Entanglement, coherence, and redistribution of quantum resources in double spontaneous downconversion processes

Entanglement, coherence, and redistribution of quantum resources in double spontaneous downconversion processes

David Edward Bruschi York Centre for Quantum Technologies, Department of Physics, University of York, YO10 5DD Heslington, UK    Carlos Sabín Instituto de Física Fundamental (CSIC),Serrano 113-bis, 28006 Madrid, Spain    Gheorghe Sorin Paraoanu Low Temperature Laboratory and Centre for Quantum Engineering, Department of Applied Physics, Aalto University School of Science, FI-00076 Aalto, Finland
Abstract

We study the properties of bi-squeezed tripartite Gaussian states created by two spontaneous parametric down-conversion processes that share a common idler. We give a complete description of the quantum correlations across of all partitions, as well as of the genuine multipartite entanglement, obtaining analytical expressions for most of the quantities of interest. We find that the state contains genuine tripartite entanglement, in addition to the bipartite entanglement among the modes that are directly squeezed. We also investigate the effect of homodyne detection of the photons in the common idler mode, and analyse the final reduced state of the remaining two signal modes. We find that this measurement leads to a conversion of the coherence of the two signal modes into entanglement, a phenomenon that can be regarded as a redistribution of quantum resources between the modes. The applications of these results to quantum optics and circuit quantum electrodynamics platforms are also discussed.

I Introduction

The vacuum in quantum theory is one of the most subtle concepts in modern physics. The classical picture of a “void” or “emptiness” does not accurately capture the nature of this particular state, and new phenomena can be unveiled by systematically employing quantum mechanics Gea-Banacloche et al. (1988). In the language of quantum physics, the vacuum state is the lowest energy eigenstate of a particular field Hamiltonian. One can picture the quantum vacuum as a state with some latent structure (see e.g. Paraoanu (2015)), which can manifest, for example, through the conversion of quantum fluctuations into real excitations when some parameter in the Hamiltonian is changed (sudden quench, parametric driving, etc.). This phenomenon is generally known as dynamical Casimir effect Moore (1970), as is typically exemplified by a mirror moving in vacuum at relativistic speeds Birrell and Davies (1984). This effect can be demonstrated in the laboratory by using, for example, superconducting circuits Wilson et al. (2011); Lähteenmäki et al. (2013). In these scenarios, abrupt modifications of the boundary conditions Wilson et al. (2011) or of the speed of light in a meta-material Lähteenmäki et al. (2013) by means of an external pump result in field excitations that can be amplified and detected. These processes give rise to two-mode squeezed microwaves which display entanglement Lähteenmäki et al. (2013) and other forms of quantum correlations Johansson et al. (2013); Sabín et al. (2015); Sabín and Adesso (2015), triggering the question of their employability as resources for quantum technologies Bruschi et al. (2016); Felicetti et al. (2014); Streltsov et al. (2016); Krenn et al. (2016).

Recently, it has been reported that a new class of three-mode states can be generated in the laboratory by double-pumping a superconducting resonator Lähteenmäki et al. (2016), with one mode common to both pumps. This can be regarded as a “double dynamical Casimir effect”, since now the mirror moves under the action of two pumps with different frequencies - or, in other words, the motion of the mirror is a harmonic oscillation with the average frequency of the pumps modulated due to beating at half-difference frequency. One starts by considering three modes , and , where is the common mode (conventionally referred from now on as idler). The parametric processes are arranged such that one downconversion occurs between modes and the other between modes . This clearly leads to two-mode squeezing between modes and , and modes and respectively. However, we show that the resulting tripartite state not only does contain the standard correlations due to parametric two-mode squeezing, but it also displays coherence correlations among the modes and , even if these modes are not directly connected by the pumps. The origin of this coherence is the lack of which-path information for the photons emitted in the idler. The effect is analogous to the phenomenon of induced coherence without induced emission Zou et al. (1991); Wang et al. (1991), with the difference that in quantum optics the parametric processes use the nonlinearity of an optical crystal, while in the case of the dynamical Casimir effect the system is linear and the pump changes an electrical/optical length or a boundary condition. In the following we will not distinguish between these two cases, as they are both instances of spontaneous parametric downconversion, that is, the decay of a pump photon into a signal photon and an idler photon, triggered by vacuum fluctuations.

Motivated by these experimental advances, we study here the entanglement and the coherence of bi-squeezed tripartite Gaussian states generated by double spontaneous parametric downconversions, deriving analytical results confirmed by numerical calculations. We provide a systematic analysis of the quantum correlation properties of the aforementioned class of tripartite Gaussian states. We find that there exists genuine tripartite entanglement above a threshold value of the initial squeezing parameter as well as - and - entanglement, but no - entanglement. We also propose an experiment, similar to postselection, where we perform homodyne detection on the common idler mode, and we calculate the covariance matrix for the remaining signal modes. Surprisingly, it appears that some of the coherence between modes and is converted into entanglement after the homodyne detection of mode , providing an interesting example of redistribution of quantum resources.

We finally note that the study of tripartite systems has been first introduced in physics with the aim of understanding the fundamental quantum statistical properties of light and the nonlocal features of quantum physics, but interest in applications for the development of quantum technologies has recently witnessed a resurgence. The field is developing fast, with several experimental platforms being used recently to generate tripartite states with high efficiency: cascaded parametric downconversion setups employing two Hübel et al. (2010); Jia et al. (2012) or three separate crystals Guerreiro et al. (2014), nonlinear waveguides Krapick et al. (2016), quantum dots Khoshnegar et al. (2015), and hybrid systems comprising a Rb-85 hot atom cell and a nonlinear waveguide Ding et al. (2015). The applications include for example quantum imaging Lemos et al. (2014), interferometry Lahiri et al. (2016); Hochrainer et al. (2017a, b), and quantum computing using networks and cluster states Menicucci et al. (2008); Pysher et al. (2011); Pinel et al. (2012); Roslund et al. (2014); Chen et al. (2014). The results presented here are device-independent, therefore they can be tested on any of these experimental platforms.

The paper is organised as follows. ln Section II we introduce the essential tools from quantum optics with continuous variables as well as the covariance matrix mathematical formalism. The creation of bi-squeezed tripartite Gaussian states in systems driven parametrically by two pumps is then described in Section III. In Section IV we study the genuine multipartite entanglement generated, the bipartite entanglement across all reduced states, and the coherence properties of these states. Section V demonstrates the creation of entanglement between the signal modes under a homodyne measurement of the idler. Next, in Section VI we study a few applications of these techniques to realistic scenarios, such as experiments at low temperatures or modes with very close frequencies. We discuss our results and the perspectives of this work in a final conclusions section. For completeness, we provide details of the derivations in four appendices at the end of the paper.

We use in this work the following conventions: bold symbols stand for matrices and plain font with under scripts denote elements of vectors and matrices. In this work stands for transposition, in order to avoid confusion with temperature, denoted by , and time, denoted by .

Ii Continuous variables and Gaussian states

In this work we restrict our attention to Gaussian states of bosonic fields only. Gaussian states are a class of quantum states that enjoy remarkable properties, in particular when the transformations involved are linear unitary transformations, i.e. they are quadratic in the creation and annihilation operators Braunstein and van Loock (2005). In this case, the Gaussian character of the state is preserved and one can employ techniques from the covariance matrix formalism Adesso et al. (2014). Gaussian states of bosonic fields naturally occur in many experiments and, when applicable, offer a convenient description of the state of the electromagnetic field in the optical or microwave range. In this section we set the notations and we briefly introduce the main concepts of covariance matrix formalism, which is a powerful tool that can be used when considering unitary linear transformations between Gaussian states of bosonic fields.

ii.1 Symplectic matrices

We start by considering bosonic modes (e.g., harmonic oscillators) with annihilation and creation operators and . These operators satisfy the standard canonical commutation relations , while all other commutators vanish. It is convenient to collect all the operators and introduce the vector , where Tp stands for transpose. For example, we have or with this choice of operator ordering. We notice in passing that the techniques developed below can be extended in a straightforward fashion to an infinite number of bosonic operators. This situation occurs, for example, in quantum field theory in flat and curved spacetime Birrell and Davies (1984).

The canonical commutation relations can now be written as , where are the elements of the matrix , known as symplectic form, which has the following expression

(1)

Here are the identity matrices.

Any unitary transformation , with Hermitian generator quadratic in the creation and annihilation operators (or, equivalently, in the quadrature operators), induces a linear transformation on the (collection of) operators through the relation . The unitary operators in the expression act on each element of the vector independently and is a symplectic matrix that takes the form , where is a real function that needs to be determined, while is defined through .

The matrix is called symplectic since it satisfies or, equivalently, . We note that , see e.g. Arvind and Simon, 1995.

A symplectic matrix can always be written, with the particular choice of operator ordering in , as

(2)

The matrices and , in the case of quantum fields and curved spacetime, collect the well known Bogoliubov coefficients used extensively in literature Birrell and Davies (1984). These coefficients satisfy the well known Bogoliubov identities Birrell and Davies (1984), which read and in compact form.

We finally notice that the formal machinery introduced here is independent of the initial state of the system.

ii.2 Gaussian states

Unitary evolution and transformations, represented by a unitary operator , of bosonic systems which are initially in a state are of great importance in physics. Unitary evolution leads to a final state through the standard Heisenberg equation . If the state is a Gaussian state, and the unitary is a linear operator (see above), the Gaussian character is preserved; therefore, employing the specific results of Gaussian state formalism becomes very convenient Adesso et al. (2014).

In general, a state of bosonic modes is defined by an infinite amount of degrees of freedom. However, a Gaussian state of bosonic modes is characterised only by a finite amount of degrees of freedom. In particular, it is uniquely defined by the vector of first moments and the second moments defined by and respectively, see Adesso et al. (2014). Here, all expectation values of an operator are defined by and is the anticommutator of operators and . In this work we ignore the first moments, which can be safely set to zero without loss of generality. We make this choice since we are interested in quantum correlations, which are unaffected by the first moments. Initial vanishing moments remain zero under symplectic transformations and the second moments can be conveniently collected into the Hermitian covariance matrix . We notice that a covariance matrix represents a physical state if it satisfies in the operatorial sense Adesso et al. (2014). This amounts to computing the usual eigenvalues of the matrix and checking of they are positive.

We can now recast the Heisenberg equation into a relation between covariance matrices. Let the initial state be represented by the covariance matrix and the final state by the covariance matrix . We have already seen that any quadratic unitary can be represented by a symplectic matrix . Then, the Heisenberg equation takes the form , which reduces the problem of usually untreatable operator algebra to matrix multiplication of matrices.

Williamson’s theorem Williamson (1936a, b); Arnold (1978) guarantees that any covariance matrix can be put in diagonal form by a symplectic matrix. This means that, given a covariance matrix it is always possible to find a symplectic matrix such that , where the diagonal matrix is called the Williamson form of and are called the symplectic eigenvalues of . The symplectic eigenvalues are obtained as the eigenvalues of the matrix . The purity of the state is given by , and the state is pure if (or, equivalently, for all ).

A covariance matrix is a Hermitian matrix that can be written in the form

(3)

where the matrices and satisfy and .

ii.3 Useful properties of the covariance matrix

In this subsection we provide some useful insight on some properties enjoyed by the elements of the covariance matrices. We start by introducing the symplectic eigenvalues . These eigenvalues can be written as , where is the local temperature of the one-mode reduced state. The reason that the symplectic eigenvalues have this form results from the fact that every single-mode reduced state of a Gaussian state is a thermal state up to local operations Adesso et al. (2014). Notice that if a state is a thermal state then it coincides with its Williamson form, i.e., .

An important operation is the process of “tracing out” a particular subsystem. In this language, this operation just amounts to deleting the rows and columns corresponding to the system one wishes to trace out Adesso et al. (2014).

As a useful application, we now show how we can employ the covariance matrix to compute quantities of interest. Let be the number expectation value of mode . Without loss of generality, let us assume that the first moments vanish, i.e., . Then it is easy to show that , which highlights the role of the covariance matrix when computing physically relevant quantities.

ii.4 Entanglement in Gaussian states

The quantitative characterisation of entanglement is a central task in many areas of quantum science. For example, entanglement is at the core of quantum computation Lloyd and Braunstein (1999); Ladd et al. (2010), quantum cryptography and quantum communication Gisin et al. (2002). For two modes in general, and for Gaussian states in particular, the task has been fully solved in an unambiguous way, and two-mode entanglement has been completely characterised Adesso et al. (2014).

It has been shown that every measure of entanglement for two mode symmetric Gaussian states is a function of the smallest symplectic eigenvalue of the partial transpose Adesso et al. (2014). This establishes the PPT criterion as the paramount criterion for two-mode symmetric Gaussian states, i.e. states for which the determinants of the reduced single modes states are the same. One starts from the two mode state and obtains the partial transpose as , where the partial transposition matrix takes the form

(4)

One then computes the symplectic eigenvalues of the partial transpose as the eigenvalues of the matrix . These eigenvalues come in two pairs of identical eigenvalues and we denote the smallest one as . If then there is entanglement.

The choice of a particular measure is a matter of convenience or of the problem at hand, since all measures are (decreasing) monotonic functions of . We employ here the negativity defined as

(5)

and the logarithmic negativity defined as

(6)

We can also choose the entanglement of formation for symmetric states defined as

(7)

where we have introduced the functions for convenience of presentation.

ii.5 Coherence in Gaussian states

The role of quantum coherence in emergent quantum technologies such as quantum thermodynamics, quantum metrology or quantum biology is currently the subject of intense research – see the recent review Streltsov et al. (2016), and so far there is no uniquely accepted measure of coherence. Quantum coherence amounts to superposition with respect to a fixed orthonormal basis. A state is maximally incoherent (or mixed) if it is diagonal in the chosen basis. From here one can already see that the concept of coherence is linked to a choice of basis, therefore when using any measure of coherence one has to be specific. We choose to employ in the following two measures of coherence, a bipartite one defined operationally and based on interferometry, and a global one based on entropy. The meaning of these measures is rather different: the first one refers only to two modes and characterises what occurs if these modes are combined by a beam-splitter. The second one measures how close is the state from a maximally mixed state, thus it provides a global measure of coherence that cumulates the information about all the possible correlations.

ii.5.1 First-order bipartite quantum coherence

Given two modes and , we call the correlation (first-order) bipartite coherence, sometimes denoted by in optics Walls and Milburn (2008). This definition applies in general to any state, and therefore it can be used as well for Gaussian states. This measures corresponds to a simple interferometric setup, where we collect the photons in the modes and , add a phase difference between their paths, and let them interfere. We will witness the formation of an interference pattern only if the quantity is non-zero. This quantity can be normalized by the power in each mode, and in this case we recover the standard definition of first-order amplitude correlation function from quantum optics applied to modes and , namely

(8)

Finally, we highlight a connection with many-body physics, where one often finds useful to employ the so-called single-particle density matrix , see Penrose and Onsager (1956); Nozières (1995), defined as

(9)

The single-particle density matrix is an essential tool in the study of phase localization Castin and Dalibard (1997); Paraoanu (2008a) and fragmentation of Bose-Einstein condensates Mueller et al. (2006); Paraoanu (2008b) - where the vanishing of the off-diagonal element is used as a criterion for fragmentation (the single coherent wavefunction or order parameter associated with condensation breaks into a Fock state).

ii.5.2 Relative entropy of coherence

A measure of quantum coherence for -mode Gaussian states has been recently introduced Xu (2016) as , where is the relative entropy and is a tensor product of reduced thermal states of each mode . This measure is thus defined only in terms of the covariance matrix and displacement vectors. The von Neumann entropy of the system in terms of the symplectic eigenvalues is given by:

(10)

where and are the symplectic eigenvalues of , while the mean occupation value is:

(11)

Here and are the -th element of the reduced correlation matrix and the first statistical moment of the mode, respectively. In this work, the latter will always be equal to 0. It is possible to obtain an analytical expression in closed form Xu (2016):

(12)

ii.6 An example: Two-mode squeezing

To get a clear picture of the covariant matrix formalism, let us consider an useful example, that of two-mode squeezing. In the experiments discussed further, two-mode squeezed states are produced by a single parametric process, i.e. by the action of each pump acting separately. Let . The unitary operator that implements two mode squeezing is and it is easy to show that in this (simplified) case its symplectic representation is

(13)

Notice that we have chosen a special case where the transformation is real, for the sake of simplicity and without any loss of generality. We can define the vector of new operators. The two mode squeezing transformation reduces to its well-known form

(14)

In the usual Fock state formalism, the two-mode squeezed state of two modes and has the form

(15)

In the covariance matrix formalism, we can easily see that the two-mode squeezed state (15) takes the form

(16)

This explicitly shows how Gaussian states in the Fock state formalism reduce to simple matrices in the Gaussian state formalism. In particular, simple analytical formulae are known for calculating fidelities Paraoanu and Scutaru (2000) and distance measures Paraoanu and Scutaru (1998).

We now proceed and compute the spectrum of the matrix , which is the set of the symplectic eigenvalues of the partial transpose of the state (16). It is easy to show that they are . We see that the smallest symplectic eigenvalue has the expression . This implies that the logarithimic negativity reads , see Ohliger et al. (2010). The coherence for two-mode squeezed states can be calculated as well. In the case of the interference-based bipartite coherence, we find that , which is a consequence of the peculiar structure of the state in the number basis Eq. (15). The entropy of coherence however gives a non-zero result which grows monotonically with :

(17)

Once more, this underlines the power of the covariance matrix formalism, where simple analytical expressions can be obtained for the relevant quantities.

Iii Generation of bi-squeezed tri-partite Gaussian states

We now move to the physical system of interest, see Fig. 1. This consists of three bosonic modes with frequencies respectively. The three modes are modulated parametrically by two pump fields at the frequencies and . Systems of this type have been studied experimentally, both in the optical and in the microwave frequency range. To encompass all the physical realizations, we use a semi-abstract, device-independent representation Krenn et al. (2016) which shows the mode as common to two parametric processes and occurring in a parametrically device pumped at and . In practice, this can be realized by overlapping the paths of the idler photons of two different optical crystals, by using a single nonlinear crystal in a multimode cavity (e.g. bismuth borate (BIBO) in a ring cavity Roslund et al. (2014), periodically poled KTiOPO(PPKTP) with zzz quasi-phase- matching Chen et al. (2014); Pysher et al. (2011)), or a single superconducting resonator with double-modulated electrical length Lähteenmäki et al. (2016).

Figure 1: Generic representation of the system of interest and the mesurement configuration. We use a semi-abstract representation for the parametric generation of photons in the three modes , , and . This can be realized using either one resonator with mode as idler common to the two pumps and or two different nonlinear crystals with idlers aligned with each other. Each mode can be measured by a homodyne detection scheme using oscillators , , . One can either measure the 3x3 correlation matrix or one can perform only a -quadrature measurement in mode , to be left with a state .

The total Hamiltonian for this type of configuration can be constructed by adding to the harmonic-oscillator Hamiltonian of the three modes, two parametric perturbations corresponding to each pump Walls and Milburn (2008). One obtains the total Hamiltonian in the form

(18)

Here and have dimensions of frequencies and describe the parametric coupling of the pumping fields into the modes , representing the parametric analog of the Rabi frequency of driven two-level systems. This prescription is very general, irrespective to the particular physical system employed or to weather the modulation is done on the boundary conditions or in the bulk of the material or device (see e.g. Supporting Information in Lähteenmäki et al. (2013) and Johansson et al. (2010)). Consider now the unitary transformation . Assuming that the frequencies of the pumps and are chosen such that the energy conservation conditions and are satisfied, the Hamiltonian (18) can be transformed into , where we have defined

(19)

The effective Hamiltonian is now time-independent and describes the evolution of the system in a triple rotating frame (with frequencies ). Suppose now that the system is pumped for a finite time , as it was done in the time-domain experiments in previous work Lähteenmäki et al. (2016). Then, introducing the two mode squeezing operators and and the corresponding two-mode squeezing parameters and we get

(20)

Here we implicitly assume that the resonators or cavities have a high enough quality factor, ensuring that absolute values of the parametric coupling is larger than the decay rate. Also, after preparation, the measurement is realized on a timescale smaller than the relaxation time. These conditions are easily met in the present optical or superconducting-circuits setups. For example, in the experiments realized with a SQUID-based modulated resonator with decay rate of 1 MHz Lähteenmäki et al. (2016) the correlations were measured in time-domain, under the double parametric excitation of the system with 1 s microwave pulses. Thus, for this system the conditions above are easily satisfies for s. These results can be readily extended to larger timescales and the signal can be enhanced with the use of higher-Q resonators. Moreover, we emphasize that the same structure comprising two two-mode squeezing operators can be recovered also in frequency space for continuous pumping of systems with dissipation in the input-output formalism Lähteenmäki et al. (2016).

To investigate systematically the correlations induced by this operator, we collect the creation and annihilation operators of these modes in the vector . We assume that the initial state is a thermal state at temperature , since temperature is always present in any real system. As mentioned before, in this case the state of the system coincides with its Williamson form, i.e., .

Next, we proceed to construct the final state of interest , represented by the covariance matrix , that we obtain by applying the operator to the thermal state . For simplicity, we assume that the squeezing parameters and are both real. This is not a loss of generality: indeed, if the pumps have nonzero phases, , we can obtain the evolution from Eq. 20 with real and by redefining , , such that and .

In general, it is possible to compute the state the symplectic matrix representing the operator (20) in the Fock state formalism. However, the results can be extremely difficult to manage analytically.

Here we use a recently-developed technique Bruschi et al. (2013a); Moore and Bruschi (2016) (see also Brown et al. (2013) for an alternative approach) to obtain a more convenient representation of the operator (20), based on the Lie algebra structure of the group Arnold (1978). In Appendix A we show that it is possible to re-write the operator (20) as

(21)

where the real squeezing parameters and phase have the exact expression

(22)

as functions of the new parameters and . Here is a beam-splitter transformation. The result is remarkable, because in general it is not possible to obtain simple analytical solutions when trying to factorize an exponential of multiple-mode operators using the well-known Hausdorff-Baker-Campbell approach to decoupling exponentials. We emphasise that the unitary operators (20) and (21) are equivalent, and the final state obtained under their action is also the same.

We note that an alternative technique to decouple Eq. (20) has been developed Braunstein (2005). This yields a global passive operation, followed by a set of single mode squeezers, followed by another global passive transformation. Differently from this, the decomposition (21) comprises a series of two mode squeezers, which also gives a direct operational meaning as a sequence achievable in experiments. Specifically, the factorized representation (21) can be used as well as a heuristic tool in designing novel experiments, since the bi-squeezed Gaussian state obtained by double parametric pumping can be created also by pumping first one pair of modes, then another pair, and finally performing a beam-splitter transformation. For example, in quantum optics it might be convenient to even use two different crystals for realizing the two squeezing operations.

In order to obtain the correlation matrix, we start by applying a beam splitting on modes and which, in symplectic geometry, has the form

(23)

Notice that this would be a trivial operation if the state was the vacuum; however, the initial state is thermal and the beamsplitter can have a non-trivial effect.

We proceed by applying a two mode squeezing on modes and and a two mode squeezing on modes and . These have the form

(24)

and

(25)

where we have introduced , and , for compactness and simplicity of presentation.

The final state, when acting on the thermal state as well as all the reduced states, can be obtained analytically, see Appendix B for the full expressions of each matrix element. Here we report only the structure of these states, which is essential for the ensuing calculations. The three-mode state reads

(26)

The final two-mode and single-mode states read

(27)

The reduced two-modes and single-mode covariance matrices were obtained using the trace-out prescription from Section II C, namely eliminating one, and respectively two, modes from the three-mode matrix (26). Finally, notice that all reduced single mode states are thermal.

Iv Characterising bi-squeezed tripartite Gaussian states

In this section we present a full description of the bi-squeezed tripartite Gaussian states create by the double parametric pumping described in the previous section. In particular, we focus on the entanglement properties, showing that the state has so-called genuine tripartite entanglement, and on the phenomenon of induced coherence between modes and due to the indistinguishability of the photons in the common idler .

iv.1 Number expectation values

We can now turn to computing the final number expectation value for all three modes. We have

(28)

iv.2 Purity of all reduced states

We wish to understand the correlation structure of the whole state. A rough understanding can be already given by computing the purity of all the reduced states.

Let us start with the purity of the initial global tri-partite thermal state, which remains unchanged under our unitary transformations. We then list the initial purities of the thermal state which read

(29)

We now find that the purities of all reduced states of our given state are

(30)

We see that local purities have changed from the values in (IV.2) to the ones in (IV.2), therefore we expect some correlations between the different modes. We proceed to study this in the next section.

iv.3 Tripartite entanglement

Here we look at the nature of (quantum) correlations in the tripartite state of interest in this work. We study the global (genuine) correlations as well as the bipartite correlations across all bipartite reduced states.

A measure of the tripartite entanglement can be obtained through a suitable average of the entanglement of all the bipartitions of the system. For instance, we can consider the tripartite negativity defined by

(31)

where is the negativity of the bipartition as provided by the partial transposition with respect to the mode . In Fig. 2 we plot all the and the resulting .

Figure 2: Tripartite negativity , , and vs. squeezing parameter for , , and .

We notice that there is need for a certain amount of squeezing before genuine multipartite entanglement can be created.

iv.4 Bipartite entanglement and coherence

iv.4.1 Bipartite entanglement in the “” and “” subsystems

As previously discussed, we now need to compute the smallest symplectic eigenvalue for each reduced state. This eigenvalue provide us with a quantification of entanglement.

We start by the reduced state of modes and . We can compute the smallest symplectic eigenvalue of the partial transpose and we find

(32)

Similarly, for the reduced state of modes and we can compute the smallest symplectic eigenvalue of the partial transpose,

(33)

These eigenvalue can now be used, together with Eqs. (5) and (7), to compute the negativities , as well as the entanglement of formation , in the reduced states respectively .

iv.4.2 Bipartite entanglement in the “” subsystem

Next, we calculate the reduced state of modes and . We can compute the smallest symplectic eigenvalue of the partial transpose and we find

(34)

It is easy to show that . This, in turn can be written as , where is the purity of the final reduced state. Since the purity of any state, in this language, satisfies we conclude that

(35)

which implies that there can never be any entanglement between the modes and , as expected.

The bipartite negativities between modes i and j, where are plotted in Fig. (3). We observe that while is different from 0 and grows with the initial squeezing, the negativity is 0 for any value of the initial squeezing - as expected. Next, we proceed to study the issue of coherence.

iv.4.3 Bipartite coherence and relative entropy of coherence

We are now ready to discuss some peculiar aspects of the “ac” subsystem. We proceed to show that, although the modes a and c have not been directly squeezed and therefore there is no entanglement between them, we still witness the appearance of nontrivial bipartite coherence correlations . This term can be obtained in a simple way as . We find

(36)

The mechanism by which these correlations are established reminds of the standard which-way information concepts from interferometry. In standard interferometry (or each time we deal with a linear superposition of states) the absence of information about the path that the photon takes (or, equivalently, the information about which specific wave-function within the superposition that constitutes tho total wave-function of the particle is “actualized”) results in the formation of an interference pattern. In this case, given a boson occupying mode , we cannot know from which downconversion process (corresponding to either pump or ) it originates. The first-order coherence of modes , can be readily obtained,

(37)

and using and as determined from (26) we get

(38)

Finally, the single-particle density matrix is

(39)

From Fig. (3) we see that a nonzero degree of bipartite coherence between and exists, and it increases with the squeezing. Also the relative entropy of coherence in can be calculated from Eqs. (10-12), and this quantity is nonvanishing as well.

This underlines the fundamental difference between the two-mode correlations produced by a single pump (which produce entanglement but no bipartite coherence) and those produced between the extremal modes in the double dynamical Casimir effect (which have coherence but no entanglement).

Figure 3: Bipartite negativities of the reduced states , , , coherence measurement and interference-based coherence of the a-c reduced state vs. squeezing parameter for , , and . Note that due to squeezing in each parametric process separately, while .

V Homodyne measurement of the common idler and coherence-to-entanglement conversion

In this section we compute the resulting state of modes and after a perfect homodyne detection of mode . In particular, we analyse the coherence and entanglement of the resulting state. In order to reach this goal, we employ the formalism of homodyne detection developed in Ref. [Spedalieri et al., 2013]. The technical details of this can be found in Appendix C and we omit them here in order to focus on correlations between modes and after homodyne detection.

After some lengthy algebra (see also Giedke and Ignacio Cirac (2002)), one has the final state of modes and after homodyne detection of the quadrature of mode , which reads

(40)

A simple inspection of the state (40) after homodyne detection allows the identification of the differences with respect to the reduced state . We see that the coherence of the latter is now decreased. In particular, we expect entanglement to be present in the new state (40) since it contains nonvanishing elements in the upper-right part of the state. To prove analytically that there is entanglement requires lengthy formulas, but we will see later that in the case of frequencies that are very close to each other, analytical insight can be gained. The smallest symplectic eigenvalue of the partial transpose of the state (40) can be computed and has a lengthy expression, which we reproduce in Appendix D.

The computation of the negativity of the state (40) after homodyne detection is straightforward and follows step-by-step what has been done above.

In Figure 4 we plot the negativity , the bipartite coherence , and the entropy of coherence as a function of the squeezing parameter for typical values of frequencies and temperature encountered in experiments with superconducting circuits. The result is that after this projective measurement of quadrature in the common idler mode , the entanglement becomes nonzero at the expense of a reduction in bipartite coherence. In other words the state has become more squeezed in the two modes but has lost in interferometric visibility. In Figure 5 we also compare with the other elements of the correlation matrix corresponding to the state.

Figure 4: Negativity , bipartite coherence , and entropy of coherence versus squeezing parameter of the state for , , and .
Figure 5: Elements of the state as a function of the squeezing parameter, for the same parameters as in the previous figures. Here we have and = =, .

Note that the entropy of coherence also changes, but not significantly. This occurs because the entropy of coherence is a measure of the total coherence-like resources available (it quantifies globally how different the state is from a mixed state) but it doesn’t contain any information about how these resources are distributed between the states forming the basis (the number of particle eigenstates). In contrast, the bipartite coherence concentrates on only one resource, the visibility in an interference experiment realized with two modes, and ignores the information about the other possible correlations.

The result can be regarded as a phenomenon of redistribution of tripartite quantum resources into a bipartite resource (bipartite entanglement), when one mode is eliminated by measurement. Similar effects have been studied before with discrete variables in spin chains, where entanglement can be localized between two spins by the measurement of the other spins Verstraete et al. (2004). With superconducting circuits, in a tripartite system consisting of two superconducting qubits coupled to a resonator with only one quanta of excitation present, it was shown that a null measurement on the number of particles in the resonator results in the creation of entanglement between the qubits Li et al. (2008, 2009), a technique that can be thought of as a particular form of dissipation engineering Plenio et al. (1999).

To get a better grasp of this, let us consider a W state, which is a tripartite state with one excitation distributed in an equal superposition over three modes, . These states display a different type of entanglement from GHZ states Greenberger et al. (1990), and their nonlocal features have been studied intensely Paraoanu (2011). It is easy to check that bi-squeezed tripartite states truncated to the subspace of at most two excitations have indeed a W structure Lähteenmäki et al. (2016). If any of the modes is traced out, the remaining two modes are entangled with concurrence 2/3. Consider now a measurement on any of the three modes: if the result is , then we have projected the W state into a maximally entangled Bell state