In-Situ Observation of the Phase of a Microwave Field using Single-Atom Nonlinear Optics

Transcription

1 1 [Submitted to Nature] In-Situ Observation of the Phase of a Microwave Field using Single-Atom Nonlinear Optics George C. Cardoso 1, Prabhakar Pradhan 1 & Selim M. Shahriar 1,2 1 Department of Electrical and Computer Engineering, Northwestern University, 2145 Sheridan Rd, Evanston, IL Research Laboratory of Electronics, Massachussets Institute of Technology, Cambridge, MA 2139 Measuring the amplitude and absolute phase of a microwave field at a specific point of space and time, has many potential applications, including precise qubit rotations 1,2,3 and wavelength quantum teleportation 4,5,6. One possible approach for such a measurement is the use of high-speed diodes; however, the mesoscopic size of such a device will itself affect the very field it is meant to detect. Alternatively, one could place spin polarized atoms at the location of interest, and detect the effect of the microwave field on these atoms. However, it is generally understood that atomic state populations are not sensitive to the absolute phase of the field under such an excitation. Several experiments 7,8,9,1 have demonstrated indirect signatures of carrier field phase-dependence for ultrashort pulses, but these do not demonstrate the ability to map the phase of the field. Here, we show how a realtime, localized phase mapping can indeed be performed using a novel type of single-atom non-linear optics, based on self-interference between the positive and negative frequency components of the microwave field. To illustrate the single-atom non-linear optics, we consider an ideal two-level system where a ground state > is coupled to a higher energy state 1>. We assume that the > 1> transition is magnetic dipolar, the magnetic field in the microwave is of the form B=B cos(ωt+φ), and the atom is initially in the ground state. In the dipole ) approximation, the Hamiltonian can be written as H = ε ( σ σ z ) / 2 + g( t) σ x, where g(t)=-g o [exp(iωt+iφ)+c.c.]/2, σ i are the Pauli matrices, and ε=ω corresponding to resonant excitation. We perform a rotating wave transformation by operating on the state vector ξ(t)> with the unitary operator:

2 2 ) Q = ( σ + σ z ) / 2 + exp( + iωt + iφ)( σ σ z ) / 2. (2) ~ so that the new state vector is ξ > Qˆ ξ( t) >. Given the periodic nature of the interaction, the state vector can be written in the form: ~ an ξ >= exp( n( i2ωt i2φ )) n b. (4) = n Without the rotating wave approximation (RWA) and to the lowest order in η (g /4ω), we can ignore terms with n >1. Considering g to have a time-dependence of the form g = g M [1 exp( t / τ sw )], where τ sw >>ω -1, g -1 M, we can use adiabatic elimination, which is valid to first order in η. We get the following expressions for the amplitudes of the two levels : C t) = Cos( g t / 2) 2η Σ Sin( g / 2) (6a) C ( t i ( ωt+ φ ) * 1 = ie [ Sin( g t / 2) + 2ηΣ Cos( g t / 2)] where we have defined Σ ( i / 2) exp[ i(2ω t + 2φ )], and t 1 t 1 g = g o ( t') dt' = g[1 ( ) (1 exp( t / τ sw ))] t (7) τ sw To lowest order in η, this solution is normalized at all times. Note that if one wants to produce this excitation in an ensemble of atoms using a π /2 pulse and measure the population of the state 1> after the excitation terminates ( at g (τ ) τ /2= π /2), the result would be a signal given by 1 C g ( τ ), ) 2 = [1+2η sin(φτ )], (8) 2 ( φ where we have defined the phase of the field at t=τ to be φ τ ωτ+φ. This signal contains information of both the amplitude and phase of the electromagnetic field (RF) B(t). The second term of Eq. 8 above is related to the Bloch-Siegert shift 11,12 and henceforth we will call it the Bloch-Siegert oscillation (BSO) signal. For η =, we have the conventional Rabi flopping that is obtained with the RWA 12. For a stronger coupling field, where the RWA cannot be made, the second term of Eq. 8 becomes important, 7,8,9 and the population will depend now both on the Rabi frequency and the phase of the coupling field. Indirect evidence of the phenomenon was observed in GaAs for optical transitions and named carrier-wave Rabi flopping 8,7,9. In this work, we demonstrate how to get an explicit map of the phase of the driving field using the BSO. This result can be appreciated best by considering, for example, an atom of 87 Rb, caught in a dipole force trap, where the states > 5 2 S 1/2 : F=1,m=1> and 1> 5 2 S 1/2 : F=2,m=2> forms the two level system. These states differ in frequencies by GHz. When illuminated by resonant, right-circularly polarized light at a frequency of Hz (78 nm), state 1> couples only to the state 2> 5 2 P 3/2 : F=3,m=3>, which in turn can decay only to state 1>. This cycling transition can thus be used to pump the system into state 1>. When a right-circularly polarized microwave field at GHz is applied, state 1> couples only to state >, even when the RWA approximation breaks down. The microwave field is turned on (6b)

3 adiabatically with a switching time-constant τ SW, starting at t=. After an interaction time of τ, chosen so that g o '(τ)τ=π, the population of state 1> can be determined by coupling this state to the state 2> with a short (faster than 1/ω and 1/g M ) laser pulse, and monitoring the resulting fluorescence 2. We have simulated this process explicitly for g o =.1ω, and τ SW =.1τ. These numbers are easily achievable experimentally. The laser pulse width, τ LP, is chosen to be 1-12 sec, in order to satisfy the constraint that τ L <<1/ω and τ L <<1/g M. In order to optimize the signal, the laser Rabi-frequency, Ω L is chosen to be such that Ω L τ L =π, so that all the populations of state 1> is excited to state 2> at the end of the pulse. After the laser pulse is turned off, the fluorescence is collected for a duration longer than the spontaneous-decay lifetime (γ -1 = 32 nsec) of state 2>. Under this condition, our simulation verifies that the detector signal is essentially proportional to the population of state 1>, as given by eqn. 8, with the proportionality constant determined by the efficiency of the detection system. Note that the speed of the detection system is limited fundamentally by the spontaneous decay rate γ, of state 2>. As such, it is impossible in this explicit scheme to monitor the phase of the microwave field on a time scale shorter than its period. If one were interested in monitoring the phase of a microwave field of a lower frequency (so that γ >>ω), it would be possible to keep track of its phase on a timescale much shorter than its period. Here, we report the experimental realization of this process in the Zeeman sublevels of the Rb ground state 5S 1/2, F=1, m=-1, and 1. An RF field couples the sublevels with m =1, as detailed in the inset of Fig. 1. Our simulations show that the dynamics of this three-level system is qualitatively the same as for a pure two-level system, i.e., the RF phase information is contained in the atomic population. More generally, the phase signature is likely to appear in the population of the atomic levels, no matter how many levels are involved, as long as the Rabi frequency is strong enough for the RWA to break down. The atoms are provided by an effusive atomic beam apparatus. Here the single atom concept is still valid, but the signal is stronger, as it is a sum of the signal of many atoms. The RF field is applied to the atoms by a coil, and the interaction time τ is set by the time of flight of the individual atoms in the RF field before they are probed by a strongly focussed, circularly polarized laser beam, as shown in Fig. 1. For a given atom, the RF field interacts for a time τ, and then is excited, at time t, by the probe beam that couples the optical sublevel 5S 1/2, F=1, m=-1 to an excited sublevel in 5P 3/2. The average atomic speed is 5 m/s, so that a typical atom takes about.6 µs to cross the probe beam, long enough for the occurrence of some optical pumping, such that even though the BSO signal amplitude is proportional to the initial population difference between the Zeeman sublevels, no additional optical pumping was necessary in this experiment. The typical atom stays within the probe beam for less than.3 Rabi period of the RF driving field (for g/2π = 4 MHz). As already discussed, within the Rabi cycle, the BSO signal is maximum for g o τ/2 = (2n+1)π/2, where n =,1,2,, so that there is at least one maximum of the BSO signal within the region of the probe. Although the accuracy in the detection of the phase is inversely proportional to the time of flight though the probe beam, the phase error induced by finite beam diameter still does not affect the BSO signal strongly. Atoms with different velocities have different interactions time τ with the RF field, and will produce a spread in the BSO signal amplitude within the probe region. But the phase of the BSO signal is the 3

4 4 same for all the atoms, since it corresponds to the phase seen by the atoms at the detection point. Thus, there is no wash-out of the BSO signal due to the velocity distribution in the atomic beam. The use of a very strong RF field assures that the Rabi frequency will be strong enough for the appearance of the nonlinear effects, even when the Zeeman sublevels are degenerate and the RF field interacts nonresonantly with them. For this reason, no bias magnetic field was necessary in the experiment. External and stray magnetic fields were shielded by an annealed µ-metal enclosure In Figure 2 we show the spectrum of the BSO signal. The RF field was tuned to.5 MHz, with a power of about 1 W, corresponding to a Rabi frequency of about 4 MHz on the transitions 5S 1/2, F=1, m =1 5P 3/2, F=1, m =. The probe power was.5 mw focused to spot of about 3 µm, giving a Rabi frequency of about 1 Γ, where is Γ the linewidth of the optical transition. In Figure 3 we show the actual phase information signal in the atomic fluorescence. For this measurement, the second harmonic of the driving field was used to trigger a 1 MHz digital oscilloscope while the fluorescence signal was averaged 256 times. When the probe beam is tuned to the F=1-F = transition, the population at m=-1 is probed. When the probe is tuned to F=1- F =1, the population of m=-1 and m= are probed. That results in an effective detection of the complement of the population of m=1. On the other hand, when the probe beam is locked to the F=1-F =2 transition all the three Zeeman sublevels of F=1 are simultaneously probed and the phase information is not present, since the total population of level F=1 is a constant. In Figure 4 we show that the fluorescence signal is phase locked to the second harmonic of the driving field. First we placed a delay line of.4 µs on the cable of the reference field used to trigger the oscilloscope and recorded the fluorescence (Fig. 4a). Then we put the.4µs delay line on the BSO signal cable, and recorded the fluorescence (Fig. 4b). The phase difference between the signals recorded in 4a and 4b is approximately.8µs, as expected for a phase locked fluorescence signal. The data was recorded for the probe resonant with the transition F=1 F =1, but the same results were observed for F=1 F =. Recently, it has been shown that an RF field can be frequency-locked to a beat note generated from frequency combs produced by ultrashort pulses 1, thus realizing an optical clock. In principle, the beat note can also be phase-locked to the RF field. The ability to measure the absolute phase of the RF field, as demonstrated here, can thus be used to monitor as well as stabilize the phase, and thereby the shape, of the ultra-short pulse, a matter of great interest in applications of ultrashort pulses. If an extension of this concept can also be used to teleport the wavelength of an oscillator, given the presence of degenerate distant entanglement, even in the presence of unknown fluctuations in the intervening medium, with potential applications to improving the spatial resolution of the global positioning system. 13,14 Acknowledgements: We wish to acknowledge support from DARPA under the QUIST program, ARO under the MURI program, and NRO. Correspondence and requests for materials should be addressed to shahriar@ece.northwestern.edu.

5 5 APD I o sin (2ωt +2φ +δ) 87 Rb F=2 F=1 F= Probe Lens F=1 RF Probe Atomic Beam RF B o sin (ωt +φ) Figure 1: Simplified experimental set-up. The RF of frequency ω is fed by a power amplifier connected to the resonant coil. The 1 mm cross section rubidium atomic beam passes through the symmetry axis of the RF coil whose the magnetic field is along the direction of travel of the atoms. A circularly polarised probe laser beam is focused down to 3 µm in diameter through a gap in the middle of the RF coil, and propagates perpendicularly to the direction of travel of the atomic beam. The atomic fluorescence is collected by the lens L and detected by an avalanche photodiode (APD). The phase signature appears in the fluorescence signal encoded in a oscillation at a frequency 2ω due to the Bloch-Siegert effect. In the picture, δ is an additional phase delay due to the APD circuits and cabling. Inset: Diagram of the relevant sublevels of the D2 line of 87 Rb. The numbers on the left represent the total angular momentum of the respective levels. The strong driving RF couples to the ground state Zeeman sublevels. The probe beam must be resonant with an appropriate optical transition for the observation of the phase-locked signal, as discussed in the main text.

6 Figure 2: Bloch-Siegert oscillation spectra. RF at.5 MHz and Rabi frequency around 4 MHz. (a) Probe beam resonant with the 5S 1/2, F=1 5S 3/2, F = transition. The signal appears at 1 MHz with a linewidth less than 1 khz (resolution limited by the spectrum analyzer). (b) Probe beam blocked. The dip structure around 1 khz is an artifact due to the amplifier gain curve. Inset: Spectrum for same configuration and RF power around 1 MHz. Notice the 2 MHz harmonic. 6

7 7 Atomic Fluorescence (arb. units) Time (µs) Figure 3: Time-dependence of the fluorescence signal at 2ω when the probe beam is resonant to different excited states. The black circles, red line and noisy blue line correspond to the probe locked to the transitions F=1 F =, F=1 F =1 and F=1 F =2, respectively of the 5S 1/2 5P 3/2 87 Rb transition.

8 8 Atomic Fluorescence (arb. units) (a) (b).79t Time (µs) Figure 4: Demonstration of phase-locked fluorescence. T is the period of the Bloch-Siegert oscillation. In (a) a.4t delay line was placed in the reference field cable. In (b) the.4t delay line was placed in the florescence signal cable. The figure shows that signal (b) is about.8t ahead of (a) showing that the atomic fluorescence carries phase information which is locked to the absolute RF field phase. The thin sinusoidal curves are fittings to the experimental data and was used for period and delay determination.

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