Abstract
LandauZener (LZ) transition has received renewed interest as an alternative approach to control singlequbit states. An LZ transition occurs when a system passes through an avoided crossing that arises from quantum mechanical coupling of two levels, taking the system to a coherent superposition of the two states. Then, multiple LZ transitions induce interference known as LandauZenerStückelberg (LZS) interference whose amplitude strongly depends on the velocity or adiabaticity of the passage. Here, we study the roles of LZ transitions and LZS interference in coherent charge oscillations of a oneelectron semiconductor double quantum dot by timedomain experiments using standard rectangular voltage pulses. By employing density matrix simulations, we show that, in the standard setup using rectangular pulses, even a small distortion of the pulse can give rise to LZ transitions and hence LZS interference, which significantly enhances the measured oscillation amplitude. We further show experimentally that the nature of the coherent charge oscillations changes from Rabitype to LZS oscillations with increasing pulse distortion. Our results thus demonstrate that it is essential to take into account LZS interference for both precise control of charge qubits and correct interpretation of measurement results.
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Introduction
A quantum twolevel system forms the basis for studying and controlling the dynamics of quantum coherent phenomena in terms of quantum information physics, and is a building block of a quantum bit, or a qubit^{1}. The standard architecture of qubit operation is based on the premise that any singlequbit gate can be constructed from a sequence of pulses that change the system’s effective Hamiltonian nonadiabatically. The evolution of the qubit state is described by successive rotations of the state vector on the Bloch sphere, and the state is read out by projecting it on an appropriately chosen measurement basis, typically Pauli matrix σ_{ z }. In the solid state, various quantum twolevel systems have been implemented^{2,3,4,5,6,7,8,9}, and quantum coherent oscillations driven by pulsing the system have been reported as a demonstration of successful qubit operations.
Recently, LandauZener (LZ) transitions^{10,11,12} have received renewed interest as an alternative approach to control singlequbit states^{13}. An LZ transition occurs when a system passes through an avoided crossing that arises from quantum mechanical coupling of two levels. Except for the fully adiabatic or nonadiabatic limits, an LZ transition takes the system to a coherent superposition of the two states with the relative amplitudes determined by the avoided crossing gap and the velocity, or adiabaticity, of the passage. By appropriately tuning the velocity, a single passage can create a coherent superposition with equal weights, serving as a 50:50 beam splitter for the incoming state. Successively sweeping back and forth through the avoided crossing thus induces multiple LZ transitions, resulting in the interference between the outgoing states. This phenomenon, known as LandauZenerStückelberg (LZS) interference^{10,11,12}, has been proposed as a method to control qubits without the need for sharp pulses^{13}, and observed in various quantum twolevel systems in the solid state based on Josephson junctions^{14,15}, coupled donor states^{16}, and semiconductor double quantum dots (DQDs)^{7,17,18,19,20}.
In ref.^{20}, coherent oscillations arising from LZS interference (hereafter LZS oscillations) between singlet and triplet spin states in a DQD were demonstrated by sweeping through an avoided crossing induced by nuclear spins, where the long time scale (of the order of 10 ns) set by the small gap facilitated the precise velocity control. On the other hand, in semiconductor charge qubits^{5,6,7}, due to the large avoided crossing gap, typically 10 μeV, the relevant time scale is in the range of several tens of picoseconds, where accurate velocity control becomes a greater challenge. Consequently, LZS interference in charge qubits has been studied in the context of photonassisted interdot tunneling under continuous microwave irradiation^{16,17}. Timedomain observation of LZS interference has been reported only recently in ref.^{18}, where the velocity was controlled by varying the amplitude of a fixedlength short (150 ps) pulse or shaping a rectangular pulse with a lowpass filter.
In this paper, we study the roles of LZ transitions and LZS interference in coherent charge oscillations of a oneelectron semiconductor DQD by timedomain experiments using standard rectangular voltage pulses. By employing density matrix simulations, we show that, in the standard setup using rectangular pulses, even a small distortion of the pulse can give rise to LZ transitions and hence LZS interference, which significantly enhances the measured oscillation amplitude. We further show experimentally that the nature of the coherent charge oscillations changes from Rabitype to LZS oscillations with increasing pulse distortion. Our results thus demonstrate that it is essential to take into account LZS interference for both precise control of charge qubits and correct interpretation of measurement results.
LandauZenerStückelberg Interference
We consider a oneelectron state in a DQD and take the basis to be L〉 and R〉, in which the electron occupies the left and right dot, respectively. Using Pauli matrices σ_{ x } and σ_{ z }, the Hamiltonian of the system can be written as^{5}
where ε(t) = E_{L}(t) − E_{R}(t) is the detuning between the energy levels in the left and right dots, and T_{C} is the interdot tunnel coupling energy. Here we assume without loss of generality that the initial state is prepared in R〉 with ε set to a large negative value ε_{0} at t < 0 [Fig. 1(a)] (see Methods). In the standard qubit operations, coherent Rabitype oscillations between R〉 and L〉 are triggered by switching ε nonadiabatically to a value ε′ close to the resonance ε = 0, and then the state is projected onto the measurement basis for readout by switching ε back to ε_{0}. The important observation here is that the qubit manipulation for ε′ > 0 inevitably involves passages through the avoided crossing at ε = 0. Therefore, if ε changes at a finite velocity v ≡ dε/dt, each passage will induce an LZ transition with the asymptotic probability \({P}_{{\rm{LZ}}}=\exp (2\,\pi \delta )\), where \({\rm{\delta }}={T}_{C}^{2}/\hslash v\) (\(\hslash \) is Planck’s constant divided by 2π)^{20}, which then gives rise to LZS interference when ε is set back to ε_{0} for the readout [Fig. 1(c)]. Note that δ → 0(∞) corresponds to the nonadiabatic (adiabatic) limit, for which P_{LZ} → 1(0).
The impacts of LZ transitions and LZS interference on the qubit manipulation can be better understood by using the Bloch sphere representation. It is known that the dynamics at an LZ transition can be expressed by a unitary operation \({U}_{{\rm{LZ}}}={R}_{z}({\tilde{{\rm{\phi }}}}_{{\rm{S}}}){R}_{x}({\theta }_{{\rm{LZ}}}){R}_{z}({\tilde{{\rm{\phi }}}}_{{\rm{S}}})\) that represents successive rotations of the Bloch vector for the incoming state^{15}. Here, \({R}_{x}({\theta }_{{\rm{LZ}}})=\exp ({\rm{i}}{\theta }_{{\rm{LZ}}}{\sigma }_{x}/2)\) and \({R}_{z}({\tilde{{\rm{\phi }}}}_{{\rm{S}}})=\exp ({\rm{i}}{\tilde{{\rm{\phi }}}}_{{\rm{S}}}{\sigma }_{z}/2)\) describe rotations around the x and zaxes by angles θ_{LZ} and \({\tilde{{\rm{\phi }}}}_{{\rm{S}}}\), respectively. θ_{LZ} and \({\tilde{{\rm{\phi }}}}_{{\rm{S}}}\) are given by \({\theta }_{{\rm{LZ}}}=2{\sin }^{1}\sqrt{{P}_{{\rm{LZ}}}}\) and \({\tilde{{\rm{\phi }}}}_{{\rm{S}}}=\text{arg}[\Gamma (1{\rm{i}}\delta )+\delta (ln\delta 1)]\pi /4\), where Γ is the Gamma function^{15}. Using these representations, LZS interference can be expressed as U_{LZ}R_{ z }(φ_{p})U_{LZ}, where \({{\rm{\phi }}}_{{\rm{p}}}={\hslash }^{1}\int [{E}_{{\rm{L}}}(t){E}_{{\rm{R}}}(t)]\,dt\) is the phase accumulated during the z rotation between the first and second LZ transitions. In this idealized model, the z component of the Bloch vector changes only via the two x rotations, but their effects can be constructive or destructive^{21}, depending on the z rotation angle—that is, the phase accumulated—between them. This phasedependent action of the two x rotations is the essence of LZS interference.
To illustrate these, we computed the trajectories of the Bloch vector driven by trapezoidal pulses with slightly different lengths [Fig. 1(d) and (e)]. The parameters T_{C} and v were chosen in such a way that P_{LZ} ∼ 0.5. The use of a trapezoidal pulse shape allows us to decompose the dynamics into three steps: the first LZ transition [Fig. 1(a)], phase accumulation [Fig. 1(b)], and the second LZ transition [Fig. 1(c)]. The trajectories in the time domains corresponding to the three steps are shown in red, green, and blue, respectively, in Fig. 1(d) and (e). The simulations confirm that both the first and second LZ transitions can be approximately viewed as combinations of x and z rotations. In these simulations, the pulse lengths were chosen for the effects of the two x rotations to be the most destructive [Fig. 1(d)] or the most constructive [Fig. 1(e)]. Consequently, after the second LZ transition the direction of the Bloch vector spans almost the full range from the south to north poles. Note that this is possible because we set P_{LZ} ∼ 0.5 (i.e., θ_{LZ} ∼ π/2), which makes each x rotation serve as a π/2 pulse. It is instructive to compare these results with the case for a nonadiabatic rectangular pulse with the same height [Fig. 1(f)]. Since θ_{LZ} = 0 in this case, the trajectory stays near the south pole. The amplitude of this Rabitype oscillation determined by ε′ and T_{C} alone is much lower than 1 for an offresonant pulse as shown in Fig. 1(f). These observations indicate that the finite adiabaticity of the pulse can lead to LZS interference, where the amplitude of the measured oscillations may significantly exceed that expected for an ideal nonadiabatic pulse.
Measurements using rectangular pulses
Figure 2(a) schematically shows the device structure and experimental setup. A DQD is formed in a twodimensional electron gas at the interface of a GaAs/AlGaAs heterojunction by applying negative voltages to the surface Schottky metal gates. We use gate voltages V_{L} and V_{R} to vary the electron number in the DQD and V_{C} to tune the interdot tunnel coupling energy T_{C}. We focus on the oneelectron regime of the DQD. Highfrequency voltage pulses are applied to the drain electrode of the DQD, which provides fast control of the dot energy levels and hence the detuning ε(t). Experiments were performed at a lattice temperature of 20 mK. A magnetic field of 0.2 T was applied perpendicular to the sample to eliminate unwanted level degeneracy.
First, we present results for the standard singlequbit operation exhibiting Rabitype oscillations. For this experiment, the gate voltages were tuned to allow for strong dotlead coupling so that the charge state in the DQD could be readout by measuring the current flowing through the DQD as in ref.^{5} (for details, see Methods). Rectangular voltage pulses with amplitude corresponding to a change in the detuning of Δε ≡ ε′ − ε_{0} = 65 μeV were applied to the drain electrode at repetition frequency f_{rep} = 50 MHz. Figure 2(c) shows the average number of pulseinduced tunneling electrons, n_{p} = I_{p}/ef_{rep}, plotted as a function of pulse duration t_{p} and detuning ε′ during the pulseon period. The data show behavior characteristic of Rabitype oscillations. From the oscillation frequency at the resonance (ε′ = 0), T_{C} is estimated to be 4.25 μeV. We focus on the asymmetry with respect to ε′ = 0; clearly, the oscillations are much more pronounced at ε′ > 0. Similar asymmetry has been seen not only in semiconductor charge qubits^{5,6} but also in superconducting charge qubits^{8}. Recently, effects of LZS interference on coherent charge oscillations were theoretically studied in ref.^{22}, where it was argued that the asymmetry was a clear signature of coherent LZS oscillations. However, as will be shown below, the asymmetry can arise from the finite slope of the rising edge of the pulse alone, i.e., without LZS interference. To elucidate the effects of LZS interference in Rabitype oscillations, below we examine the state evolution during the falling time as well as the rising time of the pulse.
We simulate the time evolution of the system by solving the master equation for a reduced density matrix ρ with the effects of decoherence taken into account in the Lindblad formalism:
Here, L is the Lindblad operator defined as \(L=\sqrt{{\rm{\Gamma }}}({\rm{ES}}\rangle \langle {\rm{ES}}{\rm{GS}}\rangle \langle {\rm{GS}})\), Γ (=1/T_{2}) is the decoherence rate of the charge state, and GS〉 (ES〉) is the instantaneous ground (excited) state of the twolevel system. We model the pulse distortion by taking the convolution of a rectangular pulse with an exponential function \({e}^{t/{t}_{d}}\) [Fig. 2(b)], where t_{d} represents the finite response time of the dot potential to the applied pulse. To distinguish the effects of LZ transitions and LZS interference, we compare the probabilities of finding the state in L〉 for state projection at different times, (i) t = t_{p} and (ii) t_{p} + 2 ns, i.e., just before and sometime after the pulse is turned off [see arrows in Fig. 2(b)]. Note that LZS interference is involved only in the latter. In the following, we refer to (i) and (ii) as “singlepassage” and “doublepassage” models, respectively.
Figure 2(d) and (e) show the results for the double and singlepassage models, respectively, calculated for t_{d} = 30 ps, T_{C} = 4.25 μeV, Γ = 1 ns^{−1}, and Δε = 65 μeV. These parameters were chosen to fit the coherent oscillations at ε′ = 0 (for details, see Methods). The figures plot the probability of finding the state in L〉, which we denote by P_{L}^{S} (P_{L}^{D}) for the single (double) passage model. The two models yield similar results—both reproduce the observed behavior well, including the asymmetry with respect to ε′ = 0. (Although not shown, the asymmetry is absent for a perfectly rectangular pulse with t_{d} = 0.) It should be noted, however, that the contrast between ε′ > 0 and ε′ < 0 is much more pronounced in P_{L}^{D}. It is particularly noteworthy that for P_{L}^{D} pronounced oscillations persist up to large ε′. Figure 2(f) compares the line cuts of P_{L}^{S} and P_{L}^{D} at t_{p} = 0.32 ns. In both traces, the oscillation amplitude decreases with increasing ε′, but much more slowly in P_{L}^{D} [solid trace in Fig. 2(f)]. This is because LZS interference not only enhances the oscillation amplitude but also renders it less dependent on ε′. Note that in the experiment the amplitude decays faster with increasing ε′ [Fig. 2(c)]. This is because in reality Γ increases with ε′, due to additional decoherence induced by background charge fluctuations^{5,6}.
Figure 3(a) and (b) compare the trajectories of the Bloch vector for pulses with the same t_{p} (=0.32 ns) but at slightly different ε′ values corresponding to the minimum and maximum in P_{L}^{D} [marked “A” and “B” in Fig. 2(d) and (f)]. The trajectories for t < t_{p} (t > t_{p}) are shown in red (blue). The system follows similar trajectories for t < t_{p}: the state initially located at the south pole (R〉) is moved toward the equator by the x rotation that accompanies the first LZ transition before exhibiting a z rotation driven by the offresonant pulse. In contrast, the trajectories for t > t_{p} are entirely different: in Fig. 3(a) the state remains near the south pole, whereas in Fig. 3(b) it shifts further toward the equator, where it makes a greater circle. The z projections of these trajectories are plotted in Fig. 3(c) as a function of t. Note that in the doublepassage model the measured signal corresponds to the time average of the z component in the pulseoff period, labeled Z_{A} and Z_{B} in Fig. 3(c). As seen in Fig. 3(c), the high contrast between the maxima (Z_{B}) and minima (Z_{A}) of the oscillations that is to be measured in the experiment arises from the postpulse dynamics shown in blue. This indicates that, although the overall features of the oscillations in Fig. 2(d) share those of Rabitype oscillations, the dominant contributor to the oscillation amplitude is LZS interference. The existence of such postpulse dynamics in a semiconductor charge qubit is corroborated by the observation of Ramsey fringes in ref.^{23}. Note that t_{p} is fixed in the present case, so the interference phase is tuned via ε′, which determines the z rotation frequency.
Measurements using largely distorted pulses
For the parameters used above, P_{LZ} values for the rising and falling edges of the pulse are in the range of 0.25 to 0.3. The signature of LZS interference is expected to be further strengthened when P_{LZ} becomes closer to 0.5. Since highfrequency voltage pulses are applied to the drain electrode of the DQD in our experimental setup, the actual pulse shape experienced by the electron in the DQD could be modified by the capacitive component around the lead to the DQD. As we show below, the pulse shape becomes largely distorted when the isolation between the dot and lead is increased by applying a larger negative gate voltage. In the following measurement, the dotlead coupling was set in a weak coupling regime, where the current flowing through the DQD is no longer measurable. The charge state of the DQD was therefore detected by measuring the current I_{QPC} that flows through the QPC charge sensor attached nearby the DQD.
Figure 4(a) shows the pulseinduced QPC current I_{QPC} as a function of pulse duration t_{p} and detuning ε′ (Δε = 145 μeV and f_{rep} = 250 MHz) (for details, see Methods). We find that the gross features of the data are largely different from those in Fig. 2(c). Now the coherent oscillations at ε′ = 0 are barely visible. Oscillatory behavior is seen only at ε′ > 0, i.e., only when the state passes through the avoided crossing. Furthermore, the oscillations maintain high visibility over a wider range of ε′ (>0). Consequently, the data possess the general features of LZS oscillations^{18,20}. Figure 4(b) shows the probability of finding the state in L〉, P_{L}^{D}, as a function of ε′ and t_{p}, calculated in the doublepassage model with the state projection at t = t_{p} + 2 ns. With appropriately chosen parameters (t_{d} = 100 ps, T_{C} = 7.5 μeV, and Γ = 1 ns^{−1}), the simulation well reproduces the experimentally observed behavior. Here, the long t_{d} of 100 ps is essential to the LZS oscillationlike behavior. The corresponding P_{LZ} values are in the range of 0.45 to 0.5. Because the P_{LZ} values become closer to 0.5, LZS interference plays an even more important role in the measured coherent oscillations. To highlight the signature of LZS interference, line cuts of the plots in Fig. 4(a) and (b) at t_{p} = 0.3 ns are shown in Fig. 4(c). As the simulation (solid line in the upper panel) shows, the ε′ dependence of the oscillation amplitude is much weaker than in Fig. 2(f). The weak ε′ dependence is evident also in the experiment [circles in the lower panel of Fig. 4(c)], though the oscillation amplitude is much reduced.
Discussion
When the state evolution is well described by the actions of successive x and z rotations, the probability of finding the state in L〉 after the pulse (with the initial state being R〉) can be expressed as
where P_{1} and P_{2} are the asymptotic LZ transition probabilities at the rising and falling edges of the pulse, respectively^{24}. Owing to the phasedependent term, P oscillates as a function of ε′ and t_{p}, with the maxima (minima) indicating that the interference between the outgoing states is the most constructive (destructive)^{21}. In our experiment, P_{1} and P_{2} are not equal and both vary with the detuning as a result of the nontrapezoidal pulse shape. The amplitude of the interference fringe \(F=2\sqrt{{P}_{1}{P}_{2}(1{P}_{1})(1{P}_{2})}\) reaches a maximum value F = 0.5 for P_{1} = P_{2} = 0.5, where P oscillates between 0 and 1. In our experiment, the F value estimated from the t_{d} of 100 ps (30 ps) that provides a good fit to the experiment in the weak (strong) dotlead coupling regime is about 0.5 (less than 0.3). As shown by the dashed line in the upper panel of Fig. 4(c), the densitymatrix calculation for t_{d} = 100 ps without dephasing (i.e., Γ = 0) shows oscillations between nearly 0 and 1. This confirms that the simple analysis based on the asymptotic probabilities P_{1} and P_{2} captures the essence of the physics.
In summary, we studied the impact of LZS interference on coherent charge oscillations in a oneelectron DQD. By numerical simulation, we found a significant enhancement of the oscillation amplitude due to LZS interference when the system traverses the avoided crossing. Our results demonstrate that LZS interference is inherent to charge qubits. This indicates that appropriate tuning of the pulse shape and an analysis including afterpulse dynamics are essential for the precise control of charge qubits and correct interpretation of the measurement results.
Methods
Experimental procedure
The device was fabricated from a GaAs/AlGaAs heterostructure wafer with electron density of 2.3 × 10^{15} m^{−2} and mobility of 180 m^{2}/Vs. The twodimensional electron gas formed at the heterointerface is located 10 nm below the surface. Rectangular highfrequency voltage pulses with amplitude corresponding to a change in detuning Δε and duration time t_{p}, applied to the drain electrode of the DQD, allow for a fast control of the dot energy levels. We employed two different detection schemes for reading out the charge state of the DQD, depending on the strength of the dotlead coupling. In the strong coupling regime, we measured dot current I_{p} flowing from the source to drain through the DQD as in ref.^{5}. In the weak coupling regime, I_{p} is no longer measurable, so we used a nearby quantum point contact (QPC) charge sensor set in the linear tunneling regime^{6}. We employed a lockin technique to improve the signaltonoise ratio by chopping the pulses at 100 Hz.
The energy level alignments of the DQD during the pulse on and off periods are depicted in Fig. 5(a) for the cases of strong and weak couplings. Note that the polarity of the pulse is opposite in the strong and weakcoupling cases and, accordingly, the sign of detuning is reversed between them. We therefore use different sign definitions of detuning as ε = ±(E_{L} − E_{R}) for the strong (+) and weak (−) couplings, in such a way that ε_{0} < 0 and the system passes through the avoided crossing for ε′ > 0 in both cases. This allows us to use the Hamiltonian of the same form.
As shown in Fig. 5(a), the DQD is initialized in R〉 in both strong and weakcoupling cases in spite of the opposite polarity of the pulse. For the strongcoupling case, this happens because of the large sourcedrain bias applied during the pulseoff period and the large ratio between the dotlead coupling Г_{R(L)} and the interdot coupling Г_{i} (≪Г_{R}, Г_{L})^{5}. Note that during the pulseon period the DQD is effectively isolated from the electrodes by the Coulomb blockade. After the pulse, the electron can tunnel out to the drain electrode and contribute to I_{p} only if the final state is L〉. In the weakcoupling regime, the small tunneling rate Г_{L} through the left barrier prevents the electron from escaping to the drain electrode during the pulseon period.
Coherent charge oscillations
Figure 5(b) shows I_{p} vs t_{p} measured at ε′ = 0 in the strong dotlead coupling regime for different interdot tunnel coupling strength T_{C}. The oscillations at ε′ = 0, where LZ transitions are not involved, demonstrate coherent Rabitype oscillations between R〉 and L〉. The oscillation frequency changes as T_{C} is increased by setting V_{C} less negative. By fitting the oscillations with a damped cosine function \({n}_{{\rm{p}}}({t}_{{\rm{p}}})=AB\,\exp ({t}_{{\rm{p}}}/{T}_{2})\cos ({\rm{\Omega }}{t}_{{\rm{p}}})+C{t}_{{\rm{p}}}\), T_{C} (≡\(\hslash \)Ω/2) is estimated to be 2.6 ~ 4.6 μeV. (A, B, C, and T_{2} are fit parameters.) The decoherence rate Γ (=1/T_{2}) obtained from the fitting is ~1 ns^{−1} almost independent of T_{C}, indicating that it is limited by decoherence due to cotunneling between dot and leads^{5,6}.
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Acknowledgements
The authors thank Y. Tokura and T. Fujisawa for valuable discussions. Part of this work was financially supported by a MEXT GrantinAid for Scientific Research on Innovative Areas (JSPS KAKENHI: JP21102003) and the Funding Program for WorldLeading Innovative R&D on Science and Technology (FIRST).
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T.O., K.H. and K.M. planned the experiments. K.H. fabricated the device and T.O. performed the measurements. T.O. and K.M. analyzed the data and prepared the manuscript.
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Ota, T., Hitachi, K. & Muraki, K. LandauZenerStückelberg interference in coherent charge oscillations of a oneelectron double quantum dot. Sci Rep 8, 5491 (2018). https://doi.org/10.1038/s41598018234682
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DOI: https://doi.org/10.1038/s41598018234682
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