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Journal of Physics B Atomic Molecular and Optical Physics

Photon-spin-dependent contribution to the P , T -odd Faraday rotation effect for atoms

S. D. Chekhovskoi, D. V. Chubukov, Leonid V. Skripnikov, Alexander N. Petrov, Leonti N Labzowsky

Published Jan 27, 2021• 3 citations
Publisher

Abstract

Abstract The photon-spin-dependent (PS) contribution to the ordinary Faraday effect—optical rotation in an external magnetic field H ⃗ is well known. This contribution is proportional to factor ( s ⃗ p h ⋅ H ⃗ ) where s ⃗ p h is the photon spin. Recently a P , T -odd Faraday effect—optical rotation in an external electric field E ⃗ —was discussed as a promising tool for observation of the electron electric dipole moment in the intra-cavity absorption spectroscopy. In the present paper, we introduce the PS contribution to the P , T -odd Faraday rotation effect. This contribution is proportional to ( s ⃗ p h ⋅ E ⃗ ) and has to be taken into account in the calculations of the P , T -odd Faraday effect.

DOI
10.1088/1361-6455/abe0a6

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Abstract The photon-spin-dependent (PS) contribution to the ordinary Faraday effect—optical rotation in an external magnetic field H ⃗ is well known. This contribution is proportional to factor ( s ⃗ p h ⋅ H ⃗ ) where s ⃗ p h is the photon spin. Recently a P , T -odd Faraday effect—optical rotation in an external electric field E ⃗ —was discussed as a promising tool for observation of the electron electric dipole moment in the intra-cavity absorption spectroscopy. In the present paper, we introduce the PS contribution to the P , T -odd Faraday rotation effect. This contribution is proportional to ( s ⃗ p h ⋅ E ⃗ ) and has to be taken into account in the calculations of the P , T -odd Faraday effect.
We investigate the process of photon absorption by atoms or molecules shortly interacting with a laser beam in the dipole approximation. Assuming that the interaction time $\ensuremath{\tau}$ is much smaller than the lifetime of the corresponding excited state, we examine the absorption probability as a function of $\ensuremath{\tau}$. In addition, we incorporate Doppler broadening due to nonzero temperature of the atoms (molecules). It is demonstrated that in the case of a zero detuning and without Doppler broadening the absorption probability is quadratic in $\ensuremath{\tau}$. Once Doppler broadening is taken into account or the laser beam is off from the resonant frequency, the absorption probability becomes linear in $\ensuremath{\tau}$. Our findings are expected to be important for experimental studies in optical cells or cavities where atoms or molecules traverse continuous laser beams. The experimental prospects of searching for the electric dipole moment of the electron are discussed in detail.
The interaction of electrons and photons lies at the heart of quantum physics. The \nmost notable phenomena which are described by quantum physics but which obviously \ninvalidate a classical description of the electromagnetic field - the photoelectric \neffect, the Compton effect, or the antibunching of photons emitted from a single atom, \nto mention a few - are intimately related to the interaction of electrons and photons. \nElectrons possess an internal degree of freedom, the spin. The spin S of an electron \ncan be described as an internal angular momentum, leading to a magnetic moment \nm due to the electric charge of the electron. Stern and Gerlach have shown that \nthe projection of m onto a quantization axis (defined by an external magnetic field \nin their experiment), e.g., along the z axis, is either mz = g�μB/2 (\\spin up") or \nmz = g�μB=2 (\\spin down"), where �μB � 9:2741 � 10-24 J/T is the Bohr magneton \nand g is the g-factor of the electron (g = 2 for free electrons). \n \n An electron spin \nis therefore usually referred to as a two-level system. Due to this property, the \nspin of electrons which are localized in semiconductor quantum dots has recently \nattracted significant attention regarding the implementation of quantum information \nprocessing: Electron spins can be used as carriers of quantum information. One \ncan define that the spin pointing "up" corresponds to the logical value "0" and \nthe spin pointing "down" corresponds to "1". Moreover, because the electron spin \nis a quantum mechanical property one can form arbitrary coherent superpositions of \n"up" and "down" with a single spin. Moreover, because the electron spin \nis a quantum mechanical property one can form arbitrary coherent superpositions of \n"up" and "down" with a single spin. A system with this property is called a quantum \nbit (qubit). The additional possibilities due to quantum superpositions of qubits are \nexploited, e.g., in the algorithms introduced by Shor and Grover to solve certain tasks \nmuch more efficiently than with a classical computer (i.e., the prime factorization of \nlarge numbers for Shor's algorithm and the search within an unstructured database for \nGrover's algorithm). While quantum computation has presently only been achieved \nin prototypical experiments with few qubits, the implementation of efficient largescale \nquantum computation with many qubits still remains an extremely demanding \ntask. Yet, other quantum mechanical properties of qubits, such as entanglement, \nhave already been exploited experimentally with photons in quantum communication \nschemes, for example, quantum teleportation and quantum data compression. \nIn this thesis, we investigate the interaction of electrons and photons in semiconductor \nquantum dots. Optical transitions in quantum dots enable a direct link between \nelectron spins and photon polarizations due to conservation laws. We show that \n \nentanglement can be transferred from electron spin qubits to qubits defined by the \nphoton polarization, enabling the measurement of entangled spin states via photons. \nThe mechanism under study can also be used for the production of entangled photons, \nfor instance for the implementation of quantum communication protocols. In contrast \nto the presently used sources of entangled photons, the photon source we propose here \nis deterministic, providing entangled photons on demand. It has been demonstrated \nin several recent experiments that the most obvious way to achieve such a transfer of \nentanglement - using the recombination of biexcitons in a single quantum dot - fails \nin the presently available quantum dot structures. We have analyzed the problems \nof this approach. As a solution, we propose schemes based on charged excitons in \nsingle or coupled quantum dots. We discuss the generation of entangled two- and \nfour-photon states. \nIn addition to the transfer of quantum states, photons can also be used to probe \nelectron spin states. We investigate in detail different methods to optically measure \nthe decoherence time of a single electron spin in a quantum dot. The decoherence time \nof a spin establishes the time scale during which coherent manipulation is possible. \nMeasurements of the electron spin decoherence time are therefore highly desirable \nin view of the implementation of spin-based quantum information processing. We \nshow that the schemes we propose can be implemented with current experimental \ntechniques. \nWe then study the magneto-optical effect called Faraday rotation. Using the technique \nof time-resolved Faraday rotation, a recent experiment has demonstrated the \ncoherent transfer of spin between quantum dots coupled by molecules. We calculate \nthe Faraday rotation signal for a coupled dot system and show that a two-site \nHamiltonian with a transfer term captures the essential features observed in this experiment. \nWe also present results for a system of two coupled dots doped with a \nsingle electron. \nWe finally show that the coupled states of two qubits can be detected via the optical \ninteraction with a cavity in the dispersive regime. We present a Schrieffer-Wolff \ntransformation which removes the coupling of the two qubits to the cavity in leading \norder. The different two-qubit states lead to a different spectral shift of the cavity \nline. For a sufficiently low cavity linewidth, this enables the direct readout of a \ntwo-qubit system.
Nonlinear in laser power, near-resonant light-atom interaction leads to a number of spectacular modifications of a well-known effect -polarization plane rotation in a magnetic field applied along the direction of light propagation.In particular, the rotation can be enhanced by up to nine orders of magnitude compared to the linear case.Faraday effect [ 1 ] is the rotation of the plane of linear light polarization by a medium in a magnetic field applied in the direction of light propagation.In 1898, Macaluso and Corbino eJ, studying absorption spectra of the alkali atoms in the presence of magnetic fields, discovered that the Faraday effect in the vicinity of resonance absorption lines also had a distinct resonant character.The essence of the Macaluso-Corbino effect can be illustrated by considering a simple case of a F= 1 ~ F'=O transition (Fig. 1 ).Linearly polarized light incident on the sample can be resolved into two counter-rotating circular components ( cr~.In the absence of a magnetic field, the M=±l sublevels are degenerate and the optical resonance
When an off-resonant light field is coupled with atomic spins, its polarization can rotate depending on the direction of the spins via Faraday rotation, which has been used for monitoring and controlling the atomic spins. We observed Faraday rotation by an angle of more than 10 deg for a single nuclear spin of 1/2 of a 171Yb^{171}\mathrm{Yb}171Yb atom in a high-finesse optical cavity. By employing the coupling between the single nuclear spin and a photon, we have also demonstrated that the spin can be projected or weakly measured through the projection of the transmitted single ancillary photon.
A theory for the nonlinear (with respect to the electric field) Faraday effect has been developed and applied to a simple model. Within the framework of this model, our analysis shows that the ratio of the nonlinear Verdet constant to the linear Verdet constant is proportional to the intensity of the incident light and depends on the frequency of light, the excitation energy of the model molecule, and the temperature of the medium. This ratio is small under ordinary conditions. However, under certain circumstances it may become large enough for experimental observation.