Radiation Force on an Anomalously Dispersive Medium L. J. Wang NEC Research Institute, 4 Independence Way, Princeton, NJ 08540, USA

P. D. Lett NIST, 100 Bureau Drive, Stop 8424, Gaithersburg, MD 20899, USA (September 10, 2001) We consider the radiation force on a highly anomalously dispersive medium, caused by a probe pulse that propagates superluminally in the medium. We show that, this radiation force can also emerge at a velocity that exceeds c, being infinity, or negative. This apparent “force-at-distance” is not at odds with causality, owing to the conservation of momentum. PACS: 03.65.Sq., 42.25.Bs, 42.50.Vk

It is well known by now that with the aid of gain lines, a transparent medium can become anomalously dispersive [1–6]. If there is no appreciable gain or loss and the dispersion is linear in frequency, an incident pulse described by a sufficient smooth envelope E(t) becomes simply E(t − L/Vg ) after propagating a distance L. Here, Vg is the group velocity given by Vg = c/ng , with ng = n + ω dn/dω being the group velocity index. Experimentally, when the group index becomes negative, the peak of an exit pulse will appear on the far side of the medium before the peak of the incident pulse arrives at the entrance side [3,4]. The effect can be well explained by invoking the process of “rephasing” in an anomalously dispersive medium [5]. Furthermore, using an operational definition of the signal velocity, it was recently shown [6] that such a superluminal group velocity is not at odds with causality owing to quantum fluctuations.

amine the radiation force on such a medium. We start by considering the experimental situation illustrated in Fig.1(a). For simplicity, we consider a cloud of cold atoms in a trap as the medium (region-II). The trapped atoms are initially in a state |1. Two Raman pumping beams E1 and E2 whose frequencies are detuned from the single photon transition |1 → |0 by an amount ∆0 that exceeds the residual Doppler amount [Fig.1(b)] are introduced opposite to each other from the side such that their mechanical effects cancel. The two Raman pumping beams are shifted by a small relative amount of 2∆ and the residual mechanical effect can be ignored. Raman Pump-II

y x

(a)

In this letter, we consider the radiation force of a pulse propagating through such an anomalously dispersive medium. Traditionally, radiation force was first studied in the context of the momentum of electromagnetic fields, both in vacuum and in a medium. Particularly, it played a heavy role in the debate of whether the Minkowski form or the Abraham form for the momentum should be assigned [7,8]. In modern days, radiation forces played a center role in the cooling, trapping of neutral atoms and in the realization of the Bose condensate [9]. Of course, the radiation force, sometimes referred to as the Abraham force [10] has also been extended to a dispersive medium. Recently, it was applied to the case of the ultra-high normally dispersive (“slow light”) medium and interesting wave mixing processes were studied [11]. For long, it has been thought that anomalous dispersion can only occur in an absorptive medium [10]. For an absorptive medium, the majority of the radiation pressure comes from the momentum transfer due to absorption and the Abraham force is no longer important. Because it was only recently shown that anomalous dispersion can also occur in a transparent medium [1–4], we herein ex-

z A

B

C

Region-II

Region-I

z

Region-III

Raman Pump-I

(b)

(c) ∆0

2∆ω

Probe E1 1

E2

EP 2

Absorption Im(χ) & Refractive Index

0

n(ω)

Im(χ)

FIG. 1. (a) Schematic setup. Region-II represents a cold atom cloud. Pulse-A is the incident pulse in a vacuum (Region-I). When the group velocity index becomes negative, ng < 0, a rephasing pulse (pulse-B) is initiated from the exit surface and propagates backward inside the medium at a velocity c/|ng |. (b) Energy level diagram for realizing such a gain-assisted transparent anomalous dispersion. (c) Refractive index and gain as functions of probe frequency.

1

It is easy to show that [3,4] the linear susceptibility can be expressed as a function of the probe frequency ω: χ(ω) =

M M + ω − ∆ω + iγ ω + ∆ω + iγ

F(z, t) =

∂P × B. ∂t

ˆ e−i(ω0 t−ko z) ξ(t − z/Vg ) + c.c E(z, t) = x n(ω0 ) ˆ e−i(ω0 t−ko z) ξ(t − z/Vg ) B(z, t) = y + c.c c ˆ e−i(ω0 t−ko z) P0 (t, z) + c.c. P(z, t) = x

 ∂P ˆ e−i(ω0 t−ko z) − iω0 χ(ω0 ) ξ(t − z/Vg ) (z, t) = ε0 x ∂t  d[ω χ(ω)]  ∂ξ(t − z/Vg )  + + c.c . (7)  dω ∂t ω0 Using eqs.(6) and (7), one hence readily obtain from Eq.(5) for this radiation force: Fz (z, t) ≈

2 ε0 d (ng − 1) |ξ(t − z/Vg )|2 . c dt

(8)

Here, we have assumed that the medium is dilute such that the refractive index n(ω0 ) ≈ 1 and further n(ω) ≈ 1 + χ(ω)/2. The physical implication of Eq.(8) can be readily seen. First, for a medium that is dilute and non-dispersive, it follows immediately that ng − 1 = n − 1 ≈ χ(ω0 )/2. This implies that when an electromagnetic pulse propagates through a weakly polarizable medium, the radiation force arises from the moving energy potential of the polarized dipoles: ε0 χ(ω0 )|ξ(t − z/c)|2 as discussed in ref. [7]. This force is often very small. When the medium is normally dispersive with a large group index, the pulse inside the medium is moving at a greatly reduced group velocity Vg c. This results in a spatial compression of the pulse inside the medium [11]. Consequently, the gradient of the dipolar potential ∇ |ξ(t − z/Vg )|2 is now enhanced by a factor of ng − 1 compared to the vacuum case, giving the effect of “atom surfing” [11]. However, a less intuitive situation occurs when the medium’s dispersion is anomalous. A special case is that

(2)

0

In this case, by combining with the Maxwell equations, it is easy to show that the first term in Eq.(2) simply becomes 1 ∂B ∇(P · E) + P × . 2 ∂t

(6)

Here, ξ(t − z/Vg ) is the envelope function of the probe pulse inside the medium and P (z, t) ≈ ε0 χ(ω0 ) ξ(t−z/Vg ) is the envelope of the polarization. In the absence of spatial dispersion, the first term in Eq.(5) simply becomes ε0 χ(ω0 )∇|ξ(t−z/Vg )|2 . However, because the medium is spectrally dispersive, the second term in Eq.(5) is more complex. Following the method outlined in ref. [10], we can write:

Here, P is the dipolar polarization and E and B are the respective total electro- and magnetic fields at the point z and time t. If the medium has no spatial dispersion, then its susceptibility is only a function of time:  ∞ P(z, t) = dt χ(t )E(z, t − t ). (3)

P · (∇E) =

(5)

Since the optical frequency far exceeds any possible mechanical response of the medium, this radiation force is to be averaged over time. To do so, we expand the optical pulse’s electromagnetic fields and the induced polarization about the carrier frequency ω0 in a complex form:

(1)

where γ > 0 is the linewidth and M > 0 is the two photon matrix element. The expressions for γ and M are not important at this stage and their parameter dependence will be examined later. The radiation force on the medium is simply the Lorentz force on a dipole. Suppose that we examine a small volume dV located at a point z inside the medium, the force on the volume dV is merely F(z, t) dV , where F(z, t) is the force density, i.e., the force on a unit volume of the medium at point z in space and at time t. This small test volume need not be filled with “test atoms,” for if it is small enough, the absence of dV cannot affect the overall propagation of the light pulse in regions nearby. Further, in the case of a dilute gaseous medium, as is in the case of a cold atom cloud, we need not be concerned with the effect of the “local fields.” The reason is simply that in a linear region far below breakdown, the charges are bound and hence no “surface” dipole at the optical frequency can be formed inside the medium to cancel out the incident electromagnetic fields. Of course, it is perfectly allowed to examine the effect on “test atoms” immersed inside the medium. Particularly, in the case of test atoms, depending the general detuning of the probe pulse relative to the transition frequency of the test atoms, both positive and negative “ponderomotive” forces can be formed. We further ignore the residual gain at the probe beam frequency (the frequency region between the two gain lines in Fig.1(c)). Technically, this can be easily achieved by adding some residual absorbing atoms in the cloud. Under these conditions, we can write the radiation force on the medium: F(z, t) = P · (∇E) +

∂ 1 ∇(P · E) + (P × B) . 2 ∂t

(4)

From Eq.(2), we readily obtain the radiation (Abraham) force 2

ng (ω0 ) = n(ω0 )+ω0 dn/dω|ω0 = 0. This is readily achievable in the experiments [3–5]. In this case, the group velocity Vg = c/ng becomes infinity. Physically, inside the medium, at any given moment in time, the field intensity is uniform or independent of z as shown in Fig.2(a). Hence, there is no spatial gradient of the electromagnetic field intensity. However, from Eq.(8), we see that the medium still experiences a force. In Fig.2(b), we plot this force Fz (z, t) = −

2 ε0 d |ξ(t, z = 0)|2 , c dt

it is uniform in space (z-direction). However, the effect becomes again intuitive if it is examined from the viewpoint of momentum conservation. Namely, let us suppose that we place an absorptive plate at the exit surface of the medium. Here, since the pulse is advanced such that the transit time through the medium vanishes, the absorptive plate will receive an extra momentum “kick” compared to a pulse transmitting through a vacuum. Owing to the conservation of momentum, the medium must experience a force backward. It is also easy to show that at a latter stage on the trailing edge of the pulse, this force on the medium becomes positive (along the pulse propagation direction) again.

(9)

as a function of time, normalized to its peak value.

Second, we notice that in this case, the radiation force is independent of space, as if a “force-at-distance” is at play. In other words, every atom inside the whole medium experiences the same force. This is certainly different from the “atom surfing” case as in the “slow light” situation. But this situation, just as that analyzed in ref. [6], is not at odds with causality.

Normalized intensity |ξ(t)|

2

1 (a)

Medium

0.8 0.6 0.4

Furthermore, we examine the case when the group velocity becomes negative [3,4]. In this case, owing to the effect of “rephasing” [5], the pulse’s peak exits the medium before the incident pulse’s peak reaches the entrance surface, as illustrated in Fig.3(a).

t=-2/3 τ

0.2 0 -4

-3

-2

-1

0

1

2

3

4

z/cτ

(b)

Medium

(a)

1

0.5 Intensity

0.8

z

Force F (z,t) (a.u.)

1

0

-0.5

0.6 0.4

t=-τ

0.2

-1

0

-4

-3

-2

-1

0

1

2

3

4

-4

-3

-2

-1

0

1

2

3

4

2

3

4

Distance z/cτ

t/τ 1.2

(b)

0.8

z

Force F (z,t)

FIG. 2. (a), Pulse intensity as a function of distance. The “snapshot” is plotted for a time t = −2/3 τ , where τ is half of the 1/e width of the intensity, and t = 0 for the moment when the peak reaches z = 0. (b) Force on the medium as a function of time. Note here that the force is negative at the earlier part of the pulse and becomes positive at the latter part, with a total momentum transfer (the integral of the force) that is zero. This force Fz (z, t) is independent on the distance z, in this case.

0.4

(III) 0

(II) (I)

-0.4 -0.8 -1.2 -4

-3

-2

-1

0

1

t/τ

Here we immediate observe two interesting features. First, even when there is no intensity and potential gradient inside the medium, the radiation force can still occur. In this case, the simple intuitive picture that there is a moving potential inside the medium whose spatial gradient produces a force (“atom surfing” [11]) does not apply. Here, indeed the potential varies with time, but

FIG. 3. (a), Pulse intensity as a function of distance. The “snapshot” is plotted for a time t = −τ . (b) Force density Fz (z, t) inside the medium as a function of time. Curves (I), (II), and (III) correspond to the entrance z = 0, the center z = 0.5L, and the exit z = L, respectively. Here, the group velocity is taken as Vg = −c/3 and the medium’s thickness is taken as L = cτ /2.

3

At the mean time, a rephasing peak is formed inside the medium and propagates backward at a velocity Vg = c/|ng |. This results in the unusual situation as illustrated in Fig.3(b). Here the atoms near the exit surface of the medium z = L experiences a force against the light pulse propagation direction the earliest. Under the conditions given by parameters in Fig.3(a) and (b), on the leading edge of the incident pulse at a moment of t = −3τ , far earlier than the peak of this pulse has reached the medium’s entrance, a sizable force has already existed at the medium’s exit. Again, this should be explained by invoking the conservation of momentum. Namely, at this moment, the exit pulse is leaving the medium and hence possesses a momentum that is larger than if the pulse is propagating through a vacuum, at a velocity c. This change of momentum is to be balanced via the fact that a negative momentum is transferred to the medium near the exit surface. Furthermore, this counterintuitive effect is not at odds with causality, owing to the necessarily associated gain lines and excess quantum fluctuation they bring [6]. Simply put, because of the existence of the gain lines to enable the anomalously dispersive spectral regions, excess noise must appear. These excess noise terms causes a fluctuation in the intensity of the light field inside the medium. Consequently, the radiation force on the atomic medium is also fluctuating. Hence, it renders it impossible to transmit a signal faster than c by observing the forces on the atoms near the exit surface. However, the averaged field, at substantial level, still causes a force at the exit surface earlier than the case of propagation through vacuum. The detailed analysis is more complex and is beyond the scope of this letter.

ACKNOWLEDGMENTS

We wish to thank S. E. Harris for helpful discussions. LJW’s email is [email protected].

[1] R. Y. Chiao, Phy. Rev. A 48, R37 (1993); A. M. Steinberg and R. Y. Chiao, Phys. Rev. A. 49, 2071 (1994). [2] R. Y. Chiao and A. M. Steinberg, in Progress in Optics, edited by E. Wolf, (Elsevier, Amsterdam, 1997), p.345. [3] L. J. Wang, A. Kuzmich, and A. Dogariu, Nature (London) 406, 277 (2000). [4] A. Dogariu, A. Kuzmich, and L. J. Wang, Phys. Rev. A 63, 053806 (2001). [5] A. Dogariu, A. Kuzmich, H. Cao, and L. J. Wang, Opt. Exp. 8, 344 (2001). [6] A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, Phys. Rev. Lett. 86, 3929 (2001). [7] J. P. Gordon, Phys. Rev. A 8, 14 (1973), and references therein. [8] K. K. Thornber, Phys. Lett. A 43, 501 (1973). [9] H. J. Metcalf and P. van der Staten, Laser Cooling and Trapping (Springer-Verlag, New York 1999). [10] L. D. Landau and E. M. Lifshitz Electrodynamics of Continuous Media (Pergamon, Oxford, 1960) Chap. IX, and references therein. [11] S. E. Harris, Phys. Rev. Lett. 85, 4035 (2000).

4