Simulation of Laser Manipulation of Bloch Vector in Adiabatic Regime

Simulations of the Bloch vector manipulation of two operations (one that utilizes the geometric phase (geometric operation) and the other that does not utilize the geometric phase (simple operation) in the adiabatic regime enabled us to study the tolerance of the geometric phase utilized operation against the laser noise. We simulated the Bloch vector manipulation by the laser field with the amplitude, the frequency, and the constant phase are proportionally, integrally, and derivatively controlled. The manipulation of the Bloch vector is a /2 rotation around 2-axis. There are few advantage by using the operation of the geometric phase over the simple operation. It has an advantage over only when the geometric phase operation is utilized such that the magnitude of the frequency of the noise is large, and that the noise correlation time is small.


Intrtoduction
The realization of quantum computers will require long coherence time for materials and low error rate for control operations.[1][2][3] The low error rate for quantum computers requires a series of accurately controllable quantum operations.The field that manipulates a Bloch vector (a two-level atom) usually has noise components.Therefore, a noise-free control method (operation) is needed for the accurate control of the Bloch vector.To control the Bloch vector accurately, it is necessary to know the tolerance of the control operation when the field is noisy.After the Berry's adiabatic geometrical phase (GP) [4] has been introduced, many papers have appeared concerning to the basic science and applications [5,6].The GP operations have been considered to be less sensitive than the operations without using GP against noises of the control fields.Therefore, the GP utilized Bloch vector manipulation was proposed as the low error rate manipulating method.However, no clear conclusion seems to have been obtained about the usefulness of the GP utilized operation [7][8][9][10].Also, there seems to have been a few discussions which noise has an influence on the GP operation among the amplitude, the frequency, and the phase noises of the optical field.One simple way to see the tolerance of the GP operation is to simulate both GP and dynamical operations and compare their tolerances.Here, we numerically studied and compared the tolerance of a GP operation a dynamical operation against the laser noise in the adiabatic regime.The simulation was performed while imaging a very universal model (a two-level atom).There are so many real physical systems corresponding to a two-level atom: Gas (Rb gas, Cs gas), rare-earth ion doped crystals, and semiconductor quantum dots even in the optical regime [11][12][13].

Calculation Procedure
To make clear which of the two operations (the GP operation and the simple operation) is less affected by the laser field noise, we performed the following calculation.We consider a two-level atom irradiated by an optical field.The wave function of the two-level atom with the two energy eigenstates ) (r The excitation optical field has the amplitude 0 E , the center angular frequency . For simplicity, in the numerical calculations, the optical pulses have square envelopes with the same amplitude if the optical fields are noise-free fields.Following the procedure of Meystre and Sargent III [14], we obtain the solutions for the probability amplitudes ) (t c a and ) (t c b for the excitation optical field with the duration t as: with

Simulation of laser manipulation of Bloch vector in adiabatic regime
Here, 0 R is the Rabi angular frequency, the detuning , and . The real optical field usually has the amplitude, the frequency, and the phase noises simultaneously.In the simulation, however, we intentionally introduce only one of the noises of the laser field during the Bloch vector manipulation.The simulation will clarify which noise will affect the accurate control of the Bloch vector.
We simulate some noises of the proportionally, integrally, and derivatively (PID) controlled laser field by using the random numbers with the help of the PID loop.
Each laser field is temporally divided into N pieces of sub-fields with the pulse duration /N.Within the sub-field, the noise is constant.The parameter N /  ( is the pulse width) corresponds to the noise correlation time.The Rabi angular frequency 0 R and the detuning  can be defined in the unit of the Rabi angular frequency noiseless R on resonance given by the noise-free optical field: The fluctuation of the phase factor  around 0  is given by Here, we call In the simulation, uniform random numbers have been often used to simulate the noise of the control field [9].We consider that the random numbers defined in the following are more realistic to describe the noise of the control field.
Using a linear congruential method, we first generate uniform random numbers by the relation . We also used another set of parameters for the random numbers.We obtained similar simulation results, and the conclusion does not change.The PID controlled random numbers which we use here are generated using the random numbers n  's which are created by the uniform random numbers n x with their values between -1 and 1: . The random numbers n  's thus generated simulates the noise characteristics of free-running lasers better than the uniform random numbers.The (n+1)-th PID random numbers , and in the following relations.)are given by the following relations: .At this point, one might consider that the noises generated by above-mentioned process are physically meaningless because the noises are not generated by a particular set of equations with a particular set of parameters.For example, one might consider that we should use the Langevin-type equations to generate noises with finite correlation times.However, the laser noise properties of the amplitude, phase, and frequency are different depending on the property of the laser system.Thus, the simulation results with specific noises which are generated by a particular set of equations with a particular set of parameters only represent one specific case.
The random numbers n e 's created by our method deviates from the starting value zero as the number n increases and behaves like the steps of a drunk [16].To correct and reduce the deviation of the random number from the initial value, the PID control is performed.The PID controlled random numbers reproduce the jitter correction process of a laser system.The testing two operations rotates the Bloch vector around 2-axis by an angle /2.One operation rotates simply the Bloch vector around 2-axis with a pulse area /2 by a laser field (Here, we call this operation as the simple operation.).The GP operation on resonance optical pulse ( 0   ) that rotates the Bloch vector around 2 axis for an angle of  is given by the optical pulse sequences:  A ) corresponding to the simple operation (the GP operation) is given by the products of N (4N) matrixes.To estimate the tolerance of the operations against the noisy optical fields, the fidelity is often used.The definition of the fidelity F between two wave vectors . However, the fidelity between two wave vectors  and   gives the same fidelity as that between  and  .Therefore, the fidelity defined above cannot distinguish between  and   . Thus, when the noise field is strong, the fidelity may be not a good measure to estimate the tolerance of operations.Here, instead of using the fidelity, we use the following error value for the tolerance estimation.The matrix elements ) of the two operations thus obtained are different from those for the noise-free optical pulse.Let the difference be denoted by We define the error as .The error thus defined is not affected by the initial state of the Bloch vector, and gives a measure for the tolerance of the operations against the noise of the optical field.Then, m error  was averaged for ~10 5 times.

Numerical Results and Discussions
First, we show and discuss the results when Kp = 1. Figure 1     .The tendency is observed that cross N becomes small as the noise amplitude  A becomes large.We presently have no clear idea to explain why this inversion of the magnitude occurs.Above results are for the case where the feedback coefficient 1  Kp .We also simulated Bloch vector manipulation and calculated the errors    Figure 6 shows the N dependences of averaged errors    GP error and    Simple error for the phase noise with  A = 0.003.The squares, triangles, and circles correspond to the averaged errors for Kp = 0.3, 1, and 1.7, respectively.The solid and dashed curves correspond to the errors with the simple operation and those with the GP operation, respectively.We immediately notice that the GP operation is very weak against the phase noise.Also, it is observed that the errors    fluctuates very fast in the time domain (the case of Kp = 1.7), the amplitude will be larger than the settled amplitude (the initially settled amplitude without noise) in a short time region, and the amplitude will be smaller than the settled amplitude in the next short time region.The fluctuation of the deviation of the amplitude from the settled amplitude will change rapidly and the error of the rotation angle will be cancelled and will not accumulate.Thus, the final rotation angle will not deviate largely from the settled angle.
If the noise component of the phase or the frequency changes rapidly in the time domain (case of Kp = 1.7), the rotation direction of the Bloch vector shifts from the settled direction and fluctuates.However, due to the fast fluctuation of the rotation axis around the initially aimed direction, roughly speaking, the Bloch vector will rotate around the initially aimed direction.Thus, the final rotation angle and rotation axis will not deviate largely from the settled angle and the rotation axis.This consideration will also explain why the averaged errors for In the past, it has been discussed that due to the non-cyclic evolution (not a closed loop) of a Bloch vector by the noisy control field, the tolerance of the GP operation becomes worse [9]. Figure 3 suggests, however, that the GP operation is better than a simple operation in some cases even a non-cyclic evolution may occur.
states with the angular frequency b  ) with the transition angular frequency given by b a A , and  A as noise amplitudes.The random numbers R e ,  e , and  e are proportionally, integrally, and derivatively (PID) [15] controlled random numbers.

K
determines the main characters of the PID controlled random numbers, and shows the feedback strength.The parameters in the calculation shows the N dependence of the averaged error    error for the amplitude noise of the optical field with the noise amplitudes R A = 0.003 and 0.1.The dashed curves are averaged errors for the simple operation and the solid curves are those for the GP operation.For any N values calculated, the amplitude noise only causes the rotation angle jitter, we conclude that the GP operation is not useful when the optical field has the amplitude noise.This tendency is observed also for R A = 0.01 and 0.03.The N dependence of    error the phase noise with the noise amplitude  A = 0.003 and 0.1 is shown in Fig.2.The dashed curve is the averaged errors for the simple operation and the solid curve is that for the GP operation.Unlike the case of GP operation is weak against the phase noise.This tendency is observed also for  A = 0.01 and 0.03.

Figure 3 Fig. 1 .
Fig. 1.N dependence of <error> for the amplitude noise.Square plots are <error> for GP gates with the noise amplitudes 0.1 and 0.003.The circles are <error> for the simple gate with the noise amplitudes 0.1 and 0.003.

Fig. 2 .Fig. 3 .
Fig. 2. N dependence of <error> for the phase noise.Square plots are <error> for GP gates with the noise amplitudes 0.1 and 0.003.The circles are <error> for the simple gate with the noise amplitudes 0.1 and 0.003.

A
calculated.However, for  A = 0.1 and N = 1000, vector manipulation, both the frequency and the constant phase fluctuations cause the shift of the rotation axis.When the detuning  changes its value monotonically from (-) sign to (+) sign, the rotation axis directs from downward to upward.The phase sift, however, causes the rotation axis to turn around 3-axis periodically as  increases.The differences of the role for the rotation axis shift may cause the drastic difference of the N dependence for  = 0.1, we see that as the number of sub-fields N becomes large, the ratio  for  A = 0.001 and 0.03.Here, we define the number of sub-fields when  noise amplitudes of 0.003, 0.01, 0.03, and 0.1.Figures4 and 5show the N dependences of errors  003.To avoid confusion, the results of the errors by the simple and GP operations are separated, and Figs. 4 and 5 correspond to the case of the simple operation, and GP operation,

Fig. 4 N
Fig.4 N dependences of averaged errors    Simple error of the simple operation with the amplitude noise R A = 0.003.The squares, triangles, and circles represent the data with Kp = 0.3, 1 and 1.7, respectively.

Fig. 5 N
Fig.5 N dependences of averaged errors    GP error of the GP operation with the amplitude noise R A = 0.003.The squares, triangles, and circles represent the data with Kp = 0.3, 1 and 1.7, respectively.
similar N dependences as those for Kp = 1, and in most cases  = 0.3 is about 10 times larger than the corresponding errors for Kp = 1 and 1.7.
.3 are about 10 times larger than the corresponding errors for Kp = 1 and 1.7, the tendency observed for the case of the amplitude noise seen in Figs.4 and 5.

Figure 7
 A = 0.003.The squares, triangles, and circles correspond to the errors for = 0.3, 1, and 1.7, respectively.The solid curves and dashed curves are for those with the simple operation and the GP operation, respectively.We immediately notice that the GP operation is very weak against the phase noise.The tendency observed for the case of the amplitude and phase noises seen in Figs.4, 5, and 6 is also observed for the case of the frequency nose, and errors  for Kp = 0.3 are about 10 times larger than the corresponding errors

Fig. 6 NA
Fig.6 N dependences of averaged errors    GP error

Fig. 7 N
Fig. 7 N dependences of averaged errors    GP error

Kp = 1
.7 has similar magnitudes as those for Kp = 1.Above considerations will qualitatively explain the reason that the errors  3 are larger than the corresponding errors for Kp = 1 and 1.7.