1404:. This is the principle behind noise reduction in an SU(1,1) interferometer. The first parametric amplifier produces two quantum entangled fields that are used for phase sensing and are the input to the second parametric amplifier where the amplification takes place. In a scenario of no internal losses inside the interferometer, the output noise for a destructive quantum interference is still similar to the case of a SU(2) interferometer. Overall, there is an amplification in the signal with no change in the noise (as compared to SU(2) interferometer). This leads to the improvement in phase sensitivity over SU(2) interferometry.
63:
230:
1417:
inferred that an SU(1,1) interferometer is robust against losses due to inefficient detectors because of the disentanglement of states at the second parametric amplifier before the measurements. However, the internal losses in an SU(1,1) interferometer limit its sensitivity below the
Heisenberg limit
54:
and precision measurements for achieving sensitivity in measurements beyond what is possible with classical methods and resources. Interferometry is a desired platform for precise estimation of physical quantities because of its ability to sense small phase changes. One of the most prominent examples
1442:
was challenging to realize experimentally due to very low photon numbers expected at the output (for ideal sensitivity) and also the theory did not take into account the internal losses that could affect the phase change sensitivity of the interferometer. Subsequently, modifications to the scheme
1429:
showed that in the presence of any internal losses, an SU(1,1) interferometer could not achieve the
Heisenberg limit for configurations with no input fields at both the ports or a coherent state input in one of the ports (this configuration was considered for the Theory section above). For a case
1509:
2. “Truncated” SU(1,1) interferometer with no second parametric amplifier and rather using a photocurrent mixer to realize the superposition of the fields. Such a configuration opens the possibility to implement SU(1,1) interferometry in experiments where fewer optical elements help minimize the
1505:
1. SU(1,1) interferometry with one parametric amplifier and a beam splitter replacing the second amplifier: The signal to noise ratio improvement in this configuration was found to be essentially the same as that of the original SU(1,1) interferometry sensitivity improvement over conventional
1392:
amplification at the second parametric amplifier. Similar amplification could also be implemented in a SU(2) interferometer where both the signal and the noise (vacuum quantum fluctuations) gets amplified. However, the difference comes in the noise performance of an SU(1,1) interferometer.
79:. In this type of interferometry, the input field is split into two by a beam splitter which then propagates along different paths and acquires a relative phase difference (corresponding to a path length difference). Considering one of the beams undergoing a phase change as the
1412:
The reduced sensitivity of an interferometer can be mainly due to two types of losses. One of the sources of losses is the inefficient detection protocol. Another type of loss affecting the sensitivity is the internal loss of an interferometer. Theoretical studies by Marino
87:, the relative phase is estimated after the two beams interfere at another beam splitter. The estimation of the phase difference is done through the detection of the intensity change at the output after the interference at the second beam splitter.
90:
These standard interferometric techniques, based on beam-splitters for the splitting of the beams and linear optical transformations, can be classified as SU(2) interferometers as these interferometric techniques can be naturally characterized by
1443:
were studied taking into account the losses and other experimental imperfections. Some of the initial experiments proving the predicted scaling of the SU(1,1) interferometer and other experiments modifying the scheme are discussed below.
894:
74:
of electromagnetic waves. Although the design and layout for these types of interferometers can vary depending upon the type of application and corresponding suitable scheme, they all can be mapped to an arrangement similar to that of a
1097:
The first feature indicates a high correlation of the output photon numbers (intensities) and the second feature shows that there is an enhancement of the signal strength for a small phase change as compared to SU(2) interferometers.
1353:
of sensitivity in conditions where a SU(2) interferometer approaches the shot-noise limit. It is also shown in Ref. that with no coherent state injection, the SU(1,1) interferometer approaches the
Heisenberg limit of sensitivity.
1501:
Modifications to SU(1,1) interferometers have been proposed and studied with the goal of finding an experimentally ideal scheme with the desired characteristics of SU(1,1) interferometry. Some of the schemes explored include:
45:
is an important technique in the field of optics that have been utilised for fundamental proof of principles experiments and in the development of new technologies. This technique, primarily based on the interference of
559:
151:
is the mean number of particles (photons for electromagnetic waves) entering the input port of the interferometer. The shot noise limit can be overcome by using light that utilizes quantum properties such as
1314:
935:
774:
1021:
733:
642:
1425:
did not take into account the internal losses and for a moderate gain of the parametric amplifier produced a low number of photons which made it difficult for its experimental realization. Marino
1273:
213:. The advantage that comes with the parametric amplifiers is that the input fields can be coherently split and interfere that would be fundamentally quantum in nature. This is attributed to the
475:
394:
1459:
with Rb-85 vapor cells for parametric amplification. The experiment verified the increase in the fringe size due to the amplification of the signal. Later, experiments performed by
Hudelist
1430:
with coherent state input at both the input ports, it was shown that the interferometer is robust against internal losses and is one of the ideal schemes for achieving the
Heisenberg limit.
1400:
for effective noise cancellation. For an input state with a strong correlation with the internal modes of an amplifier, the quantum noise can be cancelled at the output through destructive
233:
Schematic of a balanced SU(1,1) interferometer with a coherent state injection in one of the input ports and no input (vacuum) in the other port. Parametric amplifiers have gains G,g.
1952:
Gupta, Prasoon; Schmittberger, Bonnie L.; Anderson, Brian E.; Jones, Kevin M.; Lett, Paul D. (8 January 2018). "Optimized phase sensing in a truncated SU(1,1) interferometer".
594:
129:
1144:
312:
277:
1183:
95:(Special Unitary(2)) group. Theoretically, the sensitivity of conventional SU(2) interferometric schemes are limited by the vacuum fluctuation noise, also called the
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1390:
1347:
781:
194:
1072:
1050:
1092:
497:
149:
221:. Theoretically, SU(1,1) interferometers can achieve the Heisenberg limit of sensitivity with fewer optical elements than conventional interferometers.
237:
To briefly understand the benefit of using a parametric amplifier, a balanced SU(1,1) interferometer can be considered. Treating the input fields as
1758:
Kong, Jia; Hudelist, F.; Ou, Z. Y.; Zhang, Weiping (18 July 2013). "Cancellation of
Internal Quantum Noise of an Amplifier by Quantum Correlation".
1683:
Ou, Z. Y. (14 February 2012). "Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer".
502:
1801:
Plick, William N; Dowling, Jonathan P; Agarwal, Girish S (6 August 2010). "Coherent-light-boosted, sub-shot noise, quantum interferometry".
1506:
interferometers. This study showed that the improvement is mainly due to the entangled fields generated at the first parametric amplifier.
1052:
The two output intensities of an SU(1,1) interferometer are in phase whereas in a SU(2) interferometer the two outputs are out of phase.
242:
1723:
Marino, A. M.; Corzo Trejo, N. V.; Lett, P. D. (24 August 2012). "Effect of losses on the performance of an SU(1,1) interferometer".
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23:
400:
319:
62:
1024:
47:
1534:
Ou, Z. Y.; Li, Xiaoying (1 August 2020). "Quantum SU(1,1) interferometers: Basic principles and applications".
2019:
157:
26:
for splitting and mixing of electromagnetic waves for precise estimation of phase change and achieves the
165:
92:
27:
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1908:
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1820:
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1732:
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1608:
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1401:
1397:
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This shows two main distinguishing features of an SU(1,1) interferometer from SU(2) interferometers:
238:
153:
102:
1451:
The boost in the photon numbers from a coherent state injection was proposed and studied by Plick
1995:
1961:
1836:
1810:
1569:
1543:
1104:
282:
247:
1149:
889:{\displaystyle I_{2}^{o}=(2G^{2}g^{2}+1)(\left\vert \alpha \right\vert ^{2})(1+\nu \cos \phi )}
1987:
1934:
1783:
1642:
1624:
71:
51:
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1916:
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1700:
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1322:
218:
214:
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The improvement of phase measurement sensitivity in an SU(1,1) interferometer is not by a
1895:
Hudelist, F.; Kong, Jia; Liu, Cunjin; Jing, Jietai; Ou, Z.Y.; Zhang, Weiping (May 2014).
171:
1975:
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1897:"Quantum metrology with parametric amplifier-based photon correlation interferometers"
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The outputs are amplified as compared to that of a SU(2) interferometer when the gain
2013:
1592:
1573:
203:
70:
Conventional interferometers are based on the wave nature of the light and hence the
1999:
1185:
of a SU(2) interferometer is calculated to be (see Ref. for detailed calculation):
1840:
1779:
1854:
Jing, Jietai; Liu, Cunjin; Zhou, Zhifan; Ou, Z. Y.; Zhang, Weiping (4 July 2011).
1319:
This means that the sensitivity in phase measurements is improved by a factor of
1744:
1704:
161:
96:
1620:
209:
in which the beam splitters in conventional interferometers were replaced by
55:
of the application of this property is the detection of gravitational waves (
1645:; Ou, Z. Y. (31 March 2016). "Nonlinear interferometers in quantum optics".
1438:
The originally proposed configuration of an SU(1,1) interferometer by Yurke
1991:
1938:
1787:
554:{\displaystyle \left\vert G\right\vert ^{2}-\left\vert g\right\vert ^{2}=1}
314:, the output quantum fields from a parametric amplifier can be written as:
1628:
1983:
1666:
1493:
with SU(1,1) interferometry over the conventional SU(2) interferometry.
196:
with the change in the mean number of photons entering the input port.
1920:
199:
1880:
1856:"Realization of a nonlinear interferometer with parametric amplifiers"
1855:
1565:
1396:
A quantum amplifier (in this case a parametric amplifier) can utilize
1421:
The original scheme for an SU(1,1) interferometer proposed by Yurke
1146:
for an SU(1,1) interferometer compared to the signal to noise ratio
1966:
1548:
1815:
1463:
showed that there is an enhancement in the signal by a factor of
164:), at the unused input port. In principle, this can achieve the
56:
596:
at the first parametric amplifier with initial input intensity
1309:{\displaystyle \left\vert \alpha \right\vert ^{2}>>1}
930:{\displaystyle \left\vert \alpha \right\vert ^{2}>>1}
769:{\displaystyle \left\vert \alpha \right\vert ^{2}>>1}
1016:{\displaystyle \nu ={\frac {2G^{2}g^{2}}{2G^{2}g^{2}+1}}}
728:{\displaystyle I_{1}^{o}=2G^{2}g^{2}I_{0}(1+\cos \phi )}
637:{\displaystyle I_{0}=\left\vert \alpha \right\vert ^{2}}
66:
Schematic of a conventional Mach-Zehnder interferometer.
1455:. Such a scheme was experimentally implemented by Jing
1349:
in an SU(1,1) interferometer and hence can achieve the
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137:
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1599:(1 June 1986). "SU(2) and SU(1,1) interferometers".
1268:{\displaystyle R_{SU(1,1)}/R_{SU(2)}\approx 2G^{2}}
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1101:From these properties, the signal to noise ratio
470:{\displaystyle a_{2}^{o}=ga_{1}^{in}+Ga_{2}^{in}}
389:{\displaystyle a_{1}^{o}=Ga_{1}^{in}+ga_{2}^{in}}
30:of sensitivity with fewer optical elements than
1529:
1527:
1525:
1523:
1447:Coherent state boosted SU(1,1) interferometer
8:
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217:processes in parametric amplifiers such as
50:, has been widely explored in the field of
1510:error due to experimental imperfections.
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202:interferometers were first proposed by
32:conventional interferometric techniques
1718:
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1714:
1418:during its physical implementation.
7:
1678:
1676:
644:, the output intensities will be:
14:
1647:Advances in Optics and Photonics
1497:Modified SU(1,1) interferometers
589:{\displaystyle |\alpha \rangle }
168:of sensitivity which scales as
1780:10.1103/PhysRevLett.111.033608
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1:
1833:10.1088/1367-2630/12/8/083014
211:optical parametric amplifiers
124:{\displaystyle 1/{\sqrt {N}}}
1139:{\displaystyle R_{SU(1,1)}}
77:Mach-Zehnder interferometer
2036:
1745:10.1103/PhysRevA.86.023844
1705:10.1103/PhysRevA.85.023815
499:is the amplitude gain and
307:{\displaystyle a_{2}^{in}}
272:{\displaystyle a_{1}^{in}}
83:and the other beam as the
1178:{\displaystyle R_{SU(2)}}
22:is a technique that uses
1621:10.1103/PhysRevA.33.4033
24:parametric amplification
16:Interferometry technique
1860:Applied Physics Letters
1760:Physical Review Letters
1803:New Journal of Physics
1487:
1486:{\displaystyle 2G^{2}}
1386:
1385:{\displaystyle 2G^{2}}
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1342:{\displaystyle 2G^{2}}
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99:limit which scales as
72:classical interference
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20:SU(1,1) interferometry
1901:Nature Communications
1595:; McCall, Samuel L.;
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48:electromagnetic waves
1984:10.1364/OE.26.000391
1667:10.1364/AOP.8.000104
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1402:quantum interference
1398:quantum entanglement
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154:quantum entanglement
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1976:2018OExpr..26..391G
1913:2014NatCo...5.3049H
1872:2011ApPhL..99a1110J
1825:2010NJPh...12h3014P
1772:2013PhRvL.111c3608K
1737:2012PhRvA..86b3844M
1697:2012PhRvA..85b3815O
1659:2016AdOP....8..104C
1613:1986PhRvA..33.4033Y
1558:2020APLP....5h0902O
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189:{\displaystyle 1/N}
1921:10.1038/ncomms4049
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1067:{\displaystyle 2.}
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1045:{\displaystyle 1.}
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1725:Physical Review A
1685:Physical Review A
1601:Physical Review A
1566:10.1063/5.0004873
1358:Noise performance
1087:{\displaystyle G}
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492:{\displaystyle G}
144:{\displaystyle N}
119:
52:quantum metrology
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1607:(6): 4033–4054.
1597:Klauder, John R.
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28:Heisenberg limit
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2020:Interferometry
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1960:(1): 391–401.
1954:Optics Express
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1653:(1): 104–155.
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660:
656:
631:
626:
623:
620:
615:
610:
606:
585:
582:
578:
566:coherent state
550:
547:
542:
537:
534:
531:
526:
521:
516:
513:
510:
488:
464:
461:
456:
452:
448:
445:
440:
437:
432:
428:
424:
421:
416:
411:
407:
383:
380:
375:
371:
367:
364:
359:
356:
351:
347:
343:
340:
335:
330:
326:
301:
298:
293:
289:
266:
263:
258:
254:
239:quantum fields
226:
223:
185:
181:
177:
140:
118:
112:
108:
43:Interferometry
39:
36:
15:
13:
10:
9:
6:
4:
3:
2:
2032:
2021:
2018:
2017:
2015:
2001:
1997:
1993:
1989:
1985:
1981:
1977:
1973:
1968:
1963:
1959:
1955:
1948:
1945:
1940:
1936:
1931:
1926:
1922:
1918:
1914:
1910:
1906:
1902:
1898:
1891:
1888:
1882:
1877:
1873:
1869:
1866:(1): 011110.
1865:
1861:
1857:
1850:
1847:
1842:
1838:
1834:
1830:
1826:
1822:
1817:
1812:
1809:(8): 083014.
1808:
1804:
1797:
1794:
1789:
1785:
1781:
1777:
1773:
1769:
1766:(3): 033608.
1765:
1761:
1754:
1751:
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1742:
1738:
1734:
1731:(2): 023844.
1730:
1726:
1719:
1717:
1715:
1711:
1706:
1702:
1698:
1694:
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1690:
1686:
1679:
1677:
1673:
1668:
1664:
1660:
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1648:
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1567:
1563:
1559:
1555:
1550:
1545:
1542:(8): 080902.
1541:
1537:
1536:APL Photonics
1530:
1528:
1526:
1524:
1520:
1513:
1511:
1507:
1503:
1496:
1494:
1478:
1474:
1470:
1462:
1458:
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1128:
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1007:
1004:
999:
995:
989:
985:
981:
974:
970:
964:
960:
956:
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947:
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924:
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908:
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877:
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658:
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477:
462:
459:
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450:
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438:
435:
430:
426:
422:
419:
414:
409:
405:
396:
381:
378:
373:
369:
365:
362:
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354:
349:
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341:
338:
333:
328:
324:
315:
299:
296:
291:
287:
264:
261:
256:
252:
244:
241:described by
240:
231:
224:
222:
220:
216:
212:
208:
205:
201:
197:
183:
179:
175:
167:
163:
159:
155:
138:
116:
110:
106:
98:
94:
88:
86:
82:
78:
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64:
60:
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1957:
1953:
1947:
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1900:
1890:
1863:
1859:
1849:
1806:
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1796:
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1759:
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1728:
1724:
1688:
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1600:
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41:
38:Introduction
19:
18:
1907:(1): 3049.
1434:Experiments
162:NOON states
1967:1802.04314
1549:2004.12469
1514:References
1094:is large.
1025:visibility
97:shot-noise
1816:0911.5714
1574:216553469
1288:α
1250:≈
948:ν
909:α
881:ϕ
878:
872:ν
847:α
748:α
720:ϕ
717:
622:α
584:⟩
581:α
525:−
243:operators
215:nonlinear
85:reference
2014:Category
2000:25337227
1992:29328316
1939:24476950
1788:23909323
1301:>>
922:>>
761:>>
131:, where
1972:Bibcode
1930:3916837
1909:Bibcode
1868:Bibcode
1841:4980455
1821:Bibcode
1768:Bibcode
1733:Bibcode
1693:Bibcode
1655:Bibcode
1629:9897145
1609:Bibcode
1554:Bibcode
1023:is the
200:SU(1,1)
1998:
1990:
1937:
1927:
1839:
1786:
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1572:
1461:et al.
1457:et al.
1453:et al.
1440:et al.
1427:et al.
1423:et al.
1415:et al.
940:where
568:input
564:For a
479:where
225:Theory
207:et al.
156:(e.g.
1996:S2CID
1962:arXiv
1837:S2CID
1811:arXiv
1570:S2CID
1544:arXiv
1275:(for
896:(for
735:(for
204:Yurke
93:SU(2)
81:probe
1988:PMID
1935:PMID
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1625:PMID
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1980:doi
1925:PMC
1917:doi
1876:doi
1829:doi
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1008:1
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1000:2
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334:o
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184:N
180:/
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