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a long standing goal in optics is the development and improvement of precision optical metrology . in the first paper on weak values in 1988 @xcite , aharanov ,
albert and vaidman suggested that the weak value effect might be used as an amplifier in order to measure ( in the case they were considering ) the value of a small magnetic field by looking at the anomalously large deflection of a beam of atoms traversing a stern - gerlach apparatus .
the general validity of this weak value effect was later shown experimentally in an optical context by ritchie _
@xcite , who replaced the magnetic spin with transverse polarization , and brunner _ et al . _
@xcite , who illustrated the pervasiveness of the weak value effect in common optical telecom networks .
more recently , the amplification properties of this weak value effect have been exploited in similar optical systems to precisely measure beam deflection @xcite , phase shifts @xcite , frequency shifts @xcite , time delays @xcite , and even temperature shifts @xcite , by using either polarization or which - path degrees of freedom .
although these experiments can be described using classical wave optics @xcite , the analysis using quantum techniques provides additional insight and allows for future extension to cases with no classical counterpart .
hence , we shall continue to use a quantum approach in this work as well .
our theoretical analysis begins with the rochester setup @xcite , where the tilt of a moving mirror within an interferometer is detected from the signal on a split - detector .
while this setup has a sub - picoradian resolution with only milliwatts of laser power , there are a number of ways this can be improved to yield even greater sensitivity .
a generic shortcoming of weak - value - related metrological techniques is the fact that only a small fraction of the events are `` post - selected '' , while the vast majority of events are intentionally thrown away .
the main goal of the current work is to investigate how this situation can be further improved if those events are recycled .
this will be done by taking photons which are not post - selected and reinjecting them back into the interferometer , so that eventually , every photon can be post - selected in principle .
we will see that this strategy does indeed lead to an improvement in the signal - to - noise ratio of the desired parameter , effectively given by the power increase of the split - detection signal .
moreover , since the existing single - pass weak - value amplification already achieves the sensitivity of standard measurement techniques ( such as homodyne detection ) but with lower technical noise @xcite , the improvements from recycling should _ exceed _ the sensitivity of the standard techniques .
we note that because we employ not just a single pass , but many passes of a given photon through the interferometer , the simple weak value formula used in the first paper on the subject @xcite will no longer suffice , and we must develop a theoretical formalism for multiple passes that will account for the amplification of the deflection , as well as the probability of reaching the detector after some number of traversals .
while the recycling scheme is an important advance in its own right ( and can be generically applied to all weak value amplification schemes ) , it also lends itself to further enhancement if combined with other precision metrology techniques currently in use .
for example , the inclusion of a spatial filter or parity - flipping element to zeno - stabilize the beam , or the use of a squeezed reference beam @xcite could significantly reduce degradation effects and quantum noise , respectively .
the recycling technique , therefore , sets the stage for combined weak value / quantum light amplification strategies for future research .
furthermore , though our present work focuses on a novel pulsed recycling method , possible extensions to continuous wave operation may allow for the use of power recycling @xcite and signal recycling @xcite techniques , both of which are in use in modern gravitational wave detectors @xcite .
this paper is organized as follows . in section
[ sec : qual ] we give a heuristic estimation of the expected gains from a recycling setup based on qualitative power considerations and accessible laboratory conditions . in section  [ sec : collimated ] we analytically compute the exact recycling solution for a particular setup under the assumption that the beam can stay perfectly collimated . in section  [ sec : diverging ] we relax this assumption with a numerical treatment including diffraction effects .
we summarize our conclusions in section  [ sec : conclusion ] .
our baseline for comparison will be the rochester continuous wave ( cw ) sagnac interferometric scheme described in @xcite .
we wish to improve the detected signal - to - noise ratio ( snr ) by using a combination of pulsed laser operation with the same average power output and a design that sends the undetected portion of each pulse back into the interferometer .
such a setup is illustrated in figure  [ fig : setup ] for reference .
however , before committing to a particular recycling design we can make fairly general estimations about the increases in sensitivity that we expect from any similar recycling scheme .     between the clockwise ( @xmath0 ) and counter - clockwise ( @xmath1 ) traveling paths .
the piezo - driven mirror imparts a transverse momentum kick @xmath2 that differs by a sign for the @xmath0 and @xmath1 paths .
a split detector is placed at the dark port to measure a resulting pulse deflection .
the h - polarized part of the pulse that exits the bright port is rotated again by the active pc back to v before being confined by the pbs and mirror to return the pulse to the interferometer through the pc , now _
inactive_. ]      the constraints on how much we can increase the power collected by the dark port detector in order to improve the measurement sensitivity depend crucially on the relative time scales involved , which include :    1 .   the pulse duration @xmath3 being emitted by the laser . for typical lasers
this can vary between @xmath4 ns and @xmath5  fs reasonably , which correspond to pulse lengths of @xmath6 m and @xmath7 m , respectively .
2 .   the repetition period @xmath8 of the laser . for typical lasers
the repetition rate @xmath9 can vary from @xmath4  hz to several ghz reasonably .
3 .   the traversal period @xmath10 of the interferometer setup .
this is determined by the physical size of the setup . as an upper - bound estimate , a @xmath11
m long recycling setup will have a total period of @xmath12 ns .
the gating time @xmath13 for adding new pulses to the interferometer .
this will determine the minimum inter - pulse spacing @xmath14 inside the interferometer .
this also must be strictly less than the time between pulses emitted by the laser @xmath15 so that every new laser pulse can be injected . for a pockels cell , @xmath16 ns .
these are summarized in table  [ tab : timescales ] .
.relevant time scales for a recycling experiment . [
cols="<,<,>",options="header " , ]
and @xmath17 . in order of
( blue , solid ) , ( red , dashed ) , ( purple , dot - dashed ) , and ( brown , dotted ) we show weak initial diverging lenses with extreme focal lengths @xmath18 and no lens .
the ( black , dashed ) curve shows the scaling given by power considerations in eq .  .
here @xmath19 refers to the snr of a single unrecycled pulse with an optimally chosen focal length of @xmath20 , while @xmath21 is the accumulated snr over @xmath22 traversals for the indicated lens choices .
, width=288 ]    by assuming a collimated beam , we have so far neglected beam propagation effects in the analysis , as well as any lens effects that could further change the detection physics . in order to incorporate these effects ,
we now alter our measurement operators and pursue a numerical approach .
we find that these effects may slightly enhance the snr gains from recycling before saturation due to the finite detector size , but do not fundamentally alter the basic power - scaling behavior .
for that reason , it will be sufficient to illustrate only the weak - value regime as an example .    extending the collimated analysis in section  [ sec : collimated ] to include beam propagation leads to a replacement of the measurement operators with @xmath23 , where @xmath24 , @xmath25 is the carrier momentum of the pulse , and @xmath26 is the propagation length from piezo to 50:50 beam splitter @xcite
. the number density will then involve the composite measurement operator @xmath27 adding a diverging lens with focal length @xmath28 also modifies the initial state with an operator @xmath29 .
hence , powers of the following modified operators will appear in the full solution ,    [ eq : recursiveparts ] @xmath30    which can be simplified recursively , as detailed in appendix  [ sec : recursiveappendix ] .
the effect of a diverging lens is considered for comparison with the unrecycled experiment in ref .
@xcite , where such a lens was able to enhance sensitivity .
table  [ tab : numericalparameters ] shows the parameters which describe the laser and experimental geometry .
our choice of @xmath25 corresponds to the 780 - 800 nm lasers used in @xcite , and the 3-meter interferometer length @xmath26 is taken from the generous upper bound estimate discussed in section  [ sec : qual ] as a worst case scenario for beam divergence effects . for the weak - value regime of small @xmath31 and @xmath32
such that @xmath33 , we found that it was more computationally efficient to expand the sine and cosine functions in to second order in @xmath2 and fourth order in @xmath32 , as shown in appendix  [ sec : truncation ] . to test the validity of this truncation
, we initially set the interferometer length @xmath26 to zero so that a comparison could be made with the previously calculated collimated solutions .
we restrict our attention to the snr gains achieved by recycling a single pulse for @xmath22 traversals , since adding more pulses leads to a simple scaling of the single pulse result .
the snr gains for different choices of initial diverging lens are shown in fig .
[ fig : snrplateaus ] , where they are compared to the ideal power - scaling curve that we expect from our qualitative considerations given by . in all cases ,
the expected gains roughly follow the qualitative power scaling rule for a large number of traversals before saturating due to the beam growing larger than the finite size of the split - detector .
note that the beam divergence mitigates the snr decay that was observed for the collimated case , even without beam stabilization due to flipping or zeno reshaping .
by investigating the optical design shown in fig .  [
fig : setup ] , we have shown how a single optical pulse can be trapped inside the interferometer until the photons all exit the dark port and are `` post - selected , '' greatly boosting the sensitivity of the precision measurement .
the added power accumulated at the detector within a fixed duration of time is the dominant source of sensitivity gain .
further increases are achievable by trapping multiple pulses in the interferometer simultaneously .
the number of trapped pulses is limited by the length of the pulses , the gating frequency of current pockels cells , and the physical size of the interferometer .
we carefully analyzed the case of a collimated beam and showed that repeated post - selections cause a walk - off effect in the recycled pulse , which tends to diminish the snr .
however , we also showed that these walk - off effects can be easily corrected by zeno reshaping , or by a parity flip , which reflects the beam around its optic axis on each traversal .
somewhat surprisingly , the gains with parity correction can even exceed those expected from the power scaling . including propagation effects
does not destroy the sensitivity gain shown for the collimated case , but instead can produce additional enhancement .    while these sensitivity gains alone are a substantial improvement over the original idea , the combination of these techniques with other established metrology techniques  such as the use of a squeezed reference beam
 could further increase the sensitivity beyond that indicated here .
we acknowledge support from the us army research office grants no .
62270phii : stir , and no . w911nf-09 - 0 - 01417 , as well as the national science foundation grant no . dmr-0844899 .
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using equations and from above , an exact recursive simplification can be constructed .
written explicitly in the momentum basis we have @xmath34 where the functions @xmath35 are given by the recursion relation @xmath36 the overall measurement operator is then given simply by multiplying the unitary operators @xmath37 and @xmath38 from the right .
note that each pulse retains the phase and momentum information accumulated from previous traversals .
given an input gaussian profile as in , we find @xmath39 with @xmath40    the solution with no initial diverging lens is given by taking limit @xmath41 , which modifies equation   with the replacement @xmath42 .
similarly , setting @xmath43 recovers the collimated solution as expected .
truncating the measurement operators that include propagation effects to second order in @xmath2 and fourth order in @xmath32 for the weak value regime produces the expressions ,    [ eq : approximateexpansions ] @xmath44{\langle p \rvert } , \nonumber \\   { \hat{\bm u}}_\ell^2 { \hat{\bm m}}_+ & = \int \ ! \mathrm{d}p{\lvert p\rangle}e^{-ip^2\ell / k_0}\ , \times \\   & \qquad \left[1 - \frac{\phi^2}{8 } + \frac{\phi^4}{384 } + \frac{ik\phi}{2}\partial_p - \frac{ik\phi^3}{48}\partial_p
\right.\nonumber \\   & \qquad\quad\left .
+ \frac{k^2}{2}\partial_p^2 - \frac{k^2\phi^2}{16}\partial_p^2\right]{\langle p \rvert } .
\nonumber\end{aligned}\ ] ]
we consider the use of cyclic weak measurements to improve the sensitivity of weak - value amplification precision measurement schemes . 
 previous weak - value experiments have used only a small fraction of events , while discarding the rest through the process of `` post - selection '' . 
 we extend this idea by considering recycling of events which are typically unused in a weak measurement . here 
 we treat a sequence of polarized laser pulses effectively trapped inside an interferometer using a pockels cell and polarization optics . in principle 
 , all photons can be post - selected , which will improve the measurement sensitivity . 
 we first provide a qualitative argument for the expected improvements from recycling photons , followed by the exact result for the recycling of collimated beam pulses , and numerical calculations for diverging beams . 
 we show that beam degradation effects can be mitigated via profile flipping or zeno reshaping . 
 the main advantage of such a recycling scheme is an effective power increase , while maintaining an amplified deflection .