Coupling molecular spin states by photon-assisted tunneling
Abstract
Artificial molecules containing just one or two electrons provide a powerful platform for studies of orbital and spin quantum dynamics in nanoscale devices. A well-known example of these dynamics is tunneling of electrons between two coupled quantum dots triggered by microwave irradiation. So far, these tunneling processes have been treated as electric dipole-allowed spin-conserving events. Here we report that microwaves can also excite tunneling transitions between states with different spin. In this work, the dominant mechanism responsible for violation of spin conservation is the spin-orbit interaction. These transitions make it possible to perform detailed microwave spectroscopy of the molecular spin states of an artificial hydrogen molecule and open up the possibility of realizing full quantum control of a two spin system via microwave excitation.
In recent years, artificial molecules in mesoscopic systems have drawn much attention due to a fundamental interest in their quantum properties and their potential for quantum information applications. Arguably, the most flexible and tunable artificial molecule consists of coupled semiconductor quantum dots that are defined in a 2-dimensional electron gas using a set of patterned electrostatic depletion gates. Electron spins in such quantum dots exhibit coherence times up to s (1), about times longer than the relevant quantum gate operations (2); (3), making them attractive quantum bit (qubit) systems (4).
The molecular orbital structure of these artificial quantum objects can be probed spectroscopically by microwave modulation of the voltage applied to one of the gates that define the dots (5). In this way, the delocalized nature of the electronic eigenstates of an artificial hydrogen-like molecule was observed (6); (7). More recently, electrical microwave excitation was used for spectroscopy of single spins (8); (9); (10) and coherent single-spin control (8); (10), via electric dipole spin resonance (EDSR).
Here, we perform microwave spectroscopy (6); (7); (11) on molecular spin states in an artificial hydrogen molecule formed by a double quantum dot (DD) which contains exactly two electrons. In contrast to all previous PAT experiments, we observe not only the usual spin-conserving tunnel transitions, but also transitions between molecular states with different spin quantum numbers. We discuss several possible mechanisms and conclude from our analysis that these transitions become allowed predominantly through spin-orbit (SO) interaction. The possibility to excite spin-flip tunneling transitions lifts existing restrictions in our thinking about quantum control and detection of spins in quantum dots, and allows universal control of spin qubits without gate voltage pulses.
I Device and excitation protocol
Fig. 1a displays a scanning electron micrograph of a sample similar to that used in the experiments. It shows the metal gate pattern that electrostatically defines a DD and a quantum point contact (QPC) within a GaAs/(Al,Ga)As two-dimensional electron gas. An on-chip Co micro-magnet (magnet) indicated in blue in Fig. 1a generates an inhomogeneous magnetic field across the DD, which adds to the homogeneous external in-plane magnetic field (see the appendix A for more sample details), but is not needed for the molecular spin spectroscopy. The sample was mounted in a dilution refrigerator equipped with high-frequency lines. The gate voltages are set so that the DD can be considered as a closed system (the interdot tunneling rates are times larger than the dot-to-lead tunneling rates), and the tilt of the DD potential is tuned by the dc-voltages and , applied to the left and right side gates. Working near the turn-on of the first conductance plateau, the current through the QPC, , depends upon the local charge configuration and provides a sensitive meter for the absolute number of electrons in the left and right dot, respectively (12); (13).
First, we excite the DD as indicated in the top panel of Fig. 1b, by adding to continuous-wave microwave excitation at fixed frequency GHz. When the photon energy of the microwaves matches the energy splitting between the ground state and a state with a different charge configuration, a new steady-state charge configuration results, which is visible as a change in the QPC current, . The excitation is on-off modulated at 880 Hz and lock-in detection of reveals the microwave-induced change of the charge configuration (see the appendix A for further experimental details). The lower panel of Fig. 1b shows as a function of and near the to boundary of the charge stability diagram. Sharp red (blue) lines indicate microwave-induced tunneling of an electron from the right to the left dot (left to right), labeled as (), respectively (see Fig. 1c). Sidebands can result from multi-photon absorption. At the boundaries with the and charge states, no energy quantization is observed, since here electrons tunnel to and from the electron-state continuum of the leads. At first sight, the observations in Fig. 1b thus appear to be well explained by the usual spin-conserving PAT processes.
Surprisingly, the position in gate voltage of the resonant lines exhibits a striking dependence on the in-plane magnetic field, . This is clearly seen in Figs. 2a and 2b, which display the measured PAT spectrum along the DD detuning axis (dashed black arrow in Fig. 1b) as a function of for 20 GHz and 11 GHz excitation, respectively. Since the gate constitutes an open-ended termination of the transmission line, the excitation produces negligible AC magnetic fields at the DD, and is therefore expected to give rise to only electric-dipole allowed spin-conserving transitions, with no dependence. Furthermore, there is a pronounced asymmetry between the position of the red and blue PAT lines.
In these figures, the detuning axis was calibrated for all magnetic fields by introducing a reference line (see also Fig. 1d, black arrow) that facilitates interpretation of the spectra despite residual orbital effects of the magnetic field. This line was produced by interspersing the microwaves every 5 s with 200 ns gate voltage pulses along the detuning axis (see top panel in Fig. 1d), leading to singlet-triplet mixing as described in Ref. (2). The short gate voltage pulses do not noticeably alter the position of the PAT lines (compare Figs. 1b and 1d). The reference peaks visible at around eV in Figs. 2a and 2b were aligned by shifting all data points at a given by the same amount in detuning (see the appendix for the full details of this post-processing step).
Ii Interpretation of the photon-assisted tunneling spectra
The complexity of the PAT spectra shown in Fig. 2a,b can be understood in detail if we allow for non-spin conserving transitions. The two diagrams in Fig. 2c show the energies of all relevant DD-states (four -states and one singlet -state) as a function of for two different (fixed) magnetic fields, i.e. the spectrum of the DD along the two horizontal dotted lines in Fig. 2b (14). Note that the only difference between the two diagrams is the splitting between the three triplet -states.
First we explain the resonances observed along the upper dotted line in Fig. 2b ( T). In the corresponding (upper) diagram in Fig. 2c, we plot the ground state energy for all with a thick line. If the microwave excitation is off-resonance with all transitions, the system will be in this ground state. For instance, at eV, there is no state available 11 GHz above the ground state (gray arrow) and the system stays in . However, when decreasing the detuning, at some point becomes energetically accessible (red arrow) and, since we allow for non-spin-conserving transitions, is populated due to the microwave excitation. For this PAT transition, the spin projection on the quantization axis is changed by . The resulting change of steady-state charge population (increased population of , or ) is detected by the QPC and yields the red peak in Fig. 2b. Decreasing further, there are two more resonances detectable: (i) the - transition (dotted red arrow, ), although the signal will be weakened due to the fact that is not unambiguously the ground state anymore. Note that a transition - could appear in nearly the same detuning position, as will be discussed below. (ii) the - transition (blue arrow), where stands for the hybridized singlet, results in a negative (blue, ,) signal from the charge detector since the ground state is now and the excited state is . We see that this simple analysis explains both the positions and the signs of the resonances observed in the data.
A similar analysis can be made for other magnetic fields. For instance, for the spectrum plotted in the lower diagram of Fig. 2c we find two resonances with (red arrows), and one with (blue arrow). Note that the ‘blue’ transition now connects the ground state to the other branch of the hybridized compared to the high magnetic field case. Indeed, the singlet anti-crossing is directly probed, resulting in the two blue curved lines observed in the data around (Fig. 2b). The fading out of the blue signal at low fields can be understood from pumping into the metastable state : the microwaves excite the system from to , from where it relaxes quickly to . However, relaxation from back to the ground state is slow due to the small energy difference of this transition and the small phonon density of states at low energies (13); (15). This pumping weakens the detector signal, since has the same charge configuration as the ground state.
In order to verify this interpretation, we calculate in Figs. 2d,e the position and intensity of the spectral lines at fixed microwave frequency, based on the energy level diagram of Fig. 2c. In the simulations, all single-photon transitions between the ground state and the excited states are allowed by including a matrix element (see appendix A). The input parameters for the calculation of the resonant positions are the interdot tunnel coupling , the absolute electron -factor , a magnetic-field contribution from the magnet parallel to as well as a magnetic field gradient between the dots (in Fig. 3, we show how , and can be extracted from the experimental spectra). The color scale represents the calculated steady-state that results from microwave excitation, orbital hybridization and phonon absorption and emission at 100 mK. All the PAT transitions visible in the simulation also appear in the experiment, with excellent agreement in both the position and relative intensity of the spectral lines. Especially, the vanishing signal due to spin pumping is also predicted by the simulations which include phonon relaxation.
When we zoom in on the boxed region of Fig. 2b, we see an additional horizontal blue feature at T (Fig. 2f) that also appears in the calculated spectra of Fig. 2e. This feature is due to a triplet resonance from to that becomes detectable by relaxation into the meta-stable state. In the detuning range where this line appears, the state lies energetically only slightly above the state, so relaxation back to the ground state is suppressed, again by the small phonon density of states at low energies (see Fig. S2a of the appendix). The triplet resonance is expected to appear at , where is Planck’s constant and the Bohr magneton. The inset of Fig. 2f shows the magnetic fields corresponding to the center of the measured triplet resonance line for three excitation frequencies (see Figs. S2b,c of the appendix for the spectra), which are in good agreement with the expected positions (black dashed lines in Fig. 2f) based on the values and determined in the next section from other features of the PAT spectra. Surprisingly, the measured triplet resonance exhibits a finite slope in the spectra. A longitudinal magnetic field gradient gives rise to such a detuning dependence, but the required in our simulations to reproduce the observed slope is mT/50 nm, an order of magnitude larger than the gradient we calculate for the magnet. The magnitude of the slope remains a puzzle.
Iii Extracting artificial molecule parameters
We now show how , and , the parameters used for all simulations, can be extracted independently from the experimental spin-flip PAT spectra. For this analysis we only use the relative distance between PAT lines at fixed magnetic field, in order to be independent from the calibration of the detuning axis by means of the reference line. Fig. 3a shows as a function of using the 20 GHz data. is defined as the difference in detuning between the red and PAT lines. For a fixed , both increase linearly with the Zeeman energy and therefore allow fitting of . (Note that for the linearity is only exact for sufficiently large , at which the singlet anti-crossing does not affect the energy; see the appendix for a detailed discussion). A least-squares fit to the data gives (Fig. 3a). From the linear behavior of , we also deduce that there is negligible dynamic nuclear polarization in the experiment.
Knowing precisely, we make use of the blue anti-crossing in Figs. 2b and 2f in order to determine (and ). Fig. 3b shows the difference in detuning between the blue and the red lines in Fig. 2f. Assuming the line corresponds to the - transition (as shown below), then
(1) |
where the first term is the detuning position of the - transition and the second the one of the - transition. The best fits are obtained with eV and mT. This value for matches very well our simulations of the stray field of the magnet at the DD location (see appendix A).
An important question left open so far is whether the red line involves predominantly transitions from to or to . The transition to does not require a change in the (total) spin and is thus expected to be excited more strongly than that to . However, relaxation from back to will be stronger as well, so it is not obvious what steady-state populations will result in either case. Furthermore, given the small energy difference between and , the two transitions are not resolved in Fig. 2. Fig. 3a helps to answer this question: The observation that indicates that the line originates from the transition to and not to . For the former we expect , with the exchange energy, whereas the latter would result in (in both scenario’s, causes an additional fixed offset in both , but it does not contribute to their difference). This interpretation is consistent with the increase of with larger interdot tunnel coupling, hence larger (Fig. 3a inset; note that the slopes are not affected). It is further supported by the data in Fig. S3b.
So far only single-photon processes were considered, but at higher microwave power, also multi-photon lines emerge (Fig. 4a), mostly for the - transition (green dashed lines in Fig. 4a). Like the single-photon - line, their position in detuning is -independent (see appendix for details).
Iv Identification of the spin-flip mechanisms
Having shown the power of spin-flip PAT for detailed molecular spin spectroscopy, we now discuss the mechanisms responsible for this process as confirmed by our simulations. As a first possibility, the transitions from to the triplet states can take place through a virtual process involving : the state is coupled to by the interdot tunnel coupling, and an (effective) magnetic field gradient across the DD couples the spin part of all the states to each other (13); (16). Here has a contribution from the effective nuclear field and from the magnet. The transition matrix element from to is , assuming . In the following, we use the -dependence of this process as a fingerprint and focus on the red line in Figs. 2a and 2b, as we can follow it over the entire magnetic field range. The intensity of this line is constant in and even if the microwave amplitude is varied, we observe no -dependence in the area under this peak (Fig. 4b). Before we conclude that the transition rate is magnetic field independent, we recall that the observed PAT lines reflect the steady-state change in the charge configuration resulting from stimulated photon emission and absorption and spontaneous relaxation. In order to rule out that a field-independent steady state is reached from a field dependence of relaxation and excitation that cancel each other, we verify that the spontaneous relaxation rate is field-independent as well (see appendix). These observations suggest that the coupling mechanism is magnetic field independent and thus virtual processes involving do not give a strong contribution to the transition rates.
More recently, two mechanisms were considered that provide a direct, -independent matrix element between and the triplet states: (i) the hyperfine contact Hamiltonian is of the form . Thus, nuclear spins, , in the barrier regions, where the spatially overlaps with each of the triplet states, can flip-flop with the electron spin, , simultaneously with charge tunneling (17). (ii) The SO Hamiltonian is of the form and can directly couple states which differ in both orbital and spin (18); (19). (When the orbital part of the initial and final state are the same, the SO Hamiltonian does not provide a direct matrix element and the transition rate becomes -dependent (20); (21); (22); (8); (23).) The ratio of the SO mediated rate and the hyperfine mediated rate can be estimated as , which is a few thousand in the experiment (see the appendix). Here is the single-dot level spacing, the number of nuclei in contact with one dot, the hyperfine coupling strength, the interdot distance and the spin-orbit length. We therefore believe that SO interaction is the dominant spin-flip mechanism for the observed PAT transitions. The presence of a magnetic field independent matrix element between and the triplets is confirmed by the observation that the intensity of the reference signal shows no field dependence.
Finally, we extract from Fig. 4a the fitted linewidth as a function of driving power (Fig. 4c). For small , we find a width GHz, similar to that observed earlier for spin-conserving PAT processes (6). For stronger driving, both the and the lines are further broadened, up to mV. If these lines were power broadened, their width would imply transition rates in excess of 1 GHz. However, we do not believe that this is the case, since in measurements with short microwave bursts the populations saturated only on a long (10 s) timescale (data not shown). Presumably charge or gate voltage noise is responsible for the broadening instead.
We have shown that in our DD system, all spin states have at least weakly allowed electric dipole transitions to . In materials with high SO interaction like InAs, the effect of the non-spin conserving PAT will be even stronger. In materials with weak SO interaction, a strong gradient magnetic field can be used to facilitate spin-flip tunneling transitions. In all cases, this opens the possibility of realizing full quantum control of the spin space via off-resonant (microwave) Raman transitions through the excited state, which enables a variety of new approaches to manipulating and even defining qubits in DDs. Furthermore, such control enables new measurement techniques that do not rely on Pauli spin blockade (24). An example is a measurement that distinguishes parallel from anti-parallel spins while acting non-destructively on the subspace, by coupling resonantly the and states to followed by charge readout. This constitutes a partial Bell measurement and leads to a new method for producing and purifying entangled spin states (25).
Appendix A Methods
a.1 Sample fabrication
30 nm thick TiAu gates are fabricated on a 90 nm deep (Al,Ga)As/GaAs two-dimensional electron gas (2DEG) by means of ebeam lithography. The double dot axis is aligned along the [110] GaAs crystal direction (z-direction), which is parallel to the external magnetic field direction (Fig. 1a). The 2DEG is Si -doped (40 nm away from the hetero-interface), exhibits an electron density of cm and a mobility of cm/Vs at 1 K in the dark. The grounded, 275 nm thick, 2 m wide and 10 m long Co magnet is evaporated on top of a 80 nm thick dielectric layer, aligned along (magnetic easy axis) and placed nm away from the closest dot center. We calculate (26) that at the double dot position the magnet adds mT to and generates a magnetic field gradient of mT/50 nm and a transverse gradient of mT/50 nm at saturation ( T).
a.2 Measurement
The sample is mounted in an Oxford KelvinOx 300 dilution refrigerator at 30 mK. Left and right side gate voltages, and , are set by low-pass filtered dc lines and dB attenuated coaxial lines combined with bias-tees with a cutoff frequency of 30 Hz. The pre-amplified current through the quantum-point contact is read out by a lock-in amplifier locked to the 880 Hz on-off modulation of the microwaves. The bias across the double dot is set to 0 V. Voltage pulses to the left and right side gates are generated with a Sony Textronix AWG520. The microwaves are generated with a HP83650A and combined with the pulses to the right side gate. Microwave bursts and detuning pulses are synchronized to ensure that the microwave excitation is switched off during the detuning pulses that generate the reference signal (see Fig. 1d).
a.3 Simulation
The Hamiltonian describing the two-spin system near the - transition is taken to be a five-state system, with four spin states and a spin singlet (16) in the presence of an external magnetic field and a magnetic field gradient , which includes both the quasi-static nuclear field and the field from the magnet. This is given by , were is the projector onto the subspace, is the detuning due to the difference in gate potentials from the left and right gates and the tunnel coupling with the spin-conserving tunnel coupling and the spin-orbit coupling set to % of .
To find the signal we expect theoretically from the experiment, we add a weak, rapidly oscillating term to the Hamiltonian: . We diagonalize with , then make a rotating frame transformation in which levels are grouped into bands (defined by a projector ) where the states in a band are much closer in energy than , while the energy difference between states in band and are within 2/3rds of . Each band rotates at a rate , and we can then make a rotating wave approximation, keeping terms due to that couple adjacent bands, i.e., our perturbation in the rotating frame and rotating wave approximation is . Next, we add dissipation and dephasing by including relaxation due to coupling of the electron charge to piezoelectric phonons in a two-orbital (Heitler-London-like) model.
Appendix B Calibration of the detuning axis
The photon-assisted tunneling (PAT) spectra in Figs. 2a and 2b of the main article are measured both with a high energy resolution along the double dot (DD) detuning axis and over a wide external magnetic field range. In the experiment, we observe a monotonous, reproducible drift of the stable charge regions predominantly along the right side gate voltage as we change the magnetic field. Changing the voltages applied to the left and to the right side gate accordingly, we partially compensate for this drift. We then record an 11 GHz PAT spectrum as displayed in Fig. S1a. In order to precisely calibrate the detuning axis for all , a reference signal is generated together with the PAT spectrum by interspersing the microwaves every 5 s by a 200 ns detuning pulse with amplitude towards negative detuning (Fig. S1b). 200 ns are found to be sufficient to mix the state entirely with the state at their anti-crossing , so that a Pauli spin blocked signal is observed at a detuning . The magnitude of is chosen such that the reference signal appears at a detuning position far away from the PAT signal. The pulses do not alter the detuning position of the PAT resonances. In addition to mixing, mixing of the with and is observed due to the pulsing. The former gives rise to a positive background on the left of the reference signal in Fig. S1b. The latter generates a weak second reference line that overlaps with the reference line for T, but shifts towards negative detuning as is increased.
In a post-processing step, we separately fit the position of the peaks for all and shift every row of the spectrum, such that the peaks are vertically aligned at as shown in Fig. S1c. Thus, is at the anti-crossing, by definition. All data points at a given are shifted by the same amount in detuning. This -detuning axis is well-defined but ‘moves’ with respect to the -detuning axis as a function of , since . The lever arm for the voltage to energy conversion is read from the voltage distance of the second and third PAT line at GHz (see Fig. 4a), which equals the photon energy (7). For the analysis of all spectra we used the same lever arm.
For better readability of the PAT spectra, we finally convert the -scale to the -scale found in literature, for which is defined by the to anti-crossing. To do so, we additionally shift all data points at a given by towards positive detuning (Fig. S1d). However, since holds true only for , where is the interdot tunnel coupling, the detuning axis conversion fails for low magnetic fields. As a result the PAT resonance-lines bend towards positive for T in Fig. S1d and in the spectra shown in the main article (Fig. 2a,b).
Appendix C Triplet spin resonance
In Fig. 2f of the main article, a PAT feature is observed that is due to a transition from the ground state to the excited state. The state can relax via spontaneous phonon emission to the singlet bonding state, a superposition of and . This state is metastable, since the spontaneous phonon relaxation is suppressed by the small energy difference to the ground state (see Fig. S2a), which makes the transition detectable by . A peculiarity of the to resonance is its slope in the PAT spectrum, which might be a result from a gradient magnetic field along the magnetic field direction as discussed in the main article. Note that in the same detuning range, we observe also direct PAT transitions from the ground state to the singlet anti-bonding state and singlet bonding state at lower and higher magnetic fields, respectively.
Here we investigate the position of the to resonance, as a function of the microwave frequency . The Figs. S2b and S2c show raw PAT spectra (without any post-processing step applied as explained above) recorded with GHz and GHz, respectively. The dashed lines mark the magnetic field, at which the electron spin resonance condition is fulfilled. Here we use the absolute electronic g-factor and longitudinal magnetic field offset mT as determined by the Figs. 3a and 3b of the main article. Alternatively, we might use the to resonance feature to determine . If we use the center magnetic field of this feature as the resonant field, we calculate from the microwave frequency dependence in good agreement with the found in Fig. 3a in the main article.
Appendix D The PAT transition
In the main article, we discuss whether the red () PAT resonance is dominantly due to a transition from the ground state to the state or to the state. It is difficult to spectroscopically resolve these transitions, since they differ only by the exchange energy . In Fig 3a of the main article, we use , the difference in detuning between the red and the lines, to assign the PAT resonance. Here, we support this argument by calculating the expected functions for both extreme scenarios: a pure / and a pure / transition. For the calculation, we use the determined , eV and mT values. The result is displayed in Fig. S3a for a microwave frequency GHz. Obviously, the scenario of a pure singlet transition results in , whereas is found for a transition to the triplet state. In the experiment, we clearly observe and therefore the transition is dominantly a singlet transition.
In both scenarios, the increases non-linearly towards high , since the resonance becomes sensitive to the curved singlet anti-crossing at , where is the Zeeman energy. This curvature is not observed in the experiment as the transition fades out at T. In both scenarios, the functions are linear, which holds true only if . The choice of a high and an appropriate -range allows to extract from by a simple linear least-squares fit as demonstrated in the main article.
As a final step, we analyze the splitting of the and function quantitatively. Their offsets depend upon the exchange energy , which we can experimentally vary by or indirectly by . For , the detuning position of the resonance becomes strongly affected by . As shown in the inset of Fig. S3b, the / transition (green arrow) shifts more towards negative detuning than the potential / transition (violet arrow). We measure for various and determine the by a linear fit at a sufficiently high magnetic field range. This procedure turned out to be impractical with , because the line fades out at high magnetic fields. Note that the linear extrapolation of the transition to zero field (red arrow) exhibits a different detuning position than the / transition (violet arrow), because the linear extrapolation follows the dashed blue line in the inset of Fig. S3b, i.e. the linear extrapolated detuning position is not affected by the hybridization of the singlets. The detuning position of the / PAT transition, however, is affected by the singlet hybridization, since it lowers the energy of the initial state . In summary, the value of is always larger then zero, but also depends upon the nature of the PAT resonance.
In the inset of Fig. S3b, for the / transition is drawn. Obviously, is considerably smaller, if the / PAT resonance dominates over the / resonance. The analysis of is complicated by the remanence of the magnet, which is not exactly known, but should be smaller than the fully magnetized field of mT. Due to the remanence, the linear extrapolation of the / transition towards zero external magnetic field, leaves an additional offset on . This offset, however, is independent from and . In Fig. S3b, the extrapolated values are plotted as a function of the microwave frequency for two tunnel couplings (filled circles and open squares). Fitting the filled circles with the well-known eV, we determine a reasonable fit by assuming the PAT resonance to be purely singlet (green line). The fit with a potential / transition (violet line) fails at GHz. The only fit parameter used here is the remanence of the magnet, which was found to be mT for the fit function assuming a / transition, and mT assuming a / transition. The latter is very unlikely, since only a maximum magnetization of mT was found at an external magnetic field of 2 T.
As a final check, we take values into account, which were determined when the DD was tuned to a smaller eV (open squares). These data points cannot be fitted by the fit function that assumes a / transition at all, since the values observed are already smaller than the remanence of mT, which would stay valid for the altered tunnel coupling. Obviously, this leads to a contradiction, since all would become negative after subtracting the remanence. Only the assumption of a purely singlet PAT transition in combination with the smaller remanence of mT, as fitted above, allows reasonable fitting. Our conclusion from the main article is therefore further supported.
Appendix E Spin flip-tunneling mechanism
As noticed in the main text, there is a direct matrix element between the and the triplet states. This can occur due to the nuclear spins in the barrier between the dots and due to spin-orbit (SO) interaction. We look at a toy model to examine the relative importance of these two processes. Specifically, we consider the hopping matrix element for a single electron with spin between two wavefunctions associated with an electron on the left () and on the right () via the perturbation:
(2) | |||||
where we have absorbed the Rashba () and Dresselhaus () terms into a single spin-orbit interaction with a characteristic spin-orbit length m. We recall that is the effective electron mass, eV, is the unit cell volume and is the nuclear spin at .
We now wish to estimate the spin-flip tunneling for the single electron case, given by averaging over the orbital dipole:
(3) |
This can be evaluated explicitly for where is the inter-dot axis (at an angle with the axis from the spin-orbit interaction in Eq. 2) and is the transverse-longitudinal wavefunction. As tunneling occurs only along the -axis, matrix elements with are zero. We find two tunneling matrix elements:
(4) | |||||
(5) | |||||
(6) | |||||
(7) |
We remark that the rms value for is given by
(8) | |||||
That is, it is the rms value for a single dot, , multiplied by . Also, the size of the single-particle wavefunction, , is related to the orbital energy scale of a single dot by . Thus, the relative strength of the two tunneling terms (including spin flip) is
(9) |
where is the number of spins in a single quantum dot.
We explore briefly how this ratio varies with dot size and spacing . Specifically, , so a larger dot reduces the strength of spin-orbit tunneling compared to hyperfine-assisted tunneling. On the other hand, increasing the distance increases the relative strength of spin-orbit tunneling to hyperfine-assisted tunneling. Setting in eV, , nm, eV and m, we calculate the ratio of the matrix elements to be . We remark that in external field parallel to prevents any SO spin-charge flips. All spin charge flips occur only via .
In the bases (, , , , ) the two electron Hamiltonian that ignores the small term has the form
(10) |
where, e.g., the Larmor precession frequency of an electron spin in the left dot is .
Appendix F Simulations of the PAT spectra - relaxation
The simulated spectra in Figs. 2c,d of the main article include the effect of the phonon-mediated relaxation. In addition to the explanations of the simulations in appendix A, we continue here on the coupling of the electron spin to the phonon bath. We thereby neglect deformation phonons as the energy scales examined in the experiment (7-22 GHz) are much smaller than the characteristic frequency scale of a phonon on the length scale of the dot GHz. To determine the coupling, we take as an ansatz for the electronic wavefunctions the Fock-Darwin states, given by Gaussians, and calculate the coupling after orthogonalizing the states with the perturbation (27), where for the energy scales we are working with. We then use Fermi’s golden rule to calculate excitation and relaxation from thermal and spontaneous emission of phonons. Finally, we numerically solve the superoperator for the steady state and compare the expectation value of with and without the excitation , mimicking the effect of the lock-in detection.
Appendix G Measurement of spontaneous relaxation
In the main article, we state that the spontaneous relaxation rate from to is found to be magnetic field independent. Relevant for the PAT process is the spontaneous relaxation at a fixed energy difference between the and state, which is set by the photon energy. Thus, when changing the magnetic field, the detuning has to be changed accordingly. The measurement of the spontaneous relaxation rate is done as follows: Starting from , we populate the by % via a ns detuning pulse (28) with amplitude in the absence of microwaves, and monitor the decay back to . The relaxation rate can be extracted from the time averaged lock-in signal , where is the time spent in Pauli blockade between the pulses. , which is independent from , is extracted from the fit in the inset of Fig. S4. In order to cover various values of the detuning and the magnetic field, we next fix s and record as a function of for three different magnetic fields. We observe that s and thus are essentially independent of (Fig. S4). This holds true for all and hence for all - energy splittings. Note that regardless of , this energy splitting is given by alone. This reflects exactly the situation in the PAT experiment, for which the microwave frequency alone sets the energy splitting and thus also the required phonon energy for the spontaneous relaxation process.
Appendix H Power dependence
In Fig. 4a of the main article, the power dependence of the PAT sidebands is shown for two magnetic fields. In Fig. S5a-e, GHz-spectra measured with a series of magnetic field values are plotted, to ease keeping track of the resonances as they shift in detuning with the external magnetic field. Here, the spectra are plotted such that is the anti-crossing for all (compare Fig. S1c). The reference peaks due to pulsing to the anti-crossing are all aligned at mV, which is the pulse amplitude. The amplitude of the microwaves is estimated from the attenuation of the high-frequency circuit at room temperature.
PAT sidebands at larger detuning appear when is increased as expected for PAT. Oscillation of the PAT amplitude as a function of is hardly visible, because we cannot reach sufficiently high and because the PAT lines become broadened as a function of power. One sideband emerges at high mV already at low . This resonance stays at a constant for all and is therefore a transition from to (). As is increased, the other transitions move towards lower while keeping the distance along . They are due to PAT transition with . At T, the PAT transition overlaps with the 2-photon PAT transition. This is expected, since GHz equals the Zeeman energy at this magnetic field in our double dot.
Acknowledgements.
We gratefully acknowledge discussions with S. M. Frolov, M. Laforest, D. Loss, Yu. V. Nazarov, K. C. Nowack, M. Shafiei and thank H. Keijzers for help with the sample fabrication and R. Schouten, A. van der Enden and R. G. Roeleveld for technical support. This work is supported by the ‘Stichting voor Fundamenteel Onderzoek der Materie (FOM)’ and a Starting Investigator grant of the ‘European Research Council (ERC)’. L.R.S, F.R.B., V.C. and T.M. performed the experiment, W.W. grew the heterostructure, T.M. fabricated the sample, L.R.S., J.D., J.M.T and L.M.K.V. developed the theory, J.M.T. did the simulations, all authors contributed to the interpretation of the data and commented on the manuscript, and L.R.S., J.D., J.M.T. and L.M.K.V. wrote the manuscript.References
- Bluhm, H. et al. Long coherence of electron spins coupled to a nuclear spin bath. http://arxiv.org/abs/1005.2995v1 (2010).
- Petta, J. R. et al. Coherent manipulation of coupled electron spins in semiconductor quantum dots. Science 309, 2180 (2005).
- Koppens, F. H. L. et al. Driven coherent oscillation of a single electron spin in a quantum dot. Nature 442, 766 (2006).
- Hanson, R., Kouwenhoven, L. P., Petta, J. R., Tarucha, S. & Vandersypen, L. M. K. Spins in few-electron quantum dots. Rev. Mod. Phys. 79, 1217 (2007).
- van der Wiel, W. G. et al. Electron transport through double quantum dots. Rev. Mod. Phys. 75, 1 (2003).
- Oosterkamp, T. H. et al. Microwave spectroscopy of a quantum-dot molecule. Nature 395, 873 (1998).
- Petta, J. R., Johnson, A. C., Marcus, C. M., Hanson, M. P. & Gossard, A. C. Manipulation of a single charge in a double quantum dot. Phys. Rev. Lett. 93, 186802 (2004).
- Nowack, K. C., Koppens, F. H. L., Nazarov, Y. V. & Vandersypen, L. M. K. Coherent control of a single electron spin with electric fields. Science 318, 1430 (2007).
- Laird, E. A. et al. Hyperfine-mediated gate-driven electron spin resonance. Phys. Rev. Lett. 99, 246601 (2007).
- Pioro-Ladriere, M. et al. Electrically driven single-electron spin resonance in a slanting Zeeman field. Nat. Phys. 4, 776 (2008).
- Petersson, K. D. et al. Microwave-driven transitions in two coupled semicondoctor charge qubits. Phys. Rev. Lett. 103, 016805 (2009).
- Field, M. et al. Measurements of coulomb blockade with a noninvasive voltage probe. Phys. Rev. Lett. 70, 1311 (1993).
- Johnson, A. C. et al. Triplet-singlet spin relaxation via nuclei in a double quantum dot. Nature 435, 925 (2005).
- Koppens, F. H. L. et al. Control and detection of singlet-triplet mixing in a random nuclear field. Science 309, 1346 (2005).
- Meunier, T. et al. Experimental signature of phonon-mediated spin relaxation in a two-electron quantum dot. Phys. Rev. Lett. 98, 126601 (2007).
- Taylor, J. M., Petta, J. R., Johnson, A. C., Marcus, C. M. & Lukin, M. D. Relaxation, dephasing and quantum control of electron spins in double quantum dots. Phys. Rev. B 76, 035315 (2007).
- Stopa, M., Krich, J. J. & Yacoby, A. Inhomogeneous nuclear spin flips: Feedback mechanism between electronic states in a double quantum dot and the underlying nuclear spin bath. Phys. Rev. B 81, 041304R (2010).
- Nadj-Perge, S. et al. Disentangling the effects of spin-orbit and hyperfine interactions on spin blockade. Phys. Rev. B 81, 201305R (2010).
- Danon, J. & Nazarov, Y. V. Pauli spin blockade in the presence of strong spin-orbit coupling. Phys. Rev. B 80, 041301R (2009).
- Khaetskii, A. V. & Nazarov, Y. V. Spin-flip transitions between zeeman sublevels in semiconductor quantum dots. Phys. Rev. B 64, 125316 (2001).
- Fujisawa, T., Austing, D. G., Tokura, Y., Hirayama, Y. & Tarucha, S. Allowed and forbidden transitions in artificial hydrogen and helium atoms. Nature 419, 278 (2002).
- Golovach, V. N., Borhani, M. & Loss, D. Electric-dipole-induced spin resonace in quantum dots. Phys. Rev. B 74, 165319 (2006).
- Golovach, V. N., Khaetskii, A. & Loss, D. Spin relaxation at the singlet-triplet crossing in a quantum dot. Phys. Rev. B 77, 045328 (2008).
- Ono, K., Austing, D. G., Tokura, Y. & Tarucha, S. Current rectification by Pauli exclusion in a weakly coupled double quantum dot system. Science 297, 1313 (2002).
- Taylor, J. M. et al. Solid-state circuit for spin entanglement generation and purification. Phys. Rev. Lett. 94, 236803 (2005).
- Goldman, J. R., Ladd, T. D., Yamaguchi, F. & Yamamoto, Y. Magnet designs for a crystal-lattice quantum computer. Appl. Phys. A 71, 11 (2000).
- Kittel, C. Quantum Theory of Solids (Wiley, 1987).
- Barthel, C., Reilly, D. J., Marcus, C. M., Hanson, M. P. & Gossard, A. C. Rapid single-shot measurement of a singlet-triplet qubit. Phys. Rev. Lett. 103, 160503 (2009).