Entanglement indicators for quantum optical fields: three-mode multiport beamsplitters EPR interference experiments
We generalize a new approach to entanglement conditions for light of undefined photons numbers given in [Phys. Rev. A 95, 042113 (2017)] for polarization correlations to a broader family of interferometric phenomena. Integrated optics allows one to perform experiments based upon multiport beamsplitters. To observe entanglement effects one can use multi-mode parametric down-conversion emissions. When the structure of the Hamiltonian governing the emissions has (infinitely) many equivalent Schmidt decompositions into modes (beams), one can have perfect EPR-like correlations of numbers of photons emitted into “conjugate modes” which can be monitored at spatially separated detection stations. We provide entanglement conditions for experiments involving three modes on each side, and three-input-three-output multiport beamsplitters, and show their violations by bright squeezed vacuum states. We show that a condition expressed in terms of averages of observed rates is a much better entanglement indicator than a related one for the usual intensity variables. Thus the rates seem to emerge as a powerful concept in quantum optics, especially for fields of undefined intensities.
pacs:03.67.Mn, 03.67.Ud, 03.65.Ta, 42.50.-p
New methods of correlation analysis in multi-photon quantum optics were introduced in  and . They allow to construct much better entanglement witnesses, and new more effective Bell inequalities. The inequalities are devoid of theoretical loopholes (i.e. additional assumptions, apart from local-realism/causality). What is very important generalized methods were devised, which allow to find other entanglement conditions, beyond the ones explicitly derived in the papers. The “headline” result of this line of research is the new approach to polarization correlations of quantum light for states of undefined intensities. This leads to a revision of the concept of quantum Stokes observables.
In , one can find a derivation which shows that the standard âtextbookâ Bell inequalities [3, 4] for quantum optical fields, which address the case of states of undefined intensities (or for simplicity, photon numbers), can be replaced by new ones, which do not rest on certain additional assumptions, which were necessary to derive the textbook ones. To this end, one has to use redefined observables, which we shall call rates. In the case of polarization measurements (say discriminating between H and V polarized photons) such a rate registered at by a detector in, say, channel H is the ratio of the number of photons registered by it, to the number of photons counted by both detectors (for a given run, not averages). The eigenvalues of such rate observables are rational numbers between one and zero. Because of that, in a smooth way one can re-write any known Bell inequalities for pairs of particles, to get Bell inequalities for optical fields in terms of the rates . It is worth noting that the new inequalities, which do not require additional assumptions, can detect entanglement in situations in which the âstandardâ ones [3, 4] fail.
The above results lead to a reconsideration of the usage of the Stokes parameters for quantum optical fields . Despite the fact that the parameters were introduced in 1852(!), they are used in quantum optics without any modification. They are the differences of the average intensities (or photon numbers) of light exiting polarization analyzers, measured in three complementary arrangements (horizontal-vertical, diagonal-antidiagonal and right-left-handed circular polarization analysis), and the total average intensity. If the photon numbers are undefined, the instances when their high values are registered contribute more to the parameters. Redefined quantum Stokes parameters, introduced in , describe the averaged measured polarization with influence of intensity fluctuations removed. Note that the quantum state describes a statistical ensemble of equivalently prepared systems. The average value of an observable is taken over such a statistical ensemble. In an experiment, this implies many repetitions (runs) with the average of the results of the runs giving the experimental value of the observable. Each Stokes observable is redefined as the ratio of the difference of photon numbers at the two exits of a polarizing beam splitter to their sum. It is this ratio that is to be averaged, both in the experiment and theory, and not the numerator and the denominator separately, as in the conventional approach. As a result, in each run the registered intensity of light does not influence the weight of a given run in calculating the average polarization. This approach is most useful in the case of observation of polarization correlations at two or more separate detection stations. In order to measure the correlations of new Stokes parameters, one does not require any new registration techniques, when compared with measurements of old Stokes observables. All that is needed is a different analysis of the data.
The new Stokes parameters allow to re-formulate entanglement conditions, so that they, in the case of the (four-mode, bright) squeezed vacuum (i.e. the output of a strongly pumped down conversion source), allow one to detect entanglement via polarization measurements for stronger pumping and hence for larger mean photon numbers, as well as for higher losses. The approach also allows one to re-write (or map) any entanglement condition for two qubits in such a way that we get a condition for polarization of quantum optical fields, which employs the modified Stokes parameters. Thus we can now construct plethora of new entanglement conditions for correlations of quantum light. The new ideas of replacing intensities via rates in correlation functions, lead to stronger visibility of some non-classical phenomena, in the case of quantum states of undefined photon numbers, see  in which the working example is the Hong-Ou-Mandel dip , under strong pumping condition.
Here we extend this approach beyond polarization effects. We hope that the results will contribute to the experimental search of non-classical phenomena with integrated optics methods, which allow observations stable multichannel interference effects. With the recent progress in photon number resolved detection, our new entanglement conditions may play an important role in the field. We shall study entanglement conditions for bright multi-mode quantum optical fields of undefined intensities (essentially, photon numbers). Extensions of the approach to Bell inequalities for multiport experiments will be presented elsewhere.
The gedanken but already feasible experiments which we study involve pairs of spatially separated multi-port beam-splitter interferometers, of the kind studied in . Multi-port techniques were tested by Walker [8, 9]. Ideas concerning their use to observe effects related with higher dimensional entanglement one can find in [10, 11]. The fact that the multi-port interferometers can perform any unitary transformations of finite dimensional single photon states was shown by Reck et al. In 2000, it was shown that two-photon higher dimensional entanglement leads to stronger violation of local realism than two-qubit one (numerical results of , confirmed analytically in  and ). Also, two-particle higher dimensional entanglement has specific traits which are not present in qubit systems .
The work of Reck et alprovides an operational blueprint for any finite dimensional unitary transformations of single particle states. If the description is limited of just states which allow superpositions of a (single) particle to be in a particular beam, and all other degrees of freedom are treats as irrelevant, then the Reck et altransformations are produced with the use suitably interconnected beam splitters and phase shifters, forming -input--output multiports. Such multi-port devices give optical beams (modes) coupling via a unitary relation between the input and output modes. If the state of a single particle (photon) of being in input beam , where is denoted by , and the states of being in output beams are denoted by , then the unitary transformation describing the action of multi-port, in the form of transformation of the basis states is given by , where is the related unitary matrix. This implies for the second quantized description that for the transformation of photon creation operators related to the modes reads . Note that the basis property of the considered single photon states implies the following commutation relations , and . Identical relations also hold for the ‘out’ operators. Such devices, as they are generalizations of Mach-Zehnder () interferometers in principle allow to experimentally/operationally test basic single and two (three, …, etc.) photon interference and quantum information processes, see e.g., . Here we want to study the extensions of such experiments to the second quantized optical fields, having in mind especially states of light with undefined photon numbers. We investigate non-separability criteria. As our working example of an entangled quantum optical state, we shall take the six-mode bright squeezed vacuum. Such states allow perfect EPR correlations, by which we understand perfect correlations for at least two complementary measurement arrangements. Using this property one can derive entanglement conditions, as separable states cannot have EPR correlations. Entanglement conditions for quantum optical files derivable from EPR correlations were given for the case of four-mode squeezed (polarization entangled) vacuum in , see also [19, 20]. They are in the form of conditions for correlations of standard polarization measurements, that is for Stokes parameters. One can easily generalize the EPR correlations to higher number of modes, see further.
With the ongoing improvements in parametric down-conversion techniques, the birth of integrated optics, and laser imprinting methods to build such devices, the multi-port interferometry experiments, such as the ones suggested in , are becoming feasible. As a matter of fact, important tests of exactly such configurations were recently done, see Schaeff et al. The schemes discussed here involve parametric down-conversion for higher pump powers, in the case of which we do not have only spontaneous emissions of pairs of correlated photons, but superpositions of multi-pair emissions. Therefore new phenomena need to be studied. At least one should check to what extent the features of two-photon correlations are still present in the case of stronger fields.
We take as our example a six-mode squeezed vacuum state which can be produced with the use of a parametric down conversion crystal. However, our intention is not a proposal of feasible experiment, but introduction of new entanglement conditions for multimode optical states of undefined photon numbers. The operational situation which we study serves only as an example here. With the current rapid development of integrated optics, which now includes not only integrated multimode interferometers [22, 23, 24, 25, 26, 27], but also sources [28, 29, 30, 31, 32, 33, 34], the schematic configuration which we present may become feasible.
We shall consider such sets of settings of the local multiport interferometers, which in the case of single photons allow transformations to full sets of mutually unbiased state bases. We shall limit here our considerations to the first non-trivial step, that is to . Higher ‘dimensional’ experiments of a similar kind have been studied in . As a by-product of our considerations we shall also get complementarity relations for multi-port interferometry of general quantum optical fields. We shall see that many of the traits of complementarity relations for single particle multi-port measurements still hold for fields of undefined photon numbers. If one replaces intensities at the outputs of such devices by rates (ratios between the observed intensity at the given output divided by the total intensity), such relations are quite elegant.
The broader implication of our results is that they question the usual paradigm that the quantum coherence properties of optical fields can be best revealed by intensity correlation functions, see any textbook of Quantum Optics. The results presented here show that, in the case in which one can use correlations between rates, instead of the usual intensity correlations, one often gains in the visibility of non-classical phenomena. This finding will be additionally supported in forthcoming publications.
2 EPR correlations: sources, states and measurements
The entanglement indicators of Ref.  are for polarization correlations. They involve measurements of three mutually unbiased, complementary polarizations: e.g., horizontal/vertical, diagonal/anti-diagonal and circular right/left handed. Here, we shall derive generalizations of such entanglement conditions for the multi-mode cases.
The construction of entanglement indicators of  and  takes as its starting point the fact that for the four-mode BSV one can observe perfect EPR correlations (in many pairs of polarization measurements bases), and that separable states do not have this property. They can be only classically correlated. We shall extend this idea to the multimode case.
Here, as our starting point we take multi-mode emissions in the down-conversion process [7, 17]. The emissions from the parametric down-conversion source are directionally correlated due to the phase matching conditions. In the type-I parametric down-conversion, the pairs of photons of the same frequency are emitted into a cone, in such a way that one can register coincidences into pairs of directions along the cone which lay in the same plane as the pump field, for details see . One can select in principle several pairs of such directions, and collect their radiations.
The interaction Hamiltonian which describes such an arrangement has the following form:
where and are the creation operators of th signal-idler mode pair, and is a coupling constant proportional to the pumping power. The modes are directed (via optical fibers, etc.) to a detection station ‘Alice’, while modes to ‘Bob’. Notice that the Hamiltonian can be put in the following form: If one takes a unitary matrix , one has . Further if one defines , and , then one can write the Hamiltonian down in an equivalent form:
This symmetry of implies an invariance of the perfect EPR correlations. Such a transformation can be done using a specific pairs of ‘conjugate’ multi-port interferometers, one for Alice one for Bob. In other words, as the squeezed vacuum state resulting from the application of Hamiltonian on initial vacuum reveals perfect correlations, such correlations also occur after the pair of local mode transformations.
Notice that one can consider the transformations matrices of modes which are associated with unitary transformations leading to unbiased bases for a -dimensional Hilbert space.
Consider . For three pairs of Schmidt modes, the emitted photon pairs are prepared in the following entangled state:
Here, the sum is taken over all combinations of nonnegative integers . The parameter describes ‘gain’ and is dependent on and the interaction time (basically equal the length of the non-linear crystal divided by the speed of light).
Our measurement devices consist of an unbiased symmetric multi-port beam-splitter  and detectors behind the beam splitters. The unbiased multi-port beam-splitter is defined as an -input and -output interferometric device, of the property that light entering via only a single port is split to all output ports, of the intensity into each exit. Following the one photon case  one can relate with each exit th complex roots of unity, values . The three-output case is shown in figure 1.
3 Entanglement criteria for three-port beamsplitters EPR interference
We shall study here an approach which uses the specific value assignment as in figure 1 (see ). Consider four unitary transformations which lead to the unbiased (complementary) bases in a -dimensional Hilbert space. Note that generally when the dimension of a Hilbert space is an integer power of a prime number, the number of mutually unbiased bases is known to be [36, 37]. We put , while the other three, indexed with , have matrix elements which lead to the following transformations of the bases [36, 37]:
where now . With such transformations one can relate a multi-port beam-splitter (interferometer) which couples the input beams the creation operators, , with the output ones, in the following way:
and we define .
As our (local) observables we shall define, for each of the four complementary (local) multiports ,
where is the rate operator for exit mode of multiport . In the formula for the rate is the photon number operator for the mode. The symbol stands for the operator of the total number of photons (it is invariant with respect to unitary mode transformations). Finally, we have the projection operator , where is the vacuum state of the three modes. Thanks to the use of the above, the operator acts only in the non-vacuum part of the Fock space of photons. This trick makes the operator properly defined.
As the squeezed vacuum state has EPR correlations in measured numbers of photons, therefore one has
where indices mark operators for Alice and Bob, respectively. This notation will be used whenever we want to distinguish the two local situations.
We will show that for separable states the condition (8) does not hold. Instead of it, one has
To this end, we use the following two formulas. In A, we show the following operator relation
while in B we also show that for any separable state, , one has
The condition is necessary for a state to be separable.
It is easy to check, just by retracing our calculations in the appendices, that an analog condition involving photon numbers rather than rates reads:
where This is a (direct) generalization of the condition of  to , in the case of which we have summation over up to , on the right hand side we also replace 3 by 2, and (standard) Stokes parameters, , replace , that is we have . The index numbers three fully complementary polarization measurements.
Note, that (11) is by itself a separability condition. Its stronger version can be put as
3.1 Comparison: entanglement conditions employing intensities vs. the ones for rates
In C, we give an analysis of noise resistance of the above entanglement criteria. This is given for our ‘reference’ state, the bright squeezed (six-mode) vacuum. The considered noise is the one for photon losses. We assume that all detectors in the two multi-port experiment of Figure 1 are of finite efficiency, . The critical efficiencies for the two conditions are very interesting.
For the criterion (12) based on photon numbers, we obtain requirement of , for all values of . Notice that it is a very telling result. It means that, with even ideal detections we cannot get an experiment revealing entanglement of bright squeezed vacuum by splitting the radiation of the source on both sides into four directions (each beam) and then directing each of the branches to four pairs of conjugate complementary interferometers, see figure 2 (if this is unclear for the Reader, please note that in the case of polarized beams, as considered in  an equivalent arrangement would be to split the beam leading say to Alice into three of identical intensities, and measuring in each of the three beams, or branches, three fully complementary polarizations, and a similar action on Bob’s side). Such splitting, if one recalls the rule that a passive optical device can be permuted (if this does not lead to a different interferometric setup), acts in each branch in the same way as if the detection efficiency in the branch is . Therefore, we see that the intensity based criterion even for perfect efficiency inherits the usual complementarity features of single photon experiments. If one tries to make simultaneously measurements in all full complementary situations, the entanglement criterion is worthless. However, this is not so for the criterion based on the rates (9). For a very low gain , the critical efficiency is by a whisker below (this is reflecting the fact that the experiment in such a regime is effectively a two-photon one, and standard Bohrian complementarity applies). But for very high one has robustly , and as a matter of fact can be as low as . Thus we have not only a better resistance to losses, but additionally, in principle we can detect entanglement of very bright squeezed vacuum by beam-splitting its radiation to each side into four channels and making all the measurements at the same time. This hints that with the use of the rates we are probing deeper into the nature of multi-photon light.
4 Complementarity relations
In Section 3, we show a separability condition based on the local operator with the specific measurement assignments (the power of ). To this end, we use the following relation (for the derivation, see B)
Note that this is a complementarity relation for the four possible, mutually exclusive interferometric measurements involving beams. The interferometers are such that they perform unitary transformations leading to mutually unbiased bases for single photon state. If, say then for all we have .
5 Summary and closing remarks
For -mode quantum optical fields of undefined intensities, we formulate the entanglement criteria inspired by properties of EPR correlations, which are generalizations of the ones presented in  and . The first ones are based on intensities, and the second one use rates. As an example, we consider a six-mode bright squeezed vacuum state. Such optical states have EPR-like correlations of numbers of photons registered in conjugated modes. With the help of multi-port beam-splitter techniques, we are able to see such correlations. In case of inefficient detection, our approach in terms of rates is able to detect entanglement for a wider range of parameters describing the state (pumping strength) and detection efficiency. As the critical efficiencies are quite moderate, and generation of squeezed six-mode squeezed vacuum seems feasible, our entanglement conditions can find application in experiments. This is mainly due to the fact that integrated optics techniques allow now to produce stable multiport interferometers, this allows to put the ideas of  in practice. Such arrangements, like the one of figure 1 may have various quantum informational applications. These applications may go beyond single photon at single detection station paradigm, as we show that EPR correlations, which reveal non-classicality are observable also in the case of undefined photon numbers.
We have derived both conditions which are more traditional, that is based on correlation of intensities, and conditions, inspired by , which use correlations of rates. The latter ones are capable to detect entanglement where the former fail.
The other important consequence is that the presented results confirm our conjecture that correlation functions involving rates rather than intensities can become a useful tool in quantum optics, and that at least in some cases they outperform the standard ones based on intensities. We expect that one can find benefits by using the rates in various cases, e.g., quantum steering and etc. The results can be generalized to all for which mutually unbiased bases are known to exits, and other methods of detecting entanglement, see our forthcoming manuscripts. The approach with rates is also very handy in the case of formulation of Bell inequalities, see e.g.  for the case of polarization correlations..
Appendix A Derivation of (10)
Let us generalize for while our considerations to prime mode case. Let be a row matrix as , and then its “column Hermitian conjugate” involves the annihilation operators. One can put , where the matrix is given by
which is an analog of for a -dimensional Hilbert space (of single photon states). The matrices form the unitary generalizations of Pauli operators for any (prime) dimensional Hilbert space. For we put , which we shall denote the matrix as . If one defines by (where necessary, all formulas here are modulo , with respect to indices), one has and for . The set of ’s is just a permutation of the set of ’s. One has .
Using the above algebraic relations we can put:
Note that the and commute with each other and with any operator of the form . We make the following transformations (on the way of which we use the following relations: , and ):
For , as in this case , the formula (LABEL:EQ:STOKES_d) reads
Therefore, we have (10).
Appendix B Derivation of (11)
Here we derive the relation (11) for any separable states, namely
First we notice that as separable states are of the form of a convex combination , we can search the maximum of LHS of (19) using
One can further simply the derivations by considering only tensor products of pure states, what we do further on. Note that
Thus in order to find the maximum we must know the general properties of and . Specifically, what will be needed will be the upper bound for
where , as by Cauchy inequality
To this end we can use the algebraic relations already established in the derivation of (10), as given in (LABEL:EQ:STOKES_d). We take any pure state , and consider . Notice that if one takes the average of with respect to and inserts between the pairs of conjugated operators, one gets . This in turn be rearranged using the same algebraic steps of the first three of the equalities of (LABEL:EQ:STOKES_d), as they involve only the properties of matrices. Below we put these manipulations explicitly for :
Thus, we get
The upper bound of middle term is
while the last term is, due to the fact that for and for one has , given by
Thus we get
Appendix C Losses induced noise
Here we study noise robustness of the entanglement criteria. As our model of noise we shall take losses of photon counts due to inefficiency of the detection. We shall assume that all detectors have the same efficiency.
In the case of a theoretical description of detection/collection losses, an inefficient detector can be emulated by a perfect one with a beam-splitter of transmissivity in front of it, so that the probability that photons are counted while reach beam-splitter reads
The numerical results to be presented here consider the six-mode bright squeezed vacuum state (3):
We cut off the sequence at to . With the losses model which we adopt one has
where and implies that the fractions take value 0 when the denominator is equal to zero. Here denotes the average over . To calculate the right hand side of (9) we notice that
Because of the symmetry of the state, and the detection efficiency, for each pair of conjugated unbiased bases one has the same value of of .
These averages are summed with weights defined by the superposition coefficients:
where is the average over the state . Plugging this into inequality (9) we get that if
we detect entanglement. For large we find that the LHS of (34) is less than 0 for . For , we get requirement of .
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