30 years of squeezed light generation

Ulrik L. Andersen,1, 2 Tobias Gehring,1 Christoph Marquardt,1, 2, 3 and Gerd Leuchs2, 3

1Department of Physics, Technical University of Denmark, Fysikvej, DK-2800 Kongens Lyngby, Denmark2Max-Planck-Institut for the Science of Light, Gunther-Scharowsky-Straße 1, Bau 24, 91058 Erlangen, Germany

3Institute for Optics, Information and Photonics,University Erlangen-Nurnberg, Staudtstr. 7/B2, 90158 Erlangen, Germany

(Dated: April 14, 2016)

Squeezed light generation has come of age. Significant advances on squeezed light generation hasbeen made over the last 30 years – from the initial, conceptual experiment in 1985 till todays top-tuned, application-oriented setups. Here we review the main experimental platforms for generatingquadrature squeezed light that has been investigated the last 30 years.

INTRODUCTION

The year of light 2015 marks the 1000th anniversary ofthe seven volume treatise on optics “Kitab al-Manazir”written by the scientist Ibn al-Haytham and it marks the150th anniversary of Maxwell’s equations. It is also aspecial year for the experimental quantum optics com-munity: 2015 is the year in which we celebrate the 30thanniversary of the first generation of squeezed light.

At the beginning of the eighties there was already anenormous literature on squeezed light on the theory side.Up to that time the experimental efforts had been invain. To illustrate this we would like to cite from thetalk Marc D. Levenson gave at the seventh InternationalLaser Spectroscopy Conference (ICOLS VII) on Maui inthe summer of 1985 after years and years of working onthe topic: “(Squeezed) states have eluded experimentaldemonstration, at least so far. From an experimental-ist’s point of view squeezed state research can be bestdescribed as a series of difficulties that must somehow beovercome”. What follows in the proceedings are nine sec-tions, titled “First Difficulty” all the way up to “NinthDifficulty”, nothing more nothing less [1].

Then in the Fall 1985, the first signature of squeezedlight was observed in a groundbreaking experiment bySlusher, Hollberg, Yurke, Mertz and Valley [2] usingthe process of four-wave-mixing in an atomic vapor ofSodium atoms. Despite the fierce competition betweena number of groups in the USA using different techno-logical platforms, the group of Slusher et al. won thesqueezing race and witnessed the long-sought-after effectof squeezing – a true quantum effect of light.

In strong competition with the atomic vapor technol-ogy for generating squeezed light via four-wave-mixing in1985 was the fiber based approach exploiting the third-order Kerr type nonlinearity of SiO2 as well as the ap-proach based on the second-order nonlinearity of a ferro-electric crystal. These two technologies finally succeededin generating squeezed light in the spring and summer of1986 [3, 4]. Soon thereafter, in December 1986, anothertechnology for squeezed light production was presented.Based on a current noise suppression technique, Machida

et al. managed to observe squeezing in the output of adiode laser [5]. The four experiments are presented inFig. 1.

But what is a squeezed state? Consider first the wavefunction of an optical state in the position and momen-tum representations, φ(x) and φ(p), where x and p cor-respond to the amplitude and phase quadratures of light.The norm squared of these wave functions, |φ(x)|2 and|φ(p)|2, are the marginals of the state’s Wigner functionand represent the probability distributions for the am-plitude and phase quadrature outcomes. For the vac-uum and the coherent state, the two wave functions arerotationally symmetric Gaussians with identical widthswhich means that the variance associated with the mea-surement of any quadrature will be all the same and of-ten normalized to unity: V (x) = V (p) = 1. These statescan be portrayed in phase space by depicting the crosssections of their respective Wigner functions as shown inFig. 2. A quantum state is called squeezed if the varianceof a quadrature amplitude is below the variance of a vac-uum or a coherent state (e.g. V (x) < 1). This comes atthe expense of having the conjugated quadrature variancebeing above the variance of the vacuum (e.g. V (p) > 1)in order to obey Heisenberg’s uncertainty relation. Typ-ical examples of squeezed states are the squeezed vac-uum and squeezed coherent states as shown in Fig. 2a.These states remain Gaussian as the squeezing transfor-mation is a Gaussian map. However, squeezed states canbe also non-Gaussian – one example is the simple su-perposition of vacuum, |0〉, and a single photon state,|1〉: |Φ〉 = a|0〉+ b|1〉 which is squeezed by 1.3 dB belowthe vacuum noise limit [6] or the two-photon state |2〉:|Φ〉 = a|0〉 + b|2〉 which is squeezed by 2.6 dB. They areillustrated in Fig. 2b and 2c, respectively.

The original definition of squeezed states refers to thesqueezing of the quadrature amplitudes but squeezing ofother quantities has also been studied in the literature.This includes squeezing of the photon numbers – knownas photon number squeezing – where the photon numberdistribution is squeezed below the distribution of a co-herent state, as well as the squeezing of the polarizationStokes parameters which is a form of two-mode squeez-

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FIG. 1. The figure summarizes the first three squeezing experiments. (a) Slusher et al. generated squeezing in an atomic Nabeam. The Na atoms were pumped in the cavity formed by the mirrors PM1 and PM2. Squeezing was generated by four-wave-mixing in the pumped Na atoms inside the standing-wave cavity formed by the mirrors SM1 and SM2. The measuredsqueezing was 0.3 dB below the vacuum noise in the squeezed quadrature, see figure on the right. The vacuum noise referenceis shown in the figure by the dim trace labelled “VAC+AMP”. (b) Shelby et al. also generated squeezing by four-wave-mixing,but within a 114 m long optical fiber coiled up and cooled to 4.2 K in liquid Helium. The squeezing was measured by self-homodyne detection where the phase between the local oscillator and the squeezed beam was shifted by a single-ended cavity.The measured squeezing shown on the right was about 0.6 dB below the vacuum noise. DVM: Digital voltmeter. (c) Wuet al. used parametric down-conversion in a Magnesium doped Lithium Niobate crystal embedded in a standing-wave cavityand pumped at the second harmonic of the degenerate signal and idler fields. The results are displayed on the right, wherethe root-mean-square noise voltage measured by the spectrum analyzer is shown versus the phase of the local oscillator. Thesqueezing was about 3.5 dB below the vacuum noise (dashed line). (d) Machida et al. suppressed the photocurrents driving asemiconductor laser in order to produce squeezing. This was done in the ”test laser” while the ”reference laser” was used asa mean to calibrated the shot noise level. The spectrum of the output of the ”test laser” is shown. A maximum squeezing of0.2dB was measured.

FIG. 2. Phase space diagrams of (a) a vacuum (brown, solid),a 6 dB squeezed vacuum (grey, solid), a coherent (brown,dashed) and a 6 dB squeezed coherent state (grey, dashed);(b) |Φ〉 = a |0〉 + b |1〉 with a ≈ 0.87 and b ≈ 0.5 (blue); (c)|Φ〉 = a |0〉+ b |2〉 with a ≈ 0.95 and b ≈ −0.30 (green). Thestates are shown by their standard deviation in the respectivequadrature.

ing [7, 8] reminiscent of quadrature squeezing for brightbeams while similar to photon number squeezing for dimbeams [9].

The first encounter of squeezed states in the litera-

ture (although not coined squeezed state at that time)appeared in 1927 in a paper by Earle Kennard [10]. Hetreated the evolution of a generic Gaussian wave packet ofa harmonic oscillator under the constraints of the Heisen-berg uncertainty relation by which squeezed states wereborn. The theory of Kennard was finalized in Copen-hagen where he had discussions with Heisenberg andBohr who are acknowledged in his paper. Many yearslater the theory was formalized by introducing the fa-mous squeezing operator [11], and further generalizationswere made by Takahashi [12], and Miller and Miskin [13].A detailed review of squeezed states was presented byYuen who proposed to coin it “two-photon coherentstates” [14] closely related to the “contractive states” ofa mechanical system [15]. Hollenhorst talked about the”squeeze” operator [16] while Caves finally suggested thename that is used today: “squeezed states” [17]. A num-ber of review papers on squeezed states have been writtenover the years [18–25], but most of these papers are de-voted to the theoretical treatment of squeezed states. Inthe present Review, we will focus on experimental devel-opments in generating squeezed light during the last 30

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years.Since the first milestone-marking experiments on

squeezed light generation in 1985 and 1986, a numberof other groups world-wide have embarked on route to-wards generating stronger squeezing using different tech-nologies. The squeezing technology has been continu-ously improved with low-loss optical components, high-efficiency detectors and low-noise electronics leading tovery large squeezing degrees of 9 dB using atomic en-sembles, 7 dB using optical fibers and 13 dB using fer-roelectric crystals. This constitutes significant improve-ments compared to the initial experiments where 0.3 dB,0.6 dB and 3.5 dB, respectively, were observed in thesesystems. On the occasion of the 30th anniversary of thefirst squeezed light experiment, we will review the mainachievements on generating squeezed light during the lastthree decades starting from the initial pioneering exper-iments to the current state-of-the-art.

SQUEEZED LIGHT FROM PARAMETRICDOWN-CONVERSION

Parametric down-conversion (PDC) has a long-standing reputation as a generator of squeezed light. Itwas first experimentally realized by Wu et al. in 1986 [3]– just one year after the first demonstration of squeezedlight. It was a scientific breakthrough as this new sourceof squeezed light exhibited a significant improvement inthe generation of squeezed light (3.5 dB squeezing) com-pared to the atomic squeezing experiment from 1985where 0.3 dB squeezing was observed.

In PDC, a pump photon with frequency ωp, incidenton a dielectric with a χ(2) nonlinearity, breaks up intotwo new photons; a signal photon of frequency ωs andan idler photon of frequency ωi where ωp = ωi + ωs.For degenerate PDC, the idler and signal photons are in-distinguishable in frequency (ωi = ωs) and polarization;otherwise the process is non-degenerate.

The χ(2) nonlinearity is in general very weak and inorder to observe a significant induced polarization of themedium and hence efficient squeezing, different consider-ations have to be taken into account. First, the concen-trated field in the crystal has to be relatively high whichis either solved by using high power pulsed lasers or byplacing the crystal inside a cavity. Experiments usingthese two approaches are discussed in the following sec-tions. Second, the momentum of the involved fields hasto be conserved, that is kp = ks +ki where kp is the wavevector for the incident pump beam while ks and ki are thewave vectors associated with the signal and idler beams,respectively. This is the phase matching condition, andcan be achieved by temperature and wavelength tuningand/or periodically poling the non-linear crystal.

In the frequency and polarization degenerate case, cor-responding to Type I phase matching, the system Hamil-

tonian can be written in the form

H = κ(a2 − a†2) , (1)

where a is the annihilation operator for the signal (oridler) field, and κ is the non-linear coupling parame-ter. The unitary evolution of the input signal under thisHamiltonian is U = exp(−iκ(a2 − a†2)) which is exactlythe form of the so-called squeezing operator.

Optical parametric oscillation

An optical parametric oscillator (OPO), where theparametric down-conversion process takes place insidea cavity, has proven to be the most efficient source ofquadrature squeezed light. In most cavity configurations,the signal and idler modes are resonant thereby enhanc-ing the effective non-linearity but also enriching the dy-namics. E.g. for degenerate PDC, the cavity is intro-ducing a critical condition (a pitch-fork bifurcation) inthe system which is the well-known threshold for opti-cal parametric oscillation. For an ideal system withoutlosses, infinite squeezing is expected around this thresh-old point. If the degenerate signal and idler modes ofthe OPO are seeded with a bright beam, the bifurcationpoint disappears and the OPO works as a squeezing am-plifier also known as a phase-sensitive amplifier.

The experiment of Wu et al. [3] (see Fig. 1c), in 1986made use of a sub-threshold OPO where a MgO:LiNbO3

crystal was placed in linear optical cavity (see Fig. 1 c).A frequency doubled laser beam at 532 nm served as thepump field to drive the system close to threshold, therebygenerating squeezed vacuum in the down-converted fieldat 1064 nm. The resulting squeezed vacuum state wasanalyzed by means of homodyne detection comprising abright local oscillator (LO) at 1064 nm, a balanced beamsplitter and two photodiodes. The difference of the pho-tocurrents was recorded and fed into a spectrum analyzerthat displays the power spectral density correspondingto the variance of the quadratures of the squeezed vac-uum. By scanning the phase of the LO, all quadraturevariances could be traced out as shown on the graph inFig. 1c. The dashed line represents a calibration of thevacuum noise limit, and thus squeezed light productionwas clearly witnessed.

Polzik et al. changed the configuration to a bow-tieshaped ring cavity with a KNbO3 crystal and gener-ated frequency tunable squeezed light of 3.8 dB for spec-troscopy [26]. Using a MgO:LiNbO3 crystal as a mono-lithic cavity design where the end-facets of the crystalwere curved and coated with high-reflective mirror coat-ings, Breitenbach et al. [27] obtained 6 dB squeezing whileLam et al. achieved approximately 7 dB vacuum squeez-ing in an unlocked system [28]. Such a monolithic sys-tem for non-classical light generation was pioneered by

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Sizmann et al. [29, 30] for the inverse process of up-conversion as discussed below. Stable squeezing of 6 dBfor hours of operation was achieved by Schneider et al.using a semi-monolithic cavity system that was seededwith a displacement beam [31].

For several years, the achievable squeezing degree lev-eled around 6 dB which was caused by the (at thattime inevitable) intra-cavity losses, detector losses andphase noise. A turning point for squeezed light genera-tion via OPO occurred around the year 2006 where the6 dB squeezing limit was surpassed first in a ring-cavityconfiguration with a periodically poled KTiOPO4 (PP-KTP) crystal (achieving 7.2 dB in 2006 [32] and 9 dB in2007 [33]) and later in a linear-cavity configuration us-ing a LiNbO3 crystal (achieving the magic 10 dB squeez-ing) [34]. These experiments have been further opti-mized and squeezing levels of more than 10 dB have beenachieved in all cavity configurations shown in Fig. 3 ex-cept the linear cavity with crystal in the middle (Fig. 3a).Thereby 12.7 dB squeezing was measured with a PP-KTP crystal using a monolithic cavity [35] (see Fig. 4a),12.3 dB with a semi-monolithic cavity [36] and 11.6 dBwith a ring cavity [37]. Moreover, by reducing technicalnoise at very low frequencies, squeezed light has been ob-served in the audio frequency regime which is importantfor applications in Gravitational wave detection [38, 39].

The technical advances necessary for reaching highsqueezing levels concern all three types of noise mech-anisms mentioned above: intra-cavity losses, detectionlosses and phase noise. Intra-cavity losses were espe-cially reduced by introducing PPKTP crystals which ex-hibit low optical absorption at the (fundamental) squeez-ing wavelength [40] and a negligible amount of pump-induced absorption (a problem that cannot be neglectedin other crystals [32]). Furthermore, the developmentof low-loss coatings helped to reduce intra-cavity loss sothat escape efficiencies of more than 97 % (defined by thecoupling rate divided by the intra-cavity loss rate) havebeen achieved [33–35]. Detection losses are mainly dueto an imperfect quantum efficiency of the photo diodesand the visibility between local oscillator and squeezedbeam at the homodyne detector’s beam splitter. Whilephoto diodes for visible wavelength based on Silicon witha quantum efficiency close to 100 % were already avail-able for a long time, e.g. [26], high efficiency InGaAsphoto diodes for infrared wavelengths became availableonly about 8 years ago [41]. Visibilities of the interfer-ence contrast at the homodyne detector close to 100 %were also very important to measure high squeezing val-ues. Reducing the phase noise was achieved by filteringthe pump beam with a ring cavity [34] and by optimizedfeedback systems for cavity and phase locks [33].

Squeezing in an OPO cavity was not only achieved be-low threshold, but also above, separately in both signaland idler beam [42]. Using a whispering gallery moderesonator (WGM) made of LiNbO3 with a record-low

threshold power of only a couple of µW , 1.2 dB squeez-ing could be observed. This squeezing in only one ofthe nondegenerate OPO beams is hardly important forapplications but relevant for better understanding theprocess. In the above-threshold operation, classical noiseintroduced by the pump is a critical problem for observ-ing squeezed light. When exploiting the combined ac-tion of both signal and idler in the degenerate or non-degenerate regime, WGMs are a promising source ofsizeable squeezed light usable in real-world applicationsas they are compact, tunable and operational with lowpump powers.

Parametric up-conversion

Parametric up-conversion, where a fundamental in-put beam undergoes a frequency doubling process, isthe inverse process of down-conversion at the doubledfrequency. This process has also shown to be a viablestrategy for producing squeezed light. The first experi-ments were carried out by Sizmann et al. and by Kurz etal. [29, 30] in a double resonating system where both thefundamental as well as the frequency doubled mode wassupported by a solid monolithic cavity. A single resonantsystem has been also realized and used for demonstratingsqueezing of the frequency doubled mode [43]. However,since squeezing of this mode for a single resonant systemis theoretically limited to a finite value, this strategy hasnot been followed by many groups.

Single-pass optical parametric amplification

An alternative way of addressing the small χ(2) non-linearity is to significantly increase the pump amplitudeby using ultra-short pump pulses. This will effectivelyincrease the non-linearity and thus the production ofsqueezed light. However, the pulse shape also leads tocomplications as the amount of squeezing strongly de-pends on the temporal mode shape overlap of the pumpand the LO pulse. Despite of these complications relatedto the pulsing operation, squeezing from a pulsed para-metric down-conversion source was observed already in1987 by Grangier et al. [44], and significantly improvedin Refs. [45] and [46] in the group of Kumar. In thelatter work (shown in Fig. 4b), a mode-tailored local os-cillator was produced by injecting an auxiliary beam intothe squeezing crystal in a polarization mode orthogonalto the signal mode, but still actively amplified using aKTP Type II phase matched crystal. This resulted in aspatio-temporal mode of the LO near identical to that ofthe squeezed signal mode. Due to this mode optimizationstrategy of the LO, a record-breaking squeezing value of5.8 dB was measured in 1994. After more than 20 years ofresearch, this is still the quadrature squeezing record in

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FIG. 3. Different cavity geometries used to produce squeezedlight by parametric down-conversion. (a) Linear cavity withcrystal in the middle, (b) semi-monolithic linear cavity whereone end-face of the crystal is curved. (c) monolithic linearcavity formed by the two end-faces of the crystal. (d) Bow-tie travelling-wave cavity. Coatings at the signal and idlerwavelengths: AR: anti-reflective coating, HR: high-reflectivecoating, PR: partially reflective coating.

pulsed optical parametric amplification systems. Pulsedsqueezed light experiments based on optical parametricamplification have been extended to include periodicallypoled crystals [47].

By confining the pump field in the nonlinear crystalover a long distance by means of a crystalline waveg-uide structure (thereby circumventing diverging beamsdue to diffraction), it is possible to increase the effectivenonlinearity even further. Using such systems single-passpulsed squeezing [48–50] and even single-pass continuouswave squeezing has been observed [51, 52].

SQUEEZED LIGHT FROM OPTICAL FIBERS

As an alternative to second order nonlinear processesalso third order nonlinearities can serve to generatesqueezed light. Third order nonlinear processes are muchweaker than their second order counterparts, howeverthey also exist in amorphous materials, rendering thechoice of the physical system more flexible. The weakinteraction can successfully be compensated by a longinteraction length. This is why squeezing inside opticalfibers is a practical alternative. Optical fibers also offerlow loss over long distances. One of the highly activecompetitors to the first squeezing experiment was in facta setup using an optical fiber. At IBM Shelby et al., [4],used a 114 m long optical fiber and injected high powercontinuous wave laser beams. In hind side this experi-ment can be regarded as quite heroic as several obstaclesthat nowadays are considered interesting physics in theirown right had to be identified and overcome (see intro-duction).

Squeezing in optical fibers relies on four-wave-mixingand the nonlinear optical Kerr effect. Third order non-

linearities lead to a situation where the refractive indexof the material in which the light is propagating dependson the intensity of the light itself:

n = n0 + n2 · I . (2)

The Kerr effect transforms a coherent state from a pumplaser into a squeezed state (see Fig. 5). This is best ex-plained by regarding the acquired phase shifts in phasespace. Regions in phase space with higher amplitude areassociated with an increased phase shift as a direct con-sequence of the nonlinear refractive index. Hence the cir-cular uncertainty region of the coherent state gets trans-formed into a squeezed state (this is true in a first orderapproximation which practically can be applied as highintensities are needed for a considerable phase shift [54]).The nonlinear Kerr effect does not need sophisticatedphase matching strategies and thus is one of the sim-plest means to produce squeezed light. The challengeshere stem from the fact that third order nonlinearitiesare weak and high power levels are needed that also cantrigger other unwanted nonlinear interactions.

The first experiment in 1986 (see Fig. 1b) used acontinuous-wave pump to generate the squeezing. As therequired pump powers are significant it was soon realizedthat this leads to a number of different effects inside thefiber that hinders or reduces the generation of squeez-ing. The most important negative effect here is photon-phonon coupling inside the fiber. Guided Acoustic WaveBrillouin scattering [1] introduces noise at acoustic fre-quencies up to the GHz regime, thus the fiber has tobe cooled and these frequencies have to be avoided inmeasurements. At the required power levels stimulatedBrillouin scattering backscattered most of the launchedlight. To stay below the Brillouin threshold a phase mod-ulation scheme generating several different wavelengthshad to be used. The measurement itself cannot easilyrely on homodyne technology as the intensities involvedare already very high. Thus it is technically not possibleto employ an even brighter LO reference beam. InsteadShelby et al. used a phase shifting cavity [56, 57]. Byreflecting off a cavity resonance the bright carrier can bephase shifted compared to the radio frequency sidebands.A detuning from the cavity enables to measure a rotatedprojection of the squeezing ellipse in phase space.

Later experiments could lower the technical difficultiessignificantly by using short pulses [44, 58, 59]. The shorttime scales and high peak intensities of these pulsed sys-tems reduce the average power required into regimes thatare more favorable for the detection system and do notsuffer from stimulated Brillouin backscattering. In orderto avoid dispersive pulse spreading in the optical fibersone can go into a regime of optical solitons [59, 60]. Inaddition, in the soliton regime the correlations betweenwavelength components can be used to generate squeezedbeams by spectral filtering [61, 62].

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FIG. 4. Squeezing records in different systems. (a) 12.7 dB squeezing was generated by parametric down-conversion in amonolithic PPKTP cavity using a frequency-doubled 1064 nm continuous-wave laser beam as pump. The modecleaner improvesthe spatial beam profile of the local oscillator and thus enhances the interference contrast in the homodyne detector. [35] (b)A single-pass non-degenerate OPA (NOPA) consisting of a KTP crystal pumped with a frequency-doubled, mode-locked,q-switched laser was used to generate 5.8 dB squeezing. The process was Type II phase matched which means that brightsqueezing was produced in two orthogonal polarization modes. This was used to form a squeezed vacuum state and a brightlocal oscillator by means of a balanced beam splitter [46]. (c) 6.8 dB polarization squeezing was generated by the optical Kerreffect in a polarization maintaining fiber using a pulsed 1500 nm laser. The squeezing was measured with a Stokes parametermeasurement scheme. To compensate for the birefringence of the fiber, the two orthogonal pulses were shifted in time prior tofiber injection [41]. (d) A double lambda scheme is employed to generate strongly correlated beams via four-wave-mixing. Apump and a probe beam interacts in a cloud of Rubidium atoms and subsequently forms correlations between the probe and itsconjugate beam. The results are recorded on a spectrum analyzer (SA) and a maximum of 9.2 dB squeezing was measured [53].

FIG. 5. Squeezing of a coherent state by the nonlinear optical Kerr effect in quadrature phase space. The acquired phase shiftvaries with amplitude

√I and transforms the quantum state uncertainty into an ellipse [55].

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The Kerr effect is photon-number preserving and thesqueezing is thus not measurable in direct detection asnoted above. Measuring other quadratures by homodynetechniques often suffers from the signal beam being al-ready very intense. As a viable alternative to spectralfiltering (see above) one may utilize interferometric tech-niques to be able to measure the squeezing [63].

One interferometric solution is to use a cavity as pre-sented above. Another successful solution uses a Sagnacinterferometer employing either a balanced or unbalancedbeam splitter. Using the balanced beam splitter the twocounter propagating beams get squeezed. At the outputboth beams interfere destructively resulting in a vacuumsqueezed beam and constructively resulting in a brightbeam that can be used as a local oscillator [58]. With anunbalanced beam splitter one of the counter propagat-ing beams is bright and gets squeezed. The other (weak)beam interferes at the output and shifts the squeezedbeam in phase space. For certain power levels this inter-ference leads to directly measurable amplitude squeezing[64–66]. To introduce more flexibility into the setup thatallows for independent control of the relative intensitiesand phase of the two pulses, a linear version of the asym-metric Sagnac interferometer was suggested and demon-strated [67].

Interferometric setups can be largely simplified whenemploying polarization. Polarization squeezing denotesthe reduction of the quantum uncertainty of the polar-ization of the light field. The concept of polarizationsqueezing relies on a Heisenberg uncertainty relation inpolarization variables that uses Stokes operators and isclose to what one uses to describe the uncertainty of theatomic spin. For intense beams (as used in Kerr squeez-ing) polarization squeezed beams can be well approxi-mated by quadrature variables.

Polarization squeezing can be achieved by a single passof optical pump pulses on the two polarization axes of apolarization maintaining optical fiber. When compensat-ing for the birefringence inside the fiber the two orthog-onal polarized squeezed beams interfere at the output ofthe fiber. The resulting polarization squeezing can thenbe determined by a Stokes measurement. The simplicityof the setup and very good spatial and spectral overlapof the two interfering beams led to a measured squeezingof around 7 dB [41, 68] (see Fig. 4c). A detailed theo-retical analysis shows that indeed Brillouin and Ramanscattering are limiting the squeezing in practice [69].

Photonic crystal fibers (PCFs) offer new possibilitiesin generating squeezed states. Here the light is guidedby an effective refractive index through a microstruc-ture around a solid core or a photonic crystal band gapthat enables guiding inside a hollow core. These type offibers offer much higher effective nonlinearities and flexi-bility in dispersion management. Squeezing in PCFs hasbeen shown early after their development [70–72]. Theinvolved powers are lower than those needed in standard

fibers and wavelengths outside the telecom band can beutilized. The strong nonlinearities and dispersive prop-erties of PCFs however require a careful balancing andso far the best squeezing results still have been achievedwith standard fibers.

SQUEEZED LIGHT FROM ATOMICENSEMBLES

As mentioned in the introduction, the first evidence ofsqueezing the optical field was obtained through the non-linear interaction between light and atomic vapor. Themotivation for using an atomic ensemble for squeezedlight generation at the early days of quantum optics wasthat the intrinsic nonlinearity associated with light-atominteraction can be very large – significantly larger thanusing χ(2) nonlinear crystals. The simplest kind of anonlinearity is atomic saturation which is observed whenthe ensemble is illuminated by laser light resonant withthe atomic transition. However, it is a four-wave mix-ing mechanism that leads to the formation of squeezedlight. The process is enabled by a Λ-shaped atomic en-ergy level configuration where two ground states are cou-pled to a single excited state with optical pump beamsthat are either frequency degenerate or non-degenerate.The pump beams enable a cycling from one ground stateto the other and back under the emission of two pho-tons that are frequency non-degenerate by an amountgiven by the frequency difference of the two ground statelevels. These two modes (occupied by the photons) areknown as Stokes and Anti-Stokes and they form a two-mode squeezed state in which squeezing can be observedby using a local oscillator of frequency located betweenthe two Stokes modes.

In the experiment of Slusher et al. [2] (see Fig. 1a)the nonlinear interaction took place between laser lightand a beam of Sodium atoms. To enhance the effect,the atomic beam was embedded in two cavities resonantfor the Stokes and Anti-Stokes sidebands. As a result ofthe non-linear interaction, correlations between the twosidebands were established and thus squeezed light couldbe measured. The results of a homodyne measurementare shown in Fig. 1a. A modest squeezing degree of 0.3 dBwas observed in this experiment.

This small amount of squeezing measured in this ini-tial experiment was mainly due to other detrimentalnon-linear processes such as Raman scattering and flu-orescense occurring simultaneously with the four-wave-mixing process. These processes lead to incoherent emis-sion of noisy modes into the squeezed modes and therebya degradation of the squeezed state. These effects havebeen limiting the amount of squeezing in atomic sys-tems for a number of years. However, by consideringintensity-difference squeezing (also known as twin beamsqueezing, see below) of a double-Λ system, many of the

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parasitic processes cancel out, and thus strong squeezingcan be revealed. This led the group of Paul Lett to ob-serve close to 9 dB squeezing in the intensity differenceof the two output modes in four-wave-mixing in atomicvapor of Rubidium [73, 74]. A pump and a probe beamare incident onto the atomic ensemble, and interact effi-ciently via the four-wave-mixing process. The probe andthe conjugate are amplified in the process establishingquantum correlations that are revealed through intensity-difference detection. The noise of the intensity differencewas squeezed by 8.8 dB below shot noise level. Usingimproved techniques, the intensity-difference squeezingwas increased to 9.2dB which is the squeezing record forsqueezing in atomic ensembles [53]. This experiment isshown in Fig. 4d. A similar system has been used also toproduce single-mode squeezing with a squeezing degreeof 3 dB [75].

It is also possible to observe squeezing through a Fara-day rotation nonlinearity in atomic systems [76, 77]. Herea polarization mode orthogonal to the polarization of thepump is squeezed as a result of a cross-phase modulationnonlinearity. Using this atomic nonlinerity, 0.9 dB [76]and 2.9 dB [77] squeezing have been measured.

The benefits of a fiber (strong confinement over a longdistance) and atomic vapor (high nonlinearity) can bemerged in a single system by filling a hollow core PCFwith gaseous or liquid materials. This offers promis-ing new possibilities of efficient nonlinear interaction forthe generation of squeezing largely avoiding Brioullinand Raman scattering. First results have demonstratedsqueezing in fibers filled with high pressure gas [78] andatomic vapour [79].

SQUEEZED LIGHT FROM SEMICONDUCTORLASERS

Going back to the days of vacuum tubes, it was a majorinvention that current controlling or amplifying vacuumtubes can work in a sub shot noise regime provided thatthe current is in the space charge limited regime [80].This leads to electrons arriving more evenly spaced thanexpected for shot noise. If one only could transfer eachof these electrons into one photon, then one would havesub shot noise light. An obvious candidate for such atransfer is a light emitting diode or a light emitting laserdiode. The latter type is preferable if one wants to ul-timately collimate the light. Machida, Yamamoto andItaya followed this approach and in 1986 succeeded inmeasuring squeezed light emitted directly from a laserdiode, the noise reduction being 0.33 dB [5]. In the fol-lowing years the same group tried to improve the set-upand to optimize the parameters of operation to minimizethe noise. Eventually they managed to observe a verylarge noise reduction by positioning the detector and thelaser face to face [81]. Other groups who first collimated

the light beam before detecting the squeezing could atfirst not come any where near reaching similar numbers.The noise reduction in the collimated light emission fromlaser diodes lies around 3-4 dB [82]. Ultimately it becameclear that in order to observe larger noise reduction, onehas to measure the whole emission, as might be guessedfrom the argument above, i.e. one has to measure allmodes. There are strong correlations between differentspatial modes [83]. Some of these modes with negligiblepower help to significantly reduce the noise (for detailssee [82] and references there in).

A SHORT NOTE ON MULTIMODE SQUEEZING

The field of multimode squeezing and entanglement isa highly topical field due to the associated interestingapplications in quantum communication, quantum com-puting and quantum sensing. Multimode squeezing willbe briefly discussed in this section. Although we havemostly been referring to “single mode” squeezing in pre-vious sections, as a matter of fact, all squeezed states aremultimode states possessing multi-mode quantum corre-lations. Squeezing is a result of quantum correlations orentanglement between sideband frequency modes. As aresult, their joint measurement with a single local oscil-lator (located in between the sidebands) will reveal theso-called single-mode squeezing. This was experimentallyaddressed in Ref. [84] where the entanglement of the twosidebands was directly measured. When the sidebandsare “widely” separated, either in distinct frequency orpolarization modes, the modes become easier to access.In some cases, only the intensity correlations can easilybe measured due to complications in performing homo-dyne detection. For such measurements, only squeezingbetween a single pair of quadratures have been witnessedand this is often referred to as twin-beam squeezing. Suchan experiment was demonstrated for the first time al-ready in 1987 in an OPO operating above threshold [85]holding the world record in squeezing for many years.Later, the same effect has been realized in a number ofother systems including seeded OPO [86], new versionsof the above-threshold OPO (in a bulk cavity [42, 87] anda micro cavity [88]), seeded [89] and unseeded single-passOPA [90], waveguide OPA [91] as well as fiber [92] andatomic based systems [73]. There are also a number ofrecent studies on squeezing and entanglement between alarge number of different modes which includes standardtwo-mode squeezing [93, 94], N-mode entanglement [95–99], entanglement between spatial modes [100–103], fre-quency modes [104–106], temporal modes [107], mixedmodes [108, 109] and different colors [110]. Using thesemultimode squeezing processes it is in principle possibleto generate large cluster states that are the main resourcein linear quantum computing [111], or to produce spa-tial correlations that can be used for squeezing-enhanced

9

imaging [112].

FINAL REMARKS

Squeezed light has come a long way since its firstdemonstration 30 years ago. Significant advancementshave been made from the initial 0.3 dB squeezing till to-days near 13 dB squeezing. It is, however, interesting tonote that the experimental platforms of nonlinear crys-tals, fibers and atomic ensembles used in 1985 and 1986are the same as those used today for generating highlyefficient squeezing. The advancements have mainly beenof technical nature, that is, successful development oflow-noise electronics for phase locking, low loss opticalcomponents and high efficiency photo diodes have led tolargely improved systems.

In addition to these technical advancements, withinthe last few years there have been demonstrations of theproduction of squeezed light in new systems. Using sin-gle emitters such as a single ion in a high finesse cav-ity [113] or a single semiconductor quantum dot [114]small degrees of squeezed light has been generated. Inthese accounts, the intrinsic strong nonlinearity betweena two-level system and light was employed. Recently, itwas also shown that squeezed light can be produced on amicron-sized platform exploiting the nonlinear couplingbetween light and a mechanical oscillator. Here a sus-pended mechanical oscillator is embedded into a cavityand displaced by the radiation pressure force producedby the interacting circulating light. The displacementof the mechanical oscillator will in turn shift the phaseof the circulating field. In essence, this corresponds toan intensity dependent phase shift which is reminiscenceof the optical Kerr effect that is known to squeeze thefield. Experiments have been performed in a photoniccrystal cavity supporting phononic and photonic modessimultaneously [115], and in a bulk cavity setup con-taining a mechanical membrane [116]. Another route tosqueezed light generation on micro-platforms is to exploitthe Kerr non-linearity of CMOS compatible materialssuch as Silicon Nitride (SiN). E.g. there has been recentreports on intensity difference squeezing [88] and singlemode squeezing [117] generated in SiN micro-ring cavi-ties. All these recent demonstrations of squeezed lightin completely new and miniaturized settings might alsorepresent a guide towards the future trends in squeezedlight generation: Generating squeezing on smaller andpotentially more rugged platforms allow for up-scalingand eventually real-life applications.

Initially, squeezed light was envisaged to enable en-hanced communication rates [96, 118–120] and improveddetection of weak forces such as gravitational waves [17,121, 122] [123]. The latter was demonstrated first at theGEO600 gravitational wave detector [124] and later atthe LIGO detector [125]. These applications proposed

more than 35 years ago are still some of the most promi-nent applications of squeezed light. However, in ad-dition to these applications, squeezed states have alsobeen shown to be the resource of quantum teleporta-tion [126, 127], continuous variable quantum comput-ing [111], quantum error correction coding [128, 129],phase estimation [130] and tracking [131], fundamen-tal tests of quantum mechanics (such as the Einstein-Podolsky-Rosen gedanken experiment) [8, 23, 93], quan-tum imaging [112, 132] of e.g. biological samples [133],clock synchronization [134] and magnetometry [135, 136].Moreover, in recent years, a squeezed light source hasbeen the working horse for quantum state engineering,in particular non-Gaussian state generation using themethod of photon subtraction [137–140] as required forvarious quantum processing protocols [141–145]. Thus,a plethora of new and exciting applications of squeezedlight have appeared since the initial proposals, and it willbe exciting to observe where these application studies willtake us the next 30 years.

ACKNOWLEDGEMENTS

We acknowledge support from the Lundbeck Foun-dation, the Danish Council for Independent Re-search(Sapere Aude 0602-01686B and 4184-00338B) andthe H.C. Ørsted postdoc programme.

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