Comment to mirhosse@optics.rochester.edu

Weak-value amplification of the fast-light effect in rubidium vapor

Mohammad Mirhosseini,1, ∗ Gerardo Viza,2 Omar S. Magana-Loaiza,1

Mehul Malik,1, 3 John C. Howell,1, 4 and Robert W. Boyd1, 4

1The Institute of Optics, University of Rochester, Rochester, New York 14627, USA2Department of Physics, University of Rochester, Rochester, New York 14627, USA

3Institute for Quantum Optics and Quantum Information (IQOQI),Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria

4Department of Physics, University of Ottawa, Ottawa ON K1N 6N5, Canada(Dated: January 27, 2015)

We use weak-value amplification to enhance the polarization-sensitive fast-light effect from in-duced Raman absorption in hot rubidium vapor. We experimentally demonstrate that projectingthe output signal into an appropriate polarization state enables a pulse advancement of 4.2 µs,which is 15 times larger than that naturally caused by dispersion. More significantly, we show thatcombining weak-value amplification with the dispersive response of an atomic system provides aclear advantage in terms of the maximum pulse advancement achievable for a given value of loss.This technique has potential applications for designing novel quantum-information-processing gatesand optical buffers for telecommunication systems.

PACS numbers: 42.30.Ms, 42.50.Ar, 42.30.Va

Due to Kramers-Kronig relations, a sharp change inthe absorption or the transmission of an optical mediumresults in a large modification of the group index [1].Controlling the group velocity using slow and fast-lightis an enabling technology with many applications in pho-tonics [2–5]. Additionally, fast-light provides a uniquetestbed for studying the fundamental physics behind su-perluminal pulse propagation. While slow light can beachieved with no appreciable loss using electromagneti-cally induced transparency (EIT) [6], one needs to oper-ate close to the center of an absorption line to achievefast-light [7]. In this situation, the large amount of ab-sorption limits both the applications and the magnitudeof the observed fast-light effect.

A weak measurement is a generalized form of quantummeasurements, in which a weak unitary interaction is fol-lowed by a strong projective measurement [8, 9]. Unlikethe standard measurements, the result of a weak mea-surement, known as a weak value, can be beyond therange of eigenvalues of the measured operator [10]. Thisproperty, known as weak-value amplification (WVA), hasbeen used before to sensitively measure a variety of ef-fects, such as a transverse beam deflection [11–14], phase[15], velocity [16], and time delay [17]. Further, it hasbeen suggested that the weak-value amplification can beused to enhance nonlinear optical effects in the few pho-tons regime [18].

In this work, we amplify the negative time delay asso-ciated with the “superluminal” pulse propagation of anoptical pulse in hot rubidium vapor. The fast-light effectis caused by an induced Raman absorption profile of therubidium hyperfine structure in a pump-probe nonlinearinteraction. Due to this effect, the polarization of an op-tical pulse is weakly coupled with its arrival time. By

appropriately preparing and post-selecting the polariza-tion states of the pulse, we can effectively engineer thedispersion properties of the medium [17, 19], and thusamplify the weak coupling between polarization and ar-rival time. Using this technique, we were able to advancethe peak of an optical pulse by an amount that is up to15 times larger than the original fast-light advancement.

The fast-light effect due to absorption and the enhance-ment due to weak-value amplification are both lossy pro-cesses. Here, we study how the achieved temporal ad-vancement scales as a function of loss due to atomic ab-sorption and compare it to the scaling as a function ofloss due to post-selection in WVA. Remarkably, we findthat for a given value of loss, an optimized combination ofboth these processes provides a larger time advance thanthat obtained by just increasing the atomic absorptionitself. In light of the ongoing debate on the usefulness ofWVA [20–22], we find this result to be both timely andsignificant.

Tunable group delay from atomic response. In a dis-persive medium the group velocity and the phase velocityare not the same. The group velocity and the group indexcan be calculated from the standard results

vg = c/ng, ng = n+ ωdn

dω. (1)

Slow and fast-light correspond to the situations whereng >> 1 and ng < 1, respectively. Due to the Kramers-Kronig relation, a sharp change in the absorption coeffi-cient can lead to a substantial change in the group index.A large pulse advance in fast light can be achieved by op-erating in a wavelength close to the center of an absorp-tion line (See Fig. 1). However, the large amount of lossin this region limits the amount of maximum negativedelay that can be achieved in practice.

arX

iv:1

412.

3019

v1 [

quan

t-ph

] 9

Dec

201

4

2

-4 -2 0 2 4

absorptionresonance

index of refraction

group index

-4 -2 0 2 4

-4 -2 0 2 4

fast light

slow lightslow light

SignalCoupling

a) b)

d)c)

FIG. 1: a) The absorption profile of a Lorentzian lineshape.b) The refractive index associated with the absorption linecan be calculated using the Kramers-Kronig relations. c) Thegroup index for the same line. d) Schematic diagram of athree-level Lambda system.

We use a nonlinear-process to induce a polarization-sensitive absorption line in an atomic vapor. Considera three-level atomic Λ system, where levels 1 and 2are connected via the signal field

Ωp

2 e−i(ωpt−kz) + c.c

and level 2 and 3 are connected by a strong couplingfield Ωc

2 e−i(ωct−kz) + c.c. The detunings are defined as

∆ = ∆p = (ω2 − ω1) − ωp, ∆c = (ω2 − ω3) − ωc andδ = ∆p − ∆c. In the case where the coupling and thesignal have orthogonal polarizations the susceptibility atthe signal frequency can be calculated as

χ(∆, δ,Ωc) = βδ − iγ

(δ − iγ)(∆− iΓ/2)− |Ωc|2/4. (2)

Here, γ and Γ are the excited-state and the ground-state decoherence rates, respectively. The factor β isequal to Nµ2/~ε0, where N is the number density andµ is the transition dipole moment between levels 1 and2. This formula can explain many interesting results in-cluding the EIT effect. For large single photon detuningvalues, ∆ >> Γ, this expression can be approximated bya Lorentzian line shape. In this case the susceptibilitycan be written as

χ(∆ >> Γ) = β|Ωc|2

4∆2

δ′ + iγ′

δ′2 + γ′2, (3)

In the above δ′ = δ− δ0, γ′ = γ+ γ0, where δ0 = |Ωc|2∆4∆2+Γ2

and γ0 = |Ωc|2Γ8∆2+2Γ2 .

It should be emphasized that in this configuration, thepolarization component of the signal that is orthogonalto the polarization of the signal beam experiences an ad-vancement in time with respect to propagation in vac-uum, whereas the component of the signal polarized par-allel to the coupling beam experiences no modification

in the group delay while traveling through the medium.When the signal field is comprised of both polarizationcomponents, the differential time advance can be en-hanced by a projective measurement in the polarization[17, 19]. It is well known that the amplification of a timedelay obtained in this manner is in fact an interferenceeffect that can be fully understood using classical electro-magnetic theory. However, expressing this phenomenonwithin the weak-value formalism leads to a simpler andmore elegant description that is easier to understand.

Weak-value amplification. We cast the propagation ofan optical pulse through the atomic vapor in the languageof quantum state measurement. The polarization and thetemporal state of the signal beam before the cell can bedescribed as

|Ψin〉 =1√

(1 + T )

(|H〉+

√T |V 〉

)⊗ |f(t)〉e−iω0t, (4)

where we have assumed a quasi-monochromatic single op-tical mode with a pulse shape described by |f(t)〉. Thehorizontal and vertical polarization states are shown as|H〉 and |V 〉 respectively. Since the polarization compo-nent |H〉 attenuates upon propagation, it is weighted by alarger factor to pre-compensate for the effect of loss. Thepower transmission efficiency of propagation through thecell for the horizontally polarized beam is denoted by T .We now consider a case where the coupling beam is po-larized in the vertical direction. In this situation, thestate of the signal beam in the output can be describedas [6, 23]

|Ψout〉 =

√T

(1 + T )(|H〉|f(t+ t0)〉+ |V 〉|f(t)〉) e−iω0t.

(5)Here, t0 = −(ng − 1)Lc is the absolute value of the groupdelay for a propagation length L. We have dropped thecommon delay time between the two polarization statesin order to simplify the notation. It is seen that hor-izontal polarization experiences attenuation and an ad-vancement in time compared to the vertical componentof the field. Additionally, we have assumed the opticalpath length in the medium for the two polarization com-ponents are equal. This results in the convenient phasedifference of zero between the two polarization compo-nents in the output. In practice, a non-zero phase dif-ference can always be pre-compensated by changing thepolarization state of the input signal beam.

We use the weak value formalism for the case wherethe advancement time t0 is much smaller than the tem-poral duration of the pulse f(t). Although our formalismclosely follows that of Ref. [17], the large amount of timeadvance and the tunability provided by the atomic sys-tem in our experiment offers a degree of control absentfrom previous realizations. Furthermore, we provide acomparison of the loss vs. time advance from WVA tothe one obtained from the KK relations. This analysis

3

FIG. 2: Schematic of the experimental setup. The continuous wave laser beam is divided to two copies using a non-polarizingbeam splitter. The probe is frequency shifted using an acousto-optic modulators. The polarization state of the pump andthe probe are controlled using wave plates. The post-selection is done using a polarizer and a fast detector. The outputphoto-current is analyzed by an oscilloscope.

provides theoretical evidence for the efficacy of combin-ing weak value amplification from the natural dispersivetime advance from atomic response.

Assuming a Gaussian pulse shape, a post-selection inthe a linear polarization state with an angle θ with re-spect to the frame of the experiment results in

| ΨPS〉 ≈

√T

(1 + T )(cos θ|H〉+ sin θ|V 〉)⊗|f(t+Awt0)〉e−iω0t.

(6)Here, ΨPS is the polarization and the temporal state ofthe post-selected beam.

The weak value Aw corresponds to the temporal shiftof the optical pulse and can be calculated using the for-mula

Aw =〈Φθ | A | Ψin〉〈Φθ | Ψin〉

=cos θ

sin θ + cos θ, (7)

where |Φθ〉 is the post-selection polarization state andthe measurement operator is A = |H〉〈H|. It can beseen in Fig. 3 that choosing the post-selection angle θclose to 45 results in a large amplification factor Aw.More interestingly, it is possible to achieve a negativeamplification and hence convert a time delay to a timeadvance and vice versa.

Experiment. We realize the Raman absorption profileusing warm atomic rubidium vapor. A sketch of the ex-perimental setup is depicted in Fig. 2. The beam from a795-nm narrow-line-width tunable diode laser is passedthrough a tapered-fiber amplifier to obtain a high powerbeam. The frequency of the laser is tuned to have thecoupling beam near resonant with the |2〉 to |3〉 transi-tion. The signal beam is obtained by frequency-shiftingpart of the the laser beam by 3.035 GHz by double-passing it through a tunable acousto-optic modulator.An 8 cm vapor cell is heated using strip heaters insidea teflon tube enclosed by antireflection-coated windowsat each end to achieve temperature stability. The cell isshielded from stray magnetic field by a Mu-metal tubing.

0 5 10 15 20 25Time (μs)

0

0.2

0.4

0.6

0.8

1

Opt

ical

Pow

er (a

.u.)

−100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0

−20

−10

0

10

20

Post−Selection Angle (deg)

Ampl

ifica

tion

Fact

or

FIG. 3: Top: Measured optical power as a function of time.The cyan and orange curves show the detected optical powerfor the |V 〉 and |H〉 sates. The red and blue curve correspondto post-selection angles θ = −40 and θ = −50. Bottom:The amplification factor as a function of post-selection an-gle. The amplification factor is calculated from data by per-forming a Gaussian fit.The error bars correspond to the 95 %confidence interval.

The vapor cell contains both rubidium isotopes in theirnatural abundance. In addition, we also have 20 Torrneon in the cell, which acts as a buffer gas. The tem-perature of each vapor cell is about 80C, resulting in anumber density of about 1012cm−3.

4

The coupling and the signal beams are aligned co-linearly to achieve the absorption dip. We use an atomicprism to filter out the coupling beam after the cell [24].This prism contains isotopically pure 87Rb and is heatedto 100C to achieve a large dispersion (dn/dλ). The cou-pling and the signal beams propagate at different anglesonce they exit the prism. We collect the signal beam andproject it on an arbitrary superposition of the two beamsusing a polarizer. The pulses are detected using a fastphoto-detector diode.

We have chosen to work in the center of the absorptionline (δ′ = 0) to achieve a maximal time advance t0 =0.28µs. The top part in Fig.3 presents measured opticalpower as a function of time for 4 different post-selectionangles. It can be seen that the post-selection can greatlymodify the amount of delay, and even flip the sign of itresults for two different post-selection angles. We havecalculated the amplification factor, by fitting a Gaussianwave form to the pulse and measuring the position of itscenter. The value of amplification factor as a functionof post-selection angle from the experiment is plotted inthe bottom part of Fig.3. It can be seen that the delaygets drastically amplified as θ approaches −45.

The scaling of loss and group-delay. The analysis andthe experimental results above suggest that the WVA canbe combined with the dispersive response of an atomicsystem to provide extra control over the value of groupdelay. However, to get an appreciable amplification fac-tor one needs to post-select on a state that is nearly or-thogonal to the input state. In this situation, the effi-ciency of the process is significantly reduced. This is,in fact, a universal property associated with weak valuesand the usefulness of WVA in presence of this additionalloss has been a topic of debate recently [20–22].

Before analyzing the effect of loss from post-selection,we investigate the relation between loss and group delayfrom the atomic response. For operation at the center ofthe absorption line, δ′ = 0, we have (see the appendix)

T = exp (−2γ′t0). (8)

In deriving this relation we have used Eq. 3 and the ap-proximation n ≈ 1+ 1

2χ. It is evident that increasing theabsolute value of time advance via increasing the non-linear interaction results in an exponential decrease inthe transmission efficiency. Consequently, the maximumachievable delay from atomic response for a given value ofminimum acceptable transmission efficiency can be cal-culated as

tatom = − lnT

2γ′. (9)

Equation 9 calculates the maximum amount of time ad-vance that can be achieved from the atomic response for

a given value of transmission efficiency. As we showedearlier, the WVA amplification can be employed to in-crease the amount of group delay at the cost of a furtherincrease in the loss. For a given value of loss, the maxi-mum value of achievable time advance can be calculatedby optimizing the absorption from the atomic responseand the loss from post-selection. Performing the analysiswe find the maximum value of time advance as (see theappendix)

tWVA =1

2γ′maxθ

(cos θ

sin θ + cos θln

[2 sin2

(θ + π

4

)T

− 1

]).

(10)

10−3 10−2 10−1 1000

9

6

3

Power Transmission Efficiency

1.20.04 0.08

1.7

FIG. 4: Maximum achievable time advance as a function oftransmission efficiency (calculated from theory)

The solutions of Eq. 10 have been calculated numer-ically and plotted in Fig. 4, along with the solutionsfor Eq. 9. It is evident that the WVA procedure pro-vides a slightly lower time advance for large values oftransmission efficiency. However, as the transmission ef-ficiency decreases, the time advance obtained by usingWVA grows rapidly, crossing the advancement obtainedfrom the atomic response at T ≈ 5%. For all values ofT lower than this value, WVA provides a larger time ad-vance than that obtained from the atomic response alone.This showcases a clear instance where WVA provides anadvantage for the estimation of a small interaction pa-rameter.

Conclusions We have used weak-value amplifica-tion to enhance the fast-light effect caused byelectromagnetically-induced absorption in warm Rubid-ium vapor. By appropriately preparing and post-selecting the polarization state of an optical pulse, wehave obtained an advancement in time that is 15 timeslarger than that obtained from the atomic resonance.The enhancement from WVA can also be tuned to con-vert a time advance into a time delay and vice versa.

5

Additionally, we have shown that when the total trans-mission through the system is lower than 5%, the use ofWVA provides a clear enhancement in the amount of timeadvance possible. Our technique has the potential to al-low better control over photonic technologies that use fastand slow light, such as quantum-information-processinggates and optical buffers for telecommunication systems.

We gratefully acknowledge valuable discussions withS. Gillmer, J. Vornehm, Z. Shi, J. Ellis and D. Gau-thier. This work was supported by the DARPA/DSO In-Pho program, the Canadian Excellence Research Chair(CERC) program, Army Research Office grant numberW911NF-12-1-0263. OSML acknowledges the supportfrom CONACYT and the Mexican Secretarıa de Edu-cacion Publica (SEP).

∗ Electronic address: mirhosse@optics.rochester.edu[1] R. W. Boyd, Nonlinear Optics (Academic Press, New

York, 2003), 3rd ed.[2] R. M. Camacho, P. K. Vudyasetu, and J. C. Howell, Na-

ture Photon. 3, 103 (2009).[3] K. F. Reim, J. Nunn, V. O. Lorenz, B. J. Sussman, K. C.

Lee, N. K. Langford, D. Jaksch, and I. A. Walmsley,Nature Photon. 4, 218 (2010).

[4] Z. Shi, R. Boyd, R. Camacho, P. Vudyasetu, and J. How-ell, Phys. Rev. Lett. 99 (2007).

[5] R. W. Boyd and D. J. Gauthier, in Progress in Optics(Elsevier, New York, 2002), p. 497.

[6] M. Fleischhauer, A. Imamoglu, and J. Marangos, Rev.Mod. Phys. 77, 633 (2005).

[7] E. E. Mikhailov, V. A. Sautenkov, Y. V. Rostovtsev, andG. R. Welch, JOSA B 21, 425 (2004).

[8] Y. Aharonov, D. Albert, and L. Vaidman, Phys. Rev.Lett. 60, 1351 (1988).

[9] J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, andR. W. Boyd, Rev. Mod. Phys. 86, 307 (2014).

[10] J. Dressel, S. Agarwal, and A. N. Jordan, Phys. Rev.Lett. 104, 240401 (2010).

[11] P. Dixon, D. Starling, A. Jordan, and J. Howell, Phys.Rev. Lett. 102 (2009).

[12] N. Ritchie, J. Story, and R. Hulet, Phys. Rev. Lett. 66,1107 (1991).

[13] O. Hosten and P. Kwiat, Science 319, 787 (2008).[14] O. S. Magana-Loaiza, M. Mirhosseini, B. Rodenburg, and

R. W. Boyd, Phys. Rev. Lett. 112, 200401 (2014).[15] N. Brunner and C. Simon, Phys. Rev. Lett. 105, 010405

(2010).[16] G. I. Viza, J. Martinez-Rincon, G. A. Howland,

H. Frostig, I. Shomroni, B. Dayan, and J. C. Howell,Opt. Lett. 38, 2949 (2013).

[17] N. Brunner, V. Scarani, M. Wegmuller, M. Legre, andN. Gisin, Phys. Rev. Lett. 93, 203902 (2004).

[18] A. Feizpour, X. Xing, and A. M. Steinberg, Phys. Rev.Lett. 107, 133603 (2011).

[19] D. Solli, C. McCormick, R. Chiao, S. Popescu, andJ. Hickmann, Phys. Rev. Lett. 92 (2004).

[20] C. Ferrie and J. Combes, Phys. Rev. Lett. 112, 040406(2014).

[21] A. N. Jordan, J. Martinez-Rincon, and J. C. Howell,Phys. Rev. X 4, 011031 (2014).

[22] G. C. Knee and E. M. Gauger, Phys. Rev. X 4, 011032(2014).

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APPENDIX

Using the susceptibility in Eq. 3 and assumingn ≈ 1 + 1

2χ we get

n = Re [n] + i Im [n] = 1 + β|Ωc|2

8∆2

δ′ + iγ′

δ′2 + γ′2. (11)

The group delay can be calculated using the real part ofthe group index. Assuming ∆ >> Γ, and ∆ >> |Ωc| wehave

ng|δ′=0 = 1 + β|Ωc|2

8∆2

ω

γ′2. (12)

The group delay can be calculated using the group indexas

Lng

c . The value of group delay includes the propaga-tion time in vacuum. The differential time advance canbe found as

t0 = βL

c

|Ωc|2

8∆2

ω

γ′2(13)

We find the value of absorption at the center of theLorentzian line as

α =ω

cIm [n]

∣∣∣δ′=0

=β

8c

|Ωc|2

∆2

ω

γ′. (14)

Subsequently, the power transmission efficiency is

T = exp (−2αL) = exp (−2γ′t0). (15)

The relation above can be inverted to find the maximumachievable time advance for a minimum acceptable trans-mission efficiency

tatom = − lnT

2γ′. (16)

The alternative strategy is to initially achieve a time ad-vance t0 from the atomic response, at the cost of a reduc-tion in the transmission efficiency for one of the polariza-tion components to T . The input beam is described bythe state

|Ψin〉 =1√

1 + T

(|H〉+

√T |V 〉

)⊗ |f(t)〉e−iω0t, (17)

6

and the state after the post-selection

| ΨPS〉 =

√T

1 + T(cos θ|H〉+ sin θ|V 〉)⊗|f(t+Aw t0)〉e−iω0t.

(18)As a result, the total transmission efficiency is

T = |〈ΨPS |ΨPS〉|2 =2T

1 + Tsin2

(θ +

π

4

)(19)

The total time advance is equal to

t0 = Aw t0 =cos θ

sin θ + cos θt0. (20)

Using the relation t0 = − ln T2γ′ we have

t0(θ) =1

2γ′cos θ

sin θ + cos θln

[2 sin2

(θ + π

4

)T

− 1

]. (21)

And finally, we maximize the time advance to find theoptimal post-selection angle

tWVA =1

2γ′maxθ

(cos θ

sin θ + cos θln

[2 sin2

(θ + π

4

)T

− 1

]).

(22)