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[Submitted to Nature]
In-Situ Observation of the Phase of a Microwave Field
using Single-Atom Nonlinear Optics
George C. Cardoso1, Prabhakar Pradhan1 & Selim M. Shahriar1,2
1
Department of Electrical and Computer Engineering, Northwestern University, 2145
Sheridan Rd, Evanston, IL 60208
2
Research Laboratory of Electronics, Massachussets Institute of Technology,
Cambridge, MA 02139
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
rotations1,2,3 and wavelength quantum teleportation4,5,6. 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 experiments7,8,9,10 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 |0> is coupled to a higher energy state |1>. We assume that
the |0> Æ|1> transition is magnetic dipolar, the magnetic field in the microwave is of
the form B=B0cos(ωt+φ), and the atom is initially in the ground state. In the dipole
)
approximation, the Hamiltonian can be written as H = ε (σ 0 − σ z ) / 2 + g (t )σ x , where
g(t)=-go[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
)
Q = (σ 0 + σ z ) / 2 + exp(+iωt + iφ )(σ 0 − σ z ) / 2.
(2)
~
so that the new state vector is | ξ (t ) >≡ Qˆ | ξ (t ) > . Given the periodic nature of the
interaction, the state vector can be written in the form:
∞
 an 
~
(4)
| ξ (t ) >= ∑  exp(n(−i 2ωt − i 2φ )) .
n = −∞  bn 
Without the rotating wave approximation (RWA) and to the lowest order in η≡(g0 /4ω),
we can ignore terms with |n|>1. Considering g0 to have a time-dependence of the form
g 0 (t ) = g 0 M [1 − exp(−t / τ sw )] , where τ sw >>ω-1, g-10M , 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 :
(6a)
C 0 (t ) = Cos ( g 0′ (t )t / 2) − 2ηΣ ⋅ Sin( g 0′ (t )t / 2)
C1 (t ) = ie − i (ω t +φ ) [ Sin( g 0′ (t )t / 2) + 2ηΣ * ⋅ Cos( g 0′ (t )t / 2)]
(6b)
where we have defined Σ ≡ (i / 2) exp[−i (2ωt + 2φ )] , and
g 0′ (t ) =
t
1
t
g o (t ' )dt ' = g 0 [1 − ( ) −1 (1 − exp(−t / τ sw ))]
∫
t0
τ sw
(7)
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 0′ (τ )τ /2= π /2), the
result would be a signal given by
1
| C0 ( g 0′ (τ ),φ ) | 2 = [1+2η sin(φτ)],
(8)
2
where we have defined the phase of the field at t=τ to be φτ≡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 shift11,12 and
henceforth we will call it the Bloch-Siegert oscillation (BSO) signal. For η = 0, we
have the conventional Rabi flopping that is obtained with the RWA12. 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 flopping8,7,9. 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 87Rb,
caught in a dipole force trap, where the states |0>≡52S1/2:|F=1,m=1> and |1>
≡52S1/2:|F=2,m=2> forms the two level system. These states differ in frequencies by
6.68347 GHz. When illuminated by resonant, right-circularly polarized light at a
frequency of 3.844.1014 Hz (780 nm), state |1> couples only to the state |2>≡
52P3/2:|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 6.68347 GHz is applied, state |1> couples only to state |0>, even
when the RWA approximation breaks down. The microwave field is turned on
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adiabatically with a switching time-constant τSW, starting at t=0. After an interaction
time of τ , chosen so that go'(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/g0M) laser pulse,
and monitoring the resulting fluorescence2. We have simulated this process explicitly
for go=0.1ω, and τSW=0.1τ. These numbers are easily achievable experimentally. The
laser pulse width, τLP, is chosen to be 10-12 sec, in order to satisfy the constraint that
τL<<1/ω and τL<<1/g0M. In order to optimize the signal, the laser Rabi-frequency, ΩL is
chosen to be such that ΩLτL=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 5S1/2, F=1, m=-1, 0 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 5S1/2, F=1, m=-1 to an
excited sublevel in 5P3/2. The average atomic speed is 500 m/s, so that a typical atom
takes about 0.06 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 0.3 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
goτ/2 = (2n+1)π/2, where n =0,1,2,…, 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
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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
0.5 MHz, with a power of about 10 W, corresponding to a Rabi frequency of about 4
MHz on the transitions 5S1/2, F=1, |m|=1 Æ 5P3/2, F=1, |m|=0. The probe power was 0.5
mW focused to spot of about 30 µm, giving a Rabi frequency of about 1000 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 100 MHz digital oscilloscope while
the fluorescence signal was averaged 256 times. When the probe beam is tuned to the
F=1-F’=0 transition, the population at m=-1 is probed. When the probe is tuned to F=1F’=1, the population of m=-1 and m=0 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 0.4 µs on the cable of the
reference field used to trigger the oscilloscope and recorded the fluorescence (Fig. 4a).
Then we put the 0.4µs 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 0.8µs, 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’=0.
Recently, it has been shown that an RF field can be frequency-locked to a beat
note generated from frequency combs produced by ultrashort pulses10, 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
[email protected].
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Io sin (2ωt +2φ +δ)
APD
87
F=2
F=1
Rb
F=0
Lens
Probe
Probe
F=1
RF
Atomic Beam
RF
Bo 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 30 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 87Rb. 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.
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Figure 2: Bloch-Siegert oscillation spectra. RF at 0.5 MHz and Rabi frequency
around 4 MHz. (a) Probe beam resonant with the 5S1/2, F=1 – 5S3/2, F’=0
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 100 kHz is an artifact due to the amplifier gain curve. Inset:
Spectrum for same configuration and RF power around 10 MHz. Notice the 2
MHz harmonic.
Atomic Fluorescence (arb. units)
7
0
2
4
6
8
10
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’=0, F=1
– F’=1 and F=1 – F’=2, respectively of the 5S1/2 – 5P3/2 87Rb transition.
Atomic Fluorescence (arb. units)
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(a)
0.0
(b)
1.0
0.79T
2.0
3.0
4.0
Time (µs)
Figure 4: Demonstration of phase-locked fluorescence. T is the period of the
Bloch-Siegert oscillation. In (a) a 0.4T delay line was placed in the reference
field cable. In (b) the 0.4T delay line was placed in the florescence signal cable.
The figure shows that signal (b) is about 0.8T 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.
9
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