PHYSICAL REVIEW A 87, 012132 (2013) Effects of detuning on tunneling and traversal of ultracold atoms through vacuum-induced potentials Fazal Badshah, 1 Muhammad Irfan, 1 Sajid Qamar, 2 and Shahid Qamar 1,* 1 Department of Physics and Applied Mathematics, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad 45650, Pakistan 2 Department of Physics, COMSATS Institute of Information Technology, Islamabad, Pakistan (Received 10 August 2012; published 31 January 2013) In this paper, we study the tunneling and traversal of ultracold two-level atoms through the potential induced by the vacuum cavity mode. In particular, we discuss the effects of off-resonant interaction between the cavity mode and atomic transition on tunneling time of the ultracold atoms through a high-Q mazer cavity. The phase time which may be considered as an appropriate measure of the time required for the atom to cross the cavity, exhibits some interesting features in the presence of off-resonant interaction. For example, switching between the sub and superclassical behaviors in phase time occurs for proper choice of detuning. Similarly, negative phase time appears for the transmission of atoms in both excited and ground states in the presence of off-resonant interaction. DOI: 10.1103/PhysRevA.87.012132 PACS number(s): 03.65.Xp, 42.50.−p, 03.75.Lm I. INTRODUCTION Tunneling is one of the most fundamental and important phenomenon in quantum mechanics, which provides the physical basis for many useful semiconductor devices and scanning tunneling microscope. Soon after the stimulating work of MacColl and Hartman [1,2] on the dynamics of wave packets through potential barriers, many tunneling time definitions were introduced [3–5]. Among all, phase time [2,6,7] is widely studied and well established, which measures how long it takes for the peak of the transmitted wave packet to emerge from the exit of the barrier. It is related to the energy derivative of the phase shift, and has been studied using numerical, experimental, and analytical methods [5,8–15], in quite detail. More recently, interesting effects related to the tunneling problem were studied. These include bounds and enhancement for the Hartman effect derived from the causality principle [16], superluminal tunneling as a weak measurement effect [17], the speedup effect due to the entanglement between the spin and the spatial degree of freedom in a magnetic field [18], and the reshaping mechanism of quantum tunneling [19]. The interaction of ultracold atoms with a high-Q microwave cavity has been a problem of considerable interest in recent years. The quantum theory of induced emission due to the quantized motion of the ultracold atoms passing through a micromaser cavity was established in a seminal paper by Scully et al. [20]. The drastic change in the atom-field coupling which results when the cold atom enters the cavity leads to an entirely different kind of emission known as microwave amplification via z-motion-induced emission of radiation (mazer). The dressed state analysis of the problem shows that the interaction of cold atoms with a single-mode cavity is equivalent to a combination of potential barrier and a well [21–23]. Recently, Arun and Agarwal [24] discussed the tunneling of ultracold two-level atoms through the vacuum-induced potential. It was shown that phase tunneling time for ultracold atoms exhibit both superclassical and subclassical behaviors * shahid_qamar@pieas.edu.pk which can be understood in terms of the momentum depen- dence of the transmission amplitudes. The passage of the atoms through the cavity involves a coherent addition of the transition amplitudes corresponding to both barrier and well; as a result it is unique. In our earlier study [25] we discussed the passage of ultracold three-level atoms through a high-Q bimodal cavity. It was shown that the presence of dark states and interference effects in cascade atomic configuration affect the phase tunneling time. In some recent studies, the effects of detuning have also been investigated in the context of the emission probability for two-level ultracold atoms passing through a high-Q microwave cavity [26]. It was shown that detuning adds a potential step effect that is not present for the resonant case. It results in a well-defined acceleration or deceleration (depending upon the sign of the detuning) of the excited atom that contributes a photon inside the cavity. The use of positive detuning provides a well-controlled cooling mechanism. It was also shown that the photon emission can be completely blocked by appropriate choice of the detuning [26]. The problem of mazer action is closely related to the velocity selection of ultracold atoms [27]. It was shown that the velocity selection for ultracold atoms can be very easily tuned and enhanced using off-resonant interaction [28–30]. In the earlier studies, the tunneling times of ultracold atoms were discussed in the resonant cases where the mode frequency is equal to the atomic transition frequency [24,25]. In this paper we discuss the effects of off-resonant interaction on the tunneling or traversal time for ultracold two-level atoms passing through a high-Q cavity. The atoms, which are assumed to be initially in their excited state, after interaction with the cavity field (initially in a vacuum state) may be transmitted or reflected while remaining in the same state or making a transition to the ground state. We calculate the phase tunneling time for both situations using stationary phase approximation. Our results show some interesting features of phase time in the presence of off-resonant interaction. For example, negative phase time is obtained for transmission of the atoms in both excited as well as in the ground state. In particular, for a proper set of parameters, we find that change in the sign of detuning switches the tunneling time 012132-1 1050-2947/2013/87(1)/012132(7) ©2013 American Physical Society