as mentioned before , the increase in phase measurement sensitivity in the nci is not due to noise reduction but due to signal enhancement . in order to demonstrate this ,
we experimentally compare it with a ci , which works at the snl . as the snl is related to the phase - sensing photon number .
the schematic diagrams of an su(1,1 ) nci and a conventional mz interferometer ( ci ) are shown in fig .
beam splitters are replaced by parametric amplifiers ( pa1 , pa2 ) , which amplify the incoming 
idler input field is labelled as   and is in vacuum . in the present realization of nci ,
the parametric amplifier is based on non - degenerate four - wave mixing in hot rb-85 atomic vapour cells29 ( see methods for the details of the experimental arrangement ) .
output relation for a parametric amplifier is given by : where g is the amplitude gain and g = g1 .
with an overall phase shift  in the idler arm , we obtain the output fields of the interferometer : with ( )=ge+g , ( )=gg(1+e ) as the phase - sensitive gains . for a coherent state | injected in the signal input and vacuum in the idler input
, we obtain the output intensity in the idler port or one of the outputs of the nci as where  g(+1 ) is the intensity of the phase - sensing field ( idler field in fig .
1a ) with || as the input intensity . for > > 1 , g . a conventional mz interferometer shown in fig .
1b , on the other hand , has a fringe given by with 50:50 beam splitters and a coherent state | input .
since the sensitivity of an interferometer is related to the photon number inside the interferometer , for a fair comparison between nci and ci , we set = , which can be achieved by adjusting , independently .
therefore , we see from equations ( 3 ) and ( 4 ) that , under the condition of the same phase - sensing intensity , the output of the nci is 2 g times that of the ci .
however , as is well known , for any amplifier , the amplification of signal is also accompanied by the amplification of the noise .
so , the signal - to - noise ratio at best does not change in the process31 .
fortunately , it is not the case for the nci discussed here . from equation ( 2 ) , at dark fringe when =(2n+1) , we have . however , since   is in vacuum , the output of the nci is at vacuum noise level , just like the ci at dark fringe .
thus , the signal in the nci is increased while the noise level is not and we should expect an enhancement of 2 g in signal - to - noise ratio of the nci over the ci .
what happens here for the nci is that although each amplifier increases signal and noise together , the first parametric amplifier ( pa1 ) produces two fields ( two arms of the nci ) that are quantum mechanically entangled323334 so that their noises are also correlated , and a destructive interference occurs at the second parametric amplifier ( pa2 ) when the interferometer is operated at dark fringe so that most of the amplified noise from pa1 are cancelled and only the vacuum noise is left35 .
therefore , the nci increases the signal while it keeps the noise unchanged , leading to an enhancement in signal - to - noise ratio for phase measurement as compared with the ci .
figure 1c shows the interference fringes observed for both nci and ci with the same phase - sensing intensity = .
the fringe size of the nci is bigger than that of the ci because of the amplification at pa2 , which leads to an enhancement in the phase signal .
when operated at dark fringe with a small phase change : =(2n+1)+ ( <<1 , n = integer ) , we obtain from equations ( 3 ) and ( 4 ) the change in the output intensities of the two interferometers due to  as : so , the signal is enhanced by bsinci/ici=2 g for the same  and = . the fringe enhancement in fig
the noise performance of the nci is measured by homodyne detection ( hd ) method at the idler output port , as shown in fig .
2c shows the noise level at the output of the nci when the phase is scanned .
the minimum noise occurs at dark fringe when there is a destructive quantum interference at pa2 .
this is in contrast to the trace in yellow when one arm of the nci is blocked and there is no interference effect .
the trace in yellow corresponds to two uncorrelated amplifiers in series , as depicted in fig .
respectively show the noise levels for individual amplifiers when the other one is not active .
notice that at some phases , the output noise level ( the trace in dark blue ) of the nci is even lower than the amplified noise levels of individual amplifiers ( the traces in green and dark brown ) as a result of noise cancelation .
2c respectively show the measured noise levels at dark fringe setting for the nci and the ci under the same condition when the fringes in fig .
the observed noise level for the nci is 3.30.3  db higher than that of the ci .
this is not what is predicted in previous paragraphs , that is , the noise level of the nci is the same as that of the ci at the vacuum noise level .
the higher measured output noise of the nci is due to internal losses of the interferometer ( more on this later ) .
30 that the signal due to a small phase shift  in hd of an interferometer has the same form as that in direct intensity detection ) , we still obtain an enhancement of 4.10.3  db in signal - to - noise ratio for the nci over the ci , that is , snrnci / snrci=4.1  db=2.5 . from equation
( 5 ) , we know that the signal is proportional to  , whereas the noise measured in the experiment is the noise power .
so , we have the phase measurement sensitivity as   in rms value . as the ci has a phase measurement sensitivity at snl3 , our nci beats the shot noise level by a factor of = 1.6 in rms value .
the simple theory in the previous part does not include losses in the system and predicts the continuous increase of the snr with the gain of the amplifiers .
losses , however , will introduce extra vacuum noise , which is also amplified by pa1 and pa2 but is not correlated and can not be cancelled , thus leading to higher noise level for the nci than the ci but at a manageable level , as observed in fig .
2c . the extra amplified vacuum noise will increase with the gain of the amplifiers in the same way as the phase signal .
3 where we plot in blue diamond shape the ratio of the output noise level of the interferometer at dark fringe ( the light blue trace in fig .
2c ) as a function of the average noise gain of individual amplifiers ( the traces in green and dark brown in fig .
since the vacuum noise level is also the output noise level of the ci at dark fringe , this is also the noise level ratio of the nci to the ci . shown also in fig .
3 in black square shape are the ratio of the signal level of the nci to that of the ci , or the signal enhancement factor bs extracted from the fringes similar to fig . 1c .
therefore , the ratio of the black square to the blue diamond gives the enhancement of the signal - to - noise ratio of the nci over the ci , which is shown as red triangle in fig .
3 . as seen in the figure , both the signal enhancement factor ( black ) and the noise level ( blue ) at the output increase with the amplifier gain in about the same rate at high gain .
so , the signal - to - noise improvement factor ( red ) levels out at a gain of ~4  db , in qualitative agreement with ref . 30
next , we characterize the effect of losses on the performance of the nci by introducing some extra losses .
first , we place an attenuator with loss l in front of the homodyne detector and vary the attenuation .
the signal enhancement factor bs and noise n above vacuum noise level after the attenuator will change with the loss l by where bs0 and n0 are the respective values before the attenuator .
so , the signal - to - noise ratio after the attenuator becomes the solid lines in fig .
4a are fitted to equations ( 6 ) and ( 7 ) with bs0=7.5  db and n0=3.2  db .
so , the signal - to - noise improvement factor does not change much at small loss for large n0 .
this is well known for amplification process : the amplified noise is above the vacuum level so that the noise will decrease under loss just like the signal , making snr less sensitive to external losses .
this is quite different from the squeezed state scheme56 in which the improvement will suffer under loss . for large loss , however , snr will drop with the increase of the loss because the noise level is mostly vacuum noise already and wo n't decrease further . for the effect of internal loss , we place a variable attenuator in the paths of both arms of the nci .
we need to readjust the gain of pa2 to re - balance the interferometer so that it produces fringe visibility close to 100% . in the meantime , gain balancing also leads to optimized output noise levels shown fig .
4b , the output noise level ( blue diamond ) increases as the loss increases  more uncorrelated vacuum noise enters the interferometer and gets amplified .
therefore , the improvement factor in signal - to - noise ratio decreases more quickly with internal loss ( fig .
the effect of internal loss on the nci is complicated so we can not fit the data to a theoretical prediction .
as mentioned before , the increase in phase measurement sensitivity in the nci is not due to noise reduction but due to signal enhancement . in order to demonstrate this ,
we experimentally compare it with a ci , which works at the snl . as the snl is related to the phase - sensing photon number .
the schematic diagrams of an su(1,1 ) nci and a conventional mz interferometer ( ci ) are shown in fig .
beam splitters are replaced by parametric amplifiers ( pa1 , pa2 ) , which amplify the incoming 
idler input field is labelled as   and is in vacuum . in the present realization of nci ,
the parametric amplifier is based on non - degenerate four - wave mixing in hot rb-85 atomic vapour cells29 ( see methods for the details of the experimental arrangement ) .
output relation for a parametric amplifier is given by : where g is the amplitude gain and g = g1 .
with an overall phase shift  in the idler arm , we obtain the output fields of the interferometer : with ( )=ge+g , ( )=gg(1+e ) as the phase - sensitive gains . for a coherent state | injected in the signal input and vacuum in the idler input
, we obtain the output intensity in the idler port or one of the outputs of the nci as where  g(+1 ) is the intensity of the phase - sensing field ( idler field in fig .
1a ) with || as the input intensity . for > > 1 , g . a conventional mz interferometer shown in fig .
1b , on the other hand , has a fringe given by with 50:50 beam splitters and a coherent state | input .
since the sensitivity of an interferometer is related to the photon number inside the interferometer , for a fair comparison between nci and ci , we set = , which can be achieved by adjusting , independently .
therefore , we see from equations ( 3 ) and ( 4 ) that , under the condition of the same phase - sensing intensity , the output of the nci is 2 g times that of the ci .
however , as is well known , for any amplifier , the amplification of signal is also accompanied by the amplification of the noise .
so , the signal - to - noise ratio at best does not change in the process31 .
fortunately , it is not the case for the nci discussed here . from equation ( 2 ) , at dark fringe when =(2n+1) , we have . however , since   is in vacuum , the output of the nci is at vacuum noise level , just like the ci at dark fringe .
thus , the signal in the nci is increased while the noise level is not and we should expect an enhancement of 2 g in signal - to - noise ratio of the nci over the ci .
what happens here for the nci is that although each amplifier increases signal and noise together , the first parametric amplifier ( pa1 ) produces two fields ( two arms of the nci ) that are quantum mechanically entangled323334 so that their noises are also correlated , and a destructive interference occurs at the second parametric amplifier ( pa2 ) when the interferometer is operated at dark fringe so that most of the amplified noise from pa1 are cancelled and only the vacuum noise is left35 .
therefore , the nci increases the signal while it keeps the noise unchanged , leading to an enhancement in signal - to - noise ratio for phase measurement as compared with the ci .
figure 1c shows the interference fringes observed for both nci and ci with the same phase - sensing intensity = .
the fringe size of the nci is bigger than that of the ci because of the amplification at pa2 , which leads to an enhancement in the phase signal .
this is the key difference between an nci and a ci . when operated at dark fringe with a small phase change : =(2n+1)+ ( <<1 , n = integer ) , we obtain from equations ( 3 ) and ( 4 ) the change in the output intensities of the two interferometers due to  as : so , the signal is enhanced by bsinci/ici=2 g for the same  and = . the fringe enhancement in fig .
the noise performance of the nci is measured by homodyne detection ( hd ) method at the idler output port , as shown in fig .
2c shows the noise level at the output of the nci when the phase is scanned .
the minimum noise occurs at dark fringe when there is a destructive quantum interference at pa2 .
this is in contrast to the trace in yellow when one arm of the nci is blocked and there is no interference effect .
the trace in yellow corresponds to two uncorrelated amplifiers in series , as depicted in fig .
respectively show the noise levels for individual amplifiers when the other one is not active .
notice that at some phases , the output noise level ( the trace in dark blue ) of the nci is even lower than the amplified noise levels of individual amplifiers ( the traces in green and dark brown ) as a result of noise cancelation .
2c respectively show the measured noise levels at dark fringe setting for the nci and the ci under the same condition when the fringes in fig .
the observed noise level for the nci is 3.30.3  db higher than that of the ci .
this is not what is predicted in previous paragraphs , that is , the noise level of the nci is the same as that of the ci at the vacuum noise level .
the higher measured output noise of the nci is due to internal losses of the interferometer ( more on this later ) .
30 that the signal due to a small phase shift  in hd of an interferometer has the same form as that in direct intensity detection ) , we still obtain an enhancement of 4.10.3  db in signal - to - noise ratio for the nci over the ci , that is , snrnci / snrci=4.1  db=2.5 . from equation
( 5 ) , we know that the signal is proportional to  , whereas the noise measured in the experiment is the noise power .
so , we have the phase measurement sensitivity as   in rms value . as the ci has a phase measurement sensitivity at snl3 , our nci beats the shot noise level by a factor of = 1.6 in rms value .
the simple theory in the previous part does not include losses in the system and predicts the continuous increase of the snr with the gain of the amplifiers .
losses , however , will introduce extra vacuum noise , which is also amplified by pa1 and pa2 but is not correlated and can not be cancelled , thus leading to higher noise level for the nci than the ci but at a manageable level , as observed in fig .
2c . the extra amplified vacuum noise will increase with the gain of the amplifiers in the same way as the phase signal .
3 where we plot in blue diamond shape the ratio of the output noise level of the interferometer at dark fringe ( the light blue trace in fig .
2c ) as a function of the average noise gain of individual amplifiers ( the traces in green and dark brown in fig .
since the vacuum noise level is also the output noise level of the ci at dark fringe , this is also the noise level ratio of the nci to the ci . shown also in fig .
3 in black square shape are the ratio of the signal level of the nci to that of the ci , or the signal enhancement factor bs extracted from the fringes similar to fig . 1c .
therefore , the ratio of the black square to the blue diamond gives the enhancement of the signal - to - noise ratio of the nci over the ci , which is shown as red triangle in fig .
3 . as seen in the figure , both the signal enhancement factor ( black ) and the noise level ( blue ) at the output increase with the amplifier gain in about the same rate at high gain .
so , the signal - to - noise improvement factor ( red ) levels out at a gain of ~4  db , in qualitative agreement with ref . 30 .
next , we characterize the effect of losses on the performance of the nci by introducing some extra losses .
first , we place an attenuator with loss l in front of the homodyne detector and vary the attenuation .
the signal enhancement factor bs and noise n above vacuum noise level after the attenuator will change with the loss l by where bs0 and n0 are the respective values before the attenuator .
so , the signal - to - noise ratio after the attenuator becomes the solid lines in fig .
4a are fitted to equations ( 6 ) and ( 7 ) with bs0=7.5  db and n0=3.2  db .
so , the signal - to - noise improvement factor does not change much at small loss for large n0 .
this is well known for amplification process : the amplified noise is above the vacuum level so that the noise will decrease under loss just like the signal , making snr less sensitive to external losses .
this is quite different from the squeezed state scheme56 in which the improvement will suffer under loss . for large loss , however , snr will drop with the increase of the loss because the noise level is mostly vacuum noise already and wo n't decrease further . for the effect of internal loss , we place a variable attenuator in the paths of both arms of the nci . we need to readjust the gain of pa2 to re - balance the interferometer so that it produces fringe visibility close to 100% . in the meantime , gain balancing also leads to optimized output noise levels shown fig .
4b , the output noise level ( blue diamond ) increases as the loss increases  more uncorrelated vacuum noise enters the interferometer and gets amplified .
therefore , the improvement factor in signal - to - noise ratio decreases more quickly with internal loss ( fig .
the effect of internal loss on the nci is complicated so we can not fit the data to a theoretical prediction
we have demonstrated an su(1,1 ) nci with 4  db improvement in signal - to - noise ratio in phase measurement over a ci operated under the same condition .
the improvement is mostly limited by the loss inside the interferometer and is less sensitive to external loss than the squeezed state method .
the improvement in signal - to - noise ratio comes from the increase in signal rather than the reduction of noise .
furthermore , if we inject squeezed states into the unused idler input port of pa1 ( see fig .
1a ) , reduction of the noise of the interferometer is possible , leading to further improvement of signal - to - noise ratio . therefore , the current scheme and the squeezed state scheme are complementary to each other . notice that our nci involves amplification of the input field whereas a conventional mz interferometer ( ci ) does not .
however , according to refs 21 , 25 it is the photon number of the phase probing field that counts for the limit of sensitivity .
therefore , when we compare the performance of our nci with the ci , we make sure that the light fields probing the phase change have the same intensity in both nci and ci .
when we reduce the intensity of the injected coherent state , this interferometer will approach to the originally proposed su(1,1 ) interferometer by yurke et al.19 , which can have a phase measurement sensitivity reaching the heisenberg limit .
3 , the internal loss of the interferometer is the main limiting factor in achieving higher sensitivity at higher gain .
30 and demkowicz - dobrzanski et al.36 then , it is natural to ask what kind of phase measurement sensitivity limit we can reach in our new interferometer . to answer that
unfortunately , our measurement technique , that is , the hd method , does not allow us to perform an analysis with fisher information method , which requires photon counting technique .
phase estimate with fisher information method for this new type of interferometer will be the topic in future researches .
what we did in current work is to compare the performance of our new interferometer with a conventional mz interferometer , which has the well - known snl ( 1/-dependence for phase measurement uncertainty ) , and show a 4-db improvement in signal - to - noise ratio .
the use of a nonlinear process to mix the signal and idler fields means that the two fields may have different frequencies . indeed , in our experiment , the frequency difference between the signal and the idler fields is ~6  ghz .
this opens up a new way to construct hybrid interferometers involving different waves and thus broadens the scope of precision measurement involving interferometry .
5 . the two parametric amplifiers ( pa1 , pa2 ) employed for the nci as shown in fig .
1a are based on non - degenerate four - wave mixing process in two hot rb-85 atomic vapour cells temperature - stabilized at 117  c ( ref .
29 ) ; see inset of fig . 5 for energy diagram of relevant components ) .
they are respectively pumped by a vertically polarized beam ( p1 , p3 ) at a maximum power of 400  mw with a waist of 500  m .
the pump beams are from a ti : sapphire laser frequency - stabilized to a stable reference cavity .
the pump beams are detuned about 0.8  ghz above the transition line of rb-85 f=2f at 795  nm .
the pump power can be adjusted with a combination of polarization beam splitters ( pbs ) and half - wave plates . a horizontally polarized seed beam ( also known as 
signal ) with waist of 200  m is injected with an angle of 0.4 relative to the pump .
signal beam is 3.04  ghz red shifted from the pump beam by an acousto - optic modulator in double - pass configuration .
the pump and signal beams are combined with a glan - laser pbs before being sent into the atomic cell . at the output of the atomic cell , the seed signal beam
is amplified with a gain that depends on the power and frequency detuning of the pump beam .
the amplification of the signal beam is accompanied by a conjugate beam ( called  idler ) with a frequency of 3.04  ghz blue shift from the pump at the other side of the pump beam .
the second cell is identical to the first one except that both the signal and the idler fields have inputs that are from the first cell .
a 4-f imaging system ( l1 , l2 ) is used to mode match the fields of the two cells .
a ci of mz type is formed in the path of the idler field with a pair of flipping pbs ( pbs1 , pbs2 ) and a beam - redirecting mirror . in this way
we block the signal arm and the pump to cell 2 ( p3 ) when we operate the ci .
a pzt - mounted mirror ( pzt ) is placed in both interferometers as the phase modulator so that we can compare the performance of both interferometers under the same phase change .
since the pbs splits the idler beam into two equal parts in the ci and reduces the phase - sensing intensity to half , we need to increase the injected signal at cell 1 to bring the phase - sensing intensity of the ci to the same level as that of the nci ( ips=60  w with iseed=16  w ) . a regular silicon photo - detector ( d ) is placed in the idler output port of the nci , which is also the output port for the ci .
this detector records the fringes of the nci and the ci in a digital scope , as shown in fig .
the noise performance of the nci is measured by hd ( with a 98% homodyne efficiency ) at the output of the idler side of the nci under different circumstances .
the local oscillator of 726  w for hd is obtained from cell 1 with another seed ( 196  w ) and pump ( p2=300  mw)34 .
the local oscillator goes through the same imaging system to match its mode with those in the interferometer .
5 . the two parametric amplifiers ( pa1 , pa2 ) employed for the nci as shown in fig .
1a are based on non - degenerate four - wave mixing process in two hot rb-85 atomic vapour cells temperature - stabilized at 117  c ( ref .
29 ) ; see inset of fig . 5 for energy diagram of relevant components ) .
they are respectively pumped by a vertically polarized beam ( p1 , p3 ) at a maximum power of 400  mw with a waist of 500  m .
the pump beams are from a ti : sapphire laser frequency - stabilized to a stable reference cavity .
the pump beams are detuned about 0.8  ghz above the transition line of rb-85 f=2f at 795  nm .
the pump power can be adjusted with a combination of polarization beam splitters ( pbs ) and half - wave plates . a horizontally polarized seed beam ( also known as 
signal ) with waist of 200  m is injected with an angle of 0.4 relative to the pump .
signal beam is 3.04  ghz red shifted from the pump beam by an acousto - optic modulator in double - pass configuration .
the pump and signal beams are combined with a glan - laser pbs before being sent into the atomic cell . at the output of the atomic cell , the seed signal beam
is amplified with a gain that depends on the power and frequency detuning of the pump beam .
the amplification of the signal beam is accompanied by a conjugate beam ( called  idler ) with a frequency of 3.04  ghz blue shift from the pump at the other side of the pump beam .
the second cell is identical to the first one except that both the signal and the idler fields have inputs that are from the first cell .
a 4-f imaging system ( l1 , l2 ) is used to mode match the fields of the two cells .
a ci of mz type is formed in the path of the idler field with a pair of flipping pbs ( pbs1 , pbs2 ) and a beam - redirecting mirror . in this way
we block the signal arm and the pump to cell 2 ( p3 ) when we operate the ci .
a pzt - mounted mirror ( pzt ) is placed in both interferometers as the phase modulator so that we can compare the performance of both interferometers under the same phase change .
since the pbs splits the idler beam into two equal parts in the ci and reduces the phase - sensing intensity to half , we need to increase the injected signal at cell 1 to bring the phase - sensing intensity of the ci to the same level as that of the nci ( ips=60  w with iseed=16  w ) . a regular silicon photo - detector ( d ) is placed in the idler output port of the nci , which is also the output port for the ci .
this detector records the fringes of the nci and the ci in a digital scope , as shown in fig .
the noise performance of the nci is measured by hd ( with a 98% homodyne efficiency ) at the output of the idler side of the nci under different circumstances .
the local oscillator of 726  w for hd is obtained from cell 1 with another seed ( 196  w ) and pump ( p2=300  mw)34 .
the local oscillator goes through the same imaging system to match its mode with those in the interferometer .
 conventional interferometers usually utilize beam splitters for wave splitting and recombination . 
 these interferometers are widely used for precision measurement . 
 their sensitivity for phase measurement is limited by the shot noise , which can be suppressed with squeezed states of light . here 
 we study a new type of interferometer in which the beam splitting and recombination elements are parametric amplifiers . 
 we observe an improvement of 4.10.3  db in signal - to - noise ratio compared with a conventional interferometer under the same operating condition , which is a 1.6-fold enhancement in rms phase measurement sensitivity beyond the shot noise limit . 
 the improvement is due to signal enhancement . 
 combined with the squeezed state technique for shot noise suppression , this interferometer promises further improvement in sensitivity . 
 furthermore , because nonlinear processes are involved in this interferometer , we can couple a variety of different waves and form new types of hybrid interferometers , opening a door for many applications in metrology . 
Results
Comparison between different types of interferometers
Experimental results
Discussions
Methods
Detailed experimental arrangement
Author contributions
Additional information