A Quantum Heat Machine from Fast Optomechanics

We consider a thermodynamic system in which the working fluid is a quantized harmonic oscillator subjected to periodic squeezing operations at a rate much larger than its resonance frequency. This device could potentially be constructed using pulsed optomechanical interactions. It can operate as a heat pump, a heat engine, or a refrigerator. We find that the oscillator can be transiently cooled to temperatures below that of the cold bath. In addition, we show that the heat engine and refrigerator behaviors vanish when the system--bath coupling is treated using the conventional rotating wave approximation, an effect not seen in previous optomechanical engine proposals.

The union of thermodynamics and quantum mechanics has proved extremely fruitful since its earliest days. Recent years have seen a rapid acceleration of this progress [1,2], including important developments such as the generalization of the second law of thermodynamics to the quantum realm [3], as well as the first general proof of the third law of thermodynamics [4]. Tools from quantum information theory [5] have also clarified the role of information in thermodynamical processes, as demonstrated by, for example, the quantum Szilard thermodynamic engine [6].
Quantum systems with engineered Hamiltonians and system-bath interactions-such as atoms in optical cavities, superconducting circuits, and opto-or electromechanical devices [7]-are promising tools with which to experimentally study quantum thermodynamics [8][9][10][11][12][13][14]. These systems are frequently modelled using the Born-Markov master equation (BMME), or its equivalent Langevin equations. The BMME provides a convenient description of the system's behavior as it interacts with its environment over timescales comparable to or greater than a characteristic internal time (τ S ); however, it fails to capture shorter timescales correctly [7]. This inadequacy may be traced to the treatment of the heat bath in the BMME. If one begins with an independent oscillator model of the environment-the most general microscopic model of a linear, passive heat bath [15,16]-one obtains the BMME by making the rotating wave approximation (RWA) [39]. Essentially, this consists of neglecting nonenergy-conserving terms in the Hamiltonian, the absence of which 'washes out' dynamics faster than τ −1 S . It is wellknown in many branches of quantum mechanics that the RWA can mask interesting physical phenomena [17][18][19], or introduce non-physical artifacts into the theoretical description of a system [20][21][22][23][24].
In this Letter we study the influence of the RWA on predictions in quantum thermodynamics. Specifically, we consider an oscillator subjected to a periodic train of impulsive squeezing operations at a rate much larger than its natural oscillation frequency. The oscillator couples to a hot bath between squeezers, whilst imperfect squeezing operations provide a cold bath. We prove that within the approximation of the BMME the oscillator can only operate as a heat pump. However, without the RWA we find consistently richer behaviors, including additional refrigerator and heat engine regimes. Thus we predict the existence of thermodynamic cycles not permitted by the RWA. Our scheme also provides an alternative to standards techniques of cooling the oscillator such as optomechanical sideband cooling [25], distinguished by its ability to cool below the cold bath temperature. Together, these phenomena indicate the emergence of non-trivial thermodynamical behavior beyond the RWA.
For concreteness we will consider a mechanical oscillator, such as a micro-or nano-mechanical resonator. Such devices have been reported to have Q factors up to (9.8 ± 0.2)×10 7 [26], with Q > 10 5 being readily achieved in many materials and resonator geometries [27]. As such, they are weakly coupled to their thermal environment and generally amenable to treatment using the Markovian limit of the independent oscillator model. Similar conclusions may be drawn for analogous systems e.g. electronic circuits [28] or optical cavities.
The equations of motion derived from the independent oscillator model are [40] (e.g. [7,29]) where X and P are the dimensionless position and momentum operators obeying [X, P ] = 2i, ω M is the resonance frequency, and Γ = ω M /Q is the decay rate. Note that these equations are asymmetric under rotations in phase space because the loss (−ΓP ) and thermal noise (ξ (t)) terms couple only to P . This is in contradistinction to the BMME (here written as the equivalent Langevin where loss and noise affect X and P equally. The asymmetry of Eqn (1) is strongly manifest in the short-time behavior of the system. This is most plainly seen from the covariance matrix with V = V T and { · · · } denoting the real part of the expectation value. For all times t ≥ 0, V is related to its initial value V 0 by a linear transformation where M (t) is a square matrix encoding the homogeneous part of the dynamics (cf. Supplemental Information). The aggregated effect of thermal noise is described by the added noise covariance matrix V FF (t), which depends upon M (t) and , wheren H is the equilibrium occupancy of the hot bath. For short evolution times (t ω −1 M ) this added noise (calculated according to Eqn (1)) is to leading non-trivial order in each matrix element. The first (second) diagonal element describes the noise added to X (P ). Thus, over short timescales the noise introduced by the environment is 'squeezed', in the sense that the diagonal elements of V FF are markedly unequal (see Fig. 1 b)). Strikingly, this phenomenon is absent in the BMME prediction, where, again taking t ω −1 M , V (RWA) FF = (2n H + 1) Γt1, with 1 being the identity matrix.
These observations indicate that there is a link between the short-time behavior of Eqn (1) and squeezing. We therefore consider the possibility of manipulating the dissipative dynamics using squeezing operations which are applied much more frequently than ω M .
Our proposed protocol is divided into three broad steps (see Fig. 1 a) & d )): an initial imperfect squeezing operation S 1 , a short time of evolution whilst in contact with the hot bath, and a second imperfect squeezer S 2 . This sequence is repeated at a rate of ω ap = 2π/t.
As seen in Fig. 1 c), each imperfect squeezer S j (j = 1, 2) is modeled by subdividing it into two further steps. The first is a unitary squeezing operation S j , distinguished by the lack of prime. The second is a beamsplitter interaction between the system and a thermal state of covariance V C = (2n C + 1) 1, with transmissivity 1 − and effective thermal occupancyn C <n H .
We select the first squeezer (S 1 ) such that it performs the operation X → µ −1 X & P → µP , where µ is the squeezing strength. The momentum becomes antisqueezed for µ > 1 and squeezed for µ < 1. S 2 is then chosen to be S 2 = RS −1 1 R T where R is a rotation matrix with angle ω M t. This choice ensures that if the oscillator is decoupled from both hot and cold baths (Γ = 0, = 0, under which conditions M = R) the evolution induced by the protocol is completely passive, with the state effectively experiencing only free evolution. A typical thermodynamic device (engine, pump, or refrigerator) operates by performing a cyclical process in which the working fluid is returned to the same state at the end of each cycle. We will refer to this as the 'steadystate'. Repeated application of our squeezing protocol will gradually force any initial state towards a zero-mean Gaussian 'steady-state' with covariance V SS [41] defined by the self-consistency condition

Environment Squeezer
The matrix M hom = (1 − ) S 2 M (t) S 1 is the homogeneous component of the evolution, and is the aggregate effect of the noise entering from the baths. It is critical to note that V FF and V C are strongly modified by S 2 . This allows us to manipulate the squeezing of V add by adjusting the evolution time t and squeezing strength µ. Eqn (4) can be recognised as a Sylvester equation which is readily solved using standard numerical techniques. Solutions show that the steady-state is very well approximated by the thermal state V SS = (2n SS + 1) 1, wherē n SS is the effective occupancy (cf. Supplemental Material). Representative calculations ofn SS as a function of the squeezing strength µ and the squeezing application rate ω ap = 2π/t are given in Fig. 2 a) & b). We use ω M = 1 MHz & Q = 10 6 -as might be expected for SiN or SiC microstrings [30]-andn H = 4 × 10 4 (T H ≈ 300 mK, achievable in a 3-He cryostat), and will continue to use these parameters unless otherwise stated. Fig. 2 clearly shows that the squeeze-rotate-squeeze protocol can reduce the temperature of the oscillator to well belown H , opening up a new method of cooling mechanical oscillators to study their quantum behavior, and as a preparatory step for quantum sensing and information processing protocols [7]. To understand this phenomenon we employed a combination of analytical and numerical techniques (see Supplemental Material), to find thatn where γ = ω ap /π is the effective decay rate to the cold bath, and µ opt is the squeezing magnitude which minimizes the energy added by the baths during each cycle. The functional form of Eqn (5) is reminiscent of that of the standard quantum limit for position measurement on a free mass, where an optimal interaction strength exists that balances measurement noise with quantum back-action noise [7]. In this case the balance is between attenuating the noise added to P and amplifying the noise added to X during each timestep. These processes can be linked to the µ −2 and µ 2 /µ 4 opt terms of Eqn 5 respectively. When the squeezing strength reaches µ opt the noises added to X and P are equal, and the total noise energy is minimised.
In the case of {Q,n H } 1 {ω M t, } we find Together, Eqns (5) and (6) provide an excellent approximation ton SS , as seen in Fig. 2 c). In the limiting case ofn C 1 andn H n C , µ opt reduces to a simple ratio of heating rates, viz.
. This shows that the oscillator may be made arbitrarily cold-even colder than n C -as γ → 0 or Γ → 0.
As clearly shown in Fig. 2 a),n SS can be reduced well belown H even when = 0. The second law of thermodynamics implies that this must be accompanied by a net heat flux into the hot bath, because otherwise the system would extract work from a single thermal bath and reduce the entropy of the universe. The system is acting as a heat pump, taking work from the squeezers and pushing it into the hot bath.
In order to determine if this behavior persists with imperfect squeezers we calculate the work (W ), heat from the cold bath (Q C ), and heat from the hot bath (Q H ) during a cycle of evolution in the steady-state. A positive number indicates an influx of energy to the oscillator, normalized to units of mechanical quanta. It is clear that because the perfect squeezers S j are unitary (isentropic) they are associated with W , with the other operations corresponding to heat exchange. Thus where V (2) and V (3) are the covariance matrices imme- diately after S 1 and immediately before S 2 respectively (cf. Fig. 1 d )). Given the process is cyclical, the heat transferred to the cold bath is Q C = − (W + Q H ).
The system can indeed act as a heat pump when the cold bath is present, fulfilling Q H < 0 & W > 0. It can also act as a heat engine (Q H > 0 & W < 0); or as a refrigerator (Q C > 0 & W > 0). These regimes are shown in Fig. 3. There is also a fourth region in which the work performed on the system is positive but insufficient to reverse the flow of heat from the hot bath to the oscillator. Together, these form a 'phase diagram' which is vastly richer than the RWA predicts.
The heat pump behavior is straightforward to understand by considering the limit as the time between squeezers (t) tends to zero. The damping rate of the system into the hot bath then becomes 2Γ P 2 , where P 2 is boosted above its 'steady-state' value by a factor of (1 − ) µ 2 during the first squeezing interaction. Thus heat pumping occurs whenever this boosted loss rate overwhelms the noise coming in from the bath. This phenomenon is also behind the remarkable fact that the oscillator may be cooled to temperatures lower than that of the cold bath, unlike with other cooling techniques such as sideband cooling (see Supplemental Material and [31]).
In the heat engine region the system is extracting work from the hot bath and dumping entropy into the cold bath. This occurs when the state V (3) is more squeezed than V (2) . The momentum-damped Langevin equations (Eqn (1)) permit this if (see Supplemental Material) When { , n C } 1 n H this becomes n H >n SS µ 2 i.e. the 'apparent occupancy' of the momentum must be less than the hot bath occupancy, such that more heat flows from the bath to the oscillator than in the reverse direction.
Refrigeration-removing heat from the cold bath-only occurs whenn C is sufficiently large. The sign of Q C may be determined by considering the two loss steps involving the cold bath, as shown in the Supplemental Material. For small we find the condition for refrigeration is By explicitly calculating the steady-state covariance matrix in the RWA we were able to derive no-go theorems (Supplemental Material) which show that the heat engine and refrigerator phases of operation are forbidden in the RWA. The former proof is valid in all parameter regimes satisfying basic requirements of physicality, whilst the latter is valid in the 1 and Γ ω M ω ap regime considered throughout this Letter.
Finally, we can consider the performance of the available thermodynamic cycles. The relevant coefficients of performance are defined by where the thermodynamic limits are expressed in terms of the Carnot efficiency η = 1−T C /T H . A representative calculation is provided for a cross-section through Fig. 3 d ), as shown in Fig. 4 a). This demonstrates that (for these parameters) the peak engine efficiency is approximately one third of the Carnot limit. It also confirms that the heat pumping efficiency is maximised when the squeezing strength µ is below the value µ opt which minimizes the steady-state occupancy. Conversely, in the RWA case the heat pump efficiency improves monotonically with µ. This difference arises because the RWA does not have a refrigerating phase (or an engine phase). Calculations for different bath temperatures (e.g. Fig. 4 b)) show the expected trends: namely that the heat engine becomes less efficient asn C increases, whereas COP fridge improves withn C .
It is worthwhile to identify the limitations of the theories employed in this study. Eqns (1) are not of Lindblad form, and so are not guaranteed to be completely positive for all initial states and (hot) bath temperatures [32,33]; however, they are valid in the high-Q andn H 1 regime considered throughout this Letter (n H ≥ 10 2 ). No such restriction exists for the cold bath temperature in our model. We have also approximated all squeezing interactions as occurring instantaneously, rather than the conventional case of parametric amplification over many mechanical periods (e.g. [34]). The extremely rapid mechanical squeezing operations needed by this protocol could potentially be realised by combining the pulsed optomechanics interface of [35] with the arbitrary-quadrature pulsed optomechanical interaction of [36], using a squeezed optical state as an ancilla.
We have proposed and modeled a thermodynamic system based on a momentum-damped mechanical oscillator subjected to rapid squeezing operations. Our calculations indicate that such a system can operate as a heat pump, a refrigerator, or a heat engine. Importantly, if lossy evolution is modeled by the BMME the latter two effects vanish. This indicates the emergence of rich-and potentially useful-quantum thermodynamical phenomena beyond the RWA.
The authors thank Mr Kiran Khosla for discussions concerning pulsed optomechanics, and Prof. Ronnie Kosloff for discussion of Markovian Langevin equations. This work was funded by the Australian Research Council (ARC), CE110001013. JSB is supported by an Australian Government Research Training Program Scholarship. WPB is an ARC Future Fellow (FT140100650).

SUPPLEMENTAL MATERIAL This Supplemental Material discusses the;
• detailed dynamics of the oscillator, as predicted by the linear oscillator bath model and the Born-Markov Master Equation (BMME); • derivation of the steady-state occupancy; • derivation of the bounds on thermodynamic cycles in the linear oscillator model; • no-go theorems for heat engines and refrigerators in the rotating wave approximation (RWA); and • alternative model of lossy squeezing, showing its similarities to the simpler model used in the main text.

Damped Harmonic Motion Linear Oscillator Model
Consider the equations of motion of a harmonic oscillator of (angular) frequency ω M subject to momentum-dependent damping at rate Γ.Ẋ = +ω M P, (11a) Note that the loss appears only in Supp. Eqn. (11b), as does the noise operatorξ (t). These equations are typically employed to describe classical systems, but may be readily-if not straightforwardly-quantized (e.g. [37]).
Integrating the equations of motion is straightforward because of their linearity. One obtains Here the modified resonance frequency is σω M with σ = 1 − Γ 2 /4ω 2 M (we have assumed an underdamped oscillator, Γ < 2ω M ), and the effect of the thermal noise is captured by the increments δX and δP . The solutions (12) may be recast as a matrix equation of the form where and with X = (X P ) T .
In the high-temperature, high-Q limit the noise operatorξ (t) becomes Markovian and Gaussian. We shall henceforth restrict our attention to this case. The noise correlation function becomes Gaussian noise and quadratic operations (i.e. squeezing, rotations, linear loss, and beamsplitters) imply that the steady-state will be characterised by its first and second moments. It is thus appropriate to consider the covariance matrix, given by From Supp. Eqn (13) we obtain the solution which is valid because the added noise is Markovian and uncorrelated to the system. The aggregate added noise V FF is Note that for t ω −1 M the variance of δP goes linearly in t, the covariance of δX & δP goes quadratically, and the variance of δX scales cubically, viz.
The long-time evolution yields the steady-state covariance matrix which is easily recognizable as a thermal state with occupancyn H ≈ k B T H / ω M (for the relevant case of k B T H ω M , as phonons are bosonic excitations).

Born-Markov Master Equation
It is instructive to consider the oscillator's dynamics under the BMME. Solving Eqns (2) (main text) requires knowledge of the noise correlation functions, viz.
Using these yields the evolution of the covariance matrix, which is entirely analogous to Supp. Eqn (16) except that where R (ω M t) is a rotation through an angle of ω M t, and 1 is the identity matrix. Note that V (RWA) FF is proportional to the identity matrix, and at short times both diagonal elements of V (RWA) FF grow at first order in t; this is very different behavior from the independent oscillator model, where δX 2 initially only increases at third order in t.
Despite this, the long-time behavior in the RWA is identical to the independent oscillator model, viz.

Approximation of Steady-State Occupancy
In the main text we defined the 'steady-state' covariance matrix as Eqn (4), reproduced here for convenience.
We can cast this in the typical form of a Sylvester equation by noting that M −1 hom does exist for any non-negative, finite time t. Thus This can be solved with standard computational packages, or indeed analytically. The general solution to Supp. Eqn (18) is [38] vec where ⊗ is the Kronecker product and vec {· · ·} is the vectorization operation. As seen in Supp. Fig. 5, the steady-state Wigner function is essentially symmetrical between position and momentum. Therefore, the state is thermal and characterized by an effective occupancȳ n SS can in principle be calculated entirely analytically, but the resulting expression is immensely unwieldy. Instead, we will find an approximate expression which is accurate in the region of interest ({Q,n H } 1, { , n C , ω M t} 1). For simplicity, let us first consider the perfect squeezing ( = 0) case. Each matrix may be calculated analytically. We then expand vec {V SS } to second order in t, calculate the determinant, and truncate the result at O t 2 and O (Γ). Discarding small terms (which are not boosted by µ) then gives Consider the coefficient of µ 2 in Supp. Eqn. (19). This can in fact be related directly to properties of the added noise. To see this, let us ask which value of µ minimizes the energy contained by the added noise V add . In this case ( = 0) we have V add = S 2 V FF S T 2 ; thus we wish to select the value of µ which minimizes Tr S T 2 S 2 V FF . Calculating this yields which is the inverse of the coefficient of µ 2 in Supp. Eqn. (19). Similarly, consider the µ = 1 value ofn SS | =0 , which is approximatelyn H . If we rewriten SS asn then-guided by our numerical calculations-we can make the educated guess that the same form holds when the squeezers become imperfect.
The first factorn SS | µ=1 needs to be generalized to there being two baths. The most likely estimate is the typical equilibrium occupancy expected from detailed balance, viz.
The effective coupling rate to the cold bath is γ, which is taken to be γ = 2 /t = ω ap /π because the loss of 2 occurs over every timestep t.
Secondly, we replace µ opt | =0 with the value of µ which minimizes the trace of V add including noise from the cold bath. This may be analytically found to be , which simplifies (for small t and ) to , which in turn can be reduced to the expression appearing in the main text ifn C 1, and t → 0 asn H → ∞.
By this route we arrive at Eqn (6) of the main text. As seen in Supp. Fig. 7, this is an excellent approximation in the parameter regime of interest.
Note that none of the bounds of thermodynamic cycles (below) rely on this particular form ofn SS ; they simply require a symmetrical steady-state Wigner function.
As seen in Supp. Fig. 6, the occupancyn SS can be reduced to less thann C . This is because the oscillator can inject energy into the cold bath, much as it injects heat into the hot bath in the RWA: this can be seen by calculating the occupancy with Γ = 0.

Bounds on Thermodynamic Cycles
Consider a complete cycle of the squeeze-rotate-squeeze protocol with lossy squeezing as described in the main text. Let us divide up the protocol into the following steps (cf. Fig. 1 of main text): The squeezing steps perform work, allowing us to identify the total work (in quanta) as Similarly,

Criterion for Heat Engine Phase
The system behaves as a heat engine if it receives heat from the hot bath (Q H > 0) and produces output work (W < 0). Any excess entropy is dumped into the cold bath.
Our numerical calculations show that the engine phase overlaps very well with the parameter regime where V (3) is more squeezed than V (2) i.e. where the asymmetry of the lossy evolution step is strong enough to actually increase the asymmetry of the Wigner function over a short timestep.
Firstly, let us establish that the degree of squeezing present in an arbitrary (single mode, 2 × 2) covariance matrix can be expressed as the ratio of its eigenvalues, κ. For a vacuum state operated on by S j (µ) we have κ = P 2 / X 2 = µ 4 . It is easily seen that κ is in fact a function of which is directly proportional to the product of the energy (∝ Tr {V }) and purity |V | −1/2 of the state. Thus we can use K as a proxy for the level of squeezing.
To simplify the calculation of K (2) and K (3) we will consider the high-Q limit and set σ = 1 − Γ 2 /4ω 2 M = 1. We expand the resulting inequality around small t to first order, yielding where the diagonal elements of V are V XX and V PP , and the off-diagonal elements are both V XP . Since V (2) is position squeezed the right hand side of this inequality is a negative number. Thus we get where we used the fact that V where η is the squeezing parameter after being degraded by the cold bath. Then we obtain Substituting the explicit form of the variance yields (Eqn (8) of main text) (2n H + 1) > (1 − ) (2n SS + 1) µ 2 + (2n C + 1) , which in most cases of interest ({ , n C } < 1 n H ) can be replaced by the simplified condition n H > µ 2n SS .

Criterion for Refrigeration Phase
In the refrigeration regime, the resonator uses input work (W > 0) to extract heat from the cold reservoir (Q C > 0) and push it into the hot reservoir (Q H < 0).
Using the fact that V SS = (2n SS + 1) 1, we obtain In the limit that 1 this becomes equal to Eqn (9) given in the main text.
• Conclude that the quadratic condition on is never satisfied for physical values of ; thereby see that λ R < 1 ∀ ∈ (0, 1). • Use the definition of λ to see that λ > 1 for all Γt of interest, and thus this λ R is not physical.
• Conclude that the term in Supp. Eqn (22) is always positive.
With this we can see that which prompts us to consider the simpler inequality equivalent to considering the right hand side with cos 2ω M t = 1. This new condition factorizes rather straightforwardly, yielding which is obviously satisfied. Thus we conclude that b > 0 always.
Since we have established that a, b, c, and d are all positive we know B is positive. Consider then Supp. Eqn (20), which is quadratic in µ 2 . There are only solutions to Supp. Eqn (20) when the determinant of the left hand side, B 2 − 4, is non-negative. This gives B ≥ 2. Rearranging this yields the condition which is seen to be quadratic in λ = e Γt . The determinant of the left hand side is negative, thus there are no real roots. This means that this conditions is always satisfied, and B is always at least 2.
We may thus calculate that µ 4 + Bµ 2 + 1 < 0 is satisfied when This clearly has no solution for real µ because both the upper and lower limits of Supp. Eqn (24) are negative. This implies W (RWA) < 0 cannot be satisfied, and hence that there is no heat engine phase in the RWA. Note that we have not taken any limits during this derivation, nor have we approximated V SS as diagonal, so our no-go theorem holds for arbitrary ∈ (0, 1) and any positive Γ < ω M /2.
The sign of each coefficient is controlled by the terms in square brackets. α does have real zeros, but these only occur for negative values of µ 2 , which are unphysical. Thus α > 0 for all µ > 1.
The (square) bracketed term in β is quadratic in µ 2 . Calculating its determinant makes it clear that β < 0 for all physical µ.
This means that the left hand side of Supp. Eqn (25) is always positive, whilst the right is always negative, thereby ensuring that the condition cannot be met. We conclude that there is no refrigeration in the RWA (at least in the small , high-Q regime). Our numerical calculations support this conclusion.

Alternative Lossy Squeezing Model
As discussed in the main text, a conceptually clean model of a lossy squeezer is to subdivide each squeezer S j into a perfect squeezing step S j followed by a beamsplitter operation with a cold thermal mode. However, one might suspect that in a physical implementation of the squeezer S j the cold bath temperature would potentially be a function of µ due to effects such as pump depletion. We will therefore consider a slightly more complicated model, and show that it reproduces essentially the same predictions for low to moderate µ.
Instead of subdividing S into one perfect squeezer and one beamsplitter we will consider subdividing it into 2N substeps of perfect squeezing followed by loss on a beamsplitter. Each sub-squeezer squeezes by an amount µ k = µ 1/N , such that the net squeezing parameter (in the absence of loss) is µ = Π N k=1 µ k . Similarly, each attenuation step is characterized by k = 1 − (1 − ) 1/N , such that the total attenuation is 1 − . After one substep (k) of squeezing and loss we have (1 − k ) k (2n C + 1) S k S T k + k (2n C + 1) 1 . . .
Taking the N → ∞ limit then yields the full model of the squeezer The homogeneous component of the evolution is the same as the simple model, but the added noise is now a function of µ in addition to andn C . We have not employed this model through the main text because it yields results which are qualitatively very similar to the simpler two-step squeezer model in the regime of interest. An example of this is shown in Supp. Fig. 7. Comparison of the steady-state occupancynSS as a function of squeezing parameter µ for three noise models. Line A shows the result for perfect squeezing operations; line B incorporates the simple lossy squeezing model employed in the main text; and line C is calculated using the more complicated loss model described above. Note that B and C agree well at low-to-moderate µ. Parameters are given in the main text (lines A and B are identical to Fig. 2 panel c) of main text).