July 1, 2014 / Vol. 39, No. 13 / OPTICS LETTERS
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Many-atom–cavity QED system with homogeneous atom–cavity coupling Jongmin Lee, Geert Vrijsen, Igor Teper, Onur Hosten,* and Mark A. Kasevich Physics Department, Stanford University, Stanford, California 94305, USA *Corresponding author: [email protected] Received May 21, 2014; accepted May 22, 2014; posted May 27, 2014 (Doc. ID 212573); published June 30, 2014 We demonstrate a many-atom–cavity system with a high-finesse dual-wavelength standing wave cavity in which all participating rubidium atoms are nearly identically coupled to a 780-nm cavity mode. This homogeneous coupling is enforced by a one-dimensional optical lattice formed by the field of a 1560-nm cavity mode. © 2014 Optical Society of America OCIS codes: (020.0020) Atomic and molecular physics; (120.3940) Metrology; (140.4780) Optical resonators; (300.6260) Spectroscopy, diode lasers. http://dx.doi.org/10.1364/OL.39.004005
0146-9592/14/134005-04$15.00/0
the atoms [Fig. 1(b)]. In the former case the atoms are localized at the maxima of the probe mode profile, attaining a maximal coupling strength; and in the latter they are localized at the minima, showing that properly positioned atoms can be nearly invisible to light circulating inside the cavity. The work presented here builds on some of our previous results [11,14,15], but the description is self-contained. At the heart of our experimental apparatus is a compact cavity assembly schematically shown in Fig. 1(a), constructed from Zerodur glass (low helium permeability
Ion Pump
(a)
(b) Indalloy
MOT beam
Zerodur Glass
End-cap
In-phase 3D MOT
Cavity mirror
High-temp. . Epoxy
Out-of-phase
2D MOT 87Rb
dispenser
(c)
(d) 1
6S1/2
1560 nm
(87Rb D2)
0 50 Time (µs)
1560nm Lattice
5P3/2 5P1/2 780nm
0
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1366nm 1529nm
780 nm
Cavity Ring-down Measurement
There has been growing interest in collective interactions of large ensembles of atoms with cavity fields, in addition to experiments [1,2] pursuing cavity quantum electrodynamics (cQED) with individual atoms. Topics studied include cavity-aided entanglement generation (spin squeezing) for quantum-enhanced metrology [3–5]; opto-mechanics with atoms, where collective motional degrees of freedom are coupled to cavity fields [6,7]; cavity-enhanced atomic quantum memories for quantum information processing [8]; and ultra-narrow-linewidth lasers using narrow-transition ultra-cold atoms as the gain medium for metrological purposes [9,10]. Important in many such systems is the inhomogeneity of the coupling strength between the participating atoms and the cavity field, which degrades the coherence of the interactions and complicates the dynamics and the analysis of the basic physics by obscuring the relevant system parameters [11]. Limiting our attention to Fabry–Perot cavities, in which the modes are standing waves, homogeneous coupling of atoms to the relevant cavity field (the probe mode) can be achieved by tightly trapping the atoms with a spatial period that is commensurate with the wavelength of the probe. Thus far, experimentally realized trapping configurations have involved incommensurate trap-probe periods, resulting in inhomogeneous atom– probe couplings that often require the definition of effective, averaged coupling constants [7]. Two recent efforts came to our attention that investigate commensurate dual-wavelength cavity designs. One utilizes a traveling-wave cavity to trap and probe atoms in which there is no particular atom registration [12]; and the other utilizes a standing wave cavity, for the different purpose of investigating atomic self-organization [13]. In this Letter, we present the realization and some characteristics of a many-atom cQED system employing probe and trapping modes that are commensurate. We use a dual-wavelength cavity with high finesse at both 780 nm, used to probe the D2 transition in 87 Rb, and 1560 nm, used to trap the atoms in a far-detuned onedimensional lattice. At the central region of the cavity, depending on the exact wavelength relationship, these commensurate probe and trapping wavelengths allow an in-phase as well as an out-of-phase registration of
780nm Probe 1560nm Lattice
100
5S1/2
Fig. 1. (a) Schematic of the cavity assembly. (b) In-phase and out-of-phase atom registrations at the cavity center. Black curves, lattice potential resulting from the 1560 nm light; red curves, 780-nm probe light intensity. (c) Ring-down measurements of the 780 and 1560 nm cavity modes after an abrupt turn-off of the incident light using an AOM. The measured ring-down times are τ780 19.9 μs and τ1560 13.3 μs. (d) Energy level diagram of 87 Rb atoms. © 2014 Optical Society of America
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and a small thermal expansion coefficient [≃10−8 ∕K]). This assembly includes a current-controlled rubidium dispenser (Alvatec) and a two-dimensional magneto-optical trap (MOT), which prepares a collimated beam of atoms that passes through a 1.5 mm aperture into the main chamber to load a three-dimensional (3D) MOT. Extension tubes attached to the main chamber hold the low-loss cavity mirrors. The Zerodur glass contacts, except the mirror end-caps, are held together with a sodium silicate molecular bonding agent, and the glass-metal contacts for the ion pump and the dispenser are sealed with Indalloy. The mirror end-caps, to which the mirror substrates are glued, are attached to the extension tubes with a thin layer of high-temperature epoxy (Epotek 353ND). The epoxy contacts of the chamber can endure a baking temperature of ∼120°C, permitting us to reach vacuum pressures lower than ∼10−10 mbar, maintained by a 5-L/s ion pump. The 9.9 cm radius-of-curvature mirrors, coated for high reflectivity at both 780 and 1560 nm (by Research Electro-Optics), form a 10.73 cm long near-confocal optical cavity with modes that overlap with the 3D MOT at its center. The cavity length can be tuned via its temperature. The Rayleigh range is 4.93 cm for both wavelengths. The Gouy phase shifts on the two modes cause the registration to change as one moves away from the cavity center. (At one Rayleigh range atoms localize at the slopes of the probe mode.) However, the atomic cloud size of only 0.5 mm justifies taking every lattice site to be identical. Key system parameters are given in Table 1. The trap and the probe light are generated from the same 1560 nm (New Focus Vortex) external-cavity diode laser (Fig. 2). This scheme eliminates the need to lock the 780 nm light independently to the cavity which would inevitably require a finite intra-cavity power introducing unnecessary ac-Stark shifts and decoherence on the atoms. The 1560 nm laser is first actively frequency stabilized (∼5 MHz lock bandwidth) to a reference cavity via the Pound–Drever–Hall (PDH) method [16]. A 1560 nm diode laser is injection-locked to the light transmitted by the reference cavity to remove amplitude fluctuations, and its output is amplified to ∼250 mW by an erbium-doped fiber amplifier (EDFA) and split into two paths. The first path gets frequency-doubled to 780 nm in a periodically poled lithium niobate (PPLN) waveguide followed by injection into a 780 nm diode Table 1. Parameters: Finesse, F ; Mode Waist, ω0 ; Spot Size at Mirrors, ωm ; Free Spectral Range, νfsr ; Maximum Atom–Cavity Coupling Strength, g0 ; Cavity FWHM Linewidth, κ; Atomic FWHM Linewidth, Γ; and Single Atom Cooperativity, 4g20 ∕κΓ Parameters F ω0 ωm νfsr g0 κ Γ 4g20 ∕κΓ
780-nm Cavity
1560-nm Cavity
175; 000 111 μm 164 μm
117,000 157 μm 231 μm
1.3964 GHz 2π × 142 kHz — 2π × 7.98 kHz 2π × 11.96 kHz 2π × 6.06 MHz 1.68 —
Fig. 2. Laser stabilization and generation of spectral lines. (a) The 1560 nm master laser is locked to a reference cavity, and the reference cavity is locked to the main cavity. A 780 nm light generated by frequency doubling is used to probe the main cavity. EOM, electro-optic modulator; AOM, acoustooptic modulator; PPLN, periodically poled Lithium Niobate; DBS, dichroic beam splitter; λ∕4, quarter-wave plate; “A,” amplifier; “C,” controllable attenuator; FR, Faraday rotator; OI, optical isolator; and EDFA, erbium doped fiber amplifier. (b) Schematic of the generated optical frequencies with respect to the cavity modes (dotted lines). νt is the frequency of the optical lattice; νp , frequency of the probe; m, an integer multiplier; and δm , modulation frequency of the 1560-nm carrier.
laser. The second path is phase-modulated with an electro-optic modulator (EOM) at δm ∼ 40 MHz, and one of the sidebands is locked to the in-vacuum cavity via the PDH method (∼100 kHz lock bandwidth), using an acousto-optic modulator (AOM) and the piezo-electric transducer of the reference-cavity as the feedback elements. This sideband serves as the trap light whose power can easily be manipulated by changing the modulation depth. Once locked, the 1560 nm master laser follows a resonance of the reference cavity, which in turn follows a resonance of the in-vacuum cavity as its length drifts. The frequency-doubled 780 nm probe light is then also automatically stabilized to the cavity. A sideband of the 780 nm slave laser (generated by an EOM at ∼6 GHz) serves as the probe light that couples near-resonantly to the trapped atoms, facilitating the measurement of the cavity resonance frequency in presence of intra-cavity atoms; both of these modes are TEM00 . The cavity frequency measurement is accomplished via a heterodyne beatnote measurement between the 780 nm carrier and sidebands as the probe sideband is frequency swept over the cavity resonance (similar to the PDH method). Not shown in figures is a microwave horn to drive the ∼6.835-GHz 87 Rb hyperfine transitions. In a typical experiment, atoms trapped in the 3D MOT are further cooled to ∼15 μK by switching to a far-detuned MOT (6Γ red-detuned, ∼25 ms) then by polarization-gradient cooling (PGC) (27Γ red-detuned,
July 1, 2014 / Vol. 39, No. 13 / OPTICS LETTERS (a) Atom Registration ξ (a.u.)
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∼7 ms) in the presence of a weak 1560-nm optical lattice. Near the end of PGC we adiabatically increase the lattice power 100 times and load 104 –105 atoms, all in jF 1i, distributed over 1000 lattice sites, with a 1∕e lifetime of 1.1 s, mainly limited by background gas collisions. The PGC in the full lattice is hindered by the presence of strong red Stark-shift gradients on the 5P excited levels (more than an order of magnitude larger than those experienced by the 5S ground states) because of the 1560 nm induced 5P − 4D coupling [Fig. 1(d)]. With the onset of the full lattice (∼40 W circulating power), we observe a 100 kHz level differential drift between the 780 nm and the 1560 nm mode frequencies because of the heating of the mirror coatings. Nevertheless, equilibrium is attained in hundred millisecond timescales and can be accounted for in measurements. Even without heating effects, the measured free spectral range (FSR) at 1560 nm is 7 kHz smaller than that at 780 nm, which we disregard for simplicity. We note that when probe and trap mode frequencies are not exactly commensurate, one still expects good registrations at the cavity center provided the mirrors are identical, with the caveat that registration changes away from the cavity center. The linearly polarized probe interacts dispersively with the atoms by coupling the jF 2i ground states to the 5P 3∕2 excited states with an effective frequency detuning Δ (averaged over excited hyperfine states that are Starkshifted by the trapping light), which we typically set at about −1 GHz. The resulting refractive index of the atoms shifts the probe mode frequency. The effects of absorption are minor. Note that jF 1i atoms also give rise to probe shifts, but at a smaller order of magnitude because of the larger detuning, which we will omit for simplicity. The amount of shift produced by an atom at position ri can be expressed in terms of the positiondependent atom–cavity coupling constant gi ≡ gri , as δνi P g2i ∕Δ. The shift produced by all the atoms is then δν i δνi . We utilize this cavity shift to show that we can trap the atoms entirely either at the anti-nodes or at the nodes of the probe mode, corresponding to the in-phase and out-of-phase registrations, respectively. In particular we probe the atoms that are in jF 2; mf 0i which are transferred from j1; 0i by a microwave pulse. Fig. 3(a)P shows theP inferred atomic registration parameter ξ i g2i trap ∕ i g2i mot for three different probe mode frequencies separated by one FSR. This parameter compares the coupling strength of an unlocalized P 2ensemble to that of a registered ensemble. Here i gi mot P δνmot Δmot , i g2i trap δνΔ, where δνmot and δν are the observed cavity shifts before and after the atoms are loaded into the optical lattice, respectively; Δmot and Δ are the effective detunings for the corresponding cases. For every change in the frequency of the probe in steps of an FSR, the wavelength of the probe changes by half a wavelength resulting in the overlap with the lattice near the cavity center to alternate between in-phase and outof-phase registrations. The finite temperature of the atomic cloud leads to an imperfect registration, as hotter atoms explore larger volumes in the trapping sites. This especially leads to a finite coupling for the out-of-phase registration. The measured
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Fig. 3. (a) Registration parameter versus probe detuning from the F 2 to F 0 3 transition [m 3; 2; 1 cf. Fig. 2(b)]. Registration ratio: 6.2 0.2. (b) Differential ac Stark shift of the clock states as a function of the estimated trap depth: −0.74 Hz∕μK, measured by the microwave transition frequency. (c) Rabi oscillations with a period of 10.73 and 57 ms coherence time measured via the cavity shift.
ratio of 6.2 0.2 of the in-phase to out-of-phase atom registration parameters provides a direct measure of the temperature of the trapped atoms at the employed lattice depth. Assuming a thermal distribution in the lattice sites, and utilizing the measured differential ac Stark shift of the hyperfine clock sates [Fig. 3(b)] to infer the lattice depth, we arrive at ∼70 μK for the temperature of the atoms in a lattice depth of 870 μK corresponding to a 598 Hz transverse and 265 kHz axial trap frequencies. For this configuration, each atomic “pancake” in the lattice has a 22 μm transverse and a 50 nm axial rms width, to be compared with the probe waist of 111 μm and the probe standing wave period of 390 nm. Because of this finite size, the mean coupling strength hg2 i for the inphase registration is 74% of the value that would be obtained for perfectly localized atoms. The rms spread in coupling is 24% over the ensemble. However, for atomprobe interaction times much larger than the axial trap period of 3.8 μs, the axial inhomogeneity mostly averages
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out as the atoms oscillate back and forth. Then, the relevant spread (e.g., for ac Stark shift inhomogeneities) is predominantly because of the transverse distribution, and becomes 14%. In the case of an out-of-phase registration, similar calculations show a mean coupling strength of 12% of the maximal. The dispersive cavity measurements described here can be used as a sensitive spectroscopic tool. For example, Fig. 3(c) shows Rabi oscillations resulting from a particular microwave drive resonantly coupling the clock transition. The measured quantity is the population in the upper clock state after a given duration of microwaves, observed by cavity shifts, showing a coherence time of 57 ms. With similar methods, Ramsey oscillation sequences with echo, implemented inside the lattice (π∕2-pulse, T/2-delay, π-pulse, T/2-delay, π∕2-pulse), show 1∕e coherence times up to T 205 ms. In summary, we demonstrated homogeneous coupling of atoms to a standing wave cavity mode and the potential use of the dispersive measurements as a spectroscopic tool. Although in the described version of the apparatus, the cavity read-out noise is at the atomic shot noise for 104 atoms, with further improvements we expect the possibility of generating spin-squeezed states potentially up to 20 dB below shot noise variance. Because of the homogeneous atom–cavity coupling, the generated squeezed states could be released into free space to be used as input states to atom interferometric sensors for enhanced sensitivity. In addition, the described apparatus might enable atom counting in mesoscopic ensembles with an actual quantized signal, which so far has been challenging [17]. This work was funded by DARPA and the MURI on Quantum Metrology sponsored by the Office of Naval
Research. We are thankful to the Fejer group for providing the PPLN. References 1. H. Mabuchi and A. C. Doherty, Science 298, 1372 (2002). 2. H. J. Kimble, Phys. Scr. T76, 127 (1998). 3. M. Schleier-Smith, I. D. Leroux, and V. Vuletić, Phys. Rev. Lett. 104, 073604 (2010). 4. I. D. Leroux, M. Schleier-Smith, and V. Vuletić, Phys. Rev. Lett. 104, 073602 (2010). 5. Z. Chen, J. Bohnet, S. R. Sankar, J. Dai, and J. K. Thompson, Phys. Rev. Lett. 106, 133601 (2011). 6. F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, Science 322, 235 (2008). 7. T. Purdy, D. Brooks, T. Botter, N. Brahms, Z. Ma, and D. Stamper-Kurn, Phys. Rev. Lett. 105, 133602 (2010). 8. H. Tanji, S. Ghosh, J. Simon, B. Bloom, and V. Vuletić, Phys. Rev. Lett. 103, 043601 (2009). 9. D. Meiser, J. Ye, D. R. Carlson, and M. Holland, Phys. Rev. Lett. 102, 163601 (2009). 10. J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, Nature 484, 78 (2012). 11. I. Teper, G. Vrijsen, J. Lee, and M. A. Kasevich, Phys. Rev. A 78, 051803 (2008). 12. S. Bernon, T. Vanderbruggen, R. Kohlhaas, A. Bertoldi, A. Landragin, and P. Bouyer, New J. Phys. 13, 065021 (2011). 13. K. J. Arnold, M. P. Baden, and M. D. Barrett, Phys. Rev. Lett. 109, 153002 (2012). 14. A. K. Tuchman, R. Long, G. Vrijsen, J. Boudet, J. Lee, and M. A. Kasevich, Phys. Rev. A 74, 053821 (2006). 15. R. Long, A. K. Tuchman, and M. A. Kasevich, Opt. Lett. 32, 2502 (2007). 16. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983). 17. H. Zhang, R. McConnell, S. Cuk, Q. Lin, M. Schleier-Smith, I. D. Leroux, and V. Vuletic, Phys. Rev. Lett. 109, 133603 (2012).
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Many-atom–cavity QED system with homogeneous atom–cavity coupling Jongmin Lee, Geert Vrijsen, Igor Teper, Onur Hosten,* and Mark A. Kasevich Physics Department, Stanford University, Stanford, California 94305, USA *Corresponding author: [email protected] Received May 21, 2014; accepted May 22, 2014; posted May 27, 2014 (Doc. ID 212573); published June 30, 2014 We demonstrate a many-atom–cavity system with a high-finesse dual-wavelength standing wave cavity in which all participating rubidium atoms are nearly identically coupled to a 780-nm cavity mode. This homogeneous coupling is enforced by a one-dimensional optical lattice formed by the field of a 1560-nm cavity mode. © 2014 Optical Society of America OCIS codes: (020.0020) Atomic and molecular physics; (120.3940) Metrology; (140.4780) Optical resonators; (300.6260) Spectroscopy, diode lasers. http://dx.doi.org/10.1364/OL.39.004005
0146-9592/14/134005-04$15.00/0
the atoms [Fig. 1(b)]. In the former case the atoms are localized at the maxima of the probe mode profile, attaining a maximal coupling strength; and in the latter they are localized at the minima, showing that properly positioned atoms can be nearly invisible to light circulating inside the cavity. The work presented here builds on some of our previous results [11,14,15], but the description is self-contained. At the heart of our experimental apparatus is a compact cavity assembly schematically shown in Fig. 1(a), constructed from Zerodur glass (low helium permeability
Ion Pump
(a)
(b) Indalloy
MOT beam
Zerodur Glass
End-cap
In-phase 3D MOT
Cavity mirror
High-temp. . Epoxy
Out-of-phase
2D MOT 87Rb
dispenser
(c)
(d) 1
6S1/2
1560 nm
(87Rb D2)
0 50 Time (µs)
1560nm Lattice
5P3/2 5P1/2 780nm
0
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1366nm 1529nm
780 nm
Cavity Ring-down Measurement
There has been growing interest in collective interactions of large ensembles of atoms with cavity fields, in addition to experiments [1,2] pursuing cavity quantum electrodynamics (cQED) with individual atoms. Topics studied include cavity-aided entanglement generation (spin squeezing) for quantum-enhanced metrology [3–5]; opto-mechanics with atoms, where collective motional degrees of freedom are coupled to cavity fields [6,7]; cavity-enhanced atomic quantum memories for quantum information processing [8]; and ultra-narrow-linewidth lasers using narrow-transition ultra-cold atoms as the gain medium for metrological purposes [9,10]. Important in many such systems is the inhomogeneity of the coupling strength between the participating atoms and the cavity field, which degrades the coherence of the interactions and complicates the dynamics and the analysis of the basic physics by obscuring the relevant system parameters [11]. Limiting our attention to Fabry–Perot cavities, in which the modes are standing waves, homogeneous coupling of atoms to the relevant cavity field (the probe mode) can be achieved by tightly trapping the atoms with a spatial period that is commensurate with the wavelength of the probe. Thus far, experimentally realized trapping configurations have involved incommensurate trap-probe periods, resulting in inhomogeneous atom– probe couplings that often require the definition of effective, averaged coupling constants [7]. Two recent efforts came to our attention that investigate commensurate dual-wavelength cavity designs. One utilizes a traveling-wave cavity to trap and probe atoms in which there is no particular atom registration [12]; and the other utilizes a standing wave cavity, for the different purpose of investigating atomic self-organization [13]. In this Letter, we present the realization and some characteristics of a many-atom cQED system employing probe and trapping modes that are commensurate. We use a dual-wavelength cavity with high finesse at both 780 nm, used to probe the D2 transition in 87 Rb, and 1560 nm, used to trap the atoms in a far-detuned onedimensional lattice. At the central region of the cavity, depending on the exact wavelength relationship, these commensurate probe and trapping wavelengths allow an in-phase as well as an out-of-phase registration of
780nm Probe 1560nm Lattice
100
5S1/2
Fig. 1. (a) Schematic of the cavity assembly. (b) In-phase and out-of-phase atom registrations at the cavity center. Black curves, lattice potential resulting from the 1560 nm light; red curves, 780-nm probe light intensity. (c) Ring-down measurements of the 780 and 1560 nm cavity modes after an abrupt turn-off of the incident light using an AOM. The measured ring-down times are τ780 19.9 μs and τ1560 13.3 μs. (d) Energy level diagram of 87 Rb atoms. © 2014 Optical Society of America
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and a small thermal expansion coefficient [≃10−8 ∕K]). This assembly includes a current-controlled rubidium dispenser (Alvatec) and a two-dimensional magneto-optical trap (MOT), which prepares a collimated beam of atoms that passes through a 1.5 mm aperture into the main chamber to load a three-dimensional (3D) MOT. Extension tubes attached to the main chamber hold the low-loss cavity mirrors. The Zerodur glass contacts, except the mirror end-caps, are held together with a sodium silicate molecular bonding agent, and the glass-metal contacts for the ion pump and the dispenser are sealed with Indalloy. The mirror end-caps, to which the mirror substrates are glued, are attached to the extension tubes with a thin layer of high-temperature epoxy (Epotek 353ND). The epoxy contacts of the chamber can endure a baking temperature of ∼120°C, permitting us to reach vacuum pressures lower than ∼10−10 mbar, maintained by a 5-L/s ion pump. The 9.9 cm radius-of-curvature mirrors, coated for high reflectivity at both 780 and 1560 nm (by Research Electro-Optics), form a 10.73 cm long near-confocal optical cavity with modes that overlap with the 3D MOT at its center. The cavity length can be tuned via its temperature. The Rayleigh range is 4.93 cm for both wavelengths. The Gouy phase shifts on the two modes cause the registration to change as one moves away from the cavity center. (At one Rayleigh range atoms localize at the slopes of the probe mode.) However, the atomic cloud size of only 0.5 mm justifies taking every lattice site to be identical. Key system parameters are given in Table 1. The trap and the probe light are generated from the same 1560 nm (New Focus Vortex) external-cavity diode laser (Fig. 2). This scheme eliminates the need to lock the 780 nm light independently to the cavity which would inevitably require a finite intra-cavity power introducing unnecessary ac-Stark shifts and decoherence on the atoms. The 1560 nm laser is first actively frequency stabilized (∼5 MHz lock bandwidth) to a reference cavity via the Pound–Drever–Hall (PDH) method [16]. A 1560 nm diode laser is injection-locked to the light transmitted by the reference cavity to remove amplitude fluctuations, and its output is amplified to ∼250 mW by an erbium-doped fiber amplifier (EDFA) and split into two paths. The first path gets frequency-doubled to 780 nm in a periodically poled lithium niobate (PPLN) waveguide followed by injection into a 780 nm diode Table 1. Parameters: Finesse, F ; Mode Waist, ω0 ; Spot Size at Mirrors, ωm ; Free Spectral Range, νfsr ; Maximum Atom–Cavity Coupling Strength, g0 ; Cavity FWHM Linewidth, κ; Atomic FWHM Linewidth, Γ; and Single Atom Cooperativity, 4g20 ∕κΓ Parameters F ω0 ωm νfsr g0 κ Γ 4g20 ∕κΓ
780-nm Cavity
1560-nm Cavity
175; 000 111 μm 164 μm
117,000 157 μm 231 μm
1.3964 GHz 2π × 142 kHz — 2π × 7.98 kHz 2π × 11.96 kHz 2π × 6.06 MHz 1.68 —
Fig. 2. Laser stabilization and generation of spectral lines. (a) The 1560 nm master laser is locked to a reference cavity, and the reference cavity is locked to the main cavity. A 780 nm light generated by frequency doubling is used to probe the main cavity. EOM, electro-optic modulator; AOM, acoustooptic modulator; PPLN, periodically poled Lithium Niobate; DBS, dichroic beam splitter; λ∕4, quarter-wave plate; “A,” amplifier; “C,” controllable attenuator; FR, Faraday rotator; OI, optical isolator; and EDFA, erbium doped fiber amplifier. (b) Schematic of the generated optical frequencies with respect to the cavity modes (dotted lines). νt is the frequency of the optical lattice; νp , frequency of the probe; m, an integer multiplier; and δm , modulation frequency of the 1560-nm carrier.
laser. The second path is phase-modulated with an electro-optic modulator (EOM) at δm ∼ 40 MHz, and one of the sidebands is locked to the in-vacuum cavity via the PDH method (∼100 kHz lock bandwidth), using an acousto-optic modulator (AOM) and the piezo-electric transducer of the reference-cavity as the feedback elements. This sideband serves as the trap light whose power can easily be manipulated by changing the modulation depth. Once locked, the 1560 nm master laser follows a resonance of the reference cavity, which in turn follows a resonance of the in-vacuum cavity as its length drifts. The frequency-doubled 780 nm probe light is then also automatically stabilized to the cavity. A sideband of the 780 nm slave laser (generated by an EOM at ∼6 GHz) serves as the probe light that couples near-resonantly to the trapped atoms, facilitating the measurement of the cavity resonance frequency in presence of intra-cavity atoms; both of these modes are TEM00 . The cavity frequency measurement is accomplished via a heterodyne beatnote measurement between the 780 nm carrier and sidebands as the probe sideband is frequency swept over the cavity resonance (similar to the PDH method). Not shown in figures is a microwave horn to drive the ∼6.835-GHz 87 Rb hyperfine transitions. In a typical experiment, atoms trapped in the 3D MOT are further cooled to ∼15 μK by switching to a far-detuned MOT (6Γ red-detuned, ∼25 ms) then by polarization-gradient cooling (PGC) (27Γ red-detuned,
July 1, 2014 / Vol. 39, No. 13 / OPTICS LETTERS (a) Atom Registration ξ (a.u.)
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(c) 1 Atomic Population
∼7 ms) in the presence of a weak 1560-nm optical lattice. Near the end of PGC we adiabatically increase the lattice power 100 times and load 104 –105 atoms, all in jF 1i, distributed over 1000 lattice sites, with a 1∕e lifetime of 1.1 s, mainly limited by background gas collisions. The PGC in the full lattice is hindered by the presence of strong red Stark-shift gradients on the 5P excited levels (more than an order of magnitude larger than those experienced by the 5S ground states) because of the 1560 nm induced 5P − 4D coupling [Fig. 1(d)]. With the onset of the full lattice (∼40 W circulating power), we observe a 100 kHz level differential drift between the 780 nm and the 1560 nm mode frequencies because of the heating of the mirror coatings. Nevertheless, equilibrium is attained in hundred millisecond timescales and can be accounted for in measurements. Even without heating effects, the measured free spectral range (FSR) at 1560 nm is 7 kHz smaller than that at 780 nm, which we disregard for simplicity. We note that when probe and trap mode frequencies are not exactly commensurate, one still expects good registrations at the cavity center provided the mirrors are identical, with the caveat that registration changes away from the cavity center. The linearly polarized probe interacts dispersively with the atoms by coupling the jF 2i ground states to the 5P 3∕2 excited states with an effective frequency detuning Δ (averaged over excited hyperfine states that are Starkshifted by the trapping light), which we typically set at about −1 GHz. The resulting refractive index of the atoms shifts the probe mode frequency. The effects of absorption are minor. Note that jF 1i atoms also give rise to probe shifts, but at a smaller order of magnitude because of the larger detuning, which we will omit for simplicity. The amount of shift produced by an atom at position ri can be expressed in terms of the positiondependent atom–cavity coupling constant gi ≡ gri , as δνi P g2i ∕Δ. The shift produced by all the atoms is then δν i δνi . We utilize this cavity shift to show that we can trap the atoms entirely either at the anti-nodes or at the nodes of the probe mode, corresponding to the in-phase and out-of-phase registrations, respectively. In particular we probe the atoms that are in jF 2; mf 0i which are transferred from j1; 0i by a microwave pulse. Fig. 3(a)P shows theP inferred atomic registration parameter ξ i g2i trap ∕ i g2i mot for three different probe mode frequencies separated by one FSR. This parameter compares the coupling strength of an unlocalized P 2ensemble to that of a registered ensemble. Here i gi mot P δνmot Δmot , i g2i trap δνΔ, where δνmot and δν are the observed cavity shifts before and after the atoms are loaded into the optical lattice, respectively; Δmot and Δ are the effective detunings for the corresponding cases. For every change in the frequency of the probe in steps of an FSR, the wavelength of the probe changes by half a wavelength resulting in the overlap with the lattice near the cavity center to alternate between in-phase and outof-phase registrations. The finite temperature of the atomic cloud leads to an imperfect registration, as hotter atoms explore larger volumes in the trapping sites. This especially leads to a finite coupling for the out-of-phase registration. The measured
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Fig. 3. (a) Registration parameter versus probe detuning from the F 2 to F 0 3 transition [m 3; 2; 1 cf. Fig. 2(b)]. Registration ratio: 6.2 0.2. (b) Differential ac Stark shift of the clock states as a function of the estimated trap depth: −0.74 Hz∕μK, measured by the microwave transition frequency. (c) Rabi oscillations with a period of 10.73 and 57 ms coherence time measured via the cavity shift.
ratio of 6.2 0.2 of the in-phase to out-of-phase atom registration parameters provides a direct measure of the temperature of the trapped atoms at the employed lattice depth. Assuming a thermal distribution in the lattice sites, and utilizing the measured differential ac Stark shift of the hyperfine clock sates [Fig. 3(b)] to infer the lattice depth, we arrive at ∼70 μK for the temperature of the atoms in a lattice depth of 870 μK corresponding to a 598 Hz transverse and 265 kHz axial trap frequencies. For this configuration, each atomic “pancake” in the lattice has a 22 μm transverse and a 50 nm axial rms width, to be compared with the probe waist of 111 μm and the probe standing wave period of 390 nm. Because of this finite size, the mean coupling strength hg2 i for the inphase registration is 74% of the value that would be obtained for perfectly localized atoms. The rms spread in coupling is 24% over the ensemble. However, for atomprobe interaction times much larger than the axial trap period of 3.8 μs, the axial inhomogeneity mostly averages
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OPTICS LETTERS / Vol. 39, No. 13 / July 1, 2014
out as the atoms oscillate back and forth. Then, the relevant spread (e.g., for ac Stark shift inhomogeneities) is predominantly because of the transverse distribution, and becomes 14%. In the case of an out-of-phase registration, similar calculations show a mean coupling strength of 12% of the maximal. The dispersive cavity measurements described here can be used as a sensitive spectroscopic tool. For example, Fig. 3(c) shows Rabi oscillations resulting from a particular microwave drive resonantly coupling the clock transition. The measured quantity is the population in the upper clock state after a given duration of microwaves, observed by cavity shifts, showing a coherence time of 57 ms. With similar methods, Ramsey oscillation sequences with echo, implemented inside the lattice (π∕2-pulse, T/2-delay, π-pulse, T/2-delay, π∕2-pulse), show 1∕e coherence times up to T 205 ms. In summary, we demonstrated homogeneous coupling of atoms to a standing wave cavity mode and the potential use of the dispersive measurements as a spectroscopic tool. Although in the described version of the apparatus, the cavity read-out noise is at the atomic shot noise for 104 atoms, with further improvements we expect the possibility of generating spin-squeezed states potentially up to 20 dB below shot noise variance. Because of the homogeneous atom–cavity coupling, the generated squeezed states could be released into free space to be used as input states to atom interferometric sensors for enhanced sensitivity. In addition, the described apparatus might enable atom counting in mesoscopic ensembles with an actual quantized signal, which so far has been challenging [17]. This work was funded by DARPA and the MURI on Quantum Metrology sponsored by the Office of Naval
Research. We are thankful to the Fejer group for providing the PPLN. References 1. H. Mabuchi and A. C. Doherty, Science 298, 1372 (2002). 2. H. J. Kimble, Phys. Scr. T76, 127 (1998). 3. M. Schleier-Smith, I. D. Leroux, and V. Vuletić, Phys. Rev. Lett. 104, 073604 (2010). 4. I. D. Leroux, M. Schleier-Smith, and V. Vuletić, Phys. Rev. Lett. 104, 073602 (2010). 5. Z. Chen, J. Bohnet, S. R. Sankar, J. Dai, and J. K. Thompson, Phys. Rev. Lett. 106, 133601 (2011). 6. F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, Science 322, 235 (2008). 7. T. Purdy, D. Brooks, T. Botter, N. Brahms, Z. Ma, and D. Stamper-Kurn, Phys. Rev. Lett. 105, 133602 (2010). 8. H. Tanji, S. Ghosh, J. Simon, B. Bloom, and V. Vuletić, Phys. Rev. Lett. 103, 043601 (2009). 9. D. Meiser, J. Ye, D. R. Carlson, and M. Holland, Phys. Rev. Lett. 102, 163601 (2009). 10. J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, Nature 484, 78 (2012). 11. I. Teper, G. Vrijsen, J. Lee, and M. A. Kasevich, Phys. Rev. A 78, 051803 (2008). 12. S. Bernon, T. Vanderbruggen, R. Kohlhaas, A. Bertoldi, A. Landragin, and P. Bouyer, New J. Phys. 13, 065021 (2011). 13. K. J. Arnold, M. P. Baden, and M. D. Barrett, Phys. Rev. Lett. 109, 153002 (2012). 14. A. K. Tuchman, R. Long, G. Vrijsen, J. Boudet, J. Lee, and M. A. Kasevich, Phys. Rev. A 74, 053821 (2006). 15. R. Long, A. K. Tuchman, and M. A. Kasevich, Opt. Lett. 32, 2502 (2007). 16. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983). 17. H. Zhang, R. McConnell, S. Cuk, Q. Lin, M. Schleier-Smith, I. D. Leroux, and V. Vuletic, Phys. Rev. Lett. 109, 133603 (2012).
