rus
Issue #1/2023
V. M. Petrov, D. A. Koroteev, D. A. Semisalov, V. S. Strashilin, D. S. Khlusevich, M. I. Yakovlev, M. V. Parfenov
Integrated Optical C-NOT Gates: Estimation of the Main Parameters for Practical Design
Integrated Optical C-NOT Gates: Estimation of the Main Parameters for Practical Design
DOI: 10.22184/1993-7296.FRos.2023.17.1.58.70
The influence of deviation of the beam splitter parameters on the operation of a quantum photon gate in an integrated optical version is considered. It is shown that the required accuracy is quite achievable for electro-optical control in the X-splitter geometry. The estimated length of the device demonstrates its possible implementation even on the substrates with a length of 3 inches.
The influence of deviation of the beam splitter parameters on the operation of a quantum photon gate in an integrated optical version is considered. It is shown that the required accuracy is quite achievable for electro-optical control in the X-splitter geometry. The estimated length of the device demonstrates its possible implementation even on the substrates with a length of 3 inches.
Теги: c-not gate c-not гейт photonic gates quantum logic gate success probabilities вероятности успешного срабатывания photonic integrated circuits квантовый логический вентиль фотонные гейты фотонные интегральные схемы
Integrated Optical C-NOT Gates: Estimation of the Main Parameters for Practical Design
V. M. Petrov1, D. A. Koroteev2, D. A. Semisalov2, V. S. Strashilin2, D. S. Khlusevich2, M. I. Yakovlev2, M. V. Parfenov3
Saint-Petersburg State University, Saint-Petersburg, Russia
National Research University ITMO, Saint-Petersburg, Russia
A. F. Ioffe Physical and Technical Institute, Saint-Petersburg, Russia
The influence of deviation of the beam splitter parameters on the operation of a quantum photon gate in an integrated optical version is considered. It is shown that the required accuracy is quite achievable for electro-optical control in the X-splitter geometry. The estimated length of the device demonstrates its possible implementation even on the substrates with a length of 3 inches.
Keywords: photonic integrated circuits, photonic gates, C-NOT gate, quantum logic gate, success probabilities
Received: December 25, 2022
Accepted: January 25, 2023
Introduction
The development and production of the implementable, reliable, energy-efficient quantum computers is an urgent task. It is possible to determine the following main process platforms that are used to solve this problem: ions/atoms in the electromagnetic traps, superconducting circuits, quantum dots/impurity atoms, and optical (photonic) integrated circuits.
The quantum dual-mode photon gates (hereinafter referred to as the gates) ensure transformation of the quantum state of one photon depending on the state of another photon. In this sense, the role of quantum gates is very similar to the role of a semiconductor transistor in electronics, in which the charge-carrier flow is controlled by the flow of other charge carriers. The gate that provides execution of the controlled NOT algorithm (note: Controlled NOT gate – C-NOT gate) is one of the most popular in the modern quantum computing problems.
In this paper, we will consider the design aspects of photonic gates based on the available domestic hardware components in an integrated optical version.
There are different types of hardware components used to produce the gates. First, it is the so-called 3D-version. In this case, the volumetric, discrete elements are used, for example, ordinary beam splitting cubes [1–3] or three-dimensional acoustic and optical modulators as the controlled beam splitters [4]. Another type is the integrated optical design, when the entire device is made in the form of an integrated optical structure on one substrate in the form of a single chip [5–8]. We have opted for an integral design for the following reason: it provides mechanical stability compared to the 3D or fiber options that is important when manufacturing the interferometers. In addition to the monolithic, integrated-optical design, the small size of interferometer makes it possible to align its arms with very high accuracy that leads to an increased stability. The well-proven production technology of integrated optical chips has already allowed us to develop the highly sensitive electric field intensity sensors [9], broadband microwave integrated optical modulators [10], broadband quantum noise sources [11], and optical comb generators for the quantum communication systems [12].
An important area of research is the substrate material on which the gate is produced. The most popular types include lithium niobate LiNbO3 [13, 14], the high-quality quartz glass [8], the silica-on-silicon structures or doped silica films, that is the silicon-based waveguides [15]. The research works on the use of the A3-B5 type compounds [16] and polymers [17] are rather persistent and active.
In this paper, we have proposed a C-NOT gate circuit based on the integrated optical beam splitters and studied the influence of amplitude and phase errors of count-down ratios on the gate operation.
Integrated optical gate circuit
The main element of the gate is the beam splitter. In the field of integrated optics, a beam splitter can be implemented on the basis of various elements: X- or Y-couplers, Mach-Zehnder interferometers (MZIs). In this paper, we consider the case when the beam splitter is based on еру X-couplers. We have to note that such classification is rather conditional, since two tandem couplers can be considered as one MZI.
The C-NOT gate is one of the main quantum logic gates used in the quantum computations. The main feature of this gate is the inversion of dependent modes, if the control qubit value is equal to one. The controllable-NOT operation matrix is as follows:
. (1)
The gate circuit [1] in the integrated optical version ensuring the algorithm execution (1), is shown in Fig. 1.
All gate elements are located on a substrate made of lithium niobate LiNbO3 with a Z-cut. The operating wavelength of the gate is λ = 810 nm. Such wavelength selection is due to the availability of entangled photon pairs consisting of a semiconductor laser λ = 405 nm and a nonlinear crystal of nonlinear optical crystals of barium beta-borate, β-BaB2O4 (BBO), performing the parametric down-conversion at a wavelength of 810 nm. The waveguides Ain and Fin and Aout and Fout are inactive. The circuit contains 5 beam splitters BS1–BS5. The beam splitters BS1 and BS5 have the count-down ratios of 50:50, i. e. 1 / 2. The beam splitters BS2–BS4 have the count-down ratios of 1/3. For this case, the circuit additionally indicates which channel receives 33.33…% of the incident power, and which channel receives 66.66…%.
Impact assessment of non-ideal count-down ratios
The gate operation is significantly affected by deviation of the count-down ratio values from the given ones. The accuracy with which the beam splitter divides the photon flux significantly affects the most important specifications of the gate, namely the fidelity and probability. Even a minimal deviation from the required value of any of the count-down ratios leads to the loss of gate performance.
The QuTiP computational physics software library [18] has been used to simulate and study various specifications of a quantum optical system. Based on the KLM protocol, each qubit is represented as two optical modes, which provide for a photon with vertical or horizontal polarization depending on the mode.
Having used the formal description of the Fock states, it is possible to describe one mode condition through the matrix representation:
, , . (2)
The state of two modes in the two-channel encoding protocol is a computational qubit and is recorded as follows:
, (3, а)
. (3, б)
The impact of a beam splitter on the state of a system consisting of two independent modes α and b can be represented as a sum of Hamiltonian functions:
, (4, а)
, (4, б)
, (4, в)
where ϕ is the phase difference between the transmitted and reflected beams. In turn, the system can be represented as a unitary operator:
, (5)
where θ is an angle determining the beam part to be transmitted or reflected:
. (6)
Application of the beam splitter operator to the system state leads to a fairly accurate description of the physical impact of the transmitted beam. It should be noted that such calculation does not consider the time component, since the system is not changed over time, and the photons achieve the beam splitter simultaneously to an approximation of a fraction of their coherence length.
The gate circuit (Fig. 1) used in this paper is based on a two-channel encoding protocol. In the given circuit of tandem beam splitters, the Bin and Cin channels are the control modes cH and cV, the Din and Ein channels are the dependent modes tH and tV, the Ain and Fin channels are the unoccupied auxiliary modes vc and vt.
We have considered the case when the input system state | 0, 0, 1, 0, 1, 0 ⟩ is affected by the beam splitters with a slight deviation from the theoretical model. Therefore, we can get the following probability distribution of the output states.
To describe the effects occurred when the parameters of dual-mode beam splitters deviate from the given values, we have simulated the system with due regard to the possible deviations in the reflection coefficient and phase affecting the initial system state.
The unitary matrix describing the beam splitter is as follows [19]:
. (7)
In this case, θ is an angle given by the reflection and transmission coefficients, and ϕ is a change in the phase component (see (4), (6)). Thus, any change in these physical variables will occur according to the cosinusoidal/sinusoidal law and on an exponential basis.
a). Impact on a State With a Photon in One Mode
We will consider the beam splitters BS1 and BS5 with a count-down ratio of 1 / 2. We will consider the state as an input state described by two optical channels.
As can be seen in the graphs (Fig. 3), any change in the beam splitter count-down ratio of 50 / 50 leads to a deviation in the probability of the system location in one of the output states or. In this case, a 50% probability is achieved with an insignificant deviation from the theoretical model, and the deviation is almost linear on a scale under study. The phase component deviation has insignificant impact on the system with a photon in one mode and approaches the required distribution with the relatively small deviations.
Next, we will consider the beam splitters BS2 –BS4 with a count-down ratio of 1 / 3.
As can be seen in the graph (Fig. 4), any change in the reflection/transmission coefficient of the beam splitter 33 / 66 also leads to a deviation in the probability of the system location in one of the output states or. In this case, the deviation from the theoretical model increases the probability of one or another state.
Figure 5 shows the dependences diagrams of the probability of a given state for the same beam splitters and deviation ϕ from the given value.
It can be seen that the phase component deviation, as in the case of the 50 / 50 beam splitter, has insignificant impact on the system with a photon in one mode and upon decrease, leads the system to the theoretical distribution.
b). Impact on a State With a Photon in Two Modes or with a Superposition of the Horizontal and Phase Components
Now, we will consider the state as an input state described by two optical channels.
Let us consider the beam splitters BS1 and BS5 with a count-down ratio of 1 / 2 (Fig. 6, 7). Changes in the reflection/transmission coefficient of the beam splitter 50 / 50 leads to a deviation of the probability of the system location in the system states | 0, 2 ⟩ and | 2, 0 ⟩ and occurrence of an additional state | 1, 1 ⟩. Moreover, as the deviation from the theoretical model is becoming more significant, the Hong-Ou-Mandel effect [19] ceases to be observed, when two photons incident on the 50 / 50 beam splitter get out through one mode, since the probability of the state | 1, 1 ⟩ is increased.
Now let us consider the beam splitters BS2–BS4 with a count-down ratio of 1/3 (Fig. 8, 9). It can be seen in the graph (Fig. 8) that any change in the reflection / transmission coefficient of the beam splitter 1 / 3, on the contrary, brings it closer to the values of the beam splitter 1 / 2 under certain conditions that reduces the probability of the system location in the state | 1, 1 ⟩.
Assessment of the gate
geometric dimensions
The geometric dimensions of the gate are important since they determine the dimensions of the required lithium niobate substrates that are typically manufactured with the standard diameters of 3 or 5 inches (7.62 or 12.7 mm).
The overall gate width (Y-axis) is determined by the distance between the six input waveguides. Moreover, it should be considered that at the subsequent gate assembly stage, the optical fibers in the form of a four-fiber assembly should be glued to its input and output planes to be connected to the B-E channels (we should remember that the A and F channels are not used). The V-groove-based optical fiber microassembly has a minimum standard packaging pitch of 250 µm. Thus, the minimum total gate width required to accommodate 6 waveguides (8 intervals of 250 µm, with due regard to the gap to the blank edge) is 2.0 mm. In practice, the gate blank width can be 3 mm due to the blank operation convenience, mechanical strength, etc.
The total length of the gate L (along the photon propagation direction, X-axis) is determined by the lengths of the linear sections δL, the lengths of the rounded (S-shaped) sections ΔX, as well as the lengths of the X-coupler sections ΔL (Fig. 1). In the longitudinal direction, we have six S-shaped sections and three X-coupler sections.
In the Wolfram Matematica software, we have performed mathematical modeling of the f electrically controlled beam splitters and S-shaped sections of waveguides in order to determine their minimum allowable lengths and radii of curvature. It has been found that the minimum length of the X-coupler section ΔL must be at least 8 mm. In this case, the control voltage value does not exceed 8–10 V. The minimum allowable length of the waveguide S-shaped sections should be at least 6 mm. In this case it is possible to neglect the losses related to the leaking modes in the curved integrated optical waveguides [21]. The length of all linear sections between the remaining elements can be reduced to zero, except for the input and output linear sections δL that can be 2 mm from each edge.
Thus, the minimum gate length is 64 mm. This is obtained as: (3 × 8 mm)+ (6 × 6 mm) + (2 × 2 mm) = 64 mm. This formula shows that even on a standard substrate with a diameter of 3 inches can accommodate 3–4 blanks for the gate.
Discussion
The results presented in this paper confirm the assumption about the possible practical implementation of a quantum photon gate based on the integrated optics. The main specifications of a quantum device that determine its performance are optical losses, adjustment accuracy of the count-down ratio in the individual beam splitters, and geometric dimensions. A separate publication will be devoted to the issue of optical losses. In this paper, it is shown that the adjustment accuracy of the count-down ratios within the range of ±0.1% – 0.5% almost does not make any parasitic contribution to the gate operation.
It is found that imperfection of each of the beam splitters makes various contributions to the violation of the two-photon interference conditions. Thus, for example, deviations from the set parameters in the BS2 and BS4 beam splitters have a minimal impact on the gate operation. Next, in order of increasing influence, there are the beam splitters BS1 and BS5, and, finally, the beam splitter BS3. It should also be noted that the required adjustment accuracy of the reflection/transmission coefficients of the integrated-type beam splitters is implementable.
Another important result is the practical assessment of the gate dimensions. The integrated optical elements used on a lithium niobate substrate have a noticeable length. Therefore, from a practical point of view, it is important to evaluate the entire longitudinal dimension of the device in order to determine the substrate dimensions. We have shown that even the substrates with a diameter of 3 inches can accommodate several blanks.
About the authors
Petrov V. M., Doctor of Physical and Mathematical Sciences (radiophysics), d.f.-m.s. (optics), Professor, Department of General Physics, St. Petersburg State University, St. Petersburg, Russia.
ORCID: 0000 0002 8523 0336
Koroteev D. A., student, Department of Photonics, National Research University ITMO, Saint-Petersburg, Russia.
ORCID: 0000-0002-5489-4017
Semisalov D. A., student, Department of Photonics, National Research University ITMO, Saint-Petersburg, Russia.
ORCID: 0000-0003-1757-6519
Strashilin V., student, Department of Photonics, National Research University ITMO, Saint-Petersburg, Russia.ORCID: 0000-0002-0655-0199
Khlusevich D. S. student, Department of Photonics, National Research University ITMO, Saint-Petersburg, Russia.
ORCID: 0000-0002-8298-9451
Yakovlev M. I., student, Institute of Laser Technologies, National Research University ITMO, Saint-Petersburg, Russia.
ORCID: 0000-0003-1757-6519
Parfenov M. V., Junior Researcher, Department of Quantum Electronics, Physicotechnical Institute named after A. F. Ioffe Saint-Petersburg, Russia.
ORCID: 0000-0003-3867-9007
AUTHORS CONTRIBUTION
Petrov V. M.: statement of the problem, discussion of the results; Koroteev D. A.: simulation of leaks and losses in gates; Semisalov D. A.: calculation of the curvature of waveguides and gate beamsplitters; Strashilin V.S.: calculation of the influence of non-ideal gate division coefficients; Khlusevich D. S.: calculation and participation in the manufacture of gate photomasks; Yakovlev M. I.: development and production of an experimental stand for studying gates; Parfenov M. V.: fabrication of waveguides, measurement of waveguide characteristics.
V. M. Petrov1, D. A. Koroteev2, D. A. Semisalov2, V. S. Strashilin2, D. S. Khlusevich2, M. I. Yakovlev2, M. V. Parfenov3
Saint-Petersburg State University, Saint-Petersburg, Russia
National Research University ITMO, Saint-Petersburg, Russia
A. F. Ioffe Physical and Technical Institute, Saint-Petersburg, Russia
The influence of deviation of the beam splitter parameters on the operation of a quantum photon gate in an integrated optical version is considered. It is shown that the required accuracy is quite achievable for electro-optical control in the X-splitter geometry. The estimated length of the device demonstrates its possible implementation even on the substrates with a length of 3 inches.
Keywords: photonic integrated circuits, photonic gates, C-NOT gate, quantum logic gate, success probabilities
Received: December 25, 2022
Accepted: January 25, 2023
Introduction
The development and production of the implementable, reliable, energy-efficient quantum computers is an urgent task. It is possible to determine the following main process platforms that are used to solve this problem: ions/atoms in the electromagnetic traps, superconducting circuits, quantum dots/impurity atoms, and optical (photonic) integrated circuits.
The quantum dual-mode photon gates (hereinafter referred to as the gates) ensure transformation of the quantum state of one photon depending on the state of another photon. In this sense, the role of quantum gates is very similar to the role of a semiconductor transistor in electronics, in which the charge-carrier flow is controlled by the flow of other charge carriers. The gate that provides execution of the controlled NOT algorithm (note: Controlled NOT gate – C-NOT gate) is one of the most popular in the modern quantum computing problems.
In this paper, we will consider the design aspects of photonic gates based on the available domestic hardware components in an integrated optical version.
There are different types of hardware components used to produce the gates. First, it is the so-called 3D-version. In this case, the volumetric, discrete elements are used, for example, ordinary beam splitting cubes [1–3] or three-dimensional acoustic and optical modulators as the controlled beam splitters [4]. Another type is the integrated optical design, when the entire device is made in the form of an integrated optical structure on one substrate in the form of a single chip [5–8]. We have opted for an integral design for the following reason: it provides mechanical stability compared to the 3D or fiber options that is important when manufacturing the interferometers. In addition to the monolithic, integrated-optical design, the small size of interferometer makes it possible to align its arms with very high accuracy that leads to an increased stability. The well-proven production technology of integrated optical chips has already allowed us to develop the highly sensitive electric field intensity sensors [9], broadband microwave integrated optical modulators [10], broadband quantum noise sources [11], and optical comb generators for the quantum communication systems [12].
An important area of research is the substrate material on which the gate is produced. The most popular types include lithium niobate LiNbO3 [13, 14], the high-quality quartz glass [8], the silica-on-silicon structures or doped silica films, that is the silicon-based waveguides [15]. The research works on the use of the A3-B5 type compounds [16] and polymers [17] are rather persistent and active.
In this paper, we have proposed a C-NOT gate circuit based on the integrated optical beam splitters and studied the influence of amplitude and phase errors of count-down ratios on the gate operation.
Integrated optical gate circuit
The main element of the gate is the beam splitter. In the field of integrated optics, a beam splitter can be implemented on the basis of various elements: X- or Y-couplers, Mach-Zehnder interferometers (MZIs). In this paper, we consider the case when the beam splitter is based on еру X-couplers. We have to note that such classification is rather conditional, since two tandem couplers can be considered as one MZI.
The C-NOT gate is one of the main quantum logic gates used in the quantum computations. The main feature of this gate is the inversion of dependent modes, if the control qubit value is equal to one. The controllable-NOT operation matrix is as follows:
. (1)
The gate circuit [1] in the integrated optical version ensuring the algorithm execution (1), is shown in Fig. 1.
All gate elements are located on a substrate made of lithium niobate LiNbO3 with a Z-cut. The operating wavelength of the gate is λ = 810 nm. Such wavelength selection is due to the availability of entangled photon pairs consisting of a semiconductor laser λ = 405 nm and a nonlinear crystal of nonlinear optical crystals of barium beta-borate, β-BaB2O4 (BBO), performing the parametric down-conversion at a wavelength of 810 nm. The waveguides Ain and Fin and Aout and Fout are inactive. The circuit contains 5 beam splitters BS1–BS5. The beam splitters BS1 and BS5 have the count-down ratios of 50:50, i. e. 1 / 2. The beam splitters BS2–BS4 have the count-down ratios of 1/3. For this case, the circuit additionally indicates which channel receives 33.33…% of the incident power, and which channel receives 66.66…%.
Impact assessment of non-ideal count-down ratios
The gate operation is significantly affected by deviation of the count-down ratio values from the given ones. The accuracy with which the beam splitter divides the photon flux significantly affects the most important specifications of the gate, namely the fidelity and probability. Even a minimal deviation from the required value of any of the count-down ratios leads to the loss of gate performance.
The QuTiP computational physics software library [18] has been used to simulate and study various specifications of a quantum optical system. Based on the KLM protocol, each qubit is represented as two optical modes, which provide for a photon with vertical or horizontal polarization depending on the mode.
Having used the formal description of the Fock states, it is possible to describe one mode condition through the matrix representation:
, , . (2)
The state of two modes in the two-channel encoding protocol is a computational qubit and is recorded as follows:
, (3, а)
. (3, б)
The impact of a beam splitter on the state of a system consisting of two independent modes α and b can be represented as a sum of Hamiltonian functions:
, (4, а)
, (4, б)
, (4, в)
where ϕ is the phase difference between the transmitted and reflected beams. In turn, the system can be represented as a unitary operator:
, (5)
where θ is an angle determining the beam part to be transmitted or reflected:
. (6)
Application of the beam splitter operator to the system state leads to a fairly accurate description of the physical impact of the transmitted beam. It should be noted that such calculation does not consider the time component, since the system is not changed over time, and the photons achieve the beam splitter simultaneously to an approximation of a fraction of their coherence length.
The gate circuit (Fig. 1) used in this paper is based on a two-channel encoding protocol. In the given circuit of tandem beam splitters, the Bin and Cin channels are the control modes cH and cV, the Din and Ein channels are the dependent modes tH and tV, the Ain and Fin channels are the unoccupied auxiliary modes vc and vt.
We have considered the case when the input system state | 0, 0, 1, 0, 1, 0 ⟩ is affected by the beam splitters with a slight deviation from the theoretical model. Therefore, we can get the following probability distribution of the output states.
To describe the effects occurred when the parameters of dual-mode beam splitters deviate from the given values, we have simulated the system with due regard to the possible deviations in the reflection coefficient and phase affecting the initial system state.
The unitary matrix describing the beam splitter is as follows [19]:
. (7)
In this case, θ is an angle given by the reflection and transmission coefficients, and ϕ is a change in the phase component (see (4), (6)). Thus, any change in these physical variables will occur according to the cosinusoidal/sinusoidal law and on an exponential basis.
a). Impact on a State With a Photon in One Mode
We will consider the beam splitters BS1 and BS5 with a count-down ratio of 1 / 2. We will consider the state as an input state described by two optical channels.
As can be seen in the graphs (Fig. 3), any change in the beam splitter count-down ratio of 50 / 50 leads to a deviation in the probability of the system location in one of the output states or. In this case, a 50% probability is achieved with an insignificant deviation from the theoretical model, and the deviation is almost linear on a scale under study. The phase component deviation has insignificant impact on the system with a photon in one mode and approaches the required distribution with the relatively small deviations.
Next, we will consider the beam splitters BS2 –BS4 with a count-down ratio of 1 / 3.
As can be seen in the graph (Fig. 4), any change in the reflection/transmission coefficient of the beam splitter 33 / 66 also leads to a deviation in the probability of the system location in one of the output states or. In this case, the deviation from the theoretical model increases the probability of one or another state.
Figure 5 shows the dependences diagrams of the probability of a given state for the same beam splitters and deviation ϕ from the given value.
It can be seen that the phase component deviation, as in the case of the 50 / 50 beam splitter, has insignificant impact on the system with a photon in one mode and upon decrease, leads the system to the theoretical distribution.
b). Impact on a State With a Photon in Two Modes or with a Superposition of the Horizontal and Phase Components
Now, we will consider the state as an input state described by two optical channels.
Let us consider the beam splitters BS1 and BS5 with a count-down ratio of 1 / 2 (Fig. 6, 7). Changes in the reflection/transmission coefficient of the beam splitter 50 / 50 leads to a deviation of the probability of the system location in the system states | 0, 2 ⟩ and | 2, 0 ⟩ and occurrence of an additional state | 1, 1 ⟩. Moreover, as the deviation from the theoretical model is becoming more significant, the Hong-Ou-Mandel effect [19] ceases to be observed, when two photons incident on the 50 / 50 beam splitter get out through one mode, since the probability of the state | 1, 1 ⟩ is increased.
Now let us consider the beam splitters BS2–BS4 with a count-down ratio of 1/3 (Fig. 8, 9). It can be seen in the graph (Fig. 8) that any change in the reflection / transmission coefficient of the beam splitter 1 / 3, on the contrary, brings it closer to the values of the beam splitter 1 / 2 under certain conditions that reduces the probability of the system location in the state | 1, 1 ⟩.
Assessment of the gate
geometric dimensions
The geometric dimensions of the gate are important since they determine the dimensions of the required lithium niobate substrates that are typically manufactured with the standard diameters of 3 or 5 inches (7.62 or 12.7 mm).
The overall gate width (Y-axis) is determined by the distance between the six input waveguides. Moreover, it should be considered that at the subsequent gate assembly stage, the optical fibers in the form of a four-fiber assembly should be glued to its input and output planes to be connected to the B-E channels (we should remember that the A and F channels are not used). The V-groove-based optical fiber microassembly has a minimum standard packaging pitch of 250 µm. Thus, the minimum total gate width required to accommodate 6 waveguides (8 intervals of 250 µm, with due regard to the gap to the blank edge) is 2.0 mm. In practice, the gate blank width can be 3 mm due to the blank operation convenience, mechanical strength, etc.
The total length of the gate L (along the photon propagation direction, X-axis) is determined by the lengths of the linear sections δL, the lengths of the rounded (S-shaped) sections ΔX, as well as the lengths of the X-coupler sections ΔL (Fig. 1). In the longitudinal direction, we have six S-shaped sections and three X-coupler sections.
In the Wolfram Matematica software, we have performed mathematical modeling of the f electrically controlled beam splitters and S-shaped sections of waveguides in order to determine their minimum allowable lengths and radii of curvature. It has been found that the minimum length of the X-coupler section ΔL must be at least 8 mm. In this case, the control voltage value does not exceed 8–10 V. The minimum allowable length of the waveguide S-shaped sections should be at least 6 mm. In this case it is possible to neglect the losses related to the leaking modes in the curved integrated optical waveguides [21]. The length of all linear sections between the remaining elements can be reduced to zero, except for the input and output linear sections δL that can be 2 mm from each edge.
Thus, the minimum gate length is 64 mm. This is obtained as: (3 × 8 mm)+ (6 × 6 mm) + (2 × 2 mm) = 64 mm. This formula shows that even on a standard substrate with a diameter of 3 inches can accommodate 3–4 blanks for the gate.
Discussion
The results presented in this paper confirm the assumption about the possible practical implementation of a quantum photon gate based on the integrated optics. The main specifications of a quantum device that determine its performance are optical losses, adjustment accuracy of the count-down ratio in the individual beam splitters, and geometric dimensions. A separate publication will be devoted to the issue of optical losses. In this paper, it is shown that the adjustment accuracy of the count-down ratios within the range of ±0.1% – 0.5% almost does not make any parasitic contribution to the gate operation.
It is found that imperfection of each of the beam splitters makes various contributions to the violation of the two-photon interference conditions. Thus, for example, deviations from the set parameters in the BS2 and BS4 beam splitters have a minimal impact on the gate operation. Next, in order of increasing influence, there are the beam splitters BS1 and BS5, and, finally, the beam splitter BS3. It should also be noted that the required adjustment accuracy of the reflection/transmission coefficients of the integrated-type beam splitters is implementable.
Another important result is the practical assessment of the gate dimensions. The integrated optical elements used on a lithium niobate substrate have a noticeable length. Therefore, from a practical point of view, it is important to evaluate the entire longitudinal dimension of the device in order to determine the substrate dimensions. We have shown that even the substrates with a diameter of 3 inches can accommodate several blanks.
About the authors
Petrov V. M., Doctor of Physical and Mathematical Sciences (radiophysics), d.f.-m.s. (optics), Professor, Department of General Physics, St. Petersburg State University, St. Petersburg, Russia.
ORCID: 0000 0002 8523 0336
Koroteev D. A., student, Department of Photonics, National Research University ITMO, Saint-Petersburg, Russia.
ORCID: 0000-0002-5489-4017
Semisalov D. A., student, Department of Photonics, National Research University ITMO, Saint-Petersburg, Russia.
ORCID: 0000-0003-1757-6519
Strashilin V., student, Department of Photonics, National Research University ITMO, Saint-Petersburg, Russia.ORCID: 0000-0002-0655-0199
Khlusevich D. S. student, Department of Photonics, National Research University ITMO, Saint-Petersburg, Russia.
ORCID: 0000-0002-8298-9451
Yakovlev M. I., student, Institute of Laser Technologies, National Research University ITMO, Saint-Petersburg, Russia.
ORCID: 0000-0003-1757-6519
Parfenov M. V., Junior Researcher, Department of Quantum Electronics, Physicotechnical Institute named after A. F. Ioffe Saint-Petersburg, Russia.
ORCID: 0000-0003-3867-9007
AUTHORS CONTRIBUTION
Petrov V. M.: statement of the problem, discussion of the results; Koroteev D. A.: simulation of leaks and losses in gates; Semisalov D. A.: calculation of the curvature of waveguides and gate beamsplitters; Strashilin V.S.: calculation of the influence of non-ideal gate division coefficients; Khlusevich D. S.: calculation and participation in the manufacture of gate photomasks; Yakovlev M. I.: development and production of an experimental stand for studying gates; Parfenov M. V.: fabrication of waveguides, measurement of waveguide characteristics.
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