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arXiv:0901.4477 · Physical Review A

Smooth quantum-classical transition in photon subtraction and addition processes

Alexandre V. Dodonov, Salomon Sylvain Mizrahi

Published Feb 18, 2009• 12 citations
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Abstract

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.

DOI
10.1103/physreva.79.023821

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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 aaa does indeed describe a single photon subtraction, while the creation operator a\ifmmode†\else\textdagger\fi{a}^{\ifmmode\dagger\else\textdagger\fi{}}a\ifmmode†\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 aaa and a\ifmmode†\else\textdagger\fi{a}^{\ifmmode\dagger\else\textdagger\fi{}}a\ifmmode†\else\textdagger\fi. Complimentarily, we analyze in detail such a smooth quantum-classical transition as a function of beam intensity for both processes.
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.
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.
We have recently realized experimental schemes to implement the action of single-photon creation and annihilation operators onto completely classical and fully incoherent thermal light states (Parigi et al 2007 Science 317 1890). By applying alternated sequences of the creation and annihilation operators we observed that the resulting states depend on the order in which the two quantum operators are applied, thus obtaining the most direct experimental test of non-commutativity. Here we provide an extensive and detailed discussion of the main experimental issues related to the realization of these schemes.
We propose an experiment to directly prove the commutation relation between bosonic annihilation and creation operators, based on the recent experimental success in single-photon subtraction and addition. We devise a single-photon interferometer to realize coherent superpositions of two sequences of photon addition and subtraction. Depending on the interference outcome, the commutation relation is directly proven or a highly nonclassical state is produced. Experimental imperfections are assessed to show that the realization of the scheme is highly feasible.
The subtraction of a single photon from a multimode quantum field is analyzed as the conditional evolution of an open quantum system. We develop a theory describing different subtraction schemes in a unified approach and we introduce the concept of subtraction modes intrinsic to the process. The matching between the subtraction modes and the modes of the field defines different possible scenarios for the photon subtraction. In particular, our framework identifies: the conditions of pure photon subtraction, the quantum states of the field modes conditioned to the photon subtraction, the mode with the highest fidelity with a single-photon state when the subtraction is performed on multimode squeezed light. We use our theory to analyze the photon subtraction from a highly multimode quantum resource - a train of quantum squeezed or correlated optical pulses. Performing the photon subtraction optimally on various multimode light field has the potential to implement a number of quantum information protocols in a multiplexed and scalable way.
The non-Gaussian operations effected by adding or subtracting a photon on entangled optical beams emerging from a parametric down-conversion process have been suggested to enhance entanglement. Heralded photon addition or subtraction is, as a matter of fact, at the heart of continuous-variable entanglement distillation. The use of such processes has recently been experimentally demonstrated in the context of the generation of optical coherent-state superpositions or the verification of canonical commutation relations. Here, we carry out a systematic study of the effect of local photon additions and subtractions on a two-mode squeezed vacuum state, showing that the entanglement generally increases with the number of such operations. This is analytically proven when additions or subtractions are restricted to one mode only, while we observe that the highest entanglement is achieved when these operations are equally shared between the two modes. We also note that adding photons typically provides a stronger entanglement enhancement than subtracting photons, while photon subtraction performs better in terms of energy efficiency. Furthermore, we analyze the interplay between entanglement and non-Gaussianity, showing that it is more subtle than previously expected.