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Physical Review A

Spectral properties of trapped two-level ions interacting with quantized fields

Navid Yazdanpanah, Mohammad Kazem Tavassoly

Published Jun 14, 2017• 4 citations
Publisher

Abstract

In this paper we study the full quantum approach to the interaction of $N$ trapped two-level ions with $N$ quantized fields in such a way that each of the trapped ions separately interacts with one of the quantized fields. A semiclassical model of this type of interaction has been already studied, using classical laser fields instead of quantized fields, by Cirac and Zoller [Phys. Rev. Lett. 74, 4091 (1995)]. During each interaction, the excitation of a trapped ion occurs due to the occurrence of resonance between the trapped ion and one of the modes of multimode quantized field. In this way, we are able to find the quantized radiation field, which propagates from each excited trapped ion after its deexcitation. It is demonstrated that such a propagated field is a function of the position of ion within the trap, ion number, trapping and ionic transition frequencies, spontaneous emission rate, and time evolution of trapped ionic operators. In the continuation, to achieve the explicit form of such a field we propose a method to evaluate the temporal evolution of operators of the system by utilizing the Heisenberg operator approach. Via the propagated field of each trapped ion, the spectral properties such as intensities as well as power spectra of their radiations are evaluated. Furthermore, the quantum statistics of field and in particular, its photon antibunching and sub-Poissonian statistical properties are investigated. Finally, resonance spectroscopy of the trapped ions is obtained by which one can allocate a characteristic spectrum to each of them.

DOI
10.1103/physreva.95.063815

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In this paper we study the full quantum approach to the interaction of NNN trapped two-level ions with NNN quantized fields in such a way that each of the trapped ions separately interacts with one of the quantized fields. A semiclassical model of this type of interaction has been already studied, using classical laser fields instead of quantized fields, by Cirac and Zoller [Phys. Rev. Lett. 74, 4091 (1995)]. During each interaction, the excitation of a trapped ion occurs due to the occurrence of resonance between the trapped ion and one of the modes of multimode quantized field. In this way, we are able to find the quantized radiation field, which propagates from each excited trapped ion after its deexcitation. It is demonstrated that such a propagated field is a function of the position of ion within the trap, ion number, trapping and ionic transition frequencies, spontaneous emission rate, and time evolution of trapped ionic operators. In the continuation, to achieve the explicit form of such a field we propose a method to evaluate the temporal evolution of operators of the system by utilizing the Heisenberg operator approach. Via the propagated field of each trapped ion, the spectral properties such as intensities as well as power spectra of their radiations are evaluated. Furthermore, the quantum statistics of field and in particular, its photon antibunching and sub-Poissonian statistical properties are investigated. Finally, resonance spectroscopy of the trapped ions is obtained by which one can allocate a characteristic spectrum to each of them.
Following the path of Bužek et al. in [Phys. Rev. A 56 2352 1997], we investigate the full intensity-dependent interaction between a motional two-level ion which is trapped in a f-deformed oscillator (so possessing vibrational modes too) with a single-mode nonlinear quantized field in the Lamb-Dicke limit. After finding the explicit form of the entire ion-field state vector in a general framework, we particularly restrict ourselves to an appropriate nonlinearity function (describing intensity-dependent interaction) corresponding to the center-of-mass motion of a trapped ion which explicitly depends on the Lamb-Dicke parameter as well as another well-known nonlinearity function in the form n. Then, we evaluate the effects of two different initial quantized field states as well as the considered nonlinearity function (via varying the Lamb-Dicke parameter) on a few physical properties of the system. For this purpose, we pay attention to the amount of entanglement, mean number of vibrational quanta (phonons), population inversion of the two-level ion and Q-distribution function. Finally, by comparing our results with the previously considered “linear” one which is pointed in the above Ref., we establish that, by tuning the Lamb-Dicke parameter, one can adjust the mentioned nonclassical properties of the system appropriately.
To achieve more accuracy in a system containing N trapped two-level ions interacting with a single-mode quantized field, in this paper we will consider the dipole–dipole interaction (DDI) between the trapped ions. In more detail, we first suggest an appropriate form of a quantized DDI Hamiltonian between two trapped ions that has arisen from the spontaneous emission of the excited one. Accordingly, we have considered the spontaneous emission as the origin of dissipation in our proposed model. Then, the case in which the trapped ions interact with a single-mode quantized field in the Lamb–Dicke regime as well as the first vibrational sideband is considered, in which the DDI between the trapped ions is also taken into account. Using the state vector, which is analytically obtained for the whole system under our consideration, we evaluate the amount of entanglement between the field and the two trapped ions by utilizing the linear entropy. We observe that the entanglement depends on the distance between the two trapped ions; i.e., decreasing the distance led to decrement in the amount of entanglement. In the continuation, we extend the proposed model at first to three (N=3) and then to more trapped ions (N>3), and the most general entangled state vectors related to the outlined systems are obtained, for which the same scenario (for two trapped ions) is repeated. Comparing the latter cases (N≥3) with the previous one (N=2), we conclude that the entanglement is sensitive to the number of interacting trapped ions in addition to the distances between them in such a way that by increasing the number of trapped ions, the degree of entanglement between them and the field is increased. These results help one to control the entanglement between the field and the trapped ions in a more accurate manner by tuning the distances between the trapped ions and their numbers.
In this paper, we first study the interaction between a trapped two-level ion with a field reservoir in a general form. Then, we use thermal and squeezed vacuum reservoirs to continue our investigation. We try to find the equation of motion for the whole system density operator by applying the Weisskopf theory. Next, we evaluate the explicit form of the density matrix elements analytically. In the obtained density matrix elements, we arrive at some physical parameters of the trapped ion system as well as the applied field reservoirs by which one can control the transition probabilities between the trapped ionic states. We also introduce some hermitian operators for the trapped ion system and calculate the expectation values of their time evolution by considering the mentioned two reservoirs. In the continuation, we design a system for measuring the hermitian trapped ion’s operators theoretically. Finally, we find a way to transport the information of the trapped ion system by our proposed quantum processor.
In this paper, the dynamical properties of a trapped three-level ion which interacts with a two-mode quantized field in the presence of a Kerr medium are studied. The ion-field coupling is also considered to be intensity-dependent (nonlinear interaction). In this respect, the Λ, V and ladder-type three-level ions are considered. Via solving the time-dependent Schörodinger equation, the state vectors of the corresponding ion-field systems are analytically obtained after choosing special initial conditions. In the continuation, the mean phonon number as a criterion of trapping energy, the population inversion as a criterion of exchanging energy (between the ion and the field) and the linear entropy as a criterion of entanglement between the trapped ion and the field are studied. To get more general insight about the interaction conditions, the effect of intensity-dependent ion-field coupling (nonlinearity regime) on the dynamics of system is investigated. Finally, by comparing the obtained results for the three configurations of trapped three-level ions with the previously considered two-level one, we conclude that the degree of entanglement for three-level ions is more than the two-level ion. Also, among the considered trapped ions, the ladder-type ion possesses the most amount of entanglement with the quantized field, whereas the Λ-type possesses the least. We also conclude that, the general temporal behaviour of the other considered quantities (mean phonon number and population inversion) are the same for all three types of trapped three-level ions.
The interaction of a two-level atom with a single-mode radiation field through multiphoton transitions is studied in terms of the dressed-state formalism. It is shown that when only certain dressed states are initially excited, population trapping occurs. We particularize these results for some well-known models of one- and two-photon transitions. Photon statistics of the cavity mode is analyzed in terms of the quasiprobability distribution Q(\ensuremath{\alpha}). It is shown that when population trapping occurs some two-time correlation functions become stationary, which allows one to obtain the resonance spectrum as the Fourier transform of the two-time dipole autocorrelation function.
The statistical properties of the system consisting of a trapped ion and a single_mode quantized radiation field are investigated under the conditions of carrier excitation (ωL=ω0), red excitation ［ωL=ω０-jν(j=1,2,3)］ and blue excitation ［ωL=ω０+jν(j=1,2,3)］.The time_evolution of the state of the system,the photon_and phonon_state exchange conditions and the corresponding populations of the ion are discussed.The corresponding averages and variances for photon and phonon numbers,as well as the second_order correlation function for the photon and phonon fields etc,have been obtained.