A hybrid source of quantum light for generation of frequency tunable Fock states
Aleksa Krstić Priyanshu Tiwari Florian Höhe Frank Setzpfandt Ulf Peschel Joachim Ankerhold Sina Saravi


APS/123-QED




aleksa.krstic@uni-jena.de

 Institute of Applied Physics, Abbe Center of Photonics, Friedrich-Schiller University Jena, Albert-Einstein-Straße 15, 07745 Jena, Germany

 

 Institute of Applied Physics, Abbe Center of Photonics, Friedrich-Schiller University Jena, Albert-Einstein-Straße 15, 07745 Jena, Germany

 
 Institute for Complex Quantum Systems and IQST, University of Ulm, Albert-Einstein-Allee 11, 89069 Ulm, Germany
 


 Institute of Applied Physics, Abbe Center of Photonics, Friedrich-Schiller University Jena, Albert-Einstein-Straße 15, 07745 Jena, Germany
Fraunhofer Institute for Applied Optics and Precision Engineering, Albert-Einstein-Straße 7, 07745 Jena, Germany


Institute of Condensed Matter Theory and Solid State Optics, Friedrich-Schiller University Jena, Max-Wien-Platz 1, 07743 Jena, Germany


Institute for Complex Quantum Systems and IQST, University of Ulm, Albert-Einstein-Allee 11, 89069 Ulm, Germany
 


 Institute of Applied Physics, Abbe Center of Photonics, Friedrich-Schiller University Jena, Albert-Einstein-Straße 15, 07745 Jena, Germany



             


We propose a scheme for quantum-light generation in a nonlinear cavity hybridized with a 2-level system. We theoretically show that, when excited by a series of controlled pump pulses, the hybrid source can generate various Fock states with high probabilities, such as near-on-demand generation of 1- and 2-photon states, and above 50% probability for generation of Fock states with up to 7 photons. More importantly, the tailorable nature of the nonlinear cavity and its pumping allows for generating Fock states with arbitrary frequencies, even with a fixed 2-level system, creating fundamentally new opportunities in all areas of quantum technologies.




A hybrid source of quantum light for generation of frequency tunable Fock states

    Sina Saravi
    April 29, 2024
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Introduction.

To support the realization of optical quantum technologies <cit.>, development of efficient and tailorable sources of quantum light, such as squeezed light and Fock states, is of major interest.
While squeezed light can reliably be generated using nonlinear parametric processes <cit.>, generation of optical Fock states with photon numbers n≥ 2 is notoriously difficult in practice and is an ongoing topic of research. The majority of proposals and implementations rely on either of the two general approaches: using atomic or atom-like solid-state quantum systems <cit.>, with successful implementations mainly based on temporal demultiplexing of single photons from a single quantum dot <cit.>, or using nonlinear parametric sources, where Fock states can be heralded by conditional measurement of squeezed-light states
<cit.>.
Yet both approaches have their limitations. Although capable of deterministic generation, atomic and solid-state systems have very limited tunabilities <cit.>, especially in their spectra, which result in states with limited tailorability in spectral/modal properties.

In contrast, nonlinear sources are widely tunable in spectral, modal, and polarization degrees of freedom, specially in nanostructured platforms <cit.>, yet they only generate Fock states probabilistically, with decreasing probability for higher photon-number states <cit.>.

Motivated by these complementary advantages and disadvantages of nonlinear and atom-like systems, we aim for a hybrid system combining the two, that exploits both their advantages towards creating an ideal source of Fock states, one that is on-demand and tunable.
Such hybrid systems have attracted attention recently, with proposals for enhancing atom-cavity interaction <cit.>, single-photon generation with enhanced purity <cit.>, generating equally weighted superposition of Fock states <cit.>, and generating Schrödinger cat states <cit.>.

In this Letter, we propose a hybrid source, consisting of a nonlinear cavity and a two-level system (2LS), for generation of Fock states with enhanced probabilities and arbitrary frequencies. To this end, we use a sequence of optical pump pulses for the nonlinear cavity, with controlled amplitudes, phases, and temporal delays, and leverage the complex interplay between squeezed-light generation and its interaction with a 2LS. Importantly, the properties of the generated Fock states, specially their frequency, can be adjusted by tuning the parameters of the nonlinear process. We also study this system's performance in the presence of cavity losses. 



Basic principles. 
The hybrid source is depicted schematically in Fig. <ref>(a), in which a nonlinear crystal with second-order susceptibility  and a 2LS, are embedded in an optical cavity.
The nonlinear cavity, without the 2LS, when pumped by an optical field of frequency ω_p=ω_s+ω_i, can generate two-mode squeezed vacuum (TMSV) into two separate resonance modes of the cavity, referred to as signal and idler modes, of frequencies ω_s and ω_i, respectively. The 2LS has the transition frequency ω_0, which is resonant with the idler mode, i.e., ω_i=ω_0.
The TMSV state generated by the squeezing operator exp-i (r + H.c.) can be written as a superposition of perfectly photon-number correlated states as |TMSV⟩=ζ∑_n=0^∞ (iexpiϕtanhζ)^n|n,n⟩ <cit.>. Here,  and  are the creation operators for the idler and signal modes, respectively. We adopt the convention that a state of the form |m,n⟩ contains m idler, and n signal photons and also refer to states where n=m as multi-pair states. Additionally, ζ and ϕ are the magnitude and phase, respectively, of the complex parametric gain r=ζexpiϕ, determined by the properties of the nonlinear cavity and the pump field <cit.>. 

In Fig. <ref>(b), we show the probability of finding the first few multi-pair states in a TMSV state, as a function of the parametric gain. We observe that, as the parametric gain increases, these probabilities rise to a maximum value and then decrease. This is a consequence of lower-order |n,n⟩ states "seeding" the generation of the higher-order ones. This property of squeezed-light sources sets a fundamental limit on the probability of generating any given multi-pair state. 
In principle, a combination of heralding and multiplexing approaches can be used to enhance the generation probability of a specific Fock state. Yet, this demands enormous resources in practice, even for the simplest case of realizing an on-demand source of single photons <cit.>. 
 
Hybrid dynamics. The hybrid system dynamics can be described with the following Hamiltonian in the interaction picture,
Ĥ(t)=ħΓℰ(t)+i ħΩ/2 + H.c. <cit.>.
Here, Γ is a nonlinear coupling constant, representing the nonlinear efficiency of the cavity for TMSV generation. Ω is the single-photon Rabi frequency, and  is the atomic lowering operator for the 2LS.
ℰ(t) is the temporal envelope of the pump field, which is made of a sequence of temporally distinct pulses.
Each pump pulse is enumerated by the number j, and is characterized by a parametric gain
r_j=ζ_jexpiϕ_j=Γ∫_jtℰ(t), where the integration extends over the duration of the j-th pulse.
Without loss of generality, we use Gaussian pump pulses with temporal widths fixed to ≈5×10^-32π/Ω, much shorter than the temporal period of few-photon Rabi oscillations between the idler photons and the 2LS, which allows us to isolate the dynamics of nonlinear generations from the Rabi oscillations. Essentially, with every shot of a pump pulse, the nonlinear Hamiltonian induces a quick change in the system's state, followed by much slower Rabi oscillations. For numerical calculations, we also used  library in Python <cit.>.
The details of the analytical derivations and numerical procedures are outlined in the Supplementary <cit.>.

To understand the hybrid dynamics, an example with a two-pulse pump sequence is shown in Fig. <ref>(c-f). We plot the temporal evolution of the first few multi-pair states, where we show their probabilities and their relative phases with respect to the |0,0⟩ state.
In Fig. <ref>(c,d), we study the case of the nonlinear cavity without the 2LS, essentially a TMSV source. 
We use two pump pulses of equal gain magnitudes, but with a relative phase difference of π, with r_1=-r_2≈0.58.
The sharp changes in all the plots correspond to the points in time where a pump pulse is incident on the nonlinear cavity.
As we see in this case, these two pulses cancel each others parametric gains, resulting in a perfect destructive nonlinear interference <cit.>, creating the TMSV with the first pulse and going back to the vacuum state with the second pulse.
In Figs. <ref>(e,f), we show the results for the same two-pulse configuration as in Figs. <ref>(c,d), but now in the presence of the 2LS in the nonlinear cavity.
Firstly, we see that for each individual multi-pair state generated by the first pump pulse, the hybrid system experiences Rabi oscillations between composite states |n,n,g⟩ and |n-1,n,e⟩, where g (e) indicates the ground (excited) state of the 2LS.
The frequency of each Rabi oscillation is proportional to √(n) <cit.>.
This scaling of the Rabi frequency with photon number, results in a complex set of relative phases between individual multi-pair states by the time the second pump pulse arrives, which consequently changes the nature of nonlinear interferences. This then results in a significantly different final state, as can be seen, compared to the obtained vacuum state in the case without the 2LS.
Interestingly, we can see in Fig. <ref>(e), that the probability of generating a single pair of signal and idler photons reaches a value close to 0.5, which is twice the value than can be achieved in a conventional TMSV source, as was seen in Fig. <ref>(b).



In fact, the final state varies significantly with the parametric gain, relative phase, and the delay time between the pump pulses. To see this, in Fig. <ref>, we show the probability of finding a single-photon state in the signal mode (calculated by adding the probabilities of the |1,1,g⟩ and |0,1,e⟩ states): (a) as a function of the delay time T_1 and gain magnitude ζ_2 of the second pulse, for a fixed relative phase ϕ_2-ϕ_1=π; (b) as a function of the phase ϕ_2 and magnitude ζ_2 of the parametric gain of the second pulse, for a fixed delay time T_1=0.612π/Ω. The other parameters were identical to the configuration in Figs. <ref>(e,f). In Fig. <ref>(a), we see that the single-photon generation probability attains a maximum of 0.5 for a delay of T_1=0.612π/Ω and a parametric gain of ζ_2≈0.55. In Fig. <ref>(b), we see that phases ϕ_2≠π always result in a lower single-photon generation probability. 
Further simulations (see Supplementary <cit.>) also showed that, within the considered ranges of ζ_2 and T_1, the highest single-photon probabilities are always obtained for a relative phase of π between two consecutive pump pulses.

Three-pulse configuration. 
While the high-dimensional space of control parameters in a multi-pulse sequence offers many ways to tune desired outputs, even configurations with a limited number of varying parameters are capable of generating high-photon-number Fock states, with generation probabilities far surpassing what is possible in a conventional TMSV source. 
To demonstrate this, we calculate the output for a configuration with three pump pulses. We use the parametric gains and delay times of the pulses as "control knobs" for tuning the system towards desired final states, namely to maximize the generation probability of different Fock states in the signal mode of the cavity.
We fixed the relative phase between the pump pulses to π, such that ϕ_1=0, ϕ_2=π, and ϕ_3=0. This choice is guided by our study shown in Fig. <ref>, to maximize the single-photon probability in the signal mode. This does not necessarily mean that this phase choice is a global optimum for maximizing the generation probability of other Fock states, yet it is a reasonable choice, and mainly an attempt to limit the space of control parameters in our optimization study.

For the parametric gain magnitudes, we consider values of ζ≤15dB, which are experimentally achievable in TMSV sources <cit.>. From this point onward, we express parametric gain magnitudes in units of dB, calculated as ζ[dB]=-10log_10(exp-2ζ), for easier comparison with experimental works on TMSV sources. We also restrict the total interaction time to ≤32π/Ω, as the duration of coherent interactions between a quantum emitter and an optical cavity is limited in practice <cit.>.
In the next section, we will briefly address the feasibility of realizing our optimized pulse sequences, in terms of the required interaction time, with respect to such realistic effects.



Under the expressed boundary conditions for our optimization with three pump pulses, we show in Figs. <ref>(a,b,c), the pulse configurations that result in the maximum achievable generation probability for the Fock states |1⟩, |2⟩ and |3⟩ in the signal mode, respectively.
The probability of generating an n-photon Fock state in the signal mode is calculated from summing up the mutually exclusive probabilities of generating the |n,n,g⟩ and |n-1,n,e⟩ states. As the signal mode does not interact with the 2LS, the Fock state in it can eventually escape a realistic cavity of finite quality factor for further utilization.
The left panels in Figs. <ref>(a-c) show the temporal evolution of the probabilities, while the right panels show the final distribution of the probabilities for the signal-mode Fock states, where we find very high probabilities for generation of these distinct Fock states. Especially Figs. <ref>(a,b), show a near-on-demand generation of single- and two-photon Fock states, with 0.98 and 0.93 probabilities, respectively. 
In Fig. <ref>(d), we show the maximum probabilities for generation of Fock states with photon numbers up to n_s=9, that can be obtained in our three-pulse optimization. With it, we also show the corresponding maximum probabilities attainable using a conventional TMSV source for comparison, which clearly demonstrates the enhancement achieved in the hybrid system. As an example, the probability for generating the n_s=9 state is enhanced more than 10-fold compared to the TMSV source.
The exact pulse sequence parameters for attaining each of the Fock states shown in Fig. <ref>(d) are given in the Supplementary <cit.>.



Effects of loss. 
The obtained results up to now are only strictly valid in an idealized system, with no cavity losses, 100% coupling of the 2LS radiation into the cavity mode, and no internal dephasing of the 2LS, which are not fully achievable in practice <cit.>. Longer interaction times, give more time to these effects to negatively affect the system dynamics. To take the first step into investigating our scheme under non-ideal conditions, we focus on the effects of having a finite quality factor for the cavity.
We assume equal losses for both signal and idler cavity modes, characterized by the decay rate γ_c, and solve the Lindblad master equation
∂_tρ̂=1/iħ[Ĥ,ρ]-∑_j1/2(ĉ_j^†ĉ_jρ+ρĉ_j^†ĉ_j-2ĉ_jρĉ_j^†), 
where ĉ_j=√(γ_c)â_j with j={i,s} are the jump operators <cit.>.
We simulated the lossy hybrid system for the same sets of pump parameters as in Fig. <ref>(a-c). The results are shown in Fig. <ref>, where we only show the temporal evolution of the |1,1,g⟩ [(a)], |2,2,g⟩ [(b)], and |3,3,g⟩ [(c)] states for three values of the decay rate γ_c={0.001,0.008,0.03}Ω. As expected, the cavity loss reduces the probabilities of obtaining the target states, with the higher-order states suffering more due to their higher photon number. 
To emphasize this, the probabilities for obtaining a Fock state with n_s=1,2,3 photons in the signal mode immediately after the third pulse are shown in Fig. <ref>(d), where each set of bars corresponds to a particular value of γ_c and the values shown are obtained by tracing-out the idler and 2LS degrees of freedom.
In the supplementary <cit.>, we present a rough calculation to connect the used γ_c values to the Purcell factor of an optical cavity, considering common range of solid-sate-emitter parameters for the 2LS <cit.>, where in our estimation, state-of-the-art micro/nanostructured optical cavities can reach these requirements <cit.>.

Discussion and conclusion.
Our results are obtained by considering only a portion of the full control-parameter space of the hybrid system. Further expanding the search to include more pulses, continuous relative phase variations, higher parametric gains, and frequency detuning between the cavity modes and the 2LS, could lead to further enhancing the generation probabilities, to potentially reach near-deterministic generation of higher Fock states, or even other classes of non-Gaussian states for continuous variable quantum information processing <cit.>.
Moreover, we only investigated loss with configurations optimized under lossless conditions. With an optimization under lossy conditions, one might identify pulse sequences that make the process more robust to losses.

Importantly, the hybrid source inherits the tunability of a TMSV source. Specially, the frequency spectrum of the output photons in the signal mode can be fully controlled by the energy conservation relation ω_s=ω_p-ω_0, only by tuning the frequency spectrum of the pump pulses and the resonances of the nonlinear cavity, even while the 2LS is unchanged.
This opens the door to fundamentally new opportunities for optical quantum technologies, starting with near-deterministic generation of Fock states for different applications in quantum computing, communication, and sensing, which commonly operate in very different wavelength ranges. Moreover, with this scheme, one could in principle realize many indistinguishable sources of Fock states by using many frequency-distinguishable solid-state emitters <cit.>.

Finally, although we had photonic systems in mind, the predicted dynamics are independent of the frequency range of operation and could potentially be adapted to other quantum information processing platforms, e.g. for microwave quantum-state generation in superconducting circuits <cit.>, motional quantum-state generation in trapped ions <cit.>, and generating phonon-photon excitations in optomechanical systems <cit.>.


This research is supported by funding from the Carl-Zeiss-Stiftung (CZS Center QPhoton), the German Research Foundation (DFG) under the project identifier 398816777-SFB 1375 (NOA), and the German Federal Ministry of Education and Research (BMBF) under the project identifiers 13N14877 (QuantIm4Life) and 13N16108 (PhoQuant).
J. A. and F. H. acknowledge financial support by the German Science Foundation (DFG) through AN336/18-1 and AN336/13-1. S. S. acknowledges funding by the Nexus program of the Carl-Zeiss-Stiftung (project MetaNN).






