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Supplemental Material to
‘Cross Modulation of Two Laser Beams at the Individual-Photon Level’
Kristin M. Beck,1 Wenlan Chen,1 Qian Lin,1 Michael Gullans,2, 3 Mikhail D. Lukin,2 and Vladan Vuletić1
1
Department of Physics and Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2
Department of Physics, Harvard University, Cambridge, MA 02138, USA
3
Joint Quantum Institute, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
(Dated: August 13, 2014)
THEORETICAL DESCRIPTION
The system consists of atoms with a four-state N -type level structure |f i ↔ |di ↔ |ci ↔ |ei as described in the
text. Including the decay, the effective Hamiltonian for this system can be written as
X
Hef f /~ =
c |k| a†k ak + (ωc − iκ/2)b† b + iE(b† − b)
k
+ (ωf d − iΓ/2)
X
|dij hd| + (ωce − iΓ/2)
j
+
X
X
|eij he| + (ωf c − iγ/2)
j
X
|cij hc|
(1)
j
Ω/2 eiωdc t |cix hd| + gg a† (x) |f ix hd| + gs b† |cix he| + h.c.
x
Here, c is the speed of light, k is the wavenumber of the signal (free-space) field, ωc is the cavity frequency and κ is
the decayqrate of the cavity. The
signal (free-space) and cavity fields can be written as
q electric field operators for the
P ikx
~ck0
~ωc
−1/2
Ês (x) =
ak and b are bosonic annihilation operators, ck0
ke
0 V a(x) and Êc =
0 V b, where a(x) = N
is the center frequency of the signal field, and V is the quantization volume. Additionally, E is the amplitude of the
cavity input field, ωµν is the atomic transition energy between states µ and ν, Ω is the classical Rabi frequency for
the coupling field, Γ is the linewidth of the excited states |di and |ei, γ is decoherence rate of two stable ground states
|f i and |ci, and gs , gc are the bare couplings of the atomic transition to the two fields. We take the gate and signal
fields to be resonant with the atoms so that ck0 = ωf d and ωc = ωce .
The use of this effective Hamiltonian is sufficient to describe the steady state for the case of weak coherent state
input fields gs hai Ω2 /Γ and gc hbi κ. In this limit, we can take the approach of [Ref. [1]] to calculate the
two-time correlation function between the fields
g (2) (x, τ ) =
hb† (t)a† (x, t + τ )a(x, t + τ )b(t)i
ha† (x, t)a(x, t)ihb† (t)b(t)i
(2)
In this limit, we also can write the density matrix as a product state ρ = |χ(τ )i hχ(τ )| and truncate the available
states in the system at the level of two excitations from the state with zero photons and all atoms in |f i, which we refer
x
to as |f, 0, 0i. The one-excitation states are |f, 1x , 0i = a† (x) |f, 0, 0i, |f, 0, 1i = b† |f, 0, 0i, |cx , 0, 0i ≡ σcf
|f, 0, 0i,
x
x
(2)
and |dx , 0, 0i ≡ σdf |f, 0, 0i, where σµν ≡ |µix hν|. The two-excitation states that are relevant for g are |f, 1x , 1i ≡
x
a† (x) |f, 0, 1i, |cx , 0, 1i ≡ b† |cx , 0, 0i, |dx , 0, 1i ≡ b† |dx , 0, 0i, and |ex , 0, 0i ≡ σec
|cx , 0, 0i.
d|χi
We then expand |χ(t)i in these states and find the evolution according to i dt = Hef f |χi applying the boundary
condition that the free space input field is a weak coherent state. The only terms in Hef f which create excitations are
the driving fields, which are perturbative implying that the amplitude of the one-excitation states are proportional to
E and the two-excitation amplitudes are proportional to E 2 .
To calculate g (2) (τ ) we take the picture where the detection corresponds to a quantum jump from the steady state
|χss i into the state a(x, t) |χss i for τ < 0 and b(t) |χss i for τ > 0 [Ref. [1]]. To find g (2) (τ ) we can then simply evolve
the operator ns (t) or nc (x, t) for a time τ under Hef f starting from the jump state.
To find the steady state we expand |χ(t)i in the zero-, one- and two- excitation states
|χ(x, t)i = |f, 0, 0i + A10 (x) |f, 1x , 0i + A11 (x) |cx , 0, 0i + A12 (x) |dx , 0, 0i + A13 |f, 0, 1i
+ A21 (x) |f, 1x , 1i + A22 (x) |cx , 0, 1i + A23 (x) |dx , 0, 1i + A24 (x) |ex , 0, 0i
(3)
2
where we neglect the one- and two- excitation states in the normalization because they are perturbative. The equations
of motion are for the Aji are found from i d|χi
dt = Hef f |χi.
√
(∂t + c ∂x )A10 (x) = −igs N A12 (x),
(4)
√
∂t A12 (x) = −Γ/2A12 (x) − igs N A10 (x) − iΩ/2A11 (x),
(5)
∂t A11 (x) = −γ/2A11 (x) − iΩ/2A12 (x),
∂t A13
(∂t +
c ∂x )A21 (x)
=
−κ/2A13
+ E,
EA10 (x)
(6)
√
=
−κ/2A21 (x)
=
−(κ + γ)/2A22 (x) − igc A24 (x)
−Γ/2 A24 (x) − igc A22 (x)
(7)
N A23 (x),
− igs
√
∂t A23 (x) = −(Γ + κ)/2A23 (x) − igs N A21 (x) − iΩ/2A22 (x) + EA12 (x),
∂t A22 (x)
∂t A24 (x)
=
+
−
iΩ/2A23 (x)
+
EA11 (x),
(8)
(9)
(10)
(11)
These eight equations are the only ones relevant for g (2) (t), they give the steady state
N
x
2gs2 N x
1
Ā0 (x) = α exp −
=
α
exp
−
Γ + Ω2 /γ c
2(1 + Ω2 /γΓ) L
√
2gs N Ω 1
Ā11 (x) = − 2
Ā (x),
Ω + γΓ 0
√
i2gs N 1
Ā (x),
Ā12 (x) = −
Γ + Ω2 /γ 0
E
Ā13 =
,
κ/2
1
Ā21 (x)
η
N x
=
+
exp
−
+ O(κ/Γ)
1+η 1+η
2ζ L
Ā10 (x)Ā13
(12)
(13)
(14)
(15)
(16)
where α is the amplitude of the input coherent state, N is the number of atoms, N = 4gs2 N L/cΓ is the optical depth,
L is the length of the medium, η = 4gc2 /κΓ is the cooperativity, and we have defined
γΓ
Ω2 /κΓ + γ/κ
ζ = 1+ 2
1+
(17)
Ω
1+η
a correction factor that accounts for finite γ and cooperativity η.
When τ < 0 the free space photon is detected first leading to a quantum jump into the state
|χJ i = p
a(L, τ ) |χss i
hχss | a† (L, τ )a(L, τ ) |χss i
= |f, 0, 0i +
Ā21 (L)
|f, 0, 1i
Ā10 (L)
(18)
Now
|Ã1 (t)|2
hχJ (t)| b† (τ )b(τ ) |χJ (t)i
= 3 2
†
hχss | b (τ )b(τ ) |χss i
Ā1 3
2
η −κ< |τ |/2
−N /2ζ
= 1− 1−e
e
1+η
g (2) (τ ) =
(19)
(20)
where κ< = κ and Ã13 (t) is found from Eq. 7 with the initial condition Ã13 (0) = Ā21 (L)/Ā10 (L).
For τ > 0 the procedure is the same, except we have to evolve Eqs. 4-6 starting from the initial conditions
A10 (x, τ ) = Ā21 (x), A11 (x, τ ) = Ā22 (x), and A12 (x, τ ) = Ā23 (x). This corresponds to the state |χJ i ∝ b |χss i. The result
can be expressed in the same form as Eq. 19 with κ< replaced by κ> = Ω2 /Γ + γ.
SCHEME TO PRODUCE POLARIZATION-ENTANGLED STATES
Continuous entanglement of two light beams can be achieved in the following way: prepare the atomic ensemble in
the |F = 3, mF = 3i and |F = 3, mF = −3i states with equal population in a small magnetic field oriented at small
3
angle to the cavity mode, and apply the signal beam along the magnetic field. The scheme we demonstrate in the
manuscript then works independently for both circular polarizations, so that |σ+ is |σ+ ic (the subscript indicates the
signal and cavity mode, respectively, and σ± indicates circular photon polarization) becomes |σ+ is |0ic or |0is |σ+ ic ,
|σ− is |σ− ic becomes |σ− is |0ic or |0is |σ− ic , and the states |σ+ is |σ− ic and |σ− is |σ+ ic are unchanged. When linearly
polarized beams are sent into both the signal and cavity paths and we post-select for detecting a photon in each mode,
the output is the entangled (Bell) state √12 (|σ− is |σ+ ic + |σ+ is |σ− ic ).
[1] H. J. Carmichael, R. J. Brecha, and P. R. Rice, Optics Communications 82, 73 (1991).