SUPPLEMENTARY INFORMATION

Transcription

1 satisfy the condition 31 ω LO,a ω a = ω b ω LO,b. (4) doi: /nature07751 Tunable delay of Einstein-Podolsky-Rosen entanglement A. M. Marino 1, R. C. Pooser 1, V. Boyer 1, & P. D. Lett 1 1 Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland, Gaithersburg, Maryland 0899, USA MUARC, School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 TT, United Kingdom 1 Envelopes of the correlation functions The cross-correlation between the quadratures of the fields is given by G XaXb (τ) = X a (t)x b (t + τ), (1) where here the brackets indicate time averaging. The measurement of the quadratures is done with homodyne detectors, which give a signal of the form X a (t) = E LO e i(ωlo,at+θ) Ea(t)e iωat + ELOe +i(ωlo,at+φ) E a (t)e iω at () X b (t) = E LO e i(ωlo,bt+θ) Eb(t)e iωbt + ELOe +i(ωlo,bt+φ) E b (t)e iωbt, (3) where E LO and ω LO are the amplitude and frequency of the local oscillators, respectively, and θ is the relative phase between the probe or conjugate and the corresponding local oscillator (phase of the homodyne detector). The center frequencies for modes a and b are given by ω a and ω b. In our experiment these frequencies correspond to the centre of the gain profiles for the probe and the conjugate, respectively. We find that the optimum squeezing is obtained at frequencies detuned from the centre of the gain profiles. The local oscillators, which are generated with a seeded fourwave mixing process 19, are tuned to the optimum squeezing and as a result their frequencies are different from the the center frequencies for modes a and b. The correlations present in the vacuum twin beams, modes a and b, are between sidebands with frequencies on opposite sides of the frequencies of the carriers of the probe and conjugate, ω a and ω b. In order to measure these correlations the frequencies of the local oscillators have to 1

2 (a) 0 (b) 0 Cross-Correlation [a.u.] Cross-Correlation [a.u.] Delay [ns] Delay [ns] Supplementary Figure 1: Envelope of cross-correlation function. (a) Cross-correlation as the phase θ of the homodyne detectors is scanned between 0 and π/. (b) The envelope of the crosscorrelation function, black curve, is obtained from the absolute value of the cross-correlation functions measured for different values of θ. a.u., arbitrary units. In our experiment this condition is automatically satisfied because the local oscillators are also generated with the four-wave mixing process. Taking these relations into account and performing the time averaging, we find that the correlation function takes the form G XaXb (τ) = ELORe{e i( τ θ) } E a (t)e b (t + τ), (5) where ELO E a (t)e b (t + τ) gives the envelope of the cross-correlation function. This is the quantity we are interested in calculating. As Eq. (5) shows, the offset of the local oscillator frequencies from the centre of the gain profile of the four-wave mixing process introduces oscillations in the correlation functions at a frequency. In addition, different phases θ of the homodyne detectors, corresponding to different quadratures, introduce an offset between the time origin of the oscillations and the envelope. This makes it possible to take data for different quadratures and obtain an accurate representation of the envelope of the cross-correlation function. Supplementary Figure 1(a) shows a set of typical curves that we obtain for the cross-correlation as we scan θ from 0 to π/. We take the absolute value of these curves and then calculate the envelope, as shown in Supplementary Fig. 1(b). Once we obtain the envelope of the cross-correlation, we use the mid-point between the values τ at half the maximum to measure the relative delay between the probe and the conjugate. In the paper, the cross-correlation functions are calculated after separating the dc portions

3 of the photocurrents, which are used to normlise them. As a result, the figures show the crosscorrelation functions between the fluctuations of the probe and the conjugate. The quantum correlations between the beams are contained in these fluctuations. Inseparability and EPR criteria The entanglement properties of a two-mode Gaussian state can be completely characterised through its mean values and correlation matrix, or second order moments. For the case of the electromagnetic field we look at the noise properties of different combinations of the quadratures of the fields. In particular, the inseparability criterion allows for a rescaling of the individual quadratures when calculating the joint quadratures, that is ˆX g, = 1 ( ˆXa g ˆX b ) Ŷ g,+ = 1 (Ŷa + gŷb), (7) such that the standard quantum limit (SQL) of these joint quadratures is now (1 + g )/. Here, the individual quadratures have been normalised such that their SQL is equal to one. Experimentally, g corresponds to a variable electronic gain in one of the homodyne detectors. Given these quadratures, the inseparability criterion states that for a Gaussian state ˆX g, + Ŷ g,+ < ( 1 + g ) (8) is a necessary and sufficient condition for the state to be inseparable or entangled 16,1. We can normalise Eq. (8) to the SQL obtained when rescaling the individual quadratures such that it takes the form I = ˆX g, N + Ŷ g,+ N <, (9) where the N subindex indicates that the variances have been normalised to the corresponding SQL. The variances shown throughout the paper have been normalised in this way, such that the inseparability criterion in the form of Eq. (9) is appropriate. When performing the measurements, the variable electronic gain in the detector for the probe is used to minimise ˆX g, N = Ŷ g,+ N and thus obtain the optimum inseparability parameter I. (6) In order to directly use the measured noise properties of our system in the inseparability criterion the following conditions need to be satisfied 16,1 ˆX a 1 ˆX b 1 = Ŷ a 1 Ŷ b 1 (10) 3

4 ( ˆX a 1)( ˆX b 1) ˆX a ˆXb = ( Ŷ a 1)( Ŷ b (11) ˆX a Ŷ a + Ŷa ˆX a = ˆX b Ŷ b + Ŷb ˆX b = 0 (1) ˆX a Ŷ b = ˆX b Ŷ a = 0. (13) If these conditions are not satisfied the inseparability criterion can still be used, however, the correlation matrix has to be transformed to the appropriate form. The set of measurements that we perform for each delay include the noise of the joint quadratures and of the probe and conjugate individually as we scan the phase θ of the homodyne detectors, as well as the corresponding SQL levels. We find that our system behaves as a phase insensitive amplifier. That is, the noise on each beam by itself is phase insensitive, such that the single beam noise ( ˆX (a,b) θ ) is independent of θ, and both joint quadratures have the same noise properties, ˆX = Ŷ +. These results are sufficient to show that conditions (10-13) are satisfied. The fact that the noise on each beam, individually, is phase independent implies that ˆX a = Ŷ a and ˆX b = Ŷ b, and thus condition (10) is satisfied. This property also reduces condition (11) to ˆX a ˆXb = ŶaŶb. These quantities can be obtained from the variances of the joint quadratures, which are of the form ˆX = ( ˆX a + ˆX b )/ ˆX a ˆXb (14) Ŷ + = ( Ŷ a + Ŷ b )/ + ŶaŶb. (15) Given the phase insensitivity of the process, both for the single beams and the joint quadratures ( ˆX = Ŷ + ), it follows from the relations above that ˆX a ˆXb = ŶaŶb, thus satisfying condition (11). Conditions (1-13) imply that there are no cross-quadrature correlations in a single beam and between beams, respectively. In order to verify condition (1) we look at ( ˆX θ (a,b) ) and find that for mode a, for example, ˆX a Ŷ a + Ŷa ˆX a = ( The phase insensitivity of a single beam implies that ( ˆX a Ŷ a + Ŷa ˆX a = 0 and similarly for mode b. ˆX θ=45 a ) ( ˆX a + Ŷ a )/. (16) ˆX θ=45 a ) = ˆX a = Ŷ a, such that Condition (13) can be verified through the joint quadratures when the phase of the homodyne 4

5 detectors is θ = 45. After taking condition (1) into account we find that ( ˆXθ=45 a ) θ=45 ˆX b = ( ˆX a + ˆX b ˆX a ˆXb ) ( Ŷ a + Ŷ b ŶaŶb ) 1 ˆX a Ŷ b + ˆX b Ŷ a. (17) The first term in parenthesis on the right hand side corresponds to the minimum noise level of the oscillations obtained when measuring the joint quadratures (see Supplementary Fig. ) while the second one corresponds to the maximum noise level. In Supplementary Fig., the noise at θ = 45 corresponds to the intersection of the red and blue curves, while the dashed line gives the mean value. As can be seen, the noise level at θ = 45 is equal to the mean value, which implies that ˆX a Ŷ b + ˆX b Ŷ a = 0. This, combined with the fact that ˆX a Ŷ b = ˆX b Ŷ a as a result of the phase insensitivity of the system proves that condition (13) is satisfied. The EPR criterion requires the conditional variances. In particular, for the case in which we want to use the measurements on the quadratures of mode b to estimate the properties of mode a, the conditional variance can be calculated according to 17 V Xa X b = ( ˆX a g ˆX b ) gmin, (18) where g is a parameter used to minimise ( ˆX a g ˆX b ). Experimentally, g corresponds to the variable electronic gain used in the homodyne detector for the probe. The optimum value of g is such that the minimum value for the conditional variance is given by V Xa X b = ˆX a ˆX a ˆXb ˆX b. (19) The same procedure can be used to calculate the conditional variance V Ya Yb. These conditional variances are then used to calculate E ab. It is also possible to estimate the properties of mode b through measurements on mode a. The EPR criterion can be asymmetric, such that E ab E ba. 15 We use the same data that was optimised for the inseparability criterion to calculate the conditional variances and thus the EPR parameter E ab. The variances ˆX a and ˆX b are given by the noise measurements performed on the individual beams normalised to the corresponding SQL, while ˆX a X b can be obtained from the variance of the joint quadratures. Experimentally we 5

6 Noise Power [db] Time [s] Supplementary Figure : Inseparability criterion. The inseparability criterion, Eq. (9), requires that there are no cross-quadrature correlations between beams. This is the case if the mean value between the maximum and minimum noise levels, dashed curve, is equal to the noise level for a homodyne detector phase θ = 45, or the point where the red and blue curves cross. measure the noise optimised with the variable electronic gain and normalised to the corresponding SQL, that is ˆX N = ˆX a + g ˆX b g ˆX a ˆXb 1 + g, (0) where the N subscript indicates that it is normalised. We thus have that ˆX a ˆXb = 1 ( ) 1 g ˆX a + g ˆX b 1 + g g ˆX N. (1) The experimental value of g can be directly estimated by comparing the noise levels that are measured by the homodyne detectors when the twin beams are blocked (i.e., the unnormalised SQL for the probe and the conjugate), such that g = ˆX b SQL ˆX a SQL. () We follow the same procedure with Ŷ + N to calculate the conditional variance for the phase quadrature, V Ya Y b. In our experimental setup, the delayed probe and its local oscillator follow similar paths in order to obtain good mode matching and phase stability for the homodyne detection. As a result, the local oscillator for the probe goes through the slow light medium and experiences the same gain as the delayed probe. This change in local oscillator power is equivalent to a change in the 6

7 electronic gain of the homodyne detector. Thus, the value of g is a combination of the optical gain of the local oscillator and the variable electronic gain of the homodyne detector for the probe and is given by Eq. (). 31. Marino, A. M., Stroud, C. R., Wong, V., Bennink, R. S. & Boyd, R. W. Bichromatic local oscillator for detection of two-mode squeezed states of light. J. Opt. Soc. Am. B 4, (007). 7