$\delta$-Quench Measurement of a Pure Quantum-State Wave Function

The measurement of a quantum state wave function not only acts as a fundamental part in quantum physics but also plays an important role in developing practical quantum technologies. Conventional quantum state tomography has been widely used to estimate quantum wave functions, which usually requires complicated measurement techniques. The recent weak-value-based quantum measurement circumvents this resource issue but relies on an extra pointer space. Here, we theoretically propose and then experimentally demonstrate a direct and efficient measurement strategy based on a $\delta$-quench probe: by quenching its complex probability amplitude one by one ($\delta$ quench) in the given basis, we can directly obtain the quantum wave function of a pure ensemble by projecting the quenched state onto a postselection state. We confirm its power by experimentally measuring photonic complex temporal wave functions. This new method is versatile and can find applications in quantum information science and engineering.

Figure 1

Schematic comparison of quantum measurement strategies. To determine the wave function in the basis $\{|a⟩\}$, conventional quantum state tomography (QST) needs a collection of projection measurement in $\{|a⟩\}$ and its mutually unbiased bases (MUB) $\{..,\{|b_v⟩\},\dots \}$ followed by a multiparameters fitting process. Weak-value-based quantum measurement couples (${\widehat{U}}_{{\widehat{\pi }}_a,\widehat{p}}$ with ${\widehat{\pi }}_a=|a⟩⟨a|$) the quantum state with a pointer state ($|0{⟩}_{\widehat{p}}$) then directly reads $\psi (a)$ from the pointer state measurement conditioned on a postselection projection. Our proposed $\delta$-quench measurement ($\delta$ QM) directly reads $\psi (a)$ sequentially by first quenching the quantum state with the operator $\widehat{M}(a,\theta )$ and then analyzing the response in projection measurement of the quenched state onto one fixed postselection state($|b_0⟩$) belonging to a MUB of basis $\{|a⟩\}$.

Figure 2

Experimental setup for the $\delta$-quench measurement of the temporal wave function. (a) Signal photons are produced by attenuating a laser beam with central wavelength $\lambda =780\mathrm{nm}$ (carrier frequency ${\omega }_0=2\pi c/\lambda$ with $c$ as the light speed) and linewidth of around 50 kHz. The average power of the photon stream is about $10\mu \mathrm{W}$. In the state preparation stage, we use an acousto-optic modulator (Brimrose, TEF-110-30-780, not shown in the figure) for preparing the amplitude of complex envelope $|\psi (t)|$, and a fiber electro-optical phase modulator (modulation bandwidth 20 GHz, EO-Space, PM-0K5-20-PFU-PFU-795-UL, not shown in the figure) for the phase envelope $\phi (t)$. A second EOPM of the same model is used to realize the $\delta$-like phase quench ($e^{i\theta }$). The postselection projection measurement is conducted with a high finesse optical cavity (finesse $F=20000$, linewidth of 72 kHz, customized by Stable Laser System), whose resonance frequency is tuned to be ${\omega }_0$. The transmitted photons are eventually detected by an avalanche photon detector (Thorlabs, APD120A/M) and then recorded by an oscilloscope (Tektronics, DPO4104B). (b) The temporal length of the photonic quantum wave function is $2\mu \mathrm{s}$ and the measurement time-bin width is $0.1\mu \mathrm{s}$. EOPM: electro-optical phase modulator, APD: avalanche photon detector.

Figure 3

The measured response factors and temporal wave functions. (a1)–(d1) Experimentally measured response factor $p_{1,2}$ for the prepared temporal wave function with different envelopes. (a2)–(d2) Directly calculated real and imaginary components of temporal wave function $\mathrm{Re}[\psi (t)]$ (pink triangles) and $\mathrm{Im}[\psi (t)]$ (green diamonds). (a3)–(d3) Normalized intensity envelopes of temporal wave function $|\psi (t){|}^2$. (a4)–(d4) Phase envelopes obtained by equation $\phi (t)=\arctan (\mathrm{Im}[\psi (t)]/\mathrm{Re}[\psi (t)])$. Dashed lines in (a1)–(d1) and (a2)–(d2) are just guide lines to connect neighboring data points. Solid lines in (a3)–(d3) are the normalized intensity profiles of quantum state measured before being quenched. Solid lines in (a4)–(d4) are the theoretical phase envelopes derived from the preparation equipments. Only the measured phases in the nonzero amplitude region are presented in (a4)–(d4) considering that the phase is not well defined when the amplitude is zero. Measurement experiments are repeated 8 times and the error bars denote the statistical variance of 1 s.d.

Figure 4

The performance of the $\delta$-quench measurements. (a)–(c) Fidelities dependence on quench depth of the amplitude, phase, and the overall wave function, respectively. (d) The magnitude of the response factor as a function of the quench instant $t_0$ and quench depth $\theta$. The quench operation is set to ${\theta }_1=-{\theta }_2=\theta$ and quantum wave function with the rectangle amplitude and the triangle relative phase shown in Fig. 3 is measured. The theoretical curves in (a)–(c) are the numerically calculated results. The open squares are the mean fidelities of 8 repeatedly measured raw waveforms with the same $\theta$; the attached error bars denote the 1 s.d. statistical variance of repeated experiments. Solid diamonds without an error bar are the fidelities of waveforms obtained by accumulating all the raw photon counts data (see Supplemental Material [34]).