Coupling molecular spin states by photon-assisted tunneling
L. R. Schreiber, F. R. Braakman, T. Meunier, V. Calado, J. Danon, J. M. Taylor, W. Wegscheider, L. M. K. Vandersypen
aa r X i v : . [ qu a n t - ph ] O c t Coupling molecular spin states by photon-assisted tunneling
L. R. Schreiber, ∗ F. R. Braakman, T. Meunier,
1, 2
V. Calado, J.Danon, J. M. Taylor, W. Wegscheider,
5, 6 and L. M. K. Vandersypen Kavli Institute of Nanoscience, TU Delft, 2600 GA Delft, The Netherlands Institut N´eel, CNRS and Universit´e Joseph Fourier, Grenoble, France Dahlem Center for Complex Quantum Systems, Freie Universit¨at Berlin, Germany National Institute of Standards and Technology, University of Maryland, USA Institute for Experimental and Applied Physics, University of Regensburg, Germany Solid State Physics Laboratory, ETH Zurich, Switzerland (Dated: November 1, 2018)Artificial molecules containing just one or two electrons provide a powerful platform for studies oforbital and spin quantum dynamics in nanoscale devices. A well-known example of these dynamicsis tunneling of electrons between two coupled quantum dots triggered by microwave irradiation. Sofar, 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 differentspin. 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 ofthe molecular spin states of an artificial hydrogen molecule and open up the possibility of realizingfull quantum control of a two spin system via microwave excitation.
In recent years, artificial molecules in mesoscopic sys-tems have drawn much attention due to a fundamental in-terest in their quantum properties and their potential forquantum information applications. Arguably, the mostflexible and tunable artificial molecule consists of cou-pled semiconductor quantum dots that are defined in a2-dimensional electron gas using a set of patterned elec-trostatic depletion gates. Electron spins in such quan-tum dots exhibit coherence times up to 200 µ s [1], about10 − times longer than the relevant quantum gateoperations [2, 3], making them attractive quantum bit(qubit) systems [4].The molecular orbital structure of these artificial quan-tum objects can be probed spectroscopically by mi-crowave modulation of the voltage applied to one of thegates that define the dots [5]. In this way, the delocal-ized nature of the electronic eigenstates of an artificialhydrogen-like molecule was observed [6, 7]. More re-cently, electrical microwave excitation was used for spec-troscopy of single spins [8–10] and coherent single-spincontrol [8, 10], via electric dipole spin resonance (EDSR).Here, we perform microwave spectroscopy [6, 7, 11] onmolecular spin states in an artificial hydrogen moleculeformed by a double quantum dot (DD) which containsexactly two electrons. In contrast to all previous PAT ex-periments, we observe not only the usual spin-conservingtunnel transitions, but also transitions between molecu-lar states with different spin quantum numbers. We dis-cuss several possible mechanisms and conclude from ouranalysis that these transitions become allowed predomi-nantly through spin-orbit (SO) interaction. The possibil-ity to excite spin-flip tunneling transitions lifts existingrestrictions in our thinking about quantum control and ∗ [email protected] detection of spins in quantum dots, and allows universalcontrol of spin qubits without gate voltage pulses.
1. DEVICE AND EXCITATION PROTOCOL
Fig. 1a displays a scanning electron micrograph ofa sample similar to that used in the experiments. Itshows the metal gate pattern that electrostatically de-fines a DD and a quantum point contact (QPC) withina GaAs/(Al,Ga)As two-dimensional electron gas. Anon-chip Co micro-magnet ( µ magnet) indicated in bluein Fig. 1a generates an inhomogeneous magnetic fieldacross the DD, which adds to the homogeneous externalin-plane magnetic field B (see the appendix A for moresample details), but is not needed for the molecular spinspectroscopy. The sample was mounted in a dilution re-frigerator equipped with high-frequency lines. The gatevoltages are set so that the DD can be considered as aclosed system (the interdot tunneling rates are 10 timeslarger than the dot-to-lead tunneling rates), and the tiltof the DD potential is tuned by the dc-voltages V L and V R , applied to the left and right side gates. Working nearthe turn-on of the first conductance plateau, the currentthrough the QPC, I QP C , depends upon the local chargeconfiguration and provides a sensitive meter for the ab-solute number of electrons ( n L , n R ) in the left and rightdot, respectively [12, 13].First, we excite the DD as indicated in the top panelof Fig. 1b, by adding to V R continuous-wave microwaveexcitation at fixed frequency ν = 11 GHz. When thephoton energy of the microwaves matches the energysplitting between the ground state and a state with adifferent charge configuration, a new steady-state chargeconfiguration results, which is visible as a change in theQPC current, ∆ I QP C . The excitation is on-off modu-lated at 880 Hz and lock-in detection of ∆ I QP C reveals -1185 -1180 -595 d c - b i a s V L ( m V ) dc-bias V R (mV) c da b -1185 -1180 -600-595 d c - b i a s V L ( m V ) dc-bias V R (mV) (1,1) (0,2)(0,1) (1,2) ε I QPC V R V L µ m ac-V R ac-V L ac-V R ac-V L µ s -600 ∆ I QPC (a. u.) µ F (1,1) (0,2) µ F ε > ε < ∆ n = +1 ∆ n = -1 FIG. 1:
Photon-assisted tunneling in a 2-electron dou-ble quantum dot. a , Scanning-electron micrograph topview of the double dot gate structure with Co micromagnet(blue). The voltages applied to the left V L and right V R sidegates (red) control the detuning ε of the double dot potential.The double dot charge state is read out by means of the cur-rent I QPC running through a nearby quantum point contact(white arrow). b , Charge stability diagram around the 2-electron regime at B = 1 . n L , n R ) indicate the absolutenumbers of electrons in the left and right dot, respectively.During measurements, 11 GHz microwaves with 880 Hz on-off modulation are applied to the right side gate. The toppanel displays schematically one cycle of the ac signal appliedto the right side gate. Along the detuning axis (dashed ar-row) PAT-lines are observed. c , In the conventional pictureof PAT, the first sidebands seen in b should appear when thedetuning of the (0 ,
2) and (1 ,
1) states matches the photonenergy and interdot tunneling is induced. Further sidebandsare then interpreted as multi-photon transitions. µ F is thechemical potential of the left and right electron reservoir. d ,Same as in b , but the microwaves are interrupted every 5 µ sby a 200 ns, P ε = 2 mV detuning pulse applied to the leftand right side gates (see the schematic in the top panel). Thepulses generate a reference line (black arrow) due to mixingat the ST + anti-crossing. the microwave-induced change of the charge configura-tion (see the appendix A for further experimental de-tails). The lower panel of Fig. 1b shows ∆ I QP C as afunction of V L and V R near the (1 ,
1) to (0 ,
2) boundaryof the charge stability diagram. Sharp red (blue) lines in-dicate microwave-induced tunneling of an electron fromthe right to the left dot (left to right), labeled as ∆ n = +1(∆ n = − ,
1) and (1 ,
2) charge states, no energy quan-tization is observed, since here electrons tunnel to and from the electron-state continuum of the leads. At firstsight, the observations in Fig. 1b thus appear to be wellexplained by the usual spin-conserving PAT processes.Surprisingly, the position in gate voltage of the reso-nant lines exhibits a striking dependence on the in-planemagnetic field, B . This is clearly seen in Figs. 2a and 2b,which display the measured PAT spectrum along the DDdetuning ε axis (dashed black arrow in Fig. 1b) as a func-tion of B for 20 GHz and 11 GHz excitation, respectively.Since the gate constitutes an open-ended termination ofthe transmission line, the excitation produces negligibleAC magnetic fields at the DD, and is therefore expectedto give rise to only electric-dipole allowed spin-conservingtransitions, with no B dependence. Furthermore, thereis a pronounced asymmetry between the position of thered and blue PAT lines.In these figures, the detuning axis was calibrated forall magnetic fields by introducing a reference line (seealso Fig. 1d, black arrow) that facilitates interpretationof the spectra despite residual orbital effects of the mag-netic field. This line was produced by interspersing themicrowaves every 5 µ s with 200 ns gate voltage pulsesalong the detuning axis (see top panel in Fig. 1d), lead-ing to singlet-triplet mixing as described in Ref. [2]. Theshort gate voltage pulses do not noticeably alter the po-sition of the PAT lines (compare Figs. 1b and 1d). Thereference peaks visible at around ε = 200 µ eV in Figs. 2aand 2b were aligned by shifting all data points at a given B by the same amount in detuning (see the appendix forthe full details of this post-processing step).
2. INTERPRETATION OF THEPHOTON-ASSISTED TUNNELING SPECTRA
The complexity of the PAT spectra shown in Fig. 2a,bcan be understood in detail if we allow for non-spin con-serving transitions. The two diagrams in Fig. 2c showthe energies of all relevant DD-states (four (1 , S (0 , ε for twodifferent (fixed) magnetic fields, i.e. the spectrum of theDD along the two horizontal dotted lines in Fig. 2b [14].Note that the only difference between the two diagramsis the splitting between the three triplet T (1 , B = 2 . ε with a thick line. If the mi-crowave excitation is off-resonance with all transitions,the system will be in this ground state. For instance, at ε = 150 µ eV, there is no state available 11 GHz above the S (0 ,
2) ground state (gray arrow) and the system stays in S (0 , T + (1 ,
1) becomes energetically accessible (red ar-row) and, since we allow for non-spin-conserving transi-tions, is populated due to the microwave excitation. Forthis PAT transition, the spin projection on the quan-tization axis is changed by ∆ m = +1. The resulting -100 0 100 -1000100-1000100 ene r g y ( µ e V ) ε ( µ eV) ε ( µ eV) B ( T ) ε ( µ eV)0 100 200123 ε ( µ eV) B ( T ) ε ( µ eV) a b cd e ∆ m = + ∆ m = - ∆ m = - ∆ m = ∆ m = + ∆ m = ∆ m = - ∆ n T - (1,1)T (1,1)T + (1,1) f ∆ I QPC (arb. units.) B ( T ) ε ( µ eV) ν = 11 GHz ν = 11 GHz ν = 20 GHz ν = 20 GHz E z = h ν S(0,2)B = 2.5 TB = 1.5 T S(1,1) P ε ∆ m = + ∆ m = ∆ m = - ∆ m = - r e f e r en c e li ne r e f e r en c e li ne ν (GHz) B ( T ) FIG. 2:
Photon-assisted-tunneling spectra and simulations. a,b , Microwave induced change of the QPC current∆ I QPC as a function of the double dot detuning ε and the external magnetic field B for 20 GHz ( a ) and 11 GHz ( b ) frequency,respectively. singlet-triplet mixing due to 2 mV detuning pulses generates a reference signal that is used to calibrate thedetuning axis (see lower panel in Fig. 2f). c , Eigenenergies vs. double dot detuning ε of the 2-electron spin states in the (1 , ,
2) charge regime for two external magnetic fields B = 2 . B = 1 . S (0 , S (1 ,
1) and T (1 ,
1) character of the eigenstates is indicated by blue, green and red color, respectively. The molecularspin ground state is indicated by thick lines. The vertical arrows indicate PAT transitions for a constant microwave frequencyinvolving spin flips. The transition indicated by a dashed arrow is suppressed, because the initial state lies above the groundstate. The red circle in the lower panel indicates the detuning position of the reference signal, that is generated by a detuningpulse with amplitude P ε to the ST + anti-crossing. d,e , Simulated PAT spectra for 20 GHz ( a ) and 11 GHz ( b ) frequency,respectively. The color indicates the change of the population of the steady-state charge state ∆ n as would be observed inan on-off lock-in detection. A finite temperature of 100 mK and spontaneous relaxation via the phonon bath are taken intoaccount. f , I QPC scanned with higher resolution in the anti-crossing region (black rectangle in Fig. 2b) at 11 GHz excitation.The green dashed lines indicate the expected detuning positions of the PAT transitions. The horizontal black dashed lineindicates the magnetic field, at which the electron spin resonance condition is fulfilled E z = gµ B ( B + b ) = hν . The graph isconcatenated from two scans that overlap at 1.9 T. The inset displays three magnetic fields (blue dots), at which the center ofthe horizontal blue triplet resonance line is observed, as a function of the microwave frequency ν . The dashed black line givesthe expected position of the triplet resonance. change of steady-state charge population (increased pop-ulation of (1 , I QP C >
0) is detected by the QPCand yields the red peak in Fig. 2b. Decreasing ε fur-ther, there are two more resonances detectable: (i) the S (0 , S (1 ,
1) transition (dotted red arrow, ∆ m = 0),although the signal will be weakened due to the fact that S (0 ,
2) is not unambiguously the ground state anymore.Note that a transition S (0 , T (1 ,
1) could appear innearly the same detuning position, as will be discussedbelow. (ii) the T + (1 , S transition (blue arrow), where S stands for the hybridized S (0 , − S (1 ,
1) singlet, re-sults in a negative (blue, ∆ I QP C < m = −
1) signalfrom the charge detector since the ground state is now(1 ,
1) and the excited state is (0 , I QP C > I QP C < S com-pared to the high magnetic field case. Indeed, the singletanti-crossing is directly probed, resulting in the two bluecurved lines observed in the data around ε = 0 (Fig.2b). The fading out of the blue signal at low fields canbe understood from pumping into the metastable state S (1 , T + (1 , S (0 , S (1 , S (1 ,
1) back to the ground state isslow due to the small energy difference of this transitionand the small phonon density of states at low energies[13, 15]. This pumping weakens the detector signal, since S (1 ,
1) has the same charge configuration as the groundstate.In order to verify this interpretation, we calculate inFigs. 2d,e the position and intensity of the spectrallines at fixed microwave frequency, based on the en-ergy level diagram of Fig. 2c. In the simulations, allsingle-photon transitions between the ground state andthe excited states are allowed by including a matrix el-ement | T ± (1 , ih S (0 , | (see appendix A). The inputparameters for the calculation of the resonant positionsare the interdot tunnel coupling t c , the absolute elec-tron g -factor | g | , a magnetic-field contribution b fromthe µ magnet parallel to B as well as a magnetic fieldgradient ∆ ~B = (∆ B ⊥ x , , ∆ B k ) between the dots (in Fig.3, we show how t c , | g | and b can be extracted fromthe experimental spectra). The color scale represents thecalculated steady-state ∆ n that results from microwaveexcitation, orbital hybridization and phonon absorptionand emission at 100 mK. All the PAT transitions visiblein the simulation also appear in the experiment, with ex-cellent agreement in both the position and relative inten-sity of the spectral lines. Especially, the vanishing signaldue to spin pumping is also predicted by the simulationswhich include phonon relaxation.When we zoom in on the boxed region of Fig. 2b,we see an additional horizontal blue feature at B ≈ T + (1 ,
1) to T (1 ,
1) that becomes detectable by relax-ation into the meta-stable S (0 ,
2) state. In the detuningrange where this line appears, the S (0 ,
2) state lies en-ergetically only slightly above the T + (1 ,
1) state, so re-laxation back to the T + (1 ,
1) ground state is suppressed,again by the small phonon density of states at low ener-gies (see Fig. S2a of the appendix). The triplet resonanceis expected to appear at E z = gµ B ( B + b ) = hν , where h is Planck’s constant and µ B the Bohr magneton. Theinset of Fig. 2f shows the magnetic fields correspond-ing to the center of the measured triplet resonance linefor three excitation frequencies (see Figs. S2b,c of theappendix for the spectra), which are in good agreementwith the expected positions (black dashed lines in Fig.2f) based on the values | g | and b determined in the nextsection from other features of the PAT spectra. Sur-prisingly, the measured triplet resonance exhibits a finiteslope in the B ( ε ) spectra. A longitudinal magnetic fieldgradient ∆ B k gives rise to such a detuning dependence,but the ∆ B k required in our simulations to reproducethe observed slope is ∆ B k &
80 mT/50 nm, an order ofmagnitude larger than the gradient we calculate for the µ magnet. The magnitude of the slope remains a puzzle.
3. EXTRACTING ARTIFICIAL MOLECULEPARAMETERS
We now show how | g | , t c and b , the parameters usedfor all simulations, can be extracted independently fromthe experimental spin-flip PAT spectra. For this analysis B (T) a b
30 40 50-500 ∆ ε ' ( µ e V ) t c = 8.7 µ eV B ( µ eV)
11 GHz ∆ε + ∆ε - B (T) ∆ ε + ( µ e V ) g=0.382 ∆ ε + ( µ e V ) FIG. 3:
Analysis of the photon-assisted-tunnelingspectra. a , The detuning difference ∆ ε between the spinconserving line ∆ m = 0 and the ∆ m = ± B at 20 GHz (seeFig 2a), in order to fit the absolute effective electron g-factor g from the slopes of the linear fits (solid lines). inset , ∆ ε forthe blue ∆ m = +1 line at 11 GHz for different tunnel cou-plings. The offset of the curves increases with increasing t c (red to black circles) while g remains constant. b , ∆ ε betweenthe ∆ m = 0 line and the ∆ m = − ,
1) to (0 ,
2) transitionfrom Fig. 2c. The fit of the anti-crossing (red line) allows fora precise determination of t c . we only use the relative distance ∆ ε between PAT linesat fixed magnetic field, in order to be independent fromthe calibration of the detuning axis by means of the ref-erence line. Fig. 3a shows ∆ ε ± as a function of B usingthe 20 GHz data. ∆ ε ± is defined as the difference indetuning between the red ∆ m = ± m = 0 PATlines. For a fixed ν , both ∆ ε ± increase linearly with theZeeman energy and therefore allow fitting of | g | . (Notethat for ∆ ε + the linearity is only exact for sufficientlylarge B , at which the singlet anti-crossing does not af-fect the T + (1 ,
1) energy; see the appendix for a detaileddiscussion). A least-squares fit to the ∆ ε + data gives | g | = 0 . ± .
004 (Fig. 3a). From the linear behaviorof ∆ ε + , we also deduce that there is negligible dynamicnuclear polarization in the experiment.Knowing | g | precisely, we make use of the blue anti-crossing in Figs. 2b and 2f in order to determine t c (and b ). Fig. 3b shows the difference in detuning ∆ ε ′ betweenthe blue ∆ m = − m = 0 lines in Fig. 2f.Assuming the ∆ m = 0 line corresponds to the S (0 , S (1 ,
1) transition (as shown below), then∆ ε ′ = t c − ( hν − gµ B ( B + b )) hν − gµ B ( B + b ) − p ( hν ) − (2 t c ) , (1)where the first term is the detuning position of the T + (1 , S transition and the second the one of the S (0 , S (1 ,
1) transition. The best fits are obtained with t c = 8 . ± . µ eV and b = 109 ±
16 mT. This value for b matches very well our simulations of the stray field ofthe µ magnet at the DD location (see appendix A).An important question left open so far is whetherthe red ∆ m = 0 line involves predominantly transitionsfrom S (0 ,
2) to S (1 ,
1) or to T (1 , S (1 ,
1) does not require a change in the (total) spinand is thus expected to be excited more strongly thanthat to T (1 , S (1 ,
1) backto S (0 ,
2) will be stronger as well, so it is not obviouswhat steady-state populations will result in either case.Furthermore, given the small energy difference between S (1 ,
1) and T (1 , ε + > ∆ ε − indicates that the ∆ m = 0line originates from the transition to S (1 ,
1) and not to T (1 , ε ± = E z ± J , with J ( ε, t c ) = t c ε + O ( t c ) the exchange energy, whereas thelatter would result in ∆ ε + . ∆ ε − (in both scenario’s, b causes an additional fixed offset in both ∆ ε ± , but it doesnot contribute to their difference). This interpretation isconsistent with the increase of ∆ ε + with larger interdottunnel coupling, hence larger J (Fig. 3a inset; note thatthe slopes are not affected). It is further supported bythe data in Fig. S3b.So far only single-photon processes were considered,but at higher microwave power, also multi-photon linesemerge (Fig. 4a), mostly for the S (0 , S (1 ,
1) transi-tion (green dashed lines in Fig. 4a). Like the single-photon S (0 , S (1 ,
1) line, their position in detuning is B -independent (see appendix for details).
4. IDENTIFICATION OF THE SPIN-FLIPMECHANISMS
Having shown the power of spin-flip PAT for detailedmolecular spin spectroscopy, we now discuss the mech-anisms responsible for this process as confirmed by oursimulations. As a first possibility, the transitions from S (0 ,
2) to the triplet (1 ,
1) states can take place through avirtual process involving S (1 , S (0 ,
2) is cou-pled to S (1 ,
1) by the interdot tunnel coupling, and an(effective) magnetic field gradient ∆ ~B = (∆ B ⊥ , , ∆ B k )across the DD couples the spin part of all the (1 ,
1) statesto each other [13, 16]. Here ∆ ~B has a contributionfrom the effective nuclear field and from the µ magnet.The transition matrix element from S (0 ,
2) to T ± (1 ,
1) is ∝ t c ∆ B ⊥ B , assuming E z ≫ J . In the following, we use the B -dependence of this process as a fingerprint and focuson the red ∆ m = +1 line in Figs. 2a and 2b, as we canfollow it over the entire magnetic field range. The inten-sity of this line is constant in B and even if the microwaveamplitude E is varied, we observe no B -dependence inthe area under this peak (Fig. 4b). Before we concludethat the transition rate is magnetic field independent, werecall that the observed PAT lines reflect the steady-statechange in the charge configuration resulting from stim-ulated photon emission and absorption and spontaneousrelaxation. In order to rule out that a field-independentsteady state is reached from a field dependence of re-laxation and excitation that cancel each other, we verify
50 100 150 200123 ε ( µ eV) E ( m V ) E ( m V ) pea k a r ea ( a r b . un i t s ) E (mV) bc -4010 a B = . T B = . T F W H M ( G H z ) ∆ m=0 2ph- ∆ m=0 3ph- ∆ m=0 ∆ m=+1 E (mV)
B = 1 T ∆ m = +1 ∆ I QPC
050 50 100 150 ε ( µ eV) ∆ I Q P C E = 0.76 mV
FIG. 4:
Power-dependence of the photon-assisted tun-neling spectra and spontaneous relaxation. a ∆ I QPC as a function of the double dot detuning ε and the microwaveamplitude E for 11 GHz and B = 1 . B = 2 . P ε = 1 . m = 0(∆ m = +1) PAT transitions. The reference signal stemmingfrom the pulse is marked by orange arrows. The voltage am-plitude E is measured at the end of the coaxial lines at roomtemperature. The uppermost panel displays a linecut mea-sured at 1 T. The red line is a least-squares fit by a sum offour lorentzian peaks to the data. b The fitted Lorentzianpeak area of the ∆ m = +1 line from a is plotted as a func-tion of microwave amplitude E and magnetic field B . c Thefitted Lorentzian linewidth at half maximum (FWHM) of the∆ m = +1 and the lines ∆ m = 0 from a is plotted as a func-tion of microwave amplitude E for B = 1 T. The error barsin b , c and are determined from the Lorentzian least-squaresfit (see the inset of Fig. 4a). that the spontaneous relaxation rate is field-independentas well (see appendix). These observations suggest thatthe coupling mechanism is magnetic field independentand thus virtual processes involving S (1 ,
1) do not give astrong contribution to the transition rates.More recently, two mechanisms were considered thatprovide a direct, B -independent matrix element between S (0 ,
2) and the (1 ,
1) triplet states: (i) the hyperfine con-tact Hamiltonian is of the form P j δ ( r − r j ) ~I j · ~S . Thus,nuclear spins, ~I j , in the barrier regions, where the S (0 , ,
1) triplet states, canflip-flop with the electron spin, ~S , simultaneously withcharge tunneling [17]. (ii) The SO Hamiltonian is of theform p x,y S x,y and can directly couple states which differin both orbital and spin [18, 19]. (When the orbital partof the initial and final state are the same, the SO Hamil-tonian does not provide a direct matrix element and thetransition rate becomes B -dependent [8, 20–23].) Theratio of the SO mediated rate and the hyperfine medi-ated rate can be estimated as E √ N A dλ SO , which is a fewthousand in the experiment (see the appendix). Here E is the single-dot level spacing, N the number of nu-clei in contact with one dot, A the hyperfine couplingstrength, d the interdot distance and λ SO the spin-orbitlength. We therefore believe that SO interaction is thedominant spin-flip mechanism for the observed PAT tran-sitions. The presence of a magnetic field independentmatrix element between S (0 ,
2) and the (1 ,
1) triplets isconfirmed by the observation that the intensity of thereference signal shows no field dependence.Finally, we extract from Fig. 4a the fitted linewidth asa function of driving power (Fig. 4c). For small E , wefind a width & m = 0 and the ∆ m = +1 lines are furtherbroadened, up to E ≈ . µ s)timescale (data not shown). Presumably charge or gatevoltage noise is responsible for the broadening instead.We have shown that in our DD system, all (1 ,
1) spinstates have at least weakly allowed electric dipole tran-sitions to S (0 , ,
1) spin space via off-resonant (microwave) Ramantransitions through the excited S (0 ,
2) state, which en-ables a variety of new approaches to manipulating andeven defining qubits in DDs. Furthermore, such controlenables new measurement techniques that do not rely onPauli spin blockade [24]. An example is a measurementthat distinguishes parallel from anti-parallel spins whileacting non-destructively on the S (1 , − T (1 ,
1) sub-space, by coupling resonantly the T + (1 ,
1) and T − (1 , S (0 ,
2) followed by charge readout. This con-stitutes a partial Bell measurement and leads to a newmethod for producing and purifying entangled spin states[25].
Appendix A: Methods1. Sample fabrication
30 nm thick TiAu gates are fabricated on a 90 nmdeep (Al . ,Ga . )As/GaAs two-dimensional electron gas (2DEG) by means of ebeam lithography. The doubledot axis is aligned along the [110] GaAs crystal direc-tion (z-direction), which is parallel to the external mag-netic field direction ~B (Fig. 1a). The 2DEG is Si δ -doped (40 nm away from the hetero-interface), exhibitsan electron density of 2 . × cm − and a mobil-ity of 2 . × cm /Vs at 1 K in the dark. Thegrounded, 275 nm thick, 2 µ m wide and 10 µ m long Co µ magnet is evaporated on top of a 80 nm thick dielectriclayer, aligned along ~B (magnetic easy axis) and placed ≈
400 nm away from the closest dot center. We calcu-late [26] that at the double dot position the µ magnet adds b ≈
110 mT to B z and generates a magnetic field gradi-ent of ∆ B k ≈ B ⊥ ≈ − B z &
2. Measurement
The sample is mounted in an Oxford KelvinOx 300dilution refrigerator at 30 mK. Left and right side gatevoltages, V L and V R , are set by low-pass filtered dc linesand ≈
60 dB attenuated coaxial lines combined with bias-tees with a cutoff frequency of 30 Hz. The pre-amplifiedcurrent through the quantum-point contact is read outby a lock-in amplifier locked to the 880 Hz on-off modula-tion of the microwaves. The bias across the double dot isset to 0 µ V. Voltage pulses to the left and right side gatesare generated with a Sony Textronix AWG520. The mi-crowaves are generated with a HP83650A and combinedwith the pulses to the right side gate. Microwave burstsand detuning pulses are synchronized to ensure that themicrowave excitation is switched off during the detuningpulses that generate the reference signal (see Fig. 1d).
3. Simulation
The Hamiltonian describing the two-spin system nearthe (1 , ,
2) transition is taken to be a five-state sys-tem, with four (1 ,
1) spin states and a (0 ,
2) spin singlet[16] in the presence of an external magnetic field B anda magnetic field gradient ∆ ~B , which includes both thequasi-static nuclear field and the field from the µ magnet.This is given by H = gµ B P ( ~S · ( ~B + ∆ ~B ) + ~S · ( ~B − ∆ ~B )) P − ε | S (0 , ih S (0 , | + H t , were P is the projec-tor onto the (1 ,
1) subspace, ε is the detuning due to thedifference in gate potentials from the left and right gatesand the tunnel coupling H t = t c ( | S (1 , i h S (0 , | ) + t SO ( | T + (1 , i h S (0 , | ) + t SO ( | T − (1 , i h S (0 , | ) + h . c . with t c the spin-conserving tunnel coupling and t SO thespin-orbit coupling set to 5 % of t c .To find the signal we expect theoretically from the ex-periment, we add a weak, rapidly oscillating term to theHamiltonian: ε → ε + Ω cos( νt ). We diagonalize H with Ω = 0, then make a rotating frame transformationin which levels are grouped into bands n (defined by aprojector P n ) where the states in a band n are muchcloser in energy than ~ ν , while the energy difference be-tween states in band n and n + 1 are within 2/3rds of hν . Each band rotates at a rate nν , and we can thenmake a rotating wave approximation, keeping terms dueto δ that couple adjacent bands, i.e., our perturbation inthe rotating frame and rotating wave approximation is V = Ω P n P n | (0 , S ih (0 , S | P n +1 + h . c . . Next, we adddissipation and dephasing by including relaxation due tocoupling of the electron charge to piezoelectric phononsin 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 ahigh energy resolution along the double dot (DD) detun-ing axis and over a wide external magnetic field range. Inthe experiment, we observe a monotonous, reproducibledrift of the stable charge regions predominantly alongthe right side gate voltage as we change the magneticfield. Changing the voltages applied to the left and tothe right side gate accordingly, we partially compensatefor this drift. We then record an 11 GHz PAT spectrumas displayed in Fig. S1a. In order to precisely calibratethe detuning axis for all B , a reference signal is generatedtogether with the PAT spectrum by interspersing the mi-crowaves every 5 µ s by a 200 ns detuning pulse with am-plitude P ε towards negative detuning (Fig. S1b). 200 nsare found to be sufficient to mix the S (0 ,
2) state entirelywith the T + (1 ,
1) state at their anti-crossing ε ST + , sothat a Pauli spin blocked T + (1 ,
1) signal is observed at adetuning ε = ε ST + + P ε . The magnitude of P ε is chosensuch that the reference signal appears at a detuning po-sition far away from the PAT signal. The pulses do notalter the detuning position of the PAT resonances. Inaddition to ST + mixing, mixing of the S with T (1 , T − (1 ,
1) is observed due to the pulsing. The formergives rise to a positive ∆ I QP C background on the left ofthe reference signal in Fig. S1b. The latter generates aweak second reference line that overlaps with the ST + reference line for B → B is increased.In a post-processing step, we separately fit the positionof the ST + peaks for all B and shift every row of thespectrum, such that the peaks are vertically aligned at ε ∗ = P ε as shown in Fig. S1c. Thus, ε ∗ = 0 is at the ST + anti-crossing, by definition. All data points at agiven B are shifted by the same amount in detuning.This ε ∗ -detuning axis is well-defined but ‘moves’ withrespect to the ε -detuning axis as a function of B , since ε ∗ = ε − ε ST + ( B ). The lever arm for the voltage toenergy conversion is read from the voltage distance ofthe second and third ∆ m = 0 PAT line at ν = 11 GHz(see Fig. 4a), which equals the photon energy hν [7]. Forthe analysis of all spectra we used the same lever arm.For better readability of the PAT spectra, we finally a bc d ε ( µ eV) B ( T ) -1180 -1178 -1176123 V R (V L ) (mV) B ( T ) -1180 -1178 -1176123 V R (V L ) (mV) B ( T ) ε * ( µ eV) B ( T ) ∆ I QPC (a.u.) ST + ST - Fig. S 1:
Calibration of the detuning axis for the PATspectra. a , PAT spectrum as measured without pulses (seeinset of Fig. 1b) with a microwave frequency of 11 GHz. Thevoltage applied to the right side gate V R and left side gate V L is swept simultaneously to follow the detuning axis indicatedin Fig 1b. b , same PAT spectrum, but with 200 ns pulses thatintersperse the microwave bursts every 5 µ s (see top panelof Fig. 1d). A red reference line appears at positive DDdetuning, when the detuning pulse reaches exactly the ST + anti-crossing. Also a weak line due to mixing at the ST − anti-crossing is observed. Both lines are marked by arrows.The more red background is due to ST mixing. c , PATspectrum with calibrated detuning axis. The horizontal linesof the spectrum are shifted so that the reference pulse appearsat the detuning position, which equals the pulse amplitude P ε converted into energy (see text) along the detuning axis. Here ε ∗ = 0 is equal to the ST + anti-crossing for every magneticfield. d , To convert the ε ∗ scale of the detuning axis into the ε scale, where ε = 0 equals the S (1 , S (0 ,
2) anti-crossing, thespectrum is sheared by the Zeeman-energy using the electrong-factor that is independently determined as explained in themain article. At low magnetic fields this transformation ofscales is wrong and gives rise to a curvature of the spectrum,which is a function of the tunnel coupling. convert the ε ∗ -scale to the ε -scale found in literature, forwhich ε = 0 is defined by the S (1 ,
1) to S (0 ,
2) anti-crossing. To do so, we additionally shift all data pointsat a given B by | g | µ B | B | towards positive detuning (Fig.S1d). However, since ε ST + = | gµ B B | holds true only for | gµ B B | ≫ t c , where t c is the interdot tunnel coupling, thedetuning axis conversion fails for low magnetic fields. Asa result the PAT resonance-lines bend towards positive ε for B → b -1182 -11811.21.31.41.51.6 V R (mV) B ( T ) ν = 7.5 GHz ∆ I QPC (arb.units) a B = 2 TT - (1,1)T (1,1)T + (1,1) S(0,2)S(1,1) -100 0 100-50050 ene r g y ( µ e V ) ε ( µ eV) c -1182 -11802.02.12.22.32.4 ν = 13 GHz V R (mV) B ( T ) ∆ I QPC (arb. units)
Fig. S 2:
Triplet resonance in the PAT spectra. a ,Energy eigenstates of the two-electron spin states plottedwith the same color code as in Fig. 2c of the main arti-cle. Microwaves are resonant to the T + (1 , T (1 ,
1) transi-tion. Spontaneous relaxation from T (1 ,
1) to the metastable S (0 ,
2) enables the detection via ∆ I QPC . b , The PAT spec-trum as measured with a microwave frequency of 7.5 GHzexhibits a nearly horizontal blue feature as visible in Fig. 2f.Also at this microwave frequency the line starts from the mag-netic field, at which we expect the electron spin resonancecondition to be fulfilled (black dashed line) hν = gµ ( B + b ). c , PAT spectrum recorded at 13 GHz with corresponding lineindicating electron spin resonance. Appendix C: Triplet spin resonance
In Fig. 2f of the main article, a PAT feature is observedthat is due to a transition from the T + (1 ,
1) ground stateto the T (1 ,
1) excited state. The T (1 ,
1) state can relaxvia spontaneous phonon emission to the singlet bondingstate, a superposition of S (1 ,
1) and S (0 , T + (1 , I QP C . A peculiarity of the T + (1 , T (1 ,
1) resonance is its slope in the PAT spectrum,which might be a result from a gradient magnetic fieldalong the magnetic field direction as discussed in themain article. Note that in the same detuning range,we observe also direct PAT transitions from the T + (1 , T + (1 ,
1) to T (1 ,
1) resonance, as a function of the microwave fre-quency ν . The Figs. S2b and S2c show raw PAT spectra (without any post-processing step applied as explainedabove) recorded with ν = 7 . ν = 13 GHz, re-spectively. The dashed lines mark the magnetic field,at which the electron spin resonance condition hν = gµ B ( B + b ) is fulfilled. Here we use the absolute elec-tronic g-factor | g | = 0 .
382 and longitudinal magnetic fieldoffset b = 109 mT as determined by the Figs. 3a and3b of the main article. Alternatively, we might use the T + (1 ,
1) to T (1 ,
1) resonance feature to determine | g | .If we use the center magnetic field of this feature as theresonant field, we calculate | g | = 0 . ± .
005 from themicrowave frequency dependence in good agreement withthe | g | = 0 .
382 found in Fig. 3a in the main article.
Appendix D: The ∆ m = 0 PAT transition
In the main article, we discuss whether the red (∆ m =0 , ∆ n = 1) PAT resonance is dominantly due to a transi-tion from the S (0 ,
2) ground state to the S (1 ,
1) state orto the T (1 ,
1) state. It is difficult to spectroscopicallyresolve these transitions, since they differ only by the ex-change energy J ( ε ). In Fig 3a of the main article, weuse ∆ ε ± ( B ), the difference in detuning between the red∆ m = ± m = 0 lines, to assign the PAT res-onance. Here, we support this argument by calculatingthe expected ∆ ε ± ( B ) functions for both extreme scenar-ios: a pure S (0 , S (1 ,
1) and a pure S (0 , T (1 , | g | = 0 . t c = 8 . µ eV and b = 109 mT values.The result is displayed in Fig. S3a for a microwavefrequency ν = 20 GHz. Obviously, the scenario of apure singlet transition results in ∆ ε + ( B ) > ∆ ε − ( B ),whereas ∆ ε + ( B ) = ∆ ε − ( B ) is found for a transition tothe triplet state. In the experiment, we clearly observe∆ ε + ( B ) > ∆ ε − ( B ) and therefore the ∆ m = 0 transitionis dominantly a singlet transition.In both scenarios, the ∆ ε − ( B ) increases non-linearlytowards high B , since the resonance ∆ m = − E z ≈ hν − t c , where E z is the Zeeman energy. This curva-ture is not observed in the experiment as the ∆ m = − B ≈ . ε + ( B ) functions are linear, which holds true onlyif E z ≪ t c − hν . The choice of a high ν and an ap-propriate B -range allows to extract | g | from ∆ ε + ( B ) bya simple linear least-squares fit as demonstrated in themain article.As a final step, we analyze the splitting of the ∆ ε + ( B )and ∆ ε − ( B ) function quantitatively. Their offsets de-pend upon the exchange energy J ( ε, t c ), which we can ex-perimentally vary by t c or indirectly by ν . For hν → t c ,the detuning position of the ∆ m = 0 resonance be-comes strongly affected by J . As shown in the in-set of Fig. S3b, the S (0 , S (1 ,
1) transition (greenarrow) shifts more towards negative detuning ε thanthe potential S (0 , T (1 ,
1) transition (violet arrow).We measure ∆ ε + ( B ) for various ν and determine the ∆ε ν (GHz) ∆ ε ( T ) + ∆ε + ∆ε - ∆ε + ∆ε - calculation 20 GHz S(1,1) T (1,1) ∆ ε + ( µ e V ) B (T) ∆ ε + ( µ e V ) ∆ε +0 a b + Fig. S 3:
Analysis of the ∆ m = 0 PAT resonance.a , ∆ ε + and ∆ ε − calculated for the parameters | g | = 0 . t c = 8 . µ eV, b = 109 mT, ν = 20 GHz and two possiblescenarios: The ∆ m = 0 line is resonant between the S (0 , S (1 ,
1) (red and orange line) and T (1 ,
1) (greenand blue line). Zoom-in for low-magnetic fields in the inset. b ,The detuning offset ∆ ε +0 between the transition ∆ m = +1 for B → m = 0 tran-sitions from S (0 ,
2) to S (1 ,
1) (green arrow) and to T (1 , ν . The error bars are determined from thelinear extrapolation B → t c = 8 . µ eV (filled circles) and at t c ≈ µ eV (open rect-angles). The green and violet lines are least-squares fits tothe measured offsets assuming a pure S (0 ,
2) to S (1 ,
1) andto T (1 ,
1) transition, respectively. ∆ ε +0 = ∆ ε + ( B →
0) by a linear fit at a sufficiently highmagnetic field range. This procedure turned out to be im-practical with ∆ ε − ( B ), because the ∆ m = − S (0 , /T + (1 ,
1) transition to zero field (redarrow) exhibits a different detuning position than the S (0 , T (1 ,
1) transition (violet arrow), because the lin-ear extrapolation follows the dashed blue line in the insetof Fig. S3b, i.e. the linear extrapolated detuning posi-tion is not affected by the hybridization of the singlets.The detuning position of the S (0 , T (1 ,
1) PAT tran-sition, however, is affected by the singlet hybridization,since it lowers the energy of the initial state S (0 , ε +0 is always larger then zero,but also depends upon the nature of the ∆ m = 0 PATresonance.In the inset of Fig. S3b, ∆ ε +0 for the S (0 , S (1 , ε +0 is considerablysmaller, if the S (0 , T (1 ,
1) PAT resonance dominatesover the S (0 , S (1 ,
1) resonance. The analysis of ∆ ε +0 is complicated by the remanence of the µ magnet, whichis not exactly known, but should be smaller than the fullymagnetized field of b = 109 mT. Due to the remanence,the linear extrapolation of the S (0 , T + (1 ,
1) transitiontowards zero external magnetic field, leaves an additionaloffset on ∆ ε +0 . This offset, however, is independent from ν and t c . In Fig. S3b, the extrapolated ∆ ε +0 values areplotted as a function of the microwave frequency ν for twotunnel couplings (filled circles and open squares). Fit-ting the filled circles with the well-known t c = 8 . µ eV, we determine a reasonable fit by assuming the ∆ m = 0PAT resonance to be purely singlet (green line). The fitwith a potential S (0 , T (1 ,
1) transition (violet line)fails at ν = 5 GHz. The only fit parameter used here isthe remanence of the µ magnet, which was found to be ≈
70 mT for the fit function assuming a S (0 , S (1 , ≈
140 mT assuming a S (0 , T (1 , b = 109 mT was found at anexternal magnetic field of 2 T.As a final check, we take ∆ ε +0 values into account,which were determined when the DD was tuned to asmaller t c = 4 µ eV (open squares). These data pointscannot be fitted by the fit function that assumes a S (0 , T (1 ,
1) transition at all, since the ∆ ε +0 /gµ B val-ues observed are already smaller than the remanence of ≈
140 mT, which would stay valid for the altered tun-nel coupling. Obviously, this leads to a contradiction,since all ∆ ε +0 would become negative after subtractingthe remanence. Only the assumption of a purely singlet∆ m = 0 PAT transition in combination with the smallerremanence of ≈
70 mT, as fitted above, allows reasonablefitting. Our conclusion from the main article is thereforefurther supported.
Appendix E: Spin flip-tunneling mechanism
As noticed in the main text, there is a direct matrixelement between the S (0 ,
2) and the (1 ,
1) triplet states.This can occur due to the nuclear spins in the barrierbetween the dots and due to spin-orbit (SO) interaction.We look at a toy model to examine the relative impor-tance of these two processes. Specifically, we consider thehopping matrix element for a single electron with spin ~S between two wavefunctions associated with an electronon the left ( | L i ) and on the right ( | R i ) via the perturba-tion: V = ~ m ∗ λ SO [ − α ( S ˜ z p ˜ y − S ˜ y p ˜ z ) + β ( p ˜ y S ˜ y + p ˜ z S ˜ z )] ++ Av X j δ ( r − r j ) ~I j · ~S (E1)where we have absorbed the Rashba ( α ) and Dresselhaus( β ) terms into a single spin-orbit interaction with a char-acteristic spin-orbit length λ SO ∼ µ m. We recall that m ∗ is the effective electron mass, A ≈ µ eV, v is theunit cell volume and I j is the nuclear spin at ~r j .We now wish to estimate the spin-flip tunneling for thesingle electron case, given by averaging over the orbitaldipole: h R | V | L i (E2)This can be evaluated explicitly for | L ( R ) i = exp( − ( z ± a/ / σ ) / (2 πσ ) / φ ( x, y ) where z is the inter-dot axis(at an angle θ with the axis ˜ z from the spin-orbit interac-tion in Eq. E1) and φ ( x, y ) is the transverse-longitudinal0wavefunction. As tunneling occurs only along the z -axis,matrix elements with p y are zero. We find two tunnelingmatrix elements: t SO = ~ m ∗ σ a λ SO e − a / σ ~n · ~S (E3) t nuc = gµ B ~B nuc , f · ~S (E4) ~n = − cos( θ )[( α − β ) cos( θ ) + ( α + β ) sin( θ )]ˆ z − sin( θ )[( β − α ) sin( θ ) + ( α + β ) cos( θ )]ˆ y (E5) gµ B ~B nuc , f = Av X j | ψ L ( r j ) || ψ R ( r j ) | ~I j (E6)We remark that the rms value for ~B nuc , f is given by gµ B q | ~B nuc , f | = A s v X j | ψ L | | ψ R | I ( I + 1) ≈ e − a / σ A s v X j | ψ L | (E7)That is, it is the rms value for a single dot, A/ √ N , mul-tiplied by exp( − a / σ ). Also, the size of the single- particle wavefunction, σ , is related to the orbital energyscale of a single dot by ∆ ≈ ~ m ∗ σ . Thus, the relativestrength of the two tunneling terms (including spin flip)is | t SO || t nuc | = ∆ A/ √ N | ~n × ~B ext | | B ext | a λ SO (E8)where N is the number of spins in a single quantum dot.We explore briefly how this ratio varies with dot size σ and spacing a . Specifically, ∆ √ N ∝ σ − , so a larger dotreduces the strength of spin-orbit tunneling compared tohyperfine-assisted tunneling. On the other hand, increas-ing the distance a increases the relative strength of spin-orbit tunneling to hyperfine-assisted tunneling. Settingin A = 100 µ eV, N = 4 × , a = 75 nm, ∆ = 1000 µ eVand λ SO = 10 µ m, we calculate the ratio of the matrixelements | t SO || t nuc | to be ≈
60. We remark that in externalfield parallel to ~n prevents any SO spin-charge flips. Allspin charge flips occur only via t nuc .In the bases ( T − (1 , | ↓↑i , | ↑↓i , T + (1 , S − (0 , t nuc term has the form B z ∆ B ⊥ x − i ∆ B ⊥ y − ∆ B ⊥ x − i ∆ B ⊥ y − t SO,y / √ ∆ B ⊥ x + i ∆ B ⊥ y − ∆ B k − ∆ B ⊥ x − i ∆ B ⊥ y ( − it SO,z − t c ) / √ − ∆ B ⊥ x + i ∆ B ⊥ y B k ∆ B ⊥ x − i ∆ B ⊥ y ( − it SO,z + t c ) / √ − ∆ B ⊥ x + i ∆ B ⊥ y B ⊥ x + i ∆ B ⊥ y − B z − t SO,y − t SO,y / √ it SO,z − t c ) / √ it SO,z + t c ) / √ − t SO,y / √ − ε (E9)where, e.g., the Larmor precession frequency of an elec-tron spin in the left dot is B z − ∆ B k . Appendix F: Simulations of the PAT spectra -relaxation
The simulated spectra in Figs. 2c,d of the main arti-cle include the effect of the phonon-mediated relaxation.In addition to the explanations of the simulations in ap-pendix A, we continue here on the coupling of the electronspin to the phonon bath. We thereby neglect deformationphonons as the energy scales examined in the experiment(7-22 GHz) are much smaller than the characteristic fre-quency scale of a phonon on the length scale of the dot c ph /l dot ∼ −
120 GHz. To determine the coupling,we take as an ansatz for the electronic wavefunctions theFock-Darwin states, given by Gaussians, and calculatethe coupling after orthogonalizing the states with the per-turbation V ph = P e,k f ( k z ) q ~ ρω e,k e i~k · ~r β e ( a e,k − a † e, − k ) [27], where f ( k z ) ≈ | S (0 , ih S (0 , | with and without theexcitation Ω, mimicking the effect of the lock-in detec-tion. Appendix G: Measurement of spontaneousrelaxation
In the main article, we state that the spontaneous re-laxation rate from T + (1 ,
1) to S (0 ,
2) is found to be mag-netic field independent. Relevant for the PAT processis the spontaneous relaxation at a fixed energy differ-ence between the T + (1 ,
1) and S (0 ,
2) state, which is setby the photon energy. Thus, when changing the mag-netic field, the detuning ε has to be changed accord-ingly. The measurement of the spontaneous relaxation1 ∆ I Q P C / ∆ I P ε ( µ eV) τ ( µ s) τ = 5 µ sno microwaves Fig. S 4:
Spontaneous relaxation rate.
Normalized QPCcurrent averaged over τ = 5 µ s immediately after full mixingat the ST + anticrossing for various magnetic fields at zero mi-crowave power. The spontaneous spin relaxation after mixingis measured at a distance P ε to the ST + mixing point (seescheme in Fig. 2c). The dependence on the averaged time in-terval τ is displayed in the inset for B = 1 . P ε = 2 mVtogether with a least-squares fit (red solid line). rate is done as follows: Starting from S (0 , T + (1 ,
1) by 50 % via a 200 ns detuning pulse[28] with amplitude P ε in the absence of microwaves,and monitor the decay back to S (0 , s can be extracted from the time averaged lock-in signal ∆ I QP C ( τ ) = ∆ I τ R τ exp ( − t Γ s ) dt , where τ isthe time spent in Pauli blockade between the pulses.∆ I = ∆ I QP C (0), which is independent from B , is ex-tracted from the fit in the inset of Fig. S4. In order tocover various values of the detuning and the magneticfield, we next fix τ = 5 µ s and record ∆ I QP C as a func-tion of P ε for three different magnetic fields. We observethat ∆ I QP C (5 µ s) and thus Γ s are essentially indepen-dent of B (Fig. S4). This holds true for all P ε and hencefor all T + (1 , S (0 ,
2) energy splittings. Note that re-gardless of B , this energy splitting is given by P ε alone.This reflects exactly the situation in the PAT experiment,for which the microwave frequency alone sets the energysplitting and thus also the required phonon energy forthe spontaneous relaxation process. Appendix H: Power dependence
In Fig. 4a of the main article, the power dependence ofthe PAT sidebands is shown for two magnetic fields. InFig. S5a-e, ν = 11 GHz-spectra measured with a series ofmagnetic field values are plotted, to ease keeping track ofthe resonances as they shift in detuning with the externalmagnetic field. Here, the spectra are plotted such that ε ∗ = 0 is the ST + anti-crossing for all B (compare Fig.S1c). The reference peaks due to pulsing to the ST + anti-crossing are all aligned at ε ∗ = P ε = 1 . E is estimated from the attenuation of the high-frequencycircuit at room temperature. PAT sidebands at larger detuning appear when E isincreased as expected for PAT. Oscillation of the PATamplitude as a function of E is hardly visible, becausewe cannot reach sufficiently high E and because the PATlines become broadened as a function of power. One side-band emerges at high ε ∗ ≈ . E . Thisresonance stays at a constant ε ∗ for all B and is there-fore a transition from S (0 ,
2) to T + (1 ,
1) (∆ m = 1). As B is increased, the other transitions move towards lower ε ∗ while keeping the distance along ε ∗ . They are due toPAT transition with ∆ m = 0. At B ≈ m = 1PAT transition overlaps with the 2-photon ∆ m = 0 PATtransition. This is expected, since ν = 11 GHz equalsthe Zeeman energy at this magnetic field in our doubledot. Acknowledgments
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 withthe sample fabrication and R. Schouten, A. van der En-den and R. G. Roeleveld for technical support. This workis supported by the ‘Stichting voor Fundamenteel On-derzoek der Materie (FOM)’ and a Starting Investigatorgrant of the ‘European Research Council (ERC)’.L.R.S, F.R.B., V.C. and T.M. performed the experi-ment, W.W. grew the heterostructure, T.M. fabricatedthe sample, L.R.S., J.D., J.M.T and L.M.K.V. devel-oped the theory, J.M.T. did the simulations, all authorscontributed to the interpretation of the data and com-mented on the manuscript, and L.R.S., J.D., J.M.T. andL.M.K.V. wrote the manuscript.2 -0.5 0.0 0.5 1.0 1.5123 ε * (mV) E ( m V ) -0.5 0.0 0.5 1.0 1.5123 ε * (mV) E ( m V ) -0.5 0.0 0.5 1.0 1.5123 ε * (mV) E ( m V ) -0.5 0.0 0.5 1.0 1.5123 ε * (mV) E ( m V ) -0.5 0.0 0.5 1.0 1.5123 ε * (mV) E ( m V ) -4010 ∆ I QPC
B = 1.0 T B = 1.5 T B = 2.0 TB = 2.5 T B = 3.0 T a b cd e
Fig. S 5:
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