Conditional phase-shifter in refractive index enhancement scheme

Optics Communications 283 (2010) 556–560

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Conditional phase-shifter in refractive index enhancement scheme D.E. Sikes *, D.D. Yavuz Department of Physics, University of Wisconsin-Madison, 1150 University Ave., Madison, WI 53706, USA

a r t i c l e

i n f o

Article history: Received 8 September 2009 Received in revised form 21 October 2009 Accepted 21 October 2009

PACS: 42.50.p 42.50.Gy 42.65.An 42.65.Dr 42.65.Pc 42.79.Ta

a b s t r a c t We propose a conditional phase-shifter between two weak laser beams in an alkali vapor. We begin by enhancing the refractive index of a weak, far-detuned probe beam while maintaining vanishing absorption. The enhancement is a result of interfering an absorptive and an amplifying Raman resonance in a four-level atomic system. Then, a conditional switching beam ac stark shifts the narrow Raman resonances, modifying the phase accumulation experienced by the probe beam. We find that the scheme works with acceptable fidelity at the tens of photons level, and that the energy requirement can be reduced further at the expense of noise. Published by Elsevier B.V.

Keywords: Cross phase modulation Low power switch Refractive index enhancement

Traditionally photons have been considered to be noninteracting particles, but advances in nonlinear and quantum optics have shown that significant light–light interactions can occur in highly polarizable media [1,2]. An important such interaction is cross phase modulation because of its potential applications to optical information processing. Cross phase modulation is a nonlinear interaction where the phase of a light field is modified by an amount determined by the intensity of another light field. The key challenge is to develop a technique to achieve a cross phase modulation phase shift of p radians between two weak light fields with minimal dissipation of the fields. Realization of this phase shift with single photon pulses could lead to development of quantum logic via a quantum phase gate [3]. In this article we suggest an implementation of a phase-shifter where a conditional switch beam induces a cross phase modulation on a probe beam propagating in a refractive index enhanced medium. Before proceeding further, we would like to discuss some important results in the nonlinear interaction of weak light beams in atomic media exhibiting electromagnetically induced transparency (EIT) [4]. EIT schemes with cold atoms are a popular choice to implement cross phase modulation of weak light pulses because of the resonantly enhanced giant Kerr nonlinearity and the sup* Corresponding author. E-mail address: [email protected] (D.E. Sikes). 0030-4018/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.optcom.2009.10.085

pression of the absorption of the beam [5,6]. The double-EIT suggestion for cross phase modulation by Lukin and Imamog˘lu [7], where two light pulses with matched slow group velocities interact strongly, has generated much interest over the last decade [8–12]. The fidelity of the double-EIT scheme as a phase-shifter has been investigated as well [13,14]. Also of interest in weak light interaction is the photonic switching achieved by the absorptive analog of the giant Kerr nonlinearity [15], which has shown much experimental progress [16–20]. Further related work is listed in Refs. [21–27]. We propose a cross phase modulation scheme in a refractive index enhanced atomic medium. In a traditional atomic medium a light beam tuned near an atomic resonance experiences a large refractive index but also large absorption. Scully has suggested that an atomic system can be prepared such that quantum interference results in an enhanced refractive index with vanishing absorption [28]. Others have developed this idea in several proposed schemes and proof of principle experiments [29–36]. The phase-shifter described here is built around a refractive index enhancement scheme in a far-off resonant atomic system [33,34], that was recently demonstrated experimentally [35]. We use a four-level system as shown in Fig. 1. The system consists of a ground state jgi, two excited Raman states j1i and j2i, and an excited upper state jei. For simplicity, we take the two Raman states to be degenerate. The system is initially prepared with all atoms pumped into the

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δωs

δωs

Es E c1

|e〉〉

Es

v¼

hN

0

j1 ¼

δω1

|1〉

E c2 δω2

jj1 j2 jj2 j2 : þ dx1 þ jc1 dx2  jc2

ð1Þ

The strength of the Raman coupling of the states j1i and j2i by the probe and control beams is related to the constants

Ep Ep

!

|2〉

| 〉 |g〉 Fig. 1. The schematic of the proposed scheme. A weak far-off resonant probe beam Ep , and two strong control lasers, Ec1 and Ec2 , two photon couple the ground state jgi to the excited Raman states j1i and j2i. A weak switch beam Es one photon couples the degenerate excited Raman states j1i and j2i to the excited upper state jei. Activation of the switch beam induces a conditional ac stark shift on j1i and j2i which alters the two-photon detunings dx1 and dx2 and modifies the propagation of the probe beam.

ground state. The probe beam, Ep , is weak and largely detuned from any one photon resonance. The strong control beams, Ec1 and Ec2 , each interact with the probe beam in a two photon configuration which couples jgi with the Raman states j1i and j2i. We choose the states and the polarization of the laser beams appropriately such that there is no cross coupling, due to angular momentum selection rules. Since the order at which the probe beam is involved in each Raman process is different, such a scheme achieves an amplifying and an absorptive resonance for the probe laser. We achieve refractive index enhancement with vanishing absorption by using the interference of these two resonances. The amplifying and absorptive resonances can be manipulated by modifying the parameters of Ec1 and Ec2 , respectively. In particular, the two resonances can be precisely tuned by choosing the frequencies of the two control lasers. The two-photon detuning dx1 (dx2 ) corresponds to the amplifying (absorptive) resonance on the probe beam. The degree of interference between the resonances is characterized by D ¼ dx1 þ dx2 , the size of the separation between the two resonances when scanning the probe frequency. In this article, we propose using this refractive index enhancement scheme as a conditional phase-shifter simply by adding a switch beam, Es . The switch beam is weak and couples the states j1i and j2i to jei with a one photon detuning of dxs as shown in Fig. 1. The switch beam interacts with the probe beam by ac stark shifting the energy level of states j1i and j2i. This shift in the energy levels affects how strongly the Raman resonances of the probe beam interfere by changing the two-photon detunings. For the ideal case of equal dipole matrix elements, l1e ¼ l2e , the change in the detunings equally shifts the positions of the resonances relative to the probe beam frequency. Thus the strength of the interference is modified but the probe beam remains tuned at the position of vanishing absorption. The key idea behind this suggestion is that when the switch beam is applied the interference of the resonances is modified resulting in a change in the phase accumulation of the probe beam while maintaining vanishing absorption. We proceed with a detailed discussion of the scheme. When the single photon detunings from the excited state are large, the probability amplitude of the excited state can be adiabatically eliminated. In the perturbative limit where most of the population stays in the the ground state and neglecting power broadening, the steady state susceptibility is given by [33]

lge l1e Ec1 ; 2h xe  x1  xp 2

j2 ¼

lge l2e Ec2 ; 2 h xe  xg  xp 2

ð2Þ

where the quantities l are the relevant dipole matrix elements and c1 and c2 are the respective (amplitude) decay rate of the excited Raman states j1i and j2i. For simplicity, we assume balanced parameters for both the amplifying and absorptive resonances such that j1 ¼ j2 and c1 ¼ c2 , which we hereby refer to as j and c. Fig. 2 plots the susceptibility, v, of the probe beam as its frequency, xp , is 0 scanned across the Raman resonances. pffiffiffiffiffiffiffiffiffiffiffiffiffiThe real part v is related to00 0 the index of refraction by n ¼ 1 þ v and the imaginary part v corresponds to the gain/absorption. The phase accumulation is x 0 x / ¼ cp v2 l and the power absorption coefficient is a ¼ cp v00 l, where l is the medium length. At dx1 ¼ dx2 ¼ D=2 the probe beam is tuned at the midpoint between the resonances and constructive interference in v0 gives enhanced index of refraction while destructive interference of v00 gives vanishing absorption [33]. Also we note that the group velocity of the probe beam when tuned to vanishing absorption is v g  c because a local extremum of v0 , i.e. zero slope, coincides with v00 ¼ 0 as seen in Fig. 2. As a result, there will not be a group velocity mismatch between the probe beam and an off-resonant switch beam. The key idea of our phase-shifter is to induce a conditional change in the separation of the Raman resonances, D, which results in a significant shift in the nonlinear phase accumulated by the probe beam. Comparison of the plots in Fig. 2, where D ¼ 10c; 5c; c; c, respectively, shows there is a strong dependence of the refractive index on D. When the probe beam is tuned at the point for vanishing absorption the phase accumulation as a function of the separation of the two resonances is given by

/¼

D hxp jjj2 Nl 2   ; c 0 c2 þ D2 2

ð3Þ

and is plotted in Fig. 3. The largest phase accumulation occurs at 2 D ¼ 2c and has the value /max ¼ hx2pcjcjj0 Nl. In the region jDj  c the variation of / is steepest but the values of / are small because the resonances are overlapping causing near complete cancellation of the nonlinear interaction. The most efficient phase shift is achieved by changing the value of D near the point D ¼ 0 where the slope is greatest since a steeper variation in / allows a small change in D to create a larger shift in the phase. In the region jDj  c the system becomes two isolated resonances and / drops as 1=D. Before we proceed with an analysis of the role of the switch beam, we first discuss how the fidelity of the phase-shifter is affected by spontaneously emitted photons. A Heisenberg–Langevin analysis of the refractive index enhancement scheme has been conducted in Ref. [37] and this quantum treatment of the probe beam is consistent with the semiclassical derivation of Eqs. (1)– (3). The key result of this analysis is that due to the presence of the amplifying resonance there are spontaneously emitted photons into the probe beam, which we refer to as noise photons. The number of noise photons added to the probe beam depends on the gain of the amplifying resonance and is given by

f¼

hxp jjj2 Nl c   ; 2 c0 c þ D2 2

ð4Þ

and is plotted as a function of the separation between the resonances D in Fig. 3. Note that the number of noise photons emitted into the probe beam is greatest when the amplifying and absorp-

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05 0.5

05 0.5

0

0

χ

1

χ

1

05 -0.5

-0.5 05

-1 -10

-5

0

ωp/γ

5

-1 -10

10

0.5

0.5

0

0

χ

1

χ

1

-0.5

-5

0

ωp/γ

5

10

-0.5

-1 1 -5

0

5

ωp/γ

1 -1 -5

0

ωp/γ

5

No oise photon n number

2

2

ζ

1.5

1.5

1

1

φ

0.5

0.5

0

0

-0.5 -1 -20

-0.5

-15

-10

-5

0

Δ/γ

5

10

15

-1 20

No onlinear Ph hase Accumulation ((radians)

Fig. 2. The real part v0 (blue solid line) and the imaginary part v00 (red dashed line) of the nonlinear susceptibility as a function of the probe beam frequency. Between the plots the separation of the resonances is varied to the respective values D ¼ 10c; 5c; c; c. Note that the sign of v0 changes accordingly when the sign of D changes. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 3. The nonlinear phase accumulation / (blue solid line) and the number of spontaneous noise photons f (red dashed line) of the probe beam as a function of the separation between the resonances D. The peak of the noise curve coincides with the point of steepest variation in the phase accumulation, thus there is a tradeoff between the efficiency of the phase-shifter and noise emission. The plot is given hx jjj2 Nl  for the case /max ¼ 1 radian by choosing the prefactor pc0 ¼ 2c. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

tive resonances are closest, jDj  c. This result is unfortunate because it is ideal to operate the phase-shifter in the region jDj  c where the variation of the phase is greatest but the noise added to the probe beam is unacceptably high. Thus there is a trade-off in the efficiency of the phase-shifter and the noise of the probe beam. Next we examine how the presence of a weak switch beam in the scheme effectively shifts the phase of the probe beam. The switch beam ac stark shifts the excited Raman states j1i and j2i. For simplicity, we assume equal dipole matrix elements, l1e ¼ l2e , and therefore assume equal shift of both states. We define the size of this stark shift in frequency space as Ds =2. The energy level shift modifies the two photon detunings such that dx1ð2Þ ! dx1ð2Þ þ Ds =2. Thus the switch beam is the conditional mechanism to change the separation of the Raman resonances

such that D ¼ D0 ! D ¼ D0 þ Ds . Using these two conditional values for D, Eqs. (3) and (4) give the conditional phase shift, d/, and the effective noise photon number, feff :

! D0 þDs D0 hxp jjj2 Nl 2 2 d/ ¼  2   2 ; c0 c2 þ D0 þ2 Ds c2 þ D20 feff ¼

hxp jjj2 Nl c  2 ; c0 c2 þ Dmin 2

ð5Þ ð6Þ

where Dmin ¼ minfjD0 j; jD0 þ Ds jg is the value when the resonances are closest together, giving an upper bound on the noise photon number. Both d/ and feff are functions of the variables D0 and Ds . The value D0 can be set by appropriately changing the frequencies of the control lasers. The change in the separation of the resonances due to the switch beam, Ds , in the limit that dxs  Ce , where Ce is the (amplitude) decay rate of the the excited upper state jei, is given by

Ds ¼

X2s 2dxs

;

ð7Þ

where Xs ¼ l1e Es = h is the Rabi frequency of the switch beam. We note that if we considered the complex detuning of the switch beam, dxs ! dxs  jCe , the presence of the switch beam would cause nonlinear absorption due to power broadening of the Raman 2 s Ce . However, the equally modified resonances will states, c ! c þ X 4dx2s continue to destructively interfere just as the original resonances did before, resulting in complete cancellation of nonlinear absorption of the probe beam. Power broadening will modify Eqs. (5) and (6) through the modified decay rate, c. In this paper, for simplicity, we will focus our attention to the limit dxs  Ce such that power broadening is negligible. Using the Wigner–Weisskopf result 3 [38], l21e ¼ 3px0 h3c Ce , the conditional change in the separation of the 1e resonances can be reduced to

 2 !   3 ks Ce 1 ; 4p A d xs s



Ds ¼ ns

ð8Þ

where s is the switch pulse duration and A is the cross sectional area of the switch beam. In the remainder of this paper, we will take

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D.E. Sikes, D.D. Yavuz / Optics Communications 283 (2010) 556–560 k2s

ideal quantities for these parameters, A ¼ 2p (atomic cross section) and s ¼ 1=c and assume dxs ¼ 10Ce . With these assumptions, Eq. (8) reduces to

Ds ¼ ncns

0.6 Switch ON

0.4

ð9Þ

1 D0 ¼  ncns : 2

ð10Þ

0.2

φ/π

where we have the numerical factor n ¼ 0:15. We define a figure of merit for the effectiveness of our phaseshifter: signal-to-noise ratio, SNR ¼ d/=feff . The SNR depends critically on the initial choice for the separation of the two resonances. As we will show later, the maximum SNR, for a given number of switch photons is obtained when

0

-0.22 -0.4 Switch OFF

-0.6 0

0.1

0.2

0.3

0.4

0.5

propagation length (mm)

Using Eqs. (9) and (10), the Eqs. (5) and (6) reduce to:

h  xp jjj2 Nl nns d/ ¼   ; 2cc0 1 þ nns 2 4

ð11Þ

h  xp jjj2 Nl 2   ; 2cc0 1 þ nns 2 4 nns SNR ¼ : 2

feff ¼

ð12Þ ð13Þ

The SNR depends linearly on the size of the stark shift as seen in Fig. 4. For an effective phase-shifter the lower bound is SNR ¼ p, where there is a p radian phase shift and a single noise photon. The value of SNR can be increased beyond this lower bound by sufficiently increasing the value of the stark shift. For a single switch beam photon, ns ¼ 1, SNR ¼ 0:075. A SNR ¼ p is obtained for ns ¼ 42 photons in the switch beam. Unfortunately, these results show that our scheme can not be used as an effective phase-shifter at the single photon level in free space. The plot in Fig. 5 shows a numerical example of our phase-shifter in a real system based on the analytical steady states solutions of Ref. [33] for the enhanced refractive index at vanishing absorption. We simulate the four-level system in a medium of ultra-cold 87 Rb atoms with density N ¼ 2  1014 /cm3 and length l ¼ 0:5 mm based on techniques using either cold atom traps (e.g. magnetooptical or dipole traps) or hollow-core optical fibers [39,40]. The excited electronic state 5P1=2 (D1 line) is used for the state jei and jF ¼ 2; mF ¼ 2i, jF ¼ 2; mF ¼ 2i, and jF ¼ 2; mF ¼ 0i hyperfine states of the electronic ground state 5S1=2 are used for states j1i, j2i, 8 7

|SNR|

6 5 4 3

δφ

2

ζeff

1 0

0

20

40

60

80

100

number of switch photons, ns Fig. 4. The phase shift d/ (green solid line), the effective noise feff (cyan dashed line), and the absolute value of the Signal-to-Noise ratio jSNRj (blue dotted line) as functions of the number of photons in the switch beam, ns , with the initial separation of the resonance optimally set to D0 ¼  12 ncns . The numerical factor is n ¼ 0:15 for the switch beam parameters dxs ¼ 10Ce , A ¼ k2 =2p, and s ¼ 1=c. The plot is given for the case d/ ¼ p radians when jSNRj ¼ p by choosing the prefactor hxp jjj2 Nl  ¼ 0:55p. (For interpretation of the references to color in this figure legend, 2cc0 the reader is referred to the web version of this article.)

Fig. 5. The phase accumulation / for the probe beam of the unshifted case (blue solid line) and the shifted case (red dashed line) as a function of the propagation length of the probe beam in the medium. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

and jgi, respectively [41]. The excited state decay rate is Ce ¼ 2p 5:75 MHz and we assume a Raman line width of c ¼ 2p  10 kHz. The probe, switch, and control beams have polarizations rþ , p, and r , respectively and wavelengths 795 nm. The probe beam is red-detuned from the excited state by 10 GHz. The power of the control beams are 75 mW and are focused to a beam diameter 1 mm. To satisfy the condition that SNR ¼ p, we set ns ¼ 42 photons per atomic cross section based on the results of Fig. 4. The plots in Fig. 5 show the accumulated phase of the probe beam as a function of the propagation length for both cases when the switch beam is present and absent. Note that the phase accumulation of the two cases are symmetric about / ¼ 0 which follows from Eqs. (10) and (11) that D ¼ 12 ncns and D ¼  12 ncns when the switch beam is present or absent, respectively. These opposite sign phase accumulations achieve a net phase difference of  p radians at a propagation length of 0.5 mm. Finally, we show how we derived the condition on the initial separation of the resonances D0 , given by Eq. (10), to maximize the SNR for a given set of switch beam parameters. We analytically optimized the SNR with respect to D0 , treating Ds as a constant. This resulted in the condition D0 þ Ds ¼ D0 to maximize the SNR. This condition physically corresponds to a configuration where the noise of the shifted case equals the noise of the unshifted case. This dependence of the optimal value of D0 on Ds can also be seen in the plots in Fig. 6, which show the phase shift, effective noise, and jSNRj as functions of the initial separation between the resonances D0 for various values of Ds ¼ 1:5c; 6c; 12c. The top axis of each of the plots is normalized to its respective value of Ds . The general result seen is that for a given value Ds the maximum jSNRj occurs at D0 ¼ Ds =2. Also in the limit that D0 ¼ 1 the jSNRj asymptotically approaches this maximum value. In the regime that D0 ¼ 1 the nonlinear response of the atomic medium is greatly reduced and would require greater resources for achieving the phase shift. Thus, the key result of Fig. 6 is that the initial system configuration of D0 ¼ Ds =2 yields the highest signal-to-noise ratio. However, we do note that when the condition dxs  Ce is not satisfied, power broadening of Raman linewidth c in the shifted case must be included and a more general expression for the optimal value of D0 will depend on Ds and the ratio of the excited state linewidth and the switch beam detuning Ce =dxs . In summary we have developed a scheme for a low intensity phase-shifter and estimated its fidelity. The lower bound energy cost for a p radian phase shift with acceptable fidelity is on the order of tens of photons. There is an inherent trade-off where fidelity can be increased at the expense of energy cost of the switch beam. We believe our phase-shifter is ideally suited for use in high

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Δ0/Δs 3.5 3

-10

Δs =1.5 γ

-5

Acknowledgements

0

10

5

We would like to sions. This work was of Scientific Research dation (WARF), and funds.

2.5 2 1.5

thank Nick Proite for many helpful discussupported with funds from Air Force Office (AFOSR), Wisconsin Alumni Research FounUniversity of Wisconsin-Madison start-up

References

1 0.5 0 -0.5 -20

-10

-3

-2

Δs =6 γ

3

0 -1

0

10 1

20 2

3

|SNR|

2 1

ζeff

0

δφ -1 -1 -20

-10

-1.5 6

-1

0

-0.5

0

20

10

0.5

1

1.5

Δs =12 γ

4 2 0 -2 -20

-10

0

Δ0/γ

10

20

Fig. 6. The phase shift d/ (green solid line), the effective noise feff (cyan dashed line), and the absolute value of the Signal-to-Noise ratio jSNRj (blue dotted line) as functions of the initial separation of the resonances D0 . Between the plots the value of the conditional change in the separation of the resonances Ds is varied to 1:5c; 6c, and 12c, respectively. The bottom axis expresses D0 in terms of system constant c, the Raman transition linewidth. The top axis expresses D0 in terms of system variable Ds . The key result seen here is that jSNRj is optimized when D0 ¼ Ds =2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

fidelity optical information processing of weak beams of 102–103 photons corresponding to an energy of 0.01–0.1 fJ, where the constraints on beam focusing and pulse duration can be relaxed to more practical values.

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