Optical spin squeezing: bright beams as high-flux entangled photon sources

Optical spin squeezing: bright beams as high-flux entangled photon sources

Abstract

In analogy with the spin-squeezing inequality of Wang and Sanders [Physical Review A 68, 012101 (2003)], we find inequalities describing macroscopic polarization correlations that are obeyed by all classical fields, and whose violation implies entanglement of the photons that make up the optical beam. We consider a realistic and exactly-solvable experimental scenario employing polarization-squeezed light from an optical parametric oscillator (OPO) and find polarization entanglement for postselected photon pairs separated by less than the OPO coherence time. The entanglement is robust against losses and extremely bright: efficiency can exceed that of existing “ultra-bright” pair sources by at least an order of magnitude. This translation of spin-squeezing inequalities to the optical domain will enable direct tests of discrete variable entanglement in a squeezed state.

Introduction - Spin squeezing inequalities, in which squeezing of an ensemble implies entanglement of the constituent particles, are powerful tools for understanding the relationship between macroscopic and microscopic quantum features Gühne and Tóth (2009). The work of Sørensen et al. Sørensen et al. (2001) showed that squeezing of a spin-1/2 ensemble implies entanglement in the ensemble. Wang and Sanders Wang and Sanders (2003) considered symmetric spin systems and showed that squeezing implies entanglement in every reduced two-atom density matrix. Similar results have been found for larger-spin systems, for other kinds of squeezing, and for multi-partite entanglement Vitagliano et al. (2011); Korbicz et al. (2005, 2006); Gühne and Tóth (2009). Spin squeezing has been produced in a number of experiments Appel et al. (2009); Leroux et al. (2010); Gross et al. (2010); Chen et al. (2011); Sewell et al. (2012), implying entanglement of macroscopic numbers of atoms. To date, there has been no direct observation of the implied entanglement.

Here we present a result analogous to that of Wang and Sanders, but for optical fields. To our knowledge, this is the first spin-squeezing-type inequality in the optical domain i.e., the first demonstration that optical continuous-variable (CV) non-classicality implies discrete variable (DV) entanglement upon projection to photon pairs. Production and detection of optical squeezing is a well-developed technology, with quadrature squeezing levels reaching 12.3 dB Mehmet et al. (2011). Simultaneously, efficient detection of photons is routine in quantum optics laboratories, as is quantum state tomography of entangled pairs James et al. (2001); Adamson et al. (2007). Together, these offer the possibility to test the predicted relations between macroscopic squeezing and microscopic entanglement.

We also give a practical implementation and show that the efficiency of narrowband entangled pair generation exceeds the state-of-the-art by at least an order of magnitude, a promising result for quantum networking.

Scenario - We consider a beam with a single-spatial mode and stationary statistical properties, i.e., a continuous-wave beam. We suppose that the and modes of this beam are in a product state, with being a strong coherent state, and being a weak non-classical state. We also assume the state is invariant under , i.e., a phase shift of the mode or equivalently . This includes an important class of practical nonclassical states, for example squeezed thermal states and even cat states. Because of the product structure, there is no entanglement of the , modes. The coherent state contains many photons but no entanglement, while the entanglement content of the non-classical state is limited, due to its low brightness.

Nonclassicality and entanglement - The nonclassicality and entanglement properties are related through the first- and second-order correlation functions

 R(1)i,j(τ)≡⟨^a†i(t)^aj(t+τ)⟩ (1)

and

 R(2)ij,mn(τ)≡⟨^a†i(t)^a†j(t+τ)^an(t+τ)^am(t)⟩, (2)

respectively, where is the annihilation operator for the mode . Note that we have inverted the last two indices in the definition of , in keeping with the convention in photonic quantum state tomography James et al. (2001).

A simple non-classicality condition is found using the Cauchy-Schwarz inequality 1 or by systematic derivations Shchukin and Vogel (2005); Vogel (2008): classical fields obey

 |R(2)HH,VV(τ)|2 ≤ R(2)HV,HV(τ)R(2)VH,VH(τ) (3a) |R(2)HV,VH(τ)|2 ≤ R(2)HH,HH(τ)R(2)VV,VV(τ) (3b)

whereas quantum fields can violate these inequalities. We now show that Eqs. (3a),(3b) imply polarization entanglement when DV detection methods, i.e. photon counting, are used.

For single-photon detection with polarization , a vector in the basis, Glauber theory indicates the average rate of detections as

 W(1)p=Tr[ΠpR(1)(0)], (4)

where is a projector onto . It is conventional to define the one-photon observable density matrix (ODM) , so that relative probabilities of detection are given by the Born rule

 P(1)p=Tr[ΠpR(1)(0)]. (5)

In the same way, the pair polarization , where , are unit polarization vectors, gives the rate at which photon pairs arrive separated by time

 W(2)r(τ)dτ=Tr[ΠrR(2)(τ)]dτ. (6)

Again, this is usually expressed via the Born rule where is the two-photon ODM. Recovery of from is the subject of quantum state tomography James et al. (2001).

Using the invariance, we find an ODM similar to that found for symmetric atomic states Wang and Sanders (2003):

 R(2)∝⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝R(2)HH,HH00R(2)HH,VV0R(2)HV,HVR(2)HV,VH00R(2)VH,HVR(2)VH,VH0R(2)VV,HH00R(2)VV,VV⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (7)

where all elements are functions of . This describes a mixture of a state in the {} subspace and another in {}.

By inspection, positivity under partial transposition of the above density matrix is equivalent to the two conditions of eq. (3a),(3b). Thus non-classical polarization correlations imply DV entanglement of the photons in the extracted modes, for stationary beams with the symmetry described above. In the following section, we will describe a feasible experimental scenario that allows to violate the spin-squeezing-type inequalities with available technologies.

Practical implementation - Continuous wave non-classical polarizations have been produced by combining two bright squeezed beams of orthogonal polarization Korolkova et al. (2002); Bowen et al. (2002), by optical self-rotation Ries et al. (2003) and by combining a coherent state (-polarized) with -polarized squeezed vacuum Predojević et al. (2008). We consider the last case, which manifestly shows symmetry under .

For the squeezed vacuum, we consider a sub-threshold OPO, as described by Collett and Gardiner Collett and Gardiner (1984). The field operator is expressed via a Bogoliubov transformation of the vacuum input and loss reservoir operators and , respectively

 ^aV(ω)=f1(ω)^a1(ω)+f2(ω)^a†1(−ω)+f3(ω)^a2(ω)+f4(ω)^a†2(−ω). (8)

The coefficients

 f1(ω) = [η2−(1−η−iω/δν)2+μ2]A−1(ω), (9a) f2(ω) = [2ημ]A−1(ω), (9b) f3(ω) = [2√η(1−η)(1−iω/δν)]A−1(ω), (9c) f4(ω) = [2μ√η(1−η)]A−1(ω), (9d) A(ω) = (1−iω/δν)2−μ2, (9e)

are functions of the experimental parameters of the OPO: the cavity FWHM bandwidth , the photon flux of the squeezed vacuum state , and the cavity escape coefficient , i.e. the ratio between the transmission of the output coupler and the sum of both the intracavity losses and the transmission of the output coupler (). is the pump power expressed as a fraction of the threshold power. We take the phase of the pump field equal to for simplicity.

The time-domain correlation functions required for  (2) can be computed as Fourier integrals, to find

 R(2)HH,HH = ΦC2 (10a) R(2)HV,HV = ΦCαμ, (10b) R(2)HH,VV(τ) = ΦCα[coshx+μsinhx]e−δν|τ| (10c) R(2)HV,VH(τ) = ΦCα[μcoshx+sinhx]e−δν|τ| (10d) R(2)VV,VV(τ) = α2{β+[(1−μ2)cosh(2x) (10e) +2μsinh(2x)]e−2δν|τ|}

where

 x = μδν|τ|, (11a) α = ημδν1−μ2, (11b) β = 1πδν(1−μ2){μ4(1−η−πδν)+ (11c) +μ2[πδν+2η(1+η)−1]+6η2−9η+4}

and is the photon flux of the coherent state.

Entanglement under realistic conditions - We now show that it is possible to achieve either high entanglement or high rates of entangled pairs with feasible experimental values.

We quantify the entanglement associated with a pair extracted from a polarization squeezed state by means of the concurrence Wootters (1998)

 C=max(0,√λ1−√λ2−√λ3−√λ4), (12)

where are the eigenvalues of in decreasing order and is a Pauli matrix. The relevant experimental parameters are the time interval between detections and the average photon fluxes of the coherent and squeezed state, and respectively. Changing the other parameters do not change significatively the concurrence, so we fix them to typical experimental values, specifically  MHz and , from Predojević et al. (2008).

Figure 1 shows that the state is entangled for any choice of and , provided that the two photons are detected within the coherence time of the squeezed state, while the concurrence goes to zero when . However, the concurrence does not change much in a wide range of time separation , which goes from the time resolution of actual single photon detectors (some tens of picoseconds) to hundreds of nanoseconds ().

We estimate the entangled pair flux by averaging the concurrence with the corresponding photon flux:

 W(2)=∫+∞−∞dτTr[R(2)(τ)]C(τ), (13)

and we plot it in Fig. 2 compared to concurrence: the experimental parameters and can be suitably chosen in order to obtain a Bell-like state with high concurrence ( inside the innermost (yellow) surface in Fig. 1).

However, there are some cases where high entanglement flux can be more important than maximal entanglement. For example, non-maximally entangled spin-1/2 states which violate a Bell inequality can be useful for teleportation Horodecki et al. (1996). A “typical” state statisfying this requirements

 R(2)typ≈⎛⎜ ⎜ ⎜⎝0.601000.38800.0670.067000.0670.06700.388000.264⎞⎟ ⎟ ⎟⎠, (14)

obtained with squeezed beam flux  photons/s (2.6% OPO threshold), coherent beam flux  photons/s and arrival-time difference  ns, can combine a high rate of entangled pairs with easily detectable concurrence: the state of Eq. (14) has and ebit/s, well above the ebits/s that can be reached by states with high concurrence (. Such a state can be used for teleportation with up to 88 fidelity Hu (2013) and can be generated feasibly with current technology: in fact, it only needs 2.3 dB of squeezing, well within existing capabilities.

The ability to trade brightness against entanglement purity may be advantageous also in applications of quantum non-locality. Hu et al. Hu (2013) calculate the achievable Clauser-Horne-Shimony-Holt inequality violation for states with the form of . Using that result, and the fact that statistical significance (in standard deviations) scales as , where is the acquisition time and is the rate of detections within a coincidence window of width , we find the figure of merit to describe how quickly a Bell inequality violation acquires statistical significance. As shown in Fig. 2, the largest occur for bright, modestly-entangled states, , and in some regions entanglement dilution (increasing while keeping constant) increases .

Comparison to other sources - DV polarization entanglement is the preferred embodiment for many quantum information tasks, e.g. free-space quantum key distribution Steinlechner et al. (2012) and optical quantum computing Walther et al. (2005). Our source incorporates a coherent state, which gives a high brightness but also less strict photon-number correlations than true pair sources such as parametric down-conversion. A conservative estimate of the pairs-to-singles flux ratio is . As an example, , , as in Eq. (14), gives pairs per photon, whereas an ideal entangled pair source would have . This imperfect correlation would prevent a loophole-free Bell test with this state, for example. In contrast, the brightness is attractive for quantum networking with atoms, an application currently limited by the spectral brightness of narrow-band sources Haase et al. (2009); Piro et al. (2011). Cavity-enhanced sources of polarization-entangled photons have demonstrated detected (inferred) spectral brightness per pump power of 70 (1221) pairs/(s MHz mW) Wolfgramm et al. (2008, 2010) and 50 (5500) pairs/(s MHz mW) Zhang et al. (2011). The spectral brightness we predict here is /(s MHz) for an 8 MHz bandwidth. The required pump power for the OPO (about 10 % of threshold) depends on the implementation: for a doubly-resonant OPO, e.g. Bowen et al. (2002), the power is \SI5mW, while for a triply-resonant OPO, e.g. Martinelli et al. (2001), it can be \SI50\microW. The source described here is thus one to three orders of magnitude brighter than existing sources.

Discussion - Even though the polarization squeezed state is a product of an entangled state (squeezed vacuum) and a classical one (coherent) with orthogonal polarization, our result shows that the contribution of both initial states is fundamental for the pairwise entanglement of the final state. In fact, the maximum concurrence corresponds to the case that most resembles a Bell state, in which it is equally probable to detect two -polarized or two -polarized photons (, showing that the coherent state plays an important role in the generation of polarization entangled pairs.

Conclusions - We have derived a “spin-squeezing inequality” for photons, analogous to the result of Wang and Sanders Wang and Sanders (2003) for spins in symmetric states. The result shows that non-classical macroscopic polarization correlations imply microscopic entanglement of the photons in the beam. Considering polarization-squeezed light from a sub-threshold OPO, we find exact expressions for the entangled state, and show that an experimental demonstration of DV entanglement associated to squeezing, not yet practical with atoms, is feasible with photons. We predict polarization-entangled photon sources robust against losses and brighter than existing “ultra-bright” sources by one to three orders of magnitude, of considerable interest for quantum networking applications.

Acknowledgements.
We thank Jaroslaw Korbicz, Alejandra Valencia, and Philipp Hauke for helpful discussions. This work was supported by the Spanish MINECO under the project MAGO (Ref. FIS2011-23520), by the European Research Council under the project AQUMET and by Fundació Privada CELLEX.

Footnotes

1. For example, with a pure classical state where are c-numbers. Rearranging we find which is constrained by inequality (3a) after similar RHS rearrangements.

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