Optomechanical entanglement at room temperature: a simulation study with realistic conditions
Kahlil Y. Dixon, Lior Cohen, Narayan Bhusal, Christopher Wipf, Jonathan P. Dowling, Thomas Corbitt
OOptomechanical entanglement at room temperature: a simulation study with realisticconditions
Kahlil Dixon , Lior Cohen , Narayan Bhusal , Christopher Wipf , Jonathan P. Dowling , , , , and Thomas Corbitt Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA LIGO Laboratory, California Institute of Technology, Pasadena, California 91125, USA NYU-ECNU Institute of Physics at NYU Shanghai,3663 Zhongshan Road North, Shanghai, 200062, China. CAS-Alibaba Quantum Computing Laboratory, CAS Center for Excellence in Quantum Information and Quantum Physics,University of Science and Technology of China, Shanghai 201315, China. and National Institute of Information and Communications Technology,4-2-1, Nukui-Kitamachi, Koganei, Tokyo 184-8795, Japan (Dated: July 24, 2020)Quantum entanglement is the key to many applications like quantum key distribution, quan-tum teleportation, and quantum sensing. However, reliably generating quantum entanglement inmacroscopic systems has proved to be a challenge. Here, we present a detailed analysis of pon-deromotive entanglement generation which utilizes optomechanical interactions to create quantumcorrelations. We numerically calculate an entanglement measure – the logarithmic negativity – forthe quantitative assessment of the entanglement. Experimental limitations, including thermal noiseand optical loss, from measurements of an existing experiment were included in the calculation,which is intractable to solve analytically. This work will play an important role in the developmentof ponderomotive entanglement devices.
I. INTRODUCTION
Entanglement is the most common and important re-source for various quantum technologies, from quantummetrology [1, 2], to quantum communication [3] andquantum computing [4, 5]. It is well known that quan-tum light sources, in particular entangled sources, requirenon-linear interaction. To date, most of these sources arebased on all-optical nonlinear processes in crystals [6],which are good enough for most applications but insuf-ficient for applications with very short wavelengths [7].Recently, efforts have been devoted to explore differentavenues to generate quantum entanglement [8–11]. Oneapproach is to use strong light-matter interaction withsingle atoms [8] or single quantum dots [9]. While thismethod is very efficient, its production is limited to a sin-gle entangled photon pair at a time. A reliable source ofentanglement for short wavelength which provides multi-photon entanglement is still in need.Radiation pressure — the force electromagnetic radia-tion exerts on a material surface — is a significant sourceof noise in optical metrology [12]. The light’s momen-tum causes fluctuations in the mirror’s position, yieldingphase noise in the electromagnetic wave. However, thisinteraction creates quantum correlations that can be ex-ploited to produce non-classical light. It has been shownthat when an electromagnetic wave is incident on a mir-ror, it generates a squeezed light, i.e., the electromag-netic wave experiences an optical nonlinearity [13]. Thisnonlinearity can also generate entanglement between thelight and the mirror [14]. Moreover, if two light sourcessimultaneously interact with the mirror both of them en-tangle with the mirror, and thus, may entangle with eachother [15–19]. This form of bipartite optical entangle-ment generation has been demonstrated experimentally using a vibrating silicon oxide membrane [20]. This workconsiders a cantilever µ mirror in-place of the silicon ox-ide membrane. This oscillating mirror has higher-ordermodes that should strongly affect entanglement.To observe the effect of the quantum back action be-tween the two light fields, we consider an homodynequadrature variance measurement of two output opticalfields from a single cavity double optical spring with a µ mirror. Here, we report the experimental feasibility ofobservable ponderomotive entanglement at room temper-ature and lower temperature. This work identifies experi-mental configurations that will yield observable entangle-ment using programs and measurements that have beenpreviously tested and reported [21–23]. We numericallyevaluate the amount of entanglement, and investigate thedependence of entanglement of various experimental pa-rameters such as temperature, side-band frequency, cav-ity length, and loss. II. METHODSA. Experimental considerations
The experimental consideration is shown in Fig. 1.We chose this configuration because it allows for a stableoptomechanical system with no external feedback, whichcould disrupt the entanglement. It utilizes a single op-tomechanical cavity, that acts as a double optical spring.This setup will convert the squeezing effects of the opticalspring cavity into entanglement. Measuring this form ofentanglement requires dual homodyne detection to prop-erly measure the squeezing correlations. The two lasersare frequency locked in order to maintain their relativedetunings with respect to the cavity resonance. The laser a r X i v : . [ qu a n t - ph ] J u l field are arranged with orthogonal relative polarizations. LASER L A S E R Half-wave PlatePolarizing BeamsplitterPhoton DetectorSuspended 1g OpticsSuspended 250g Optics
FIG. 1. Schematic diagram of the proposed experiment. Thescheme uses two frequency locked laser fields that are pre-pared in two orthogonal polarizations before entering the cav-ity. A balanced homodyne measurement is employed to con-struct the full quadrature covariance matrix. The schemeutilizes polarizing beamsplitters to isolate orthogonal polar-izations.
In order to accurately predict entanglement generationfrom the µ mirror cavity we take into account: tempera-ture, cavity loss, laser power, and optical spring detun-ings. Concurrently, we restrict the optical detuning tomaintain a stable optical spring; while including more re-alistic models for input noises. Other variables pertain-ing to the optomechanical cavity, such as the thermalnoise from the µ mirror motion (at room temperature),was taken from experimental data of the same setup [22]. B. Measuring entanglement
To determine whether or not the output fields are en-tangled we need to choose a convenient measure. Themain measure we will use here is the logarithmic neg-ativity entanglement measure [24, 25]. The variancematrix assembled from the quadrature operators will bethe main output to measure. The variance matrix canbe written as follows: V = (cid:104) X ∗ X (cid:105) + (cid:104) X ∗ Y (cid:105) + (cid:104) X ∗ X (cid:105) + (cid:104) X ∗ Y (cid:105) + (cid:104) Y ∗ X (cid:105) + (cid:104) Y ∗ Y (cid:105) + (cid:104) Y ∗ X (cid:105) + (cid:104) Y ∗ Y (cid:105) + (cid:104) X ∗ X (cid:105) + (cid:104) X ∗ Y (cid:105) + (cid:104) X ∗ X (cid:105) + (cid:104) X ∗ Y (cid:105) + (cid:104) Y ∗ X (cid:105) + (cid:104) Y ∗ Y (cid:105) + (cid:104) Y ∗ X (cid:105) + (cid:104) Y ∗ Y (cid:105) + (1)were (cid:104) u ∗ v (cid:105) + = (cid:104) u ∗ v + v ∗ u (cid:105) , or in block form: V = (cid:18) V V V V (cid:19) . (2)
1. Logarithmic negativity
The logarithmic negativity is useful for measuringcontinuous-variable (CV) entanglement and is monotone
Parameter Variable name Stable and E N (cid:54) = 0Temperature T P . P . L s ppm Carrier detuning d . d − . Q L n . for Gaussian beams (see appendix for an alternate entan-glement measure and results). The information-theoreticmeaning of logarithmic negativity in terms of exact en-tanglement cost of quantum Gaussian states was estab-lished in [26–28]. Conveniently, the logarithmic negativ-ity can be calculated from the variance matrix [29]: E N = max[ 0 , − ln (cid:113) η − (cid:112) η − V ] (3)where η = det V + det V − V . (4) C. Computational resources
While the quantum Langevin approach is more conve-nient for an analytical approach, to experimentally andcomputationally develop a simulation, sideband opera-tor propagation is preferred; due to its more intuitivetreatment of the optics and higher modularity [30, 31].The simulation assumes an input field and cavity con-figuration specified by some parameter configuration ξ and outputs the homodyne measurement of the quadra-tures. It solves for the output quadratures via successivetransformation of the input sideband quadratures. Tocalculate the effect of the micromirror on the input side-bands, measurement data from previous work with theoptomechanical cavity is used to simulate the cantilever’seffects. These data allow our simulation to consider thecantilever’s higher harmonic modes’ effect on the entan-glement.These programs are written to calculate the entangle-ment measures over different parameter spaces. Thereare nine adjustable parameters; highlighted in the tableI. Past experiments identified configurations that wouldyield observable single mode squeezing for a single inci-dent beam. These results narrowed our search for opti-mal parameters. III. RESULTS
The last column in table I represents a parameter setthat generates the highest logarithmic negativity and sta-ble optical spring. After the simulations are preformedthe output is used to calculate the variance matrices andentanglement measures. All subsequent figures will usethe parameters in the table above unless otherwise speci-fied. For example at room temperature and frequency ofabout 20 kHz, the program predicts an output variancematrix, V where E N ( V ) = 0 .
104 and where: V = . − . − . − . − .
38 156 . .
76 45 . − .
06 63 .
76 26 .
61 18 . − .
80 45 .
07 18 .
47 13 . (5)(note that unsqueezed shot noise would have a variancematrix of I where I is the identity matrix)Further analyzing the entanglement yields, we foundthat the E N was maximum at 20kHz for the above pa-rameters. This maximum appears to decrease slightly infrequency as temperature decreases as shown in figure 2.In the figure the sharp drops to zero E N are at the res-onance frequencies of higher order mechanical modes ofthe cantilever. FIG. 2. Logarithmic negativity measure ( E N ) of the two out-put optical fields as a function of temperature and frequency.Conveniently, these parameters yield entanglement at roomtemperature for a range of frequencies. The sharp drops in E N are due to the higher order optical spring resonances ofthe cantilever micromirror (yaw resonance at 4.3 kHz andtranslation and yaw 54 kHz are the most visible). Top andbottom most dotted lines indicate 295K and 4K respectively. Not only is the double optical spring cavity capableof entangling the two fields, it is able to do so at roomtemperature. Cooling the micromirror increases both thedegree of entanglement and the frequencies over which itis produced. However, there is no significant advantageto cooling the micromirror below 4K, for frequencies of1kHz and above. This is a result of the thermal noisebeing pushed well below the quantum back action level,as shown in figure 4. At about 4K, the logarithmic nega-tivity maximizes at approximately E N = 0 . FIG. 3. E N vs Loss and frequency at room temperature.Encircled black region denotes absolute zero logarithmic neg-ativity. For small changes in loss the maximum E N is rel-atively constant. The three higher order harmonics are allvisible here: yaw,ya-transverse, roll-transverse. The yellowline indicates current experimental losses. Lower losses also aid the entangler; figure 3 shows theentanglement increases as loss decreases. The behaviorof the classical to quantum noise ratio well follows thatof the entanglement at all temperatures.To maximize entanglement we consider changing theoptomechanical cavity length. Figure 5 shows the de-pendence between cavity length and E N .Together figures 4 and 5 pertain to the fundamen-tal concepts behind optomechanical entanglement gen-eration. The cavity length changes due to the inputlaser power. This has quantum fluctuations due to theHeisenberg uncertainty principle. This creates a funda-mental uncertainty in the overall cavity length, which inturn strongly effects the properties of the output light.This technique manipulates quantum radiation pressurenoise into an entanglement source. When this noise isgreater than the classical noise, in this case thermal noise,the entanglement should thrive; figure 4 confirms this.Furthermore, entanglement will be limited if the cavitylength fluctuations are too small relative to the overallcavity length. Moreover, the dampening effects becomemore dominant as the cavity length increases thus widen-ing the resonances that destroy entanglement. This isshown in figure 5.We would like to quantify how difficult it is to exper-imentally verify the existence of the simulated entangle-ment. We simulate a noisy variance matrix measure-ment by creating a set of variance matrices normally dis-tributed about the initial output variance matrix at each Side-band Frequency (Hz) -3 -2 -1 E N a nd Q / T N o i se T=77KT=77KE N Q/T Noise
FIG. 4. The ratio of quantum to thermal noises in the system.The Logarithmic negativity result at 77K has been includedto show agreement. While this confirms our hypothesis aboutthe quantum radiation pressure noise working in oppositionof the thermal noises to yield entanglement, in conjunctionwith our other results it also shows that such an entanglementgeneration technique heavily relies on the other experimentalparameters as well. frequency. (While we shall only show the experimentsnoise sensitivity as a function of Gaussian spread andfrequency, it is possible to vary any of the parameters inthe table for the noise analysis.) The resulting entangle-ment uncertainties are plotted in figure 6.With measurement certainty on the order of 1% theoutput noise in the measurement is several times that ofthe expected maximum entanglement. When measuringat or near the peak E N frequency, the double homodyneprecision must be on the order of 0.1%.
20 40 60 80 100
Cavity Length (mm)
Log . N e g a t i v i t y E N FIG. 5. Logarithmic negativity versus the cavity length fordifferent ambient temperatures at 20kHz. No benefit to en-tanglement generation will be seen when cooling below 4K(unless operating at the cantilever’s higher harmonic frequen-cies). Furthermore, cooling below 1K avoids losses to entan-glement due to harmonic effects the cavity length effects onthe entanglement (the drop at 10cm) are present at all tem-peratures. Entanglement does not improve at cavity lengthsshorter than 1mm. Side-band Frequency (Hz)
Log . N e g a t i v i t y E N FIG. 6. We can simulate the effects of Gaussian noise inthe double homodyne measurement by creating a normal dis-tribution of variance matrices and calculating the standarddeviation in the resulting negativity. Due to the numeric in-stabilities in the entanglement measure, entanglement veri-fication requires high precision measurement of the outputquadratures (about 0.1%) (67% confidence interval) ; the log-arithmic negativity is highly sensitive to Gaussian noise. Thegrey shaded region denotes the uncertainty in the output en-tanglement. All parameters for this calculation match thosein the table previous.
IV. CONCLUSION
All optical circuits and devices are subject to quan-tum radiation pressure effects. These effects correlateincident light; which implies potential for new entangle-ment devices. The effects are strong enough to be ma-nipulated into generating bipartite optical entanglement.Moreover, this entanglement persists at room tempera-ture with realistic losses, stable optical spring detunings,and accessible circulating powers. With experimentallystable parameters, we predict a maximum logarithmicnegativity of E N = 0 .
2; while considering parametersclose to reported experiments yields average logarithmicnegativity of E N = 0 . ACKNOWLEDGEMENT
K.D., L.C., N.B. and J.P.D. would like to acknowledgethe Air Force Office of Scientific Research, the Army Re-search Office, the Defense Advanced Research ProjectsAgency, and the National Science Foundation. This ma-terial is based upon work supported by the National Sci-ence Foundation under Grant No. PHY-1806634. Wewould like to thank X,Y,Z for important discussions.
V. APPENDIXA. Logarithmic negativity
Negativity is an ”easy-to-compute” measure of entan-glement defined as follows: N ( ρ ) = || ρ Γ A || −
12 (6),where ρ is the density matrix, A is the dimension of thesubsystem, and ρ Γ A is the partial transpose of ρ withrespect to subsystem A [32, 33]. Written with the samedependence the logarithmic negativity is the following: E N = log || ρ Γ A || . (7) B. Duan’s measure of inseparability
Since the logarithmic negativity is strongly dependenton our normalization we compute a second entanglement measure as a sanity check. We chose this measure be-cause it does not vary with choice of variance matrixnormalization. This entanglement monotone is an alter-native to the negativity based measure for CV entangledbeams [34]. The calculation/ determining of the ” a ” pa-rameter is dependent on calculations done on/with thevariance matrix ” V ”; however, the only variance matri-ces of certain forms can be used [34]. Fortunately, Duanproved that non standard form variance matrices can betransformed into their standard forms following a fewsteps and solving a few equations. The variance matrixof the standard form (the goal) shall be written as fol-lows: V (cid:48)(cid:48) = n c n c c m c m . (8)These matrix elements are computed from the elementsof what we shall call the ”substandard form of the vari-ance matrix”; this form shall be written as follows: V (cid:48) = n cn c (cid:48) c mc (cid:48) m . (9)The standard form is calculated from the substandardform by solving the following system of equations for theparameters r and r : √ r r | c | − | c (cid:48) |√ r r = √ α n α m − (cid:112) ( β n β m ) (10) β n α n = β m α m (11)where α n = nr − β n = nr − α m = mr −
1, and β m = mr −
1. Then, to apply r and r : n = nr , n = n/r , m = mr , m = mr , c = c √ r r , c = c (cid:48) √ r r .Finally, our original variance matrix ” V ” will be referencein block form as follows: V = (cid:18) V V V T V (cid:19) = (cid:18) A BB T C (cid:19) . (12). Calculating V (cid:48) from V was done using the followingequations.det A = n , det C = m , det B = cc (cid:48) , (13)det V = ( nm − c )( nm − c (cid:48) ) (14)After attaining the proper form, the following inequal-ities need to be broken for there to be entanglement inthe system: | c | ≤ (cid:112) ( n − m −
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