Abstract:
Transducers and methods of making the same include a substrate having a cavity with a diameter that supports whispering gallery modes at a frequency of an input signal. A focusing structure in the cavity focuses the electric field of the input signal. A resonator directly under the focusing structure has a crystalline structure that generates an electro-optic effect when exposed to electrical fields. An electric field of the input signal modulates an output signal in the resonator via the electro-optic effect.

Description:
BACKGROUND 
       [0001]    Technical Field 
         [0002]    The present invention relates to conversion between a microwave and an optical domain and, more particularly, to a transducer to convert single-photon microwave signals to optical signals. 
         [0003]    Description of the Related Art 
         [0004]    Various communication protocols rely on optical fibers because of their low loss, high bandwidth, low background noise, and the ease of routing. Optical fibers can also be used for sending quantum information in the form of single photons or coherent states. On the other hand, many viable quantum processing architectures operate at microwave frequencies. The high amplitude stability of microwave structures allows precise controls on quantum bits (qubits) that enable high-fidelity gate operations. However, microwave photons are more difficult to use for long-range communication purposes, due to high thermal background noise and high loss when such signals propagate in waveguides. 
         [0005]    Existing approaches to converting between microwave signals and optical signals are complicated, difficult to implement in solid state systems, or difficult to optimize. Some existing transducers use electro-optic crystalline optical resonators to perform microwave-to-optical conversion. One of the problems of using such resonators is that other coexisting non-linear properties, such as pyro-electricity and piezo-electricity, impede the microfabrication processes of the microwave resonator. Another problem is that microfabrication can likewise contaminate the crystalline electro-optic optical resonators and reduce the quality factor. These resonators also have a high microwave loss and are difficult to align at cryogenic temperatures. 
       SUMMARY 
       [0006]    A transducer includes a substrate having a cavity with a diameter that supports whispering gallery modes at a frequency of an input signal. A focusing structure in the cavity focuses the electric field of the input signal. A resonator directly under the focusing structure has a crystalline structure that generates an electro-optic effect when exposed to electrical fields. An electric field of the input signal modulates an output signal in the resonator via the electro-optic effect. 
         [0007]    A quantum computing device includes a qubit configured to provide a first signal at a first frequency. A transducer coupled to the qubit and includes a substrate having a cylindrical cavity with a diameter that supports whispering gallery modes at the first frequency. There is a central pin in the cavity. A resonator is positioned directly under the central pin. The resonator has a crystalline structure that generates an electro-optic effect when exposed to electrical fields. An electric field of the input signal modulates a second signal at a second frequency in the resonator via the electro-optic effect. A waveguide is optically coupled to the resonator and is configured to convey the modulated second signal away from the resonator. 
         [0008]    A method for forming a transducer includes fabricating a resonator on a first substrate, resonant at a first frequency, by depositing a straining material on a resonator material to strain a crystalline structure of the resonator material to generate an electro-optic effect when exposed to electrical fields. A second substrate is fabricated with a cavity. The cavity has a diameter that supports whispering gallery modes at a second frequency. The second substrate is aligned over the first substrate such that a focusing structure in the microwave cavity aligns with the optical resonator. 
         [0009]    These and other features and advantages will become apparent from the following detailed description of illustrative embodiments thereof, which is to be read in connection with the accompanying drawings. 
     
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
         [0010]    The disclosure will provide details in the following description of preferred embodiments with reference to the following figures wherein: 
           [0011]      FIG. 1  is a cross-sectional diagram of a microwave-to-optical transducer in accordance with the present principles; 
           [0012]      FIG. 2  is a top-down diagram of a portion of a microwave-to-optical transducer in accordance with the present principles; 
           [0013]      FIG. 3  is a bottom-up diagram of a portion of a microwave-to-optical transducer in accordance with the present principles; 
           [0014]      FIG. 4  is a detailed cross-sectional diagram of a portion of a microwave-to-optical transducer in accordance with the present principles; 
           [0015]      FIG. 5  is a schematic diagram of a microwave-to-optical transducer in accordance with the present principles; 
           [0016]      FIG. 6  is a detailed cross-sectional diagram of a strain-induced electro-optic optical resonator in accordance with the present principles; 
           [0017]      FIG. 7  is a block/flow diagram of a method of fabricating a microwave-to-optical transducer in accordance with the present principles; 
           [0018]      FIG. 8  is a top-down diagram of a portion of an alternative microwave-to-optical transducer in accordance with the present principles; and 
           [0019]      FIG. 9  is a detailed cross-sectional diagram of a portion of an alternative microwave-to-optical transducer in accordance with the present principles. 
       
    
    
     DETAILED DESCRIPTION 
       [0020]    Embodiments of the present invention provide coupling between single-photon microwave signals and single-photon infrared/optical signals via the electro-optic effect using superconducting microwave and optical cavities. Each cavity incorporates an electro-optic material, with the electro-optic effect being induced by a straining material. Coupling takes place at the quantum level, with signal levels being about a single photon. The present embodiments may be implemented on one chip that may be fabricated using standard semiconductor fabrication processes. 
         [0021]    Referring now to the drawings in which like numerals represent the same or similar elements and initially to  FIG. 1 , a cross-sectional view of a microwave-to-optical transducer  100  is shown. A bottom substrate  102  is shown as having, e.g., a quantum computing device  104  (a “qubit”) that provides, e.g., single-photon level microwave signals along a superconducting channel  106  to a transducing cavity  130 . It is specifically contemplated that the bottom substrate  102  may be formed from silicon, but any other appropriate substrate material may be used in its place. After converting the microwave signal to an optical signal in the transducing cavity  130 , the optical signal couples with a waveguide  108  and is transmitted to its destination. The cavity  130  is capacitively coupled to the superconducting channel  106 , which may be either a microwave transmission line, or to other resonating structures (e.g., the qubit  104  itself). 
         [0022]    A top substrate  120  includes a cylindrical cavity  112  and a central pin  114 . In one embodiment, the cavity may have a radius of about 2.5 mm and the central pin  114  may have a radius of about 2 mm and a height of about 2 mm. It is specifically contemplated that the top substrate  120  may be formed from silicon, but any other appropriate substrate material may be used in its place. The sidewalls of the cavity  112  and the central pin  114  are coated with a superconducting film. The cavity  112  joins with a similar cavity  111  on the bottom substrate  102  to form a microwave resonator  122 , which is connected to ground. The bottom cavity  111  has an exemplary depth of 0.67 mm and an exemplary radius of 1.98 nm. It should be noted that the top substrate  120  should not come into contact with the bottom substrate, at least in regions with superconducting films or the channel  106 , to prevent damage to those structures. The central pin  114  approaches, without touching, an optical resonator  110  on a pedestal  115  in the bottom substrate  102 . It is specifically contemplated that the optical resonator  110  is formed from silicon and silicon-germanium, with the silicon-germanium providing a strain on the silicon material. In one embodiment, the optical resonator  110  may have a radius of about 2 mm and a thickness of about 0.1 mm. This strain creates the electro-optic effect in the silicon as it deforms the crystalline structure of the silicon. 
         [0023]    The integrated design of the present embodiments minimizes alignment errors between the optical resonator  110  and the waveguide  108  as the coupling between the optical resonator  110  and the waveguide  108  is defined by microfabrication. Such alignment errors would otherwise occur if the optical couplers were not integrated into the device, for example in systems that use prisms for coupling. In particular, in a cryogenic environment at millikelvin temperatures, misalignment errors due to different thermal expansion coefficients of the different materials can be reduced or avoided entirely. 
         [0024]    During operation, microwave signals from the qubit  104  couple to the microwave resonator  122 , where a standing wave forms on the outer and inner circumferences of the cavity, with strong fields at the boundaries and negligible field strength in the middle of the cavity. The superconducting film creates a low-loss resonator with a very high Q. At the junction of the central pin  114  with the optical resonator  110 , the fields of the microwave modes modulate an optical signal in the optical resonator  110 . With the aid of optical pump signals applied to the optical resonator, a microwave signal can be converted into an optical signal at a single photon level. 
         [0025]    In one embodiment, the microwave resonator can be formed in an on-chip, transmission-line cavity or a coplanar waveguide cavity. A center pin or a high-voltage electrode of a transmission-line cavity or a coplanar waveguide cavity has a circular shape that can deliver a microwave signal to the optical resonator. 
         [0026]    The optical resonator  110  may be formed in the shape of a disc, as shown, or as a ring, in both embodiments supporting whispering gallery modes at multiple frequencies. The diameter of the optical resonator  110  is selected to provide three modes at frequencies ω op −ω q  for a red-sideband, ω op  for a carrier, and ω op +ω q  for a blue-side band, with ω q  being the microwave frequency of the microwave resonator  122 . In one embodiment, ω op /2π may be about 193 THz (1550 nm wavelength) and ω q  may be about 10 GHz. This embodiment may be achieved by choosing the free spectral range to be ω q , which is determined by the refractive index and diameter of the optical resonator  110 . Using the sideband modes, a three-wave mixer is realized that couples microwave photons and optical photons. 
         [0027]    Referring now to  FIG. 2 , a top-down view of the bottom substrate  102  is shown. The superconducting qubit  104  and the superconducting channel  106  are formed in or on the substrate  102 . The lower cavity  111  is formed in the bottom substrate  102  by any appropriate process, including, e.g., micromachining or etching. A superconducting film is deposited over the internal surfaces of the lower cavity  111 . The superconducting film may include, for example, aluminum, niobium, titanium, indium, or any other material or alloy that demonstrates superconducting properties in a desired temperature range. The superconducting film may be deposited by, e.g., sputtering or by thermal evaporation in a vacuum chamber. The optical resonator  110  is formed from, e.g., a saw-tooth silicon disc or ring with a layer of silicon-germanium to provide strain to the crystalline structure of the silicon, as described in further detail below. The optical waveguide  108  couples with the optical resonator  110  to transmit the optical signal off-chip and, in one embodiment, the optical waveguide  108  is positioned less than one micron away from the optical resonator  110  to promote coupling. 
         [0028]    Referring now to  FIG. 3 , a bottom-up view of the top substrate  120  is shown. The upper cavity  112  is formed in the top substrate  120  by any appropriate process, including, e.g., micromachining or etching. The central pin  114  is formed by the micromachining process as well, and a superconducting film is deposited over the surfaces of the upper cavity  112  and the central pin  114 . A ridge  302  is formed along the outer edge of the facing circle of the central pin  114 . The ridge  302  concentrates the fields of the whispering gallery modes along this edge for coupling with the optical resonator  110 . The ridge  302  may be formed by any appropriate process, including, micromachining or etching. When the top substrate  120  is placed above the bottom substrate  102 , the ridge  302  aligns with the outer edge of the optical resonator  110 . 
         [0029]    Referring now to  FIG. 4 , a more detailed cross-sectional view of the connection between the pin  114  and the optical resonator  110  is shown. The ridge  302  is positioned slightly above the optical resonator, with small gap between the two structures to prevent plasmonic loss of the optical signal through the superconducting film. A small portion internal portion of the face of the central pin  114  is cut away, with a cutaway depth of about 0.5 mm and a cutaway radius of about 1.9 mm. In addition, superconducting surfaces  402  are shown with a heavier line weight, having had a superconducting film deposited on them. 
         [0030]    It should be noted that the central pin  114  is also recessed with respect to the sidewalls of the cavity  112 . The depth of the recess provides room for the optical resonator  110  as well as a small additional gap to prevent plasmonic modes between the pin  114  and the resonator  110 . The cavity  112  and gap may be filled with air or may be in a vacuum or an appropriate inert gas. 
         [0031]    It is to be understood that the present invention will be described in terms of a given illustrative architecture having a wafer; however, other architectures, structures, substrate materials and process features and steps may be varied within the scope of the present invention. 
         [0032]    It will also be understood that when an element such as a layer, region or substrate is referred to as being “on” or “over” another element, it can be directly on the other element or intervening elements may also be present. In contrast, when an element is referred to as being “directly on” or “directly over” another element, there are no intervening elements present. It will also be understood that when an element is referred to as being “connected” or “coupled” to another element, it can be directly connected or coupled to the other element or intervening elements may be present. In contrast, when an element is referred to as being “directly connected” or “directly coupled” to another element, there are no intervening elements present. 
         [0033]    A design for an integrated circuit chip may be created in a graphical computer programming language, and stored in a computer storage medium (such as a disk, tape, physical hard drive, or virtual hard drive such as in a storage access network). If the designer does not fabricate chips or the photolithographic masks used to fabricate chips, the designer may transmit the resulting design by physical means (e.g., by providing a copy of the storage medium storing the design) or electronically (e.g., through the Internet) to such entities, directly or indirectly. The stored design is then converted into the appropriate format (e.g., GDSII) for the fabrication of photolithographic masks, which typically include multiple copies of the chip design in question that are to be formed on a wafer. The photolithographic masks are utilized to define areas of the wafer (and/or the layers thereon) to be etched or otherwise processed. 
         [0034]    Methods as described herein may be used in the fabrication of integrated circuit chips. The resulting integrated circuit chips can be distributed by the fabricator in raw wafer form (that is, as a single wafer that has multiple unpackaged chips), as a bare die, or in a packaged form. In the latter case the chip is mounted in a single chip package (such as a plastic carrier, with leads that are affixed to a motherboard or other higher level carrier) or in a multichip package (such as a ceramic carrier that has either or both surface interconnections or buried interconnections). In any case the chip is then integrated with other chips, discrete circuit elements, and/or other signal processing devices as part of either (a) an intermediate product, such as a motherboard, or (b) an end product. The end product can be any product that includes integrated circuit chips, ranging from toys and other low-end applications to advanced computer products having a display, a keyboard or other input device, and a central processor. 
         [0035]    Reference in the specification to “one embodiment” or “an embodiment” of the present principles, as well as other variations thereof, means that a particular feature, structure, characteristic, and so forth described in connection with the embodiment is included in at least one embodiment of the present principles. Thus, the appearances of the phrase “in one embodiment” or “in an embodiment”, as well any other variations, appearing in various places throughout the specification are not necessarily all referring to the same embodiment. 
         [0036]    It is to be appreciated that the use of any of the following “/”, “and/or”, and “at least one of”, for example, in the cases of “A/B”, “A and/or B” and “at least one of A and B”, is intended to encompass the selection of the first listed option (A) only, or the selection of the second listed option (B) only, or the selection of both options (A and B). As a further example, in the cases of “A, B, and/or C” and “at least one of A, B, and C”, such phrasing is intended to encompass the selection of the first listed option (A) only, or the selection of the second listed option (B) only, or the selection of the third listed option (C) only, or the selection of the first and the second listed options (A and B) only, or the selection of the first and third listed options (A and C) only, or the selection of the second and third listed options (B and C) only, or the selection of all three options (A and B and C). This may be extended, as readily apparent by one of ordinary skill in this and related arts, for as many items listed. 
         [0037]    Referring now to  FIG. 5 , an abstract diagram of electro-optic modulation of an optical carrier is shown. An optical beam  508  is shown passing through semi-transparent, partially mirrored plate  502 , through an electro-optic modulator (EOM) region  506 , and reflecting from a mirror  504 . The mirrors  502  and  504  form a Fabry-Perot cavity. In the present embodiments, the EOM region  506  is a resonator at the optical wavelengths. An inductor  514  and the plates of capacitor  512  form a resonator at a microwave frequency, with the output of qubit  510  being injected into the resonator through capacitor  516 . As the microwave signal oscillates in the resonator, charges build and switch on capacitor plates  512  around the EOM region  506 . These charges create an oscillating electric field that causes a phase shift in the optical signal. 
         [0038]    The phase shift is caused by a change in the refractive index in the EOM  506  caused by the external electric field, E j . This change is characterized as: 
         [0000]      Δ n==− ½ n   3   r   ij   E   j  
 
         [0000]    where n is the refractive index of the medium of the EOM  506  and r ij  is the electro-optic coefficient. The phase shift is characterized as: 
         [0000]    
       
         
           
             Δφ 
             = 
             
               
                 Δ 
                  
                 
                     
                 
                  
                 kL 
               
               = 
               
                 
                   
                     ω 
                     a 
                   
                   
                     
                       c 
                       / 
                       Δ 
                     
                      
                     
                         
                     
                      
                     n 
                   
                 
                  
                 L 
               
             
           
         
       
     
         [0000]    where L is the inductance of inductor  514  and ω a  is the frequency of the optical signal. A change in the frequency is characterized by: 
         [0000]    
       
         
           
             
               Δω 
               a 
             
             = 
             
               
                 Δφ 
                 τ 
               
               = 
               
                 
                   
                     
                       ω 
                       a 
                     
                     
                       
                         c 
                         / 
                         Δ 
                       
                        
                       
                           
                       
                        
                       n 
                     
                   
                    
                   
                     L 
                     τ 
                   
                 
                 = 
                 
                   
                     
                       ω 
                       a 
                     
                      
                     
                       
                         Δ 
                          
                         
                             
                         
                          
                         n 
                       
                       n 
                     
                   
                   = 
                   
                     
                       - 
                       
                         1 
                         2 
                       
                     
                      
                     
                       ω 
                       a 
                     
                      
                     
                       n 
                       2 
                     
                      
                     
                       r 
                       ij 
                     
                      
                     
                       E 
                       j 
                     
                   
                 
               
             
           
         
       
     
         [0000]    where τ is the optical round-trip time and c is the speed of light. The indices i and j are the indices of the electro-optic material crystal axis. 
         [0039]    This embodiment specifically makes use of the Pockels effect, where the resonant frequency of the optical resonator  110  is modulated using the microwave field from the microwave cavity  112 . The electro-optic effect is caused in the optical resonator  110  by depositing a straining material that breaks the crystal symmetry of a substrate. The coupling between the microwave signal and the optical signal is described by a coupling Hamilton: 
         [0000]        ĥ′=             g ( {circumflex over (b)}   +   +{circumflex over (b)} )( â   −   +   +â   +   +â   +   + )( â   −   +â+â   + ) 
         [0000]    where â − (â −   + ), â(â + ), and â + (â +   + ) are the annihilation (creation) operators for red- sideband, carrier, and blue-sideband modes in the optical cavity  110  respectively, {circumflex over (b)}({circumflex over (b)} + ) is the annihilation (creation) operator for the microwave photons of the qubit  104 , and g is the coupling strength between the optical and microwave photons. After applying the rotating-wave approximation, the Hamiltonian Ĥ=Ĥ O +Ĥ C  of the electro-optic device  100 , including the red- and blue-sideband modes, becomes: 
         [0000]        Ĥ   O =         ω −   â   −   +   â   − +         ω op   â   +   â+             ω   +   â   +   +   â   + +         ω q   {circumflex over (b)}   +   {circumflex over (b)} 
 
         [0000]        Ĥ   C   =             g ( â   −   +   â{circumflex over (b)}   +   +â   −â   +   {circumflex over (b)}+â   +   +   â{circumflex over (b)}+â   +   â   +   {circumflex over (b)}   + ) 
         [0000]    where ω op  is the optical carrier frequency, ω −  and ω +  are the red and blue sideband frequencies, and ω q  is the microwave frequency of the qubit  104 . 
         [0040]    The coupling Hamiltonian, Ĥ C , shows the three-wave mixing among the optical photons at the carrier and the sidebands and the microwave photons of the superconducting qubit  104 . By applying a strong pump tone at ω + =ω op +ω q , the operators â +  can be replaced by the classical drive {circumflex over (α)} +  (c-number), where |α + | 2  represents the average number of pump photons at ω + , which provides an effective coupling rate between the quantum microwave node {circumflex over (b)} and the fundamental optical mode â at a rate Ω R =g|α + |. In one embodiment, with realistic parameters, this rate may be about 10 MHz, with a coupling strength of g˜10 kHz and α=1000 corresponding to 10 6  photons in the resonators. This sets an upper bound on the speed of the communication channel. In general the coupling strength can be estimated as: 
         [0000]    
       
         
           
             
               g 
               
                 2 
                  
                 π 
               
             
             ≡ 
             
               
                 - 
                 
                   1 
                   2 
                 
               
                
               
                 f 
                 a 
               
                
               
                 n 
                 2 
               
                
               r 
                
               
                 
                   V 
                   ZPV 
                 
                 d 
               
             
             ∼ 
             
               0.1 
               - 
               
                 100 
                  
                 
                     
                 
                  
                 kHz 
               
             
           
         
       
     
         [0000]    where V ZPV  is the superconducting cavity zero-point voltage (having a range of about 0.1 μV to about 1 μV), d is the thickness of the optical resonator  110  (having a range of about 1 μm to about 100 μm), f a  is an optical communications frequency (e.g., about 193 THz), n is the number of pump photons, and r is the electro-optic coefficient of the electro-optic material. It should be noted that the pump signal can be provided through the optical waveguide  108  described above. 
         [0041]    Referring now to  FIG. 6 , a cross-sectional view of detail on the structure of an optical resonator  110  is shown. The substrate  102  has a ring  602  of additional material, e.g., silicon, formed on it. The material of the ring  602  is patterned on a top surface, with another material  604 , e.g., silicon germanium, being deposited in the gaps. The additional material  604  is selected to have a different lattice structure than that of the ring  602 , which causes a strain in the lattice structure of the ring  602 . It is this strain that makes the ring  602  susceptible to the electro-optic effect. The straining material  604  may deposited by any appropriate deposition process including, e.g., chemical vapor deposition, physical vapor deposition, and atomic layer deposition. 
         [0042]    It is contemplated that other embodiments of an optical resonator  110  may be employed. As noted above, the ring  602  is only one structure, and a disc embodiment may be used instead as long as it supports whispering gallery modes at the optical frequencies in question. In addition, different materials may be used. The above-described embodiment uses strain in the crystalline lattice structure of the resonator  110  to create the Pockels electro-optic effect, but it should be noted that some materials that naturally lack crystalline inversion symmetry also exhibit this effect and may be used instead. The optical resonator  110  may be formed by any appropriate fabrication technique, including machining, micro-fabrication, etching, etc. 
         [0043]    Referring now to  FIG. 7 , a method of forming a microwave-to-optical transducer is shown. Block  702  fabricates the optical resonator  110 . In particular, block  702  forms the optical resonator in or on the bottom substrate  102  by, e.g., depositing resonator material in block  704  (or, alternatively, etching the resonator material from the bulk substrate  102 ), patterning the resonator surface in block  706  as described above to form ridges, and depositing a straining material  604  in block  708  to create a strain in the crystalline structure of the optical resonator  110 . Alternatively, block  702  can fabricate the optical resonator  110  from a material that naturally exhibits the electro-optic (Pockels) effect. 
         [0044]    Block  710  constructs the microwave cavity in the top substrate  120 . Block  712  machines the microwave cavity  112  in the top substrate  120  by any appropriate micro-fabrication technique, including micro-machining or etching. The microwave cavity  112  may be, for example, a microwave coaxial cavity (as shown above), a microwave coplanar waveguide, a microwave microstrip cavity, etc., and is formed with a smooth internal surface at a diameter that supports whispering gallery modes at the microwave frequency of the qubit  104 . Block  714  forms the ridge  302  on the face of the central pin  114  by, e.g., machining the surface of the pin  114  or by an etching process to concentrate the electric fields of the microwave signal onto the optical resonator  110 . Block  716  forms a superconducting film over the internal surface of the microwave cavity  112  and the outer surface of the central pin  114 . 
         [0045]    Block  716  fabricates the qubit(s)  104  on the lower substrate  102 . It should be noted that the qubit(s)  104  may be made with superconducting material and may be integrated with the same substrate  102  as described above, or may be formed in a separate package and subsequently connected or attached to the device. 
         [0046]    Block  718  forms a waveguide in, e.g., the bottom substrate  102 , that couples to the optical resonator  110  and provides communication of modulated signals from the optical resonator  110  to other devices on- or off-chip. Block  720  forms a superconducting coupling path  106  that couples the qubit(s)  104  to the microwave electric fields in the microwave cavity  112 . The coupling path may include, e.g., a microwave antenna or superconducting channel. Block  722  assembles the transducer, placing the top substrate  120  over the bottom substrate  102  and aligning the central pin  114  of the microwave cavity  112  above the optical resonator  110 , such that electric fields from the whispering gallery modes on the central pin  114  are applied to the optical resonator  110 . 
         [0047]    Referring now to  FIG. 8 , a top-down view of an alternative bottom substrate  802  is shown. The qubit  104 , superconducting channel  106 , and waveguide  108  are shown as being placed similarly to the embodiment of  FIG. 2 . However, instead of having a lower cavity  111  with the optical resonator  110  being placed over top, this embodiment has a ring optical resonator  804  placed directly on the bottom substrate  802  with no lower cavity at all. This embodiment may alternatively have a disc oscillator as the optical resonator  804 . 
         [0048]    Referring now to  FIG. 9 , a more detailed cross-sectional view of the connection between the pin  114  and the optical resonator  804  is shown. As can be seen, the resonator  804  rests directly on the bottom substrate  102  and is shown in cross section directly under the ridge  302 . 
         [0049]    Having described preferred embodiments of an integrated microwave-to-optical single-photon transducer (which are intended to be illustrative and not limiting), it is noted that modifications and variations can be made by persons skilled in the art in light of the above teachings. It is therefore to be understood that changes may be made in the particular embodiments disclosed which are within the scope of the invention as outlined by the appended claims. Having thus described aspects of the invention, with the details and particularity required by the patent laws, what is claimed and desired protected by Letters Patent is set forth in the appended claims.