Quantum Time Synchronization for Satellite Networks

Establishing accurate time standards across the globe is very important. The success of any real-time task depends on maintaining time synchronization in its system. Space technology helps in enabling global dissemination of time. In this paper, we propose a model that uses satellites to transfer the time information extracted from three qubits that are precisely synchronized using quantum synchronization. By applying an external field with wavelength 813.32 nm we can synchronize the three qubits (each carried in another satellite) to oscillate at the same frequency. We can ideally achieve a precision of 1.6 × 1015 signals per second, and show the corresponding Allan deviation curve to analyze the stability of our system for different noise strengths. We introduce the possibility of using quantum synchronization on satellite-carried clocks to distribute accurate time and frequency standards.


INTRODUCTION
Punctuality is a trait desirable not just in humans but in the devices that have been incorporated into our daily life. It is not wrong to equate time to money because millions of dollars can be lost if some applications are delayed by mere milliseconds. Ensuring such timeliness requires that all clocks 978-1-6654-9032-0/23/$31.00 ©2023 IEEE tracking the system must at least coincide. Not following the same timetable and functioning as a coordinated unit could cause a system to fail. In extreme cases, it can be lifethreatening, for example, in manufacturing industries where humans work along with automated processes dealing with hazardous chemicals or heavy machinery. Therefore, it is not enough to have one accurate clock but maintain this accuracy and synchronicity across all the concerned clocks.
To maintain time synchronicity around the globe, we need reference systems, such as Coordinated Universal Time (UTC), consisting of the most accurate and stable clocks. Global navigation satellite systems (GNSS) require a similar level of precision to sustain navigation performance. Hence, they use a super stable and accurate ensemble of atomic clocks. Due to their timing capabilities and inexpensiveness (compared to sophisticated atomic clocks), many applications use GNSS-receiving devices as a reference standard. For example, the Precision Time Protocol (PTP) is a time synchronization protocol for Local Area Network (LAN) that uses GNSS user equipment as the Grand Master Clock for tuning all other clocks in the network.
With the widespread use of GNSS time receivers, it is essential to investigate the source of this timekeeping, i.e., the atomic clocks and enhance their properties to provide a more precise global reference timing. An atomic clock uses the resonant oscillation frequencies of atoms as its clock resonator. It measures an indefinite time period by comparing it to a periodic signal of a known frequency. Atoms inside the atomic clock behave as a qubit. A qubit is any particle, like an electron, proton, photon or atom, assumed to be in two distinct states or remain in a linear combination of these eigenstates. Recently researchers have shown that entanglement can provide clock synchronization by entangling several qubits located at different network nodes [1]. Entanglement is a property where particles are correlated in such a way that performing a measurement on one particle will give us a probability of 100% to predict the measurement of the other particle. But generally, entanglement does not occur with 100% certainty in real-life scenarios, as experimental conditions are not optimal, and the setups are prone to errors [2].
The motivation for this work is to explore the possibilities of a more precise and stable clock that is the basis for providing time synchronicity at a global scale. We hope to design a more practically realizable time synchronization system that will benefit different industrial and research sectors such as satellite navigation, communication networks, radio systems, stock (and investment) markets, exploring fundamental sciences etc. We propose using quantum synchronization for precise time simultaneity. In quantum synchronization, individual local phase oscillators can synchronize their phase oscillation frequencies with each other or to that of an external oscillator, provided each oscillator is self-sustained. We assume our self-sustained oscillators to be qubits [3]. Further, a set of synchronized qubits can show a concurrence close to one, representing entanglement between particles [4]. Particularly, Roulet and Bruder [5] showed that quantum synchronization implies entanglement between the oscillators, but the converse statement is not necessarily true. Thus, our method can be beneficial and practical in realizing an accurate and stable timekeeping device as a global reference standard.
The structure of the remaining paper is as follows. In section 2, we discuss the theoretical background that forms the basis of this work. Section 3 provides the detailed description of the proposed idea. We analyze and further discuss our proposal in section 4. In section 5, we present different applications that could benefit from our solution. Lastly, section 6 summarizes and concludes our work.

Clock Technology
Clocks use oscillators to produce a periodic signal with a specific frequency. The underlying design and physical features of the oscillator can effect stability of this frequency and its generated time period. We cover some of the main classes of oscillators as given in the following: • Quartz crystals: These popular oscillators are used in many electronic devices, including GNSS receivers that uses it to enable radio frequency (RF) signal processing and to create the clock. Aging and environmental conditions affects the quality of quartz oscillator in terms of accuracy and stability. Considering environmental conditions (like temperature), its absolute frequency accuracy can be around 10 −6 to 10 −7 and stability around 10 −10 to 10 −12 [6]. • Atomic Standards: Its fundamental concept is coherently stimulating transitions between two energy stages (say E low and E up ) of the chosen atom and check if the transition took place. Such atomic transition's resulting frequency (ν) is: where h is Planck's constant. The interrogation signal needed for inducing atomic transition is produced by a local oscillator (generally quartz crystal) of the atomic clock unit. By combining this local quartz oscillator with a group of atoms, atomic clocks can obtain high stability. Rubidium clocks have frequency drift rates above 10 −10 per month leading to absolute accuracy to few portions in 10 −9 , while Cesium-based clocks are more accurate with values as low as 5 × 10 −13 [6]. • Optical clocks: Commercially available atomic clocks operate in radio frequencies (GHz) and are limited by this operation frequency. Optical clocks have shown to outperform them as they operate at frequencies in the THz range. They have produced fractional uncertainty of less than 10 −18 under laboratory conditions [7]. Due to their superior ability to segment time into considerably smaller units than Cesium clocks, they have long been a subject of debate in attempts to define the second even more precisely.

Quantum Basics
Quantum science focuses on the laws of physics describing particles whose fundamental physical properties are quantised. In this paper, we consider a qubit as the basic quantum unit. A qubit is any particle that can be found in one of the two (energy) eigenstates or in a linear superposition of these two states. Eigenstates represent the basis states in which the particle can exist. We consider an atom, whose states we denote as the lower state (|1⟩) and the upper state (|0⟩).
Before showing synchronization of each qubit's oscillation frequency we must understand what oscillation for a qubit means. For this purpose we use the Bloch Sphere which can visually show how a qubit performs phase rotation while evolving in time. We can represent the lower state of a particular qubit on the Bloch sphere using a state vector from the centre of the sphere to its lowest point. Similarly, we can represent any pure state using the state vector from the centre to any point on the sphere's surface. In contrast, we can illustrate mixed states using the state vector representing any point inside the Bloch sphere. With Hadamard operation, we can transform our qubits into a superposition of two energy eigenstates. The Hadamard operator is a unitary operator which changes the position of a state vector from the lower state to a point on the equatorial plane of the Bloch sphere, as shown in Figure 1. Now, we let the state evolve in time, which can be interpreted as a state vector rotating on the equatorial plane of the Bloch sphere. It thus acquires a phase factor that depends on the system's Hamiltonian.
To have perfectly synchronized clocks, we must let each qubit of the system evolve at a similar phase so that the phase difference between them would be close to zero. A system of qubits oscillating at similar frequencies will give similar time information and thereby we can achieve time synchronization. The method of quantum synchronization can provide precisely synchronized oscillation frequencies of individual qubit that are fundamental in nature.

Figure 1. Phase rotations on Bloch Sphere
Quantum entanglement is an intrinsic correlation of particles in a way that, knowing the measurement outcomes of one particle, can let us predict the measurement outcome of other. Let us consider a set of two qubits q 1 and q 2 with their wave functions as ψ 1 and ψ 2 respectively. Using Dirac's notation, we can represent the wavefunctions of the qubits as: One of the possible entangled states of these two qubits can be written as: where c 1 and c 2 are complex numbers. On measuring ψ Entangled , we find it to be in the state of |0⟩ q1 ⊗ |0⟩ q2 with a probability of c 2 1 or in the state |1⟩ q1 ⊗ |1⟩ q2 with probability c 2 2 . We can observe that whenever we measure the system, it randomly chooses one of the two states in which either the qubits are in the lower or in the upper state. This type of correlation has never been observed in the classical world description. These observations are not restricted by the distance between the qubits. Researchers have also experimentally demonstrated synchronized time in atomic clocks using entanglement over a significant distance between satellites [8].
The term synchronization was first coined by Huygens in 1673 [9]. He observed two pendulum clocks with slightly different frequencies oscillating at a common frequency, provided they are connected by an elastic material. In our daily life, synchronizing our watches can be understood as setting it equal to a reference clock at a particular point of time. Then we can rely on the frequency accuracy and stability of our watch until we do another synchronization. Basically, the systems that are to be synchronized should be self-sustained. They should show limit cycle oscillations i.e, their dynamics should show a closed loop on the phase space [9]. Further, these limit cycle oscillators have tendency to attract the neighbouring trajectories on the phase space as shown in the Figure 2. In the past few years, the study of synchronization in the quantum realm has been important for comprehending correlations and for applications in quantum networks [10]. In the quantum regime, the synchronization takes place between phase oscillation frequency of particles, such that the phase of individual oscillators like qubits start oscillating with a common frequency. Satellites [11], electrical grids [12], and clocks [13] are just a few examples of the many use cases of such synchronous dynamics.

OUR PROPOSAL
Theoretically, GPS requires at least three satellites to cover the whole earth for any given time. Therefore, our proposed framework comprises three satellites for full global coverage, as shown in Figure 3. Each satellite carries a qubit that will function as a clock. The three qubits in our space system can oscillate at distinct frequencies represented as ω q1 , ω q2 and ω q3 . These oscillations depend on the energy difference between the two quantum states of the respective qubit. Different types of physical protocols can help us to extract the information of time from these fundamental oscillations. They must show similar oscillation frequencies to have similar timely information. However, it is difficult in a real-life scenario where quantum systems are always prone to errors, and the environment is driven and dissipative. Thus we propose to use quantum synchronisation methods to provide global reference time by synchronising the oscillation frequency of individual qubits to a particular frequency, which can be the oscillation frequency of an external optical field. We must consider that quantum synchronisation only occurs when the difference between the oscillation frequencies of the external field and individual qubits is significantly less. To be particular, it depends on the external field strength f and the difference of the oscillation frequencies ω EF − ω q , such that we can observe that field strength enhances the quantum synchronization up to a particular limit [14]. Thus, for our proposed architecture, we must use a monochromatic coherent optical field between the satellites. This external field would synchronize the individual qubits to a particular frequency.
It is important to note that due to lengthy communication distances, we use the laser as the light source for most satellite interconnectivity. It can create an effective linkage for communication between satellites because of its unique properties. These properties include the production of monochromatic radiation with well-defined wavelengths and a focused, highly directed light beam. As per communication is concerned, these qualities are necessary to provide improved safety and less beam deterioration. A practical case scenario for these systems has already been demonstrated for satellite communication [15]. Hence, in this work, we want to use these already investigated configurations to establish quantum synchronization between qubits at three satellites.
Space optical communication systems mostly employ solidstate lasers because they can create connections over distances larger than 40 000 km [16]. The most frequently used solid-state lasers for these long-distance connections are crystal and semiconductor lasers, also known as laser diodes. Different kinds of lasers are categorized according to their material used for the photon creation. The type of laser is selected according to the established link's parameters, such as length, height, environmental circumstances (losses present or absent), and the needed receiver's power level. We should mainly investigate if these wavelengths are comparable to the atomic transitions that we would utilize for synchronizing atomic qubits' phase oscillation frequency. Specifically, we can use satellite communication links similar to PASTEL and OPALE, which are optical payloads on SPOT4 and ARTEMIS, respectively. Their setups are identical and were developed under the ESA project [15]. These optical payloads, which essentially weigh 160 Kg, consist of a telescope with a 25 cm lens diameter; an optical bench with a very fine pointing system; a communication sensor; a laser diode (Aluminium Gallium Arsenide (AlGaAs)) with output power 60 mW producing a coherent monochromatic light of wavelength 847 nm, and a thermal control system for accurate temperature control. These satellite links can be crucial in allowing the optical drive to connect with other system satellites. We covered the technical aspects of forming an inter-satellite optical link because we need to establish a laser connection between the satellites of our model to synchronize their respective qubits.
Another important factor in solidifying this architecture is to have an atomic qubit that can synchronize with the external field frequency. We suggest using Thulium atoms, a member of the Lanthanides group with atomic number 69, as it is currently being studied for making optical lattice clocks [17]. Our preference for an optical lattice clock over the conventional atomic clock is because it has surpassed the primary Caesium criteria in stability and precision [18]. The fractional frequency uncertainty of optical clocks for a variety of atoms, such as Sr [19], Yb [20], Yb+ [21], and Al+ [22], can reach a low level of 10 −18 .
Optical lattice clock investigated by [17] used the inner-shell magnetic dipole transition |J = 7/2, F = 4, m F = 0⟩ → |J = 5/2, F = 3, m F = 0⟩. Here, F is the total angular momentum, i.e. F = I + J, (where I is the nuclear spin angular momentum and J is the total electronic angular momentum) and m F is the magnetic quantum number associated with F Z . [17] used the fine-structure components of the atomic states to trap atoms in spatially distributed optical potentials using a wavelength of 813.32 nm. This wavelength is called the magic wavelength. It is the wavelength at which the system can perform resonant Rabi oscillation after interacting with the electric field component of the optical field. This locking ultimately provides a clock transition frequency at a wavelength of 1.14 µm. The results allow the development of lanthanide-based optical clocks with a relative uncertainty of 10 −17 .
Thus, our proposed model assumes that Thulium based atomic qubits confined in an optical lattice structure are our clock qubits. The laser diode (AlGaAs) produces an optical field as our external oscillator for synchronizing qubits. A detailed description of quantum synchronization for our system is described in the following.

Quantum Synchronization
Our atomic qubits are supplied with an external optical field f . These qubits are assumed to behave according to the free Hamiltonian H 0 given by: where Ω is the Rabi Oscillation frequency, ℏ is Planks constant and σ z is one of the Pauli's Spin-operators: where |0⟩ and |1⟩ are the upper and lower energy eigenstates, Figure 3. Three satellites, which ideally provide coverage over whole Earth, carry qubits that are synchronized by high energy laser fields respectively.
We use the one-drive coupling method [3] to synchronize the phase oscillation frequencies of the individual qubits with the frequency of the external optical field. The interaction between a two-level atom and an external field can be represented by the Rabi Hamiltonian as [23]: φ j is the phase of the driving laser relative to zero of the clock being synchronized. Extending the protocol for a system of three qubits will lead us to the driven Tavis-Cummings Hamiltonian [24]: where the first four terms are describing the photons in the resonator and the three qubits. g gives the coupling between the qubits and the photons, and the last term represents the driving field. In presence of dissipation, the system dissipation rate is λ whose quality factor is nearly Q = ω 0 /λ ∼ 100. The driving force amplitude is expressed as f = ℏλ √ n p , where n p is the number of photons in the resonator at the resonance ω = ω 0 (when g=0). Generally measurement outcomes of such an interaction is not a single probability but a statistical ensemble of probabilities. To deal with such a situation we will use density matrix formalism of quantum mechanics. Hence, the evaluation of such a system considering its dissipative nature can be described by the Master Equation [25], [26], [27] : where ρ is the density matrix of the corresponding atomic system, ℏ is Plank's constant,â is the annihilation operator andâ † is the creation operator. To analyse the system properties we can use the spectral density of laser driven qubits defined as [25]: If the qubits are oscillating with frequencies Ω 1,2,3 the corresponding spectral density will contain peaks at these frequencies respectively. Once the qubits are synchronized with the laser frequency ω, the spectral density will show a dominance at ω. In order to simulate S(ν) we first reshaped the master equation (8) as a matrix vector multiplication: where consists of all the entries of the matrixρ. The components can be expressed as where i = ⌊ a N ⌋, j = a mod N , m = ⌊ b N ⌋ and n = b mod N . Assuming that the stateρ converges to a steady state much faster than the time evolution of the Hamiltonian, we get the (steady-state) solution by determining theρ or ⃗ ρ satisfyinġ Hence, the null space of V V V (t, g) has to be determined, which can be done by a single value decomposition. Since there is only one unique solutionρ, the null space obtained is always one-dimensional. The solution has to be normalized so that tr(ρ) = 1, since this is a general criteria for a density matrix. The solutionsρ(t, g) for different t, g can be used for calculating the spectral density by equation (9).
We calculated the spectral density for an area of frequencies ν and couplings g as depicted in Fig. 4, by using four photons, ω = ω 0 = 1.0, a dissipation rate λ = 0.001 and Rabi-Oscillation frequencies Ω 1 = 1.1, Ω 2 = 1.2 and Ω 3 = 0.9. We see that there exist couplings in the range g = 0.45 to g = 0.5, where mainly the frequency of the external field is dominant, rather than the Rabi Oscillation frequencies of the qubits, which means that the qubit oscillations synchronized with the external field.
As previously discussed, optical lattice clocks use the electric field strength to trap atoms in the spatially distributed periodic potential of two counter-propagating optical fields. Quantum synchronization can help us in synchronizing the atomic qubits' oscillation frequencies in these potential traps (of optical lattice) and thus provide precise and stable time measurements. We suggest extending the experimental configuration provided by [28] from one fixed place to our proposed system of three satellites.

ANALYSIS & DISCUSSION
For stability analysis of our system, we would assume that a perfectly synchronized clock is producing 'ticks' that are uniform in time. The interval between ticks of the perfect clock is τ , so that the ticks occur at times 0, τ, kτ, (k + 1)τ, etc. Here k is some integer. The time of the clock under test is reading its time, each time the standard clock emits a tick, and the time-differences are x k , x k+1 , ... at time k τ , (k + 1) τ , ..., respectively.
The frequency y of a clock is defined as the fractional frequency difference between the clock under test and a perfect clock operating at some nominal frequency. For example, if the physical frequency of a clock under test is F then, where F 0 is the physical frequency of the perfect device taken for comparison. The above equation would provide the same outcome for any value of k if the test device had a frequency that was fixed in relation to the perfect clock.
It should be noted that the test device need not to operate at the same frequency as the ideal clock. If the frequency difference were to be any constant number, we would still have the same outcome for every value of k. Using the above definition we can also say that frequency of a clock is a dimensionless parameter. Since instantaneous frequency cannot be measured, every frequency or fractional frequency measurement necessarily entails a sample period or a timewindow during which the oscillators are monitored. Let's imagine we want to calculate the fractional frequency y(t), starting at some moment t and ending at a later time t + τ . The difference in these two time deviations with respect to τ will provide the average fractional frequency over that period: In the period between measurements, the estimated average frequency of the clock being tested in relation to our ideal clock is This equation calculates the frequency difference of the test device between two successive equal measurement intervals with no dead time in between. The two-sample standard deviation, sometimes referred to as the Allan deviation, is calculated by taking the square root of the two-sample zero dead-time variance. If there are N time difference data with indices 1, 2, . . . , N , then the Allan variance at averaging time τ is defined as the average of the N − 2 calculations as: (15) and the square root of this value is the Allan deviation. Fig.  5 shows the Allan deviation for a signal having the transition wavelength λ = 1.14µm of the qubits and a gaussian distributed random noise with a standard deviation of 0.05, i.e. the phase of the oscillation mainly takes random values in [ωt − 0.05, ωt + 0.05], where ω = 2π/λ. Table 1 shows the average number of days required for an Allan deviation of 10 −16 for several strengths of noises (given by the standard deviation of the gaussian noise function). We see that the number of days increases fast. The realistic strength and type of the noise is not yet known, in future studies one has to check wether the noise strengths stated above are realistic.

APPLICATIONS
There has been an increase in time-sensitive applications spanning large distances that need accurate time-tracking • Conventional radios need precise time synchronization of their clock to broadcast their signal. They are equipped with narrowband receivers and amplitude-modulated time signals to synchronize with an accurate reference clock. These radio clocks have low-cost signal processing and can only identify the start time with some realistic inaccuracy of few milliseconds. They start correctly right after a full synchronization and steadily deteriorate from there until the next synchronization. Our proposed technique can provide more precise and dependable time synchronization for a greater geographical area. As a result, we can minimize the transmission delay of broadcasting across the globe.
• Precise timing helps in the investigation of physical constants and the clarification of the issue of whether or not they maintain their properties across time. For instance, Shimon Kolkowitz of the University of Wisconsin-Madison is creating a gravity experiment using an optical lattice clock, a more accurate atomic clock. Kolkowitz is working to create a timepiece with two optical lattices in which one optical lattice will collapse while the other is horizontally accelerated at the same pace. In doing so, he may test Einstein's equivalence principle from general relativity, which states that acceleration due to gravity and acceleration due to nongravitational forces are equivalent.
• It is essential for trading clocks to be exactly synchronized, especially in the area of high-speed trading with stock exchanges. Investigators need this to distinguish the sequence in which trade requests in the banking sector are received and carried out. Thus, a synchronous time information can help us in identifying market abuse. Such precise time monitoring is needed because traders can use sophisticated, powerful and speedy computer networks to control the pricing of the assets and can possibly manipulate the market.

Future goals
It may take some time (perhaps over a decade) to practically implement quantum synchronized clocks in space. But once we achieve this we can provide highly accurate time and frequency information which will help in realizing many applications in the near future, such as: • Services based on geodesy and astronomy need the faithful comparison of distant optical clocks' frequency. With quantum synchronized optical clocks enabling high resolution international clock comparisons, a new concept of time and an all-optical distribution of the SI-second can be made possible. The geological processes of the Earth can also be better understood with better clocks.
• Work on fundamental physics theories will be enabled with the development of stable and accurate clocks. Physics researchers could learn if time is actually continuous or essentially granular with the aid of more precise clocks.
• By disseminating clock frequency over the novel optical fiber connections we can use the enhance clock for detecting dark matter, gravitational waves, earthquake etc.
• An improved clock technology will also support the evolution of new and high performing GNSS system which further encourages the development of universal time dissemination.

CONCLUSION
We proposed a model consisting of three satellites, each carrying one qubit representing an optical clock. Initially, each qubit oscillates at different frequencies. After applying an external laser field of a particular oscillation frequency, all the qubits start oscillating at this specific frequency. However, to ensure a synchronized frequency of oscillation amongst the qubits, we must maintain an inter-satellite laser link of sufficiently high energy between them. We simulated the spectral density inside the resonators, which proved we could achieve synchronization for the three-qubit setup. Using a laser field of 813.32nm results in the clock's transition frequency at a wavelength of 1.14 µm. This means that ideally we can achieve a precision of 1.6534·10 15 time-signals in one second. To evaluate the stability of our system, we plotted an Allan deviation curve. However, using an exact noise model of the Thulium-based optical clock is beyond the scope of this work. Therefore we calculated the number of days required for a stable Allan deviation for the gaussian noise model with different noise strengths to show which noise strength would be required to achieve good clock stability. With increasing noise strength, the number of days required for good stability increases significantly.
Our focus in this work is to introduce the idea of using quantum synchronization to precisely synchronize the oscillation frequencies of long-distant clocks and thereby extract accurate time data from them that we can transfer around the globe. A perfectly synchronized timekeeping can enable future use cases requiring ultra-low latency communications. The purpose of this paper is to present a starting point for applying quantum synchronization over satellites to possibly achieve accurate global time standards.