Microwave-resonator-detected excited-state spectroscopy of a double quantum dot
Ming-Bo Chen, Shun-Li Jiang, Ning Wang, Bao-Chuan Wang, Ting Lin, Si-Si Gu, Hai-Ou Li, Gang Cao, Guo-Ping Guo
MMicrowave-resonator-detected excited-state spectroscopy of a double quantum dot
Ming-Bo Chen,
1, 2
Shun-Li Jiang,
1, 2
Ning Wang,
1, 2
Bao-Chuan Wang,
1, 2
TingLin,
1, 2
Si-Si Gu,
1, 2
Hai-Ou Li,
1, 2
Gang Cao,
1, 2, ∗ and Guo-Ping Guo
1, 2, 3, † Key Lab of Quantum Information, CAS, University of Science and Technology of China, Hefei, China CAS Center for Excellence in Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, Anhui 230026, China Origin Quantum Computing Company Limited, Hefei, Anhui 230026, China (Dated: February 17, 2021)As an application in circuit quantum electrodynamics (cQED) coupled systems, superconduct-ing resonators play an important role in high-sensitivity measurements in a superconducting-semiconductor hybrid architecture. Taking advantage of a high-impedance NbTiN resonator, weperform excited-state spectroscopy on a GaAs double quantum dot (DQD) by applying voltagepulses to one gate electrode. The pulse train modulates the DQD energy detuning and gives riseto charge state transitions at zero detuning. Benefiting from the outstanding sensitivity of the res-onator, we distinguish different spin-state transitions in the energy spectrum according to the Pauliexclusion principle. Furthermore, we experimentally study how the interdot tunneling rate modifiesthe resonator response. The experimental results are consistent with the simulated spectra basedon our model.
Semiconductor quantum dots are promising candidatesfor quantum computation [1, 2]. Conventionally, qubitsdefined in quantum dots are studied by direct currenttransport measurements [3–5], but the readout fidelityis limited by the small tunneling current in the few-electron regime. External electrometers such as quan-tum point contacts and single electron transistors arealso commonly used in quantum dot devices [6–9], butthe short-range Coulomb interaction will be an obstaclefor the electrode layout when scaling up [10–12]. In addi-tion, both readout methods inevitably introduce couplingto the environment that damps the quantum state coher-ence [13–15], or leads to back action that causes delete-rious inelastic transitions between the quantum dots andthe leads [16, 17].Alternatively, gate-based electrometers using on-chipsuperconducting resonators [18, 19] and lumped elementresonators [20–23] have been developed with high readoutfidelity. In these designs, only a single gate electrodeis required for readout, significantly simplifying the gatestructure. In particular, resonators have the advantage ofperforming quantum non-demolition readout [24] and canbe utilized as mediators for long-distance coupling andentanglement between qubits, which provide a desirableroute for scalable semiconductor quantum computing [25,26].In recent years, research on circuit quantum elec-trodynamics (cQED) hybrid devices with semiconduc-tor quantum dots has been greatly promoted with thehelp of high-impedance resonators that improve the cou-pling strength [27–31]. With the application of high-impedance resonators, various measurements for charac-terizing qubits have been performed, such as detecting of ∗ [email protected] † [email protected] stability diagram, spectroscopy of valley states [32] anddistinguishing of two-electron spin states [33].Here, we use a high-impedance NbTiN superconduct-ing resonator to perform excited-state spectroscopy ona GaAs double quantum dot (DQD). The energy spec-trum is dispersively measured by applying a train of50% duty-cycle square pulses to the DQD in the four-electron regime [34–37]. According to the resonator re-sponse, we can probe the pulse-induced electron-statetransitions [18, 32] and identify the dynamics of inter-dot transitions between different spin and orbital states.Utilizing the tunability of the semiconductor DQD, wefurther investigate the DQD parameter dependence ofthe resonator response and numerically calculate the in-fluence of the interdot tunneling rate. The combinationof low-frequency pulsed-gate technology and advancedgate-based photonic probe method provides a powerfultool for detecting singlet-triplet qubits and characteriz-ing multi-level quantum dot systems. Setup .—As shown in Fig. 1(a)–(c), the device is com-posed of a DQD and a quarter-wavelength superconduct-ing resonator, fabricated on a GaAs/AlGaAs heterostruc-ture. The two-dimensional electron gas is removed every-where except for a small mesa region that hosts the DQD.The DQD is defined by applying DC voltages to top gateelectrodes L, R, U and D. Gates L and R control thetunneling barriers between the dots and the leads. GatesU and D control the interdot tunneling rate 2 t . Theplunger gate labeled P is zero biased and idle. In ourmeasurement, the source and drain are grounded.To couple the resonator photons to the qubit, the leftplunger gate of the DQD is galvanically connected to thevoltage antinode of the superconducting resonator. Theresonator is fabricated from 11-nm-thick NbTiN with anarrow center conductor and remote ground planes usingelectron-beam lithography, deposited on a substrate witha 15-nm-thick Al O layer. The center conductor, witha width of 260 nm and a length of 300 µ m, is grounded a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b at one end and capacitively coupled to a drive line at theother end. The choice of NbTiN material and extremelysmall cross section [38] leads to a large kinetic inductance L k ≈
320 pH µ m − . Then the characteristic impedance Z r ≈ . Ω is far beyond that of conventional 50 Ω res-onators [39, 40]. This high-impedance resonator helpsimprove the coupling strength g between the DQD andresonator which is proportional to √ Z r [41].Our device is cooled in a dilution refrigerator with abase temperature of ∼
23 mK. We measure the responseof the resonator using a vector network analyzer withprobe frequency f p . The microwave photons are attenu-ated approximately 70 dB before reaching the drive line,reflected by the resonator, and then amplified at 4 K androom temperature sequentially. Measuring the reflectionsignal S , we determine that the resonator frequencyis f r = ω r / π = 6 .
04 GHz with a photon decay rate of κ/ π = 35 . H = (cid:126) Ω σ z / (cid:126) ω r a † a + (cid:126) g c ( a † σ − + aσ + ) , (1)where (cid:126) Ω = (cid:112) ε + (2 t ) is the energy difference betweenthe lowest two charge states of the DQD, ε is the energydetuning, σ z and σ ± are the Pauli operators in the DQD Drive line 1 μm RL UD P (a) (b)(c)
100 μm
DQD (d) V L ( m V ) (2,2) (2,3) V R (mV)(1,2) (1,3) ε -1330-1336 -784 -778 0.860.820.78 | S |500 nm 260 nm FIG. 1. Hybrid device and stability diagram. (a) False-colorscanning electron micrograph of the device, showing the res-onator (azure) and gate electrodes (yellow) of the DQD. Theresonator capacitively couples to the drive line at one endand grounded at the other end. The DQD region is indicatedby a red rectangle. (b) DQD defined by metallic top gates.The left plunger gate highlighted in blue is connected to theNbTiN superconducting resonator. Pulses are applied to gateR to perform the excited-state spectroscopy. (c) NbTiN res-onator with a narrow width of 260 nm. (d) Transition linebetween charge configurations (2,2) and (1,3) measured bythe resonator. The detuning ε indicated by an arrow can becontrolled by the gate voltage V R . Dashed lines indicate theelectron tunneling between the DQD and the leads. eigenstate basis, a † and a are the photon creation and an-nihilation operators, g c is the effective coupling strengthgiven by g c = g · t/ (cid:126) Ω [41], with g being the overallcoupling rate, and (cid:126) is the reduced Plank constant. Thefirst (second) term represents the free Hamiltonian of thequbit (resonator). The third term describes the charge-cavity electric dipole coupling. Due to the coupling rate g c , a change in the DQD charge state results in a fre-quency shift ∆ f r on the resonator, varying the amplitudeand phase of the reflection signal [39, 40]. Specifically, inthe dispersive limit g c (cid:28) ∆, where ∆ = Ω − ω r is the de-tuning between the DQD and resonator, the Hamiltoniancan be simplified to H ≈ (cid:126) Ω σ z / (cid:126) ( ω r + g ∆ σ z ) a † a, (2)and the frequency shift ∆ f r is proportional to g / ∆.Therefore, we can probe the tunneling dynamics of theDQD according to the resonator response.The DQD is configured and characterized by the res-onator reflection amplitude | S | in response to the gatevoltages. Figure 1(d) shows the charge stability diagramnear the (2,2)–(1,3) charge transition obtained by mea-suring | S | as a function of V L and V R with probe fre-quency f p = f r , where ( m, n ) denotes m ( n ) electrons inthe left (right) quantum dot. The detuning ε , indicatedby a yellow arrow in the diagram, is defined as the elec-trochemical potential difference between the two dots.It can be controlled by gate R with a voltage-to-energyconversion ratio of α/h ≈ . / mV [43]. With ap-propriate gate voltages, the electrochemical potentials ofthe two dots are aligned and ε = 0, leading to a co-herent interdot transition that increases | S | . Fitting | S | with the input-output theory [29, 33, 44], we ob-tain the coupling strength g / π ≈ . γ/ π ≈
72 MHz and the tunneling rate2 t/h ≈ . Experiment .—To perform the excited-state spec-troscopy on the DQD, we apply a 50% duty-cycle squarepulse train to gate R with width τ and peak-to-peakvoltage V pp [Fig. 2(a)]. We sweep the detuning en-ergy ε through the (2,0)–(1,1) transition line by varying V R , while the applied pulse train modulates the detun-ing periodically. Then, the resonator response ∆ | S | is measured as a function of V pp and ε for τ = 30 ns(repetition rate 1 / τ ≈ .
67 MHz) and tunneling rate2 t/h ≈ . f p = f r .Figure 2(b) shows the spectrum after subtracting the (a) Δ | S | (c) A B C (d) (e) (f) (g) D (1,1)2 t (2,0)(2,0) e t (2,0)(2,0) e (1,1) (2,0)(2,0) e (1,1)2 t t’ (2,0) (2,0) e (1,1) V pp( V ) ε / h (GHz) Δ| S |(2,0) (1,1) A BD C (b) -160 160 00.050.020.10.150.20.25 τ ti m e V PP t / h = 5.8 GHz E n e r gy εT (1,1) T (2,0) T (2,0) T (1,1) S (1,1) S (1,1) S (2,0) S (2,0) ε / h (GHz) -160 160-80 0 80 FIG. 2. (a) Energy levels of two-electron spin states as a function of detuning ε . A pulse train is applied to gate R which isgenerated from an arbitrary wavefunction generator with width τ and peak-to-peak voltage V pp . (b) Reflection signal ∆ | S | measured as a function of detuning ε and pulse voltage V pp after subtracting the background in the Coulomb blockade regime.(c) Horizontal line cut of the spectrum at V pp = 0 . background in the Coulomb blockade regime. In the spec-trum, three transition lines are clearly observed, whichcan be divided into four parts for explanation conve-nience, labeled A, B, C and D. The relevant dynamicprocesses are schematically illustrated in Fig. 2(d)–(g).When the pulse voltage V pp is small, two slanting lineslabeled A and B are observed. Line A (B) results fromalignment of the electrochemical potentials of chargestates (2,0) and (1,1) during the lower (upper) voltagelevel of the pulse train. In these situations, the energy de-tuning ε is compensated by the pulse voltage ± V pp / t . With increasing V pp , line A splits intolines C and D at V pp = 0 .
13 V. The pulse voltage is largeenough that it enables the transition between the excitedstate (2 , e and (1,1) during the upper voltage level ofthe pulse train, schematically shown in Fig. 2(f). We candirectly determine the energy splitting ∆ E = 330 µ eVbetween the ground state and excited state of (2,0) fromthe energy difference between lines B and C.More interestingly, the amplitude of line A is muchlarger than that of line B in the spectrum for a smallpulse voltage V pp . The different resonator responses canbe explained as a result of Pauli spin blockade consider-ing the spin degrees of freedom. Figure 2(a) shows theschematic energy diagram of the two-electron spin states.In the (2,0) configuration, the ground-state configurationfor ε <
0, two electrons favor the spin-singlet state, de-noted S(2,0). The triplet states T(2,0) are energeticallyinaccessible for a small pulse voltage ( V pp < .
13 V). For ε >
0, the separated (1,1) charge configuration is domi-nant. All the spin states are accessible in the absence of an external magnetic field: the singlet, denoted S(1,1),and three triplets, denoted T , ± (1,1). These spin statescould be strongly mixed in the (1,1) configuration dueto the coupling of GaAs nuclear spins within the spin-dephasing time 3 −
10 ns [48, 49]. Since the pulse fre-quency is much smaller than the dephasing rate, we sup-pose that all four spin states are fully mixed during thepulse period of large negative voltage.For line A, the ground state is S(2,0) during the highervoltage level of the pulse train, schematically shown inFig. 2(d). With a subsequent lower voltage level pulse,the transition between S(2,0) and S(1,1) occurs. In con-trast, for line B, the ground charge state is (1,1) dur-ing the lower voltage level in Fig. 2(e) and all the spinstates S(1,1) and T , ± (1,1) are accessible. Nevertheless,a subsequent higher voltage level pulse only allows theS(1,1)–S(2,0) transition while the T(1,1)–S(2,0) transi-tion is forbidden due to spin blockade [50, 51]. Therefore,when performing the spectroscopy with a small voltagepulse train, only the singlet state contributes to the res-onator response at the interdot transition, which stronglyreduces the amplitude of line B in comparison to that ofline A. Figure 2(c) shows a horizontal cut of the energyspectrum through lines A and B at V pp = 0 . V pp is larger than 0 .
13 V, thethree triplet states T , ± (2,0) are no longer inaccessible.The transition between T(2,0) and T(1,1) with a tunnel-ing rate of 2 t (cid:48) is allowed when the two electrochemicalpotentials are in alignment by applying the pulse train (d)(c) (b)(a) t / h = 7.5 GHz 2 t / h = 5.0 GHz Δ| S | V pp ( V ) Δ| S | × -3 -4 V pp ( V ) ε / h (GHz) -160 160-80 0 80 ε / h (GHz) -160 160-80 0 802 t / h = 7.5 GHz 2 t / h = 5.0 GHz Δ| S | V pp ( V ) Δ| S | V pp ( V ) ε / h (GHz) -160 160-80 0 80 ε / h (GHz) -160 160-80 0 80 Simulation Simulation a . u . a . u . FIG. 3. (a)–(b) Excited-state spectroscopy for 2 t/h =7 . . [Fig. 2(f)], labeled C in the excited-state spectrum. LineD, which corresponds to the S(2,0)–S(1,1) transition, isalso influenced by the participation of the excited stateT(2,0). Because of the large peak-to-peak pulse voltage V pp , electrons can be loaded into S(2,0) or T(2,0) dur-ing the higher voltage level of the pulse train [Fig. 2(g)].Considering that only the electrochemical potentials ofS(1,1) and S(2,0) are aligned during the lower voltagelevel, triplet states are not involved in the transition.Consequently, the resonator signal | S | of line D is sup-pressed compared to that of line A. Therefore, the disper-sive photonic probe with pulsed-gate technology providesan effective readout method for singlet-triplet qubits.Next, we investigate the 2 t dependence of the excited-state spectrum by tuning the voltage bias on gate U.The results are shown in Fig. 3(a) and (b) for 2 t/h = 7 . . t/h > f r , the qubit energy (cid:126) Ω = (cid:112) ( ε + (2 t ) ) is always larger than the resonance fre-quency hf r . The response amplitude to the interdot elec-tron transition decreases rapidly with increasing 2 t (seeappendix for more details). In Fig. 3(a), 2 t/h = 7 . f r = 6 .
04 GHz,leading to a small signal-to-noise ratio. In contrast, when2 t/h < f r , the DQD energy matches the resonator fre-quency at ε = ± (cid:112) ( hf r ) − (2 t ) and the frequency shift∆ f r reaches a maximum, which results in a good res-onator response in Fig. 3(b).Moreover, the resonator response to the (2,0) e –(1,1)transition in Fig. 3(b) is much stronger than that inthe case of 2 t/h = 5 . . t (cid:48) at the (2,0) e –(1,1) transition is larger than 2 t . For2 t/h = 5 . t (cid:48) is closer to the resonance frequency f r . As a result, the resonator ismore sensitive to the (2,0) e –(1,1) tunneling process, witha strong enhancement of the amplitude of the middletransition line in Fig. 3(b). To examine our explana-tion, we theoretically simulate the experiment as shownin Fig. 3(c) and (d) based on our model (see appendix),and we achieve good agreement with the expeirment. Discussion and summary .—The resonator-detectedspectrum has a clean background because the resonatoronly responds to the elastic interdot tunneling that ap-proaches the resonance frequency. The resonator main-tains a good response in the few-electron regime. To fur-ther improve the readout quality, suppressing microwaveleakage to gate electrodes by using delicate filtering cir-cuits [52, 53] and improving the signal-to-noise ratio bythe application of Josephson parametric amplifiers [54]are demanded.In summary, we have demonstrated the energy spec-trum measurement of a four-electron GaAs DQD usinga high-impedance superconducting NbTiN resonator byapplying 50% duty-cycle pulses to a gate electrode. Weattribute the different resonator responses to electrontransitions in the spectrum to different combinations ofaligned spin states and the effect of the Pauli exclusionprinciple. By tuning the voltage bias on the barrier gate,the influence of the interdot tunneling rate is also inves-tigated and verified based on our model.Our results provide a useful means of sensitive read-out for singlet-triplet qubits in cQED architectures. Thisgate-based readout method is not only available for gate-defined DQD devices, but also can be easily extended toother cavity-coupled hybrid systems for spectroscopicallyprobing qubits and characterizing energy levels.
ACKNOWLEDGMENTS
This work was supported by the National Key Re-search and Development Program of China (Grant No.2016YFA0301700), the National Natural Science Founda-tion of China (Grants No. 61922074, 11674300, 61674132,11625419 and 11804327), the Strategic Priority ResearchProgram of the CAS (Grant No. XDB24030601), theAnhui initiative in Quantum Information Technologies(Grant No. AHY080000). This work was partially car-ried out at the University of Science and Technology ofChina Center for Micro and Nanoscale Research and Fab-rication.
APPENDIX1. Theoretical method
In absence of an external magnetic field, all threetriplets T ± and T are degenerate. Then, for simplicity,the Hamiltonian for a two-electron double quantum dotsystem that conserves electron spins can be described inthe singlet and triplet states basis [S(1,1), S(2,0), T(1,1)and T(2,1)] as H = t t ε t (cid:48) t (cid:48) ε + ∆ E . (3)Here, t ( t (cid:48) ) is the interdot coupling rate between sin-glet states, S(1,1) and S(2,0) [triplet states, T(1,1) andT(2,0)], and ∆ E is the energy splitting between S(2,0)and T(2,0).Applying a slow-frequency pulse train onto the DQD,the detuning energy is time-periodic, ε ( τ ) → (cid:26) ε + αV pp / , nτ ≤ τ < (2 n + 1) τ ,ε − αV pp / , (2 n + 1) τ ≤ τ < (2 n + 2) τ , with an integer n and a repetition rate of 1 / τ , where 2 τ is the pulse period, V pp is the peak-to-peak pulse voltageand α is voltage-to-energy conversion ratio of the pulse.To simplify the contribution from the electron reservoircoupling to the DQD electron states, we assume that allthe accessible two-electron spin states can be loaded withequal probabilities when the electrochemical potential ofthe dot lies below the electron reservoir. Hence for the(1,1) charge configuration, all the spin states can be ini-tialized with a probability of one fourth. For the (2,0)charge configuration, however, the initialization dependson the pulse amplitude. For a small driving amplitude V pp , only S(2,0) is accessible. If V pp is large enough,which in our experiment is 0 .
13 V, triplet states T , ± (2,0)can also be accessed. Since the pulse frequency is muchslower than the decoherence rate [48], the cavity reflec-tion signal can be treated as an average response to thestationary state occupation for the two different pulselevels. Given the input-output theory [29, 33, 44], wethen simulate the exited-state spectra for the parame-ters used in the experiment as shown in Fig. 3(c) and(d). The relevant parameters are the coupling strength g / π ≈ . κ i , κ e , κ ) / π = (4 . , . , . γ/ π ≈
72 MHz, and interdot tunnelingrates 2 t/h = 7 . µ eV, consid-ering an inhomogeneous broadening of the excited-statespectrum induced by the slow uctuations of the detuning[55, 56]. Because of the difficulty in extracting t (cid:48) andthe triplets-resonator coupling strength g (cid:48) , we assumethat 2 t (cid:48) = 6 GHz for 2 t = 5 GHz and 2 t (cid:48) = 10 GHz for2 t = 7 . g (cid:48) ≈ g . The results in Fig. 3(c) and(d) agree with the experiment qualitatively well, whichreveal that our measurement can also be used to estimatethe tunneling rate between excited states.
2. Influence of other system parameters
According to Hamiltonian (2), the resonator frequencyis related to the coupling strength g c and resonator-DQDenergy difference ∆, which are both functions of the in-terdot tunneling rate 2 t . To explain the different signal-to-noise ratios in Fig. 3, we calculate the frequency shiftdependence on 2 t and ε by solving the Hamiltonian.As shown in Fig. 4(a), the two dashed lines colored inred and white correspond to the values of 2 t in Fig. 3(a)and (b), respectively. When 2 t/h > f r , the qubit en-ergy Ω = (cid:112) ( ε + (2 t ) ) is always larger than the res-onance frequency hf r and the frequency shift ∆ f r de-creases rapidly with increasing 2 t . Consequently, for2 t/h = 7 . f r = 6 .
04 GHz, lead-ing to a small signal-to-noise ratio in Fig. 3(a). When2 t/h < f r , the DQD energy matches the resonator fre-quency at ε = ± (cid:112) ( hf r ) − (2 t ) . Near both reonsnacepoints, ∆ f r reaches a maximum, which results in a goodresonator response in Fig. 3(b). Therefore, the excited-state spectrum for 2 t = 5 GHz is much better than thatfor 2 t = 7 . g also has an influence on the resonatorresponse. Based on the numerical results in Fig. 4(b),we determine the 2 t dependence of ∆ f r at zero detuningfor different coupling strengths g . Obviously, a larger g leads to a larger ∆ f r as well as a larger amplitude | S | . t / h (GHz) Δ f r ( M H z ) (b) >52 t / h (GHz) ε / h ( GH z ) Δ f r (MHz) (a) -10 -5
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