Suppressing deleterious effects of spontaneous emission in creating bound states in cold atom continuum

TL;DR

This paper proposes a method using vacuum-induced coherence to suppress spontaneous emission, enabling the creation of bound states in the continuum in cold atoms, which was previously hindered by decay effects.

Contribution

It introduces a novel approach employing vacuum-induced coherence to counteract spontaneous emission, facilitating the experimental realization of bound states in continuum in cold atom systems.

Findings

Vacuum-induced coherence can cancel spontaneous emission effects.

Microwave dressing of molecular states achieves necessary conditions for VIC.

Proposes a feasible experimental setup for BIC in cold atoms.

Abstract

In a previous paper [B. Deb and G. S. Agarwal, Phys. Rev. A 90, 063417 (2014)], it was theoretically shown that, magneto-optical manipulation of low energy scattering resonances and atom-molecule transitions could lead to the formation of a bound state in continuum (BIC), provided there is no spontaneous emission. We find that even an exceedingly small spontaneous decay from exited molecular states can spoil the BIC. In this paper, we show how to circumvent the detrimental effect of spontaneous emission by making use of vacuum-induced coherence (VIC) which results in the cancellation or suppression of spontaneous emission. VIC occurs due to the destructive interference between two spontaneous decay pathways. An essential condition for VIC is the non-orthogonality of the transition dipole moments associated with the decays. Furthermore, the interference between decay pathways requires that the spacing between the two decaying states must be comparable to or smaller than the square root of the product of the two spontaneous linewidths. We demonstrate that these conditions can be fulfilled by microwave dressing of two appropriately chosen molecular excited states, opening a promising prospect for the experimental realization of BIC of cold atoms.

Equations

$$\mid e_1⟩$$

$$\mid e_2⟩$$

$$\mid 3⟩\equiv \mid c⟩\otimes \mid N_1{\omega }_1,N_2{\omega }_2⟩\equiv \mid c,N_1,N_2⟩$$

$$\mid 2⟩\equiv \mid e_2⟩\otimes \mid N_1{\omega }_1,(N_2-1){\omega }_2⟩\equiv \mid e_2,N_1,N_2-1⟩$$

$$\mid 1⟩\equiv \mid e_1⟩\otimes \mid (N_1-1){\omega }_1,N_2{\omega }_2⟩\equiv \mid e_1,N_1-1,N_2⟩$$

$$\mid E⟩\equiv \mid E{⟩}_b\otimes \mid N_1{\omega }_1,N_2{\omega }_2⟩\equiv \mid E,N_1,N_2⟩$$

$$H_1^0$$

$$\displaystyle+\Big{(}E_{2}+N_{1}\hbar\omega_{1}+\left(N_{2}-1\right)\hbar\omega_{2}\Big{)}\mid e_{2},N_{1},N_{2}-1\rangle\langle e_{2},N_{1},N_{2}-1\mid$$

$$\displaystyle+\Big{(}E_{3}+N_{1}\hbar\omega_{1}+N_{2}\hbar\omega_{2}\Big{)}\mid c,N_{1},N_{2}\rangle\langle c,N_{1},N_{2}\mid$$

$$\displaystyle+\int\limits_{E_{th}}^{\infty}dE\Big{(}E+N_{1}\hbar\omega_{1}+N_{2}\hbar\omega_{2}\Big{)}\mid E,N_{1},N_{2}\rangle\langle E,N_{1},N_{2}\mid\bigg{]}$$

$$V_1=\sum _{N_1,N_2}$$

$$+\underset{E_{th}}{\overset{\infty }{\int }}dE{\Lambda }_{1E}\mid e_1,N_1-1,N_2⟩⟨E,N_1,N_2\mid$$

$$+\underset{E_{th}}{\overset{\infty }{\int }}$$

$$\mid 1k⟩\equiv \mid f⟩\otimes \mid N_1{\omega }_1,N_2{\omega }_2,k⟩\equiv \mid f,N_1,N_2,k⟩$$

$$\mid 2k^{{}^{\prime }}⟩\equiv \mid f⟩\otimes \mid N_1^{{}^{\prime }}{\omega }_1,N_2^{{}^{\prime }}{\omega }_2,k^{{}^{\prime }}⟩\equiv \mid f{N}_1^{{}^{\prime }},N_2^{{}^{\prime }},k^{{}^{\prime }}⟩$$

$$⟨1k\mid 2k^{{}^{\prime }}⟩={\delta }_{N_1,N_1^{{}^{\prime }}}{\delta }_{N_2,N_2^{{}^{\prime }}}\delta \left(k-k^{{}^{\prime }}\right)$$

$$H_2^0=\sum _{N_1,N_2}\underset{0}{\overset{\infty }{\int }}kdk\mid 1k⟩⟨1k\mid +\sum _{N_1^{{}^{\prime }},N_2^{{}^{\prime }}}\underset{0}{\overset{\infty }{\int }}{k}^{{}^{\prime }}d{k}^{{}^{\prime }}\mid 2k^{{}^{\prime }}⟩⟨2k^{{}^{\prime }}\mid$$

$$V_2=\sum _{N_1,N_2}\underset{0}{\overset{\infty }{\int }}dk{V}_{1k}\mid 1⟩⟨1k\mid +\sum _{N_1^{{}^{\prime }},N_2^{{}^{\prime }}}\underset{0}{\overset{\infty }{\int }}d{k}^{{}^{\prime }}{V}_{2k^{{}^{\prime }}}\mid 2⟩⟨2k^{{}^{\prime }}\mid +h.c$$

$$⟨e_n\mid H_{eff}\mid e_n⟩=\left(E_n-\hslash {\omega }_n+E_n^{{}^{\mathrm{sh}}}\right)-\frac{i\hslash }{2}\left({\Gamma }_n+{\gamma }_n\right),n=1,2$$

$$⟨c\mid H_{eff}\mid c⟩=E_3+E_F^{{}^{\mathrm{sh}}}-\frac{i\hslash }{2}{\Gamma }_F$$

$$⟨e_1\mid H_{eff}\mid e_2⟩=\alpha -\frac{i\hslash }{2}\left[{\gamma }_{{}_{LIC}}+{\gamma }_{{}_{VIC}}\right]$$

$$⟨e_n\mid H_{eff}\mid c⟩={\beta }_n+\hslash {\Omega }_{nc}^{{}^{\prime }}-\frac{i\hslash }{2}{\Gamma }_{nF},n=1,2$$

$$⟨c\mid H_{eff}\mid e_n⟩=⟨e_n\mid H_{eff}\mid c⟩$$

$$E_3+E_F^{{}^{\mathrm{sh}}}=-{\left(k_c{a}_s\right)}^{-1}\frac{\hslash {\Gamma }_F}{2}$$

$${\beta }_n=\mathcal{P}\underset{0}{\overset{\infty }{\int }}dE\frac{{\Lambda }_{nE}^{{}^{\prime }}{V}_{3E}^{{}^{\prime \ast }}}{E_3-E}$$

$$\alpha$$

$$H^{eff}=\frac{\hslash {\Gamma }_F}{2}[A+iB]$$

$$\displaystyle\mathbf{A}=\left(\begin{array}[]{c c c}{\delta}_{1}&\delta&q_{1}\sqrt{g_{1}}\\ \delta&{\delta}_{2}&q_{2}\sqrt{g_{2}}\\ q_{1}\sqrt{g_{1}}&q_{2}\sqrt{g_{2}}&-(k_{c}a_{s})^{-1}\end{array}\right)$$

$$\displaystyle\mathbf{B}=-\left(\begin{array}[]{c c c}g_{1}+{\tilde{\gamma}}_{1}&g_{12}+\eta&\sqrt{g_{1}}\\ g_{12}+\eta&g_{2}+{\tilde{\gamma}}_{2}&\sqrt{g_{2}}\\ \sqrt{g_{1}}&\sqrt{g_{2}}&1\end{array}\right)$$

$$x^3+G_2{x}^2+G_{1}x+G_0=0$$

$$G_2=$$

$$G_1=$$

$$G_0=$$

$${\eta }^2-{\tilde{\gamma }}_1{\tilde{\gamma }}_2=0$$

$$\eta =\pm \sqrt{{\tilde{\gamma }}_1{\tilde{\gamma }}_2}$$

$$\displaystyle{X}=C\left(\begin{array}[]{c }x_{1}\\ x_{2}\\ 1\end{array}\right)$$

Full text

I introduction

The idea of a bound state in continuum (BIC) was first put forward by J. von Neuman and E. Wigner in 1929 neuman1929 , in very early days of quantum mechanics. Such a state is particularly unusual and counter-intuitive as it is a localised square-integrable state despite the energy eigenvalue of the state being above the continuum threshold for the usual negative-energy bound states. To create such a state, Neuman and Wigner considered the amplitude modulation of a free-particle wave function, resulting in a local potential which could support a BIC. Since then, this idea of local potential was put under extensive theoretical investigations in a number of works jain1975 ; gazdy1977 ; molina2012 . Two-particle BIC has been shown to have a deep connection with two-body resonance scattering theory fonda1960 ; friedrich1985a ; friedrich1985b where BIC is recognised to be related to a resonance of zero width. Owing to its fundamental significance in quantum mechanics, BIC has been re-examined by several authors robnik1986 ; nockel1992 ; duclos2001 ; ordonez2006 ; prodanovic2006 ; weimann2013 ; rivera2016 . The fact that the underlying concept in BIC pertains to wave phenomenon has motivated several theoretical and experimental works towards its possible realization and applications in various types of material systems such as superlattices herrick1977 ; stillinger1977 , photonic crystal structures marinica2008 ; hsu2013b , acoustic waves porter2005 ; linton2007 , two-particle Hubbard model zhang2012 , optical systems capasso1992 and so on. In recent times, experimental realisation of BIC in extended photonic structures is reported by several workers plotnik2011 ; hsu2013a ; kodigala2017 . BIC has found an important application in creating an exotic laser system called BIC laser kodigala2017 . There is an excellent review in literature written by Hsu et. al. hsu2016 , where the study of BIC is categorically summarized on the basis of different theoretical approaches and corresponding experimental implementations.

In the context of atomic and molecular systems, BIC is yet to be realized. In the past, the possibility of creating a BIC in atomic and molecular systems was discussed by several workers friedrich1985a ; stillinger1974 ; stillinger1975 . With the recent advent of high-precision spectroscopy and coherent control of ultracold atoms and molecules, the realization of a BIC for ultracold atoms appears to be promising. Recently, the creation of a BIC by magneto-optically controlling ultracold collisions of atoms has been proposed by two of us deb2014 . This proposal has made use of two excited molecular states without considering any spontaneous emission from these states. A generic feature of atomic or molecular system is the spontaneous decay of excited states into lower-energy states via vacuum-field induced electric dipole transitions.

In this paper, we study the effects of spontaneous emission on the atomic BIC as proposed in the Ref. deb2014 . Our model system consists of three molecular bound states which interact with the continuum states of ground-state atom-atom collisions in the presence of external magnetic and optical fields. To include the spontaneous emission in the model, we consider that the continuum of vacuum electromagnetic modes interact with the excited molecular states. Our results show that spontaneous emission is a tremendous hindrance to the forming of the proposed BIC. In this work, we show how to overcome this challenge by almost completely nullifying the detrimental effect of spontaneous emission by making use of vacuum induced coherence (VIC) agarwal_book . We show that, VIC agarwal_book ; Fleischhauer1992 ; zhu1995 ; zhu1996 ; das2012 ; ficek_book which is basically an interference phenomenon between two spontaneous emission pathways, can play a decisive role in the formation of BIC in a realistic model that we present in this work.

One of the key conditions for VIC to take place is the non-orthogonality of the dipole moments involved in two spontaneous emission pathways. Another essential condition is that the two decaying excited states should be energetically close enough compared to the geometric mean of the two spontaneous emission linewidths. Coherently controlled ro-vibrational level structure of an excited molecule driven by a pair of photoassociation (PA) lasers has been shown to facilitate for the fulfillment of the required non-orthogonality conditiondas2012 , thus providing a testing ground for the theory of VIC agarwal_book . For instance, as we schematically show in Fig.1, let us choose a pair of ro-vibrational excited states $\mid 1\rangle$ and $\mid 2\rangle$ of a molecule with same rotational ($J_{1}=J_{2}$) but two different vibrational ($v_{1}\neq v_{2}$) quantum numbers. Then obviously, the two dipole moments for electric dipole transitions from these states to a common lower state are non-orthogonal. In this context, it is worth mentioning that the similar idea of non-orthogonality between two molecular dipole transitions has been suggested in the context of incoherent pumping tscherbul2014 . The typical vibrational energy separation in low lying molecular states is in infra-red domain of the electromagnetic spectrum. In photoassociation (PA) spectroscopy of cold atoms, highly excited vibrational states with energy spacing lying in the microwave domain can be populated. Since microwave frequency is much greater than typical linewidth (a few MHz) of the molecular excited states accessible by PA, the second condition for VIC regarding energy spacing between the states will not be fulfilled. This difficulty can be circumvented by dressing the two vibrational states with a micro-wave (MW) field inducing magnetic M1 transitions. As a result, two dressed states $\mid e_{1}\rangle$ and $\mid e_{2}\rangle$ which are the coherent superpositions between $\mid 1\rangle$ and $\mid 2\rangle$ will be formed. The spacing between these dressed states is Rabi frequency $\Omega$ if the MW field is tuned on resonance. Now, if the intensity of the MW field is such that $\Omega\simeq\sqrt{\gamma_{1}\gamma_{2}}$ where $\gamma_{1(2)}$ is the spontaneous linewidth of $\mid 1\rangle$ $\left(\mid 2\rangle\right)$ then the two decay pathways from the dressed levels are likely to interfere giving rise to VIC.

The predicted bound state of two ultracold atoms in continuum is basically a multichannel resonance state with zero width. Ideally, this is perfect BIC in the absence of spontaneous emission. Our results show that, even a vanishingly small spontaneous emission is sufficient to completely wash away any signature of BIC formation. Nonetheless, here we show that, the BIC can be absolutely protected against such highly destructive spontaneous emission by fulfilling the condition of VIC as we have just discussed above. We further show that if the VIC condition deviates even by 0.1 percent, the BIC can be destroyed completely.

The paper is organized in the following way. In the next section II, we present our model. The Hamiltonian describes interacting molecular bound and collisional continuum states. Our model takes into account the interaction of the excited molecular bound states with the vacuum of background electromagnetic field continuum to mimic the effects of spontaneous emission. We then derive the analytical solution of the model and the condition for VIC in section III. We use Feshbach projection operator method to eliminate the collisional and vacuum continua and thus to obtain a complex effective Hamiltonian describing three interacting bound states. The eigenvalues of the effective Hamiltonian are in general complex. A real eigenvalue of the Hamiltonian implies the formation of a BIC. Our analytical results show that, unless VIC condition is fulfilled, the Hamiltonian does not yield any real eigenvalue. In section IV we discuss the possibility of detection of BIC by photoassociative spectroscopy. We present and analyze our numerical results in section V. The paper is concluded in section VI.

II The Model

We consider a magnetic Feshbach resonance between two ground-state (S + S) atoms under two-channel approximation mies2000 ; chin2010 . The Feshbach-resonant state as well as the continuum of scattering states can be coupled to molecular bound states in an excited potential by laser light bauer2009 ; rempe1 . In the excited manifold, we consider two vibrational states $\mid v_{1}\rangle$ and $\mid v_{2}\rangle$ with same rotational quantum number $J_{1}=J_{2}=J$. These two excited states can be coupled by a microwave field due to magnetic dipole (M1) transition. The dressed states formed due to such coupling are

[TABLE]

where $\tan 2\theta=\Omega_{m}/{\Delta_{m}}$, $\Omega_{m}=-\frac{1}{\hbar}\langle v_{1}J_{1}\mid\vec{\mu}.{\cal\vec{B}}\mid v_{2}J_{2}\rangle$ is the microwave Rabi coupling with ${\Delta_{m}}=\omega_{12}-\omega_{m}$ being detuning of the microwave field of frequency $\omega_{m}$ from the vibrational spacing $\omega_{12}=\omega_{v_{2}}-\omega_{v_{1}}$, where $\hbar\omega_{v_{i}}$ is the eigen energy of the state $\mid v_{i}\rangle$. The purpose of this microwave coupling is to control the energy gap between the dressed states $\mid e_{1}\rangle$ and $\mid e_{2}\rangle$. The intensity of the microwave should be set at a level such that $\Omega_{m}\simeq\sqrt{\gamma_{1}\gamma_{2}}$ where $\gamma_{1}$ and $\gamma_{2}$ are the two spontaneous emission linewidths. For homonuclear cold molecules composed of alkali atoms, the spontaneous linewidth of the excited states is typically $\gamma/(2\pi)\simeq 5$ MHz. For hetero-nuclear or homonuclear cold molecule composed of alkaline earth-type atoms the linewidth may be much smaller. The intensity of microwave dressing field should be set at a value such that $\Omega_{m}$ is comparable to $2\pi\gamma$, since near resonance for the microwave transition the two dressed states will be separated by about $\Omega_{m}$. Here we are dressing two excited states. Usually, a two-level system consisting of a ground-state (lossless) and an excited state with a small spontaneous emission linewidth is dressed by Rabi coupling the two states with a strong field. The dressing is effective if $\Omega_{m}$ exceeds $\gamma$. Now, in our case if the microwave field is too strong, then $\Omega_{m}$ would be too large and so would be spacing between $\mid e_{1}\rangle$ and $\mid e_{2}\rangle$. Therefore, the two excited states should be optimally Rabi coupled so that $\Omega_{m}\simeq\gamma$. Under such condition, the two decay paths will interfere destructively leading to the cancellation or inhibition of the spontaneous emission. As we will discuss later, this condition is important for the achievement of BIC. Fig.2 shows that the bare continuum of scattering states $\mid E\rangle_{b}$ in the open channel as well as the bound state $\mid c\rangle$ are coupled to the two dressed states $\mid e_{1}\rangle$ and $\mid e_{2}\rangle$ by two lasers $L_{1}$ and $L_{2}$, respectively.

Next, we consider the following product states of the atomic kets and photon number kets

[TABLE]

[TABLE]

[TABLE]

[TABLE]

as the basis of our analysis, where $N_{1}$, $N_{2}$ represent the photon numbers in the two lasers $L_{1}$, $L_{2}$, respectively. In absence of any spontaneous radiative decay, the Hamiltonian of the system can be expressed as $H=H^{0}_{1}+V_{1}$, where

[TABLE]

and

[TABLE]

Here, $E_{j}$ is the energy of the state $\mid e_{j}\rangle(j=1,2)$ and $E_{3}$ is the energy of $\mid c\rangle$. $\omega_{j}(j=1,2)$ denotes the frequency of the $L_{j}$ laser. $\Omega_{13}$ $\left(\Omega_{23}\right)$ represents the bound-bound coupling between $\mid 3\rangle$ and $\mid 1\rangle$ $\left(\mid 2\rangle\right)$. $\Lambda_{1E}$ and $\Lambda_{2E}$ are the free-bound coupling parameters, $V_{3E}$ is the coupling between $\mid 3\rangle$ and $\mid E\rangle$ due to spin-dependent interactions. Explicitly, $\Omega_{i3}=(1/2)G_{i3}\sqrt{N_{i}+1}$, $(i=1,2)$ where $G_{i3}=-(1/\hbar)\langle i\mid{\mathbf{D}}_{i}\cdot{\mathbf{E}}_{i}\mid 3\rangle$ with ${\mathbf{E}}_{i}$ being the laser field amplitude corresponding to the zero photon; and ${\mathbf{D}}_{i}$ being the molecular transition dipole moment between states $\mid e_{i}\rangle$ and $\mid c\rangle$.

Now, we consider spontaneous radiative decay from $\mid e_{1}\rangle$ and $\mid e_{2}\rangle$. Usually, these states decay to nearly all continuum and bound states of the open and closed channels. However, since both MW-dressed these states have the same molecular angular momentum and components of the two vibrational states, it is expected that both these states will be coupled to one or multiple final states by vacuum fields. As a model system, we assume that $\mid e_{1}\rangle$ and $\mid e_{2}\rangle$ predominantly decay to a common external state $\mid f\rangle$ as shown in Fig.2 and neglect all other spontaneous decay pathways. We describe these two decay processes by introducing the following photon continuum channels haan1987

[TABLE]

[TABLE]

where $k$ represents the energy of the spontaneously emitted photon. The two radiative decay channels $\mid 1k\rangle$ and $\mid 2k^{{}^{\prime}}\rangle$ are normalized in the following way

[TABLE]

When spontaneous decay is incorporated, the Hamiltonian becomes $H=H^{0}+V$ with $H^{0}=H^{0}_{1}+H^{0}_{2}$ and $V=V_{1}+V_{2}$ where

[TABLE]

[TABLE]

and the basis are expanded to {\Big{[}}\mid 1\rangle,\mid 2\rangle,\mid 3\rangle,\mid E\rangle,\mid 1k\rangle,\mid 2k^{{}^{\prime}}\rangle{\Big{]}}. We treat laser fields semiclassically which is equivalent to a quantum treatment if the fields are in coherent states with the average photon numbers being large. This was discussed extensively especially by C. Cohen-Tannoudji cohen1977 ; cohen_book who examined quantized dressed state and how the semiclassical description emerges in the limit of large photon numbers.

Next, following the Feshbach projection operator technique, we eliminate all the continuum states and find the effective Hamiltonian $H_{eff}$ in the atomic subspace {\Big{[}}\mid e_{1}\rangle,\mid e_{2}\rangle,\mid c\rangle{\Big{]}}. $H_{eff}$ is non-hermitian. The imaginary part of the Hamiltonian describes interaction of the subspace with the rest of the complete space. A BIC refers to an eigenstate of this $H_{eff}$ having real eigenvalue. In the following, we write all the matrix elements $H_{eff}$ of the effective Hamiltonian following the derivations done in appendix A.

[TABLE]

Here, $E^{{}^{\rm{sh}}}_{1}$ \Big{(}E^{{}^{\rm{sh}}}_{2}\Big{)} is the laser-induced shift of the state $\mid e_{1}\rangle$ \Big{(}\hskip 2.84526pt\mid e_{2}\rangle\hskip 2.84526pt\Big{)} and $E^{{}^{\rm{sh}}}_{F}$ is magnetic field induced shift of the state $\mid c\rangle$ which are shown explicitly in appendix A. $\gamma_{{}_{LIC}}$ and $\gamma_{{}_{VIC}}$ are the terms that arise respectively due to the laser-induced and vacuum-induced coherence with the latter existing only in case of non-orthogonal transition dipole moments. We assume all the vacuum induced shifts to be negligible. The shifted energy $E_{3}+E^{{}^{\rm{sh}}}_{F}$ of the state $\mid c\rangle$ can be related to the magnetic field dependent scattering length $a_{s}({\cal B})$ of the colliding atom-pairs in the limit $E_{3}\to 0$ as deb2014

[TABLE]

where $k_{c}$ is the wave number related to the collisional energy $E={\hbar^{2}k_{c}^{2}}/{2\mu}$, $\mu$ being the reduced mass of the atom pairs and $\Gamma_{F}$ is the width of FBR. $\beta_{n}$ and $\alpha$ are defined as the following principal value integrals consist of laser and magnetic coupling parameters

[TABLE]

[TABLE]

$\Lambda^{{}^{\prime}}_{nE}$, $\Omega^{{}^{\prime}}_{nE}$ and $V^{{}^{\prime}}_{3E}$ are the coupling parameters in the regime of classical field approximation as mentioned in appendix A. For notational convenience we introduce dimensionless parameters $\delta_{n}=\left(E_{1}-\hbar\omega_{L_{n}}+E^{{}^{\rm{sh}}}_{n}\right)/\left(\hbar\Gamma_{F}/2\right),\hskip 11.38109pt$ $\delta=\alpha/\left(\Gamma_{F}/2\right),\hskip 11.38109pt$ $g_{n}=\Gamma_{n}/\Gamma_{F}$,$\hskip 11.38109ptg_{12}=\gamma_{{}_{LIC}}/\Gamma_{F}$,$\hskip 11.38109pt{\eta}=\gamma_{{}_{VIC}}/\Gamma_{F}\hskip 5.69054pt$and$\hskip 5.69054pt\tilde{\gamma}_{n}=\gamma_{n}/\Gamma_{F}\hskip 5.69054pt$for$\hskip 5.69054ptn=1,2$. We also introduce the Fano-Feshbach asymmetry parameters $q_{n}=\left({\beta+\hbar\Omega^{{}^{\prime}}_{n3}}\right)/{\left(\hbar\Gamma_{nF}/2\right)}$ where $n=1,2$. In terms of these parameters we write $H_{eff}$ in matrix form as follows.

[TABLE]

[TABLE]

[TABLE]

III Solution: The effect of VIC

A BIC is characterized by an eigenstate of $H_{eff}$ with real eigenvalue. The effective Hamiltonian may have one or more real eigenvalues when the eigenvectors of B with zero eigenvalues become simultaneous eigenvectors of A with real eigenvalues. First we determine the condition for at least one zero eigenvalue of B. The secular equation for the B matrix is

[TABLE]

where

[TABLE]

In the absence of spontaneous radiative decay i.e $\tilde{\gamma}_{1}=0$, $\tilde{\gamma}_{2}=0$ and $\eta=0$, we reproduce the same secular equation which was derived in a previous work deb2014 . It is evident from Eq. (25) that $\bf B$ can have one zero eigenvalue only when $G_{0}$ vanishes which gives

[TABLE]

Thus,

[TABLE]

We consider only one root $\eta=\sqrt{\tilde{\gamma}_{1}{\tilde{\gamma}}_{2}}$, which may be interpreted as the vacuum coherence induced decay between the two excited states. The corresponding eigenvector is given by

[TABLE]

where

[TABLE]

and

[TABLE]

$C=\left(x_{1}^{2}+x_{2}^{2}+1\right)^{\frac{1}{2}}$ is the normalization constant. The state given by (33) will be a BIC if it becomes an eigenvector of $\bf A$ with real eigenvalue $\lambda$.

[TABLE]

Consequently, we get the following three equations to be satisfied

[TABLE]

[TABLE]

[TABLE]

We can assume $(k_{c}a_{s})^{-1}\rightarrow 0$ if the magnetic field is tuned closed to Feshbach resonance. The quantities ${\tilde{\gamma}}_{1}$, ${\tilde{\gamma}}_{2}$, $q_{1}$ and $q_{2}$ pertain to a particular system under consideration. For a particular set of values of these quantities together with the values of $g_{1}$, $g_{2}$ and $\delta$, Eqs. (37), (38) and (39) can be solved for $\delta_{1}$, $\delta_{2}$ and $\lambda$.

It is important to note that in the absence of VIC, i.e $\eta=0$, the secular equation Eq. (25) of matrix $\bf B$ cannot have a zero eigenvalue due to the presence of spontaneous emission as $G_{0}$ remains non-zero. Thus, the presence of spontaneous emission alone, rules out the possibility of a BIC. Only when VIC is present and nullifies the effect of spontaneous emission through Eq. (28) making $G_{0}=0$, a BIC is obtained.

The existence of the VIC parameter $\eta$ crucially depends on the non-orthogonality of the transition dipole moments associated with the spontaneous emissions. In case of molecular bound states, two different vibrational levels with same rotational quantum number facilitate such non-orthogonality. In section V, we show that in the most common realistic situations where the spontaneous decay is present without any VIC, a BIC cannot be found. Only when the VIC term is non-zero and Eq. (28) is fulfilled, a BIC can be created.

It is also important to note that an effective destructive interference between the two spontaneous emission pathways will occur if the frequency spacing between two decaying states is much less than $\eta$. To ensure this condition, we consider microwave dressed states $\mid e_{1}\rangle$ and $\mid e_{2}\rangle$ whose energy difference can be tuned by the microwave dressing field to the desired level.

IV Photoassociative detection of BIC Via vanishing width line in Fano spectrum

The probability of the Photoassociative transition $\mid E\rangle_{b}\rightarrow\mid e_{n}\rangle(n=1,2)$ is modified due to the formation of BIC and it can be expressed as

[TABLE]

where dressed continuum state $\mid E+\rangle=\Omega_{+}\mid E\rangle_{b}$ and $\Omega_{+}$ denotes a Møller operator deb2014 . Here, the quantity

[TABLE]

is the photoassociation probability per unit collision energy. Now,

[TABLE]

Using the standard form for the inverse of a matrix, we find

[TABLE]

where   $\tilde{E}=E/E_{F}$,  $E_{F}=\hbar\Gamma_{F}/2$,   $\tilde{H}^{{}^{\prime}}_{eff}=\mathbf{A}+i\mathbf{B}$ and ${\mathscr{A}}$ is the transpose of the co-factor matrix of $(\tilde{E}-\tilde{H}_{eff})$. From Eq. (43), the denominator part

[TABLE]

where $\tilde{E}_{i}$ denotes an energy eigenvalue of $\tilde{H}_{eff}$, may become zero for a real $\tilde{E}_{i}$ only including the numerator part is also zero for that $\tilde{E}_{i}$. The system exhibits an exceptionally sharp Fano-like spectrum fano1 ; fano2 when the imaginary part of a complex eigenvalue of $H_{eff}$ is extremely small i.e one of the eigenvalues of $H_{eff}$ become real.

V Results and discussions

In Fig.3, we show the scaled photoassociative absorption spectra ${\cal S}_{1}\left(\tilde{E}\right)=S_{1}(E)E_{F}$ as a function of the scaled energy $\tilde{E}=E/E_{F}$ for a particular set of parameters as given in the caption. When the parameters are set to match a perfect BIC condition, no spectrum can be obtained as the width of the resonance becomes zero. For this reason, we slightly deviate the value of $g_{1}$ from the required value to satisfy BIC condition. This allows a very small imaginary part in the real root and show up an ultra-narrow resonance line.

In the first panel, with the given set of parameters, the exceedingly narrow feature of the spectrum indicates the presence of BIC corresponding to an eigenvalue $\tilde{E}=1.30-10^{-6}i$ in absence of any spontaneous decay. The BIC is sustained in presence of spontaneous emissions ${\tilde{\gamma}}_{1}={\tilde{\gamma}}_{2}=\gamma=0.01$, if we include the VIC term satisfying the condition (28). The corresponding curves are hardly distinguishable as they almost exactly coincide. In the second panel, we show that if the condition (28) deviates slightly, it affects the BIC drastically with the height of the spectrum being severely decreased. The BIC is destroyed showing a widened peak when we switch off the VIC term. Therefore, VIC parameter has a decisive role in obtaining BIC in presence of spontaneous emissions.

The significance of VIC is shown in another way in Fig.4. It appears that the ultra-narrow nature of the spectra is drastically broadened if we change the value of $\eta$ to even $1\%$ from that satisfying BIC condition (28). The physical origin of such spectral behavior may lie in the strong dependence of the decoherence of the ”Fano” coherence koyu2018 or excited molecular dark state saha2014 on the VIC. In particular, the decoherence has been shown to be highly sensitive to the variation of the alignment between the two dipole moments koyu2018 , or in other words, to the degree of non-orthogonality of the dipole moments. Here, we plot the spectrum considering another set of parameters given in the caption. Since $\gamma=1$, the BIC condition (28) is satisfied when $\eta=1$ and the eigenvalue is $\tilde{E}=6.76-10^{-6}i$. The width of the spectrum happens to be $W\simeq 1\times 10^{-6}$ which is shown in panel-(a). We take three values of $\eta$ that gradually deviates from unity and show how the nature of the spectrum changes so dramatically. From panel-(b) to (d) in Fig.4, the value of $\eta$ is considered to be $0.999$, $0.99$ and $0.9$ for which the width become $2.0\times 10^{-5}$, $0.025$ and $0.25$, respectively. The width of the peak is calculated as the difference between the two values of $\tilde{E}$ where the value of ${\cal S}_{1}\left({\tilde{E}}\right)$ decreases to $1/e$ of its maximum value. Subsequently, the height of the resonance peaks decrease by two or three orders of magnitude. In panel-(e), we show the movement of the imaginary part of $\tilde{E}_{1}$, as $\eta$ deviates from unity at which BIC appears. As $\eta$ deviates gradually form unity to $0.999$, $0.99$ and $0.9$, the imaginary part of $\tilde{E}_{1}$ changes to $-10^{-3}$, $-10^{-2}$ and $-10^{-1}$, respectively, leading to the drastic change in spectral width and height as shown in Fig.4. In Fig.5, we show the variation of width ($W$) as a function of $\eta$ in the close vicinity of $\eta=1$.

It is important to note that a BIC can serve as a tool for the efficient production of Feshbach molecules deb2014 . In general, a BIC is different from a Feshbach molecular state as it is in linear superposition of all the excited as well as the ground bound states and the eigenvalue may lie above the threshold of collisional continuum. But a Feshbach molecule is a negative-energy bound state lying below the threshold. Generally, a Feshbach molecule is produced by an inelastic collision involving three-body process, but for the formation of a BIC, three-body process is not essential. It is evident from Eqs. (34) and (35), the more we offset the ratio $g_{1}/g_{2}$ from $\gamma_{1}/\gamma_{2}$, the more does the superposition lean towards $\mid c\rangle$ which is a multichannel quasi-bound state. The wavefunction corresponding to a BIC expected to be very similar to that of the usual bound states having an exponential decay tail at large inter-atomic separation.

VI Conclusions

In conclusion, we have demonstrated how to suppress the deleterious effects of spontaneous emission in creating diatomic bound state in continuum by manipulating the quantum states of a pair of interacting cold atoms with external fields. We have shown that the suppression crucially depends on the use of appropriately tailor-made vacuum-induced coherence, the effectiveness of which essentially relies on the availability of two non-orthogonal transition dipole moments for two closely-spaced excited states. We emphasize that the non-orthogonality condition can be fulfilled by employing two molecular excited states having same rotational quantum number and differing only in vibrational quantum number. The energy spacing between these two states is required to be comparable to the geometric mean of the two spontaneous emission linewidths. We have suggested that this requirement on energy spacing can be fulfilled by dressing the two excited states with a microwave field. In passing, we remark that a variant of our model may be possible by considering the orbital Feshbach resonance between ground and metastable atoms kato2013 ; pagano2015 ; hofer2015 and coupling ground-state molecular bound states with the resonant states. Metastability of the excited states will give some advantage. We hope to address this issue in our future communication. Finally, we stress that with the recent development in the technology of coherent control over the states of cold atoms and cold molecules, the prospect for the realization of our proposed cold-atom BIC appears to be quite promising.

VII Acknowledgment

Dibyendu Sardar is grateful to CSIR, Government of India, for a support.

Appendix A Derivation of the effective Hamiltonian

Following section II, in the classical field limit we can replace $\Omega_{i3}$ by $\Omega_{i3}^{{}^{\prime}}=(1/2)G_{i3}\sqrt{\langle N_{i}\rangle}$, where $\left\langle N_{i}\right\rangle$ is the average photon number of $i-$th laser. Similarly, $\Lambda_{iE}^{{}^{\prime}}\propto\sqrt{\langle N_{i}\rangle}$ while $V^{{}^{\prime}}_{ik}=\left\langle V_{ik}\right\rangle$. However, the spontaneously emitted field is treated quantum mechanically as a one-photon state. So, the states $\mid 1k\rangle$ and $\mid 2k^{{}^{\prime}}\rangle$ defined by Eqs. (5) and (6) reduce to $\mid f,k\rangle$ and $\mid f,k^{{}^{\prime}}\rangle$, respectively. The semiclassical Hamiltonian in the rotating wave approximation is then given by

[TABLE]

where

[TABLE]

and

[TABLE]

Next, we define the projection operators for our model as

[TABLE]

[TABLE]

where $P$ and $Q$ satisfy the conditions $PP=P$, $QQ=Q$, $PQ=QP=0$ and $P+Q=1$. For the reduced Hamiltonian $H^{{}^{\prime}}$ obtained above, the resolvent operators $G(z)=(z-H^{{}^{\prime}})^{-1}$ and $G_{0}(z)=(z-H^{{}^{\prime}}_{0})^{-1}$ are related as

[TABLE]

and

[TABLE]

where

[TABLE]

where $z$ is a variable in the complex energy plane. The effective Hamiltonian in subspace defined by $P$ is given by

[TABLE]

Combining (52) and (53) we write

[TABLE]

From (A) and (48), the first term in (A) can be shown to be

[TABLE]

Using (A) and (48), The second term in (A) comes out to be

[TABLE]

The deduction of the last term in (A) is cumbersome and make use of Eq. (A-49) and gives

[TABLE]

The Eq. (A) consists of two parts. The terms which contain the $k$-integrals represent the effect of vacuum interaction. We consider it as the first part. The last nine terms constitute the second part which represents the effect of the applied lasers and the magnetic field. We first do the $k$-integrals having simple pole structure. For that we consider $k$ to be slightly complex $k\rightarrow k+i\epsilon$ and use the prescription $\lim\limits_{\epsilon\to 0}\frac{1}{k-z-i\epsilon}={\cal P}\frac{1}{k-z}+i\pi\delta(k-z)$, where $\cal P$ denotes Cauchy principal value. The vacuum coupling parameters $V^{{}^{\prime}}_{1k}$ $\left(V^{{}^{\prime}}_{2k^{{}^{\prime}}}\right)$ can be assumed to have a significant value very close to $k=k_{1f}$ $\left(k_{2f}\right)$, the energy difference between the electronic states $\mid f\rangle$ and $\mid e_{1}\rangle$ $\left(\mid e_{2}\rangle\right)$. So, we can replace $V^{{}^{\prime}}_{1k}$ $\left(V^{{}^{\prime}}_{2k^{{}^{\prime}}}\right)$ by $V^{{}^{\prime}}_{1f}$ $\left(V^{{}^{\prime}}_{2f}\right)$ and take them outside the integrals. On doing so, the principal part becomes zero. Thus, in this way we neglect all the vacuum induced shifts and we are then left only with the vacuum induced widths from the imaginary part of the $k$-integrals. Since the vacuum interaction is very weak, this can be considered a good approximation. The width coming from the imaginary part of the diagonal terms may be identified as the spontaneous decay rates.

[TABLE]

$\gamma_{{}_{VIC}}$ represent vacuum induced coherence (VIC) coming from the off-diagonal terms given by

[TABLE]

We have assumed that both the lasers have the same phase and so that $\gamma_{{}_{V}IC}$ is independent of laser phase. It can be shown that this term only exists if the two continuum $\mid f,k\rangle$ and $\mid f,k^{{}^{\prime}}\rangle$ becomes non-orthogonal for some value of $k$.

The second part of Eq. (A) contains only $E$-integration and gives rise to laser and magnetic field induced shifts and widths as well as coherence between them. Similar to the prescription done in doing $k$-integration we here consider $E$ to be slightly complex $E\rightarrow E+i\epsilon$ and use $\lim\limits_{\epsilon\to 0}\frac{1}{E-z-i\epsilon}={\cal P}\frac{1}{E-z}+i\pi\delta(E-z)$. If we consider the presence of a Feshbach resonance at $E=E_{3}$, then the bound-continuum coupling terms $\Lambda^{{}^{\prime}}_{1E}$, $\Lambda^{{}^{\prime}}_{2E}$ as well as $V^{{}^{\prime}}_{3E}$ assume sharp variation in the neighbourhood of $E_{3}$. Thus, significant contribution to the integrals come from that neighbourhood and we can replace $z$ by $E_{3}$. We sort out some important terms as follows. $E^{{}^{\rm{sh}}}_{1}$, $E^{{}^{\rm{sh}}}_{2}$ and $E^{{}^{\rm{sh}}}_{F}$ are shifts of the eigenvalues of $H^{{}^{\prime}}_{0}$ and are given by

[TABLE]

[TABLE]

[TABLE]

$\beta_{1}$, $\beta_{2}$ and $\alpha$ are complex quantities which appear in the off-diagonal part and are given by

[TABLE]

[TABLE]

The laser induced widths $\Gamma_{1}$, $\Gamma_{2}$ and the width $\Gamma_{F}$ of the FBR are given by

[TABLE]

where $\Lambda^{{}^{\prime}}_{1F}$, $\Lambda^{{}^{\prime}}_{2F}$ and $V^{{}^{\prime}}_{F}$ are the values of $\Lambda^{{}^{\prime}}_{1E}$, $\Lambda^{{}^{\prime}}_{2E}$ and $V^{{}^{\prime}}_{3E}$ respectively, calculated at $E=E_{3}$. $\gamma_{{}_{LIC}}$ represents the coherence between the two lasers which is given by

[TABLE]

$\Gamma_{1f}$ and $\Gamma_{2f}$ represents cross coupling between the lasers and the magnetic field

[TABLE]

For the sake of calculational simplicity, we assume $\Gamma_{nF}$, $\gamma_{{}_{LIC}}$, $\gamma_{{}_{VIC}}$, $\alpha$, $\beta$ and $\Omega_{n3}$ are real quantities.