## Abstract

A double quantum dot device is a tunable two-level system for electronic energy states. A dc electron current was used to directly measure the rates for elastic and inelastic transitions between the two levels. For inelastic transitions, energy is exchanged with bosonic degrees of freedom in the environment. The inelastic transition rates are well described by the Einstein coefficients, relating absorption with stimulated and spontaneous emission. The most effectively coupled bosons in the specific environment of the semiconductor device used here were acoustic phonons. The experiments demonstrate the importance of vacuum fluctuations in the environment for quantum dot devices and potential design constraints for their use for preparing long-lived quantum states.

Electronic quantum devices allow the quantum mechanical properties of electrons confined to small regions in a solid to be explored. Existing devices include semiconductor resonant tunneling diodes (1) (based on quantum mechanical confinement), superconducting Josephson junction circuits (2) (based on macroscopic phase coherence), metallic single-electron transistors (3) (based on quantization of charge), and molecular electronic devices (4). The principle of operation in circuits of these devices is based on controlling energy states, for instance, by means of an external (gate) voltage. Thermal energy is always a source for unwanted transitions and errors. Even at zero temperature, however, vacuum fluctuations in the environment can give rise to transitions between states of nonequal energy by spontaneous emission of an energy quantum. Such inelastic transitions cause errors in many proposed schemes for quantum circuits. Here we studied inelastic transitions in a fully controllable, two-level quantum system realized in a double quantum dot device. We can relate the transition rates involving emission to absorption rates by the Einstein coefficients over the full energy and temperature range we studied. At the lowest temperature (23 mK), we directly measured the energy-dependent rate for spontaneous emission and determined that, in our specific semiconductor device, this energy is emitted into the environment formed by acoustic phonons.

Our double quantum dot (Fig. 1A) is fabricated in the two-dimensional electron gas (2DEG) of an AlGaAs-GaAs semiconductor heterostructure (5). The source and drain are large 2DEG regions that serve as leads for contacting current and voltage wires. The two dots, L and R, are separated from each other and from the leads by potential barriers induced by negative voltages applied to the three metallic gates. Tunneling between the different regions is sufficiently strong to detect current but weak enough that the number of electrons in each dot is a well-defined integer. The energy states in such fully confined regions are discrete, 0D states, resembling discrete atomic states (6, 7). The discrete energies include contributions from single-electron charging energies (arising from Coulomb interactions) and from quantum-mechanical confinement. The lowest energy state for one additional electron in the L dot is labeled in Fig. 1, B to D, as*E*
_{L}, and similarly*E*
_{R} for the R dot. Figure 1C illustrates the resonance condition, *E*
_{L} =*E*
_{R}, in which case an electron can tunnel elastically from an occupied state in the source via*E*
_{L} and *E*
_{R} to an empty state in the drain. Such tunneling sequences of single electrons are regulated by the Coulomb charging energies (3, 7). When the two states are not aligned, *E*
_{L} ≠*E*
_{R}, only inelastic transitions are allowed for which some energy needs to be exchanged with the environment. A measured off-resonance current, therefore, directly provides information about the coupling between electrons on the dots to degrees of freedom in the environment. The inelastic rates can be analyzed with well-developed methods in quantum optics (8, 9).

A typical current spectrum versus ɛ ≡ *E*
_{L} −*E*
_{R} at our lowest lattice temperature*T = *23 mK (10) is shown in Fig. 1E. The gate voltages *V*
_{GR} and*V*
_{GL} are swept simultaneously such that the respective energies are like those illustrated in Fig. 1, B to D; that is, ɛ = 0 occurs in the middle between the Fermi energies of source and drain, μ_{S} and μ_{D}, and |ɛ| =*eV*
_{SD} (its maximum) corresponds to having the states *E*
_{L} and *E*
_{R}aligned to one of the Fermi energies. To analyze the large asymmetry, we decomposed the total current *I*
_{tot}(ɛ) =*I*
_{el}(ɛ) + *I*
_{inel}(ɛ > 0) into a symmetric part *I*
_{el}(ɛ) =*I*
_{el}(−ɛ) (dashed curve) and the remaining asymmetric part *I*
_{inel}(ɛ > 0) (dotted-dashed curve). At *T* = 0, I_{el}(ɛ) is due to elastic tunneling and has a Lorentzian line shape*I*
_{el}(ɛ) =*I*
_{el,max}ω^{2}/(ω^{2} + ɛ^{2}) (11). The full width at half maximum, 2ω, can be tuned by the central gate voltage*V*
_{GC} roughly from 4 to 20 μeV. From measurements of *I*
_{el}(ɛ) at positive and negative *V*
_{SD}, it is possible to extract values for the tunnel couplings Γ_{L}, Γ_{R}, and *T*
_{c} (11,12).

The remaining current, *I*
_{inel}(ɛ > 0), which is nonzero only for ɛ > 0, is due to inelastic tunneling. In Fig. 1E,*I*
_{inel} is nonzero over an energy range of ∼100 μeV, despite the thermal energy *kT* (23 mK) = 2 μeV being much smaller (*k* is Boltzmann's constant). (The irregular fine structure is discussed below.) In general, we found that*I*
_{inel} vanishes when one of the levels,*E*
_{L} or *E*
_{R}, crosses one of the two Fermi energies. In the specific case of Fig. 1E,*E*
_{L} and *E*
_{R} cross the Fermi energies simultaneously, implying that *I*
_{inel} is cut off at ɛ = *eV*
_{SD}. Below this cutoff, the value of *I*
_{inel} was not influenced by the value of *V*
_{SD} (13). For*T* = 0, we can write the condition for a nonzero inelastic current as μ_{S} > *E*
_{L} >*E*
_{R} > μ_{D} = μ_{S} −*eV*
_{SD}. The amount of inelastic current depends on the transition rates as *I*
_{inel}(ɛ) =*e*[Γ_{L}
^{−1} + Γ_{i}
^{−1}(ɛ) + Γ_{R}
^{−1}]^{−1}. When the inelastic rate Γ_{i}(ɛ) from *E*
_{L} to*E*
_{R} is much smaller than the rates through the outer barriers, then *I*
_{inel}(ɛ) =*e*Γ_{i}(ɛ).

The effect of a nonzero temperature on the current is shown in Fig. 2A. A higher temperature *T*enhances *I*
_{tot} on both the emission (ɛ > 0) and the absorption (ɛ < 0) side. The absorption spectrum shows an exponential temperature dependence,*e*
^{ɛ/kT} (dashed lines), for absolute energies larger than the elastic current measured at 23 mK, that is |ɛ| > *w*.

To analyze the temperature dependence, we assumed boson statistics for the degrees of freedom in the environment. The average occupation number *〈n〉* of environmental modes at energy ɛ is given by the Bose-Einstein distribution function: 〈*n*〉 = 1/(*e*
^{ɛ/kT} − 1). The rates for absorption, *W*
_{a}, and emission, *W*
_{e}, can be expressed very generally by *W*
_{a} =*B*
_{a}ρ and *W*
_{e}= *A + B*
_{e}ρ, where the Einstein coefficients stand for spontaneous emission *(A)* and stimulated emission *(B*
_{e}) and absorption*(B*
_{a}), and ρ is the energy density (8). From the Einstein relations, *B*
_{a}= *B*
_{e} = *A〈n〉/ρ* (8), we obtain(1)
To test whether the inelastic current follows emission and absorption statistics, we calculated the full current spectrum from Eq. 1. First, we obtained the spontaneous emission rate from*A*(ɛ) = *I*
_{inel}(ɛ > 0,*T* = 23 mK)/*e.* The trace at 23 mK is effectively at zero temperature for ɛ ≫ 2 μeV because then 〈*n*〉 ≪ 1. The emission current at higher temperatures follows from *I*
_{inel}(ɛ > 0,*T*) = *e*(〈*n*〉 + 1)*A*(ɛ), whereas the absorption current follows from*I*
_{inel}(ɛ < 0, *T*) =*e*〈*n*〉*A*(−ɛ). The reconstructed current spectrum is shown in Fig. 2B. The central part of the curves (|ɛ| < 10 μeV) is kept blank because Eq. 1 does not include the *T* dependence of *I*
_{el}. The calculated current reproduces the measured current well up to 200 mK. Even the small steplike feature seen at ɛ ∼ 30 μeV is reflected by a shoulderlike feature at ɛ ∼−30 μeV in the measured and in the calculated absorption spectra (indicated by arrows). For*T* > 200 mK, the measured current significantly exceeds the calculated current, which is probably due to thermally excited electrons. (Equation 1 only describes the *T* dependence of the environment. The thermal excitations in the electron leads are not included.) Further confirmation of the applicability of the Einstein relations to our quantum dot system follows from the prediction [*I*
_{inel}(ɛ > 0) −*I*
_{inel}(ɛ < 0)]/*eA*(|ɛ|) = [*W*
_{e} −*W*
_{a}]/*A* = 1, which is valid independent of *T.* A plot of the normalized rates,*W*
_{a}/*A* and*W*
_{e}/*A*, versus*kT*/|ɛ| for various ɛ and *T* up to 200 mK is shown in Fig. 2C. The measured data closely follow the prediction [*W*
_{e} −*W*
_{e}]/*A* = 1; that is, the normalized rates, *W*
_{a}/*A* and*W*
_{e}/*A*, differ by 1 over the temperature range *T* < 200 mK without fitting any parameter.

The inelastic rate for a two-level system coupled to a bosonic environment at *T* = 0 is expected to have a*T*
_{c}
^{2} dependence (14,15). Still, without identifying the bosonic environment, we can test this dependence on the elastic tunnel coupling*T*
_{c} between the two dots. The inelastic current increases with *T*
_{c}, as shown inFig. 3A. For the largest coupling, we obtain a saturation where the elastic current peak can no longer be distinguished. By fitting the elastic current part to a Lorentzian line shape (11) we can obtain rough estimates for*T*
_{c} as long as the current is less than the saturation value. We found that with these fitted values, the inelastic current scales as *T*
_{c}
^{α} with an exponent α = 2.5 ∼ 3, perhaps somewhat greater than expected. The effect of the increased coupling on the symmetric part of the current at low *T* is shown in Fig. 3B. For small tunnel coupling, we always obtain Lorentzian line shapes. For increasing couplings, the data still fit to a Lorentzian tail on the absorption side. However, we generally found significant deviations for small ɛ, implying that for large coupling the elastic and inelastic rates can become of the same order. This effect may form a significant limitation for the coherence time in coupled quantum devices (16).

The importance of fluctuations in the environment on electron tunneling through quantum devices has been recognized for a long time. Environmental studies on Coulomb blockade devices with two or more tunnel junctions have only discussed effects due to absorption (3, 17). For emission it is required that electrons are first pumped to a higher energy state, as has recently been done in a superconducting Cooper pair transistor under microwave irradiation (18). In the case of a double dot, pumping occurs when*E*
_{L} > *E*
_{R} and an electron tunnels in from the left reservoir to*E*
_{L}. A double dot thus offers a unique two-level system that is pumped by a dc voltage without the induction of heating currents. It is therefore possible to reach an out-of-equilibrium situation so close to *T* = 0 that vacuum fluctuations become the main source for generating electron transport.

To identify whether photons, plasmons, or phonons form the bosonic environment (19), we measured spontaneous emission spectra while placing the double dot in different electromagnetic environments. In the regime from 10 to 100 μeV, the typical wavelengths are 1 to 10 cm for photons and 0.3 to 30 cm for 2DEG plasmons. We tested the coupling to the photonic environment by placing the sample in microwave cavities of different size (20). To check the coupling to plasmons, we measured different types of devices with largely different dimensions of the 2DEG leads, gate pads, and bonding wires. Both types of variation had no effect on the emission spectra; even the fine structure was reproduced.

The third option of acoustic phonons is the most likely possibility (21). Phonon emission rates have been calculated for single dots (22). For a double dot system, we can obtain the general energy dependence (15). For a deformation potential, we expect a rate dependence of ɛ^{D−2} (ɛ for 3D phonons and constant for 2D phonons) and for piezoelectric interaction of ɛ^{D−4} (1/ɛ for 3D phonons and 1/ɛ^{2} for 2D surface acoustic waves) (23). InFig. 3C traces measured on two different types of devices are compared. Here, the emission current is plotted versus ɛ on a log-log scale. Ignoring the bumps, we found an energy dependence between 1/ɛ and 1/ɛ^{2}. This result implies that the dominant emission mechanism is the piezoelectric interaction with 2D or 3D acoustic phonons. Note that a 1/ɛ or 1/ɛ^{2} dependence should be avoided in coherent devices, because the inelastic rate becomes large near resonance (ɛ ∼ 0) (16).

The bumps observed in both type of devices suggest the existence of resonances, for instance, due to a finite size in the phonon environment. The bumps are particularly clear in the derivative of the current to energy (dotted curves in Fig. 3A). The large bump in Fig. 3A at ɛ = 30 μeV corresponds to a frequency of *f* = ɛ/*h* = 7.3 GHz. For 3D phonons, this yields a wavelength λ^{3D} =*s*
^{3D}/*f* = 640 nm (*s*
^{3D} = 4800 m/s is the 3D sound velocity), whereas for 2D surface acoustic waves λ^{2D} =*s*
^{2D}/*f* = 380 nm (*s*
^{2D} = 2800 m/s). Both wavelengths, more or less, fit with the dimensions of the two quantum dot devices. We have not yet been able to control these resonances by studying devices with a variety of gate dimensions. However, we predict that it will be possible to gain control over the phonon environment by making 3D phonon cavities in hanging bridges (24) or by creating a 2D phonon band gap with a periodic gate geometry (25).

↵* To whom correspondence should be addressed. E-mail: leo{at}qt.tn.tudelft.nl