A quantum-mechanical photon-pair generator includes first, second, third, and fourth Josephson junctions electrically connected in a bridge circuit having first, second and third resonance eigenmodes, and a source of magnetic flux configured to provide, during operation, a magnetic flux through the bridge circuit to cause coupling between the first, second and third resonance eigenmodes when the third resonance eigenmode is excited. The photon-pair generator further includes first, second and third electromagnetic resonators having eigenmodes in resonance with the first, second and third resonance eigenmodes of the bridge circuit, respectively. The third frequency of the third resonance eigenmode is equal to a sum of a first frequency of the first resonance eigenmode plus a second frequency of the second resonance eigenmode such that, during operation, a photon having the third frequency is split into two quantum-mechanically entangled photons having the first and second frequencies, respectively.

BACKGROUND

The currently claimed embodiments of the present invention relate to superconducting quantum mechanical devices, and more specifically, to a quantum-mechanical photon-pair generator.

Single photon generators can be very useful in a variety of quantum information processing applications such as, for example, quantum cryptography, quantum communication, and quantum computing with photons. Superconducting qubits can be used under certain conditions as single-microwave-photon generators. For example, superconducting qubits can be excited to their first excited state and then stimulated on demand to emit their excitation in the form of a microwave photon into a resonator or a transmission line.

In order to construct a non-degenerate parametric device (the Josephson parametric converter (JPC)), which is capable of amplifying and/or mixing microwave signals at the quantum limit, a Josephson ring modulator (JRM) is incorporated into two microwave resonators at an rf-current anti-node of their fundamental eigenmodes. The Josephson ring modulator (JRM) is a nonlinear dispersive element based on Josephson tunnel junctions that can perform three-wave mixing of microwave signals at the quantum limit. The JRM has four nominally identical Josephson junctions arranged in Wheatstone-like bridge configuration. The performance of the JPC including power gain, dynamical bandwidth, and dynamic range, are strongly dependent on the critical current of the Josephson junction of the JRM, the specific realization of the electromagnetic environment (i.e. the microwave resonators), and the coupling between the JRM and the resonators. However, the JPC couples the JRM to only two resonators.

SUMMARY

An aspect of the present invention is to provide a quantum-mechanical photon-pair generator including a first Josephson junction, a second Josephson junction electrically connected to the first Josephson junction, a third Josephson junction electrically connected to the second Josephson junction and a fourth Josephson junction electrically connected to the third Josephson junction and the first Josephson junction such that the first, second, third and fourth Josephson junctions are connected in a bridge circuit having a first resonance eigenmode, a second resonance eigenmode and a third resonance eigenmode. The quantum-mechanical photon-pair generator further includes a source of magnetic flux arranged proximate the bridge circuit, the source of magnetic flux is configured to provide, during operation, a magnetic flux through the bridge circuit to cause coupling between the first, second and third resonance eigenmodes when the third resonance eigenmode is excited. The quantum-mechanical photon-pair generator also includes a first electromagnetic resonator electrically connected to the first and fourth Josephson junctions at a first node therebetween and to the second and third Josephson junctions at a second node therebetween; a second electromagnetic resonator electrically connected to the first and second Josephson junctions at a third node therebetween and to the third and fourth Josephson junctions at a fourth node therebetween; and a third microwave resonator connected to the first and fourth Josephson junctions at the first node therebetween and to the second and third Josephson junctions at the second node therebetween. The first electromagnetic resonator has an eigenmode in resonance with the first resonance eigenmode of the bridge circuit. The second electromagnetic resonator has an eigenmode in resonance with the second resonance eigenmode of the bridge circuit. The third electromagnetic resonator has an eigenmode in resonance with the third resonance eigenmode of the bridge circuit. The third frequency of the third resonance eigenmode is equal to a sum of a first frequency of the first resonance eigenmode plus a second frequency of the second resonance eigenmode such that, during operation, a photon having the third frequency is split into two quantum-mechanically entangled photons having the first and second frequencies, respectively.

In an embodiment, the first electromagnetic resonator, the second electromagnetic resonator and the third electromagnetic resonator are microwave resonators. In an embodiment, the first frequency of the first resonance eigenmode, the second frequency of the second resonance eigenmode, and the third frequency of the third resonance eigenmode are in the microwave frequency range.

In an embodiment, the quantum-mechanical photon-pair generator further includes a first capacitor (Cc) connected in parallel with the first Josephson junction; a second capacitor (Cc) connected in parallel with the second Josephson junction; a third capacitor (Cc) connected in parallel with the third Josephson Junction; and a fourth capacitor (Cc) connected in parallel with the fourth Josephson junction. In an embodiment, the quantum-mechanical photon-pair generator further includes a fifth capacitor (Ca) connected in parallel with the bridge circuit at the first node and the second node of the bridge circuit; and a sixth capacitor (Cb) connected in parallel with the bridge circuit at the third node and the fourth node of the bridge circuit.

In an embodiment, the first frequency and the second frequency depend on capacitance values of the fifth capacitor (Ca) and the sixth capacitor (Cb) and the first frequency and the second frequency are selected by selecting capacitance values of the fifth capacitor (Ca) and the sixth capacitor (Cb), respectively. In an embodiment, a capacitance value of the first capacitor, a capacitance value of the second capacitor, a capacitance value of the third capacitor, and a capacitance value of the fourth capacitor are selected so that the third frequency of the third resonance eigenmode is equal to the sum of the first frequency of the first resonance eigenmode plus the second frequency of the second resonance eigenmode.

In an embodiment, the quantum-mechanical photon-pair generator further includes a seventh capacitor (Ccc) connected to a third resonator feedline coupled to the third resonator and to the first node of the bridge circuit; an eighth capacitor (Cca) connected to a first resonator feedline coupled to the first resonator and to the first node of the bridge circuit; a ninth capacitor (Ccb) connected to a second resonator feedline coupled to the second resonator and to the third node of the bridge circuit; and a tenth capacitor (Ccc) connected to a third resonator feedline coupled to the third resonator and to the second node of the bridge circuit. The seventh, eighth, ninth and tenth capacitors are selected to satisfy a frequency hierarchy conduction such that: a coupling constant between the first, the second and third eigenmodes is less than decay rates of the first and second eigenmodes to corresponding external feedlines, and the coupling constant between the first, the second and third eigenmodes is greater than a decay rate of the third eigenmode to a corresponding external feedline.

In an embodiment, the first, second and third microwave resonators are coplanar strip-line resonators or micro-strip resonators. In an embodiment, the first, second and third microwave resonators are compact lumped-element resonators or three-dimensional cavities. In an embodiment, the source of magnetic flux is a current-carrying element to provide an electromagnetic source of magnetic flux that flux-biases the bridge circuit. In an embodiment, the source of magnetic flux is a magnetic material to provide an electromagnetic source of magnetic flux that flux-biases the bridge circuit. In an embodiment, the source of magnetic flux provides a half of flux quantum (φ0/2). In an embodiment, the first frequency, the second frequency and the third frequency are in the microwave frequency range.

Another aspect of the present invention is to provide a quantum-mechanical non-linear circuit including a first Josephson junction; a second Josephson junction electrically connected to the first Josephson junction; a third Josephson junction electrically connected to the second Josephson junction; and a fourth Josephson junction electrically connected to the third Josephson junction and the first Josephson junction, such that the first, second, third and fourth Josephson junctions are connected in a bridge circuit. The quantum-mechanical non-linear circuit further includes a first capacitor (Cc) connected in parallel with the first Josephson junction; a second capacitor (Cc) connected in parallel with the second Josephson junction; a third capacitor (Cc) connected in parallel with the third Josephson Junction; a fourth capacitor (Cc) connected in parallel with the fourth Josephson junction; a fifth capacitor (Ca) connected in parallel with the bridge circuit at a first node between the first Josephson junction and the fourth Josephson junction and a second node between the second Josephson junction and the third Josephson junction; and a sixth capacitor (Cb) connected in parallel with the bridge circuit at the third node between the first Josephson junction and the second Josephson junction and a fourth node between the third Josephson junction and the fourth Josephson junction. The quantum-mechanical non-linear circuit has a first resonance eigenmode, a second resonance eigenmode and a third resonance eigenmode. The third frequency of the third resonance eigenmode is equal to a sum of a first frequency of the first resonance eigenmode plus a second frequency of the second resonance eigenmode such that, during operation, a photon having the third frequency is split into two quantum-mechanically entangled photons having the first and second frequencies, respectively.

In an embodiment, when in operation, the first resonance eigenmode, the second resonance eigenmode and the third resonance eigenmode are coupled by applying a magnetic field to generate a magnetic flux through the bridge circuit when the third resonance eigenmode is excited. In an embodiment, the first frequency and the second frequency depend on capacitance values of the fifth capacitor (Ca) and the sixth capacitor (Cb), respectively, and the first frequency and the second frequency are selected by selecting capacitance values of the fifth capacitor (Ca) and the sixth capacitor (Cb), respectively. In an embodiment, a capacitance value of the first capacitor, a capacitance value of the second capacitor, a capacitance value of the third capacitor, and a capacitance value of the fourth capacitor are selected so that the third frequency of the third resonance eigenmode is equal to the sum of the first frequency of the first resonance eigenmode plus the second frequency of the second resonance eigenmode.

In an embodiment, the quantum-mechanical non-linear circuit further includes a seventh capacitor (Ccc) connected to the first node of the bridge circuit; an eight capacitor (Cca) connected to the first node of the bridge circuit and to the fifth capacitor (Ca); a ninth capacitor (Ccb) connected to the third node of the bridge circuit and to the sixth capacitor (Cb); and a tenth capacitor (Ccc) connected to the second node of the bridge circuit. The seventh, eighth, ninth and tenth capacitors are selected to satisfy a frequency hierarchy conduction such that: a coupling constant between the first, the second and third eigenmodes is less than decay rates of the first and second eigenmodes to corresponding external feedlines, and the coupling constant between the first, the second and third eigenmodes is greater than a decay rate of the third eigenmode to a corresponding external feedline.

Another aspect of the present invention is to provide a method for generating a quantum-mechanical entangled photon-pair. The method includes inputting an electromagnetic wave at a third frequency into a quantum-mechanical non-linear circuit having a first resonance eigenmode, a second resonance eigenmode and a third resonance eigenmode, the third frequency corresponding to a frequency of the third eigenmode; and applying a magnetic field to generate a magnetic flux in the quantum-mechanical non-linear circuit to couple the first eigenmode, the second eigenmode and the third eigenmode such that, a photon having the third frequency is split into two quantum-mechanically entangled photons having a first frequency of the first eigenmode and a second frequency of the second eigenmode, respectively, the third frequency being equal to a sum of the first frequency plus the second frequency.

In an embodiment, the method further includes selecting the first frequency and the second frequency by selecting capacitance values of first capacitors in the quantum-mechanical non-linear circuit. In an embodiment, the method further includes selecting capacitance values of second capacitors in the quantum-mechanical non-linear circuit so that the third frequency of the third resonance eigenmode is equal to the sum of the first frequency of the first resonance eigenmode plus the second frequency of the second resonance eigenmode. In an embodiment, the method further includes selecting capacitance values of third capacitors in the quantum-mechanical non-linear circuit so as to satisfy a frequency hierarchy conduction such that a coupling constant between the first, the second and third eigenmodes is less than a decay rate of the first eigenmode to a corresponding external feedline and a decay rate of the second eigenmode to a corresponding external feedline, and the coupling constant between the first, the second and third eigenmodes is greater than a decay rate of the third eigenmode to a corresponding external feedline.

The present quantum-mechanical photon-pair generator is based on coupling a bridge circuit to three electromagnetic resonators. quantum-mechanical photon-pair generator is configured to down-convert a higher frequency photon entering one port of the generator into a pair of quantum-mechanical entangled photons having lower frequencies. The pair of quantum-mechanical entangled photons are generated on-demand and thus can be useful in remote quantum-mechanical entanglement of superconducting qubits, quantum communication, quantum cryptography, etc.

DETAILED DESCRIPTION

FIG. 1is a schematic diagram of a quantum-mechanical photon-pair generator100, according to an embodiment of the present invention. The quantum-mechanical photon-pair generator100includes a first Josephson junction102, and a second Josephson junction104electrically connected to the first Josephson junction102. The quantum-mechanical photon-pair generator100also includes a third Josephson junction106electrically connected to the second Josephson junction104and a fourth Josephson junction108electrically connected to the third Josephson junction106and the first Josephson junction102such that the first Josephson junction102, the second Josephson junction104, the third Josephson Junction106, and the fourth Josephson junction108are connected in a bridge circuit110having a first resonance eigenmode (X-mode), a second resonance eigenmode (Y-mode) and a third resonance eigenmode (Z-mode).

The term “bridge circuit” as used in this specification is intended to refer to a circuit that has at least four Josephson junctions connected similar to the arrangement of resistors in a Wheatstone bridge circuit. However, the “bridge circuit” as used herein is a quantum mechanical circuit, not a classical electrical circuit.

FIGS. 2A-2Cshow a bridge circuit110having a first resonance eigenmode, a second resonance eigenmode and a third resonance eigenmode, according to an embodiment of the present invention.FIGS. 2A-2Cshow the rf voltage polarity on the bridge nodes corresponding to the first, second, and third eigenmodes.FIG. 2Ashows the bridge circuit110operating in the first eigenmode (X-mode).FIG. 2Bshows the bridge circuit110operating in the second eigenmode (Y-mode).FIG. 2Cshows the bridge circuit110operating in the third eigenmode (Z-mode). In some embodiments, the first, second and third eigenmodes are mutually orthogonal to each other. The bridge circuit110which can be a Josephson Ring Modulator (JRM) in some embodiments is a nonlinear dispersive circuit based on Josephson tunnel junctions (e.g., four Josephson junctions) that can perform three-wave mixing of electromagnetic signals (e.g., microwave signals) at the quantum limit. However, the broad concepts of the current invention are not limited to only four Josephson junctions. Additional Josephson junction could be included on one or more of the legs of the bridge circuit110according to some embodiments.

The quantum-mechanical photon-pair generator100also includes a source of magnetic flux112arranged proximate the bridge circuit110. The source of magnetic flux112is configured to provide, during operation, a magnetic flux through the bridge circuit110to cause coupling between the first resonance eigenmode (X-mode), the second resonance eigenmode (Y-mode) and the third resonance eigenmode (Z-mode) when the third resonance eigenmode (Z-mode) is excited.

In an embodiment, the source of magnetic flux112is a current-carrying element to provide an electromagnetic source of magnetic flux that flux-biases the bridge circuit110. In an embodiment, the source of magnetic flux112is a magnetic material to provide an electromagnetic source of magnetic flux that flux-biases the bridge circuit110. In an embodiment, the source of magnetic flux112provides a half of flux quantum ((φ0/2).

The quantum-mechanical photon-pair generator100further includes a first electromagnetic resonator114electrically connected to the first Josephson junction102and fourth Josephson junction108at a first node1therebetween and to the second Josephson junction104and third Josephson junction106at a second node2therebetween. The quantum-mechanical photon-pair generator100also includes a second electromagnetic resonator116electrically connected to the first Josephson junction102and the second Josephson junction104at a third node3therebetween and to the third Josephson junction106and the fourth Josephson junction108at a fourth node4therebetween. The quantum-mechanical photon-pair generator100also includes a third electromagnetic resonator118connected to the first Josephson junction102and the fourth Josephson junction108at the first node1therebetween and to the second Josephson junction104and the third Josephson junction106at the second node2therebetween. InFIG. 1, for purposes of illustration, the reference numerals114,116and118corresponding to the first, second and third resonators are shown generally pointing to respective external resonator feedlines114A,116A and118A that are coupled to the first electromagnetic resonator114, the second electromagnetic resonator116and the third electromagnetic resonator118. These resonator feedlines114A,116A and118A carry the input and output signals into and out of the respective resonators. However, as it is understood, the first, second and third electromagnetic resonators114,116and118include other elements. For example, the first electromagnetic resonator114is represented inFIG. 3Aby the corresponding mode it generates, the second electromagnetic resonator116is represented inFIG. 3Bby the corresponding mode it generates, and the third resonator118is represented inFIG. 3Cby the corresponding mode it generates, as will be explained further in the following paragraphs.

The first electromagnetic resonator114has an eigenmode in resonance with the first resonance eigenmode (X-mode) of the bridge circuit110. The second electromagnetic resonator116has an eigenmode in resonance with the second resonance eigenmode (Y-mode) of the bridge circuit110. The third electromagnetic resonator118has an eigenmode in resonance with the third resonance eigenmode (Z-mode) of the bridge circuit110. The third frequency fPof the third resonance eigenmode (Z-mode) is equal to a sum of a first frequency fSof the first resonance eigenmode plus a second frequency fIof the second resonance eigenmode such that, during operation, a photon having the third frequency fPis split into two quantum-mechanically entangled photons having the first frequency fSand the second frequency fI, respectively. The first frequency fSis sometimes referred to as the signal frequency, the second frequency fIis sometimes referred to as the idler frequency, and the third frequency fPis sometimes referred to as the pump frequency. In an embodiment, the first frequency fS, the second frequency fIand the third frequency fPare in the microwave frequency range, for example.

In an embodiment, the first electromagnetic resonator114, the second electromagnetic resonator116and the third electromagnetic resonator118are microwave resonators operating in the microwave frequency range. However, as it must be appreciated, the first electromagnetic resonator114, the second electromagnetic resonator116and the third electromagnetic resonator118can also be configured to operate in another frequency range, for example, below or above the microwave frequency range.

In an embodiment, the quantum-mechanical photon-pair generator100further includes a first capacitor (Cc)122connected in parallel with the first Josephson junction102, and a second capacitor (Cc)124connected in parallel with the second Josephson junction104. In an embodiment, the quantum-mechanical photon-pair generator100also includes a third capacitor (Cc)126connected in parallel with the third Josephson Junction106, and a fourth capacitor (Cc)128connected in parallel with the fourth Josephson junction108. Although one capacitor is shown connected in parallel with each of the Josephson junctions102,104,106and108, two or more capacitors can also be connected in parallel with each of the Josephson junctions102,104,106and108. For example, the two or more capacitors can be connected in series or in parallel, or both and then connected as whole in parallel with each of the Josephson junctions102,104,106and108. In addition, although the capacitors102,104,106and108are shown as having a same capacitance Cc, the capacitors102,104,106and108can also have different capacitances and/or each have a different number of capacitors.

In an embodiment, the quantum-mechanical photon-pair generator100further includes a fifth capacitor (Ca)132connected in parallel with the bridge circuit110at the first node1and the second node2of the bridge circuit110, and a sixth capacitor (Cb)134connected in parallel with the bridge circuit110at the third node3and the fourth node4of the bridge circuit110. Although the capacitors132and134are shown inFIG. 1as single capacitors, two or more capacitors can also be used instead of a single capacitor. For example, the two or more capacitors can be connected in series or in parallel, or both and as whole be used as capacitor132or capacitor134.

In an embodiment, the first frequency fSand the second frequency fIdepend on capacitance values of the fifth capacitor (Ca)132and the sixth capacitor (Cb)134. In an embodiment, the first frequency fSand the second frequency fIare selected by selecting capacitance values of the fifth capacitor (Ca)132and the sixth capacitor (Cb)134, respectively.

In an embodiment, a capacitance value of the first capacitor122, a capacitance value of the second capacitor124, a capacitance value of the third capacitor126, and a capacitance value of the fourth capacitor128are selected so that the third frequency fPof the third resonance eigenmode is equal to the sum of the first frequency of the first resonance eigenmode plus the second frequency fIof the second resonance eigenmode.

FIGS. 3A-3Cshow the circuits corresponding to the various modes (first resonance eigenmode, second resonance eigenmode, and third resonance eigenmode) and their associated resonance frequencies, according to an embodiment of the present invention. For example, the first frequency of the first resonance mode fS, the second frequency fIof the second resonance mode, and the third frequency fPof the third resonance mode can be expressed as follows:

fs=12⁢π⁢Lj⁡(Cc+Ca)(1)fI=12⁢π⁢Lj⁡(Cc+Cb)(2)fP=12⁢π⁢Lj⁢Cc(3)
where Ljis the inductance of each of the Josephson junctions. In equations (1), (2) and (3), the inductances Ljof the Josephson junctions are assumed to be equal. In addition, in equations (1), (2) and (3), the coupling capacitors Ccc, Ccaand Ccbare ignored for simplicity. They do affect the resonance frequencies to a limited degree, but their effect can be taken into account using common microwave simulation tools. Their main purpose is to couple the different external ports to the corresponding three electromagnetic resonators that include the bridge circuit110. As shown in equations, (1), (2) and (3), the frequencies fS, fI, and fP, are inversely proportional to the square root of the capacitances Ca, Cb, and Cc. Therefore, increasing the capacitances will decrease the respective frequencies.

In an embodiment, the inductance Ljis selected such that the quantum-mechanical photon-pair generator100acts as a quantum mechanical device with discrete energy levels that can work at the single photon levels and which increases the coupling constant g3between the modes. In an embodiment, the inductance Ljis selected to be in the range between 5 nH and 30 nH. Given the desired frequencies fSand fI, one can find the appropriate capacitance values Caand Cbof the capacitors132and134, respectively, which yield these frequencies. In an embodiment, the capacitance values Caand Cbof the capacitors132and134, respectively are selected in the range between 10 fF and 1 pF. The capacitance value Ccof the first, second, third and fourth capacitors122,124,16and128are selected so that the condition fP=fI+fSis satisfied.

In an embodiment, the quantum-mechanical photon-pair generator100also includes a seventh capacitor (Ccc)142connected to the third resonator feedline118A and to the first node1of the bridge circuit110. In an embodiment, the quantum-mechanical photon-pair generator100further includes an eighth capacitor (Cca)144connected to the first resonator feedline114A and to the first node1of the bridge circuit110. In an embodiment, the quantum-mechanical photon-pair generator100also includes a ninth capacitor (Ccb)146connected to the second resonator feedline116A and to the third node3of the bridge circuit110. In an embodiment, the quantum-mechanical photon-pair generator100also includes a tenth capacitor (Ccc)143that couples the third resonator feedline118A and 180-degree hybrid line118B to the third resonator118corresponding to the third eigenmode. The tenth capacitor (Ccc)143is connected to the third resonator feedline118A, to the 180-degree hybrid line118B, and to the second node2of the bridge circuit110. In an embodiment, the tenth capacitor (Ccc)143is provided so as to keep a symmetry with respect to nodes1and2when exciting the third mode (common mode).

In an embodiment, the seventh capacitor (Ccc)142, the eighth capacitor (Cca)144, and the ninth capacitor (Ccb)146are selected to satisfy a frequency hierarchy conduction such that:1. a coupling constant between the first, the second and the third eigenmodes g3is less than decay rates γSand γIof the first and second eigenmodes, respectively, to corresponding external feedlines (g3<γI˜γS); and2. the coupling constant between the first, the second and the third eigenmodes g3is greater than a decay rate γPof the third eigenmode to a corresponding external feedline (g3>γP).

Example Implementation: The following values can be selected to generate a two-photon quantum-mechanical entangled photon-pair.Critical current of the Josephson junctions I0=40 nA.Inductance of the Josephson junction at zero flux bias Lj0=Φ0/I0=8 nH.Inductance of the Josephson junction at half flux quantum Lj=√{square root over (2)}Φ0/I0=11.3 nH.Magnetic flux generated by the source of magnetic flux112, Φext=Φ0/2.fs=5 GHz.fI=6 GHz.Ca=72 fF (selected knowing the desired output photon frequency fs).Cb=45 fF (selected knowing the desired output photon frequency fI).fP=11 GHz (fP=fs+fI).γI/2π=γS/2π=100 MHz (decay rates of the first and second eigenmode to corresponding external feedlines).γP/2π=10 MHz (decay rate of the third eigenmode to a corresponding external feedline).g3˜50 MHz (coupling constant between the first, the second and the third eigenmodes).

As it can be understood from the above paragraphs, an aspect of the present invention is also to provide a quantum-mechanical non-linear circuit200. The quantum-mechanical non-linear circuit200includes the first Josephson junction102; the second Josephson junction104electrically connected to the first Josephson junction102; the third Josephson junction106electrically connected to the second Josephson junction104; and the fourth Josephson junction108electrically connected to the third Josephson junction106and the first Josephson junction102, such that the first, second, third and fourth Josephson junctions102,104,106and108are connected in the bridge circuit110.

The quantum-mechanical non-linear circuit200further includes the first capacitor (Cc)122connected in parallel with the first Josephson junction102; the second capacitor (Cc)124connected in parallel with the second Josephson junction104; the third capacitor (Cc)126connected in parallel with the third Josephson Junction106; and the fourth capacitor (Cc)128connected in parallel with the fourth Josephson junction108.

The quantum-mechanical non-linear circuit200also includes the fifth capacitor (Ca)132connected in parallel with the bridge circuit110at the first node1between the first Josephson junction102and the fourth Josephson junction108and the second node2between the second Josephson junction104and the third Josephson junction106. The quantum-mechanical non-linear circuit200further includes the sixth capacitor (Cb)134connected in parallel with the bridge circuit110at the third node3between the first Josephson junction102and the second Josephson junction104and a fourth node4between the third Josephson junction106and the fourth Josephson junction108.

The quantum-mechanical non-linear circuit200has the first resonance eigenmode, the second resonance eigenmode and the third resonance eigenmode. The third frequency (fP) of the third resonance eigenmode is equal to the sum of a first frequency (fS) of the first resonance eigenmode plus the second frequency (fI) of the second resonance eigenmode such that, during operation, a photon having the third frequency (fP) is split into two quantum-mechanically entangled photons having the first frequency (fS) and second frequency (fI).

In an embodiment, when in operation, the first resonance eigenmode, the second resonance eigenmode and the third resonance eigenmode are coupled by applying a magnetic field using the source of magnetic flux112to generate the magnetic flux through the bridge circuit110when the third resonance eigenmode is excited.

In an embodiment, the first frequency (fS) and the second frequency (fI) depend on capacitance values of the fifth capacitor (Ca)132and the sixth capacitor (Cb)134, respectively, and the first frequency (fS) and the second frequency (fI) are selected by selecting capacitance values of the fifth capacitor (Ca)132and the sixth capacitor (Cb)134, respectively.

In an embodiment, the capacitance value of the first capacitor122, a capacitance value of the second capacitor124, a capacitance value of the third capacitor126, and a capacitance value of the fourth capacitor128are selected so that the third frequency (fP) of the third resonance eigenmode is equal to the sum of the first frequency (fS) of the first resonance eigenmode plus the second frequency (fI) of the second resonance eigenmode.

In an embodiment, the quantum-mechanical non-linear circuit200further includes a seventh capacitor (Ccc)142connected to the first node1of the bridge circuit110; an eight capacitor (Cca)144connected to the first node1of the bridge circuit110and to the fifth capacitor (Ca)144; and a ninth capacitor (Ccb)146connected to the third node3of the bridge circuit110and to the sixth capacitor (Cb)134. In an embodiment, the quantum-mechanical non-linear circuit200also includes a tenth capacitor (Ccc)143that is connected to the second node2of the bridge circuit110.

The seventh capacitor142, the eighth capacitor144and ninth capacitor146are selected to satisfy a frequency hierarchy conduction such that:1. a coupling constant between the first, the second and the third eigenmodes g3is less than decay rates γSand γIof the first and second eigenmodes, respectively, to corresponding external feedlines (g3<γI˜γS); and2. the coupling constant between the first, the second and the third eigenmodes g3is greater than a decay rate γPof the third eigenmode to a corresponding external feedline (g3>γP).

FIG. 4is a flow diagram of a method for generating a quantum-mechanical entangled photon-pair, according to an embodiment of the present invention. The method includes inputting an electromagnetic wave at a third frequency (fP) into a quantum-mechanical non-linear circuit (e.g., circuit200) having a first resonance eigenmode, a second resonance eigenmode and a third resonance eigenmode, the third frequency (fP) corresponding to a frequency of the third eigenmode, at S400. The method also includes applying a magnetic field to generate a magnetic flux (for example, using the source of magnetic flux112) in the quantum-mechanical non-linear circuit (e.g., circuit200) to couple the first eigenmode, the second eigenmode and the third eigenmode such that, a photon having the third frequency (fP) is split into two quantum-mechanically entangled photons having a first frequency (fS) of the first eigenmode and a second frequency (fI) of the second eigenmode, respectively, the third frequency (fP) being equal to a sum of the first frequency (fs) plus the second frequency (fI), at S402.

In an embodiment, the method includes selecting the first frequency (fS) and the second frequency (fI) by selecting capacitance values of capacitors (Ca)132and (Cb)134in the quantum-mechanical non-linear circuit200. In an embodiment, the method includes selecting the capacitance values of capacitors (Ca)132and (Cb)134in the quantum-mechanical non-linear circuit200given the desired first frequency (fS) and second frequency (fI) of the two quantum-mechanically entangled photons. In an embodiment, the capacitance values of capacitors (Ca)132and (Cb)134in the quantum-mechanical non-linear circuit200can be selected so that the desired first frequency (fS) is equal to the second frequency (fI).

In an embodiment, the method further includes selecting capacitance values of capacitors (Cc)122,124,126and128in the quantum-mechanical non-linear circuit200so that the third frequency (fP) of the third resonance eigenmode is equal to the sum of the first frequency (fS) of the first resonance eigenmode plus the second frequency (fI) of the second resonance eigenmode, i.e., fP=fS+fI.

In an embodiment, the method also includes selecting capacitance values of capacitors Ccc142, Cca144and Ccb146in the quantum-mechanical non-linear circuit200so as to satisfy a frequency hierarchy conduction such that a coupling constant g3between the first, the second and third eigenmodes is less than a decay rate γSof the first eigenmode to a corresponding external feedline and a decay rate γIof the second eigenmode to a corresponding external feedline, and the coupling constant g3between the first, the second and third eigenmodes is greater than a decay rate γPof the third eigenmode to a corresponding external feedline.

A quantum-mechanical photon-pair generator that emits quantum-mechanical entangled photon-pairs on demand can be useful in remote quantum entanglement of superconducting qubits, quantum communication, quantum cryptography.