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Abstract: We propose the concept of quantum timing jitter in single-photon detection, based on the observation of a finite rise time in the quantum probabilisticΒ ...
CLEO 2018 Β© OSA 2018

Concept of quantum timing jitter and non-Markovian limits in single photon detection Li-Ping Yang1, Hong X. Tang2, and Zubin Jacob1* 1

Birck Nanotechnology Center and Purdue Quantum Center, Department of Electric and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA 2 Department of Electrical Engineering, Yale University, New Haven, Connecticut 06511, USA *[email protected]

Abstract: We propose the concept of quantum timing jitter in single-photon detection, based on the observation of a finite rise time in the quantum probabilistic signal transduction (absorption). We place a fundamental limit on this quantum jitter, which is governed by the bandwidth of the fielddetector interaction spectrum. We also shed light on the fundamental differences between linear and h o r ( s ) nonlinear detector outputs for single photon Fock state vs. coherent state pulses. Β©2018TheAut OCIS codes: (040.5570) Quantum detectors, (270.5585) Quantum information and processing Recently, single-photon detectors with high efficiency (> 90%), low dark count rate (< 1 mHz), and reduced timing jitter (< 20 ps) have raised substantial interest due to their widespread applications in quantum information processing, imaging, sensing/ranging and astronomy [1]. Current single-photon detectors typically work by outputting a classical electrical signal that results from the amplification of a weak quantum signal generated within the detector after the transduction event. The previous photo-detection theories, like Mandel- Glauber-Kelley- Kleiner [2] theories are limited by first-order perturbation theory while Mollow-Scully-Lamb [3] models are based on quasi-single mode assumption, lose their validity in ultra-short single pulse detection. These theories cannot place any limit on the practical engineering metrics of single-photon detectors. While the classical response of the amplifier stage currently dominates the characteristics of the single photon detector, it is important to note that the initial transduction process from photons to detector modes is fundamentally probabilistic in nature. As absorption mode volumes are decreased and amplifiers are improved, it is plausible that quantum limits of single photon transduction to detector modes will eventually manifest itself. Our major contribution in this paper is to propose the concept of the quantum timing jitter and to place a fundamental limit on it. Currently timing jitter of a single photon detector is defined using the rising edge of the classical deterministic output voltage/current. However, rising edges can also be defined for quantum signals during the initial single photon transduction event in a statistical sense, which will set fundamental limits to the timing jitter performance. We explore the fundamental limits in photo-detection starting from an exactly solvable model---a twolevel atom interacting with a multi-mode single-photon. In previous Markovian theories, the rise time of the quantum probabilistic output signal 𝑃 𝑑 = 𝐢 𝑑 % [𝐢 𝑑 the excitation probability amplitude of the atom] can be infinitely fast implying there exists no quantum timing jitter in principle. In our non-Markovian theory, 7

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CLEO 2018 Β© OSA 2018

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Figure 1. (a) Schematic of a two-level atom, which functions as a narrow-band photo-detector, in a cavity excited by single-photon Fock-state pulses with different envelopes. Here, 𝜏9 is the length of the propagating pulse with center frequency πœ”: , Ξ³ the Markov spontaneous decay rate of the atom with energy splitting πœ”< = πœ”: , and ΞΊ the loss rate of the cavity. (b) In contrast to the sharp time-stamping of a counting event in an ideal photo-detector, the finite rise/fall time of the quantum output signal leads to an intrinsic jitter of a realistic detector. Our non-Markovian theory places limits on this rise time which can be infinitely fast in Markovian approaches [4]. In (c) and (d), we contrast between a classical deterministic signal sequence and a quantum probabilistic output sequence. For 0.06the classical deterministic 0.06 =1/100 =1/10 0.3 f=1/100 f =1/10 f 0.06𝑇0.3 f the same temporal profile, the signal=1/10 (c) with perfect period and clicks happen at the same position each time when =1/100 0.06 0.3 f f =1/100 0.3 f=1/10 0.04 f 0.04 the classical output pulses cross the threshold. Thus, the click events are deterministic and no timing jitter exists. But 0.2 0.04 0.2 0.04 0.2 for quantum signal sequence (d), each pulse describes the excitation probability. The atom could be excited at any 0.2 0.02 0.02 0.1 unphysical 0.1 unphysical Non-Markov Non-Markov Non-Markov Non-Markov (c) 0.02 results (a) (c) (a) time pulse, which in the quantum jitter. 0.1 during each unphysical rise time rise time 0.02 Non-Markov Non-Markov (c) 0.1 unphysical (a)

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t t t t Figure 3. The output waveforms 𝑦(𝑑) of linear (a) and nonlinear (b) photodetectors are displayed. The solid green line and dotted red line denote the single-photon Fock-state pulse and single-photon coherent-state pulse, respectively. The output waveform 𝑦(𝑑) of the linear photodetector for Fock-state puse and coherent pulse are the same. However, for the nonlinear photodetector, the maximum of 𝑦(𝑑) for the coherent-state pulse is much smaller than that of the Fock state pulse. [1] R. H. Hadfield, Nat. Photonics 3, 696 (2009); M. Eisaman, J. Fan, A. Migdall, and S. Polyakov, Rev. Sci. Instrum. 82, 071101 (2011). [2] L. Mandel, Proc. Phys. Soc. 72, 1037 (1958); R. J. Glauber, Quantum theory of optical coherence: selected papers and lectures (John Wiley & Sons, 2007) Chap. 2.4; P. Kelley and W. Kleiner, Phys. Rev 136, A316 (1964). 
.
 [3] B. Mollow, Phys. Rev. 168, 1896 (1968); M. O. Scully and W. E. Lamb Jr., Phys. Rev. 179, 
368 (1969). 
 [4] Y. Wang, J. Minar, L. Sheridan, and V. Scarani, Phys. Rev. A 83, 063842 (2011); B. Q. Baragiola, R. L. Cook, A. M. Branczyk, and 
J. Combes, Phys. Rev. A 86, 013811 (2012).