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Sending Quantum Correlations through Dispersive Media Paul D. Lett1, Jeremy Clark1, Ryan Glasser1, Tian Li1, Quentin Glorieux2, Ulrich Vogl3, Kevin Jones4 1 Joint Quantum Institute, National Institute of Standards and Technology and the University of Maryland, Gaithersburg, Maryland 20899 USA 2 Laboratoire Kastler Brossel, Université Pierre et Marie Curie, 4 Place Jussieu, 75005 Paris, France 3 Institut of Optics, Information and Photonics University Erlangen-Nuremberg, 91058 Erlangen, Germany 4 Department of Physics, Williams College, Williamstown, Massachusetts 01267 USA
[email protected]
Abstract: We send one half of a bipartite entangled state through a dispersive medium and examine the effects of normal and anomalous dispersion on the quantum entanglement, correlations and arrival time of quantum mutual information. OCIS codes: (270.0270) Quantum Optics; (270.65700) Squeezed states
1. Introduction The propagation of classical information through dispersive media is well-understood. Normal dispersion leads to “slow light” effects, while anomalous dispersion leads to apparent superluminal propagation or “fast light” effects. These fast light effects are fully consistent with causality and even though the effects can be somewhat counterintuitive they logically and consistently obey the familiar laws of physics. It is easy to argue that, in keeping with causality, classical information cannot be advanced by sending it through an anomalously dispersive medium, even though it is rather difficult to prove the negative experimentally. On the other hand, there seems to be nothing to prevent quantum correlations, even in the form of quantum entanglement, from being advanced in an anomalously-dispersive medium, as no superluminal signaling is enabled in this way. Nonetheless, it seems natural to argue that quantum information, just like classical information, should not be able to be advanced. We explore the advancement of quantum correlations by sending one half of a bipartite entangled state through an anomalously dispersive medium. In this way we investigate the physical mechanisms that seem to act to constrain the system. Dispersion in an optical system, normal or anomalous, is generated in conjunction with gain or loss. Causal dispersion relationships, analogous to the familiar Kramers-Kronig relations for linear media, can be written for nonlinear optical interactions as well. In particular, we can take advantage of the dispersion near a four-wavemixing (4WM) gain line in Rb vapor and the resulting rapid variation in the refractive index of the gas to create fast and slow light conditions. The gain or loss involved in the process results in noise being added to the optical signal. This noise affects our ability to discern an advance in any sort of signal sent through the medium. The region of linear dispersion near a gain or loss feature is typically rather small, and generating information whose spectrum fully fits within such a linearly-dispersive region is difficult. Classical analytic signals carry the information everywhere; the earliest leading edge carries all of the same information as in a peak that follows. Nonanalytic points carry new information but also entail an infinitely broad spectrum, so that some frequency components always travel around the linear dispersive region, leading to precursors that travel precisely at the speed of light. We address these difficulties by explicitly looking only at narrowband signals, generated by quantum fluctuations, and measured by homodyne detection. Again taking advantage of a 4WM process in Rb vapor, we generate quantum-entangled “twin beams,” whose correlations are stronger than classically possible [1]. We then send one of these twin beams through either a fast or slow light medium and examine the effect on the intensity correlations, quantum entanglement, and quantum mutual information. While it is not surprising that we find that information, quantum or otherwise, cannot be advanced through a fast light medium, the manner in which these measures behave can tell us something about the mechanisms that prevent such advancements. Four-wave mixing (4WM) near atomic resonance lines in Rb vapor can generate strongly correlated twin beams of light [1]. The process can also be used to create a nearly quantum-noise-limited phase-insensitive amplifier, whose gain is associated with a rather sharp dispersive feature. The dispersion near a 4WM gain feature can be used to generate either slow- or fast- light conditions [2,3]. Dispersion is associated with gain or loss in the system, each of which will add noise to a signal. It is this added noise that seems to provide the physical means by which nature prevents the advance of information.
QTu3A.1.pdf
Research in Optical Sciences © OSA 2014
2. Fast and slow light results A simple 4WM gain feature operates as a phase insensitive amplifier. Such an amplifier, even when operating at the lowest noise level permitted by quantum mechanics, will necessarily degrade an optical signal. We operate near a 4WM gain line under conditions that result in a gain of approximately 1.2. The dispersion is large enough to observe an advance of 3.7 ns for fluctuations in the measurement bandwidth of 100 kHz to 2 MHz. This advance is measured by looking at the intensity cross correlations between the two twin beams, before and after one of the beams has passed through a fast light medium with anomalous dispersion. The advance is shown in Fig. 1. In addition, we show that the two beams retain the property of quantum entanglement even after passing through the fast light medium, although the entanglement has been degraded by the added noise.
Fig. 1 Persistence of correlations associated with entanglement in the presence of anomalous dispersion. a) We observe up to -3 dB of intensity-difference or phase-sum squeezing when the second (fast light) four-wave mixing process is suppressed. b) In the presence of a small phase-insensitive gain, giving rise to anomalous dispersion, the squeezing reduces to -2.3 dB, which is still sufficient to show entanglement. c) Average normalized cross-correlation functions for the correlated and anti-correlated joint quadratures. Parts d) and e) provide closer looks at the peak correlation and anti-correlations, respectively.
In Fig. 2 we plot the calculated quantum mutual information between the two beams as a function of delay, again both before and after one of the beams is passed through a fast light medium. In our experiment, where we are measuring the continuous random fluctuations of the probe and conjugate beams, there is no imposed “signal" as such. The fluctuations on one beam, however, carry information about the fluctuations on the other. The quantum mutual information between the two beams can be obtained from the same basic data as used to demonstrate entanglement. The mutual information quantifies the total (classical plus quantum) correlations between the two beams [4]. We exploit the fact that any bipartite Gaussian state can be completely characterized by the variances and covariances of the field quadratures [5]. In good agreement with the squeezing and cross-correlation measurements, we observe an advancement of 3.7(1) ns of the peak of the delay-dependent mutual information, paired with a degradation due to uncorrelated noise added by the fast light cell. This degradation appears to prevent us from observing an advance of the leading edge of the fast light mutual information (red curve in Fig. 2).
QTu3A.1.pdf
Research in Optical Sciences © OSA 2014
Fig. 2 Comparison of the computed quantum mutual information between the probe and conjugate as a function of the relative delay betweent he beams for the cases of fast and slow light. When considering the fast light advancement of the conjugate (red trace), we observe an advance in the peak of the mutual information of 3.7 (1) ns. There is no statistically significant advance of the leading edge of the mutual information in the case of fast light propagation. Repeating the same analysis for slow light propagation of the probe, we observe a delay of both the leading and trailing edge of the mutual information by hundreds of standard deviations (green trace).
The inability to advance the quantum mutual information is in contrast to the advance of the correlations corresponding to quantum entanglement. While this result is not terribly surprising, it also contrasts with the results found when one beam is passed through a normal-dispersion medium. The green curve in Fig. 2 shows the quantum mutual information delay under slow light conditions that result in an equivalent amount of degradation of the mutual information. Fig. 2 demonstrates that under these conditions the arrival of quantum mutual information can be delayed significantly with respect to its arrival time without passing through the slow light medium. The fast and slow light cases are apparently not symmetric. 3. Conclusion We have demonstrated an advancement in time of the quantum fluctuations of one mode of an entangled state of light passing through a fast light medium while preserving entanglement between the modes. We showed that the peak of the quantum mutual information between the modes can be advanced in time, but that added noise associated with the dispersion apparently prevents us from observing an advance of the leading edge, that is, a true advancement of information. In contrast, in a slow light medium operating under conditions which produce a similar reduction in the peak of the mutual information, the leading and trailing edges of the mutual information can both be significantly delayed. 4. References [1] McCormick, C. F., Boyer, V., Arimondo, E. & Lett, P. D., “Strong relative intensity squeezing by 4-wave mixing in Rb vapor,” Opt. Lett. 32, 178 (2007). [2] Marino, A., Pooser, R., Boyer, V. & Lett, P. D., “Tunable delay of Einstein-Podolsky-Rosen entanglement,” Nature 457, 859 (2009). [3] Glasser, R. T., Vogl, U. & Lett, P. D., “Stimulated generation of superluminal light pulses via four-wave mixing,” Phys. Rev. Lett. 108, 173902 (2012). [4] Ollivier, H. and Zurek, W. H., “Quantum Discord: A Measure of the Quantumness of Correlations,” Phys. Rev. Lett. 88, 017901 (2001). [5] Serafini, A., Illuminati, F., and Siena, S. D., “Symplectic invariants, entropic measures and correlations of Gaussian states,” J. Phys. B. 37, L21 (2004).
