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Characterisation and feasibility study for superradiant lasing in 40Ca atoms

A. Gogyan, G. Kazakov, M. Bober, and M. Zawada

Open Access Open Access

Abstract

Superradiant active clocks operating on narrow linewidth clock transitions are predicted to achieve precision orders of magnitude higher than any currently existing optical atomic clocks. We introduce a theory of superradiant lasing and implement it for the example of 40Ca atoms. The presented model, however, is valid for any two- or three-level system in an optical lattice. We perform a feasibility analysis and suggest a set of parameters for the experimental fulfillment of superradiant lasing in Ca. Moreover, we present an overview of different magic wavelengths for the 4s2 1S0 ↔ 4s4p 3P1 (mJ = 0) transition in Ca for different polarizations and a robustness analysis of these magic conditions. We also report the magic-zero wavelengths for the 4s4p 3P1, mJ = 0 state.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Bad-cavity (or superradiant) lasers, relying on the effect of superradiance [1] are under active consideration during the last years. Coherence in such lasers is primarily stored in the gain medium, rather than the cavity field and their gain is characterized by a much narrower linewidth than the optical cavity mode. In contrast to "conventional" good-cavity lasers, in bad-cavity lasers the fluctuations of cavity length — the main limitation for short-term stability of conventional lasers [2] — are highly suppressed. This robustness stands behind the idea and concept of an active optical frequency standard [3–5], where a stable periodic signal is formed by the collective atomic emission from a narrow optical transition, like microwave emission in an active hydrogen maser, a well-known microwave frequency standard.

This approach paves the way beyond the fundamental limitations of the common passive optical atomic clocks. The linewidths of clock lasers stabilized to high-Q optical cavities is limited by the cavity thermal noise and mechanical vibrations and at the moment is on the level of about 10-20 mHz [6,7]. The superradiant atomic clocks linewidth, however, in principle depends on the natural linewidth of the ultra-narrow transition multiplied by the cooperativity parameter of cavity QED [8–10]. From the perspective of designing optical atomic sensors the crucial advantage of active optical atomic clocks is the fact that the phase of the atoms is intrinsically encoded in the emitted light.

The superradiant lasing of cold atoms in a cavity was observed recently [9,11,12]. In these experiments, cold atoms are coupled to the EM field of a cavity of low to medium finesse. In this paper we present the theory of superradiant (SR) pulse production in earth-alkaline atoms in a cavity and perform a feasibility analysis for en experimental implementation using the example of $^{40}$Ca atoms, reported in [11]. The presented model, however, is valid for any two- or three-level system in an optical lattice, for instance in clock transitions of $^{88}$Sr and $^{87}$Sr. We show that the superradiant pulse properties are visibly affected by inhomogeneous optical pumping procedure.

We have found that the transition from quadratic to linear scaling of the amplitude of the superradiant signal with the number of emitters may occur at a much lower number of atoms than necessary for transition to the regime of damped Rabi oscillations between the atomic polarization and the cavity field, as observed in [12]. Understanding these processes is critical for the construction of continuous active superradiant lasers, particularly in the scheme proposed in [10], where groups of atoms prepared in the upper lasing state will be delivered into the superradiant cavity either in separated clouds or in continuous beam. Finite-time pumping well resembles the introduction of such atoms into the superradiant cavity mode up to the absence of the field in the cavity at the beginning of the pumping. The main difference would be that the pumped atoms are spontaneously establishing the coherence, while the new atoms are introduced to the initial cavity with already existing field. For instance, this field leads to a reduction of the delay of the produced SR pulse.

We also report an analysis of the magic wavelengths for the $4s^2~{}^1S_0 \leftrightarrow {4s4p~{}^3P_1 (m_J=0)}$ transition in Ca atoms performed for different polarization. The influence of imperfect polarization on the magic wavelength robustness is especially important for non $J=0 \rightarrow J=0$ transitions. We expand the set of possible magic wavelengths as compared to Ref. [13] and characterise their robustness of the magic condition. Additionally, we report the magic-zero (tune-out) wavelengths [14] for the $4s4p~{}^3P_1, m_J=0$ state.

2. Model

We consider a simplified model of three-level atoms in a relatively low-finesse cavity of frequency $\omega _c$ and decay rate $\kappa$ (Fig. 1(a)) that is tuned to the lasing transition ${^3P_1}, m_J = 0 \rightarrow {^1S_0}, m_J = 0$ ($2\rightarrow 1$ in Fig. 1(b)). A magnetic field perpendicular to the cavity axis $z$ is applied so that the degeneracy is removed and transitions from $^3P_1, m_J \neq 0$ are out of resonance with the cavity mode. A laser beam at the magic wavelength, i.e. a beam that leads to equal ac Stark shifts for the states of lasing transition, is coupled to the cavity to create an intra-cavity optical lattice potential. This ensures that the Lamb-Dicke regime is fulfilled.

 figure: Fig. 1.

Fig. 1. (a) Magic wavelength lattice confines atoms in a cavity in lattice sites. Applied magnetic field is parallel to the $x$-axis, the superradiant pulse is generated along the cavity axis $z$. (b) Level diagram of $j$-th atom, interacting with a clock transition laser field $\Omega ^j$, cavity mode $g^j$. Atoms are either initially prepared at the state 2 or are prepared at state 3 and incoherently pumped to state 2 by the rate $W(t)$.

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We suppose atoms are initially prepared in an intermediary metastable state $^3P_0$ (state 3 in Fig. 1(b)) and are pumped incoherently by a strong pulse of rate $W(t)$ to state 2 (Fig. 1(b)), as is the case in the experimental realisation [11]. During the incoherent pumping, a part of the atoms is transferred to the $^3P_1, m_J = \pm 1$ states as well. However, these atoms do not participate in the process of SR pulse production. The atom number calculations in [11] also does not include atoms in the $m_J = \pm 1$ states. Additionally, the decay from these states is slower than typical timescales for SR production, so we neglect the population transfer to these states and consider all atoms are pumped to the $^3P_1, m_J = 0$ state.

While interacting with the cavity mode $g$, photons are coherently and simultaneously emitted, thus giving rise to a SR pulse that travels out of the cavity along the $z$ axis. In general a weak laser field of Rabi frequency $\Omega$ may be applied at the clock transition in order to trigger the SR pulse. The SR pulse is described by the flux of the output field defined by

$$\frac{dn_{\textrm{out}}(t)}{dt} = \langle \hat{c}^{\dagger}_{\textrm{out}}(t) \hat{c}_{\textrm{out}}(t)\rangle,$$
where $n_{\textrm {out}}(t)$ is the mean photon number leaving the cavity through the mirror, $\hat {c}_{\textrm {out}}(t)$ is the corresponding photon annihilation operator. $\hat {c}_{\textrm {out}}(t)$ is connected with the input $\hat {c}_{\textrm {in}}(t)$ and intracavity $\hat {c}(t)$ fields annihilation operators by the input-output formulation [15]
$$\hat{c}_{\textrm{out}}(t) + \hat{c}_{\textrm{in}}(t)=\sqrt{\kappa}\hat{c}(t)$$
and satisfies the commutation relation $[\hat {c}_{\textrm {out}}(t),\hat {c}_{\textrm {out}}^{\dagger }(t^{\prime })]=[\hat {c}_{\textrm {in}}(t),\hat {c}_{\textrm {in}}^{\dagger }(t^{\prime })]=\delta (t-t^{\prime })$. We assume there are no incoming photons to the cavity, i.e. $\langle \hat {c}^\dagger _{\textrm {in}} \hat {c}_{\textrm {in}}\rangle = 0$. From here for 1 we find
$$\frac{dn_{\textrm{out}}(t)}{dt} =\kappa \langle \hat{c}^{\dagger}(t) \hat{c}(t) \rangle.$$
The cavity field operator is obtained from the dynamics of the system that is given by the master equation for the whole density matrix $\rho$ for the atoms and cavity mode:
$$\frac{d\rho }{dt}=-\frac{i}{\hbar }[\hat{H},\rho ]+\frac{d\rho }{dt}|_{rel},$$
where the second term on the right hand side (rhs) accounts for all relaxations in the system, including the incoherent pumping from state 3 to the state 2. With the use of the Lindblad operator $L[\hat {O}]\rho =\hat {O} \rho \hat {O}^{\dagger }-(\hat {O}^{\dagger }\hat {O}\rho +\rho \hat {O}^{\dagger }\hat {O})/2$ it is written as
$$\frac{d\rho }{dt}|_{rel}=\kappa L[\hat{c}]\rho +\sum_j \left(\frac{\gamma}{2} L[\hat{\sigma}_{12}^j]\rho + W(t)L[\hat{\sigma}^j_{23}]\rho\right) +\frac{d\rho }{dt}|_{deph},$$
where $\hat {\sigma }^j_{ik} = |i_j \rangle \langle k_j|\otimes \prod _{l\neq j} \hat {\mathbf {1}}_l$ is the projection operator acting on the state of $j$th atom, the first term on the rhs of this equation represents the cavity output coupling, the second one describes the spontaneous decay with the rate $\gamma /2$ to the ground state 1, $\gamma$ is the linewidth of the state $^3P_1, m_J = 0$ and the third term accounts for the incoherent pumping from state 3 to state 2 with the rate $W(t)$. The last term stands for the inelastic collisions and dephasing. We solve the equations of motion for the relevant expectation values of atomic and field operators using $\langle \dot {\hat {O}} \rangle = Tr[\dot {\rho } \hat {O}]$. In the mean-field approximation we factorize the atomic and field operators expectation values as $\langle \hat {\sigma }_{ik}^j \hat {c} \rangle = \langle \hat {\sigma }_{ik}^j \rangle \langle \hat {c} \rangle$.

Within the rotating wave approximation the Hamiltonian of the considered system of $N$ atoms within the cavity has the form

$$\hat{H} = \hbar \sum_{j=1}^N \bigl [\Delta_j \hat{\sigma}_{22}^j + (g^j\hat{\sigma}_{21}^j \hat{c} + \Omega_j\hat{\sigma}_{21}^j + H.c.) \bigr ],$$
where $\Delta _j = \omega _{21}^j - \omega _c$ is the detuning of the cavity from the atomic transition of frequency $\omega _{21}^j$, $\Omega _j$ is the Rabi frequency for the $j$-th atom of the laser field applied at the lasing transition. The peak value of the atom-cavity coupling constant $g^j$ is defined as $g_{0} = \sqrt {\frac {6 c^3 \gamma }{\omega _{21}^2 w_c^2 L_c}}$, where $w_c$ and $L_c$ are the cavity mode waist and length. In general, $g^j = g(x,y,z)$ depends on the spatial coordinates of the $j$-th atom and is given by
$$g(x,y,z) = g_{0}\exp{\biggl( -\frac{x^2+y^2}{w_c^2}\biggr)}\cos{(kz)},$$
however, we can spatially average the coupling coefficient over the radial distribution of atoms, which leads to the following expression
$$\tilde{g}(z) = g_0 \frac{w_c^2}{w_c^2+w_r^2/2} \cos{kz_0},$$
where $z_0$ is the axial position of the trap site, $k$ is the wavenumber of the $^1S_0 \rightarrow ^3P_1$ transition and $w_r$ is the atomic cloud size in the radial direction. Namely, the density of atoms in a single trap site is proportional to
$$n = n_0 \exp \left(-\frac{2 r^2}{w_r^2} - \frac{2 (z-z_0)^2}{w_z^2} \right),$$
where $w_z$ is the characteristic width of the atoms in the lattice site in the axial direction. Thus, to find the flux of the output SR pulse, we need to calculate the combined equations for atomic populations and the photon annihilation operator. To do it numerically, we partition the atoms into $M$ groups and assume that all parameters of the atom and cavity within one group are the same. In this case $j$ varies over all $M$ groups and the expectation value of the cavity annihilation operator is
$$\langle \dot{\hat{c}}\rangle = i \sum_{j = 1}^{M} [ (i\kappa + \Delta^j) \langle \hat{c} \rangle + N_j \tilde{g}^j \langle \hat {\sigma}_{21}^{(j)}\rangle \langle \hat{c}\rangle ],$$
where $N_j$ is the number of atoms in the group $j$. The stability analysis of the base model is elaborated and addressed in detail in [16].

3. Superradiant pulse

We numerically solve Eq. (10) along with the equations for the atomic projection operators and calculate the flux 3 using the experimental parameters from [11]. We take a cavity finesse of $F=2200$ and a decay rate $\kappa = \pi \times 2260$ kHz. The cavity waist for the $^1S_0 \rightarrow ^3P_1$ transition 657 nm wavelength is $w_c = 190~\mu$m. A lattice of magic wavelength 800.8 nm and the depth of 20 MHz is applied and the trap axial and radial frequencies are around 1.9 MHz and 1.8 kHz respectively. We assume that the atoms experience a normally distributed frequency shift of $3.4(4.3)$ Hz, following the error budget from [17]. The pumping field shape is taken to be

$$W(t) = \frac{W_0}{2} \bigl (\tanh{[C_{\textrm{s}} (t-t_1) ]}-\tanh{[C_{\textrm{s}}(t-t_2)]} \bigr ),$$
where the peak value of the pumping field Rabi frequency is $W_0 = 2\pi \times 25~$kHz and the duration is $t_2 - t_1 = 50~\mu$s, the pulse is turned on (off) at $t_1$ ($t_2$). The pulse slope is defined by $C_{\textrm {s}}$ and we take $C_{\textrm {s}} = 200 \gamma$. In the experiment, this pumping field intensity is sufficient to transfer $\sim 100\%$ of atomic population in around 24 $\mu$s from state 3 to state 2 and other Zeeman sublevels of the ${^3P_1}$ state [11], which do not take part in the superradiance. We assume that on the typical timescales of the SR pulse the collisional losses and dephasing do not have a considerable influence on the experiment.

Figure 2(a) shows the SR pulse flux for atom numbers $N = 69 000, 57 000, 49 000, 41 000, 34 000$. As seen from the figure, the pulse width decreases as the number of atoms is increased and at the same time the peak value of the SR increases. The parameters for the numerical calculations are the same as in [11] and we obtain similar results, however, there is a mismatch in the number of atoms. To understand the underlying reason of this mismatch, we present in Fig. 2(b) the atomic populations of the ground (blue) and excited (red) states for $N = 69 000$ atoms in the cavity lattice sites. The magenta dashed curve is the sum of the populations of the ground and excited states and the magenta curve shows how the population of state 3 is transferred to state 2.

 figure: Fig. 2.

Fig. 2. (a) Generated SR pulse flux for different values of the atom number (inset shows the number of atoms). (b) Populations of the states 1 (blue - $^1S_0$), 2 (red - $^3P_1$) and 3 (magenta - $^3P_0$) for $N = 69 000$ atoms. The solid curves show the averaged populations over all lattice sites. Populations of ground and excited states in different lattice sites is within the range between the blue and red dotted curves, correspondingly. The cyan dotted curve is the sum of the populations of the ground and excited states. The black dashed curve is the scaled shape of the pumping pulse $W(t)$. See the text for the rest of parameters.

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Due to the mismatch between the nodes and antinodes of the magic wavelength lattice (800.8 nm) and the cavity mode at the lasing transition (657 nm), there are different values of the atom-cavity coupling 8 in different lattice sites. There is also frequency shift distribution of the lasing transition frequency, as was mentioned previously. As a result, the populations of the ground and excited states change differently in different axial or radial positions. Populations of individual atomic groups of $N_j$ atoms of Eq. (10) are within the blue and red dotted curves for ground and excited states, respectively, and the solid blue and red curves are averaged populations over all lattice sites. As is seen, when the pumping field is applied, the population of the excited state grows fast and simultaneously, the population is depleted to the ground state due to the spontaneous losses. Before the SR pulse is produced around the time moment $55~\mu$s, on average around $\sim 10\%$ of atoms is already depleted to the ground state, and additionally, around $\sim 40\%$ of atoms remain in the excited state after pulse production. As a result, on average only $\sim 50\%$ of atoms participate in the production of the SR pulse. Atoms for which the atom-cavity coupling has the highest value and the frequency shift of the lasing transition from the resonance is zero, have the biggest influence on the production of the pulse. The ground state population of those atoms (blue dotted curve) reaches up to $\sim 90\%$. In [11], the number of atoms is deduced by fitting the experimental data with a simplified model for the SR pulse production and hence, atoms that do not contribute to the pulse production, are not taken into account.

In Fig. 3 we show how the SR pulse depends on the number of atoms for the same parameters as in Fig. 2. The red curves show the case when atoms are pumped from state 3 to state 2 with the rate $W(t)$. The cyan curves correspond to the limiting case of instantaneous pumping of state 2, which is equivalent to neglecting the process of pumping and assuming the atoms are initially prepared in state 2. The dependence of the peak value of the SR pulse on the atom number is shown in Fig. 3(a). When the pumping process is taken into account, the peak value of the pulse increases quadratically for small atom numbers, however, the increase becomes linear as the number of atoms is increased further. A quadratic fit is shown as a magenta dashed curve. A similar quadratic behaviour is also observed in $^{87}$Sr atoms [9] for small numbers of atoms. The atoms number in Fig. 3 corresponds to the regime where $\sqrt {N}g_0\;<\;\kappa$. In this regime, if we neglect the influence of the pumping of atoms, we get the characteristic quadratic dependence for SR. The black dashed curve is a quadratic fit to the solid cyan curve. The total number of photons in the SR pulse leaving the cavity is shown in Fig. 3(b) and again, in the presence of pumping field this dependence is linear only for small number of atoms, see the red solid curve and the magenta dashed linear fit to it. Figure 3(c) shows the decrease of the FWHM of the SR pulse and Fig. 3(d) represents the temporal position of the peak value of the generated pulses. The latter shows that the pulse is produced at earlier times, when the number of atoms is increased. Time is required to transfer population from excited state 3 to state 2 and the sooner the pulse is generated, the less population is transferred to state 2, as a result, less atoms contribute to the production of the SR pulse. While the pumping process has almost no influence on the width of the produced pulse, its delay, obviously, depends on how the atoms are pumped to the state 2. This feature was observed experimentally in [11], where an offset is observed in the delay time dependence on atom number.

 figure: Fig. 3.

Fig. 3. Influence of the number of atoms on (a) the peak value of the pulse; (b) the photon number; (c) the pulse width; (d) the temporal position of the pulse peak value. The red curves correspond to the case when atoms are pumped from the state 3 to the state 2 with the pumping rate of $W(t)$ and the cyan curves correspond to the emission when the atoms are pumped instantaneously. The magenta and black dashed curves are functions fitted to the red and cyan curves, correspondingly. In (a) a quadratic function is fitted, while in (b) a linear function is fitted. Other relevant parameters are the same as in Fig. 2.

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If we consider the regime where $\sqrt {N}g_0\;>\;\kappa$ even for instantaneous pumping we observe a linear dependence of the peak photon number on the number of atoms, as is shown in Fig. 4. However, it seems experimentally challenging to trap a large enough number of atoms in a 1D magic wavelength lattice, keeping atomic density low enough to sustain the superradiance, to observe the linear dependence with instantaneous pumping. In the superradiance experiment presented in [9] the number of $^{87}$Sr atoms in lattice reaches to $2 \times 10^5$ and in the experiment that studied collisional effects [18] the number of $^{88}$Sr atoms in lattice reaches $3\times 10^6$. Nevertheless, while the theory is elaborated for cold atoms in a lattice, similar quadratic and linear regimes were observed and presented experimentally for a cloud of $^{88}$Sr atoms in [12].

 figure: Fig. 4.

Fig. 4. SR pulse peak value dependence on the number of atoms for the regime $\sqrt {N}g_0\;>\;\kappa$ for instantaneous pumping. Parameters are the same as in Fig. 3. Dashed curves are quadratic and linear fits for small and large number of atoms, correspondingly.

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Similar dependence on the cavity finesse is shown in Fig. 5 when the atoms are prepared in state 2 and no pumping field is applied. The number of atoms is taken as $N = 69 000$ and the rest of parameters are the same as in Figure 2. It is evident, that while the peak value of the SR pulse increases with increasing finesse, the number of the photons in a pulse saturates. The width of the pulse also reaches a limited value, which means that the peak value of the pulse saturates as well. For a large enough cavity finesse, the generated photons stay in the cavity for a longer period of time and the peak value of the pulse also saturates.

 figure: Fig. 5.

Fig. 5. Impact of the cavity finesse on (a) the peak value of the pulse; (b) photon number; (c) pulse width; (d) temporal position of the pulse peak value. The number of atoms is taken $N = 69 000$. The rest of the parameters are the same as in Fig. 2. The black dashed curves are $a/x + b$ function fits.

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4. Magic wavelength

In the above calculations we have considered Ca atoms trapped by an optical lattice operating at the magic wavelength of 800.8(22) nm measured with a time domain Ramsey-Bordé atom interferometer by Degenhardt et al. [13] and used in the actual superradiant experiment by Laske et al. [11]. Here we expand the analysis of magic wavelengths for $^1S_0 \leftrightarrow {^3P_1 (m_J = 0)}$ transition in bosonic isotopes of Ca, such as $\rm ^{40}Ca$, considering the robustness of the magic conditions with respect to imperfect polarization.

4.1 Ac Stark shifts

The magic wavelength is a wavelength where the ac Stark shifts of both the upper and lower clock states are the same. The ac Stark shift $\Delta ^S_i$ of the state $|i\rangle$ in an external monochromatic electromagnetic field can be written (in a Gaussian system of units) as [19]:

$$\Delta^S_i(\lambda)=-\alpha(i,p,\lambda) \frac{E_0^2}{4}=-\alpha(i,p,\lambda) \frac{2 \pi I}{c},$$
where $E_0$ is the amplitude of the electromagnetic field, $I$ is the intensity of the light, $c$ is the speed of light, and $\alpha (i,p,\lambda )$ is the polarizability of the state $|i\rangle$ depending on the polarization of the light $p$ and wavelength $\lambda$. For the sake of definiteness, we consider in this section a running-wave laser field, where $I=E_0^2c/(8\pi )$ (for a standing-wave laser field the light shift should be multiplied by factor of 4). For atoms with no hyperfine structure, such as bosonic isotopes of Ca, this shift may be expressed as [19]
$$\alpha(i,p,\lambda)=\frac{3c^3}{2} \sum_{k,m_k}\frac{A_{Jki}(2 J_k+1)}{\omega^2_{Jki}(\omega^2_{Jki}-\omega^2)} \left( \begin{array}{ccc} J_i & 1 & J_k \\ m_i & p & -m_k \end{array} \right)^2 .$$
Here $A_{Jki}$ is the spontaneous transition rate between the states $|k\rangle$ and $|i\rangle$, with respective total electronic angular momenta $J_k$ and $J_i$, $\omega _{Jki}$ is the transition frequency, $\omega =2\pi c/\lambda$ is the frequency of the light, $J_{i,k}$ and $m_{i,k}$ are the electronic momenta and their projections of the states $|i\rangle$ and $|k\rangle$ respectively, and the Wigner 3-j symbol is given in round brackets. For the calculations we use all the values of $A_{Jki}$ that are reported in [19].

We consider two particular cases of polarization that can be attained with linearly polarized laser light, namely $\pi$-polarization and $xy$-polarization, see Fig. 6 for details.

 figure: Fig. 6.

Fig. 6. Realisation of "pure" types of linear polarization. (a): $\pi$-polarization, where the amplitude $\vec {E}_0$ of electric component of the laser field is polarized along the quantization axis defined by the external magnetic field $\vec {B}$; (b) and (c): $xy$-polarization, where $\vec {B}\perp \vec {E}_0$.

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First, we suppose that the polarization is perfect, i.e., the electric field of the trapping laser light is either parallel, or perpendicular to the quantization axis. The calculated ac Stark shifts for $4s^2~{}^1S_0$ and $4s4p~{}^3P_1, m_J=0$ states of Ca atom corresponding to these configurations are presented in the Figure 7. The magic wavelengths, where the $^1S_0$ and $^3P_1,m_J=0$ ac Stark shifts are identical are equal to $799.2$ nm, $456.2$ nm, $428.87$ nm and 428.82 nm for $xy$-polarization, and to $468.3$ nm, $434.3$ nm and $428.9$ nm for $\pi$-polarization. Note that the first magic wavelength, $799.2$ nm, differs slightly from the experimental value of $800.8$ nm found in [13], which can be explained by the inaccuracies of the values of Einstein coefficients and the limited set of states listed in [19], lack of auto-ionizing states, continuum, etc.

 figure: Fig. 7.

Fig. 7. Ac Stark shifts for ${}^1S_0$ (black) and ${}^3P_1,m_J=0$ (red) states of Ca atom in the running-wave laser field with $xy$- (a) and $\pi$-polarization (b), whose intensity $I=10~\textrm {kW/cm}^2$, versus frequency $\omega$ (bottom scale) or wavelength $\lambda$ (top scale) of the field. Blue circles indicate “magic” points, where the ac Stark shifts are equal, and green diamonds indicate “zero-magic” points, where the ${}^3P_{1}(m_J=0)$ state is not trapped.

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Moreover, using the same calculations, a sequence of magic-zero (tune-out) wavelengths for the $4s4p~{}^3P_1, m_J=0$ state is determined: 1388.5 nm, 558.8 nm, 435.7 nm and 429.17 nm for $xy$-polarization, and 1380.0 nm, 439.0 nm, 429.2 nm for $\pi$-polarization.

4.2 Influence of imperfect polarization and other characteristics of the magic wavelength

The ac Stark shift depends on the polarization of the light field if angular momenta of the upper and the excited states are not both zeros. In an ideally linear polarization of the field any predominance of the left-rotating wave over the right-rotating one, or vice versa, is absent. However, in a real experiment a deviation of the alignment of the magnetic field $\vec {B}$ from the polarization vector $\vec {E}_0$ and/or propagation vector $\vec {k}$ is always present. In this section we consider the robustness of the magic condition with respect to such misalignment.

Let us introduce the angle $\varphi$ between $\vec {E}_0$ and $\vec {B}$: $\varphi =0$ for ideal $\pi$-polarization, and $\varphi =\pi /2$ for ideal $xy$-polarization. polarizability for linearly polarized light at arbitrary $\varphi$ can be expressed as

$$\alpha_\varphi(i,\phi,\lambda)=\frac{\alpha(i,1,\lambda)+\alpha(i,-1,\lambda))}{2}\sin^2{\varphi} +\alpha(i,0,\lambda)\cos^2 \varphi.$$
Therefore, the contribution of the polarizability with undesirable component is proportional to the squared deviation of the angle $\varphi$ from 0 or $\pi /2$.

To characterize the magic wavelength, the following set of parameters is used: $\lambda _m$ — magic wavelength, $U/I$ — trap depth to intensity ratio, $(d \Delta _{LS})/(U d\lambda )$ — sensitivity of differential ac Stark shift $\Delta _{LS}$ to variation of the wavelength at unit trap depth ratio, and $(d^2 \Delta _{LS})/(U d\varphi ^2)$ — sensitivity of the differential ac Stark shift $\Delta _{LS}$ to the variation of the polarization angle $\varphi$ at unit trap depth ratio. Here the differential light shift $\Delta _{LS}=\Delta ^S_{e}-\Delta ^S_{g}$, where the indices $e$ and $g$ correspond to the $^3P_1,m_J=0$-state and $^1S_0$-state, respectively. These parameters for different wavelengths are summarized in Table 1.

Tables Icon

Table 1. Characterization of magic wavelengths for the $^1S_0 \leftrightarrow {^3P_1 (m_J=0)}$ transition in bosonic isotopes of Ca

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All calculated magic wavelengths can be used experimentally. Even without a power build-up cavity a reasonable trap depth of a few tens to a few hundreds of $\mu$K (for a waist between 30 and 100 $\mu$m) can be obtained for all reported magic wavelengths. Taking into account both frequency and polarization effects the calculated magic wavelengths around 800 nm, 456 nm and 468 nm seem to be most promising from the point of view of experimental condition robustness. For the magic wavelengths around 800 nm, 456 nm and 468 nm the ac Stark shift uncertainties $\Delta _{LS}$ are on the order of a few mHz (3, 6 and 15 mHz, respectively) for reasonable experimental conditions with a 35 $\mu$K deep trap and with the trap laser frequency instability below 1 MHz. On the other hand, a much larger uncertainty in $\Delta _{LS}$ is expected due to variations of the polarization angle. While a precise polarizer and a power build-up cavity with a birefringence element can be used for polarization control, it would be challenging to align the polarization with respect to the magnetic field with accuracy better than 10 mrad. For a 35 $\mu$K deep optical lattice trap the uncertainty $\Delta _{LS}$ for the polarization control at the level of 10 mrad would be around -23 Hz, 16 Hz and -19 Hz, for the magic wavelengths of around 800 nm, 456 nm and 468 nm, respectively. Moreover, since the single photon ionization barrier for calcium is at 562.86 nm, before exploiting the optical traps at the 456 nm and 468 nm magic wavelengths one needs to measure the relevant photoionization cross-sections.

5. Conclusions

We have presented a semi-classical theory of superradiant pulse production. The theory includes the axial and radial distribution of the cavity coupling rate for the clock transition, and normally distributed frequency shift due to external factors, e.g. collisional frequency shifts. We theoretically reproduced the experimental results reported in Ref. [11] and presented an analysis of the experimental parameters.

The new features in Ca superradiance for the regime where $\sqrt {N}g_0\;<\;\kappa$, similar to the ones observed also in Sr [12] for the regime $\sqrt {N}g_0\;>\;\kappa$, are explained: we show the influence of the pumping process on the features of the produced SR pulse. In particular, it may lead to a quadratic dependence of the SR pulse peak value on the number of atoms for small number of atoms and to a linear dependence for a larger number of atoms, while still operating in the $\sqrt {N}g_0\;<\;\kappa$ regime.

Additionally, a study of the magic wavelengths for the $^1S_0 \leftrightarrow {^3P_1 (m_J=0)}$ transition in bosonic isotopes of Ca is presented, considering the robustness of the magic conditions with respect to imperfect polarization present in real experiments. Moreover, the magic-zero (tune-out) wavelengths for the $4s4p{}^3P_1, m_J=0$ state are reported.

If the (re)pumping of the Ca atoms is performed via the $4s5s^3S_1$-state (31539.495 cm ${}^{-1}$), one has to take into account that this state is above the single photon ionization limit for all the wavelengths shorter than 562.86 nm and the feasibility of the pumping depends on the ionization cross-section. The only magic wavelength longer than this energy is around 800 nm. This wavelength is robust with respect to fluctuations of polarization, and is characterized by a negligible scattering rate. It lies, however, far from the resonances and thus requires a strong laser field to form a suitably deep trap.

Funding

The EMPIR Programme cofinanced by the Participating States and from the European Union’s Horizon 2020 Research and Innovation Programme (EMPIR 17FUN03 USOQS); the European Union’s Horizon 2020 Research and Innovation Programme No 820404, (iqClock project); Narodowe Centrum Nauki (Quantera Q-Clocks, 2017/25/Z/ST2/03021); Fundacja na rzecz Nauki Polskiej (Project TEAM/2017-4/42).

Acknowledgments

Authors thank to Andreas Hemmerich and Torben Laske for providing details of their experimental set-up. The UMK research was performed at the National Laboratory FAMO (KL FAMO) in Toruń, Poland, and was supported by a subsidy from the Polish Ministry of Science and Higher Education.

Disclosures

The authors declare no conflicts of interest.

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Figures (7)
Fig. 1.
Fig. 1. (a) Magic wavelength lattice confines atoms in a cavity in lattice sites. Applied magnetic field is parallel to the $x$-axis, the superradiant pulse is generated along the cavity axis $z$. (b) Level diagram of $j$-th atom, interacting with a clock transition laser field $\Omega ^j$, cavity mode $g^j$. Atoms are either initially prepared at the state 2 or are prepared at state 3 and incoherently pumped to state 2 by the rate $W(t)$.

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Fig. 2.
Fig. 2. (a) Generated SR pulse flux for different values of the atom number (inset shows the number of atoms). (b) Populations of the states 1 (blue - $^1S_0$), 2 (red - $^3P_1$) and 3 (magenta - $^3P_0$) for $N = 69 000$ atoms. The solid curves show the averaged populations over all lattice sites. Populations of ground and excited states in different lattice sites is within the range between the blue and red dotted curves, correspondingly. The cyan dotted curve is the sum of the populations of the ground and excited states. The black dashed curve is the scaled shape of the pumping pulse $W(t)$. See the text for the rest of parameters.

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Fig. 3.
Fig. 3. Influence of the number of atoms on (a) the peak value of the pulse; (b) the photon number; (c) the pulse width; (d) the temporal position of the pulse peak value. The red curves correspond to the case when atoms are pumped from the state 3 to the state 2 with the pumping rate of $W(t)$ and the cyan curves correspond to the emission when the atoms are pumped instantaneously. The magenta and black dashed curves are functions fitted to the red and cyan curves, correspondingly. In (a) a quadratic function is fitted, while in (b) a linear function is fitted. Other relevant parameters are the same as in Fig. 2.

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Fig. 4.
Fig. 4. SR pulse peak value dependence on the number of atoms for the regime $\sqrt {N}g_0\;>\;\kappa$ for instantaneous pumping. Parameters are the same as in Fig. 3. Dashed curves are quadratic and linear fits for small and large number of atoms, correspondingly.

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Fig. 5.
Fig. 5. Impact of the cavity finesse on (a) the peak value of the pulse; (b) photon number; (c) pulse width; (d) temporal position of the pulse peak value. The number of atoms is taken $N = 69 000$. The rest of the parameters are the same as in Fig. 2. The black dashed curves are $a/x + b$ function fits.

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Fig. 6.
Fig. 6. Realisation of "pure" types of linear polarization. (a): $\pi$-polarization, where the amplitude $\vec {E}_0$ of electric component of the laser field is polarized along the quantization axis defined by the external magnetic field $\vec {B}$; (b) and (c): $xy$-polarization, where $\vec {B}\perp \vec {E}_0$.

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Fig. 7.
Fig. 7. Ac Stark shifts for ${}^1S_0$ (black) and ${}^3P_1,m_J=0$ (red) states of Ca atom in the running-wave laser field with $xy$- (a) and $\pi$-polarization (b), whose intensity $I=10~\textrm {kW/cm}^2$, versus frequency $\omega$ (bottom scale) or wavelength $\lambda$ (top scale) of the field. Blue circles indicate “magic” points, where the ac Stark shifts are equal, and green diamonds indicate “zero-magic” points, where the ${}^3P_{1}(m_J=0)$ state is not trapped.

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Tables (1)

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Table 1. Characterization of magic wavelengths for the 1 S 0 ↔ 3 P 1 ( m J = 0 ) transition in bosonic isotopes of Ca

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Equations (14)

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d n out ( t ) d t = ⟨ c ^ out † ( t ) c ^ out ( t ) ⟩ ,
c ^ out ( t ) + c ^ in ( t ) = κ c ^ ( t )
d n out ( t ) d t = κ ⟨ c ^ † ( t ) c ^ ( t ) ⟩ .
d ρ d t = − i ℏ [ H ^ , ρ ] + d ρ d t | r e l ,
d ρ d t | r e l = κ L [ c ^ ] ρ + ∑ j ( γ 2 L [ σ ^ 12 j ] ρ + W ( t ) L [ σ ^ 23 j ] ρ ) + d ρ d t | d e p h ,
H ^ = ℏ ∑ j = 1 N [ Δ j σ ^ 22 j + ( g j σ ^ 21 j c ^ + Ω j σ ^ 21 j + H . c . ) ] ,
g ( x , y , z ) = g 0 exp ⁡ ( − x 2 + y 2 w c 2 ) cos ⁡ ( k z ) ,
g ~ ( z ) = g 0 w c 2 w c 2 + w r 2 / 2 cos ⁡ k z 0 ,
n = n 0 exp ⁡ ( − 2 r 2 w r 2 − 2 ( z − z 0 ) 2 w z 2 ) ,
⟨ c ^ ˙ ⟩ = i ∑ j = 1 M [ ( i κ + Δ j ) ⟨ c ^ ⟩ + N j g ~ j ⟨ σ ^ 21 ( j ) ⟩ ⟨ c ^ ⟩ ] ,
W ( t ) = W 0 2 ( tanh ⁡ [ C s ( t − t 1 ) ] − tanh ⁡ [ C s ( t − t 2 ) ] ) ,
Δ i S ( λ ) = − α ( i , p , λ ) E 0 2 4 = − α ( i , p , λ ) 2 π I c ,
α ( i , p , λ ) = 3 c 3 2 ∑ k , m k A J k i ( 2 J k + 1 ) ω J k i 2 ( ω J k i 2 − ω 2 ) ( J i 1 J k m i p − m k ) 2 .
α φ ( i , ϕ , λ ) = α ( i , 1 , λ ) + α ( i , − 1 , λ ) ) 2 sin 2 ⁡ φ + α ( i , 0 , λ ) cos 2 ⁡ φ .
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