PPhase shifts vs time delays: Sagnac and Hong-Ou-Mandel
S.J. van Enk
Oregon Center for Optics, Department of PhysicsUniversity of Oregon, Eugene, OR 97403
We point out that the Sagnac effect can be measured by means of the Hong-Ou-Mandel effect.The latter is not sensitive to phase shifts, and thus the Hong-Ou-Mandel Sagnac effect hinges onthe fact that the Sagnac effect is, fundamentally, a time delay, not a phase shift.
I. INTRODUCTION
The Sagnac effect refers to a difference in roundtriptimes between co- and counter-propagating waves in arotating ring interferometer [1–5]. The effect is relativis-tic in the sense that a Galilean-invariant theory predicts anull effect [6]. The effect is universal in at least two ways.It exists for any type of waves, including classical lightwaves and quantum de Broglie waves. Moreover, the timedifference is independent of the propagation speed of thesignal and hence is the same for any frequency (Fourier)component of the wave. In particular, for light wavespropagating in a co-moving medium, the time differencedoes not depend on the medium’s refractive index.When nearly monochromatic CW laser waves are usedin a Sagnac interferometer, the Sagnac time differencemanifests itself as a phase shift, or, equivalently, as afringe shift in the interference pattern (and in a ring cav-ity, as in commercial laser gyros, the effect is transformedinto a frequency shift). In fact, in any interference experi-ment with monochromatic light, time delays, phase shifts,and fringe shifts are all equivalent. In general, though,they are not. Here we point out that there are states ofthe (quantized) electromagnetic field whose interferencebehavior is very differently affected by time delays thanby phase shifts. More precisely, there are states that areinsensitive to constant (i.e., the same for each Fouriercomponent) phase shifts, but that do sense constant timedelays. Such states then, can be used to directly test onefundamental aspect of the Sagnac effect, that it is a con-stant time shift, not a constant phase shift. In particular,one example of an effect that is sensitive to time delaysbut not to phase shifts, is the Hong-Ou-Mandel effect [7].Interestingly, the inverse Hong-Ou-Mandel effect is sensi-tive to both, and is in fact twice as sensitive to phase shiftsas classical light waves are (as was pointed out in Ref. [8],although there the effect was treated for monochromaticlight, and hence did not distinguish between time shiftsand phase shifts).
II. THEORYA. Sagnac
Consider a ring interferometer, enclosing an area A , ro-tating at a frequency Ω (see Figure 1). There are two setsof modes of interest, a ( ω ) and b ( ω ), co-propagating and BS a in b in c out d out Ω FIG. 1: Sagnac interferometer rotating at a frequency Ω.Light propagates through a co-rotating medium in both thecounter-clock-wise and clock-wise directions. The initial com-plex amplitudes of the two light waves at the source are in-dicated by a in and b in , respectively. After one roundtrip thewaves exit the interferometer by impinging on a (co-rotating)50/50 beamsplitter, resulting in output waves with amplitudes c out and d out . counter-propagating, respectively. These modes propa-gate from their source to the detection point, whose lo-cations as seen from the rotating frame of reference co-incide. The modes make a single roundtrip through aco-moving medium with refractive index n ( ω ) and takea time τ ± ( ω ) to complete this roundtrip. We can dis-tinguish two contributions to this total time: the first ispresent even when there is no rotation, and is determinedby n ( ω ) and the distance traveled; this contribution is thesame for co- and counter-propagating waves. The secondterm contains the Sagnac time delays, τ ± S , which dependon both ω and Ω. Their difference, ∆ τ S = τ + S − τ − S does not depend on the refractive index of the medium nor on ω , and is given by ∆ τ S = 4 A Ω c . (1)More precisely, this is the time delay as measured fromthe lab frame, assumed to be an inertial frame of refer-ence. There is an extra time dilation factor for the timedelay as measured in the rotating frame of reference, butto first approximation (using Ω R/c as small expansionparameter, with R the distance of the detector/source a r X i v : . [ phy s i c s . op ti c s ] A p r from the rotation axis) we can neglect this correction.We assume the modes impinge on a 50/50 beamsplit-ter after one roundtrip, exit the ring interferometer, afterwhich measurements are performed. We can express the“output” amplitudes of the modes (after the beamsplit-ter, denoted by c and d ) in terms of the “input” am-plitudes (at the source, denoted by a and b for co- andcounter-propagating modes, resp.) through c out ( ω ) = a in ( ω ) e iωτ + ( ω ) + ib in ( ω ) e iωτ − ( ω ) √ ,d out ( ω ) = b in ( ω ) e iωτ − ( ω ) + ia in ( ω ) e iωτ + ( ω ) √ , (2)and measurements are performed on the output modes c and d . Assume now we measure the intensities of modes c and d (integrating over all frequencies ω ). We have (cid:104) I c,d (cid:105) = 12 (cid:90) dω (cid:104) I a ( ω ) (cid:105) + (cid:104) I b ( ω ) (cid:105)± Im (cid:90) dωe iω ∆ τ S (cid:104) b ∗ in ( ω ) a in ( ω ) (cid:105) . (3)The last term describes interference caused by the Sagnactime difference. If we replace the frequency-dependentphase factor e iω ∆ τ S by a frequency-independent phasefactor e iφ (i.e., a pure phase shift), the interference term isstill nonzero in general. Thus, in general, classical wavesare sensitive to both phase shifts and time delays (andthe sensitivity is the same for monochromatic waves).We can easily switch to a quantum description by pro-moting the complex amplitudes to annihilation operators(in the Heisenberg picture). In particular, if the inputstate of modes a and b is given by a general pure state | ψ (cid:105) in = (cid:90) dω (cid:90) dω (cid:48) (cid:88) n,m a nm ( ω, ω (cid:48) ) × ( a † in ( ω )) n ( b † in ( ω (cid:48) )) m | vac (cid:105) , (4)then the output state of modes c and d can be foundby inverting and taking the hermitian conjugate of therelations (2), which results in | ψ (cid:105) out = (cid:90) dω (cid:90) dω (cid:48) (cid:88) n,m a nm ( ω, ω (cid:48) ) × e inωτ + ( ω ) (cid:32) c † out ( ω ) + id † out ( ω ) √ (cid:33) n × e imω (cid:48) τ − ( ω (cid:48) ) (cid:32) d † out ( ω (cid:48) ) + ic † out ( ω (cid:48) ) √ (cid:33) m | vac (cid:105) , (5)where the invariance of the vacuum state | vac (cid:105) under the(photon-number-preserving) unitary evolution was used.It is easy to see what sort of input states are insen-sitive to constant (differential) phase shifts, a in ( ω ) (cid:55)→ exp( iφ a ) a in ( ω ) and b in ( ω ) (cid:55)→ exp( iφ b ) b in ( ω ), namely,states with definite photon numbers in both input modes a in and b in , as there is only a physically irrelevant over-all phase shift of the input (and output) state. But suchstates are, in general, sensitive to time delays (except thevacuum, of course). B. Hong-Ou-Mandel
The prime example of a state that is sensitive to timedelays—but not to constant phase shifts— in exactly thissame setup is the state with two spectrally identical pho-tons, one in each input mode, as this is the state thatfeatures in the Hong-Ou-Mandel effect measuring thatvery time delay. Indeed, any state of the form | Ψ (cid:105) in = (cid:90) dω (cid:90) dω (cid:48) φ ( ω ) φ ( ω (cid:48) ) a † in ( ω ) b † in ( ω (cid:48) ) | vac (cid:105) , (6)(with normalization (cid:82) dω | φ ( ω ) | = 1) leads to an outputstate with the property that at zero time delay one neverfinds one photon in each output mode, as the last line inthe following equation shows: | Ψ (cid:105) out = i (cid:90) dω (cid:90) dω (cid:48) φ ( ω ) φ ( ω (cid:48) ) e iωτ + ( ω )+ iω (cid:48) τ − ( ω (cid:48) ) × (cid:104) c † out ( ω ) c † out ( ω (cid:48) ) + d † out ( ω ) d † out ( ω (cid:48) )+ (cid:16) − e i ( ω (cid:48) − ω )∆ τ S (cid:17) c † out ( ω ) d † out ( ω (cid:48) ) (cid:105) | vac (cid:105) . (7)Thus, the ( ω -independent) time delay ∆ τ S can be mea-sured directly by measuring coincidence counts betweenthe two output modes (where, to mention the same pointonce more, an ω -independent phase shift would give nosuch effect).If we define K = (cid:90) dω | φ ( ω ) | e − iω ∆ τ S , (8)then the probability to detect one photon in each outputmode is P = 12 (cid:0) − | K | (cid:1) . (9) C. Inverse Hong-Ou-Mandel
For completeness let us analyze what happens when weuse the output of a standard (i.e., in an inertial frame ofreference) Hong-Ou-Mandel experiment as input to theSagnac interferometer. That is, assume our input statehas the form | Φ (cid:105) in = (cid:90) dω (cid:90) dω (cid:48) φ ( ω ) φ ( ω (cid:48) ) × (cid:104) a † in ( ω ) a † in ( ω (cid:48) ) + b † in ( ω ) b † in ( ω (cid:48) ) (cid:105) | vac (cid:105) . (10)Then the output state will be | Φ (cid:105) out = (cid:90) dω (cid:90) dω (cid:48) φ ( ω ) φ ( ω (cid:48) ) e iωτ + ( ω )+ iω (cid:48) τ + ( ω (cid:48) ) × (cid:104) i (cid:110) e − i ( ω + ω (cid:48) )∆ τ S (cid:111) c † out ( ω ) d † out ( ω (cid:48) )+ 12 (cid:110) − e − i ( ω + ω (cid:48) )∆ τ S (cid:111) × (cid:16) c † out ( ω ) c † out ( ω (cid:48) ) − d † out ( ω ) d † out ( ω (cid:48) ) (cid:17) (cid:105) | vac (cid:105) . (11)The last term vanishes when ∆ τ S = 0. The time delaycan be measured now by counting the cases where bothphotons appear in one and the same output mode. Thephase factor in the last line now depends on the sum of thetwo (dummy) frequencies ω and ω (cid:48) , where the difference appeared in the last line of (7): this is why the Hong-Ou-Mandel effect is not sensitive to constant phase shifts, butthe inverse Hong-Ou-Mandel effect is twice as sensitive tophase shifts as classical light waves are [recall that in theclassical case there appears just a phase factor e iω ∆ τ S ,see Eq. (3)].More precisely, in terms of the quantity K , defined in(8), the probability to find two photons in the same mode(either c or d ) is P = 12 (cid:0) − Re( K ) (cid:1) . (12)We always have P ≥ P with equality only if K is real. III. DISCUSSION
When contemplating experimental implementation ofthe Hong-Ou-Mandel experiment on a rotating platform,there are a number of issues that spring to mind. For-tunately, some of those problems have been dealt withsuccessfully in the only single -photon implementation ofthe Sagnac effect so far [9]. The main features of that ex-periment was that a fiber setup was used, with the single-photon source (at 1550 nm) placed outside the rotating platform (any Doppler shifts can be neglected to first-order approximation). By winding the fiber loop manytimes around the platform, the Sagnac effect, being pro-portional to the area enclosed, is enhanced by the wind-ing number, so that a small value of Ω is tolerable. Infact, a very clean fringe was observed with a visibility ofmore than 99%, and the maximum time delay was about100ps, a time delay easily measurable with the Hong-Ou-Mandel effect. (Remembering the present context, notethis single-photon experiment was sensitive to both con-stant phase shifts and time delays. The input state usedwas of the form | ψ (cid:48) (cid:105) in = (cid:90) dωφ (cid:48) ( ω )[ a † in ( ω ) + b † in ( ω )] | vac (cid:105) , (13)idealizing it as pure input state.)The Hong-Ou-Mandel effect is more complicated, asit needs two (spectrally) identical photons (More pre-cisely, in the rotating frame of reference they need tobe identical spectrally. In a frame of reference relativeto which the beamsplitter is moving, perfect interferencetakes place for spectrally different wave packets, thanksto the Doppler effect (see [10])). In practice, of course, thetwo photons will be different. That will mean there willalways be coincidence counts in the output. Similarly,a beamsplitter is never exactly 50/50, and that leads tocoincidence counts as well, even with identical photonsas input. But these effects are all independent of the ro-tation rate Ω. Performing a control experiment at zerorotation rate provides, therefore, a standard solution, byallowing one to subtract off the effects of imperfections.Finally, creating a state of the form | Φ (cid:105) in is in principlemerely just as complicated as creating a state of the form | Ψ (cid:105) in , as the former is created from the latter, by imping-ing it on a 50/50 beamsplitter. In the present context,it has to be added that that 50/50 beamsplitter shouldbe at rest in an inertial frame. This will make sure allinevitable imperfections in the input state will again beindependent of the rotation rate Ω. [1] E. J. Post, Sagnac effect , Rev. Mod. Phys. , 475494(1967)[2] G.B. Malykin, The Sagnac effect: correct and incorrectexplanations , Phys. Usp. , 1229 (2000).[3] R. Anderson, H.R. Bilger, and G.E. Stedman, Sagnaceffect: A century of Earth rotated interferometers ,” Am.J. Phys., , pp. 975-985 (1994),[4] J. Anandan, Sagnac effect in relativistic and nonrelativis-tic physics , Phys. Rev. D , 338-346 (1981).[5] Guido Rizzi and Matteo Luca Ruggiero, A Direct Kine-matical Derivation of the Relativistic Sagnac Effect forLight or Matter Beams , General Relativity and Gravita-tion , 2129-2136 (2003).[6] Dennis Dieks and Gerard Nienhuis, Relativistic aspectsof nonrelativistic quantum mechanics , Am. J. Phys. , 650655 (1990).[7] C. K. Hong, Z. Y. Ou, and L. Mandel, Measurement ofsubpicosecond time intervals between two photons by in-terference , Phys. Rev. Lett. , 2044-2046 (1987).[8] Aziz Kolkiran and G. S. Agarwal, Heisenberg limitedSagnac interferometry , Optics Express, , 6798-6808(2007).[9] G. Bertocchi, O. Alibart, D.B. Ostrowsky, S. Tanzilli andP. Baldi, Single-photon Sagnac interferometer , J. Phys.B: At. Mol. Opt. Phys. , 1011-1016 (2006).[10] M.G. Raymer, S.J. van Enk, C.J. McKinstrie, and H.J.McGuinness, Interference of two photons of differentcolor , Optics Communications,283