The statistical properties of photon sequences originating from single-molecule emitters differ from those produced by generic light sources. Let us start with the latter. A typical light source (such as a lightbulb) is composed of numerous independent lightemitting atoms or molecules. A sufficiently sensitive photodetector may be used to record the arrival time of each individual photon. The simplest assumption one could adopt to describe this kind of measurement is that the arrival times for different photons are completely uncorrelated (see, however, the discussion at the end of this section). If the properties of the light source (such as, e.g., its brightness) do not depend on time then we should expect that the probability of detecting a photon during an infinitesimal time interval dt is proportional to dt , with the proportionality constant being independent of time or the arrival times of other photons. Let us call this probability λdt , where λ is the probability of detecting a photon per unit time (which will be referred to as the photon count rate). To estimate λ we could measure the number of photons n detected over a certain period of time t and divide it by t : λ ≈ n/t. (7.1) But, since photon arrival times are random, the exact number of photons found during any finite time interval t will fluctuate around its mean value λt , as illustrated in Fig.7.1. A complete description of this process thus requires the probability wn(t) that a specified number of photons, n, is observed during the time interval t . To obtain this probability, it is helpful to think of the number of detected photons, n, as the “state” of the system at time t . At t = 0 we have n = 0. Each time a new photon is detected, the system undergoes a first-order irreversible “reaction” n → n + 1. The overall process thus can be described by the kinetic scheme: 0 λ→ 1 λ→ 2 λ→ 3 λ→ 4 → · · · . Accordingly, wn(t) is described by a system of differential equations of the form dwn(t)/dt = λwn−1(t) − λwn(t), n > 0 (7.2) and dw0(t)/dt = −λw0(t), (7.3) with the initial conditions wn(0) = 0, n > 0 and w0(0) = 1. The solution to Eq. 7.3 is w0(t) = e−λt . (7.4) Substituting this into the differential equation for w1 (Eq. 7.2) we can find the time dependence of w1(t), then substitute the result into the equation for w2(t) and so on. Suppose we have found wn−1(t) and now wish to solve for wn(t). It is convenient to use the ansatz wn(t) = ρn(t)e−λt .
The statistical properties of photon sequences originating from single-molecule emitters differ from those produced by generic light sources. Let us start with the latter. A typical light source (such as a lightbulb) is composed of numerous independent lightemitting atoms or molecules. A sufficiently sensitive photodetector may be used to record the arrival time of each individual photon. The simplest assumption one could adopt to describe this kind of measurement is that the arrival times for different photons are completely uncorrelated (see, however, the discussion at the end of this section). If the properties of the light source (such as, e.g., its brightness) do not depend on time then we should expect that the probability of detecting a photon during an infinitesimal time interval dt is proportional to dt , with the proportionality constant being independent of time or the arrival times of other photons. Let us call this probability λdt , where λ is the probability of detecting a photon per unit time (which will be referred to as the photon count rate). To estimate λ we could measure the number of photons n detected over a certain period of time t and divide it by t : λ ≈ n/t. (7.1) But, since photon arrival times are random, the exact number of photons found during any finite time interval t will fluctuate around its mean value λt , as illustrated in Fig.7.1. A complete description of this process thus requires the probability wn(t) that a specified number of photons, n, is observed during the time interval t . To obtain this probability, it is helpful to think of the number of detected photons, n, as the “state” of the system at time t . At t = 0 we have n = 0. Each time a new photon is detected, the system undergoes a first-order irreversible “reaction” n → n + 1. The overall process thus can be described by the kinetic scheme: 0 λ→ 1 λ→ 2 λ→ 3 λ→ 4 → · · · . Accordingly, wn(t) is described by a system of differential equations of the form dwn(t)/dt = λwn−1(t) − λwn(t), n > 0 (7.2) and dw0(t)/dt = −λw0(t), (7.3) with the initial conditions wn(0) = 0, n > 0 and w0(0) = 1. The solution to Eq. 7.3 is w0(t) = e−λt . (7.4) Substituting this into the differential equation for w1 (Eq. 7.2) we can find the time dependence of w1(t), then substitute the result into the equation for w2(t) and so on. Suppose we have found wn−1(t) and now wish to solve for wn(t). It is convenient to use the ansatz wn(t) = ρn(t)e−λt .
In optical quantum information processing, two of the most basic elements are the sources of quantum mechanical states of light, and the devices that can detect these states. In this chapter, we narrow this down to photon sources and photodetectors. We will describe first how detectors work, starting from abstract ideal detectors, via a complete description of realistic detectors in terms of POVMS, to a brief overview of current photodetectors. Subsequently, we will define what is a single-photon source, and how we can determine experimentally whether a source produces single photons or something else. Having laid down the ground rules, we will survey some of the most popular ways photons are produced in the laboratory. Finally, we take a look at the production of entangled photon sources and quantum non-demolition measurements of photons. A mathematical model of photodetectors Photodetectors are devices that produce a macroscopic signal when triggered by one or more photons. In the ideal situation, every photon that hits the detector contributes to the macroscopic signal, and there are no ‘ghost’ signals, or so-called dark counts. In this situation we can define two types of detector, namely the ‘photon-number detector’, and ‘detectors without number resolution’. First, the photon number detector is a (largely hypothetical) device that tells us how many photons there are in a given optical mode that is properly localized in space and time. This property is called ‘photon-number resolution’.
The generation of non-classical states of light [1] is an important scientific challenge with many potential applications. For example, squeezed states of light [2] have generated much interest in the past decade, because measurements can be performed with noise levels that are lower than those available with classical light. A particularly novel non-classical source of light is a triggered single-photon source: a source that has the property to emit with a high degree of certainty one (and only one) photon at a user-specified time. Such a deterministic source of single photons can be important for quantum information processing [3], quantum cryptography [4] and certain quantum computation problems [5]. A single quantum emitter such as a molecule or an atom is a good potential source because it can emit only one photon at a time. The emitted light is antibunched [6, 7] and consists of single photons separated by random time intervals, which depend on the excited state lifetime and on the pumping rate. Thus triggered emission of single photons can be obtained with controlled excitation of a single emitter. The first demonstration of a single photons source [8] using a single molecule in a solid is based on a rapid adiabatic passage method [9] to excite the molecule. In the effective spin picture, the Bloch vector follows the driving magnetic field adiabatically if the laser-molecule frequency detuning is swept slowly through the resonance. Since the passage inverts the field, the system goes from the ground to the excited state. From this fluorescent state the molecule can emit only a single photon. The molecular resonance is swept by Stark effect at fixed laser frequency. This is done easily by applying a radio frequency electric field to the molecule. The results showed photon states with statistics that differ radically from those of coherent states. But this method requires cryogenic temperatures (2K) since narrow zero phonon lines are needed for the adiabatic following. At room temperature a source of triggered single photons based on the concept of pulsed optical excitation of a single highly fluorescent molecule has been demonstrated [10]. A short pulse of green laser light pumps the molecule from the ground state to a vibronically excited level of the first electronic excited state. After fast (ps) relaxation to the lowest electronic excited state, the molecule subsequently emits a single photon. A frequency-doubled mode-locked Nd-YAG laser is used to pump highly stable single terrylene molecules contained in a p-terphenyl molecular crystal where they are protected from exposure to diffusing quenchers (such as oxygen). The result shows that the photons are emitted after the laser pulses at times separated by the laser repetition time within the lifetime of terrylene in p-terphenyl. Since the pump pulse width is very short compared to the fluorescence lifetime, the probability of two (or more) photon emission per pulse is always zero. At the maximum laser power, 90% of the laser pulses lead to a single photon emission. The parameters of this source (repetition rate and single photon generation probability) were limited only by the laser system used; nevertheless, this performance already surpasses that of previous work. The high performance combined with the vanishing two photon emission probability and the simplicity of this source suggests that it can be considered for a variety of quantum optical experiments and for secure quantum cryptography [11]. Semiconductor quantum dots (QD) are other potential systems for single photon generation [12]. QDs are often referred to as “artificial atoms” because many of their optical properties, such as ultranarrow transitions, arise from discrete, atomic-like energy levels [13, 14]. To reinforce this analogy a study of the photon antibunching in this system was needed. QDs differ from atoms and molecules since the creation of multiple excitons is possible in QDs. Consequently, if two electron-hole pairs can radiatively recombine, the simultaneous emission of two photons from a QD is then possible. Thus different regimes of photon statistics were expected because the probability of multiple exciton creation depends on excitation intensity. The fluorescence intensity correlation function of a single CdSe quantum dot has been investigated [15] using a coincidence setup (start-stop experiment). A strong photon antibunching has been observed over a large range of intensities (0.1-100 kW/cm2). The lack of coincidence at zero time delay indicates a highly efficient Auger ionization process [16], which suppresses multi-photon emission in these colloidal quantum dots. This result shows that room temperature single photon sources can be achieved with colloidal nanoparticles.
Single molecule fluorescence experiments yield a stream of photocounts, whose statistical properties contain valuable information about processes within the molecule and in its microscopic environment. The photon statistics display features specific of single quantum systems and are therefore best discussed in the frame of a quantum mechanical theory of radiation. The present summary of a lecture given in April 2002 at the Hofgeismar Spring School on single molecules presents the main concepts used in quantum theory of light, together with a few useful references, and discusses some illustrations and applications to single molecule measurements drawn from the recent literature.
Correction: March 31, 2023 We remove the form D = 1-a(p)a(p) in the note as it does not equal the time evolution measure of (1). We wish to only point out that sinc(T/2 |(En-Em)|) in the expression for D has an analogue in the spatial behaviour of a particle in an infinite well where: sinc(T/2 |En-Em\) from D --> a(k) sqrt[3.14/(hbar L)] [n3.14 + kL] / (n3.14) i.e. is proportional to the momentum wavefunction a(k) multiplied by a linear function in k. We point out that even though this analogy exists, the summations involved in D do not map over to the single particle case. Thus there is not a complete analogy. In particular, one could sum over k, but would not generally sum over n, the energy level. --------------------------------------------------------------------------------------- In (1) a measure to determine time evolution of a quantum system is given as: D = 1- Integral dt (0,T) dt’ (0,T) <W(x,t)|W(x,t’)><W(x,t)|W(x,t’)>. Using W(x,t) = Sum over n cn exp(-iEnt) Wn(x) (1) shows: D= 1 - Sum (n,m) |cncm||cncm| sinc[wnm T/2] sinc[wnm T/2] The authors argue that this measure may be applied to qubit systems. Their result is based on the form exp(-iEnt)Wn(x) and the idea of a fixed time interval T. Here we argue that an identical form appears in the spatial realm for a system which instead of a finite time interval contains a finite length. The relevant system with a finite length is that of an infinite well potential with length L because the wavefunction is 0 outside this length. In such a case, instead of exp(-iEnt)Wn(x) we relate exp(-ipx)Wn(x) to a(p), the momentum wavefunction, and show that measure of time evolution is essentially of the same mathematical form as: 1- a(p)a(p) for an infinite well of length L. 1- a(p)a(p) has the interpretation of the probability not to find a particle in an infinite well in a momentum state of p for a given energy level with average momentum of n 3.14/L.
Optical techniques in single molecule imaging rely heavily on photon counting for data acquisition. Extraction of information from the recorded readings is often done by means of statistical signal processing, however this requires a full knowledge of the photoelectron statistics. In addition to counting statistics we include a specific form of random signal variations namely reorientational dynamics, or wobble to derive the general probability density function of the number of detected photons. The relative importance of the two factors is dependent upon the total number of photons in the system and results are given in all regimes.
How much detected thermal light is needed to identify several emitting sites? This working paper studies binary sources on a known grid of addressable source-image modes, using one fixed passive optical mask, photon counting, and a predetermined observation length. It distinguishes the original one-reference apparatus, complete Fourier reference collection, and a separate apparatus with one added vacuum input. All recorded photons count toward the measurement cost. The central result establishes the optimal quadratic detected-photon order at unit brightness for every fixed support limit on every grid, without geometric restrictions or primality assumptions. A reciprocal transmission mask attains this order using target counts alone, while recorded reference photons remain included in the cost. The proof reconstructs short cyclic arcs from two rational-probe values, then uses contour integration to isolate entire clusters of poles. This gives uniform quantitative separation without requiring a fixed minimum distance between individual poles. A bounded candidate list removes the grid-size logarithm from the final comparisons. On prime grids, a shorter finite-field argument recovers the support from two integer edge profiles. The quadratic law also holds uniformly over every nonempty reference subset when known brightness remains in a fixed interval bounded away from zero. The contour argument extends to a general stability theorem for rational interaction observables: under explicit locality and conditioning assumptions, quantitative distinguishability within clusters yields quantitative separation of the global polynomial data. For irregular registered dictionaries, this gives a separation margin proportional to the minimum site spacing, with an optimal spacing exponent. This extension assumes exactly resolved distance channels; their optical acquisition and robustness to unknown position errors require further analysis. Explicit dependence on the support limit also yields a growing-support regime: the photon cost remains n2+o(1)log(1/δ) whenever s2log(2s)=o(logn). The accompanying computational bound is stated in real arithmetic, and the dependence on sparsity is deliberately conservative. The manuscript retains explicit ramp-mask constructions for every configuration of at most four emitters, for short-span sources, and for sources with distinct cyclic pair distances. The last class has no span restriction and contains an asymptotically overwhelming fraction of fixed-size supports. For four emitters, ordinary-correlation rigidity and cyclic separation together distinguish reflected configurations. An explicit passive mask also attains the exact optimal Fourier-intensity separation under a prescribed transmission floor. In both the original and complete-reference experiments, sufficiently faint light produces a different law: at fixed sparsity, the optimal photon order grows as the grid size raised to twice the support limit. For at most two emitters, the paper determines the optimal detected-photon order throughout the known-brightness range, from arbitrarily faint light through unit brightness, on odd grids in the original apparatus and on every grid with complete reference collection. The result includes the dependence on grid size, brightness, and error probability, including an additional logarithmic information cost between the faint and bright limits. At fixed confidence, the quadratic regime begins at brightness of order (logn)/n2. A mask combining an index ramp with a quadratic-residue code attains the sharp order. On even physical grids, one known vacuum input and odd-dimensional Fourier optics give the same law; omitting one target output preserves the detector count. This is a distinct apparatus. For the specified two-scale transmissions on an infinite even-grid family, an exact lower bound using the complete count record proves that changing phases or decoders alone cannot remove their intermediate-brightness limitation. A collected-light model makes the physical scope explicit. For a known collection operator, binary-source identifiability is equivalent to injectivity of the induced thermal covariance map. Covariance collisions give identical quantum states and therefore prevent discrimination by any subsequent receiver. A passive multiport establishes population identifiability when the map is injective. The paper also derives the exact throughput for uniform passive mode separation from the smallest Gram eigenvalue and accounts separately for detected and incident photons. These results connect the assumed mode access to a specified collection model; they do not establish a general quantum-optimal receiver or a complete astronomical implementation. Reference excess variance supplies a normalization that identifies the spatial source despite an unknown finite common-profile thermal-mode spectrum and independent unknown Poisson backgrounds. The finite-sampling guarantee includes the empty source under explicit mean and nonzero-signal bounds. Population identifiability of the mode spectrum is distinguished from stable estimation. When background means are known, the arm means recover source mass and improve a guaranteed scale-free precision radius. Further results include exact thermal count variances, confidence exponents, calibration bounds, a photon penalty for discarding detector labels, and an exact target-only localization obstruction when singleton brightness is unrestricted and unknown. The results have analytical proofs, supported by 34 reproducible finite and symbolic verification suites. These checks provide corroboration, not experimental validation or external peer review. The unrestricted fixed-sparsity unit-brightness problem is resolved on all grids. Remaining questions include the optimized even-grid intermediate-brightness transition in the original apparatus, transitions for three or more emitters, sharp growing-support rates, sharp reference-coverage dependence at vanishing brightness, and optimal covariance-only scale-free precision. Continuous off-grid recovery requires a tolerance-based criterion and additional stability analysis. The work makes no literature-priority claim.