RAPID COMMUNICATIONSPHYSICAL REVIEW A 93, 060301(R) (2016)

Fock-state stabilization and emission in superconducting circuits
using dc-biased Josephson junctions
J.-R. Souquet and A. A. Clerk
Department of Physics, McGill University, 3600 rue University, Montreal, Quebec, Canada H3A 2T8(Received 1 February 2016; published 21 June 2016)
We present and analyze a reservoir-engineering approach to stabilizing Fock states in a superconducting
microwave cavity which does not require any microwave-frequency control drives. Instead, our system makes
use of a Josephson junction biased by a dc voltage which is coupled to both a principal storage cavity and asecond auxiliary cavity. Our analysis shows that Fock states can be stabilized with an extremely high fidelity. We
also show how the same system can be used to prepare on-demand propagating Fock states, again without theuse of microwave pulses.
DOI: 10.1103/PhysRevA.93.060301

Introduction. The ability to prepare, stabilize, and, ultimately, transmit nontrivial quantum states is crucial to a varietyof tasks in quantum information processing [1]. In the presenceof dissipation, state stabilization can be accomplished either
via measurement-plus-feedback schemes (see, e.g., [2–9]) or
autonomously, using quantum reservoir engineering (QRE)
techniques [10]. There has been considerable progress in implementing QRE ideas in superconducting circuits, including
experiments which have stabilized qubit states [11–13] aswell as photonic Fock states [14] inside microwave frequency
cavities. These schemes are typically complex, requiring theuse of several high-frequency microwave control tones.
In this work, we focus on the stabilization and emission of
Fock states in superconducting quantum circuits. Such states
have applications in quantum communication, cryptography,
and information processing (see, e.g., [15]) and are a crucial
ingredient in linear optical quantum computation [16]. We takean alternate approach here to Fock-state stabilization, which
requires no high-frequency microwave control tones and only
a dc voltage and a single Josephson junction. This leads toconsiderable simplicity over existing protocols and negates
any issues involving the microwave control tones corrupting
the generated Fock state (a problem that is typically solved
by using elaborate fast frequency-tunable qubits [17,18]). Our
approach also represents a fundamentally different approach toreservoir engineering which utilizes a highly nonlinear drivingof a cavity, rather than nonlinear cavity-cavity interactions.
The starting point for our scheme is a setup discussed in a
number of recent theoretical studies [19–21] and experiments[22–25], where a microwave cavity is coupled to a dc-biased
Josephson junction. The driven junction does not act likea qubit, but rather as a highly nonlinear driving element.
That such a setup can produce nonclassical photonic states
was first pointed out in Ref. [20], which showed that states
could be produced having suppressed number fluctuations anda vanishing g (2) intensity correlation function. While thesestates violate a classical Cauchy-Schwarz inequality, they
are mixtures of the vacuum and the one-photon Fock stateand have a limited fidelity with a pure Fock state [26]. As
a result, they do not exhibit any negativity in their Wigner
functions [27].
Here, we show how optimally coupling the junction to asecond cavity (as depicted in Fig. 1) lets one transcend the2469-9926/2016/93(6)/060301(6)limitations of the single-cavity system and prepare singlephoton Fock states with an extremely high fidelity. In addition,
it can easily be adapted to prepare higher Fock states. The second cavity acts like an engineered dissipative reservoir whichfreezes the system dynamics in the desired Fock state. Similar
two-cavity-plus-junction systems have recently been realized
experimentally [22] and have also been discussed theoretically[28–31], largely in the context of generating cavity-cavity
correlations. Armour et al. [29] found numerically that sucha system could generate cavity states with weakly negative
Wigner functions; they did not, however, discuss Fock-statestabilization or the particular mechanism we elucidate andoptimize.
In addition to efficient and high-fidelity Fock-state stabilization, we show that our setup can also play anothercrucial role: it can act as an efficient means for producing
propagating Fock states on demand. In circuit QED setups,
propagating single-photon states are usually generated bydriving microwave cavities coupled to a qubit with highfrequency control pulses (see, e.g., [32]). Among the many
challenges in the standard approach is the requirement that thegenerated photon should be far detuned from the frequencyof the control pulse [17,18,33]. Our system is capable ofon-demand Fock-state generation without any microwave
control pulses: one simply needs to pulse a dc control voltage.
We note that recent work by Leppäkangas et al. studied analternative approach to generating propagating single photons
using a biased Josephson junction [34,35]. Contrary to ourwork, this setup is based on using Coulomb blockade physics
and does not lead to the stabilization of an intracavity Fockstate; it also does not allow for the production of higher Fockstates.
Biased junction as a nonlinear drive. To set the stage for our
two-cavity system, we start by quickly reviewing the physics
of the one-cavity version, as studied in [19,20]. The system
consists of a Josephson junction in series with a LC resonator(frequency ωc , impedance Z) and a dc voltage source Vdc . We
focus exclusively on temperature and voltages small enoughthat superconducting quasiparticles are never excited. The
Hamiltonian has the form Ĥ = ωc (â † â + 1/2) − EJ cos(φ̂),
where â is the cavity photon annihilation operator, EJ is the
Josephson energy, and φ̂ is the phase difference across thejunction; unless otherwise noted, we set  = 1 throughout.

060301-1©2016 American Physical Society

RAPID COMMUNICATIONS

J.-R. SOUQUET AND A. A. CLERKPHYSICAL REVIEW A 93, 060301(R) (2016)

FIG. 1. Schematic of the proposed setup. A dc-biased Josephson
junction of energy EJ is in series with two cavities. Each cavity has
an impedance close to RK = h/e2 and is damped via a coupling to atransmission line.

The voltage drop across the junction d φ̂/dt is given by thedifference of Vdc and the cavity voltage√ V̂cav . This leads
to φ̂ = 2[eVdc t + λ(â + â † )], where λ = π Z/RK sets themagnitude of the cavity’s zero-point flux fluctuations and
RK = h/e2 is the resistance quantum. The cavity is also
coupled to a transmission line, which is treated (as standard)
as a Markovian reservoir and which gives rise to an energy
damping rate κ. Working in an interaction picture at the cavity
frequency, the Hamiltonian of the cavity plus biased junction
takes the form [19,20,30,36–38]
Ĥ = −

EJ 2ieVdc t(eD̂[α(t)] + e−2ieVdc t D̂ † [α(t)]),2(1)where D̂[α(t)] is the cavity displacement operator [corresponding to a time-dependent displacement α(t)]. It is definedin terms of the photon annihilation operator â as†∗D̂[α(t)] = eα(t)â −α (t)â , α(t) = 2λeiωc t .

(2)Note that we consider the case where intrinsic phase fluctuations of the junction (due to charging effects or noise from ashunt resistor) play no role. For more details on the derivationof this Hamiltonian, see, e.g., Refs. [19,20].
From the point of view of the cavity, Ĥ describes a highly
nonlinear (but coherent) cavity drive [20]: each term tunnels
a Cooper pair and also displaces the cavity state by ±α(t). To
see clearly how these displacements can result in Fock-state
generation, we follow a different route from [20] and expressD̂[α(t)] directly in the Fock basis (see, e.g., [39]):D̂[α(t)] =

∞ 
∞


wn,n+l [λ] |nn + l|e−ilωc t

n=0 l=0+∞(−1)l |n + ln|wn+l,n [λ]eilωc t .

(3)

l=1

Here, the transition amplitude wn,n+l [λ] is nothing more thana generalized Frank-Condon factor. For l  0 we have

2
2
n!
L(l)
(4)
wn+l,n [λ] = e−2λ (2λ)l (n+l)!
n (4λ ),
where L(k)n is a Laguerre polynomial [40] and wn+l,n [λ] =wn,n+l [−λ].
As is well known, Frank-Condon factors are highly nonlinear functions of the magnitude of the displacement (here setby λ) and can even exhibit zeros [41] corresponding to zerosof the Laguerre polynomials; the behavior of relevant factors
is shown in Fig. 2. We let λ̃n+l,n denote the smallest value of
λ which makes wn+l,n [λ] vanish:
wn+l,n [λ̃n+l,n ] = wn,n+l [λ̃n+l,n ] = 0.

(5)

FIG. 2. Matrix elements wn+l,n [λ] for junction-driven cavity
Fock-state transitions as a function of the zero-point voltage fluctuation amplitude λ for different values of n and l. These functions
are highly nonlinear with respect to λ and can cancel for specificvalues. Roots of wn+l,n [λ], denoted λ̃n+l,n , are indicated.

We note that in comparison to the Bessel function expansion
used in Ref. [20], the decomposition in Eq. (3) makes it easier
to identify values of λ yielding zeros and also highlights the
direct connection to Franck-Condon physics.
The route to preparing single Fock states now seems clear:by tuning the value of both Vdc and λ (via the cavity impedanceZ), one can arrange for the effective driving of the cavity by
the junction to shut off when a given Fock state is reached[14]. For concreteness, suppose we chose Vdc = k/2e (k aninteger), so that to leading order, Cooper pairs can only tunnel
by emitting or absorbing k cavity photons. If we take ωc 
EJ ,κ, we can restrict attention to these processes and make arotating-wave approximation (RWA) to our full Hamiltonian.
Within the RWA we have∞
EJ 
wn+k,n [λ]|n + kn| + H.c. (6)ĤRWA = (−1)k+1
2 n=0
Consider the simplest case where k = 1, and each resonantly
tunneling Cooper pair√emits or absorbs a single cavity photon.
If we then set λ = 1/ 2 ≡ λ̃2,1 (see Fig. 2, left panel), there
is no matrix element in Eq. (6) for a transition from one to two
cavity photons. As first discussed in Ref. [20], the junctioninduced drive now can add at most one photon to the cavity,
implying that the system effectively acts like a driven two-levelsystem.
The cavity steady state is found by solving the Lindblad
master equation for the density matrix of the cavity ρ̂c forρ̂˙c = 0,ρ̂˙c = −i[ĤRWA ,ρ̂c ] + κL[â]ρ̂c ,(7)which includes the dissipation from the (zero-temperature)transmission line; here, L[â]ρ̂ = â ρ̂ â † − 12 {ρ̂ n̂ + n̂ρ̂}. Note
this dissipator naturally implies that the lifetime of the nth
Fock state decreases with n. When λ is set to λ̃2,1 , the stationary
intracavity state can be termed nonclassical in that it results ina vanishing g (2) intensity-intensity correlation function [20].
This simply reflects the fact that there is zero probability forhaving two or more photons in the cavity. We are still far,however, from our goal of producing a single-photon Fockstate. As the cavity is effectively a driven two-level system,
population inversion is impossible, and at best the steady state060301-2

RAPID COMMUNICATIONSFOCK-STATE STABILIZATION AND EMISSION IN . . .

PHYSICAL REVIEW A 93, 060301(R) (2016)is an incoherent mixture with an equal probability of vacuum
and single photon. We stress that such a state exhibits nonegativity in its Wigner function.
Fock-state stabilization. Heuristically, the poor performance of the single-cavity setup is easy to understand: even if
we eliminate the matrix element for |1 → |2 transitions, thejunction-driven cavity continues to oscillates back and forth
between the vacuum and the |1 Fock state. To achieve true
Fock-state generation, one needs to shut off the oscillation
dynamics when the system is in the |1 state. As we now
discuss, this can be achieved rather simply by coupling thejunction to a second “auxiliary” cavity (see Fig. 1) whose
damping rate κaux is taken to be sufficiently large. We have so
far not included the effects of low-frequency voltage fluctuations on the dynamics. The primary effect of these fluctuations
is to cause dephasing (see, e.g., Ref. [22]), causing coherencesin the density matrix to decay. As these coherences already decay due to the large auxiliary cavity damping, we are relatively
robust against this noise. This is discussed further (along withfull master-equation simulations including dephasing) in the
Supplemental Material [42].
In what follows, we denote quantities for the main “storage”
cavity with a subscript s, while auxiliary cavity quantities
have a subscript aux; Fock states of the two-mode system
are denoted |n,m = |ns ⊗ |maux . Working in an interaction
picture with respect to the free-cavity Hamiltonians, the system
Hamiltonian isĤ = − E2J e2ieVdc t D̂[αs (t),αaux (t)] + H.c.

(8)

FIG. 3. Schematic depictions of biased-junction cavity pumping
processes. The boxed digit indicates the number of Cooper pairs thathave tunneled, whereas dots in the parabolas indicate the intracavity
photon number. (a) Single-cavity setup for an impedance yielding
λ = λ̃2,1 . Cooper-pair tunneling can take the cavity between the zeroand one-photon states, but transitions to the two-photon state areblocked. (b) Two-cavity setup for Fock-state stabilization, where
λs = λ̃2,1 . Starting from vacuum, Cooper-pair tunneling can cause
oscillations between the |0,0 and |1,1 photon Fock states. Photon
decay from the aux cavity, however, freezes the system into the desired|1,0 state. When the storage-cavity photon decays (due to κs ), thecycle repeats.

Let us make the above picture more quantitative. We startby applying the RWA to Eq. (8),⎞⎛∞EJ ⎝ 
ĤRWA =
wn +1,ns [λ]|ns + 1ns |⎠2 n =0 s
s
⎛
⎞
∞

⊗⎝
wnaux +1,naux [λ]|naux + 1naux |⎠ + H.c.
naux =0

Here, D̂[αs (t),αaux (t)] = D̂s [αs (t)] ⊗ D̂aux [αaux (t)] is the tensor product of displacement operators for each cavity, withrespective displacement amplitudes αj (t) = 2λj eiωj t .
For the auxiliary cavity to play the desired role, we tune thevoltage so that Vdc = (ωs + ωaux )/2e. Cooper-pair tunnelingthus requires simultaneously emitting a single photon to eachcavity (or absorbing a photon from each cavity). We also tune
the storage-cavity impedance so that λs = λ̃2,1 . We choose
λaux such that w1,0 [λaux ] is large (see Fig. 2), but we stress thatno special tuning of λaux is needed. The resulting dynamics
is sketched in Fig. 3(b). Similar to the single-cavity system,
the effective driving from the biased junction couples |0,0
to |1,1 but not to states with a higher photon number. We
would seem yet again to have an effective two-level system
and might expect coherent oscillations between these twostates. However, the large damping rate κaux of the auxiliary
cavity prevents this: if the system is in the |1,1 state, κaux willcause a rapid decay to |1,0. In the absence of storage-cavity
damping, the system is then effectively stuck: Cooper-pairtunneling against the voltage is impossible (as there are no
photons in the aux cavity), while tunneling with the voltage
is impossible as there is no matrix element connecting |1,0
and |2,1. Including storage-cavity damping does not ruin thephysics: if the storage-cavity photon leaks out, one is back inthe vacuum state, and the process starts again. One thus sees
the possibility for having a steady state that has a high fidelity
with the state |1,0, i.e., a stabilized single-photon state in thestorage cavity.

(9)
Each cavity is also coupled to its own zero-temperature bath,and the reduced density matrix ρ̂ of the two cavities obeys the
Lindblad master equation:ρ̂˙ = −i[ĤRWA ,ρ̂] + κaux L[âaux ]ρ̂ + κs L[âs ]ρ̂.

(10)Consider first the ideal case, where λs is tuned perfectly
to equal λ̃2,1 . In the relevant limit κsκaux , the steady-state
probability that the system is in the desired state |1,0 is
1,0|ρ̂|1,0

+ κs

,

=

(EJ w1,0 [λs ]w1,0 [λaux ])2
,
κaux
(11)

where plays the role of an effective pumping rate from |0,0
to |1,0 and we have dropped terms as small as κs /κaux . The
probability to be in the desired state tends to 1 in the limit κaux  κs . The large |1,0 population here is analogous
to the population inversion possible in a driven three-levelsystem [43]. We note in passing that the fidelity between thestate of the storage cavity and the desired |1 Fock state is justgiven by the probability to have a single photon [26]; hence,
the above expression is a lower bound on the fidelity.
The above process can be efficient even if the storagecavity impedance is not perfectly tuned. Suppose λs = λ̃2,1 + εwith ε
1. Assuming again κsκaux , one finds that the060301-3

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J.-R. SOUQUET AND A. A. CLERKPHYSICAL REVIEW A 93, 060301(R) (2016)

FIG. 4. Steady-state probability ps [n] for the storage cavity to be
in a Fock state |ns , as obtained from the RWA master equation inωs ,ωaux , and λaux = 1/2; λs and κs
Eq. (10). We take EJ = κaux
are indicated in the plots. For each n, the rightmost bar represents
λ closest to perfect tuning. (a) Probabilities when λs is tuned closeto λ̃2,1 (for stabilizing |1). (b) The same for λs tuned close to λ̃3,2(for stabilizing |2). Even with imperfect tuning, the target Fock stateis stabilized with a high fidelity. (c) and (d) Corresponding Wigner
function W [α] for the storage-cavity state, showing that negativevalues are obtained; note that W [α] is rotationally invariant in allcases.

now effectively get stuck in the state |n,0. Numerical results
for the case n = 2 are shown in Figs. 4(b) and 4(d).
We stress that the general scheme here can be viewed as
an example of reservoir engineering [10], with the biased
junction and auxiliary cavity acting as an effective dissipativeenvironment which stabilizes the storage cavity in the desired
Fock state. In that respect, our protocol has similarities to the
Fock-state stabilization scheme described and implemented inRef. [14]. In that work, the engineered reservoir also involved
an auxiliary cavity but used a qubit and two microwave control
tones (or more if the target Fock state had n > 1). In ourwork, the auxiliary cavity is still there, but the microwave
control tones and qubit have been replaced with a Josephson
junction biased by a dc voltage. Note that conversely toRef. [14], reaching higher Fock states does not require
additional resources.
Itinerant Fock states on demand. The above protocol forstabilizing Fock states can naturally be used to produce
propagating Fock states: after the desired Fock state has beenstabilized, one simply turns off the dc bias voltage when theitinerant photon is desired, and the storage-cavity Fock state
will be emitted into the transmission line coupled to it (ina temporal mode with an exponential profile [45,46]). An
alternate strategy is to exploit the transient dynamics of ourscheme and produce an itinerant Fock state with a pulsed
dc voltage. In this case, one optimizes parameters to have ahigh-fidelity intracavity Fock state at an intermediate time, as
opposed to in the long-time steady state. The voltage is thenturned off at this intermediate time. This makes it possible
to achieve a high-fidelity Fock state even if the tuning ofprobability to be in |1,0 is modified to be1,0|ρ̂|1,0+ κs + 4ε2

2 /κ

.

(12)

s

Because of the imperfect tuning of λs , it is no longer
advantageous to have  κs (i.e., large EJ ), as transitions tohigher states will corrupt the dynamics. Equation (12) suggeststhat an optimal choice would be to have κs = 2ε (while still
maintaining κaux  κs ).
To complement the above analytical results, we have
performed a full numerical simulation of the RWA masterequation in Eq. (10) using the QUTIP package [44]. Figure 4(a)shows the stationary storage-cavity photon-number distribution for different choices of λs . The probability to be in thedesired |1 Fock state is ∼0.70 even with a 10% mismatch
between λs and its ideal value and jumps to ∼0.86 if themismatch is reduced to 2%. These fidelities are high enoughto give rise to storage-cavity Wigner functions that exhibit
large amounts of negativity [as shown in Fig. 4(c)]. They
also compare favorably both to the single-cavity scheme ofRef. [20] (where even in the ideal case the fidelity can never begreater than 0.5 and negative Wigner functions are impossible)
and to Ref. [14], where the experimentally measured fidelity
was ∼0.67.
Our protocol can also be used to stabilize higher Fock stateswith a good fidelity. One keeps the voltage set to Vdc = (ωs +ωaux )/2e but now tunes the storage-cavity impedance such thatλs = λ̃n+1,n for some chosen n > 1. The system dynamics will

FIG. 5. Time dependence of storage-cavity photon-number occupancies ps [n,t], where the bias Vdc = (ωaux + ωs )/2e is turned on
at t = 0 and the cavities start from vacuum. Dashed line: κs chosen
to optimize the steady-state value of ps [1,t] (thick dashed line) with
the Fock-state stabilization protocol. Solid line: alternate choice of κswhich optimize ps [1,t] (thick solid line) at intermediate times withthe pulsed protocol. In this latter protocol, the voltage can be turned
off near toff , resulting with high probability in the production of a
propagating Fock state in the transmission line coupled to the storage
cavity. In both cases, we assume the realistic situation where thecavity impedance has not been tuned perfectly (here, λs = 0.95λ̃21
and λaux = 0.13λ̃21 ). The inset shows ps [n,t → ∞] for the pulsed
protocol.

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PHYSICAL REVIEW A 93, 060301(R) (2016)

the storage-cavity impedance is not perfect, i.e., λs = λ̃2,1 .
In addition, by choosing parameters such that the oscillation
dynamics associated with Cooper-pair tunneling is slightly
overdamped, one can have a protocol which is relatively
insensitive to the timing of the voltage pulse.
In Fig. 5, we show a comparison between the steady-statestabilization protocol and this pulsed protocol, where λs deviates by 5% from the ideal value λ̃2,1 needed for single-photon
generation. By using a value of κs smaller than the choicethat optimizes the steady-state single-photon probability, one
obtains a much higher fidelity with a single-photon Fockstate at intermediate times. We find that at an optimal turnoff
time, a state fidelity of ∼0.92 with the |1 Fock state can beobtained (see thick solid line in Fig. 5). Further, the maximum
probability for having a single storage-cavity photon is a rather
broad function of time, meaning that one does not need precise
control of the shutoff time of the dc voltage. This is in stark
contrast to standard protocols for preparing a Fock state using

two-level system dynamics (e.g., in the one-cavity version of
our system), where one needs precise control of the durationof the pulses control.
Note that in the above simulations, the voltage is abruptly
turned on. More realistic situations where the voltage is slowly
ramped on yield a similar result as the tunneling processes areresonant only when 2eVaux + s .
Conclusion. We have shown how a system where a voltagebiased Josephson junction is coupled to two cavities can beused to stabilize Fock states with a high efficiency. While ourapproach does require one to carefully tune the impedance ofthe main storage cavity, it does not require any microwavefrequency control tones. We also discussed how the same
setup could be used to produce itinerant single photons on
demand.
Acknowledgments. This work was supported by NSERC.
We acknowledge fruitful discussions with P. Hofer, M.
Hofheinz, F. Portier, and J. Viennot.

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060301-6

Supplemenary material for “Fock-state stabilization and emission in superconducting circuits using
dc-biased Josephson junctions”
J.-R. Souquet, A. A. Clerk
Department of Physics, McGill University, 3600 rue University, Montreal, Quebec, H3A 2T8, Canada(Dated: May 25, 2016)

In this supplemental section, we study the effects of decoherence due to voltage fluctuations from the low-frequency
environment and show that they have a rather limited impact on the results presented in the main text.
The
low-frequency voltage fluctuations can be well-modelled asstemming from an environment with a roughly frequencyindependent impedance at relevant frequencies [? ? ]. For
temperatures T > ω, the spectral density of the voltage
fluctuations are then given by S[ω] ' 2ZkB T with Z theimpedance of the low frequency environment (see, e.g., [? ]).
As typical dephasing rates due to the voltage noise areworst comparable to T , we can safely use a Markovian approximation to describe the voltage-noise induced dephasing(i.e. effectively treat the spectral density as being white). We
note that in the experiment of Ref.[? ] (which had relatively large amounts of voltage noise), the dephasing rate due
to voltage noise was estimated as Γϕ ∼ 150 MHz. In thisworst-case scenario, a white noise approximation should besafe for temperatures down to ∼ 75 mK. Within the white
noise approximation, the dephasing rate is simply given byΓϕ = S[0]. Note even using the numbers from Ref.[? ],this rate is much smaller than the Josephson energy and theauxiliary cavity damping rate we used in our simulations(EJ = κaux ' 1GHz).
To include the voltage-noise dephasing in our master equation treatment, we start by adding a classical fluctuating voltage δVcl (t) to the bias applied to the Josephson junction, suchthat 4e2 hδVcl (t)δVcl (t0 )i = Γϕ δ(t − t0 ). The Hamiltonian
from Eq.(9) of the main text now reads:

FIG. 1. (a-c) Steady-state probability ps [n] for the storage cavity to
be in a Fock state |ni , as obtained from the RWA master equation(Eq.(10) of the main text) with the addition of a decoherence term
describing voltage fluctuations (c.f. Eq. (3)); results for various decoherence rates Γϕ are shown. (d) Time dependence of storage cavity photon number occupancy ps [1, t], for various decoherence rates
Γϕ . Even for very large decoherence rates (Γϕ ' EJ ' 1GHz), thelow-frequency voltage fluctuations of the environment have negligible effects on the results presented in the main text.

new gauge, the storage cavity has a fluctuating frequency:
ĤRWA = 2eδVcl (t)â†s âs +EJ X
wnaux +1,naux [λaux ]2 n ,nsaux

wns +1,ns [λs ](|naux + 1ihnaux | ⊗ |ns + 1ihns | + H.c.). (2)ĤRWA =

EJ 2iee2Rt

dt0 δVcl (t0 )Xwnaux +1,naux [λaux ]ns ,naux

wns +1,ns [λs ](|naux + 1ihnaux | ⊗ |ns + 1ihns | + H.c.) (1)It is convenient to make a change of gauge which converts
this fluctuating voltage into an equivalent fluctuating cavity
frequency; such a fluctuating frequency will cause an equivalent decoherence of coherences in the cavity density matrix. The gaugeR change is done using the unitary operatortÛ = exp(−2ie dt0 δVcl (t0 )â†i âi ) with i = s or aux. One
can gauge the voltage noise into either the storage or cavity
frequency, as both choices lead to equivalent decoherence of
relevant density matrix coherences. We chose i = s. In the

Averaging over these frequency fluctuations in the standard manner (for white, Gaussian fluctuations) leads to an additional dephasing superoperator in the master equation thatcauses the decay of coherences. It has the form:



1 2
dρ
= Γϕ n̂s ρ̂n̂s −n̂s ρ̂ + ρ̂n̂2s .
(3)
dt noise2
where n̂s is the photon number operator for the storage cavity.
In Figs. 1(a-c), we plot the steady-state probability ps [n] forthe storage cavity to be in a Fock state |ni for different valuesof Γϕ and different values of λ and κ corresponding to thoseof Figs. 4a, 4b and 5 of the main text. In Fig. 1a we plot theprobability of stabilizing one photon for a 10% mismatch in λ(corresponding to light blue bars in Fig.4a of the main text).

2
Even for values much larger than we expect for Γϕ (that isΓϕ ' EJ ' 1GHz), voltage fluctuations have a rather negligible effect on our results. In fact, large amounts of dephasing
have a positive effect, in that they suppress transitions intohigher unwated Fock states.
In Fig. 1b, we plot the probability of stabilizing one photon
for a 2% mismatch in λ (corresponding to the middle bars in
Fig.4a of the main text). We see that as we decrease the mismatch in impedance, the effects of the low-frequency environment get smaller. In Fig. 1c, we plot the probability of stabilizing two photons for a 10% mismatch in λ (corresponding to
the light green bars in Fig.4b of the main text) and conclude
that decoherence still has a minor effect on stabilizing higher
Fock states. Finally in In Fig. 1(d), we plot the time-dependent
number occupancy of the storage cavity ps [1, t] (correspond-

ing to the light green curve of Fig.5 of the main text), for different values of Γϕ . Looking back to Eq.(12) of the main
text, we see that voltage fluctuations serve to effectively decrease Γ, the effective pumping rate from |0, 0i to |1, 0i. Similarly, they decrease the effective transition rate from |n, 0i
to |n + 1, 0i. In principle, this implies that one could use
a smaller value of κs (as the principle reason for having a
not-too-small κs is to prevent the population of higher Fockstates). If one makes this change, the results with voltage fluctuations become even closer to those presented in the maintext.