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Physical Chemistry Chemical Physics

Theory of single photon on demand from a single molecule source

Yong He, Eli Barkai

Published Jan 1, 2006• 11 citations
Publisher

Abstract

We consider the theory of single photon on demand from a two level atom or molecule source. Using optical Bloch equations and the generating function formalism we investigate three approaches to single photon control: (i) the square laser pulse; (ii) the square modulation of absorption frequency; and (iii) the rapid adiabatic following approach investigated in the experiments of Brunel et al., Phys. Rev. Lett., 1999, 83, 2722. We discuss the conditions for obtaining the maximum of the probability of emission of a single photon and a pair of photons, under the constrains of finite field strength and finite interaction time with excitation field. We obtain analytical expression for the probability of emitting zero, one, and two photons for the square pulse, and discuss semi-classical and strongly quantum limiting cases. Numerical results obtained from the generating function formalism are compared with experimental results showing that the two level system approach is suitable for the description of cryogenic temperature single molecules, and that experiments were conducted very close to the optimal conditions.

DOI
10.1039/b606198b

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We consider the theory of single photon on demand from a two level atom or molecule source. Using optical Bloch equations and the generating function formalism we investigate three approaches to single photon control: (i) the square laser pulse; (ii) the square modulation of absorption frequency; and (iii) the rapid adiabatic following approach investigated in the experiments of Brunel et al., Phys. Rev. Lett., 1999, 83, 2722. We discuss the conditions for obtaining the maximum of the probability of emission of a single photon and a pair of photons, under the constrains of finite field strength and finite interaction time with excitation field. We obtain analytical expression for the probability of emitting zero, one, and two photons for the square pulse, and discuss semi-classical and strongly quantum limiting cases. Numerical results obtained from the generating function formalism are compared with experimental results showing that the two level system approach is suitable for the description of cryogenic temperature single molecules, and that experiments were conducted very close to the optimal conditions.
We investigate the theory of a few photons on demand from a single two-level emitter excited by a square laser pulse. Using the generalized optical Bloch equations, we find exact expressions for the probability of generating zero, one, two, and three photons, the average ⟨N⟩⟨N⟩⟨N⟩, and Mandel's QQQ parameter, which exhibit rich physical behaviors. The exact solution is used to design control fields that yield an optimal single- or two-photon source, under the constraints of finite laser-field strength and finite interaction time.
The photon emission from a single molecule driven simultaneously by a laser and a slow electric radio frequency (rf) field is studied. We use a non-Hermitian Hamiltonian approach which accounts for the radiative decay of a two-level system modeling the single-molecule source. We apply the renormalization group method for differential equations to obtain long time solution of the corresponding Schrodinger equation, which allows us to calculate the average waiting time for the first photon emission. Then, we analyze the conditions for suppression and enhancement of photon emission in this dissipative two-level system. In particular we derive a transcendental equation, which yields the nontrivial rf field control parameters, for which enhancement and suppression of photon emission occurs. For finite values of radiative decay rate an abrupt transition to the state when both situations are indistinguishable is found for certain values of the rf field parameters. Our results are shown to be in agreement with the available experiments [Ch. Brunel et al., Phys. Rev. Lett. 81, 2679 (1998)].
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.
The photon emission from a single molecule driven simultaneously by a laser and a slow electric radio frequency (rf) field is studied. We use a non-Hermitian Hamiltonian approach which accounts for the radiative decay of a two level system modeling a single molecule source. We apply the renormalization group method for differential equations to obtain long time solution of the corresponding Schrdinger equation, which allows us to calculate the average waiting time for the first photon emission. Then, we analyze the conditions for suppression and enhancement of photon emission in this dissipative two-level system. In particular we derive a transcendental equation, which yields the non-trivial rf field control parameters, for which enhancement and suppression of photon emission occurs. For finite values of radiative decay rate an abrupt transition from the molecule's localization in its ground state to delocalization is found for certain critical values of the rf field parameters. Our results are shown to be in agreement with the available experiments [Ch. Brunel et al, Phys. Rev. Lett. 81, 2679 (1998)].
There are many ways of calculating photon statistics in quantum optics in general and single molecule spectroscopy in particular such as the generating function method, the quantum jump approach or time ordering methods. In this paper starting with the optical Bloch equation, within the paths interpretation of Zoller, Marte and Walls we obtain the photon statistics from a sequence of laser pulses expressed by means of quantum trajectories. We find general expressions for Pn(t) - the probability of emitting n photons up to time t, discuss several consequences and show that the interpretation of the quantum trajectories (i) emphasizes contribution to the photon statistics of the coherence paths accumulated in the delay interval between the pulses and (ii) allows simple classification of the terms negligible under certain physical constraints . Applying this method to the concrete example of two square laser pulses we find the probabilities of emitting 0,1 and 2 photons, examine several limiting cases and investigate the upper and lower bounds of P0(t), P1(t) and P2(t) for a sequence of two strong pulses in the limit of long measurement times. Implication to single molecule non-linear spectroscopy and theory of pairs of photons on demand are discussed briefly.
The detection of photons emitted from a single molecule under the influence of electromagnetic radiation is considered. Utilizing a generating function formalism, we derive several exact results for the statistics of such emitted photons within the framework of the temporally modulated optical Bloch equations. Additionally, it is shown how these results reduce to previously obtained limiting behaviors. An appealing feature of this formulation is the inclusion of both photon bunching and anti-bunching effects within a single theoretical framework that is well suited for numerical analysis. Several examples are considered to demonstrate the feasibility of the approach in calculations. In most cases, these results verify known phenomena. In one case, we report a result that was missed by prior approximate treatments. This new effect centers around the fact that a chromophore will display anti-bunching behavior in the limit of fast modulation of the resonant absorption frequency.