9. CAPÍTULO 2
9.1 El período del liberalismo radical 1863-1876
9.1.1 La educación en el período del liberalismo
When a single atom is prepared in an excited state|eiit can spontaneously decay to the ground level |gi and emit a single photon. Due to conservation of angular momentum in spontaneous emission the polarization of the emitted photon is correlated with the final quantum state |gi of the atom. For a simple two-level atom, after spontaneous emission, the system is in a tensor product state of the atom and the photon. But for multiple decay channels to different ground states the resuling state of atom and photon is entangled.
The physical process of spontaneous emission can not be explained by a semiclassical treatment of the light field but only by a quantum field approach (this can be found in [47]). I do not intend to give a sophisticated treatment of spontaneous emission, but I rather give a phenomenological approach, which will be sufficient to understand atom-photon entanglement in the context of the present work.
2.4.1 Weisskopf-Wigner theory of spontaneous emission
Let us consider a single atom at time t = 0 in the excited state |ei and the field modes in the vacuum state |0i. In the “dressed state picture” the state of the system is then given by |e,0i=|ei ⊗ |0i, the product of the atomic state |ei and the vacuum state |0i of the electromagnetic field. Due to coupling of the atom to vacuum fluctuations of the electromagnetic field (this can be found in [48]) the atom will decay with a characteristic time constant τ, called the natural lifetime, to the ground state |gi and emit a single photon into the field mode k. For t → ∞ the sytem atom+photon will be in the state
2.4 Atom-photon entanglement
g, 1 e, 0
t=0 t > 0
Figure 2.6: Spontaneous emission of a photon. At time t = 0 the atom is in the excited state |ei. After spontaneous emission the atom passes to the ground state
|g,1i, and the electromagnetic field mode is occupied with one photon. The time evolution of the system is governed by the time-dependent Schr¨odinger equation
|Ψ(˙ t)i=−i ¯
hH|Ψ(t)i, (2.34)
where H denotes the Hamiltonian describing the interaction of a single two-level atom with a multi-mode radiation field. In the rotating-wave approximation the simplified Hamiltonian H is given by H =X k ¯ hωkˆa†kaˆk+ 1 2¯hωegσˆz+ ¯h X k gk(ˆσ+ˆak+ ˆa†kσˆ−). (2.35)
This Hamiltonian consists of three parts. The first term in Eq. (2.35) describes the energy of the free radiation field in terms of the creation and destruction operators ˆa†k
and ˆak, respectively. The second term ¯hωegσˆz/2 desrcibes the energy of the free atom, whereby ˆσz is given by|eihe|−|gihg|. The third term finally characterizes the interaction energy of the radiation field with the two-level atom. In detail ˆσ+ and ˆσ− are operators which take the atom from the lower state to the upper state and vice versa. Hence, ˆ
a†kσˆ− describes the process in which the atom makes a transition from the upper to the lower level and a photon in the modek is created, whereas ˆσ+ˆak describes the opposite process in which the atom is excited from the lower to the upper level and a photon is annihilated.
In the Weisskopf-Wigner approximation the eigenstate vector is given by (see [49], pp. 206) |Ψ(t)i=e−Γt/2|e,0i+|giX k Wke−ikr0 " 1−ei∆t−Γt/2 iΓ/2 + ∆ # |1ki. (2.36) Here, the form of the probability amplitude of the state|e,0isignals that an atom in the excited state |ei in vacuum decays exponentially with the lifetime τ = 1/Γ and emits a photon of angular frequency ωk. The probability amplitude of the state |1ki describes the temporal occupation of the modes k of the radiation field, where Wk denotes the
2 Theory of atom-photon entanglement ∆m = +/−1 ∆m = 0 z z θ θ
Figure 2.7: Emission characteristics of light emitted from dipole transitions with ∆m = 0,±1.
overlap between the atomic states|giand|eiin the field modek, and ∆ =ωk−ωeg the detuning in respect to the atomic transition frequency ωeg. For times long compared to the radiative decay the first term in (2.36) is negligible and the state of the system is given by a linear superposition of single-photon states with different wave vectors.
2.4.2 Properties of the emitted photon
Because atomic states are eigenstates of the total angular momentum, the modes of the electromagnetic field after spontaneous emission are also eigenstates of angular momen- tum. Therefore the polarization state of a photon spontaneously emitted from an atomic dipole depends upon the change in angular momentum ∆mof the atom along the dipole axis, and the direction of the emission. For ∆m = 0,±1, the polarization state of a spontaneously emitted photon is
|Π0i= sinθ |πi for ∆m = 0 (2.37)
|Π±1i=
√
1 + cos2θ/√2 |σ±i for ∆m =±1, (2.38)
where θ is the spherical polar angle with respect to the dipole (quantization) axis, and
|πiand|σ±idenote the polarization state of the photon. Note that along a viewing axis
parallel to the dipole (θ = 0) only|σ±i-polarized radiation is emitted.
2.4.3 Spontaneous emission as a source of atom-photon
entanglement
Until now we considered a single atom in free space which spontaneously emits a photon from a two-level transition. According to Weisskopf and Wigner the state of the system atom+photon is a simple tensor product state of the form |gi|Π∆mi, where the atom is in the ground state |giand the photon in the state |Π∆mi. For multiple decay channels
2.4 Atom-photon entanglement 0,0 S1/2 2 P3/2 2 σ+ σ− 1,−1 1,0 1,+1 780 nm F=1 F=2 F=0
Figure 2.8: Atomic level structure in 87Rb used to generate atom-photon entanglement.
Provided the emission frequencies forσ+,σ− and π polarized transitions are
indistinguishable within the natural linewidth of the transition, the polar- ization of a spontaneously emitted photon will be entangled with the spin state of the atom.
of spontaneous emission to different ground states of atomic angular momentum F, the resulting (unnormalized) state of photon and atom is
|Ψi=X
∆m
C∆m|F∆mi|Π∆mi, (2.39)
where C∆m are atomic Clebsch-Gordon coefficients for the possible decay channels and
|F∆mi denote the respective atomic ground states. The state in (2.39) can not be represented as a tensor product, only as a linear superposition of different product states. Therefore the spin degree of freedom of the atom and the polarization of the photon are entangled.
In the current experiment we excite a single 87Rb atom to the 2P3
/2,|F = 0, mF = 0i state by a short optical π-pulse. In the following spontaneous emission the atom decays either to the|1,−1iground state while emitting a|σ+i-polarized photon, or to the|1,0i
state while emitting a |πi-polarized photon, or to the |1,+1i ground state and emits a
|σ−i-polarized photon. Provided these decay channels are spectrally indistinguishable,
a coherent superposition of the three possibilities is formed and the magnetic quantum numbermF of the atom is entangled with the polarization of the emitted photon resulting in the atom-photon state
|Ψi= √1 2 q (1 + cos2θ)/2|1,−1i|σ+ i+|1,−1i|σ−i+ sinθ|1,0i|πi , (2.40)
where the first index in the atomic basis state |F, mFi denotes the total angular mo- mentum F and the second index indicates the respective magnetic quantum number
2 Theory of atom-photon entanglement
If we now put a single photon detector in the far-field region of the atom and detect the spontaneously emitted photon along the quantization axisz - defined by the optical axis of the detection optics - then the |πi-polarized light is not detected (see Fig. 2.7). Thus the resulting atom-photon state is maximally entangled:
|Ψ+i= √1
2(|1,−1i|σ
+
i+|1,+1i|σ−i). (2.41)
2.4.4 Experimental proof of atom-photon entanglement
To verify atom-photon entanglement in an experiment, one has to disprove the alter- native description of the system being in a statistical mixture of seperable states. Ex- perimentally one has to determine the diagonal density matrix elements in at least two complementary measurement bases (see sect. 2.3.3). The choice of the measurement basis can be realized in two ways. First, the atomic and photonic spin-state is ro- tated by an active unitary transformation into the new measurement basis. Second, the atomic and photonic spin-state stay unchanged, but the spin-analyzer is rotated by a respective passive transformation. For photons, the state-measurement can be realized relatively simple by a rotable birefringent waveplate followed by a polarizer. For atomic spin-states |↑iz =|1,−1iand |↓iz =|1,+1i active rotations can be realized by suitable optical Raman laser pulses that perform the transformation:
|↑iz+eiφ|↓iz → |↑iz (2.42)
|↑iz−eiφ|↓iz → |↓iz, (2.43) where φ is the relative geometric phase between the basis states |↑iz and |↓iz.
In the present experiment the atomic state detection is realized by a suitable Stimulated- Raman-Adiabatic-Passage (STIRAP) laser pulse which transfers, e.g. the superposition state |↑iz+eiφ|↓iz to the hyperfine ground state F = 2, whereas the orthogonal super- position state |↑iz −eiφ|↓iz does not couple to the laser field and remains “dark”. By scattering light from a preceding laser pulse which couples only to F = 2 both states can be identified with nearly perfect efficiency (see chapter 5).