The effects of the N atom collective Lamb shift on single photon superradiance

Physics Letters A 373 (2009) 1283–1286

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The effects of the N atom collective Lamb shift on single photon superradiance Marlan O. Scully a,b , Anatoly A. Svidzinsky a,b,∗ a b

Institute for Quantum Studies and Department of Physics, Texas A&M University, College Station, TX 77843, United States Applied Physics and Materials Science Group, Engineering Quad, Princeton University, Princeton, NJ 08544, United States

a r t i c l e

i n f o

Article history: Received 11 December 2008 Accepted 5 February 2009 Available online 20 February 2009 Communicated by P.R. Holland

a b s t r a c t The problem of single photon collective spontaneous emission, a.k.a. superradiance, from N atoms prepared by a single photon pulse of wave vector k0 has been the subject of recent interest. It has been shown that a single photon absorbed uniformly by the N atoms will be followed by spontaneous emission in the same direction [M. Scully, E. Fry, C.H.R. Ooi, K. Wodkiewicz, Phys. Rev. Lett. 96 (2006) 010501; M. Scully, Laser Phys. 17 (2007) 635]; and in extensions of this work we have found a new kind of cavity QED in which the atomic cloud acts as a cavity containing the photon [A.A. Svidzinsky, J.T. Chang, M.O. Scully, Phys. Rev. Lett. 100 (2008) 160504]. In most of our studies, we have neglected virtual photon (“Lamb shift”) contributions. However, in a recent interesting paper, Friedberg and Mannassah [R. Friedberg, J.T. Manassah, Phys. Lett. A 372 (2008) 2514] study the effect of virtual photons investigating ways in which such effects can modify the time dependence and angular distributions of collective single photon emission. In the present Letter, we show that such virtual transitions play no essential role in our problem. The conclusions of [M. Scully, E. Fry, C.H.R. Ooi, K. Wodkiewicz, Phys. Rev. Lett. 96 (2006) 010501; M. Scully, Laser Phys. 17 (2007) 635; A.A. Svidzinsky, J.T. Chang, M.O. Scully, Phys. Rev. Lett. 100 (2008) 160504] stand as published. However, the N atom Lamb shift is an interesting problem in its own right and we here extend previous work both analytically and numerically. © 2009 Elsevier B.V. All rights reserved.

In recent work, we have investigated the temporal and spatial evolution of cooperative superradiant emission from a single photon Dicke-like state [1–3]. Specifically, we have studied the N-atom state N 1 

|ψ+  = √

N j =1

e ik0 ·r j |b1 b2 · · · a j · · · b N ,

(1)

where k0 is the wave vector of the incident photon (k0 = ω/c, h¯ ω = E a − E b ), r j is the radius vector to the atom j, |b1 b2 · · · a j · · · b N  is a Fock state in which atom j is in the excited state a and all other two level atoms being in the ground state b. The symmetric state |ψ+  was prepared via a conditional process and its subsequent evolution has been investigated. This problem has been the subject of interest at several conferences and published papers. Most recently the N atom collective Lamb shift associated with single photon states such as Eq. (1) has been studied [4–7]. In particular, the treatment of [1–3] has been extended to include virtual photon processes [4,5]. Furthermore, in Refs. [5,6] the question as to whether such virtual photon processes are important in cal-

*

Corresponding author at: Institute for Quantum Studies and Department of Physics, Texas A&M University, College Station, TX 77843, United States. E-mail address: [email protected] (A.A. Svidzinsky). 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.02.027

culating the time evolution and directionality of radiation emitted by an ensemble of atoms described by Eq. (1) has been broached. The issue of what to keep and what to ignore in modeling a problem is, of course, of paramount importance. Theoretical physics is the art of approximation, examples include the ideal gas, the two level atom and the rotating wave approximation. Each of these serves a useful role in providing quick insight and can tell us which calculations or experiments we should invest in. In this sense, it is important to understand the effects of virtual photons in the evolution of single photon superradiance. In particular, Friedberg and Manassah have noted that single photon decay properties can be seriously flawed by neglecting virtual photons [5,6]. In the present note, we address the issue and show that the virtual photon contribution is small as concern the time evolution of the state |ψ+  given by Eq. (1) (see also Ref. [7]). There are however, other single photon cooperative spontaneous emission problems in which the virtual photon contributions are very substantial and the neglect of these contributions can lead to qualitatively incorrect answers, see, e.g. Fig. 3. An initial state decays via Weisskopf–Wigner spontaneous emission processes accompanied by “Lamb shift” virtual transitions. The main result of the present Letter is that in the large sample limit R  λ (λ = 2π c /ω is the wavelength of the emitted photon, R is the radius of the atomic cloud) the effect of virtual photons is small for evolution of the |ψ+  state and it approx-

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Table 1 Decay rate of |ψ+ , i.e. the fastest decaying state, calculated with (exp) and without (sin) Lamb shift virtual transitions.

Small sample limit R  λ Large sample R  λ

sin kernel

exp kernel

Γ ≈ Nγ γ Γ ≈ 2(3N k R )2

Γ ≈ Nγ γ Γ ≈ 2(3N k R )2

0

0

imately decays into the ground state without coupling to other states. In the small sample limit R  λ the decay rate of the |ψ+  state is also practically unaffected by virtual photon contributions. Table 1 compares decay rates of |ψ+  for small and large atomic samples; and Figs. 1 and 2 show the spatial and temporal evolution of radiation from the |ψ+  state calculated with and without taking into account virtual photons. Table 1 and Figs. 1 and 2 (see also Eq. (24) and Fig. 3) make it clear that virtual photons are not important for the decay of |ψ+ . Evolution of the atomic state vector

Ψatom (t ) =

N 

β j (t )|b1 b2 · · · a j · · · b N 

j =1

is given by the following equations for β j (t ) ( j = 1, . . . , N) (see Appendix A)

β˙ j (t ) = −

N  k

−



dt  gk2 β j  (t  )e i (νk −ω)(t −t )+ik·(r j −r j )

Fig. 1. Radiation pattern at frequency ω for R = 0.2λ, 2λ and 5.125λ produced by decay of the state (1) calculated taking into account virtual photons (solid line) and omitting them (dots). θ is the angle between k0 and radiation direction, while P γ (θ) is probability of photon emission at angle θ .

j  =1 0 N 

 k

t

t



dt  gk2 β j  (t  )e i (νk +ω)(t −t )−ik·(r j −r j )

j  =1, j  = j 0

− ( N − 1)

 k

t gk2



dt  β j (t  )e i (νk +ω)(t −t ) ,

(2)

0

where gk is the atom field coupling constant. In Eq. (2) we do not take into account polarization effects. The last two terms in Eq. (2) describe contributions from counter rotating terms and contain the sum of frequencies νk + ω . In the usual Markovian approximation, Eq. (2) reduces to an equation with exponential kernel [4,5]

β˙ j (t ) = −γ β j (t ) + i γ

N  exp(ik0 |r j − r j  |) β j  (t ), k 0 |r j − r j  | 

(3)

j = j

where γ is the single atom decay rate given by γ = V ph k20 gk2 /π c 0 in which V ph is the quantization volume and c is the speed of light. If we ignore virtual contributions then Eq. (2) yields an equation with sinusoidal kernel N  sin(k0 |r j − r j  |) β˙ j (t ) = −γ β j  (t ). k 0 |r j − r j  | 

(4)

j =1

We solve the evolution equations (3) and (4) numerically for N = 10 000 atoms randomly distributed in a sphere of radius R = 0.2λ, 2λ and 5.125λ. Initially atoms are prepared in the state |ψ+ . Fig. 1 compares radiation patterns obtained using the exponential and sinusoidal kernels, while in Fig. 2 we plot the probability P + (t ) to find atoms in the state |ψ+  as a function of time. From Figs. 1 and 2 we see that the results with and without virtual photons are essentially identical. Next, we provide analytical analysis which explains why virtual processes, for all practical purposes, do not affect evolution of the |ψ+  state. For a dense cloud one can treat the atom distribution as continuous and rewrite Eq. (2) as

Fig. 2. Probability P + (t ) to find atoms in the state (1) as a function of time obtained using the evolution equations (3) with exp kernel (red solid line) and (4) with sin kernel (blue dash line). Initially atoms are prepared in the state (1). Simulations are made for 10 000 atoms randomly distributed in a sphere with R = 0.2λ, 2λ and 5λ.

˙ t , r) = − β(

V ph N

(2π )3 V



 dk

dr



t

dt  gk2 β(t  , r )

0

      × e i (νk −ω)(t −t )+ik·(r−r ) + e i (νk +ω)(t −t )−ik·(r−r ) − ( N − 1)

V ph

(2π )3



t dk gk2



dt  β(t  , r)e i (νk +ω)(t −t ) ,

(5)

0

where r integration is over the volume V = 4π R 3 /3 of spherical atomic cloud. Having in mind the initial condition 1

β(0, r) = √ e ik0 ·r N

(6)

as per Eq. (1), we look for a solution of Eq. (5) in the form

β(t , r) = β(t )e ik0 ·r . Then Eq. (5) yields

(7)

M.O. Scully, A.A. Svidzinsky / Physics Letters A 373 (2009) 1283–1286

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For a finite (but large) atomic sample such that k0 R  1, following [2], we write the delta function in terms of the magnitude of k and solid angle unit vector Ωˆ k as

δ(k ± k0 ) = δ(k − k0 )  ∼ =

δ(Ωˆ k − Ωˆ ±k0 )

2R

1 2π

k2



δ(Ωˆ k − Ωˆ ±k0 )

e i (k−k0 )x dx

k2

−2R

(14)

,

where R is the radius of the atomic cloud. Then Eq. (8) yields (we approximate gk2 ≈ gk2 ) 0

Fig. 3. Probability that atoms are excited P (t ) as a function of time calculated using the evolution equations (3) and (4) with exp (red) and sin (blue) kernels for √ N= 10 000 and R = 0.5λ. Solid lines show the result for the initial state β j (0) = 1/ N, while dash lines correspond to the initial state (1).

˙ t) = − β(

0



˙ t) = − β(

(2π

)3



 dk

V

dr

t

dx e

−2R

0

0



V ph

t

0

V ph



t dk gk2

(2π )3



dt  β(t  )e i (νk +ω)(t −t ) .

(8)

0

and after integration over time obtain

˙ t) = − β(

and we find

˙ t ) = −Ω β(



t



dt β(t ) − Ω 0

2





dt β(t )e

i (νk0 +ω)(t  −t )

−

− ( N − 1)

(2π )3

t

 dk gk2





dt β(t )e

i (νk +ω)(t  −t )

,

dx β t −

0

2cV

−2R



2R

V ph N gk2

dx β t −

0

2cV

−2R

− ( N − 1)

(9)



2R

V ph N gk2

0

V ph



V ph

(2π )3

N

V ph V

gk0

(10)

is an effective Rabi frequency and we recall that V ph is the photon quantization volume and V is the volume of the atomic cloud. The last two terms in Eq. (9) describe virtual photon processes. Because these terms contain integrals over rapidly oscillating functions we may omit them. Therefore the effect of virtual processes is negligible for the present problem and we obtain

˙ t ) = −Ω 2 β(

t

dt  β(t  ).

c

−

x

−

c

t

kˆ 0 · r



c kˆ 0 · r



c

ˆ

e −2ik0 (x+k0 ·r)



dt  β(t  )e i (νk +ω)(t −t ) .

(16)

0

where

Ω=

x

dk gk2

0



(15)

∞



dr e −i (k±k0 )·r = (2π )3 δ(k ± k0 )

t



dt  β(t  )e i (νk +ω)(t −t ) ,



dk e izk = π δ( z),

For an infinitely large atomic sample integration over r gives the delta-function

2

ic (k+k0 )(t  −t )+i (k−k0 )kˆ 0 ·r

where kˆ 0 is a unit vector in the direction of k0 . Next, we replace integration over k by a delta function

0



+e

dk gk2

(2π )3

dt  β(t  )

0

      × e i (νk −ω)(t −t )+i (k−k0 )·(r−r ) + e i (νk +ω)(t −t )−i (k+k0 )·(r−r ) − ( N − 1)

t

i (k−k0 )x

ic (k−k0 )(t  −t )+i (k−k0 )kˆ 0 ·r

− ( N − 1)

dt  gk2 β(t  )

2R

dk

2π V

× e V ph N

∞

V ph N gk2

(11)

The last two terms in Eq. (16) contain integrals over rapidly oscillating functions. One can disregard such terms, which shows that the virtual photon processes give a small contribution, and find

˙ t) = − β(

V ph N gk2 2cV



2R

dx β t −

0

−2R

x c

−

kˆ 0 · r



c

(17)

.

We are here concerned with emission times t ∼ 1/Γ such that t  R /c. Thus in Eq. (17) we replace t − the result of Ref. [2]

˙ t ) = −Γ β(t ), β(

x c

−

kˆ 0 ·r c

→ t and obtain (18)

where

0

2V ph N R gk2

3N

Differentiation Eq. (11) over time yields a harmonic oscillator equation

Γ =

¨ t ) + Ω 2 β(t ) = 0 β(

Next, we show that for t  1/Γ every state decays with the same rate for both (exponential and sinusoidal) kernels irrespective of the size and shape of the atomic cloud. Namely, we show that the function

(12)

and, thus, atomic state undergoes harmonic oscillations with frequency Ω [3] 1

β(t , r) = √

N

cos(Ω t ) exp[ik0 · r].

0

cV



(13)

P (t ) =



=

2(k0 R )2

2

dr β(t , r) ,

γ.

(19)

(20)

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M.O. Scully, A.A. Svidzinsky / Physics Letters A 373 (2009) 1283–1286

which determines the probability that atoms are excited at time t has equal time derivatives at t = 0 for both kernels. From Eq. (3) for a continuous atom distribution we obtain

∂|β(t , r)|2 N = iγ ∂t V −





exp(ik0 |r − r |) ∗ dr β (t , r)β(t , r ) k0 |r − r |

exp(−ik0 |r − r |) k0 |r − r |

β(t , r)β ∗ (t , r ) ,

Acknowledgements We thank R. Friedberg and J. Manassah for useful and stimulating comments. We gratefully acknowledge the support of the Office of Naval Research (Award No. N00014-07-1-1084 and N0001408-10948) and the Robert A. Welch Foundation (Award A-1261).

(21) Appendix A. Derivation of eigenvalue equation

or

 ∂|β(t , r)|2 N sin(k0 |r − r |) = −γ dr ∂t V k0 |r − r |   ∗ × β(t , r)β (t , r ) + β ∗ (t , r)β(t , r )

  cos(k0 |r − r |) + β(t , r )β ∗ (t , r) − β ∗ (t , r )β(t , r) . ik0 |r − r | (22)

We consider a system of two level (a and b) atoms, initially one of them is in the excited state a and E a − E b = ω . Initially there are no photons. Atoms are located at positions r j ( j = 1, . . . , N). In the dipole approximation the interaction of atoms with photons is described by the Hamiltonian (we disregard polarization effects)

ˆ int = H

N 

Integrating Eq. (22) over r we find





k

sin(k0 |r − r |) k |r − r |

∂ P (t ) N = −γ dr dr ∂t V 0   × β(t , r)β ∗ (t , r ) + β ∗ (t , r)β(t , r )

  cos(k0 |r − r |)  ∗ ∗  + β(t , r )β (t , r) − β (t , r )β(t , r) . (23) ik0 |r − r | The factor [β(t , r )β ∗ (t , r) − β ∗ (t , r )β(t , r)] in the last term of

cos(k |r−r |)

Eq. (23) changes sign under transformation r ↔ r , while k |0r−r | 0 remains the same. Therefore, due to symmetry, the last term in Eq. (23) vanishes after integration over dr dr and Eq. (23) reduces to

 ∂ P (t ) N sin(k0 |r − r |) = −γ dr dr ∂t V k0 |r − r |   ∗  × β(t , r)β (t , r ) + β ∗ (t , r)β(t , r ) .

×



gk σˆ j e −i ωt + σˆ j e i ωt †

j =1



† aˆ k e i νk t −ik·r j



 + aˆ k e −i νk t +ik·r j ,

(A.1)

where σˆ j is the lowering operator for atom j, aˆ k is the photon operator and gk is the atom–photon coupling constant for the k mode. Please note that we do not make the rotating wave approximation in Eq. (A.1). We look for a solution of the Schrödinger equation for the atoms and the field as a superposition of Fock states

Ψ=

N 

β j (t )|b1 b2 · · · a j · · · b N |0 +

j =1

+





γk (t )|b1 b2 · · · b N |1k 

k

αmn,k (t )|b1 , b2 , . . . , am , . . . , an , . . . , b N |1k ,

(A.2)

m
(24)

Eq. (24) has the same form for the exponential and sinusoidal kernels. As a result, for early times the slope of P (t ) is the same irrespective of which kernel is used in calculations. This property is valid for any initial state and any size and shape of the atomic cloud. Because the slope of P (t ) is the same for both kernels at t = 0 the state evolution at later moments of time can be very close. This is the case for the state (1). In the small sample limit R  λ the |ψ+  state (which for R  λ coincides with the symmetric state βs (r) = 1) is an eigenstate for the sin kernel and decays at a rate Γ+ = N γ . For the exp kernel the fastest decaying eigenstate is β00 (r) = sin(π r /2R )/r and Γ0 = 96N γ /π 4 ≈ 0.986N γ [4]. Because Γ0 ≈ Γ+ and overlapping between states β+ (r) and β00 (r) is almost 1: β+ (r)|β00 (r) = 0.993 the |ψ+  state decays essentially the same way for both kernels. It is important to mention that there are single photon states for which the virtual photon transitions substantially modify the state decay. A state in which atoms are uniformly excited√without the phase factors e ik·r j of Eq. (1), that is β j (0) = 1/ N, is an example of such state. For this the probabilNinitial condition 2 ity that atoms are excited P (t ) = j =1 |β j (t )| calculated using sin and exp kernels is shown in Fig. 3. There we see that √ the virtual photon contributions are substantial. The β j (0) = 1/ N state is an example of a trapped state which for a large atomic sample decays many orders of magnitude slower then the |ψ+  state. In conclusion: Friedberg and Manassah [5,6] have emphasized that different atoms can acquire different phase factors at different points in the medium due to virtual processes and this can destroy the validity of Eq. (4) governed by the sin kernel. We find, however, that for evolution of the |ψ+  state the effects of the N atom collective Lamb shift are small.

where αmn,k = αnm,k . States in the first sum correspond to zero number of photons, while in the second sum the photon occupation number is equal to one and all atoms are in the ground state b. The third term corresponds to presence of two excited atoms inside the cloud and one (virtual) photon with “negative” energy. Substitution of Eq. (A.2) into the Schrödinger equation yields the following equations for β j (t ), γk (t ) and αmn,k (t ) (we put h¯ = 1)

β˙ j (t ) = −i





gk γk (t ) exp −i (νk − ω)t + ik · r j



k

−i

 k

γ˙k (t ) = −i

N 

N 

gk

α j j ,k (t )e ik·r j e −i(νk +ω)t ,

(A.3)

j  =1, j  = j





gk β j (t ) exp i (νk − ω)t − ik · r j ,

j =1





α˙ mn,k (t ) = −igk βn (t ) exp i (νk + ω)t − ik · rm + (n ←→ m).

(A.4) (A.5)

Integrating Eqs. (A.4) and (A.5) over time with initial conditions

γk (0) = 0, αmn,k (0) = 0 and substituting the answer into Eq. (A.3) we obtain Eq. (2). References [1] [2] [3] [4] [5] [6] [7]

M. Scully, E. Fry, C.H.R. Ooi, K. Wodkiewicz, Phys. Rev. Lett. 96 (2006) 010501. M. Scully, Laser Phys. 17 (2007) 635. A.A. Svidzinsky, J.T. Chang, M.O. Scully, Phys. Rev. Lett. 100 (2008) 160504. A.A. Svidzinsky, J.T. Chang, Phys. Rev. A 77 (2008) 043833. R. Friedberg, J.T. Manassah, Phys. Lett. A 372 (2008) 2514. R. Friedberg, J.T. Manassah, Phys. Lett. A 372 (2008) 5734. A.A. Svidzinsky, J.-T. Chang, Phys. Lett. A 372 (2008) 5732.