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Physica Scripta

Nonlinearity and nonclassicality in single-qubit laser operation

S. Ya. Kilin, Alexander B. Mikhalychev

Published Apr 1, 2014• 5 citations
Publisher

Abstract

We investigate an incoherently pumped single-qubit laser as a system, providing optical nonlinearity for the generation of nonclassical states. We show that, in the general case of arbitrary values of coupling strength and other system parameters, the stationary state of an incoherently pumped single-qubit laser is a nonlinear coherent state, being the phase-averaged eigenstate of the deformed annihilation operator of a specific form. The manifestation of the inherent nonlinearity of the system is found in the form of the stationary state nonclassicality, present for any values of the system parameters. Another important implication of inherent system nonlinearity and nonclassicality is the impossibility of describing the lasing effect by means of field-independent spontaneous and induced transition probabilities, as in the case of a conventional laser.

DOI
10.1088/0031-8949/2014/t160/014021

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We investigate an incoherently pumped single-qubit laser as a system, providing optical nonlinearity for the generation of nonclassical states. We show that, in the general case of arbitrary values of coupling strength and other system parameters, the stationary state of an incoherently pumped single-qubit laser is a nonlinear coherent state, being the phase-averaged eigenstate of the deformed annihilation operator of a specific form. The manifestation of the inherent nonlinearity of the system is found in the form of the stationary state nonclassicality, present for any values of the system parameters. Another important implication of inherent system nonlinearity and nonclassicality is the impossibility of describing the lasing effect by means of field-independent spontaneous and induced transition probabilities, as in the case of a conventional laser.
The stationary state of a single-atom (single-qubit) laser is shown to be a phase-averaged nonlinear coherent state---an eigenstate of a specific deformed annihilation operator. The solution found for the stationary state is unique and valid for all regimes of the single-qubit laser operation. We have found the parametrization of the deformed annihilation operator which provides superconvergence in finding the stationary state by iteration. It is also shown that, contrary to the case of the usual laser with constant Einstein coefficients describing transition probabilities, for the single-atom laser the interaction-induced transition probabilities effectively depend on the field intensity.
Based on the approximation of strong correlations between an atom and an intracavity field, which implies the equal probabilities of finding the atom in the ground state and n photons in the field and of finding the atom in an excited state and n −1 photons in the field, it is shown that the conditional states of a field generated by a single-atom laser are described by the diagonal part of the generalized coherent Mittag-Leffler state. The quasi-distributions P and Q of the intracavity-field probability amplitude are found, and the boundedness of the Glauber function on a segment is shown. The possibility of inversionless lasing is demonstrated, and the absence of a lasing threshold is found for some region of parameters. The regimes of generation of the amplitude-squeezed states of the field are studied and the parameters of the system providing the maximum squeezing are determined. It is shown that the atom-field states are entangled at weak pump intensities.
We investigate the dynamics of a single-qubit single-mode laser with continuous incoherent pumping of a qubit, the field mode being initially prepared in a coherent state. Analysis of partial differential equations for quasiprobability distributions helps us distinguish two stages of evolution (coherent and incoherent ones) with their characteristic features. The system can exhibit bistabilitylike behavior, similar to that reported for a single-atom laser with continuous coherent pumping, and is capable of generating Schr\"odinger cat states. Quantitative analysis of the field state nonclassicality shows that its maximum is reached at intermediate stages of evolution.
We show how a pair of superconducting qubits coupled to a microwave cavity mode can be used to engineer a single-atom laser that emits light into a nonclassical state. Our scheme relies on the dressing of the qubit-field coupling by periodic modulations of the qubit energy. In the dressed basis, the radiative decay of the first qubit becomes an effective incoherent pumping mechanism that injects energy into the system, hence turning dissipation to our advantage. A second, auxiliary qubit is used to shape the decay within the cavity, in such a way that lasing occurs in a squeezed basis of the cavity mode. We characterize the system both by mean-field theory and exact calculations. Our work may find applications in the generation of squeezing and entanglement in circuit QED, as well as in the study of dissipative few- and many-body phase transitions.
We have introduced the notion of ‘four-photon nonlinear coherent states’ (FPNCSs) as right eigenstates of the fourth power of the generalized annihilation operator A4. It has been shown that there are four possible sets of such states which all can be expressed as the deformed Schrödinger cat type states derived from the superposition of four nonlinear coherent states (NCSs) which are π/2 out of phase. The nonclassical properties of the FPNCSs would be studied for two well-known quantum systems, a Kerr-like medium and a trapped laser-driven ion far from the Lamb-Dicke regime. We have found that the depth or the domain of the nonclassical features of FPNCSs in terms of sub-Poissonian photon statistics, higher order squeezing as well as amplitude-squared squeezing are higher than the corresponding standard states. Specifically, the proposed states in this paper exhibit sub-Poissonian statistics over an extensive range of real amplitudes which is in contrast with the lower level of sub-Poissonian characteristics of four-photon standard coherent states. Also the depth or domain of the nonclassical features of the FPNCSs can be modified by changing the magnitude of the nonlinearity parameters. Analysis of the quasi-probability distributions like Glauber–Sudarshan P-function, Husimi Q-function and Wigner distribution function verifies the nonclassical nature of the presented states. Studying the cavity field evolution in the interaction of a single two-level atom with an even NCS in the framework of f-deformed Jaynes-Cummings model shows that such a system can potentially be considered as a possible source for generating the proposed states.
We consider a class of nonlinear coherent states, which are right-hand eigenstates of the product of the boson annihilation operator and a nonlinear function of the number operator. Such states may appear as stationary states of the center-of-mass motion of a trapped and bichromatically laser-driven ion far from the Lamb-Dicke regime. Besides coherence properties, they exhibit nonclassical features such as amplitude squeezing and self-splitting, which is accompanied by pronounced quantum interference effects. \textcopyright{} 1996 The American Physical Society.