Nuclear spin physics in quantum dots: An optical investigation

The mesoscopic spin system formed by the ${10}^4–{10}^6$ nuclear spins in a semiconductor quantum dot offers a unique setting for the study of many-body spin physics in the condensed matter. The dynamics of this system and its coupling to electron spins is fundamentally different from its bulk counterpart or the case of individual atoms due to increased fluctuations that result from reduced dimensions. In recent years, the interest in studying quantum-dot nuclear spin systems and their coupling to confined electron spins has been further fueled by its importance for possible quantum information processing applications. The fascinating nonlinear (quantum) dynamics of the coupled electron-nuclear spin system is universal in quantum dot optics and transport. In this article, experimental work performed over the last decade in studying this mesoscopic, coupled electron-nuclear spin system is reviewed. Here a special focus is on how optical addressing of electron spins can be exploited to manipulate and read out the quantum-dot nuclei. Particularly exciting recent developments in applying optical techniques to efficiently establish nonzero mean nuclear spin polarizations and using them to reduce intrinsic nuclear spin fluctuations are discussed. Both results critically influence the preservation of electron-spin coherence in quantum dots. This overall recently gained understanding of the quantum-dot nuclear spin system could enable exciting new research avenues such as experimental observations of spontaneous spin ordering or nonclassical behavior of the nuclear spin bath.

Figure 1

Reciprocal interaction between electron and nuclear spins. The electron-spin state is initialized through optical pumping.

Figure 2

(a) $1\mu \mathrm{m}\times 1\mu \mathrm{m}$ atomic force microscopy image of InAs dots on GaAs. (b) $40\mathrm{nm}\times 34\mathrm{nm}$ cross-sectional scanning-tunneling microscopy image of a GaAs dot in AlGaAs. From 102. (c) Schematic energy level diagram for an InAs QD in GaAs, where the growth axis is along the $Oz$ direction.

Figure 3

Sample A: (a) Scheme of InAs QDs embedded into a charge-tunable device as in 192, where for a voltage $V_{g1}$ applied to the top gate the electronic level of the dot is above the Fermi energy of the highly $n$-doped back contact. The QD contains no conduction electron. For a gate voltage $V_{g2}$ the electronic level of the dot is now below the Fermi sea and an electron can tunnel into the dot. (b) The charging of a single InAs QD with electrons is accompanied by discrete jumps in the emission energy when going from the neutral exciton $X^0$ (one electron, one hole) to the charged exciton $X^{-}$ (two electrons, one hole), etc. until the wetting layer (WL) is charged. Sample B: (c) Left: Charge fluctuations (a doping hole or electron tunnel into and out of the dot) in nonintentionally doped dots allow the observation of neutral excitons $X^0$, charged excitons $X^{+}$, and biexcitons $2X^0$ in photoluminescence (PL) spectra that are integrated over seconds, i.e., over times much longer than the charge fluctuation times (17) Right: In addition to the fine structure, the emission intensity of each transition as a function of optical excitation power allows one to distinguish between different exciton complexes containing two, three, or four optically generated charge carriers.

Figure 4

Optical selection rules for interband transitions involving valence-band electrons with angular momentum $J^{\mathrm{ve}}=3/2$ in units of $\hslash$, the corresponding photon polarization ${\sigma }^{+}$ or ${\sigma }^{-}$ is indicated. Valence states with $J^{\mathrm{ve}}=1/2$ are well separated in energy and are not shown here. Absorption of a ${\sigma }^{+}$ polarized photon by a $J_z^{\mathrm{ve}}=-3/2$ valence electron results in the promotion of this electron to the conduction state $m_s=-1/2$ and a heavy hole $J_z=+3/2$ is left behind in the valence band. The energy separation between valence heavy- and light-hole states ${\Delta }_{\mathrm{HL}}$ is typically several tens of meV in InAs dots in GaAs.

Figure 5

(a)–(d) A QD exchanges an electron with the reservoir via a virtual two electron state, and (e) calculation of the cotunneling rate as a function of gate voltage after 50 for tunneling rates $0.1{\mathrm{ns}}^{-1}$ (solid curve) and $0.02{\mathrm{ns}}^{-1}$ (dashed curve).

Figure 6

(a) Gray-scale plot showing exciton PL spectra recorded for an individual InGaAs dot. The spectra are recorded at $B_z=2.5\mathrm{T}$ using unpolarized detection. (b) Power dependences of the Zeeman splitting measured at $B_z=2\mathrm{T}$ for ${\sigma }^{+}$ and ${\sigma }^{-}$ excitation polarizations. From 182. (c) Trion $X^{-}$ absorption at zero magnetic field with Lorentzian fit of linewidth $2\mu \mathrm{eV}$, and (d) strong deviation from Lorentzian line shape at $B_z=4.5\mathrm{T}$. From 127. (e) Zeeman splitting for a single GaAs interface fluctuation QD. Because of dynamic nuclear polarization the Zeeman splitting for ${\sigma }^{+}$ and ${\sigma }^{-}$ polarized excitations are different. Randomizing the nuclear spin orientation with a radio-frequency field (rf on) results in the same Zeeman spliting for both laser polarizations. From 73.

Figure 7

(a) Sketch of the strain distribution in an InAs/GaAs QD and splitting of energy levels in zero field due to axially symmetric quadrupolar interaction for a nuclear spin $I=9/2$. (b) Reduction factor of nuclear polarization along $z$ due to the inclination $\theta$ of the quadrupolar main axis. In the approximation of high spin temperature, it is given by $⟨I_z⟩/⟨I_z^0⟩=(I+3/4)(I-1/2){cos}^2\theta /[I(I+1)]+3(I+1/2)/[4I(I+1)]$. (c) A nucleus with nonspherical, prolate charge distribution is equivalent to a spherically symmetric charge distribution plus some positive charge shared between the two polar regions and a band of equal negative charge added around the equator. This addition has no dipole moment but does have electric quadrupole moment. From 195.

Figure 8

Electron spin relaxation induced by nuclei in a QD. The calculation is done for $N={10}^5$ nuclei per dot. From 53.

Figure 9

(a) Circular polarization dynamics of the positively charged exciton $X^{+}$ luminescence for $B=0$ for an ensemble of InAs dots. $X^{+}$ is formed by a photocreated electron-hole pair and a resident hole. Since unpolarized holes form a zero spin singlet, the spin-polarized electron is not coupled to the holes by the exchange interaction, which can dominate the hyperfine interaction (24; 53). The inset displays the $I^{+}$ and $I^{-}$ luminescence dynamics. (b) Schematics of the electron-spin precession around (left) the effective nuclear field $\delta B_n$ and (right) the total field $B_{\mathrm{tot}}=B_z+\delta B_n$ if an external magnetic field $B_z$ is applied.

Figure 10

Circular polarization dynamics of the $X^{+}$ luminescence in an ensemble of InAs dots for $B_z=0$ and $B_z=100\mathrm{mT}$. The inset displays the calculated time dependence of the average electron spin $⟨\mathbf{S}(t)⟩/S_0$. From 148 and 24.

Figure 11

Single InAs dot measurement. (a) Polarization-resolved spectra of $X^{+}$ luminescence for two different magnetic fields. (b) Circular polarization of $X^{+}$ line vs external magnetic field $B_z$. The excitation polarization is provided by a 50 kHz photoelastic modulator to avoid the buildup of a nuclear polarization through optical pumping. From 25.

Figure 12

Time dependence of the singlet (two-electron-spin) return probability in an electrically defined GaAs double QD. The data are fitted using a semiclassical model (171) which is similar to the Merkulov-Efros-Rosen approach. The fits correspond to $\delta B_n=2.3\mathrm{mT}$ and a corresponding $T_2^{*}=10\mathrm{ns}$. From 163.

Figure 13

(a) Electron-spin dephasing time $T_2^{*}=T_{\Delta }$ in colloidal $n$-type ZnO nanocrystals as a function ${}^{67}\mathrm{Zn}$: (circle) 4.1% natural abundance, (diamond) 6.8%, and (square) 9.6% ${}^{67}\mathrm{Zn}$ (134). (b) Spin-echo decay curve measured by electron paramagnetic resonance spectroscopy of ZnO nanocrystals ($d\sim 4.0\mathrm{nm}$, ${}^{67}\mathrm{Zn}=4.1\%$) at 5 K. From 194.

Figure 14

Comparison of experimental (dotted line) and theoretical curves of time-resolved photoluminescence circular polarization of $X^{+}$ for a transverse magnetic field $B_x=750\mathrm{mT}$. The damping reflects the electron-spin dephasing in the ensemble of InAs dots. The theoretical curves (gray line) are given with $\Delta g/g=0.07$ or without $\Delta g/g=0$ electron $g$-factor fluctuations. From 135.

Figure 15

(a) Schematics of the mode locking of electron spins in an ensemble of spins precessing around a transverse magnetic field at different frequencies due to the $g$-factor fluctuations from dot to dot (see text). (b) Faraday rotation amplitude at negative delay in an ensemble of $n$-doped InAs QDs as a function of the time interval between subsequent pump pulses measured at $B_x=6\mathrm{T}$ and $T=6\mathrm{K}$. From 85 and 53.

Figure 16

Electron-spin pumping in a single dot. (Left) Four-level system describing the singly charged QD. The fluctuations of the hyperfine field lead to a slowly varying coherent coupling of the spin ground states with a rate ${\gamma }_{hf}$ and the heavy-hole–light-hole mixing yields a diagonal transition with a rate $\gamma$. (Right) Absorption of the single dot as a function of the magnetic field $B_z$. The absorption decreases with increasing $B_z$ due to optical spin pumping which becomes more efficient since the efficiency of the hyperfine field coupling of the ground state decreases. The inset shows the corresponding raw laser scans from 0 T (top) to 300 mT (bottom). From 9.

Figure 17

Hole-spin pumping in a single dot. (a) A laser with ${\sigma }^{+}$ polarization drives the $|\Downarrow ⟩\leftrightarrow |\Uparrow \Downarrow ,\downarrow ⟩$ transition. No transmission dip is observed demonstrating that the hole population is shelved into state $|\Uparrow ⟩$. (b) Simultaneous excitation with both ${\sigma }^{+}$ and ${\sigma }^{-}$ at the same frequency. A large transmission dip is observed, as the additional ${\sigma }_{-}$ excitation leads to a repumping of the electron spin and thereby suppresses electron-spin pumping. From 76.

Figure 18

(a) PL spectra of a single $X^{+}$ line in zero magnetic field measured under copolarized (${\sigma }^{+}/{\sigma }^{+}$) and cross-polarized (${\sigma }^{+}/{\sigma }^{-}$) excitation and detection configuration. (b) Evolution of the corresponding circular polarization as a function of the total PL intensity for an incident laser excitation varied from 150 nW to $250\mu \mathrm{W}$.

Figure 19

(a), (b) Energy levels and relaxation rates according to Eq. (21) for a spin $I=9/2$ and $\theta =0.05$. The anticrossings are marked by circles of different colors in (a), and the corresponding relaxation rates are shown in (b) in the same color.

Figure 20

(a) Density plots of $X^{+}$ PL intensity from a single QD at $T=1.7\mathrm{K}$ as a function of an increasing or decreasing magnetic field $B_z$, detected over a 1 meV energy range around $E_0=1.3143\mathrm{eV}$. The quasiresonant ${\sigma }^{-}$ polarized excitation is $\sim 60\mathrm{meV}$ above the trion line in zero field. (b) Absolute Overhauser shift of the Zeeman splitting for two excitation powers, and (c) circular polarization as a function of $B_z$. The measurement at $0.025P_{\mathrm{sat}}$ has been shifted and slightly reduced in size for clarity. Adapted from 116.

Figure 21

(a) Density plots of $X^{-}$ PL intensity from the same QD of Fig. 20 at $T=1.7\mathrm{K}$ as a function of an increasing or decreasing magnetic field $B_z$, detected around $E_0=1.305\mathrm{eV}$. The quasiresonant ${\sigma }^{+}$ or $\pi$ polarized excitation is $\sim 40\mathrm{meV}$ above the trion line in zero field. (b) Overhauser shifts of the Zeeman splitting for both excitation polarizations with superimposed fits of the model using the same parameters (solid lines). The measurements under $\pi$ polarization, performed in positive fields, have been duplicated antisymmetrically in negative fields. (c) Circular polarization under ${\sigma }^{+}$ excitation as a function of $B_z$ showing no correlation to the Overhauser shift in contrast to $X^{+}$. The inset illustrates schematically $X^{-}$ emission, followed by the capture of a second resident electron defining the correlation time ${\tau }_c^e\sim 20\mathrm{ps}$ in the model. Adapted from 116.

Figure 22

Overhauser shift $\propto ⟨{\widehat{I}}_z⟩$ in a single InAs/GaAs QD as a function of the measured circular polarization from the $X^{-}$ PL line (essentially $\propto ⟨S_z^e⟩$ of the electron left behind) around $E_0=1.265\mathrm{eV}$ at different longitudinal magnetic fields. The polarization of the quasiresonant excitation is varied by stepwise rotating a quarter-wave plate over a total angle of $\pi$, which achieves the complete polarization cycle $\pi \to {\sigma }^{+}\to \pi \to {\sigma }^{-}\to \pi$. Solid lines represent fits of Eq. (20) with ${\tau }_c^e=31\pm 2\mathrm{ps}$, $g_e^{\star }=0.54$, and the product $f_e{T}_d$ adjusted for each field as indicated. The curves and experimental points at different magnetic fields have been translated for clarity.

Figure 23

Overhauser shift in a single InAs/GaAs QD as a function of the laser or PL circular polarization for an $X^{+}$ PL line around $E_0=1.348\mathrm{eV}$ at two longitudinal magnetic fields. The polarization of the quasiresonant excitation is changed by stepwise rotating a quarter-wave plate as in Fig. 22. Solid lines represent fits of Eq. (20) with ${\tau }_c^e=31\pm 2\mathrm{ps}$, $|g_e|=0.5$, $|g_e^{\star }|=0.45$, and the product $f_e{T}_d$ as indicated. The curves and experimental points for both magnetic fields have been translated for clarity.

Figure 24

(a) Spin splitting in a single InAs/GaAs QD for an $X^{+}$ PL line as a function of temperature. The squares (circles) are measured following ${\sigma }^{+}$ (${\sigma }^{-}$) polarized laser excitation and the triangles following linear laser excitation, for which DNP is absent. (b) Schematics of the electron-nuclei level splitting and the change of broadening due to temperature raising. (c) Evolution of the Overhauser shift as a function of $⟨S_z^e⟩$ deduced from the PL circular polarization ($⟨S_z^e⟩=-{\rho }_c/2$) for different temperatures. Solid lines are a fit of the model to the experimental data with the corresponding correlation time ${\tau }_c^e$ and relaxation time $T_d$ as indicated. From 186.

Figure 25

(a) Overhauser shift and (b) circular polarization of a single $X^{-}$ PL line from an InAs/GaAs QD as a function of a small longitudinal magnetic field for ${\sigma }^{+}$ and ${\sigma }^{-}$ circularly polarized excitation. From 122.

Figure 26

Density plot of PL spectra of three different QDs ( labeled $A$, $B$, and $C$) around $E_0=1.36\mathrm{eV}$ as a function of a longitudinal magnetic field under ${\sigma }^{-}$ excitation at about 1LO phonon energy (37 meV) above $E_0$. The specific critical fields at which the nuclear polarization collapses vary from dot to dot and allow one to recognize different excitonic lines originating from the same QD.

Figure 27

(a) PL spectra of $X^0$ and $X^{+}$ from the same InAs/GaAs QD in zero external magnetic field under the same experimental conditions, yet measured in $({\pi }^x,{\pi }^y)$ and $({\sigma }^{+},{\sigma }^{-})$ basis, respectively. The same Overhauser shift ${\delta }_n\equiv \hslash {\omega }_{\mathrm{OS}}^e$ generated under ${\sigma }^{+}$ circular polarization is evidenced for both $X^0$ and $X^{+}$. (b) Evolution of circular (top) and linear (bottom) polarization of $X^0$ PL as a function of the measured Overhauser shift which increases in absolute value when the power of the ${\sigma }^{\pm }$ excitation increases. Solid lines follow the simple theoretical analysis given by 17.

Figure 28

(a) Top: PL at 4.2 K from a single QD vs applied bias driven with excitation at $X^0$ energy. $X^{-}$ PL appears in a narrow range of voltage. Bottom: Energy dependence vs bias for the QD vacuum state $|0⟩$ and the single-electron state $|e⟩$, showing a crossing where the ground state changes. $X^0$ and $X^{-}$ cross at lower bias on account of the hole (Coulomb interaction). For the chosen bias region (hybridization region), automatic cycling takes place when a laser is tuned to the $|0⟩\leftrightarrow |X^0⟩$ transition. (b) Left: Experimentally measured signal intensity, polarization degree, and Overhauser shift vs laser energy (detuning) for a ${\sigma }^{+}$ pump and an external field of $+0.5\mathrm{T}$ at fixed bias in the center of the hybridization region. Right: Comparison with theory from 109.

Figure 29

(a) Overhauser field $B_N$ in the dot for $B_z=2.5\mathrm{T}$ as a function of laser energy. (b) PL transition energy as a function of laser energy. (c) Energy level diagram of a positively charged dot in a magnetic field $B_z$. Long thick arrows show “allowed,” and thin arrows show “forbidden” optical transition. From 36.

Figure 30

Comparison of the dragging spectra of the blueshifted Zeeman branch for the $X^{-}$ trion as a function of laser detuning for a sample with 25 nm thick tunnel barrier (a) and the prediction of the rate equation (28) (b). For the following parameters: $N=3.2\times {10}^4$, where $N$ is the number of nuclear spins, ${\tilde{A}}^i=120\mu \mathrm{eV}/N$, $B_z=4.5\mathrm{T}$, $A_{\mathrm{nc}}^i=0.012{\tilde{A}}^i$, $\hslash \Gamma =0.58\mu \mathrm{eV}$, ${\Omega }_{\mathrm{L}}=0.7\Gamma$, step size $\Delta \delta =0.23\mu \mathrm{eV}$, and dwell time $t_c=0.2\mathrm{s}$. From 92.

Figure 31

Differential transmission as a function of gate voltage showing a dragging plateau for a charge-tunable sample with a tunnel barrier thickness of (a) 25 nm at $B_z=4.5\mathrm{T}$ and (b) as (a), but measured in resonance fluorescence for tunnel barrier thickness 35 nm at the $B_z=4.0\mathrm{T}$ sample.

Figure 32

Energy level diagram depicting the nuclear spin-flip assisted transitions for the $X^{-}$ trion higher energy Zeeman branch in a finite magnetic field applied along the growth direction $z$: a resonant laser field couples dipole allowed and dipole forbidden transitions (straight and diagonal arrows, respectively) of the trion-nuclear spin manifold. The lower states of the manifold are electron spin-up states $|\uparrow ⟩$ split by the sum of nuclear Zeeman energy and the Overhauser field, ${\delta }_{\mathrm{GS}}={\omega }_{\mathrm{Z}}^{\mathrm{n}}-\tilde{A}/N$, according to their nuclear spin projection $I_z$ along $z$. The upper states are $X^{-}$ trion states (an individual hole and two electrons in a spin-singlet state) split by the nuclear Zeeman energy through ${\delta }_{\mathrm{ES}}={\omega }_{\mathrm{Z}}^{\mathrm{n}}$. Nuclear spin polarization occurs through spin-flip assisted diagonal transitions (diagonal arrows) followed by spin preserving radiative decay (wavy arrows). Finite laser detunings lead to an imbalanced competition between the bidirectional nuclear spin diffusion processes within the manifold (horizontal arrows): the coupled trion-nuclear spin system reaches steady state by locking to the laser resonance. From 92.

Figure 33

Faraday rotation signal from an ensemble of single-electron-charged QDs resonantly excited with a periodic laser pulse train. The data are taken in Voigt geometry at $B_x=6\mathrm{T}$. The top panel shows the Faraday signal when the ensemble is excited with a single pulse train. Remarkably, excitation with a two-pulse train leads to bursts in the Faraday signal at time delays that are integer multiples of the time delay of the two laser pulses, where electron spins recover a high degree of polarization. Even after the second laser pulse is turned off, the system continues to produce bursts, suggestive of memory times exceeding minutes. From 83.

Figure 34

(a) Schematic of the pulse sequences used in the buildup and decay time measurements of DNP in zero external magnetic field. An optical modulator is used as a fast switch for the PL excitation laser, where o and c denote the open and closed states, respectively. This creates pump (probe) pulses of respective lengths ${\tau }_{\mathrm{pump}}$ (${\tau }_{\mathrm{probe}}$), separated by a waiting time ${\tau }_{\mathrm{wait}}$. A mechanical shutter blocks the pump-induced luminescence from reaching the spectrometer, while letting the probe-induced luminescence pass. (b) Buildup of nuclear spin polarization measured by varying ${\tau }_{\mathrm{pump}}$ at fixed ${\tau }_{\mathrm{wait}}$ (0.5 ms) and ${\tau }_{\mathrm{probe}}$ (0.2 ms) for ${\sigma }^{-}$ (${\sigma }^{+}$) excitation. (c) DNP decay curves obtained by varying ${\tau }_{\mathrm{wait}}$ at fixed ${\tau }_{\mathrm{pump}}$ (50 ms) and ${\tau }_{\mathrm{probe}}$ (0.5 ms). Exponential fits yield time constants ${\tau }_{\mathrm{buildup}}=9.4\mathrm{ms}$ and ${\tau }_{\mathrm{decay}}=1.9\mathrm{ms}$. Adapted from 141.

Figure 35

Measurements in zero external magnetic field. (a) Timing diagram for the gate voltage switching experiment: DNP is established at a gate voltage $V_1$, corresponding to the center of the $X^{-1}$ stability plateau. During the period ${\tau }_{\mathrm{wait}}$, the QD gate voltage is switched to a value $V_2$ and the nuclei are left to decay. (b) Measurement of DNP decay, with $V_2$ corresponding to the center of the $X^0$ stability plateau. The gray (black) data points represent DNP decay under ${\sigma }^{+}$ (${\sigma }^{-}$) excitation. For comparison, the light gray curve shows the mean of the data presented in Fig. 34b. (c) The same measurement as in (b), but over a longer time scale. The exponential trace (light gray) indicates a decay time constant of 1 h. Adapted from 140.

Figure 36

(a) Nuclear spin polarization after free evolution of the nuclear spins during ${\tau }_{\mathrm{wait}}=20\mathrm{ms}$ at a gate voltage $V_2$ for (a) $B_{\mathrm{ext}}=0$ and (b) $B_{\mathrm{ext}}=1.5\mathrm{T}$. For odd numbers of QD electrons, DNP decays in a ms time scale, while the decay takes seconds for an even number of electrons [gray (black) data points correspond to QD excitation with ${\sigma }^{+}$ (${\sigma }^{-}$) polarized light. Adapted from 140.

Figure 37

(a) Buildup of DNP in external magnetic fields on the order of the final Overhauser field. Experiments were performed with the procedure and external parameters described in the main text at magnetic fields $B_1=1.1\mathrm{T}$, $B_2=1.2\mathrm{T}$, $B_3=1.3\mathrm{T}$, and $B_4=1.4\mathrm{T}$. (b) Simulations according to the classical nonlinear rate equation (19). The magnetic fields used for the simulation are $B_1=1.22\mathrm{T}$, $B_2=1.24\mathrm{T}$, $B_3=1.26\mathrm{T}$, and $B_4=1.28\mathrm{T}$. Adapted from 140.

Figure 38

Decay of DNP in an external magnetic field of $B_{\mathrm{ext}}=1\mathrm{T}$. The nuclear spin polarization was initialized with a 100 ms, ${\sigma }^{+}$-polarized pump pulse at a gate voltage $V_1$ corresponding to the center of the $1e^{-}$ plateau, resulting in an initial Overhauser shift ${\omega }_{\mathrm{OS}}^e({\tau }_{\mathrm{wait}}=0)\approx 55\mu \mathrm{eV}$. Immediately after this nuclear spin initialization, the gate voltage was switched to a value $V_2$. (Left) Measurement of ${\omega }_{\mathrm{OS}}^e$ in $\mu \mathrm{eV}$ as a function of waiting time ${\tau }_{\mathrm{wait}}$ and gate voltage $V_2$ at $T=1.7\mathrm{K}$ (upper panel) and $T=5\mathrm{K}$ (lower panel). The dashed, gray lines indicate the transition between the QD charging states. (Right) Simulations according to Eq. (18). Adapted from 140.

Figure 39

(a) Decay of the Overhauser field with negligible cotunneling for the case of a resident electron ($V_{\mathrm{wait}}=530\mathrm{mV}$) at 200 mK (squares) and 4 K (circles). The inset shows the same data in a linear-linear plot. (b) Demonstration of the spatially limited nuclear spin diffusion: By sequential polarization of the nuclear spins, the polarization can be saturated, suppressing further nuclear spin diffusion. From 125.

Figure 40

(a) Schematics of optical transitions and polarization rules for an $X^{+}$ trion in a transverse magnetic field $B_x$. (b) Typical density plot of $X^{+}$ PL intensity from a single QD against magnetic field $B_x$ and detection energy around $E_0=1.355\mathrm{eV}$. The excitation is linearly polarized and the detection either ${\pi }_x$ or ${\pi }_y$ as indicated. (c) Depolarization curves (Hanle effect) of $X^{+}$ lines from three different QDs under ${\sigma }^{+}$ quasiresonant excitation. QD1 measurements are vertically shifted for clarity. From 117.

Figure 41

(a) Comparison of $X^{+}$ PL density plots measured under ${\sigma }^{+}/{\sigma }^{+}$ or ${\sigma }^{+}/{\pi }_x$ configurations of excitation or detection polarizations. The latter enables one to resolve the transition splittings for fields above $\sim 0.5\mathrm{T}$ and thus to infer the nuclear field. (b) Sketch of the nuclear polarization generated almost parallel to $B_x$ giving rise to a nuclear field $B_{n,x}\approx -B_x$ responsible for the anomalous Hanle effect. (c) Splitting of $X^{+}$ ${\pi }_x$- and ${\pi }_y$-polarized transitions evidencing the cancellation of the external field for $B_x<B_x^c$. Normal splittings are recovered above $B_x^c$ when the nuclear polarization is eventually destroyed. Adapted from 117.

Figure 42

Optically detected nuclear magnetic resonance in semiconductor QDs. (a) NMR spectra of a single interfacial, GaAs QD: The Zeeman splitting of the QD in a longitudinal magnetic field of 1 T changes as a function of the frequency of an applied transverse rf magnetic field when the frequency matches the nuclear Larmor precession frequency. Adapted from 28. Frequency offset for ${}^{75}\mathrm{As}:7.274\mathrm{MH}z$, and for ${}^{69}\mathrm{Ga}:10.193\mathrm{MH}z$. (b) Hanle curves measured on an ensemble of self-assembled InAs QDs for ${\sigma }^{+}$-polarized excitation in the absence and presence of an rf field applied along the $z$ axis. The radio frequency is scanned over all the In and As nuclear magnetic resonances for the relevant magnetic field range. For comparison, the Hanle curve detected for excitation with polarization modulation at $f_{\mathrm{mod}}=100\mathrm{kH}z$ is shown. (c) Effect of fixed radio-frequency irradiation (with rf field along the $y$ axis), which reveals distinct nuclear resonances for the low-field range indicated by the dashed rectangle (curves are shifted vertically for clarity). (d) Magnetic field dependences of resonant frequencies extracted from the inset in (a). The dependences of NMR for $m_I=\pm 1/2$ states in ${}^{71}\mathrm{Ga}$ and ${}^{75}\mathrm{As}$ nuclei calculated with and without quadrupole interaction are shown by the solid and dashed lines, respectively. Adapted from 69.

Figure 43

(a) Schematic diagram of the experimental procedure for adiabatic demagnetization of QD nuclear spins in a charge-tunable InGaAs/GaAs QD. The nuclei are optically pumped at $B_{\mathrm{ext}}=B_{\mathrm{i}}$ via quasiresonant excitation of the $X^{-}$ trion while the QD is in charge state $q_{\mathrm{QD}}=1e$. Immediately after the pumping pulse, the electron is ejected from the QD. $B_{\mathrm{ext}}$ is then linearly ramped at a rate ${\gamma }_{\mathrm{B}}$ to a final value $B_{\mathrm{f}}$, at which the nuclear polarization $P_{\mathrm{nuc}}$ is measured with a short linearly polarized optical pulse. (b) Demagnetization as a function of the final magnetic field $B_{\mathrm{f}}$, for a monotonic decrease from $B_{\mathrm{i}}=2\mathrm{T}$ (black) and for a return from an intermediate field $B_{\mathrm{f}}^{\prime }=-1\mathrm{T}$ (gray). (c) Remanent polarization at zero field normalized to the polarization generated at 2 T, when $B_{\mathrm{i}}$ is varied. (d) Nuclear polarization following a complete round-trip as indicated. Adapted from 142.

Figure 44

(a) Photo-induced circular dichroism (PCD) signal as a function of pump-probe delay, when the $p$-doped QDs are excited by a periodic train of ultrashort pulses. (b) Normalized PCD amplitude at negative pump-probe delay $t=-130\mathrm{ps}$ (i.e., reflecting the hole spin polarization) vs the applied longitudinal magnetic field $B_z$, performed on a QD sample with one hole per dot. The HWHM equals 2.5 mT. The solid line is a fit using the model developed by 61. (c) Solid curve: normalized PCD amplitude at negative pump-probe delay $t=-130\mathrm{ps}$ (i.e., reflecting the electron spin polarization) vs the applied longitudinal magnetic field, performed on QDs charged with one electron. The HWHM equals 47 mT. Similar measurement on QDs charged with one hole (dashed curve) is added on the same figure to directly compare the efficiency of hyperfine-induced dephasing for electrons and holes. From 47.

Figure 45

Measurement of the electron- and hole-nuclear spin interaction in a neutral InP dot at $B_z=6\mathrm{T}$. The angle of a $\lambda /2$ plate is varied to change the polarization of the pump laser resulting in a change of nuclear spin polarization. From 34.