Deterministic single-atom source of quasi-superradiant $N$-photon pulses

Introduction.–Recent experiments with trapped atoms and fiber-integrated, optical micro- and nano-cavities have pushed the field of cavity quantum electrodynamics (cavity QED) into a new realm of single-atom–photon coupling strengths, corresponding to unprecedentedly large single-atom cooperativities Gehr et al. (2010); Haas et al. (2014); Barontini et al. (2015); Thompson et al. (2013); Tiecke et al. (2014); Samutpraphoot et al. (2020), while also offering the possibility of integrated quantum networks for quantum communication or simulation of quantum many-body systems Tan and Rohde (2019); Slussarenko and Pryde (2019); Flamini et al. (2018); O’Brien et al. (2009); Kok et al. (2007); Kimble (2008); O’Brien (2007). A further, well-known capability provided by such large coupling strength is the generation of single photons with high fidelity through cavity-enhanced atomic spontaneous emission. Efficient single-photon sources are of course central to many efforts to realize optical quantum computation and communication.

Beyond this, however, lies an even greater, and still outstanding challenge to produce a similarly efficient source of pulses containing exactly $N$ ($\geq 2$) optical photons. Such highly nonclassical states of light are of fundamental interest to quantum optics and constitute a starting point for the engineering of yet more complex quantum states. They are also essential for newly-emerging, more resource-efficient photonic architectures for universal quantum computation and quantum error correction using individual, higher-dimensional systems Sabapathy and Weedbrook (2018); Michael et al. (2016); Hu et al. (2019) (cf. multiple two-state systems), as well as for optimal capacity of a quantum communication channel Yamamoto and Haus (1986); Caves and Drummond (1994), and Heisenberg-limited quantum metrology (interferometry) Slussarenko et al. (2017); Pan et al. (2012); Nagata et al. (2007); Mitchell et al. (2004); Pezzé and Smerzi (2013); Campos et al. (2003); Holland and Burnett (1993).

Recent proposals and proof-of-principle demonstrations of optical $N$-photon sources, typically using parametric down conversion or quantum dots, are intrinsically probabilistic and low-yield in nature Muñoz et al. (2014); Waks et al. (2006); Krapick et al. (2016); Moebius et al. (2016); Fischer et al. (2017); Khoshnegar et al. (2017); Loredo et al. (2019). The use of a known number of (effective) two-level atoms emitting into a cavity or photonic waveguide has also been proposed Brown et al. (2003); González-Tudela et al. (2015, 2017); Paulisch et al. (2019), but the required many-body control and repeatability is still very challenging.

Here, we propose a deterministic $N$-photon source that requires just a single atom and makes use of its entire, multilevel energy structure in a manner that reduces the effective system dynamics to a simple and transparent form. In particular, we demonstrate that a single alkali atom coupled strongly to a cavity mode and subject to Raman transitions between sublevels of a ground $F$ hyperfine state can replicate the collective emission of $N=2F$ initially excited, two-level atoms into the cavity mode. In this way, a “superradiant” pulse of precisely $N$ photons can be extracted, through the cavity mode, from a single atom. Moreover, an initial, coherent superposition state of the atom’s ground sublevels is also preserved in the emission process, enabling the generation of a light pulse in an arbitrary superposition of Fock states; as particular examples, we consider $0N$-state and binomial-code-state pulses of light.

Key to the reduction of the single-atom, multilevel dynamics is a very large detuning of the laser and cavity fields from an entire excited state hyperfine manifold of the atom, such that the excited-state hyperfine splittings can be ignored; alternatively, such that the total electronic angular momentum, $J$, is a good quantum number. Such large detuning from the atomic transition in turn demands a very large atom-cavity coupling strength and, as we show here, the experimental configurations of Gehr et al. (2010); Haas et al. (2014); Barontini et al. (2015); Thompson et al. (2013); Tiecke et al. (2014); Samutpraphoot et al. (2020) attain the requisite strength for our scheme to be feasible and efficient.

The potential for making use of the multilevel energy structure of an alkali atom to prepare $N$-photon states has been considered previously, using either adiabatic passage with time-dependent laser and atom-cavity coupling strengths Parkins et al. (1993, 1995), or cavity-mediated optical pumping between atomic ground state sublevels Gogyan et al. (2012). However, in contrast to the present scheme, these approaches assume near-resonant laser and cavity fields and consider just a single $F\leftrightarrow F^{\prime}$ transition. This limits the range of validity of the approaches and means that Clebsch-Gordan coefficients between $m_{F}$ and $m_{F^{\prime}}$ sublevels play a nontrivial and restricting (with regards to choice of $F$ and $F^{\prime}$) role in the scope and performance of the scheme.

Engineered Tavis-Cummings Dynamics.–We consider a single alkali atom tightly confined inside an optical cavity. The atom couples to a $\pi$-polarized cavity mode (annihilation operator $\hat{a}$) and is also driven by either a $\sigma_{+}$- or $\sigma_{-}$-polarized laser field (Fig. 1). We define the atomic dipole transition operators

where $q=\{-1,0,1\}$ and $\mu_{q}$ is the dipole operator for $\{\sigma_{-},\pi,\sigma_{+}\}$-polarization, normalized such that $\langle\mu\rangle=1$ for a cycling transition. The master equation for the density operator, $\hat{\rho}$, of our system in the interaction picture is ($\hbar=1$)

where $\kappa$ is the cavity field decay rate, $\gamma$ the free-space atomic spontaneous emission rate, and $\mathcal{D}[\hat{O}]\hat{\rho}=2\hat{O}\hat{\rho}\hat{O}^{\dagger}-\hat{\rho}\hat{O}^{\dagger}\hat{O}-\hat{O}^{\dagger}\hat{O}\hat{\rho}$. Setting the zero of energy at the lower ground hyperfine level, and assuming (for the moment) zero magnetic field, the Hamiltonian is

Here, $\Delta_{\text{c}}=\omega_{\text{c}}-\omega_{\pm}$ is the detuning between the cavity and laser frequencies, $\Omega=|\Omega|e^{i\phi}$ the Rabi frequency of the $\sigma_{\pm}$-polarized laser field, $g$ the atom-cavity coupling strength, $\omega_{\text{GHS}}$ the ground state hyperfine splitting [$F_{\uparrow}$ ($F_{\downarrow}$) denotes the upper (lower) hyperfine ground state], and $\Delta_{F^{\prime}}=\omega_{\pm}-\omega_{F^{\prime}}$ the detuning of the laser from the $F_{\downarrow}\leftrightarrow F^{\prime}$ transition. Note that, given the large coupling strengths and detunings that we consider here, we assume that the light fields couple all hyperfine ground and excited states. Consistent with this, we also assume that all atomic decays of a given polarization are into a common reservoir Birnbaum et al. (2006).

If we now assume also, more specifically, that the detunings of the fields (cavity and laser) are much larger than the excited state hyperfine splitting, such that this splitting can essentially be neglected, then, in addition to being able to adiabatically eliminate the atomic excited states and neglect atomic spontaneous emission, we obtain a tremendously simplified effective model of the atom-cavity dynamics in the form of an anti-Tavis-Cummings or Tavis-Cummings model (anti-TCM or TCM, depending on the polarization of the laser field) for a collective spin $F$ Masson et al. (2017); Zhiqiang et al. (2017); Masson and Parkins (2019), where $F$ (either $F_{\uparrow}$ or $F_{\downarrow}$) is determined by the initial state of the atom; i.e., our master equation reduces to

with

where $\{\hat{S}_{\pm},\hat{S}_{z}\}$ are the spin-$F$ angular momentum operators and, for example, for the $D_{1}$ line of ${}^{87}{\rm Rb}$, the effective parameters are

Here, we now assume an external magnetic field giving rise to a shift $\omega_{\text{Z}}$ of the $m_{F}$ levels. The detuning $\Delta$ depends on the choice of $F$; for $F=F_{\uparrow}$, we take $\Delta=\Delta_{F^{\prime}}+\omega_{\text{GHS}}$, where the choice of $F^{\prime}$ in $\Delta_{F^{\prime}}$ makes little difference due to the very large detuning assumed (in practice, we pick the lowest $F^{\prime}$). The same forms of expressions for $\{\omega,\omega_{0},\lambda\}$ are obtained for the $D_{1}$ and $D_{2}$ lines of other alkali atoms, but with slightly different numerical factors.

The essence of our scheme follows clearly and simply from the dynamics described by (4) and (5). With the choice $\hat{\mathcal{H}}_{\pm}$ and corresponding initial atom-cavity state $|F,m_{F}=\mp F\rangle|0\rangle_{\rm cav}$, the system evolves irreversibly to the state $|F,m_{F}=\pm F\rangle|0\rangle_{\rm cav}$ with emission from the cavity of a pulse of exactly $2F$ photons. Additionally, in the regime of interest to us, where $\kappa\gg\sqrt{F}\lambda$ and $\omega\simeq\omega_{0}\simeq 0$ (via tuning of $\Delta_{\text{c}}$ and $\omega_{\text{Z}}$), the effective model describes resonant, cavity-mediated superradiant emission of the collective spin; i.e., the emitted $2F$-photon pulse will have a characteristic $\text{sech}^{2}$-shaped temporal profile Eberly (1972).

Output photon number.–A preliminary way to quantify the quality of our state generation scheme is to compute the time evolution of the output photon flux from the cavity and the mean number of emitted photons, $\overline{N}=2\kappa\int_{0}^{\infty}dt\langle\hat{a}^{\dagger}(t)\hat{a}(t)\rangle$. We consider first the case of a ${}^{87}{\rm Rb}$ atom initially prepared in the ground state $|F=2,m_{F}=-2\rangle$ and coupled to the laser and cavity fields via the $D_{1}$ line. With this system, we expect an output pulse of exactly 4 photons.

We solve the master equation numerically for the full model, (2–3), and compare results with the solution for the effective anti-TCM, (4–5), for two sets of cavity QED parameters: (i) $\{\kappa,g,\gamma\}/2\pi=\{50,250,5.7\}\text{ MHz}$ and (ii) $\{\kappa,g,\gamma\}/2\pi=\{0.5,2,0.0057\}\text{ GHz}$. The first set corresponds to the fiber microcavity system of Gehr et al. (2010); Haas et al. (2014); Barontini et al. (2015), while the second set is relevant to the nanocavity system of Thompson et al. (2013); Tiecke et al. (2014); Samutpraphoot et al. (2020). The results are shown in Fig. 2, with two different detunings $\Delta$ considered for each set. The agreement between the full and reduced models is clearly very good, and the predicted $\text{sech}^{2}$-shaped pulse is confirmed, with a duration on the order of $(F\lambda^{2}/\kappa)^{-1}$. The atomic state populations in the full model also show the expected evolution, with a smooth transfer of population along the $F=2$ hyperfine level to the final state $|F=2,m_{F}=+2\rangle$ (see SI). A very small fraction of population may be transferred to the $F=1$ ground state via off-resonant processes, but, in fact, the effective superradiant emission simply continues from within this level and any population there is ultimately driven back (also by an off-resonant process) into the $F=2$ level and so to the final (dark) state $|F=2,m_{F}=+2\rangle$.

Similar results are shown in Fig. 3 for a ${}^{133}{\rm Cs}$ atom initially prepared in the $|F=4,m_{F}=-4\rangle$ ground state, also operating on the $D_{1}$ line. For the parameters used, there is a slight discrepancy between pulse shapes for the two models, but the mean photon number obtained from the full model is still very close to the expected value of 8. The discrepancy arises primarily from the larger excited state hyperfine splitting in ${}^{133}{\rm Cs}$, which means that a larger detuning is required to ensure closer agreement with the TCM; a similar discrepancy is also observed in ${}^{87}{\rm Rb}$ for smaller detunings (see SI).

The integrated photon flux obtained from the master equation, however, does not tell us about the variance in the photon number of the output pulse. To get the variance we perform quantum trajectory simulations Carmichael (1993a); Mølmer et al. (1993) and record the times and total number of photon counts in each trajectory. The histogram of photon detection times gives us again the output photon flux, which is shown for comparison in Figs. 2 and 3, along with the photon number distribution and its variance for the output pulse. The distributions are clearly very close to an ideal number state. Note that for these simulations we do not use the full model (owing to the stiffness of the numerical integration caused by the large detunings and ground state hyperfine splitting), but rather use the anti-TCM (or TCM) with spontaneous emission added, in the form of an effective Lindblad operator acting just within the relevant ground state (see SI). Spontaneous emission to the other hyperfine ground state is therefore neglected, but the numerical results from the master equation support this as a good approximation.

Finally, we note that shorter or longer output pulses can be obtained by changing the detuning $\Delta$ and/or laser Rabi frequency $\Omega$. Also, in the results presented here, we assume an instantaneous switch-on of the laser field. This can be relaxed to allow for a smooth initial ramp of the laser field to its peak value, which can be used to tailor the shape of the output $2F$-photon pulse (see the SI for additional examples).

$0N$-states and other superpositions.–Instead of starting with an atom in an end state of the Zeeman ladder, we also have the option to start with a superposition of Zeeman substates; for example, $|\psi_{\text{oat}}\rangle_{F}=(|F,-F\rangle+|F,+F\rangle)/\sqrt{2}$, which can be created using a one-axis twisting (oat) scheme Chalopin et al. (2018), or an arbitrary superposition, created using a scheme such as in Law and Eberly (1998). With our proposed system, these atomic superposition states are directly mapped onto photonic states of the output light pulse. So, for example, the initial state $|\psi_{\text{oat}}\rangle_{F}$ leads to an output pulse in a coherent superposition of vacuum and $N=2F$ photons, i.e., an $0N$-state, $|\Psi\rangle_{\text{pulse}}=(|0\rangle_{\text{out}}+|N\rangle_{\text{out}})/\sqrt{2}$, which is a basic resource in schemes proposed for universal quantum computation Sabapathy and Weedbrook (2018). The output photon flux and photon number distribution for initial atomic state $|\psi_{\text{oat}}\rangle_{F=2}$ are shown in Fig. 4 for ${}^{87}{\rm Rb}$ with cavity QED parameters relevant to the fiber microcavity.

As a further example, in Fig. 4 we also consider an initial state of ${}^{87}{\rm Rb}$ of the form $|\psi_{\text{bc}}\rangle_{F=2}=(|2,-2\rangle+\sqrt{2}|2,0\rangle+|2,+2\rangle)/2$, yielding an output pulse state $|\Psi\rangle_{\text{pulse}}=(|0\rangle_{\text{out}}+\sqrt{2}|2\rangle_{\text{out}}+|4\rangle_{\text{out}})/2$. Such a state is of particular interest, as it constitutes a superposition of states $|0_{\text{L}}\rangle=(|0\rangle+|4\rangle)/\sqrt{2}$ and $|1_{\text{L}}\rangle=|2\rangle$, which are logically encoded (binomial code) states of a qubit for a quantum computation scheme protected up to one photon loss Michael et al. (2016). Note that in mapping general atomic ground-state superpositions onto the states of the output light pulses, one must pay attention to the relative phases between the different components and the phase $\phi$ of the effective cavity-spin coupling. For an exact mapping of relative phases, we require in our model that $\phi=\mp\pi/2$ (depending on the sign of $\Delta$), which can of course be chosen through the phase of the laser field (see SI for further details). Alternatively, $\phi$ can also be incorporated as the relative phase between neighbouring $m_{F}$ levels in the initial atomic state.

Quantum state tomography.–The photon number distribution of the output pulse is not sufficient to confirm that the desired output state has been generated. To verify that the target quantum state has indeed been generated, we implement quantum state tomography on simulated, pulsed-homodyne measurements, obtained via the method of homodyne quantum trajectories (see SI for details). That is, we reconstruct the Wigner function of the pulse by measuring marginals of the Wigner function for a set of homodyne phase angles $\theta$ and then applying the inverse Radon transform to these marginals Vogel and Risken (1989). For these simulations, we again use the TCM with effective spontaneous emission added. Results of these reconstructions are shown in Fig. 5. Our reconstructions can clearly be assigned to the predicted, ideal state. For a better comparison, we remove most of the noise from the simulated results by smoothing with a Gaussian blur, which reveals some discrepancy in the heights of the maxima and minima between the ideal case and our reconstruction. This effect can also be observed in the untreated marginals, where we observe some noise in the outer peaks of these marginals that can be attributed to atomic spontaneous emission.

Alternatively, the density matrix itself can be reconstructed using maximum likelihood estimation Hradil (1997); Lvovsky (2004) on the marginals. We start with a normalized identity matrix as the initial guess and go through $500$ iterations. The resulting fidelities of the estimated density matrices for the Fock state, $0N$-state and the binomial-code state from Fig. 5 are $0.922$, $0.959$, and $0.968$, respectively. Almost identical fidelities have also been obtained using the input-output formalism for quantum pulses Kiilerich and Mølmer (2019) (see SI). This illustrates that, even though the superposition states are somewhat more complex, the initial atomic superposition states are, in a certain sense, closer to the final atomic state, and therefore more robust against spontaneous emission.

Conclusion and outlook.–We have proposed a single-atom, deterministic source of optical number-state, $0N$-state, and binomial-code-state pulses. The scheme does not require time-dependent atom-laser or atom-cavity coupling strengths or detunings, or specific $F\leftrightarrow F^{\prime}$ atomic transitions, and should be feasible with recently-demonstrated, fiber-integrated micro- and nano-cavity QED setups. Some other potential features of the scheme are worth noting. For the case of number-state pulses, it is a simple matter to generate a stream of such pulses by simply switching the polarization of the laser field at the end of each pulse and cycling the atom back and forth between the end states $|F,\pm m_{F}\rangle$. In this context, one may also increase $N$ by adding more atoms; e.g., with two identically-prepared ${}^{87}{\rm Rb}$ atoms coupled collectively to the cavity mode, the effective spin in the TCM is simply doubled, enabling the generation of 8-photon pulses. Finally, we have assumed throughout this work that the cavity is essentially one-sided, so that pulses are emitted in just one direction into the output fiber. We could equally well assume a symmetric cavity, in which case our scheme could be equated to a 50/50 beamsplitter with the incident state in one input port determined by the initial state of the atom. This would provide a straightforward means of producing an entangled state of light fields propagating in opposite directions away from the cavity.

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