Broad-band spectral control of single photon sources using a nonlinear photonic crystal cavity
Abstract
Motivated by developments in quantum information science, much recent effort has been directed toward coupling individual quantum emitters to optical microcavities. Such systems can be used to produce single photons on demand, enable nonlinear optical switching at a single photon level, and implement functional nodes of a quantum network, where the emitters serve as processing nodes and photons are used for long-distance quantum communication. For many of these practical applications, it is important to develop techniques that allow one to generate outgoing single photons of desired frequency and bandwidth, enabling hybrid networks connecting different types of emitters and long-distance transmission over telecommunications wavelengths. Here, we propose a novel approach that makes use of a nonlinear optical resonator, in which the single photon originating from the atom-like emitter is directly converted into a photon with desired frequency and bandwidth using the intracavity nonlinearity. As specific examples, we discuss a high-finesse, TE-TM double-mode photonic crystal cavity design that allows for direct generation of single photons at telecom wavelengths starting from an InAs/GaAs quantum dot with a nm transition wavelength, and a scheme for direct optical coupling of such a quantum dot with a diamond nitrogen-vacancy center at nm.
Nonlinear optical frequency conversion is widely used in fields as diverse as ultrahigh-resolution imaging Campagnola et al. (1999) and telecommunications, as it allows for the generation of light in parts of the spectrum for which there are no convenient sources. Common implementations include optical parametric oscillators to make tunable femtosecond lasers in the infrared, and conversion of the 1064 nm Nd:YAG laser to make green laser sources via second-harmonic generation. Recently, nonlinear photonic crystal cavities McCutcheon et al. (2007); Rodriguez et al. (2007); Bravo-Abad et al. (2007); Liscidini and Andreani (2006); Cowan and Young (2005) have emerged as promising systems in which similar nonlinear functionalities can be achieved at micron scales, which would enable the miniaturization of optical devices onto integrated platforms. While the majority of such work focuses on conversion of classical fields, these systems are now being applied to quantum optics and quantum information science VanDevender and Kwiat (2004); Langrock et al. (2005); Tanzilli et al. (2005); VanDevender and Kwiat (2007).
In recent years, there has also been a concerted research effort to develop on-demand single-photon sources using single quantum emitters strongly coupled to resonant optical microcavities (cavity QED) Michler et al. (2000); Pelton et al. (2002); McKeever et al. (2004). The strong coupling of the emitter to a resonant cavity results in preferential emission into the cavity mode of a single photon with frequency near the atomic resonance. Connecting pairs of such systems would form the basis for distributed quantum networks, where the emitters serve as processors and photons carry information between the nodes Cirac et al. (1997). In practice, however, the photon emission occurs at wavelengths determined by the atomic resonance frequency. This is impractical, as it does not exploit the low-loss telecom frequency band for long-distance transmission and requires all emitters in a quantum network to be identical.
Here, we describe a novel approach to generate single photons with controllable wavelength and bandwidth. Our approach makes use of an integrated nonlinear optical cavity in which optical emission is directly frequency-shifted into the desired domain using intracavity nonlinear optical processes. This cavity-based generation technique is quite robust in that the maximum efficiency does not depend on an explicit phase-matching condition Boyd (1992), as would occur in an extended nonlinear crystal or fiber, but rather only on the ratio of the cavity quality factor to mode volume (). 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 quantum dots to the telecom band. We also present a similar GaP-based design for direct coupling between a nitrogen-vacancy center in diamond Dutt et al. (2007); Santori et al. (2006); Gaebel et al. (2006); Hanson et al. (2006) and an InAs/GaAs quantum dot Srinivasan and Painter (2007); Hennessy et al. (2007); Englund et al. (2007), which could enable practical realization of a heterogeneous quantum network. In addition to effective wavelength control of single photons VanDevender and Kwiat (2004); Langrock et al. (2005); Tanzilli et al. (2005); VanDevender and Kwiat (2007), our approach enables the manipulation of their bandwidth, which is important for fast communication.
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 Cirac et al. (1997); McKeever et al. (2004), a single three-level atom (or any other quantum dipole emitter) is resonantly coupled to one mode (here denoted ) of an optical cavity. The emitter is initialized in metastable state , and an external laser field with controllable Rabi frequency couples to excited state . The transition between and ground state is resonantly coupled to cavity mode (frequency ), with a single-photon Rabi frequency . The relevant decay mechanisms (illustrated with gray arrows in the figure) are a leakage rate for photons to leave cavity mode , and a rate that spontaneously emits into free space rather than into the cavity. Conventionally, in absence of an optical nonlinearity, the control field creates a single atomic excitation at some desired time in the system, which via the coupling 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 Cirac et al. (1997).
In our system, the cavity is also assumed to possess a second mode with frequency , and our goal is to induce the single photon to exit at this frequency rather than . This can be achieved, provided that the cavity medium itself possesses a second-order () nonlinear susceptibility, by applying a classical pump field to the system at the difference frequency . The induced coherent coupling rate between modes and is denoted . The field need not correspond to a cavity mode. Mode has a photon leakage rate, which we separate into an “inherent” rate, , and a “desirable” (extrinsic) rate, . characterizes the natural leakage into radiation modes and also absorption losses, and can be expressed in terms of the (unloaded) cavity quality factor as . characterizes the out-coupling rate into any external waveguide used for photon extraction. The total leakage of mode is then .
More quantitatively, the effective Hamiltonian for the system (in a rotating frame) is given by
(1) |
where describes the coherent part of the system evolution (for simplicity we take to be real), and is a non-Hermitian term characterizing the losses. are atomic operators, while is the photon annihilation operator for mode . The vacuum Rabi splitting can be written in the form , where is the dipole matrix element of the - transition, and is the electric field amplitude per photon at the emitter position . The electric field per photon in mode is determined by the normalization
(2) |
where is the dimensionless electric permittivity of the material. The nonlinearity parameter is given by Rodriguez et al. (2007)
(3) |
The amplitudes appearing above are normalized by Eq. (2), while is the classical pump amplitude. Importantly, one can compensate for a small nonlinear susceptibility or field overlap (phase matching) simply by using larger pump amplitudes to achieve a desired strength.
For a system initialized in , there can never be more than one excitation, and the system generally exists as a superposition of having the system in state or (with no photons) or having a photon in one of the modes (and the emitter in ),
(4) |
The system is initialized to with all other and the time evolution is given by . In this effective wave-function approach, provided that is always driven, as due to losses, which can be connected with population leakage out of one of the aforementioned decay channels. In the limit that is small and varies slowly, all other adiabatically follow (see the Methods section), and one finds
(5) |
Physically, we can identify as the cavity-enhanced total decay rate of , where the first (second) term corresponds to direct radiative emission (emission into mode ). Similarly, the denominator corresponds to the new total decay rate of mode in the presence of an optical nonlinearity, as it yields a new channel for photons to effectively “decay” out of mode into at rate . It is clear that some optimal value of exists for frequency conversion to occur. In particular, for no nonlinearity () this probability is non-existent. On the other hand, for , one finds , which indicates that the leakage from mode into is so strong that no cavity-enhanced emission occurs. Note that the use of time-varying control and pump fields allows for arbitrary shaping of the outgoing single-photon wavepacket at frequency , provided only that the photon bandwidth is smaller than (physically, the photon cannot leave faster than the rate determined by the cavity decay, see Methods). This feature is particularly useful in two respects. First, in practice can be much larger than , 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 is produced and extracted into the desired out-coupling waveguide is given by
(6) |
where characterizes the branching ratio in mode of nonlinearity-induced leakage to inherent losses, and is the inherent cavity cooperativity parameter for mode in absence of nonlinearity. The first term on the right denotes the probability for to decay into mode , the second term the probability that a photon in mode couples into mode , and the third term the probability that a photon in mode out-couples into the desired channel (see Methods for an exact calculation). depends on the pump amplitude , with the optimal value yielding the maximum in . For large , the maximum probability is
(7) |
Considering an emitter placed near the field maximum of mode ,
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 is incident upon the system, converted into a photon in mode , and coherently absorbed by an atom with the aid of an impedance-matched pulse , causing its internal state to flip from to . Generally, by time-reversal arguments Gorshkov et al. (2007), it can be shown that the probability for single-photon storage is the same as that for generation.
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 must have a high cooperativity () to ensure strong coupling of the emitter (see Fig. 1). For mode , a high factor (small ) is important to maximize the nonlinear coupling parameter, , and hence reduce the pump power needed in order to reach the optimum nonlinear coupling strength, . The cavity should also be composed of a 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 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 tensor (e.g., from an off-chip laser), one of the cavity modes must have a TM polarization. ) 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 (
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 Hennessy et al. (2007); Englund et al. (2007). In addition, they have been used as platforms for classical nonlinear optical generation and switching McCutcheon et al. (2007); Tanabe et al. (2005). 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 Sauvan et al. (2005); Zain et al. (2008); McCutcheon and Lončar (2008); Notomi et al. (2008); Deotare et al. (2009) due to their exceptional cavity figures of merit ( and ), relative ease of design and fabrication, and potential as a platform to realize novel optomechanical effects Povinelli et al. (2005); Eichenfield et al. (2008). 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.
Example implementations
As a first example, we design a GaAs photonic crystal nanobeam cavity with modes at 950 nm and 1425 nm suitable for directly generating single photons at telecom wavelengths from InAs/GaAs quantum dots. To achieve such a large spectral separation, we couple the fundamental TE cavity mode to a higher-order TM cavity mode (see inset Fig. 2). Crucially, the photonic crystal lattice tapering McCutcheon and Lončar (2008); Notomi et al. (2008); Eichenfield et al. (2008) is effective in enhancing the factors of both TE and TM modes. Details of the cavity parameters and optimization are provided in the Methods section. For this cavity, the coupling field () must have a wavelength m in order to efficiently drive the difference-frequency process. GaAs is an attractive nonlinear cavity material because it has a reasonably large strength Singh (1986), 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 TE and TM 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 TM 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. Note that the fundamental TE-TM mode overlap is nearly ideal, and would be appropriate for applications requiring relatively small frequency shifts.
We now calculate the probability to convert a single photon from 950 to 1425 nm in our system. The optimized cavity design (see Methods) simultaneously yields high quality factors and small mode volumes, which allows for extremely high cooperativities for each mode (). From Eq. (7) we find that this enables an internal conversion probability of up to 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 to be large (i.e. overcoupled). The branching parameter scales as , and so to increase the extraction ratio , the pump power () must also be increased to maintain the optimal . Essentially, achieving good extraction efficiency requires one to intentionally increase the losses in mode (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 as a function of pump power and extraction ratio . For a given , the power can be chosen to maximize the probability, reflecting the optimal value of for frequency conversion. The probability rises rapidly with , reaching a maximum at relatively low powers (visible as the sharp ridge in the contours). Three fixed 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 , an extraction probability of 0.7 can be realized with a coupling laser power of 3 mW focused in a diffraction-limited focal spot. We note that the absorption of GaAs at 2.85 m is negligible, and so there will be no pump-induced heating. For this particular cavity design, the outgoing converted photon can be shaped to have a bandwidth of up to MHz.
By exploiting the scaling properties of Maxwell’s equations, it is straightforward to design a similar cavity in GaP which supports modes at 637 nm and 950 nm. GaP is a nonlinear wide bandgap semiconductor which is transparent at 637 nm and has recently shown promise for the microcavity enhancement of diamond NV emission Fu et al. (2008); Rivoire et al. (2008). Accounting for the exact refractive index dispersion and strength of GaP, we calculate that the internal frequency conversion probability is 0.99, and the extraction probability is 0.7 for 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 Kurtsiefer et al. (2000). 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.
Outlook
We have shown that high-fidelity, 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 . As realistic implementations, we propose two different high-cooperativity, double-mode photonic crystal nanocavities to enable highly efficient coupling from 950 nm – 1425 nm and 637 nm – 950 nm, respectively. Single-photon conversion between these wavelengths would allow, in the first instance, coupling of InAs/GaAs quantum dot emission Srinivasan and Painter (2007); Hennessy et al. (2007); Englund et al. (2007) into low-loss optical fibers in the telecom spectrum. The second example would facilitate a direct optical connection between two types of solid-state emitters currently of great interest: a nitrogen-vacancy center in diamond Dutt et al. (2007); Santori et al. (2006); Gaebel et al. (2006); Hanson et al. (2006) and an InAs/GaAs quantum dot. Integrated together, the two designs could allow for cascaded frequency conversion of NV center emission into the telecom band. Further design improvements should lead to larger frequency spans and also lower pump power requirements (e.g., by allowing 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 TE mode with the fundamental TM mode, which has a larger factor than the TM mode studied here. The TE-TM modes have a larger spatial overlap, reducing the coupling power required for high probability frequency conversion.
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 Barclay et al. (2005). 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 Cabrillo et al. (1999); Bose et al. (1999); Duan and Kimble (2003). 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 Ekert (1991). 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. Finally, it would be intriguing to combine these ideas with cavities that exhibit opto-mechanical coupling Eichenfield et al. (2008), which would potentially allow the photonic frequencies to be dynamically and rapidly tuned.
Methods
Derivation of nonlinear conversion efficiency
The state amplitudes of the wave-function given in Eq. (4) evolve under the interaction Hamiltonian of Eq. (1) through the following equations,
(8) |
These equations describe both coherent evolution (terms proportional to ) and population loss in the system (terms proportional to ). The population loss in the system can be connected to direct radiative emission of the excited state (at a rate ), radiation leakage and absorption losses of mode (), and absorption and leakage out of mode (, of which is successfully out-coupled to a waveguide). In general the efficiency of extracting a single photon of frequency out into the waveguide is thus
(9) |
For arbitrary , Eqs. (8) and (9) can be evaluated numerically. However, in certain limits one can find approximate solutions. In particular, when and its rate of change are small compared to the natural oscillation and decay rates of the system, the state amplitudes will follow the instantaneous value of . Formally, we can adiabatically eliminate these states, setting for . Then, one finds
(10) |
while the other , with the proportionality coefficients being functions of . The resulting substitution of the solutions of into Eq. (9) 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 predicted from state does not exceed the rate that a photon can leak out through the cavity mode .
In the effective wave-function approach used here, the population leakage out of mode 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 to photons propagating in a single direction in a waveguide with the following Hamiltonian (in a rotating frame),
(11) |
Here denotes the set of wavevectors of the continuum of waveguide modes, is the velocity of waveguide fields, is the coupling strength between cavity and waveguide modes, and denotes the position along the waveguide where the cavity is coupled to it (for simplicity we set from this point on). Since we are now explicitly accounting for the waveguide degrees of freedom, we add a term to the effective wave-function of the system. The equations of motion of the total system are identical to Eq. (8), except that
(12) | |||||
(13) |
where . Compared to Eq. (8), we have now included the coupling of mode to the waveguide, and accordingly have replaced in the equation for since the leakage into the waveguide should be accounted for by the new coupling terms. The equation for can be formally integrated; assuming that the waveguide initially is unoccupied, , one has
(14) |
Substituting this into the equation for and performing the Wigner-Weisskopf approximation Scully and Zubairy (1997), one recovers the expression for in Eq. (8) by identifying . The one-photon wave-function Scully and Zubairy (1997) is given by , where is the step function. The wave-function shape is thus directly proportional to . Under adiabatic elimination,
(15) |
and thus for a desired (and properly normalized) pulse shape one needs only to solve Eqs. (15) and (10) to obtain the corresponding external field . It is straightforward to show that the normalization is given by provided that . This normalization reflects the probability that a single photon ends up in the waveguide.
Nonlinear cavity design
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. The 950-1425 nm cavity nanobeam has width = 420 nm and depth = 307.5 nm, and the hole spacing tapers from = 360 nm in the mirror to = 337 nm in the center. The holes were made elliptical to give an additional design parameter to separately optimize the TE and TM mode factors. The elliptical hole semi-axes are 84 nm and 108 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 and for the TE mode, and and for the TM mode ( is the mode volume normalized by ). The factor = 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 (1425 nm) = 3.38 and (950 nm) = 3.54 Palik (1985).
The nonlinear parameter is determined by calculating the volume integral of Eq. (3) using the exact mode fields, and , extracted from our 3D-FDTD calculation. Because the mode fields are oriented along the and -axes, respectively, as defined in Fig. 2(c), the classical field which drives the differency frequency generation, , must be polarized along . This field has a frequency , which corresponds to a wavelength m. The relevant nonlinear susceptibility tensor elements are pm/V and pm/V Singh (1986); Shoji et al. (1997).
We assume the classical field is constant over the spatial extent of the cavity modes, which allows to be taken in front of the integral for , giving
(16) |
To justify this assumption, we simulated a Gaussian beam with m 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. = -1 m to +1 m). In the calculation, the magnitude of for a given beam power, , is then determined from the relation , where is the focal spot radius.
Acknowledgements
MWM would like to thank NSERC (Canada) for its support, and DEC acknowledges support from the Caltech CPI. The authors also gratefully acknowledge useful discussions with Jelena Vučković.
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