Recently Parigi et al. ( Science , 317 (2007) 1890 ) measured the statistics of a photon-subtracted and a photon-added thermal field. They showed that the measurements agree with the theoretical predictions of the beam splitter (BS) model. We show the equivalence of the quantum trajectory approach and the BS model in photon subtraction by deriving the BS model from the generalized photon counting operators. Our photon counting operators, corresponding to measurements using a resolving detector and a nonresolving detector, are generalization of the standard quantum jump approach. Our model is exact from weak quantum fields to the classical limit of strong fields. We show that our generalized photon counting operators reproduce exactly the results of the BS model when the reflection probability of the beam splitter is equated with the absorption probability of a photon in the standard quantum jump model of cavity field damping. Our theory can explain the recent experimental results of Parigi et al. and shows how similar experiments can be made to test the photon subtraction for the whole range of electromagnetic field intensities ranging from quantum to classical limit. We propose new experiments to test the generalized photon counting operators in the intermediate regime between the quantum and classical limits.
Recently Parigi et al. ( Science , 317 (2007) 1890 ) measured the statistics of a photon-subtracted and a photon-added thermal field. They showed that the measurements agree with the theoretical predictions of the beam splitter (BS) model. We show the equivalence of the quantum trajectory approach and the BS model in photon subtraction by deriving the BS model from the generalized photon counting operators. Our photon counting operators, corresponding to measurements using a resolving detector and a nonresolving detector, are generalization of the standard quantum jump approach. Our model is exact from weak quantum fields to the classical limit of strong fields. We show that our generalized photon counting operators reproduce exactly the results of the BS model when the reflection probability of the beam splitter is equated with the absorption probability of a photon in the standard quantum jump model of cavity field damping. Our theory can explain the recent experimental results of Parigi et al. and shows how similar experiments can be made to test the photon subtraction for the whole range of electromagnetic field intensities ranging from quantum to classical limit. We propose new experiments to test the generalized photon counting operators in the intermediate regime between the quantum and classical limits.
Recently Parigi et al. [Science 317, 1890 (2007)] implemented experimentally the photon subtraction and addition processes from and to a light field in a conditional way, when the required operations were produced successfully only upon the positive outcome of a separate measurement. It was verified that for a low-intensity beam (quantum regime) the bosonic annihilation operator a does indeed describe a single photon subtraction, while the creation operator ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ describes a photon addition. Nonetheless, the exact formal expressions for these operations do not always reduce to these simple identifications, and in this connection here we deduce the general superoperators for multiple photon subtraction and addition processes and analyze the statistics of the resulting states for classical field states having an arbitrary intensity. We obtain closed analytical expressions and verify that for classical fields with high intensity (classical regime) the operators that describe photon subtraction and addition processes deviate significantly from simply a and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$. Complimentarily, we analyze in detail such a smooth quantum-classical transition as a function of beam intensity for both processes.
Time development of electromagnetic fields in closed cavities under continuous detection of photons continues to be a subject of confusing controversy. Recently Dodonov et al. [Phys. Rev. A, 75, 013806, 2007] argued that their model of quantum superoperators (E model) invalidates some of the predictions of the previously introduced photon counting model of Srinivas and Davies [J. Mod. Optic. 28, 981, 1981] (SD model). Both the SD and the E models are based on two postulated quantum jump superoperators: (1) the one-count operator corresponding to the absorption of a single photon and (2) the no-count operator. In this work we develop a stochastic difference equation that describes the dissipative coupling of the cavity field and the detector. The difference equation is based on non-perturbative treatment of the cavity-detector coupling. In spite of being non-integrable due to the coupling of the detector with an external reservoir it can be used to derive the exact forms of the quantum jump superoperators. When applied to a particular photon counting measurement our theory gives predictions identical with those of the SD model which should be considered a non-perturbative and ab-initio result. It is pointed out that available experimental results coincide with the results given by the ab-initio SD model. We summarize some of the key characteristics of cavity fields and photon counting processes to demonstrate that the results given by the SD model are consistent with the principles of quantum mechanics while those given by the E model are not.
Analysis of quantum optical experiments and the simulation of optical devices require detailed quantum mechanical models, especially in the case of weak optical fields. In this thesis the quantum dynamics of cavity fields are investigated and new tools for modeling cavity fields interacting with an energy reservoir are developed. Using the quantum trajectory approach the field dynamics during photon detection processes are investigated. Two experimentally feasible detector models, the resolving and the non-resolving detector scheme, are derived and applied to single photon detection and coincidence photon detection experiments. Furthermore, equivalence of the cavity field model to the beam splitter based single photon subtraction and addition schemes is shown. In addition to the detection schemes described above, a reduced model for fields in a non-ideal cavity interacting with a dissipative and amplifying reservoir through multiple two state systems is derived. The reduced model can be used to describe e.g. light emitting diodes and lasers depending on the relative strengths of the losses and energy injection. In these cases the model reproduces fields that approach a thermal or a coherent field, respectively. The derived models can be applied to wide variety of cavity field experiments. The reduced field model can be applied to modeling the optical fields of semiconductor devices or to describe cavity field based quantum information processing experiments. Furthermore, fundamental quantum optics experiments of single photon addition, single photon subtraction, coincidence detection, and their combinations can be analyzed using the derived models.
In atomic-scale or quantum physics, an electromagnetic field is composed of photons, optical packets so small that a typical laser pointer with 1mW of power emits billions of them each second. Being able to take away or add a single photon in a light field at will would be useful for accurate engineering of quantum states. Such pure single photons are ideal for carrying and manipulating information in emerging quantum technologies, but generating them still is very challenging. In quantum physics, subtraction or addition of a photon is not a deterministic process. Instead, the probability is proportional to the number of photons originally in the electromagnetic field. This is described by the equation â|n〉 = √ n|n − 1〉 and â†|n〉 = √ n+ 1|n + 1〉, where â and ↠are annihilation and creation operators, respectively, representing the action of subtracting and adding a photon in a field composed of n photons.1 Such single-photon operations were limited to pure theoretical discussions until 2004, when Grangier and coworkers found a simple way to subtract a photon2 from a field. Passing through a beam splitter of high transmittivity, some photons can be split from the initial field. A ‘click’ detected in the photodetector in the reflected channel means one photon has been subtracted from the field: see Figure 1(a). When the initial field is composed of a small number of photons, the effect is dramatic and may even completely change the characteristics of the field. For example, let us take an initial single-mode squeezed state generated by ‘degenerate’ parametric down-conversion, which is an optical nonlinear process that converts a pump photon into two twin photons with half the pump photon energy. (This state is well known for its quantum noise squeezed into one quadrature at the expense of expanded noise in the other quadrature.) In order to visualize the noise properties and characteristics of the squeezed state, a probability-like function, such Figure 1. Setups to subtract (a) and add (b) a single photon from/to a light field. ρin and ρout denote the density operators of the input and output fields; BS is a low-reflectivity beam splitter; PDC is a nonlinear crystal where parametric down-conversion takes place; â and ↠denote on/off photodetectors.
Thermal radiation played a pivotal role in the preliminary development of quantum physics where it helped resolve the apparent incongruity of the ultraviolet catastrophe. In contemporary physics, thermal state generation and manipulation finds new application in fields such as quantum imaging and quantum illumination and as a practical realization of Maxwell's demon. These applications often go hand in hand with photon subtraction operations which probabilistically amplify the mean photon number (MPN) of thermal light as a result of its super-Poissonian photon statistics. In this article, we introduce an operation for thermal states of light based on a generalized photon subtraction scheme. Displaced-photon subtraction (DPS) makes use of coherent state displacement followed by a subsequent anti-displacement in combination with single-photon detection to probe the MPN of a thermal state. We find regimes in which the output of a successful DPS is amplified, unchanged, or attenuated relative to the unconditioned output state. The regime of operation is controlled via the magnitude of the coherent displacement. A theoretical description of generalized photon subtraction of a displaced thermal state is derived via a two-mode moment-generating function (MGF) and used to describe generalized DPS. We perform a proof of principle experimental implementation of DPS for the case of a balanced beam splitter for which results demonstrate good agreement with the model.
Abstract We study feasible and effective techniques to improve the efficiency and fidelity of photon subtraction in order to enable practical quantum resource engineering based on it. We use thermal light for our investigation and consider non-negligible beam splitting beyond the conventional setup, also taking into account photon detection errors. We find that the output is still a photon-subtracted thermal state when a non-negligible amount of light is reflected from the input beam and measured by the photon detector, but it has an effective mean photon number different than that of the input. It enables a strategy for achieving a target photon-subtracted thermal state by employing a stronger input light and leads to substantial improvement in the probability of successful photon subtraction events. We calculate the fidelity of the output state and analyze how it worsens with larger beam splitting ratios and photon counting errors. Using our understanding of the limiting factors in the conventional routine, we propose a new method for photon subtraction that can achieve high-fidelity output with decent efficiency, and using only mediocre photon detectors. We experimentally verify our solutions that are valuable for improving photon subtraction and lowering the experimental barrier.