Entangled State vs Product State and Classically Correlated State

(1) Product state:

In general, a product state describes the behavior of a system composed of two or more independent sub-systems. Usually, the measurement of a product state involves the jointly- detection of two or more independent particles, such as the coincidence measures between two or more particle counting detectors within a certain time window. The experimental setup is similar to that of the EPR correlation measurement, however, the measurement only deals with independent particles. The independent particle system can be characterized as a product state, i.e., the state of the system is factorizable into a product of states of two or more sub-systems. For example

The density matrix p in Eq. (9.2.1) characterizes a quantum system composed of two independent sub-systems of p and p. In Eq. (9.2.1), each independent sub-system can be in any state. The simplest case is a product of two pure states:

where {|o)} and {)} are two sets of orthogonal vectors for subsystems 1 and 2, respectively, and

(2) Entangled state:

Differing from product states, entangled states describe the behavior of entangled quantum systems. The entangled two-particle state was mathematically formulated by Schrodinger. Consider a pure state for a system composed of two spatially separated subsystems,

where, again, {|o)} and (|f>)} are two sets of orthogonal vectors for subsystems 1 and 2, respectively, and p is the density matrix. If c(a, b) does not factor into a product of the form f(a) x g(b), then it follows that the state does not factor into a product state for subsystems 1 and 2:

the state was defined by Schrodinger as the entangled state.

Following this notation, the first classic example of a two-particle entangled state, the EPR state of Eq. (8.9.1) and Eq. (8.9.2), is thus written as:

where we have described the EPR entangled system as the coherent superposition of the momentum eigenstates as well as the coherent superposition of the position eigenstates of each particle. One clear property of the EPR state is its independence of vector bases. The two ^-functions in Eq. (9.2.5) represent, respectively and simultaneously, the perfect position-position correlation and moment urn-moment urn correlation. Although the two distant particles are interaction-free, the superposition selects only the eigenstates which are specified by the (5-function. We may use the following statement to summarize the surprising feature of the EPR state: the values of the momentum and the position for neither interaction-free single subsystem is determinate. However, if one of the subsystems is measured to be at a certain value of momentum and/or position, the momentum and/or position of the other one is 100% determined, despite the distance between them.

It is necessary to emphasize that Eq. (9.2.5) is true, simultaneously, in the conjugate momentum and position space.

(3) Classically correlated state:

There exist many different types of classically correlated states. The following are two typical classically correlated states that have been used to simulate the EPR state of Eq. (9.2.5):

or

Differing from entangled states, Eq. (9.2.6) and Ecj. (9.2.7) cannot be simultaneously true for one system as we have discussed earlier.

(4) Entangled states in spin variables:

A different classic example of entangled two-particle system was suggested by Bohm. Instead of using continuous space-time variables, Bohm simplified the entangled two-particle state to discrete spin variables. EPR-Bohm state is a singlet state of two spin 1/2 particles:

where the kets |t) and |f) represent states of spin “up” and spin “down”, respectively, along an arbitrary direction. Again, for this state, the spin of neither particle is determined; however, if one particle is measured to be spin up along a certain direction, the other one must be spin down along that direction, despite the distance between the two spin 1/2 particles. Similar to the original EPR state, Eq. (9.2.8) is independent of the choice of the spin directions and the eigenstates of the associated non-commuting spin operators. It is easy to show that Ecj. (9.2.8) is true, simultaneously, in the three orthogonal spin representations:

The above two equations are very different from the classically correlated state. It is easy to show that classically correlated states are coordinate dependent. The state does not hold the same form if choosing the other orthogonal spin directions.

The most widely used entangled two-particle states might have been the “Bell states” (or EPR-Bohm-Bell states). Bell states are a set of polarization states for a pair of entangled photons. The four Bell states which form a complete orthonormal basis of two-photon state are usually represented as

where |0) and |1) represent two arbitrary orthogonal polarization bases, for example, | 0) =

| H) (horizontal) and 11) = |R) (vertical) linear polarization, respectively. We will have a detailed discussion on Bell states later.

Entangled Biphoton State

The state of a signal-idler photon pair created in the nonlinear optical process of SPDC is a typical EPR state. Roughly speaking, the process of SPDC involves sending a pump laser beam into a nonlinear material, such as a non-centrosymmetric crystal. Occasionally, the nonlinear interaction leads to the annihilation of a high frequency pump photon and the simultaneous creation of a pair of lower frequency signal-idler photons into an entangled two-photon state:

where uij, kj (j = s, i, p) are the frequency and wavevector of the signal (s), idler (i), and pump (p), a| and a] are creation operators for the signal and the idler photon, respectively, and Фо is a normalization constant. We have assumed a CW monochromatic laser pump, i.e., uip and kp are considered as constants. The two delta functions in Eq. (9.3.1) are technically named as phase matching condition:

The names signal and idler are historical leftovers. The names probably came about due to the fact that in the early days of SPDC, most of the experiments were done with non- degenerate processes. One radiation was in the visible range (and thus easily detected, the signal), and the other was in IR range (usually not detected, the idler). We will see in the following discussions that the role of the idler is not less than that of the signal. The SPDC process is referred to as type-I if the signal and idler photons have identical polarizations, and type-II if they have orthogonal polarizations. The process is said to be degenerate if the SPDC photon pair have the same free space wavelength (e.g. A, = As = 2Ap), and nondegenerate otherwise. In general, the pair exit the crystal non-collinearly, that is, propagate to different directions defined by the second equation in Eq. (9.3.2) and the Snell’s law. Of course, the pair may also exit collinearly, in the same direction, together with the pump.

The state of the signal-idler pair can be derived, quantum mechanically, by the first order perturbation theory with the help of the nonlinear interaction Hamiltonian. The SPDC interaction arises in a nonlinear crystal driven by a pump laser beam. The polarization, i.e., the dipole moment per unit volume, is given by

where уА"1) is the mth order electrical susceptibility tensor. In SPDC, it is the second order nonlinear susceptibility yC) that plays the role. The second order nonlinear interaction Hamiltonian can be written as

where the integral is taken over the interaction volume V.

It is convenient to use the Fourier representation for the electrical fields in Eq. (9.3.4):

Substituting Eq. (9.3.5) into Eq. (9.3.4) and keeping only the terms of interest, we obtain the SPDC Hamiltonian in the interaction representation:

where h.c. stands for Hermitian conjugate. To simplify the calculation, we have also assumed the pump field to be plane and monochromatic with wave vector k;, and frequency uip.

It is easily noticeable that in Eq. (9.3.6), the volume integration can be done for some simplified cases. At this point, we assume that V is infinitely large. Later, we will see that the finite size of V in longitudinal and/or transversal directions may have to be taken into account. For an infinite volume V. the interaction Hamiltonian Eq. (9.3.6) is written as

It is reasonable to consider the pump field classical, which is usually a laser beam, and quantize the signal and idler fields, which are both in single-photon level:

where a.t(k) and a(k) are photon creation and annihilation operators, respectively. The state of the emitted photon pair can be calculated by applying the first order perturbation

By using vacuum |0) for the initial state in Eq. (9.3.9), we assume that there is no input radiation in any signal and idler modes, that is, we have a SPDC process.

Further assuming an infinite interaction time, evaluating the time integral in Eq. (9.3.9) and omitting altogether the constants and slow (square root) functions of w, we obtain the entangled two-photon state of Eq. (9.3.1) in the form of integral:

where Фц is a normalization constant which has absorbed all omitted constants. Eq. (9.3.10) has been used in chapter 3 for the calculation of second-order correlation function.

Three widely used SPDC setups,

FIGURE 9.3.1 Three widely used SPDC setups, (a) Type-I SPDC. (b) Collinear degenerate type-II SPDC. Two rings overlap at one region, (c) Non-collinear degenerate type-II SPDC. For clarity, only two degenerate rings, one for e-polarization and the other for o-polarization, are shown.

The way of achieving phase matching, i.e., the way of achieving the delta functions in Eq. (9.3.10) basically determines how the signal-idler pair "looks”. For example, in a negative uniaxial crystal, one can use a linearly polarized pump laser beam as an extraordinary ray of the crystal to generate a signal-idler pair both polarized as the ordinary rays of the crystal, which is defined as type-I phase matching. One can alternatively generate a signal- idler pair with one ordinary polarized and another extraordinary polarized, which is defined as type II phase matching. Fig. 9.3.1 shows three examples of SPDC two-photon source. All three schemes have been widely used for different experimental purposes. Technical details can be found from text books and research references in nonlinear optics.

The two-photon state in the forms of Eq. (9.3.1) or Eq. (9.3.10) is a pure state, which describes the behavior of a signal-idler photon pair mathematically. Does the signal or the idler photon in the EPR state of Eq. (9.3.1) or Eq. (9.3.10) have a defined energy and momentum regardless of whether we measure it or not? Quantum mechanics answers: No! However, if one of the subsystems is measured with a certain energy and momentum, the other one is determined with certainty, despite the distance between them.

In the above calculation of the two-photon state we have approximated an infinite large volume of nonlinear interaction. For a finite volume of nonlinear interaction, we may write the state of the signal-idler photon pair in a more general form:

where

where e is named as parametric gain, f is proportional to the second order electric sus- ceptibilit.y x^2' and is usually treated as a constant; L is the length of the nonlinear interaction; the integral in к is evaluated over the cross section A of the nonlinear material illuminated by the pump, p is the transverse coordinate vector, itj (with j = s,i) is the transverse wavevector of the signal and idler, and /(| p) is the transverse profile of the pump, which can be treated as a Gaussion in most of the experimental conditions. The functions f(AzL) and htr(ki + «2) can be approximated as «5-functions for an infinitely long (L ~ 00) and wide (A ~ 00) nonlinear interaction region. The reason we have chosen the form of Eq. (9.3.12) is to separate the “longitudinal” and the “transverse” correlations. We will show that S(u>p — u>s — ш,) and f(AzL) together can be rewritten as a function of u)soji. To simplify the mathematics, we assume near со-linearly SPDC. In this situation,

I ^s,i I ^ I I*

Basically, function f(AzL) determines the “longitudinal” space-time correlation. Finding the solution of the integral is straightforward:

Now, we consider f(AzL) with S(u>pu>su>i) together, and taking advantage of the <5- function in frequencies by introducing a detuning frequency v to evaluate function f(A~_L):

The dispersion relation к(ш) allows us to express the wave numbers through the detuning frequency //:

where us and щ are group velocities for the signal and the idler, respectively. Now, we connect Дг with the detuning frequency v

where D = Lfus 1 /и,. We have also applied kp — k(oj°s) — k(u>f) = 0 and 1 ks i sj |. The “longitudinal” wavevector correlation function is rewritten as a function of the detuning frequency v f(AzL) = f(vDL). In addition to the above approximations, we have inexplicitly assumed the angular independence of the wavevector к = п(0)ш/с. For type II SPDC, the refraction index of the extraordinary-ray depends on the angle between the wavevector and the optical axis and an additional term appears in the expansion. Making the approximation valid, we have restricted our calculation to near-collinear process. Thus, for a good approximation, in the near-collinear experimental setup:

Type-I degenerate SPDC is a special case. Due to the fact that us = щ, and hence,

D = 0, the expansion of k(u>) should be carried out up to the second order. Instead of (9.3.17), we have

where

The two-photon state of the signal-idler pair is then approximated as where the normalization constant has been absorbed into /((/).

 
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