Patent Publication Number: US-2019190474-A1

Title: Low-noise josephson junction-based directional amplifier

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. provisional application Ser. No. 61/891,226, entitled “LOW-NOISE JOSEPHSON JUNCTION-BASED DIRECTIONAL AMPLIFIER,” filed Oct. 15, 2013, the entire contents of which are incorporated by reference herein. 
    
    
     GOVERNMENT SUPPORT 
     This invention was made with government support under W91 INF-09-01-0514 awarded by The United States Army Research Office. The government has certain rights in the invention. 
    
    
     BACKGROUND 
     Quantum information processing uses quantum mechanical properties to extend the capabilities of information processing. For example, security of information transfer may be enhanced, the amount of information encoded in a communication channel may be increased, and the number of operations required to perform certain computations may be reduced. Just as in conventional information processing where information is stored in one or more bits, quantum information is stored in one or more quantum bits, known as “qubits.” A qubit may be implemented physically in any two-state quantum mechanical system, such as photon polarization, electron spin, nuclear spin, or various properties of a superconducting Josephson junction, such as charge, energy, or the direction of a current. 
     One type of qubit based on the plasma oscillation of a superconducting Josephson junction is a circuit known as a “transmon.” Operations on a transmon, such as quantum state initialization, quantum gate operations and quantum measurements, may be implemented by transmitting and receiving microwave photons with the transmon. Microwave photon detectors are used to measure the photons received from the transmon and it would be desirable to provide the detection with high quantum efficiency and low noise. 
     SUMMARY 
     The following is a non-limiting summary of some embodiments of the present application. 
     Some embodiments are directed to a low-noise directional amplifier that includes a first port and a second port; a first coupler and a second coupler, wherein the first port and the second port are coupled to the first coupler; a first phase preserving amplifier connected to the first coupler and the second coupler, and a second phase preserving amplifier connected to the first coupler and the second coupler. 
     In some embodiments, the first port is an input port configured to receive at least one input signal and the second port is an output port configured to output at least one signal. 
     In some embodiments, low-noise directional amplifier also includes a third port coupled to a cold load and a fourth port coupled to a cold load. 
     In some embodiments the low-noise directional amplifier comprises fewer than four ports. 
     In some embodiments, the first coupler is a 3 dB coupler. 
     In some embodiments, a reflection gain amplitude of the first phase preserving amplifier is the same as a reflection gain amplitude of the second phase preserving amplifier and a transmission gain amplitude is the same as a transmission gain amplitude of the second phase preserving amplifier. 
     In some embodiments, the reflection gain amplitude of the first phase preserving amplifier is greater than or equal to unity and less than the reciprocal of a transmission amplitude of the second coupler. 
     In some embodiments, the first phase preserving amplifier and the second phased preserving amplifier are each a Josephson Parametric Converter (JPC). 
     In some embodiments, a phase of a pump signal of the first phase preserving amplifier is different from a phase of a pump signal of the second phase preserving amplifier. 
     In some embodiments, the difference between the phase of the pump signal of the first phase preserving amplifier and the phase of the pump signal of the second phase preserving amplifier is pi divided by two radians. 
     In some embodiments, the difference between the phase of the pump signal of the first phase preserving amplifier and the phase of the pump signal of the second phase preserving amplifier determines whether the low-noise directional amplifier is non-reciprocal. 
     In some embodiments, a transmission of a signal from the first port to the second port is substantially 100% when no pumps are applied to the first phase preserving amplifier and the second phase preserving amplifier. 
     In some embodiments, the low-noise directional amplifier is non-reciprocal and does not include a circulator. 
     In some embodiments, the low-noise directional amplifier is at least part of an integrated circuit. 
     Some embodiments are directed to an integrated circuit that includes a low-noise directional amplifier and a qubit coupled to the low-noise directional amplifier such that the low-noise directional amplifier is configured to measure a state of the qubit. The low-noise directional amplifier includes a first port and a second port; a first coupler and a second coupler, wherein the first port and the second port are coupled to the first coupler, a first phase preserving amplifier connected to the first coupler and the second coupler; and a second phase preserving amplifier connected to the first coupler and the second coupler. 
     In some embodiments, the low-noise directional amplifier is configured to measure the state of the qubit at the quantum noise limit. 
     In some embodiments, the qubit is one of a plurality of qubits; and the low-noise directional amplifier is one of a plurality of low-noise directional amplifiers, each low-noise directional amplifier of the plurality of low-noise directional amplifiers connected to at least one of the plurality of qubits. 
     Some embodiments are directed to a method of amplifying a microwave signal. The method includes acts of: receiving the signal at an input port of a directional amplifier; amplifying at least a portion of the signal using both a first parametric amplifier and a second parametric amplifier to create an amplified signal; and transmitting the amplified signal out an output port of the directional amplifier. 
     In some embodiments, the method further includes act an act of splitting the signal into at least two portions prior to the act of amplifying. 
     In some embodiments, the first parametric amplifier is pumped using a first microwave pump with a first phase; and the second parametric amplifier is pumped using a second microwave pump with a second phase, wherein the different between the first phase and the second phase is pi/2 radians. 
     The features and advantages of the present invention will be more readily understood and apparent from the following detailed description, which should be read in conjunction with the accompanying drawings, and from the claims. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
       The accompanying drawings are not intended to be drawn to scale. In the drawings, each identical or nearly identical component that is illustrated in various figures is represented by a like numeral. For purposes of clarity, not every component may be labeled in every drawing. In the drawings: 
         FIG. 1  shows a schematic of an integrated circuit according to some embodiments; 
         FIG. 2  illustrates a schematic of a low-noise directional amplifier according to some embodiments; 
         FIG. 3  illustrates a Josephson Parametric Converter according to some embodiments; 
         FIG. 4A  illustrates a schematic of an unshunted Josephson Ring Modulator according to some embodiments; 
         FIG. 4B  illustrates a schematic of a shunted Josephson Ring Modulator according to some embodiments; 
         FIG. 5  illustrates the paths waves may take through a directional amplifier according to some embodiments; 
         FIG. 6A  illustrates a first path of a wave through a directional amplifier according to some embodiments; 
         FIG. 6B  illustrates a second path of a wave through a directional amplifier according to some embodiments; 
         FIG. 6C  illustrates a third path of a wave through a directional amplifier according to some embodiments; 
         FIG. 6D  illustrates a fourth path of a wave through a directional amplifier according to some embodiments; and 
         FIG. 7  illustrates a method of amplifying a signal according to some embodiments. 
     
    
    
     DETAILED DESCRIPTION 
     Conventional dispersive electronic circuits, such as those formed from capacitors and inductors, are reciprocal, meaning an output signal that is re-directed back toward the circuit will follow the same paths as the input signal that created the output signal and be transmitted out the input port. This reciprocal behavior is known in optics as Helmholtz Reciprocity. Reciprocity is a symmetry of a system under the interchange of the source and the observer or, in the case of a circuit with an input and an output port, the interchange of the input port and the output port. The inventors have recognized and appreciated that the reduction of noise in a microwave amplifier for detecting low levels of microwave radiation is limited by the principle of reciprocity since noise coming from the final stages of the circuit may find its way back to the device under study with increased intensity. The problem may be particularly important for measurements done on devices at very low temperatures, as is the case with superconducting qubits, and the final stages of the electronics operate at room temperature, which is how conventional superconducting qubit systems are operated. Consequently, the inventors have recognized and appreciated that placing one or more devices that break the reciprocity of the circuit may be used within the circuit to prevent, by their valve-like properties, noise from the room temperature circuitry reaching the low temperature device. Devices that amplify a signal, and are non-reciprocal are referred to as “directional” amplifiers. 
     The inventors have further recognized and appreciated that conventional non-reciprocal components in microwave electronics are based on the use of ferrites in high magnetic fields. One example of a ferrite-based non-reciprocal component is a circulator. These components are bulky and may be difficult to incorporate into an integrated circuit. Consequently, it may be difficult to integrate a conventional ferrite-based non-reciprocal component with superconducting qubits on a single chip. Moreover, the magnetic field produced from the ferrites that cause the non-reciprocal behavior may negatively impact the performance of superconducting devices because a material&#39;s superconductivity loses many of its useful properties in even a modest magnetic field. Accordingly, the inventors have recognized and appreciated that there is a need for non-reciprocal device where the non-reciprocity is not based on the magnetic field of a ferrite, but instead is created using components that may be formed in an integrated circuit. 
     Some embodiments are directed to a non-reciprocal microwave circuit component that is based on a principle of “active non-reciprocity.” which is a technique for creating non-reciprocal microwave components with one or more devices that are actively controlled with one or more input signals. For example, active non-reciprocity may be based on techniques that are phase sensitive, such as parametric up-conversion and down-conversion, which are sensitive to the phase of the pump signal used. For example, if a signal is first up-converted to a higher frequency and then the resulting signal is down-converted back to the original frequency, the overall process phase shifts the signal by an overall phase that depends on the phase difference between the two pump signals used in the two frequency conversion acts. This overall phase shift depends on the direction of the phase gradient between the two frequency conversion pumps. 
     Some embodiments combine the aforementioned phase shifting technique with two additional techniques: 1) arranging the up-conversion and down-conversion processes to produce overall gain on the received signal, and 2) configuring beam-splitters (sometimes referred to as couplers) with particular characteristics to create a wave interference that transforms the non-reciprocity in phase into a non-reciprocity in amplitude. The result of such embodiments is that signals going in one direction through the device will be amplified whereas signals traveling in the opposite direction through the device will remain substantially the same. The resulting device is a non-reciprocal amplifier referred to as a “directional amplifier.” In some embodiments, because of the feedback loop created by the arrangement of components, one or more losses are introduced via, e.g., a coupler or a lossy channel, to ensure that the feedback is stable. For example, the reflection-gain amplitude, r, of the up-conversion and down-conversion processes may be less than the reciprocal of the transmission coefficient, a, of a back coupler. Accordingly, the reverse gain amplitude may be increased as the amount of loss introduced between the up-conversion and down-conversion process is increased. However, despite the introduction of losses, embodiments have the capability to perform at or near the quantum limit, meaning the amount of noise added to the signal during the amplification process is at or near the minimum amount of noise that is required to be added by the principle of quantum mechanics. As used throughout the present application, “low-noise” refers to devices that operate at or near the quantum noise limit. 
     In some embodiments two Josephson Parametric Converters (JPCs) are wired together to provide a low-noise directional amplifier. By the principles of quantum mechanics, any amplification must introduce at least a minimum amount of noise. In some embodiments, the low-noise directional amplifier may introduce noise that is at or near the minimal noise required by quantum mechanics. This minimal noise is the noise equivalent to a half photon. Embodiments may be used to read out one or more superconducting qubits, where embodiments have been experimentally used to improve the signal-to-noise ratio in the qubit measurements while not degrading the performance of the qubit significantly by its back-action noise. In particular, the inventors have used at least one embodiment to observe quantum jumps of the superconducting qubit, which are the hallmarks of efficient, low-noise readout and amplification circuitry. Conventionally, observation of quantum jumps in a superconducting qubit has been done by employing ferrite-based circulators and isolators. At least one embodiment of the present application allows quantum jumps to be observed without such non-reciprocal elements in their pre-amp stages. 
       FIG. 1  shows a schematic of an integrated circuit (chip  100 ) according to some embodiments. The chip comprises at least one superconducting qubit  101  and at least one low-noise directional amplifier  103 . A single chip  100  may include a plurality of superconducting qubits  101  and a plurality of directional amplifiers  103 . Each directional amplifier of the plurality of directional amplifiers is connected to at least one superconducting qubit  101 . The chip  100  may include microwave circuitry that connects the plurality of superconducting qubits together such that two or more superconducting qubits may interact with one another. Interactions between superconducting qubits may be used, for example, to implement quantum gates and/or entangle two or more superconducting qubits  101 . 
     The chip  100  may receive input microwave signals from at least one control circuit. The at least one control circuit may provide microwave signals to control the qubit  101  and/or the low-noise directional amplifier  103 . For example, the control circuit may send microwave signals to the qubit  101  that implement one or more quantum control gates. The control circuit may also transmit the pump microwaves for use in driving one or more parametric amplifiers in the low-noise directional amplifier  103 . In some embodiments, the at least one control circuit may include one or more additional low-noise directional amplifiers. However, any suitable control electronics may be used. 
     In some embodiments, the chip  100  has at least one output port for outputting amplified microwave signals from the directional amplifier  103 . The output signals may be directed to additional amplifiers and/or detection electronics used to record the measured values of the state of the superconducting qubit  101 . Any suitable detection electronics may be used. 
     The superconducting qubit  101  may be any suitable device comprising at least one Josephson Junction. For example, the superconducting qubit  101  may be a charge qubit, a flux qubit, a phase qubit, or a transmon qubit. In some embodiments, the superconducting qubit  101  may include at least one Josephson Junction inside a microwave resonator. The resonator may be, for example, a strip line resonator or a three-dimensional cavity. 
     The low-noise directional amplifier  103  may be any suitable directional amplifier that is cable of being integrated into chip  100  and is non-reciprocal so as to reduce noise feedback reaching the superconducting qubit  101 . Embodiments of low-noise directional amplifiers are discussed in more detail below. 
     In some embodiments, the entire chip  100  is held at low temperature to reduce noise and to maintain the superconductor below its critical temperature. For example, the chip  100  may be held in a dilution refrigerator at temperatures on the order of tens to hundreds of millikelvin. In some embodiments, the temperature of the chip  100  may be maintained at approximately 10 millikelvin (10 milli-degrees above absolute zero) such that thermal noise is reduced and is not capable of destroying the quantum information stored in the qubit  101  and amplified by the directional amplifier  103 . The temperature of the chip  100  may be held at these low temperatures using cryogenic techniques known in the art. For example, a dilution refrigerator using liquid Helium may be used to cool the chip  100  to the selected temperature. 
       FIG. 2  illustrates a low-noise directional amplifier  103  according to some embodiments. The directional amplifier  103  has at least two external ports that may be used to connect to external devices, such as qubit  101 —a first port  200   a  acting as an input port, and a second port  200   b  acting as an output port. Additional external ports may also be used. For example, in the embodiment shown in  FIG. 2 , two additional “cold load” ports, port  200   c  and port  200   d  are part of the device. However, ports  200   c  and  2004  are not used to send or receive signals—they are connected to “cold loads” in that the loads connected to those ports are kept at millikelvin temperatures. The directional amplifier  103  may also include internal ports for connecting a first component of the directional amplifier  103  to a second component of the directional amplifier. For example, port  210   a  connects a front coupler  201  to a first parametric amplifier  202 , port  210   b  connects the front coupler  201  to a second parametric amplifier  204 , port  210   c  connects the first parametric amplifier  202  to a back coupler  203 , and port  210   d  connects the second parametric amplifier to the back coupler  203 . The adjective “front” refers to the fact that the front coupler  201  provides the two external ports (an input port  200   a  and an output port  200   b ) that connect the directional amplifier  103  to external components. The adjective “back” refers to the fact that the back coupler  203  does not connect to any external components that send a signal to or receive a signal from the directional amplifier  103 . 
     The embodiment of directional amplifier  103  shown in  FIG. 2  comprises four main components: a first coupler  201  (referred to as a front coupler), a second coupler  203  (referred to as a back coupler), a parametric amplifier  202  and a second parametric amplifier  204 . In some embodiments, the front coupler  201  is a symmetric coupler that acts like a beamsplitter for microwave input signals. For example, front coupler  201  includes the input port  200   a  that is configured to receive the signal to be amplified, and may transmit or reflect an input signal to the ports  210   a  and  210   b , respectively. Similarly, while output port  200   b  is configured to output the amplified signal, a signal representing noise and/or reflections from external components subsequent to the directional amplifier  103  may be received by the directional amplifier  103  via output port  200   b  and may be transmitted or reflected to ports  210   b  and  210   a , respectively. Based on reciprocity, it is also possible that any signal received by ports  210   a  and  210   b  from other portions of the directional amplifier  103  can transmit or reflect the signal out either input port  200   a  or output port  200   b . Some embodiments are configured such that, for the entire system of the directional amplifier  103  receiving a signal via input port  200   a , the probability amplitude of a signal being output via input port  200   a  is substantially equal to zero. This reduction of the probability amplitude reduces feedback to the system being measured/amplified by the directional amplifier  103  (e.g., qubit  101 ). 
     The couplers  201  and  203  may have any suitable transmission and reflection amplitudes. For example, for a signal incoming to port  200   a  of the front coupler  201 , the “transmission amplitude” of the front coupler  201  represents the probability amplitude that the signal will leave via internal port  210   a , and the “reflection amplitude” of the front coupler  201  represents the probability amplitude that the signal will leave via internal port  210   b . In some embodiments, the front coupler  201  and the back coupler  203  may be symmetric couplers, meaning the amplitude of the transmission and reflection amplitudes are equal. For example, the transmission amplitude for a signal incoming to input port 1 of the front coupler  201  may be 1/sqrt(2) and the reflection amplitude for a signal incoming to input port 1 of the front coupler  201  may be i/sqrt(2), where i=sqrt(−1). The back coupler  203  is may have arbitrary transmission amplitude α and arbitrary reflection amplitude iβ, where α and β are both real numbers. In some embodiments, the back coupler may also be a symmetric coupler such that α=β=1/sqrt(2). Symmetric couplers are sometimes referred to as 3 dB couplers. However, embodiments are not limited to any particular transmission or reflection amplitudes for the front coupler  201  and the back coupler  203 . The back coupler  203  may be implements in any suitable way. For example, the back coupler may be used to introduce a lossy channel between the first parametric amplifier  202  and the second parametric amplifier  204 , in which case a lossy microwave transmission connection may be used instead of a hybrid coupler. 
     The parametric amplifiers  202  and  204  may be any suitable phase-preserving amplifier. For example, in some embodiments, a Josephson Parametric Converter (JPC) may be used. The JPC is a non-degenerate phase-preserving amplifier based on a ring of Josephson junctions, and is discussed in more detail in connection with  FIG. 3  and  FIG. 4  below. The two parametric amplifiers  202  and  204  utilize a coherent pump signal to perform the amplification via three-wave mixing. In some embodiments, the phase of the pump for each amplifier may be unequal, but held at a constant relationship with each other. For example, the difference between the two pump phases may be held constant, represented by the formula: Δφ=φ 1 −φ 2 =C, where Δφ is the difference between the pump phase φ 1  of the first amplifier  202  and the pump phase of the second amplifier φ z  of the second amplifier  204 . In some embodiments, Δφ=π/2, which increases the forward gain of the amplifier  103 . In some embodiments, the reflection coefficient for the overall device may be tuned to be substantially zero so that components, such as qubit  101 , are not disturbed by reflections from directional amplifier  103 . In other embodiments, the phase may be tuned such that a reverse gain through the directional amplifier  103  is negligible at the expense of having a non-zero reflection coefficient. Such an embodiment may be useful in application where reducing the reverse gain is important and where the corresponding non-zero reflection coefficient will not create problems for other components of the system. 
       FIG. 3  illustrates an example JPC  202  according to some embodiments. The central element of JPC  202  is a Josephson ring modulator  402 , which is discussed in more detail in connection with  FIG. 4 . Signals are input and output from the JPC  202  via ports  301 - 303 . Because JPC  202  is a parametric amplifier for microwave frequencies, the nomenclature of optical parametric amplifiers that implement three-wave mixing is adopted. Accordingly, port  301  corresponds to the “signal mode,” which may include, for example, microwave radiation with a small intensity, which will be amplified by the JPC  202 ; port  302  corresponds to the “idler mode,” which may be an empty mode with no microwave radiation present (i.e., a vacuum state); and port  303  corresponds to the “pump mode,” which may include, for example, microwave radiation with a much larger intensity than the intensity of the microwave radiation in the signal mode. The microwave radiation in the pump mode is what provides the energy to amplify the radiation in the signal mode. In some embodiments, port  301  of  FIG. 3  corresponds to port  200   a  of  FIG. 2  and port  302  of  FIG. 3  corresponds to port  200   b  of  FIG. 2 . In this way, the idler modes of the two JPCs  202  and  204  are connected via the back coupler  203 . 
     The JPC  202  includes two transmission line resonators that support one-half wave at the operational frequency, e.g., the length of the resonator is substantially equal to the length of one-half of the operational wavelength. Any suitable transmission line resonator may be used, such as, for example, stripline resonators. In some embodiments, JPC  202  performs non-degenerate amplification, which means the wavelength of the idler mode is different from the wavelength of the signal mode. In non-degenerate embodiments, the JPC  202  supports two fundamental modes of different frequency—a first frequency ω s  associated with a signal and a second frequency wt associated with an idler. The modes may be determined by the length of two half-wave microstrip resonators of the JPC. To apply gain, the ring is pumped with a coherent non-resonant pump P at the sum frequency ω p =ω s +ω i . In non-degenerate embodiments, the first transmission line resonator comprises portion  321  and portion  322  and supports the shorter wavelength idler mode than the second transmission line resonator, which comprises portion  323  and portion  324  and supports the longer wavelength signal mode. The two transmission line resonators cross each other at a voltage node, where the Josephson Ring Modulator (JRM) is disposed. 
     Each portion of the transmission line resonators is associated with a respective coupling capacitor: portion  321  of the first transmission line resonator is associated with coupling capacitor  331 , portion  322  of the first transmission line resonator is associated with coupling capacitor  332 , portion  323  of the second transmission line resonator is associated with coupling capacitor  333 , and portion  324  of the first transmission line resonator is associated with coupling capacitor  334 . Coupling capacitors  331  and  332 , associated with the idler mode, are coupled to the port  302  associated with the idler mode and coupling capacitor  333  is coupled to the port  301  associated with the signal mode. Coupling capacitor  334  is shorted to ground  343  or a 50Ω cold load. 
     In some embodiments, the idler mode and the pump mode are mixed at a 180 degree hybrid coupler  305 . The frequency of the pump radiation, which is equal to the sum of the frequency of the signal radiation and the frequency of the idler radiation, is not resonant with the JPC  202 . While not illustrated in  FIG. 3 , in some embodiments, instead of connecting coupling capacitor  334  to ground, the port  301  associated with the signal mode may be input into a hybrid coupler along with a 50Ω cold load, the two outputs of the hybrid coupler being connected to coupling capacitor  333  and coupling capacitor  334 . 
     The JPC  202  is based on a Josephson ring modulator  310  (JRM), which provides the nonlinearity that results in the three-wave mixing process that amplifies the radiation in the signal mode. The JRM  310  including at least four superconducting tunnel junctions (Josephson junctions), which is flux biased with a flux of Φ 0 /2, where Φ 0 =h/2e is the quantum flux The Josephson ring modulator  310  acts as a nonlinear medium and mixes the frequencies by converting pump photons into signal and idler photons. In particular, amplification of the signal is achieved via down-conversion of the pump P into microwave photons at the signal frequency ω s . The signal mode is well isolated from the pump and idler modes, keeping the signal free from noise from the pump and idler photons. The phase acquired during the transmission gain process of the JPC is non-reciprocal and depends on the phase of the pump P. 
       FIG. 4A  illustrates a schematic of a first JRM  400  according to some embodiments. The JRM  400  includes for input/output ports (1-4) and four Josephson junctions  410 - 413  in a Wheatstone bridge-like configuration. In some embodiments, the four Josephson junctions  410 - 413  have substantially the same properties.  FIG. 4B  illustrates a schematic of a second JRM  401  according to some embodiments. The JRM  401  includes for input/output ports (1-4), four Josephson junctions  410 - 413  in a Wheatstone bridge-like configuration, and four shunting Josephson junctions  420 - 423 . The shunting Josephson junctions  420 - 423  may be larger than the four Josephson junctions  410 - 413 . Including the shunting Josephson junctions  410 - 413  may increase the tenability of the JRM  401  relative to the unshunted JRM  400 . In bother JRM  400  and JRM  401 , the Josephson junctions  410 - 413  are what introduce the nonlinearity into the JPC  202 . 
     Though a particular type of JPC based on microstrip resonators is illustrated above, some embodiments may use other types of JPCs. For example, compact resonator JPCs, based on resonators created from capacitors and resonators rather than a microstrip of a particular length, and capacitively and inductively shunted JPCs, where the capacitive elements of the JPC are parallel plate capacitors and the inductance is governed primarily by the inductance of the Josephson junctions. Moreover, the parametric amplifiers  202  and  204  are not limited to JPCs at all. Any suitable phase-preserving amplifier may be used. 
     Whereas  FIG. 2  is a schematic illustrating the physical ports and connections of directional amplifier  103 ,  FIG. 5  is a schematic representation of directional amplifier  103  where each line represents a path that a wave may take through the system. For example, the input port 1 is associated with two lines: a first line representing a wave flowing into the directional amplifier  103  and a second line representing a wave flowing back out from the directional amplifier  103 . The lines in  FIG. 5  are marked with an arrows representing the direction of the wave flow associated with each line. 
       FIG. 5  illustrates the same main components of the directional amplifier  103  as well as the relevant variables associated with each component&#39;s effects on waves that enter the directional amplifier  103 . For example, the front coupler  201  is labeled with the reflection amplitudes and transmission amplitudes associated with each line. Because front coupler  201  is a symmetric (3 dB) coupler, the transmission amplitude for microwaves entering port 1 and exiting port 1′ and the transmission amplitude for microwaves entering port 1′ and exiting port 1 and the transmission amplitude for microwaves entering port 2 and exiting port 2′ and the transmission amplitudes for microwaves entering port 2′ and exiting port 2 are all equal to 1/sqrt(2). Whereas the reflection amplitude for microwaves entering port 1 and exiting port 2′, the reflection amplitude for microwaves entering port 2′ and exiting port 1, the reflection amplitude for microwaves entering port 2 and exiting port 1′, and the reflection amplitude for microwaves entering port 1′ and exiting port 2 are all equal to i/sqrt(2), where i=sqrt(−1). 
     Similarly, the back coupler  203  is associated with its own set of transmission amplitudes, α, and reflection amplitudes, iβ, where α and β are both real numbers. The transmission and reflection amplitudes of the back coupler  203  may be tuned to desired values as illustrated below. 
     The first parametric amplifier  202  and the second parametric amplifier  204  are each associated with a reflection gain amplitude and a transmission gain amplitude at the resonant frequency, which are is denoted as r and s, respectively. The reflection and transmission gain amplitudes satisfy the relation r 2 −s 2 =1. The gain amplitudes are complex numbers, the absolute value squared of the gain amplitudes representing the actual gain a signal will experience when reflected by or transmitted through the parametric amplifiers. Some embodiments, such as the one shown in  FIG. 5 , use the same r and s for both the first parametric amplifier  202  and the second parametric amplifier  204 , however, embodiments are not limited to having the same gain amplitudes. Other embodiments may use different gains for the two amplifiers. Any suitable value of r and s may be used. In some embodiments, the reflection gain amplitude r is limited to be within the range 1≤r&lt;α −1 . A reflection gain amplitude within this range may result in a more stable feedback loop within the directional amplifier  103 . 
     The first parametric amplifier  202  and the second parametric amplifier  204  are also each associated with a phase that is determined by the phase of the pump used to drive the amplifiers. The first parametric amplifier  202  has a first phase, φ 1 , and the second parametric amplifier  202  has a first phase, φ 2 . The difference between the first phase the second phase may be set to any value. For example, in some embodiments, the difference between the first phase the second phase is equal to pi/2 radians (or 90 degrees). i.e., φ 1 −φ 2 =π/2. 
     There are four paths through the directional amplifier  103  illustrated in  FIG. 5  that ultimately lead from input port 1 to output port 2, that will be illustrated in connection with  FIG. 6A-D . Each figure shows the same directional amplifier  103  as illustrated in  FIG. 5 , but with arrows highlighting a particular path through the amplifier  103 . 
     In the first path, illustrated in  FIG. 6A , a signal incoming to port 1 is transmitted with transmission amplitude 1/sqrt(2) through the front coupler  201  to port 1′. Upon reaching the first parametric amplifier  202 , the signal is reflected, with a gain amplitude r, back to the front coupler  201 . At the front coupler  201 , the signal is reflected with reflection amplitude i/sqrt(2) to the output port 2, where the signal exits the device  103 . 
     In the second path, illustrated in  FIG. 6B , a signal incoming to port 1 is reflected with reflection amplitude i/sqrt(2) in the front coupler  201  to port 2′. Upon reaching the second parametric amplifier  204 , the signal is reflected with a gain amplitude r back to the front coupler  201 . At the front coupler  201 , the signal is transmitted with transmission amplitude 1/sqrt(2) to the output port 2, where the signal exits the device  103 . 
     In the third path, illustrated in  FIG. 6C , a signal incoming to port 1 is transmitted with transmission amplitude 1/sqrt(2) through the front coupler  201  to port 1′. Upon reaching the first parametric amplifier  202 , the signal is transmitted with a gain amplitude s and a phase φ 1  to the second parametric amplifier  204 . At the second parametric amplifier  204 , the signal is transmitted with a gain amplitude of s and a phase of φ 2  to the front coupler. At the front coupler  201 , the signal is transmitted with transmission amplitude 1/sqrt(2) to the output port 2, where the signal exits the device  103 . 
     In the fourth path, illustrated in  FIG. 6D , a signal incoming to port 1 is reflected with reflection amplitude i/sqrt(2) in the front coupler  201  to port 2′. Upon reaching the second parametric amplifier  204 , the signal is transmitted with a gain amplitude s and a phase φ 2  to the first parametric amplifier  202 . At the first parametric amplifier  202 , the signal is transmitted with a gain amplitude of s and a phase of φ 1  to the front coupler. At the front coupler  201 , the signal is reflected with reflection amplitude 1/sqrt(2) to the output port 2, where the signal exits the device  103 . 
     These four paths add coherently such that the paths interfere with one another, resulting in the desired amplification of the received microwave signal. It is possible to express the effect of the directional amplifier  103  in terms of the various parameters of the components of the directional amplifier  103  using a “scattering matrix,” which maps how signal input into any of the four ports of the device is transformed by the directional amplifier  103  into an output signal that is output from the four ports. Thus, a scattering matrix S is defined as: 
     
       
         
           
             
               
                 
                   
                     [ 
                     S 
                     ] 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             S 
                             11 
                           
                         
                         
                           
                             S 
                             12 
                           
                         
                         
                           
                             S 
                             13 
                           
                         
                         
                           
                             S 
                             14 
                           
                         
                       
                       
                         
                           
                             S 
                             21 
                           
                         
                         
                           
                             S 
                             22 
                           
                         
                         
                           
                             S 
                             23 
                           
                         
                         
                           
                             S 
                             24 
                           
                         
                       
                       
                         
                           
                             S 
                             31 
                           
                         
                         
                           
                             S 
                             32 
                           
                         
                         
                           
                             S 
                             33 
                           
                         
                         
                           
                             S 
                             34 
                           
                         
                       
                       
                         
                           
                             S 
                             41 
                           
                         
                         
                           
                             S 
                             42 
                           
                         
                         
                           
                             S 
                             43 
                           
                         
                         
                           
                             S 
                             44 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     1 
                   
                   ) 
                 
               
             
           
         
       
     
     where S ij  represents the scattering amplitude for a signal input into port j and output from port i. In some embodiments, when all the components of the directional amplifier  103  are taken into account, the scattering matrix simplifies into the form illustrated below. In said embodiments, the back coupler&#39;s transmission and reflection amplitudes are set such that α=β=1/sqrt(2) and the phase difference between the two parametric amplifiers is set such that Δφ=π/2. In this case, the scattering matrix of the directional amplifier  103  simplifies to 
     
       
         
           
             
               
                 
                   
                     [ 
                     S 
                     ] 
                   
                   = 
                   
                       
                     
                       ( 
                       
                         
                           
                             0 
                           
                           
                             
                               i 
                                
                               
                                 H 
                               
                             
                           
                           
                             
                               
                                 
                                   H 
                                   - 
                                   1 
                                 
                                 2 
                               
                             
                           
                           
                             
                               - 
                               
                                 
                                   
                                     H 
                                     - 
                                     1 
                                   
                                   2 
                                 
                               
                             
                           
                         
                         
                           
                             
                               i 
                                
                               
                                 G 
                               
                             
                           
                           
                             0 
                           
                           
                             
                               i 
                                
                               
                                 
                                   
                                     G 
                                     - 
                                     1 
                                   
                                   2 
                                 
                               
                             
                           
                           
                             
                               i 
                                
                               
                                 
                                   
                                     G 
                                     - 
                                     1 
                                   
                                   2 
                                 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 
                                   
                                     G 
                                     - 
                                     1 
                                   
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 i 
                               
                                
                               
                                 
                                   
                                     H 
                                     - 
                                     1 
                                   
                                   2 
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 
                                   
                                     G 
                                   
                                   + 
                                   
                                     H 
                                   
                                 
                                 2 
                               
                             
                           
                           
                             
                               
                                 
                                   G 
                                 
                                 - 
                                 
                                   H 
                                 
                               
                               2 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 
                                   
                                     G 
                                     - 
                                     1 
                                   
                                   2 
                                 
                               
                             
                           
                           
                             
                               i 
                                
                               
                                 
                                   
                                     H 
                                     - 
                                     1 
                                   
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   G 
                                 
                                 - 
                                 
                                   H 
                                 
                               
                               2 
                             
                           
                           
                             
                               - 
                               
                                 
                                   
                                     G 
                                   
                                   + 
                                   
                                     H 
                                   
                                 
                                 2 
                               
                             
                           
                         
                       
                        
                       
                           
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
     where the following parameter definitions are used: 
     
       
         
           
             
               
                 
                   
                     
                       g 
                       + 
                       h 
                     
                     = 
                     
                       G 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     3 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       g 
                       - 
                       h 
                     
                     = 
                     
                       H 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     4 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     g 
                     = 
                     
                       
                         
                           1 
                           + 
                           
                             s 
                             2 
                           
                         
                       
                       
                         1 
                         - 
                         
                           s 
                           2 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     5 
                   
                   ) 
                 
               
             
             
               
                 and 
               
               
                 
                     
                 
               
             
             
               
                 
                   h 
                   = 
                   
                     
                       
                         
                           2 
                         
                          
                         
                           s 
                           2 
                         
                       
                       
                         1 
                         - 
                         
                           s 
                           2 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     6 
                   
                   ) 
                 
               
             
           
         
       
     
     The above Equations 2-6 illustrate that, when the front coupler and back coupler amplitudes are fixed and the phase difference is fixed, the entire scattering matrix of the directional amplifier  103  may be written such that the only parameter is the transmission gain amplitude s of the two parametric amplifiers. Thus, it is possible to determine how the directional amplifier  103  behaves when the device is “off” by setting s=0. This may be done by turning off the pump signal or otherwise preventing the pump signal from entering the parametric amplifiers  202  and  204 . When the directional amplifier  103  is off, the scattering matrix reduces to: 
     
       
         
           
             
               
                 
                   
                     [ 
                     S 
                     ] 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             0 
                           
                           
                             i 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             i 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               - 
                               1 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               - 
                               1 
                             
                           
                         
                       
                       ) 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     7 
                   
                   ) 
                 
               
             
           
         
       
     
     The scattering matrix of Equation 7 indicates that when the directional amplifier  103  is turned off, the transmission amplitude from the input port 1 to the output port 2 is unity. Accordingly, the device connected to port 1 from which the signal is obtained (e.g., a superconducting qubit  101 ), may be measured using alternative measurement means without disconnecting the directional amplifier  103  or using a switch to switch between the directional amplifier and the alternative measurement means. 
     In the “high gain limit,” where the parametric amplifiers are pumped with as much gain as physically possible, s=1. Using Equations 5 and 6, the limit from below of the scattering matrix may be calculated to determine how the directional amplifier acts in this high gain limit as s approaches unity. In the high gain limit: 
     
       
         
           
             
               
                 
                   
                     G 
                   
                   = 
                   ∞ 
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     8 
                   
                   ) 
                 
               
             
             
               
                 and 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     H 
                   
                   = 
                   
                     3 
                     
                       2 
                        
                       
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     9 
                   
                   ) 
                 
               
             
           
         
       
     
     Thus, the scattering matrix for the directional amplifier becomes: 
     
       
         
           
             
               
                 
                   
                     [ 
                     S 
                     ] 
                   
                   = 
                   
                       
                     
                       ( 
                       
                         
                           
                             0 
                           
                           
                             
                               i 
                                
                               
                                 
                                   9 
                                   8 
                                 
                               
                             
                           
                           
                             
                               i 
                               4 
                             
                           
                           
                             
                               - 
                               
                                 i 
                                 4 
                               
                             
                           
                         
                         
                           
                             
                               i 
                                
                               
                                 G 
                               
                             
                           
                           
                             0 
                           
                           
                             
                               
                                 
                                   
                                     
                                       G 
                                       - 
                                       1 
                                     
                                   
                                 
                                 
                                   
                                     2 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       G 
                                       - 
                                       1 
                                     
                                   
                                 
                                 
                                   
                                     2 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 i 
                               
                                
                               
                                 
                                   
                                     G 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                   2 
                                 
                               
                             
                           
                           
                             
                               1 
                               4 
                             
                           
                           
                             
                               - 
                               
                                 
                                   
                                     G 
                                   
                                   + 
                                   
                                     
                                       9 
                                       8 
                                     
                                   
                                 
                                 2 
                               
                             
                           
                           
                             
                               
                                 
                                   G 
                                 
                                 - 
                                 
                                   
                                     9 
                                     8 
                                   
                                 
                               
                               2 
                             
                           
                         
                         
                           
                             
                               i 
                                
                               
                                 
                                   
                                     G 
                                     - 
                                     1 
                                   
                                   2 
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 1 
                                 4 
                               
                             
                           
                           
                             
                               
                                 
                                   G 
                                 
                                 - 
                                 
                                   
                                     9 
                                     8 
                                   
                                 
                               
                               2 
                             
                           
                           
                             
                               - 
                               
                                 
                                   
                                     G 
                                   
                                   + 
                                   
                                     
                                       9 
                                       8 
                                     
                                   
                                 
                                 2 
                               
                             
                           
                         
                       
                        
                       
                           
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     7 
                   
                   ) 
                 
               
             
           
         
       
     
     Thus, in the high gain limit, the theoretical gain for a signal input via port 1 of directional amplifier  103  is infinite while the reverse gain of the device is limited to |S 12 | 2 =9/8, which is very near unity. Accordingly, in some embodiments, the directional amplifier  103  has potentially unlimited forward gain while limiting the amount of reverse gain through the device. Also, input port 1 and output port 2 are perfectly matched such that S 11 =S 2 2=0, indicating that there is no reflection signal even at the high gain limit. 
     The above embodiments thus illustrate an embodiment of a non-reciprocal directional amplifier capable of quantum-limited operation with, in principle, no limitation on the forward gain and 9/8 limit on the reverse gain. 
     In other embodiments, it may be desirable to reduce the reverse gain (e.g., element S 21  of the scattering matrix) through the directional amplifier  103 . Accordingly, the scattering matrix may be tuned such that a reverse gain through the directional amplifier  103  is negligible at the expense of having a non-zero reflection coefficient. Such an embodiment may be useful in application where reducing the reverse gain is important and where the corresponding non-zero reflection coefficient will not create problems for other components of the system. 
       FIG. 7  illustrates a method  700  of amplifying a signal according to some embodiments. At act  702 , a microwave signal is received at an input port of a directional amplifier such as the directional amplifier according to some of the embodiments illustrated above. At act  704 , the received microwave signal is split at a hybrid coupler into two portions. The signal may further be split into additional portions at a first parametric amplifier and/or a second parametric amplifier based on whether the respective amplifier reflects or transmits the received signal. 
     At act  706 , at least a portion of the signal is amplified by both the first parametric amplifier and the second parametric amplifier. In some embodiments, the first parametric amplifier is pumped using a first microwave pump with a first phase and the second parametric amplifier is pumped using a second microwave pump with a second phase. In some embodiments, the difference between the first phase and the second phase is pi/2 radians. 
     While embodiments of the low-noise directional amplifier are illustrated above as being used to measure a superconducting qubit, embodiments may be used in a variety of applications. For example, embodiments may be used to initialize and/or perform quantum gate operations on superconducting qubits. 
     While embodiments of the low-noise directional amplifier may be used to measure the state of a superconducting qubit, embodiments may be used in a variety of applications. For example, embodiments may be used to initialize and/or perform quantum gate operations on superconducting qubits. Embodiments may also have applications outside of quantum information processing. For example, embodiments may be used in any situation where detection of low intensity microwave radiation is being performed, such as microwave telescopes used in astronomy or the detection of radar signals. 
     Having thus described and illustrated several aspects of at least one embodiment of a low-noise directional amplifier it is to be appreciated that various alterations, modifications, and improvements will readily occur to those skilled in the art. Such alterations, modifications, and improvements are intended to be part of this disclosure, and are intended to be within the spirit and scope of the invention. While the present teachings have been illustrated in conjunction with various embodiments and examples, it is not intended that the present teachings be limited to such embodiments or examples. On the contrary, the present teachings encompass various alternatives, modifications, and equivalents, as will be appreciated by those of skill in the art. Those skilled in the art will readily appreciate that all parameters, dimensions, materials, and configurations described are provided as non-limiting examples and that the actual parameters, dimensions, materials, and/or configurations will depend upon the specific application or applications for which the inventive teachings is/are used. Those skilled in the art will also recognize, or be able to ascertain using no more than routine experimentation, many equivalents to the specific inventive embodiments described. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the invention, inventive embodiments may be practiced otherwise than as specifically described and claimed. Inventive embodiments of the present disclosure may be directed to each individual feature, system, system upgrade, and/or method described. In addition, any combination of two or more such features, systems, and/or methods, if such features, systems, system upgrade, and/or methods are not mutually inconsistent, is included within the inventive scope of the present disclosure. 
     Further, though some advantages of the described embodiments may be indicated, it should be appreciated that not every embodiment will include every described advantage. Some embodiments may not implement any features described as advantageous. Accordingly, the foregoing description and drawings are by way of example only. 
     The indefinite articles “a” and “an” as used herein, unless clearly indicated to the contrary, should be understood to mean “at least one.” 
     The phrase “and/or,” as used herein, should be understood to mean “either or both” of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with “and/or” should be construed in the same fashion, i.e., “one or more” of the elements so conjoined. Other elements may optionally be present other than the elements specifically identified by the “and/or” clause, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, a reference to “A and/or B”, when used in conjunction with open-ended language such as “comprising” can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc. 
     As used herein, “or” should be understood to have the same meaning as “and/or” as defined above. For example, when separating items in a list, “or” or “and/or” shall be interpreted as being inclusive, i.e., the inclusion of at least one, but also including more than one, of a number or list of elements, and, optionally, additional unlisted items. Only terms clearly indicated to the contrary, such as “only one of” or “exactly one of,” will refer to the inclusion of exactly one element of a number or list of elements. In general, the term “or” as used shall only be interpreted as indicating exclusive alternatives (i.e. “one or the other but not both”) when preceded by terms of exclusivity, such as “either,” “one of” “only one of,” or “exactly one of.” 
     As used herein, the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, “at least one of A and B” (or, equivalently, “at least one of A or B.” or, equivalently “at least one of A and/or B”) can refer, in one embodiment, to at least one, optionally including more than one, A, with no B present (and optionally including elements other than B); in another embodiment, to at least one, optionally including more than one, B, with no A present (and optionally including elements other than A); in yet another embodiment, to at least one, optionally including more than one, A, and at least one, optionally including more than one, B (and optionally including other elements); etc. 
     All transitional phrases such as “comprising,” “including,” “carrying,” “having,” “containing,” “involving,” “holding,” “composed of,” and the like are to be understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases “consisting of” and “consisting essentially of” shall be closed or semi-closed transitional phrases, respectively.