Electron Spin Excited States Spectroscopy in a Quantum Dot Probed by QPC Back-action
HaiOu Li, Ming Xiao, Gang Cao, Cheng Zhou, RuNan Shang, Tao Tu, GuangCan Guo, HongWen Jiang, GuoPing Guo
EElectron Spin Excited States Spectroscopy in a Quantum Dot Probed by QPCBack-action
HaiOu Li, Ming Xiao, ∗ Gang Cao, Cheng Zhou, RuNan Shang, Tao Tu, GuangCan Guo, HongWen Jiang, and GuoPing Guo † Key Laboratory of Quantum Information, Chinese Academy of Sciences,University of Science and Technology of China, Hefei 230026, People’s Republic of China Department of Physics and Astronomy, University of California at Los Angeles,405 Hilgard Avenue, Los Angeles, CA 90095, USA (Dated: April 2, 2019)The quantum point contact (QPC) back-action has been found to cause non-thermal-equilibriumexcitations to the electron spin states in a quantum dot (QD). Here we use back-action as anexcitation source to probe the spin excited states spectroscopy for both the odd and even electronnumbers under a varying parallel magnetic field. For a single electron, we observed the Zeemansplitting. For two electrons, we observed the splitting of the spin triplet states | T + (cid:105) and | T (cid:105) andfound that back-action drives the singlet state | S (cid:105) overwhelmingly to | T + (cid:105) other than | T (cid:105) . Allthese information were revealed through the real-time charge counting statistics. The spin states of few electrons in quantum dots(QD) have been demonstrated as potential candidates forqubits [1, 2]. The Zeeman states for odd number of elec-trons and the spin singlet-triplet states for even numberof electrons form the basis of qubit operation and de-tection. Also we know that any detection to the qubitstates would necessarily have back-action [3, 4]. Recentlywe found that the back-action from a quantum point con-tact (QPC) would excite the spin singlet to triplet statesand degrade the fidelity of qubit operation [5]. However,this study was limited to the non-degenerate spin statesat zero magnetic field.To know the influence of back-action on all relevantspin states would be very helpful for spin based qubits.In this work we applied a parallel magnetic field to lift thespin states degeneracy and explored their spectroscopy.By studying the non-equilibrium part of the real-timecharge counting statistics under strong back-action con-dition, we observed the excitation from an up to a downspin state for one single electron, and the excitation fromthe spin singlet state | S (cid:105) to triplet states | T + (cid:105) and | T (cid:105) for two electrons. We revealed the linear dependence ofthe exchange energy on the magnetic field. More im-portantly, we found that back-action overwhelmingly ex-cites | S (cid:105) to | T + (cid:105) state. This implies that spin qubitsbased on | S (cid:105)−| T (cid:105) are more immune of back-action than | S (cid:105) − | T + (cid:105) .The GaAs sample is the same as in earlier experiments[5, 6]. Fig. 1 is a scanning electron microscopy (SEM)image of the QD-QPC structure. The QD is configuredto be in thermal equilibrium with the electron reservoiron the right side. Random telegraph signal (RTS), thesingle electron tunneling back and forth between the QDand the right side reservoir due to thermal fluctuations, ∗ Electronic address: [email protected] † Electronic address: [email protected]
FIG. 1: (a) A SEM picture showing the geometry of our sam-ple. The dotted circle is the QD and the arrowed line in-dicates the QPC channel. (b) RTS statistics Γ out and Γ in .Black open circles for 1 e and red closed triangles for 2 e . has been recorded and analyzed. Fig. 1 (b) shows theRTS statistics Γ out and Γ in for the last two electrons.As we found earlier, the QPC back-action caused severenon-thermal-equilibrium effect. For the ( n − e ↔ ne tunneling, the back-action drives the n th electron out ofthe QD even when the electron addition energy µ n << the reservoir Fermi level E F [6]. As a result, Γ out showsa strong saturation tail for both 1 e and 2 e when µ n − E F <<
0. What’s more, those slowly relaxing excitedstates provide additional tunneling channels and give riseto extra features [5]. For 2 e , we see a strong side peakin Γ out and an extra elevated plateau in Γ in at about − . meV , arising from the spin singlet-triplet excitation.For 1 e , no extra feature is seen since the spin up and downstates are degenerate and indistinguishable.Now we apply a parallel magnetic field to lift the en-ergy degeneracy between the two Zeeman states for theodd number of electrons, and between the three tripletstates for the even electron numbers. This will enableus to probe the detailed spin excited states spectroscopyand study the effect of back-action on each individualspin states. Fig. 2 (a) shows the total tunneling rateΓ total for 2 e with varying magnetic field strength B . Theblack line at the bottom is at B = 0 T . Due to the strongback-action, we see both the ground state | S (cid:105) and the ex-cited states | T (cid:105) , located at µ − E F = 0 and − . meV , a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec FIG. 2: (a) Γ total for 2 e at different magnetic field. Frombottom to up, B increases from 0 T to 8 T by 1 T each step.All traces except 8 T are shifted on the y-axis for comparison.(b) Up: extracted energy of | T (cid:105) and | T + (cid:105) with respect to | S (cid:105) .Red triangles for | T (cid:105) and blue circles for | T + (cid:105) . Solid linesare the predicted energy for all the spin singlet and tripletstates. Down: Red triangles are the extracted energy shiftof | T (cid:105) with magnetic field, which represents the shift of theexchange energy, i.e, ∆ J ( B ). Red line is the linear fitting.Blue closed circles are the energy shift of | T + (cid:105) . Blue opencircles are the energy difference between | T + (cid:105) and | T (cid:105) , whichagrees well with the predicted Zeeman splitting assuming g-factor is 0 . out versus magnetic field and electron energy.On the top and bottom, an original trace at B = 8 T and 0 T is respectively presented. (d) 3D plot of Γ in . respectively. At this point, we don’t know | T (cid:105) refers towhich one of the three energy degenerate configurations | T + (cid:105) , | T (cid:105) or | T − (cid:105) . As we increase the magnetic field,the peak at − . meV shifts closer to the singlet state | S (cid:105) , at a speed comparable to that of the Zeeman split-ting. This suggests that the | T + (cid:105) state plays the majorrole. The | T (cid:105) state, which stays more constant, is alsovisible at large magnetic field ( B ≥ T ). But its peakamplitude is much smaller than that of | T + (cid:105) . The thirdstate | T − (cid:105) is not visible in this experiment. This phe-nomena is also presented in Fig. 2 (c) and (d), whichare the 3D diagrams for the numerical derivative of Γ out and Γ in . The triplet state splits into two with increas-ing magnetic field. The one obviously shifting towards the singlet state is | T + (cid:105) , and the other one staying moreconstant is | T (cid:105) .In Fig. 2 (b) we explicitly extracted the energy of eachstates with respect to the singlet energy, as functions ofthe magnetic field. On the upper part of the figure theblue circles are the energy of | T + (cid:105) and the red trianglesare that of | T (cid:105) . Only at B ≥ T we are able to collectdata for | T (cid:105) . The solid lines show the standard pictureof the spin singlet-triplet energy spectrum. At B = 0, thethree triplet states are degenerate with a certain separa-tion from | S (cid:105) , called exchange energy J . With increasing B , | T (cid:105) is assumed invariant while | T + (cid:105) and | T − (cid:105) shiftby the amount of ± gµ B B respectively. Experimentally,we found that the separation between | T + (cid:105) and | T (cid:105) fitsvery well with the predicted Zeeman splitting using a g-factor − . | T (cid:105) isnot exactly invariant under magnetic field. It slightlydecreases with field, which means that the exchange en-ergy J ( B ) is actually a function of magnetic field. Inthe lower part of Fig. 2 (b) we presented ∆ J ( B ), de-fined as J ( B ) − J (0). The linear fitting gives a slope( − . ± . meV /T . If we don’t consider the de-crease of exchange energy with field, then the extractedenergy shift of | T + (cid:105) itself is too large to be explained asthe Zeeman splitting, shown as the blue closed circles inthe lower part of Fig. 2 (b). Thus the field dependenceof J ( B ) must be taken into account. The decrease ofexchange energy could be due to the confinement of theelectron wavefunctions by magnetic field, which isolatesthe two electrons and decreases their interaction.One striking finding in the above experiment is thatthe | T + (cid:105) state is much more prominent than | T (cid:105) . Nomatter for Γ total , Γ out or Γ in , the | T + (cid:105) state is always themajor structure while | T (cid:105) is barely visible until it is sep-arated from | T + (cid:105) by a large enough magnetic field. Usingthe phenomenological model that we developed to ana-lyze the back-action driven tunneling rates [5], we couldquantitatively compare the excitation rate from | S (cid:105) to | T + (cid:105) and | T (cid:105) . Simulation on the data taken at 8 T firstreveals that the back-action driven rate for the electronsto tunnel out of the QD is 0 .
4% of the maximum RTStunneling rate for the ground state | S (cid:105) (Λ out = 30 Hz and Γ ∗ S = 8 kHz ). Γ ∗ for the two singlet states | T + (cid:105) and | T + (cid:105) varies a little (Γ ∗ T + = 15 kHz and Γ ∗ T = 12 kHz ),and their relaxation time also shows small change (1 . ms and 1 . ms ). However, the back-action driven excitationrate from | S (cid:105) to | T + (cid:105) is 6 times large as that from | S (cid:105) to | T (cid:105) : Λ ST + = 600 Hz and Λ ST = 100 Hz . It is thisbig difference that makes the | T + (cid:105) state dominate theexcited states spectroscopy.Thus, we made the conclusion that QPC back-actionprefers driving the spin singlet | S (cid:105) up to | T + (cid:105) otherthan | T (cid:105) state. It is uaually assumed that due to thespin-phonon selection rules, the | T (cid:105) − | S (cid:105) interaction is FIG. 3: (a) Γ total for 1 e at different magnetic field. Frombottom to up, B varies from 0 T to 8 T . B increases from 0 T to 6 T by 1 T each step, and then increases from 6 . T to 8 T by0 . T each step. All traces except 8 T are shifted on the y-axisfor comparison. (b) Extracted energy of |↓(cid:105) with respect to |↑(cid:105) . The solid line is the predicted Zeeman splitting. (c) 3Dplot of Γ out versus magnetic field and electron energy. Onthe top and bottom, an original trace at B = 8 T and 0 T isrespectively presented. (d) 3D plot of Γ in . suppressed in the lowest order, which makes the relax-ation from | T (cid:105) to | S (cid:105) longer than the other two tripletstates [7]. On the other hand, this also means that thephonon mediated back-action between | T (cid:105) and | S (cid:105) is weaker. This could possibly explain our observation. Ei-ther | T (cid:105) − | S (cid:105) or | T + (cid:105) − | S (cid:105) has been chosen as thebasis for qubit operation. For instance, the spin swapoperation has been demonstrated with the | T (cid:105) − | S (cid:105) states [2], and | T (cid:105)−| S (cid:105) has been utilized to implement aspin beam-splitter [8]. Our finding implies that the back-action should have less impact on spin qubits based onthe former mechanism.For the odd number of electrons, at zero magnetic fieldthe two spin states are energy degenerate and are notdistinguishable. Now with a parallel magnetic field wecan study them separately. Fig. 3 (a) shows Γ total for1 e as a function of magnetic field. The splitting of thespin down state is overall weak, and only visible at B ≥ . T . Nonetheless, in Fig. 3 (b) we extracted the energyseparation between |↑(cid:105) and |↓(cid:105) states at large field, whichagrees well with the predicted Zeeman splitting. Thisverified the spin nature of these two states. Fig. 3 (c)and (d) show the 3D pictures for the numerical derivativeof Γ out and Γ in . Although noisy, the linear splitting ofthe two spin states with magnetic field is not missed,even at small field. The weak signal for the spin downstates is possibly due to the large thermal energy ( T =240 mK ). Unlike the separation of | T + (cid:105) from | S (cid:105) by alarge exchange energy, the two spin states for a singleelectron are too close to each other, unless at extremelylarge magnetic field. Thus it is not easy to distinguishthe two spin states for odd electron numbers.In conclusion, the detailed spin states spectroscopy forone and two electrons in a single QD was studied un-der strong back-action condition. The real-time chargecounting statistics could detect the excitation from a spinup to down state for a single electron, and the exci-tation from spin singlet to different triplet states. Werevealed that the spin exchange interaction linearly de-creases with magnetic field, and found that the QPCback-action mainly drives | S (cid:105) to the | T + (cid:105) state.This work was supported by the NFRP 2011CBA00200and 2011CB921200, NNSF 10934006, 11074243,10874163, 10804104, 60921091.state.This work was supported by the NFRP 2011CBA00200and 2011CB921200, NNSF 10934006, 11074243,10874163, 10804104, 60921091.