Decoherence

We have so far neglected the presence of spin-wave decoherence forσˆgsby settingγgs= 0(e.g., neglecting spin-exchange collisions). Practically, this is a good approximation, given that the interaction timesδtw,rfor the writing and reading processes are fast compared to the relevant coherence times. However, if the delayτ between the storage and retrieval processes is longer than the coherence timeτmof the spin waves, we must also include various spin-wave dynamics induced by the decoherence mechanisms. Here, we will review two major dissipative contributions^q to spin-wave coherence in our experiment: (1) Inhomogeneous broadening and (2) motional dephasing.

2.6.1 Inhomogeneous broadening of spin waves

In our experiment, inhomogeneous broadenings of spin waves are dominated by two factors: (1) Inhomoge- neous light shifts between|g� − |s�(in the case of a nano-fiber trap in chapter 10), and from (2) inhomo- geneous Zeeman broadening (see, e.g., chapters 4–5). Here, we describe a one-dimensional model^r for the second type of decoherence mechanism. This model, however, should also be applicable to the first type of decoherence. The Zeeman decoherence model described in this section is similar to the one developed by Daniel Felinto¹⁴⁷.

We assume an initial atom-field stateρˆ_aγ1 = ˆρ_a(0)⊗ρˆγ1(0), whereρˆ_a(0) =pm_Fg|Fg, mF��Fg, mF|is the initial atomic state with Zeeman populationspmF, andρˆγ1(0) =|0�k1��0�k1|is the initial vacuum state for field 1 (with all other modes traced over). In the single-excitation limit, a photoelectric event in field 1 (with unit detection efficiency) heralds an atomic stateρˆa(tw) = |ψ(tw)�a�ψ(tw)|containing a single spin-wave excitation, with

|ψ(tw)�a= 1

√NA

��

i

e^{i(�kw·�ri−k1zi)}|g1,· · ·, gi−1,˜si(tw), gi+1,· · ·, gNA�

�

, (2.88)

wheree^{i(�kw·�ri−k1zi)}gives the phase-matching condition(�kw−�kr)·�ri−(k1−k2)zi= 0during the reading process (section 2.6.2) andtwis the time at the end of the write pulse. Here, given the initial atomic stateρˆ_a(0) populated with multiple Zeeman sublevels, the single-atom spin flips˜i(tw)corresponds to a superposition of Zeeman-dependent hyperfine spin flips given by

|s˜i(tw)�= 1

√Nc

�

m_Fg,m_Fs

�pm_Fg

�ξ^qm^w_Fg^,q1,m_Fse^iµB(^ggmFg−gsm_Fs)^Bz(�ri)tw/�|sm_Fs�, (2.89)

qDispersive van der Waals interaction between the atomic dipolesgvdW ∼ _4π�^d₀²⁰_r3 ij

lead to a van der Waals dephasing for suffi- ciently high density. Generally, for high phase-space density, one may additionally consider elaborate collisional processes and radiation trapping¹⁹⁴(due to imperfect optical pumping). Alternatively, the atoms can be trapped in an optical lattice to reduce collisional dephas- ing¹¹⁶.

rThe spatial variation of the magnetic field across the transverse directions can be neglected, given the aspect ratio of the atomic sam- ple (2w0/L�2×50µm/3mm�1). The inhomogeneous Zeeman broadening is thus dominated by the longitudinal inhomogeneity Bz(z)of magnetic field.

whereNc = �

m_Fg,m_Fsξ_m^qw_Fg^,q1_,mFs is the normalization constant, gg,s are the respective g-factors for the ground states {|g�,|s�},µB is the Bohr’s magneton, Bz(�ri) = B(�r� i)·ˆz is the magnetic field projected alongzˆat position�ri, andξ^q_m^w_Fg^,q1_,mFs is the generalized effective coupling constant for the specific excitation pathway (see Eq. 2.49). Here,qw,1are the polarization helicities for the writing laser and the field 1.

This expression can be easily derived fromξ=tanh²(i�∞ 0 dt^��L

0 dzχp(z, t^�))(Eq. 2.35) by generalizing the parametric coupling constantχpto accommodate Zeeman-dependent transition constants (which depends on the parameters{Fg, Fs, mFg, mFs, qw, q1} via the various Cleabsh-Gordan coefficients). Alternatively, Eq. 2.89 can be derived from the mapping HamiltonianHˆ_tot^(map). In writing Eqs. 2.88–2.89, I assumed the effective single-mode model for parametric interactionξ_m^qw_Fg^,q1_,mFs in section 2.3.2.2, where the beam-waist of the writing laser is substantially larger than that of the quantum field. The Larmor precession of a single-atom spin-flip after a delayτis then described by

�s˜i(tw+τ)|˜si(tw)�= 1 Nc

�

m_Fg,m_Fs

pm_Fgξ^q_m^w_Fg^,q1_,mFse^iµB(^ggmFg−gsm_Fs)^{Bz(�ri)τ /}�. (2.90)

To understand the collective dynamics ofρˆ_a(τ), we now calculate the overlap function_a�ψ(tw+τ)|ψ(tw)�a

between the initial collective state and the final collective state, where the retrieval efficiencyηris ideally proportional to|a�ψ(tw+τ)|ψ(tw)�a|²(section 2.5). If we assume that the atoms are stationary so that the phase-matching condition is preserved, using Eqs. 2.88–2.90, we obtain the following overlap,

a�ψ(tw+τ)|ψ(tw)�a = 1 NANc

�

m_Fg

pm_Fgξ_m^qw_Fg^,q1_,mFg+qw−q1

×

� L 0

dzn(z)e^iµB(^(gg−gs)mFg−gs(qw−q1))^{Bz(z)τ /}�, (2.91) where we made the continuum approximation for the summation�

i →�

dzn(z)and assumed pure polar- ization states (qw,1) for the writing laser and the field 1 (i.e.,mFs =mFg +qw−q1).

Assuming a flat-top atomic distributionn(z) =NA/Land a quadratically inhomogeneous Zeeman shift Bz(z) = (4δBz/L²)(z−L/2)², we obtain

a�ψ(tw+τ)|ψ(tw)�a= 1 Nc

√L

�

m_Fg

pm_Fgξ_m^qw_Fg^,q1_,mFg+qw−q1fd(τ), (2.92)

where the decay amplitudefd(τ)for the Raman transitionmFg →mFs =mFg +qw−q1is given by

fd(τ) =−

√iπerf��

−iαm_Fgτ�

2√αm_Fgτ , (2.93)

with a Zeeman-dependent decay constantαm_Fg = µB((gg −gs)mFg −gs(qw −q1))δBz/�. The func- tion erf(x) = ^√2_π�x

0 dx^�e^−x�2 is the error function. In Fig. 2.7, we show the result of our calculation

0 10 20 30 40 50 0.0

0.2 0.4 0.6 0.8 1.0

(4)

(5) (3) (2)

 

r

 ( s)

(1) pc

 ( s)

0.1 1 10

0.00 0.05 0.10 0.15

50

Figure 2.7:Spin-wave decoherence due to inhomogeneous Zeeman broadening.We show the normalized retrieval efficiency based on the theoretical expressionηr∼ |a�ψ(tw+τ)|ψ(tw)�a|²(Eq. 2.93) as a function of storage timeτ, for (1)δBz = 10, (2)20, (3)30, (4)40, and (5)50mG. A typical experimental value for the inhomogeneous Zeeman broadening isδBz = 20mG (red line), after nulling the magnetic fields with bias coils (based on our measurement of off-resonant Raman spectroscopy¹⁹⁵). We assumed the length of the ensemble to beL= 3mm. The atomic cloud is initially prepared in the ground state|g�with uniform Zeeman distributionpm_Fg = 1/Fg. In this calculation,{|g�,|s�,|e�}denote the levels{|6S1/2, F = 4�,|6S1/2, F = 3�,|6P3/2, F = 4�}. (inset) We measure the conditional probabilitypcto detect a single-photon in field 2 as a function ofτ. The expected Zeeman dephasing isτd � 30µs (red line) atδBz � 10mG (based on Raman spectroscopy), whereas the measured coherence time is onlyτd �12µs (red points). This result is consistent with the theoretical prediction for motional dephasing (red dashed line) in Fig. 2.8. For reference, we also plot the spin-wave coherence measurements atδBz � 30mG (black points), which are consistent with Zeeman dephasing (black line). The vertical axes of the theory (black, red) lines forηrare scaled to fit the experimental data for the photoelectric detection probabilitypcof the field 2.

of the retrieval efficiency ηr after storage timeτ. Note that the second-order approximation Bz(z) = (4δBz/L²)(z−L/2)² is reasonable for our experiment, since the Helmholtz bias coils can in principle cancel inhomogeneous Zeeman broadening only up to the first-order. Collective Larmor rephasing (oscilla- tion inηrat longτ), also known as dark-state polariton collapse and revival^119,120, is suppressed in our case of inhomogeneous Zeeman broadening by the additional factor of2√αm_Fgτin the denominator.

Generally, the temporal profile of the spin-wave dephasingηr(τ)due to Zeeman broadening depends on both the number densityn(z)and the inhomogeneous magnetic fieldBz(z)across the sample. Practically, the atomic density is approximately a Gaussian distributionn(z)∼e^−4z2/L2in our experiment. If we assume that the first-order (linear) Zeeman shift is dominant, withBz(z) = (δBz/L)z, the equivalent expression for�s(t˜ w+τ)|s(t˜ w)�(see, e.g., Eq. 2.91) simply depicts a Fourier transform ofn(z), which results in a Gaussian decay ofηr. Thus, the temporal profile ofηr(τ)may provide some information about the spatial inhomogeneity of the Zeeman levels, given the measured atomic densityn(z).

2.6.2 Motional dephasing of spin waves

As seen in Eq. 2.88, the spin-wave dynamics is also dictated by the spatial coherence�kw·�ri−k1z1 of the spin-wave, which preserves the phase-matching condition required for collective enhancement. Here, we provide a simple model for motional dephasing

We first assume a linear atomic motion by�ri(tw+τ) =�ri(tw) +�viτduring the storage timeτfor thei^th atom. The final collective state after motional dephasing can thus be written as

|ψ(tw+τ)�a= 1

√NA

��

i

e^{iδk(yi(tw)+vy,iτ)}|g1,· · ·, gi−1,˜si, gi+1,· · · , gNA�

�

, (2.94)

whereδk = |�kw−�k1| � kwsinθw1 is the net momentum transfer to the spin-wave (with a small angle approximation θw1 � 1 between thek-vectors of the writing laser and field 1 on the y −z plane) and vy,i=�vi·yˆis the atomic velocity projected alongy.ˆ

As we discussed in section 2.6.1, the retrieval efficiency is proportional to the overlap|a�ψ(tw+τ)|ψ(tw)�a|². If we assume a continuum limit for the momentum distributiong(v)of the thermal atoms, the overlap is given by

a�ψ(tw+τ)|ψ(tw)�a =

�

dvg(v)e^iδkvτ. (2.95)

For thermal atoms atTd, as in our case of laser-cooled Cesium atomsTd�100µK, the atomic motion follows the Maxwell-Boltzmann distribution g(v) = e^−mv2/2kBTd. The retrieval efficiency ηr(τ) ∼ |a�ψ(tw+ τ)|ψ(tw)�a|²is then (Fourier-transformingg(v))

ηr=e^−τ2/τd2, (2.96)

whereτd=λs/2πvswithvs=�

2kBTd/m. Here, we defined the spatial coherence lengthλs= 2π/δkfor the spin-wave with momentum transferδk.

In Fig. 2.8, we show the temporal profile of the normalized retrieval efficiency ηr as a function of storage timeτ (Eq. 2.96) for various temperatures Td and for two different anglesθw1 � 3^◦ (θcs � 2^◦) between the two fields (thereby, varying the coherence lengthλsof the spin-wave). I note that there are two relevant factors which fully characterize the motional dephasingτd: (1) Atomic motionvsand (2) spin-wave coherence lengthλs. By decreasing the temperatureTdof the atomic sample, we can reduce the mean velocity vs = �

2kBTd/m, and thereby increase the coherence time, as shown in Fig. 2.8. By placing an optical lattice along the momentum transferδ�k� yˆ^s, the atomic motion can also be dramatically reduced115–117,196

(see also Fig. 1.3 in chapter 1). On the other hand, to improveτc, it is also possible to increase the spin-wave coherence lengthλs = 2π/δkfor a givenvs(Fig. 2.8) by moving to a collinear geometry (�kw � �k1) with

sIn this case, it is important to also consider the inhomogeneous light shifts between|g� − |s�from the optical lattice, as discussed in the previous section.