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Dressed-AtomModel

Z. Ficek, Mohamed Ridza Wahiddin

Published Apr 19, 2016• 0 citations
Publisher

Abstract

In this chapter, we present a powerful method of solving the Bloch equations or the master equation of a driven system called the dressed-atom model. The method is valid in situations where the Rabi frequency of the applied driving field is much larger than the spontaneous emission rate of the atom, . Under such conditions, one can make the secular approximation that consists of dropping terms oscillating in time with frequencies 2 and higher. These terms, if kept in the master equation, would make corrections to the dynamics of the system of the order of /, and thus are negligible. Although limited in the range of parameters for which it can be used, the dressed-atommodel provides a physical insight into the properties and dynamics of the system. Within this model, one can explicitly calculate energy states and transition rates between them in a relatively simple way. The knowledge of the energy states and transition rates is for most of the problems enough to fully understand the underlying physics. There are two mathematically different approaches to the dressed-atommodel, but giving the same results, depending on whether we treat the driving field classically or quantum mechanically. These are the semiclassical and quantum dress-atom models. In the following we explain in details these two dressed-atommodels.

DOI
10.1201/b16875-17

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In this chapter, we present a powerful method of solving the Bloch equations or the master equation of a driven system called the dressed-atom model. The method is valid in situations where the Rabi frequency of the applied driving field is much larger than the spontaneous emission rate of the atom, . Under such conditions, one can make the secular approximation that consists of dropping terms oscillating in time with frequencies 2 and higher. These terms, if kept in the master equation, would make corrections to the dynamics of the system of the order of /, and thus are negligible. Although limited in the range of parameters for which it can be used, the dressed-atommodel provides a physical insight into the properties and dynamics of the system. Within this model, one can explicitly calculate energy states and transition rates between them in a relatively simple way. The knowledge of the energy states and transition rates is for most of the problems enough to fully understand the underlying physics. There are two mathematically different approaches to the dressed-atommodel, but giving the same results, depending on whether we treat the driving field classically or quantum mechanically. These are the semiclassical and quantum dress-atom models. In the following we explain in details these two dressed-atommodels.
We use a basis of Bloch wave functions to solve the problem of spontaneous emission from an atom inside a standing-wave cavity. In this way we incorporate the effects of the atom’s quantized center-mass-motion in the periodic light-force potential of the one-quantum dressed states during the spontaneous emission process. We calculate emission spectra and the momentum distribution of the atom. Under strong-coupling conditions, “vacuum” Rabi spectra become asymmetric, with an increased relative weight given to the red side of the spectrum because of the transfer of energy to the center-of-mass motion of the atom. Because of the nonperturbative interaction between the atom and the cavity mode, many momentum quanta are transferred to the atom from the cavity mirrors during the irreversible emission of just one photon. Thus, the atom is diffracted by its own radiation reaction field. We analyze these effects using the Bloch-state basis. Use of the secular approximation provides a simple rate equation description of transitions from the Bloch states excited in the initial state to the atom-cavity ground state. We compare this approach with other ways of doing the calculations and discuss the accuracy of the secular approximation. We consider the extension of these calculations to treat the effects of photon statistics on the momentum distribution of atoms traversing a standing-wave cavity.
In the field of atom optics, the basis of many experiments is a two-level atom coupled to a light field. The evolution of this system is governed by a master equation. The irreversible components of this master equation describe the spontaneous emission of photons from the atom. For many applications, it is necessary to minimize the effect of this irreversible evolution. This can be achieved by having a far detuned light field. The drawback of this regime is that making the detuning very large makes the time step required to solve the master equation very small, much smaller than the time scale of any significant evolution. This makes the problem very numerically intensive. For this reason, approximations are used to simulate the master equation, which are more numerically tractable to solve. This paper analyzes four approximations: The standard adiabatic approximation, a more sophisticated adiabatic approximation (not used before), a secular approximation, and a fully quantum dressed-state approximation. The advantages and disadvantages of each are investigated with respect to accuracy, complexity, and the resources required to simulate. In a parameter regime of particular experimental interest, only the sophisticated adiabatic and dressed-state approximations agree well with the exact evolution.
The eigenstates of the Jaynes-Cummings Hamiltonian (the dressed atomic states) are widely used to describe the interaction between a single two-level atom and a single mode of the electromagnetic field. They provide a particularly useful basis when the secular approximation is justified. For an atom interacting fog with n photons in a cavity, the secular approximation requires that the n-photon Rabi frequency 2 n g be much larger than the atomic and cavity linewidths. Normally this condition is met when n is large and the coupling constant g is much smaller than the atomic and cavity linewidths. In this paper we allow g to be larger than the linewidths, and we allow the photon number to be small. We show that when the cavity is driven by a coherent field, the usual secular approximation cannot be used. The natural basis states become are now the eigenstates of the Jaynes-Cummings Hamiltonian plus the external field interaction. We solve this eigenvalue problem by separating the field and atomic parts of the Hamiltonian using a displacement and squeeze in the Hilbert-space of the field. We find that the Rabi frequencies are renormalized downwards as the strength of the driving field is increased. They eventually vanish at a critical field strength. At this critical field strength we observe a spontaneous symmetry breaking in the steady-state solution of the master equation for a single atom interacting with a driven cavity mode.
One of the simplest nontrivial problems involving the atom–field interaction is the coupling of a two-level atom with a single mode of the electromagnetic field. A two-level atom description is valid if the two atomic levels involved are resonant or nearly resonant with the driving field, while all other levels are highly detuned. Under certain realistic approximations, it is possible to reduce this problem to a form which can be solved exactly; allowing essential features of the atom-field interaction to be extracted. In this chapter we present a semiclassical theory of the interaction of a single two-level atom with a single mode of the field in which the atom is treated as a quantum two-level system and the field is treated classically. A fully quantum mechanical theory will be presented in Chapter 6. A two-level atom is formally analogous to a spin-1/2 system with two possible states. In the dipole approximation, when the field wavelength is larger than the atomic size, the atom–field interaction problem is mathematically equivalent to a spin-1/2 particle interacting with a time-dependent magnetic field. Just as the spin-1/2 system undergoes the so-called Rabi oscillations between the spin-up and spin-down states under the action of an oscillating magnetic field, the two-level atom also undergoes optical Rabi oscillations under the action of the driving electromagnetic field. These oscillations are damped if the atomic levels decay. An understanding of this simple model of the atom–field interaction enables us to consider more complicated problems involving an ensemble of atoms interacting with the field.
A qubit--harmonic-oscillator open system coupled to thermal baths was considered and the master equation describing the evolution of the open system was deduced when the qubit transition frequency is $\ensuremath{\lesssim}0.1$ the oscillator frequency. The master equation is valid in all the qubit-oscillator state space and holds for all values of the qubit-oscillator coupling including the ultrastrong and deep strong coupling regimes. It only requires an oscillator frequency much larger than the relaxation rates. The qubit-oscillator coupling can enhance or decrease both the relaxation rates and the frequency shifts induced by the thermal baths. It was found that weak, sinusoidal qubit driving forces the qubit-oscillator open system to behave like a driven qubit whose evolution is governed by equations similar to those of the Bloch vector in the optical Bloch equations and whose transition frequency decreases with increasing qubit-oscillator coupling strength. Finally, it was shown how one can reach the adiabatic regime by using sinusoidal qubit driving with large driving frequency and the concepts of $\ensuremath{\pi}$ and $\ensuremath{\pi}/2$ pulses were generalized to manipulate transitions between dressed states.
The analysis in Chapter 3 used a phenomenological form of the permittivity to describe active materials. A proper understanding of optical amplification requires a quantum-mechanical approach for describing the interaction of light with atoms of an active medium [1]. However, even a relatively simple atom such as hydrogen or helium allows so many energy transitions that its full description is intractable even with modern computing machinery [2, 3]. The only solution is to look for idealized models that contain the most essential features of a realistic system. The semiclassical two-level-atom model has proven to be quite successful in this respect [4]. Even though a real atom has infinitely many energy levels, two energy levels whose energy difference nearly matches the photon energy suffice to understand the interaction dynamics when the atom interacts with nearly monochromatic radiation. Moreover, if the optical field contains a sufficiently large number of photons (> 100), it can be treated classically using a set of optical Bloch equations. In this chapter, we learn the underlying physical concepts behind the optical Bloch equations. We apply these equations in subsequent chapters to actual optical amplifiers and show that they can be solved analytically under certain conditions to provide a realistic description of optical amplifiers. It is essential to have a thorough understanding of the concept of a quantum state [5]. To effectively use the modern machinery of quantum mechanics, physical states need to be represented as vectors in so-called Hilbert space [6].