Simulation of Laser Manipulation of Bloch. Vector in Adiabatic Regime

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1 Advaces i Applied Physics, Vol. 2, 214, o. 2, HIKAI Ltd, Simulatio of Laser Maipulatio of Bloch Vector i Adiabatic egime yuzi Yao Murora Istitute of Techology, 27-1, Mizumoto-cho Murora , Japa Hiroyuki Shiojima Kurume Natioal College of Techology Kurume , Japa Copyright 214 yuzi Yao ad Hiroyuki Shiojima. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract Simulatios of the Bloch vector maipulatio of two operatios (oe that utilizes the geometric phase (geometric operatio) ad the other that does ot utilize the geometric phase (simple operatio) i the adiabatic regime eabled us to study the tolerace of the geometric phase utilized operatio agaist the laser oise. We simulated the Bloch vector maipulatio by the laser field with the amplitude, the frequecy, ad the costat phase are proportioally, itegrally, ad derivatively cotrolled. The maipulatio of the Bloch vector is a /2 rotatio aroud 2-axis. There are few advatage by usig the operatio of the geometric phase over the simple operatio. It has a advatage over oly whe the geometric phase operatio is utilized such that the magitude of the frequecy of the oise is large, ad that the oise correlatio time is small. Keywords: Bloch Vector, geometric phase, PID cotrol 1 Itrtoductio The realizatio of quatum computers will require log coherece time for materials ad low rate for cotrol operatios. [1-3] The low rate for quatum computers requires a series of accurately cotrollable quatum operatios. The field that maipulates a Bloch vector (a two-level atom) usually

2 54 yuzi Yao ad Hiroyuki Shiojima has oise compoets. Therefore, a oise-free cotrol method (operatio) is eeded for the accurate cotrol of the Bloch vector. To cotrol the Bloch vector accurately, it is ecessary to kow the tolerace of the cotrol operatio whe the field is oisy. After the Berry s adiabatic geometrical phase () [4] has bee itroduced, may papers have appeared cocerig to the basic sciece ad applicatios [5, 6]. The operatios have bee cosidered to be less sesitive tha the operatios without usig agaist oises of the cotrol fields. Therefore, the utilized Bloch vector maipulatio was proposed as the low rate maipulatig method. However, o clear coclusio seems to have bee obtaied about the usefuless of the utilized operatio [7-1]. Also, there seems to have bee a few discussios which oise has a ifluece o the operatio amog the amplitude, the frequecy, ad the phase oises of the optical field. Oe simple way to see the tolerace of the operatio is to simulate both ad dyamical operatios ad compare their toleraces. Here, we umerically studied ad compared the tolerace of a operatio a dyamical operatio agaist the laser oise i the adiabatic regime. The simulatio was performed while imagig a very uiversal model (a two-level atom). There are so may real physical systems correspodig to a two-level atom: Gas (b gas, Cs gas), rare-earth io doped crystals, ad semicoductor quatum dots eve i the optical regime [11-13]. 2 Calculatio Procedure To make clear which of the two operatios (the operatio ad the simple operatio) is less affected by the laser field oise, we performed the followig calculatio. We cosider a two-level atom irradiated by a optical field. The wave fuctio of the two-level atom with the two eergy eigestates u a(r) (the excited states with the agular frequecy a ) ad u (r b ) (the groud states with the agular frequecy b ) with the trasitio agular frequecy give by a b is writte by ( r, t) c ( t)exp( i t) u ( r) c ( t)exp( i t) u ( r) (1) a a a The excitatio optical field has the amplitude E, the ceter agular frequecy ad the costat phase : E ( t) E cos( t ). For simplicity, i the umerical calculatios, the optical pulses have square evelopes with the same amplitude if the optical fields are oise-free fields. Followig the procedure of Meystre ad Sarget III [14], we obtai the solutios for the probability amplitudes c a (t) ad c b (t) for the excitatio optical field with the duratio t as: ca( t) ca() A(,, ) t (2) cb ( t) cb () with b b b

3 Simulatio of laser maipulatio of Bloch vector i adiabatic regime 55 t t t t t [cos i si ]exp i i si exp( i)exp i A(,, t). t t t t t i si exp( i)exp i [cos i si ]exp i Here, is the abi agular frequecy, the detuig, ad 2 2. The real optical field usually has the amplitude, the frequecy, ad the phase oises simultaeously. I the simulatio, however, we itetioally itroduce oly oe of the oises of the laser field durig the Bloch vector maipulatio. The simulatio will clarify which oise will affect the accurate cotrol of the Bloch vector. We simulate some oises of the proportioally, itegrally, ad derivatively (PID) cotrolled laser field by usig the radom umbers with the help of the PID loop. Each laser field is temporally divided ito N pieces of sub-fields with the pulse duratio /N. Withi the sub-field, the oise is costat. The parameter / N ( is the pulse width) correspods to the oise correlatio time. The abi agular frequecy ad the detuig ca be defied i the uit of the abi agular frequecy oiseless o resoace give by the oise-free optical field: (1 A e ), (3) oiseless ( / ) A e. (4) oiseless oiseless oiseless The fluctuatio of the phase factor aroud is give by 2 A e. (5) Here, we call e, ad A, A, ad A as oise amplitudes. The radom umbers e, e are proportioally, itegrally, ad derivatively (PID) [15] cotrolled radom umbers. I the simulatio, uiform radom umbers have bee ofte used to simulate the oise of the cotrol field [9]. We cosider that the radom umbers defied i the followig are more realistic to describe the oise of the cotrol field. Usig a liear cogruetial method, we first geerate uiform radom umbers 27 by the relatio x 1 ax b, (mod c), with a 153, b 7391, c 2. We also used aother set of parameters for the radom umbers. We obtaied similar simulatio results, ad the coclusio does ot chage. The PID cotrolled radom umbers which we use here are geerated usig the radom umbers s which are created by the uiform radom umbers x with their values betwee -1 ad 1: 1 x 1. The radom umbers s thus geerated simulates the oise characteristics of free-ruig lasers better tha the uiform radom umbers. The (+1)-th PID radom umbers e, 1, e, 1, ad e, 1 (We write them by e 1 i the followig relatios.) are give by the followig relatios: e e Y, (6) 1 1

4 56 yuzi Yao ad Hiroyuki Shiojima with e e e 1 TD e e 1 e Y Y 1 KP 2 2 (7) TI The magitude of K P determies the mai characters of the PID cotrolled radom umbers, ad shows the feedback stregth. The parameters i the calculatio are K P 1, T I 1, ad T D. 1. The mius sig of K P appears to reduce the deviatio e from zero. The variables Y, e whe. At this poit, oe might cosider that the oises geerated by above-metioed process are physically meaigless because the oises are ot geerated by a particular set of equatios with a particular set of parameters. For example, oe might cosider that we should use the Lagevi-type equatios to geerate oises with fiite correlatio times. However, the laser oise properties of the amplitude, phase, ad frequecy are differet depedig o the property of the laser system. Thus, the simulatio results with specific oises which are geerated by a particular set of equatios with a particular set of parameters oly represet oe specific case. The radom umbers e s created by our method deviates from the startig value zero as the umber icreases ad behaves like the steps of a druk [16]. To correct ad reduce the deviatio of the radom umber from the iitial value, the PID cotrol is performed. The PID cotrolled radom umbers reproduce the jitter correctio process of a laser system. The testig two operatios rotates the Bloch vector aroud 2-axis by a agle /2. Oe operatio rotates simply the Bloch vector aroud 2-axis with a pulse area /2 by a laser field (Here, we call this operatio as the simple operatio.). The operatio o resoace optical pulse ( ) that rotates the Bloch vector aroud 2 axis for a agle of is give by the optical pulse sequeces: # 1;pulse area / 2,, # 2;, / 2, ad # 3; / 2, [13]. We thus cosider the case of / 2. The 2 2 matrix A ( A ) correspodig to the simple operatio (the operatio) is give by the products of N (4N) matrixes. To estimate the tolerace of the operatios agaist the oisy optical fields, the fidelity is ofte used. The defiitio of the fidelity F betwee two wave vectors 2 ad is defied as F [17]. However, the fidelity betwee two wave vectors ad gives the same fidelity as that betwee ad. Therefore, the fidelity defied above caot distiguish betwee ad. Thus, whe the oise field is strog, the fidelity may be ot a good measure to estimate the tolerace of operatios. Here, istead of usig the fidelity, we use the followig value for the tolerace estimatio. The matrix elemets m a i, k (i,k=1,2, m, ) of the two operatios thus obtaied are differet m from those for the oise-free optical pulse. Let the differece be deoted by a i, k. m m 2 We defie the as, a 1,2,. The thus defied is ot i k i affected by the iitial state of the Bloch vector, k ad gives a measure for the m tolerace of the operatios agaist the oise of the optical field. The, was averaged for ~1 5 times.

5 Simulatio of laser maipulatio of Bloch vector i adiabatic regime 57 3 Numerical esults ad Discussios First, we show ad discuss the results whe Kp = 1. Figure 1 shows the N depedece of the averaged for the amplitude oise of the optical field with the oise amplitudes A =.3 ad.1. The dashed curves are averaged s for the simple operatio ad the solid curves are those for the operatio. For ay N values calculated, ~ 3. As the amplitude oise oly causes the rotatio agle jitter, we coclude that the operatio is ot useful whe the optical field has the amplitude oise. This tedecy is observed also for A =.1 ad.3. The N depedece of the phase oise with the oise amplitude A =.3 ad.1 is show i Fig. 2. The dashed curve is the averaged s for the simple operatio ad the solid curve is that for the operatio. Ulike the case of the amplitude oise, for ay N calculated. Also, as N icreases, the differece betwee ad becomes larger. Thus, the operatio is weak agaist the phase oise. This tedecy is observed also for A =.1 ad.3. Figure 3 shows the N depedece of the averaged for the frequecy oise with the oise amplitude A =.1 ad.3. The dashed curves are the averaged s for the simple operatio ad the solid curves are those for the operatio. Whe A =.3, clearly larger tha is for.3 A.1 A Fig. 1. N depedece of <> for the amplitude oise. Square plots are <> for gates with the oise amplitudes.1 ad.3. The circles are <> for the simple gate with the oise amplitudes.1 ad.3. Fig. 2. N depedece of <> for the phase oise. Square plots are <> for gates with the oise amplitudes.1 ad.3. The circles are <> for the simple gate with the oise amplitudes.1 ad.3..3 A.3 A.3 A.1 A.1 A.1 A Fig. 3. N depedece of <> for the detuig oise. Square plots are <> for gates with the oise amplitudes.1 ad.3. The circles are <> for the simple gate with the oise amplitudes.1 ad.3.

6 58 yuzi Yao ad Hiroyuki Shiojima ay N we calculated. However, for A =.1 ad N = 1, is smaller tha. For A =.3, ot show i Fig.3, ~ whe N 1. I the Bloch vector maipulatio, both the frequecy ad the costat phase fluctuatios cause the shift of the rotatio axis. Whe the detuig chages its value mootoically from (-) sig to (+) sig, the rotatio axis directs from dowward to upward. The phase sift, however, causes the rotatio axis to tur aroud 3-axis periodically as icreases. The differeces of the role for the rotatio axis shift may cause the drastic differece of the N depedece for ad. I most cases is larger tha. Exceptio is the case of the frequecy oise. The Fig.4 N depedeces of averaged s becomes smaller tha of the simple operatio with the whe amplitude oise A =.3. The squares, N is large eough. We see that for A triagles, ad circles represet the data with Kp =.3 ad N 1, is =.3, 1 ad 1.7, respectively. larger tha. However, for A =.1, we see that as the umber of sub-fields N becomes large, the ratio / becomes small, ad fially becomes smaller tha whe N = 1. This iversio of the relative magitudes betwee ad is observed also for A =.1 ad.3. Here, we defie the umber of sub-fields whe = as N cross. The tedecy is observed that N cross becomes small as the oise amplitude A becomes large. We presetly have o clear idea to explai why this iversio of the magitude occurs. Above results are for the case where the feedback coefficiet Kp 1. We also simulated Bloch vector maipulatio ad calculated the s ad for Kp =.3 ad 1.7 at oise amplitudes of.3,.1,.3, ad.1. Figures 4 ad 5 show the N depedeces of s ad for the amplitude oise with A =.3. To avoid cofusio, the results of the s by the simple ad operatios are separated, ad Figs. 4 ad 5 correspod to the case of the simple operatio, ad operatio, Fig.5 N depedeces of averaged s of the operatio with the amplitude oise A =.3. The squares, triagles, ad circles represet the data with Kp =.3, 1 ad 1.7, respectively.

7 Simulatio of laser maipulatio of Bloch vector i adiabatic regime 59 respectively. The averaged s ad for Kp =.3 ad 1.7 show similar N depedeces as those for Kp = 1, ad i most cases is larger tha. The s ad for Kp = 1 ad 1.7 take similar values. Also, the ( ) for Kp =.3 is about 1 times larger tha the correspodig s for Kp = 1 ad 1.7. Figure 6 shows the N depedeces of averaged s ad for the phase oise with A =.3. The squares, triagles, ad circles correspod to the averaged s for Kp =.3, 1, ad 1.7, respectively. The solid ad dashed curves correspod to the s with the simple operatio ad those with the operatio, respectively. We immediately otice that the operatio is very weak agaist the phase oise. Also, it is observed that the s ad for Fig.6 N depedeces of averaged s Kp =.3 are about 1 times larger tha ad for the phase the correspodig s for Kp = 1 ad oise with A =.3. The squares, 1.7, the tedecy observed for the case of the amplitude oise see i Figs. 4 ad 5. Figure 7 shows the N depedeces of s ad for the frequecy oise with A =.3. The squares, triagles, ad circles correspod to the s for =.3, 1, ad 1.7, respectively. The solid curves ad dashed curves are for those with the simple operatio ad the operatio, respectively. We immediately otice that the operatio is very weak agaist the phase oise. The tedecy observed for the case of the amplitude ad phase oises see i Figs. 4, 5, ad 6 is also observed for the case of the frequecy ose, ad the s ad for Kp =.3 are about 1 times larger tha the correspodig s for Kp = 1 ad 1.7. Why the s ad for Kp =.3 are about 1 times larger tha the correspodig s triagles, ad circles correspod to the s for Kp =.3, 1, ad 1.7, respectively. The dashed curves ad solid curves correspod to the averaged s with the simple operatio ad those with the operatio, respectively. Fig. 7 N depedeces of averaged s ad for the frequecy oise with the oise amplitude A =.3. The squares, triagles, ad circles correspod to the s for Kp =.3, 1, ad 1.7, respectively. The dashed curves ad solid curves are the s of the simple operatio ad the operatio, respectively.

8 6 yuzi Yao ad Hiroyuki Shiojima for Kp = 1 ad 1.7. We discuss this reaso qualitatively. Figure 8 shows series of the PID cotrolled radom umbers for Kp =.3, 1, ad 1.7. To avoid cofusio, the radom umbers with Kp =.3 (1.7) are shifted vertically by 5. (-5.). If we assume that the horizotal axis represets the time, these series of radom umbers show the temporal oise behaviors. The series of the radom umbers with Kp = simulates the oise of free-ruig laser. From Figure 8, we immediately otice that: Whe the feedback factor Kp = 1, the series of the radom umbers is well cotrolled, although the oise caot be Fig.8 Series of the PID cotrolled radom umbers for Kp =.3, 1, ad 1.7. To avoid cofusio, the radom umbers with Kp =.3 (1.7) are shifted by 5. (-5.). perfectly cacelled. The series of the radom umbers with Kp = 1.7 has very sharp oisy compoet due to the excess feedback. I the time domai, the oise has high frequecy compoets. The series of the radom umbers with Kp =.3 chages its value slowly due to the weak feedback. I the time domai, the oise has slow frequecy compoets. The amplitude oise causes the fluctuatio of the pulse area (the rotatio agle). The phase ad frequecy oises cause the fluctuatio of the rotatio axis (the directio of the torque vector). [18] Therefore, if the oise compoet of the amplitude chages slowly i the time domai (case of Kp =.3), the amplitude will be larger tha the settled amplitude (the iitially settled amplitude without oise) i a certai time regio, ad i aother time regio, the amplitude will be smaller tha the settled amplitude. The fluctuatio of the deviatio of the amplitude from the settled amplitude will chage slowly ad the of the rotatio agle accumulates. Thus, the fial rotatio agle will deviate largely from the settled agle. I the similar way, if the oise compoet of the phase or the frequecy chages slowly i the time domai (the case of Kp =.3), the rotatio directio of the Bloch vector shifts from the settled directio ad fluctuates upward ad dowward (the frequecy oise), or rightward ad leftward (the phase oise) i the Bloch sphere. If the temporal chage of the rotatio directio fluctuates slowly, the ifluece of the deviatio of the rotatio directio accumulates. Thus, the fial rotatio agle ad rotatio axis will deviates largely from the settled agle ad rotatio axis. O the other had, if the oise compoet of the amplitude of the optical field

9 Simulatio of laser maipulatio of Bloch vector i adiabatic regime 61 fluctuates very fast i the time domai (the case of Kp = 1.7), the amplitude will be larger tha the settled amplitude (the iitially settled amplitude without oise) i a short time regio, ad the amplitude will be smaller tha the settled amplitude i the ext short time regio. The fluctuatio of the deviatio of the amplitude from the settled amplitude will chage rapidly ad the of the rotatio agle will be cacelled ad will ot accumulate. Thus, the fial rotatio agle will ot deviate largely from the settled agle. If the oise compoet of the phase or the frequecy chages rapidly i the time domai (case of Kp = 1.7), the rotatio directio of the Bloch vector shifts from the settled directio ad fluctuates. However, due to the fast fluctuatio of the rotatio axis aroud the iitially aimed directio, roughly speakig, the Bloch vector will rotate aroud the iitially aimed directio. Thus, the fial rotatio agle ad rotatio axis will ot deviate largely from the settled agle ad the rotatio axis. This cosideratio will also explai why the averaged s for Kp = 1.7 has similar magitudes as those for Kp = 1. Above cosideratios will qualitatively explai the reaso that the s ad for Kp =.3 are larger tha the correspodig s for Kp = 1 ad 1.7. I the past, it has bee discussed that due to the o-cyclic evolutio (ot a closed loop) of a Bloch vector by the oisy cotrol field, the tolerace of the operatio becomes worse [9]. Figure 3 suggests, however, that the operatio is better tha a simple operatio i some cases eve a o-cyclic evolutio may occur. 4 Summary We umerically studied the tolerace of two operatios agaist oisy optical laser fields for the case of the Bloch vector rotatio with a agle of /2 aroud 2-axis. We foud that the operatio is especially weak agaist the phase oise. We also foud that operatio has a advatage over the simple operatio agaist the frequecy oise whe the amplitude oise is large ad the correlatio time is small. Thus, if the optical field has oly the frequecy oise, the tolerace of the operatio is better tha that of the simple operatio whe the oise correlatio time is small. efereces [1] M. A. Nielse ad I. L. Chuag, Quatum Computig ad Quatum Iformatio (Cambridge Uiversity Press, Cambridge, UK, 2). [2] G. eeti, G. Casati, ad G. Strii, Priciples of Quatum Computatio ad

10 62 yuzi Yao ad Hiroyuki Shiojima Quatum Iformatio, Volume I Basic Cocepts I (World Scietific, Sigapore, 24). [3] M. Le Bellac, A short Itroductio of Quatum Iformatio ad Quatum Computatio, (Cambridge Uiv., 26). [4] M. V. Berry, Proc. oy. Soc. Lodo, A392 (1985) [5] Giuseppe Falci, osario Fazio, G. Massimo Palma, Jes Siewert ad Vlatko Vedral, Nature 47 (2) [6] A. Bohm, A, Mostafazadeh, H. Koizumi, Q. Niu, ad J. Zwaziger, The geometric Phase i Quatum Systems (Spriger, 23). [7] J. A. Joes, V. Vedral, A. Ekert, ad G. Castagoli, Nature 43, (2) [8] A. Ekert, M. Ericsso, P. Hayde, H. Iamori, J. Joes, D. Oi, ad V. Vedral, J. Mod. Opt. 47, (2) [9] A. Blais, ad A. -M. S. Tremblay, Phys. ev. A67, (23) [1] O. Oreshkov, T. A. Bru, ad D. A. Lidar, Phys. ev. Lett. 12, (29) [11] A. P. Heberle, J. J. Baumberg, ad K. Köhler, Phys. ev. Lett. 75 (1995) [12] C. Moroe, D. M. Meekhof, B. E. Kig, W. M. Itao, ad D. J. Wielad, Phys. ev. Lett. 75 (1995) [13] M. Tia, Zeb W. Barber, J. A. Fischer, ad Wm. adall Babbitt, Phys. ev. A69 (24) 531(). [14] P. Meystre, ad M. Sarget III, Elemets of Quatum Optics 3rd ed. (Spriger, 1999) p [15] Y. Cheg-Chig, Autotuig of PID Cotrollers: elay Feedback Approach 2d ed. (Spriger, 26). [16] F. eif, Fudametals of Thermal ad Statistical Physics, (McGraw-Hill, Aucklad, 1965), p.4.

11 Simulatio of laser maipulatio of Bloch vector i adiabatic regime 63 [17] ef. 2, p.15. [18] ef. 14, p.95, Fig eceived: November 12, 214; Published: December 5, 214