Abstract
The search for new control methods over lightmatter interactions is one of the engines that advances fundamental physics and applied science alike. A specific class of lightmatter interaction interfaces are setups coupling photons of distinct frequencies via matter. Such devices, nontrivial in design, could be endowed with multifunctional tasking. Here we envisage for the first time an optomechanical system that bridges optical and robust, highfrequency xray photons, which are otherwise notoriously difficult to control. The xrayoptical system comprises of an optomechanical cavity and a movable microlever interacting with an optical laser and with xrays via resonant nuclear scattering. We show that optomechanically induced transparency of a broad range of photons (10 eV–100 keV) is achievable in this setup, allowing to tune nuclear xray absorption spectra via optomechanical control. This paves ways for metrology applications, e.g., the detection of the ^{229}Thorium clock transition, and an unprecedentedly precise control of xrays using optical photons.
Introduction
In cavity optomechanics^{1}, the coupling of electromagnetic radiation to mechanical motion degrees of freedom^{2} can be used to connect quantum system with different resonant frequencies. For instance, via a common movable microlever, an optical cavity can be coupled to a microwave resonator to bridge the two frequency regimes^{3,4,5,6}. Going towards shorter photon wavelengths is highly desirable and timely: in addition to improved detection, xrays are better focusable and carry much larger momenta, potentially facilitating the entanglement of light and matter at a singlephoton level. Unfortunately, a direct application of the sofar employed interface concept for a device that mediates an optical and an xray photon is bound to fail. First, the required highperformance cavities are not available for xrays. Second, exactly the potentially advantageous high momentum carried by an xray photon renders necessary a different paradigm. We note here that xrays are resonant to transitions in atomic nuclei which can be regarded as xray cavities with good quality. The rapidly developing field of xray quantum optics^{7,8,9,10,11,12} has recently reported so far key achievements and promising predictions for the mutual control of xrays and nuclei^{13,14,15,16,17,18,19,20,21,22}.
Here we present an innovative solution for coupling xray quanta to an optomechanical, solidstate device which can serve as a node bridging optical and xray photons in a quantum network. We demonstrate theoretically that using resonant interactions of xrays with nuclear transitions, in conjunction with an optomechanical setup interacting with optical photons, an opticalxray interface can be achieved. Such a device would allow to tune xray absorption spectra and eventually to shape xray wavepackets or spectra for single photons^{19, 23,24,25,26,27,28} by optomechanical control. The role of the xray cavity is here adopted by a nuclear transition with long coherence time that eventually stores the highfrequency photon. Our calculations show that optomechanically induced transparency of xrays can be achieved in the opticalxray interface paving the way for both metrology^{29} and an unprecedently precise control of xrays using optical photons. In particular, a metrologyrelevant application for the nuclear clock transition of ^{229}Th, which lies in the vacuum ultraviolet (VUV) region, is presented.
The optomechanicalnuclear system under investigation is illustrated in Fig. 1a. An optomechanical cavity of length L driven by an optical laser has an embedded layer in the tip of the microlever containing Mössbauer nuclei that interact with certain sharply defined xray frequencies. The nuclei in the layer have a stable or very longlived ground state, and a first excited state that can be reached by a resonant xray Mössbauer, i.e., recoilless, transition. Typically, this type of nuclear excitation or decay occurs without individual recoil, leading to a coherent scattering in the forward direction^{30}. Another type of excitation including the nuclear transition together with the motion of the microlever, i.e., phonons, can also be driven by red or bluedetuned xrays. The nuclear twolevel system can be therefore coupled to the mechanical motion of the microlever of mass M. The term “phonon” is used here to describe the vibration of the center of mass of the cantilever, visible in the tip displacement y. According to the specifications^{1} of various mechanical microlever designs, the phonons in this setup are expected to be in the MHz regime. We choose to label the space coordinate with y since the notation x will be used in the following for the xray field. An effective model of nuclear harmonic oscillator interacting with xrays can be constructed to describe the hybridization^{31, 32} of the xraynucleioptomechanical systems. To this end the wellknown optomechanical Hamiltonian^{1, 33,34,35,36,37} is extended to include also the xray interaction with the nuclear layer embedded in the tip of the microlever. Since the nuclear transition widths are very narrow (10^{−9}–10^{−15} eV), we assume that the nuclei interact with a single mode of the xray field.
The full Hamiltonian of the system sketched in Fig. 1a is a combination of the optomechanical Hamiltonian^{1, 33, 36} and nuclear interaction with xray photons, which can be written in the interaction picture and linearized version as (see Methods and Supplementary Information for detailed derivation) as
Here, ω _{ m } is the optomechanically modified oscillation angular frequency of the microlever, Δ_{ c } is the effective optical laser detuning to the cavity frequency obtained after the linearization procedure, and G the coupling constant of the system. The operators \({\widehat{a}}^{ {\dagger } }\) \((\widehat{a})\) and \({\widehat{b}}^{ {\dagger } }\) \((\widehat{b})\) act as cavity photon and phonon creation (annihilation) operators, respectively. As further notations in Eq. 1, Δ = ω _{ x } − ω _{ n } is the xray detuning with ω _{ n } the nuclear transition angular frequency and ω _{ x } (k _{ x }) is the xray angular frequency (wave vector), respectively. Ω is the Rabi frequency describing the coupling between the nuclear transition currents^{38} and the xray field, Y _{ZPF} is the zeropoint fluctuation, ħ the reduced Planck constant, and e and g denote the nuclear excited and ground state, respectively. The linearization procedure leading to the Hamiltonian in Eq. (1) was performed in the reddetuned regime, namely, cavity detuning Δ_{ c } = −ω _{ m }, which results in the socalled “beamsplitter” interaction^{1} with the optomechanical coupling strength G. We use the master equation involving the linearized interaction Hamiltonian to determine the dynamics of the interface system and the nuclear xray absorption spectra as detailed in Methods.
Figure 2 demonstrates the xray/VUV absorption spectra for several nuclear targets, together with an illustration of the corresponding LambDicke parameter η = k _{ x } Y _{ZPF}. We consider nuclear transitions from the ground state to the first excited state in ^{229}Th, ^{73}Ge and ^{67}Zn, with the relevant nuclear and optomechanical parameters presented in Table 1. The chosen optomechanics setup parameters^{39} are M = 0.14 μg, the inherent phonon frequency ω _{0} = 2π × 0.95 MHz, the optomechanical damping rate γ _{0} = 2π × 0.14 kHz, the optical cavity decay rate κ = 2π × 0.2 MHz, cavity frequency ω _{ c } ~ 10^{15} Hz and the optomechanical coupling constant \({G}_{0}=\frac{{\omega }_{c}}{L}{Y}_{{\rm{ZPF}}}\sqrt{{\bar{n}}_{{\rm{cav}}}}=2\pi \times 3.9\) Hz. These parameters have been experimentally demonstrated^{39}. The required optomechanical system is a 25mmlong FabryPérot cavity made of a highreflectivity mirror pad (reflectivity >0.99991) that forms the endface^{39}. A realistic estimate of the optical thickness values for the nuclear xray absorption is presented in the Supplementary Information.
For a comprehensive explanation, we begin with the case in the absence of optomechanical coupling, i.e., G = 0. Green lines in Fig. 2 illustrate a central nuclear absorption line with detuning Δ = 0 corresponding to m = n, and sidebands that occur with excitation or decay of phonons in the system m = n ± 1, n ± 2, …. The width of the peaks is determined by the value of \(s={\rm{\Gamma }}\mathrm{/2}+\kappa +{\gamma }_{m}\) (see Methods) of the order of MHz, similar in scale with the inhomogeneous broadening of the nuclear transition, which we neglect in the following. In order to resolve the sidebands, a constraint has to be imposed on the oscillation frequency of the microlever, i.e., the microlever frequency ω _{ m } > s, and the FranckCondon coefficients \({F}_{n}^{m}\mathop{ > }\limits_{ \tilde {}}0.1\) for at least the first phonon lines m = n ± 1 (see Methods). As a consequence, nuclear species with large LambDicke parameters η allow the observation of xray absorption sidebands. For example, compared to ^{229}Th (η = 9.92 × 10^{−9}) in Fig. 2a which presents only the zero phonon line, the spectra of ^{73}Ge (η = 1.69 × 10^{−5}) in Fig. 2b show an observable sideband as indicated by the red arrow. Moreover, there are several sidebands appearing in the spectrum of ^{67}Zn (η = 11.87 × 10^{−5}) depicted in Fig. 2c. Further Mössbauer nuclei with suitable first excited states whose decay rates are lower than the phonon angular frequency of around 6 MHz are for instance^{30} ^{45}Sc, ^{157}Gd and ^{181}Ta.
We are now ready to discuss the results including the optomechanical coupling, G > 0, illustrated by the blue and red dashed lines in Fig. 2. Remarkably, the optomechanical coupling introduces a dip at the center of each line. As illustrated also in Fig. 1b, the line splittings are caused by the optomechanical coupling G, which links different phonon Fock states via the beam splitter interaction^{1}. We stress here that the nuclear xray absorption is only modified by the optomechanical coupling and does not have to do with xray recoil which is not occuring in our scheme. The depth and the spacing of the dips are proportional to the input optical laser power which modifies the strength G. Figure 2 shows that the absorption gradually goes to zero with increasing laser power P. The diagonalization of the Hamiltonian shows that the two split peaks around the zero phonon line are approximately positioned at \({\rm{\Delta }}=\pm \sqrt{{(G\sqrt{m+v+2mv}+s)}^{2}2{s}^{2}}\). These two eigenvalues correspond to transitions between the ground state g, v, n〉 and the two eigenstates \(\sqrt{\frac{(1+m)v}{(1+v)m}}e,v1,m+1\rangle \mp \sqrt{\frac{m+v+2\,mv}{(1+v)m}}e,v,m\rangle +e,v+1,m1\rangle \) (see Methods). These eigenstates result in an analog of the socalled optomechanically induced transparency^{34,35,36} in the xray domain and offer means of controlling xray spectra. This is a new mechanism compared to typical target vibration experiments of Mössbauer samples^{23,24,25,26}, in the classical phonon regime. The width of the splitting indicates that, with sufficient phonon numbers, the compelling optomechanical coupling can be accomplished by an optical laser. This feature may render control of xray quanta by means of weak optical lasers possible. In order to demonstrate this possibility, laser power parameters of few nW are used in the calculation to implement full transparency of xrays around the nuclear resonance (see blue dasheddotted and red dashed lines in Fig. 2).
Since the natural nuclear linewidths are far more narrow than present xray sources, the suitable solution for resolving the phonon sidebands of the keV xray or VUV resonance energies is to employ a Mössbauer drive setup. ^{67}Zn Mössbauer spectroscopy for instance is a wellestablished technique with exceptionally high sensitivity for the gammaray energy. This has been exploited^{40} for precision measurements of hyperfine interactions ^{67}Ga decay schemes, which populate excited states in ^{67}Zn. The decay cascade will eventually populate the first excited level, which then releases single photons close to the resonance energy of the nuclear layer on the microlever. Assuming 50 mCi source activity and a solid angle corresponding to a 20 × 20 μm^{2 67}Zn layer placed 10 cm away, the rate of xray photons close to the resonance is approx. 40 Hz. The finetuning for matching the exact resonance energy is achieved by means of the Doppler shift using a piezoelectric drive with μm/s velocities^{40}.
While the 7.8 eV transition of ^{229}Th is not traditionally regarded as a Mössbauer case, studies have shown that when embedded in VUVtransparent crystals, thorium nuclei are expected to be confined to the LambDicke regime^{41,42,43}. In this regime one expects clear parallels to nuclear forward scattering techniques as known from traditional Mössbauer transitions. The uniquely low lying state and the very narrow transition width of approx. 10^{−19} eV makes ^{229}Th a candidate for a stable and accurate nuclear frequency standard^{29}. The most important step in this direction would be a precise measurement of the nuclear transition frequency, at present considered to be 7.8 ± 0.5 eV^{44}. However, two major difficulties have been encountered in such measurements. First, the extremely narrow linewidth of 10^{−5} Hz makes very difficult both the excitation and the detection of fluorescence for this transition. Second, the isomeric transition has a disadvantageous signal to background ratio and strong fake signals from the environment have been so far impairing experiments^{45,46,47}.
The VUV spectra of ^{229}Th illustrated in Fig. 2a reveal that our chipscale system could be used to determine the nuclear clock transition energy^{43, 44, 48}. For this exceptional case with VUV nuclear transition energy, the excitation could be achieved with VUV lasers at present in development^{49}. Two important advantages arise in the VUVoptomechanical interface: (i) the width \(s\gg {\rm{\Gamma }}\) broadens the VUV absorption linewidth by 10 orders of magnitude, namely, \(s\sim {10}^{10}\,{\rm{\Gamma }}\), facilitating the excitation and speeding up the nuclear target’s decoherence. (ii) The VUV spectra are optomechanically tunable. This can offer a clear signature of nuclear excitation circumventing false signals which unavoidably appear from either crystal sample^{45} or surrounding atmosphere^{46, 47}.
We have put forward the theoretical formalism for optomechanically induced transparency of xrays via optical control. In particular, our results show that the induced transparency may be achieved for nuclear transitions, with possible relevance for metrological studies, e.g., detection of nuclear clock transition. The opposite situation, of xray photons controlling the optomechanical setup, may open new possibilities for connecting quantum network devices^{50} on atomic and mesoscopic scales.
Methods
The full Hamiltonian of the system sketched in Fig. 1a is a combination of the optomechanical Hamiltonian and nuclear interaction with xray photons^{1, 33, 36} (see also Supplementary Information),
Here, ω _{0} denotes the inherent phonon, ω _{ c } the resonant cavity, and ω _{ n } the nuclear transition angular frequency, respectively, and Ω is the Rabi frequency describing the coupling between the nuclear transition currents^{38} and the xray field. The operators \({\widehat{x}}^{ {\dagger } }\) \((\widehat{x})\) act as xray photon creation (annihilation) operators, respectively. The optomechanical coupling constant is given by G _{0} = ω _{ c } Y _{ZPF}/L, where Y _{ZPF} denotes the zeropoint fluctuation. Typically, the Hamiltonian expression above is transformed in the interaction picture and linearized with respect to the cavity photon number at equilibrium^{1, 36}, i.e., the balance between external pumping and cavity loss. It is therefore convenient to neglect external cavity driving terms by classical optical fields in the Hamiltonian of the system^{1, 35, 36}. We will see below that one can effectively attribute the modified properties of the system to the new optomechanical coupling constant G. By an unitary transformation to the rotating frame^{1} (see Supplementary Information), we obtain the Hamiltonian in the interaction picture
where Δ_{ c } = ω _{ l } − ω _{ c } is the optical laser detuning to the cavity frequency, ω _{ l } the optical laser angular frequency and Δ = ω _{ x } − ω _{ n } the xray detuning. The final step is to linearize the Hamiltonian by performing the transformation \(\widehat{a}\to \sqrt{{\overline{n}}_{cav}}+\widehat{a}\), where \({\overline{n}}_{cav}\) is the averaged cavity photon number, and \(\widehat{a}\) becomes the photon number fluctuation^{1, 36}. The expression \({\overline{n}}_{cav}+\langle v{\widehat{a}}^{ {\dagger } }\widehat{a}v\rangle \) gives the photon number of the full cavity field. We neglect the first order terms of \({\widehat{a}}^{ {\dagger } }{\widehat{b}}^{ {\dagger } }\) and \(\widehat{a}\widehat{b}\) in the rotating wave approximation, and the second order terms proportional to \({\widehat{a}}^{ {\dagger } }\widehat{a}\). The zero order terms \({\overline{n}}_{{\rm{cav}}}({\widehat{b}}^{ {\dagger } }+\widehat{b})\) may be omitted^{1} after implementing an averaged cavity length shift \(\delta L=\hslash {\omega }_{c}{\overline{n}}_{cav}/(Lm{\omega }_{0}^{2})\) and the averaged cavity angular frequency shift \(\delta {\omega }_{c}=\hslash {\omega }_{c}^{2}{\overline{n}}_{cav}/({L}^{2}m{\omega }_{0}^{2})\), leading to the effective detuning^{1} Δ_{ c } → Δ_{ c } + δω _{ c }. We focus on the reddetuned regime, namely, cavity detuning Δ_{ c } = −ω _{ m }, which results in the socalled “beamsplitter” interaction^{1}. We obtain the linearized Hamiltonian given in Eq. (1) with the new coupling constant \(G={G}_{0}\sqrt{{\overline{n}}_{{\rm{cav}}}}\). The effective phonon angular frequency ω _{ m } = ω _{0} + δω _{0} is introduced where \(\delta {\omega }_{0}=4{G}^{2}(\frac{{\omega }_{0}}{{\kappa }^{2}+16{\omega }_{0}^{2}})\) is the optomechanically modified oscillation angular frequency of the microlever^{1}. The zeropoint fluctuation of the microlever’s mechanical motion can then be written as \({Y}_{{\rm{ZPF}}}=\sqrt{\hslash /(2M{\omega }_{m})}\).
We use the master equation \({\partial }_{t}\widehat{\rho }=\frac{1}{i\hslash }[\widehat{H},\widehat{\rho }]+{\widehat{\rho }}_{dec}\) involving the linearized interaction Hamiltonian to determine the dynamics of the interface system (see Supplementary Information for the explicit form of each matrix). Decoherence processes are described by \({\widehat{\rho }}_{dec}\) including the spontaneous nuclear decay characterized by the rate Γ, the inherent mechanical damping rate of the microlever γ _{0} and the optical cavity decay rate κ. The density matrix elements \({\rho }_{\beta d\nu }^{\alpha c\mu }={A}_{\alpha c\mu }^{\ast }{A}_{\beta d\nu }\) correspond to the state vector \(\psi \rangle={A}_{gv1n+1}g,v1,n+1\rangle+{A}_{gvn}g,v,n\rangle+\) \({A}_{gv+1n1}g,v+1,n1\rangle\) \(+{A}_{ev1m+1}e,v1,m+1\rangle +{A}_{evm}e,v,m\rangle +{A}_{ev+1m1}e,v+1,m1\rangle \) where the system is initially prepared^{39, 51} in the nuclear ground state with \({\overline{n}}_{{\rm{cav}}}\) fluctuated cavity photons at the level of v and n phonons g, v, n〉, and the nuclear excited state with m phonons e, v, m〉 is reached by xray absorption, as illustrated in Fig. 1b. Four additional states with n ± 1 and m ± 1 phonons are coupled by the beam splitter interaction. In the reddetuned regime^{1} the mechanics of the optically tunable microlever can be described as \({\partial }_{t}^{2}y+{\gamma }_{m}{\partial }_{t}y+{\omega }_{m}^{2}y=0\), where y(t) denotes the displacement of the microlever as illustrated in Fig. 1(a), and the optomechanical damping rate shift is given by \(\delta {\gamma }_{0}=4{G}^{2}(\frac{1}{\kappa }\frac{\kappa }{{\kappa }^{2}+16{\omega }_{0}^{2}})\). The effective optomechanical damping rate γ _{ m } = γ _{0} + δγ _{0}. A relevant quantity is the average number of photons inside the cavity, which depends on the optical laser power \(P\) and is given by ref. 1 \({\overline{n}}_{{\rm{cav}}}=\frac{\kappa P}{\hslash {\omega }_{l}[{({\omega }_{l}{\omega }_{c})}^{2}+{(\kappa \mathrm{/2})}^{2}]}\).
The xray absorption spectrum of the interface system is determined by the offdiagonal terms of the Hamiltonian \(\widehat{H}\), i.e., \(\langle e,v,m\widehat{H}g,v,n\rangle =\frac{\hslash {\rm{\Omega }}}{2}\langle m{e}^{i{k}_{x}{Y}_{{\rm{ZPF}}}({\widehat{b}}^{ {\dagger } }+\widehat{b})}n\rangle \). The phase term η = k _{ x } Y _{ZPF} is the socalled LambDicke parameter, and for \(\eta \sqrt{n} < 1\), \({F}_{n}^{m}=\langle m{e}^{i\eta ({\widehat{b}}^{ {\dagger } }+\widehat{b})}n\rangle \) denotes the FranckCondon coefficient^{33}
Typically, only low nuclear excitation is achieved in nuclear scattering with xrays, such that the master equation in the perturbation region \({\rm{\Gamma }}\mathrm{/2}+\kappa +{\gamma }_{m} > G\gg {\rm{\Omega }}\) can be used, corresponding to the stable regime. We note here that nuclear scattering experiments and simulations have confirmed in this low excitation regime the validity of the semiclassical limit for xraynucleus interaction^{52}. The steady state solution reads
where the total decoherence rate notation \(s={\rm{\Gamma }}\mathrm{/2}+\kappa +{\gamma }_{m}\) was introduced. By replacing \({F}_{n}^{m}\) with \({F}_{n}^{m}\), the sum of the imaginary part of Eq. (6) for corresponding transitions, namely, \({\sum }_{m=n6}^{n+6}{\rm{Im}}[{\rho }_{gvn}^{evm}({\rm{\Delta }})]\), provides the xray absorption spectrum. Eq. (6) shows that the xray absorption is directly dependent on the numbers of photons \({\overline{n}}_{{\rm{cav}}}\) and averaged number of phonons m and n and their statistics.
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Acknowledgements
The authors would like to thank Markus Aspelmeyer for fruitful discussions. WTL is supported by the Ministry of Science and Technology, Taiwan (Grant No. MOST 1052112M008001MY3). WTL is also supported by the National Center for Theoretical Sciences, Taiwan. AP gratefully acknowledges funding by the EU FETOpen project 664732.
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W.T.L. and A.P. contributed equally to this work. W.T.L. performed the numerical calculations. W.T.L. and A.P. discussed the results and wrote the manuscript text.
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Liao, W., Pálffy, A. Optomechanically induced transparency of xrays via optical control. Sci Rep 7, 321 (2017). https://doi.org/10.1038/s4159801700428w
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