PRL 112, 133904 (2014)

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PHYSICAL REVIEW LETTERS

Quantum-Limited Amplification via Reservoir Engineering
A. Metelmann* and A. A. Clerk
Department of Physics, McGill University, 3600 rue University, Montréal, Quebec H3A 2T8, Canada(Received 8 November 2013; published 4 April 2014)
We describe a new kind of phase-preserving quantum amplifier which utilizes dissipative interactions
in a parametrically coupled three-mode bosonic system. The use of dissipative interactions provides afundamental advantage over standard cavity-based parametric amplifiers: large photon number gains are
possible with quantum-limited added noise, with no limitation on the gain-bandwidth product. We showthat the scheme is simple enough to be implemented both in optomechanical systems and in superconducting microwave circuits.
DOI: 10.1103/PhysRevLett.112.133904

PACS numbers: 42.65.Yj, 03.65.Ta, 07.10.Cm, 42.50.Wk

Introduction.—The past few years have seen a resurgence of interest in amplifiers working near the fundamental limits set by quantum mechanics [1], in contexts
varying from quantum information processing in circuits[2–4], to radio astronomy [5], to ultrasensitive force
detection (e.g., for gravity wave detection [6]). The
standard paradigm for a quantum-limited, phase-preserving
amplifier is the nondegenerate parametric amplifier(NDPA) [7–10], which is based on a coherent interaction
involving three bosonic modes (pump, signal, and idler).
This interaction simply converts a pump mode photon into
two photons, one in the signal mode, the other in the idler
mode. The result is that weak signals incident on the signal
mode are amplified, with the minimum possible added
noise. There has been remarkable progress in realizing such
amplifiers using superconducting circuits [11–18]. This inturn has enabled a number of breakthroughs, from themeasurement of mechanical motion near the quantum limit[19], to the measurement of quantum jumps of a superconducting qubit [20,21] and the implementation of quantum feedback schemes [22,23].
Despite their many advantages, standard cavity-based
parametric amplifiers suffer from the limitation of having afixed gain-bandwidth product: as one increases the gain of
the amplifier, one also reduces the range of signal frequencies over which there is amplification. This is a fundamental consequence of the amplification mechanism, which
involves introducing effective negative damping to the signal
mode. The consequent reduced damping rate determines
the amplification, but also sets the amplification bandwidth(see, e.g., [3]). This tradeoff between gain and bandwidth
can severely limit the utility of cavity-based parametricamplifiers in many applications. Traveling-wave parametricamplifiers (TWPAs) [24,25] do not use cavities and are inprinciple not limited in the same way. In practice, however,good device performance and bandwidth of TWPAs islimited by the requirement of phase-matching (though see
Ref. [26] for recent progress in the microwave domain).
In this work, we introduce a new approach for quantumlimited amplification based now on three localized bosonic
0031-9007=14=112(13)=133904(5)modes. Unlike a NDPA, our scheme explicitly involves
dissipative (i.e., non-Hamiltonian) interactions between themodes. We show that this approach allows a large gain withquantum limited noise, but crucially is not limited by afixed gain-bandwidth product: the gain can be arbitrarily
large without any corresponding loss of bandwidth. Notethat non-Hamiltonian evolution is also utilized in a very
different way in probabilistic amplifiers [27–30], which canstochastically amplify signals without adding noise.
Our approach is related to reservoir engineering [31],where one constructs a nontrivial dissipative reservoir that
relaxes a system to a desired target state (e.g., an entangledstate [32–36]). Here, we instead construct an engineered
reservoir which mediates a dissipative amplification process. Our mechanism can also be interpreted as a kind of
coherent feedback process [37–40], where the amplification is the result of an autonomous quantum nondemolition(QND) measurement combined with a feedback operation.
Our scheme is simple enough to be realized using existing
experimental capabilities, either with three-mode optomechanical systems (where a mechanical mode couples to twoelectromagnetic cavity modes) [41,42], or with superconducting circuits [16,43,44].
Model.—While our scheme is amenable to many possible realizations, we focus here for concreteness on athree-mode optomechanical system. Two cavity modes(frequencies ω1 and ω2 ), are coupled to a single mechanicalmode ωM , cf. Fig. 1. The cavity photons interact with themechanical mode via radiation pressure forces, and thesystem is described by the Hamiltonian Ĥ ¼ ĤS þ Ĥdiss .
Here, ĤS is the coherent system Hamiltonian (ℏ ¼ 1),
X
fωj þ gj ðb̂ þ b̂† Þgâ†j âj þ ωM b̂† b̂;
(1)
ĤS ¼
j¼1;2

where b̂ (âj ) is the annihilation operator for the mechanicalresonator (cavity j), and gj is the optomechanical couplingstrength for cavity j. Ĥdiss describes the damping of all
three modes (each by independent baths), and the laser
drives on the two cavity modes; these are treated at the level

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PHYSICAL REVIEW LETTERS

FIG. 1 (color online). (a) Schematic showing the optomechanical realization of the dissipative amplification scheme. Twodriven cavities (1,2) are both coupled parametrically to a third
auxiliary mechanical mode. The mechanics mediates a dissipativeinteraction between modes 1 and 2. Signals incident on either
cavity are amplified in reflection. (b) Alternate realization, where
two pump modes ω1;P , ω2;P are used to generate the required
interaction Hamiltonian, Eq. (2); this setup could be directly
implemented using superconducting microwave circuits [46].

of standard input-output theory [3,45], resulting in cavity(mechanical) damping rates κj (γ).
In what follows, the two cavity modes will play roles
similar to “signal” and “idler” modes in a NDPA, whereas
the mechanics will be used to mediate an effective interaction between them. To achieve this, we assume a strong
coherent drive on each cavity, detuned to the red (blue)
mechanical sideband for cavity 1 (2), (i.e., drive frequencies
ωL;1=2 ¼ ω1=2 ∓ωM Þ. We work in an interaction picturewith respect to the free Hamiltonians, and perform displacement transformations: âj ≡ āj eiωM t þ d̂j , where āj is theaverage classical amplitude of cavity j due to the laser drive;
we take these to be real without loss of generality. Assuming
the standard experimental situation where gj are small andāj are large, we linearize ĤS , resulting in
ĤS ¼G1 ðd̂1 b̂† þ d̂†1 b̂Þ þ G2 ðd̂2 b̂ þ d̂†2 b̂† Þ þ ĤCR : (2)Here, Gj ¼ gj āj are the many-photon optomechanicalcouplings, and ĤCR describe nonresonant interaction processes. We focus on the good-cavity limit ωM ≫ κ j , γ, where
the effects of ĤCR will be negligible. We will thus start bydropping ĤCR for transparency; i.e., we make the rotating
wave approximation (RWA); full results beyond the RWA
are presented in the figures and in the Supplemental
Material [46].
If G1 ¼ 0, the Hamiltonian of Eq. (2) describes anoptomechanical NDPA, with the mechanics acting as idler;
this was recently realized by Massel et al. [47]. One might
guess that turning on the beam-splitter interaction withcavity 1 by making G1 ≠ 0 would simply act to laser cooland optically damp the mechanical mode [48,49], but not
fundamentally change the amplification physics. This is not
the case: as we show below, the coherence between thecontrol lasers leads to a completely new mechanism. For
G1 ≥ G2 , the interactions in Eq. (2) have been discussed asa means to generate photonic entanglement [34,50–55];
amplification was not discussed. In contrast, we focus
on the case G1 ¼ G2 ; while this only leads to minimal

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intracavity entanglement [52], it is optimal in allowing themechanics to mediate amplifying interactions between thetwo cavities.
Dissipative interactions.—If the mechanical resonator
was strongly detuned (in the interaction picture) from thetwo cavity modes by a frequency Δ, then standard adiabatic elimination of the mechanics would yield the2~ d̂1 d̂2 þH:c:, with G∼G~
NDPA Hamiltonian, ĤPA ¼ G=Δ.
In contrast, we are interested in the resonant case, where
the induced interactions are more subtle. As the system
is linear, one can exactly solve the Heisenberg-Langevinequations corresponding to Eq. (2), and use these to derive
effective equations for the cavity modes with the mechanics
eliminated [46]. We first consider the simple limit whereγ ≫ κ, G; this results in effectively instantaneous induced
interactions. We also specialize to the ideal case whereκ1 ¼ κ 2 ≡ κ (see [46] for κ 1 ≠ κ 2 ). Introducing the effective
coupling rate Γ ¼ 4G2 =γ, the resulting Langevin equations
for the cavity modes are
pﬃﬃﬃ
pﬃﬃﬃ
ðκ þ ΓÞΓd̂1 − d̂†2 − κd̂1; in þ i Γb̂in ;d̂_ 1 ¼ −
2
2

(3a)

pﬃﬃﬃ
pﬃﬃﬃ
ðκ − ΓÞΓd̂2 þ d̂†1 − κ d̂2; in þ i Γb̂†in :d̂_ 2 ¼ −
2
2

(3b)

The operators d̂j;in (b̂in ) describe the quantum andthermal noise incident on the two cavities (the mechanics);
they have zero mean and correlation functionshôin ðtÞô†in ðt0 Þi ¼
hô†in ðtÞôin ðt0 Þi þ δðt − t0 Þ ¼ δðt − t0 Þðn̄To þ 1Þ, where o ¼ dj ,b, and n̄To is the thermal occupancy of each bath.
The mechanical resonator gives rise to two effects in
Eqs. (3). First, it gives rise to an additional positive
damping Γ of mode 1, and an additional negative damping−Γ of mode 2; each effect corresponds simply to one of thetwo terms in Eq. (2). In contrast, the joint action of bothinteraction terms gives rise to terms in Eqs. (3) reminiscentof a NDPA, where d̂1 is driven by d̂†2 and vice versa. Notecrucially the opposite sign of this term in Eq. (3a) versus
Eq. (3b); this difference implies that these terms cannot
be derived from an NDPA interaction Hamiltonian ĤPA .
Instead, they correspond to an effective dissipative parametric interaction. Such terms can be obtained from
Lindbladian dissipators in a quantum master equation[46]; they are also sometimes referred to as a phaseconjugating interaction [56]. On their own, such terms
cause a coherent rotation between d̂1 and d̂†2 , and as such no
amplification. However, when combined with the mechanically induced damping and antidamping terms, one finds a
striking result: the linear system described by Eqs. (3)always gives rise to exponential decay in the time domain at
a rate κ=2, irrespective of the value of Γ [46]. Thus, unlikea standard paramp, the mechanically induced cavity-cavity
interactions here do not give rise to a slow system decayrate, and do not cause any instability (i.e., the linear systemis stable for all values of Γ). This conclusion holds evenwhen γ=κ is finite: the system decay rates are independentof G [46].

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PRL 112, 133904 (2014)

Scattering properties.—While the mechanically induced
interactions do not yield any net antidamping, they dononetheless enable amplification. We use standard inputoutput theory to calculate the scattering matrix S½ω which
relates output and input fields. For simplicity, we firstneglect internal cavity losses. Introducing the cooperativity
C ¼ 4G2 =ðκγÞ, and defining the input and output vectorsD̂l ≡ ðd̂1;l ; d̂†2;l ; b̂l ÞT (l ∈ fin; outg), we find in the limitγ ≫ κ, ω,D̂out ½ω ¼ S½ωD̂in ½ω;0~2
2C−1−ω~ 2
ð1−iωÞ

B
B −2C
S½ω ¼ B ð1−i
~ 2
ωÞ
@
pﬃﬃ
2i C
1−iω~

2C
~ 2
ð1−iωÞ
ω~− 2Cþ1þ~ 2
ð1−iωÞ
pﬃﬃ
2i C
1−iω~

2

(4)
pﬃﬃ 1
2i C
1−iω~
pﬃﬃ C
−2i C C;C
1−iω~ A

(5)−1

~ ¼ 2ω=κ. Note that at ω ¼ 0, the above result
where ωholds for any value of γ; the full expression of S½ω forarbitrary γ is given in the Supplemental Material [46]. For
C > 1, S½ω implies that signals incident on either cavity ina bandwidth ∼κ around resonance will be amplified andreflected. For concreteness, we focus on signals incident
on cavity 1 (see Supplemental Material [46] for the similarcase of signals incident on cavity 2). The amplitude gain
for such a signal at resonanceis simply the reflection
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
coefficient S11 ½0 ¼ 2C − 1 ≡ G1 ½0. Clearly, the gain
can be made arbitrarily large by increasing C with nocorresponding reduction of bandwidth (which remains ∼κ).
This is in stark contrast to a standard NDPA, and is a directconsequence of the behavior discussed above: the mechanically induced interactions do not induce any net negative
damping of the system.
While for simplicity we have focused on the case where
the mechanical damping γ is large, the same physics holds
for an arbitrary κ=γ ratio. In the limit of large C, the photon
number gain is well approximated as
G1 ½ω ≡ jS11 ½ωj2 ≃C2:½1 þ ð2ω=γÞ2 ½1 þ ð2ω=κÞ2 2(6)

The effective bandwidth of the gain interpolates between κfor γ=κ ≫ 1, and γ for γ=κ ≪ 1. Our general conclusions
still hold: the gain can be arbitrarily large by increasing C,
and there is no fundamental limitation on the gainbandwidth product in this system.
Added noise.—Our scheme can also achieve a quantumlimited added noise. This follows immediately from the S
matrix in Eq. (5). As usual, we define the added number ofnoise quanta of the amplifier by first calculating the noise
spectral density of the amplifier output (i.e., d̂1; out ½ω).
The contributions to this noise from the mechanical andcavity 2 input noises constitute the amplifier added noise.
Expressing this as an equivalent amount of incident noisein the signal defines the number of added noise quantan̄add ½ω; the quantum limit on this quantity in the large-gain
limit is n̄add ½ω ≥ 1=2 [3]. We find at zero frequency

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pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ð G1 ½0 þ 1Þ2 1
1
þG1 ½0
n̄add ½0 ¼
þ n̄Td2 þð1 þ 2n̄Tb Þ
2
G1 ½0
G1 ½0
2 þ 2n̄Td2 þ 2n̄Tb
1
1
Tp
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
þO:(7)¼ þ n̄d2 þ
2
G1 ½0
G1 ½0
Thus, if cavity 2 is driven purely by vacuum noise, then in
the large-gain limit our amplifier approaches the standard
quantum limit on a phase-preserving linear amplifier. On
some level, this is surprising. The ideal performance of a
NDPA can be attributed to the fact that it has only a single
additional degree of freedom beyond the signal mode [3]. In
contrast, our system has two additional degrees of freedom(i.e., idler mode and mechanical mode); one might haveexpected that the presence of an extra mode would imply
extra noise beyond the quantum limit. That this is not thecase highlights the fact that the mechanical mode acts only asa means to mediate an effective dissipative coupling.
It is also worth stressing that in the large G1 limit, thecontributionof mechanical thermal noise is suppressed by apﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
factor 1= G1 ½0. This is in stark contrast to the optomechanical NDPA of Ref. [47]. In that system, the mechanicalmode acts as the idler; as such, quantum-limited performance is only possible if the mechanical resonator is atzero temperature, irrespective of the amplifier gain.
To illustrate the effectiveness of our scheme, we show in
Fig. 2 expected results for the gain and added noise for a
realization based on a microwave-cavity optomechanicalsystem, similar to those in Refs. [57,58]. While such
experiments typically have a small mechanical damping
rate γ (and, hence, small bandwidth), one could use a third
auxiliary mode to both laser cool the mechanical mode andenhance its linewidth [52]; we have assumed this situation.
One could also use a GHz-frequency, low-Q mechanicalresonator (similar to, e.g., Ref. [59]) to achieve bandwidths∼10–100 MHz [46].
Connection to QND measurement.—To provide further
intuition on the mechanism underlying our scheme, it isuseful to consider the dynamics in terms of canonicallyconjugate quadrature operators. We introduce these operators in our interaction picture in the standard way: d̂j ≡
pﬃﬃﬃ
pﬃﬃﬃ
ðX̂j þ iP̂j Þ= 2 and b̂ ¼ ðÛ þ iV̂Þ= 2. The interaction
Hamiltonian in Eq. (2) (with G1 ¼ G2 ¼ G) then becomespﬃﬃﬃ(8)Ĥint ¼ 2GðÛX̂þ þ V̂ P̂− Þ;
where we have introduced
joint cavity quadrature
operators
pﬃﬃﬃ
pﬃﬃﬃ
X̂ ¼ ðX̂1  X̂2 Þ= 2, P̂ ¼ ðP̂1  P̂2 Þ= 2.
Equation (8) lets us understand the importance of having
G1 ¼ G2 : for this choice, X̂þ and P̂− are QND observables.
They commute with the Hamiltonian and are thus conserved quantities. The QND interaction allows themechanical resonator to “measure” both of these joint
cavity quadratures: the V̂ (Û) quadrature of the mechanicaloutput field will contain information on X̂þ (P̂− ).
A QND measurement on its own will not generate
amplification. The interaction in Eq. (8) does more: it also
performs a kind of coherent feedback operation, where

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FIG. 2 (color online). (a) Black curves: photon number gain versus cooperativity C for parameters corresponding to a microwavecavity optomechanical realization of the dissipative amplification scheme. We take ωM =ð2πÞ ¼ 20 MHz and κ=ð2πÞ ¼ 1 MHz (solid
curve); the latter includes internal loss κint =ð2πÞ ¼ 10 kHz. The dotted line includes the effect of asymmetric cavity damping (seelegend); the dashed line shows instead the effects of G1 ≠ G2 (see legend). We assume that the mechanical resonator is coupled to a third
auxiliary cavity which is used to both cool and optically damp it, leading to a total mechanical damping rate of γ=ð2πÞ ¼ 200 kHz. Red
dashed-dotted curve: bandwidth (defined as the full-width at half-maximum of G1 ½ω) versus C, same parameters as the solid curve.
(b) Amplifier added noise n̄add versus C. Solid curve: mechanics and cavity 2 driven by vacuum noise only. Dashed-dotted curve:
mechanics now driven by thermal noise. Dashed curve: both mechanics and cavity 2 driven by thermal noise. Other parameters identicalto solid black curves in (a) and with n̄Tb;d1;2 as denoted in the graph. All curves are produced without making the RWA (though they are
well described by the RWA theory).

the results of the “measurement” are used to displace theunmeasured quadratures X̂− and P̂þ . For example, via thefirst term in Eq. (8), the mechanical V̂ quadrature measuresX̂þ : at zero frequency (and ignoring noise), thepHeisenberg
ﬃﬃﬃ
equations of motion (EOM) yield V̂ ¼ −ð2 2G=γÞX̂þ .
But via the second term in Eq. (8), we see that V̂ is a force
on the X̂− quadrature.
Again, the EOMs at zero frequency
pﬃﬃﬃ
yield X̂− ¼ ð2 2G=κÞV̂ ¼ −2CX̂þ . This directly translates
into the (ω ¼ 0) input-output relations
X̂þ; out ¼ −X̂þ; in ;(9a)X̂−;out ¼ 4CX̂þ; in − X̂−;in ;(9b)where we neglect mechanical noise contributions. Thus, thejoint measurement plus feedback operation has made X̂−an amplified copy of X̂ þ , while leaving the QND observableX̂þ unperturbed. In an analogous fashion, P̂þ becomes anamplified copy of P̂− . If we now express d̂1;out in terms ofjoint quadratures, we can immediately understand how we
obtain amplification. For large C, we have
1X
1ðX̂σ þ iP̂σ Þout ≃ ðX̂− þ iP̂þ Þoutd̂1;out ¼
2 σ¼2
1≃ ð4CÞðX̂þ þ iP̂− Þin ¼ 2Cðd̂1 þ d̂†2 Þin :(10)
2Thus, the QND measurement-plus-feedback operations on
the joint quadrature operators directly let us understand thestructure of the scattering matrix, and the observed amplification. Note that somewhat analogous QND interactions
play a crucial role in the construction of continuous variable
cluster states [60].
The QND form of Eq. (8) also explains the absence of
any induced damping of the cavities by the mechanics, see[46]. When G1 ≠ G2 , the QND nature of the interaction islost (i.e., Xþ , P− are no longer conserved), and thus for
fixed G2 =G1 , G1 ½0 saturates as a function of C1 . The sameis true when κ1 ≠ κ2 . One finds that in this case, the gain
G1 ½0 saturates at a value ½ðκ 1 þ κ 2 Þ=ðκ 2 − κ1 Þ2 in the large

C limit [46]. We stress that even with small coupling or
damping rate asymmetries, one can achieve very large gains[see Fig. 2(a)] with no loss of bandwidth. One can evensignificantly increase the amplification bandwidth over thesymmetric case, yielding amplitude-gain bandwidth products which far exceed κ (see Supplemental Material [46]).
Superconducting circuit realization.—Our scheme could
also be realized in a superconducting circuit, where therequired interactions in Eq. (2) are realized using Josephson
junctions. Here, the role of the mechanical mode would
now also be played by a microwave cavity mode, allowing
γ to be large. Further details on such realizations are presented in the Supplemental Material [46], where we showthat they offer advantages over conventional Josephsonparamps, such as Ref. [61]. Using similar parameters tothat work, our scheme can achieve quantum-limited amplification with a bandwidth of ∼47 MHz and a amplitude
gain-bandwidth product of ∼1900 MHz, a factor of 3.8
larger than the device reported in Ref. [61]; unlike
Ref. [61], this performance does not require a low-Q signal
cavity. An optomechanical system using a high-frequency,
low-Q mechanical resonator (like in the experiment ofRef. [59]) could also attain similar performance.
Conclusion.—We have described a new method forquantum-limited phase-preserving amplification which utilizes dissipative interactions; unlike standard cavity-based
parametric amplifiers, it does not suffer from any fundamental limitation on the gain-bandwidth product. The
scheme can be implemented both with optomechanics
and with superconducting circuits.
This work was supported by the DARPA ORCHID
program through a grant from AFOSR.

*

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014508 (2013).

133904-5

Supplemental Material for “Quantum-Limited Amplification via Reservoir Engineering”
A. Metelmann and A. A. Clerk
Department of Physics, McGill University, Montréal, Quebec, Canada H3A 2T8

GAIN AND NOISE ON THE LEVEL OF LINEAR LANGEVIN EQUATIONSStarting from the system Hamiltonian in RWA approximation, cf. Eq. (2) without ĤCR , and by using input-output
theory [1] to include the dissipative environment, we derive the quantum Langevin equations for the system. We
define the mode operator D̂ ≡ (dˆ1 , dˆ†2 , b̂)T and include the intrinsic losses D̂ξ = [ξˆ1int , ξˆ2int† , 0]T with the loss rates
T
ex
ˆ
ˆ†
κint
j (j ∈ 1, 2) for both cavity modes, as well as the input fluctuations D̂in = [d1,in , d2,in , b̂in ] with the rates κj for theexcavities and γ for the mechanical system. Here κj is associated with the coupling of each cavity to the waveguide
int
used to drive it and to extract signals. Hence, the total decay rates for the photonic system become κj = κex
j + κj .
All fluctuations are correlated to thermal baths as denoted in the main text. As usual, we work in frequency spaceand end up with the quantum Langevin equations

 p int p ex
0
0
− κ1
κ
0
0
−
1
p
p
e 
(S.1)
D̂[ω] = χ[ω]0− κex
0  D̂in [ω] + 
0− κint0  D̂ξ [ω] ,
2
2
√
0
0
− γ
0
0
0
introducing the susceptibility matrix
−1χ1 [ω]−10iG1e0χ2 [ω]−1 −iG2  ,
χ[ω]
=
iG1
iG2
χM [ω]−1


(S.2)


−1

κ −1
and χM [ω] = −iω + γ2
for the cavities and the mecontaining the free susceptibilities χj [ω] = −iω + 2j
chanical resonator. In this section, we concentrate on a symmetric parameter setting, which means that we assume
κ1 = κ2 ≡ κ as well as G1 = G2 ≡ G. Following from the assumption of equal decay rates, the free susceptibilities
for both cavity modes coincide, i.e., χ1 [ω] = χ2 [ω] ≡ χ[ω], and the three eigenvalues of the susceptibility matrix,
calculated for zero frequency, are1,2 = 2/κ,

3 = 2/γ.

(S.3)

Hence, they are independent of G and the time-dependent solutions are damped and will oscillate as D(t) = E1 e−κ/2t +E2 e−κ/2t + E3 e−γ/2t . From this we see that the system is stable irrespective of G, because no antidamping is present,ei.e., the eigenvalues of χ[ω]are always real and positive.
Moreover, we want to calculate the scattering matrix of the system; to keep the resulting expressions√compact
we neglect intrinsic losses for this derivation. We start from the input-output relations dˆj,out = dˆj,in + κdˆj and√b̂out = b̂in + γ b̂, which connect the input signals to the respective output signals. Hence, the scattering matrix reads




√
4ω 2
2C − (1 − i 2ω
)
1
+
2
γ
κ
2C
2i C




2ω
2ω
2ω
2ω 2
2ω 2
(1
−
i
)(1
−
i
)
(1
−
i
)(1
−
i
)
(1
−
i
)(1 − i 2ω
) 

γ
κ
γ
κ
γ
κ




2
√


2C + (1 − i 2ω
) 1 + 4ω
2

,
γ
κ
C
2C
2i
(S.4)
S[ω] = 

−
−
−
2ω
2ω 2
2ω
2ω 2
2ω
2ω 

(1 − i γ )(1 − i κ )(1 − i γ )(1 − i κ )(1 − i γ )(1 − i κ ) 

√
√


2i C
2i C
2


1
−
2ω
2ω
2ω
2ω
(1 − i 2ω
)(1
−
i
)
(1
−
i
)(1
−
i
)
1
−
i
γ
κ
γ
κ
γ
where D̂out [ω] = S[ω]D̂in [ω] and from which we derived the result in Eq. (5) of the main text for the limit of γ  κ, ω.
The first (second) row of the scattering matrix equals the output signal of cavity 1 (2). The diagonal elements coincidewith the reflection coefficient and the off-diagonal terms correspond to the added noise by the amplifier.
Now we include again√intrinsic losses, i.e., we have then κ = κex + κint and the slightly modified input-output
relations dˆj,out = dˆj,in + κex dˆj for the cavity modes. Additionally, we want to discuss if differences arise if the input

2
signal is applied either to cavity 1 or cavity 2. The respective gain for both cases yields


2

q
C
κex
κex  12e, (S.5)
∓
G
[0]
−
1
⇒
G
[0]
=
|S
[0]|
=
1
+
Sjj [ω] =1 − 2
j
j
jj
2
κ 1 − i 2ωκ
1 − i 2ω 1 − i 2ω
κ
γ

κ

p
where the tilded Gej [0] denotes the gain for κint 6= 0 and Gj [0] = ±(2C ∓ 1) equals the photon number gain without
intrinsic losses on resonance. The bandwidth of an output signal either at cavity 2 or cavity 1 is equal and, in theregime of large gain, there are no significant differences between the maximal obtained gain.
In the range of intermediate amplification (i.e., C not much larger than one) the gain maxima differ slightly:
 ex

8κex
κ − κintee
G1 [0] − G2 [0] = −
C.
(S.6)
κ
κex + κint
Thus, for κex > κint the amplification gain is slightly larger if the signal is applied to mode 2. Keep in mind howeverthat the gain scales with C 2 and hence the difference in the amount of gain between the two modes is negligible in
the limit of large amplification. To amplify an input signal incident on cavity 1 one needs C > 1; this condition is
independent of the amount of intrinsic losses. In contrast, amplification of a signal incident on cavity 2 is slightly moresensitive to intrinsic losses and it exhibits a different threshold for the cooperativity and the decay rates. Amplificationof signals incident on cavity 2 only occurs if C > κint /κex , which is a weak condition as one generally wants the couplingκ to dominate, i.e., κint  κex .
We find a similar behavior for the corresponding output noise, which we calculate by solving at first the system
of Langevin equations in Eq. (S.1) and afterwards, we use the input-output relation to obtain the output operatorsdˆj,out . The corresponding noise spectra are then obtained from1S̄j,out [ω] =2dΩh{dˆj,out (ω), dˆ†j,out (Ω)}i.
2π

(S.7)

To evaluate this, we need the definitions of the noise correlation functions hôin (ω)ô†in (Ω)i = hô†in (ω)ôin (Ω)i + 2πδ(ω +Ω) = 2πδ(ω + Ω)(n̄To + 1), where o = dj , b. Finally, for an input signal incident on cavity j the corresponding output
noise on resonance becomes
2
2


q

q



q
1
1
1
κint κex
κex
2κex
1
int
T
T
S̄j,out [0] =Gej [0] n̄ex
+
G
[0]
−
1
n̄
+
G
[0]
+
1
n̄
+
G
[0]
+
1
+
+
+
n̄
+
,
j
j
j
dj
dj
dj̄
b
2
κ2
2
κ
2
κ
2
(S.8)
int int
with the definition n̄Tdj = (κex n̄ex
j + κ n̄j )/κ, j ∈ 1, 2. Here, cavity j equals the signal mode and cavity j̄ canbe understood as the corresponding idler mode, while the mechanical oscillator is just an auxiliary mode. The firstterm in Eq. (S.8) is simply the amplified noise of the signal incident on cavity j and the second term originates fromthe losses inside of this cavity, a contribution which also appears in a standard paramp setup. The remaining terms
in Eq. (S.8) are important. They correspond to the added noise by the idler cavity mode as well aspthe mechanicalmode. The noise arising due to the latter auxiliary mode scales only linear with the cooperativity, cf. Gj [0] ∼ C, andhence is much smaller than the other contributions. Thus, for a large cooperativity the added noise of the amplifieris quantum limited, as discussed in the main text.
The output noise emanating from cavity 1 or 2 differ slightly from each other in the regime of intermediate amplification; similar to the gain maxima, cf. Eq. (S.6), the deviations scale linearly with the cooperativity C. The more
relevant question is whether the number of added noise quanta is different for both cavities. Therefore, we consider
the added noise quanta (without intrinsic losses)






S̄j,out [0]
1
4C 2
4C
1
1
T
T
n̄j,add ≡− n̄Tdj +
=
n̄
+
n̄
+
,
(S.9)
+
dj̄
b
Gj [0]
2(2C ∓ 1)2
2
(2C ∓ 1)2
2

which sets the arising noise in relation to the resulting gain and the signal’s input noise. In the above equation, the−(+) sign refers to j = 1(2). The mechanic’s and the idler’s noise contribution to the added noise are the same
whether one uses cavity 1 or cavity 2 as the signal mode, but the difference in the respective gain leads to


 



2n̄Td +1
2n̄Tb +1
16C 2
1
T
T
n̄1,add − n̄2,add =C 2n̄d +1 + 2n̄b +1 =++O 3 ,(S.10)(4C 2 − 1)2
C
C2
C

3
where we assumed n̄Td1 = n̄Td2 ≡ n̄Td for the cavity baths. Therewith we see, that in the limit of a large cooperativity
the difference between the added noise quanta is negligible.
Note, we are interested in the regime of large gain and additionally, we want to have κex  κint . The influenceof intrinsic losses on the gain and noise properties is then not significant. Hence, we neglect them in our further
discussions (though they are easily to include as described in this section).

ADIABATICALLY ELIMINATION OF THE MECHANICAL MODE AND MASTER EQUATION

For the case that the damping γ is large, we can perform an adiabatic elimination of the mechanical mode. We startagain from the system Hamiltonian in RWA, i.e., Eq. (2) of the main text without ĤCR , and calculate the quantum
Langevin equations in time-space. Afterwards, we derive the stationary solution for the mechanical operator (cf. third
row in Eq. (S.1) for ω = 0 and G1 = G2 ≡ G) and obtainb̂ ' −2iG ˆ
2(d1 + dˆ†2 ) − √ b̂in .
γ
γ(S.11)

Inserting this into the Langevin equations for the cavity operators in time-space, we obtain Eqs. (3) of the main text.
In the large γ regime our system can as well be described by a master equation approach. After the eliminationof the mechanical mode we obtain a Markovian master equation, which has standard Lindblad form. It contains thecavity decay terms as well as the dissipative parametric amplification contribution:
i
1 h
d
ρ̂ =
Ĥeff , ρ̂ + ΓL[dˆ1 + dˆ†2 ]ρ̂ + κL[dˆ1 ]ρ̂ + κL[dˆ2 ]ρ̂,dti~

(S.12)

with the Lindblad super-operator L[ô]ρ̂ = ôρ̂ô† − 21 ô† ôρ̂ − 12 ρ̂ô† ô and Γ = 4G2 /γ, while Ĥeff just describes the drivingof cavity 1 (or 2) by an input signal. Expanding the first Lindblad superoperator in Eq. (S.12), the terms involving asingle cavity operator yield the Γ-dependent damping and antidamping terms in Eqs. (3) of the main text. In contrast,
the terms involving both cavity 1 and 2 operators give the phase-conjugated interaction between the two cavities.
The above master equation can as well be expressed in terms of joint quadrature operators, which we defined in
the main text (see text below Eq. (8) of the main paper). We find:noL[dˆ1 + dˆ†2 ]ρ̂ = L[X̂+ + iP̂− ]ρ̂ = L[X̂+ ]ρ̂ + L[P̂− ]ρ̂ + i P̂− ρ̂X̂+ − X̂+ ρ̂P̂− .
(S.13)
Here, L[X̂+ ] and L[P̂− ] are standard measurement superoperators: they describe the measurement of the X̂+ andP̂− quadratures by the mechanics at the rate Γ. The remaining terms give rise to the effective feedback dynamics
described in the main text.

UNEQUAL DECAY RATESAs discussed in the main text, if κ1 6= κ2 , the QND nature of our system is lost (i.e., X̂+ and P̂− are not conserved
quantities), as can easily be seen by writing the Langevin equations in terms of quadraturesdκ1 + κ2
κ1 − κ2X̂− = −X̂− −X̂+ − ΓX̂+ ,dt
44dκ1 + κ2
κ1 − κ2
P̂+ = −P̂+ −P̂− − ΓP̂− ,
dt
4
4

d
κ1 + κ2
κ1 − κ2
X̂+ = −
X̂+ −
X̂− ,
dt
4
4
d
κ1 + κ2
κ1 − κ2P̂− = −
P̂− −
P̂+ ,
dt
4
4

(S.14)

where we ignored noise terms. The joint quadratures X̂+ and P̂− do not directly couple to the mechanical mode, but
due to the mixing of the joint quadrature operators for unequal decay rates the QND nature gets lost and X̂+ and P̂−are not longer conserved quantities. For κ1 = κ2 the quadratures X̂+ and P̂− decouple completely from the systemand the QND measurement is restored.
We like to briefly discuss the modifications of the gain and the noise in the case of unequal decay rates. The
corresponding expressions are derived in the simplest manner by starting again at the level of Langevin equations in4
Eq. (S.1). For concreteness, we also focus on the case where the signal is incident on cavity 1 (similar results follow
for the opposite case). We find for the photon number gain:  2ω
"
#2
2ω
δκ
−i 2ω
−
1
−i
+
1
+
+
1
+ C(2 + δκ
−i
[C
−
1]
2C − 1 + δκ
κ
κ
κ
γ
κ )
κ

⇒ G1 [0] =,(S.15)
S11 [ω] =δκ−i 2ω + 1 −i 2ω + 1 + δκ −i 2ω + 1 + C δκκ [C + 1] + 1
κ

κ

κ

γ

κ

with κ2 = κ1 + δκ ≡ κ + δκ. When δκ 6= 0, the lack of QND structure means that the optomechanical interactions
can cause a net damping or antidamping of the system, even leading to instability. From Eq. (S.15) we directly seethat one requires δκ > −κ/[C + 1], to ensure the system is stable. For a large cooperativity, the stability condition
is approximately δκ > 0, implying that the decay rate for cavity 2 has to be larger as for cavity 1 (κ2 > κ1 ). The
difference between the decay rates δκ in Eq. (S.15) is multiplied with the cooperativity C, and hence even small
deviations from the symmetric κ1 = κ2 case can suppress the gain. Moreover, for C → ∞ the resulting gain saturatesat ((κ1 + κ2 )/(κ2 − κ1 ))2 , and hence cannot be arbitrarily large as in the case for equal decay rates, cf. Fig. 2 of themain text. Even with this gain saturation at large C, the bandwidth λ of the amplification remains finite in the limitof large gain and is even increased if κ1 6= κ2 . Based on Eq. (S.15) and in the limit κ  γ we can approximate thebandwidth as 2 
λ
δκ
δκ
δκ
≈1+C
=1+C
+O
,
(S.16)γκ + δκκ
κ2in the second step we performed an expansion for small deviations δκ. Assuming for example δκ/κ = 1/C the resulting
gain is reduced by a factor of 4, while the bandwidth is increased by a factor of 2.
With the reduction of the gain we also obtain a reduction in the noise. The symmetrized output noise becomes
#2
#2
"
 #2
 
"
 " √

2 C 1 + δκ
2C − 1 + δκ
δκ
2C
1
1
1
T
T
T
κ
κ [C − 1]
+
1
+
+
.
n̄
+
n̄
+
n̄
+
S̄1,out [0] =
d1
d2
b
δκ
δκ
δκ
2
κ
2
2
κ [C + 1] + 1
κ [C + 1] + 1
κ [C + 1] + 1(S.17)In the limit of a large cooperativity and small deviations δκ we can estimate for the number of added noise quanta


 2


1
1
1 T
3 δκ
δκ 1
T
T
T
T
n̄add = n̄d2 + +n̄ + n̄b + 1 +,,(S.18)n̄d2 + 2n̄b ++O
2 C d22C
2 κ
κ2 C 2
which clearly shows that for a large cooperativity we still reach the quantum limit. A small deviation in the decay
rates has no significant effect on the added noise by the amplifier. From the expansion in Eq. (S.18) we see that higher
order contributions scale with δκ/C and thus are suppressed for the large cooperativity we require to have large gain.
Additionally, it is worth noting that for unequal decay rates one also has the possibility of an effective normalmode splitting in certain parameter regimes. In general, mode splitting in optomechanical systems appears in thestrong coupling regime, where the interaction with the mechanical system dominates the dissipative interaction,i.e., G  κ, γ [2]. There a cavity mode hybridize with the mechanical mode and the resulting two normal modes
perform coherent oscillations. The corresponding reflection spectra shows then two peaks at the frequencies of thenormal modes. The situation is quite different here; for simplicity, consider the large γ limit. For a phase-conjugatedcoupling of the two modes alone (i.e., Eqs. (3) of the main text without the ±Γ damping and antidamping terms), one
finds analogous behavior (i.e., the phase conjugated coupling gives rise to oscillatory dynamics in the time domain).
However, for symmetric κ the additional damping and antidamping terms completely cancel this oscillatory tendency,
as discussed in the main text. Introducing a damping asymmetry spoils this cancellation, and hence oscillations (andmode splitting) are again possible. For unequal κ, we find that mode splitting occurs whenG2 >

κ21 κ22,8(κ22 − κ21 )forκ2 > κ1 and γ  κ1,2 .

(S.19)In the regime of mode splitting, signal amplification still occurs, but now the gain exhibits two peaks as a function of
frequency.
UNEQUAL COUPLING STRENGTHS

While the main text focuses on G1 = G2 , it is also interesting to consider the case of unequal couplings. Similar tothe situation where the decay rates are unequal, the QND nature of the Hamiltonian is lost. The result is that one
can not longer achieve an arbitrarily large gain, as a large enough cooperativity can make the system unstable.

5
We start from the system Hamiltonian in Eq. (2) in the main text and assume the same driving and dissipation
terms as in the main text. With the former used description based on quantum Langevin equations and input-output
theory we derive the photon number gain



2
2ω
2

2
2
− 4ω
−i 2ω
2
κ2 − 1
γ + 1 − i κ C1 (rG − 1) + C1 (rG + 1)
G2
C1 (rG+ 1) − 1hi, rG ≡,⇒G[0]=
S11 [ω] =1

2
1 + C1 (1 − rG )G1
−i 2ω + 1
−i 2ω + 1 −i 2ω + 1 + C (1 − r2 )
κ

γ

κ

1

G

(S.20)
pwhich gives us the stability condition rG < 1 + 1/C1 , or equivalent G21 > G22 − κγ/4 as in Ref. 3. Here, we defined
the cooperativity as C1 = 4G21 /(κγ) and for rG → 1, i.e., G2 → G1 , we recover the result for equal coupling strengths.
The consequences of unequal coupling strengths are similar to the case for unequal decay rates discussed in the lastsection. In the limit of a large cooperativity, the maximal gain saturates at (G22 + G21 )2 /(G22 − G21 )2 . However, thebandwidth λ remains finite and is even increased compared to the case for equal strengths G1 = G2 . For κ  γ we
can approximate the frequency dependent gain resulting from Eq. (S.20) and obtain for the corresponding bandwidth


4 2
2
λ ≈ γ 1 + C1 [1 − rG
] =γ+G1 − G22 = γ + Γopt ,κ(S.21)

i.e., the bandwidthis just the mechanical damping plus the optical damping Γopt . Thus, if we choose for example
√
2
C
and
assume a large cooperativity, the resulting gain is G1 ≈ 4C1 , while the bandwidth is increased to=1/
1 − rG1√
λ ≈ γ C1 . Nevertheless, we have to care about possible mode splitting as in the case for unequal decay rates. Note,that when G1 6= G2 , we have basically the same system studied in Ref. 3, and the arising normal modes here are thesame as those discussed in that work. The eigenvalues of the susceptibility matrix, i.e., Eq. (S.2), are now given by:


s
2
2
1
1
1
1
1
4C
1
2 ] ,
 + ∓1 = ,
2,3 =−−[1 − rG(S.22)
2]
κ
1 + C1 [1 − rG
γ
κ
κ γ
κγ
and if 2,3 have a non-zero imaginary part, two additional peaks show up in the corresponding reflection spectra. For
rG = 1 we recover the real eigenvalues of Eq. (S.3), which are independent of the coupling strength G = G1 = G2 .
Hence, in the symmetric parameter regime (i.e., for equal coupling strengths and equal decay rates) we find no mode
splitting at all. Otherwise, to avoid the effect of mode splitting for G1 6= G2 we have to consider the condition
G21 − G22 < (κ − γ)2 /4. For γ  κ we can approximate this condition as Γopt < κ√and hence the maximal obtained
bandwidth is λmax ' κ, cf. Eq. S.21. Thus, for the upper example, i.e., Γopt = γ C1 , the maximal amplitude-gain
bandwidth product is 2κ2 /γ.
The zero-frequency noise of the output signal reads
√
2
 
2
 
2


2
2C1 rG
2i C1
C1 (rG+ 1) − 1
1
1
1
T
T
T
S̄1,out =
n̄
+
+
n̄
+
+
n̄
+
,
(S.23)
d1
d2
b
2)
2)
2)
1 + C1 (1 − rG2
1 + C1 (1 − rG2
1 + C1 (1 − rG2which is reduced as the gain. Finally, in the limit of large gain and for small deviations between the coupling strengths
the added noise quanta yield





1
1
1
3
T
T
T
T
T
2 1
n̄add = n̄d2 + +n + n̄b + 1 +n̄d2 + 2n̄b +[1 − rG ] + O (1 − rG ) , 2 ,(S.24)
2 C1 d2
C1
2
C1
which clearly shows that we still can reach the quantum limit.
COUNTER-ROTATING TERMS

We now consider the effects of the nonresonant, counter-rotating terms on our dissipative parametric amplification
scheme; these are described by:
n



o
ĤCR =G dˆ†1 b̂† + dˆ2 b̂† ei2ωM t + dˆ1 b̂ + dˆ†2 b̂ e−i2ωM t .
(S.25)
This Hamiltonian describes Stokes (anti-Stokes) processes for the photons in cavity 2(1) which are neglected in RWA.
The full non-RWA theory is exactly solvable by working in an interaction picture where the total Hamiltonian istime-independent.

6

FIG. S.1: Influence of counter-rotating terms on the relevant system quantities gain, added noise quanta and bandwidth.
The solid lines show the ideal symmetric case, while the dashed-dotted (dashed) line includes the effect of asymmetric cavity
damping (coupling strengths), for parameters see legend. The horizontal dotted lines correspond to the respective RWA result.
We choose κ/(2π) = 1 MHz, which includes internal loss κint /(2π) = 10kHz, and a cooperativity of C = 25. We assume themechanical mode to be optically cooled, leading to the effective damping γ/(2π) = 200 kHz. (a) Photon number gain as afunction of ωM /κ. (b) Added noise quanta versus ωM . The lower lines correspond to zero temperature baths (without intrinsic
losses); and the lines above to the the case where we include thermal noise on both the cavities and the mechanics, with n̄Tb,d2
as denoted in the graph. (c) Bandwidth versus the mechanical frequency, which is defined as the full-width at half-maximumof G1 [ω]. Note, in the asymmetric cases the resulting bandwidth is significantly increased, but by the price of a reduction inphoton number gain, cf. graph (a).

Taking the counter-rotating terms into account, we perform our standard calculation based on input-output theory
to derive the photon number gain on resonancehi
1− 4i ωγM
2C − 1 + C δκωMωMp
κ 1 + (−4i κ+δκ+1)(−4i κ +1)ih.
(S.26)
G1 [0] =
i γ
1
C δκ
+
1
+
1
−
ωM
ωM
κ
4 ωM
(−4i
+1)(−4i
+1)
κ+δκ

κ

Using this, we can estimate that if the decay rates are unequal the counter-rotating terms have more influence thanin the ideal, symmetric case. Similar behavior is found for unequal coupling strengths. But for δκ = 0 the terms inthe square brackets in Eq. (S.26) disappear and we obtain
2

G1 [0] =(2C − 1)2 + γ16 ω12M

2

1 + γ16 ω12M

2= (2C − 1) + 4C(C − 1)

∞
X
n=1

n

(−1)



γ/4
ωM

2n
,

(S.27)

therewith we have a scaling with γ/ωM and the cooperativity C remains a linear factor and does not enhance higher
order terms. It is interesting to note that these terms do not depend on the decay rate κ; ωM  γ is the only relevant
condition to keep the unwanted side-band processes off-resonant. We stress that this conclusion only holds for thegain, and for the ideal symmetric case G1 = G2 and κ1 = κ2 . For the number of added noise quanta the RWA andnon-RWA results require ωM  κ to coincide even in the symmetric case, as shown in Fig. S.1, which compares thefull theory against the approximate RWA results. The added noise in non-RWA contains as well noise incident on
cavity 1 due to next sideband contributions. Moreover, deviations from the ideal QND situation require ωM  κ if
one wishes the RWA to be valid.

SUPERCONDUCTING CIRCUIT REALIZATION OF DISSIPATIVE AMPLIFICATIONAs mentioned in the main text (and sketched in Fig. 1(b)), the dissipative amplification scheme we describe could
also be readily implemented in superconducting circuits utilizing Josephson junctions. The basic idea is to use thenonlinearity of a Josephson junction (essentially a nonlinear inductance) to realize the kind of parametric interactions
required in our basic interaction Hamiltonian (Eq. (2) of the main text). A key advantage over the optomechanicalrealization is that all three bosonic modes (signal, idler and auxiliary “b” mode) will be microwave modes. As such,having the auxiliary mode damping rate (and hence the amplification bandwidth) be in the range of several MHz is
not difficult to achieve.
While several routes are possible, we imagine a implementation using the Josephson parametric converter (JPC),
as pioneered in Ref. 4. The JPC is a symmetric circuit involving four Josephson junctions in a ring geometry that

7

FIG. S.2: (a) Black curves: photon number gain versus cooperativity C for parameters corresponding to a superconducting
circuit realization of the dissipative amplification scheme [5]. We take ωM /(2π) = 1 GHz, ω1 = ω2 = 10 GHz and κ/(2π) =γ/(2π) = 100 MHz. The dotted line depicts the zero frequency gain G1 [0]; the grey dashed line shows the RWA result andthe solid line corresponds to the actual maximum G1max . Red dashed-dotted curve: bandwidth (defined as the full-width at
half-maximum of G1 [ω]) versus C, same parameters as solid curve. (b) Amplifier added noise n̄add versus C. Solid curve: vacuum
noise only. Dashed curve: thermal noise with n̄Tb,d1,2 = 1 as denoted in the graph. The shown results include the relevant
counter-rotating terms, i.e., the leading sidebands associated with all counter-rotating terms in Eq. S.31, which lead to a shiftof the gain maximum G1max from the zero frequency result G1 [0], as well as a decrease of the bandwidth for higher values of C.

realizes an interaction involving three bosonic modes of the form




ĤJPC = Λ â1 + â†1 â2 + â†2 â3 + â†3 ,(S.28)

where âj (j = 1, 2, 3) are the annihilation operators for each mode. We now imagine a circuit having two JPCs with
one shared mode:









X 
ĤSC =
ωj,P â†j âj + ωj dˆ†j dˆj + ωM b̂† b̂ + g1 dˆ1 + dˆ†1 b̂ + b̂† â1 + â†1 + g2 dˆ2 + dˆ†2 b̂ + b̂† â2 + â†2 .
j∈1,2(S.29)
Here, the lowering operators of the five modes are â1,2 , dˆ1,2 , b̂, and g1,2 are proportional to the Josephson energies ofeach JPC. The operator b̂ (with resonance frequency ωM ) describes the shared common mode between the two JPCs; itwill play an analogous role to the b̂ mode in the main text (the mechanical mode in an optomechanical realization), andwill be used to mediate a dissipative interaction. The modes dˆ1,2 (resonance frequencies ωj , j = 1, 2) will play the roleof signal and idler modes, also in complete analogy to the main text. Finally, the modes âj (resonance frequencies ωj,P )will be pump modes: by strongly driving these at the appropriate frequency, we can realize parametric interactions
between dˆ1,2 and b̂ given in the central interaction Hamiltonian of the main text, Eq. (2). We take the pump mode
frequencies to be ω1,P = ω1 − ωM and ω2,P = ω2 + ωM , and assume that each pump mode is strongly driven on
resonance. The strong drive lets us linearize the above Hamiltonian by replacing the pump-mode operators âj bytheir average value āj . Working in an interaction picture with respect to the free mode Hamiltonians, and taking
g1 = g2 ≡ g as well as ā1 = ā2 ≡ ā for simplicity, we find:ĤSC = G dˆ1 b̂† + dˆ†1 b̂ + dˆ2 b̂ + dˆ†2 b̂† + ĤSC,CR ,(S.30)noĤSC,CR = G dˆ†1 b̂† + dˆ2 b̂† ei2ωM t + dˆ†1 b̂ei2ω1,P t + dˆ†2 b̂† ei2ω2,P t + dˆ†1 b̂† ei2ω1 t + dˆ†2 b̂ei2ω2 t + h.c. ,(S.31)
where G = gā is the pump-enhanced parametric interaction strength. The terms in ĤSC,CR describe non-resonantinteractions and will have minimal impact if the associated frequencies are much larger than the damping rates κ ofthe modes dˆj , b̂. In this limit, we see that the dual JPC system achieves the same Hamiltonian given in Eq. (2) of themain text, i.e., the basic interaction Hamiltonian which gives rise to dissipative amplification.
While the interaction Hamiltonian in the RWA is identical to that derived in the main text for the optomechanicalrealization, the non-RWA terms in Eq. (S.31) are not identical to the non-RWA terms arising in the optomechanicalrealization, c.f. Eq. (S.25). In particular, the terms oscillating at twice the pump-mode frequencies ωj,P do not appear
in the optomechanical setup. We find that the impact of these terms on the perfect dissipative amplification physics
arising from the Hamiltonian in Eq.(2) is more severe than other non-RWA terms. In practice, this means that the
maximum achievable photon number gain will be more limited by non-RWA terms in the JPC realization than the
optomechanical realization.

8
Nevertheless, this circuit realization of our dissipative amplification scheme could allow one to surpass the current
state of the art in Josephson junction paramps [5, 6]. As demonstrated by Mutus et al. [6], with a lumped-element
Josephson-junction parametric amplifier, it is possible to achieve a bandwidth of ∼ 50 MHz for a gain of 20 dB, byusing a highly damped microwave cavity (Q ∼ 10, κ ∼ 1GHz). Our proposed setup can match this performance easily
starting with a much higher Q microwave cavity. As shown in Fig. S.2, with C ∼ 6 we obtain a gain of 20 dB, while
the bandwidth is approximately κ/2, for a bandwidth of 50 MHz and κ = 100 MHz. The fact that our scheme allows
such performance with a higher Q (Q1 ∼ 100 versus Q ∼ 10 in Ref. 6) has may significant practical advantages. For
example, using low-Q resonances means that very large pump powers are required for amplification, which in turn could
lead to unwanted effects, e.g., the resulting large cavity amplitudes can lead to additional dissipative processes, as wellas higher nonlinearities in the Josephson potential can become important. In contrast, the dissipative amplification
scheme shows a good performance for higher Q values, without damaging the bandwidth of the amplified signal. The
results shown in Fig. S.2, can be further improved with small changes in parameters. For example, by now choosingωM = 2.5 GHz, κ = 250 MHz, γ = 125 MHz, we obtain a bandwidth of 90 MHz for a gain of 20 dB. This results in a
amplitude-gain bandwidth product that is a factor of two better than in Ref. 6, while still having a signal cavity withQ ∼ 40. Another way of increasing the bandwidth is possible by choosing asymmetric coupling strengths, e.g. taking
the same parameters as in Fig. S.2 but with G1 /G2 = 0.94 results in a bandwidth of 120 MHz for 21 dB gain.
Perhaps even more advantageous than the ability to match existing amplifier performance with large cavity Q isthe ability to achieve much higher gains without any consequent decrease in bandwidth. For example, as shownin Fig. S.2, a small increase in C allows to increase the gain from 20 dB to 30 dB, while maintaining a bandwidthof 50 MHz. Such higher gains are desirable as they greatly reduce the requirements on the following amplifier andany associated insertion losses; this is particularly important in state-of-the-art experiments requiring near quantumlimited performance. Consider for example applications in continuous-measurement based quantum feedback, as
implemented in recent experiments [7, 8]. Such protocols have a measurement efficiency η = 1/2n̄add , where n̄add
is the total added noise including the effects of following amplifiers. A standard HEMT following amplifier adds
anywhere from 20 to 50 noise quanta (when one includes typical insertion losses). A paramp gain of 20 dB meansthat this noise, referred back to the signal, yields 50/100 = 0.5 noise quanta on top of the paramp noise contribution.
Thus, even if the paramp is quantum limited, the total added noise is twice the quantum limit value, implyingη ∼ 0.5. In contrast, using a gain of 30 dB (as is achieved by our scheme with no bandwidth degradation) improves
this efficiency to η = 0.95. Such an improvement in measurement efficiency can dramatically increase the power ofmeasurement-based feedback protocols (see, e.g., Refs. 7, 8).
Finally, note that the optimal high-bandwidth performance shown in Fig. S.2 could in principle also be reachedin microwave-cavity optomechanical systems, if one now made use of a high-frequency, low-Q mechanical resonator(i.e., with a damping rate γ ∼ κ ∼ 100 MHz). Resonators of this sort were recently used in pioneering experiments
by O’Connell et al. [9], where they were coupled to microwave-frequency superconducting qubits. Similar approaches
could be used to achieve to couple them to microwave-frequency cavities, leading to a possible realization of ourscheme. Note that in such a system, the fact that mechanical and cavity frequencies are comparable implies that thesame additional counter-rotating terms found in the JPC setup will be relevant, i.e., Eq. S.31.

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