Squeezed state generation using cryogenic InP HEMT nonlinearity

Ahmad Salmanogli

doi:10.1088/1674-4926/44/5/052901

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 inFig. 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 inFig. 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 (g_{m2_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.

Figure 1.(Color online) Schematic of the system containing two external oscillators coupling through a nonlinear device, inset figure: nonlinear device internal circuit.

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The high-frequency model of an InP HEMT transistor^[25,27] coupled with two oscillators is shown inFig. 2. Some elements in the nonlinear transistor model are created owing to the high-frequency effect, such as C_gs, C_gd, L_g, and L_d and some other components, such as resistors, as shown inFig. 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 inFig. 2. In fact,Ī_g²,Ī_j²,Ī_d², andṼ_i² model the thermal noise ofR_g,R_gd,R_ds, andR_gs, respectively. A noise model is applied to the circuit to demonstrate the effects of the contributed resistors. Additionally, we ignored the Ls effect and mergedR_s withR_gs to simplify algebra. In the circuit,i_ds andĪ_ds² are the dependent current sources containing the transistor’s nonlinearity and related thermal noise, which is a critical factor in generating noise. Finally,C_f,C_in,V_RF,φ₁, andφ₂ 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.

Figure 2.Step response of the circuit.

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2.2.　Designed system Hamiltonian

This section analyzes the circuit shown inFig. 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 inTable 1. First, we theoretically derive the total Lagrangian of the system to obtain the quantum properties of the circuit illustrated inFig. 2. For the analysis, the coordinate variables are defined asφ₁ andφ₂ (node flux), as shown inFig. 2, and the momentum conjugate variables are defined byQ₁ andQ₂ (loop charge). The total Lagrangian of the circuit is derived as^[29]:

Table 1. Values for the small signal model of the 2 × 50μm InP HEMT at 5 K^[31-33].

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Table 1. Values for the small signal model of the 2 × 50μm InP HEMT at 5 K^[31-33].

	Stands for	Value
R_g	Gate resistance	0.3 Ω
L_g	Gate inductance	75 pH
L_d	Drain inductance	70 pH
C_gs	Gate–source capacitance	69 fF
C_ds	Drain–source capacitance	29 fF
C_gd	Gate–drain capacitance	19 fF
R_gs	Gate-source resistance	4 Ω
R_gd	Gate–drain resistance	35 Ω

1 $\begin{array}{l} L_{c} = & \frac{C_{in}}{2} {({\dot{φ}}_{1} - V_{RF})}^{2} + \frac{C_{1}}{2} {\dot{φ}}_{1}^{2} - \frac{1}{2 L_{1}} φ_{1}^{2} + \bar{I_{g}^{2}} φ_{1} \\ + \frac{C_{gs}}{2} {({\dot{φ}}_{1} - \bar{V_{i}^{2}})}^{2} + \frac{C_{gd} + C_{f}}{2} {({\dot{φ}}_{1} - {\dot{φ}}_{2})}^{2} + i_{ds} φ_{1} \\ + (\bar{I_{ds}^{2}} + \bar{I_{d}^{2}}) φ_{2} + \frac{C_{2}}{2} {\dot{φ}}_{2}^{2} - \frac{1}{2 L_{2}} φ_{2}^{2} + \bar{I_{j}^{2}} (φ_{2} - φ_{1}) . \end{array}$

In this equation, the dependent current source is defined asi_ds =g_mV_gs +g_m2V_gs²₊g_m3V_gs^{3 [22,30]}, whereg_m is the intrinsic transconductance of the transistor andg_m2 andg_m3 are the second-and third-order transconductance. These terms (g_m2,g_m3) bring nonlinearity to the circuit. Moreover, thermally generated noises by the resistors and the current source are defined as ${\bar{I}}_{g}^{2} = 4 K_{B} T / R_{g}$ , ${\bar{I}}_{d}^{2} = 4 K_{B} T / R_{d}$ , ${\bar{I}}_{j}^{2} = 4 K_{B} T / R_{j}$ , ${\bar{I}}_{ds}^{2} = 4 K_{B} T γ g_{m}$ , and ${\bar{I}}_{i}^{2} = 4 K_{B} T / R_{i}$ , whereK_B,T, andγ 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 $Q_{i} = \partial L_{c} / \partial (\partial φ_{i} / \partial t)$ fori = 1, 2 represented as:

2 $\begin{matrix} Q_{1} = (C_{in} + C_{1} + C_{gs} + C_{f} + C_{gd}) {\dot{φ}}_{1} - (C_{f} + C_{gd}) {\dot{φ}}_{2} \\ - C_{in} V_{RF} + g_{m} φ_{2} + 2 g_{m}_{2} φ_{2} {\dot{φ}}_{1} + 3 g_{m 3} φ_{2} {\dot{φ}}_{1}^{2} - C_{gs} \bar{V_{i}^{2}}, \\ Q_{2} = (C_{2} + C_{f} + C_{gd}) {\dot{φ}}_{2} - (C_{f} + C_{gd}) {\dot{φ}}_{1} . \end{matrix}$

Applying Legendre transformation, the classical Hamiltonian is expressed as:

3 $\begin{array}{l} H_{c} = & {\frac{C_{A}}{2} {\dot{φ}}_{1}^{2} + \frac{1}{2 L_{1}} φ_{1}^{2} + \frac{C_{B}}{2} {\dot{φ}}_{2}^{2} + \frac{1}{2 L_{2}} φ_{2}^{2}} \\ + {- C_{c} {\dot{φ}}_{1} {\dot{φ}}_{2} + g_{m 2} φ_{2} {\dot{φ}}_{1}^{2} + 2 g_{m 3} φ_{2} {\dot{φ}}_{1}^{3}} \\ + {- φ_{1} (\bar{I_{g}^{2}} - \bar{I_{j}^{2}}) - φ_{2} (\bar{I_{d}^{2}} + \bar{I_{j}^{2}} + \bar{I_{ds}^{2}}) - \frac{C_{gs}}{2} \bar{V_{i}^{2}}}, \end{array}$

whereC_A =C_in +C₁ +C_gs +C_f +C_gd,C_B =C_gd +C_f +C₂, andC_C =C_f +C_gd. In Eq. (3), the first term relates to the LC resonance Hamiltonian affected by the coupling elements. It is clearly shown thatC_A andC_B 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 $(\partial φ_{i} / \partial t)$ in terms of the momentum conjugates (Q_i) represented in the matrix form as:

4 $[\begin{matrix} {\dot{φ}}_{1} \\ {\dot{φ}}_{2} \end{matrix}] = \frac{1}{C_{M}^{2}} {[\begin{matrix} C_{B} & C_{C} \\ C_{C} & C_{A} + C_{N} \end{matrix}] [\begin{matrix} Q_{1} \\ Q_{2} \end{matrix}] + [\begin{matrix} C_{B} & C_{C} \\ C_{C} & C_{A} + C_{N} \end{matrix}] [\begin{matrix} C_{in} & 0 \\ 0 & 0 \end{matrix}] [\begin{matrix} V_{RF} \\ 0 \end{matrix}] - [\begin{matrix} C_{B} & C_{C} \\ C_{C} & C_{A} + C_{N} \end{matrix}] [\begin{matrix} 0 & g_{m} \\ 0 & 0 \end{matrix}] [\begin{matrix} φ_{1} \\ φ_{2} \end{matrix}]},$

whereC_M² =C_B(C_A +C_N) –C_C² andC_N = 2g_m2φ_{2_dc} + 6g_m3φ_{2_dc} $(\partial φ_{i} / \partial t)$ |_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 C_N. 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:

5 $H_{L} = {\frac{1}{2 C_{q 1}} Q_{1}^{2} + \frac{1}{2 L_{1}} φ_{1}^{2} + \frac{1}{2 C_{q 2}} Q_{2}^{2} + \frac{1}{2 L_{2^{'}}} φ_{2}^{2}} + {\frac{1}{2 C_{q 1 q 2}} Q_{1} Q_{2} + g_{12} Q_{1} φ_{2} + g_{22} Q_{2} φ_{2}} + {V_{q 1} Q_{1} + V_{q 2} Q_{2} + I_{p 2} φ_{2} - \bar{I_{gs}^{2}} φ_{1}},$

whereC_q1,C_q2,C_q1q2,g₁₂,g₂₂,L₂’,V_q1,V_q2, andI_p2 are defined in Appendix A, and the dc terms are ignored for simplicity. In fact, these coefficients are essentially the function ofg_m,C_N,C_C, andV_RF 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 areZ₁ = (L₁/C_q1)^0.5,Z₂ = ( $L_{2^{'}}$ /C_q2)^0.5, and the associated frequencies areω₁ = (L₁C_q1)^−0.5,ω₂ = ( $L_{2^{'}}$ /C_q2)^−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 asQ₁Q₂,Q₂φ_1, andQ₁φ₂ show the coupling between oscillators in the circuit design. Also, in the third term in Eq. (5), some terms such asV_q1Q₁, ${\bar{I}}_{gs}^{2}$ φ_1,V_g2Q_2, andI_p2φ₂ in the equation declare the RF source and thermal noise coupling to the contributed oscillators. In this equation, ${\bar{I}}_{gs}^{2} = {\bar{I}}_{g}^{2} - {\bar{I}}_{j}^{2}$ . 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 asQ₁ = –i(a₁ –a₁⁺)(ħ/2Z₁)^0.5,φ₁ = (a₁ +a₁⁺)(ħZ₁/2)^0.5 andQ₂ = –i(a₂ –a₂⁺)(ħ/2Z₂)^0.5,φ₂ = (a₂ +a₂⁺)(ħZ₂/2)^0.5, where (a_i,a_i⁺) 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:

6 $\begin{array}{l} H_{L} = & {ℏ ω_{1} (a_{1}^{+} a_{1} + \frac{1}{2}) + ℏ ω_{2} (a_{2}^{+} a_{2} + \frac{1}{2})} + {- \frac{ℏ}{2} \frac{1}{C_{q 1 q 2} \sqrt{Z_{1} Z_{2}}} (a_{1} - a_{1}^{+}) (a_{2} - a_{2}^{+}) - \frac{i ℏ}{2} g_{12} \sqrt{\frac{Z_{2}}{Z_{1}}} (a_{1} - a_{1}^{+}) (a_{2} + a_{2}^{+}) - \frac{i ℏ}{2} g_{22} (a_{2} - a_{2}^{+}) (a_{2} + a_{2}^{+})} \\ + {- i V_{q 1} \sqrt{\frac{ℏ}{2 Z_{1}}} (a_{1} - a_{1}^{+}) - i V_{q 2} \sqrt{\frac{ℏ}{2 Z_{2}}} (a_{2} - a_{2}^{+}) + I_{p 2} \sqrt{\frac{ℏ Z_{2}}{2}} (a_{2} + a_{2}^{+}) - \bar{I_{gs}^{2}} \sqrt{\frac{ℏ}{2 Z_{2}}} (a_{1} + a_{1}^{+})} . \end{array}$

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:

7 $H_{N} = {g_{m 2} + 6 g_{m 3} {\dot{φ}}_{1_dc}} φ_{2} {\dot{φ}}_{1}^{2} .$

Using Eq. (4), the nonlinear Hamiltonian is given by:

8 $\begin{matrix} H_{N} = g_{m 2_N} [{\frac{C_{B}^{2}}{C_{M}^{4}} φ_{2} Q_{1}^{2} + \frac{C_{C}^{2}}{C_{M}^{4}} φ_{2} Q_{2}^{2} + \frac{2 C_{B} C_{C}}{C_{M}^{4}} φ_{2} Q_{2} Q_{1} - \frac{2 g_{m} C_{B}^{2}}{C_{M}^{4}} φ_{2}^{2} Q_{1} - \frac{2 g_{m} C_{B} C_{C}}{C_{M}^{4}} φ_{2}^{2} Q_{2} + \frac{g_{m}^{2} C_{B}^{2}}{C_{M}^{4}} φ_{2}^{3}}_{NL} \\ \times {\frac{- 2 g_{m} C_{B}^{2} C_{in} V_{RF}}{C_{M}^{4}} φ_{2}^{2} + \frac{2 C_{B}^{2} C_{in} V_{RF}}{C_{M}^{4}} Q_{1} φ_{2} + \frac{2 C_{B} C_{C} C_{in} V_{RF}}{C_{M}^{4}} Q_{2} φ_{2} + \frac{C_{B}^{2} V_{in}^{2} V_{RF}^{2}}{C_{M}^{4}} φ_{2}}_{L}], \end{matrix}$

whereg_{m2_N} = g_m2 + 6g_m3 $(\partial φ_{i} / \partial t)$ |_dc. The linear part of Eq. (8) can directly attach to Eq. (5) to make the modified linear Hamiltonian given by:

9 $\begin{array}{l} H_{L} = & {\frac{1}{2 C_{q 1}} Q_{1}^{2} + \frac{1}{2 L_{1}} φ_{1}^{2} + \frac{1}{2 C_{q 2}} Q_{2}^{2} + (\frac{1}{2 L_{2^{'}}} + \frac{1}{2 L_{2 N}}) φ_{2}^{2}} + {\frac{1}{2 C_{q 1 q 2}} Q_{1} Q_{2} + (g_{12} + g_{12 N}) Q_{1} φ_{2} + (g_{22} + g_{22 N}) Q_{2} φ_{2}} \\ + {V_{q 1} Q_{1} + V_{q 2} Q_{2} + (I_{p 2} + I_{p2N}) φ_{2} - \bar{I_{gs}^{2}} φ_{1}} . \end{array}$

As can be clearly seen in Eq. (9), the nonlinear Hamiltonian can change the coupling between oscillators in the circuit. The most important factor isg_{m2_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 L_2N. The term brought from nonlinearity (L_2N) 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:

10 $\begin{array}{l} H_{t} = & [{ℏ ω_{1} (a_{1}^{+} a_{1} + \frac{1}{2}) + ℏ ω_{2} (a_{2}^{+} a_{2} + \frac{1}{2})} \\ + {- \frac{ℏ}{2} \frac{1}{C_{q1q2} \sqrt{Z_{1} Z_{2}}} (a_{1} - a_{1}^{+}) (a_{2} - a_{2}^{+}) - \frac{i ℏ}{2} g_{12}' \sqrt{\frac{Z_{2}}{Z_{1}}} (a_{1} - a_{1}^{+}) (a_{2} + a_{2}^{+}) - \frac{i ℏ}{2} g_{22}' (a_{2} - a_{2}^{+}) (a_{2} + a_{2}^{+})} \\ + {{- i V_{q 1} \sqrt{\frac{ℏ}{2 Z_{1}}} (a_{1} - a_{1}^{+}) - i V_{q 2} \sqrt{\frac{ℏ}{2 Z_{2}}} (a_{2} - a_{2}^{+}) + I_{p 2}' \sqrt{\frac{ℏ Z_{2}}{2}} (a_{2} + a_{2}^{+}) - \bar{I_{g s}^{2}} \sqrt{\frac{ℏ}{2 Z_{2}}} (a_{1} + a_{1}^{+})}]}_{L} \\ + [- ℏ g_{13} {(a_{1} - a_{1}^{+})}^{2} (a_{2} + a_{2}^{+}) + ℏ g_{14} (a_{2} + a_{2}^{+}) {(a_{2} - a_{2}^{+})}^{2} - ℏ g_{15} (a_{1} - a_{1}^{+}) (a_{2} - a_{2}^{+}) (a_{2} + a_{2}^{+}) \\ + {ℏ g_{16} {(a_{2} + a_{2}^{+})}^{3} + i ℏ g_{17} (a_{1} - a_{1}^{+}) {(a_{2} + a_{2}^{+})}^{2} + i ℏ g_{18} (a_{2} - a_{2}^{+}) {(a_{2} + a_{2}^{+})}^{2}]}_{NL}, \end{array}$

whereI_P2’ =I_P2 +I_P2N,g₁₂’ =g₁₂ +g_12N, andg₂₂’ =g₂₂ +g_22N. Also, g₁₃ = (1/2Z₁)( $\sqrt{(ℏ / 2 Z_{2})}$ )g_{m2_N}C_B²/C_M⁴,g₁₄ = (1/2Z₂)( $\sqrt{(ℏ / 2 Z_{2})}$ )g_{m2_N}C_C²/C_M⁴,g₁₅ = (1/ $\sqrt{(Z_{2} Z_{1})}$ )( $\sqrt{(ℏ / 2 Z_{2})}$ )g_{m2_N}C_BC_C/C_M⁴,g₁₆ = (Z₂/2)( $\sqrt{(ℏ Z_{2} / 2)}$ )g_m²g_{m2_N}C_B²/C_M⁴,g₁₇ =Z₂( $\sqrt{(ℏ / 2 Z_{1})}$ )g_mg_{m2_N}C_B²/C_M⁴, andg₁₈ =Z₂( $\sqrt{(ℏ / 2 Z_{2})}$ )g_mg_{m2_N}C_BC_C/C_M⁴. It is clearly shown in coefficients fromg₁₃ tog₁₈ where the effect ofg_{m2_N} is dominant. In other words, the system’s nonlinearity in this work is strongly changed and controlled by the second-order transconductanceg_{m2_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 asa² anda⁺² in the exponent. The squeezed-coherent stateS(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 asa² anda⁺² in the Hamiltonian may generate squeezing. The Hamiltonian in Eq. (10), the squeezed state, can be generated by:

11 $S (ζ) = \exp [ζ_{1} (a_{2}^{2} - a_{2}^{+ 2}) + (\frac{ζ_{2}^{*}}{2} a_{2}^{2} - \frac{ζ_{2}}{2} a_{2}^{+ 2})] t .$

In this equation the squeezing parameters are defined asζ₁ = –0.5g₂₂’ +g₁₈Re{A₂} + jg₁₄Im{A₂} andζ₂ =2A₁^*(–g₁₇–jg₁₅), whereA₁ andA₂ 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 containingg_m2 andg_m3, which is directly connected to LC₂. The important point about the squeezing strength parametersζ₁ andζ₂ is that each depends ong_{m2_N}. In Eq. (11),ζ₁ andζ₂ 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:

12 $S_{2} (ζ) = \exp [(\frac{ζ_{t 1}^{*}}{2} a_{1} a_{2} - \frac{ζ_{t 1}}{2} a_{1}^{+} a_{2}^{+}) + (\frac{ζ_{t 2}^{*}}{2} a_{1} a_{2} - \frac{ζ_{t 2}}{2} a_{1}^{+} a_{2}^{+})] t,$

whereζ_t1 = jA₂g₁₅ andζ_t2 = jA₁g₁₃. In the same way, the squeezing parameters are strongly dependent on g_{m2_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 ast₀ < {1/κ₁, 1/κ₂}, whereκ₁ andκ₂ are the first and second oscillators’ decay rates. By selectingt₀, the system is forced to generate squeezing before the resonator decaying, by which the squeezing is destroyed^[1]. Some more information is introduced aboutt₀ 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 functiong²(τ = 0) = 1 + (V(n) – <n>)/<n>, whereV(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 statisticsg²(τ = 0) < 1. Of course,g²(τ = 0) < 1 is not a necessary condition for squeezing the state; however, ifg²(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 inFig. 3. As seen inFig. 3(a), which illustrates the quadrature operator’s variance versusg_m with the nominal reference of 0.25, the squeezing appears in the resonator and reaches the maximum value forg_m around 45 mS. However, the amount of the squeezing is decreased wheng_m exceeds 60 mS. More clearly, the optimum amount of squeezing occurs between 38 and 60 mS. It may relate to the fact thatg_m directly manipulatesĪ_ds² by which the noise exceeds in the system, and due to that, the squeezing is strongly limited wheng_m is increased. Additionally, inFig. 3(a), the dashed graph shows ∆y² > 0.25, indicating that this operator for each value ofg_m 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 includeC_N,C_N^’, andg_{m2_N}. To know about the factor's effects and compare with other elements, the contributed values are calculated forg_m = 45 mS and represented asC_N = 3.3 pF,C_N^’ = 72 mA/V, andg_{m2_N} = 677 mA/V². These values show nonlinear effects in the transistor, which changes the electrical properties of the circuit. For instance, one can compareC_N withC_gd orC_gs, which indicates thatC_N is greater than the internal capacitances andg_{m2_N} is comparable withg_m2.

Figure 3.(Color online) (a) Quadrature variance versusg_m. (b) Second-order correlation function versusg_m.C_f = 0.02 pF,g_m3 = 1200 mA/V³,g_m2 = 200 mA/V²,V_RF = 1μV,κ =κ₁=κ₂= 0.001ω.

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In addition, the second-order correlation functiong²(τ = 0) behavior can be considered, as illustrated inFig. 3(b). The figure shows perfect consistency with the quadrature operators’ variance aroundg_m = 40 mS. The value ofg²(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 ofg²(τ = 0) from bunching (g²(τ = 0) >1) to antibunching (g²(τ = 0) <1) indicates squeezing in the system. The results shown inFig. 3(b) reveal that squeezing occurs only for small values ofg_m. In other words, the transistor's current amplification factor (g_m) should be maintained at a low level to generate squeezing at cryogenic temperatures. Nonetheless, this is clearly shown in Eq. (11) thatζ₁ andζ₂ are strongly manipulated byg_{m2_N}, which is a fundamental function ofg_m3, and thatC_N 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 ofg_m2,g_m3, and thatC_N 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 ofg_m3 as a nonlinearity factor on the quadrature operator variance and photon bunching and antibunching is illustrated inFig. 4.Fig. 4(a) shows that by increasingg_m3, the quadrature variance increases. This contributed to the increase in the squeezing strength parameters. In addition, the figure shows that increasingg_m3 leads to maintaining ∆x² < 0.25 for largerg_m. In the same way, the effect ofg_m3 increasing on the second-order correlation function is depicted inFig. 4(b). This reveals that increasingg_m3, and thatC_N 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 ofg_m3 causes an increase ing_m to 120 mS, in which the second-order correlation shows antibunching. This contributes to the fact that increasingg_m3 changesC_N andg_{m2_N}, strengthening the squeezing behavior.

Figure 4.(Color online) (a) Quadrature variance versusg_m, (b) second-order correlation function versusg_m for different g_m3(mA/V³);C_f = 0.02 pF,g_m2 = 200 mA/V²,V_RF = 1μV,κ = 0.001ω.

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Additionally, the effects of other parameters such asC_f,V_RF,g_m2, and oscillator decay rateκ are analyzed in this study. The results of the simulations are depicted inFig. 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 ofg_m. In this simulation, it is additionally, the effects of other parameters such asC_f,V_RF,g_m2, and oscillator decay rateκ are analyzed in this study. The results of the simulations are depicted inFig. 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 ofg_m3, and thatC_N 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 ofg_m. In this simulation, it is assumed that the two oscillators had the same decay rateκ₁ =κ₂ =κ. As expected,Fig. 5 shows that increasingC_f,V_RF, andg_m2 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 inFig. 5(b) reveals that increasing the feedback capacitor causes antibunching for a largerg_m. 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 rateκ₁ =κ₂ =κ. As expected,Fig. 5 shows that increasingC_f,V_RF, andg_m2 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 inFig. 5(b) reveals that increasing the feedback capacitor causes antibunching for a largerg_m. 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.

Figure 5.(Color online) Second-order correlation function versusg_m. (a)g_m2 effect. (b) Feedback capacitance effect (C_f). (c) LC resonator decay rate effect (κ). (d) Input RF source amplitude effect (V_RF).g_m3 = 1200 mA/V³.

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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 asC_q1,C_q2,C_q1q2,g₁₂,g₂₂,V_q1,V_q2 andI_P2 are given by:

$\begin{matrix} \frac{1}{C_{q 1}} = \frac{2 C_{B}^{2} (C_{N} + 0.5 C_{A})}{C_{M}^{4}} - \frac{C_{C}^{2} C_{B}}{C_{M}^{4}}, \\ \frac{1}{C_{q 2}} = \frac{2 C_{C}^{2} (C_{N} + 0.5 C_{A})}{C_{M}^{4}} + \frac{C_{A}'^{2} C_{B}}{C_{M}^{4}} - \frac{2 C_{C} (C_{N} + C_{A}) C_{B}}{C_{M}^{4}}, \\ \frac{1}{C_{q1q2}} = \frac{2 C_{C} C_{B} (C_{N} + 0.5 C_{A})}{C_{M}^{4}} + \frac{C_{C} (C_{N} + C_{A}) C_{B}}{C_{M}^{4}} - \frac{C_{C}^{2} C_{B}}{C_{M}^{4}}, \\ \frac{1}{L_{p 2}} = \frac{2 g_{m}^{2} C_{B}^{2} (C_{N} + 0.5 C_{A})}{C_{M}^{4}} + \frac{2 g_{m} C_{N}' C_{B}}{C_{M}^{2}}, \\ g_{12} = \frac{- 2 g_{m} C_{B}^{2} (C_{N} + 0.5 C_{A})}{C_{M}^{4}} + \frac{C_{N}' C_{B}}{C_{M}^{2}} - \frac{3 g_{m} C_{C}^{2} C_{B}}{2 C_{M}^{4}}, \\ g_{22} = \frac{- 2 g_{m} C_{B} C_{C} (C_{N} + 0.5 C_{A})}{C_{M}^{4}} + \frac{C_{N}' C_{C}}{C_{M}^{2}} + \frac{g_{m} C_{C}^{3}}{C_{M}^{4}} - \frac{g_{m} (C_{N} + C_{A}) C_{C} C_{B}}{C_{M}^{4}}, \\ V_{q 1} = \frac{2 C_{B} C_{C} C_{in} V_{RF} (C_{N} + 0.5 C_{A})}{C_{M}^{4}} - \frac{C_{C}^{2} C_{in} C_{B} V_{RF}}{C_{M}^{4}}, \\ V_{q 2} = \frac{2 C_{B} C_{C} C_{in} V_{RF} (C_{N} + 0.5 C_{A})}{C_{M}^{4}} - \frac{C_{C}^{3} C_{in} V_{RF}}{C_{M}^{4}}, \\ I_{p 2} = \frac{- 2 g_{m} C_{B}^{2} C_{in} V_{RF} (C_{N} + 0.5 C_{A})}{C_{M}^{4}} + \frac{g_{m} C_{B} C_{in} C_{C}^{2} V_{RF}}{C_{M}^{4}} - \frac{C_{B} C_{in} C_{N}' V_{RF}}{C_{M}^{2}} - \bar{I_{ds}^{2}}, \end{matrix}$

where andC_N’ = 2g_m2 $(\partial φ_{i} / \partial t)$ |_dc +12g_m3 [ $(\partial φ_{i} / \partial t)$ |_dc]².

Appendix B

In this appendix, we try to give some information aboutt₀. The main article discusses thatt₀ 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 inFig. 2 is demonstrated inFig. B1 (Dip Trace software is used to draw the schematic), and the related transfer function is derived as:

Figure 1.(Color online) Schematic of the system containing two external oscillators coupling through a nonlinear device, inset figure: nonlinear device internal circuit.

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$\frac{V_{out} (s)}{V_{RF} (s)} = \frac{g_{m} L_{p 2} L_{1} S^{2}}{L_{p 2} L_{1} C_{p 1} C_{p 2} S^{4} + L_{p 2} L_{1} C_{p 2} S^{3} + (L_{1} C_{p 1} + L_{p 2} C_{p 2}) S^{2} + L_{1} S + 1},$

whereL_p2 =L_2N||L₂,C_p1 =C_gs +C₁ + (C_gd +C_f)A_v0,C_p2 =C_N +C₂. As can be seen in the expressions, the second oscillator’s inductance and capacitance are affected by the nonlinearity effects asL_2N andC_N, and the first oscillator is just influenced by the gain of the circuitA_v0–g_mr₀, wherer₀ is the resistance generated due to the channel length modulated effect. The step response of the transfer function expressed in Eq. (B1) is illustrated inFig. B2. It is shown that the settling time is around 80 ns; this time is very close tot₀, which we selected based on the oscillator’s decay rates. In fact,t₀ is selected around the settling time for the system.

Figure 2.Step response of the circuit.

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Category: Articles

Received: Nov. 11, 2022

Accepted: --

Published Online: Jun. 15, 2023

The Author Email: Salmanogli Ahmad (salmanogli@cankaya.edu.tr)

DOI:10.1088/1674-4926/44/5/052901

Table 1. Values for the small signal model of the 2 × 50μm InP HEMT at 5 K[31-33].

Table 1. Values for the small signal model of the 2 × 50μm InP HEMT at 5 K[31-33].

Table 1. Values for the small signal model of the 2 × 50μm InP HEMT at 5 K^[31-33].

Table 1. Values for the small signal model of the 2 × 50μm InP HEMT at 5 K^[31-33].