Is Wave Function Collapse Necessary? Explaining Quantum Nondemolition Measurement of a Spin Qubit within Linear Evolution

The measurement problem dates back to the dawn of quantum mechanics. Here, we measure a quantum dot electron spin qubit through off-resonant coupling with a highly redundant ancilla, consisting of thousands of nuclear spins. Large redundancy allows for single-shot measurement with high fidelity $\approx 99.85\%$. Repeated measurements enable heralded initialization of the qubit and backaction-free detection of electron spin quantum jumps, attributed to burstlike fluctuations in a thermally populated phonon bath. Based on these results we argue that the measurement, linking quantum states to classical observables, can be made without any “wave function collapse” in agreement with the Quantum Darwinism concept.

Figure 1

(a) Energy level diagram of the nuclear spins (short green arrows), the electron (solid blue circle and arrow), and the optically excited trion, containing one hole (open red circle and arrow) and two electrons. (b) Conduction band energy diagram of the semiconductor structure, showing the GaAs quantum dot, AlGaAs barriers, and the doped AlGaAs layers. (c) The solid (dashed) curve shows schematically the nuclear magnetic resonance spectrum in the presence of a spin-up (spin-down) electron. Vertical arrows show the bare nuclear frequency ${\nu }_{\mathrm{N}}$ and the frequency ${\nu }_{\mathrm{N}}-a/(2h)$ of the detuned radiofrequency (rf) pulse. (d) The electron spin qubit projection is first copied into multiple nuclear spin ancillae by the rf pulse. The total nuclear polarization is then measured from the hyperfine shifts $E_{\mathrm{hf}}$ in the time-averaged PL spectra. (e) Timing diagram showing optical pump and probe, rf pulses, and the switching of the QD between the neutral ($0e$) and electron-charged ($1e$) states.

Figure 2

(a) A random set of 16 probe PL spectra following the detuned rf $\pi$ pulse applied to a charged ($1e$) QD. The changes in the doublet splitting $\mathrm{\Delta }{E}_{\mathrm{PL}}$ are due to the changes $\mathrm{\Delta }{E}_{\mathrm{hf}}$ in the hyperfine shift $E_{\mathrm{hf}}$. The bimodality in $\mathrm{\Delta }{E}_{\mathrm{PL}}$ (and $\mathrm{\Delta }{E}_{\mathrm{hf}}$) corresponds to the two $s_{\mathrm{z}}$ states of the electron. (b) Histogram of the single-cycle NMR signals $\mathrm{\Delta }{E}_{\mathrm{hf}}$ measured at variable detunings from the ${}^{69}\mathrm{Ga}$ bare NMR frequency ${\nu }_{\mathrm{N}}$ (${\nu }_{\mathrm{N}}\approx 72.15\mathrm{MHz}$ at $B_{\mathrm{z}}\approx 7\mathrm{T}$). The solid (dashed) line traces the branch of the NMR resonance corresponding to the $s_{\mathrm{z}}=+1/2$ ($-1/2$) electron spin state. The dotted line shows the same single-QD resonance but measured in a neutral charge state ($0e$) via “inverse” NMR [58].

Figure 3

(a),(b) Histograms of the single-cycle NMR signals $\mathrm{\Delta }{E}_{\mathrm{hf}}$ measured at $B_{\mathrm{z}}=1.6\mathrm{T}$ on the same individual QD in a neutral charge state ($0e$, a), and in a single-electron charged state ($1e$, b). The NMR signals are produced by a single detuned rf $\pi$ pulse. Solid lines show the best model fitting. (c) Same as (b) but on a different QD and at $B_{\mathrm{z}}=5.3\mathrm{T}$, where more efficient nuclear spin polarization results in a larger separation of the histogram modes. The dashed line shows a model distribution for a randomly oriented electron spin. (d),(e) Histograms of the single-cycle NMR signals $\mathrm{\Delta }{E}_{\mathrm{hf}}$ measured at $B_{\mathrm{z}}=7\mathrm{T}$ with two rf $\pi$ pulses applied to ${}^{75}\mathrm{As}$ and ${}^{69}\mathrm{Ga}$ and delayed by $T_{\text{Evol}}$. A full 2D histogram at variable $T_{\text{Evol}}$ is shown in (e), while (d) and (f) show the cross sections at long and short $T_{\text{Evol}}$, respectively.