Single-photon filtering by a cavity quantum electrodynamics system

Transcription

1 PHYSICAL REVIEW A 77, Single-photon filtering by a cavity quantum electroynamics system Kazuki Koshino* College of Liberal Arts an Sciences, Tokyo Meical an Dental University, -8-3 Konoai, Ichikawa 7-87, Japan an PRESTO, Japan Science an Technology Agency, Honcho Kawaguchi, Saitama 33-1, Japan Receive 1 August 7; publishe 5 February 8 The nonlinear ynamics of a classical photon pulse in a cavity-qed system is investigate theoretically. It is shown that this system can work as a single-photon filter, which rastically suppresses the multiple-photon probability of the output. The output photon statistics is sensitive to the input pulse length. A suitable choice of pulse length prouces a photon pulse with the single-photon probability of.3, while the multiple-photon probability is suppresse to.1. DOI: 1.113/PhysRevA PACS numbers: 4.5.Pq, 4.5.Gy, 4.5.Ar I. INTRODUCTION Single photons are regare as a promising caniate for qubits in quantum information processing, because photons can maintain quantum coherence for a long time. Therefore, extensive efforts have been mae in generating an etecting single photons, an remarkable improvements have been reporte in this fiel 1 7. However, the generation of single photons on eman is still a ifficult experimental task, an attenuate laser pulses weak classical pulses are use as a substitute for single photons in quantum key istribution 8 1. A problem in using classical pulses is that pulses containing multiple photons may be abuse by eavesroppers 8 1. In orer to suppress the multiple-photon probability, a highly attenuate pulse must be use, in which the single-photon probability is also suppresse. For example, in orer to suppress the multiple-photon probability to less than.1, the single-photon probability must be less than.14, consiering Poissonian photon statistics of classical pulses. However, for efficient information processing, a pulse with a larger single-photon probability is greatly esire. Photon statistics can be controlle by utilizing nonlinear optical effects. In particular, for the purpose of suppressing the multiple-photon probability, the photon blockae effect is promising In systems showing the photon blockae effect, excitation by a first photon blocks absorption of another photon while the system is excite. Therefore, it is natural to expect that when a classical pulse with a short coherence length is input, the photon blockae mechanism woul filter out the multiple-photon component in the output. From this perspective, we clarify in this stuy how a classical pulse is transforme after transmission through a photon blockae system, base on a rigorous multimoe analysis It is shown that using a weak classical pulse an a photon blockae system, a photon pulse can be generate having the single-photon probability of.3 while the multiple-photon probability is suppresse to.1. Namely, a photon blockae system is shown to work as a single-photon filter, which can substantially improve the efficiency of information processing. *ikuzak.las@tm.ac.jp II. SYSTEM A. Hamiltonian As the simplest photon blockae system, we consier in this stuy a two-level atom place insie a two-sie cavity, as illustrate in Fig. 1. Setting =c=1, taking the atomic resonance as the origin of the energy, an taking the issipationless limit for simplicity, the Hamiltonian of the whole system incluing the external photon fiel is given by H = g c + c + kka k a k + /c a k + a k c + kkb k b k + /c b k + b k c, where g an represent the vacuum Rabi frequency an the cavity ecay rate, respectively. The escape rates of a cavity photon into the left- an right-han sies are ientical. an c enote the annihilation operators for the atomic excitation an the cavity photon, respectively, an a k an b k enote the annihilation operators for the external photon moe with energy k, extening in the left- an right-han sies of the cavity, respectively. c, a k, an b k are the bosonic operators satisfying c,c =1 an a k,a k =b k,b k =k k, whereas is the Pauli operator satisfying, =1. The real-space representation of a k, enote hereafter by ã r, is given by ar (r<) ar (r>) Classical Pulse (Input) Reflecte Photons (Output) ( ) cavity+atom br (r<) br (r>) Transmitte Photons FIG. 1. Schematic illustration of the situation consiere in this stuy. A classical pulse is input from the left-han sie of a twosie cavity containing a two-level atom. Photons emitte in the reflection irection are regare as the output /8/77/ The American Physical Society

2 KAZUKI KOSHINO PHYSICAL REVIEW A 77, ã r = 1/ ke ikr a k. p n;j = n rg n;j r 1,...,r n, 8 b r is efine similarly. As illustrate in Fig. 1, the r an r regions correspon to the incoming an outgoing fiels, respectively. Throughout this stuy, we consier the weak-coupling case, characterize by g. As will be shown in the Appenix, the ynamics in this case is characterize solely by the atomic ecay rate, which is given by = g /. B. State vector of input photons In this stuy, we investigate the following situation: A classical light pulse is input from the left-han sie of the cavity the r region of ã r, whereas no light pulse is input from the right-han sie the r region of b r, an photons emitte in the reflection irection the r region of ã r are regare as the output. The amplitue of the input pulse at the initial moment, = is enote by fr, where the input moe function fr is normalize as rfr =1 an is localize in the r region, an represents the mean photon number. The state vector of the input pulse is given by in = N exp rfrã r, where N is the normalization constant, given by N =exp /. The input state vector can be expane as in = N + 1 in + 1/ in +, where n in is the n-photon Fock state for the moe function fr, given by n in =n! 1/ rfrã r n. C. State vector of output photons The output state vector at the final moment =t, where t is a sufficiently long time is given by out =e iht in. From the linearity of the quantum time evolution an the photon number conservation in the present system, the output state vector takes the same form as Eq. 5, out = N + 1 out + 1/ out +. In the output state, both transmitte an reflecte photons exist. Therefore, n out can be written as n n out = j= n r g n;jr 1,...,r n ã r1 ã rj j! n j! b r j+1 b r, n 7 where g n;j r 1,...,r n is the n-photon output wave function with j reflecte an n j transmitte photons. Although the time coorinate is not explicitly shown, g enotes the wave function at the final moment. The probability of j-photon reflection for the n-photon input, hereafter enote by p n;j,is given by namely, the norm of g n;j. The probability conservation law, n p n;j =1, hols since n out n out =1. j= III. WAVE FUNCTION OF OUTPUT PHOTONS A. Rules for construction of output wave functions The n-photon output wave function g n;j can be obtaine through the correlation function in the output fiel 1. In the Appenix, the etaile calculation of g ;1 is presente for example. The results can be summarize by the following three rules: i g n;j r 1,...,r n = n C j ã r1 ã rj b r j+1 b r n, where ã r1 ã rj b r j+1 b r n is the lowest-orer component of the n-point correlation function in the output fiel. ii When calculating the correlation function, b r can be replace with ã r fr t, where fr is the input moe function. Note that fr is a c number, whereas ã r is an operator. iii ã r1 ã rn is a symmetric function of r 1,...,r n by efinition, an is given, for r 1 r n,byf r n n 1 j=1 f r j e r j r j+1 f r j+1, where f r=g 1;1 r is the wave function of the reflecte photon for the single-photon input, given by f r = fr t + e. B. Concrete forms of output wave functions By following these rules, the n-photon output wave function g n;j can be expresse in terms of fr an f r. Here we present concrete forms of the output wave functions, up to the two-photon component. When the one-photon Fock state 1 in is input, the wave functions of the transmitte an reflecte photons are given by g 1; r = f r fr t, 9 1 g 1;1 r = f r. 11 When the two-photon Fock state in is input, the output wave functions are given by g ; r 1,r = g 1; r 1 g 1; r e r 1 r f R, 1 g ;1 r 1,r = g 1;1 r 1 g 1; r e r 1 r f R, 13 g ; r 1,r = g 1;1 r 1 g 1;1 r e r 1 r f R, 14 where R=maxr 1,r. The first an secon terms on the right-han sie of each equation represent the linear an nonlinear components, respectively. These wave functions contain the full information of output photons. The reflection an transmission probabilities p n;j can be calculate by Eq. 8. The photon statistics of the 385-

3 SINGLE-PHOTON FILTERING BY A CAVITY QUANTUM PHYSICAL REVIEW A 77, Transmission/Reflection Probabilities (a).8 p 1; p 1;1 (b) output pulse can be calculate by Eqs. 8 an 17. The intensity istribution of the output pulse can be calculate by Eq.. IV. NUMERICAL RESULTS A. Input moe function To be more specific, we hereafter assume that the input light is in resonance with the atom an has a Gaussian moe function with pulse length. Denoting the initial position of the pulse by a a an a, the input moe function is given by 1/4exp r a fr = p ; p ;1 p; FIG.. Plots of the transmission an/or reflection probabilities as functions of the pulse length ; a one-photon input, b twophoton input. p n;j represents the probability of j-photon reflection for n-photon input. In b, the transmission an/or reflection probabilities of the linear case are also plotte by thin lines for reference.. 15 Then, from Eq. 9, we obtain f r = 4 1/4 expr a t erfc r a t +, 16 where the complementary error function is efine by erfcx= 1/ x exp. B. Transmission an reflection probabilities The norms of g 1; an g 1;1 give the transmission an reflection probabilities, p 1; an p 1;1. In Fig. a, p 1; an p 1;1 are plotte as functions of the pulse length. It is observe that the reflection probability p 1;1 increases monotonically as is increase, an approaches unity in the limit of. Probability Photon Number FIG. 3. Input an output photon statistics. The thin otte line plots the input photon statistics for =1. The input photon statistics is inepenent of. The soli an ashe lines show the output photon statistics for =.5 1 an 1, respectively. Thus, for efficient reflection of the single-photon component, a long pulse is avantageous. The norms of g ;, g ;1, an g ; give the probabilities p ;, p ;1, an p ;, which are shown in Fig. b; for reference, the L corresponent probabilities of the linear case, given by p n;j = n C j p 1; n j p 1;1 j, are also plotte by thin lines. p n;j agrees L well with p n;j in the small region, in which most photons cannot be absorbe by the atom an therefore the saturation nonlinearity is weak. However, for larger, p n;j eviates from p L n;j ue to the increasing nonlinear effect. It is observe that, in comparison with the linear case, the two-photon reflection probability p ; is suppresse, while the zero- an one-photon reflection probabilities p ; an p ;1 increase, emonstrating the photon blockae effect. p ; increases monotonically as is increase for two reasons: 1 the increase of the linear reflectivity p 1;1 an weakness of the blockae effect when the pulse length is large 3. Thus, for exclusion of the two-photon component in the output, a short pulse is avantageous. C. Photon statistics The photon statistics of the output pulse is reaily obtaine from p n;j. Here, the photon statistics is observe as a function of the pulse length an the input intensity. Since the n-photon probability of the input pulse in is given by n e /n!, the j-photon probability in the output pulse is given by P j, = n=j p ne. 17 n;j n! The photon statistics of the input an output pulses are plotte in Fig. 3, where the input intensity is fixe at unity an the pulse length is chosen to be.5 1 an 1.Inthe input pulse thin otte line in Fig. 3, the multiple-photon probability P m =P + P 3 + is comparable with the singlephoton probability P 1 P m =.6, P 1 =.37. In contrast, in the output pulse, the multiple-photon probability P m is rastically suppresse, while the single-photon probability P 1 is almost unchange. For example, when =.5 1 soli line in Fig. 3, P m =.1 an P 1 =.3. As expecte from Fig., the single-photon probability can be increase by using a 385-3

4 KAZUKI KOSHINO (a) (b) longer pulse, accompanie by an increase in the multiplephoton probability. For example, when = 1 ashe line in Fig. 3, P m =.4 an P 1 =.4. D. Intensity istribution Since analytic forms of the output wave functions are available, we can observe the spatial shape of the photon pulse thus generate. For this purpose, we use the intensity istribution Ir of the output pulse, given by Ir = out ã r ã r out. Since the multiphoton probability is negligibly small, Ir can be regare as the intensity istribution of a single photon. Using Eq. 6, we have Ir = I 1 r + I r +, 18 I n r = e n n out ã n! r ã r n out. 19 Namely, I n r escribes the intensity istribution of a single photon that originates from n in. In terms of the output wave function g n;j, after tracing out the variables r for the transmitte photons, I n r is given by I n r = e n n 1 rg n;1 r,r n! 1,...,r n 1. In Fig. 4a, the intensity istribution Ir of the output pulse is compare with that of the input, given by I in r = in ã r ã r in = fr, for =1 an =.5 1.Itisobserve that the output becomes much weaker than the input ri in r=1 whereas rir=.3, since a consierable fraction of the input is filtere out by this optical system. It is also observe that the output pulse is elaye in comparison with the input, ue to the absorption an emission by the I(r) Iin(r) 1 FIG. 4. a Intensity istributions of the input pulse otte line an the output pulse soli line, observe from a coorinate system moving at light velocity. The input pulse is characterize by =1 an =.5 1. b Plots of I 1 r soli line, I r ashe line, an I 3 r otte line. Ir is plotte by a thin line for reference. I(r) I1(r) I(r) I3(r) PHYSICAL REVIEW A 77, atom, which takes a time of the orer of 1. In Fig. 4b, I n r is plotte for n=1,, an 3. By evaluating ri n r/rir, it is confirme that a generate single photon originates from 1 in by 51%, in by 35%, an 3 in by 1%. After normalization, it is also confirme that I 1 r, I r, an I 3 r have similar spatial shapes. The elay of I 1 r is slightly larger than those of I r an I 3 r, since the elay mechanism absorption an emission by the atom per photon ecreases when multiple photons are involve in the ynamics. V. SUMMARY In summary, we have investigate the nonlinear ynamics of a classical photon pulse in a photon blockae system. As a photon blockae system, a two-sie cavity containing a two-level atom is consiere; the photons emitte in the reflection irection are regare as the output Fig. 1. The wave functions of the transmitte an reflecte photons are obtaine analytically. Using these output wave functions, the transmission an/or reflection probabilities Fig., output photon statistics Fig. 3, an spatial shape of the output pulse Fig. 4 are observe. It is emonstrate that a cavity- QED system can function as a single-photon filter. When the mean photon number is unity, the single-photon probability P 1 an the multiple-photon probability P m are comparable in the input P 1 =.37 an P m =.6. After transmission through the filter, the multiple-photon probability is rastically suppresse whereas the multiple-photon probability is nearly unchange when the pulse length =.5 1, P 1 =.3, an P m =.1. Such a single-photon filter woul be useful for secure an efficient information processing. ACKNOWLEDGMENTS The author is grateful to K. Eamatsu an S. Kono for fruitful iscussions. This research was supporte by the Research Founation for Opto-Science an Technology, an by a Grant-in-Ai for Creative Scientific Research Grant No. 17GS14. APPENDIX: CALCULATION OF OUTPUT WAVE FUNCTION 1. Heisenberg equations From the Hamiltonian of Eq. 1, the Heisenberg equation for a k is given by a k = ika k i /c. A1 Denoting the initial an final moments by an t, the operator at time t is represente in two ways, a k = a k e ik i / ce ik, A 385-4

5 SINGLE-PHOTON FILTERING BY A CAVITY QUANTUM PHYSICAL REVIEW A 77, a k = a k te ik t + i / ce ik. A3 Using Eq. an the above two forms of a k, ã is recast into the following two forms: ã = ã i /c, t A4 ã = ã t t + i /c. A5 Equating the right-han sies an introucing a new label r=t, we obtain the input-output relation, as given by ã r t = ã r t i ct r, A6 where rt. Thus, the output fiel operator at time t is expresse in terms of the input fiel operator at the initial moment an the cavity-moe operator at t r. Similarly, the input-output relation for b r is given by b rt = b r t i ct r. A7 The Heisenberg equations for the atomic operator an the cavity-moe operator c are erive from the Hamiltonian of Eq. 1. Using Eq. A4 an its counterpart for b, they are recast into the following forms: = ig1 c, A8 c = c ig i ã + b. A9 Throughout this stuy, a weak coupling case g is consiere. In this case, Eq. A9 can be solve aiabatically. The cavity operator is given by c = ig/ i 1/ ã + b. A1 Substituting Eq. A1 into Eqs. A6 A8, we obtain the following equations: ã r t = b r t t r, b rt = ã r t t r, A11 A1 = 1 ã + b, A13 where is the atomic ecay rate, efine by Eq. 3. These are the basic equations for calculation of the correlation function in the output fiel.. Relation between the output wave function an the correlation function Here, we investigate the relation between the output wave function an the n-point correlation function. The n-point correlation function is efine by out ã r1 ã rj b r j+1 b r n out. We enote the lowest-orer component of this correlation function, which is proportional to n,byã r1 ã rj b r j+1 b r n. From Eqs. 6 an 7, the following equations can be confirme: ã r1 ã rj b r j+1 b r n = ã r 1 ã rj b r j+1 b r n n out n!, A14 g n;j r 1,...,r n = ã r 1 ã rj b r j+1 b r n n out. A15 j! n j! Combining Eqs. A14 an A15, we obtain g n;j r 1,...,r n = n C j ã r1 ã rj b r j+1 b r n. This is the first rule presente in Sec. III A. A16 3. Calculation of the correlation function Next, we procee to calculate the correlation function in the lowest orer. For example, we consier the two-point correlation function ã r1 b r. In orer to calculate the correlation function, it is convenient to work in the Heisenberg representation. Namely, we use the output fiel operators given by Eqs. A11 an A1, an the initial state vector given by Eq. 4. Then, ã r1 b r is recast into the following form: ã r1 b r = fr tt r 1 + t rt R, A17 where r=minr 1,r an R=maxr 1,r. The following two facts shoul be remarke in erivation of the above equation: i the input-fiel operators ã r an b r commute with t if r+t, an ii the input-fiel operators can be replace with c numbers ã r fr an b r since the input state vector in is an eigenstate of the input-fiel operators. Equation A17 emonstrates that the fiel correlation function ã r1 b r is compose of the input moe function, fr, an the atomic correlation functions, an. The atomic correlation function can be calculate by using Eqs. 4 an A13. The equation of motion for t is given by = f. A18 Note that the thir-orer term has been remove in the above equation, since we are concerne with only the linear component. By integrating Eq. A18, we obtain where f is efine in Eq. 9. = f t, A

6 KAZUKI KOSHINO From Eqs. 4 an A13, the equation of motion for for is given by = f. A Note that the fourth-orer term has been remove. The solution of Eq. A, satisfying the initial conition of =, is given by PHYSICAL REVIEW A 77, = e. A1 Combining Eqs. A16, A17, A19, an A1, an removing the perturbation coefficient, we finally obtain g ;1 r 1,r = f r 1 f r fr t e r 1 r f R. A Thus, Eq. 13 is reprouce. Extension to general g n;j is straightforwar. The results can be summarize by the three rules presente in Sec. III A. 1 O. Benson, C. Santori, M. Pelton, an Y. Yamamoto, Phys. Rev. Lett. 84, 513. Z. Yuan, B. E. Karynal, R. M. Stevenson, A. J. Shiels, C. J. Lobo, K. Cooper, N. S. Beattie, D. A. Ritchie, an M. Pepper, Science 95, 1. 3 M. Keller, B. Lange, K. Hayasaka, W. Lange, an H. Walther, Nature Lonon 431, J. McKeever, A. Boca, A. D. Boozer, R. Miller, J. R. Buck, A. Kuzmich, an H. J. Kimble, Science 33, N. Namekata, Y. Makino, an S. Inoue, Opt. Lett. 7, A. Tomita an K. Nakamura, Opt. Lett. 7, C. Langrock, E. Diamanti, R. V. Roussev, Y. Yamamoto, M. M. Fejer, an H. Takesue, Opt. Lett. 3, B. Huttner, N. Imoto, N. Gisin, an T. Mor, Phys. Rev. A 51, G. Brassar, N. Lutkenhaus, T. Mor, an B. C. Saners, Phys. Rev. Lett. 85, N. Lutkenhaus, Phys. Rev. A 61, W.-Y. Hwang, Phys. Rev. Lett. 91, X.-B. Wang, Phys. Rev. Lett. 94, A. Imamoglu, H. Schmit, G. Woos, an M. Deutsch, Phys. Rev. Lett. 79, P. Grangier, D. F. Walls, an K. M. Gheri, Phys. Rev. Lett. 81, M. J. Werner an A. Imamoglu, Phys. Rev. A 61, 1181R S. Rebic, A. S. Parkins, an S. M. Tan, Phys. Rev. A 65, K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, an H. J. Kimble, Nature Lonon 436, K. Koshino an H. Ishihara, Phys. Rev. Lett. 93, J. H. Shapiro an R. S. Bonurant, Phys. Rev. A 73, J. H. Shapiro, Phys. Rev. A 73, K. Koshino, Phys. Rev. Lett. 98, Q. A. Turchette, C. J. Hoo, W. Lange, H. Mabuchi, an H. J. Kimble, Phys. Rev. Lett. 75, When a resonant continuous wave is input, the normalize secon-orer correlation function g of the output is given by g =1 e in this system. Therefore, the photon blockae mechanism becomes weaker when