Binary Quantum Communication using Squeezed Light: Theoretical and Experimental Frame Work

The aim of this paper is to develop framework to generate squeezed light for binary quantum communication. Both theoretical and experimental models to generate squeezed state using optical parametric amplifier (OPO), which is implemented around He-Ne laser, are described in details. The results will be used as a guide line to investigate the performance of squeezed light-based quantum communication over noisy channel and this issue will be presented in accompanying paper [1].

Binary quantum communication may be realized using two quantum states that can carry classical information of logic 0 and 1 [25]. Since the states are in general not orthogonal and even they are so, the orthogonality will be rapidly destroyed due to the noise effects that added by the channel during the transmission of the state. Therefore, unavoidable errors will be occurred in the receiver side because of the overlap between the received states. Moreover, the channel is a freespace channel. Therefore, the states are affected by the thermal noise added by the environment and the dissipation. Hence, there is a need to investigate the best performance of discriminating noisy non-orthogonal states [26]. The realization of an optimal quantum receiver with minimum measurement errors is a crucial topic for the effective implementation of quantum communication channel. Receivers based on homodyne detection have great advantages for the discrimination between states in the presence of noise [27]. Moreover, coherent and squeezed states are Gaussian states then homodyne receivers have been demonstrated to represent the optimum Gaussian method for the discrimination of the binary quantum communication [28,29]. These issues will be addressed in details in this work.

SQUEEZED STATE
Define quadrature operators and which are Hermitian operators representing the real and imaginary parts of the annihilation operator. The variances of the quadrature operators are equal to 1/2. Thus, they are symmetric and the minimum uncertainty principle may be allowed for the vacuum and the coherent state after the following relation [30] (1) So the squeezing may be achieved by squeezing one quadrature at the expense of stretching the other keeping the Heisenberg relation unchanged as shown in Figure 1.
The squeezed vacuum state of a single-mode field is defined by applying the squeezed operator on the vacuum state (2) The squeezed operator is a unity operator given by [31] ( with the annihilation and creation operators and , respectively. Further, is a complex parameter. r is the squeezed parameter varying from 0 to representing the amplitude and is the phase defining the squeezing quadrature. The action of the squeezed operator on the annihilation and creation operators can be expressed as [31] N o v 1 7 , 2 0 1 3 The action of the squeezed operator on the quadrature operators in order to evaluate the variance and the mean of a single mode coherent squeezed state [31] (5a) (5b)

For
, which is the case in this work, Eqns. (5a) and (5b) become This indicates clearly that the variance is squeezed in one quadrature by and stretched in the other by . The uncertainty principle is hold unchanged. The mean photon number is given by [32] Eqn. (7) indicates that (i) Increasing the squeezed amplitude r will increase the mean photon number.
(iii) When , the squeezed vacuum state is obtained which is characterized by an average number of photons depends on r.
Finally, the Wigner function of the squeezed coherent state along the q quadrature can be evaluated as [32] (8) The squeezed coherent state can also be obtained by shifting the squeezed vacuum state using the displacement operator as shown in Figure 2 (9) N o v 1 7 , 2 0 1 3

GENERATION of SQUEEZED STATE USING OPO: EXPERIMENTAL FRAME WORK
This section is directed to perform a setup for generating squeezed states of light using OPO. The setup consists of four main parts as shown in Figure 3. (i) Nd-YAG laser is used to generate the first harmonic (1064 nm) and the second harmonic (SHG) at 532 nm. The fundamental laser system is built through a ring cavity.
(ii) Squeezed cavity involves two curved mirrors and a nonlinear crystal.
(iii) A Pound-Drever-Hall (PHD) system controls the cavity length of the OPO.
(iv) Balanced homodyne detection (BHD) is used to detect the squeezed state of light with a strong reference as a local oscillator (LO).

First and Second-Harmonic Generation
This subsection involves two principal parts: The setup design includes different computations: the cavity dimensions, position of the multiple devices, the spot size of the beam, and the expected power generated by the setup. N o v 1 7 , 2 0 1 3 (ii) Experimental setup to generate the first and the second harmonics.

Preliminary Calculations
The first and the second harmonics are generated by a ring resonator as shown in Figure 4. It consists of four mirrors: two are planes and the other are curved. A periodically poled lithium niobate (PPLN) crystal is used to generate the second harmonic while the first harmonic is generated using an Nd-YAG rod. In Figure 4, , , , , , the Brewster plate length , and . The starting point is to calculate the refractive indices of the barium borosilicate glass (BK7) and the Nd-YAG rod using Sellmeier equation [33,34] (10) A, B1, B2, B3, C1, C2, and C3 are the Sellmeier coefficients. λ is the wavelength related to the material in . Eqn. (10) is the dispersion equation describing the relation between the refractive index and the wavelength for a given material. Table  1 displays the coefficients of Sellmeier equation for the Nd-YAG rod and the BK7. where is the Brewster angle. The FSR is given by (12) The calculation yields: hence . Thus, the frequency spacing between two modes is 102 MHz.
Next, it is fundamental to determine the stability of the cavity for the fundamental transversal mode. TEM00 which has a Gaussian transversal profile and represents a Gaussian beam. This can be derived from Gaussian beam propagation through the resonator under the condition of self-reproduction after one complete round trip. This requires the computing of the round-trip ABCD matrices along the ring resonator. Take the reference plane at the thermal lens in Figure 4 and going from right to left. The matrix form is either lens matrix or free space matrix as follow [35] (13a) where z represents the propagation distance and where f is the focal length of a thin lens. Hence, the round-trip matrix is a product of matrices. i refers to the medium in the resonator and each matrix corresponds to a medium where the beam propagates. (14) where, A, B, C and D are the matrix coefficients and is shown in Table 2.
Since the matrix coefficients are function of the focal length of the thermal lens then the stability, the waist and the inverse radius of curvature depend also on this parameter.
The position of the crystal used to generate the second harmonic should be given at the maximum focusing of the first harmonic. Therefore, there is a need to determine the propagation of the beam along the ring cavity. The start point is to calculate the waist and the inverse of the radius of curvature of the propagation at the reference plane and then in each point of the cavity as follow

Experimental Setup to Generate the Second Harmonic
The experimental setup to generate the first and the second harmonics is shown in Figure 5. The optical devices in Figure 5 have the following specifications: (i) : Plane mirror with high reflectivity to the first harmonic.
(ii) : Output coupler plane mirror with reflectivity of to the first harmonic.
(iii) : Curved mirror has a radius of curvature with transmission power of to the second harmonic and high reflectivity to the first harmonic. N o v 1 7 , 2 0 1 3 : Output coupler mirror has a radius of curvature with transmission power of to the second harmonic and high reflectivity to the first harmonic. (v) : Brewster Plate to minimize the reflection of the p polarized light. It is a (BK7) plate with length of .

(vi)
FI: Faraday insulator of 20 mm is used to prevent the antireflection of the beam into the source.
(vii) PPLN: Nonlinear crystal with 5% magnesium-oxide MgO doped PPLN. MgO:PPLN offers high-efficiency wavelength conversion. The PPLN has a length of 10 mm and 1mm thickness. The crystal is inserted in a PC10 clip then it is placed in a PV10 oven which is controlled by an OC1 temperature controller.

(viii)
Active medium is a Neodymium-doped Yttrium Aluminum Garnet (Nd-YAG) with chemical structure of ( Nd:Y3Al5O12 ). It is a rod of a length of 30 mm and 2 mm diameter as shown in Figure 2.11d. Note that during operation, the active medium is equivalent to a thermal lens.
(ix) PD1 and PD2 are used to detect the optical signals by measuring their currents which are converted to voltages with a load resistance of 1kΩ. To prevent saturation, optical filters are used to attenuate the intensity of the light beams for both PDs. PD1 and PD2 are of silicon types having an efficiency of 0.4 A/W at 1064 nm and 0.23 at 532 nm. Figure 6 shows the Nd-YAG rod and its equipments. The Nd-YAG rod is side pumped by semiconductor laser diodes at 808 nm. They provide a pump current range up to 24A as shown in Figure 6a and 6b. A control unit is used to control the pump current and the system cooling which is as important as the rest of the system as shown in Figure 6c.
Without a sufficient cooling, the laser will break down soon. The stability and the efficiency of the laser are quite dependent on cooling. The cooling system is an open-loop cooling system with tap water flowing across the rod. The pump current determines the characteristics of the Nd-YAG laser. As the pump current increases the focal length of the induced thermal lens decreases as show in Figure 6e.
In this part of the experiment, the work involves the following steps (i) Determination of the temperature at which the quasi-phase matching is achieved. This is done by fixing the pump current and increasing the temperature applied on the crystal and marking the second-harmonic power. (ii) Measuring the first and the second-harmonic powers by increasing the pump current while maintaining the temperature on the PPLN crystal fixes.
The measures are given by using PD1 and PD2. The photocurrent detected at the PDi can be written as (23) where i = 1 and 2 which correspond to the first (1ω) and the second harmonic (2ω), respectively, is the responsivity of the PDi in A/W and is the optical power incident on the PDs. The incident power is calculated by measuring the voltage Vi across the detector load resistor RL=1kΩ (24) where are the filter power transmission coefficients used to reduce optical-power intensity incident on PD1 and PD2 respectively.

Squeezed Cavity Design
Two curved mirrors are used with their specifications are shown in Table 3. These mirrors should form a squeezed cavity of length Lsq as shown in Figure 7. A lithium niobate doped with 5% magnesium oxide MgO (Mg-LNBO3) is placed inside the cavity. The ensemble forms an OPO to generate the squeezed light after pumping by the second-harmonic beam.

Calculation of the Two Refractive Indices
The crystal is 2×2×10 mm, X cut, θ=90 o , φ=0 from Covesion Ltd. [36]. Its two faces are both anti-reflected coating at 1064 nm and 532 nm. It is a nonlinear crystal with two refractive indices, extraordinary ne and ordinary no, which are calculated by Sellmeier equation as follow [37] ( 25) where , , , , and are Sellemeier coefficients given in Table 4, is the wavelength of the laser light in and is expressed as (26) Note that there are two wavelength, and representing the first and the second harmonics, respectively. The refractive index in Eqn. (25) depends on the wavelength and the temperature because the process to achieve phase matching through the Mg-LNBO3 is the NCPM.

Free Spectral Range and Squeezed Resonator Stability
First, compute the effective length and then the The is computed from Eqn. (12).
Next, In order to test the stability of the squeezed resonator, one should start to compute the matrix of Figure 8. The reference plane is taken at the mirror Ms1 and using Eqns. (13a) and (13b) for the space and lens matrices, respectively. Thus, the round-trip matrix for the squeezed cavity may be written as a product of all matrices elements (28) where the matrices are described in Table 5. Thus, the stability may be calculated from Eqn. (15).

First-Harmonic Propagation
The start point is to calculate the propagation at each medium N o v 1 7 , 2 0 1 3

Second-Harmonic Propagation
Take the reference plane at the mirror MS2, then the radius of curvature of the beam is the same as that of the mirror MS2. The waist of the second harmonic is calculated from the waist of the first harmonic for the OPO by the formula [38] (39) Next, calculate the threshold power required for the OPO. The threshold intensity is given by [39] ( 40) where . Since, the pump beam is a single mode Gaussian beam with waist , then, the threshold power for a Gaussian beam is given by [81] (41)

Generation of Squeezed Light
The generation of the squeezed light is achieved inside the nonlinear crystal by SPDC. SPDC is a wave mixing process in which the pump decay into two beams at lower frequency namely signal and idler. The crystal is pumped by the second harmonic of a frequency and some photons of the pump are converted into pairs of identical photons of frequency . The Hamiltonian of this process is given by [40] ( 42) where , and are the annihilation and creation operators of the signal and pump fields respectively and is the second-order susceptibility. The first and the second terms, in Eqn. (42), represent the energy of the signal and the pump, respectively. The third term represents the interaction between the states. However, the pump is a coherent strong field N o v 1 7 , 2 0 1 3 that have the state . Thus, and correspond to and respectively. Hence, the Hamiltonian in Eqn. (42) becomes (43) where . The constant term is dropped because it is irrelevant to the system. Now, transform to the interaction picture yields (44) Since the generated signal has a frequency of then (45) Using the associated evolution operator yields Assuming then Eqn. (46) has an identical form to the squeezed operator of Eqn. (3).

Squeezed and Anti-squeezed Variance
Spontaneous parametric down conversion (SPDC) process is used to generate a squeezed state via an OPO. The Hamiltonian of the OPO is given by [40] (47) where is a complex constant depends on the nonlinearity and the pump intensity of the OPO as in Eqn. (46). The equation of motion of the OPO is described by [41] (48a) , and are the cavity decay rates due to the input and output mirrors reflectivities, is the round trip time, is the intra-cavity losses of the OPO, is the internal loss for a single pass and is the total cavity losses. represents the signal beam, where and represent the vacuum fluctuations associated with the losses. Take the Fourier transform (FT) of Eqns. (47a) and (48b) yields where is the detection frequency and all and are function of . Re[E] and Im[E] denote the real and imaginary parts of the complex number E, respectively. Therefore, the spectral density of the noise quadrature components represented by its squeezed and anti-squeezed variances can be calculated from [40] (50a) Hence the noise variance at the OPO output is [42] (52a) N o v 1 7 , 2 0 1 3 where the efficiency of the photodetectors, the efficiency of the homodyne detection and is the escape efficiency, P and are the pump and threshold powers.
The most important factor in Eqns. (52a) and (52b) is the escape efficiency because by increasing this factor, the losses inside the cavity is reduced. Obviously, the escape efficiency can be increased by reducing the reflectivity of the OPO front face. However, this is at the expense of a higher OPO threshold. Thus, a further increase in the escape efficiency is only feasible with a more efficient SHG source or more powerful pump laser. Now, for power reflectivities of 0.90 and 0.99 for MS1 and MS2, respectively, and a round trip time , the power spectral losses are: , , Hz with internal loss . Therefore the overall losses . Hence the escape efficiency . On the other hand, for high-homodyne efficiency and detection . The threshold power Pth=1mW.
The quadrature variance is squeezed from the vacuum noise by the subtracted term in Eqn. (52a) and stretched by the added term in Eqn. (52b). Moreover, Multiple parameters determine the degree of squeezing, the frequency , the pump power, and the escape efficiency.

Pound-Drever-Hall (PDH) Control System
PDH is a control system used to maintain the stability of the laser of the squeezed cavity, which is so sensitive to the environment and the mechanical vibration. PDH consists of an error signal that is proportional to the difference in frequency between the laser light and the cavity resonance. This is done by examining the light reflected from the cavity. Hence, a stable laser frequency may be built by this technique. Therefore, the length of the squeezed cavity can be adjusted through a piezoelectric mounted on the mirror MS2. The piezoelectric is controlled electrically as shown in Figure  8. From the setup of Figure 8, the laser source is first phase modulated by a nonlinear crystal then the optical-modulated signal is transmitted to the optical resonator (which represents here the squeezed cavity). Next the light beam is reflected back by the mirror MS2 and detected by a photodetector. Radio frequency (RF) generator is mixed with the detected signal through an electrical mixer. The mixed signal is used to derive the piezoelectric. In fact all the process of the PDH technique is based on the reflected power of the laser light beam inside the squeezed resonator. N o v 1 7 , 2 0 1 3

PDH Technique
The idea of this technique is based on the reflected beam. It consists to modulate the phase of the incoming laser beam through a lithium niobate (LNB) crystal. It will result sidebands with a definite phase relationship to the incident and the reflected beam. Then interfere of these sidebands with the reflected beam yields a beat pattern at the modulation frequency. The phase of this pattern will be measured to give an indication on the phase of the reflected beam.
The LNB crystal has a length and . The voltage is applied transversally on the crystal. The phase modulation depth β is equal to [43] (53) where at (the extraordinary index of refraction for the LNB crystal), is the coefficient of the susceptibility matrix, is the distance between the two surfaces of the crystal and is the applied voltage.
The electric field is phase modulated with a frequency of by the electro-optical modulator [44] (54) where is the modulation frequency. From Eqn. (53), the modulation depth is estimated to be . Note that which indicates narrow band phase modulation and therefore the high-order terms in Eqn. (55a) can be neglected Eqn. (55b) displays the carrier at frequency and the two sidebands at frequencies . The reflected beam consists of several reflected sub beams, each one with its appropriate frequency as follow (56) The power in the reflected beam is measured by the photodetector which is proportional to the square of the reflected electric field amplitude. If the power in the carrier is , then the power in each sideband is and the reflected power will be (57) The interesting terms in Eqn. (57) are the oscillating terms at the modulation frequency because they sample the phase of the reflected carrier. Additionally, term arises from the interference between the carrier and the sidebands, and terms come from the sidebands interfering with each other.

PDH Error Signal Measurement
The reflected power in Eqn. (57) is measured by the photodetector as shown in Figure 8. Then it mixed with a sinewave signal at the modulation frequency which is supposed here high frequency about 250 MHz. Hence the real part in Eqn. (56) will be vanished and after the LPF only the imaginary term will be remained which represents the error signal of the system as indicated in Eqn. (58) [44] (58)

Homodyne Detection System
A homodyne detection scheme is proposed in Figure 9 to detect the quadrature squeezed states of the laser light beam. N o v 1 7 , 2 0 1 3 For balanced homodyne detection, the current to voltage converter circuit, V1 and V2 may be written as (59a) (59b) Thus, the output of the difference amplifier V0 can be expressed as From Eqn. (60), V0 is proportional to which is proportional to the difference number of photons and [45] (61a) and are the annihilation operators of the coherent state and the LO field, respectively, which may be described by Here , and , are the complex amplitudes and the standard deviations of the coherent state and the LO, respectively. Assuming the mode is in the coherent state with , Eqn. (61b) yields The squeezed state and the coherent field are derived from the same laser. Thus, they have the same frequency so . Then Eqn. (63) can be written as The homodyne current is proportional to . Thus, a measurement to the quadrature squeezing is given and the trace of the homodyne measurement can be written as [46] (65) The complete setup to generate squeezed light is shown in Figure 10. N o v 1 7 , 2 0 1 3

CONCLUSIONS
Mathematical and experimental frame work has been presented for generating non-classical quantum state using an optical parametric oscillator via a spontaneous parametric down conversion technique. The aim is to investigate squeezed states with quantum noise in one quadrature below the standard quantum limit at the expense of the other. The setup involves four main parts: generation of Nd-YAG second harmonic via a ring resonator, squeezed cavity with a nonlinear crystal inside to generate the squeezed state, Pound-Drever-Hall technique to stabilize the laser in the squeezed cavity and balanced homodyne receiver with high efficiency to detect the squeezed state.