Much recent effort has focused on coupling individual quantum emitters to optical microcavities in order to produce single photons on demand, enable single-photon optical switching, and implement functional nodes of a quantum network. Techniques to control the bandwidth and frequency of the outgoing single photons are of practical importance, allowing direct emission into telecommunications wavelengths and “hybrid” quantum networks incorporating different emitters. Here, we describe an integrated approach involving a quantum emitter coupled to a nonlinear optical resonator, in which the emission wavelength and pulse shape are controlled using the intra-cavity nonlinearity. Our scheme is general in nature, and demonstrates how the photonic environment of a quantum emitter can be tailored to determine the emission properties. As specific examples, we discuss a high Q-factor, TE-TM double-mode photonic crystal cavity design that allows for direct generation of single photons at telecom wavelengths (1425 nm) starting from an InAs/GaAs quantum dot with a 950 nm transition wavelength, and a scheme for direct optical coupling between such a quantum dot and a diamond nitrogen-vacancy center at 637 nm.
© 2009 Optical Society of America
In recent years, there has been a concerted research effort towards achieving strong coupling between single quantum emitters and high-finesse, resonant optical microcavities (cavity QED) [1, 2, 3]. The strong coupling results in preferential emission into the cavity mode of a single photon with frequency near the emitter resonance. This is beneficial to generate single photons on demand, realize large optical nonlinearities at a single-photon level [4, 5] for low-power switching or quantum logic gates, and facilitate communication between distant nodes of a quantum network [6, 7]. In state-of-the-art systems, however, the operating (emission) wavelength is determined by the material properties of the quantum emitter – namely, the atomic or excitonic resonance frequency. With the exception of quantum dots, this wavelength is fixed and cannot be engineered. For example, the operating wavelength of systems based on Cs or Rb atoms is limited to 852 nm and 780 nm, respectively. In practice, however, the ability to substantially shift the emission frequency would open up a number of important possibilities, including direct emission into low-loss telecom frequency bands for long-distance transmission of photons over existing communication channels. It would also allow direct coupling between different types of emitters, enabling hybrid quantum networks in which the best attributes of various emitters can be combined – e.g., allowing single photons generated by solid-state systems [8, 9, 10, 11, 12, 13, 14] to be coherently stored for long times in atomic gases [15, 16]. This would be an important step towards realizing fully functional quantum networks (with sources, processors, memory, etc.). Similarly, tailoring the bandwidth of the emitted photons would allow for fast communication and operation times.
Here, we demonstrate a novel approach to achieve these goals based on a quantum emitter strongly coupled to a nonlinear optical resonator, where the optical emission is directly frequency-shifted into the desired domain using efficient intracavity nonlinear optical processes. By tailoring the optical degrees of freedom (resonances) of such a system, we demonstrate that the wavelength of the emitted photons can be shifted over a large spectral range (more than 500 nm) with high efficiency. Our approach allows direct control of the single photon shape and bandwidth, which is important for efficient and rapid optical coupling between quantum nodes . As a practical implementation, we propose a system based on a doubly resonant nonlinear photonic crystal cavity  which allows for up to ~99% photon conversion efficiencies. This system facilitates nonlinear optical processes at a wavelength scale, and is therefore of interest for the realization of integrated platforms that can operate at low power levels [18, 19, 20, 21, 22, 23]. Moreover, our system allows for the integration of large arrays of single-photon sources, each tailored to emit at different wavelengths (within the ~500 nm wide wavelength range) using the same type of quantum emitter. This could facilitate the realization of multi-color single-photon sources which could be used for secure optical interconnects.
Our technique is quite robust in that the maximum efficiency does not depend on an explicit phase-matching condition , as would occur in an extended nonlinear crystal, but rather only on the ratio of the cavity Quality factor to mode volume (Q/V). This figure of merit can be extremely high in realistic photonic crystal structures. As an example, we demonstrate a novel double mode TE-TM cavity design in a GaAs photonic crystal that is well-suited for the conversion of photons from the near infrared to the telecom band. We also present a similar GaP-based design suitable for direct coupling between a nitrogen-vacancy center in diamond [8, 9, 10, 11] and an InAs/GaAs quantum dot [12, 13, 14], which could enable practical realization of a hybrid quantum network.
Finally, we emphasize that our proposal has a number of advantages compared to previous schemes for spectral control of single photons [25, 26, 27, 28, 29], in that here one can simultaneously shift the emission wavelength over hundreds of nm and enhance the emission bandwidth on an integrated platform. In contrast, techniques based on electromagnetically induced transparency in atomic vapors  cannot enhance the photon bandwidth over the atomic linewidth, and wavelength tailoring is practical only very near the atomic resonant wavelength where the atoms have a significant optical response (≲1 nm tuning range). On the other hand, macroscopic nonlinear crystals have been used to achieve wavelength conversion over hundreds of nm [26, 27, 28, 29], but these approaches offer no bandwidth control and require separate single-photon sources as inputs. Moreover, they require stringent phase-matching conditions and high peak-power laser pulses for efficient conversion. Our nanophotonics platform overcomes these shortcomings, thereby enabling highly efficient conversion and bandwidth control of single photons in mode volumes smaller than a cubic wavelength.
2. The concept of single-photon spectral control
We first discuss the general protocol for generating single photons on demand at arbitrary frequencies using a nonlinear double-mode cavity, and introduce a simple theoretical model to derive the efficiency of the process. The system of interest is illustrated schematically in Fig. 1. As in standard cavity-based single-photon generation protocols [6, 3], a single three-level atom (or any other quantum dipole emitter) is resonantly coupled to one mode (here denoted a) of an optical cavity. The emitter is initialized through optical pumping in state |s〉, the specific nature of which depends on the type of emitter used. This is a standard technique for preparing the initial state of an emitter, and has been experimentally demonstrated for both quantum dots  and diamond NV centers . An external laser field with controllable Rabi frequency Ω(t) couples |s〉 to excited state |e〉. The transition between |e〉 and ground state |g〉 is resonantly coupled to cavity mode a (frequency ωa), with a single-photon Rabi frequency g 1. The relevant decay mechanisms (illustrated with gray arrows in the figure) are a leakage rate κa for photons to leave cavity mode a, and a rate γ that |e〉 spontaneously emits into free space rather than into the cavity. Conventionally, in absence of an optical nonlinearity, the control field Ω(t) creates a single atomic excitation at some desired time in the system, which via the coupling g1 is converted into a single, resonant cavity photon. This photon eventually leaks out of the cavity and constitutes an outgoing, resonant single photon generated on demand whose spatial wave-packet can be shaped by properly choosing Ω(t) .
In our system, the cavity is also assumed to possess a second mode c with frequency ωc, and our goal is to induce the single photon to exit at this frequency rather than ωa. This can be achieved, provided that the cavity medium itself possesses a second-order (χ (2)) nonlinear susceptibility, by applying a classical pump field to the system at the difference frequency ωb=ωa-ωc. The induced coherent coupling rate between modes a and c is denoted g 2. The field b need not correspond to a cavity mode. Mode c has a photon leakage rate, which we separate into an “inherent” rate, κ c,in, and a “desirable” (extrinsic) rate, κ c,ex. κ c,in characterizes the natural leakage into radiation modes and also absorption losses, and can be expressed in terms of the (unloaded) cavity quality factor as κ c,in=ωc/2Qc. κ c,ex characterizes the out-coupling rate into any external waveguide used for photon extraction. The total leakage of mode c is then κ c=κ c,in+κ c,ex.
More quantitatively, the effective Hamiltonian for the system (in a rotating frame) is given by
where Hc describes the coherent part of the system evolution (for simplicity we take g 1,2,Ω to be real), and Hloss is a non-Hermitian term characterizing the losses. σij=|i〉〈j| are atomic operators, while ai is the photon annihilation operator for mode i. The vacuum Rabi splitting g 1 can be written in the form g 1=d·E a(r)/h̄, where d is the dipole matrix element of the |g〉-|e〈 transition, and E a(r) is the electric field amplitude per photon at the emitter position r. The electric field per photon in mode i=a,c is determined by the normalization
where ε (r) is the dimensionless electric permittivity of the material. The nonlinearity parameter is given by 
The amplitudes E a,c appearing above are normalized by Eq. (2), while Eb is the classical pump amplitude. Importantly, one can compensate for a small nonlinear susceptibility χ (2) or field overlap (phase matching) simply by using larger pump amplitudes Eb to achieve a desired g 2 strength.
For a system initialized in |s〉, there can never be more than one excitation, and the system generally exists as a superposition of having the system in state |s〉 or |e〉 (with no photons) or having a photon in one of the modes a,c (and the emitter in |g〉),
The system is initialized to cs(0)=1 with all other ci(0)=0 and the time evolution is given by ċj=-(i/h̄)〈j|HI|ψ(t)〉. In this effective wave-function approach, provided that |s〉 is always driven, ∑j|cj|2→0 as t→∞ due to losses, which can be connected with population leakage out of one of the aforementioned decay channels. In the limit that Ω(t) is small and varies slowly, all other ci(t) adiabatically follow cs(t) (see Appendix), and one finds
Physically, we can identify as the cavity-enhanced total decay rate of |e〉, where the first (second) term corresponds to direct radiative emission (emission into mode a). Similarly, the denominator κa+4g 2 2/κc corresponds to the new total decay rate of mode a in the presence of an optical nonlinearity, as it yields a new channel for photons to effectively “decay” out of mode a into c at rate 4g 2 2/κc. It is clear that some optimal value of g 2 exists for frequency conversion to occur. In particular, for no nonlinearity (g 2=0) this probability is non-existent. On the other hand, for g 2→∞, one finds γ total=γ, which indicates that the leakage from mode a into c is so strong that no cavity-enhanced emission occurs. Note that the use of a time-varying control field allows for arbitrary shaping of the outgoing single-photon wavepacket at frequency ωc, provided only that the photon bandwidth is smaller than κc (physically, the photon cannot leave faster than the rate determined by the cavity decay; see Appendix). This feature is particularly useful in two respects. First, in practice κc can be much larger than γ, which enables extremely fast operation times. Second, pulse shaping is useful for constructing quantum networks, as it allows one to impedance-match the outgoing photon to other nodes of the network.
Based on the above arguments, the probability that a single photon of frequency ωc is produced and extracted into the desired out-coupling waveguide is given by
where ϕ=4g 2 2/(κaκc) characterizes the branching ratio in mode a of nonlinearity-induced leakage to inherent losses, and Cin=4g 2 1/γκa is the inherent cavity cooperativity parameter for mode a in absence of nonlinearity. The first term on the right denotes the probability for |e〉 to decay into mode a, the second term the probability that a photon in mode a couples into mode c, and the third term the probability that a photon in mode c out-couples into the desired channel (see Appendix for an exact calculation). ϕ depends on the pump amplitude Eb, with the optimal value yielding the maximum in F. For large Cin≫1, the maximum probability is
Considering an emitter placed near the field maximum of mode a, , where Qa,Va are the mode quality factor and volume, respectively, and n is the index of refraction at frequency ωa. The ratio γ/γ 0 is the spontaneous emission rate into non-cavity modes normalized by the spontaneous emission rate γ 0≡nω 3 a|d|2/(3πε 0 h̄c 3) of an emitter embedded in an isotropic medium of index n. This ratio is expected to be of order 0.1–1 for our devices of interest, and thus the efficiency essentially depends only on Qa/Va.
We have analyzed here the case of a purely radiative emitter. For certain solid-state emitters, dephasing of its electronic transitions may not be negligible. In the presence of dephasing, it is necessary to solve for the full density matrix ρ of the system (see Appendix). This yields a modified expression for the probability of frequency conversion (eq. (21)). However, we note that the effects of dephasing are likely to be small. For example, in the charged InAs quantum dot in ref. , the dephasing rate is γd=0.54 GHz, and in ref. , γd=1.17 GHz. In the specific example we consider in Section 4, the effect of these decoherence channels on the frequency conversion probability (Fig. 3) is negligible. By the same token, in diamond NV centers, the dephasing rate of the electron spin can be extremely small, on the order of ~1 MHz .
Finally, while we have focused on the case of single-photon generation here, the reverse process can also be considered, where a single incoming photon at frequency ωc is incident upon the system, converted into a photon in mode a, and coherently absorbed by an atom with the aid of an impedance-matched pulse Ω(t), causing its internal state to flip from |g〉 to |s〉. Generally, by time-reversal arguments , it can be shown that the probability F for single-photon storage is the same as that for generation.
3. Realization in a nonlinear photonic crystal cavity
In order to implement this frequency conversion scheme in a practical fashion, there are several constraints on the design of the cavity modes. For the nonlinear process to be efficient, mode a must have a high cooperativity (Q/V) to ensure strong coupling of the emitter (see Fig. 1). For mode c, a high Q factor (small κc) is important to maximize the nonlinear coupling parameter, ϕ, and hence reduce the pump power needed in order to reach the optimum nonlinear coupling strength, g 2. The cavity should also be composed of a χ (2) nonlinear material that is transparent in the desired frequency range. Finally, in order for the modes to couple efficiently via the nonlinear susceptibility of the cavity, they must have a large spatial overlap and the appropriate vector orientation, as determined by the elements of the χ (2) tensor of the cavity material (see Eq. (3)).
As a host platform for the nonlinear cavity, the III–V semiconductors are promising candidates because of their significant second-order nonlinear susceptibilities and mature nanofabrication technologies. However, the symmetry of the III–V group χ (2) tensor (χ (2) ijk≠0, i≠j≠k) requires that the dominant field components of the modes be orthogonal in order to maximize the nonlinear coupling. It further implies that if the classical field which drives the nonlinear polarization is incident from the normal direction (e.g., from an off-chip laser), one of the cavity modes must have a TM polarization.
We adopt a photonic crystal platform to realize a wavelength-scale nonlinear cavity that meets these requirements. Recently, 2D photonic crystal nanocavities have shown great promise for strongly coupling an optical mode to a quantum dot emitter [13, 14]. In addition, they have been used as platforms for classical nonlinear optical generation and switching [18, 32]. Photonic crystal cavities with Q factors of up to 700,000 have been realized in GaAs [33, 34], which shows the feasibility of using a III–V platform for our proposal. The challenge, however, is to design a nonlinear photonic crystal nanocavity which supports two orthogonal, high cooperativity modes with a large mode field overlap.
To enable a monolithic cavity design which supports both TE and TM modes, we design a photonic crystal “nanobeam” cavity – a free-standing ridge waveguide patterned with a one-dimensional (1D) lattice of holes – for which we can control both TE and TM photonic bandstructures. Recently, there has been much interest in photonic crystal nanobeam cavities [35, 36, 37, 38, 39, 40] due to their exceptional cavity figures of merit (Q and V), relative ease of design and fabrication, and potential as a platform to realize novel optomechanical effects [41, 42]. Our frequency conversion scheme can be realized in a similar structure, as shown in Fig. 2. We optimize two high cooperativity cavity modes by exploiting the different quasi-1D TE and TM photonic stopbands of the patterned nanobeam (shaded regions in the inset of Fig. 2). A key design point is that the TE and TM bandstructures can be tuned somewhat independently by varying the cross-sectional aspect ratio of the ridge. For example, in a nanobeam with a square cross-section, the two stopbands overlap completely. As the width-to-depth ratio of the waveguide is increased, the effective index of the TE modes increases relative to the TM modes, shifting the TE stopband to longer wavelengths .
4. Example implementations
Although our scheme is general in nature, and does not depend on the particular flavor of emitter or wavelength of operation, it is interesting and instructive to consider some concrete examples. As a first example, we design a GaAs photonic crystal nanobeam cavity with modes at 950 nm and 1425 nm, which would be suitable for directly generating single photons at telecom wavelengths from InAs/GaAs quantum dots designed to emit near 950 nm. We note that quantum dots emitting near 1.5 µm do exist [43, 44], as do protocols for generating single photons at this wavelength [45, 46]. However, this example shows how “top-down” engineering of the photonic environment of a given emitter can completely determine its emission properties. In a solid-state system, this gives a great deal of flexibility in tailoring the emission frequency, avoiding the complexities of changing material systems, and paving the way for multiple devices with different operating frequencies on a single platform. Moreover, this approach is particularly powerful for atomic emitters, which are fixed in their emission frequency.
To achieve such a large spectral separation (950–1425 nm), we couple the fundamental TE0 cavity mode to a higher-order TM2 cavity mode (see inset Fig. 2). Crucially, the photonic crystal lattice tapering [37, 38, 39] is effective in enhancing the Q-factors of both TE and TM modes. The choice of 1425 nm was dictated by the constraint of maintaining a high Q-factor (>105) for each mode. At wavelengths longer than 1425 nm, the TM2 mode Q factor declines rapidly. We emphasize that this example is purely illustrative, and the quantum dot photon could be converted to any wavelength between 950 and 1425 nm with a suitable adjustment of the cavity geometry, provided a coupling laser with wavelength to match the modes’ difference frequency was readily available.
The nanobeam cavities are formed by a 4-period taper in the size and spacing of the holes in the uniform photonic “mirror” on both sides of the cavity center in order to introduce a localized potential for the TE and TM modes. We assume the nanobeam is oriented such that the x and y directions as defined in Fig. 2 are commensurate with the  and  crystal axes of the GaAs. The 950–1425 nm cavity nanobeam has width w=389 nm and depth d=285 nm, and the hole spacing tapers from a 0=334 nm in the mirror to ac=312 nm in the center. The holes were made elliptical to give an additional design parameter to separately optimize the TE and TM mode Q factors. The elliptical hole semi-axes are 78 nm and 100 nm in the mirror section, and the hole size-to-spacing ratio is held constant through the taper section. This design yields cavity parameters of Q=1.4×107 and Vn=0.77 for the TE mode, and Q=1.3×105 and Vn=1.44 for the TM mode (Vn is the mode volume normalized by (λ/n)3). The factor γ/γ 0=0.10 (0.20) for the TE (TM) mode is determined by simulating the total power emitted by a non-resonant dipole source in the cavity center. We have accounted for the index dispersion of our candidate material, GaAs, for which n(1425 nm)=3.38 and n(950 nm)=3.54 .
The nonlinear parameter g 2 is determined by calculating the volume integral of Eq. (3) using the exact mode fields, Ea and Ec, extracted from our 3D-FDTD calculation. Because the mode fields are oriented along the y and z-axes, respectively, as defined in Fig. 2(c), the classical field which drives the differency frequency generation, Eb, must be polarized along x. This field has a frequency ωb=ωa-ωc, which corresponds to a wavelength λb=2.85 µm. The relevant nonlinear susceptibility tensor elements are χ (2) xyz (GaAs)=2d 14=550 pm/V and χ (2) xyz (GaP)=320 pm/V [48, 49]. Based on previous investigations of the second-order nonlinear properties of a photonic crystal cavity , we expect the bulk χ (2) tensor to be the relevant quantity of interest, and surface effects to play a minor role.
We assume the classical field is constant over the spatial extent of the cavity modes, which allows E b,x to be taken in front of the integral for g 2, giving
To justify this assumption, we simulated a Gaussian beam with λb=2.85 µm that is focused by a lens with a modest numerical aperture (NA) of 0.5 onto a ridge waveguide, and found that the average field amplitude is approximately uniform over the linear extent of our cavity modes (approx. x=-1 µm to +1 µm). In the g 2 calculation, the magnitude of E b,x for a given beam power, Pb, is then determined from the relation Pb=ε 0 cπr 2 E 2 b,x/4, where r is the focal spot radius.
In order to efficiently drive the difference-frequency process, the coupling field (Eb) must have a wavelength λb=2.85 µm, which could be achieved using a type of Er3+-doped laser , which can output tens of mW of cw power in the range 2.7–2.9 µm. GaAs is an attractive nonlinear cavity material because it has a reasonably large χ (2) strength , a high refractive index, and mature microfabrication techniques.
As evident in Fig. 2, the overlap of the two modes changes sign near the edges of the ridge compared to the middle due to the different symmetries of the TE0 and TM2 modes. However, the induced nonlinear polarization is dominated by the negatively signed anti-nodes near the middle of the ridge, and the imperfect overlap in the integral can be completely compensated for by a stronger pump beam. Thus, by selecting a higher order TM2 mode, we have gained a larger frequency conversion bandwidth at the expense of the somewhat higher pump power required to overcome the ensuing phase mismatch. For applications requiring relatively small frequency shifts, the two fundamental modes, TE0-TM0, would be a more appropriate choice, as their overlap is almost perfect .
We now calculate the probability to convert a single photon from 950 to 1425 nm in our system. The optimized cavity design simultaneously yields high quality factors and small mode volumes, which allows for extremely high cooperativities for each mode (Cin>104). From Eq. (7) we find that this enables an internal conversion probability of up to F=0.99 when waveguide extraction efficiency is not taken into account. In practice, to efficiently out-couple the frequency-converted single photon into a waveguide, we require the ratio δ=κ c,ex/κ c,in to be large (i.e. overcoupled). The branching parameter ϕ scales as Pb/(κ c,in(1+δ)), and so to increase the extraction ratio δ, the pump power (Pb) must also be increased to maintain the optimal ϕ. Essentially, achieving good extraction efficiency requires one to intentionally increase the losses in mode c (via the out-coupling waveguide), which in turn requires more pump power to maintain the critical coupling. This relationship is made clear in Fig. 3, which plots the probability F as a function of pump power Pb and extraction ratio δ. For a given δ, the power Pb can be chosen to maximize the probability, reflecting the optimal value of g 2 for frequency conversion. The probability rises rapidly with Pb, reaching a maximum at relatively low powers (visible as the sharp ridge in the contours). Three fixed δ contours are plotted in Fig. 3(b), demonstrating that efficient extraction of frequency-converted photons can be realized at modest pumping powers. For example, for δ=5, an extraction probability of 0.7 can be realized with a coupling laser power of 1.5 mW focused in a diffraction-limited focal spot. For this particular cavity design, the outgoing converted photon can be shaped to have a bandwidth of up to κc~100 MHz.
The seemingly high power of the coupling laser (1.5 mW) is required to achieve a sufficiently high electric field amplitude (Eb) in the cavity (4.2×105 V/m). However, only a small fraction of the coupling photons participate in the frequency conversion process, due to the discrepancy in size between the cavity mode and the coupling laser spot size. We have sketched a free-space, diffraction-limited coupling beam in Fig. 2 for simplicity, but if the coupling laser were coupled via the cavity waveguide, the power level required would be more than an order of magnitude smaller. Alternatively, if the difference frequency coincided with a third cavity mode, the local field enhancement in that mode would also greatly reduce the operating power. We also note that the operating power reflects in part the phase mismatch between the TE0 and TM2 cavity modes. Frequency conversion over smaller wavelength bands could be achieved at microwatt coupling power levels with the TE0-TM0 mode pair, which, as mentioned above, have an extremely high overlap function .
By exploiting the scaling properties of Maxwell’s equations, it is straightforward to design a similar cavity in GaP, a large bandgap semiconductor which could support cavity modes at 637 nm and 950 nm. Accounting for the exact refractive index dispersion and χ (2) strength of GaP, we calculate an internal frequency conversion probability on this platform of 0.99, and an extraction probability 0.7 for <4 mW coupling power. The 637–950 nm span would be sufficient to couple any pair of the most relevant quantum emitters, namely NV centers in diamond; atoms such as Cs or Rb; and InAs/GaAs quantum dots. Such a cavity could also be integral to creating a stable, room temperature single-photon source emitting in the telecom band based on frequency-converted NV center emission in diamond . Given the challenge of fabricating high-Q cavities directly in diamond , hybrid approaches involving GaP-diamond heterostructures [53, 54, 55, 56] or diamond nanocrystals [57, 58] are promising alternatives to realize cavity-coupled NV center emission. We have shown previously that, despite a small perturbation of the cavity, the strong coupling regime can still be accessed with a diamond nanocrystal positioned on top of a photonic crystal nanobeam cavity . Given that it may be difficult to span the large spectrum from 637 nm to telecom wavelengths in a single monolithic design, a 637–950 nm cavity could be the first stage of a two-step frequency conversion process involving our first example as the second stage. More generally, cascading allows our design to be extended to cover virtually any frequency span.
5. Summary and outlook
We have shown that high-efficiency, intra-cavity frequency conversion of single photons from a dipole-like emitter can be achieved using a two-mode nonlinear cavity pumped by a classical field. Our general framework is valid for conversion between arbitrary frequencies, and the efficiency depends only on the cavity parameter Q/V. Further design improvements should lead to larger frequency spans and also lower pump power requirements (e.g., by allowing ωb to correspond to a third cavity mode). Although we have emphasized large frequency shifts in this paper, a smaller shift could be readily achieved by coupling the TE0 mode with the fundamental TM0 mode, which has a larger Q factor than the TM2 mode studied here. The TE0-TM0 modes have a larger spatial overlap, reducing the coupling power required for high probability frequency conversion. In our scheme, wavelength trimming to achieve the strong coupling condition between the cavity and emitter could be readily achieved by temperature or electrostatic tuning [59, 60], and additional dynamical tuning functionality could potentially be realized by optomechanical effects .
Beyond the aforementioned applications, the techniques described here can potentially be extended to open up many intriguing opportunities. For example, the photon emission of a particular emitter could be shifted into wavelengths where high-efficiency detectors are available. It also allows coupling of atomic emitters such as Cs or Rb with solid-state emitters to create hybrid atom-photonic chips . In addition, a number of quantum entanglement schemes for atoms rely on joint photon emission and subsequent detection to probabilistically project the atomic system into an entangled state [62, 63, 64]. Such schemes rely on the indistinguishability of photons emitted from each atom, and implementing such techniques in nonlinear cavities could allow entanglement between different types of emitters. In addition, the protocol described here could be extended for generating narrow-bandwidth, entangled photon pairs with high efficiency and repetition rates, which are a valuable resource for applications such as quantum cryptography . Our scheme could also be applicable in active materials, where laser wavelengths could be converted from easily accessible regions like 1500 nm to the mid-infrared range.
Derivation of nonlinear conversion efficiency
These equations describe both coherent evolution (terms proportional to Ω(t),g 1,2) and population loss in the system (terms proportional to γ,κ a,c). The population loss in the system can be connected to direct radiative emission of the excited state |e〉 (at a rate γ|ce|2), radiation leakage and absorption losses of mode a (κa|ca|2), and absorption and leakage out of mode c (κc|cc|2, of which κ c,ex|cc|2 is successfully out-coupled to a waveguide). In general the efficiency of extracting a single photon of frequency ωc out into the waveguide is thus
For arbitrary Ω(t), Eqs. (9) and (10) can be evaluated numerically. However, in certain limits one can find approximate solutions. In particular, when Ω(t) and its rate of change are small compared to the natural oscillation and decay rates of the system, the state amplitudes c a,c,e will follow the instantaneous value of cs(t). Formally, we can adiabatically eliminate these states, setting ċi=0 for i=a,c,e. Then, one finds
while the other ci∝cs(t), with the proportionality coefficients being functions of g 1,g 2,κa,κc,Ω(t). The resulting substitution of the solutions of ci(t) into Eq. (10) allows great simplification because the integrands now become time-independent, and after some simplification yields Eq. (6). Self-consistency of the adiabatic elimination solution requires that the the effective rate of population loss ~4Ω(t)2/γtotal predicted from state |s〉 does not exceed the rate κc that a photon can leak out through the cavity mode c.
In the effective wave-function approach used here, the population leakage out of mode c can also be explicitly related to the shape of the outgoing single-photon wavepacket. For instance, we can model the linear coupling of cavity mode c to photons propagating in a single direction in a waveguide with the following Hamiltonian (in a rotating frame),
Here k denotes the set of wavevectors of the continuum of waveguide modes, v is the velocity of waveguide fields, gw is the coupling strength between cavity and waveguide modes, and zc denotes the position along the waveguide where the cavity is coupled to it (for simplicity we set zc=0 from this point on). Since we are now explicitly accounting for the waveguide degrees of freedom, we add a term ∫dkck(t)|1k〉 to the effective wave-function of the system. The equations of motion of the total system are identical to Eq. (9), except that
where δk=k-ωc/v. Compared to Eq. (9), we have now included the coupling of mode c to the waveguide, and accordingly have replaced κc→κ c,in in the equation for ċc since the leakage into the waveguide should be accounted for by the new coupling terms. The equation for ċk can be formally integrated; assuming that the waveguide initially is unoccupied, ck(0)=0, one has
Substituting this into the equation for ċc and performing the Wigner-Weisskopf approximation , one recovers the expression for ċc in Eq. (9) by identifying κ c,ex=2πg 2 w/v. The one-photon wave-function  is given by , where Θ(z) is the step function. The wave-function shape is thus directly proportional to cc(t). Under adiabatic elimination,
and thus for a desired (and properly normalized) pulse shape ψw one needs only to solve Eqs. (16) and (11) to obtain the corresponding external field Ω(t). It is straightforward to show that the normalization is given by ∫dz|ψw(z, t→∞)|2=F provided that cs(∞)→0. This normalization reflects the probability that a single photon ends up in the waveguide.
We now consider the effects of dephasing on our previous analysis. In the presence of dephasing, it is necessary to solve for the full density matrix ρ of the system, whose evolution is given by ρ̇=-(i/h̄)[Hc,ρ]+L[ρ]. Here Hc is the Hamiltonian describing the coherent evolution, and L[ρ] is the Liouvillian describing the decoherence and dissipative processes,
where γd describes the pure dephasing of the excited state.
The full density matrix equations in principle can be adiabatically eliminated as before, but the general solutions are quite cumbersome and offer little insight. To simplify the situation, we assume that cavity mode c can be effectively eliminated to yield a new effective linewidth κa→κa(1+ϕ) for cavity mode a, and from this point forward consider the reduced system consisting of states e,g, and a. We then adiabatically eliminate the density matrix elements for this system in terms of ρss. Of particular interest here are the equation of motion for the population ρss in the meta-stable state, the population of the cavity mode ρaa, and the coherence ρas in the limit of weak driving (Ω small),
We now derive the probability that the excitation initially stored in state s decays through cavity mode a. This probability is given by
In the presence of dephasing, the expression above should replace the first term on the right-hand side of Eq. 6, leading to a reduction in the single-photon conversion efficiency. In particular, note that this quantity simplifies to in the limit that γd→0. A second relevant quantity to consider is the coherence of the generated single photon. Specifically, in the ideal process (with γd=0) a definite phase relationship is established between the outgoing photon and the state s. Maintaining coherence is important in the ability to create indistinguishable photons over successive operations of the device, or being able to implement the time-reversal of the generation process (i.e., coherent photon storage). We define the coherence by 𝓒≡|ρas|2/ρaaρss, which for our system is given by
Note that for γd→0, the coherence reaches its maximum allowed value of 𝓒=1.
MWM would like to thank NSERC (Canada) for its support. DEC acknowledges support from the Gordon and Betty Moore Foundation through Caltech’s Center for the Physics of Information, and the National Science Foundation under Grant No. PHY-0803371.
References and links
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