Vacuum Rabi oscillation induced by virtual photons in the ultrastrong coupling regime

Vacuum Rabi oscillation induced by virtual photons in the ultrastrong coupling regime

C. K. Law Department of Physics and Institute of Theoretical Physics, The Chinese University of Hong Kong, Shatin, Hong Kong Special Administrative Region, People’s Republic of China

We present an interaction scheme that exhibits a dynamical consequence of virtual photons carried by a vacuum-field dressed two-level atom in the ultrastrong coupling regime. We show that, with the aid of an external driving field, virtual photons provide a transition matrix element that enables the atom to evolve coherently and reversibly to an auxiliary level accompanied by the emission of a real photon. The process corresponds to a type of vacuum Rabi oscillation, and we show that the effective vacuum Rabi frequency is proportional to the amplitude of a single virtual photon in the ground state. Therefore the interaction scheme could serve as a probe of ground state structures in the ultrastrong coupling regime.

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A single-mode electromagnetic field interacting with a two-level atom has been a fundamental model in quantum optics capturing the physics of resonant light-matter interaction. In particular, the Jaynes-Cummings (JC) model JC1 (); JC2 (), which describes the regime where the interaction energy is much smaller than the energy scale of an atom and a photon , has tremendous applications in cavity QED Kimble_review (); Haroche () and trapped ion systems Wineland (). Recently, there has been considerable research interest in the ultrastrong coupling regime where becomes comparable to and . Such a regime has been explored by experiments in various related systems with artificial atoms and cavity photon resonators, including superconducting qubit in coplanar waveguide Niemczyk () or LC resonator BSshift (), microcavities embedding doped quantum wells Gunter (); Todorov (), and two-dimensional electron gas coupled to metamaterial resonators Scalari (). In addition, theoretical investigations have also found novel phenomena in the ultrastrong coupling regime, such as the asymmetry of vacuum Rabi-splitting Cao (), photon blockade Ridolfo (), nonclassical states generation Ashhab1 (), superradiance transition Ashhab2 (), and collapse and revivals dynamics Casanova ().

A key feature in the ultrastrong coupling regime is the significant number of virtual photons existing around the vacuum-field dressed atom. These virtual photons are generated by counter-rotating terms in the Hamiltonian, and they can have direct physical consequences. For example, by modulating the atom-field coupling strength virtual photons can be released as a form of quantum vacuum radiation Liberato (). In this paper we address a different effect of the vacuum-field dressed atom, namely, a kind of vacuum Rabi oscillations that would not occur if virtual photons are absent.

Specifically, we investigate the quantum dynamics of a driven quantum Rabi model. The configuration of our system is shown in Fig. 1 in which a -type three-level atom is confined in a single-mode cavity. The atomic levels and are coupled to a cavity field of frequency . These two atomic levels and the cavity field mode constitute a Rabi model. In addition, there is an external classical field driving the transition between and the third atomic level . We note that some theoretical aspects of three-level artificial atoms in circuit QED was discussed in YouNori (), and type superconducting atoms have been demonstrated in experiments 3level_1 (); 3level_2 (); 3level_3 (). Recently Carusotto et al. have studied the dynamics of a related system in a different driving configuration Carusotto ().

The Hamiltonian of our system is given by ,


where is the Hamiltonian of the Rabi model Rabi (),


Here is the (bare) transition frequency between and , and is the transition frequency between and . The parameter denotes the atom-cavity coupling strength, and the classical driving field has a frequency and an interaction strength . In writing , we have kept counter-rotating terms because is comparable to in the ultrastrong coupling regime. Note that the coupling between the cavity mode and the level is assumed to be weak and so that it is not included in the Hamiltonian.

Figure 1: (Color online) Interaction scheme of a type three-level atom in a cavity. The atomic states and and a cavity field mode of frequency form a quantum Rabi model described by , and an external classical field of frequency drives the transition between and .

Initially the system is prepared in the ground state of , which is the lowest-energy state of the system in the absence of the driving field. Our task is to determine the dynamics after the driving field is turned on. To analyze the problem, we apply a unitary transformation to simplify the Hamiltonian. It is known that for low energy states of the Rabi model, can be transformed to into a form of Jaynes-Cummings Hamiltonian approximately by a unitary operator Gan (). Here the operator and its parameters are defined by:


Then it can be shown that is approximately given by Gan (); Zheng1 (); Zheng2 (); Oh ()


where describes a JC model in which the atomic frequency and cavity-atom interaction strength are renormalized as and , respectively.

Note that in Eq. (6) is an approximation to , and the difference describes multi-photon processes that correspond to higher order corrections Gan (); Zheng1 (); Zheng2 (); Oh (). Since is the ground state of , is an approximated ground state of in the original frame. The accuracy of such an approximation has been tested in Ref. Gan (). Specifically, if is comparable but smaller than , the ground state energy of has a good agreement with that of obtained by exact numerical calculations over a range of parameters. For example in the case , the approximated ground state energy obtained by has the percentage error about .

Now we perform the transformation for our system Hamiltonian , which becomes,


Since , we expand the hyperbolic sine and cosine operator functions in normal order up to first order in ,


Therefore the transformed Hamiltonian becomes,


where is a renormalized driving field strength, and the last term indicates a new coupling between and through the cavity field mode.

A further simplification can be made by exploiting resonance when is tuned to a certain resonance frequency defined by the undriven system. In this paper we consider the resonance at


which corresponds to the transition between to , since the square bracket term is the approximate ground state energy of by the transformation method. By the condition (11), and are resonantly coupled, but and is far away from resonance (the corresponding detuning is of order ). Therefore if is not too strong, the system is confined to the two resonantly coupled states, i.e., all off-resonant transitions may be ignored. In this way in the interaction picture is reduced to


Eq. (12) indicates that the system would execute a form of vacuum Rabi oscillations, in which behaves as an excited atom in the vacuum field, and behaves as an ground atom with a single photon. In cavity QED, such oscillations lead to vacuum Rabi splitting VR1 (); VR2 (); VR3 (). Note that the effective vacuum Rabi frequency here is , which is significant in the ultrastrong coupling regime where is comparable to .

It is useful to go back to the original frame in which the Rabi oscillations occur between the states and . Since , an initial ground state will evolve to after half of a Rabi period. If we switch off the external field at this moment, the single photon described by will be free to escape the cavity because the atom in the state does not couple to the cavity field when , i.e., the photon cannot be reabsorbed by the atom. In this way, a pulse of the driving field can generate a real photon deterministically while the atom is excited to the state.

To gain a better insight of the physical process without relying on the approximation made in Eqs. (6) and (10), we express the Hamiltonian by the eigenbasis of . Let be an eigenvector of with the eigenvalue , i.e., (the ground state is denoted by ), and consider the expansion with the coefficients . Therefore


In this way, the Hamiltonian (1) in the interaction picture becomes,


At the resonant frequency , and are resonantly coupled. If we keep only the resonant terms, then we have


Comparing with in Eq. (12) and noting that , describes the same type of resonant interaction as . However, we emphasize that in Eq. (15) is a more accurate interaction Hamiltonian than because is derived directly from the eigenbasis of without making use of the approximation in Eq. (6). In this sense, the resonant condition (11) can be improved by replacing the square bracket term by .

Figure 2: (Color online) Probability amplitude of in the ground state of as a function of the coupling strength for the case. The solid red line corresponds to exact numerical values, and the dashed blue line is obtained from the approximated ground state according to Eq. (6).

The role of virtual photons is now explicitly seen in Eq. (15) through the effective vacuum Rabi frequency . This is because is precisely the probability amplitude of a single virtual photon state in . In other words, we may interpret that the interaction described in Eq. (15) is induced or mediated by a virtual photon. In Fig. 2, we plot (solid line) as a function of for the case , and the figure shows that the magnitude of is appreciable in the ultrastrong coupling regime. As a comparison, we also plot the approximate amplitude (dashed line) obtained from . For the parameters used in Fig. 2, we see that the approximation agrees well with the exact numerical calculation up to .

We have tested our prediction of the virtual-photon-induced Rabi oscillations by solving numerically the Schrödinger equation defined by the Hamiltonian (1) with the initial state . In Fig. 3 we plot the exact numerical probability of the system in the state as a function of time. The parameter used in the figure is served as an example of ultrastrong coupling. We see the Rabi cycles as predicted by the Hamiltonians (12) or (15) for relatively weak driving fields with . At a stronger driving field with (red solid line), and there is a high frequency pattern due to counter rotating terms of the classical driving field, and the Rabi oscillations are less perfect in the sense that the maximum is smaller than one. Such a behavior is understood because the off-resonance transitions neglected in Eq. (12) or (15) would generate energy shifts which in turn could bring the driven system out of resonance. As a result, the amplitude of oscillations in is reduced. Since these energy shifts are generally proportional to , as long as is small compared with detunings associated with off-resonance transitions, it would be safe to use Eq. (15), and this is demonstrated in Fig. 3 for up to .

Figure 3: (Color online) Probability of as a function of time for (blue long dashed), (green short dashed) and (red solid). The parameters used are: , , , and the numerical ground state energy . The figure is essentially the same if in Eq. (11) is used.

Finally, it is worth noting that the Hamiltonian in Eq. (14) has higher resonances at for odd positive integers . The requirement of an odd is because has a definite parity in which the atomic state and odd photon numbers are connected. In the case , the driving field at the corresponding would resonantly excite the atom to with the emission of three real photons. The effective Hamiltonian would be of the same form of (15), but with and replaced by and , i.e., the effective Rabi frequency is proportional to . Such a three-photon resonance was also observed in our numerical calculations.

To conclude, we have shown that virtual photons in the ultrastrong coupling regime can play a key role in quantum dynamics by providing the transition matrix elements that allow the system to access relevant quantum states of interest. In our scheme, the system can exhibit a form of vacuum Rabi oscillations which can be considered as a signature of virtual photons. Since our main focus in this paper is on the interaction induced by virtual photons, decoherence effects have not been included in the discussion. However, as long as the decoherence times is sufficiently short, coherent dynamics predicted by the Hamiltonian (12) or (15) would be justified. Specifically, given a vacuum Rabi period , the cavity field damping rate and atomic decay rate , the condition () ensures that the system can execute a Rabi cycle without being affected by the damping, and this is achievable in the ultrastrong coupling regime with moderate small ’s. For the parameters used in Fig. 3, for example, would be sufficient. We emphasize that a finite interaction time within is of practical importance, since the interaction (12) or (15) is switchable via the driving field. This feature could be a tool for performing quantum operations on qubits formed by the atom or the field, as well as for deterministic single-photon generation Law (); Kimble (); Grangier (). In addition, since the effective vacuum Rabi frequency is proportional to the corresponding virtual photon amplitude, our scheme can be used to probe the ground state structure of the quantum Rabi model.

The author thanks Dr. H. T. Ng for discussions. This work is partially supported by a grant from the Research Grants Council of Hong Kong, Special Administrative Region of China (Project No. CUHK401812).


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