Two-Stage Cryogenic Cooler

#	Strategic Driver	Alignment and Targets
1	To develop a continuous closed loop cooling system to achieve lower base temperatures and higher cooling power to below four millikelvin to support quantum technology research.	The 3TCC roadmap is aligned to this strategic driver. With the supplement of a pulse tube precooler (4PTC), the system can achieve precooling temperatures of four kelvin, to allow cycling of helium down to reach as low as four millikelvin.
2	To develop a dry cryogen-free system to reduce the cost of ownership and operational complexity through precooling, automation, and diagnostics.	The 3TCC roadmap is aligned to this strategic driver. A pulse tube precooler eliminates the requirement for liquid helium baths. The closed loop system allows for lower cost due to higher efficiency and continuous operations and not limited by helium supplies due to constant reuse.
3	To develop modular upgradable architectures that can support rapid change over time between experiments and long-term scalability as quantum hardware evolves.	The 3TCC roadmap is aligned to this strategic driver. Through systems such as produced by Oxford instruments, dilution refrigeration systems can be scaled to support size of laboratory and match level of research.

Company	Country	Model	Base Temperature	Cooling power at 100 $\mathrm{[mK]}$	Diameter	System Cooldown (From Room Temperature)
BLUEFORS	Finland	SD	30 mK	250 µW	148 mm (MXC flange)	12 h
BLUEFORS	Finland	LH250	< 10 mK	> 250 µW	294 mm (MXC flange)	< 24 h
BLUEFORS	Finland	LD250	10 mK	250 µW	294 mm (MXC flange)	24 h
BLUEFORS	Finland	LD400	10 mK	400 µW	294 mm (MXC flange)	24 h
BLUEFORS	Finland	LD400sl	10 mK	500 µW	294 mm (MXC flange)	22 h
BLUEFORS	Finland	XLD400sl	< 10 mK	450 µW	500 mm (MXC flange)	23 h
BLUEFORS	Finland	XLD1000sl	10 mK	> 1000 µW	500 mm (MXC flange)	29 h
Oxford Instruments	UK	ProteoxS	< 10 mK	> 300 µW	205 mm (sample space)	20 h (without magnet)
Oxford Instruments	UK	ProteoxMX	< 10 mK	> 450 µW	360 mm (sample space)	8 h
Oxford Instruments	UK	ProteoxLX	< 7 mK	> 850 µW	530 mm (sample space)	6 h
Oxford Instruments	UK	Proteox5mK	5 mK	> 850 µW	360 mm (sample space)	4 h
Oxford Instruments	UK	KelvinoxTLM	≤ 15 mK	≥ 400 µW	N/A	12 h
Leiden Cryogenics BV	Netherlands	CF-CS110-500	< 10 mK	> 500 µW	120 mm (sample space)	8 h
Leiden Cryogenics BV	Netherlands	CF-CS110-1000	< 8 mK	> 1000 µW	120 mm (sample space)	8 h
Leiden Cryogenics BV	Netherlands	CF-CS110-1500	< 7 mK	> 1500 µW	120 mm (sample space)	8 h

Symbol	Description	Units	Reference value(s)	References
$\dot{n}_3$	$^{3} H e$ molar flow rate through the DR loop	$\mathrm{[mol\,s^{-1}]}$	$0.1 \text{–} 2\ \mathrm{mmol\,s^{-1}}$	[25, p. 182]
$T_{c}$	Mixing chamber temperature	$\mathrm{[K]}$	$10 \text{–} 25\ \mathrm{mK}$	[22] [19]
$T_{h}$	Heat reject (ambient) temperature	$\mathrm{[K]}$	$300\ \mathrm{K}$	–
$\dot{W}_\text{in}$	Wall-plug electrical power	$\mathrm{[W]}$	$10 \text{–} 25\ \mathrm{kW}$	[3]

Param	Derivative	Ref. value	Sensitivity	Normalized
$\dot{n}_3$	$\frac{84\, T_c T_h}{\dot{W}_\text{in}}$	1 $\mathrm{[mmol\,s^{-1}]}$	2.52e-2 $\mathrm{[mol^{-1}\,s^{-1}]}$	1
$T_{c}$	$\frac{\dot{n}_3\, 84\, T_h}{\dot{W}_\text{in}}$	15 $\mathrm{[mK]}$	1.68e-3 $\mathrm{[K^{-1}]}$	1
$T_{h}$	$\frac{\dot{n}_3\, 84\, T_c}{\dot{W}_\text{in}}$	300 $\mathrm{[K]}$	8.4e-8 $\mathrm{[K^{-1}]}$	1
$\dot{W}_\text{in}$	$-\frac{\dot{n}_3\, 84\, T_c T_h}{\dot{W}_\text{in}^2}$	15 $\mathrm{[kW]}$	-1.68e-9 $\mathrm{[W^{-1}]}$	-1

Symbol	Description	Units	Reference value(s)	References
$P_\text{4DR}$	Cooling power provided by the DR loop at $100\ \mathrm{mK}$	$\mathrm{[W]}$	$100 \text{–} 1000\ \mathrm{\mu W}$	[10] [13] [43]
$\dot{n}_3$	$^{3} H e$ molar flow rate through the DR loop	$\mathrm{[mol\,s^{-1}]}$	$0.1 \text{–} 2\ \mathrm{mmol\,s^{-1}}$	[25, p. 182]
$T_\text{ex}$	Temperature of the liquid leaving the last heat exchanger and entering the mixing chamber	$\mathrm{[K]}$	$3 \text{–} 24\ \mathrm{mK}$	[25, p. 166-167]

Param	Derivative	Value	Sensitivity	Normalized
$P_\text{4DR}$	$\frac{1}{190\,\dot{n}_3\,\sqrt{\frac{P_\text{4DR}}{95\,\dot{n}_3} + \frac{11\,T_\text{ex}^2}{95}}}$	420 $\mathrm{[nW]}$	1316 $\mathrm{[K\,W^{-1}]}$	0.138
$\dot{n}_3$	$-\frac{P_\text{4DR}}{190\,\dot{n}_3^{2}\,\sqrt{\frac{P_\text{4DR}}{95\,\dot{n}_3} + \frac{11\,T_\text{ex}^2}{95}}}$	1 $\mathrm{[mmol\,s^{-1}]}$	-0.552 $\mathrm{[K\,\,s\,\,mol^{-1}]}$	-0.138
$T_\text{ex}$	$\frac{11\,T_\text{ex}}{95\,\sqrt{\frac{P_\text{4DR}}{95\,\dot{n}_3} + \frac{11\,T_\text{ex}^2}{95}}}$	10 $\mathrm{[mK]}$	0.289 $\mathrm{[1]}$	0.724

Limitation	Description	Current R&D Project	Relevant FOMs
Scalability	Scalability is a critical R&D focus in quantum computing, since large-scale quantum processors require significantly more cooling power, wiring density, and thermal anchoring than conventional dilution refrigerators can support. Leading industry players are developing ultra-large, multi-PTC dilution refrigerators specifically to handle the thermal and mechanical demands of thousands-to-millions of qubits.	IBM’s “Goldeneye” is an experimental, non-commercial dilution refrigerator designed by IBM to test the feasibility of scaling cryogenic infrastructure for future fault-tolerant quantum computers. The system is 1.5 meters in diameter and uses multiple pulse-tube cryocoolers to achieve a stable base temperature of ~25 mK across a very large volume [19].	Productivity, COE, Cooling power
Scarcity of Helium Isotopes	The long-term scalability of dilution refrigeration is constrained by the global shortage of helium isotopes, especially helium-3 (³He). ³He is extremely rare in nature and is produced almost exclusively as a byproduct of tritium decay in nuclear weapons programs and certain research reactors. As a result, global supply is limited, expensive, and politically sensitive, while demand from quantum computing and similar fields continues to rise. Helium-4 (⁴He), while naturally occurring, is also supply-constrained; commercial production depends heavily on extraction from a small number of natural-gas fields, making the supply chain vulnerable to geopolitical disruptions. This scarcity has driven a wave of R&D aimed at reducing ³He inventory, improving heat-exchanger efficiency, and developing alternative cryogenic technologies that either use less helium or eliminate helium isotopes entirely.	Multiple cryogenics groups are developing low-charge dilution refrigerators, or similar equipment that significantly reduced ³He flow rates while maintaining usable cooling power for quantum technologies [15], [39].	Productivity, Cooling power
Thermal Conductance	Large-scale quantum processors require hundreds to thousands of microwave and control lines, each of which introduces unwanted heat into the dilution refrigerator. The thermal conductance of traditional coaxial wiring is now a primary scaling bottleneck for 3TCC systems usage in QC. As qubit counts increase, conventional metal coax cables overwhelm the cooling power at the 4 K, 1 K, and mK stages.	Delft Circuits is currently producing the next generation flex-cable wiring system that significantly reduce thermal load per signal line while maintaining high-bandwidth performance. This enables much higher wiring density with substantially reduced conductive heat leakage into the 4 K, 1 K, and mK stages of dilution refrigerators. The company is continuing to develop their products to enable large scale wiring required for the demands of QC [37].	Thermal conductance, Cooling Power, Productivity

Publication	H. Zu, W. Dai, and A. de Waele, “Development of dilution refrigerators - A review,” Cryogenics, vol. 121, 2022. doi:10.1016/j.cryogenics.2021.103390 [47]
Summary	This review traces the technical and historical evolution of the ³He–⁴He Dilution Refrigerators (DRs). It highlights major design improvements like heat exchange optimization and the shift to “dry” DRs. This is also an important article because it reviews commercial and custom-built options as well as the common challenges of using DRs.
Significance	Comprehensive reference to the current state of DR technologies, its theoretical basis, and applications in current industry. This is a great reference article to check the current market DRs, how the DRs have evolved over time, and the basics of how the system works.

Publication	K. Uhlig, “Dry dilution refrigerator with pulse-tube precooling,” Cryogenics, vol. 44, pp. 53–57, 2004. doi: 10.1016/j.cryogenics.2003.07.007 [46]
Summary	This article reports the design and testing on a cryogen-free (“dry”) dilution refrigerator that uses a pulse-tube cryocooler (PTC) instead of a helium bath. This article goes over the results of testing and includes a summarization of the key advantages of this approach.
Significance	Demonstrates that a PTC precooling method paired with a DR can match the traditional helium bath systems while being more practical and economical. This advancement significantly brings down the maintenance and other drawbacks of the wet systems. This article set the foundation for today’s commercial “dry” cryocoolers.

Publication	K. Uhlig, “3He/4He dilution refrigerator with pulse-tube refrigerator precooling,” Cryogenics, vol. 42, pp. 73–77, 2002. doi: S0011-2275(02)00002-4 [45]
Summary	Describes first-generation version of the DR precooled by a two-stage PTC, achieving a final temperature of 15 mK. This combines a counterflow heat exchanger with a Joule-Thomson expansion stage for condensation of the ³He–⁴He mixture. This model also demonstrates an automated cooldown, stable operation, and minimal vibrations.
Significance	Represents one of the first successful integrations of PTCs with DRs, proving that cryogen-free systems can reach the milliKelvin range. This was a large jump in the current market for cryocoolers and their cumbersome maintenance and upkeep requirements.

Patent	P. Gunmann, S. B. Olivadese, and J. M. Chow, “Thermalization of cryogenic quantum circuits,” U.S. Patent 10833240B2, Nov. 10, 2020 [8]
Summary	Focuses on efficient thermal anchoring and wiring for superconducting qubit systems within dilution refrigerators. It optimizes heat transfer between microwave lines and cold stages while suppressing vibration and electromagnetic noise.
Significance	Connects dilution-refrigerator engineering to practical quantum-computing thermal management.
CPC Codes	H10N60/01 Manufacture or treatment. H01L39/24 B32B15/018 Layered products comprising a layer of metal all layers being exclusively metallic one layer being formed of a noble metal or a noble metal alloy. H01L23/373 Cooling facilitated by selection of materials for the device or materials for thermal expansion adaptation, e.g. carbon. H01L39/025 H10N60/805 Constructional details for Josephson-effect devices. H01L23/3735 Laminates or multilayers, e.g. direct bond copper ceramic substrates. H10N60/815 Containers; Mountings for Josephson-effect devices.

Patent	G. Batey, P. G. Teleberg, A. Matthews, et al., “Cryogenic cooling apparatus and method,” U.S. Patent 9816750B2, Nov. 14, 2017 [24]
Summary	Describes a cryogen-free (“dry”) DR using a mechanical PTC and an automated sample loading system. This patent covers a more durable, faster, and safer system.
Significance	Embodies modern, commercially deployed cryogen-free dilution technology used in quantum-computing cryostats.
CPC Codes	F25D29/001 Arrangement or mounting of control or safety devices for cryogenic fluid systems. F25B9/12 Compression machines, plants or systems, in which the refrigerant is air or other gas of low boiling point using ³He-⁴He dilution. F25J1/0276 Laboratory or other miniature devices.

Patent	R. C. Black, J. P. Hilton, G. Rose, et al., “Systems, methods, and apparatus for cryogenic refrigeration,” U.S. Patent 20100281885A1, Nov. 11, 2010 [38]
Summary	Describes a cryogenic refrigeration system that integrates a PTC with a DR, specifically using the PTC as a pre-component to feed the DR via ³He-⁴He isotopes. This patent specifies the step-down multiple cooling stages from PTC to DR to the DR’s mixing chamber. Specifically cites the use in superconducting computer systems.
Significance	This patent was an innovation milestone of the dry dilution refrigerator in the industry. It was also a strategic submission for use of this specific system for the quantum-computing horizon.
CPC Codes	F25B9/12 Compression machines, plants or systems, in which the refrigerant is air or other gas of low boiling point using ³He-⁴He dilution. F25B9/10 Compression machines, plants or systems, in which the refrigerant is air or other gas of low boiling point with several cooling stages. G06F1/20 Cooling means. H05K7/20372 Cryogenic cooling; Nitrogen liquid cooling. H10N60/80 Constructional details. G06F2200/201 Cooling arrangements using cooling fluid.

Patent	F. A. Staas and A. P. Severijns, “3He/4he dilution refrigerator,” U.S. Patent 4136531A, Jan. 30, 1979 [1]
Summary	Introduces a dual-mixing chamber ³He–⁴He DR, achieving lower base temperatures by circulating ³He through a “superleak”.
Significance	Foundation of intellectual property for the continuous ³He–⁴He DR systems still used in low-temperature physics and quantum device cooling today.
CPC Codes	F25B9/12 Compression machines, plants or systems, in which the refrigerant is air or other gas of low boiling point using ³He-⁴He dilution.

			SD	SD1							SD2						SD3				SD4
				Quantum Computer							Environmental Control System						Thermal Control System				Pulse Tube Cryocooler					Dilution Refrigerator				Helium Isotopes
			Quantum Computing	Qubit Platform/Processor	Quantum Control Electronics	Classical Computer & Error Correction	Quantum Interconnects & I/O	Software Stack	Infrastructure & Power	Environmental Control System	Thermal Control System	Vacuum System	Electromagnetic Shielding System	Vibration Isolation system	Thermal Anchoring & Wiring System	Monitoring & Feedback System	Pulse Tube Cooler	Dilution Refrigeration	Helium Isotopes	Pump	Condenser	Regenerator	Pulse tube	Heat Exchanger	Valve	Condenser	Mixing Chamber	Heat Exchangers	Still	Helium 3	Helium 4
SD		Quantum Computing
SD1	Quantum Computer	Qubit Platform/Processor
		Quantum Control Electronics
		Classical Computer & Error Correction
		Quantum Interconnects & I/O
		Software Stack
		Infrastructure & Power
		Environmental Control System
SD2	Environmental Control System	Thermal Control System
		Vacuum System
		Electromagnetic Shielding System
		Vibration Isolation system
		Thermal Anchoring & Wiring System
		Monitoring & Feedback System
SD3	Thermal Control System	Pulse Tube Cooler
		Dilution Refrigeration
		Helium Isotopes
		Pump
SD4	Pulse Tube Cryocooler	Condenser
		Regenerator
		Pulse tube
		Heat Exchanger
		Valve
	Dilution Refrigerator	Condenser
		Mixing Chamber
		Heat Exchangers
		Still
	Helium Isotopes	Helium 3
	Helium Isotopes	Helium 4

$P_\text{4DR}$	Cooling power produced by 4DR, in $\mathrm{[W]}$ .
$\dot{n}_\text{3}$	Molar flow of Helium isotopes, in $\mathrm{[mol\,s^{-1}]}$ .
$T_\text{min}$	Minimum temperature at mixing chamber, in $\mathrm{[K]}$ .
$T_\text{exch}$	Temperature going into mixing chamber from heat exchangers, in $\mathrm{[K]}$ .
$95$	Constant molar enthalpy difference at dilute at the mixing chamber, in $\mathrm{[J\,mol^{-1}\,K^{-2}]}$ .
$11$	Constant molar enthalpy before entering the mixing chamber, in $\mathrm{[J\,mol^{-1}\,K^{-2}]}$ .

$\dot{Q}_\text{req}$	Required heat load for system, in $\mathrm{[W]}$ .
$A$	Area cross section of material, in $\mathrm{[cm^{2}]}$ .
$\kappa$	Thermal conductivity coefficient, in $\mathrm{[W\,cm^{-1}\,K^{-1}]}$ .
$\nabla T$	Temperature gradient per unit area, in $\mathrm{[K\,cm^{-1}]}$ .

$\dot{Q}_\text{load}$	Heat load from system, in $\mathrm{[W]}$ .
$\dot{Q}_\text{leak}$	Conduction heat load through supports, wiring, etc., in $\mathrm{[W]}$ .
$\dot{Q}_\text{parasitic}$	Non-ideal contributions heat load, in $\mathrm{[W]}$ .

$\dot{n}_\text{3}$	Molar flow of helium isotopes, in $\mathrm{[mol\,s^{-1}]}$ .
$T_\text{min}$	Minimum temperature at mixing chamber, in $\mathrm{[K]}$ .
$84$	Constant molar enthalpy difference between dilute and concentrate phase in mixing chamber, in $\mathrm{[J\,mol^{-1}\,K^{-2}]}$ .

¶ Two-Stage Cryogenic Cooler

¶ 1. Roadmap Overview

¶ 2. Design Structure Matrix (DSM)

¶ 3. Roadmap Model using OPM

¶ 4. Figures of Merit (FOM)

¶ 4.1. Minimum Achievable Temperature

¶ 4.2. Thermal Conductance

¶ 4.3. Cooling Power

¶ 4.4. Coefficient of Performance

¶ 4.5. Maximum Theoretical Efficiency

¶ 4.6. Percent Carnot Efficiency

¶ 4.7. Productivity

¶ 5. Alignment of Strategic Drivers

¶ 6. Positioning of Organization vs. Competition

¶ 7. Technical Model: Sensitivities and Tradespace

¶ 7.1. Percent Carnot Efficiency

¶ 7.2. Minimum Achievable Temperature

¶ 7.3. Tradespace

¶ 8. Financial Model: Technology Value (𝛥NPV)

¶ 9. List of R&D Projects and Prototypes

¶ 10. Key Publications, Presentations and Patents

¶ 10.1. Publications & Presentations

¶ 10.2. Patents

¶ 11. Technology Strategy Statement

¶ 12. References

$\dot{W}_\text{input}$	Input power to 3TCC, in $\mathrm{[W]}$ .
$P_\text{4DR}$	Cooling power produced by 3TCC, in $\mathrm{[W]}$ .
$\dot{W}_\text{4PTC}$	Power consumed by the pulse tube refrigerator, in $\mathrm{[W]}$ .
$\dot{W}_\text{4DR}$	Power consumed by dilution refrigerator, in $\mathrm{[W]}$ .
$\dot{W}_\text{anc}$	Power consumed by ancillary equipment, in $\mathrm{[W]}$ .

$T_{c}$	Temperature of the cold reservoir, in $\mathrm{[K]}$ .
$T_{h}$	Temperature of the hot reservoir, in $\mathrm{[K]}$ .

$COP_\text{3TCC}$	The coefficient of performance, unitless.
$COP_\text{Carnot}$	The maximum theoretical efficiency, unitless.

$C_\text{total}$	Total costs including energy and maintenance, in $\mathrm{[USD]}$ .
$t_\text{op}$	Total operating time, in $\mathrm{[hour]}$ .

Param	Derivative	Ref. value	Sensitivity	Normalized
$x_{i}$	$\frac{\partial \eta_\text{II}}{\partial x_i}$	$x_{i}^{0}$	$\left.\frac{\partial \eta_\text{II}}{\partial x_i}\right\|_{\mathbf{x^0}}$	$\frac{x_i^0}{\eta_\text{II}(\mathbf{x^0})}\cdot\left.\frac{\partial \eta_\text{II}}{\partial x_i}\right\|_{\mathbf{x^0}}$
$\dot{n}_3$	$\frac{84\, T_c T_h}{\dot{W}_\text{in}}$	1 $\mathrm{[mmol\,s^{-1}]}$	2.52e-2 $\mathrm{[mol^{-1}\,s^{-1}]}$	1
$T_{c}$	$\frac{\dot{n}_3\, 84\, T_h}{\dot{W}_\text{in}}$	15 $\mathrm{[mK]}$	1.68e-3 $\mathrm{[K^{-1}]}$	1
$T_{h}$	$\frac{\dot{n}_3\, 84\, T_c}{\dot{W}_\text{in}}$	300 $\mathrm{[K]}$	8.4e-8 $\mathrm{[K^{-1}]}$	1
$\dot{W}_\text{in}$	$-\frac{\dot{n}_3\, 84\, T_c T_h}{\dot{W}_\text{in}^2}$	15 $\mathrm{[kW]}$	-1.68e-9 $\mathrm{[W^{-1}]}$	-1

Param	Derivative	Value	Sensitivity	Normalized
$x_{i}$	$\frac{\partial T_\text{min}}{\partial x_i}$	$x_{i}^{0}$	$\left.\frac{\partial T_\text{min}}{\partial x_i}\right\|_{\mathbf{x^0}}$	$\frac{x_i^0}{T_\text{min}(\mathbf{x^0})}\cdot\left.\frac{\partial T_\text{min}}{\partial x_i}\right\|_{\mathbf{x^0}}$
$P_\text{4DR}$	$\frac{1}{190\,\dot{n}_3\,\sqrt{\frac{P_\text{4DR}}{95\,\dot{n}_3} + \frac{11\,T_\text{ex}^2}{95}}}$	420 $\mathrm{[nW]}$	1316 $\mathrm{[K\,W^{-1}]}$	0.138
$\dot{n}_3$	$-\frac{P_\text{4DR}}{190\,\dot{n}_3^{2}\,\sqrt{\frac{P_\text{4DR}}{95\,\dot{n}_3} + \frac{11\,T_\text{ex}^2}{95}}}$	1 $\mathrm{[mmol\,s^{-1}]}$	-0.552 $\mathrm{[K\,\,s\,\,mol^{-1}]}$	-0.138
$T_\text{ex}$	$\frac{11\,T_\text{ex}}{95\,\sqrt{\frac{P_\text{4DR}}{95\,\dot{n}_3} + \frac{11\,T_\text{ex}^2}{95}}}$	10 $\mathrm{[mK]}$	0.289 $\mathrm{[1]}$	0.724