Direct measurement of an one-million-dimensional photonic state

Abstract

The state vector of a pure quantum system is a set of complex probability amplitudes used for describing the system in each state of a given Hilbert space. Characterizing the state of a quantum system is crucial for fundamental studies in quantum mechanics as well as for manipulating and utilizing quantum systems for practical applications. Here we describe a scan-free direct measurement approach [1] that is capable of simultaneously measuring the entire state vector of a pure quantum system, consequently eliminating the need for scanning through each basis state. Our method involves a proper arrangement of weak [2, 3] and strong measurements. Specifically, to measure the state vector of a quantum system in one Hilbert space A, one first applies a weak measurement to the quantum system in one fixed state |b_0〉. of its complementary basis B, and then performs the strong measurement directly in A. As an example, we have designed a procedure of measuring the transverse spatial state of a photon, which is a convenient high-dimensional quantum system for study, and which has a well- understood classical analogue as the transverse complex field profile of an optical beam. In order to measure the complex probability amplitude of a photon at a position state x, one first performs a weak projection measurement of the zero momentum state p_0 = 0, followed by a strong measurement of the position state. Through such a procedure, the measured weak values are inversely proportional to the state vector of the quantum system. To demonstrate our method, we characterize photons carrying different spatial modes, such as uniform amplitude and Zernike phase profiles, circular mode carrying orbital angular momentum (OAM) [4, 5], etc. Since one can have independent control of the complex probability amplitude of the photons at each pixel of the transverse space, the dimensionality of our measured state is approximately 1.2 million, which is determined by the spatial extent of the photons (approximately 7mm in diameter) and the discrete nature of our detector array (with pixel size of 5.4 µm^2). The fidelity of the photon state in the spatial Hilbert space is calculated to be approximately 0.93. Such a high fidelity of our result demonstrates that our direct measurement technique is indeed capable of measuring the complex-value quantum state vector with very high accuracy. Our scan-free direct measurement approach opens up the possibility to characterize high dimensional quantum systems in real time for which a state-by-state scanning process would become impractically time-consuming or even infeasible. Moreover, our specific demonstration of measuring photons' transverse spatial state is also a promising new technology for classical wavefront sensing applications in fields as diverse as observational astronomy, free-space optical communication, and biomedical imaging.

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