Quantum polyspectra for modeling and evaluating quantum transport measurements: A unifying approach to the strong and weak measurement regime

Quantum polyspectra of up to fourth order are introduced for modeling and evaluating quantum transport measurements offering a powerful alternative to methods of the traditional full counting statistics. Experimental time traces of the occupation dynamics of a single quantum dot are evaluated via simultaneously fitting their second-, third-, and fourth-order spectra. The scheme recovers the same electron tunneling and spin relaxation rates as previously obtained from an analysis of the same data in terms of factorial cumulants of the full counting statistics and waiting-time distributions. Moreover, the evaluation of time traces via quantum polyspectra is demonstrated to be feasible also in the weak measurement regime even when quantum jumps can no longer be identified from time traces and methods related to the full counting statistics cease to be applicable. A numerical study of a double dot system shows strongly changing features in the quantum polyspectra for the transition from the weak measurement regime to the Zeno regime where coherent tunneling dynamics is suppressed. Quantum polyspectra thus constitute a general unifying approach to the strong and weak regime of quantum measurements with possible applications in diverse fields as nanoelectronics, circuit quantum electrodynamics, spin noise spectroscopy, or quantum optics.

Figure 1

Tunnelung events of electrons between the semiconductor QD and the charge reservoir are monitored via the resonance fluorescence of the exciton transition.

Figure 2

Power spectra $S_z^{\left(2\right)}\left(\omega \right)$ of experimental fluorescence time traces $z\left(t\right)$ (inset) for a single quantum dot at 10 T and different gate voltages. Different regimes for tunnel rates ${\gamma }_{\text{in}}$ and ${\gamma }_{\text{out}}$ are observed: (a) ${\gamma }_{\text{in}}\ll {\gamma }_{\text{out}}$, (b) ${\gamma }_{\text{in}}⪆{\gamma }_{\text{out}}$, and (c) ${\gamma }_{\text{in}}>{\gamma }_{\text{out}}$.

Figure 3

(Top two rows) Experimental bispectra $S_z^{\left(3\right)}({\omega }_1,{\omega }_2)$ at $B=10\mathrm{T}$ and model fits for the tunnel regimes (a) 360, (b) 380, and (c) 382 mV (compare Fig. 2). (Bottom two rows) Experimental trispectra $S_z^{\left(4\right)}({\omega }_1,{\omega }_2)$ and model fits. The analytical expressions of the polyspectra reveal that the bispectra are sensitive to the sign of ${\gamma }_{\text{out}}-{\gamma }_{\text{in}}$, whereas the trispectrum depends mostly on ${({\gamma }_{\text{out}}-{\gamma }_{\text{in}})}^2$.

Figure 4

Tunneling rates determined from experimental polyspectra of the full time trace (blue and red dots) in comparison with values obtained from the waiting-time distribution (open squares and triangles) [4]. Both methods arrive at the same results. The vertical lines mark the gate voltages belonging to the time traces shown in Fig. 2.

Figure 5

Three-state model of the spin-depended quantum dot dynamics.

Figure 6

Calculated power-, bi- and trispectrum of the spin-dependent quantum dot dynamics for ${\gamma }_{0\downarrow }={\gamma }_{\downarrow 0}=2.5\mathrm{kHz},{\gamma }_{0\uparrow }={\gamma }_{\uparrow 0}=0.5\mathrm{kHz}$, and ${\gamma }_{\uparrow \downarrow }=0$ (from top to bottom). The bispectrum shows deviations from a simple Lorentzian shape and a dip at ${\omega }_1={\omega }_2=0$. The trispectrum strongly deviates from those of the quantum dot at 10 T (compare Fig. 2 and 3).

Figure 7

Comparison between the measured and fitted polyspectra at $B=2$ and 4 T. Cuts with ${\omega }_2=0$ (top) and cuts with ${\omega }_1={\omega }_2$ (bottom) are shown for the bi- and trispectrum. The maximum values of all spectra have been normalized to 1. We can see excellent agreement between the measurement and the three-state model. The diagonals of the bi- and trispectrum coincidentally overlap for the quantum dot system.

Figure 8

Simulated detector time trace $z\left(t\right)$ of a single quantum dot from solving the stochastic master equation in the weak measurement regime (light blue line). The electron occupation $\mathrm{Tr}\left(\rho \right(t\left)n\right)$ (dark blue line) follows from the calculation but is not directly accessible in an experiment. The detector output has been scaled by ${\beta }^{-2}$ for comparison.

Figure 9

Comparison between the simulated (left) and analytical (right) polyspectra of the quantum dot system in the weak measurement regime. The spectra show good agreement and tunneling rates could successfully be recovered from the numerical spectra by a fitting procedure. The dashed lines in the power spectra indicate the level of white Gaussian background noise.

Figure 10

Occupation dynamics and quantum polyspectra of a double-dot system: the occupation dynamics of the second dot and corresponding polyspectra change for increasing measurement strengths $\beta$ ($\beta$ increases for lower rows). The coherent oscillations are suppressed for stronger measurements (quantum Zeno effect). For $\beta =2\sqrt{\mathrm{kHz}}$, the dynamics has changed to telegraph-like switching. The peak due to coherent oscillations at $\omega /2\pi =0.5\mathrm{kHz}$ in the power spectrum $S_z^{\left(2\right)}$ broadens and shifts to zero frequency (lower row). The signatures in $S_z^{\left(3\right)}$ and $S_z^{\left(4\right)}$ are also broadened and move towards the origin. The detector output $z\left(t\right)$ is dominated by strong white Gaussian noise and therefore not displayed in the first column except for the largest measurement strength in the last row (light blue line). The units of $S_z^{\left(3\right)}$ and $S_z^{\left(4\right)}$ are ${\mathrm{kHz}}^{-2}$ and ${\mathrm{kHz}}^{-3}$, respectively.

Figure 11

Parameter dependent approximate confined Gaussian window function $g_j$ for parameter $s=0.14T$ [69].