hJxi=JcosN 1 à
2 (4.6)
hJyi=hJzi= 0
⌦ Jx2↵
= J
2[2J(1 cos2(N 1) à
2) (J 1 2)A]
⌦ Jy2↵
= J 2
⇢ 1 + 1
2
✓
J 1
2
◆
[A+p
A2+B2cos 2 ]
⌦ Jz2↵
= J 2
⇢ 1 + 1
2
✓
J 1
2
◆
[A p
A2+B2cos 2 ]
where A = 1 cosN 2à, B = 4 sinà2cosN 2 à2 and = 12arctanBA. It is then apparent that the behaviour of such twisting Hamiltonian will always modify the degree of fluctuation in each direction, such that in our case, the y component experiences an enhancement in the fluctuation, and z component experiences an opposite reduction in the fluctuation. As a final remark, we note that such form of one-axis twisting will generate spin squeezing proportional2 to N2/3.
4.3 Squeezing Dynamics in Bosonic Environment
4.3.1 The Model
We now extend the model that we used in Chapter 3 to N identical two-level atoms with collective spin Jz =PN
l=1 z,i/2 interacting with a common bosonic environment. The total system-environment Hamiltonian can then be written as
H = HS+HB+HI +HOAT
= !0Jz+ Jz2+X
k
!kb†kbk
+ 2Jz
X
k
(g⇤kbk+gkb†k) (4.7)
2We refer interested readers to Ref. [150] for the derivation of this proportionality, as well as the derivation of the above expectation values.
where!0 is the energy bias, and HB is the collection of the bosonic modes that make up the environment. Note that, in contrast with Chapter 3, a non-linear interaction term between the qubits has been included via Jz2. The purpose of including the non-linear interaction term HOAT is to compare the squeezing generated in the presence of this term as well as the environment, and the spin squeezing generated solely by the environment. As mentioned in the previous section, the one-axis twisting Hamiltonian HOAT acting alone generates spin squeezing proportional to N2/3. However, the interaction with the environment greatly in- fluences the dynamics, and degree of squeezing is conventionally expected to significantly deteriorate. However, it has been shown in recent literature that the environment can ac- tually also assist in generating spin squeezed states [154, 158, 159]. Our goal is then to see how the environment parameters can be optimised to generate the maximum possible spin squeezing. This should be compared to what we performed in Chapter 3 where we optimised the environment parameters to generate two-qubit entanglement. It should be noted that there is a connection between spin squeezing and entanglement. In particular, Ref. [160]
has shown that for any arbitrary symmetric multiqubit state, spin squeezing implies the existence of pairwise entanglement between the qubits. Therefore, we expect the insights that we gained in Chapter 3 to be useful here as well.
We begin by finding the reduced system density matrix of the system of N qubits. This can be done in a very similar way to what we did before for two-qubits. We find that the reduced density matrix elements of the system are now given by
[⇢S(t)]mn= [⇢S(0)]mne i!0t(m n)e i (t)(m2 n2)e (t)(m n)2 (4.8)
with (t) and (t) given by Eqs. (3.21) and (3.29) of the previous chapter respectively, and the indicesmandnare given byJz|mi=m|miandJz|ni=n|ni. As usual in investigating the generation of spin squeezing, we will choose the initial state to be the spin-coherent state
| i = e i⇡2Jy|N/2i. To quantify spin squeezing, we use the spin squeezing parameter ⇠S2
4.3. SQUEEZING DYNAMICS IN BOSONIC ENVIRONMENT
introduced by Kitagawa and Ueda [144] given by
⇠2S = 2min(J~n2
?)
N . (4.9)
whereJ~n? is the spin perpendicular to the mean spin direction of the spin and the minimisa- tion is taken over all possible~n?. The details of how this calculation can be carried out for any spin state can be found in Appendix 4.A . It is important to note that a smaller value of ⇠S2 indicates a greater degree of spin squeezing.
4.3.2 Optimization of spin squeezing
Based on our results in Chapter 3, we expect that the Ohmicity parameter plays a key role in the optimisation of spin squeezing. Therefore, we concentrate on the role of s. As illustrated in Fig. 4.2, we again find that there exists an optimal value ofsfor which the spin squeezing is maximized. We observe that for a fixed temperature = 1, if there is no direct interaction between the atoms via the OAT Hamiltonian, the optimal squeezing occurs at around s= 2.5, while the addition of OAT Hamiltonian pushes the squeezing parameter to be at the minimum at around s = 2. Interestingly, the presence of the OAT Hamiltonian can have a negligible influence on the squeezing generated as seen for the s > 2.5 regime.
Thus it is really for thes <2.5 regimes that the OAT Hamiltonian gives spin squeezing that cannot be generated by the environment itself. This can be explained simply by noting that in the short time scale, the value of the decoherence factor (t) increases withs, and thus the value of (t) in the sub-Ohmic and Ohmic regimes is smaller than that for the super-Ohmic regimes. The presence of the OAT Hamiltonian causes the time required to achieve maximum squeezing to shorten considerably, therefore this allows significant squeezing to happen in the sub-Ohmic and the Ohmic regimes before the onset of decoherence. It should also be noted that the maximum spin squeezing in the presence of the one-axis Hamiltonian does not di↵er very appreciably from the maximum spin squeezing generated by the environment alone.
Figure 4.2: (Colour online) Variation of the optimised squeezing parameter with the Ohmicity parameter s in the presence of the OAT Hamiltonian (circle, blue dotted lines) and without the OAT Hamiltonian (square, black solid lines). Here N = 10,g= 0.05,!0= 0.1,!c = 10 and = 1.
For the OAT case, = 1.
Let us now also comment on the role of the system-environment coupling strength g for the generation of spin squeezing. In this case, we find that, interestingly, very small values of g lead to the greatest spin squeezing. In fact, the value of the spin squeezing parameter approaches the OAT value as the value of coupling g decreases for sub-Ohmic, Ohmic and super-Ohmic environments, regardless if there is OAT Hamiltonian term. This is an interesting result as this illustrates that the indirect atom-atom interaction (t) can have the same performance in generating spin squeezing as the OAT Hamiltonian. However, it is important to note that decreasingg also has the e↵ect of lengthening the time scale to reach the minimum possible squeezing parameter. This limits the practicality of such results since
4.3. SQUEEZING DYNAMICS IN BOSONIC ENVIRONMENT
Figure 4.3: Variation of minimum⇠S2(t) with coupling strengthg. HereN = 10,!0 = 0.1,s= 2.5 and = 1 and we consider the spin squeezing generated at time T = 1.
in such long time scales, not only is the validity of our pure dephasing model in question, but also the e↵ects of atomic losses become significant and thus this inhibits the validity of our results. Therefore, we now study the relationship between the coupling strength g and the degree of spin squeezing, in particular within a fixed time scale. For such a case, if g is too small, the squeezing generated at the end of this time is too small, while if g is too large, decoherence e↵ects are too dominant. Therefore, we now do expect an optimal value ofg for the generation of spin squeezing. Fig. 4.3 shows the variation of minimum squeezing parameter in an arbitrary finite end timeT = 1 using di↵erent coupling strengths g. In this case, we can clearly see that there exist an optimal value of coupling strength g such that the minimum squeezing parameter can be reached before the time T. To sum up, increasing the coupling strength g has dual e↵ects on the squeezing: firstly speeding up the time to
reach maximum squeezing; secondly, reducing the degree of squeezing that can be achieved.
Thus, the e↵ective generation of spin squeezing is a competition between these two factors.