# Observation of broadband entanglement in microwave radiation from the dynamical Casimir effect

A fundamental property of the quantum vacuum is that a time-varying boundary condition for the electro-magnetic field can generate photons from the vacuum, the so-called the dynamical Casimir effect (DCE) Moore1970 . The photons are created pairwise and should possess the particular quantum correlations referred to as entanglement. In particular, broadband entanglement sources are useful for two reasons: i) they can be very bright, generating a large number of entangled photons, and ii) the wide frequency content allows shaping the emitted radiation in time. Here we present the observation of entanglement generated in a broadband system using the DCE. The measured sample consists of a superconducting quantum interference device (SQUID) terminating a microwave transmission line. A magnetic flux modulates the SQUID’s Josephson inductance at approximately 9 GHz, resulting in a time-varying boundary condition for the quantum vacuum in the transmission line. We detect both quadratures of the microwave radiation emitted at two different frequencies separated by 0.7 GHz. Analysing the measured data and using a careful calibration of the power level emitted from the sample, we determine the purity and type of entanglement using two different methods. We estimate an entanglement rate of 5 Mebit/s (mega entangled bits per second).

Sources of entangled photon pairs at optical frequencies have found applications in quantum secure key distributionEkert1991Aug , quantum repeatersBriegel1998Dec and quantum sensingDixon2011 . At microwave frequencies, sources with two-mode entanglement are also proposed for entangling qubitsFelicetti2014 and continuous-variable quantum computingAndersen2015 . Using parametric amplificationEichler2011 ; Zhong2013 ; Chang2017Aug , the DCE in a mediumLahteenmaki2013 and shot-noise from a tunnel junctionWestig2017Mar , sources of entanglement were demonstrated. Unfortunately, only very few sources of entanglement are available with a wide bandwidthForgues2015Apr . Moreover, good temporal control over the photon generation process is required to also shape photon packagesRego2014 ; Silva2011 . Protocols reaching unity efficiency in transmitting and absorbing photons rely on such temporally shaped photon packagesCirac1997 ; Jahne2007 ; Korotkov2011 ; Pechal2014 , and would indeed allow for new and better control.

With the demonstration of the DCEWilson2011 by using a SQUID at the end of an open transmission line, two-mode squeezing was demonstrated with a wide bandwidth and a precise control of the vacuum boundary condition. However, without precise photon flux resolution, entanglement was not shown, and it was still unclear if broadband radiation from the DCE is entangled. Quantitative bounds for proving entanglement were developed theoreticallyJohansson2013 taking thermal photons into account.

In our wide band system the photons are distributed over a large frequency range. The generated photon flux density is relatively small compared to resonator based entanglement sourcesFlurin2012 ; Lahteenmaki2013 . Therefore, the requirements of resolving precise photon spectral densities to prove entanglement are correspondingly more difficult to meet.

Here we demonstrate entanglement of photon pairs generated by the dynamical Casimir effect, equivalent to using a single mirror with a sinusoidal displacementDeWitt1975Aug ; Fulling1976 . A flux-modulated SQUID, located at the end of a transmission line, acts as a movable microwave mirror (Fig. 1a). The flux modulates the SQUID inductance, which in turn changes the electro-magnetic boundary condition and, photon pairs are generated (see Methods). A covariance matrix and the two-mode squeezing is calculated from correlations of voltage quadratures at different pump amplitudes, i.e. at different displacement speeds of the electromagnetic boundary condition. We extract the log-negativity and the purity of entanglement from the covariance matrix and determine the type of entanglement from the two-mode squeezing below the vacuum.

To characterise the sample, we measure the current-voltage characteristic of the SQUID (Fig. 2a) and find a critical current of A and a superconducting energy gap voltage of 360 eV. From the forward (blue) and backwards (red) current sweeps, we observe a hysteresis, indicating that the SQUID is underdamped.

To obtain the necessary resolution, we use the SQUID itself to calibrate the noise, gain and response of the circuit. In Fig. 2b, we show the spectral density of the shot noise and the corresponding fit (see Methods) as a function of DC current through the SQUID, with a static magnetic flux of , which is used throughout the paper. Here is the magnetic flux quantum. This fit accurately determines the system noise temperature and gain.

To generate DCE photons, we send a sinusoidal signal to the flux line at GHz, while recording the signal using two digitisers at GHz and GHz placed symmetrically around (Fig. 1c and Methods). The effective speed of ’mirror’ displacement is given by the phase response (inset Fig. 2c), the flux amplitude and . For small amplitudes, the phase depends linearly on the flux, such that the boundary condition can be mapped to a sinusoidally moving mirror. With a flux pump amplitude exceeding 15 m, the change in phase becomes larger, which results in a larger photon spectral density; however, the motion also becomes non-linear. A power calibration of the flux pump amplitude is found in the supplementary Fig. 3.

We determine the DCE radiation power by comparing the output power with the flux pump on and off. When the pump is switched off, and the system noise is subtracted, we obtain a photon spectral density of corresponding to the vacuum fluctuations. The background noise in the system is determined by subtracting the amplifier noise as-well as the zero-point fluctuations from the total input noise. Any remaining noise signal would be due to thermal photons. This is smaller than what we could resolve, confirming a photon temperature of less than 40 mK (see Methods). As we increase the flux pump amplitude, we measure an increase in photon spectral density. Figure 2c, shows the generated photon spectral density versus flux pump amplitude.

We use two methods to probe and characterise entanglement between produced photon pairs: first, by calculating the log-negativity, and second, by comparing the quadrature noise to the vacuum. Both methods are commonly used to probe entanglement and non-separabilitySimon2000 ; Adesso2007 .

As we generate photons using the DCE, shown in Fig. 2c, we record the voltage quadratures , , and corresponding to the frequencies and . From the quadrature correlations, we can construct the covariance matrix (inset in Fig. 3a). Error values are estimated for all elements of the covariance matrix as one standard deviation. Once the covariance matrix is established, we calculate the logarithmic negativity Johansson2013 :

(1) | |||||

where is the 4x4 covariance matrix with the 2x2 sub-matrices , and . The logarithmic negativity is positive for a photon spectral density of or lower, as can be seen in Fig. 3a.

The logarithmic negativity is lower than the theoretical valueJohansson2013 (). The explanation for this is photon losses in the system before amplification and a small non-linearity in the SQUID, resulting in lower cross-correlation values. We estimate photon losses to be dB (see Methods), resulting in at the sample, which is approximately a factor two lower than the theoretically expected value. This remaining factor of two can be explained by the presence of a non-linearity in the response of the SQUID inductance to a magnetic flux (Fig. 2c). This non-linearity results in an effective pumping at higher harmonics such as GHz, producing unentangled photons at and . For larger photon spectral densities and flux pumping, m, decreases due to this non-linearity.

The right inset in Fig. 3b shows four histograms of measured and quadratures, taken at a flux pump amplitude m. The histograms show the difference between flux pump on and off. The top left histogram and bottom right histogram show squeezing along the dashed diagonals that are orthogonal to each other: photons are amplified along the diagonal dashed line and are squeezed orthogonally to it.

From the quadrature correlations, we calculate the combined quadrature fluctuations and as a function of flux pump amplitude (Fig. 3b), where the later fulfills the inseparability criterion for continuous variable systems by DuanDuan2000Mar ; Treps2005 for values below 1.

We observed a -0.09 0.02 dB squeezing below the vacuum in and . We also observed an amplification of 0.25 0.02 dB in and at a flux pump strength of m. For low flux pump powers in the more linear regime, both methods indicate entanglement.

To benchmark the entanglement generation, we calculate the entangled bits with the entropy of formationGiedke2003Sep for a given logarithmic negativity of 0.03, which is at the amplifier input (see supplementary). This corresponds to an entanglement rate of Mebit/s, in turn, corresponding to a distribution rate of entangled Bell pairsFlurin2012 . These numbers are substantially larger at the sample. There are two reasons for this: losses between the sample and the amplifier and the limited bandwidth of the amplifier. Taking losses into account and including the full bandwidth between DC and the pump frequency, we estimate available at the sample, corresponding to an entanglement rate of Mebits/s.

The above shows that the photon pairs generated by the DCE with a single boundary condition, equivalent to that of a sinusoidally moving mirror are indeed entangled. The high rate of entanglement, together with the control over the photon generation process, makes this device a versatile, bright and broadband source of entanglement opening the door towards possibilities such as temporally shaped entanglement sources.

Methods

Measurement environment.
Measurements are done in a dilution cryostat at a temperature of 10 mK.
The sample under test consists of a direct-current superconducting quantum interference device (DC-SQUID) at the end of an open transmission line (Fig. 1).
The aluminium SQUID with a loop area of 6x8 is made using shadow evaporation. The SQUID is directly connected to an on-chip transmission line which is 0.6 mm long.
An 80 nm thick niobium layer defines the ground-plane, flux pump and transmission line.
A high electron mobility transistor (HEMT) amplifier with a noise temperature of 2 K amplifies the signals in the range of 4-8 GHz.
After additional amplification and filtering, two digitisers detect the heterodyne down-converted signals at two frequencies, yielding the quadrature voltages and at frequencies GHz and GHz.
A probe signal can be launched via the circulator and is used to characterise the change in phase and magnitude of the reflected signal.
Furthermore, four low-pass filtered wires are connected across the SQUID to enable the DC characterisation.
The low-pass filtering in these lines consists of high resistance-capacitance and copper-powder filters with a total cut off frequency of 30 Hz.
An external magnetic coil is used to set the static flux () of the SQUID.
We modulate the boundary condition by sending an AC signal () to an on-chip flux pump line (Fig. 1b).
The flux pump frequency can be chosen arbitrarily, here we have presented data for GHz.

Photon number calibration. We calibrate the signal level generated by the SQUID by using the SQUID itself as a shot-noise source (Fig. 2b). By applying a current through the SQUID while a voltage is measured, shot noise is generatedKoch1980Dec that can be used to calibrate the systemSpietz2003 ; Spietz2006 . The resistance of the device for a voltage well above the gap voltage is (Fig. 2a). The difference compared to the impedance of the transmission line, , is taken into account using the following equations:

(2) |

where is Boltzmann’s constant, is the sample temperature, is the system noise temperature referred to the sample, and are the gain and the detection bandwidth respectively.
, and are spectral densities, which relate to the shot noise, Johnson noise and zero point fluctuations at frequency respectively.
is the shot noise power spectral density.
The system noise corresponds to a temperature of K at 4.1 GHz and K at 4.8 GHz, which matches the expectation for the HEMT amplifier with a noise temperature of approximately K connected via two circulators and filters (see supplementary Fig. 1 for additional information).

Thermal Photons.
The uncertainty in the background noise ( (s Hz)) is not small enough to resolve temperature below 40 mK which corresponds to

at 4.8 GHz.

Photon losses. We can obtain a rough of estimate photon losses from the fitted system noise temperature in comparison to the HEMT amplifier noise. For this, we assume then the first HEMT amplifier is the largest noise source in the chain and thus photon losses before the amplifier dominates changes in the fitted system noise. At 4.1 GHz we determine the gain factor before amplification (which is smaller than one indicating losses) by and at 4.8 GHz.

## References

- (1) Gerald T. Moore. Quantum theory of the electromagnetic field in a variable‐length one‐dimensional cavity. J. Math. Phys., 11(9):2679–2691, 1970.
- (2) Artur K. Ekert. Quantum cryptography based on bell’s theorem. Phys. Rev. Lett., 67(6):661, 1991.
- (3) H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller. Quantum repeaters: The role of imperfect local operations in quantum communication. Phys. Rev. Lett., 81(26):5932, 1998.
- (4) P. Ben Dixon, Gregory A. Howland, Kam Wai Clifford Chan, Colin O’sullivan-Hale, Brandon Rodenburg, Nicholas D. Hardy, Jeffrey H. Shapiro, D. S. Simon, A. V. Sergienko, R. W. Boyd, and John C. Howell. Quantum ghost imaging through turbulence. Phys. Rev. A, 83(5):051803, 2011.
- (5) S. Felicetti, M. Sanz, L. Lamata, G. Romero, G. Johansson, P. Delsing, and E. Solano. Dynamical Casimir Effect Entangles Artificial Atoms. Phys. Rev. Lett., 113(9):093602, 2014.
- (6) Ulrik L. Andersen, Jonas S. Neergaard-Nielsen, Peter van Loock, and Akira Furusawa. Hybrid discrete- and continuous-variable quantum information. Nat. Phys., 11(9):713–719, 2015.
- (7) C. Eichler, D. Bozyigit, C. Lang, M. Baur, L. Steffen, J. M. Fink, S. Filipp, and A. Wallraff. Observation of two-mode squeezing in the microwave frequency domain. Phys. Rev. Lett., 107(11):113601, 2011.
- (8) L. Zhong, E. P. Menzel, R. Di Candia, P. Eder, M. Ihmig, A. Baust, M. Haeberlein, E. Hoffmann, K. Inomata, T. Yamamoto, Y. Nakamura, E. Solano, F. Deppe, A. Marx, and R. Gross. Squeezing with a flux-driven josephson parametric amplifier. New J. Phys., 15(12):125013, 2013.
- (9) C. W. Sandbo Chang, M. Simoen, José Aumentado, Carlos Sabín, P. Forn-Díaz, A. M. Vadiraj, Fernando Quijandría, G. Johansson, I. Fuentes, and C. M. Wilson. Generating multimode entangled microwaves with a superconducting parametric cavity. arXiv, 2017.
- (10) Pasi Lähteenmäki, G S Paraoanu, Juha Hassel, and Pertti J Hakonen. Dynamical Casimir effect in a Josephson metamaterial. Proc. Natl. Acad. Sci. U.S.A., 110(11):4234–4238, 2013.
- (11) M. Westig, B. Kubala, O. Parlavecchio, Y. Mukharsky, C. Altimiras, P. Joyez, D. Vion, P. Roche, D. Esteve, M. Hofheinz, M. Trif, P. Simon, J. Ankerhold, and F. Portier. Emission of nonclassical radiation by inelastic cooper pair tunneling. Phys. Rev. Lett., 119(13):137001, 2017.
- (12) Jean-Charles Forgues, Christian Lupien, and Bertrand Reulet. Experimental violation of bell-like inequalities by electronic shot noise. Phys. Rev. Lett., 114(13):130403, 2015.
- (13) Andreson L. C. Rego, Hector O. Silva, Danilo T. Alves, and C. Farina. New signatures of the dynamical Casimir effect in a superconducting circuit. Phys. Rev. D, 90(2):025003, 2014.
- (14) Hector O. Silva and C. Farina. Simple model for the dynamical casimir effect for a static mirror with time-dependent properties. Phys. Rev. D, 84(4):045003, 2011.
- (15) J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi. Quantum State Transfer and Entanglement Distribution among Distant Nodes in a Quantum Network. Phys. Rev. Lett., 78(16):3221–3224, 1997.
- (16) K. Jahne, B. Yurke, and U. Gavish. High-fidelity transfer of an arbitrary quantum state between harmonic oscillators. Phys. Rev. A, 75(1):010301, 2007.
- (17) Alexander N. Korotkov. Flying microwave qubits with nearly perfect transfer efficiency. Phys. Rev. B, 84(1):014510, 2011.
- (18) M. Pechal, L. Huthmacher, C. Eichler, S. Zeytinoğlu, A. A. Abdumalikov, S. Berger, A. Wallraff, and S. Filipp. Microwave-Controlled Generation of Shaped Single Photons in Circuit Quantum Electrodynamics. Phys. Rev. X, 4(4):041010, 2014.
- (19) C. M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J. R. Johansson, T. Duty, F. Nori, and P. Delsing. Observation of the dynamical Casimir effect in a superconducting circuit. Nature, 479(7373):376–379, 2011.
- (20) J. R. Johansson, G. Johansson, C. M. Wilson, P. Delsing, and Franco Nori. Nonclassical microwave radiation from the dynamical casimir effect. Phys. Rev. A, 87(4):043804, 2013.
- (21) Emmanuel Flurin, Nicolas Roch, Francois Mallet, Michel H. Devoret, and Benjamin Huard. Generating entangled microwave radiation over two transmission lines. Phys. Rev. Lett., 109(18):183901, 2012.
- (22) Bryce S. DeWitt. Quantum field theory in curved spacetime. Phys. Rep., 19(6):295–357, 1975.
- (23) S. A. Fulling and P. C. W. Davies. Radiation from a Moving Mirror in Two Dimensional Space-Time: Conformal Anomaly. Proc. Royal Soc. A, 348(1654):393–414, 1976.
- (24) Lu-Ming Duan, G. Giedke, J. I. Cirac, and P. Zoller. Inseparability criterion for continuous variable systems. Phys. Rev. Lett., 84(12):2722, 2000.
- (25) R Simon. Peres-Horodecki Separability Criterion for Continuous Variable Systems. Phys. Rev. Lett., 84(12):2726–2729, 2000.
- (26) Gerardo Adesso and Fabrizio Illuminati. Entanglement in continuous-variable systems: recent advances and current perspectives. J. Phys. A, 40(28):7821–7880, 2007.
- (27) Nicolas Treps and Claude Fabre. Criteria of quantum correlation in the measurement of continuous variables in optics. Laser Phys., 15(1):1–8, 2005.
- (28) G. Giedke, M. M. Wolf, O. Krüger, R. F. Werner, and J. I. Cirac. Entanglement of formation for symmetric gaussian states. Phys. Rev. Lett., 91(10):107901, 2003.
- (29) Roger H. Koch, D. J. Van Harlingen, and John Clarke. Quantum-noise theory for the resistively shunted josephson junction. Phys. Rev. Lett., 45(26):2132, 1980.
- (30) Lafe Spietz, K. W. Lehnert, I. Siddiqi, and R. J. Schoelkopf. Primary electronic thermometry using the shot noise of a tunnel junction. Science, 300(5627):1929–1932, 2003.
- (31) Lafe Spietz, R. J. Schoelkopf, and Patrick Pari. Shot noise thermometry down to 10 mK. Appl. Phys. Lett., 89(18):2004–2007, 2006.

Acknowledgements

We thank C.M. Wilson, W. Wieczorek, V. Shumeiko and N. Treps for useful discussions on methods and entanglement.
We gratefully acknowledge financial support from the European Research Council, the European project PROMISCE,
the Swedish Research Council, and the Wallenberg Foundation. J.B. acknowledges partial support by the EU under REA Grant Agreement No. CIG-618353.

Author contributions

B.H.S. carried out the experiments, and the data analysis, A.B. and I.M. S. fabricated the sample, T.A. contributed to the experimental setup. G.J. contributed to the theoretical background. B.H.S, J. B. and P.D. wrote the manuscript. All authors commented on the manuscript. P.D. supervised the project.

Competing financial interests

The authors declare no competing financial interests.

## Supplementary

S1: Calibration

To prove entanglement and ensure data quality we need to take all sources of error into account.
Error estimation is necessary to clarify by which standard we have reached entanglement.
In our case, we detected a voltage signal at room temperature.
This signal is then averaged and processed to obtain a part of the signal which originated from the sample, we then correlate and process the corresponding signal which originated from the sample.
For this, we calibrated the circuit such that we can relate a signal strength generated from the sample to the detected voltage.
During measurements, we compare this signal to two different states: when the flux pump if switched on and on.
From this, we obtain a differential signal between the two states carrying the information about photons generated by the presence
of a flux pump.

We would like to know, what voltage our digitisers detects for a known amount of generated microwave power coming from our ’moving mirror’ (the SQUID at the end of the transmission line). The corresponding voltage signal we are interested in originated from the superconducting quantum interference device (SQUID) which sits at the end of a transmission line. This signal passes through a number of passive microwave components before it reaches a linear voltage ’HEMT’ amplifier. After the amplifier, it passes on to the digitiser where it gets detected (see figure S1).

To calibrate our signal, we apply a current through the SQUID which is larger than the critical current. The current passes through the two tunnel junctions of the SQUID and generates shot noise which then propagates into the lines. The generated noise power depends on several factors, in particular, it scales linearly with the applied current through the SQUID:

(SE1) | |||||

(SE2) | |||||

(SE3) | |||||

(SE4) | |||||

(SE5) | |||||

(SE6) |

where is the power spectral density, is the gain, is the bandwidth, is the frequency, is Boltzmann’s constant, , and are voltage noises produced by zero point fluctuations, shot noise and Johnson noise, respectively. Johnson noise is dependent on the temperature of the sample. and are energy ratio terms present in . We fit the detected power by using equation SE6. The result of this fit gives us the gain and system noise, which includes amplifier noise and other photon losses. Using these two numbers, we can subtract the system noise from the detected signal and divide the remaining power signal by () to obtain the corresponding photon spectral density n, which has the units of (s Hz).

Regarding the shot noise fit in figure S2, the critical current is suppressed with magnetic flux of . Additional features visible below the superconducting gap voltage of the SQUID (for less than 6A) are excluded from the fit. Here we instead determine the gain and system noise temperature from the slope and offset of the data.

However, using a finer and slower sweep, features below the critical current, when is not suppressed are fitted using the differential resistance of the device (not shown).
This fit provides a verification that the SQUID is in the ground state, meaning that no additional, i.e., thermal photons are present when applying a finite sub-critical current to the device.
Then similar to this plot below the critical current, after subtracting the system noise the photon rate goes down to , meaning that only photons due to vacuum fluctuations remain.
We find an accurate fit of the gain and system noise by using a pre-fitted, averaged and static resistance () with few shot noise data points.

S2: Estimation of errors

By fitting the shot noise at each desired detector frequency, we get the individual amplification gains and system noise temperatures.
Since neither rewiring nor switching to different samples is required, this calibration can be done interleaved with each measurement.
The ability to track drifts and capture changes to the system as they happen is used to achieve an upper bound on the noise present in the measurements.

The fitted results for two frequencies used = 4.8 GHz and = 4.1 GHz to the corresponding flux pump frequency GHz (which is switched off during calibration), before and after the measurement are:

(SE7) | |||||

(SE8) | |||||

(SE9) | |||||

(SE10) |

where and correspond to the gain at frequency at the start and at the end of the measurement and and to the gains at frequency . This corresponds for a gain drift of 0.58 dB and 0.59 dB for and respectively over a time period of 8 hours.

The corresponding average values are:

(SE11) | |||||

(SE12) | |||||

(SE13) | |||||

(SE14) |

We obtain the number of photons by dividing the power detected by the digitiser by a factor. That depends on the respective frequency and gain, i.e. and . The photon numbers at the corresponding frequencies are:

(SE15) |

(SE16) |

where and are the detected powers at room temperature at frequency 1 with the flux pump on or off, respectively. Similarly, and are the powers at frequency 2. The signal of interest is the power difference between flux pump on and off ().

To account for the error in this signal we can consider three aspects: First, the uncertainty in the gain from the fit. Second, the amount by which the gain drifted between the two measurements. Third, the overall noise present in with the same amount of averaging. The error, due to uncertainty in the gain is:

(SE17) |

where is the resulting uncertainty in photon numbers as a function of total photon numbers and the fraction of gain uncertainty . We find the amount of gain drift taking the difference between the start and end gains:

(SE18) | |||

(SE19) |

where and are our uncertainties in the gain due to drift. Together with the fit uncertainty we get:

(SE20) | |||

(SE21) |

The resulting uncertainty in the gain is given by:

(SE22) | |||

(SE23) |

which gives an overall gain accuracy within 1%.

However, noise between two on-off cycles in succession might be more dominant than the uncertainty in the gain. To investigate this, we calculate the variance for ( and ) under same conditions as the measurements.

(SE24) | |||

(SE25) |

which gives us an additional uncertainty of 0.0025 per photon in the OFF signal.
Assuming we have the same uncertainty when the flux pump is on and adding this, we get an uncertainty of for n1 and photons in n2.
Typically the number of photons in the differential signal in the region of interest is around 0.05 photons. This results on average in an statistical error of 6% in the photon number resolution.

S3: Flux pump power calibration

The flux pump amplitude is estimated by fitting the onset for which the photon spectral density visibly increases as a function of flux pump amplitude and dc flux offset (Fig. S3. This happens because the relative change Josephson inductance is largest close where the critical current gets suppressed.
Close to this point is where the mirror moves fastest and in a non-linear way.
A modulated AC flux pump eventually reaches this point with increased strength, i.e. and this is where the photon generation increases drastically.
Which are down-converted very efficiently from the pump and from higher harmonics of the pump signal.
This sudden increase in photon numbers is used as a reliable method to calibrate the flux pump strength.

S4: Entangled bits

We calculate the effective number of ebits (entropy of formationFlurin2012_2 ; Giedke2003Sep ) at the detectors input,
which corresponds to a shared number of EPR singlets needed to reconstruct a covariance matrixBennett1996 by using the following equationFlurin2012_2 :

(SE26) |

where and . Now we need to take the bandwidth and logarithmic negativity into account to obtain the potentially available ebits of the device.

The photon spectral density is expected to follow a parabolic functionJohansson2010 as a function of frequency:

(SE27) |

where is the frequency, is the peak photon rate and (/2) is the pump frequency. In the measurement we observed a peak logarithmic negativity of which yields an . The effective measurement bandwidth available to us, was limited by surrounding components such as circulators and the Hemt amplifier to 4-8 GHz. Taking a parabolic spectrum into account Mebit/s were available to us in this setup.

For low photon numbers the theoretical log-negativity is resulting in:

(SE28) |

which is valid for frequencies between 0 and the pump frequency (Fig. S4). The integral of this together with the eq. (SE26) from 0 to the pump frequency yields Mebit/s. In practice a smaller number will be measured due to losses, non-linearity and limited measurement sensitivity. In our case, we estimate the losses between HEMT amplifier and sample to be -2.1 dB. Taking this into account, we obtain a log-negativity at the sample , which corresponds to Mebit/s.

## References

- (1) Emmanuel Flurin, Nicolas Roch, Francois Mallet, Michel H. Devoret, and Benjamin Huard. Generating entangled microwave radiation over two transmission lines. Physical Review Letters, 109(18):183901, oct 2012.
- (2) Charles H. Bennett, David P. DiVincenzo, John a. Smolin, and William K. Wootters. Mixed-state entanglement and quantum error correction. Physical Review A, 54(5):3824–3851, nov 1996.
- (3) J. R. Johansson, Göran Johansson, C. M. Wilson, and Franco Nori. Dynamical Casimir effect in superconducting microwave circuits. Physical Review A, 82(5):052509, nov 2010.