Quantum bath suppression in a superconducting circuit by immersion cooling M. Lucas1 A. V. Danilov2 L. V. Levitin1 A. Jayaraman2 A. J. Casey1 L. Faoro3 A. Ya. Tzalenchuk14 S. E. Kubatkin2 J. Saunders1 and S. E. de Graaf4

2025-05-02 0 0 4.35MB 21 页 10玖币

侵权投诉

Quantum bath suppression in a superconducting circuit by immersion cooling

M. Lucas1, A. V. Danilov2, L. V. Levitin1, A. Jayaraman2, A. J. Casey1, L.

Faoro3, A. Ya. Tzalenchuk1,4, S. E. Kubatkin2, J. Saunders1, and S. E. de Graaf4∗

1Physics Department, Royal Holloway University of London, Egham, United Kingdom

2Department of Microtechnology and Nanoscience MC2,

Chalmers University of Technology, SE-412 96 G¨oteborg, Sweden

3Google Quantum AI, Google Research, Mountain View, CA, USA and

4National Physical Laboratory, Teddington TW11 0LW, United Kingdom

Quantum circuits interact with the environ-

ment via several temperature-dependent degrees

of freedom. Yet, multiple experiments to-date

have shown that most properties of superconduct-

ing devices appear to plateau out at T ≈50 mK –

far above the refrigerator base temperature. This

is for example reﬂected in the thermal state popu-

lation of qubits [1–4], in excess numbers of quasi-

particles [5], and polarisation of surface spins [6]

– factors contributing to reduced coherence. We

demonstrate how to remove this thermal con-

straint by operating a circuit immersed in liquid

3He. This allows to eﬃciently cool the decohering

environment of a superconducting resonator, and

we see a continuous change in measured physical

quantities down to previously unexplored sub-mK

temperatures. The 3He acts as a heat sink which

increases the energy relaxation rate of the quan-

tum bath coupled to the circuit a thousand times,

yet the suppressed bath does not introduce addi-

tional circuit losses or noise. Such quantum bath

suppression can reduce decoherence in quantum

circuits and opens a route for both thermal and

coherence management in quantum processors.

Thermal management is a central problem in computer

engineering. This is true for classical processors, where

inability to remove heat from transistors resulted in a

stalled clock frequency for the last 20 years [7], and this is

also true for superconducting quantum processors where

various temperature-dependent factors limit their coher-

ence. Scaling up quantum processors [8] inevitably exac-

erbates this problem and minimising the impact from all

decoherence mechanisms at play is essential for achieving

fault-tolerant quantum computing [9,10].

Cooling of devices operated in cryogenic vacuum rep-

resents a signiﬁcant challenge because all solid-state cool-

ing pathways – through quasiparticles in the supercon-

ducting material and phonons both there and in the

substrate – become ineﬃcient. A large body of ex-

perimental data indicates physical observables becoming

temperature-independent below ∼50 mK, well above the

dilution refrigerator base temperature of ∼10 mK. This

is consistently seen in qubit state population [1–4], qubit

∗sdg@npl.co.uk

coherence times [11], frequency ﬂicker noise [12,13], sur-

face electron spin polarisation [6], and qubit ﬂux noise

[14]. Improvement may be achieved by reducing the heat

load from various external sources, such as ionising radi-

ation [9,15], cosmic particles [16,17], and high frequency

photons [5,18,19], by careful shielding and ﬁltering. This

approach has had a lot of success over the years and is

still a subject of intense research and technical devel-

opment. However, further progress cannot be achieved

without taking due care of the circuit’s material environ-

ment, for which, unexpectedly, further cooling can lead

to increased noise and decoherence.

Although naively one would think that cooling a su-

perconducting circuit to the lowest possible temperature

would freeze out any noisy environment, this is only

partly true. To suppress decoherence originating from

equilibrium quasiparticles [5] or residual thermal qubit

excitations [1–4] the temperature shall be signiﬁcantly

below relevant energy scales, i.e T300 mK for a device

operating at 7 GHz. However, well below these temper-

atures other decoherence mechanisms, in particular that

associated with the dielectric environment of the devices,

come into play. Dielectrics contain defects, which act

as two-level systems (TLS) and counter-intuitively, noise

due to TLS increases upon cooling [12,20].

Here we present a radically diﬀerent route to approach

these challenges by immersion cooling of a superconduct-

ing circuit in liquid 3He. We show that 3He provides an

eﬃcient heat sink for the circuit environment and dra-

matically increases the energy relaxation rate of the TLS

bath, while otherwise appearing essentially inert to the

quantum circuit itself. This opens up multiple ways in

which signiﬁcant improvement in circuit coherence may

be achieved, both by cooling and by suppressing coher-

ence in the noisy environment. Future optimisation of

such quantum bath suppression using 3He may lead to

signiﬁcantly reduced noise also at dilution refrigerator

base temperatures.

Experimentally, our approach is to use planar super-

conducting resonators, which have emerged as a conve-

nient platform to interrogate the decohering environment

[6,12,21–26]. In particular, the amplitude of the low-

frequency 1/f frequency noise is very sensitive to the TLS

temperature [20]. Additionally, the temperature of the

surrounding spin bath reveals itself in the electron spin

resonance (ESR) spectrum measured via ﬁeld-dependent

losses of the same resonators. When the resonator is im-

arXiv:2210.03816v1 [quant-ph] 7 Oct 2022

FIG. 1. Immersion of a superconducting quantum circuit in liquid 3He. a) In vacuum the environment of the

quantum circuit is poorly thermalised to the cold plate of the refrigerator. b) When immersed in liquid 3He, the cooling of the

environment is signiﬁcantly improved by 3He acting as a heat sink. c) A superconducting resonator, used in our measurements,

taking the temperature of the decohering environment of quantum circuits. d) Experimental setup: The immersion cell

containing the sample is thermally anchored to an adiabatic nuclear demagnetisation stage that reaches T= 400 µK. The

nuclear stage is mounted to the mixing chamber plate of a dry dilution refrigerator. e) Energy relaxation pathway from the

TLS bath to the cold plate via 3He and silver sinter. The link between TLS medium and liquid 3He is the bottleneck for further

quantum bath suppression.

mersed in 3He, we observe improved thermalisation of

the TLS in the noise measurements and of the spin bath

in the ESR measurements, as illustrated in Figure 1.

Derived from recent advances in ultra-low temperature

technology and the cooling of electronic systems to sub-

mK temperatures [27,28] we construct an immersion cell

suitable for a superconducting quantum circuit. Cooling

is achieved by placing the circuit, in our case a NbN

superconducting resonator [29] on a sapphire substrate,

inside the immersion cell, as shown in Figure 1d. The

superﬂuid leak-tight copper cell with RF feedthroughs

and extensive RF ﬁltering provides a well controlled mi-

crowave environment. It is thermally anchored to the

experimental plate of an adiabatic nuclear demagnetisa-

tion refrigeration (ANDR) stage attached via a supercon-

ducting heat switch to the lowest temperature plate (10

mK) of a dry dilution refrigerator [30]. The experimental

plate of the ANDR can reach temperatures of ≈400 µK,

as measured using SQUID noise thermometry [30]. The

cell can be ﬁlled with 3He via a thin capillary. To ensure

good thermalisation of the liquid 3He to the cell’s metal

enclosure silver sinter heat exchangers are implemented

(See SI for further details). For ESR spectroscopy ex-

periments a magnetic ﬁeld (B) up to 0.5 T parallel to

the sample surface could be applied. We refer to Supple-

mental Information (SI) for details on our measurement

techniques.

Reliable thermometry is an essential pre-requisite for

interpretation of ultra-low temperature data. On-chip

ESR not only reveals the presence of unwanted surface

spins coupling to the resonator through their magnetic

moments (a source of ﬂux noise [14,31]), but also serve

as an intrinsic thermometer in the relevant temperature

range. To this end we show in Figure 2that, unlike

previous experiments on spins coupled to quantum cir-

cuits where the spin polarisation was saturated at about

T= 50 mK [6], surface spins are cooled to much lower

temperatures in the presence of 3He, with no other ap-

parent change in the ESR spectra. The measured ESR

spectrum is rather complex, consisting of many diﬀer-

ent species, and has been discussed in detail previously

[6,32]. Here we focus on the species that are most suit-

able for intrinsic thermometry at these low temperatures,

namely the two peaks labelled 1 and 3 that arise from

atomic hydrogen [6]. The hyperﬁne interaction in the

hydrogen atom results in two electronic spin transitions

separated in energy by 1.42 GHz (= 68 mK), with rel-

ative intensity that follows the Boltzmann distribution.

Thus if spins are cooled to zero temperature the tran-

sition involving the higher energy level transition (peak

3) will vanish, the trend clearly seen in ﬁgure 2in the

presence of 3He. Having established improved thermali-

sation of surface spins we now turn to the TLS bath that

couples through charge dipoles to the same circuit.

Figure 3a compares the temperature dependence of the

1/f frequency noise of a 6.45 GHz resonator with vac-

uum or 3He in the sample cell (for more data on diﬀerent

devices, see SI). Similar to many previous experiments

[12,13,33–35], in vacuum the noise increases on cooling

according to a power law T−1.5followed by saturation

FIG. 2. Cooling of surface electron spins. Continuous

wave electron spin resonance spectra of surface spins intrinsic

to a 5.85 GHz resonator, measured with an average number

of photons circulating in the resonator of hNi ≈ 200. Inset:

Normalised intensity of the hyperﬁne-split atomic hydrogen

peaks (labelled 1 and 3) versus nuclear stage noise thermome-

ter temperature. Empty symbols represent measurement in

vacuum and ﬁlled symbols in 3He. Error bars are propagated

errors from ﬁtting the peak intensities. Solid lines are the

expected peak intensities based on the thermal population of

ESR levels hyperﬁne-split by A= 1.42 GHz. The dashed line

is an estimate of the minimum sensitivity of our technique,

below which we could not detect the third peak.

to a constant level below ∼80 mK due to insuﬃcient

thermalisation.

When the cell is ﬁlled with 3He the situation is very

diﬀerent. The noise changes with fridge temperature all

the way down to 1 mK. Above 100 mK the magnitude

and temperature dependence of the noise is the same in

vacuum and in 3He, but below a certain crossover tem-

perature Tx∼80 mK the noise instead starts to de-

crease with reduced temperature according to a power

law T0.25. Remarkably, 3He immersion appears to break

the predicted [20] trend of increasing noise with cooling

(otherwise expected to persist to well below 10 µK, see

below). The noise measured at 1 mK is more than three

orders of magnitude below this expected T−1.5trend.

A further striking eﬀect of immersing the circuit in

3He is revealed in the dependence of the internal qual-

ity factor Qiof the resonators on the microwave power

(photon number, hNi), presented in Figure 4a for three

temperatures. Noticeably, 3He does not aﬀect Qiat the

single photon level, meaning that the number of TLS

present and their coupling to the resonator remains un-

changed. Both for resonators in vacuum and in 3He the

microwave excitation power increases Qi– a well-known

eﬀect of TLS saturation – but for resonators immersed

in 3He the same Qiis achieved with ∼1000 times higher

power; i.e. we ﬁnd a dramatic increase in the character-

istic TLS saturation power by three orders of magnitude.

Figure 4b showing the Qiextracted at a ﬁxed drive power

in the saturated regime indicates that there is a weak but

steady dependence (and hence cooling) down to <1 mK.

Furthermore, we also here observe a crossover occurring

around Tx∼80 mK.

A notorious challenge in noise measurements (and

more generally in operating quantum circuits requiring

long-term stability [8,36–38]) is inherent instabilities of

TLS energies on longer timescales (spectral drift). This

is particularly evident in the low power data in Figure 3a.

To circumvent this problem, we measure noise at some-

what higher photon number which saturates the most

strongly coupled ﬂuctuators [39]. Yet, we stay in a mod-

erately weak ﬁelds regime [20], as evidenced by the power

dependence of the noise shown in Figure 3b. In the low

power data in Figure 3a the measured noise level varies

on top of the general trend by a factor 2-3 during the

course of the measurement, which takes ∼6 days; how-

ever, the overall trend remains unchanged.

To understand the full body of experimental data we

ﬁrst focus on the region 100-250 mK where the noise is

well understood. Here the noise is increasing upon cool-

ing both in vacuum and in 3He, consistent with previous

observations and fully captured by the generalised tun-

neling model (GTM) [20] for interacting TLS defects.

Here both energy loss and 1/f noise arise from the

resonator coupling to a large number of coherent (near-

)resonant TLS defects. These TLS drain energy from

the resonator and dissipate it to substrate phonons, a

process that determines Qi, a measure of the average en-

ergy loss into the whole TLS bath. They also, through

their coherent coupling, mediate frequency ﬂuctuations

from the environment: the resonant coherent TLS are

subjected to thermally activated spectral diﬀusion due

to the interaction with a bath of incoherent, low energy

(EkBT) TLS (”thermal two-level ﬂuctuators”, TLF)

that incoherently ﬂip-ﬂop between two states, giving rise

to both noise and the coherent TLS width Γ2. Strongly

coupled TLS contribute more strongly to the noise and

they are also more easily saturated by microwave ﬁelds.

In the weak microwave ﬁeld regime (small hNi) the mag-

nitude of the noise is governed by the TLS dephasing

rate Γ2∝T1+µarising from the coupling to the TLF

bath (and independent of the TLS relaxation rate Γ1),

which yields a temperature dependence Sy∝T−1−2µ

[13,20] (for kBT < hν0). Here µis a small positive num-

ber characterising the nonlinear density of states of TLS

arising from their interactions. From the data in Figure

3we ﬁnd µ= 0.25, consistent with previous experiments

[12,13,33]. Since the magnitude and the temperature

dependence of the noise is the same in vacuum and in

3He above 100 mK we conclude that 3He does not inﬂu-

ence Γ2.

This leads to the remarkable conclusion that the enor-

mous change in saturation power observed, Figure 4a,

means that 3He increases the average TLS relaxation rate

Γ1∼1000 times because the critical number of photons

for saturation of the TLS bath scales as Nc∝Γ1Γ2.

We now turn to the regime at low temperatures, be-

FIG. 3. Cooling the TLS bath by immersion into 3He reduces noise. a) The magnitude of the 1/f frequency noise

power spectral density Sy(f) of a ν0= 6.45 GHz superconducting resonator evaluated at f= 0.1 Hz versus nuclear stage

temperature for two selected microwave drive powers (average photon number hNi) with the cell full of 3He (ﬁlled markers)

and empty cell (empty markers). The latter has been scaled by a factor 20 for better visualisation (see SI for unscaled version).

Each dataset is a single temperature ramp taking ≈6 days. Solid and dashed slopes show Tβin the low and high temperature

regimes respectively, with β= 0.25 and −1.5 respectively. Horizontal dashed line is a guide for the eye. b) Photon number

dependence of the noise with 3He at selected temperatures taken across shorter timescales (5 hours per temperature). Solid

lines are ﬁts to the expected dependence of the noise (∝(1 + hNi/Nc)−1/2) where the weak ﬁelds regime with a levelling-oﬀ to

a constant noise versus hNiis evident at high temperature. Full noise spectra across all timescales can be found in SI.

FIG. 4. 3He increases TLS relaxation. a) Comparison of the TLS-limited internal quality factor for a 6.26 GHz resonator

with and without 3He in the cell, for three temperatures. 3He increases the power needed to saturate to a given Qiby a factor

∼1000. b) The change in internal Q vs temperature for a ﬁxed drive power of hNi ∼ 104. c) Extracted critical photon number

Nctimes a prefactor c, from ﬁtting the Qi(N) data to 1/Qi∝ln(cNc/hNi). Solid line shows T1.25, the expected scaling of Γ2.

low Tx∼80 mK. First, we consider the implications

of a signiﬁcantly increased Γ1of the TLS bath. In di-

electrics the TLS excitation and relaxation occurs via

interaction with phonons which couple via strain ﬁeld.

The relaxation rate can be expressed using the Golden

rule formula as Γph

1= (M2∆2

0E)/(2πρ~4v5)×coth E

2kBT

[40], where Mis the deformation potential, ∆0is the

TLS tunneling matrix element, Eis the TLS energy, ρis

the density and vis the speed of sound of the material.

For a resonator in vacuum, the dissipation is through the

emission of phonons into the dielectrics hosting the TLS,

and at the relevant temperatures this process is temper-

ature independent with a rate that can be estimated to

Γph

1≈102−103Hz [20]. This is much smaller than the

TLS dephasing rate due to interactions Γ2: Previous es-

timates [13,33] yielded Γ2≈106−107Hz at T= 50−100

mK in similar devices.

Assuming the Γ2∝T1+µdependence persists to lower

temperatures means that in vacuum we would reach the

regime of relaxation limited dephasing, Γ2'2Γ1, of

the TLS bath below 10 µK, which is experimentally in-

accessible. 3He immersion increases the average Γ1to

∼105−106Hz, which increases this crossover temper-

ature to 10 −100 mK. This agrees with the observed

crossover temperature in the noise and in the depen-

dence of the quality factor on power Q(hNi). Further-

more, within the GTM 1/Qi∝ln(cNc/hNi) [41], where

cis a constant. The Q(hNi) data ﬁts remarkably well

to this logarithmic power dependence (see SI). In Figure

4c we show that the temperature dependence of cNcfol-

lows the predicted Nc∝Γ1Γ2(T)∼T1+µscaling at high

temperatures. However, below ∼80 mK, around Tx, this

trend changes abruptly, and becomes temperature inde-

pendent. In the relaxation limited regime the noise is not

expected to increase upon cooling, yet a temperature de-

pendence may be inherited from mechanisms contribut-

ing to Γ1, such as the 3He-TLS interaction. 3He immer-

sion thus prevents the TLS noise from rising more than

three orders of magnitude upon cooling to 1 mK.

A second scenario that in addition may account for

the apparent reduction in the noise is TLS saturation.

Such situation could arise because the measurement is

conducted at a ﬁxed driving power. As Γ2(and hence

Nc) becomes smaller at lower temperatures the applied

power more easily saturates the TLS because they be-

come more coherent [20,42]. The GTM predicts a univer-

sal T(1−µ)/2=T0.375 scaling of the noise in this regime,

and power broadening would also result in the observed

crossover in Ncfrom T1+µto constant in temperature

[41] (Figure 4c) as for the relaxation limited scenario.

Because we have signiﬁcantly increased the average Γ1

of TLS in the bath, one would think that this scenario

is of less relevance in 3He. Indeed, another important

observation is that for saturation in the regime Γ1Γ2

the crossover temperature Txshould depend on driving

power, contrary to our data.

Yet, in any practical device the spatial variations in

electric ﬁelds, the distribution of TLS parameters, and

the fact that not all TLS are located in proximity to

the exposed surface where they can couple to 3He means

there still will exist TLS that are not suppressed by

3He and are therefore easily saturated. This prompts de-

vice improvements where surfaces and edges with strong

electric ﬁelds should be placed in proximity to 3He.

As a ﬁrst step to understand the 3He-TLS interaction

we note the long-standing problem of the thermal bound-

ary resistance between solids and helium liquids, where

the details of the interface, such as surface roughness [43]

and the nature of the surface boundary layer, including

the presence of 1-2 layers of solid helium at the inter-

face due to van der Waals attraction [44,45], play a key

role [46]. Perhaps more closely related to this work are

earlier acoustic and thermal measurements on strongly

disordered [47] and porous [48,49] materials immersed

in helium that also found evidence of faster TLS relax-

ation. It has been suggested [50] that one mechanism

by which phonons in helium couple to TLS is via van

der Waals interaction. The upper bound for the relevant

deformation potential in 4He was deduced to be M.2

meV [47] compared to ≈1 eV for phonons in a solid. Us-

ing these numbers we can attempt to roughly estimate

the enhanced TLS relaxation rate in 3He, compared to

the sapphire substrate. For sapphire we use ρ= 4 ×103

kg/m3,v= 1 ×104m/s, M= 1 eV. Similar values are

also expected for TLS in the NbN surface oxide. For

3He we use ρ= 60 kg/m3,v= 200 m/s and M= 1 meV

[49]. This yields Γ3He

1/Γsap

1≈104– an order of mag-

nitude larger than experimentally observed. This is not

very surprising given the crudeness of the estimates and

the fact that we measure the average for the whole TLS

bath. Moreover, we note that below ∼100 mK the prop-

agating acoustic modes in 3He are that of zero sound [51].

Zero sound modes and the nuclear magnetism [52–54] of

3He oﬀers various interaction mechanisms with relevant

degrees of freedom and a much richer spectrum of low

energy excitations than in 4He [55]. To the best of our

knowledge, the TLS-3He coupling has not been studied

in detail before, and at low temperatures other types of

interactions may become as important as phonons, such

as direct interaction between surface TLS and quasipar-

ticles in 3He [50,56].

Understanding the mechanism at play is crucial for fu-

ture improvements, and two further experiments (details

in SI) suggests that phonon relaxation into 3He following

the Golden rule alone does not capture the full picture. i)

Measurements with only a thin (∼4 nm) ﬁlm of 3He cov-

ering the sample allow us to separate the two roles played

by 3He, namely to enhance TLS relaxation and to medi-

ate cooling. For a thin 3He ﬁlm we still observe the big

change in saturation power (3He-TLS interaction) but a

plateaued noise as in vacuum, indicating poor thermal-

isation. ii) Increasing the pressure of the 3He to 5 bar,

whereupon both ρand vincrease by ∼30% compared to

standard vapour pressure [55], should result in an almost

ﬁve-fold reduction of Γ1. Contrary, we observed a very

moderate increase in saturation power (<20%).

Finally we turn to the dielectric properties of 3He to

understand its compatibility with state of the art qubit

circuits. The resonator frequency shift due to ﬁlling the

cell agrees with the 3He dielectric constant εr= 1.0426

[57] within 1 part in 1000 (see SI). Liquid 4He has a low-

temperature dielectric loss tangent tan δ < 5×10−6at

9 GHz [58]. Similar values are expected for 3He, how-

ever, to the best of our knowledge the this value not

been reported at GHz frequencies. From the change

in single-photon Qiat 10 mK as the cell is ﬁlled with

3He we estimate an upper bound for the loss tangent of

tan δ1.5×10−5at 5.8 GHz, comparable to the best

substrate dielectrics used. Likely tan δis much lower as

signiﬁcant TLS-induced parameter drift occurs between

measurements, the main source of error in our estimate.

The bound on the loss tangent translates to a limit for

qubit coherence times of T1110 µs for a 6 GHz qubit,

i.e 3He is compatible with state of the art quantum cir-

cuits.

In conclusion we have shown that 3He is an eﬃcient,

low-loss cooling medium for quantum circuits and can

cool down environmental degrees of freedom of the cir-

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

QuantumbathsuppressioninasuperconductingcircuitbyimmersioncoolingM.Lucas1,A.V.Danilov2,L.V.Levitin1,A.Jayaraman2,A.J.Casey1,L.Faoro3,A.Ya.Tzalenchuk1;4,S.E.Kubatkin2,J.Saunders1,andS.E.deGraaf41PhysicsDepartment,RoyalHollowayUniversityofLondon,Egham,UnitedKingdom2DepartmentofMicrotechnologyandNanos...

展开>> 收起<<

Quantum bath suppression in a superconducting circuit by immersion cooling M. Lucas1 A. V. Danilov2 L. V. Levitin1 A. Jayaraman2 A. J. Casey1 L. Faoro3 A. Ya. Tzalenchuk14 S. E. Kubatkin2 J. Saunders1 and S. E. de Graaf4.pdf

共21页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Quantum bath suppression in a superconducting circuit by immersion cooling M. Lucas1 A. V. Danilov2 L. V. Levitin1 A. Jayaraman2 A. J. Casey1 L. Faoro3 A. Ya. Tzalenchuk14 S. E. Kubatkin2 J. Saunders1 and S. E. de Graaf4

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: