Entanglement with classical fields

We experimentally demonstrate a simple classical-field optical heterodyne method which employs postselection to reproduce the polarization correlations of a four-particle entangled state. We give a heuristic argument relating this method to the measurement of multiple quantum fields by correlated homodyne detection. We suggest that using multiple classical fields and postselection, one can reproduce the polarization correlations obtained in quantum experiments which employ multiple single-photon sources and linear optics to prepare multiparticle entangled states. Our experimental scheme produces four spatially separated beams which are separately detected by mixing with four independent optical local oscillators (LO) of variable polarization. Analog multiplication of the four beat signals enables projection onto a four-particle polarization-state basis. Appropriate band pass filtering is used to produce a signal proportional to the projections of the maximally entangled four-field polarization state, $\mid H_1)\mid H_2)\mid H_3)\mid H_4)+\mid V_1)\mid V_2)\mid V_3)\mid V_4)$, onto the product of the four LO polarizations. Since the data from multiple observers is combined prior to postselection, this method does not constitute a test of nonlocality. However, we reproduce the polarization correlations of the 32 elements in the truth table from the quantum mechanical Greenberger-Horne-Zeilinger experiments on the violation of local realism. We also demonstrate a form of classical entanglement swapping in a four-particle basis.

Figure 1

Correlated homodyne detection of two fields. Fluctuations in the quantum field $E_1\left(t\right)$ are measured by mixing with a strong classical local oscillator (LO) field $E_{LO}\left(t\right)$ using a balanced detection scheme. Using an independent LO field, fluctuations in the field $E_2\left(t\right)$ are measured. The product of the fluctuating currents $\Delta I_1$ and $\Delta I_2$ yields the correlation between the fluctuations in the quantum fields 1 and 2.

Figure 2

Experimental arrangement for the measurement of the four-field entangled state $\mid {\Psi }_{cl}^4{)}_{GHZ}$. $D_1$, $D_2$, $D_3$, and $D_4$ are detectors used to obtain the beat signal amplitudes $A_1$, $A_2$, $A_3$, and $A_4$ respectively, each of which contains two frequencies, 120 and $30\text{kHz}$.

Figure 3

Detection scheme for observing GHZ entanglement by frequency postselection. The analog multipliers $(\times )$, band pass filters, and summing amplifiers $(+)$ enable postprojection of an entangled four-field state onto a four-field product basis.

Figure 4

Measurement of elements of reality for the $YYX$ configuration using a four-classical-field entangled state. (a)(c)(e)(g) Components of physical reality predicted by GHZ entanglement exhibit nonzero signals at $300\text{kHz}$. (b)(d)(f)(h) Elements of physical reality not predicted by GHZ entanglement produce nearly zero signal at $300\text{kHz}$. (i) The normalized signal squared obtained for these elements agrees with the predictions of quantum mechanics.

Figure 5

Measurement of elements of reality for the $YXY$ configuration using a four-classical-field entangled state. (a),(c),(e),(g) Components of physical reality predicted by GHZ entanglement exhibit nonzero signals at $300\text{kHz}$. (b),(d),(f),(h) Elements of physical reality not predicted by GHZ entanglement produce nearly zero signal at $300\text{kHz}$. (i) The normalized signal squared obtained for these elements agrees with the predictions of quantum mechanics.

Figure 6

Measurement of elements of reality for the $XYY$ configuration using a four-classical field entangled state. (a),(c),(e),(g) Components of physical reality predicted by GHZ entanglement exhibit nonzero signals at $300\text{kHz}$. (b),(d),(f),(h) Elements of physical reality not predicted by GHZ entanglement produce nearly zero signal at $300\text{kHz}$. (i) The normalized signal squared obtained for these elements agrees with the predictions of quantum mechanics.

Figure 7

Demonstrating violation of local realism theory by measurement of elements of reality for the $XXX$ configuration using a four-classical-field state. (a),(c),(e),(g) Elements of physical reality predicted by GHZ entanglement exhibit nonzero signals at $300\text{kHz}$. (b),(d),(f),(h) Elements of physical reality not predicted by GHZ entanglement produce nearly zero signal at $300\text{kHz}$. (i) The normalized signal squared obtained for these elements agrees with the predictions of quantum mechanics producing strong signals only when $XXX=1$, i.e., for $V_1^{\prime }{H}_2^{\prime }{V}_3^{\prime }$, etc. This is in contradiction with local realism which predicts strong signals only when $XXX=-1$, i.e., for $H_1^{\prime },H_2^{\prime }{V}_3^{\prime }$, etc.

Figure 8

Symmetrical detection scheme for demonstrating classical-field entanglement swapping in the four-field basis. The signal measured by observer A is sent to observer B, who multiplies this signal by his own to obtain a net signal at $300\text{kHz}$. Observer B then measures correlations determined by the Bell state chosen by observer A.

Figure 9

Demonstration of entanglement swapping. Observer A sets his LO 1 and 2 polarizations at $45°$ and $45°$, respectively, to select the Bell state $\mid {\varphi }_{cl}^{+}{)}_{12}$. Observer B’s signals are then proportional to the projections of the corresponding Bell state $\mid {\varphi }_{cl}^{+}{)}_{34}$ onto the polarizations of LO’s 3 and 4. (a) Observer B sets his LO 3 and 4 polarizations at $45°$ and $45°$, respectively, yielding a nonzero signal at 300 kHz. (b) Observer B sets his LO 3 and 4 polarizations at $45°$ and $-45°$, respectively, yielding a zero signal at $300\text{kHz}$.