The linear time-dependent constants of motion of the parametric amplifier are obtained and used to determine in the tomographic-probability representation the evolution of a general two-mode Gaussian state. By means of the discretization of the continuous variable density matrix, the von Neumann and linear entropies are calculated to measure the entanglement properties between the modes of the amplifier. The obtained results for the nonlocal correlations are compared with those associated to a linear map of discretized symplectic Gaussian-state tomogram onto a qubit tomogram. This qubit portrait procedure is used to establish Bell-type’s inequalities, which provide a necessary condition to determine the separability of quantum states, which can be evaluated through homodyne detection. Other no-signaling nonlocal correlations are defined through the portrait procedure for noncomposite systems.
|EVOLUTION AND ENTANGLEMENT OF GAUSSIAN STATES|
|IN THE PARAMETRIC AMPLIFIER|
Julio A. López-Saldívar***Julio A. López-Saldívar e-mail:email@example.com, Armando Figueroa, Octavio Castaños,
Ramón López-Peña Margarita A. Man’ko, Vladimir I. Man’ko
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apdo. Postal 70-543 México 04510 D.F.
P. N. Lebedev Physical Institute, Leninskii Prospect, 53, Moscow 119991, Russia
Moscow Institute of Physics and Technology (State University) Dolgoprudnyi, Moscow Region 141700, Russia
Keywords: parametric amplifier, quantum entanglement, Gaussian states, Bell inequalities, probability representation.
The linear time-dependent invariants of multidimensional quadratic Hamiltonians in the position and momentum operators have been the subject of many research studies [1, 2, 3, 4, 5]. These constants of motion are useful to determine the propagators of the Hamiltonian systems and thus to study time-dependent problems in quantum mechanics. It has been shown that these propagators can be also obtained via the path integral formulation of quantum mechanics .
There are many specific systems of physical interest described by quadratic Hamiltonians, such as the parametric amplifier [7, 8], and others used in circuit electrodynamics based on the Josephson junction technique [9, 10, 11].
The tomographic probability representation introduced in 1996 is a generalization of the optical tomography scheme . In this new formulation of quantum mechanics, the states are described by measurable positive probabilities.
In 1935, EPR and Schrödinger defined the entanglement concept as a strange phenomena [13, 14]. In the sixties, Bell  and Clauser–Horne–Shimony–Holt (CHSH)  established that the predictions of quantum theory cannot be accounted by any local theory. In a typical Bell experiment , it has been established an inequality valid for separable states of a bipartite system with the bound 2. An experimental check of the violation of the Bell–CHSH inequality with the bound 2 was first performed in . Since the nineties the entanglement properties of a quantum system have been used as a resource to do tasks as quantum cryptography, quantum teleportation, and measurements in quantum computation. These facts lead to the growing interest in the identification of the quantum correlations or in the non-local behaviour of the quantum physics [17, 19]. The bound 2 of the Bell–CHSH inequality can be violated for entangled states of a composite system, which exhibit strong correlations of the subsystems. The paradigmatic upper bound of the CHSH inequality was proved by Cirelson  for quantum correlations in a bipartite system. On the other hand, formally there exists even an upper bound 4 discussed in , which corresponds to superquantum correlations in the systems modeling some properties of two-qubit states. A tomographic approach to test the nonlocality was proposed for the description of a correlated two-mode quantum state of light . This proposal was implemented in  using a balanced homodyne detection on temporal modes of light, in which a clear violation of Bell’s inequality was found.
A qubit-portrait scheme of qudit tomograms has been proposed in , which allows one to discuss the Bell–CHSH inequality [15, 16, 25] for two qubits within the framework of the probability representation of quantum mechanics introduced in [12, 26]. It was also proposed that the necessary condition of separability of a bipartite qudit state is the separability of its qubit portrait . This portrait method can be extended to photon-number tomograms with some modifications, which is useful to detect entanglement of two-mode light states. Specifically, the violation of the Bell–CHSH inequality indicates immediately that the state is entangled .
The models of the parametric amplifier and the frequency converter were proposed in  for two modes of the electromagnetic field, which represent the signal and idler photons. These harmonic oscillator modes are coupled by a classical field of frequency (referred as pump frequency) which may (or may not) satisfy the resonance condition [7, 29, 30], that is, for the parametric amplifier or for the frequency converter.
The model Hamiltonian  contains the main elements for the description of physical realizations of the parametric amplifier. As an example, we have a lossless nonlinear dielectric substance that couples the modes of a resonant cavity with reflecting walls . In this case, there is a pump field oscillating with a frequency equal to the sum of the frequencies of the two modes, and it is strong enough to be represented in classical terms. The two-mode nonresonant parametric amplifier has been studied to find nonclassical features as revivals and squeezing in , where it was shown that the existence of quantum revivals is possible and the correlation effects are very sensitive to the form of the initial state of the system.
Recently, fiber optical parametric amplifiers have been developed with a total amplification of 60–70 dB over an input signal [31, 32]. Also the dynamics of the entanglement of Gaussian states of systems in a reservoir model has been studied in [33, 34]. The properties of superpositions of coherent states are presented in [35, 36]. A new resonant condition technique has been developed to determine experimentally the quadrature fluctuations of the light field, that is, the covariance matrix [37, 38, 39]. Thus, the operation of the parametric amplifier is sufficiently well understood. Nevertheless, the dynamical invariants of this system were not discussed in the literature, and one of the aims of this paper is to obtain and apply the special linear-in-quadrature time-dependent constants of motion of this amplifier.
In this work, we construct the linear time-dependent constants of motion of the nondegenerated parametric amplifier for the trigonometric and hyperbolic cases. Using these constants of motion we calculate, for the first time, the amplifier symplectic tomograms associated to the dynamics of Gaussian states. The other goal of this article is to propose a discretization method of the amplifier density matrices, which allow us to evaluate the von Neumann and linear entropies to measure the entanglement between the idler and signal modes of the amplifier. A portrait map of the symplectic tomogram onto a qubit is defined to explore possible sufficient conditions to have entanglement. For a noncomposite system, through the portrait picture, we show the possible existence of superquantum correlations i.e., the violation of the Cirelson bound.
The results presented could be of interest mainly in the context of theoretical and experimental researches in the fields of the theory of entanglement and Bell non-locality.
This paper is organized as follows.
The Hamiltonian for the parametric amplifier is defined in section 2, together with their analytic solution (trigonometric case) in terms of the corresponding linear time-dependent invariants. They are constructed for special conditions on the Hamiltonian parameters, and the other solution (hyperbolic case) can be obtained by analytic continuation. In section 3, the two-mode Green function is written in terms of the time-dependent symplectic matrix () and the corresponding two-mode Gaussian states at time are presented. The evolution of the covariance matrices are also calculated. The symplectic and optical tomograms are described in section 4, making use of the covariance matrices. In section 5, the von Neumann and linear entropies for the two-mode Gaussian state in the parametric amplifier are calculated using the discrete form of the two-mode density matrix and the corresponding reduced density matrix for the subsystem. The portrait of continuous symplectic and optical tomograms in the form of a two-qubit tomogram is determined in section 6. This portrait is used to define Bell-type inequalities that is a sufficient condition for separability, and a violation of this inequality is a necessary condition to entanglement. Also in this section non-Bell correlations within a noncomposite system are studied obtaining strong correlations even larger that the Cirelson bound . In the final section, the conclusions are given, and some technical details are presented in Appendix A.
2 Hamiltonian and linear dynamical invariants for the parametric amplifier
The model of the parametric amplifier assumes that the two modes are described by harmonic oscillators of frequencies and . The two modes are coupled by an oscillating parameter called the pump with frequency , which may (or may not) satisfy the parametric resonance condition [29, 30, 7]. Therefore, the Hamiltonian can be written as
where , with being the group velocity of the light in the medium, is the medium nonlinearity and is the intensity of the pump . The sets and define the photon creation and annihilation operators for the two different electromagnetic modes. The parameter is the coupling constant between the electromagnetic modes. Notice that the operator is a constant of motion for the system.
The solution of the model Hamiltonian can be obtained by means of several procedures: either by the use of the Heisenberg equations of motion for the creation and annihilation operators, or by the use of the interaction picture together with the Wei–Norman procedure [28, 29, 41]. In this work, we use the construction of the linear time-dependent invariants of the system .
To obtain the linear in operators , , , constants of motion for the parametric amplifier, one should solve a system of differential equations given in Appendix A; their solution reads
which, together with the corresponding creation operators and , satisfy the commutation relations of boson operators with all others commutators equal to zero. We define , the detuning and which we take as a real number. In the case of parametric resonance, . This system also has a solution for the invariants where is real; in that case, the form of the invariants can be obtained by analytic continuation, i.e., changing in Eq. (2) and substituting trigonometric by hyperbolic functions, an analogous solution has been obtained for Heisenberg operators in .
At multiples of the time , the invariants take the form
i.e., they are equal to the original boson operators multiplied by a phase.
Establishing the relation between the creation and annihilation invariant operators (, , , ) with the momentum and position quadrature operators (, , and ) representing the electric and magnetic fields in the corresponding modes, one gets the matrix relation
where , , , and denote column vectors, while defines a symplectic matrix in four dimensions that satisfies the relation , with the definition
where the denotes a identity matrix. Here, , are the constants of motion with initial conditions and . For the parametric amplifier, the block matrices for in units where , take the following form:
We point out that at the times , with an integer, the invariants (, ) for modes and are determined by a rotation of the original quadrature operators and of the same mode; the angles of these rotations are () and (), respectively, i.e.,
with . These expressions are local transformations of the original quadrature operators. Then we can expect that the entanglement at these times are equal to the entanglement at , independently of the initial state.
3 Time evolution of two-mode Gaussian states
In this section, the evolution of a two-dimensional Gaussian packet in a parametric amplifier is studied. The time evolution of this state is obtained by means of the Green function 
where and are column vectors, the means the matrix transposition of , and the matrices were defined in the previous section. The Green function of the amplifier is also a Gaussian function of the two coordinates in the system, implying that the evolution of a Gaussian state will be also a Gaussian state.
The general two-mode Gaussian state is defined by 
with the normalization constant, ; is the transpose vector of , , and we took the matrix as real and symmetric,
Using the propagator in Eq. (5), the time evolution for the wave function is calculated in the integral form
giving the result for any quadratic Hamiltonian in the quadrature components of the electromagnetic field:
this expression will be used to calculate the density matrix and the reduced density matrix for one mode in order to calculate the von Neumann and linear entropies.
3.1 Covariance matrix
The evolution of the Gaussian in the parametric amplifier (even for any quadratic Hamiltonian) is also a Gaussian state as seen in the previous section. Any Gaussian state is completely determined by the covariance matrix and the mean values of the position. The covariance matrix of the general Gaussian state is calculated using the constants of motion. The covariances and dispersions between the quadrature components in the two-mode system can be put in the matrix form as
with the usual definition , in terms of the anti-commutator of two operators.
The covariance matrix at can be defined by
with , , and denoting the transpose of ; they satisfy , with , and .
By means of the expressions of the quadrature components in terms of the linear time-dependent invariants, i.e., the inverse of equation (3), it is straightforward to evaluate the covariance matrix at time by the expression
notice that . The inverse of the symplectic matrix is given by
Thus, the covariance matrix at time can be calculated for any quadratic Hamiltonian and takes the form
where, for simplicity of the notation, we do not express the time dependence in the matrices . The covariance matrix at time is given by the 22 matrices , and . The covariance matrix at time is calculated through Eq. (13) together with Eq. (4).
Some examples of Gaussian states are given: One can obtain a two-mode initial squeezed vacuum state making the substitution
where is called the squeezing parameter. This state is the result of the application of the two-mode squeeze operator to the vacuum state . This state can be written explicitly as with . For this evolving initial state, the block matrices for the covariance matrix at time can be calculated using formula (13), obtaining
This covariance matrix corresponds to the squeezed vacuum state: with the squeeze parameter given by
with . One can check that for , .
The two-mode coherent state can be expressed by taking
with and being the complex parameters for each mode. The initial coherent state can be obtained in terms of the translation operator applied to the vacuum state. The evolving coherent state has the following covariance matrix
the covariance matrix of the system also can be used to study the so-called tomographic representation of a Gaussian state determined by the symplectic or optical tomogram. One can note that the covariance matrix for the coherent state is a periodic function, with period while this behavior is not present in the squeezed state.
4 Symplectic and optical tomograms
The two-mode symplectic tomographic distribution describes the probability in the quadratures of the system in a rotated and rescaled reference frame of the original quadratures , with the definition
where , . This distribution, called the symplectic tomogram of two-mode system state, is nonnegative and normalized, i.e.,
When a pure state is non-entangled, the tomogram can be expressed as the multiplication of the distributions for each one of the modes 
when and are the partial (also called reduced) tomograms for the modes one and two and are defined as
This condition can be used to distinguish an entangled state and will be discussed later.
The optical tomogram is related to the symplectic tomogram:
which measures the quadratures in a rotated reference frame. Thus, all the information of a quantum state is contained in the optical (or symplectic) tomogram. The importance of the optical tomogram consists in the fact that it can be obtained through homodyne measurements for various systems .
If the two-mode state is a Gaussian one, the symplectic tomogram is described by a normal probability distribution:
and the dispersion matrix reads
The mean values and are expressed in terms of the corresponding expectation values of the quadrature components of both modes:
The dispersions and covariance are
The Gaussian state is completely determined by the covariance matrix and the mean values of the homodyne quadratures also in the tomographic representation. The time evolution in the Hamiltonian (1) of a Gaussian state is also a Gaussian state, as it can be seen in Eq. (8), so the time evolution of the tomogram in this system is given by Eq. (22).
The time-dependent functions: , , , , , , , , , are calculated in terms of the corresponding wave function in the standard form. Therefore, to calculate the optical and symplectic tomograms, we have to use the corresponding matrices and for the different initial states considered in this work, of course, by substituting properly the matrices with carrying the information of the evolution under the parametric amplifier.
In Fig. 1, the evolution of the tomogram for the squeezed vacuum state in the parametric amplifier is presented. The figure shows that at times and the tomogram is not the same as the one at ; in fact, the covariance matrix at those times is different. Although the entanglement properties are the same at those times. Making use of the squeeze parameter , one can check that , implying that the state and its covariance matrix are different.
In Fig. 2, the tomograms for the coherent state are displayed. The center of the wave packet moves according to the mean values of the position operators but the shape of the tomogram is the same at times and . Additionally, one can calculate the correlation between the two variables and represented by , and it is zero at those times.
5 Discretization of the density matrix, von Neumann and linear entropies
The entanglement between two input modes in the parametric amplifier has been shown to exist through different methods, as second order correlations [49, 7] and the calculation of the Duan et al. criterion . The entanglement between two modes in a symmetric Gaussian state have been described through EPR inequalities and the von Neumann entropy in . The entanglement present in the parametric amplifier has been used in quantum metrology  and entanglement swapping . Also the entanglement in other parametric processes, related to the amplification as the parametric oscillation is frequently used to generate correlated light [54, 55, 56] and implement quantum information protocols [57, 58]. The von Neumann and linear entropies are generally used to measure the entanglement between the modes of a bipartite pure system. However the analytic calculation of these quantities is, in general, not an easy matter. In this section, we provide a numerical method to calculate both entropies for a continuous variable density matrix using a discrete form. This method also can be used to define a positive map between the density matrix and other sub-matrices similar to the reduced density matrices that retain information on the entanglement of the system.
In order to compare the results and to evaluate the entanglement given by the tomographic representation discussed later, we calculate the von Neumann and linear entropies for the two-variable system.
The density matrix resulting from the time evolution of the Gaussian state described by Eq. (8) is a continuous function of the coordinates
To determine the entanglement properties from a continuous variable density matrix, the discrete form of the density operator is made. Let us take four sets of discrete numbers along the axis that define the density matrix variables, that is,
where the size of the steps is and . One can notice that, to define properly the transpose matrix, one should take the same number of elements for the coordinates and and, for simplicity, let us choose the same step between them: and . These partitions must be chosen to guarantee the normalization condition of the density matrix.
Therefore, the discrete two-mode density matrix can be expressed as
where the normalization condition is expressed in the form
Then, the corresponding definition of the partial density matrix of the mode 1 is given by
obtaining the eigenvalues of the reduced density matrix provide us with a method to calculate either the linear or the von Neumann entropies.
The eigenvectors and eigenvalues are obtained by solving the standard eigenvalue equation for the matrix . Denoting the corresponding eigenvalues as , the linear and von Neumann entropies can be calculated as
For the squeezed vacuum state the linear entropy is given by
with given by Eq. (16). Similarly, for the von Neumann entropy, one gets
In Fig. 3, we compare the analytic and numerical results for the von Neumann and linear entropies for a squeezed vacuum state evolving in the parametric amplifier. The difference between the analytic and numerical results has a maximum value of . In this figure, one can see that the same entanglement is obtained for times and , as both entropies show a periodic behavior with period .
Also in Fig 4, the von Neumann and linear entropies for the coherent state and a particular Gaussian state are presented. We have used the values , , , , , , , and . We can observe again an oscillatory behavior of period . Also one can see that in the coherent state the two modes initially are not entangled but the evolution in the parametric amplifier makes these two modes entangled.
The oscillation of the entropies imply that there is entanglement between the two Gaussian modes even when initially are not entangled due to the evolution in the parametric amplifier. This entanglement has a maximum value at times for odd and has a minimum value at times with even. We note that the minimum value for the entanglement is equal to the initial entropy, so given an initial entangled state the evolution in the amplifier can increase the entanglement between the modes.
6 Qubit portrait of symplectic tomograms
In this section, the qubit portrait of a symplectic tomogram is defined and calculated, in particular, for two-mode pure Gaussian states evolving in the parametric amplifier. This qubit portrait is the reduction of a symplectic or optical tomograms to a component probability vector, which for simply separable states is the tensor product of two-dimensional probability vectors.
This qubit portrait can be used to determine a Bell-type inequality where the violation of the parameter indicates that the bipartite system cannot be separable, i.e., it is entangled. Now, the converse statement is not true.
The spin tomograms for time-dependent Hamiltonians linear in spin variables have been constructed in . These tomograms have been also used to study qubits and qudits within the quantum information context of separable and entangled states . To define the qubit portrait for a continuous variable system, we generalize the idea developed for the spin tomogram, for which a qubit portrait of qudit states and Bell-type inequalities have been proposed in .
The spin states can be described by a probability distribution called spin tomogram denoted by , where is the projection in the direction . In general, the tomogram of a -dimensional qudit system has components corresponding to the different projections of the angular momentum operator. The qubit portrait is defined as the reduction of the tomogram of a qudit system to a two-qubit tomogram. To construct the portrait, we reduce the components to only 4; to make this, we construct 4 arbitrary sets of this components and sum them.
In an analogous form, one can reduce all the information contained in the symplectic or optical tomograms to numbers related with the probabilities to find the quadrature components and into the integration regions (), that do not overlap ; and the union of all these regions is equal to the complete two-dimensional space . Then each components of the four-dimensional probability vector will be given by
with and .
Then, we define a stochastic matrix as follows:
Here, each column vector specify the two-dimensional coordinate system where the measurements of the position operators are realized. Each one of them satisfy . It can be shown that in the case of a simply separable state, the matrix can be written as the direct product of two subsystems. In this case, one can define a Bell-type inequality.
Let us consider two stochastic matrices ,, and their tensor product
It is straightforward to show that the matrix elements of satisfy the Laplace equation