Squeezing dynamics of a nanowire system with spin-orbit interaction

We analyze the dynamics of squeezing in a ballistic quantum wire with Rashba spin-orbit interaction in the presence of both strong and weak magnetic fields and for different initial states of the system. Compared to the more standard measure of squeezing based on variances, we show that entropy squeezing is a more sensitive measure. Our results show that there is a strong relationship between the spin-orbit interaction and the strength of entropy squeezing. Furthermore, there is a relationship between the initial state and the number of squeezed components. This allows new knobs to control the strength and the component of entropy squeezing in a nanowire system.

Many recent studies have focused on the physical properties of semiconductor quantum wires because of their potential technological applications and for building quantum computing devices [1][2][3] . Progress towards the use of one dimensional nanostructures in an extensive range of prospective nanoscale devices applications and complex functional architectures has been reviewed in 4 . Various metals/alloys, semiconductors have been explored for nanodevices fabrication using doped semiconducting nanowires 5 . The electronic transport properties of semiconductor nano-objects and their applications for quantum information processing have been explored 6 . Furthermore, the optical properties of quantum dots and wires have been discussed 7 . Past work 8 has also explored the effect of a magnetic field on the spectral and spin properties of a ballistic quasi-one-dimensional electron system with Rashba effect 9 . These studies have shown the potential of nanowire systems for quantum computing applications. In this paper, we therefore study a nanowire system from a quantum information theoretic perspective. Our objective is to explore the behaviour of squeezing in a new system consisting of a nanowire with Rashba spin-orbit interaction in the presence of strong and weak magnetic fields for potential applications in quantum information processing.
Previous studies have explored the effect of magnetic fields on nanowires properties, but have not studied their effect on squeezing of the quantum states. The spectral, transport and conductance properties of ballistic quasi-one-dimensional systems in the presence of spin-orbit coupling and in-plane magnetic fields has been previously analyzed 10,11 . Calculations of the effective g factor of conduction electrons in nanowires subjected to in-plane magnetic fields in the presence of Rashba and Dresselhaus spin-orbit interactions was discussed in 12 . The electronic structure, spin and transport properties of double quantum wires subjected to an in-plane magnetic field by taking into account Rashba and Dresselhaus spin-orbit couplings has also been investigated 13 . Furthermore, it was shown that magnetic field effects on spin texturing in a quantum wire with Rashba spin-orbit interaction introduce additional complex features in spin texturing 14 . In this paper we add to the growing body of literature on nanowires interacting with magnetic fields by examining the dynamics of squeezing in the presence of magnetic fields.
Recent work has focused on atomic squeezing for its potential applications in quantum information theory. For instance, some applications of atom squeezing are in quantum teleportation, cryptography, and dense coding [15][16][17][18] . Atomic squeezing is based on the Heisenberg uncertainty relation (HUR) 19 , which provides a lower bound on quantum fluctuations. The HUR is formulated in terms of the variance (standard deviations) for the system and is not the optimal measure of information squeezing in some circumstances. An alternative approach to quantifying information squeezing is via the entropic uncertainty relation (EUR) [20][21][22][23][24] . The EUR has been applied to study entropy squeezing in various systems including the Jaynes-Cummings model and its generalizations [25][26][27][28] .
Here we compare entropy squeezing and variance squeezing in a nanowire with Rashba spin-orbit interaction interacting with strong or weak magnetic fields. We calculate the evolutions of the entropy and variance squeezing for different initial states of the system and for different strengths of magnetic field. Our results show that entropy squeezing is more sensitive to changes in the initial state and magnetic field compared to spin squeezing. The spin-orbit interaction strongly effects the strength and oscillation frequency of the dynamics of entropy squeezing in both strong and weak magnetic fields. The initial state effects which component (quadrature) is squeezed. Our results thus identify ways to control both the strength and the quadrature of squeezing in the system. The article is organized as follows; in section 2, the Hamiltonian model and the derivation of the time evolution of the density operator of the system are described. Expressions for the evolution of variance and entropy squeezing for the proposed model are calculated in section 3. The squeezing dynamics of the system for different system properties, and initial conditions are discussed in section 4. Finally, section 5 includes a summary and conclusion.

The Hamiltonian Model
In our proposed model, we consider a ballistic quantum wire with Rashba spin-orbit interaction due to the structural inversion asymmetry in a two-dimensional electron gas such as InAs presented in 8 . In the presence of a perpendicular magnetic field, we assume that the wire axis is taken along the y-direction with a parabolic lateral confining potential in the x-direction. Moreover, the external electric field → F is applied along the direction of quantum confinement to generate a Stark shift in the electron spectra. Furthermore, top gates are utilized for controlling the strength of the Rashba interaction 29 .
We first briefly review the Hamiltonian for perpendicular magnetic fields 30,31 . The Hamiltonian can be written as  where m and ω o are the effective mass and characteristic frequency of the parabolic confinement, x y z is the Pauli vector operator, g is Lande's g-factor, and μ B is the Bohr magneton. The perpendicular magnetic field → = B B (0, 0, ) corresponds to the vector potential → =Â xBe y in the Landau gauge. The last term in Eq. (1) represents the Rashba spin-orbit interaction (RSOI), where α r is the Rashba spin-orbit interaction parameter.
To simplify the Hamiltonian model, three length scales to characterize the strength of the lateral confining potential, magnetic field B and SOI are introduced,  Since the Hamiltonian is translationally invariant along the y-direction, the momentum component p y can be replaced by k. Then, the Hamiltonian in units of ω o  for a given k can be expressed, by using creation and annihilation operators of a shifted harmonic oscillator, † a and a, in the following dimensionless form: The parameters in the above equations are defined as  spin-orbit interaction in H′ leads to coupling of neighboring energy subbands. Also, the presence of the magnetic field produces a lateral shift in the wave function and the renormalization of the oscillator frequency Ω.
In the absence of an external electric field and kl o ≪ 1, we have ξ 1 = 0, and in the case of a strong magnetic field (l B ≪ l o ), Eq. (3) becomes the exactly integrable Jaynes-Cummings model 16 in the rotating-wave approximation 8 , In our work, a nanowire is equivalent to a two-qubit system. So, we can assume that the electrons occupy the lowest energy state {|0, ↑〉,|1, ↓〉} which consists of the spin degenerate ground and first excited eigenstates related to the confinement in the x direction. We consider that the energy state of spin is initially in a superposition state i The corresponding initial density operator at t = 0 is The density operator of the proposed model for t > 0 is then where C o (t) and C 1 (t) can be written as In the following section we use the above equations for the evolution of the density operator to study entropy squeezing.

Variance and Entropy Squeezing
The study of information squeezing depends on the Heisenberg uncertainty relation 20 z We can define the entropy squeezing for a two-level energy state by using the quantum information theoretic measure of entropy [25][26][27] . The information entropy is defined as By using the density operator ρ(t), we can write the information entropies of the component operators S x , S y and S z in the form z 11 11 22 22 where ρ 12 (t), ρ 11 (t), ρ 22 (t) and ρ ρ = † t t ( ) ( ( )) 21 12 are calculated from the following equations,   and 12 Here R(t) and V(t) are given by

Results and Discussion
We now discuss the effects of the initial state and spin-orbit interaction strength on entropy squeezing versus variance squeezing for a quantum wire system with different strengths of the magnetic field. Field (l B ≪ l o ). In our computation, we consider InAs with α r = 1.0 × 10 −11 eVm, g = −8, m = 0.04m o and l so = l o . In Fig. (1), we plot the time evolution of entropy squeezing and variance squeezing for a quantum wire with SOI in a strong magnetic field l B = 0.25l o and l so = l o . The initial state corresponds to an excited state with θ = 0 and the relative phase φ = 0. In Fig. 1(a), we observe that there is squeezing in E(S x ), while no squeezing occurs in the other quadrature E(S y ). In contrast, it is clear from Fig. 1(b) that no squeezing occurs in either variance V(S x ) or V(S y ). Furthermore, the qubit inversion W(t) 33 , which is defined as the difference between the final state (|C 1 (t)| 2) and the initial state (|C 0 (t)| 2 ) is plotted in Fig. 1(c). From Fig. (1), we can conclude that the information entropies have more information than the variances of the two-level energy state. Moreover, it shows that entropy squeezing factors is a best measure of information squeezing state than variance factors. We study the effect of an initial superposition state on the entropy squeezing and variance squeezing in a strong magnetic field in Fig. 2. We start with an initial superposition such that θ = π/2 and the relative phase φ = π/2. In Fig. 2, squeezing of both entropy and variance are clearly observed. However, there are differences between entropy squeezing E(S x,y ) and variance squeezing V(S x,y ). From Fig. (2), the amount of squeezing E(S x,y ) is greater than V(S x,y ). Therefore, the information entropy appears to be a more sensitive measure of squeezing. Comparing Figs (1) and (2), we also note that if the evolution starts with an excited state, the squeezing occurs only within one component. On the other hand, when the initial state is a superposition, squeezing occurs in both components. Field (l B ≫ l o ). Our computations for a weak magnetic field are similar to those for the strong magnetic field with slight differences achieved by using the following numerical parameters, Figure 3 shows the time evolutions of both entropy squeezing and variance squeezing for a quantum wire with SOI in a weak magnetic field with l B = 5l o and l so = l o . The initial states in the excited state. From Fig. 3(a), squeezing occurs in E(S x ) but not in E(S y ). Similarly, squeezing in V(S x ) is observed, while no squeezing occurs in V(S y ) as shown in Fig. 3(b). From Fig. 3, we can conclude that the amount of squeezing in E(S x ) is greater than in the variance V(S x ). For a weak magnetic field l B = 5l o , if the system starts in a superposition state θ = π/2 and φ = π/2, then squeezing occurs in both E(S x ) and E(S y )as shown in Fig. 4(a). In contrast, Fig. 4(b) shows that squeezing occurs only in the variance V(S x ). To sum up, the entropy squeezing is a more sensitive measure of squeezing than variance

Effect of Spin-orbit Interaction Strength (l so ) on Squeezing in both Strong and Weak Magnetic
Fields. The Spin-orbit Interaction strength (l so ) influences the oscillation frequencies and the strength of entropy squeezing in both strong and weak magnetic fields. Figures (5) and (6) show the time evolutions of entropy squeezing factors E(S x ) and E(S y ) for our quantum wire system in a strong magnetic field (l B = 0.25l o ) and weak magnetic field (l B = 5l o ) respectively, as the spin-orbit interaction strength is varied. The initial state is a superposition state with θ = π/2 and φ = π/2. We can see that the entropy squeezing factors E(S x ) and E(S y ) periodically oscillate with a phase difference of π/2 and their amplitudes change as the spin-orbit interaction strength varied. When the spin-orbit interaction strength (l so ) increased, the amount of entropy squeezing is increased and the number of periodic oscillations disappeared gradually. Field (B = 0). We now discuss the influence of spin-orbit interaction strength and the initial state of the system on the entropy squeezing in a quantum wire when the magnetic field vanishes (B = 0). In this case, the vector potential → = = A xBe 0 y and the Hamiltonian for the quantum wire system in Eq. (1) can be written as  (7) and (8)). Also, we plot the effect of spin-orbit interaction strength (l so ) on entropy squeezing of a quantum wire without a magnetic field (B = 0)in Fig. (9). From Figs (7-9), we can conclude that the effect of spin-orbit interaction strength and the initial state of the system on the information entropy squeezing of a quantum wire without a magnetic field are similar to the results of a weak magnetic field in (4.2) and (4.3), as expected.

Conclusion
In this paper, we have investigated the information entropy squeezing in a ballistic quantum wire with Rashba spin-orbit interaction in the presence of both strong and weak magnetic fields when the initial state is either in an excited or a superposition state. Our results show that the entropy squeezing is a more sensitive measure of information squeezing compared to variance squeezing. Furthermore, we have explored the effects of spin-orbit interaction strength and the initial state of the system on the information entropy squeezing for different strengths of the magnetic field. The results show that there is a strong relationship between the spin-orbit interaction and the strength of entropy squeezing. When the strength of the spin-orbit interaction is increased, the strength of the entropy squeezing is increased and vice versa. Additionally, there is a relationship between the initial state and the number of squeezed components. If the system starts in an excited state, the squeezing will only occur in one component or quadrature. On the other hand, when the system starts in a superposition state, squeezing can occur in both quadratures. Thus we have identified new ways to control the strength and the component of entropy squeezing in a nanowire system compared to previous work. This has potential applications in future quantum information technologies.