# Parametrics of electromagnetic searches for axion dark matter

###### Abstract

Light axion-like particles occur in many theories of beyond-Standard-Model physics, and may make up some or all of the universe’s dark matter. One of the ways they can couple to the Standard Model is through the electromagnetic portal, and there is a broad experimental program, covering many decades in mass range, aiming to search for axion dark matter via this coupling. In this paper, we derive limits on the absorbed power, and coupling sensitivity, for a broad class of such searches. We find that standard techniques, such as resonant cavities and dielectric haloscopes, can achieve -optimal axion-mass-averaged signal powers, for given volume and magnetic field. For low-mass (frequency ) axions, experiments using static background magnetic fields generally have suppressed sensitivity — we discuss the physics of this limitation, and propose experimental methods to avoid it. We also comment on the detection of other forms of dark matter, including dark photons, as well as the detection of relativistic hidden sector particles.

## I Introduction

Axion-like particles, in particular the QCD axion,
are a well-motivated dark matter (DM) candidate. They occur in many
models of beyond-Standard-Model physics, and can naturally be
light and weakly-coupled, allowing them to be stable
and difficult-to-detect. There are also a number of early-universe
production mechanisms, which can produce them in the correct abundance
to be the DM Kawasaki *et al.* (2015); Ringwald and Saikawa (2016); Co *et al.* (2018); Grilli di Cortona *et al.* (2016).

A wide range of existing and proposed experiments aim to detect
axion DM candidates. These span many decades of mass range,
and target a variety of possible couplings to the Standard Model (SM) Graham *et al.* (2015).
In this paper, we will focus on the
axion-photon-photon coupling, and address
the sensitivity limits on such experiments —
how small a DM-SM coupling could we possibly detect,
given the dimensions, timescales, sensors etc. available?
We choose the coupling partly because,
for a generic QCD
axion, this coupling must lie within a fairly narrow (logarithmic) range Grilli di Cortona *et al.* (2016); Di Luzio *et al.* (2017);
it is also a generic feature of many other axion-like-particle
models Jaeckel and Ringwald (2010). In addition, it represents a particularly easily-analysed
example of the kind of sensitivity limits we are interested in.

We derive bounds on the power absorbed by axion DM experiments, under fairly general assumptions, in terms of the magnetic field energy maintained inside the experimental volume. We also derive related limits on the achievable sensitivity for such experiments, using the tools of quantum measurement theory. For low-mass (frequency ) axions, we review why static-background-field experiments generally have suppressed sensitivity (compared to their scaling at higher frequencies), and point out that this suppression can be alleviated in a number of ways, potentially motivating new experimental concepts.

Similar sensitivity analyses can be applied directly to other forms of DM that couple to the SM photon, e.g. dark photon DM with a kinetic mixing. Analogous ideas can also be applied to other kinds of DM-SM couplings, and beyond that, to the detection of general hidden-sector states. We comment on some of these extensions later in the paper, and in the conclusions.

### i.1 Summary of results

Here, we give a brief summary of our main results. Suppose that dark matter consists of an axion-like particle of mass , with a coupling to the SM. For non-relativistic axion DM, this acts as an ‘effective current’ , where is the magnetic field. If we want to search for axions over a mass range , then for small enough that the target system is in the linear response regime, the expected time-averaged absorbed power from the axion effective current, at the least favourable axion mass, satisfies (under certain assumptions)

(1) |

where is the energy density of the axion DM, and is the time-averaged magnetic field energy in the experimental volume (ignoring magnetic fields on very small spatial scales). Here, the expectation value includes integrating over the unknown phases of the axion signal (otherwise, there could be components of the absorbed power). This bound applies to target systems for which the imaginary part of the response function is non-negative at positive frequencies — that is, the system on average absorbs energy from the axion forcing, rather than emitting. As we discuss below, is, in many circumstances, closely related to the detectability of an axion signal. The behaviour corresponds to the power vs bandwidth tradeoff that is a property of many detection schemes; covering a broader axion mass range in the same integration time necessarily leads to lower average signal power.

In the regime (where
is the frequency
of the axion oscillation),
cavity haloscopes such as ADMX Du *et al.* (2018)
and HAYSTAC Zhong *et al.* (2018); Droster and van
Bibber (2019) can attain the bound in equation 1 to . At higher frequencies (), dielectric haloscope
proposals Caldwell *et al.* (2017); Baryakhtar *et al.* (2018)
can also achieve this, taking the experimental volume
to be that occupied by the dielectric layers.

However, for , where is the length scale of the shielded experimental volume, the EM modes at frequencies are naturally in the quasi-static regime. In that case, a static-background-field experiment has

(2) |

This suppression affects low-frequency () axion DM detection
proposals such as ABRACADABRA Kahn *et al.* (2016) and DM Radio Chaudhuri *et al.* (2019). As discussed below, the
scaling of the detectability limit is similarly affected,
with the minimum detectable increased
by .

Even under the assumptions leading to equation 1, this quasi-static suppression is not inevitable. To alleviate it for static background magnetic fields, we would need to enhance the quantum fluctuations of the EM fields that couple to the axion effective current, at frequencies . Doing this, in an equilibrium setting, requires that the field fluctuations ‘borrow’ energy from some other source, e.g. a circuit component with negative differential resistance. The practicality of such concepts requires further investigation.

The quasi-static suppression can also be alleviated
by performing an ‘up-conversion’ experiment, in which
the background magnetic field is oscillating
at a frequency .
Up-conversion experiments have been proposed in the optical
range DeRocco and Hook (2018); Obata *et al.* (2018); Liu *et al.* (2018),
but the relatively small amplitude of achievable optical-frequency
fields means that they would have relatively poor
sensitivity.
Larger magnetic fields are attainable at lower frequencies;
in particular, it is routine to obtain
magnetic fields of at frequencies
in superconducting (SRF) cavities Padamsee *et al.* (1998); Grassellino *et al.* (2017).

These field strengths were noted in Sikivie (2010), which
proposed a SRF up-conversion experiment.^{1}^{1}1Microwave up-conversion experiments
were also proposed in Goryachev *et al.* (2019), but as reviewed in
section IV.4, errors in their sensitivity calculations made
them orders of magnitude too optimistic.
However, they mainly considered ,
for which static field experiments do not encounter
the quasi-static suppression.
Consequently, the only benefit of an SRF experiment would be
the higher cavity quality factor, which is unlikely
to overcome the disadvantages of smaller background magnetic field,
higher temperature (due to cooling power requirements), and
drive-related noise issues.
However, as we point out here, for ,
the lack of quasi-static
suppression may make up-conversion more competitive.
We investigate this possibility in
more detail in a companion paper Lasenby (2019).

#### i.1.1 Detectability

In the above paragraphs, we discussed the average power absorbed from the axion effective current. It is obvious that, other things being equal, a higher absorbed power makes it easier to detect axion DM. However, in comparing different experiments, other things are often not equal, and more generally, it is useful to have quantitative limits on how small a coupling can be detected.

By using quantum measurement techniques Clerk *et al.* (2010),
we could in principle detect almost arbitrarily small
, even with a small .
For example, by placing the target mode in a
large-number Fock state, we could Bose-enhance
the absorption (and emission) of axions Chou (2018).
However, such techniques are often difficult to implement;
in axion experiments, the only similar measurement
demonstrated so far is HAYSTAC’s squeezed state receiver,
which is planned to deliver
a factor 2 scan rate improvement in the forthcoming
Phase II run Droster and van
Bibber (2019).

One standard technique for signal detection
is linear, phase-invariant amplification
(in particular, it is employed by almost all
existing and proposed axion detection experiments
at microwave frequencies and below).
For different experimental setups, we can place
limits on the SNR obtained, in analogy to the
absorbed power limit from equation 1.
For example, if a linear amplifier
is employed in ‘op-amp’ mode Clerk *et al.* (2010),
and is subject to the ‘Standard Quantum Limit’ (SQL) Kampel *et al.* (2017), then the SNR obtained, averaged across a fractionally
small axion mass range , is

(3) | ||||

(4) |

where is the fractional bandwidth
of the axion DM signal,
is the oscillation
frequency of the axion effective current (assuming
that the magnetic field oscillation frequency is narrow-bandwidth),
is the damping rate for forcing at this
frequency, and is a function
of the thermal occupation number at ,
with for and
for .^{2}^{2}2
For up-conversion experiments,
the converted power in a narrow frequency band is
at most half of the value from equation 1,
and the value is of the
value from equation 4.
As occurs for the average absorbed power, there is an inverse
relationship between the average SNR and .

Another common setup has a linear amplifier isolated
from the target, e.g. using a circulator
connected to a cold load Chapman *et al.* (2017); Clerk *et al.* (2010),
to protect the target system from noise.
In this case, the SNR limit is also given by
equation 4, up to numerical factors.
Perhaps surprisingly, in both of these cases,
the improved sensitivity
for is actually physical,
and experiments using ‘down-conversion’ in this way
could theoretically achieve improved sensitivities.
However, due to a number of practical limitations,
including the relatively small values obtainable
for high-frequency magnetic fields, realising such
enhancements does not seem to be practical.
Additionally, if , then the
EM fields
are naturally in the quasi-static regime,
and the SNR is suppressed by ,
similarly to the absorbed power in equation 2.

At higher frequencies,
detectors other than amplifiers (e.g. photon counters,
bolometric detectors, quasi-particle
detectors, etc.) become easier to implement.
While we could analyse the properties of each
individually, it is the case that for
a wide range of setups, the sensitivity
is bounded by the number of axion quanta absorbed.
We can quantify this using quantum measurement theory.
The Fundamental Quantum Limit (FQL) Pang and Chen (2019); Tsang *et al.* (2011); Miao *et al.* (2017)
for signal detection is determined
by the quantum fluctuations of the EM fields
that couple to the axion signal. Using the arguments
that lead to equation 1, we can constrain the
frequency-integrated spectrum of these fluctuations.
For general states, we cannot use
this to place a bound on detectability
(the example
of squeezed states etc. shows that there is no such
general limit). However, if the sensor interacts
with the target via a damping-type interaction,
e.g. an absorptive photodetector or bolometer,
then its effects are equivalent to a passive
load, and the quantum fluctuations of the target EM
fields are the same as in an equilibrium state.
In these circumstances, the sensitivity
to axion DM, over a (fractionally small) mass range
, is bounded (at the least favourable
axion mass) by

(5) |

where is the probability of detecting the axion signal. We will refer to this limit as the PQL (‘Passive Quantum Limit’). It has an obvious interpretation in terms of photon counting, for schemes in which axions convert to single photons, but it also applies to other setups, e.g. where a signal consists of multiple quasi-particle excitations. Coherent-state excitations of the target’s EM fields leave their quantum fluctuations unchanged, so do not affect the PQL. As in the SQL case, the enhancement for small is physical, but probably not practical.

In we take small enough so that the assumptions behind equation 4 no longer hold, the maximum SNR from a linear amplifer isolated behind circulator with a cold load saturates to (as does the SQL op-amp limit). This is as we would expect, since the amplifier acts on the target like a passive load.

As we emphasised above, it is certainly possible to do
better than the PQL, by using techniques involving ‘non-classical’
EM field states.
One important example
is using linear amplifiers with correlated
backaction and imprecision noise. By optimising this
correlation, we can in theory obtain
the ‘quantum limit’ Clerk *et al.* (2010); Kampel *et al.* (2017),
for which the SNR bound is

(6) |

where the notation is as for equation 4. Unlike the SQL and PQL,
this limit does not involve —
a QL-limited experiment is inherently broadband, if
we can optimise the amplifier properties across
a wide bandwidth.
In the quasi-static limit, the SNR is again suppressed by
. SQUID amplifiers
(as proposed for e.g. the ABRACADABRA axion DM
detection experiment Kahn *et al.* (2016)) can,
in some circumstances, attain near-QL performance Clarke and Braginski (2006).
The fact that the QL-limited sensitivity can, in some
regimes, be better than the PQL, corresponds
to the amplifier back-action
enhancing the quantum fluctuations of the target EM
fields.

While some other measurement schemes, such
as backaction evasion Clerk *et al.* (2010), do not have such general
limits on their sensitivity, we could still analyse
their performance given more specific assumptions.
In this paper, we will restrict our discussion to the
amplifier and PQL limits introduced above.
One reason for doing so is that, taken together,
they apply to almost all existing
and proposed axion DM detection experiments.

As we will discuss, these limits help in understanding what can and cannot enhance an experiment’s sensitivity to axion DM, and in comparing the potential sensitivity of different kinds of experiments.

## Ii Axion DM interactions

We will suppose that dark matter consists of an axion-like
particle , with a coupling to the SM photon.
This has Lagrangian^{3}^{3}3
We take the signature, and use the convention
. Except where indicated, we use natural units
with .
In general, we will abbreviate .

(7) |

where is the potential for the axion — in general, only the mass term will be important for us.

The term is a total derivative, , so under integration by parts, the interaction term in the Lagrangian is equivalent to

(8) | ||||

(9) | ||||

(10) |

Note that, for our signature choice, the components of the usual 3-vector potential, which we will denote , are . To begin with, we will focus on the case of a spatially-constant (zero-velocity) axion DM field, . This is a good approximation, since the DM is highly non-relativistic, with (we will come back to the consequences of the axion velocity distribution in section II.4). If , then the interaction term is , and the interaction Hamiltonian density is .

### ii.1 Response dynamics

To analyse the effect of the axion oscillation on the system, we can decompose the EM vector potential as , where in the absence of an axion oscillation. In the notation of appendix A, . If we assume that is small compared to , then

(11) | ||||

(12) |

so the interaction term can be expanded as

(13) |

Thus,

(14) |

where we take the volume of integration large enough that the divergence term can be neglected. The full Hamiltonian for can be written as

(15) |

where represents the dynamics of the rest of the system, which couples only to (and not to ).

In general, will have some time dependence. To start with, we will assume that the time dependence and spatial profile factorize, , as is the case for e.g. a cavity standing mode (we will revisit this in section II.4). We can decompose , where . Then, writing , our Hamiltonian is

(16) |

Here, is the analogous decomposition of the electric field, and the conjugate momentum of is , with equal-time commutation relation . This is analogous to the Hamiltonian for a driven 1D oscillator,

(17) |

where , , , and .

If we consider a very short pulse, turning on and off much faster than the system’s dynamics, then its effect is to impulsively change by . Averaging over possible signs of the pulse, the expected energy absorbed is . In our case, since depends on the time derivative of , a delta-function pulse corresponds to a step function in , and we have .

This argument tells us the expected energy absorbed by the target from a very fast axion field ‘pulse’. However, as discussed above, we expect axion DM to be a narrow-bandwidth oscillation, with fractional bandwidth . If is small enough that the target is in the linear response regime, then the energy it absorbs from a finite-time signal is

(18) |

where is the linear response function for (if the dynamics are non-stationary in time, we can consider averaging over all possible starting times). A delta-function pulse has equal power at all frequencies, so

(19) |

Equating this to the energy absorbed from the pulse, we have

(20) |

This ‘sum rule’ is analogous to the Thomas-Reiche-Kuhn sum rule for ‘oscillator strengths’ in atomic physics Sakurai (1993).

By itself, equation 20 does give us any limit on the response in a specific frequency range, since the integrand could have large cancelling components. However, if is always , then we can bound the absorbed power from any signal. This obviously applies if the target is in its ground state (since a forcing can only add energy), or if is equivalent to its ground-state form. For a 1D oscillator, the latter is true in any state, up to non-linearities. More generally, if the target is in a mixed state, where the probability of a microstate decreases with increasing energy, then the condition also holds.

These arguments are a generalisation of the pulse-absorption argument
from Baryakhtar *et al.* (2018), which was used
to the determine the axion-mass-averaged signal power
from a dielectric haloscope.

### ii.2 Fluctuation sum rules

To apply the FQL detectability limits discussed in appendix A, we need to understand the fluctuation spectrum of . We can relate this to the response function via the Kubo formula Tong , , where is the spectral density of fluctuations. Thus, the sum rule in equation 20 implies a corresponding sum rule for ,

(21) |

We can also derive this sum rule directly from the commutation relations of the EM fields. The spectral density of fluctuations (assuming that they are stationary in time) is

(22) |

where is the Heisenberg picture operator for the system, in the absence of axion interactions (going forwards, we will drop the hats). Integrating this over ,

(23) | ||||

(24) |

We have

(25) |

since , so

(26) |

If the fluctuations of are stationary in time (as for e.g. a coherent state), then is real. So, is imaginary, and consequently, for equal times,

(27) |

reproducing equation 21.

As per the previous section, we are usually interested in the fluctuations across some narrow frequency range. In general, there can be contributions to equation 21 from positive and negative , leading to cancellations. However, for the ground state of the system, for ; for any operator ,

(28) |

So, for the ground state, we obtain the sum rule

(29) |

The same is true for coherent states, since their fluctuations on top of the c-number expectation value are the same as for the ground state.

### ii.3 Effective Hamiltonians

The above derivations relied on the conjugate momentum
of being , i.e. there being no
other terms in the Hamiltonian involving .
For example, if we were considering a dielectric
medium, where the energy density is , then
the conjugate momentum to would be ,
and we would have
.
Thus, if e.g. a resonant cavity is filled with dielectric
material, the power it is able to absorb decreases Chaudhuri *et al.* (2018).
From above, we know that once all of the dynamics are
taken in account,
.
This implies that the ‘extra’ fluctuations must be at frequencies
above the validity of the effective Hamiltonian.

Similarly, the limits of the integrals above should not be taken literally — at the very least, electroweak physics arises at some energy scale! What we can infer is that, for frequency ranges over which our description of the system is good, in the ground state, and so on.

### ii.4 Axion velocity

So far, we have taken the axion velocity to be zero.
This will not be strictly true; axion DM in the galaxy is expected
to have a virialized velocity distribution, with typical
velocity Necib *et al.* (2018).
As per equation 10, the interaction term
is , where

(30) |

Compared to the zero-velocity case, the axion velocity term results in a coupling to the scalar potential , as well as the vector potential . However, we can work in a gauge in which , in which case the extra coupling term is

(31) |

(after integration by parts). For an axion wave of definite momentum, this corresponds to the field in the axion rest frame, as expected.

Consequently, we can replace by as our forcing term. Since , and the attainable (static) magnetic fields in laboratories are significantly larger than attainable electric fields, the term dominates in almost all circumstances of interest.

A more important effect of the axion velocity distribution is that the axion signal is no longer a spatially uniform, single-frequency oscillation. If the experimental volume is significantly smaller than the axion coherence length (), then the axion field inside the volume is approximately uniform, but is incoherent over times , corresponding to a frequency spread . If the experimental volume is larger, then the axion field is incoherent over distances .

We can treat the spatial variation of the axion field, as well as any time-dependence of the spatial profile, by decomposing into spatially orthogonal modes, each with their own time dependence. Writing

(32) |

where , and and are functions only of time, we have

(33) |

We have equal-time commutation relations

(34) |

So, the cross terms in the sum rule vanish, giving

(35) |

This is as we would expect from the impulse argument above — over very short timescales, there is no dynamics coupling the spatially orthogonal target modes, so they have independent responses to pulses. We will discuss some of the consequences of this in section III.2.

## Iii Parametrics of DM detection

The sum rules derived in the previous section can be used to bound the average power absorbed from the axion effective current, in an axion DM detection experiment. The simplest case is when the field is static. Over sufficiently long integration times, so that we resolve the spectral features of the axion signal, the expected time-averaged power absorbed for an axion of mass is

(36) |

where we write .^{4}^{4}4for shorter integration times,
should be convolved by a kernel of width .
Averaging this over different axion masses , we
can use the fact that, since the axion bandwidth
is small, , integrating
over for fixed is approximately
the same as integrating over for fixed ,

(37) |

Hence,

(38) | ||||

(39) |

Consequently, the absorbed power, integrated over all axion masses, is set by the magnetic field energy in the field (ignoring magnetic fields on very small spatial scales — see appendix B).

If we are interested in looking for an axion within a specific mass range , then we can average over that mass range,

(40) |

where the inequality assumes that for all . Equality can be obtained if the response function is concentrate into the range (we will discuss some of the experimental practicalities of this in section IV). Since , the smallest absorbed power for any axion mass within the range is upper-bounded by equation 40. As expected, searching over a smaller axion mass range permits higher conversion powers within that range.

In many cases, instead of operating a single experimental configuration for the whole observation time, we ‘tune’ our experiment by operating it in different configurations, one after the other. The average power for a given axion mass is the appropriately weighted sum of the powers from the different configurations, and the corresponding limits apply.

The equations above apply to the whole experimental apparatus. However, a common experimental setup is to have a conductive shield (e.g. a EM cavity) inside a larger magnetic field. If the relevant dynamics inside and outside the cavity are independent, then we can apply the above arguments to the volume inside the cavity, replacing the total magnetic field energy by the energy inside the cavity.

### iii.1 PQL limits

From appendix A, if the quantum fluctuations of are stationary, then the formula for the probability of the axion interaction changing the state of the target system is

(41) |

where denotes the symmetrised spectral density, and we assume that is much longer than the inverse bandwidth of spectral features. If the fluctuations are equal to those in the ground state, then for . In that case,

(42) |

where the latter equality holds since is tightly concentrated around . Hence, , the expected number of quanta absorbed. More generally, if for the operational state of the detector satisfies , then the axion-mass-averaged excitation probability satisfies

(43) |

where the average is taken over a fractionally-small axion mass range , centred on . To be confident of identifying or excluding an axion signal, we need few.

In section IV.2.2, we discuss the circumstances under which we expect to satisfy the ground-state sum rule. The most obvious example, in which this limit is achievable, is the case of an absorptive, background-free photon counter. In the presence of noise sources, (such as thermal noise or detector noise), it may not be possible to attain this limit. Conversely, if a detection setup does not satisfy the ground-state sum rule, then the FQL still places limits on its sensitivity, but these will depend on how much the fluctuations exceed the PQL value.

### iii.2 SQL op-amp

If we read out our signal using
a phase-invariant, SQL-limited amplifier
coupled weakly to the target (i.e. in ‘op-amp’
mode Clerk *et al.* (2010)), then from
appendix A.1, the
SNR from an axion signal satisfies

(44) |

where summarises the effects of any additional
noise (beyond amplifier backaction, imprecision,
and zero-point fluctuations),
referred back to .
For example, given thermal noise at temperature , we have ,
where
.^{5}^{5}5
While the thermal fluctuations of
are set by ,
if the amplifier
is coupled to other degrees of freedom, then
the total effect of thermal noise
on the output may be greater.
We assume that the spatial profile
of can be treated as constant in time, to begin with.

If we are interested in a mass range (i.e. a fractional mass range ), then the quantity determining the axion-mass-averaged is

(45) |

For a single-pole resonator, . For a more complicated response function, if the minimum distance of a pole of (extended to a complex function) from the real axis is (i.e. the minimum damping rate), then . If we demand that satisfies the sum rule in equation 20, then is maximised by a single-pole response function, for given . For , the integrand in equation 45 is dominated by a bandwidth , and we have

(46) |

where

(47) |

assuming that . To relate this to the SNR, we can for simplicity take the axion signal to have a top-hat spectral form, . In this case,

(48) | ||||

(49) |

So, averaging this over the axion mass range,

(50) |

where we assume that is approximately constant over the range (if not, we can replace it by its minimum value). Writing this in terms of the expression from above,

(51) |

where , and is the quantity from equation 40. This inequality can be saturated for an amplifier with the correct coupling to a single-pole resonator. As mentioned above, for each resonant configuration, the bandwidth contributing most of the is , so we need to scan over multiple different configurations to have sensitivity over a wide axion mass range. Conversely, if is large enough that , then a single resonant configuration can cover the entire mass range approximately equally, giving

(52) |

Optimising axion-mass-averaged sensitivity
with an SQL-limited amplifier is also
discussed in Chaudhuri *et al.* (2018, 2019).
In their analysis of detection using
a flux-to-voltage amplifier (in the quasi-static
regime; see section IV.3),
they assume
that the amplifier’s coupling does not vary
as a function of frequency, as would be
required to achieve
over the relevant frequency range.
Consequently, for them, achieving sensitivity over
a bandwidth
requires ‘overcoupling’ the amplifier,
and degrading the sensitivity
on resonance (by ). These issues only make
an difference to the SNR achieved;
we give the full SQL-limited sensitivity,
for completeness.

In some circumstances, we can achieve tighter bounds by looking at the damping in more detail. The power absorbed, in response to a monochromatic forcing , is . This is also equal to the dissipated power, , where is the damping rate, and is the energy stored. The electric field energy in the response is , so

(53) |

So, if , we have a tighter bound than above (for example, in the quasi-static regime, as we discuss in section IV.3).

The above expressions are only valid when the relevant integration times are long compared to the inverse bandwidth of spectral features. Looking at the limit, if we consider a resonator with , then we need at least tuned configurations to have sensitivity across the mass range, so we can spend at most in each of them. If , then we do not resolve the resonator bandwidth, and the formulae above do not hold. In particular, if we fix , then for , we do not expect the SNR to continue improving with increasing (this limiting is often large, but may sometimes be of practical relevance Lasenby (2019)). Plugging into equation 51, which should be parametrically (though not numerically) valid, we obtain

(54) |

so the SNR limit is parametrically the same as the PQL.

This equivalence motivates the conjecture
that the quantum fluctuations of , for a target
connected to a SQL-limited amplifier, satisfy
the PQL sum rule (at least to ).
For the case of a critically-coupled amplifier,
the magnitude of the back-action noise is small
enough that this has to be true Clerk *et al.* (2010). For stronger couplings,
we would need to condition on the observed
output of the amplifier
(cf the analysis of backaction evasion in Clerk *et al.* (2008)).
We leave such an analysis to future work.
This conjecture is of mostly academic interest,
since we can work out the SQL limit
directly (as done here).

If the spatial profile of is not constant,
then as discussed in section II.4,
we can consider the time variation
in an orthogonal set of spatial profiles.
This is necessary if the axion
coherence length is smaller
than the scale of the experiment
(or the extent of the background magnetic field,
whichever is smaller),
i.e if .^{6}^{6}6In contrast to how
the fluctuation spectrum can be concentrated into a narrow-bandwidth
peak, at the expense of surrounding frequencies, the
frequency-averaged fluctuations for each orthogonal spatial
mode are fixed by its spatial profile, and cannot
be concentrated into one mode at the expense of others.
In the impulse picture,
if we imagine a set of simultaneous impulses at different spatial points, then
causality prevents the energy absorbed from depending
on their relative signs etc, whereas the reponses
to impulses at different times (picking out a specific frequency)
can interfere with each other.
The SNRs from these orthogonal spatial
profiles will add in quadrature.

### iii.3 Isolated amplifier

For the op-amp coupling considered in the previous
subsection, the amplifier’s backaction drives
the target mode out of equilibrium.
Another way to couple an amplifier
to the target is to isolate it, so that
the overall backaction is simply vacuum noise.
This is common at microwave frequencies,
where an amplifier is usually isolated behind
a circulator, with its backaction absorbed by a cold
load Chapman *et al.* (2017); Clerk *et al.* (2010); Boutan *et al.* (2018); Brubaker (2017).

Achieving good SNR in such configurations
requires stronger couplings to the target mode,
significantly affecting its damping Clerk *et al.* (2010).
To find a SNR limit,
we can consider
the ‘input’ coupling to encompass everything
that couples to , writing .
Then, the fluctuation spectrum of must
be such that , so
.
Using the analysis from Clerk *et al.* (2010), this implies
that the total noise at the amplifier
output is
.^{7}^{7}7
For an ideal quantum amplifier,
.
However, in our case for ,
whereas is even, which implies that there
must be wasted information in the correlation Clerk *et al.* (2010). In particular, by feeding back some of the output,
the back-action could be made symmetric, decreasing
by a factor .
Consequently, in our case, . Thus, .
Consequently,

(55) |

in agreement with the limits of the circulator-plus-cold-load
analyses from Chaudhuri *et al.* (2018); Lasenby (2019).
Similarly to the op-amp case, the axion-mass-averaged
is maximised, for given ,
by a single-pole resonance.
This gives an axion-mass-averaged of

(56) |

which has the same form as equation 50. Taking , we again obtain . An isolated amplifier satisfies the PQL assumptions, so this bound necessarily holds.

If the ‘intrinsic’ dissipation in the target
is significant compared to the damping from the
amplifier, it may not be possible to
attain the limit from equation 55.
Analysing a circulator plus cold load,
as in Chaudhuri *et al.* (2018); Lasenby (2019), shows that
the axion-mass-averaged
is optimised for ,
which gives a suppression of vs equation 56.

Since the temperature of the back-action noise
from the amplifier may be less than
the temperature of other parts of the target (see discussions in Chaudhuri *et al.* (2018, 2019); Lasenby (2019)), there
may no longer be a single equilibrium temperature
that describes our target system.
As analysed in Chaudhuri *et al.* (2018, 2019); Lasenby (2019),
if the dissipation in the target system is fixed,
it is beneficial to ‘overcouple’ to the amplifier,
increasing the dissipation of the overall system.
At , this would result in worse sensitivity,
but because it also reduces the thermal noise
reaching the detector, its overall effect is to improve
the SNR. From Chaudhuri *et al.* (2018, 2019); Lasenby (2019),
for ,
the resulting is suppressed by
compared to the value.

### iii.4 Quantum-limited op-amp

As discussed in appendix A.1, if it is possible to optimise the correlations between a linear amplifier’s backaction and imprecision noise, then the minimum added noise, referred back to the measured variable , is (as opposed to for uncorrelated noise). Consequently, for an amplifier connected in op-amp mode,

(57) |

Assuming thermal noise, , gives

(58) |

If we take to have top-hat form, as above, then

(59) |

This has the same form as the SQL expression in equation 52, when ; this is as expected, since in both cases, the added noise is dominated by the thermal + ZPF noise. If , then we can improve over the SQL limit by

As the above limits show, the sensitivity limit for a quantum-limited amplifier is set by how well we can isolate the system from its environment, and so reduce . There is no sensitivity/bandwidth trade-off, as occurs in the SQL and PQL cases, since as decreases, the ZPF noise also decreases; a setup that saturates the QL at all frequencies is naturally broadband.

The fact that a QL amplifier can have
better sensitivity than the PQL limit,
in some regimes, shows that the amplifier
must be enhancing the quantum fluctuations
of (similarly to how
backaction evasion effectively
drives an oscillator
into a squeezed state Clerk *et al.* (2008)).

At higher frequencies (), making use of correlated backaction/imprecision in this way is usually difficult (in particular, it is not compatible with isolation mechanisms such as circulators, as discussed above). However, at lower frequencies, SQUID amplifiers can attain near-QL performance, in some circumstances Clarke and Braginski (2006). We discuss axion detection at low frequencies () in section IV.3.

### iii.5 Up-conversion

We can generalise the power absorption calculations above to a time-dependent magnetic field. In this case,

(60) |

(61) |

where is the spectral density of . So, using equation 37,

(62) |

(63) |

where , so is the RMS value. In the case of a static field, this reproduces equation 39.

There are a number of qualitatively distinct cases, depending on the oscillation frequencies of the magnetic field and the axion signal. For a oscillation at frequency , an axion oscillation at will give a forcing at sum and difference frequencies, and . To start with, we will consider the ‘up-conversion’ case where , so that both sum and difference frequencies are