Quantum metrology of low frequency electromagnetic modes with frequency upconverters Stephen E. Kuenstner1Elizabeth C. van Assendelft1Saptarshi Chaudhuri2Hsiao-Mei Cho3

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Quantum metrology of low frequency electromagnetic modes with frequency

upconverters

Stephen E. Kuenstner,1Elizabeth C. van Assendelft,1Saptarshi Chaudhuri,2Hsiao-Mei Cho,3

Jason Corbin,1Shawn W. Henderson,3Fedja Kadribasic,1Dale Li,3Arran Phipps,4Nicholas

M. Rapidis,1Maria Simanovskaia,1Jyotirmai Singh,1Cyndia Yu,1and Kent D. Irwin1, 3

1Department of Physics, Stanford University, 382 Via Pueblo, Stanford, CA 94305.

2Department of Physics, Princeton University, Jadwin Hall Washington Road, Princeton, NJ 08544.

3SLAC National Accelerator Laboratory, 2575 Sand Hill Rd, Menlo Park, CA 94025

4Department of Physics, California State University East Bay,

25800 Carlos Bee Blvd, North Science 231, Hayward, CA 94542

(Dated: July 2024)

We present the RF Quantum Upconverter (RQU) and describe its application to quantum metrol-

ogy of electromagnetic modes between dc and the Very High Frequency band (VHF) (≲300MHz).

The RQU uses a Josephson interferometer made up of superconducting loops and Josephson junc-

tions to implement a parametric interaction between a low-frequency electromagnetic mode (be-

tween dc and VHF) and a mode in the microwave C Band (∼5GHz), analogous to the radiation

pressure interaction between electromagnetic and mechanical modes in cavity optomechanics. We

analyze RQU performance with quantum ampliﬁer theory, and show that the RQU can operate

as a quantum-limited op-amp in this frequency range. It can also use non-classical measurement

protocols equivalent to those used in cavity optomechanics, including back-action evading (BAE)

measurements, sideband cooling, and two-mode squeezing. These protocols enable experiments us-

ing dc–VHF electromagnetic modes as quantum sensors with sensitivity better than the Standard

Quantum Limit (SQL). We demonstrate signal upconversion from low frequencies to microwave

C band using an RQU and show a phase-sensitive gain (extinction ratio) of 46.9 dB, which is a

necessary step towards the realization of full BAE.

I. INTRODUCTION

The ﬁeld of Circuit Quantum Electrodynamics (Cir-

cuit QED) has made impressive strides in harnessing the

quantum-mechanical properties of superconducting cir-

cuits operating in the microwave frequency regime (typ-

ically several GHz) [4]. The techniques of Circuit QED

have advanced to the point that detecting [11, 17] and

coherently manipulating [25] a single microwave quan-

tum are routine operations, and individual control over

arrays of dozens of interacting quantum circuits is pos-

sible [1]. Much of this progress has been driven by the

desire to build a universal quantum computer capable of

performing calculations that would be impractical on any

classical computer.

The techniques of Circuit QED do not extend directly

to lower frequencies, however. Recently, there has been

growing interest in adapting quantum metrology tech-

niques to lower frequency electromagnetic modes, typi-

cally at frequencies between dc and the Very High Fre-

quency (VHF) band below 300 MHz. Quantum metrol-

ogy of low-frequency modes could oﬀer a sensitivity ad-

vantage over classical sensors, enabling experiments in-

cluding dark-matter searches [5], low-frequency nuclear

spin metrology [32], some astronomical measurements

[35], and low-frequency magnetometry, outperforming a

dc SQUID in certain applications.

One approach for quantum RF metrology that has

been developed at these frequencies is to use a qubit to

cool the MHz resonator to its ground state and stabi-

lize a Fock state with a small number of photons [16].

While this approach can be used in principle to measure

individual signal photons entering the resonator, it does

not provide a way to discriminate incoming signal pho-

tons from background thermal photons, which limits its

usefulness for certain measurements.

For example, searches for axion or axion-like dark

matter at mass below 1 µeV must detect or rule out

yoctowatt-scale or smaller electromagnetic signals over

many decades in frequency, spanning from ∼100Hz to

∼300MHz [5, 8, 14, 15, 26, 27, 29]. These signals can

be used to excite an electromagnetic resonator. If the

photon number state of the resonator is then measured,

the signal-to-noise ratio is limited by the random en-

try or departure of background thermal photons from

the resonator, since photon-counting techniques lack the

frequency resolution to distinguish thermal and signal

photons. An alternative approach that applies quantum

techniques to measure the dark-matter-induced voltage,

rather than the photon number, allows the signal fre-

quency to be determined. Frequency information is im-

portant as these circuits carry useful information sig-

niﬁcantly detuned from their resonant frequency. Far

from the resonant frequency, thermal ﬂuctuations are

suppressed to below the level of a single photon per

second per Hz of bandwidth [9, 10]. This oﬀ-resonant

signal information can be accessed using continuous-

variables readout techniques operating beyond the Stan-

dard Quantum Limit (SQL). In this case, improving the

readout performance does not substantially improve the

signal to noise ratio (SNR) on resonance (which is limited

by thermal ﬂuctuations), but it allows constant SNR to

arXiv:2210.05576v3 [quant-ph] 9 Mar 2025

be maintained over a much broader bandwidth, dramat-

ically increasing the axion search rate.

In this work, we describe the RF Quantum Up-

converter (RQU), a device that mimics the radiation-

pressure interaction in cavity optomechanics, but re-

places the low-frequency mechanical mode with a low-

frequency electromagnetic mode. The RQU uses the

nonlinearity of Josephson junctions to upconvert signals

from the sensor frequency to microwave frequencies us-

ing a three-wave mixing interaction. We experimentally

demonstrate three-wave mixing with an RQU in §VI.

This upconversion paradigm allows the RQU to take

advantage of several mature microwave Circuit QED

technologies, including high coherence microwave res-

onators [30], Josephson Parametric Ampliﬁers (JPAs)

[31], and microwave squeezers [6], while extending the

frequency range of quantum measurement techniques to

lower frequencies. They also enable the use of quantum-

limit evading techniques equivalent to those used in cav-

ity optomechanics, including back-action evading (BAE)

measurements[13]. After analyzing BAE using the RQU,

in §VI we experimentally demonstrate a phase-sensitive

gain of 46.9 dB, which is a signiﬁcant step towards the

realization of full BAE.

II. ANALOGY BETWEEN CAVITY

OPTOMECHANICS AND OP-AMP MODE

AMPLIFICATION

A. Upconverter Hamiltonian

To evaluate the RQU as a tool for quantum metrology,

we use a model in which both the RQU and its input cir-

cuit are quantized, with an interaction Hamiltonian that

couples the modes. This Hamiltonian is exactly analo-

gous to that of cavity optomechanics, but the mechanical

mode is replaced with an electromagnetic mode, referred

to in this section as the “low-frequency mode” to distin-

guish it from the microwave mode.

Cavity optomechanics treats two bosonic modes at dif-

ferent frequencies: an electromagnetic mode at ωaand a

mechanical mode at ωb, with ωa≫ωb[2]. The position

operator of the mechanical mode represents the position

of a movable mirror that forms one end of the optical

cavity. The modes are quantized with ladder operators

ˆa, ˆa†and ˆ

b,ˆ

b†, respectively. The uncoupled Hamiltonian

is:

H0=ℏωa(ˆa†ˆa+ 1/2) + ℏωb(ˆ

b†ˆ

b+ 1/2),(1)

In terms of ladder operators, the mirror position is given

by:

ˆx=xZPF(ˆ

b+ˆ

b†),(2)

where xZPF is the magnitude of the zero-point position

ﬂuctuations. The frequency of photons occupying the op-

FIG. 1. A circuit model for an RQU, which inductively cou-

ples a dc-VHF signal source (shown here as a resonator formed

by Cband Lb) to a tunable Josephson inductance LJ(ˆ

Φ). The

tunable inductor is made up of a superconducting interfer-

ometer with one or more Josephson junctions (JJs) and one

or more loops. The ﬂux Φ threading the inductor Lbas-

sociated with the low-frequency mode also couples through

a designable mutual inductance to each of the loops in the

JJ interferometer. Thus, ˆ

Φ changes the inductance LJpre-

sented by the JJ interferometer to the microwave mode, and

modiﬁes the resonance frequency of the microwave resonator

formed by the interferometer and linear reactances modeled

by circuit elements Caand La. A coupling capacitance Cc,

microwave transmission lines and a circulator allow the state

of the microwave resonator to be driven and detected via trav-

eling wave modes ˆain and ˆaout,respectively. The output mode

contains information in sidebands, as shown schematically in

the frequency domain. Low noise ampliﬁcation by a cryogenic

microwave ampliﬁer allows for eﬃcient detection of the out-

put state ˆaout.

tical mode depends on the position of the movable mir-

ror, leading to the parametric optomechanical interaction

HOM

int between the two modes:

HOM

int =−ℏg0

xZPF

ˆa†ˆaˆx, (3)

where g0is the optomechanical coupling strength, de-

scribing the frequency shift of an optical photon due to

the position ˆxof the mechanical oscillator.

An optomechanical-style coupling can be realized in

microwave superconducting resonant circuits by includ-

ing a Josephson interferometer whose inductance LJ(ˆ

Φ)

depends on the ﬂux ˆ

Φ in the low-frequency mode. The

ﬂux causes the microwave resonance frequency to vary,

just as position shifts of the moving mirror cause the

optical frequency to vary in the optomechanical setup.

This optomechanical-style coupling has been realized in

the microwave SQUID multiplexer [24] although not op-

timized to approach quantum limits. The dispersive

nanoSQUID magnetometer [22] uses a similar frequency-

tunable microwave resonator, but does not use a resonant

low-frequency circuit on its input, removing the eﬀects

of quantum backaction and limiting the quantum pro-

tocols which could be employed. There are also other

devices in which a resonator is tuned with a Josephson

junction array, including the Asymmetrically Threaded

SQUID (ATS) [21]. However, in the ATS, the lower fre-

quency signal and higher frequency signal are both cou-

pled as a current drive to the interferometer, and a ﬂux

input is used to bias and pump the interferometer for

degenerate four-wave mixing. In the RQU, which uses

optimechanical-style coupling, the low-frequency signal

is applied as a ﬂux to the Josephson interferometer and

three-wave mixing is realized. Figure 1 shows a circuit

model of such an upconverter with the associated ladder

operators.

The uncoupled Hamiltonian of the RQU is exactly the

one in equation 1, with the phonon ladder operators

b, ˆ

b†replaced by photon operators for the low-frequency

mode. The microwave and low-frequency modes are rep-

resented by harmonic oscillators with frequencies:

ωa(ˆ

Φ) = (La+LJ(ˆ

Φ))(Ca+Cc)−1

2,(4)

ωb= (LbCb)−1

2.(5)

The low frequency mode is inductively coupled to the

Josephson interferometer such that the ﬂux threading the

low-frequency resonator also threads the Josephson inter-

ferometer. Arrays of multiple Josephson junctions and

multiple loops can be used with both gradiometric and

non-gradiometric coupling to optimize circuit response

for diﬀerent applications. The tunability of LJ(ˆ

Φ) medi-

ates a parametric interaction, with ˆ

Φ playing the role of

the position operator ˆx. Analogously to equation 2, we

have:

Φ=ΦZPF(ˆ

b+ˆ

b†),(6)

where ΦZPF is magnitude of the zero point ﬂux ﬂuctua-

tions: ΦZPF =pℏωbLb/2.

In order to treat the interaction between the modes,

we include the perturbation of LJdue to the sensor ﬂux.

We analyze the behavior of the RQU in response to small,

time-varying ﬂux signals satisfying |⟨ˆ

Φ(t)⟩| ≪ Φ0, where

Φ0=h/2eis the magnetic ﬂux quantum that sets the

periodicity of the interferometer’s response to external

magnetic ﬂux. In this regime, we Taylor expand the mi-

crowave frequency to ﬁrst order in the sensor ﬂux, to

calculate the shift of the microwave resonance frequency

due to ﬂux in the sensor:

H=ℏωa(0) + dωa

dΦˆ

Φ(ˆa†ˆa+1/2)+ℏωb(ˆ

b†ˆ

b+1/2).(7)

The frequency shift per unit applied ﬂux describes the

strength of the interaction between the modes, with:

dωa

dΦ=dωa

dLJ

dΦ.(8)

The two derivatives on the RHS of equation 8 depend

on the particular design of the interferometer and low

frequency resonator, which we can calculate for a given

interferometer design. We can write the upconverter in-

teraction Hamiltonian in equation 7 in a form analogous

to the radiation pressure interaction in equation 3:

HRQU

int =−ℏg0

ΦZPF

ˆa†ˆaˆ

Φ = −ℏg0ˆa†ˆa(ˆ

b†+ˆ

b).(9)

Without loss of generality, we choose the the sign of in-

creasing ˆ

Φ to yield the minus sign in equation 7. Because

it involves products of three ladder operators, this inter-

action describes three-wave mixing. The strength of the

optomechanical-style coupling is given by:

g0≡dωa

dΦΦZPF.(10)

B. Input-Output Model

In order to operate the RQU, we control and detect

the state of the microwave resonator, which allows us

to infer the state of the low-frequency resonator. The

Hamiltonian in equation 7 only accounts for the interac-

tion between the two modes, and does not include ex-

ternal couplings or dissipation. In order to detect and

control the state of the microwave resonator, we couple

the microwave resonator to a waveguide that allows mi-

crowave photons to escape the cavity for ampliﬁcation

and demodulation. Finally, the model must also account

for the eﬀects of internal dissipation in both the RF and

microwave modes.

The total Hamiltonian, accounting for the external

coupling, dissipation, and the microwave drive is given

by:

Htot =ˆ

H0+ˆ

HRQU

int +ˆ

Hκ+ˆ

Hγ+ˆ

Hdrive,(11)

where ˆ

H0+ˆ

HRQU

int describes the dynamics of the isolated

RQU system (microwave resonator plus low-frequency

resonator, and their interaction), as described in equa-

tions 1 and 7. ˆ

Hκcaptures the eﬀects of loss in the

microwave resonator, which is dominated by loss to the

strongly coupled readout port. ˆ

Hγdescribes loss to inter-

nal dissipation in the low-frequency resonator. Finally,

Hdrive accounts for the energy supplied by the external

drive tones which probe the RQU state.

The traveling-wave modes used in this section are

shown in ﬁgure 1: the microwave resonator is coupled to

an “input” mode ˆain and an “output” mode ˆaout which

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QuantummetrologyoflowfrequencyelectromagneticmodeswithfrequencyupconvertersStephenE.Kuenstner,1ElizabethC.vanAssendelft,1SaptarshiChaudhuri,2Hsiao-MeiCho,3JasonCorbin,1ShawnW.Henderson,3FedjaKadribasic,1DaleLi,3ArranPhipps,4NicholasM.Rapidis,1MariaSimanovskaia,1JyotirmaiSingh,1CyndiaYu,1andKentD.Irw...

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Quantum metrology of low frequency electromagnetic modes with frequency upconverters Stephen E. Kuenstner1Elizabeth C. van Assendelft1Saptarshi Chaudhuri2Hsiao-Mei Cho3

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