Electron spin-flip correlations due to nuclear dynamics in driven GaAs double dots
We present experimental data and associated theory for correlations in a series of experiments involving repeated Landau-Zener sweeps through the crossing point of a singlet state and a spin aligned triplet state in a GaAs double quantum dot containing two conduction electrons, which are loaded in the singlet state before each sweep, and the final spin is recorded after each sweep. The experiments reported here measure correlations on time scales from 4 s to 2 ms. When the magnetic field is aligned in a direction such that spin-orbit coupling cannot cause spin flips, the correlation spectrum has prominent peaks centered at zero frequency and at the differences of the Larmor frequencies of the nuclei, on top of a frequency-independent background. When the spin-orbit field is relevant, there are additional peaks, centered at the frequencies of the individual species. A theoretical model which neglects the effects of high-frequency charge noise correctly predicts the positions of the observed peaks, and gives a reasonably accurate prediction of the size of the frequency-independent background, but gives peak areas that are larger than the observed areas by a factor of two or more. The observed peak widths are roughly consistent with predictions based on nuclear dephasing times of the order of 60 s. However, there is extra weight at the lowest observed frequencies, which suggests the existence of residual correlations on the scale of 2 ms. We speculate on the source of these discrepancies.
August 24, 2019
Nuclear spins in solid state systems provide a rich platform to study quantum many-body dynamics. The coupling of the electrons to the underlying nuclear environment plays an important role in spintronics Žutić et al. (2004) and utilization of electron spins for quantum computation Petta et al. (2005); Nowack et al. (2007); Kloeffel and Loss (2013). More generally, the interaction between a driven electron-spin qubit and its many-body environment leads to complex dynamical phenomena which are absent if the system is assumed to be at equilibrium with its environment Chekhovich et al. (2013); Petta et al. (2008); Foletti et al. (2009); Bluhm et al. (2010); Vink et al. (2009); Tartakovskii et al. (2007); Latta et al. (2009); Lai et al. (2006); Ono and Tarucha (2004); Greilich et al. (2007); Eble et al. (2006); Mehl et al. (2014); Chesi et al. (2015); Rančić and Burkard (2014). Thus, a qubit can be utilized as a probe for studying out-of-equilibrium physics in interacting quantum systems. Furthermore, understanding the rich system-environment dynamics is essential for physical implementations of fault-tolerant quantum information processingTaylor et al. (2005).
In semiconductor quantum dots the qubit is defined in terms of confined single or multi-electron states in a two-dimensional electron layer confined in a heterostructure. The singlet () and triplet () states of two electrons in a double quantum dot has proven to be a promising candidate for quantum information processing Petta et al. (2005); Foletti et al. (2009); Shulman et al. (2012). The wave functions of the electrons are typically spread over nm and the hyperfine interaction between the electrons and their nuclear environment may include several million nuclei. Although the fluctuations in the nuclear spin environment act as a source of decoherence for the - qubit, the difference in polarization of the nuclear spins between the dots of a double quantum dot (DQD) has been usefully exploited to produce rotations around an axis of the Bloch sphere of the qubit. Thus, the stable controllability of the nuclear field gradient is imperative for the efficient control of the qubit. This has been experimentally achieved by protocols to control the state of the nuclear spins through the hyperfine coupling between the electronic and nuclear degrees of freedom Foletti et al. (2009); Bluhm et al. (2010); Shulman et al. (2014).
Because the energy scale of the interaction between nuclei is much weaker than their hyperfine interaction with the electrons, the consequent separation of time scales allows one to perform high-fidelity quantum control of the - qubit, despite the fluctuating nuclear environment. In this article we shall focus on the anti-crossing between the singlet and triplet () states of the electrons, which is utilized for polarizing the nuclear spins. As the gate-voltage is swept through the - anti-crossing, an electron spin can be flipped either by spin-orbit (SO) or hyperfine (HF) interaction, and in either case, the electron system, starting in the state, will emerge in the state. (Note that the state has lower energy than that of , because the in GaAs.) Transitions caused by the HF interaction will also lead to spin flips in the nuclear system, which can lead to a significant change in the nuclear polarization, if the sweep protocol is repeated a sufficient number of times Foletti et al. (2009). Effects of the nuclear hyperfine field on the electronic spin state have also been employed in experiments on electron dipole spin resonance (EDSR) by various authors Laird et al. (2007); Shafiei et al. (2013); Tenberg et al. (2015).
In this article, we discuss an experiment where the electron sweep protocol is repeated 500 times, and the electronic state, singlet or triplet, is measured and recorded after each sweep. We then calculate and analyze the power spectrum, which characterizes correlations in the triplet return probabilities for pairs of sweeps that are separated by time intervals between 4 and 2000 s. Non-trivial correlations are to be expected in these measurements, because the nuclear configuration will evolve on this time scale, primarily because of Larmor precession in the external magnetic field, which occurs at different frequencies for the three nuclear species involved. More generally, the study of these correlations provides crucial insights into the many-body dynamics of the coupled electron-nuclear spin system. As the cumulative effect of 500 sweeps on the nuclear polarization is too small to have a significant effect on the triplet return probabilities studied in our experiments, the experiments may be interpreted as a probe of intrinsic correlations of the system.
In the article we present our experimental results and we develop a theoretical model to describe measurements. The model is based on the central spin problem Chen et al. (2007); Yao et al. (2006); Cywiński et al. (2009), where a two level system is coupled to a large number of spins. The two levels in our case are the and states of two electrons. The collection of nuclear spins is treated within a semi-classical approximation where the Overhauser fields due to a mesoscopic aggregate of spins can be treated as a set of Gaussian random variables. In the absence of SO, when the electron spin-flip probability is small, the correlations are dominated by spectral peaks centered at zero frequency and at the differences of nuclear Larmor frequencies of any two species. In the presence of SO, there are additional peaks at the individual Larmor frequencies, which are produced due to the interference of the static SO term and the nuclear precession. The center positions of the correlation peaks extracted from the experiments are in very good agreement with the values predicted from the known Larmor frequencies. The power spectrum also has a frequency-independent background which is well-predicted by our theory. However, our main focus will be on the areas and widths of the peaks.
A large part of this paper will be devoted to theoretical discussions of the predictions for the triplet-return correlation function and its power spectrum that may be deduced from our model. Because of the Gaussian nature of the nuclear spin fluctuation in the model, predictions for the triplet-return correlation function can be accurately obtained, if the model parameters are known. These parameters include the experimental sweep rates and the strengths of the spin-orbit and the root-mean square hyperfine fields, as well as assumptions about the time scale and form for decay of correlations in the transverse hyperfine fields for each of the three nuclear species.
Taking values of the key parameters from previous experiments Nichol et al. (2015), we find qualitative agreement between the predicted peak areas and the experimental results, but there is a systematic discrepancy in which the measured areas are typically smaller, by a factor of two or more than the predictions of the model. We believe that the most likely cause for this discrepancy is the effect on the triplet return probability due to high-frequency charge noise on the gates or in the quantum dots themselves. It is clear from previous experiments Nichol et al. (2015), that depending on the sweep rate across the - anti-crossing, the triplet return probability can be significantly affected by such noise. Although we do not have a detailed knowledge of the size and frequency dependence of the charge noise that may be relevant for the current experiments, and we have not conducted a quantitative analysis of the possible effects of charge noise on these experiments, we do include a qualitative discussion, which supports the hypothesis that charge-noise may be the principal source of the remaining discrepancies between the measured peak areas and the theoretical predictions.
In our analysis of the experimental data, we find that we can obtain a qualitative understanding of the widths and shape of the peaks in the correlation spectrum by assuming that the decay of correlations has the form of Gaussian relaxation functions, with correlation times of order 60 sec. However, the power spectrum has extra weight at the lowest nonzero frequencies studied in these experiments (Hz), which is not explained by the model. We discuss in an Appendix the form of the nuclear spin correlation functions to be expected if decay is primarily the result of inhomogeneous broadening of the nuclear Larmor frequencies. We find that the relaxation functions should be well described by a Gaussian for times that are not too long, but there will be deviations at larger times, which could possibly lead to anomalies in the triplet return correlation function at very low frequencies.
The general outline of the paper is as follows. In section II we describe in detail the theoretical model used to describe the coupled dynamics of the electron-nuclear system in the double quantum dot. Section III includes the specific multi-sweep protocol which was implemented experimentally and furthermore derives the predictions from the theoretical model for the frequency spectrum of the correlation function in this scenario. In subsection III A, we present the predictions of our model for the frequency-independent background contribution to the correlation spectrum. In subsection III B, the consequences of the model for the principal peaks in the spectrum are evaluated within a linear approximation, applicable for fast sweep rates, where the Landau-Zener triplet return probability is small. In subsection III C, we introduce a nonlinear approximation, valid for smaller sweep rates, which will prove necessary to make sensible predictions for the experiments under consideration. In subsection III D, we show that an exact solution of our model is possible for the time-dependence of the triplet-return correlation function, and we explain how these results can be used to obtain precise predictions for the areas of the leading peaks in the spectral function, if the model input parameters are known. In section IV we discuss the implications of charge noise in different frequency regimes and comment on their relevance to current experiments. The theoretical predictions of our model, at our various levels of approximation, are compared with each other and with the results of our experiments in Section V, and our conclusions are summarized in Section VI. As mentioned above, predictions for the form of relaxation for the nuclear spin correlation function due to inhomogeneous broadening are discussed in an Appendix.
The present paper has some overlap with a recent publication by Dickel et al. Dickel et al. (2015). In particular, the results of our full calculation of the triplet-return correlation function in the time domain, described in Section III D below, coincide with theoretical results described in that publication. Dickel, et al. also present experimental measurements in the time domain, which agree, at least qualitatively with their theoretical predictions. By contrast, in the present paper, we present results of an extensive series of measurements and associated theoretical predictions, analyzed in the frequency domain, which enables us to determine separately the effects of hyperfine and spin-orbit coupling on spin-flip correlations in the system.
Ii Multi-sweep experiments and theoretical model
The system of interest is a double quantum dot containing two electrons. The diameter of the dots in the device used for the experiments is around nm. For the temperatures relevant to this work only the lowest lying orbital state of the dots has any significant probability of occupation. Therefore, each dot could either be doubly or singly occupied by the two electrons, although due to Pauli’s exclusion principle the spin component of the (or ) state is forced to be a singlet. On the other hand, the state has no such constraint. The spin component of the electronic wave function is defined in terms of the projection of the spin along the externally applied uniform magnetic field, which we define to be the z-direction. The singlet and triplet states take their canonical form
in terms of the projections of the spins of individual electrons.
The energy-level diagram of the system, as a function of voltage difference between the dots (detuning ), is shown in Fig. 1. In the absence of spin non-conserving terms in the Hamiltonian, the singlet and triplet sectors are decoupled from each other. The uniform magnetic field splits the triplet states in energy, producing gaps equal to the net Zeeman energy of the electrons between the triplet states of the electrons , and the triplet state. At any given positive detuning, the electronic states in the singlet sector are an admixture of and states due to the tunneling Fasth et al. (2007); Brataas and Rashba (2011); Rudner and Levitov (2010). On the other hand, due to the total conservation tunneling has no influence on the triplet sector.
The electron spin non-conserving terms in GaAs semiconductors are due to the nuclear hyperfine and spin-orbit interactions. They couple the singlet and triplet subspaces of the two electrons. There are several factors contributing to the spin-orbit effect experienced by the electron confined in the dots, like the shape of the dots and the tunneling between them, the orientation of the dots with respect to the crystallographic axes, the spin orbit length of the host material, and magnetic field. The two-dimensional electron gas is fixed to lie in the plane of the GaAs crystal in the experiments, but the direction and strength of the in-plane magnetic field are controllable. The magnitude of the spin-orbit interaction strength can be varied by changing the direction of the magnetic field and is given by Stepanenko et al. (2012)
where is the angle between the in-plane magnetic field and the spin-orbit field direction, and is the strength of the spin-orbit term at the crossing point. (In our experiments, the axis of the DQD is either in the or direction, and the spin-orbit direction is perpendicular to the DQD axis.) The value of will depend on the details of the quantum dot system, and on the magnitude of the applied magnetic field, but not its direction in the plane, as long as the Zeeman energy is much stronger than . For the relevant size of the gate-defined quantum dots, the electronic wave function is typically spread over lattice sites. The contact hyperfine interaction with the nuclear spins of , , and generates an effective spin-spin interaction between the nuclei and the electrons given by
where represents the nuclear species, is the position of the nucleus, and is the electron index. is the volume per nuclear spin in GaAs. is the nuclear spin of species at site and is the spin of the electron. The gradient in the transverse component of the nuclear Overhauser field couples the - states and produces an anticrossing () at a particular value of detuning fixed by the magnetic field. Thus, the direction and magnitude of the external magnetic field serves as a convenient experimental control to tune the - anticrossing between spin-orbit-dominated and hyperfine-dominated regimes.
It is convenient to define a set of quantities
where and is the hyperfine coupling amplitude defined in terms of the electronic orbital and states, viz.
where, and are the orbital parts of the eigenfunctions and . Although for an individual nuclear spin is an operator that should be treated quantum mechanically, the quantities are each the sum of very many such variables, and they may be treated, with high accuracy, as classical complex amplitudes, which evolve in time as the nuclei precess about the applied magnetic field.
The effective electronic Hamiltonian at the - anticrossing may then be written in the form
and is the sum of the z-components of the Overhauser fields on the quantum dots, which gives rise to a shift in the energy of the triplet state. Specifically, we may write
where the hyperfine amplitude in the triplet state is
Without the loss of generality, the spin-orbit interaction will be chosen to be real.
In the experiment the electrons are loaded in the singlet state of the right dot () following which the voltage is swept through the - anticrossing. Assuming that one can neglect effects of high-frequency charge noise, the choice of a linear sweep protocol, , maps the problem to the famous Landau-Zener case Landau and Lifschitz (1977); Zener (1932); Stückelberg (1932); Majorana (1932) which gives the probability of the transition from the to state to be
for initial () and final () times tending to and respectively, and where
In using this relation, for each sweep, we evaluate and at the instant of time when the gate voltage passes through the level-crossing point, where the singlet and states would be degenerate in absence of . We have assumed that we can neglect the precession of the nuclei during the course of a single sweep, which should be a good approximation for the experiments under consideration. The precession frequencies of the nuclear species are in the range of a few MHz. The duration of the sweep is varied between to adjust the average . However, the nuclear configuration is only important during the shorter time interval, when the system is close enough to the crossing point for an electron spin-flip to occur. The total range in the energy difference during a sweep is , but the range where spin-flips can occur is when MHz.
In practice, there may be important corrections to the Landau-Zener transition probability due to charge noise in the sample or on the gates. We shall discuss effects of charge noise in Section V below, but we ignore them for the moment.
In our experiments, the gate voltage is returned rapidly, to avoid - transitions, to the side after each Landau-Zener sweep, the electronic spin state is measured using the spin-blockade technique Ono et al. (2002); Barthel et al. (2009), and the outcome is recorded. After this measurement, the electronic state is reinitialized for the next sweep by loading the electrons in the singlet. Successive sweeps are separated in time by a precise time interval = 4s, which includes the duration of a sweep as well as the waiting period between sweeps, during which the nuclear spins undergo free Larmor precession. Over the longer time scale of many sweeps, the nuclear spins also exhibit energy and phase relaxation due to nuclear dipole-dipole interactions and other mechanisms, which we shall take into account in an approximate way.
The measurements were carried out in a series of “runs”, each consisting of successive sweeps, labeled by , with sweep times separated by = 4 s. This protocol was repeated times, with a waiting period of milliseconds between successive runs. At the end of each set of 288 runs, a halt of seconds is implemented during which all components of the nuclear spins are expected to reach back to equilibrium. This whole procedure was then repeated times. These waiting periods are sufficiently long that at the beginning of each run, at least the transverse components of the nuclear spin configuration can be assumed to be in a random state sampled from the thermal ensemble, so the experimental runs may be considered as different realizations of the same ensemble.
For each sweep , we define a variable which is equal to 0 or 1 depending on whether the electron state has flipped from to or not. We may then define a spin-flip probability and a correlation function
where , and the angular brackets indicate an average over the 14,400 runs. Analysis of this correlation function will be the main focus of this paper.
The electron spin-flip probability in any given sweep depends on the orientations of the nuclei at the time of that sweep. As remarked above, the distribution of the nuclei before the first sweep in a run should be given by the thermal equilibrium distribution of the nuclei in the applied magnetic field. For the temperatures and fields relevant to these experiments, the net polarization in the z-direction will be very small compared to the maximum possible polarization of the nuclei, so that the distribution of perpendicular spin components should be essentially the same as in an equilibrium ensemble at zero magnetic field. During the course of 500 sweeps, there may be a change in the z-polarization of the nuclei due to the effects of dynamic nuclear polarization (DNP), but the polarization will still be very small compared to the maximum polarization. Therefore, for any single sweep the probability distribution of should be the same as in thermal equilibrium.
Since the complex variable is the sum of small contributions from a very large number of nuclei, it is clear that the equilibrium distribution will have the form of a Gaussian, whose form is completely determined by its first and second moments. Since the orientations of different spins are uncorrelated, it is easy to see that , and , where and
where is the spin and is the fractional abundance of species . For GaAs, 0.5, 0.2 and 0.3, for As, Ga, and Ga respectively, while for all species. The coupling constants, measured in eV are , , and . The quantity may be interpreted as the effective number of nuclei contributing to the transverse hyperfine field .
If we regard the real and imaginary components of as a two dimensional vector , the probability distribution of may be written as
Since the complex amplitudes are themselves each a sum of contributions from a large number of nuclei, their individual thermal distributions are also Gaussians, with replacing in the formula above.
We shall also be interested in the joint probability distributions of at several different times. Under the influence of the applied magnetic field, the macroscopic spins undergo Larmor precession. At the same time the collection of nuclear spins experience energy and phase relaxation due to dipolar and quadrupolar interactions. The time scale for phase diffusion in nuclear spins in GaAs is of the order while that of spin or energy diffusion can be of the order seconds Reilly et al. (2008). Since the value of at each time is the sum of contributions from very many nuclei, the joint distribution function of at two different times is again a Gaussian distribution. Consequently, the distribution is completely determined by its second-order correlations.
Iii Correlations in - sweeps
The off-diagonal matrix element coupling the and states has a time-independent part due to the spin-orbit effect and a time-dependent contribution from the transverse components of nuclear spins of the various species, given by
where is the Larmor frequency of species and the amplitude is assumed to vary only slowly, on a time scale of order 100s. It is the interference of the terms of different frequencies contributing to the - matrix element that is the source of the interesting temporal correlations in the electron spin-flip probability .
Let us write the two-time correlation function for in the form
where , and decays to zero on a time scale , which is the relaxation time of species arising from interactions in the nuclear spin system, etc. Here, we are assuming that the fluctuations in the nuclear orientations perpendicular to the applied magnetic field can be treated as a stationary stochastic process, which will not be significantly affected by the Landau-Zener process within a sequence of 500 sweeps. Motivated by experimental observations, we assume here a simple Gaussian form for :
A discussion of reasons for the (approximate) validity of this assumption, and of possible consequences of deviations from the assumed Gaussian behavior, will be given in the Appendix.
We now turn to predictions of our model for the correlation function defined in (16). Suppose that the nuclear configurations at the two times and are known, so that the corresponding LZ probabilities are also known. Then the conditional expectation value of the product will be given by
since the outcomes and are stochastic quantities that are independent if and only if . If we now average this result over all possible initial conditions of the nuclei, and take into account the effects of random dephasing between the two times and , we obtain the result
As argued above, this expectation value should be essentially independent of . We remark that the term proportional to is a quantum stochastic effect, which reflects the random outcome for the value of , even when the probability is specified. This will lead to a frequency-independent background contribution to the Fourier transform of .
In practice, it will be most convenient to work with a Fourier expansion of and to discuss the power spectrum of . We define
where is an integer, and we impose the restriction , where is the frequency defined by
The power spectrum is then defined as
We define a correlation function
which depends only on the time separation , and should be a continuous function of that variable. (This is because the values of evolve continuously in time, and are not affected by any intervening Landau-Zener sweeps on the time scale we are considering.) For times large compared to the dephasing times , the function will approach a limit,
Taking the Fourier transform of , after subtracting the infinite time limit, we define a function
We now wish to relate the experimentally observed power spectrum to the function . The functions differ for three reasons: because includes a contribution from the background term which is omitted from , because the experimental measurements are restricted to a discrete set of time steps rather than as a continuous function of time, and because the measurements are restricted to a finite time interval . This last restriction should be unimportant, provided that the time interval is large compared to all of the correlation times . The contribution of can be added explicitly, and the difference between the discrete sum and the continuous integral can be handled by use of the Poisson sum formula. The result is
In the absence of spin-orbit interactions, the integrals are simple Gaussian integrals, and one obtains, after a small amount of algebra:
The background in the presence of spin-orbit interaction can again be calculated at all orders in . The first average in Eq.(33) is now given by
where represents the real and imaginary part of . A similar calculation gives
iii.2 Linear Approximation for
We now discuss predictions for the function by first considering some simple cases. We begin by considering a linear approximation, which is valid in the regime of fast Landau-Zener sweeps, where . In this regime the Landau-Zener probability is small, and it can be expanded in a power series in
where is the time of the -th sweep.
In the absence of SOI, only the nuclear spin terms are responsible for correlations in the electron spin-flip probability. For small , the Fourier transform of the lowest order term in is given, for , by
On averaging over the nuclear spin configuration, the power spectrum has peaks at frequencies equal to the differences of the Larmor frequencies of any two of the species. In the absence of nuclear spin relaxation, these peaks are delta functions in frequency, but as we include nuclear relaxation phenomenologically, these peaks broaden and develop a finite line-width consistent with a Gaussian decay of correlations given by Eq. (21). On taking into account the Gaussian decay in time, the resulting expression is
where is a Gaussian of unit area, given by
The Gaussian peak around zero frequency receives contributions from all the three species additively, and thus is much stronger than the peaks at difference frequencies. If we substitute the expression (40) into (31), we obtain an approximate expression for the power spectrum, which we can compare with experiments.
In Figure 2, we show experimental data (black curve) for , with data for the singular point omitted, taken at two values of the applied magnetic field . The magenta curve is an empirical fit of the data to a set of Gaussian peaks, sitting on top of a frequency-independent background. Data are shown only in the positive half Brillouin zone, = 125 kHz, as the spectrum depends only on .
In each plot, one sees clearly three Gaussian peaks centered at non-zero frequencies, as well as the positive half of a quasi-Gaussian peak centered at , all of which sit on top of a frequency-independent background. The vertical lines are drawn at the three difference frequencies mod which fall in the positive half Brillouin zone. It can be seen that the centers of the Gaussian peaks agree with the positions of the vertical lines to a high degree of accuracy. We defer, until Sec. V, a more detailed comparison between theory and experiment, including the areas under the peaks, the relative widths of the peaks, and the height of the background.
In addition to the quasi-Gaussian peak around , the data shows enhanced values of the spectrum for the lowest non-zero values of the discrete frequency, particularly at kHz. This will be discussed further in Sec. V.
If one extends the theoretical analysis beyond the first term in the expansion of , one expects to find additional peaks at arbitrary linear combinations of the difference frequencies , reduced to the first Brillouin zone. However the areas of the higher order peaks will be relatively small for the values of of interest to us, and the widths of the peaks become larger with increasing order. It is therefore not surprising that we do not see signs of higher order peaks in the experimental data.
The presence of spin-orbit coupling allows for another mechanism for electron spin-flips besides nuclear spins. In the - matrix element, the effective SO interaction , which depends on the angle between the spin-orbit field and applied magnetic field according to Eq. (5), can be varied by changing the direction of the field in the plane of the sample. In this regime, the correlations in receive contributions from the spin-orbit term in combination with the dynamics of the nuclear spins. In the approximation where we keep only the lowest order term in the expansion of , interference of the two effects generates terms proportional to in the correlation function with peaks at the Larmor frequencies of the individual species, in addition to the terms in Eq. (39). On using the form of the - matrix element, given by
where is the predicted spectrum for , given by Eq. (40), and
Thus, the power spectrum in the presence of SOI, , has additional peaks at the bare Larmor frequencies of the three different species given by the functions . (Of course, in , the bare frequencies are measured modulo .) It is interesting to note that the widths of the additional peaks due to the presence of SOI are predicted to be narrower than the peaks at the differences of the Larmor frequencies.
If is turned on while the sweep rate is fixed, so that the values of are unchanged, the value of will increase, as follows from (11) and (38). This will lead to an increase in the weight of the delta function at zero frequency, which is proportional to , according to (29). However, the change in may be removed by an increase in the sweep rate, if desired.
Fig. 3 presents experimental results for the spectral function for two different values of the angle , which give rise to increasing values of . Vertical lines show the positions expected for the bare Larmor frequencies and the difference frequencies, which align extremely well with the positions of the experimental peaks, as expected from our model. Comparison between predicted and observed peak heights and areas, as well as the frequency independent background, will be discussed in Section V.
iii.3 Nonlinear Approximation for the Peak Areas
The formulas for the areas of the Gaussian peaks, derived in the the previous two subsections, are correct to lowest order in , i.e., order , when is small and may be adequately approximated by . This assumption is correct when the sweep rate is sufficiently large. For the experimental data to be discussed below, however, this linear approximation is not adequate.
As will be discussed in Subsection III D, an exact analytic calculation of the correlation function of Eq. (28), correct for arbitrary , is possible for our model, in the absence of charge noise, assuming that the input parameters are known. However, to extract the areas of the peaks in the frequency domain, it is necessary to take the Fourier transform numerically, and the results are not transparent. We shall therefore begin by presenting an approximate nonlinear calculation, which yields transparent analytic results that are a major improvement over the lowest order results, and which also give some physical insight into the size of the necessary corrections.
We shall be interested here in the areas under the peaks in the correlation function , and we will not pay attention to the detailed line shape. Our discussions, therefore, will be independent of the precise time dependence of the correlation functions defined in (20).
Let us write
Then we may expand as
where our approximation shall consist in omitting terms that are higher order in .
As in the previous subsections, we assume that is given by (19), where varies slowly in time, with a correlation function of form (21). The precise form of the correlation function is not important for the present purposes; we need only assume that (i) the are complex variables with a Gaussian joint probability distribution, (ii) that there are no correlations between different species, and that (iii) there exists a coherence time such that the correlation function vanishes for , but is essentially independent of time for .
For , we may expand the correlation function as
where we have omitted terms containing other combinations of frequencies, which will turn out to be higher order in our expansion. When we take the Fourier transform of , we find that the terms included in (50) give rise to narrow peaks in , centered at frequencies or , whose areas are given by the coefficients or , respectively. The width of the peaks are of order , but the areas do not depend on . Similarly, there will be a peak in centered at , with width of order , whose area will be equal to .
Now, using the approximation (49 ), we find that
The above expressions can be evaluated using the equalities
It is convenient to define the quantities
Then the results, which one finds after some algebra, (cf. the calculations in Subsection III A, above), are
while the two terms contributing to are given by
The primes over the last two summation signs signify that the sums are to be taken over and , with , while denotes the third species, not equal to or .
It should be emphasized that the expansion coefficients , etc., are all independent of the values of the frequencies and are well defined, as long as the frequencies are incommensurate with each other.
As one test of the validity of these approximations, we may calculate the value of using the expansion (49):
and we may compare the result with the exact answer. As an example, if we set , and for all three species, the exact value of is given by , while the number predicted by Eq. (61) is 0.3980. The value of in this case is 0.4523.
More generally, we expect that the expansion (49) should be reasonable as long as the individual are small, even if the sum is not.
iii.4 Full calculation
It is convenient to write
where is given by Eq. (35) and
Since the variables have a Gaussian distribution, this last expectation value can be expressed as a multivariable Gaussian integral, which can be evaluated by standard methods.
In the regime where , the values of may be assumed to be independent of time, so the evaluations require only integration over three independent complex variables. Then, in the case where , the results simplify further to give
where is the matrix
In the case where , the result for becomes
The equations above may be simplified further by using the results
As may be seen from the above equations, in the limit , the function is a quasiperiodic function, with three fundamental frequencies , corresponding to the three different values of . Then we can expand in the form
where are integers running from to . The expansion coefficients may then be obtained by taking the limit of the integral
(In practice, convergence can be improved by using a soft cutoff in the above integration.) The quantities and of Eq. (50), which give the areas of the lowest order peaks in , are given by the coefficients with or , and .
If one wishes to calculate in the regime of intermediate times, where is comparable to , then the values of at and should be treated as separate, but correlated, Gaussian variables. The expectation value in Eq. (63) would then be expressed as an integral over a Gaussian function of six complex variables, or twelve real variables. As a simpler alternative, however, one may consider the correlation function as arising from an inhomogeneously broadened line, so that
where the set of denote frequency shifts from the line center, and are the corresponding weights. Then may be evaluated by treating each as arising from a different nuclear species and replacing the indices and in formulas (64) to (69) by and .
In the regime where is comparable to , the function is no longer quasiperiodic, so the Fourier transform will no longer be a sum of sharp -functions.
Iv Effects of charge noise
A 2DEG buried around nm below the surface of a semiconductor heterostructure is susceptible to charge noise from various possible sources, including the random two-level systems in adjoining material or through the metal gates on the surface Dial et al. (2013). Effects of charge noise may be modeled by including fluctuati