sponding to the same electronic configuration ν are sep- 0i. Indeed, for the parameters considered here we find arated by integer multiples of vibron quanta, that is (see Eq. (5)) that the weights of |ν, N = 0i for the one- E = E + s~ω. This implies that the tunneling ν,s ν,0 (↑) (↓) 2 2 and two-particle states are |A | = |A | = 0.9 and 0,0 0,0 energies are also equally spaced, that is Δ (s, s ) = 0,1 (↑↓) 2 ′ ′ ′ |A | = 0.7 while the corresponding vibron numbers ε˜ +(s−s )~ω and Δ (s, s ) = ε˜ +U +(s−s )~ω, where 1 1,2 1 0,0 are w = w ≈ 0.085 and w = 4w . ε˜ = ε −λ /~ω and U is the direct interaction term V ↑,0 ↓,0 ↑↓,0 σ,0 1 1 1111 Note that the two-particle ground state carries more from the two-body Coulomb operator in Eq. (2). For the vibrons because the coupling between the conducting parameters chosen here we find U ∼ 1.67 meV. 5 transferred across the system along each turnstile cycle. In the first regime we set the chemical potentials of the μ , Q=1,2 leads such that the system is charged with two electrons and then completely depleted, hence Q = 2. For the sec- ond regime μ is pushed up to μ = 1.65 meV such that R R the discharging sequence allows only the tunneling from μ , Q=1 the two-particle configuration | ↑↓i. Then at the end of the turnstile cycle the total charge transferred across the system is Q = 1. The selected values of the chem- ical potentials for the Q = 1 and Q = 2 operations are also indicated by horizontal solid lines in Fig. 2. These 0-1e μ , Q=2 two regimes should reveal the dependence of the electron- 1-2e vibron coupling on the number of levels contributing to -1 -4 -3 -2 -1 0 1 2 3 4 the transport. The effects of the turnstile operations Q = 1, 2 on the FIG. 2. The energy differences associated to sequential tun- displacement d and average vibron number N are pre- neling processes leading to transitions between electronic con- sented in Figs. 3 (a) and (b). For the two-particle pump- figurations with N and N + 1 electrons. δ denotes the dif- ing (see Fig. 3 (a)) the displacement of the single-mode ference between the average number of vibrons for the vibra- ′ nanoresonator roughly mimics the behavior of the po- tional states |s i, |s ′i. For the simplicity of writing we do not ′ tential χ applied on the left contact. More precisely, d indicate the pairs (s, s ) corresponding to the same tunneling increases quickly as the electrons enter the system, sat- energy. The horizontal lines mark the values of the chemical urates once the charge occupation reaches the maximum potential for which one obtains various turnstile regimes - see the discussion in the text. value Q = 2 (not shown) and then drops to zero on the discharging half-periods. Note that the oscillations of the displacement match the period of the turnstile cycle, One notes that pairs of vibrational components 2t = 0.7 ns. It is also clear that the NR bounces between {|s i, |s i} which differ by the same amount of vibron ν a maximum value d ≈ 0.24 fm which does not depend ν max quanta δ have equal tunneling energies and will there- on the turnstile cycle and the equilibrium position (i.e. fore contribute simultaneously to the current. However, d = 0). In particular, we have checked that d and the their Franck-Condon tunneling amplitudes are different average charge Q vanish simultaneously. This behav- and decrease if s, s correspond to excited vibronic states. ior confirms that the electron-vibron coupling is indeed We also find that the tunneling amplitude of the ‘diag- periodically switched on and off along a turnstile cycle. onal’ transitions is much larger than the one of the ‘off- For Q = 2 turnstile operation the average vibron num- diagonal’ transitions (i.e for s 6= s ) which decreases as ber N displays a more surprising behavior: (i) It reaches s − s increases. a steady-state value N ≈ 0.5 on the first charging se- quence but then drops to a lower yet non-vanishing value during the depletion cycle. (ii) By repeating the turnstile B. The turnstile regime operation the same pattern is recovered as more vibrons are stored in the NEMS. Eventually, N reaches a quasi- stationary regime around t = 4.5 ns (not shown). We We denote by t the period of the charging/discharging therefore see that along the depletion cycles the vibrons cycles, such that the time needed for each turnstile oper- are stored in the system in spite of the fact that the ation is 2t . The value of the loss coefficient κ = 0.5 μeV. electron-vibron coupling is ineffective since Q . The GME was solved numerically on a subspace contain- S ing the lowest in energy 20 vibron-dressed states. We The single-particle turnstile operation (Q = 1) leads have checked that adding more vibronic states will not to a similar behavior of the average vibron number (see qualitatively alter the presented results. Let us mention Fig. 3 (b)). However, N reaches lower quasistationary here that the decreasing value of the FC factors for tran- values when compared to the two-particle turnstile oper- sitions between highly excited vibronic states is essential ation. A significant difference is noticed in the displace- in order to set a reasonably small cutoff N . In princi- ment oscillations. At the end of each turnstile cycle the ple one can include more states in the calculations, but NR does not return to its equilibrium position but settles the numerical effort to solve the master equation in the down to a distance d = d /2 from its equilibrium posi- max non-markovian regime increases considerably. tion. This happens because in the Q = 1 turnstile regime The periodic switching functions χ which simulate the effect of the electron vibron-coupling is only reduced L,R the turnstile operation are square-shaped and oscillate but not turned off because one electron is always present out-of-phase, as shown in Figs. 3(a) and (b). We assume in the electronic system and therefore induces a minimal that the initial state of the hybrid system |ν = 0, N = 0i. ‘deflection’ of the NR. In this sense, the single-particle The numerical simulations were performed for two turn- turnstile operation can be seen as a way to dynamically stile regimes which differ by the number of charges Q switch between electron-vibron interactions correspond- Δ (meV) N,N+1 6 0.3 1 (a) s=0 1.4 (a) 0.8 | ↑↓i, Q=2 N s=1 0.25 χ s=2 L 0.6 1.2 0.4 0.2 1 0.2 0.8 0.15 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.6 (b) s=0 0.1 0.8 |0i, Q=2 s=1 0.4 s=2 0.6 0.05 0.2 0.4 0.2 0 0 0 0.5 1 1.5 2 2.5 3 3.5 Time (ns) 0.3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 1.4 1 (b) N (c) s=0 0.8 | ↑i + | ↓i, Q=1 0.25 χ s=1 1.2 s=2 0.6 0.2 1 0.4 0.2 0.8 0.15 0.6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.1 Time (ns) 0.4 0.05 FIG. 4. The relevant N-particle populations P of the ν,s 0.2 ground and excited vibronic states: (a) P for Q = 2 oper- ↑↓,s ation; (b) P for Q = 2 operation; (c) The total population 0,s 0 0 0 0.5 1 1.5 2 2.5 3 3.5 of single-particle configurations P = P + P for Q = 1 1,s ↑,s ↓,s Time (ns) operation. 2.5 7 (c) Q 1.5 3 that an ‘effective’ temperature T of the hybrid system 1 eff can be derived from the equilibrium distribution func- 0.5 tion n (ω, T ) corresponding to the calculated average B eff 0 0 0 0.5 1 1.5 2 2.5 3 3.5 vibron number N (see e.g. ). Using this equivalence we Time (ns) realize that both turnstile operations induce a sequence of ‘heating’ and partial ‘cooling’ processes on the NR, FIG. 3. The dynamics of the vibron number N and the dis- as already proved by the vibron dynamics. To explain placement d of the nanoresonator in the two turnstile regimes this behavior we look more closely at the populations which allow the net pumping of Q electrons along each cycle: P along each turnstile cycle for ν = 0, ↑, ↓, ↑↓. We ν,s (a) Q = 2, μ = 3.5 meV, μ = −0.25 meV and (b) Q = 1, L R discuss first the two-electron turnstile operation. From μ = 3.5 meV, μ = 1.65 meV. The dotted lines indicate L R Fig. 4(a) we observe that at the end of the charging cy- the functions χL,R which simulate the periodic on and off switching of the two contacts. (c) The charge occupation and cles the hybrid system is completely described by several the transient currents JL,R for the Q = 1 turnstile operation. two-particle configurations |ϕ i (smaller contributions ↑↓,s Other parameters: t = 0.35 ns . of P were not shown). We also find that the popu- ↑↓,s>2 lations P = ρ reach a maximum value shortly 1,s σs,σs after the coupling of the source lead and then vanish as the two-particle states are filled. ing to a fixed number of particles. On the other hand, the different response of the NR displacement can be used to Fig. 4(b) shows that on the discharging cycles the re- ‘read’ the number of charges transferred across the sys- duced density matrix of the system contains both the tem along the turnstile cycles, in the presence of vibrons. ‘ground’ and ‘excited’ purely vibronic states. Moreover, In Fig. 3(c) we plot for completeness the dynamics of the occupation of the states |ϕ i on each depletion 0,s>0 the total charge Q along the single-particle turnstile half-period increases until a quasistationary regime is operation and the corresponding transient currents J . reached. This explains why the mean vibron number L,R The latter display sharp peaks, their different amplitudes N , which collects contributions of the type w P , in- v ν,s ν,s being a consequence of the different rates at which the creases along each turnstile cycle. One can also easily system is charged or depleted (note that Q drops more check that the decreasing population of the ground state abruptly on each discharging half-period). We recall here configuration |ϕ i is balanced by the presence of ex- 0,s=0 Charge Displacement (fm) Displacement (fm) Current (nA) Vibron number Vibron number Population Population Population 7 cited vibronic states. pletely described by purely vibronic states and therefore hϕ |a |ϕ i = 0. 0,s 0,s The accumulation of vibrons in the empty system In Fig. 4(c) we present for completeness the popula- (i.e., the partial cooling mechanism) can be explained by carefully counting the various vibron-assisted tunnel- tions of one-particle configurations which describe the hybrid system for the Q = 1 turnstile operation. Clearly, ing processes connecting pairs of fully interacting states ′ ′ the discharging cycles are now described by single- {ϕ , ϕ }. The chemical potentials of the leads are ν,s ν ,s selected such that all relevant tunneling processes (diag- particle states |ϕ i. The empty states |ϕ i are σ=↑,↓,s 0,s no longer accessible in this case so they were not shown. onal or off-diagonal) are active, i.e., most of the energies Δ (s, s ) are within the bias window (μ , μ ), for By comparing Figs. 4 (b) and (c) one notices a lower oc- N,N+1 R L cupation of the excited states |ϕ i which explains N = 0, 1 (see the chemical potentials for the Q = 2 set- σ=↑,↓,s>0 ting in Fig. 2). For the Q = 1 operation we have instead why the ‘jumps’ and drops of the mean vibron number ′ ′ are less pronounced. This could be expected because the Δ (s − s ) < μ < Δ (s − s ) < μ for the most 0,1 R 1,2 L important tunneling processes. When looking at Fig. 2 electron-vibron coupling is now enhanced/reduced only due to a single electron which is added/removed from we notice that some transitions are left outside the bias window, e.g. the ones corresponding to Δ (δ = 3, 4). the system. 0,1 However, these transitions have a small tunneling am- In the following we investigate in more detail the role plitude and they will not significantly contribute to the of the bias window on the partial cooling processes in transport. It is easy to see that for the first charging the Q = 2 turnstile operation. To this end the chem- cycle of the Q = 2 operation the sequence of ‘diagonal’ ical potential of the drain reservoir is pushed up to transitions e.g |ϕ i → |ϕ i → |ϕ i involves only μ = 0.68 meV such that the main ‘heating’ processes 0,0 σ=↑,0 ↑↓,0 the lowest vibronic components (s = s = 0) with small associated to the depletion cycles are forbidden, that is ′ ′ vibron numbers w (see Eq. (15)). These transitions are Δ (s, s ) < μ for some s < s (see the lowest dotted ν,0 0,1 R also the strongest, as the corresponding FC factors are horizontal line in Fig. 2). Figure 5 shows that in this case the largest ones. Note also that on the first charging the mean vibron number does not display steps on the cycle the vibron absorption is not possible as the initial discharging cycles (as in Fig. 3(b)) but rather vanishes - state is |ϕ i, such that the ‘excited’ states |ϕ i can in other words, the hybrid system eventually cools down 0,0 σ,s>0 only be populated through ‘off-diagonal’ weaker transi- to the temperature of the thermal bath T . In order to tions, for example |ϕ i → |ϕ i. Finally, the charg- capture the slow evolution of N we increased the turn- 0,0 σ,1 ing cycle brings the second electron to the system and stile period to t = 1 ns. Further insight into the vibron opens more tunneling paths involving both diagonal and dynamics is given by the populations P of the purely 0,s off-diagonal processes, e.g., |ϕ i → |ϕ i → |ϕ i or vibronic states which are also presented in Fig. 5. Af- 0,0 σ,1 ↑↓,1 |ϕ i → |ϕ i → |ϕ i. It is therefore clear that at 0,0 σ,1 ↑↓,2 ter an initial increase, the excited states |ϕ i and |ϕ i 0,1 0,2 the end of the first charging cycle the system is described are slowly depleted in favor of the ground state |ϕ i 0,0 by the two-particle electronic configuration | ↑↓i and sev- whose population increases uniformly on each discharg- eral vibronic components |s i with the associated vibron ing sequence. This behavior differs from the one shown ↑↓ numbers w . ↑↓,s in Fig. 4(b) and suggests a ‘redistribution’ of probability between various purely vibronic states. In the following Now, during the first depletion cycle this mixed struc- we explain this effect through the interplay of tunneling- ture of the reduced density matrix allows the activation of out and -in processes which involve the drain lead. multiple ‘diagonal’ and ‘off-diagonal’ tunneling out pro- cesses between (N + 1)−particle and N−particle config- The sudden drop of N right after opening the con- urations. For example the ‘diagonal’ backward sequence tact to the right reservoir is due to the ‘cooling’ transi- |ϕ i → |ϕ i → |ϕ i leaves the hybrid system in the tions |ϕ i → |ϕ ′i for δ = s − s > 0, whose ener- ↑↓,1 σ,1 0,1 σ,s 0,s first vibronic excited state whose population P ≈ 0.2 gies are still above μ (see Fig. 2). On the other hand, 0,1 R in Fig. 4(b). The small population P is due to the sim- the excited vibronic states |ϕ i are still being pop- 0,2 σ,s>0 ilar ‘off-diagonal’ transition from |ϕ i → |ϕ i. Other ulated via vibron-conserving transitions |ϕ i → |ϕ i σ,1 0,2 σ,s 0,s transitions leading to vibrational ‘cooling’ can be also and to a lesser extent by the partial ‘cooling’ transition identified. As a result the mean vibron number drops |ϕ i → |ϕ i. This scenario is confirmed by the initial σ,2 0,1 over the depletion cycle, but does not vanish due to the increase of the populations P and P . We find in- 0,1 0,2 ‘diagonal’ tunneling events. stead that the much slower vibronic relaxation involves two more sequential tunnelings, one from the reservoir At the next charging cycle the excited single-particle to the central system and another one back to it. In- states |ϕ i will be fed by both diagonal and off-diagonal σ,1 deed, given the fact that Δ (δ < 0) are below μ , elec- transitions, because when switching on the coupling to 0,1 R trons can tunnel back from the contact via ‘cooling’ tran- the left lead the reduced density matrix of the system sitions |ϕ i → |ϕ i (for s < s). Finally, the lower- reads ρ(2t ) = |ϕ ihϕ |. As a consequence, the 0,s σ,s p 0,s 0,s temperature single-particle states are depleted through population P almost doubles with respect to the first ↑↓,1 ′ ′ diagonal transitions |ϕ i → |ϕ i. charging cycle, whereas P brings a small contribution σ,s 0,s ↑↓,2 as well. The vanishing of the displacement d on each dis- Turning back to the symmetric bias setting (see Fig. 3 charging sequence is mandatory, as the system is com- (a)), it is readily seen that the tunneling-mediated cool- 8 1 1 1.6 (a) 0.8 0.8 1.4 s=0 s=1 1.2 s=2 0.6 0.6 0.8 0.4 0.4 0.6 λ =0.162 0.4 λ =0.096 0.2 0.2 0 λ =0.065 0.2 λ =0.028 0 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Time (ns) Time (ns) (b) FIG. 5. The dynamics of vibron number N and of the pop- 5.8 ulations P0,s for the the Q = 2 turnstile protocol. In contrast to Fig. 3(a) the complete ‘cooling’ of the nanoresonator is in- 5.6 sured by suppressing the tunneling-out tunneling processes. Other parameters: μ = 3.5 meV, μ = 0.68 meV, t = 0.5 L R p 5.4 ns. 5.2 ing mechanism presented above cannot be active. In this λ =0.162 λ =0.096 case, electrons are not allowed to tunnel back to the 4.8 0 λ =0.065 central system because μ lies below all transition en- 0 λ =0.028 4.6 ergies. Moreover, the cooling processes |ϕ i → |ϕ i σ,s 0,s with s < s are overcome by the heating processes such 1 2 3 4 5 6 7 Peak index that N settles down to a non-vanishing value after the onset of the discharging sequence. FIG. 6. The effect of the electromechanical coupling strength In order to check whether the electron-vibron cou- λ0 (given in meV units) on (a) the vibron number Nv and pling affects not only the dynamics of the NR but also (b) on the peak amplitude of the transient current J . The the transport properties of the electronic subsystem, we parameters correspond to the Q = 2 turnstile protocol: μ = present in Fig. 6(a) the vibron dynamics for several val- 3.5 meV, μ = −0.25 meV, t = 0.35 ns. R p ues of the electron-vibron coupling strength λ . This pa- rameter can be tuned by changing either the equilibrium distance between the electronic system and the nanores- λ because at such short times the vibrons are not yet onator (as shown in previous work ) or the NR mass M. activated. For the larger value λ = 0.162 meV a steep The amplitude of the heating and cooling cycles decreases reduction of the peak is noticed after two charging half- with λ and the hybrid system approaches the quasista- periods. A similar behavior is recovered for the output tionary regime much faster at larger values of λ . For 0 current J (not shown). By comparing Figs. 6(a) and (b) example, a considerable difference is noticed between the one infers that the attenuation of the peak amplitude is first two cycles at λ = 0.162 meV, the next cycles being 0 correlated to the emergence of the quasistationary regime rather similar. for the heating/cooling sequences. In Fig. 6(b) we collect the amplitudes associated to the We also considered other shapes for the switching func- first seven peaks of the current J and to the different tions χ and we recovered similar effects of the turnstile L L,R electron-vibron couplings considered in Fig. 6(a). The regime on the nanoresonator, i.e. heating/cooling on the peak evolution over few turnstile cycles can also be ex- charging/discharging half-periods. Figure 7 shows the vi- tracted from transport measurements and provides in- bron dynamics N and the displacement d for smoother direct insight on the vibron dynamics. In the weakly switching functions. When compared to the results pre- interacting case (λ = 0.028 meV) the amplitudes of the sented in Fig. 3(a) we noticed minor changes in the lo- peaks are nearly equal and one cannot discern the negli- cal maximum and minimum values of the average vibron gible effect of the electron-vibron coupling on the trans- number. However, the most important effect is a delay port properties. In contrast, as λ increases, the peaks of nanoresonator’s response to the switching functions, display noticeable differences. More precisely, their am- i.e. N and d do not increase/decrease immediately af- plitude gradually decreases from one cycle to another un- ter charging/discharging. If one is interested in imple- til it reaches a quasistationary value (for λ = 0.096 meV menting faster heating and cooling processes separated this value is roughly 5.5 nA). Note that the first peak of by longer ‘isotherms’ (i.e time intervals with constant the charging current J is less sensitive w.r.t. changes of vibron number N ) the square-wave driving is the most L v Vibron number Populations J peak amplitude (nA) Vibron number L 9 0.3 pumping one electron per turnstile cycle while keeping 1.4 the lowest level occupied one initializes a configuration 0.25 made by single-particle states ’dressed’ by vibrons. 1.2 χ On the other hand, the charging cycles activate the 0.2 1 electrostatic coupling and the vibron number increases. It also turns out that both turnstile operations induce 0.8 0.15 a heating of the nanoresonator when the electron-vibron coupling is turned on and at least a partial cooling when 0.6 0.1 it is turned off. 0.4 0.05 0.2 IV. CONCLUSIONS 0 0 0 0.5 1 1.5 2 2.5 Time (ns) We proposed and studied theoretically a quantum turnstile protocol for switching on and off the effect of FIG. 7. The dynamics of the vibron number N and displace- electron-vibron coupling between a biased mesoscopic ment d of the nanoresonator for smoother switching functions system and a vibrational mode. A detailed analysis of χ . Other parameters: λ = 0.096 meV, μ = 3.5 meV, L,R 0 L the vibron-assisted tunneling processes is provided by the μ = −0.25 meV, t = 0.35 ns. R p populations of the vibron-dressed states which are cal- culated within the generalized master equation method. We identify the role of various tunneling processes in effective. the vibron emission (heating) and absorption (cooling) Finally, we stress that the oscillations of the displace- processes. The turnstile charging and discharging cy- ment record the charge variations along the turnstile op- cles impose periodic variations of the nanoresonator’s dis- erations but do not discern between the vibron dynamics. placement with respect to its equilibrium value. As the In order to understand why this happens let us observe electronic system empties the displacement vanishes. In- first that if the coherences hϕ |ρ(t)|ϕ ′i are negligi- ν,s ν,s stead, a turnstile operation which allows only a partial ble then from Eq. (16) one gets a simpler formula for the depletion sets a lower bound of the displacement due to displacement: an extra electron residing in the system. 2λ l 0 0 The values of the displacement obtained in our model d ≈ n P . (17) ν ν,s ~ω are probably too small to be detected. However, d in- ν,s creases as more electrons tunnel across the system during a turnstile cycle. This could be achieved by increasing Secondly, since on the charging sequences the system set- the bias window such that more electronic configurations tles down to the two-electron configuration (i.e., n = 2) participate in transport. Alternatively, one can consider and P = 1 for all chemical potentials μ < ↑↓,s R lighter nanoresonators and therefore larger values of the Δ (s − s = 0) it follows that d cannot depend on μ , 0,1 R oscillator length l . even if each occupation P does. Eq. (17) also confirms ν,s We find that in general the average number of vibrons the doubling of the quasistationary displacement d max on the charging cycles with respect to the value attained does not vanish along the discharging cycles when the electron-vibron coupling is ineffective. In the quasista- along the depletion cycles of the Q = 1 turnstile opera- tion, as shown in Figs. 3(a) and (b). tionary regime the same amount of vibrons is emitted and absorbed along a turnstile cycle. Otherwise stated, the For the parameters selected here the coherences cor- responding to states with the same electronic configura- system undergoes periodic heating and cooling processes. ′ A complete cooling to the equilibrium temperature of the tions but different vibron numbers (i.e., hϕ |ρ(t)|ϕ i) ν,s ν,s do exist but they are indeed too small to induce a no- leads or of a thermal bath can be achieved by a suitable choice of the chemical potential of the drain reservoir. ticeable change of the various observables (not shown). In fact, we record some fast oscillations of the displace- We also show that the peak amplitude of the transient ment on the ‘steps’ of each turnstile cycle; the period of currents decreases as the strength of the electron-vibron coupling increases. Moreover, it turns out that as the these oscillations coincides with those of the coherences mentioned above but one can see from Fig. 3 that their heating/cooling cycles attain the quasistationary regime the peak amplitude gradually reduces to a value which amplitude is hardly noticeable. Based on these results we state that the quantum does not depend on the charging/discharging half-period. turnstile regime provides a dynamical switching of the Let us emphasize that the quantum turnstile dynam- electron-vibron coupling effects on the hybrid system. ics differs considerably from the normal transport regime Once the depletion process is complete the electron- when both leads are simultaneously coupled to the sys- vibron coupling is ineffective. However, the effect of the tem and for which one can only notice a heating process, latter is imprinted in the non-vanishing populations of as the average vibron number uniformly increases before the excited vibrational states ϕ . Alternatively, by reaching its stationary value. Also, in the present setting 0,s>0 Displacement (fm) Vibron number 10 the actuation of the nanoresonator is only due to the elec- 0221 and from the Romanian Core Program PN19-03 tronic current as there is no additional driving signal. In (contract No. 21 N/08.02.2019). B.T. and V.M. were other words, we consider that before the electronic sub- also supported by TUBITAK Grant No. 117F125. B.T. system is coupled to the leads the nanoresonator is in the further acknowledges the support from TUBA. static deflection mode. ACKNOWLEDGMENTS R.D., V.M. and S.S. acknowledge financial support from CNCS - UEFISCDI grant PN-III-P4-ID-PCE-2016- 1 16 M. Poot and H. S. J. van der Zant, Phys. Rep. 511, 273 C. Schinabeck, A. Erpenbeck, R. H¨artle, and M. Thoss, (2012). Phys. Rev. B 94, 201407(R) (2016). 2 17 L. Cockins, Y. Miyahara, S. D. Bennett, A. A. Clerk, and A. Mitra, I. Aleiner, and A. J. Millis, Phys. Rev. B 69, P. 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Quantum turnstile regime of nanoelectromechanical systems

Dragomir, R.; Moldoveanu, V.; Stanciu, S.; Tanatar, B.
Condensed Matter , Volume 2020 (2005) – May 21, 2020
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2469-9950
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ARCH-3331
DOI
10.1103/PhysRevB.101.165409
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Abstract

R. Dragomir and V. Moldoveanu National Institute of Materials Physics, Atomistilor 405A, Magurele 077125, Romania S. Stanciu Faculty of Physics, University of Bucharest, Atomistilor 405, Magurele 077125, Romania and National Institute of Materials Physics, Atomistilor 405A, Magurele 077125, Romania B. Tanatar Department of Physics, Bilkent University, Bilkent, 06800 Ankara, Turkey The effects of a turnstile operation on the current-induced vibron dynamics in nanoelectrome- chanical systems (NEMS) are analyzed in the framework of the generalized master equation. In our simulations each turnstile cycle allows the pumping of up to two interacting electrons across a biased mesoscopic subsystem which is electrostatically coupled to the vibrational mode of a nanores- onator. The time-dependent mean vibron number is very sensitive to the turnstile driving, rapidly increasing/decreasing along the charging/discharging sequences. This sequence of heating and cool- ing cycles experienced by the nanoresonator is due to specific vibron-assisted sequential tunneling processes along a turnstile period. At the end of each charging/discharging cycle the nanoresonator is described by a linear combination of vibron-dressed states s associated to an electronic configu- ration ν. If the turnstile operation leads to complete electronic depletion the nanoresonator returns to its equilibrium position, i.e. its displacement vanishes. It turns out that a suitable bias applied on the NEMS leads to a slow but complete cooling at the end of the turnstile cycle. Our calculations show that the quantum turnstile regime switches the dynamics of the NEMS between vibron-dressed subspaces with different electronic occupation numbers. We predict that the turnstile control of the electron-vibron interaction induces measurable changes on the input and output transient currents. I. INTRODUCTION In this theoretical study we focus on the time- dependent control of the entangled electron-vibron dy- namics of a NEMS in the quantum turnstile regime. The nanoelectromechanical systems are hybrid struc- More precisely, we show that the pumping of an inte- tures in which the electrostatic interaction between vibra- ger number of electrons along a turnstile period activates tional modes and open mesoscopic systems is expected to 1 the coupling to the vibrational mode during the charging play a role down to the quantum level . To support this cycle and then renders it ineffective on the discharging idea, the sensing properties of nanoresonators (NR) in cycle when the system is fully depleted. We recall that in the presence of electronic transport have been investi- 11–13 the turnstile setup , electrons are first injected from gated in various experimental settings. the source (left) particle reservoir while the contact to For instance, singly clamped cantilevers or AFM tips the drain (right) reservoir is closed. After this charging were shown to record single-electron tunneling from back- half-period, the left/right contact closes/opens simulta- gate contacts to the excited states of quantum dots de- neously (see the sketch in Fig. 1). posited on a substrate . In another class of experiments, In most experimental investigations on NEMS, a bias a suspended carbon nanotube (CNT) with an embedded voltage continuously supplies the charge flow through the quantum dot is actuated by microwave signals and the mesoscopic system which in turn interacts with the vibra- dips of its resonance frequency are associated to single- tional mode. Then the hybrid structure evolves under electron tunneling . Besides flexural modes, the CNTs the electron-vibron coupling until a stationary transport also develop longitudinal modes with higher frequencies regime is reached. At the theoretical level, the latter is (up to few GHz). Similarly, the vibration energy ~ω of 14,15 recovered by solving rate equations or hierarchical single-molecule junctions is around few meVs . For these quantum master equations (HQME) . Also, the single- systems, refined cooling techniques were used to reach level Anderson-Holstein model provides a sound descrip- the regime ~ω ≫ k T for which the vibrations of the 5,6 tion of the essential spectral properties of NEMS via the nanoresonator must be quantized . Lang-Firsov polaron transformation. On the other hand, the implementation of nanoelec- tromechanical systems as successful devices in quantum Let us stress that recent observation of real-time vibra- 7 8 20,21 22 sensing , molecular spintronics or nano-optomechanics tions in CNTs and pump-and-probe measurements requires an accurate tuning of the underlying electron- provide a strong motivation to scrutinize the time- vibron coupling. For example, the electron-vibron cou- dependent vibron-assisted transport. Few theoretical de- pling can be switched on and off by controlling the loca- scriptions of vibron-assisted transport properties in the tion of a quantum dot (QD) along the suspended CNT presence of pumping potentials acting on the electronic 9,10 in which it is formed . system can be mentioned. The effect of a cosine-shaped arXiv:2005.10466v1 [cond-mat.mes-hall] 21 May 2020 2 where H accounts for the two components of the S,0 NEMS, i.e. the QW accommodating several interacting electrons and the vibrational mode with frequency ω as- sociated to a molecule or a nanoresonator: X X X † † † H = ε c c + V c c c c S,0 iσ iσ ijkl ′ lσ kσ iσ iσ jσ i,σ σ,σ i,j,k,l + ~ωa a. (2) Here c creates an electron with spin σ on the single- iσ particle state ψ of the electronic system with the corre- iσ FIG. 1. Schematic view of the NEMS in the turnstile regime. sponding energy ε , the second term is the two-body iσ Source (L) and drain (R) particle reservoirs with chemical Coulomb interaction within the electronic sample and potentials μ are connected to an electronic structure (e.g. L,R a is the creation operator for vibrons. The eigenstates a quantum wire - QW). The contact regions are modulated by |ν, Ni of H are products of electronic many-body con- switching functions χ - the turnstile operation corresponds S,0 L,R to periodic out-of-phase oscillations of χ . A vibrational figurations |νi with energies E of the electronic system L,R ν mode of frequency ω interacts with the electrons, d being its and N-vibron Fock states |Ni, such that H |ν, Ni = S,0 displacement w.r.t the equilibrium position. (E + N~ω)|ν, Ni. The electron-vibron coupling V ν el−vb reads as V = λ c c (a + a), (3) el−vb i iσ driving of the contact regions has been considered within iσ 23,24 i,σ the Floquet Green’s function formalism . In a very recent paper the HQME method was adapted for a time- where λ is the electron-vibron coupling strength. 25 26 dependent setting . Avriller et al. calculated the tran- We denote by E and |ϕ i the eigenvalues and eigen- ν,s ν,s sient vibron dynamics induced by a step-like coupling of functions of the hybrid system such that molecular junctions to source and drain particle reser- voirs. H |ϕ i = E |ϕ i. (4) S ν,s ν,s ν,s In the present work we rely on the generalized master Since V conserves the electronic occupation and the el−vb equation (GME) method which was previously used to 27 spin, the fully interacting states |ϕ i can still be labeled ν,s study the turnstile regime of single-molecule magnets by a many-body configuration ν and written as: and recently extended for hybrid systems such as NEMS ( ) or cavity-QD systems . The model Hamiltonian em- (ν) bodies both the electron-electron interaction within the |ϕ i = |νi ⊗ A |Ni := |νi ⊗ |s i. (5) ν,s ν sN electronic subsystem and the spin degree of freedom. We also consider turnstile operations where more than one The ν-dependent vibrational overlap |s i contains differ- electron is transferred across the system. The reduced (ν) ent states |Ni, A being the weight of the N-vibron density operator of the hybrid system is calculated nu- sN state. If |ϕ i are obtained by numerical diagonalization merically with respect to vibron-dressed basis. As we ν,s one should truncate the indices N and s at a convenient are interested in the response of the NR to the turn- (ν) upper bound N . In this case, the coefficients A define stile pumping we also calculate its associated displace- 0 sN a finite dimensional unitary matrix which approximates ment which can be, in principle, measured. Note that the exact Lang-Firsov transformation defined by the op- this quantity is mostly derived for the classical regime of erator S = (λ /~ω)c c (a − a) (see e.g., Ref. 30). nanoresonators via the Langevin equation . i iσ iσ i,σ −S The rest of the paper is organized as follows. In Sec- The exact eigenfunctions are then |ϕ i = e |ν, Ni. ν,s tion II we introduce the model and briefly recall the main Let us stress that the electron-vibron coupling con- ingredients of the GME approach. The results are pre- stants λ depend on the single-particle wavefunctions sented in Section III, Section IV being left to conclusions. ψ of the electronic subsystem. In a recent work iσ we took this dependence into account and showed that it leads to different sensing efficiencies when a singly- II. FORMALISM clamped tip is placed above the quantum wire and swept along it. In this work the position of the NR is fixed A typical NEMS setup is sketched in Fig. 1 where a and the transport involves, for simplicity, only the low- quantum wire (QW) is capacitively coupled to a nearby est spin-degenerate single-particle state whose associated nanoresonator (NR) and tunnel-coupled to source and electron-vibron coupling strength will be denoted by λ . drain leads. The closed nanoelectromechanical system It is useful to introduce the Franck-Condon factors (FC): (i.e. not connected to particle reservoirs) is described by (ν) (ν ) ′ ′ ′ the following general Hamiltonian F := hs |s ′i = A A , |n − n | = 1, νν ;ss ν ′ ν ν sN s N N=0 H = H + V , (1) (6) S S,0 el−vb 3 where n is the number of electrons corresponding to the The GME is solved numerically with respect to the many-body configuration ν. We shall see below that for vibron-dressed basis {ϕ } of the hybrid system. Let ν,s a given pair of electronic configurations {ν, ν } one gets a us stress that choosing the fully interacting basis over ′ ′ series of vibron-assisted transitions controlled by F . the ‘free’ one {|ν, Ni} allows us to calculate the matrix νν ;ss i i H t − H t S S In view of vibron-assisted transport the electronic com- ~ ~ elements of e c e which appear in the dissi- iσ ponent of NEMS is also coupled to source (L) and drain pative kernel of the leads (see Eq. (10)). Note also that (R) particle reservoirs characterized by chemical poten- in this representation the Lang-Firsov transformation of tials μ , as shown in Fig. 1. The total Hamiltonian the tunneling Hamiltonian is not needed such that H L,R therefore becomes: does not acquire an additional operator-valued exponen- X tial. By doing so one carefully takes into account the FC H(t) = H + H + H (t), (7) S l T factors which can have both positive and negative signs, l=L,R as pointed out in Ref. . The full information on the system dynamics is em- where H is the Hamiltonian of the lead l and the tunnel- bodied in the populations of various states ing Hamiltonian reads as (h.c. denotes Hermitian conju- P (t) = hϕ |ρ(t)|ϕ i. (12) gate): ν,s ν,s ν,s X X The time-dependent currents in each lead are identified (lσ) † H (t) = dqχ (t) T c c + h.c. . (8) T l iσ from the continuity equation of the charge occupation qi qlσ l=L,R i,σ Q of the system: d d The functions χ (t) simulate the turnstile modulation of Q (t) = eTr N ρ(t) = J (t) − J (t), (13) S ϕ S L R dt dt the contact barriers between the leads and the system (lσ) and T is the coupling strength associated to a pair qi ˆ where N = c c is the particle number op- S iσ iσ i,σ of single-particle states from the lead l and the central erator, Tr stands for the trace with respect to the sample. For simplicity we assume that the tunneling pro- basis {ϕ } of the hybrid system and e is the elec- ν,s (lσ) cesses are spin conserving and that T does not de- qi tron charge. The left and right transient currents pend on σ. We describe the leads as one-dimensional J are then calculated by collecting all diagonal ele- L,R semi-infinite discrete chains which feed both spin up and ments hϕ |ρ˙(t)N |ϕ i which contain the Fermi func- ν,s S ν,s down electrons to the central system. Their spectrum is tion f . The latter appears when performing the l=L,R ε = 2t cosq , where q is electronic momentum in the q L l l partial trace of the integral kernel K(t, t − s; ρ(s)) such n o lead l and t denotes the hopping energy on the leads. ′ ′ that Tr ρ c c = δ δ δ(q − q )f (ε ) Also, L L ′ ′ ′ qlσ ll σσ l q The reduced density operator (RDO) ρ of the hybrid l q l σ system obeys a generalized master equation (GME) (for a note that from the cyclic property of the trace one ˆ ˆ derivation via the Nakajima-Zwanzig projection method has Tr {[H , ρ(t)]N } = Tr {ρ(t)[N , H ]} = 0 and ϕ S S ϕ S S see e.g., Ref. 28): Tr {L [a]ρ(t)N } = 0. ϕ κ S Other relevant observables are the average vibron num- ∂ρ(t) i ber N = Tr {ρ(t)a a} and the nanoresonator displace- = − [H , ρ(t)] − (n + 1)L [a]ρ(t) v ϕ S B κ ∂t ~ ment − n L [a ]ρ(t) − dsTr {K(t, t − s; ρ(s))} ,(9) B κ L d = l Tr {(a + a)ρ(t)}, (14) 2 0 ϕ where l = is the oscillator length and M is the where Tr is the partial trace with respect to the leads’ 0 L 2Mω degrees of freedom and we introduced the non-Markovian mass of the nanoresonator. dissipative kernel due to the reservoirs: h i III. NUMERICAL RESULTS AND DISCUSSION K(t, t − s; ρ(s)) := H (t), U [H (s), ρ(s)ρ ]U . T t−s T L t−s (10) The nanoelectromechanical system considered in our The right hand side of Eq. (9) also contains Lindblad- calculations is made of a two-dimensional quantum type operators which capture the effect of a thermal bath nanowire connected to source and drain reservoirs and described by the Bose-Einstein distribution n and by 32 a vibrational mode. The latter describes either a nearby the temperature T (κ is the loss parameter) : suspended CNT which supports longitudinal stretching modes or a vibrating molecule deposited on a substrate. † † † L [a]ρ(t) = a aρ + ρa a − 2aρa . (11) The length and width of the nanowire are L = 75 nm 2 x and L = 15 nm, while for the mass of the nanoresonator − (HS+HL+HR)t −15 In Eq. (10) U = e is the unitary evo- we set M = 2.5 × 10 kg. The turnstile operation is lution of the disconnected systems (i.e NEMS+leads). switched-on at instant t = 0. The bias applied on the Also, ρ is the equilibrium density operator of the leads. system is given by eV = μ − μ . L R 4 A. Vibron-dressed states and tunneling system and the NR increases with the particle number. Moreover, the diagonal matrix elements of the displace- ment operator are found as: In the following we express the lowest two single- particle energies of the conducting system with respect 2λ to the equilibrium chemical potential of the leads μ . 0 d := hϕ |a + a|ϕ i = n , (16) ν ν,s ν,s ν ~ω Specifically, ε = 0.875 meV and ε = 3.875 meV. 1σ 2σ We choose t = 2 meV and the vibron energy ~ω = and therefore depends only on n . 0.329 meV which is in the range of the observed longi- In view of transport calculations let us denote by tudinal stretching modes of CNTs . The value of the Δ (s, s ) = E − E ′ ′ the energy required to add N,N+1 ν,s ν ,s electron-vibron coupling parameter is λ = 0.096 meV. one electron from the leads such that the hybrid system The temperature of the particle reservoirs equals that of evolves from an N-electron state |ϕ ′ ′i to the (N + 1)- ν ,s the thermal bath. We chose k T = 4.3 μeV which corre- electron state |ϕ i. We calculate these energies for all ν,s sponds to a temperature of 50 mK. pairs of configuration {ν, ν } with a non-vanishing tun- The Hamiltonian H of the hybrid system is diag- S (lσ) neling coefficient T = hϕ |c |ϕ ′ ′if (E −E ′ ′) ′ ′ ν,s ν ,s l ν,s ν ,s νν ;ss σ onalized within a truncated subspace containing ‘free’ which describes the tunneling-in processes from the l-th states |ν, Ni obtained from the lowest-energy 16 elec- (lσ) lead. The tunneling coefficient T appears naturally ′ ′ tronic configurations and up to N = 15 vibronic states. νν ;ss in the Lindblad version of the generalized master equa- In the presence of electron-vibron coupling one gets an tion (see for example Ref. ) and controls the transport N -dimensional vibronic manifold {ϕ } associ- 0 ν,s s=0,...N processes in the quasistationary regime, that is when the ated to each electronic configuration |νi. charge occupation and mean vibron number do not de- For simplicity we set the chemical potentials of the pend on time. The argument of the Fermi function re- leads μ < ε such that the tunneling processes in- L,R 2σ veals the fact that in the quasistationary regime the en- volve only the lowest energy one- and two-particle con- ergy ε of the electron entering the sample matches the figurations. Then only four electronic configurations will l ′ ′ difference E − E between two configurations of the ν,s ν ,s contribute to the transport, namely the empty state |0i, (lσ) two spin-degenerate single-particle states | ↑ i, ↓ i and latter. Note that the tunneling amplitudes T are ′ ′ 1 1 νν ;ss the two-electron ground state | ↑ ↓ i. Henceforth we ′ 1 1 controlled by the FC factors F (see Eq. (6)). The νν ss shall drop the level index and use ↑, ↓ instead of ↑ , ↓ . same energy differences are relevant for tunneling-out 1 1 The transport through the hybrid system is then due ′ ′ processes |ϕ i → |ϕ i which are controlled by the ν,s ν ,s to the states |ϕ i, |ϕ i, |ϕ i and |ϕ i. Clearly, 0,s ↑,s ↓,s ↑↓,s f (x) = 1 − f (x). |ϕ i = |0, si such that s is simply the vibron number of 0,s Now, let us discuss the energy differences ′ ′ a Fock state, because the electron-vibron coupling does Δ (s, s ) in terms of the difference δ = s − s . N,N+1 not change the ‘empty’ states. For a mixed vibrational For tunneling-in processes one has δ > 0 if electrons state |ϕ i, s is related to the integer part of its cor- ν6=0,s have enough energy to excite more vibrons while for responding vibron number w . Indeed, using the Lang- ν,s δ < 0 the vibrations of the hybrid system are absorbed Firsov transformation one obtains the analytical result and allow tunneling of electrons from the leads at lower energies. The role of these transitions changes in the † S † −S w = hϕ |a a|ϕ i = hν, s|e a ae |ν, si ν,s ν,s ν,s case of tunneling-out processes: the system is ‘heated’ λ for δ < 0 and ‘cooled’ down if δ > 0. On the other = s + n , (15) hand, from Eq. (15) one notices that if λ /~ω ≪ 1 the ~ω 0 average vibron number are only slightly changed by the S −S λ ‘diagonal’ processes s = s . where we used the identities e ae = a + N , ~ω ˆ Figure 2 displays the tunneling energies as a function N |ν, si = n |ν, si and the fact that hν, s|a +a|ν, si = 0. S ν of δ and helps us to identify which transitions contribute On the other hand, the numerical diagonalization pro- N (ν) to the current for a symmetric bias window set by μ = 0 2 L,R vides w = N|A | which fits well to Eq. (15), ν,s N=0 sN E −E ′ ±p~ω/2, where p is an odd positive integer. For ν ν at least for the lowest vibronic components. example, the four dashed lines in Fig. (2) correspond to The vibrationally ‘excited’ states correspond to s > 0, μ = Δ ± ~ω/2 and μ = Δ ± ~ω/2. L,R 0,1 L,R 1,2 but it should be mentioned that even the lowest-energy In agreement with the analytical results obtained via states |ϕ i have a non-vanishing vibron number w ν,s=0 ν,0 the Lang-Firsov transformation, the eigenvalues corre- as they are not entirely made of a ‘free’ state |ν, N = sponding to the same electronic configuration ν are sep- 0i. Indeed, for the parameters considered here we find arated by integer multiples of vibron quanta, that is (see Eq. (5)) that the weights of |ν, N = 0i for the one- E = E + s~ω. This implies that the tunneling ν,s ν,0 (↑) (↓) 2 2 and two-particle states are |A | = |A | = 0.9 and 0,0 0,0 energies are also equally spaced, that is Δ (s, s ) = 0,1 (↑↓) 2 ′ ′ ′ |A | = 0.7 while the corresponding vibron numbers ε˜ +(s−s )~ω and Δ (s, s ) = ε˜ +U +(s−s )~ω, where 1 1,2 1 0,0 are w = w ≈ 0.085 and w = 4w . ε˜ = ε −λ /~ω and U is the direct interaction term V ↑,0 ↓,0 ↑↓,0 σ,0 1 1 1111 Note that the two-particle ground state carries more from the two-body Coulomb operator in Eq. (2). For the vibrons because the coupling between the conducting parameters chosen here we find U ∼ 1.67 meV. 5 transferred across the system along each turnstile cycle. In the first regime we set the chemical potentials of the μ , Q=1,2 leads such that the system is charged with two electrons and then completely depleted, hence Q = 2. For the sec- ond regime μ is pushed up to μ = 1.65 meV such that R R the discharging sequence allows only the tunneling from μ , Q=1 the two-particle configuration | ↑↓i. Then at the end of the turnstile cycle the total charge transferred across the system is Q = 1. The selected values of the chem- ical potentials for the Q = 1 and Q = 2 operations are also indicated by horizontal solid lines in Fig. 2. These 0-1e μ , Q=2 two regimes should reveal the dependence of the electron- 1-2e vibron coupling on the number of levels contributing to -1 -4 -3 -2 -1 0 1 2 3 4 the transport. The effects of the turnstile operations Q = 1, 2 on the FIG. 2. The energy differences associated to sequential tun- displacement d and average vibron number N are pre- neling processes leading to transitions between electronic con- sented in Figs. 3 (a) and (b). For the two-particle pump- figurations with N and N + 1 electrons. δ denotes the dif- ing (see Fig. 3 (a)) the displacement of the single-mode ference between the average number of vibrons for the vibra- ′ nanoresonator roughly mimics the behavior of the po- tional states |s i, |s ′i. For the simplicity of writing we do not ′ tential χ applied on the left contact. More precisely, d indicate the pairs (s, s ) corresponding to the same tunneling increases quickly as the electrons enter the system, sat- energy. The horizontal lines mark the values of the chemical urates once the charge occupation reaches the maximum potential for which one obtains various turnstile regimes - see the discussion in the text. value Q = 2 (not shown) and then drops to zero on the discharging half-periods. Note that the oscillations of the displacement match the period of the turnstile cycle, One notes that pairs of vibrational components 2t = 0.7 ns. It is also clear that the NR bounces between {|s i, |s i} which differ by the same amount of vibron ν a maximum value d ≈ 0.24 fm which does not depend ν max quanta δ have equal tunneling energies and will there- on the turnstile cycle and the equilibrium position (i.e. fore contribute simultaneously to the current. However, d = 0). In particular, we have checked that d and the their Franck-Condon tunneling amplitudes are different average charge Q vanish simultaneously. This behav- and decrease if s, s correspond to excited vibronic states. ior confirms that the electron-vibron coupling is indeed We also find that the tunneling amplitude of the ‘diag- periodically switched on and off along a turnstile cycle. onal’ transitions is much larger than the one of the ‘off- For Q = 2 turnstile operation the average vibron num- diagonal’ transitions (i.e for s 6= s ) which decreases as ber N displays a more surprising behavior: (i) It reaches s − s increases. a steady-state value N ≈ 0.5 on the first charging se- quence but then drops to a lower yet non-vanishing value during the depletion cycle. (ii) By repeating the turnstile B. The turnstile regime operation the same pattern is recovered as more vibrons are stored in the NEMS. Eventually, N reaches a quasi- stationary regime around t = 4.5 ns (not shown). We We denote by t the period of the charging/discharging therefore see that along the depletion cycles the vibrons cycles, such that the time needed for each turnstile oper- are stored in the system in spite of the fact that the ation is 2t . The value of the loss coefficient κ = 0.5 μeV. electron-vibron coupling is ineffective since Q . The GME was solved numerically on a subspace contain- S ing the lowest in energy 20 vibron-dressed states. We The single-particle turnstile operation (Q = 1) leads have checked that adding more vibronic states will not to a similar behavior of the average vibron number (see qualitatively alter the presented results. Let us mention Fig. 3 (b)). However, N reaches lower quasistationary here that the decreasing value of the FC factors for tran- values when compared to the two-particle turnstile oper- sitions between highly excited vibronic states is essential ation. A significant difference is noticed in the displace- in order to set a reasonably small cutoff N . In princi- ment oscillations. At the end of each turnstile cycle the ple one can include more states in the calculations, but NR does not return to its equilibrium position but settles the numerical effort to solve the master equation in the down to a distance d = d /2 from its equilibrium posi- max non-markovian regime increases considerably. tion. This happens because in the Q = 1 turnstile regime The periodic switching functions χ which simulate the effect of the electron vibron-coupling is only reduced L,R the turnstile operation are square-shaped and oscillate but not turned off because one electron is always present out-of-phase, as shown in Figs. 3(a) and (b). We assume in the electronic system and therefore induces a minimal that the initial state of the hybrid system |ν = 0, N = 0i. ‘deflection’ of the NR. In this sense, the single-particle The numerical simulations were performed for two turn- turnstile operation can be seen as a way to dynamically stile regimes which differ by the number of charges Q switch between electron-vibron interactions correspond- Δ (meV) N,N+1 6 0.3 1 (a) s=0 1.4 (a) 0.8 | ↑↓i, Q=2 N s=1 0.25 χ s=2 L 0.6 1.2 0.4 0.2 1 0.2 0.8 0.15 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.6 (b) s=0 0.1 0.8 |0i, Q=2 s=1 0.4 s=2 0.6 0.05 0.2 0.4 0.2 0 0 0 0.5 1 1.5 2 2.5 3 3.5 Time (ns) 0.3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 1.4 1 (b) N (c) s=0 0.8 | ↑i + | ↓i, Q=1 0.25 χ s=1 1.2 s=2 0.6 0.2 1 0.4 0.2 0.8 0.15 0.6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.1 Time (ns) 0.4 0.05 FIG. 4. The relevant N-particle populations P of the ν,s 0.2 ground and excited vibronic states: (a) P for Q = 2 oper- ↑↓,s ation; (b) P for Q = 2 operation; (c) The total population 0,s 0 0 0 0.5 1 1.5 2 2.5 3 3.5 of single-particle configurations P = P + P for Q = 1 1,s ↑,s ↓,s Time (ns) operation. 2.5 7 (c) Q 1.5 3 that an ‘effective’ temperature T of the hybrid system 1 eff can be derived from the equilibrium distribution func- 0.5 tion n (ω, T ) corresponding to the calculated average B eff 0 0 0 0.5 1 1.5 2 2.5 3 3.5 vibron number N (see e.g. ). Using this equivalence we Time (ns) realize that both turnstile operations induce a sequence of ‘heating’ and partial ‘cooling’ processes on the NR, FIG. 3. The dynamics of the vibron number N and the dis- as already proved by the vibron dynamics. To explain placement d of the nanoresonator in the two turnstile regimes this behavior we look more closely at the populations which allow the net pumping of Q electrons along each cycle: P along each turnstile cycle for ν = 0, ↑, ↓, ↑↓. We ν,s (a) Q = 2, μ = 3.5 meV, μ = −0.25 meV and (b) Q = 1, L R discuss first the two-electron turnstile operation. From μ = 3.5 meV, μ = 1.65 meV. The dotted lines indicate L R Fig. 4(a) we observe that at the end of the charging cy- the functions χL,R which simulate the periodic on and off switching of the two contacts. (c) The charge occupation and cles the hybrid system is completely described by several the transient currents JL,R for the Q = 1 turnstile operation. two-particle configurations |ϕ i (smaller contributions ↑↓,s Other parameters: t = 0.35 ns . of P were not shown). We also find that the popu- ↑↓,s>2 lations P = ρ reach a maximum value shortly 1,s σs,σs after the coupling of the source lead and then vanish as the two-particle states are filled. ing to a fixed number of particles. On the other hand, the different response of the NR displacement can be used to Fig. 4(b) shows that on the discharging cycles the re- ‘read’ the number of charges transferred across the sys- duced density matrix of the system contains both the tem along the turnstile cycles, in the presence of vibrons. ‘ground’ and ‘excited’ purely vibronic states. Moreover, In Fig. 3(c) we plot for completeness the dynamics of the occupation of the states |ϕ i on each depletion 0,s>0 the total charge Q along the single-particle turnstile half-period increases until a quasistationary regime is operation and the corresponding transient currents J . reached. This explains why the mean vibron number L,R The latter display sharp peaks, their different amplitudes N , which collects contributions of the type w P , in- v ν,s ν,s being a consequence of the different rates at which the creases along each turnstile cycle. One can also easily system is charged or depleted (note that Q drops more check that the decreasing population of the ground state abruptly on each discharging half-period). We recall here configuration |ϕ i is balanced by the presence of ex- 0,s=0 Charge Displacement (fm) Displacement (fm) Current (nA) Vibron number Vibron number Population Population Population 7 cited vibronic states. pletely described by purely vibronic states and therefore hϕ |a |ϕ i = 0. 0,s 0,s The accumulation of vibrons in the empty system In Fig. 4(c) we present for completeness the popula- (i.e., the partial cooling mechanism) can be explained by carefully counting the various vibron-assisted tunnel- tions of one-particle configurations which describe the hybrid system for the Q = 1 turnstile operation. Clearly, ing processes connecting pairs of fully interacting states ′ ′ the discharging cycles are now described by single- {ϕ , ϕ }. The chemical potentials of the leads are ν,s ν ,s selected such that all relevant tunneling processes (diag- particle states |ϕ i. The empty states |ϕ i are σ=↑,↓,s 0,s no longer accessible in this case so they were not shown. onal or off-diagonal) are active, i.e., most of the energies Δ (s, s ) are within the bias window (μ , μ ), for By comparing Figs. 4 (b) and (c) one notices a lower oc- N,N+1 R L cupation of the excited states |ϕ i which explains N = 0, 1 (see the chemical potentials for the Q = 2 set- σ=↑,↓,s>0 ting in Fig. 2). For the Q = 1 operation we have instead why the ‘jumps’ and drops of the mean vibron number ′ ′ are less pronounced. This could be expected because the Δ (s − s ) < μ < Δ (s − s ) < μ for the most 0,1 R 1,2 L important tunneling processes. When looking at Fig. 2 electron-vibron coupling is now enhanced/reduced only due to a single electron which is added/removed from we notice that some transitions are left outside the bias window, e.g. the ones corresponding to Δ (δ = 3, 4). the system. 0,1 However, these transitions have a small tunneling am- In the following we investigate in more detail the role plitude and they will not significantly contribute to the of the bias window on the partial cooling processes in transport. It is easy to see that for the first charging the Q = 2 turnstile operation. To this end the chem- cycle of the Q = 2 operation the sequence of ‘diagonal’ ical potential of the drain reservoir is pushed up to transitions e.g |ϕ i → |ϕ i → |ϕ i involves only μ = 0.68 meV such that the main ‘heating’ processes 0,0 σ=↑,0 ↑↓,0 the lowest vibronic components (s = s = 0) with small associated to the depletion cycles are forbidden, that is ′ ′ vibron numbers w (see Eq. (15)). These transitions are Δ (s, s ) < μ for some s < s (see the lowest dotted ν,0 0,1 R also the strongest, as the corresponding FC factors are horizontal line in Fig. 2). Figure 5 shows that in this case the largest ones. Note also that on the first charging the mean vibron number does not display steps on the cycle the vibron absorption is not possible as the initial discharging cycles (as in Fig. 3(b)) but rather vanishes - state is |ϕ i, such that the ‘excited’ states |ϕ i can in other words, the hybrid system eventually cools down 0,0 σ,s>0 only be populated through ‘off-diagonal’ weaker transi- to the temperature of the thermal bath T . In order to tions, for example |ϕ i → |ϕ i. Finally, the charg- capture the slow evolution of N we increased the turn- 0,0 σ,1 ing cycle brings the second electron to the system and stile period to t = 1 ns. Further insight into the vibron opens more tunneling paths involving both diagonal and dynamics is given by the populations P of the purely 0,s off-diagonal processes, e.g., |ϕ i → |ϕ i → |ϕ i or vibronic states which are also presented in Fig. 5. Af- 0,0 σ,1 ↑↓,1 |ϕ i → |ϕ i → |ϕ i. It is therefore clear that at 0,0 σ,1 ↑↓,2 ter an initial increase, the excited states |ϕ i and |ϕ i 0,1 0,2 the end of the first charging cycle the system is described are slowly depleted in favor of the ground state |ϕ i 0,0 by the two-particle electronic configuration | ↑↓i and sev- whose population increases uniformly on each discharg- eral vibronic components |s i with the associated vibron ing sequence. This behavior differs from the one shown ↑↓ numbers w . ↑↓,s in Fig. 4(b) and suggests a ‘redistribution’ of probability between various purely vibronic states. In the following Now, during the first depletion cycle this mixed struc- we explain this effect through the interplay of tunneling- ture of the reduced density matrix allows the activation of out and -in processes which involve the drain lead. multiple ‘diagonal’ and ‘off-diagonal’ tunneling out pro- cesses between (N + 1)−particle and N−particle config- The sudden drop of N right after opening the con- urations. For example the ‘diagonal’ backward sequence tact to the right reservoir is due to the ‘cooling’ transi- |ϕ i → |ϕ i → |ϕ i leaves the hybrid system in the tions |ϕ i → |ϕ ′i for δ = s − s > 0, whose ener- ↑↓,1 σ,1 0,1 σ,s 0,s first vibronic excited state whose population P ≈ 0.2 gies are still above μ (see Fig. 2). On the other hand, 0,1 R in Fig. 4(b). The small population P is due to the sim- the excited vibronic states |ϕ i are still being pop- 0,2 σ,s>0 ilar ‘off-diagonal’ transition from |ϕ i → |ϕ i. Other ulated via vibron-conserving transitions |ϕ i → |ϕ i σ,1 0,2 σ,s 0,s transitions leading to vibrational ‘cooling’ can be also and to a lesser extent by the partial ‘cooling’ transition identified. As a result the mean vibron number drops |ϕ i → |ϕ i. This scenario is confirmed by the initial σ,2 0,1 over the depletion cycle, but does not vanish due to the increase of the populations P and P . We find in- 0,1 0,2 ‘diagonal’ tunneling events. stead that the much slower vibronic relaxation involves two more sequential tunnelings, one from the reservoir At the next charging cycle the excited single-particle to the central system and another one back to it. In- states |ϕ i will be fed by both diagonal and off-diagonal σ,1 deed, given the fact that Δ (δ < 0) are below μ , elec- transitions, because when switching on the coupling to 0,1 R trons can tunnel back from the contact via ‘cooling’ tran- the left lead the reduced density matrix of the system sitions |ϕ i → |ϕ i (for s < s). Finally, the lower- reads ρ(2t ) = |ϕ ihϕ |. As a consequence, the 0,s σ,s p 0,s 0,s temperature single-particle states are depleted through population P almost doubles with respect to the first ↑↓,1 ′ ′ diagonal transitions |ϕ i → |ϕ i. charging cycle, whereas P brings a small contribution σ,s 0,s ↑↓,2 as well. The vanishing of the displacement d on each dis- Turning back to the symmetric bias setting (see Fig. 3 charging sequence is mandatory, as the system is com- (a)), it is readily seen that the tunneling-mediated cool- 8 1 1 1.6 (a) 0.8 0.8 1.4 s=0 s=1 1.2 s=2 0.6 0.6 0.8 0.4 0.4 0.6 λ =0.162 0.4 λ =0.096 0.2 0.2 0 λ =0.065 0.2 λ =0.028 0 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Time (ns) Time (ns) (b) FIG. 5. The dynamics of vibron number N and of the pop- 5.8 ulations P0,s for the the Q = 2 turnstile protocol. In contrast to Fig. 3(a) the complete ‘cooling’ of the nanoresonator is in- 5.6 sured by suppressing the tunneling-out tunneling processes. Other parameters: μ = 3.5 meV, μ = 0.68 meV, t = 0.5 L R p 5.4 ns. 5.2 ing mechanism presented above cannot be active. In this λ =0.162 λ =0.096 case, electrons are not allowed to tunnel back to the 4.8 0 λ =0.065 central system because μ lies below all transition en- 0 λ =0.028 4.6 ergies. Moreover, the cooling processes |ϕ i → |ϕ i σ,s 0,s with s < s are overcome by the heating processes such 1 2 3 4 5 6 7 Peak index that N settles down to a non-vanishing value after the onset of the discharging sequence. FIG. 6. The effect of the electromechanical coupling strength In order to check whether the electron-vibron cou- λ0 (given in meV units) on (a) the vibron number Nv and pling affects not only the dynamics of the NR but also (b) on the peak amplitude of the transient current J . The the transport properties of the electronic subsystem, we parameters correspond to the Q = 2 turnstile protocol: μ = present in Fig. 6(a) the vibron dynamics for several val- 3.5 meV, μ = −0.25 meV, t = 0.35 ns. R p ues of the electron-vibron coupling strength λ . This pa- rameter can be tuned by changing either the equilibrium distance between the electronic system and the nanores- λ because at such short times the vibrons are not yet onator (as shown in previous work ) or the NR mass M. activated. For the larger value λ = 0.162 meV a steep The amplitude of the heating and cooling cycles decreases reduction of the peak is noticed after two charging half- with λ and the hybrid system approaches the quasista- periods. A similar behavior is recovered for the output tionary regime much faster at larger values of λ . For 0 current J (not shown). By comparing Figs. 6(a) and (b) example, a considerable difference is noticed between the one infers that the attenuation of the peak amplitude is first two cycles at λ = 0.162 meV, the next cycles being 0 correlated to the emergence of the quasistationary regime rather similar. for the heating/cooling sequences. In Fig. 6(b) we collect the amplitudes associated to the We also considered other shapes for the switching func- first seven peaks of the current J and to the different tions χ and we recovered similar effects of the turnstile L L,R electron-vibron couplings considered in Fig. 6(a). The regime on the nanoresonator, i.e. heating/cooling on the peak evolution over few turnstile cycles can also be ex- charging/discharging half-periods. Figure 7 shows the vi- tracted from transport measurements and provides in- bron dynamics N and the displacement d for smoother direct insight on the vibron dynamics. In the weakly switching functions. When compared to the results pre- interacting case (λ = 0.028 meV) the amplitudes of the sented in Fig. 3(a) we noticed minor changes in the lo- peaks are nearly equal and one cannot discern the negli- cal maximum and minimum values of the average vibron gible effect of the electron-vibron coupling on the trans- number. However, the most important effect is a delay port properties. In contrast, as λ increases, the peaks of nanoresonator’s response to the switching functions, display noticeable differences. More precisely, their am- i.e. N and d do not increase/decrease immediately af- plitude gradually decreases from one cycle to another un- ter charging/discharging. If one is interested in imple- til it reaches a quasistationary value (for λ = 0.096 meV menting faster heating and cooling processes separated this value is roughly 5.5 nA). Note that the first peak of by longer ‘isotherms’ (i.e time intervals with constant the charging current J is less sensitive w.r.t. changes of vibron number N ) the square-wave driving is the most L v Vibron number Populations J peak amplitude (nA) Vibron number L 9 0.3 pumping one electron per turnstile cycle while keeping 1.4 the lowest level occupied one initializes a configuration 0.25 made by single-particle states ’dressed’ by vibrons. 1.2 χ On the other hand, the charging cycles activate the 0.2 1 electrostatic coupling and the vibron number increases. It also turns out that both turnstile operations induce 0.8 0.15 a heating of the nanoresonator when the electron-vibron coupling is turned on and at least a partial cooling when 0.6 0.1 it is turned off. 0.4 0.05 0.2 IV. CONCLUSIONS 0 0 0 0.5 1 1.5 2 2.5 Time (ns) We proposed and studied theoretically a quantum turnstile protocol for switching on and off the effect of FIG. 7. The dynamics of the vibron number N and displace- electron-vibron coupling between a biased mesoscopic ment d of the nanoresonator for smoother switching functions system and a vibrational mode. A detailed analysis of χ . Other parameters: λ = 0.096 meV, μ = 3.5 meV, L,R 0 L the vibron-assisted tunneling processes is provided by the μ = −0.25 meV, t = 0.35 ns. R p populations of the vibron-dressed states which are cal- culated within the generalized master equation method. We identify the role of various tunneling processes in effective. the vibron emission (heating) and absorption (cooling) Finally, we stress that the oscillations of the displace- processes. The turnstile charging and discharging cy- ment record the charge variations along the turnstile op- cles impose periodic variations of the nanoresonator’s dis- erations but do not discern between the vibron dynamics. placement with respect to its equilibrium value. As the In order to understand why this happens let us observe electronic system empties the displacement vanishes. In- first that if the coherences hϕ |ρ(t)|ϕ ′i are negligi- ν,s ν,s stead, a turnstile operation which allows only a partial ble then from Eq. (16) one gets a simpler formula for the depletion sets a lower bound of the displacement due to displacement: an extra electron residing in the system. 2λ l 0 0 The values of the displacement obtained in our model d ≈ n P . (17) ν ν,s ~ω are probably too small to be detected. However, d in- ν,s creases as more electrons tunnel across the system during a turnstile cycle. This could be achieved by increasing Secondly, since on the charging sequences the system set- the bias window such that more electronic configurations tles down to the two-electron configuration (i.e., n = 2) participate in transport. Alternatively, one can consider and P = 1 for all chemical potentials μ < ↑↓,s R lighter nanoresonators and therefore larger values of the Δ (s − s = 0) it follows that d cannot depend on μ , 0,1 R oscillator length l . even if each occupation P does. Eq. (17) also confirms ν,s We find that in general the average number of vibrons the doubling of the quasistationary displacement d max on the charging cycles with respect to the value attained does not vanish along the discharging cycles when the electron-vibron coupling is ineffective. In the quasista- along the depletion cycles of the Q = 1 turnstile opera- tion, as shown in Figs. 3(a) and (b). tionary regime the same amount of vibrons is emitted and absorbed along a turnstile cycle. Otherwise stated, the For the parameters selected here the coherences cor- responding to states with the same electronic configura- system undergoes periodic heating and cooling processes. ′ A complete cooling to the equilibrium temperature of the tions but different vibron numbers (i.e., hϕ |ρ(t)|ϕ i) ν,s ν,s do exist but they are indeed too small to induce a no- leads or of a thermal bath can be achieved by a suitable choice of the chemical potential of the drain reservoir. ticeable change of the various observables (not shown). In fact, we record some fast oscillations of the displace- We also show that the peak amplitude of the transient ment on the ‘steps’ of each turnstile cycle; the period of currents decreases as the strength of the electron-vibron coupling increases. Moreover, it turns out that as the these oscillations coincides with those of the coherences mentioned above but one can see from Fig. 3 that their heating/cooling cycles attain the quasistationary regime the peak amplitude gradually reduces to a value which amplitude is hardly noticeable. Based on these results we state that the quantum does not depend on the charging/discharging half-period. turnstile regime provides a dynamical switching of the Let us emphasize that the quantum turnstile dynam- electron-vibron coupling effects on the hybrid system. ics differs considerably from the normal transport regime Once the depletion process is complete the electron- when both leads are simultaneously coupled to the sys- vibron coupling is ineffective. However, the effect of the tem and for which one can only notice a heating process, latter is imprinted in the non-vanishing populations of as the average vibron number uniformly increases before the excited vibrational states ϕ . Alternatively, by reaching its stationary value. Also, in the present setting 0,s>0 Displacement (fm) Vibron number 10 the actuation of the nanoresonator is only due to the elec- 0221 and from the Romanian Core Program PN19-03 tronic current as there is no additional driving signal. In (contract No. 21 N/08.02.2019). B.T. and V.M. were other words, we consider that before the electronic sub- also supported by TUBITAK Grant No. 117F125. B.T. system is coupled to the leads the nanoresonator is in the further acknowledges the support from TUBA. static deflection mode. ACKNOWLEDGMENTS R.D., V.M. and S.S. acknowledge financial support from CNCS - UEFISCDI grant PN-III-P4-ID-PCE-2016- 1 16 M. Poot and H. S. J. van der Zant, Phys. Rep. 511, 273 C. Schinabeck, A. Erpenbeck, R. H¨artle, and M. Thoss, (2012). Phys. Rev. B 94, 201407(R) (2016). 2 17 L. Cockins, Y. Miyahara, S. D. Bennett, A. A. Clerk, and A. Mitra, I. Aleiner, and A. J. Millis, Phys. Rev. B 69, P. 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Published: May 21, 2020

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