Decay dynamics in a strongly driven atom-molecule coupled system
Within the framework of master equation, we study decay dynamics of an atom-molecule system strongly coupled by two photoassociation lasers. Summing over the infinite number of electromagnetic vacuum modes that are coupled to the system, we obtain an integro-differential master equation for the the system’s reduced density matrix. We use this equation to describe correlated spontaneous emission from a pair of electronically excited diatomic ro-vibrational states. The temporal evolution of emitted radiation intensity shows quantum beats that result from the laser-induced coherence between the two excited states. The phase difference between the two driving fields is found to significantly affect the decay dynamics and the beats. Our results demonstrate the possibility to control decay and decoherence in the system by tuning the relative intensity and the phase between the two lasers. We further show that, if the ground-state continuum has a shape resonance at a low energy, then the quantum beats show two distinctive time scales of oscillations in the strong coupling regime. One of the time scales originates from the energy gap between the two excited states while the other time scale corresponds to the collision energy at which free-bound Franck-Condon overlap is resonantly peaked due to the shape resonance.
Over the last two decades there has been tremendous developments in high precision spectroscopy with cold atoms. It is now possible to access low lying rotational levels of a diatomic molecule formed by photoassociation (PA) in cold atoms. For an excited long-ranged molecule (formed via narrow-line inter-combination photoassociative transitions as in cold bosonic Sr or Yb atoms) the lifetime of excited rotational levels can be as large as 10 microseconds. Such metastable molecular excited states are now experimentally accessible using optical spectroscopic techniques. This opens up the possibility of creating and studying quantum superposition states between molecular rotational states as well as superpositions between molecular states and collisional continuum of scattering states between ground-state atoms. In a PA process, the scattering state between two ground-state cold atoms become optically coupled to an excited diatomic bound (molecular) state. In the weak photoassociative coupling regime, it is interpreted as a loss process and PA spectra are detected in terms of the loss of atoms due to spontaneous emission from the excited bound state. However, in the strong-coupling regime, the continuum of the scattering states between two atoms becomes strongly-coupled leading to atom-molecule or continuum-bound dressed state quantum dynamics. A transition from the state of two colliding atoms to a diatomic bound state is generally referred to as free-bound transition. To develop a proper understanding of correlated quantum dynamics of an atom-molecule coupled system, it is important to formulate a density matrix formalism in continuum-bound dressed state picture to appropriately account for spontaneous emission and decoherence in the dynamics. The influence of spontaneous emission on a continuum-bound coupled system had been earlier discussed [1, 2, 3] in the context of autoionizing states. Early work by Agarwal et al.  treated spontaneous emission of continuum-bound coupled autoionising Fano state  within the master equation framework. We adapt such an approach to develop a master equation for the atom-molecule coupled system at ultracold temperatures. Unlike most of the the standard systems in quantum optics dealing with dissipation and decoherence from discrete levels, the master equation approach in the present context is rather involved due to the continuum of states of collision between ground-state atoms.
Here we develop a model to describe coherent effects in an atom-molecule system and demonstrate that, after having created coherent superposition between two rotational states by strongly driving two photoassociative transitions with two lasers, the superposition can be detected as rotational quantum beats in florescence light emitted from the correlated rotational levels. Considering Yb as a prototype system, we first analyze the ideal situation of the dressed continuum among two exited rotational states and the bare continuum of scattering between ground-state Yb atoms in the absence of spontaneous emission. This provides understanding of how the relative intensities and the phases between the two driving PA lasers can be used as knobs to manipulate coherence between the two excited states. We then discuss the effects of spontaneous emission and decoherence on the dynamical properties of the dressed states. Our results show that by judiciously adjusting relative intensity and the phase between the two lasers it is possible to inhibit spontaneous emission from the two correlated excited molecular states and to preserve or manipulate the coherence between the states.
Quantum beats in radiation intensity arise from coherent superposition of two long-lived excited states. Such state superpositions and their manipulations are of considerable recent interest in quantum information science. The possibility of using quantum beats as a spectroscopic measure for quantum superposition was discussed as early as in 1933 . Experimentally, spectroscopic study of quantum beats started since 1960s . The use of lasers to create quantum superposition and detect resulting quantum beats in fluorescence started in early 1970s . Forty years ago, Haroche, Paisner and Schawlow  demonstrated quantum beats in florescence light emitted from the excited hyperfine levels of a Cs atom as a signature of quantum superposition between the excited atomic states. Since then quantum beats in fluorescence spectroscopy have been studied in a variety of physical situations [9, 10]. These techniques open up new possibilities for studying excited state properties, state preparation and manipulation as well as collisional and spectroscopic aspects of ultra-cold atoms and molecules.
The paper is organized as follows: In section 2, the model is presented and discussed. We develop a master equation approach to spontaneous emission in continuum-bound atom-molecule coupled system in section 3. A solution for the master equation is presented. Numerical results are analyzed in section 4. The paper is concluded in section 5.
2 The model
Our model consists of two excited diatomic molecular ro-vibrational states and (belonging to the same molecular electronic state) coupled to the ground-state bare continuum of scattering states, by the lasers 1 and 2, respectively. Initially either or or partially both are populated due to two photoassociation lasers and of frequencies and , tuned near and transitions, respectively. The ground continuum is assumed to have only one internal molecular state with only one threshold and no hyperfine interaction. We assume that the two free-bound PA transitions between the ground-state continuum and the two excited ro-vibrational states are strongly driven so that the spontaneous emissions from these two bound states to the continuum are negligible as compared to the corresponding stimulated ones. However, these two driven bound states can spontaneously decay to other bound states in the ground electronic configuration. The model we describe in this paper may be contrasted with that in  where two excited ro-vibrational states populated by photoassociation from ground-state continuum are assumed to decay to the same continuum only. In the present paper, we primarily discuss the creation of laser-induced coherence and its implications in decay dynamics within the framework of master equation approach while the earlier work  concerns the creation of vacuum-induced coherence (VIC) with a more simplified model that is solvable by Wigner-Weisskopf method. Compared to the model used in , the present model is more realistic as it considers decay of the system outside the dressed continuum. Moreover, the present work shows exciting possibilities of manipulating excited state coherences using the relative phase between two lasers.
The Hamiltonian governing the dynamics of this system can be written as , where is the system Hamiltonian with two parts: the first part describes coherent dynamics with the two strong PA couplings. On the other hand, the second part is the interaction part of the system with a reservoir of vacuum electromagnetic modes. Explicitly, one can write
where is the Hamiltonian describing the interaction of the system with the reservoir of vacuum modes. Here are the binding energies of the bound states n; is the bare continuum state. In deriving the above Hamiltonian, we have used rotating wave approximation (RWA) . In RWA, one works in a frame rotating with the frequency of the sinusoidally oscillating field interacting with a two-level system (TLS) and neglects the counter-rotating terms that oscillate with the sum of the field and the system frequencies. It primarily relies on two conditions: (i) the system relaxation time is much larger than the time period of oscillation of the field and (ii) Rabi frequency or the system-field coupling is much smaller than the transition frequency of TLS. These conditions are in general fulfilled in most cases of a TLS interacting with a monochromatic optical field and therefore RWA can be regarded as a cornerstone for studying quantum dynamics of TLS. Nevertheless, RWA may break down in case of intense laser fields or short pulses when the Rabi frequency or the coupling becomes comparable with the system frequency. Generally, this may happen when the laser intensity is of the order of W cm or higher. In PA experiments the laser intensity is much lower, typically in the W cm or kW cm. Strong-coupling regime in ultracold PA can be reached with laser intensities higher than 1 kW cm but much lower than 1 MW cm. For driven TLS, corrections beyond RWA and in terms of Bloch-Siegert shift  have been discussed by Grifoni and Hanggi . The corrections to RWA can be formulated as a systematic expansion in terms of the ratio of Rabi frequency to the field frequency . In case of two coupled TLS, there exists a parameter regime where leading order term in the expansion vanishes rendering the next higher order term to be significant . However, such situation does not arise in our case and so RWA remains valid.
With electric dipole approximation, the laser coupling for the absorptive transition from the bare continuum to the th excited bound state is given by
where , and are the wave vector, electric field and phase of the th laser, respectively; is the center-of-mass position vector of the two atoms and is the free-bound molecular dipole moment associated with the th bound state. The electric dipole approximation here dictates that , where is the separation between the two atoms. We have thus used in writing the above equation. The operator is a raising operator, denotes the annihilation operator of the vacuum field and is the dipole coupling with , being the wave number, the transition dipole moment between th excited bound state and the ground bound state , the polarization of the field and the amplitude of the vacuum field and is the binding energy of the bound state . The Hamiltonian is exactly diagonalizable [13, 18] in the spirit of Fano’s theory . The eigenstate of is a dressed continuum expressed as
with the normalization condition . The coefficients and are derived in Ref .
By using partial-wave decomposition of the bare continuum , we have where and are the rotational and the magnetic quantum number, respectively, of the th excited bound state in the space-fixed (laboratory) coordinate system. Note that represents amplitude for free-bound transition from incident partial-wave state to the th bound state. To denote the amplitude for reverse (bound-free) transition, we use the symbol . Accordingly, we can write and where represents a unit vector along the incident relative momentum between the two atoms. Explicitly,
with being the binding energy of th excited bound state measured from the threshold of the excited state potential, is the light shift of the bound state and with is the laser frequency of -th laser and the atomic transition frequency. The two lasers interacting with the system results in an effective coupling
between the two bound states where
The term , is the stimulated linewidth of the -th bound state due to continuum-bound laser coupling. Note that the light shift is the sum over all the partial light shifts
3 Master equation
The system Hamiltonian can be written in dressed basis as
To derive master equation we work in the dressed continuum basis of the system Hamiltonian. We express bare basis in terms of dressed basis as follows
By substituting all bare basis states with there expansions in terms of dressed basis, we can write system-reservoir interaction Hamiltonian in terms of dressed basis. In the interaction picture, the effective system-reservoir interaction Hamiltonian of the driven system interacting with a reservoir of vacuum modes can be written as
where the superscript ’’ refers to interaction picture, and
Let denote the system-reservoir joint density matrix. Following Agarwal , the projection operator is defined by
wherer and are the density matrices of vacuum and the dressed system () system, respectively. With the use of this projection operator, Liouville equation under Born approximation can be expressed  as
is the density matrix in the interaction picture. Here
Tracing over the vacuum states, we obtain
From equation (22), making use of Markoff approximation, we derive the equations of motion of reduced density matrix elements in dressed basis. These are
with being the difference between -th and -th lasers and
is a function of . is the spontaneous linewidth of th excited state and is the vacuum-induced coupling between the two excited states [12, 13]. Note that in Eq. (27) we have neglected the light shift of the excited levels in comparison to the transition frequency which is in the optical frequency domain while the typical light shifts as shown in Fig.3 are of the order of MHz. The expression (27) is obtained in the following way: We first substitute equation (21) into equation (22) and express the vacuum coupling in terms of corresponding bound-bound transition dipole moment as described after equation (3). The sum over and is replaced by an integral over the infinite vacuum modes. Using standard Markoffian approximation, one can carry out first the integration over and then over the vacuum modes to arrive at the expression for as given in equation (27). The normalization condition is
The density matrix elements can be expressed in bare basis by the transformation
In interaction picture, and the equation (25) can be rewritten as
The solution of the above equation can be formally expressed as
The delta function on the right hand side is the initial value . The quantity given in equation. (26) is expressed in terms of the product of the amplitudes of the th and th bound states in energy-normalized dressed continuum of equation (4). If vacuum couplings are neglected, the bound-state probability densities are given by and the coherence terms with . It is important to note that, apart from causing spontaneous decay of the th bound-state probability with decay constant , vacuum couplings of the two excited bound states and with the ground bound-state effectively give rise to vacuum-induced coherence (VIC)  between the two excited bound states with coupling constant . Recently, atom-molecule coupled photoassociative systems are shown to be better suited for realizing VIC . Though the quantities are calculable from equation (27) when the molecular transition dipole moments are given, for simplicity of our model calculations, we have set . In fact, since we consider that both the excited bound states belong to the same vibrational level but differing only in rotational quantum number, the spontaneous linewidths and would not differ much. Furthermore, since , we have for the case considered here. The stimulated line width is a function of the collision energy for the ground state scattering between the two ground state atoms. Both in the limits and , vanishes. Let us fix an energy near which both and attain their maximum values. It is then possible to write equation (26) in the form
where and are the dimensionless quantities. The absolute value of is less than unity. Supposes, the intensities of the two lasers are high enough so that for both the excited bound states. In that case, using or the product as a small parameter, we can expand equation (30) in a time-ordered series
It is worthwhile to point out that this method of solution is similar in spirit to that of time-dependent perturbation, however it differs in essence because we have used dressed state amplitude as a small parameter and not the atom-field coupling. If a large number of terms are taken, then the expansion essentially provides solution for any time. However, numerically calculating higher order terms becomes increasingly involved because of larger number of multiple integrals in energy variable appearing in higher order terms. We therefore restrict our numerical studies to a few leading order terms as described in the next section.
5 Results and discussions
Driven by the two strong lasers, the system is prepared in a dressed continuum given by equation (4). Since this state is an admixture of the two excited bound states, it is subjected to spontaneous emission. We include spontaneous emission by considering the dressed levels to decay to a third bound level, thereby neglecting the decay of the excited states to the ground-state continuum inside the dressed-state manifold.
To discuss the effects of the phase-difference between the two lasers, the laser intensities and , and the detunings and on decay dynamics, we first rewrite the dressed-state amplitude of equation (5) in the form
The additional shift for the th excited bound state due to laser-induced cross coupling with the other () excited bound state is
Here with being the contribution to the total stimulated line width due to the cross coupling. In expression (34), the first term in the numerator corresponds to single-photon transition amplitude due to th laser while the second term describe a net 3-photon transition amplitude with 2 photons coming from the th laser and the other one from th laser.
The foregoing discussion has so far remained quite general. Now, we apply our method to ultracold Yb atoms. For numerical illustration, we use realistic parameters following the recent experimental [20, 21, 22, 23] and theoretical  works on PA with Yb. We have chosen Yb system because this offers some advantages compared to other systems. For instance, it has no hyperfine structure and the ground-state molecular potential of Yb is spin-singlet only. Furthermore, it has spin-forbidden inter-combination transitions. The total rotational quantum number is given by where is the total electronic angular momentum. For numerical work, we specifically consider a pair of Yb atoms being acted upon by two co-propagating linearly polarized cw PA lasers. The polarizations of both lasers are assumed to be same. This geometry is the same as used in Ref  for manipulation of -wave atom-atom interactions. For our numerical work, we consider that the two lasers drive transitions to the molecular bound states 1 and 2 characterized by by the rotational quantum numbers and , respectively; of the same vibrational level . The two bound states belong to (Hund’s case c) molecular symmetry meaning that the projection of on the internuclear axis being zero. Since Yb atoms are bosons, only even partial waves are allowed for the scattering between the two ground state atoms. Free-bound dipole transition selection rules then dictate that the bound state 1 can be accessed from - and -wave scattering states, while the bound state 2 is accessible from - and -wave only. Thus -wave ground scattering state is coupled to both the excited states by the two PA lasers resulting in the laser-induced coupling term . In general, -wave scattering amplitude is small at low energy. But, fortunately for Yb atoms, there is a -wave shape resonance [20, 21] in the K temperature regime leading to significant enhancement in -wave scattering amplitude at relatively short separations where PA transitions are possible. In our calculations we neglect -wave contributions.
As we prepare the system in a desired dressed continuum, the populations of the two excited bound states and the coherence between them depend on the relative intensity and phase between the two lasers. In the absence of spontaneous emission (idealized situation), the dressed state properties correspond to the initial conditions for our model. figure 1 shows variation of the initial populations and the coherence as a function of the intensity of either laser for the intensity of the other laser being fixed at 1 W cm. For all our numerical work, we set the spontaneous line width MHz .
The variation of stimulated line widths and light shifts of the two bound states of Yb as a function of collision energy for the laser intensities W cm and zero detunings are displayed in figure 2. The shift (stimulated line width ) is a sum of - and -wave partial shifts ( stimulated line widths) while the shift and the stimulated line width are made of mainly -wave partial shift and width, respectively; with no contribution from -wave. From figure 2 we notice that the shifts of both bound states as a function of energy change rapidly from negative to positive value near 194 K and the stimulated line widths of both bound states exhibit prominent peaks at that energy. This can be attributed to a -wave shape resonance [20, 24]. We have found that the -wave partial stimulated line widths of both the bound states near shape resonance are comparable. For the first bound state, the value of the -wave partial stimulated line width near the resonance is found to exceed the -wave partial line width by about 2 orders of magnitude. In figure 3, mutual light shift and stimulated line width due to the coupling between the bound states are shown as a function of collision energy in Hz. The mutual shifts and widths arise from the coupling of the -wave scattering state with the two bound states by the two lasers. Owing to the existence of the -wave shape resonance, the laser couplings of the -wave scattering state to both bound states become significant, and so are the mutual shifts and stimulated line widths.
Figures 4 and 5 exhibit intensity-dependence of the coherence . The purpose of plotting these two figures is to assert that it is possible to prepare the dressed system with a desirable coherence between the two excited bound states by judiciously selecting relative intensities and phases between the two lasers. It is interesting to note that -wave shape resonance has a drastic effect on the properties of dressed continuum. Because of this resonance, the -wave contributions to the amplitudes of transition to both the bound states are large even at a low temperature allowing an appreciable cross coupling to develop between the two bound states. For very large laser intensities at , light shifts would be so large that the system will be effectively far off resonant and therefore .
For calculating time-dependence of the density matrix elements for all the times, we need to calculate a large number of terms appearing on the right hand side of equation (33) order by order in . This is a laborious and time-consuming exercise. Instead, to demonstrate the essential dynamical features arising from quantum superposition of the two rotational states, we restrict our study of decay dynamics to relatively short times. Inserting equation (26) in equation (33), retaining the terms up to first order in , we have
where and with . Here . Similarly, the coherence term can be calculated up to the first order in . These solutions hold good for or equivalently, for both .
The decay dynamics of the populations and as a function of the scaled time are shown in figures 6 and 7, respectively. These results clearly exhibit that, when the system is strongly driven by two lasers, the decay is non-exponential and has small oscillations. The oscillation are particularly prominent for short times. In the long time limit the oscillations slowly die down. However, the oscillations can persist for long times if couplings are stronger. We have chosen the values of the laser intensities and such that the initial values of dressed population or are the same for those intensities. We notice that, though the values (or ) for a set of values for a fixed