2 E e e − − FIG. 2: Transmission probabilities calculated from (1) as a function of incoming electron energy with Γ L = Γ R = 0 . 04 ℏ ω and λ = 0 . 4 ℏ ω . Below the resonance energy ε 0 the ground state transmission functions T 00 ( ε i ) and T 01 ( ε i ) essentially vanish. The insert shows the lower sideband, where stimu- lated emission of a vibrational quantum T 10 ( ε i ) is the domi- nating transmission channel. FIG. 1: A molecule between two metal contacts is represented by a resonant state (for example the lowest unoccupied molec- ular orbital) and a vibrational potential. If the molecule is ini- tially vibrationally excited, an electron below the resonance may tunnel through the molecule by absorbing a quantum of vibration. When the molecule is initially in its vibrational ground state, transmission is not possible. pends on the ratio T$_{11}$/T$_{10}$ shown in Fig. 3 as a function of the vibrational coupling. For small coupling param- eters the ratio approaches zero and the first electron to be transmitted is therefore highly likely to induce a vi- brational decay and close the channel. For λ ≪ ℏ ω we thus have T$_{11}$ ≪ T$_{10}$ which is needed for single electron transmission. One might worry that the inelastic trans- mission amplitude may become too small for anything to happen in this case. However, the absolute amplitude of sideband transmission can easily be small if the vibra- tional lifetime of the molecule is long. For physisorbed molecules the lifetime of a vibrational state is typically on the order of nanoseconds, whereas for example with Au(111), we can use the density of states to estimate that ∼ 40 electron hit each surface atom per picosecond within 0 . 1 eV of the Fermi level. Thus, as long as T$_{10}$ is on the order of ∼ 1 × 10 − 4 there will be plenty of attempts to result in a single transmission event. and a similar expression for the coupling to the right lead Γ$_{R}$. Without the vibrational coupling, the reso- nance spectral function would then be a Lorentzian with full width at half maximum given by Γ = Γ$_{L}$ + Γ$_{R}$. We will be interested in the regime Γ , k$_{B}$T ≪ ℏ ω$_{i}$ , but we will not restrict ourselves to the classical limit Γ ≪ k$_{B}$T where the contact current can be expressed in terms of rate equations $^{8,9,14}$. Instead, we consider the transmis- sion matrix T$_{n}$ ( ε$_{i}$, ε$_{f}$ ) for an electron with initial state energy ε$_{i}$ and final state energy ε$_{f}$ . Within scattering theory, the transmission matrix can be expressed in terms of a two-particle Green function 6 which can be evaluated exactly in the wide band limit $^{15}$. The subscript n refers to the initial state of the oscillator and integrating out the final state energy, one obtains the transmission probabil- ity T$_{nm}$ ( ε$_{i}$ ) that an electron with initial state energy ε$_{i}$ is transmitted while the molecule makes the vibrational transition n → m . In the appendix we have calculated the transmission matrix in the ground and first excited state and in Fig. 2 we show the transmission probability corresponding to four different vibrational transitions of the molecule. It shows that incoming electrons with energies below ε$_{0}$ − ℏ ω have a vanishing probability of transmission ( ∑ $_{n}$T$_{0}$$_{n}$ ) when the molecule is in its vibrational ground state. This means that if a bias voltage is applied such that the Fermi level of the upstream contact is at ε$_{F}$ = ε$_{0}$ − ℏ ω no current will be observed when the molecule is in its vibrational ground state. If the molecule is vibrationally excited, e.g. by means of a IR laser, transmission becomes possi- ble through the low lying vibrational sideband ( T$_{11}$ and T$_{10}$ ). However, the first electron which is transmitted through the T$_{10}$ channel induces a transition to the vi- brational ground state and thus closes the transmission channel completely. Hence, the net effect is that applying a short laser pulse to the molecular contact can induce the transmission of a few electrons. The distribution of the number of electrons being transmitted as a result of a vibrational excitation de- FIG. 3: Ratio of elastic and inelastic transmission probabili- ties in the first vibrationally excited state. At weak coupling the elastic transmission T 11 goes to zero indicating that a vibrational excitation results in only a single electron being transmitted.