Stationary Entangled Radiation from Micromechanical Motion
Mechanical quantum systems facilitate the development of a new generation of hybrid quantum technology comprising electrical, optical, atomic and acoustic degrees of freedom. Entanglement is the essential resource that defines this new paradigm of quantum enabled devices. Here we confirm the long-standing prediction that a parametrically driven mechanical oscillator can entangle electromagnetic fields. We observe stationary emission of path-entangled microwave radiation from a micro-machined silicon nanostring oscillator, squeezing the joint field operators of two thermal modes by 3.40(37) dB below the vacuum level. Interestingly, even at room temperature, where entangled microwave states are overwhelmed by thermal and amplifier noises, we measure finite quantum correlations; a first indication that superconducting circuits could be used for quantum enhanced detection at ambient conditions.
Entanglement is the key ingredient of the emerging new quantum information technology. As an important physical resource, it has a wide range of applications in quantum information processing, communication, and sensing. Continuous variable (CV) entangled electromagnetic fields, known as Einstein-Podolsky-Rosen (EPR) states Lvovsky2009 () are spatially separated two-mode squeezed states that can be used to implement CV quantum teleportation and quantum communication between localized qubit systems Andersen2016 (). In the optical domain EPR states are typically generated using nondegenerate optical amplifiers Heidmann1987 (); Ou1992 (). At microwave frequencies Josephson parametric amplifiers can serve as a nonlinear medium to produce two-mode squeezed states Eichler2011a (); Flurin2012 (); Menzel2012 (); Ku2015 ().
Entanglement and squeezing can also result from radiation pressure and back-action between electromagnetic radiation and a mechanical resonator Fabre1994 (); Mancini1994 (). Mechanical systems could be used to entangle different types of subsystems on vastly different energy scales such as microwave and optical fields Higginbotham2018 (). Recent experiments have demonstrated single mode squeezed states of mechanical motion Wollman2015 (); Lecocq2015a (); Pirkkalainen2015a () and radiation fields at both optical Safavi-Naeini2013b (); Purdy2013b (); Sudhir2017 () and microwave Korppi2017a () frequencies. Very recently, entanglement between photons and a mechanical oscillator Palomaki2013b () and between two mechanical oscillators Riedinger2018 (); OckeloenKorppi2018 () have been realized. Our results confirm the prediction that mechanical oscillators can produce entangled radiation Paternostro2007 (); Genes2008b (); Barzanjeh2011 (); Wang2013 (); Tian2013 (). Using a highly versatile silicon-on-insulator electromechanical platform Barzanjeh2017 () we demonstrate the generation of CV entangled states between the propagating output fields of two microwave resonators separated by one meter in a millikelvin environment.
We consider a three-mode electromechanical system in which two microwave resonators with resonance frequencies and total damping rates with are capacitively coupled to a single in-plane vibrational mode of a dielectric nanostring resonator with resonance frequency and intrinsic damping rate as schematically shown in Fig. 1a. The electromagnetic field of the microwave resonators exerts radiation pressure on the mechanical resonator. In return, the vibration of the mechanical resonator mediates a retarded interaction between the microwave modes. In the presence of two strong microwave pumps with frequencies as indicated in Fig. 1b we can linearize the system and describe the physics in a reference frame rotating at the frequencies and with the Hamiltonian
where , and are the annihilation operators for the cavity and the mechanical oscillator, and are the effective and vacuum electromechanical coupling rates between the mechanical mode and cavity , respectively, while is the number of photons in the resonator due to the drive with detuning . Here we assume the regime of fast mechanical oscillations, which allows us to neglect the fast oscillating terms at . The first term in Eq. 1 describes a parametric down-conversion interaction that is responsible for entangling the microwave resonator 1 with the mechanical oscillator. The second term describes a beam-splitter interaction between the mechanical resonator and the microwave resonator 2, exchanging the state of the electromagnetic and mechanical modes. If the electromechanical coupling exceeds the decoherence rate of the mechanical resonator , with the mechanical bath occupancy , is Boltzmann’s constant and is the device temperature, the output of both microwave resonators is mapped into a two-mode squeezed thermal state Genes2008b ().
We experimentally realize the described entanglement generation scheme in a superconducting electromechanical system similar to Ref. Barzanjeh2017 (). The circuit, shown in Fig. 2a consists of a metalized silicon nanobeam resonator at MHz with an intrinsic damping rate Hz and a bath occupation , that is capacitively coupled to two high impedance superconducting coil resonators at GHz with energy decay rates MHz and waveguide coupling ratios . The vacuum electromechanical coupling strengths for this three-mode electromechanical system are Hz. The microwave resonator 1 (2) is driven from blue (red) sideband with the coherent drive power dBm at the device input, corresponding to the single cavity cooperativites of .
The output of each resonator passes through two different measurement lines as shown in Fig. 2b. After amplification the signals are filtered and down-converted to an intermediate frequency of MHz and digitized with a sampling rate of 10 MHz using an 8 bit analog-to-digital converter (ADC). The FFT based digital down conversion process extracts the quadrature voltages and for each channel. The measured quadrature voltages are converted to the unitless quadrature variables and with the scaling factors , where is the total system gain of the output channel , Hz is the digitally chosen measurement bandwidth, and the input impedance of the ADCs.
We calibrate the system gain and system noise of both measurement channels by injecting a known amount of thermal noise using two temperature controlled loads Flurin2012 (); Palomaki2013b (). The calibrators are attached to the measurement setup with two 5 cm long superconducting coaxial cables and two latching microwave switches. By measuring the noise density in V/Hz at each temperature as shown in Fig. 3a, and fitting the obtained data with the expected scaling
we accurately back out the gain dB and the number of added noise photons for each output. We use these values for all following measurements, which locates our effective points of signal detection 0.5 m from the resonator outputs and approximately 1 m apart from each other as shown in Fig. 2b. Beyond this point the generated microwaves are exposed to a higher temperature thermal bath, additional losses and to amplifier noise.
The generation of a two-mode squeezed thermal state can be verified intuitively in phase space by plotting the histograms representing the probability distribution of all possible combinations of the measured quadratures . We first measure the uncorrelated noise for each channel by performing a measurement with the microwave drives turned off. The result is shown in the insets of Fig. 3a indicating a thermal state with a variance corresponding to . In Fig. 3b we plot the 4 relevant quadrature histograms obtained when the drive tones are turned on and after subtraction of the previously measured histogram with the drives turned off. The single-mode distributions and are both slightly broadened, indicating a phase-independent increase of the voltage fluctuations, which shows that the output of each resonator is amplified. In contrast, in the histograms between different outputs and the fluctuations increase along one diagonal axes and decrease in the other, indicating a strong correlation between the two spatially separated modes.
The non-classicality of such Gaussian states can be fully characterized by the covariance matrix , a symmetric matrix with 10 independent elements Olivares2012 (). The diagonal elements are calculated from the variances of the scaled quadratures when the pumps are on and off, i.e. where , is the input quantum noise at temperature and the brackets show an average over all measurements. The off-diagonal elements of the covariance matrix are specified by the covariances of the two modes, , which are zero when the pumps are turned off. Figure 4a shows all elements of the measured covariance matrix for the two propagating output modes. The cross-mode squeezing correlation expressed by the off diagonal elements and are comparable in magnitude with the individual amplified variances given by the diagonal elements and while all other matrix elements are close to zero.
The amount of two-mode squeezing is best visualized using the quasiprobability Wigner function
with the state vector . Figure 4b shows the two relevant Wigner function projections of the measured covariance matrix in blue and the ideal vacuum state in red. The and projections clearly show cross-quadrature two mode squeezing below the quantum limit in the diagonal directions.
To quantitatively access the amount of squeezing we define the EPR operator pair and where represents a rotation of the detector phase in channel 1. For each rotation angle we evaluate the squeezing parameters and , as shown in Fig. 4c. For one common optimal rotation angle we find that both operators are squeezed by 3.43(38) dB and 3.36(37) dB below the vacuum level. The phase dependence shows squeezing and anti-squeezing as expected Olivares2012 () for a two-mode squeezer applied to a thermal Gaussian state (solid lines)
with the effective photon inputs and fully constrained by the measured output photon numbers and and the fitted squeezing parameter in agreement with the amount of correlations .
To verify the existence of entanglement between the two output modes we use the Duan criterion Duan2000 (). The two-mode Gaussian state is entangled if the parameter and it is in a vacuum state for . In Fig. 4d we show measurements of for the optimal angle as a function of the red detuned drive power and the calculated difference between the red and blue cooperativities . For small the system is predicted to be unstable and the measured values exceed the plot range. Far from this instability region (shaded in blue) at a cooperativity difference of 46, the inseparability condition is fulfilled and we find a minimum of , clearly proving that the radiation emitted from the electromechanical device is entangled before leaving the millikelvin environment. Increasing the red detuned pump the parameter eventually goes above the vacuum limit and the state becomes separable. We assign this effect to pump power dependent excitation of two-level systems which populate the microwave cavity with uncorrelated noise photons, that can result in a degradation of quantum correlations. The reported errors are the statistical error of the measured means which exceed the statistical errors of the noise measurement (pump turned off) and the error of the calibration measurements. To quantitatively understand the power dependence of the EPR parameter shown in Fig. 4d we fit the data with a full theoretical model that also takes into account the measurement bandwidth Barzanjeh2011 (). We find excellent agreement with theory over the full range using the reported and independently verified device parameters.
We quantify the degree of mechanically generated entanglement with the logarithmic negativity representing an upper bound of the distillable entanglement Olivares2012 (). For a Gaussian density operator, where with . The maximum measured entanglement in our system is and the useable distribution rate of entangled EPR pairs of 129 ebits/s (entangled bits per second) can be calculated using the entropy of formation and the bandwidth of the emitted radiation Hz.
The presented mechanical entangler offers a path forward to make use of quantum microwave systems at room temperature. On the one hand, an analogous microwave - optical implementation could be used to optically distribute entangled states between cold superconducting nodes. On the other hand, we find that our electro-mechanical system produces quantum correlations detectable even at room temperature without any noise calibration. We measure the maximum amount of quantum correlations, quantified by the quantum discord of , close to the instability region. Such generalized quantum correlations of separable states Ollivier2001 () have so far only been reported in optical detection systems and could now also be used for quantum enhanced sensing, imaging and radar at microwave frequencies Barzanjeh2015 ().
Acknowledgements This work was supported by IST Austria, the IST Austria Nanofabrication Facility, the European Union’s Horizon 2020 research and innovation program under grant agreement No 732894 (FET Proactive HOT) and the European Research Council under the grant agreement No 758053 (ERC StG QUNNECT). SB acknowledges support from the Marie Sklodowska Curie fellowship No 707438 (MSC-IF SUPEREOM). We thank Nikolaj Kuntner and Jason Jung for contributions to the digitizer software, Mike Hennessey-Wesen for developing the thermal calibration source, Georg Arnold for nanobeam simulations, and Kirill Fedorov, Mahmoud Kalaee, Oskar Painter and David Vitali for fruitful discussions.
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