A wide range of scientific advancements rely on achieving temperatures below 1 Kelvin, where thermal and electrical noise is suppressed, and unique quantum phenomena become accessible. Applications like quantum computing, communication, dark matter searches, and early universe observation necessitate millikelvin temperatures, often achieved using dilution refrigerators pre-cooled to ~4 K by pulse tube refrigerators (PTRs). Low-frequency PTRs (around 1 Hz) are typically used for pre-cooling and consume considerable power (approximately 10 kW for 1 W of cooling near 4 K). PTRs are thermoacoustic systems where heat is pumped via cyclic compression and expansion of helium gas. The acoustic network in a PTR creates traveling-wave phasing without cold moving parts, making them maintenance-free and vibrationally stable. The core component, the regenerator, stores and exchanges heat over many cycles to achieve significant temperature reduction. Despite their widespread use, PTRs are inefficient, particularly during cooldown. This inefficiency stems from the fact that current PTR designs are optimized for performance solely at their base temperature, ignoring the significant temperature variations experienced during the cooldown process. This leads to slower cooldown and wasted power at higher temperatures, directly impacting the efficiency of low-temperature experiments. A significant cooldown time increase of 3x or more offers increased measurement throughput and faster hardware modification. Oversized PTRs are common to achieve adequate cooldown speed, which results in much more cooling power at base temperature than is needed. Dynamic optimization of PTRs offers significant energy efficiency improvements, allowing the use of smaller, lower-power PTRs while preserving or improving cooldown speed, which is highly desirable for broader adoption of ultralow-temperature technologies.

While pulse tube refrigerators have a long history of use, significant inefficiencies persist, particularly regarding their optimization solely for base temperature operation. Previous work on rapid cooldown has largely focused on high-frequency coolers (tens of Hz), which operate through different acoustic mechanisms than the low-frequency systems examined in this study and thus require different optimization strategies. High-frequency PTRs directly connect to linear compressors, operating near resonance. The existing literature includes one study concerning cooldown speed in low-frequency PTRs; however, that study utilized a custom cooler that only cooled to 100 K, demonstrating limited improvement (<10%). Furthermore, this prior work lacked an analysis explaining the need for temperature-dependent acoustic optimization. No previous rapid cooldown studies have been conducted on 4 K PTRs, despite the high demand for this type of cryocooler.

This study develops analytical expressions to guide the temperature-dependent optimization of low-frequency PTRs and validates them experimentally using a common 4 K cooler (Cryomech PT407-RM with a CP2850 compressor). A simplified model of a single-stage orifice pulse tube refrigerator (OPTR) is used to analyze acoustic power delivery from the compressor to the PTR's terminating network (E2). This model treats the compressor and rotary valve system as either a pressure or flow-rate source, depending on whether a pressure relief valve is open or closed. The model includes the electrical circuit analog of the OPTR and its acoustic components. An electrical circuit analogy is used to simplify analysis, representing pressure and flow rate with voltage and current, respectively, and components like compliance and resistance with their electrical counterparts.The analysis reveals that the acoustic network must be tuned with temperature to maximize E2. The temperature dependence originates from the regenerator's attenuation of the volume flow rate, which is a function of temperature. The terminating networks (main impedance R and compliance C), consisting of a needle valve and a reservoir, are identified as key tunable components. The research replaces the manufacturer's needle valves with custom ones equipped with stepper motors for precise adjustment, allowing for dynamic control of the main impedance. Stepper motors also adjust the rotary valve speed, modifying the oscillation frequency. 'Cold impedance' refers to the setting that optimizes low-temperature performance, while 'partly open' and 'more open' represent progressively lower impedance settings. To assess cooling power (Q) at elevated temperatures, resistive heaters are mounted on the heat exchangers. Measurements quantify the cooling power at different temperatures under varying acoustic settings (cold acoustics, partly open, more open). To improve heat transfer between the stages, two thermosiphons (filled with ethane and nitrogen) are installed between heat exchangers. Cooldown speed is measured with and without dynamic acoustic optimization and heat transfer. To determine the impact of load size, experiments are conducted with varying virtual masses (through controlled heating) on the second stage. Measurements focus on the acoustic power (E2) and cooling power (Q) at various temperatures and acoustic settings, allowing for a quantitative comparison of the impacts of dynamic optimization, thermosiphon utilization, and load size on cooldown speed. Measurements of pressure oscillations, acoustic power, and temperature are taken at multiple points within the PTR. Large-stroke loss, characterized by the helium piston stroke approaching or exceeding the buffer tube length, is hypothesized as a key loss mechanism. To quantify this loss, the helium piston stroke in the first-stage buffer tube is estimated. The study analyzes the discrepancy between acoustic power and cooling power at various frequencies, associating it with loss mechanisms like large-stroke loss and boundary layer heat transport.

The study's key findings demonstrate that dynamic acoustic optimization significantly accelerates the cooldown process of pulse tube refrigerators. Specifically: 1. **Analytical Modeling:** Analytical expressions derived in the study successfully capture the temperature-dependent behavior of acoustic power in the PTR. These expressions highlight the necessity for dynamic acoustic tuning to maximize the power delivered from the compressor to the refrigerator as the temperature changes during the cooldown process. 2. **Enhanced Cooling Power:** The dynamic optimization strategy led to substantial increases in cooling power at elevated temperatures. At 295 K (ambient temperature), the first and second stages of the PTR exhibited 1.7 and 1.9 times their cold-acoustic (base-temperature optimized) cooling powers, respectively. With the implementation of a heat transfer mechanism between the stages (thermosiphons), enabling the utilization of the first stage's increased cooling power to assist the second stage, the second stage's cooling power increased to 7.3 times its cold-acoustic value at 295 K. 3. **Dramatic Cooldown Speed Improvement:** When dynamic acoustic optimization and thermosiphons were used, the cooldown speed to 4 K of the second stage increased by a factor of 3.5 compared to the cold-acoustic case. Experiments involving varying effective mass (physical and virtual) demonstrate a consistent speed increase between 2.6 and 3.5 times when heat transfer between stages is utilized. Even without the benefit of heat transfer, a speed increase of 1.7 times was achieved. 4. **Loss Mechanisms Identified:** Analysis revealed that the cooling power limitations at high temperatures stem primarily from loss mechanisms within the buffer tube, specifically a phenomenon termed 'large-stroke loss.' This loss arises from excessive helium piston movement within the buffer tube, leading to significant heat transport and reducing the overall cooling efficiency. The study suggests that redesigning the buffer tube to mitigate large-stroke loss could further enhance cooldown speed, likely surpassing the 3.5x improvement already observed.

The research successfully demonstrates the effectiveness of dynamic acoustic optimization in significantly accelerating the cooldown process of pulse tube refrigerators. The analytical framework accurately predicts the temperature-dependent behavior of acoustic power and guides the optimization strategy. The experimental results confirm the significant improvement in both cooling power and cooldown speed, with a substantial increase in the latter (1.7 to 3.5 times). The identification of large-stroke loss as a primary limiting factor suggests pathways for future improvements. The findings are relevant across diverse low-temperature research fields, offering the potential to substantially reduce cooldown times in experiments utilizing dilution refrigerators. This has significant implications for enhancing measurement throughput and shortening the time required for experimentation and hardware adjustments.

This study presents a significant advancement in cryogenic technology by demonstrating the effectiveness of dynamic acoustic optimization for pulse tube refrigerators. The results show a substantial increase in cooldown speed (1.7–3.5 times faster), achieved through temperature-dependent tuning of acoustic parameters. The observed limitations due to large-stroke loss in the buffer tube suggest avenues for future research to further improve the efficiency and cooldown speed of these critical components. The broader implications are significant for accelerating scientific discovery in fields requiring ultralow-temperature environments.

The study primarily focuses on a specific commercial PTR model. While the findings are expected to generalize to other similar low-frequency PTRs, further investigation is needed to confirm the extent of generalizability. The analysis of loss mechanisms, specifically large-stroke loss, relies on estimations and approximations. More detailed investigation could refine the understanding of these losses and identify potential strategies for mitigation.