Polarization Squeezing with Cold Atoms Vincent Josse, Aurelien Dantan, Laurent Vernac, Alberto Bramati, Michel Pinard, Elisabeth Giacobino

Polarization squeezing with cold atoms Vincent Josse, Aurelien Dantan, Laurent Vernac, Alberto Bramati, Michel Pinard, Elisabeth Giacobino

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Vincent Josse, Aurelien Dantan, Laurent Vernac, Alberto Bramati, Michel Pinard, et al.. Polarization squeezing with cold atoms. Physical Review Letters, American Physical Society, 2003, 91, pp.103601. hal-00000279v2

HAL Id: hal-00000279 https://hal.archives-ouvertes.fr/hal-00000279v2 Submitted on 12 May 2003

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The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ccsd-00000279 (version 2) : 12 May 2003 lal htcmeiieotclpmigmyresult may pumping shows optical linear analysis competitive the theoretical that on Our an clearly based of means method. model by input-output quantum interaction medium four-level the situation atomic the the describe X-like and We and light applicable between complicated. longer two-level more the no much configuration, is this model In atom atoms. cesium cold nti ae efcso h hoeia n experi- the and via a theoretical squeezing of the polarization on interaction of focus investigation been we thus mental paper has this 40% [6]. of In group reduction our noise quadrature in observed generate A and [7]. modified the squeezing of strongly fluctuations be a quantum point can the turning with light curve, the bistability at light the that, and the of cavity light is the of the by of It interaction transmitted behavior bistable the produces gives medium index. 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F=5 , d i q i hd measurement measurement phase noise l /4 PBS1 Y X G oscilloscope G Vs Vs 2 1 2

3 4 atoms from being optically pumped to the 6S1/2, F=3 D state, we superimpose a diode laser tuned to the 6S1/2, g// g// F=3 to 6P3 2, F=4 transition to the trapping beams. / g^ g^ We use a 25 cm long linear cavity built around the cell. A A The cavity is close to the hemifocal configuration with a + - waist of 260 µm. The coupling mirror has a transmission 1 2 coefficient T of 10%, the rear mirror is highly reflect- FIG. 3: Schematic energy level diagram for the X-like four- ing. Hence, the cavity is close to the bad cavity limit for level system: γ⊥ + γk = γ is the optical dipole decay rate; ∆ which the cavity linewidth (κ = 5 MHz) is larger than the is the (large) detuning from resonance. atomic linewidth (γ = 2.6 MHz). We probe the atoms with a linearly polarized laser beam detuned by about 50 MHz in the red of the 6S1/2, F=4 to 6P3/2, F=5 transi- retical frame [8, 9, 10]. Alternatively, the polarization tion. The optical power of the probe beam ranges from 5 switching threshold may be interpreted as a laser oscilla- to 15 µW. Under these conditions, the number of atoms tion threshold for the mode orthogonal to the main polar- interacting with the light beam is about 106-107. The ization mode [8]. Here, we are interested in the quantum polarization of the light transmitted by the cavity is an- fluctuations of the light, which can be strongly modified alyzed with a quarter-wave plate and a polarizing beam- via the interaction with the atoms. When the polariza- splitter (PBS1), so that Vs1 and Vs2 give the amount tion of the light is circular, the situation is analogous of respectively left-handed (I+) and right-handed (I−) to the previous experimental scheme when the incoming circular light. field was circularly polarized [6]. We will therefore focus in the following on the case for which the polarization remains linear along the x-axis. The saturation of the Vs1 ()I+ A σ± components of light causes the medium to behave as Vs2 ()I- a Kerr-like medium for the mean field Aˆx. In addition, the vacuum orthogonal Aˆy mode experiences a non linear dephasing via cross-Kerr effect [2, 8]. This system is then expected to generate quadrature squeezing for both modes; in the large detuning limit [∆ γ], the equation Transmission (a.u.) for the vacuum mode fluctuations readsÀ [8] Cavity detuning (a.u.)

dδAˆy FIG. 2: Analysis of the circular polarization content of the = [κ + i(∆c ∆0)]δAˆy (1) light transmitted by the cavity and detected by the photodi- dt − − odes shown in Fig. 1. Polarization switching occurs at point ˆ 2 sx ˆ Ax ˆ† 2κ în i∆0 [2δAy h i δAy] + δAy A. The power of the incident light is 7 µW. − 2 − Aˆ 2 √T |h xi| 2 A typical recording of the transmitted intensities I± as with ∆c the cavity detuning, ∆0 = 2Ng κ/∆T the 2 2 2 a function of the cavity length is shown in Fig. 2: start- linear atomic dephasing, sx = 2g Aˆx /∆ the satura- în |h i| ing from the left up to point A, the polarization remains tion parameter and δAy the incident field fluctuations. linear (nearly equal amount of circular polarized light on The cross-Kerr induced term has two contributions: a ˆ 2 ˆ ˆ 2 ˆ† both photodiodes), then it becomes circular. This po- dephasing ( Ax δAy) and the term in Ax δAy, larization switching was known to occur in Fabry-Perot responsible∝ for |h thei squeezing of this mode.h A similari cavities containing atomic vapors with degenerate sub- equation can be derived for the mean field Aˆx, for which levels in the ground state [9, 10]. In the experiment the the squeezing is then generated via the usual Kerr term 2 † probe beam is nearly resonant with the 6S1 2, F=4 to ( Aˆ δAˆ ). These equations are valid when the large / ∝ h xi x 6P3/2, F=5 transition: in principle, all the 20 Zeeman excess noise due to optical pumping can be neglected, sub-levels are involved in the interaction. In order to get that is for times smaller than the optical pumping time, a qualitative physical insight into this problem, avoiding in contrast with [5]. Squeezing is then predicted for noise too heavy calculations, we model the atomic medium as frequencies higher than the inverse optical pumping time. an X-like four-level system [see Fig. 3]. Using the experimental set-up described in Fig. 1, we The competitive optical pumping between the circu- have measured the quadrature noise of both modes. The lar component σ± of light makes the linear polarization signal recorded at the output of the cavity (see Fig. 2) unstable inside the cavity above some intensity thresh- is used to lock the cavity length on the regime where the old. The optical pumping is responsible for polarization polarization remains linear. After interacting with the switching and the general shape of the cavity resonance atoms, both fields are mainly reflected by BS and then curve [see Fig. 2] is well understood within this theo- mixed on PBS2 with a 10 mW local oscillator beam (LO). 3

Using the half-wave plate λ/2a, we are able to send either Their spectral noise densities satisfy uncertainty relations 2 the mean field mode or the orthogonal vacuum mode to V ˆ (ω)V ˆ (ω) ²ijk Sˆk . In our case the light is lin- Si Sj ≥ | h i| PBS3 and perform the usual homodyne detection. By ˆ ˆ 2 early polarized along the x-axis, then S0 = S1 = αx varying the relative phase of the LO with respect to the h i h i and Sˆ2 = Sˆ3 = 0, where Aˆx = αx is chosen probe beam we can detect the noise features of all the real.h Theni theh onlyi non trivialh Heisenbergi inequality quadratures of the field. In agreement with the theoreti- 4 is V ˆ (ω)V ˆ (ω) αx. Polarization squeezing is then cal predictions, we observe quadrature squeezing on both S2 S3 ≥ achieved if VSˆ2 or VSˆ3 is below the coherent state value the main mode and the orthogonal mode. The results are 2 α . The fluctuations of Sˆ2 and Sˆ3 are related to the fluc- 5% (7% after correction for optical losses, mainly due to x tuations of the quadratures of the vacuum orthogonal BS) of noise reduction for the main polarization mode at mode 6 MHz and 13% (18% after correction) for the squeezed vacuum state at 3 MHz, as shown in Fig. 4. This system is then able to produce simultaneously two squeezed † † δSˆ2 = α (δAˆ + δAˆ ) , δSˆ3 = iα (δAˆ δAˆ ) (2) modes, for which the relatives phases are intrinsically x y y x y − y fixed. Let us note that these two squeezed modes can be The physical meaning of the Stokes parameters fluc- used to generate a pair of entangled beams, which will tuations is the following: the δSˆ2 fluctuations lead to a be presented in a forthcoming publication [11]. geometric jitter of the polarization axis, whereas the δSˆ3

1.8 fluctuations are linked to ellipticity fluctuations. It can (a) be seen from Eq. (2) that these fluctuations are related 1.6 to the amplitude and phase fluctuations of Aˆy. Therefore polarization squeezing is equivalent to vacuum squeezing 1.4 on the orthogonal mode. The measurement of the Stokes parameters can be car- 1.2 ried out directly by means of two balanced photodiodes

1 and suitable combinations of half-wave and quarter-wave

plates [12]. In our set-up, however, the power of the Normalized vacuum noise 0.8 probe beam interacting with the atoms ( 10µW) is not ∼ Time(s) sufficient, so that we need a LO for the detection. The 1.4 ˆ (b) fluctuations of the vacuum mode Ay are measured using 1.3 the homodyne detection described above. Following Eq. (2) the photocurrent can be expressed in terms of the 1.2 fluctuations of Sˆ2 and Sˆ3: 1.1

1 δi cos θ δSˆ2 + sin θ δSˆ3 δSˆ (3) hd ∝ hd hd ≡ θhd

Normalized mean field noise 0.9 where θhd is the relative phase between the LO and the FIG. 4: Normalized quadrature noise for the vacuum field mean field. As θhd is varied in time, we correspondingly ˆ mode at 3 MHz (a) and for the mean field mode at 6 MHz detect the fluctuations of the Stokes parameter Sθhd . For (b). The best squeezing is 13% for the vacuum mode and 5% instance, θhd = 0 (respectively π/2) corresponds to the for the mean field mode. detection of the fluctuations of Sˆ2 (respectively of Sˆ3). Hence, in the experiment we can get the Stokes param- In the following we focus on the study of the noise eters simply by simultaneously measuring the relative of the mode with orthogonal polarization with respect phase θhd and the quadrature noise of the vacuum mode. to the main mode, commonly referred to as polarization This measurement is readily performed by setting the noise. The characterization of the quantum features of half-wave plate before PBS2 in such a way that the Aˆy the polarization state relies on the measurement of the mode is sent to the homodyne detection; the mean field quantum Stokes parameters [12]. They are defined from Aˆx goes through the other port of the beam splitter and their classical counterparts is detected together with a portion of the LO by a pho- todiode (see Fig. 1). The phase is determined via the ˆ † † † † interference signal between LO and Ax (iθ cos θhd). Sˆ0 = Aˆ Aˆx + Aˆ Aˆy , Sˆ1 = Aˆ Aˆx Aˆ Aˆy ∝ x y x − y The two signals iθ and δihd are sent to the XY chan- ˆ ˆ† ˆ ˆ† ˆ ˆ ˆ† ˆ ˆ† ˆ S2 = AxAy + AyAx , S3 = i(AyAx AxAy) nel of the oscilloscope, giving the characteristic curves − reported below. These operators obey the commutation relations In Fig. 5 the normalized quadrature noise of Aˆy, [Sˆ0, Sî] = 0 and [Sî, Sˆj ] = ²ijk2iSˆk (i = 1, 2, 3). obtained at a noise frequency of 3 MHz, is plotted 4

2 and (a) duce quadrature squeezing on the mean ﬁeld mode 1,8 on the orthogonally polarized vacuum mode. We have shown that these results can be interpreted as polariza- 1,6 tion squeezing and developed a method to measure the quantum Stokes parameters for weak beams, using a local 1,4 oscillator and a standard homodyne detection.

1,2 S3

S2 S2 This work was supported by the QIPC European 1 Project No. IST-1999-13071 (QUICOV).

Normalizedvacuumnoise Ssq 0,8

1.4 [1] A.S. Chirkin, A.A. Orlov, D.Yu. Paraschuk, Quantum S2 S2 Electron. 23, 870 (1993); A.P. Alodjants, A.M. Arake- lian, A.S. Chirkin, JEPT 108, 63 (1995); N.V. Korolkova, 1.2 A.S. Chirkin, J. Mod. Opt. 43, 869 (1996). [2] L. Boivin, H.A. Haus, Optics Lett. 21, 146 (1996); J. Heersink, T. Gaber, S. Lorenz, O. Gl¨ockl, N. Korolkova, 1 G. Leuchs, quant-ph/0302100. [3] P. Grangier, R.E. Slusher, B.Yurke, A. LaPorta, Phys. (b) Ssq=S3 Rev. Lett. 59, 2153 (1987); J. Hald, J.L. Sorensen, C.

Normalizedvacuumnoise 0.8 Schori, E.S. Polzik, Phys. Rev. Lett. 83, 1319 (1999). -1 0 1 [4] W.P. Bowen, R. Schnabel, H.A. Bachor, P.K. Lam, Phys. Normalizedinterferences: cos(qhd ) Rev. Lett. 88, 093601 (2002). [5] A.B. Matsko, I. Novikova, G.R. Welch, D. Budker, D.F. FIG. 5: Normalized quadrature noise at 3 MHz for the vac- Kimball, S.M. Rochester, Phys. Rev. A 66, 043815/1 uum mode Aˆy vs the normalized interference signal: cos θhd. (2002); I. Novikova, A.B. Matsko, G.R. Welch, private The general case is shown in curve (a): polarization squeez- communication. ◦ ing is achieved when θhd = θsq = ±30 : a linear combina- [6] A. Lambrecht, E. Giacobino, J.M. Courty, Optics Comm. tion of Sˆ2 and Sˆ3 is squeezed. In curve (b), Sˆ3 is squeezed 115, 199, (1995); A. Lambrecht, T. Coudreau, A.M. Ste- 36 (θsq = ±π/2). imberg, E. Giacobino, Europhys. Lett. , 93 (1996); A.Z. Khoury, T. Coudreau, C. Fabre, E. Giacobino, Phys. Rev. A 57, 4397 (1998). [7] L. Hilico, C. Fabre, S. Reynaud, E. Giacobino, Phys. Rev. as a function of the relative phase between the mean 46 field and the LO. In agreement with Eq. (3), it can A , 4397 (1992). ˆ [8] V. Josse, A. Dantan, A. Bramati, M. Pinard, E. Gia- be seen that the noise of S2 is given by the extreme cobino, to appear in J. Opt. B, Special Issue on Quantum points θ = 0, π on the diagram and that of Sˆ3 by the hd ± optics and Quantum entanglement, quant-ph/0302142. center point θ = π/2. In general, for an arbitrary [9] E. Giacobino, Optics Comm. 56, 249 (1985). hd ± squeezed quadrature, a linear combination of Sˆ2 and [10] S. Cecchi, G. Giusfredi, E. Petriella, P. Salieri, Phys. Rev. 49 Sˆ3 is squeezed (Fig. 5a). We find that the polarization Lett. , 1928 (1982); M.W. Hamilton, W.J. Sandle, J.T. Chilwell, J.S. Satchell, D.M. Warrington, Optics Comm. squeezing strongly depends on the operating point and 48 on the noise frequency. For instance, in Fig. 5b, we see , 190 (1983). ˆ [11] V. Josse, A. Dantan, A. Bramati, M. Pinard, E. Gia- that S3 is squeezed. cobino, to be published. [12] N. Korolkova, G. Leuchs, R. Loudon, T.C. Ralph, C. Sil- To conclude, we have demonstrated that the nearly berhorn, Phys. Rev. A, 65, 052306 (2002); N. Korolkova, resonant interaction of a linearly polarized laser beam Ch. Silberhorn, O. Glöckl, S. Lorenz, Ch. Marquardt, G. with a cloud of cold atoms in an optical cavity can pro- Leuchs, Eur. Phys. J. D, 18, 229 (2002).