SLAC-PUB-17691 Dark Matter Induced Power in Quantum Devices Anirban Das1Noah Kurinsky1 2and Rebecca K. Leane1 2

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SLAC-PUB-17691

Dark Matter Induced Power in Quantum Devices

Anirban Das,1, ∗Noah Kurinsky,1, 2, †and Rebecca K. Leane1, 2, ‡

1SLAC National Accelerator Laboratory, 2575 Sand Hill Rd, Menlo Park, CA 94025, USA

2Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA 94035, USA

(Dated: March 23, 2024)

We point out that power measurements of single quasiparticle devices open a new avenue to

detect dark matter (DM). The threshold of these devices is set by the Cooper pair binding energy,

and is therefore so low that they can detect DM as light as about an MeV incoming from the

Galactic halo, as well as the low-velocity thermalized DM component potentially present in the

Earth. Using existing power measurements with these new devices, as well as power measurements

with SuperCDMS-CPD, we set new constraints on the spin-independent DM scattering cross section

for DM masses from about 10 MeV to 10 GeV. We outline future directions to improve sensitivity

to both halo DM and a thermalized DM population in the Earth using power deposition in quantum

devices.

Introduction.— At any given moment, a powerful

stream of DM particles from the Galactic halo ﬂows

into the Earth. This Galactic DM has been extensively

searched for in direct detection experiments, which aim

to detect recoil events when DM scatters oﬀ the Standard

Model (SM) target material, thereby providing a test of

the DM-SM scattering cross section. Typically, the en-

ergy threshold of direct detection experiments assuming

nuclear recoils is about a keV, corresponding to the recoil

expected for DM with mass above about a GeV for stan-

dard analyses [1], or MeV-scale masses when exploiting

the Migdal eﬀect [2–6] or electron recoils [7–9].

Given the lack of a conclusive DM detection with di-

rect detection experiments so far, interest in novel de-

tection strategies and new devices has exploded in the

last few years [10]. In particular, the race down to in-

creasingly low thresholds has inspired use of new detec-

tors, including superconductors [11–16], superﬂuids [17–

19], polar crystals [20–22], topological materials [23], and

Dirac materials [24–27]. Superconductors show excep-

tional promise, due to their superconducting energy gaps

as low as about an meV, allowing probes of light DM.

The goal of lower threshold experiments to date has

been to push down sensitivity to lower DM masses, and

we will exploit this to test incoming halo DM down to

the MeV-scale. Lowered thresholds open up a new probe

of a DM component other than the usually-considered

halo DM. When the Galactic halo DM enters the Earth,

it scatters, loses energy, and can become gravitationally

captured. Over time, this builds up a thermalized pop-

ulation of DM particles bound to the Earth. For DM

around a few GeV that is in local thermal equilibrium,

the density of bound DM at Earth’s surface can in fact be

enormous: about 15 orders of magnitude higher than the

local DM halo density [28–36]. Unfortunately this large

density enhancement is lost on traditional direct detec-

tion experiments, as the bound DM population has a very

∗Email:anirband@slac.stanford.edu

†Email:kurinsky@slac.stanford.edu

‡Email:rleane@slac.stanford.edu

Figure 1. The qualitative diﬀerence between our proposal and

a conventional DM direct detection experiment. The noise

arises from frequent interaction between DM and the nuclei in

the detector, as opposed to once-in-a-while recoil of a nucleus

from DM scattering.

low velocity compared to halo DM, requiring thresholds

of less than about 0.05 eV at Earth’s surface.

We will demonstrate for the ﬁrst time that power

measurements using new quantum devices can be used

to detect DM with low energy depositions. This includes

sensitivity to both light DM from the halo, as well

as thermalized bound DM. As schematically shown in

Fig. 1, for thermalized DM our proposal exploits their

high DM density and is suﬃciently sensitive despite

low thermal velocities, compared to traditional direct

detection, which only measures the less frequent and

higher-velocity DM halo interactions. We point out and

will use the fact that both halo DM and thermalized

DM would produce excess quasiparticle generation in

single quasiparticle devices, and excess power produced

in athermal phonon sensors, to set new constraints on

DM with interaction cross sections larger than about

10−34 −10−28 cm2for DM masses of ∼300 MeV−10

GeV for thermalized DM. For halo DM, we will set

constraints down to about 10−29 −10−26 cm2for DM

masses of ∼10 MeV−10 GeV.

Dark Matter at Earth’s Surface.— At Earth’s po-

sition, there are two potential DM components present,

which have diﬀerent DM velocity and density assump-

tions. We will test both of these components. One is

arXiv:2210.09313v2 [hep-ph] 23 Mar 2024

DM incoming from the Galactic halo, which is usually as-

sumed for direct detection experiments. The other is the

thermalized DM component. This thermalized compo-

nent exists as once DM enters the Earth, it can thermal-

ize, and become captured and bound to the Earth. For

suﬃciently large DM-SM scattering cross sections (larger

than about 10−35 cm2), the DM rapidly thermalizes and

is said to be in local thermal equilibrium with the sur-

rounding SM matter. In this case, the DM radial proﬁle

within the Earth, nχ, is dominantly governed by the dif-

ferential equation [36]

∇nχ

nχ

+ (κ+ 1) ∇T

T+mχg

T=Φ

nχDχN

⊕

r2,(1)

where Tis the Earth’s radial temperature proﬁle at po-

sition r,R⊕is Earth’s radius, mχis the DM mass, gis

gravitational acceleration, and Φ is the incoming ﬂux of

DM particles from the Galactic halo. DχN ∼λvth and

κ∼ −1/[2(1+mχ/mSM)3/2] are diﬀusion coeﬃcients [36],

with λthe DM mean free path, vth the DM thermal ve-

locity, and mSM the SM target mass. The DM density

proﬁle is normalized by enforcing that its volume integral

equals the total number of particles expected within the

Earth [36].

Solving Eq. (1) for nχ(r) reveals that this thermalized

population of DM can be signiﬁcantly more abundant at

the Earth’s surface than the incoming halo DM particles.

For DM masses around a GeV, the local DM density can

be as high as about ∼1014 cm−3. However, as this popu-

lation is thermalized within the Earth, its velocity is low.

We approximate the thermalized DM velocity distribu-

tion as a truncated Maxwell-Boltzmann distribution,

fχ(v) = 1

e−(v/vth )2Θ(vesc −v),(2)

where N0normalizes the distribution, and v2

th =

8Tχ/πmχwith Tχ≃300 K. This velocity would require

thresholds of E<

∼0.05 eV for conventional detection

techniques. This is much lower than the reach of typ-

ical direct detection experiments, and so requires new

techniques to be detected. Our assumption of DM being

at room temperature of ∼300 K is reasonable, as even at

the largest cross sections considered the mean free path

is much larger than the size of our devices, such that DM

is not expected to thermalize with the device itself.

For halo DM, in Eq. (2)vth is replaced by the

average DM velocity in the halo v0= 230 km/s. In

this case, the relative velocity between the Earth and

DM also becomes important. Hence, for halo DM we

use the boosted velocity v→v+v⊕in Eq. (2), where

|v⊕|= 240 km/s is the Earth’s velocity in the galactic

rest frame. The halo DM density is assumed to be

0.4 GeV cm−3. We now show that quantum devices

are highly sensitive to DM with low energy depositions

through their power measurements, which includes both

the thermalized DM population, as well as light halo DM.

Scattering Rate & Energy Deposition.— As a DM

particle with velocity vscatters in the detector and trans-

fers momentum q, it deposits an amount of energy

ωq=q·v−q2

2mχ

=Ef−Ei.(3)

As a result, the target makes a transition from |i⟩to |f⟩.

For such low energy depositions, the momentum trans-

ferred is comparable to the inverse size of nuclear wave-

function in a detector crystal, and the inter-atomic forces

become important. Hence, lattice vibrations or phonon

excitations will be used to compute the DM scattering

rate. The total rate per unit target mass can be written

as [37,38]

Γ = πσχN nχ

ρTµ2Zd3vfχ(v)Zd3q

(2π)3F2

med(q)S(q, ωq) (4)

Here, fχ(v) is DM velocity distribution, ρTis the tar-

get density, σχN is the DM-nucleon scattering cross sec-

tion, µis the reduced mass of the DM-nucleon system,

Fmed(q) is a form-factor that depends on the mediator

(we assume Fmed(q) = 1), and S(q, ωq) is the dynamic

structure factor containing the detector response to DM

scattering and depends on the crystal structure of the

target material.

To compute DM scattering rates, we follow

Refs. [39,40] and use the publicly available code

DarkELF. We modify DarkELF in two main ways.

Firstly, we update the local DM density and DM veloc-

ity input to be that described in the previous section, for

halo or thermalized DM as appropriate. Secondly, the

code was developed only for materials with two atoms

per primitive cell, which is the smallest crystal unit.

Thus, we adapt it for materials like Al which has only

one atom in its primitive cell.

Detection Mechanisms and Materials.— Detect-

ing light halo DM or the captured DM population of low

thermal energy demands use of low threshold quantum

sensors that can detect ∼ O(10) meV energy deposition.

Such sensors are usually designed using superconducting

materials, which have small energy gaps [41–44]. Alu-

minum (Al) is a widely used superconductor for such a

purpose and its characterization data is readily available.

Such a small amount of energy transfer is not suﬃcient

for nuclear recoil or electronic ionization, however DM

can excite collective modes, such as phonons in the ma-

terial, resulting in an excess power. For example, in one

experimental setup, a bias circuit stabilizes the absorber

material at its transition temperature Tc, where its re-

sistance is very sensitive to any energy deposition. The

total power deposited in the detector by DM in the form

of phonons is

PDM =ϵZdω ω dΓ

dω ,(5)

where ϵis an eﬃciency factor that depends on the ex-

perimental setup. We will use this to calculate excess

power due to DM and set constraints on DM-SM inter-

actions. Volume-scaled detectors based on conventional

semiconductors, such as Si, can also be used as the ab-

sorber material to look for ambient power deposition; the

power deposited per unit volume can be obtained from

Eq. (5).

We also consider excess quasiparticle production from

DM. In a superconducting metal, the electrons are bound

into Cooper pairs through a long-range interaction with

phonons. When a DM particle scatters with a nucleus,

it may deposit its kinetic energy in the form of phonons.

If the deposited energy exceeds the energy gap ∆ of the

superconductor, these excess phonons will break some

of the Cooper pairs and release quasiparticles above the

gap. We will therefore set limits on DM-SM interactions

by calculating quasiparticle production rates from DM.

The quasiparticle generation rate Rqp by DM scatter-

ing can be written as

Rqp =ϵqp

∆Zdω ω dΓ

dω

≈PDM

9×10−23 Wµm−3Hz µm−3,(6)

where PDM is the deposited DM power above the gap in

Wµm−3, assuming a 60% quasiparticle generation eﬃ-

ciency (ϵqp = 0.6) [14,45], and using ∆ ≃340 µeV for

Al.

A conservative estimate of nqp, the steady-state quasi-

particle density, can be found using mean ﬁeld results

from Ref. [46] as follows,

dnqp

dt =−ΓR−ΓT+A≈ −¯

Γn2

qp −¯

ΓTnqp +A , (7)

with ΓR,ΓT, A as the recombination, trapping, and gen-

eration rates, respectively. With a steady state injected

power density P, we have A=P/(2∆) where ∆ is the

gap energy. In equilibrium, we thus ﬁnd

P/(2∆) = ¯

Γn2

qp +¯

ΓTnqp .(8)

The mean ﬁeld calculation assumes no trapping of QPs

with ¯

ΓT= 0, which leads to the relation nqp =pA/¯

Γ∝

√P. In case of DM scattering, A=Rqp using Eq. (6),

and ¯

Γ = 40 s−1µm3for Al. The steady-state density is

therefore

nqp ≈PDM

3.6×10−21W1/2

µm−3,(9)

which can be compared to known measurements to set

new constraints. We now discuss devices that can be

used to detect DM using power deposition.

Detecting Dark Matter with Single Quasiparticle

Devices.—

(i) Quasiparticle Tunneling in Transmon Qubits:

Quasiparticle excitations formed from broken Cooper

pairs are important to minimize in quantum devices, as

the quasiparticle background limits the operation of ap-

plications such as radiation detectors and superconduct-

ing qubits. To study the eﬀect of quasiparticle tunneling

on the decoherence of a transmon qubit, Ref. [41] con-

structed a single junction superconducting qubit made

of Al, and studied its decoherence by monitoring single-

charge tunneling rate. From the observed relaxation

rate of the qubit, they found a quasiparticle density of

0.04 ±0.01 µm−3with a thermalized distribution [41].

We convert this measurement to a power density using

Eq. (9), ﬁnding an upper limit of 3.92 ×10−24 Wµm−3.

We compare this directly with the expected DM induced

quasiparticle density in Al, and consider this an upper

limit on residual power injection. Moreover, the source

of this quasiparticle density is not known and usually

assigned to the background radiation from the environ-

ment [41]. Therefore, it is possible that the DM scat-

tering contributes to it too. We overall point out that

quasiparticles produced by DM, and therefore the DM-

SM scattering rate, can be probed using devices with

low-quasiparticle density backgrounds.

(ii) Low Noise Bolometers: Understanding our Uni-

verse deep into the infrared would reveal new secrets of

galaxy formation, exoplanets, and so much more. How-

ever, far-infrared spectroscopy requires new cryogenic

space telescopes with technologies capable of measuring

very cold objects, and therefore require low noise equiva-

lent power in their detectors. Adapting technology from

quantum computing applications, Ref. [42] developed a

quantum capacitance detector where photon-produced

free electrons in a superconductor tunnel into a small ca-

pacitive island. This setup is embedded in a resonant cir-

cuit, and therefore can be referred to as a “quantum res-

onator”. This quantum resonator measured excess power

of 4 ×10−20 W [42], making it the most sensitive existing

far-infrared detector. The volume of the absorber used

in this case was a mesh grid, roughly 60 microns square

with a 1% ﬁll factor and 60 nm thick. This corresponds

to a volume of around 1.56 µm3and thus a power den-

sity measurement of 2.6×10−20 Wµm−3. We therefore

point out that single quasiparticle devices can be used for

DM detection through their power measurements, and

will use this current measurement to set constraints on

the DM-SM scattering rate which would produce excess

power. Note that this detector has a calibrated photon

detection eﬃciency of greater than 95%, which reduces

the systematic uncertainty on the power limit. Ref. [42]

excludes the possibility that this is induced by residual

radiation and represents a true, measured excess power.

For both the devices above, we will only consider DM

detection from the superconductor ﬁlms, rather than the

substrate. In case of the transmon qubit, it is a con-

servative choice. While for the bolometer, the substrate

is connected to the ground plane made of metallic Au-

Ti, which acts as a phonon trap stopping the phonons

created in the substrate from travelling into Al; see the

Supplementary Material for more details.

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SLAC-PUB-17691DarkMatterInducedPowerinQuantumDevicesAnirbanDas,1,∗NoahKurinsky,1,2,†andRebeccaK.Leane1,2,‡1SLACNationalAcceleratorLaboratory,2575SandHillRd,MenloPark,CA94025,USA2KavliInstituteforParticleAstrophysicsandCosmology,StanfordUniversity,Stanford,CA94035,USA(Dated:March23,2024)Wepointouttha...

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SLAC-PUB-17691 Dark Matter Induced Power in Quantum Devices Anirban Das1Noah Kurinsky1 2and Rebecca K. Leane1 2

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