Microwave detection of electron-phonon interactions
in a cavity-coupled double quantum dot
Quantum confinement leads to the formation of discrete electronic states in quantum dots. Here we probe electron-phonon interactions in a suspended InAs nanowire double quantum dot (DQD) that is electric-dipole coupled to a microwave cavity. We apply a finite bias across the wire to drive a steady state population in the DQD excited state, enabling a direct measurement of the electron-phonon coupling strength at the DQD transition energy. The amplitude and phase response of the cavity field exhibit features that are periodic in the DQD energy level detuning due to the phonon modes of the nanowire. The observed cavity phase shift is consistent with theory that predicts a renormalization of the cavity center frequency by coupling to phonons.
pacs:03.67.Lx, 73.21.La, 42.50.Pq, 85.35.Gv
Phonons, the quantized lattice vibrations in a crystalline solid, are ubiquitous in condensed matter systems and impact the properties of bulk and nanostructured materials. For example, Raman scattering measurements provide a detailed probe of the phonon density of states in carbon nanotubes Dresselhaus et al. (2005) and it is now known that phonons limit the maximum current in single nanotube devices Yao et al. (2000); Javey et al. (2003); Park et al. (2004). Similarly, spin relaxation in semiconductors is generally governed by processes that involve phonon emission Khaetskii and Nazarov (2001); van der Wiel et al. (2002); Hanson et al. (2007). Electronic relaxation processes involving phonons can even be harnessed to cool mechanical degrees of freedom in nanostructures Wilson-Rae et al. (2004); Zippilli et al. (2009).
Semiconductor double quantum dots (DQDs) are well-suited for probing electron-phonon coupling Brandes and Kramer (1999); Brandes (2005); Weber et al. (2009, 2010); Roulleau et al. (2011) since inelastic interdot tunneling generally involves spontaneous emission of a phonon in order to conserve energy Fujisawa et al. (1998). Moreover, the DQD energy level difference is electrically tunable, which allows the effective electron-phonon coupling strength to be probed as a function of phonon energy. Confinement can further influence phonon modes in nanostructures and new methods of controlled nanowire placement Flohr et al. (2011); Nilsson et al. (2008) may allow for careful engineering of the phonon spectrum in semiconductors, similar to recent optomechanics experiments with suspended carbon nanotubes Aspelmeyer et al. (2014); Laird et al. (2012); Ares et al. (2016).
In this Letter, we investigate the interplay of electrons, phonons, and photons in a cavity-coupled InAs nanowire DQD which is mechanically suspended in vacuum. The interaction of photons and electrons in DQDs has been studied extensively in the circuit QED architecture, where the charge dipole of a DQD is coupled to a microwave cavity Frey et al. (2012); Delbecq et al. (2011); Petersson et al. (2012). A phonon sideband has recently been observed in DQD masers that are driven by single electron tunneling Liu et al. (2015) and related theory suggests that the detailed energy-dependence of the one-dimensional nanowire phonon spectral density should have observable consequences in the photon emission rate Gullans et al. (2015), although this has not yet been observed. Here we measure the dc current as a function of DQD energy level detuning and show that it exhibits periodic oscillations that are consistent with the phonon spectral density in a confined nanostructure Weber et al. (2010); Roulleau et al. (2011). Measurements of the cavity amplitude and phase response reveal the detailed energy dependence of the electron-phonon coupling and exhibit a response that is periodic in . We employ a microscopic theoretical model of the device which suggests that the dispersive cavity phase shift is due to a renormalization of the cavity center frequency by coupling to phonons Gullans et al. ().
The suspended nanowire DQD device is shown schematically in Fig. 1(a). A 50 nm diameter InAs nanowire Schroer and Petta (2010); Schroer et al. (2010) is manually placed across two lithographically defined Ti/Au pedestals using a long working distance optical microscope Flohr et al. (2011). The pedestal thickness results in a 15 nm vacuum gap between the nanowire and Ti/Au electrostatic back gates (green) (, , , , ) which are used to produce the double-well confinement potential. In contrast to earlier experiments Nadj-Perge et al. (2010); Petersson et al. (2012), the back gates are not coated with SiN dielectric. Instead, electrical isolation is achieved using the vacuum gap, which further confirms that the nanowire is physically suspended above the back gates. Ti/Au source (S) and drain (Cavity+D) contacts (red), with a separation of 380 nm, are deposited on top of the pedestals. A scanning electron microscope (SEM) image of a representative device is shown in Fig. 1(b).
In order to study the interaction of electrons in the DQD with photons in the cavity and phonons in the nanowire, the DQD is electric-dipole coupled to the voltage antinode of a Nb superconducting transmission line resonator (Cavity+D) with resonance frequency MHz and quality factor Liu et al. (2015, 2017). A schematic diagram illustrating the key elements of the device is shown in Fig. 1(c). The electrical connection between the DQD and cavity is shown in Fig. 1(d). Electron-phonon coupling is probed by measuring the dc current through the DQD and the cavity amplitude and phase response as a function of Stehlik et al. (2015). Measurements are performed in a dilution refrigerator with a base temperature = 10 mK.
Figure 2(a) shows the current through the device as a function of the gate voltages and for = 2.5 mV. Nonzero current is observed within finite bias triangles (FBTs), where the chemical potentials . DQD charge states are labeled by (), where () is the number of electrons in the left (right) dot. We are able to reduce the electron number to the single electron regime, although the data presented in this paper are taken with to increase the current through the device and therefore the number of photon emission events Liu et al. (2014); Stockklauser et al. (2015).
Early experiments on semiconductor DQDs showed that energy is conserved during inelastic interdot tunneling through spontaneous emission of a phonon Fujisawa et al. (1998). Measurements of the current as a function of level detuning, , probe the environment at an energy , where is the interdot tunnel coupling. Oscillations in were attributed to electron phonon coupling Fujisawa et al. (1998); Weber et al. (2010); Roulleau et al. (2011). In contrast with these previous experiments, our DQD is electric-dipole coupled to a cavity. Therefore energy can be emitted during interdot tunneling by processes that involve the emission of both a phonon and a photon Liu et al. (2014), as illustrated in Fig. 1(c). Due to these higher-order emission processes, we may expect to see signatures of electron-phonon coupling in the amplitude and phase response of the cavity.
We first measure the current through the DQD and search for features in the data indicative of transitions between discrete electronic states that are accompanied by the emission of a phonon with energy . To increase the visibility of the current oscillations at this particular charge transition we plot as a function of and in Fig. 2(b) over the same gate voltage range as Fig. 2(a). These data reveal features that are present throughout the inelastic region of the FBT. The features are parallel to the interdot charge transition axis () and therefore occur at a constant detuning, as expected for a decay mechanism that primarily depends on . Figure 2(c) shows that small oscillations are directly visible in . A plot of (inset) shows that this current modulation has a periodic spacing of approximately 60 eV. Current oscillations with a period of eV were observed at more than 10 other charge transitions in the device, at forward and reverse bias (data not shown).
To better understand the observed features in the current we model the interaction of the DQD with the lattice phonons using a spin-boson model Brandes and Kramer (1999); Brandes (2005); Weber et al. (2010)
where is the Pauli matrix acting on the two charge states of the DQD, is the dispersion of a phonon mode with index , is an electron-phonon interaction matrix element, and are bosonic operators for the phonon modes. A key quantity in this model is the phonon spectral density , which we approximate by the contribution from the lowest energy longitudinally polarized mode of the nanowire and a background term arising from other phonon modes in the system Gullans et al. (2015)
where is a constant scale factor, is the phonon speed of sound, is the spacing between the dots, is a cutoff frequency that scales with the size of each dot, and scales the relative contribution from the background. From this idealized model, we see that the spectral density exhibits oscillations with the phonon energy when the DQD spacing is an integer multiple of the phonon wavelength . This condition allows a simultaneous vibrational anti-node at the position of each DQD Brandes and Kramer (1999). Including the tunneling to the left (right) lead at rate in the presence of a source-drain bias, we can derive analytic expressions for the current through the dot Brandes and Kramer (1999); Gullans et al. (). As seen in the theory curve in Fig. 2(c), modulations in the phonon spectral density show up directly in the current because the charge relaxation rate of the DQD two-level system is directly proportional to .
Going beyond previous experiments, we now report the observation of features in the amplitude and phase response of the cavity that have the same 60 eV periodicity as the features observed in the current. The cavity transmission, , and phase response, , are investigated in Fig. 3 for the lower FBT of the same interdot transition shown in Fig. 2. These measurements are performed by driving the cavity with an input tone at MHz resulting in an intra-cavity photon number . The phase response of the cavity [Fig. 3(a)] exhibits features that are once again periodic as a function of detuning (inset), and present throughout the FBT, suggesting they are caused by an energy-dependent decay mechanism. These features are more clearly visible in [Fig. 3(b)]. Figure 3(c) shows and as a function of detuning in the lower FBT, measured along the same detuning axis as in Fig. 2(c). At positive detuning, clear features are visible in with a period of approximately eV. Data are shown over a larger range of detuning in the inset. The dip in transmission and large phase shift near are due to the dispersive interaction of the DQD and cavity photons Petersson et al. (2012). The second dip in transmission around eV may be due to a low lying excited state, although the energy scale is more consistent with the periodic features observed in , , and . The features that we observe at large detuning in and are also independent of small perturbations in the barrier gate voltage, , suggesting they are a robust consequence of coupling to the phonon bath.
It is helpful to search for correlations between the oscillations observed in the current and the oscillations observed in the cavity response. In general, the energy emitted into the environment during inelastic interdot tunneling is given by the DQD energy splitting . Energy can be released into the environment by creating phonons and cavity photons. Previous experiments show that roughly one photon is emitted into the cavity mode for every 10–10 electron tunneling events Liu et al. (2014). Therefore the electronic current primarily probes the phonon environment and should scale as . In contrast, second order processes, where a cavity photon and phonon are emitted during inelastic tunneling, scale with the phonon spectral density as . These processes are illustrated in the inset of Fig. 4(a). For the large level detunings examined here, , and . As such, there should be a correlation between the measured current and cavity response .
Figure 4(a) plots and . The y-axes offsets and data ranges have been adjusted such as to maximize overlap between the two distinct data sets. For 0.1 meV there is a strong correlation between and . Both data sets have a very similar inelastic tail. Moreover, the oscillations in occur at the same values of detuning as the oscillations in . These data give further evidence that the oscillations in the current and cavity response are due to the same electron-phonon coupling mechanism.
The cavity phase response can be modeled using a theory that takes into account the dispersive shift of the cavity (renormalization of the cavity frequency) due to the DQD-mediated coupling to nanowire phonons Gullans et al. (). Measurements of and best fits to the microscopic theory are shown in Fig. 4(b). The theoretical predictions for the dispersive cavity shift due to electron-phonon coupling are shown in the inset of Fig. 4(b). In comparison with previous theoretical analysis of these systems Gullans et al. (2015); Müller and Stace (2017), our theory accounts for small energy shifts in the cavity frequency that arise from resonant coupling of the cavity photons to the nanowire phonons when the DQD is in the excited state Gullans et al. (). The periodic oscillations in the phase arise because periodic modulations in the charge relaxation rate lead to a similar modulation in the excited state population, which shifts the cavity frequency, as seen in the inset of Fig. 4(b). The microscopic origin of the oscillations in our model is identical across all three independent measurement techniques of dc current , cavity amplitude , and phase [see Eq. (2)], which provides strong evidence that we have observed direct signatures of the electron-phonon coupling in this system.
Future work on suspended nanowires could explore the dependence of the phonon periodicity on the source-drain electrode spacing. Such experiments could help to resolve the conflicting interpretations of the cause of this periodicity in Refs. Weber et al. (2010) and Roulleau et al. (2011). More broadly, the principles of phonon spectrum engineering and measurement suggested by this work may help to minimize electron-phonon interaction processes at specific energies. For example, single spin relaxation times in III/V semiconductor quantum dots Elzerman et al. (2004); Meunier et al. (2007) are due to spin-orbit coupling and phonon emission Khaetskii and Nazarov (2000, 2001); Hanson et al. (2007). By tailoring the phonon spectrum, it may be feasible to extend electron spin lifetimes in quantum devices Trif et al. (2008).
In summary, we have shown that it is possible to create a cavity-coupled InAs nanowire DQD that is mechanically suspended above the substrate. Consistent with earlier work, we observe oscillations in the inelastic current as a function of level detuning due to electron-phonon couping in the nanowire Weber et al. (2010); Roulleau et al. (2011). Measurements of the cavity response are also sensitive to electron-phonon coupling. We couple the electronic dipole moment of an electron trapped in this DQD to the electric field of a microwave cavity and observe a periodic cavity phase response due to a dispersive interaction with nanowire phonons. A comparison of these measurements with a microscopic theoretical model of the device suggests that the coupling of phonons to photons, mediated by electron dynamics, results in a phonon renormalization of the cavity center frequency. These experiments broadly help to understand the fundamental nature of electron-phonon coupling in nanoscale systems and may provide a path toward mitigating spin decay in semiconductor quantum devices.
Acknowledgements.Supported by the Packard Foundation, the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF4535, and NSF grants DMR-1409556 and DMR-1420541. Devices were fabricated in the Princeton University Quantum Device Nanofabrication Laboratory.
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