1. Introduction
The squeezing state and its applications have been developed recently[1-3]. It has been shown that squeezing originates from nonlinearity effects in any system[1-7]. Different approaches and systems have been employed to generate squeezed states[3-5]. For instance, phase conjugate mirrors using four-wave mixing interactions have been applied to create a squeezed state[1, 8]. Another important option is using a parametric amplifier, in which three-wave mixing is used to generate the squeezed state[5, 7]. In addition, by controlling the spontaneous emission, a two-photon laser is applied to produce a squeezed state[1]. Moreover, atomic interaction with an optical wave can produce a nonlinear medium, creating a squeezed state. Additionally, other phenomena, such as third-order nonlinearity of the wave propagation in the optical fiber, can generate a squeezed state[7]. In quantum applications, the squeezed state is very important because it introduces less fluctuation in one quadrature phase than the coherent state, which is very similar to the classical state[9-13]. Quantum fluctuation in a coherent state is equal to zero-point fluctuation, in which the standard quantum limits noise reduction in a signal[1, 14]. Therefore, the noise fluctuation can be reduced below the standard limit when the system is in a squeezed state[14]. There is no classical analog for the squeezed state, and this state, in contrast to the coherent state that shows Poisson photon counting (photon bunching) statistics, may show sub-Poisson photon counting (photon antibunching)[1, 2, 13, 14]. In other words, there is no direct connection between squeezing and photon antibunching, but each is a nonclassical phenomenon[1, 3, 13]. For some quantum applications, such as quantum radar and quantum sensors[15-20], the noise effect is critical when the system tries to detect low-level backscattering signals. For signal detection, the received signals, which have minimal levels, can be easily affected by noise. Therefore, to control and limit the noise, it is necessary to prepare the key subsystems, such as the low-noise amplifier (LNA) and detector, to operate in the squeezed state by which the noise can be reduced below the zero-point fluctuation. The LNA is an electronic amplifier generally designed to amplify low-level signals while simultaneously keeping the noise at a very low level[21-24]. Today, a cryogenic LNA has been designed to operate at very low temperatures around 5 K, to strongly limit noise. So due to this fact, cryogenic LNAs are so popular in quantum applications[24-29].
With the knowledge of the points mentioned above, in this work, we attempt to design a circuit containing two external oscillators coupled to an InP high electron mobility transistor (HEMT) operating at cryogenic temperature to create the squeezed states. This type of transistor was selected because HEMT technology does not have a strong effect from freeze-out at a cryogenic temperature[24-27]. The designed circuit can be considered a core circuit for an LNA used in front-end transceivers to amplify faint signals. In this study, the nonlinear properties of the cryogenic InP HEMT play a key role, and we discuss how emerging nonlinearity can affect the state of the coupling oscillators. Additionally, a critical point will be addressed, which relates to the trade-off between squeezing generated by the nonlinear properties of the transistor and the degradation of the produced state because of the damping created by the transistor’s internal circuits.
2. Theoretical and backgrounds
2.1 System description
The circuit is schematically shown in Fig. 1, which shows two LC oscillators (resonators) coupled to each other through a nonlinear device (depicted in the inset figure). As mentioned in the previous section, the main goal is to create a squeezing state in a low-noise amplifier (LNA), which is essential in quantum sensing applications[15, 16, 25]. In fact, if such a circuit is prepared in the squeezing state, it helps minimize the noise effect. This implies that the performance of the cryogenic LNA, at which the noise strongly limits the operation, is enhanced. Therefore, the circuit shown in Fig. 1 is designed. In this circuit, the transistor is used as a nonlinear element, and the state of the oscillators may exhibit squeezing. It has been theoretically shown that nonlinearity arises because the transistor can be expressed as a nonlinear capacitor, inductor coupling to the second oscillator, and second-order transconductance (gm2_N). These factors are defined in detail in the next section and can intensely manipulate the state of the oscillators to create squeezing. Additionally, Fig. 1 schematically reveals that only the second oscillator can generate squeezing in the state; this important point will be discussed in detail later. Nonetheless, we theoretically demonstrate that coupling oscillators can generate two-mode squeezing.
The high-frequency model of an InP HEMT transistor[25, 27] coupled with two oscillators is shown in Fig. 2. Some elements in the nonlinear transistor model are created owing to the high-frequency effect, such as Cgs, Cgd, Lg, and Ld and some other components, such as resistors, as shown in Fig. 1, are created to address the thermal loss in the circuit. These elements are sources of thermal noise in the transistor, and their effects are shown as voltage and current noise sources in Fig. 2. In fact, Īg2, Īj2, Īd2, and Ṽi2 model the thermal noise of Rg, Rgd, Rds, and Rgs, respectively. A noise model is applied to the circuit to demonstrate the effects of the contributed resistors. Additionally, we ignored the Ls effect and merged Rs with Rgs to simplify algebra. In the circuit, ids and Īds2 are the dependent current sources containing the transistor’s nonlinearity and related thermal noise, which is a critical factor in generating noise. Finally, Cf, Cin, VRF, φ1, and φ2 are the feedback capacitor and coupling capacitors used to separate the input signal from the DC signals, input signal, and node flux for the input and output nodes, respectively. This circuit is wholly analyzed using quantum theory, and we will attempt to derive the contributed Lagrangian initially; then, the total Hamiltonian of the system is examined, in which the factors that cause the squeezing in the state will be addressed.
2.2 Designed system Hamiltonian
This section analyzes the circuit shown in Fig. 2 using the full quantum theory. As shown in the circuit, it includes all noises that can affect the signals, such as the thermal noises generated by the dependent current source and resistors in the circuit. The data for the nonlinear model of the InP HEMT operating at cryogenic temperatures are listed in Table 1. First, we theoretically derive the total Lagrangian of the system to obtain the quantum properties of the circuit illustrated in Fig. 2. For the analysis, the coordinate variables are defined as φ1 and φ2 (node flux), as shown in Fig. 2, and the momentum conjugate variables are defined by Q1 and Q2 (loop charge). The total Lagrangian of the circuit is derived as[29]:
Stands for | Value | |
Rg | Gate resistance | 0.3 Ω |
Lg | Gate inductance | 75 pH |
Ld | Drain inductance | 70 pH |
Cgs | Gate–source capacitance | 69 fF |
Cds | Drain–source capacitance | 29 fF |
Cgd | Gate–drain capacitance | 19 fF |
Rgs | Gate-source resistance | 4 Ω |
Rgd | Gate–drain resistance | 35 Ω |
Lc=Cin2(˙φ1−VRF)2+C12˙φ21−12L1φ12+¯I2gφ1+Cgs2(˙φ1−¯V2i)2+Cgd+Cf2(˙φ1−˙φ2)2+idsφ1+(¯I2ds+¯I2d)φ2+C22˙φ22−12L2φ22+¯I2j(φ2−φ1). |
(1) |
In this equation, the dependent current source is defined as ids = gmVgs + gm2Vgs2 + gm3Vgs3 [22,30], where gm is the intrinsic transconductance of the transistor and gm2 and gm3 are the second-and third-order transconductance. These terms (gm2, gm3) bring nonlinearity to the circuit. Moreover, thermally generated noises by the resistors and the current source are defined as ˉI2g=4KBT/Rg, ˉI2d=4KBT/Rd, ˉI2j=4KBT/Rj, ˉI2ds=4KBTγgm, and ˉI2i=4KBT/Ri, where KB, T, and γ respectively are the Boltzmann constant and operational temperature, and empirical constant[29]. The noise bandwidth is supposed to be very wide (1 Hz). Using the Legendre transformation[14, 29], one can theoretically derive the classical Hamiltonian of the circuit. For this, it is necessary to calculate the momentum conjugate variables using Qi=∂Lc/∂(∂φi/∂t) for i = 1, 2 represented as:
Q1=(Cin+C1+Cgs+Cf+Cgd)˙φ1−(Cf+Cgd)˙φ2−CinVRF+gmφ2+2gm2φ2˙φ1+3gm3φ2˙φ21−Cgs¯V2i,Q2=(C2+Cf+Cgd)˙φ2−(Cf+Cgd)˙φ1. |
(2) |
Applying Legendre transformation, the classical Hamiltonian is expressed as:
Hc={CA2˙φ21+12L1φ21+CB2˙φ22+12L2φ22}+{−Cc˙φ1˙φ2+gm2φ2˙φ21+2gm3φ2˙φ31}+{−φ1(¯I2g−¯I2j)−φ2(¯I2d+¯I2j+¯I2ds)−Cgs2¯V2i}, |
(3) |
where CA = Cin + C1 + Cgs + Cf + Cgd, CB = Cgd + Cf + C2, and CC = Cf + Cgd. In Eq. (3), the first term relates to the LC resonance Hamiltonian affected by the coupling elements. It is clearly shown that CA and CB are affected due to the transistor’s internal circuit. The second term contributes to the linear and nonlinear coupling between the LC resonators and the nonlinear circuit. Finally, the third term defines the noise effect in the system, which is originally generated by the transistor nonlinearity in the circuit. In the following, using Eq. (2), one can express the first derivative of the coordinate variables (∂φi/∂t) in terms of the momentum conjugates (Qi) represented in the matrix form as:
[˙φ1˙φ2]=1CM2{[CBCCCCCA+CN][Q1Q2]+[CBCCCCCA+CN][Cin000][VRF0]−[CBCCCCCA+CN][0gm00][φ1φ2]}, |
(4) |
where CM2 = CB(CA + CN) – CC2 and CN = 2gm2φ2_dc + 6gm3φ2_dc(∂φi/∂t)|dc is the capacitor generated due to the nonlinearity effect. In the following, we will show that this quantity strongly affects the coupled LC resonator’s frequency and impedances, and consequently, the quantum properties of the LC resonators are severely influenced by CN. By substitution of Eq. (4) into Eq. (3) one can derive the total Hamiltonian for the system; however, to study the design in detail and get to know about each factor’s impact on the system, we initially use the linearization technique to linearize the nonlinear terms in the second term of Eq. (3). Thus, the linear Hamiltonian of the system is defined as:
HL={12Cq1Q21+12L1φ21+12Cq2Q22+12L′2φ22}+{12Cq1q2Q1Q2+g12Q1φ2+g22Q2φ2}+{Vq1Q1+Vq2Q2+Ip2φ2−¯I2gsφ1}, |
(5) |
where Cq1, Cq2, Cq1q2, g12, g22, L2’, Vq1, Vq2, and Ip2 are defined in Appendix A, and the dc terms are ignored for simplicity. In fact, these coefficients are essentially the function of gm, CN, CC, and VRF by which the nonlinearity effect of the transistor is emphasized. In other words, the nonlinearity created by the transistor induces some factors by which the properties of the coupling LC resonators are strongly affected. For instance, the LC resonator impedances are Z1 = (L1/Cq1)0.5, Z2 = (L′2/Cq2)0.5, and the associated frequencies are ω1 = (L1Cq1)−0.5, ω2 = (L′2/Cq2)−0.5. The relations show that the coupling oscillator’s impedance and frequencies, especially the second LC become affected. Additionally, some terms in Eq. (2), such as Q1Q2, Q2φ1, and Q1φ2 show the coupling between oscillators in the circuit design. Also, in the third term in Eq. (5), some terms such as Vq1Q1, ˉI2gsφ1, Vg2Q2, and Ip2φ2 in the equation declare the RF source and thermal noise coupling to the contributed oscillators. In this equation, ˉI2gs=ˉI2g−ˉI2j. In the following, one can derive the linear Hamiltonian in terms of annihilation and creation operators using the quantization procedure for the coordinates and the related momentum conjugates. The quadrature operators defined as Q1 = –i(a1 – a1+)(ħ/2Z1)0.5, φ1 = (a1 + a1+)(ħZ1/2)0.5 and Q2 = –i(a2 – a2+)(ħ/2Z2)0.5, φ2 = (a2 + a2+)(ħZ2/2)0.5, where (ai, ai+) i = 1, 2 are the first and second oscillator’s annihilation and creation operators. Thus, the linear Hamiltonian in terms of the ladder operators is given by:
HL={ℏω1(a+1a1+12)+ℏω2(a+2a2+12)}+{−ℏ21Cq1q2√Z1Z2(a1−a1+)(a2−a+2)−iℏ2g12√Z2Z1(a1−a+1)(a2+a+2)−iℏ2g22(a2−a+2)(a2+a+2)}+{−iVq1√ℏ2Z1(a1−a+1)−iVq2√ℏ2Z2(a2−a+2)+Ip2√ℏZ22(a2+a+2)−¯I2gs√ℏ2Z2(a1+a+1)}. |
(6) |
In the following, it is necessary to add the nonlinearity to the Hamiltonian and derive the total Hamiltonian containing the linear and nonlinear parts. The nonlinear terms in Eq. (3) can be re-written as:
HN={gm2+6gm3˙φ1_dc}φ2˙φ12. |
(7) |
Using Eq. (4), the nonlinear Hamiltonian is given by:
HN=gm2_N[{C2BC4Mφ2Q21+C2CC4Mφ2Q22+2CBCCC4Mφ2Q2Q1−2gmC2BC4Mφ22Q1−2gmCBCCCM4φ22Q2+g2mC2BCM4φ32}NL×{−2gmC2BCinVRFC4Mφ22+2C2BCinVRFC4MQ1φ2+2CBCCCinVRFC4MQ2φ2+C2BV2inV2RFC4Mφ2}L], |
(8) |
where gm2_N = gm2 + 6gm3 (∂φi/∂t)|dc. The linear part of Eq. (8) can directly attach to Eq. (5) to make the modified linear Hamiltonian given by:
HL={12Cq1Q21+12L1φ21+12Cq2Q22+(12L′2+12L2N)φ22}+{12Cq1q2Q1Q2+(g12+g12N)Q1φ2+(g22+g22N)Q2φ2}+{Vq1Q1+Vq2Q2+(Ip2+Ip2N)φ2−¯I2gsφ1}. |
(9) |
As can be clearly seen in Eq. (9), the nonlinear Hamiltonian can change the coupling between oscillators in the circuit. The most important factor is gm2_N, which manipulates the coupling between different coordinates and their momentum conjugates. Additionally, the attachment from Eq. (8) to Eq. (5) changes the second resonator’s inductance by a factor of L2N. The term brought from nonlinearity (L2N) manipulates the second resonator’s impedance and frequency. Finally, the nonlinear terms of Eq. (8) are considered, and so, the total Hamiltonian of the system in terms of the ladder operators is given by:
Ht=[{ℏω1(a+1a1+12)+ℏω2(a+2a2+12)}+{−ℏ21Cq1q2√Z1Z2(a1−a+1)(a2−a+2)−iℏ2g′12√Z2Z1(a1−a+1)(a2+a+2)−iℏ2g′22(a2−a+2)(a2+a+2)}+{−iVq1√ℏ2Z1(a1−a+1)−iVq2√ℏ2Z2(a2−a+2)+I′p2√ℏZ22(a2+a+2)−¯I2gs√ℏ2Z2(a1+a+1)}]L+[−ℏg13(a1−a+1)2(a2+a+2)+ℏg14(a2+a+2)(a2−a+2)2−ℏg15(a1−a+1)(a2−a+2)(a2+a+2)+ℏg16(a2+a+2)3+iℏg17(a1−a+1)(a2+a+2)2+iℏg18(a2−a+2)(a2+a+2)2]NL, |
(10) |
where IP2’ = IP2 + IP2N, g12’ = g12 + g12N, and g22’ = g22 + g22N. Also, g13 = (1/2Z1)(√(ℏ/2Z2))gm2_NCB2/CM4, g14 = (1/2Z2)(√(ℏ/2Z2))gm2_NCC2/CM4, g15 = (1/√(Z2Z1))(√(ℏ/2Z2))gm2_NCBCC/CM4, g16 = (Z2/2)(√(ℏZ2/2))gm2gm2_NCB2/CM4, g17 = Z2(√(ℏ/2Z1))gmgm2_NCB2/CM4, and g18 = Z2(√(ℏ/2Z2))gmgm2_NCBCC/CM4. It is clearly shown in coefficients from g13 to g18 where the effect of gm2_N is dominant. In other words, the system’s nonlinearity in this work is strongly changed and controlled by the second-order transconductance gm2_N. Now, one can show using the total Hamiltonian of the system, which terms in the presented Hamiltonian in Eq. (10) can generate the squeezing state.
3. Results and discussions
A squeezed-coherent state is generally produced by the act of the squeezing and displacement operators on the vacuum state defined mathematically as |α,ζ > = D(α)S(ζ) |0>, where |0> is the vacuum state[14]. It is found that the coherent state is generated by the linear terms in the Hamiltonian, whereas a squeezed state needs quadratic terms such as a2 and a+2 in the exponent. The squeezed-coherent state S(ζ,α) can be analyzed as the evolution exp[Ht/jħ] under the Hamiltonian defined in Eq. (10). Based on this definition, any quadratic terms such as a2 and a+2 in the Hamiltonian may generate squeezing. The Hamiltonian in Eq. (10), the squeezed state, can be generated by:
S(ζ)=exp[ζ1(a22−a+22)+(ζ∗22a22−ζ22a+22)]t. |
(11) |
In this equation the squeezing parameters are defined as ζ1 = –0.5g22’ + g18Re{A2} + jg14Im{A2} and ζ2 =2A1*(–g17–jg15), where A1 and A2 are the strong fields (DC points) of LC1 and LC2. The DC points can be calculated using Heisenberg–Langevin equations in the steady-state[15, 16]. Also, Re{} and Im{} indicate the real and imaginary parts, respectively. Eq. (11) clearly shows that the squeezing is generated just for LC2 and does not happen for LC1. This point contributes to the nonlinearity terms in Hamiltonian expressed in Eq. (10) and also is related to the dependent current source containing gm2 and gm3, which is directly connected to LC2. The important point about the squeezing strength parameters ζ1 and ζ2 is that each depends on gm2_N. In Eq. (11), ζ1 and ζ2 are complex numbers, meaning that the squeezing parameters contain the phase which determines the angle of the quadrature to be squeezed. Additionally, we found that the system can generate two-mode squeezing. That means that the nonlinearity created by the transistor couples two oscillators so that the coupled modes become squeezed. The expression generated due to the Hamiltonian of the system for two-mode squeezing is expressed as:
S2(ζ)=exp[(ζ∗t12a1a2−ζt12a+1a+2)+(ζ∗t22a1a2−ζt22a+1a+2)]t, |
(12) |
where ζt1 = jA2g15 and ζt2 = jA1g13. In the same way, the squeezing parameters are strongly dependent on gm2_N. Finally, one can easily find that the system can generate the coherent state, which means that the state generated by the Hamiltonian expressed in Eq. (10) is a squeezed-coherent state or a two-mode squeezed-coherent state. In the following, we just focus on the squeezed-coherent state and study the parameters that can manipulate the squeezing states. For simulation, the limit for time evolution in the exponent (exp[Ht/jħ]) is defined as t0 < {1/κ1, 1/κ2}, where κ1 and κ2 are the first and second oscillators’ decay rates. By selecting t0, the system is forced to generate squeezing before the resonator decaying, by which the squeezing is destroyed[1]. Some more information is introduced about t0 in Appendix B.
In this study, quadrature variance is used to demonstrate the behavior of the state generated by the oscillators. In addition, we used the bunching and antibunching behavior of the generated photons. The second-order correlation function, g2(τ), must be calculated. For the designed system, concerning the fact that t0 limits the system, the photon counting time is sufficiently short. Thus, for such a short counting time, the variance of the photon number distribution is related to the second-order correlation function g2(τ = 0) = 1 + (V(n) – <n>)/<n>, where V(n) and <n> are the photon number variance and the average, respectively[1, 13]. It has been shown that this phenomenon is called anti-bunching, a nonclassical phenomenon for light with sub-Poissonian statistics g2(τ = 0) < 1. Of course, g2(τ = 0) < 1 is not a necessary condition for squeezing the state; however, if g2(τ = 0) > 1, the field is a classical field[1]. In other words, the squeezing state may exhibit bunching and antibunching behaviors. The following section discusses the aforementioned points with some related simulations.
The squeezing of the second oscillator regarding Eq. (11) is simulated, and the results are shown in Fig. 3. As seen in Fig. 3(a), which illustrates the quadrature operator’s variance versus gm with the nominal reference of 0.25, the squeezing appears in the resonator and reaches the maximum value for gm around 45 mS. However, the amount of the squeezing is decreased when gm exceeds 60 mS. More clearly, the optimum amount of squeezing occurs between 38 and 60 mS. It may relate to the fact that gm directly manipulates Īds2 by which the noise exceeds in the system, and due to that, the squeezing is strongly limited when gm is increased. Additionally, in Fig. 3(a), the dashed graph shows ∆y2 > 0.25, indicating that this operator for each value of gm shows bunching; in other words, the operator shows classical field behavior. We theoretically show that the important factors affecting LC2 to generate a squeezing state include CN, CN’, and gm2_N. To know about the factor's effects and compare with other elements, the contributed values are calculated for gm = 45 mS and represented as CN = 3.3 pF, CN’ = 72 mA/V, and gm2_N = 677 mA/V2. These values show nonlinear effects in the transistor, which changes the electrical properties of the circuit. For instance, one can compare CN with Cgd or Cgs, which indicates that CN is greater than the internal capacitances and gm2_N is comparable with gm2.
In addition, the second-order correlation function g2(τ = 0) behavior can be considered, as illustrated in Fig. 3(b). The figure shows perfect consistency with the quadrature operators’ variance around gm = 40 mS. The value of g2(τ = 0) around 45 mS reaches its minimum and is less than 1, which means that the second resonator exhibits antibunching. Notably, the change in the sign of g2(τ = 0) from bunching (g2(τ = 0) >1) to antibunching (g2(τ = 0) <1) indicates squeezing in the system. The results shown in Fig. 3(b) reveal that squeezing occurs only for small values of gm. In other words, the transistor's current amplification factor (gm) should be maintained at a low level to generate squeezing at cryogenic temperatures. Nonetheless, this is clearly shown in Eq. (11) that ζ1 and ζ2 are strongly manipulated by gm2_N, which is a fundamental function of gm3, and that CN plays a key role in changing the coupling between resonators. Additionally, other factors such as feedback capacitance, LC resonator decay rate, and input RF source can influence the squeezing in the system. For instance, the effect of gm2, gm3, and that CN plays a key role in changing the coupling between resonators. Additionally, other factors such as feedback capacitance, LC resonator decay rate, and input RF source can influence the squeezing in the system. For instance, the effect of gm3 as a nonlinearity factor on the quadrature operator variance and photon bunching and antibunching is illustrated in Fig. 4. Fig. 4(a) shows that by increasing gm3, the quadrature variance increases. This contributed to the increase in the squeezing strength parameters. In addition, the figure shows that increasing gm3 leads to maintaining ∆x2 < 0.25 for larger gm. In the same way, the effect of gm3 increasing on the second-order correlation function is depicted in Fig. 4(b). This reveals that increasing gm3, and that CN plays a key role in changing the coupling between resonators. Additionally, other factors such as feedback capacitance, LC resonator decay rate, and input RF source can influence the squeezing in the system. For instance, the effect of gm3 causes an increase in gm to 120 mS, in which the second-order correlation shows antibunching. This contributes to the fact that increasing gm3 changes CN and gm2_N, strengthening the squeezing behavior.
Additionally, the effects of other parameters such as Cf, VRF, gm2, and oscillator decay rate κ are analyzed in this study. The results of the simulations are depicted in Fig. 5. In this graph, the red dashed line is inserted to easily trace the bunching to the antibunching (and vice versa) entry point as a function of gm. In this simulation, it is additionally, the effects of other parameters such as Cf, VRF, gm2, and oscillator decay rate κ are analyzed in this study. The results of the simulations are depicted in Fig. 5. In this graph, the red dashed line is inserted to easily trace the bunching to the antibunching (and vice versa) entry point as a function of gm3, and that CN plays a key role in changing the coupling between resonators. Additionally, other factors such as feedback capacitance, LC resonator decay rate, and input RF source can influence the squeezing in the system. For instance, the effect of gm. In this simulation, it is assumed that the two oscillators had the same decay rate κ1 = κ2 = κ. As expected, Fig. 5 shows that increasing Cf, VRF, and gm2 causes an increase in antibunching, whereas an increase in the decay rate leads to a decrease in antibunching. In this figure, the key factor that can be freely manipulated is the feedback capacitor, by which circuit properties, such as noise, gain, and stability, can be manipulated. The graph in Fig. 5(b) reveals that increasing the feedback capacitor causes antibunching for a larger gm. This may be related to the noise figure enhancement using feedback in the circuit. In other words, using a feedback capacitor strongly enhances the noise figure of the circuit, which means that eliminating noise leads to enhanced squeezing. Assumed that the two oscillators had the same decay rate κ1 = κ2 = κ. As expected, Fig. 5 shows that increasing Cf, VRF, and gm2 causes an increase in antibunching, whereas an increase in the decay rate leads to a decrease in antibunching. In this figure, the key factor that can be freely manipulated is the feedback capacitor, by which circuit properties, such as noise, gain, and stability, can be manipulated. The graph in Fig. 5(b) reveals that increasing the feedback capacitor causes antibunching for a larger gm. This may be related to the noise figure enhancement using feedback in the circuit. In other words, using a feedback capacitor strongly enhances the noise figure of the circuit, which means that eliminating noise leads to enhanced squeezing.
The results illustrated in this study show that cryogenic InP HEMT transistor nonlinearity can generate a squeezed state. This is a significant achievement because such a system can be essential to a cryogenic detector used in quantum applications[29]. Thus, the operation of the detector or amplifier in the squeezed state implies that the noise fluctuation is limited below the zero-point changes. This is an interesting goal of this study; nonetheless, we know that this is challenging to achieve.
4. Conclusions
This article mainly emphasizes the generation of the squeezing state using the nonlinearity of the InP HEMT transistor. For this purpose, we designed a circuit containing two external oscillators coupled with a cryogenic InP HEMT transistor operating at 5 K. The circuit was analyzed using quantum theory, and the contributions of the Lagrangian and Hamiltonian functions were theoretically derived. Some key factors in the Hamiltonian arise because the transistor’s nonlinearity could generate the squeezed state. Thus, we focused on these parameters and their engineering to generate squeezing. The results show that the squeezed state occurred only for the second oscillator. This implies that the first oscillator experiences a coherent state. In addition, we theoretically demonstrate that two coupled oscillators through a cryogenic transistor can generate two-mode squeezing. Thus, as a general point, if such a cryogenic circuit could generate squeezing, then the critical noise fluctuations would be minimal by which the coherent time of a quantum system is directly manipulated. Coherent time is the duration in which the entanglement can be created between modes. As a result, by optimizing this time through minimizing the noise in the system, the entanglement can be alive for a long time.
Appendix A
In this appendix, all of the parameters used in the main article listed as Cq1, Cq2, Cq1q2, g12, g22, Vq1, Vq2 and IP2 are given by:
1Cq1=2C2B(CN+0.5CA)C4M−C2CCBC4M,1Cq2=2C2C(CN+0.5CA)C4M+CA′2CBC4M−2CC(CN+CA)CBC4M,1Cq1q2=2CCCB(CN+0.5CA)C4M+CC(CN+CA)CBC4M−C2CCBC4M,1Lp2=2g2mC2B(CN+0.5CA)C4M+2gmC′NCBC2M,g12=−2gmC2B(CN+0.5CA)C4M+C′NCBC2M−3gmC2CCB2C4M,g22=−2gmCBCC(CN+0.5CA)C4M+C′NCCC2M+gmC3CC4M−gm(CN+CA)CCCBC4M,Vq1=2CBCCCinVRF(CN+0.5CA)C4M−C2CCinCBVRFC4M,Vq2=2CBCCCinVRF(CN+0.5CA)C4M−C3CCinVRFC4M,Ip2=−2gmC2BCinVRF(CN+0.5CA)C4M+gmCBCinC2CVRFC4M−CBCinC′NVRFC2M−¯Ids2, |
where and CN’ = 2gm2(∂φi/∂t)|dc +12gm3 [(∂φi/∂t)|dc]2.
Appendix B
In this appendix, we try to give some information about t0. The main article discusses that t0 is selected less than the times that two oscillators decay with it. In fact, from a classical point of view, t0 should be in the order of the steady-state time. Therefore, in this part, we tried to calculate the step response of the circuit. For this reason, however, for simplicity, a simplified version of the circuit shown in Fig. 2 is demonstrated in Fig. B1 (Dip Trace software is used to draw the schematic), and the related transfer function is derived as:
Vout(s)VRF(s)=gmLp2L1S2Lp2L1Cp1Cp2S4+Lp2L1Cp2S3+(L1Cp1+Lp2Cp2)S2+L1S+1, |
(B1) |
where Lp2 = L2N||L2, Cp1 = Cgs + C1 + (Cgd + Cf)Av0, Cp2 = CN + C2. As can be seen in the expressions, the second oscillator’s inductance and capacitance are affected by the nonlinearity effects as L2N and CN, and the first oscillator is just influenced by the gain of the circuit Av0–gmr0, where r0 is the resistance generated due to the channel length modulated effect. The step response of the transfer function expressed in Eq. (B1) is illustrated in Fig. B2. It is shown that the settling time is around 80 ns; this time is very close to t0, which we selected based on the oscillator’s decay rates. In fact, t0 is selected around the settling time for the system.