Nondestructive measurement of electron spins in a quantum dot

We propose and implement a nondestructive measurement that distinguishes between two-electron spin states in a quantum dot. In contrast to earlier experiments with quantum dots, the spins are left behind in the state corresponding to the measurement outcome. By measuring the spin states twice within a time shorter than the relaxation time $T_1$, correlations between the outcomes of consecutive measurements are observed. They disappear as the wait time between measurements becomes comparable to $T_1$. The correlation between the postmeasurement state and the measurement outcome is measured to be $\sim 90\%$ on average.

Figure 1

(Color online) Schematic of the quantum dot throughout the nondestructive measurement scheme, for a singlet (top) or triplet (bottom) initial state. Curved arrows indicate tunnel process. The interesting feature is that the spin state is the same before and after the measurement.

Figure 2

(a) Scanning electron micrograph showing the sample design. (b) Voltage pulses applied to gate ${}^{\prime }{P}^{\prime }$ for the relaxation measurement. (c) Typical QPC response in the $400\mathrm{\mu }\mathrm{s}$ interval indicated by the rectangle in (b), for the case of singlet (left) and triplet (right). The reference of the time axis is taken $100\mathrm{\mu }\mathrm{s}$ before the measurement pulse is applied. The solid horizontal line indicates the position of the threshold. The dotted lines indicate the expected value for the dip in $\mathrm{\Delta }{I}_{QPC}$ for the case of ${}^{\prime }{S}^{\prime }$ and ${}^{\prime }{T}^{\prime }$. (d) The probability for detecting a triplet state as a function of the waiting time. Each point is an average over $500$ experiments. The solid line is an exponential fit to the data. The measurement errors $\alpha$ and $\beta$ (see text) are indicated.

Figure 3

(a) Typical QPC response for two consecutive measurements in the cases of ${}^{\prime }S{S}^{\prime }$, ${}^{\prime }T{T}^{\prime }$, ${}^{\prime }S{T}^{\prime }$, and ${}^{\prime }T{S}^{\prime }$. The threshold is the same for the two nondestructive measurement pulses. The pulse width is $20\mathrm{\mu }\mathrm{s}$ and the delay between the two measurement pulses is $60\mathrm{\mu }\mathrm{s}$. The dotted lines indicate the expected value for $\mathrm{\Delta }{I}_{QPC}$ for the cases of ${}^{\prime }{S}^{\prime }$ and ${}^{\prime }{T}^{\prime }$. (b) The recorded probabilities for each of these four events over 3000 runs, with the singlet (first graph) and mostly the triplet (second graph) as the initial state. In the third graph, the conditional probabilities $P\left(T{\mid }^{\prime }{T}^{\prime }\right)$ and $P\left(S{\mid }^{\prime }{S}^{\prime }\right)$ that the state after the first measurement corresponds to the outcome of the first measurement and the conditional probabilities $P{(}^{\prime }{T}^{\prime }{\mid }^{\prime }{T}^{\prime })$ and $P{(}^{\prime }{S}^{\prime }{\mid }^{\prime }{S}^{\prime })$ that the second measurement gives the same outcome as the first one are presented. They are extracted from the two previous graphs and the known $\alpha$ and $\beta$ with no a priori knowledge of the initial state.

Figure 4

Different events and their probabilities throughout the process of the two consecutive measurements for singlet $S$ or triplet $T$ as an initial state. State 1 corresponds to the dot configuration where only one electron is in the dot. $\alpha$ and $\beta$ are defined as the probability for the measurement to return respectively triplet and singlet if the actual state is singlet and triplet. They are obtained directly from the relaxation curve, giving $\alpha =14\%$ and $\beta =12\%$. ${\gamma }_{rel}$ is the known probability that the triplet state relaxes to singlet between the two measurement pulses separated by $60\mathrm{\mu }\mathrm{s}$. ${\gamma }_{rel}=3\%$ from the knowledge of $T_1$. $\sigma$ is the probability for an electron to tunnel out even though the QPC signal did go below the threshold and a ${}^{\prime }{S}^{\prime }$ outcome has been declared (we assume that $\sigma$ is equal for singlet and triplet initial states). Finally, $\gamma$ is the probability that, after reinitializing to a two-electron state and subsequent relaxation for $60\mathrm{\mu }\mathrm{s}$, a singlet state is present in the dot. In the experiment, $\sigma =5\%$ and $\gamma =3\%$, determined from the above probability tree, the known values of $\alpha$, $\beta$, ${\gamma }_{rel}$, and the correlation data.

Figure 5

(Color online) The probabilities for the two consecutive measurement outcomes as a function of (a) the measurement delay and (b) the measurement pulse duration. The solid lines represent exponential fits to the data.