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Estimating the Single-Photon Projection of Low-Intensity Light Sources

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Estimating the Single-Photon Projection of Low-Intensity Light Sources Estimating the single-photon projection of low-intensity light sources Jorge Rolando Chavez-Mackay,1 Peter Grünwald,2 and Blas Manuel Rodríguez-Lara1, 3 1Tecnologico de Monterrey, Escuela de Ingeniería y Ciencias, Ave. Eugenio Garza Sada 2501, Monterrey, N.L. 64849, México. 2Aarhus Universitet, Institut for Fysik og Astronomi, Ny Munkegade 120, 8000 Aarhus C, Denmark. 3Instituto Nacional de Astrofísica, Óptica y Electrónica, Calle Luis Enrique Erro No. 1, Sta. Ma. Tonantzintla, Pue. CP 72840, México. Estimating the quality of a single-photon source is crucial for its use in quantum technologies. The standard test for semiconductor sources is a value of the second-order correlation function of the emitted field below 1=2 at zero time-delay. This criterion alone provides no information regarding the amplitude of the single- photon contribution for general quantum states. Addressing this question requires the knowledge of additional observables. We derive an effective second-order correlation function, strongly connected to the Mandel-Q parameter and given in terms of both the second-order correlation and the average photon number, that provides a lower bound on the single-to-multi-photon projection ratio. Using both observables individually allows for lower and upper bounds for the single-photon projection. Comparing the tightness of our bounds with those in the literature, we find that relative bounds may be better described using the average photon number, while absolute bounds for low excitation states are tighter using the vacuum projection. Our results show that estimating the quality of a single-photon source based on additional information is very much dependent on what aspect of the quantum state of light one is interested in. I. INTRODUCTION arbitrary amplitude. In fact, g(2) does help discriminate multi- photon contributions from single- and zero-photon contribu- Single photons play an essential role in both fundamental tions within the quantum state. It provides relative bounds and applied physics. On the one hand, for example, it has for the ratio of single-to-multi-photon projection. Additional been theoretically argued [1] and experimentally verified [2] information, therein given in the form of the vacuum compo- that nonlocality may be an intrinsic characteristic of a single nent, is required to obtain absolute bounds. electromagnetic field excitation. On the other, due to their These results yield a somewhat surprising picture. The weak interaction with the environment, single photons are standard criterion g(2) < 1=2 is enough to witness the exis- good candidates for quantum key distribution where security tence of a single-photon projection within the quantum state of may be monitored in real time [3]. The full range of applica- light. These quantum states form a subset of sub-Poissonian tions beyond those mentioned here stoke the quest for a per- (i.e. nonclassical) states for which g(2) < 1. Yet, estimating fect single-photon source that emits indistinguishable single the bounds of the single-photon projection requires additional photons on demand at a high repetition rate that can be col- information. In turn, knowledge of this additional parame- lected or coupled to optical channels with high efficiency [4]. ter extends the range of applicable states. For example, for In a more realistic approach, different applications require dif- sources with 50% or more vacuum projection some classical ferent characteristics [5] but all call for a high-quality single- states can be addressed such as low-excitation coherent and photon source. The characterization of such sources is a rel- thermal states [12]. Hence, g(2) as a source of information atively open problem. The current standard is based on the about the single-photon content of a quantum state of light is second-order correlation function, g(2)(τ) [6–8] and assumes not restricted to nonclassical states. that single-photon states produce zero-delay values, τ = 0, Our aim here is to analyze how access to both the zero- that are less than one-half, g(2)(0) < 1=2 [9, 10]. In this no- time delay second-order correlation g(2) and the mean photon tation, g(2)(0) is defined via creation(annihilation) operators number N allows us to obtain knowledge about the single- a^y(^a) of a single-mode field in the steady state as photon projection (SPP) and the single-to-multi-photon pro- arXiv:2002.05937v2 [quant-ph] 9 May 2020 ha^y2a^2i jection ratio (SMPPR). We derive a new set of bounds for (2) these quantities combining the results in the literature [11, 12] g (0) = y 2 : (1) ha^ a^i and show that an effective second-order correlation function (2) (2) g~N emerges, similar to the effective second-order correlation As this manuscript deals only with g (τ = 0), we will omit (2) the argument from now on. function g~0 defined in [12]. This quantity allows us to char- Recently, two independent results shed new light on what acterize the SPP and SMPPR for a range of states beyond that g(2) < 1=2 can be estimated from low values of g(2). Zubizarreta et of the standard criterion ; in particular, classical (2) states. Comparing the two effective correlation functions we al. [11] showed that sub-Poissonian light, g < 1, implies (2) y a limited average photon number N = ha^ a^i and derived a find that the SMPPR has tighter lower bounds using g~N . In (2) (2) hard lower bound of g for any given N. In particular, a contrast, absolute bounds are described better by g~0 at least value g(2) < 1=2 implies a quantum state with N < 2. Grün- for low excitations states. In general, more states can be an- wald [12] showed that g(2) < 1=2 is sufficient to verify a alyzed with our new criteria compared to those in the litera- non-zero single-photon projection. However, it can have an ture [12]. 2 The article is organized as follows. In Sec. II, we will give The right-hand side is monotone with respect to the effective a brief review of major results from the literature [11, 12]. (2) second-order correlation function g~0 . Hence, even without Afterwards, we present our derivation of the bounds for SPP knowledge of the vacuum projection this ratio has a lower (2) and SMPPR based on g and N in Sec. III. In Sec. IV we bound based solely on g(2) < 1=2. Furthermore, when this compare the quality of the different criteria derived here and ratio is expected to be at least some value θ, the effective in [12]. We give some examples for applications in Sec. V. second-order correlation function is bounded from above by Finally, in Sec. VI, we present some conclusions and an out- look. 2(θ + 1) g~(2) ≤ : (6) 0 (θ + 2)2 II. STATE OF THE ART In contrast to these relative bounds, the absolute bounds on the SPP p1 require individual knowledge of the vacuum pro- (2) Let us briefly review the details of two recent papers [11, jection p0 and the second-order correlation function g , 12] relevant to our work. For a given average photon number q N larger than one, there is a nonzero lower bound on the value (2) 2 1 − 2~g0 of the zero-time delay second-order correlation function [11], (1 − p0) ≥ p1 ≥ (1 − p0) : (7) q (2) bNc(2N − bNc − 1) 1 + 1 − 2~g0 g(2) ≥ ; (2) N 2 as higher Fock states must be occupied. We define the floor III. SINGLE-PHOTON PROJECTION ANALYSIS function bxc as the largest integer less than x. This hard boundary means that for any given N there exists a quantum The average photon number N and the second-order corre- state with g(2) equal to the right-hand-side of Eq. (2). Despite lation function g(2) have encoded information on all the Fock- the discontinuity of the floor function, the lower bound is con- state projections p of a quantum state. Together with the tinuous and monotonically increasing. In the limit N ! 1, k completeness relation P1 p = 1, they can be used to ob- the lower bound reaches one, as, for example, coherent states k=0 k tain well-defined lower boundaries on the single-photon pro- with g(2) = 1 are independent of N. It is relevant to our jection p . Akin to [12], we define the contributions to N and discussion that the average photon number is limited by two 1 g(2) from the multi-photon projection j i, which have fixed N < 2 for g(2) < 1=2. 2 lower bounds In comparison, [12] is explicitly concerned with the SPP in a general quantum state and its relation to low g(2). We start y n2 = h 2ja^ a^j 2i ≥ 2; (8) by splitting the quantum state of light into three parts, h ja^y2a^2j i 1 p p p g = 2 2 ≥ ; (9) j i = p j0i + p j1i + qj i; (3) 2 2 0 1 2 n2 2 where the projections onto the vacuum, single- and multi- as j 2i contains no projection on vacuum and the single- photon states are given by p0, p1 and q in that order. All expec- j i = (2) photon Fock state. Both lower bounds are attained for 2 tation values discussed in [12] and this work, fg ; N; pkg, j2i j i 2 . For the general state we can write, with the latter being the Fock-state projection pk = jhkj ij , y are diagonal in the Fock basis. Thus, we can limit the dis- N = h ja^ a^j i = p1 + n2q; (10) cussion to states of the type given by Eq.
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