Maxwell’s demon (MD)^[1] is a wizard of wisdom in a thought experiment. It can separate the hot and cold gas molecule in a container box, thus creating a finite temperature difference between the two sides of the box and lowering the entropy of the system, which violates the second law of thermodynamics. However, contemporary studies have shown that the conclusion of this violation is only a paradox, because Maxwell’s demon must participate in the thermodynamic cycle as part of the heat engine, whose memory still has the information about the velocity of the molecules. According to Landauer’s principle,^[2] erasing a bit of information in the equilibrium state requires at least an energy of k_BT ln2 (where k_B is the Boltzmann constant and T is the ambient temperature) which is about 3 × 10⁻²¹ J. So, even if such a demon does exist, the second law of thermodynamics will not be violated after the erasure of the demon’s memory.^[3]

Realizing the MD experiment can help us understand the Landauer principle, which would reveal a profound relation between the information content and entropy of a thermodynamic system. In recent years, the majority of studies, both theoretically^[3,4] and, in particular, experimentally,^[5–9] have been focusing on the classical regime. Raizen et al.^[9] cooled atoms from 10 mK to 15 μK based on the principle of “Maxwell’s demon”. Toyabe et al.^[8] used organic molecules to implement the MD experiment. Koski et al.^[5,6] realized a Szilard engine^[10] in a single-electron box. These experiments demonstrated quantitatively the extraction of heat energy k_BT ln2 by creating a bit of information.

In this paper, a compound Josephson junction (CJJ) radio frequency superconducting quantum interference device (RF-SQUID)^[11–14] qubit is designed. An overdamped resistively shunted direct-current (DC)-SQUID^[15] is inductively coupled to the qubit to monitor its fluxoid state. According to the description of the MD experiment, to do the experiment we need a particle in a two-state system. In the system the particle can be easily manipulated, depending on its state, and the energy change during the manipulation can be easily calculated.

As a kind of macroscopic qubit, the CJJ RF-SQUID qubit can work as a flux qubit with two quantum states, which can be easily controlled and readout. To obtain the energy change during the experiment, an overdamped DC-SQUID magnetometer instead of unshunted DC-SQUID^[16] is used, because it can continuously monitor the state of the RF-SQUID qubit. The purpose of this design is to study the MD experiment on quantum systems.^[17–20] In this paper, we mainly describe the design and characterization of the CJJ RF-SQUID qubit. The rest of this paper is arranged as follows. In Section 2, the basic principle of the CJJ RF-SQUID qubit used in the MD experiment is introduced. In Section 3, the design and fabrication process of the qubit are presented. In Section 4, the parameters are determined by fitting the experimental data to theoretical analysis and the potential energy of the qubit is obtained. Some conclusions are drawn from the present study in Section 5, finally.

As shown in Fig. 1(a), the CJJ RF-SQUID qubit is an RF-SQUID of loop inductance L with the Josephson junction that is replaced by two Josephson junctions in a small loop of inductance l. In the limit of l/L ≪ 1 and assuming the two Josephson junctions are identical, the potential energy can be approximated well as^[12]

Fig. 1.

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Fig. 1. (color online) (a) Schematic diagram of CJJ RF-SQUID qubit with an overdamped DC-SQUID used to monitor the state of qubit. Ibias is the bias current of DC-SQUID. Vout is the output voltage of DC-SQUID. ϕ and ϕx are the total and external flux applied to the RF-SQUID loop. ϕCJJ is the external flux applied to the compound Josephson junction. ϕsq is the external flux applied to the DC-SQUID loop. ϕx tilts the potential and ϕCJJ raises and lowers the barrier between the wells. (b) Potential of the RF-SQUID as a function of ϕ. ΔUL (ΔUR) is the barrier seen from the left (right) well and ϵ is the energy difference between the two potential minima (positive in this case). (c) Sketch of energy diagrams for the operation cycle of Maxwell demon.

The potential energy is of a double-well as shown in Fig. 1(b). Quantization of fluxoid requires ϕ + θ/2π = n, n = 0, ±1, ±2, . … The left and right potential wells in the figure represent the case where the fluxoid is ‘0’ or ‘1’. When the applied flux ϕ_x = 1/2, U(ϕ) is a symmetric double-well potential, where ΔU_L = ΔU_R, and if the system is initially placed in, say, state ‘0’ at t = 0, after a time t, it could jump to state ‘1’ by incoherent tunneling.^[11] The probability with which the system remains in state “0” is given by

An ideal operation cycle for the MD experiment is shown in Fig. 1(c). The cycle starts at the potential tilt ϵ < 0 and the qubit state is ‘0’, which lead to zero entropy. The next step of the cycle is a quasi-static tilt which changes the system into the degeneracy (ϵ = 0), due to the incoherent tunneling, the qubit state is uncertain, so the entropy of the system increases by k_B ln 2, corresponding to the heat extracted from the bath, which is equal to Landauer value Q_L = k_BT ln 2. After reaching the degeneracy point, a measurement of the qubit state is performed. If the qubit is in the ‘0’ state, a fast ramp is used and the system is reset to the initial state ‘0’ for the next cycle; if the measured state is ‘1’, the system is fast reset to state ‘1’, which has the same |ϵ| as the system resetting to initial state ‘0’, for the next cycle. Such a rapid feedback drive traps the qubit to the measured state. Ideally, this drive is so fast that no state change has a chance to occur and, as a result, no heat is transferred to the reservoir.^[6]

This process will be repeated many times to obtain the average extracted heat and average work after we have designed a feedback device to realize the MD experiment. Depending on the potential tilt ϵ, the extracted heat and work can be calculated by the following equations step by step:

To minimize the effects of ambient magnetic field and cross talks between ϕ_x and ϕ_CJJ, the RF-SQUID loop was designed as a parallel gradiometer^[13,21] as shown in Fig. 2(b). The RF-SQUID had the following parameters: L = 100 pH, maximum critical current I_c = 9.8 μA, which leads to β_L0 = 2πLI_c/Φ₀ = 3.0. The overdamped DC-SQUID had the following parameters: I₀ = 18 μA, R_n = 2 Ω, where I₀ is the critical current of each junction and R_n is the shunt resistance for the DC-SQUID. In order to produce a large change in DC-SQUID output voltage when the qubit jumped from one well to the opposite well, the mutual inductance between the DC-SQUID and RF-SQUID was enhanced using an overlapping structure.

Fig. 2.

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	Fig. 2. (color online) (a) Wire bonding and (b) optical micrographs of the SQUID device. Two overdamped dc-SQUIDs are designed, and the left one is used in this paper.

With the help of InductEX,^[22] we extracted the inductance and mutual inductances of the system, and the results are listed in Table 1. In this table, L_sq is the loop inductance of the DC-SQUID, L is the loop inductance of the RF-SQUID, l is the inductance of the compound Josephson junction, M_sq is the mutual inductance between the DC-SQUID and its flux bias line, M_rf is the mutual inductance between the RF-SQUID and its flux bias line, M_CJJ is the mutual inductance between the compound Josephson junction and its flux bias line, and M is the mutual inductance between the DC-SQUID and the RF-SQUID.

Table 1.

Table 1.

Table 1. Simulation results with InductEX. . Parameters Simulation results/pH Lsq 18.1 L 104 l 7.36 Msq 0.66 Mrf 2.48 MCJJ 0.47 M 2.85

Table 1. Simulation results with InductEX. .

The fabrication process included four photolithographic layers as well as several material deposition and etching steps. In our laboratory, we used a 2-inch (1 inch = 2.54 cm) ⟨100⟩ crystalline N-type single-sided polished wafer, which had a 400-nm thick SiO₂ on its surface. The fabrication started with in situ deposition of an Nb/Al-AlO_x/Nb trilayer patterned by lift-off. Then, the second lithography determined the junction area on the trilayer film. Then, we deposited the Pt shunt resistance layer by dc sputtering. Before sputtering Pt, a 10-nm thick Ti layer was sputtered to improve adhesion. For the last step, a 300-nm thick Nb wiring layer was deposited and patterned to determine the modulation coil and connection of the resistance. The photograph of the fabricated chip with bonding wires is shown in Fig. 2(a), and the detail of the main part is shown in Fig. 2(b).

To prepare for the MD experiment, we need to know all relevant parameters of the DC-SQUID and RF-SQUID. The procedure and result are described below. All of the measurements were performed at T = 4.2 K.

4.1. Dc-SQUID characterization

Figure 3(a) shows the I–V characteristics of DC-SQUID under different values of flux bias I_sq. Derived from Fig. 3(a), the switching current of the DC-SQUID I_sw is shown in Fig. 3(b), the I–V curves of the DC-SQUID at different values of ϕ_sq are shown in Figs. 3(c) and 3(d). In this figure, ϕ_sq is obtained through I_sq divided by the period ΔI_sq = 2968 μA, from which we obtain M_sq = 0.697 pH, compared with the extracted value of 0.66 pH from InductEX. As can be seen from this figure, the parameters of the DC-SQUID are I₀ = 18.5 ± 0.1 μA, R_n = 1.94 ± 0.18 Ω and the critical current modulation is ∼ 71.9% of the total critical current. From the modulation depth and comparison with the result in Ref. [23], we can obtain the screening parameter of the DC-SQUID (β₀ = 2L_sqI₀/Φ₀) to be 0.38 ± 0.01, which leads to a self-inductance L_sq = 21.2 ± 0.6 pH. In addition, because of the asymmetry in the inductance of the two arms forming the DC-SQUID loop, I_sw(ϕ_sq) is asymmetric and there is a shift between the I_sw-maxima for opposite-polarity bias current.

Fig. 3.

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	Fig. 3. (color online) (a) DC-SQUID Vout under different values of Isq and Ibias. (b) Isw(ϕsq) curve of the DC-SQUID. Here, ϕsq is obtained through Isq divided by period ΔIsq = 2968 μA. Isw is normalized to the critical current of DC-SQUID 2I0 = 37.0 μA. (c) I–V curves of DC-SQUID under flux modulation of ϕsq. (d) I–V curves of DC-SQUID at ϕsq = 0.157 and ϕsq = 0.566.

Figure 4 shows the modulation curves versus flux bias ϕ_sq of DC-SQUID with different values of bias current I_bias. The bias currents are chosen around the critical current 2I₀ of DC-SQUID. It can be seen that the DC-SQUID is fluxsensitive near the blue cycle, where we choose the work point for the readout of the RF-SQUID.

Fig. 4.

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	Fig. 4. (color online) DC-SQUID modulation at different values of Ibias. The blue cycle is the work point used for the readout.

4.2. RF-SQUID characterization

By using bias current around I_bias/2I₀ = 0.97 and adjusting ϕ_sq around the work point shown in Subsection 4.1, we can maximize the flux sensitivity of DC-SQUID (∼ 1040 μV per Φ₀), so that it can clearly distinguish the fluxoid states of the RF-SQUID. Figure 5 shows the RF-SQUID modulation of ϕ_x under three different values of ϕ_sq (1.204 in blue color, 0.206 in red color, and −0.791 in green color), and the difference between two adjacent curves is about one flux quantum Φ₀. In this figure, ϕ_x is obtained through I_rf divided by period ΔI_rf = 865 μA, from which we can obtain M_rf equal to 2.39 pH.

Fig. 5.

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	Fig. 5. (color online) RF-SQUID modulations at different values of ϕsq. ϕx is obtained through Irf divided by period ΔIrf = 865 μA. Inset shows the detail of a hysteresis loop in the black frame.

By measuring the width of the hysteresis loop, we can calibrate ϕ_CJJ. Figure 6 shows the measured values of loop width along with the curves theoretically calculated from the qubit potential. In this figure, ϕ_CJJ is obtained through I_CJJ divided by period ΔI_CJJ = 2984 μA, from which we can obtain the mutual inductance M_CJJ to be equal to 0.69 pH. In the theoretical calculation, the parameters are L = 120 pH, I_c = 9.8 μA and ΔU = 58.0 K, which can lead to β_L0 = 3.57. Additionally, the height of the hysteresis loop at the symmetric point can be used to obtain the mutual inductance between the DC-SQUID and the RF-SQUID according to M = Δϕ_sqΦ₀/ΔI_s, where I_s = ϕΦ₀/L is the circulation current in the RF-SQUID. From the inset of Fig. 5, we can obtain ΔV ≈ 18 μV, which can lead to Δϕ_sq ≈ 0.017. Based on the dependence of the total flux in the loop ϕ on applied flux ϕ_x (here, ϕ_x of the symmetric point is adjusted to 0.5): ϕ = ϕ_x − β_L0 sin(ϕ), we can derive that ΔI_s ≈ 13.4 μA, so M ≈ 2.62 pH.

Fig. 6.

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Fig. 6. (color online) Measured width (blue points) of the hysteresis loop as a function of ϕCJJ along with the theoretical curve (in red color) calculated from the qubit potential for L = 120 pH, Ic = 9.8 μA, and ΔU = 58.0 K. ϕCJJ is obtained through ICJJ divided by period ΔICJJ = 2984 μA. Each point is obtained by averaging 200 repeats at the fixed values of Ibias and ϕsq. The inset shows the measurements of hysteresis loop at different values of ϕCJJ, here, ϕx of the symmetric point is adjusted to 0.5.

4.3. Potential characterization

By setting the bias of ϕ_x to be a fixed value around the symmetric point of the RF-SQUID, and measuring the state of the RF-SQUID in a long time, we can obtain the lifetimes of state ‘0’ (τ₀) and state ‘1’ (τ₁). Figure 7 shows the procedure of extracting τ₀ and τ₁ from a typical time record of DC-SQUID voltage, V_out. Figure 7(a) shows the original data of V_out measured at I_bias/2I₀ = 0.97, ϕ_CJJ = −0.268, ϕ_sq = 0.206, and ϕ_x = −0.4984. Figure 7(b) shows the digitized original data. To calculate the lifetime, we obtain the residence time at state ‘1’ ( ) and the residence time at state ‘0’ ( ). Figures 7(c) and 7(d) show the distributions of and , and fitting to an exponential function to determine the values of τ₁ and τ₀. In this way we can obtain the lifetimes of states ‘0’ and ‘1’ at a fixed ϕ_x. By changing ϕ_x, and repeating this procedure we can obtain the lifetimes of states ‘0’ and ‘1’ as a function of ϕ_x. The result of ϕ_CJJ = −0.268 is shown in the insert of Fig. 8. From the results of τ₀ and τ₁, we can obtain the energy difference between the values of two potential minima ϵ, which is significant for the MD experiment.

Fig. 7.

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	Fig. 7. (color online) (a) Part of the original data of Vout measured at Ibias/2I0 = 0.97, ϕsq = 0.206, ϕCJJ = −0.268, and ϕx = −0.4984. (b) Digitized original data. Distribution and exponential fitting of (c) and (d) , where and are residence times at states ‘1’ and ‘0’, respectively.

Fig. 8.

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	Fig. 8. (color online) Plot of energy difference ϵ/kBT versus δϕx (where δϕx = ϕx − 0.5). Insert shows τ1 (red) and τ0 (blue) varying with δϕx at ϕCJJ = −0.268, each point is obtained by the method shown in Fig. 7.

The CJJ RF-SQUID with overdamped DC-SQUID for continuous readout have been designed and fabricated successfully using our Nb-based lift-off process. The parameters are simulated by InductEX and verified at low temperature. Moreover, the potential energy of the RF-SQUID at ϕ_CJJ = −0.268 is obtained by monitoring its fluxoid state continuously. These experimental results lay a solid foundation for our next MD experiment.