J G Muga

X Chen

S Ibáñez

I Lizuain

A Ruschhaupt

Transitionless quantum drivings for the harmonic oscillator

Keywords: numbers: 37.10.De, 42.50.-p, 37.10.Vz

Two methods to change a quantum harmonic oscillator frequency without transitions in a finite time are described and compared. The first method, a transitionless-tracking algorithm, makes use of a generalized harmonic oscillator and a non-local potential. The second method, based on engineering an invariant of motion, only modifies the harmonic frequency in time, keeping the potential local at all times.

Introduction

Changing the external parameters of the Hamiltonian is a fundamental and standard operation to probe, control, or prepare a quantum system. In many cases it is desirable to go from an initial parameter configuration to a final one without inducing transitions, as in the expansions performed in fountain clocks [1], the release of atoms from optical lattices [2], or compressions to increase the spatial density [2]. In fact most of the current experiments with cold atoms are based on a cooling stage and then an adiabatic drive of the system to some desired final trap or regime [3]. These "transitionless" [4], or "frictionless" [5] adiabatic processes may require exceedingly large times and become impractical [2], even impossible [3], or quite simply a faster process is desirable, e.g. to increase the repetition rate of a cycle, or a signal-to-noise ratio. This motivates the generic objective of achieving the same final state as the slow adiabatic processes, possibly up to phase factors, but in a much shorter time. One may try to fulfill that goal in two different ways: (a) designing appropriate "parameter trajectories" of the Hamiltonian from the initial to the final times, or (b) applying entirely new interactions that modify the Hamiltonian beyond a simple parameter evolution of the original form, for example by adding different terms to it. In this paper we shall analyze and discuss, for the harmonic oscillator, two recently proposed methods, whose relation had not been investigated, which actually implement these two different routes. While most of the treatment is applicable to an "abstract" harmonic oscillator, we shall discuss physical implementations specific for ultracold atoms or ions. Indeed, harmonic traps and their manipulation are basic working horses of this field.

For the harmonic oscillator the parameter we consider is the trap frequency, which should go from ω 0 to ω f in a time t f , preserving the populations of the levels, P n (t f ) = P n (0). "n" labels the instantaneous n-th eigenstate of the initial and final harmonic oscillator Hamiltonians, Ĥ0 (0)|n(0) = hω 0 (n + 1/2)|n(0) ,

Ĥ0 (t f )|n(t f ) = hω f (n + 1/2)|n(t f ) . (1)
One of the methods we shall discuss here relies on a general framework set by Kato in a proof of the adiabatic theorem [6], and has been formulated recently by Berry [4]. We shall term it "transitionless-tracking" approach, or TT for short; the other one [7,8] engineers the Lewis-Riesenfeld invariant [9] by an inverse method [10] to satisfy the desired boundary conditions; we shall call this method "inverse-invariant", or II for short. In the basic version of TT the dynamics is set to follow at all intermediate times the adiabatic path defined by an auxiliary Hamiltonian Ĥ0 (t) (in our case a regular harmonic oscillator with frequency ω(t) and boundary conditions ω(0) = ω 0 and ω f = ω(t = t f )), and its instantaneous eigenvectors |n(t) , up to phase factors. Instead, in the II approach the auxiliary object is an engineered Lewis-Riesenfeld invariant I(t) set to commute with Ĥ0 (0) at t = 0 and with Ĥ0 (t f ) at t f . In both cases intermediate states may be highly non-adiabatic with respect to the instantaneous eigenstates of the Hamiltonians actually applied, ĤTT (t) and ĤII (t).

We shall provide first the equations characterizing the two approaches and then comment on possible physical implementations.

Transitionless tracking algorithm

General formalism

For the general formalism we follow [4] closely. Assume a time-dependent Hamiltonian Ĥ0 (t) with initial and final values (1), instantaneous eigenvectors |n(t) and eigenvalues

E n (t), Ĥ0 (t)|n(t) = E n (t)|n(t) . (2)
A slow change would preserve the eigenvalue and eigenvector along the dynamical evolution times a phase factor, and the adiabatic approximation of the wavefunction is

|ψ n (t) = exp - i h t 0 dt E n (t ) - t 0 dt n(t )|∂ t n(t ) |n(t) . (3)
We now seek a Hamiltonian Ĥ(t) such that the adiabatic approximation |ψ n (t) represents the exact dynamics,

ih∂ t |ψ n (t) = Ĥ(t)|ψ n (t) . (4)
Ĥ(t) (which is ĤTT if distinction with the other method is needed) is related to the corresponding unitary operator by

ih∂ t Û (t) = Ĥ(t) Û (t), (5) Ĥ(t) = ih(∂ t Û (t)) Û † (t). (6)
Choosing

Û (t) = n exp - i h t 0 dt E n (t ) - t 0 dt n(t )|∂ t n(t ) |n(t) n(0)|, (7)
we find from (6),

Ĥ(t) = n |n E n n| + ih n (|∂ t n n| -n|∂ t n |n n|) ≡ Ĥ0 + Ĥ1 , (8)
where we have simplified the notation, |n = |n(t) . It is also possible to choose other phases in (3) [4]. The simplest case is Û (t) = |n(t) n(0)|, without phase factors, which leads to Ĥ(t) = ih |∂ t n n|. Note that with this choice Ĥ0 (t) has been formally suppressed in Ĥ(t) but still plays a role through its eigenfunctions |n(t) .

Application to the harmonic oscillator

We now apply the above to the harmonic oscillator

Ĥ0 (t) = p2 /2m + ω(t) 2 x2 /2m = hω(t)(â † t ât + 1/2), (9)
where ât and â+ t are the (Schrödinger picture!) annihilation and creation operators at time t,

x = h 2mω(t) (â † t + ât ), (10)
p = i hmω(t) 2 (â † t -ât ), (11) ât
= mω(t) 2h x + i mω(t) p , (12) â
† t = mω(t) 2h x - i mω(t) p . (13)
This time dependence may be misleading and a bit unusual at first so we insist: since the frequency depends on time the "instantaneous" ladder operators ât , â † t create or annihilate different "instantaneous" states, adapted to the corresponding frequency. Thus, ladder operators with different time labels do not commute in general, although some combinations, e.g. those equivalent to powers of x and/or p, do commute, as we shall see later.

The instantaneous eigenstates |n(t) can be written in coordinate representation as

x|n(t) = 1 √ 2 n n! mω(t) πh 1/4 exp - 1 2 mω(t) h x 2 H n   mω(t) h x   , (14)
and their derivative with respect to t is

x|∂ t n(t) = 1 4 - mω(t) 2h x 2 ω ω(t) |n + mω(t) 2h x ω ω(t) √ n|n -1 . (15)
We find, using the recursion relation of Hermite polynomials and their orthogonality,

k|∂ t n =                    1 4 n(n -1) ω ω(t) k = n -2 -1 4 (n + 1)(n + 2) ω ω(t) k = n + 2 0 ( o t h e r w i s e ) , (16)
so that Ĥ1 (t) can be written as

Ĥ1 (t) = ih n |∂ t n n| ≡ ih ω ω(t) n 1 4 - mω(t) 2h x2 |n n| + mω(t) 2h x√ n|n -1 n| . (17)
Using ât = n √ n|n -1(t) n(t)|, and the relations between x, p, ât and â † t written above,

Ĥ1 (t) = ih ω ω(t) n   1 4 - mω(t) 2h x2 + mω(t) 2h xâ t   = ih 4 ω ω(t) - 1 2 ω ω(t) xp. (18)
With [x, p] = ih, we finally write the Hamiltonian Ĥ1 (t) in the following simple forms

Ĥ1 (t) = - ω 4ω (xp + px) = ih ω 4ω [â 2 -(â † ) 2 ]. (19)
In the last expression the subscript t in â and â † has been dropped because the squeezing combination â2 -(â † ) 2 is actually independent of time, so it may be evaluated at any convenient time, e.g. at t = 0. The connection with squeezing operators is worked out in the appendix. Ĥ1 is therefore a non-local operator, and does not have the form of a regular harmonic oscillator potential with an x2 term. Nevertheless the final Hamiltonian Ĥ = Ĥ0 + Ĥ1 is still quadratic in x and p, so it may be considered a generalized harmonic oscillator [11].

Physical realization

The nonlocality of Ĥ1 , with a constant prefactor, can be realized in a laboratory by means of 2-photon Raman transitions for trapped ions [12,13,14] or, as discussed below, for neutral atoms in tight dipole traps. Since we have to evaluate as well the possibility of making the prefactor in Ĥ1 time dependent, we need to provide the derivation with some detail, first for a time-independent ω.

Raman two-photon transition in a trapped ion or neutral atom

Let us consider a harmonically trapped two-level system in 1D driven by two different lasers (with coupling strengths Ω j and frequencies ω j , j = 1, 2), see Fig. 1 and Refs. [13,14]. The time dependent "Raman" Hamiltonian in the Schrödinger picture will be given by

ĤR (t) = ĤT + ĤA + Ĥint , (20)
with "trap" (T ), "atomic" (A), and interaction (int) terms

ĤT = hωâ † â, ( 21
) ĤA = hω e |e e|, (22) Ĥint
= 2 j=1 hΩ j cos (ω j t -k j x + φ j ) (|g e| + |e g|), (23)
where hω e is the energy of the excited state |e and k j = k j x the wavevector of each laser which are assumed to be pointing along the principal trap direction, the x-direction. A basic assumption in this Hamiltonian is that the same vibrational frequency, i.e., the same external potential for atomic motion is applicable to the ground and excited states |g and |e . This is the standard case for ions in Paul traps, whereas for neutral atoms it may be realized in optical traps too, at least for suitably chosen species and transitions (for example alkali-earth atoms and spin forbidden transitions), by means of "magic wavelength" compensating techniques by which the light shifts, and thus the potentials for atomic motion in the |g and |e states can be controlled and made identical [15,16].

Interaction picture

Let us now write the above Hamiltonian in an interaction picture defined by the Hamiltonian ĥ0 = ĤT + hω L |e e|, where ωL = (ω 1 + ω 2 )/2 has been introduced. The interaction Hamiltonian ĤI = e i ĥ0 t/h ( ĤR -ĥ0 )e -i ĥ0 t/h reads ĤI (t) =h ∆|e e|

+ 2 j=1 hΩ j 2 e iη j [â(t)+â † (t)] e -i(ω j -ω L )t e -iφ j |e g| + H.c. , (24)
where ∆ = ωL -ω e , and now â(t) = âe -iωt , â † (t) = â † e iωt are the time dependent Heisenberg annihilation and creation operators respectively. Note also that fast oscillating off-resonant e ±i(ω j +ω L )t terms have been neglected in the rotating wave approximation (RWA). The parameter η j = k j x 0 is known as the Lamb-Dicke (LD) parameter, where x 0 = h/2mω is the extension (square root of the variance) of the ion's ground state, i. e., x = x 0 (â + â † ).

Adiabatic elimination and effective Hamiltonian

For a general wavefunction (in the corresponding interaction picture) such as the differential equations of motion for the probability amplitudes g n (t) and e n (t) are obtained from the Schrödinger equation ih∂ t |ψ

|ψ I (t) = ∞ n=0 [g n (t)|g, n + e n (t)|e, n ] ( 2 5 )
I (t) = ĤI |ψ I (t) , i ġn (t) = 1 2 2 j=1 ∞ n =0 Ω j e i(θ j t+φ j ) n|e -iη j [â(t)+â † (t)] |n e n (t), (26)
i ėn (t) = -∆e n (t) + 1 2

2 j=1 ∞ n =0
Ω j e -i(θ j t+φ j ) n|e iη j [â(t)+â

† (t)] |n g n (t), ( 27
)
where θ j = ω j -ωL . For large detunings, i. e., for | ∆| Ω j , ω, see Fig. 1, and for an ion initially in the ground state one may assume that the excited state |e is scarcely populated and it may be adiabatically eliminated. Then, setting ė(t) = 0, e n (t) may be written as a function of the g n (t) from Eq. ( 27), and substituting this result into [START_REF] Barnett | Methods in Theoretical Quantum Optics[END_REF] there results a differential equation for the ground state probability amplitude,

i ġn (t) = sg n (t) + Ω 2 ∞ n =0 F n,n (t)g n (t), (28)
where

s = Ω 2 1 + Ω 2 2 4 ∆ , (29)
F n,n (t) = n|e -iη[â(t)+â † (t)] |n e i( δt+ φ) + n|e iη[â(t)+â † (t)] |n e -i( δt+ φ) , (30)

and where the effective two-photon Raman parameters, denoted by tildes, are given by δ

= ω 1 -ω 2 , η = η 1 -η 2 , φ = φ 1 -φ 2 , Ω 2 = Ω 1 Ω 2 4 ∆ . (31)
The equation for the ground state probability amplitude corresponds to an effective Hamiltonian Ĥeff = hs|g g| + h Ω 2 e iη[â(t)+â † (t)] e -i( δt+ φ) + H.c |g g|.

(32)

Note that the Stark-Shift produced by off resonant driving is included in s, which is a constant of motion and produces no effect on the Raman coupling between sidebands.

We have thus adiabatically eliminated the excited state |e ending with a Hamiltonian of the same form as (24) where the transitions between electronic levels are not present.

2.3.4. Two-photon Jaynes-Cummings Hamiltonian in the Raman Scheme: Vibrational RWA Using the Baker-Campbell-Hausdorff (BCH) identity, the exponential in the effective Hamiltonian (32) may be expanded in power series of η [START_REF] Orszag | Quantum Optics[END_REF][START_REF] Lizuain | [END_REF],

Ĥeff = h Ω 2   e -η 2 /2 nn (iη) n+n n!n ! â †n ân e i(n-n )ωt e -i δt e -i φ + H.c   . ( 33
)
If the effective detuning is δ = ω 1 -ω 2 = 2ω, the second blue sideband becomes resonant, and we may neglect rapidly oscillating terms in a second or vibrational RWA [START_REF] Lizuain | [END_REF]. The above Hamiltonian is then simplified to a two-photon Jaynes-Cummings-like Hamiltonian without electronic transitions. To leading order in η it takes the form

Ĥeff ≈ η2 h Ω 4 â †2 e i φ + â2 e -i φ = ih η2 Ω 4 â2 -â †2 , (34)
where, in the last step, a relative phase between the applied fields φ = φ 1 -φ 2 = -π/2 has been assumed.

Validity for time-dependent ω

Unfortunately the above formal manipulations and approximations cannot be carried out in general for a time dependent ω. The interaction picture performed in 2.3.2, in particular, assumes a constant ĥ0 . A time dependent one would require a more complex approach with time-ordering operators [19]. Similarly, the vibrational rotating wave approximation requires the stability of the frequency for times larger than a period to avoid off-resonant couplings. One may still obtain (34) for a sufficiently slowly varying ω, the criterion being that the change of the time-dependent trapping frequency in one time period T has to be much smaller than the frequency itself. We can write this condition as ω(t)T ω(t) or

ω(t) ω(t) 2 1, (35)
which turns out to be the adiabaticity condition for the harmonic oscillator. Of course, if satisfied, the whole enterprise of applying the TT method would be useless. These arguments are far from constituting a proof that the TT method cannot be implemented for the harmonic oscillator. They simply leave this as an unresolved issue.

Engineering the Lewis-Riesenfeld invariant

In this section we describe a different method for transitionless dynamics of the harmonic oscillator [7]. A harmonic oscillator such as Ĥ0 (t) in Eq. ( 9) has the following time dependent invariant [9]

Î(t) = 1 2 x2 b 2 mω 2 0 + 1 m π2 , (36)
+ ω(t) 2 b = ω 2 0 /b 3 , (37)
an Ermakov equation where real solutions must be chosen to make Î Hermitian. ω 0 is frequently rescaled to unity by a scale transformation of b [9]. Other common and convenient choice, which we shall adopt here, is ω 0 = ω(0). The eigenstates of Î(t) become, with appropriate phase factors, solutions of the time-dependent Schrödinger equation,

Ψ n (t, x) = mω 0 πh 1/4 1 (2 n n!b) 1/2 exp -i(n + 1/2) t 0 dt ω 0 b(t ) 2 (38)
× exp i m 2h ḃ b(t) + iω 0 b 2 x 2 H n mω 0 h 1/2 x b , (39)
and form a complete basis to expand any time-dependent state, ψ(x, t) = n c n Ψ n (x, t), with the amplitudes c n constant. A method to achieve frictionless, population preserving processes is to leave ω(t) undetermined first, and set b so that Î(0) = Ĥ0 (0) and [ Î(t f ), Ĥ0 (t f )] = 0. This guarantees that the eigenstates of Î and Ĥ0 are common at initial and finite times and can be done by setting

b(0) = 1, ḃ(0) = 1, b = 0, b(t f ) = γ = [ω 0 /ω f ] 1/2 , ḃ(t f ) = 0, b(t f ) = 0, (40)
and interpolating b(t) with some real function that satisfies these boundary condition. The simplest choice is a polynomial,

b(t) = 5 j=0 a j t j . (41)
Once the a j are determined from (40), ω(t) is calculated from the Ermakov equation (37), and the transitionless Hamiltonian ĤII (t) = Ĥ0 (t) is defined in terms of a local, ordinary, harmonic potential, but note that ω(t) 2 may become negative during some time interval if the process is very fast, making the potential an expulsive parabola [7,20]. This is illustrated in Figs. 234for a transition from ω(0) = 2π × 200 Hz to ω(t f ) = 2π × 2 Hz in t f = 5 ms. The II method is thus clearly distinct from from TT and implements a different Hamiltonian. Note also, by comparying the coefficients, that the invariant operator Î corresponding to ĤII is different from ĤTT , although they are both generalized harmonic oscillators.

Physical realization

The TT method only requires the time variation of a parabolic potential. Effective harmonic optical traps for neutral atoms may be formed by magnetic and/or optical means and their frequencies are routinely varied in time as part of many cold atom experiments. In magnetic traps, for example the frequency is modulated harmonically to look for collective excitation modes of a condensate [21], and ramped down adiabatically to change its conditions (critical temperature, particle number, spatial extension) [22,21], or as a preliminary step to superimpose an optical lattice [23]. Some experiments involve both time-dependent magnetic and optical traps or antitraps [24]. Purely optical traps are also manipulated in time, e.g. for adiabatic cooling of single neutral atoms [25]. In particular laser beams detuned with respect to the atomic transition form effective potentials for the ground state depending on Rabi frequency Ω and detuning ∆ as Ω 2 /4∆ by adiabatic elimination of the excited state, thus forming attractive or repulsive potentials. This effective interaction can be made time dependent by varying the laser intensity, the frequency, or both [1], since the optical frequencies are many orders of magnitude larger than Rabi frequencies or detunings, and the changes will be slowly varying in the scale of optical periods. To vary the sign of ω 2 (t), two dipole beams locked respectively on the blue and red side of the line may be superimposed. The intensity of a dipole trap can be changed by three or four orders of magnitude in 100 ns using acousto-optics or electro-optics modulators. This affects the trap radius, which can as well be controlled [2].

Discussion

We have compared and distinguished two different methods: a "transitionless-tracking" (TT) algorithm, and an "inverse-invariant" (II) method, to achieve transitionless dynamics for a fast frequency change of a quantum harmonic oscillator. They imply different driving Hamiltonians ĤTT and ĤII . The one in the II method can be implemented for ultracold atoms or ions by varying the trap frequency in time so as to enforce at initial and final times the commutativity between the Hamiltonian and an invariant of motion Î with the form of a generalized harmonic oscillator. In this method the states are at all times eigenstates of the invariant if initially so. Extending this method to other (anharmonic) potentials or systems may be difficult and remains an open question: in principle one should find in each case the corresponding state-carrying invariant and be able to engineer it. A generalization to Bose Einstein condensates has been worked out [8] without invoking the invariant concept, but it relies on scaling laws applicable to the harmonic oscillator. The TT method does not explicitly make use of invariants and provides a Hamiltonian for which the state dynamics follows exactly the adiabatic approximation of a time-dependent process defined by a reference Hamiltonian Ĥ0 . In this respect it has the advantage of being, at least formally, more generally applicable than the II approach. The feasibility of the actual realization is quite another matter and has to be studied in each case. An application example for two-level systems is discussed in [4]. For the harmonic oscillator studied here ĤTT becomes, as Î, a generalized harmonic oscillator too, but it is different from Î and plays a different role. In particular the "carrying states" are not instantaneous eigenstates of ĤTT but of the reference, harmonic oscillator Hamiltonian Ĥ0 . ĤTT includes, in addition to a harmonic term in Ĥ0 with a time dependent frequency going from the initial to the final values, a time-dependent (squeezing) non-local potential with the symmetrized product of position and momentum operators. We have found a realization for cold atoms that, unfortunately, requires a time scale larger than the times needed for an adiabatic process. Therefore the applicability of the method remains an open question. Trap expansions and compressions play such an important role in cold atom experiments that investigating further the fundamental properties of both approaches and their technical realization is definitely worthwhile.

In a similar way we note that, consistently, Ŝ(r)|n(0) = Ŝ(r) .6)

1 √ n! (â † 0 ) n |0 0 = 1 √ n! Ŝ(r)(â † 0 ) n Ŝ(-r) Ŝ(r)|0 0 = 1 √ n! (â † t ) n |0 t = |n(t) . (A

Figure 1 .

 1 Figure 1. Schematic electronic and vibrational level structure for a two-photon transition in an ion trapped with frequency ω. ω 1 and ω 2 are the laser frequencies, and ω e the transition frequency between ground and excited states. See the text for further details.

  where π = b(t)p -m ḃx plays the role of a momentum conjugate to x/b, the dots are derivatives with respect to time, and ω 0 is in principle an arbitrary constant. The scaling, dimensionless function b = b(t) satisfies the subsidiary condition b

Figure 2 .Figure 3 .Figure 4 .

 234 Figure 2. b (poynomial ansatz) versus t/t f for an expansion from ω(0) = 2π × 200 Hz to ω(t f ) = 2π × 2 Hz.

Acknowledgments

We thank M. V. Berry for discussions and acknowledge funding by Projects No. GIU07/40, FIS2009-12773-C02-01, 60806041, 08QA14030, 2007CG52, S30105, and Juan de la Cierva Program.

Appendix A. Relation to the squeezing operator

The evolution operator takes a particularly simple form when using the simplified case E n (t) = 0, so that Ĥ(t) = Ĥ1 (t). Taking into account that [ Ĥ1 (t), Ĥ1 (t )] = 0 we can write

This may be evaluated explicitly with (19) fixing the time of the creation and annihilation operators to 0,

which is a sqeezing operator with real argument r(t) = ln

and the formal properties of Ŝ, see e.g. [START_REF] Barnett | Methods in Theoretical Quantum Optics[END_REF], it is easy to prove that

In fact any combination of powers of â0 and â † 0 is mapped to the same combination of powers of ât and â † t by this unitary transformation. |0 t ≡ Ŝ|0 0 is indeed the vacuum at time t, as ât |0 t = Ŝ(r) Ŝ(-r)a Ŝ(r)|0 0 = Ŝ(r)â 0 |0 0 = 0.

(A.5)


R Fermani
email: fermani@ntu.edu.sg

T Müller

B Zhang

M J Lim

R Dumke

Heating rate and spin flip lifetime due to near field noise in layered superconducting atom chips

Keywords: numbers: 34.35.+a, 37.10.Gh, 42.50.Ct

We theoretically investigate the heating rate and spin flip lifetimes due to near field noise for atoms trapped close to layered superconducting structures with the superconductor in the Meissner state. In particular, we compare the case of a gold layer deposited above a superconductor with the case of a bare superconductor. We study a niobium-based and a YBa2Cu3O7-x (YBCO)-based chip. For both niobium and YBCO chips at a temperature of 4.2 K, we find that the deposition of the gold layer can have a significant impact on the heating rate and spin flip lifetime, as a result of the increase of the near field noise. At a chip temperature of 77 K, this effect is less pronounced for the YBCO chip.

I. Introduction

In the area of magnetic trapping of ultracold atoms, considerable attention has been recently devoted to the interaction of atomic clouds with the surfaces of both superconducting atom chips [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and superconducting solid state devices [17][18][19][20]. Technological advances will allow a new generation of fundamental experiments and applications involving the control of the interface between atomic systems and quantum solid state devices. The implementation of such technologies depends on the ability to control and efficiently manipulate atoms close to superconducting surfaces. However, below a certain separation the atom-surface coupling is strong enough that the trapping potential is modified by the interaction of the atomic magnetic moment with the near-field magnetic noise. This leads to atom heating and spin flip induced atomic loss, which is the basic limitation in metal-based atom chips [21][22][23][24][25]. The origin of near field noise lies in the thermally-induced fluctuating currents and in the resistivity of the chip materials in accordance with the fluctuation-dissipation theorem. The magnetic field noise is significantly smaller in the vicinity of a superconductor [2,3], however for the small distances involved, heating and thermally-induced spin flip may become relevant.

The reduction of the spin flip rate has been shown experimentally for different chip types: with and without a gold layer above the superconductor [10][11][12][13]. The purpose of the top gold layer in atom chips is to use them in mirror-MOTs [26]. This can be achieved by coating the chip surface with a reflective metal film. The metal film can also protect the superconductor when it is operated near the critical current density, to avoid possible quenching [7]. However, the presence of the metal film above the superconducting layer may increase the near field noise significantly.

We report our investigation of the heating rate and the thermally-induced spin flip rate of a neutral atom held close to two different superconducting structures: one with and one without a metal film above the superconducting layer. We consider the case of two different superconductors in the Meissner state: niobium (a conventional s-wave superconductor) and YBCO (a high temperature d-wave superconductor). We present a first-principles derivation of the heating rate, adopting a model for the heating mechanism similar to [23,27]. For the derivation of the spin flip rate the reader may Confidential: not for distribution. Submitted to IOP Publishing for peer review 25 February 2010 refer to earlier work [25]. The evaluation of the rates is done within the framework of the electromagnetic field quantization in absorbing dielectric media [START_REF] Vogel | Quantum Optics, An Introduction[END_REF][START_REF] Knöll | Coherence and Statistics of Photons and Atoms[END_REF]. Despite the quantum electrodynamics formalism being originally developed for dielectric and metallic media, it can be appropriately adopted to account for the electromagnetic field dissipation in superconducting materials in the Meissner state [5].

We stress that the results reported in this paper are valid only in the Meissner state. However, one should be aware that additional noise mechanisms are taking place when the superconducting layer is in the mixed state. Atomic trap position and spin flip lifetime have been reported to be influenced by the vortex dynamics in type-II superconductors [4,8,9,[13][14][15]. The vortex dynamics is affected by both temperature and transport currents. For temperatures below an activation temperature T a < T c , flux penetration happens via dendritic avalanches and the vortex density and distribution is highly unstable [13,14]. On the other hand, at higher temperatures, thermal fluctuations may provide enough energy for the vortices to oscillate around their pinning centres or to jump from one pinning centre to the other, and eventually be depinned and move freely. Vortex motion can also be induced by transport current as a consequence of the Lorentz force. All these mechanisms acting together are responsible for vortex dissipation and vortex noise and may give a relevant contribution to the heating of magnetically trapped atoms.

The paper is organized as follows. In Sec. II, we describe the model we adopt for the heating rate and we give its general formulation. We present the quantum mechanical derivation of the heating rate in Appendix A. In Sec. III, we report the formulas and simulations for the heating rate for two superconducting chip structures: with and without a metal layer. The comparison is done by considering niobium and YBCO superconductors. For completeness, we extend the comparison of the two materials by studying the spin flip lifetime in Sec. IV. Conclusions are given in Sec. V.

II. Model and Formulas

The heating of an atom harmonically trapped can be understood as a transition from a motional state of the atomic trap to a higher motional state of the same trap. This is due to fluctuations of the trap center x, as schematically represented in Fig. 2. We consider a three-dimensional harmonic oscillator with Hamiltonian H = p 2 /2 + 1 2 mω 2 x 2 -x • F where F is a force causing the trap centre to fluctuate. The force is proportional to the gradient of the magnetic potential experienced by the atom, which is a consequence of the Zeeman coupling of the atomic magnetic moment µ to a spatially varying magnetic field B(r). Fluctuations of the magnetic field B(r A ) at the atom's position r A will induce fluctuations of the trap center and this process is described by the

Hamiltonian Ĥint = -x • ∇ µ • B(r A ) , (1)
where x = j x j Xj , with Xj = (â † j + âj )/2 being the dimensionless quadrature operator and x j the average position of the trap center along the jth direction. In the following, a concise notation will be adopted by dropping the sum over j and assuming that j = x, y, z. For an atom initially in the state |n , the average trap center for a transition to a state |m = n is given by x = m| x|n which is nonzero only for

m = n ± 1, hence x j,± = 2 /M ω v,j n + 1 2 ± 1 2
, where ω v,j is the trapping frequency along the jth direction and M is the atomic mass. We obtain the rate γ j,n→n±1 for the transition n → n ± 1 along the jth trapping direction as derived in Appendix A

γ j,n→n±1 = πx 2 j,± 2 2 S F (r A , ω v,j ) , (2)
where S F (r A , ω v,j ) is the spectrum of the fluctuating force causing the shift of the trap centre. In order to obtain the heating rate, we consider the average energy of the system E(t) = n P (n, t) ω(n + 1 2 ) where the sum is performed over the motional trap states |n , and P (n, t) is the probability that at time t the system is in the state |n . The average heating rate Γ ,j ≡ Ė(t) j is given by [27] Γ

,j = n P (n, t) ω v,j (γ j,n→n+1 -γ j,n→n-1 ) , (3)
and by substituting Eq. ( 2) into the previous equation we obtain

Γ ,j = π M S F (r A , ω v,j ) , (4)
where n P (n, t) = 1. The expression of Eq. ( 4) is the most general formula we obtain for the heating rate and must be evaluated for each separate structure. In particular, the spectrum of the fluctuating force has been obtained in Appendix A as

S F (r A , ω) = π 0 c 2 (n th + 1) -→ ∇ µ • Im -→ ∇ × G(r A , r A , ω) × ← - ∇ • µ ← - ∇ , (5)
where n th = 1/(e ω/kBT -1) is the mean thermal photon number and G(r A , r A , ω) is the Green function. The Green function accounts for the spectrum of the fluctuating magnetic field according to the following expression

S B (r, r , ω) = B(r, ω) B(r , ω ) (6) = π 0 c 2 (n th + 1)Im - → ∇ × G(r, r , ω) × ← - ∇ δ(ω -ω ).
All the information regarding the trapping parameters and the geometry of the system are contained in the Green function.

III. Heating rate

We present our numerical evaluations for the heating rate of Eq. ( 4) for two different chip structures as shown in Fig. 1(a)-(b). Both structures contain a superconducting layer, either niobium or YBCO, in the Meissner state, and they are compared with an infinitely thick gold surface. We evaluate the heating rate for a 87 Rb atom held at a distance z above the chip. We consider the following trapping frequencies along the three directions in space: ω v,x = 0.1 KHz and ω v,y = ω v,z = 1 KHz.

The heating rate along the jth trapping direction for the niobium-based chip structure can be obtained from Eq. ( 4) by substituting the Green functions for isotropic media as reported in [START_REF] Dung | [END_REF], leading to

Γ ,j = α j µ 2 B M 0 c 2 (n th + 1) (7) dη η 2 4π e -2ηz 2 Im R T M ω 2 v,j c 2 + 2η 2 R T E
where α x = α y = 1 and α z = 2, and µ B is the Bohr magneton. The Fresnel coefficients R T E and R T M , for T E and T M waves, are computed according to the nature and number of layers constituting the structure. In particular, the Fresnel coefficients for the nth layer follow the recursive formula

R q n,m = r q n,m + R q m,m+1 e 2iβmdm 1 + r q n,m R q m,m+1 e 2iβmdm , (8)
where m = n + 1 denotes the consecutive layer to n,

with thickness d m , β m = k 2 m -η 2 , k 2 m = m ω 2 /c 2 ,
and m is the relative permittivity of the mth layer. The label q denotes either the TE or the TM components of the electromagnetic field. The r q n,m terms represent the Fresnel coefficients for the simplest geometry, two half spaces, and are given by

r T E n,m = β n -β m β n + β m , r T M n,m = m β n -n β m m β n + n β m . (9)
In Figure 3 we plot the heating rate Γ ,z along the z direction versus the surface distance for three different chip structures at liquid helium temperature T = 4.2 K. We scale the heating rate by the Boltzmann constant in order to facilitate comparison with relevant energy scales, however this should not be confused with the thermodynamic temperature. The heating rate is highest above the gold substrate, while the bare niobium surface yields the lowest rate. At distances of 1 µm, we estimate the heating rate for the gold substrate to be of the order of Γ ,Au 10 -7 K/s which is a measurable effect. At the same distance, the heating for the niobium-based chip is reduced but in principle still detectable with a top gold layer of 50 nm as Γ ,Nb+Au 10 -9 . However, the rate for the bare niobium chip is too small to be of experimental relevance because Γ ,Nb 10 -12 K/s. Trapping distances below 1 µm are challenging because of the van der Waals attractive potential. However, for z ∼ 100 nm, the heating rate of the niobium+gold surface approaches the heating rate of the gold substrate. This indicates that at very small distances only the effect of the metal is relevant.

For the YBCO-based chip structure, the Green function for anisotropic media as reported in [31] is adopted and the heating rate along the z-direction (perpendicular to the chip surface) is given by

Γ ,z = µ 2 B M 0 c 2 (n th + 1) (10) ∞ 0 dη η 2 4π e -2ηz Im B 11 N 2 k 2 z + 2η 2 B 11 M 1 ,
with k 2 z = ω 2 z /c 2 , while the heating rate Γ , for the plane parallel to the chip surface is given by the sum of the following expressions

Γ ,r = µ 2 B M 0 c 2 (n th + 1) (11)
∞ 0 dη η 2 4π e -2ηz 4 Im B 11 N 2 k 2 r + 3η 2 B 11 M 1 , Γ ,φ = µ 2 B M 0 c 2 (n th + 1) (12)
∞ 0 dη 4π e -2ηz Im B 11 N 2 k 2 φ + η 2 B 11 M 1 , with k 2 r,φ = ω 2 r,φ /c 2 . The scattering coefficients B 11 M 1
and B 11 N 2 play the same role of the Fresnel coefficients of Eq. ( 8) but account for the anisotropic properties of the YBCO layer. These coefficients can be obtained by following [16,31]. Our simulations for the heating rate of the YBCO chip are presented in Fig. 4. We plot the heating rate along only the z trapping direction because this term dominates. We observe that the bare YBCO surface yields the lowest heating rate, compared to the other two structures with the gold layer. However, the difference between the bare YBCO surface and the YBCO+gold chip structure is dramatically reduced at liquid nitrogen temperature T = 77 K, and is less than one order of magnitude for a wide range of surface distances. The heating rate is detectable for distances around 1 µm where Γ ,YBCO Γ ,YBCO+Au 10 -8 K/s. At smaller temperatures, the rate corresponding to the two structures differs by a few orders of magnitude, i. e. Γ ,YBCO+Au /Γ ,YBCO 10 -3 at T = 4.2 K given by the fact that the YBCO penetration depth and the skin depth of gold scale with temperature. This results in the near field spectrum being dominated by the gold. As a consequence, at very low temperatures the presence of the gold leads to a marked increase of near field noise. This increase is less pronounced at temperatures closer to the YBCO transition temperature. We conclude that the heating rate close to a bare YBCO surface and to a YBCO+gold chip for typical trapping distances of a few µm does not result in a measurable difference at 77 K. Moreover, the typical values of magnetic bias fields and temperatures used for superconducting chip experiments suggest that a hightemperature superconductor is likely to be operated in the Shubnikov phase. This involves the partial penetration of magnetic field in the form of vortices but this is not considered in the present work.

IV. Spin flip lifetime

In the following section the comparison between the bare superconducting chip and the superconductor+gold structure is extended by considering the spin flip lifetime τ of a neutral atom. As in Sec. III, we also compare the above mentioned chip structures with an infinitely thick gold substrate as a reference. The transition rate for thermally induced spin flips Γ SF = 1/τ between an initial hyperfine state |i and a final hyperfine state |f is given in its most general formulation as [25] (13) where ω SF is the spin transition frequency and i| Ŝp |f the spin matrix element for |i → |f . We restrict the calculation to a two-level system evaluating the spin flip lifetime for the transition |F = 1, m F = -1 → |F = 1, m F = 0 . We choose a typical experimental transition frequency ω SF = 2π 560 KHz [21], and we note that the near field noise spectrum does not change significantly in the rf frequency range. We evaluate the spin matrix elements via the Clebsch-Gordan coefficients and obtain for the non-zero matrix elements

Γ SF = 2cµ 2 B g 2 S 2 f | Ŝq |i i| Ŝp |f S B (r A , ω SF ) ,
| i| Ŝx |f | = | i| Ŝz |f | = 1/8 (
assuming that y is the spin quantization direction).

The spin flip lifetime close to the niobium-based chip is obtained as the inverse of the following spin flip rate [16]

Γ SF,Nb = 3π (µ B g S ) 2 32c 2 0 dη η 2 (2π) 2 e -2ηz 2 Im R T E (n th + 1) , (14)
where R T E is the Fresnel coefficient as in Eq. ( 8). The spin flip lifetime as a function of the surface distance is plotted in Fig. 5 for three different structures. The shortest spin flip lifetime is obtained for the gold surface. The longest lifetime is found close to the bare niobium surface and is two orders of magnitude longer than the lifetime near the niobium+gold chip. The spin flip rate above the YBCO-based chip can be written as [16] Γ

SF,YBCO = (µ B g S ) 2 32 0 c 2 (n th + 1) (15)
∞ 0 dη e -2ηz 8π Im 3η 2 B 11 M 1 + B 11 N 2 ω 2 SF c 2 ,
where B 11 M 1 and B 11 N 2 are the scattering coefficients of Eqs. (10)(11)(12). The spin flip lifetime τ = 1/Γ SF,YBCO is plotted in Fig. 6 for the three different structures: gold surface, YBCO surface, YBCO+gold structure. Similarly to the niobium-based chip structures, the spin flip lifetime close to the YBCO+gold structure is shorter than the one obtained near the bare superconductor, however the difference is less than an order of magnitude. A complete accounting of the total lifetime must include other experimental restrictions, such as collisions with background gas [11].

We conclude this section by plotting in Fig. 7 the heating rate and spin flip lifetime of the niobium+gold chip, of the YBCO+gold chip and of a layer of gold, as a function of the gold thickness. The niobium+gold chip has the lowest heating rate. The rate increases for the three structures with increasing the thickness of the gold until the layer thickness reaches 1 µm. The presence of the su-perconductor is then irrelevant and the heating rate is the same as what one would obtain close to a gold substrate. Typical values for the gold thickness in superconducting atom chips are around 50 -100 nm. For such values the heating rate for the YBCO+gold chip is one order of magnitude larger than the niobium+gold chip at 4.2 K. Similarly, the niobium+gold chip exhibits the longest lifetime. For the three structures, the lifetime decreases with increasing the gold thickness until it approaches the lifetime for a half-space of gold. This happens for a layer thickness of the order of 10 µm as this is of the same order of the gold penetration depth at T 4.2 K. For a layer thickness equal or greater than the penetration depth, the finite-thickness and the infinitely-thick layer of gold have the same behaviour and the presence of the superconductor is irrelevant.

V. Conclusions

We have investigated the heating rate and spin flip lifetime for a superconducting atom chip considering the cases with and without a gold layer deposited above the superconductor. All the results presented are valid for a superconductor in the Meissner state. The heating rate for trapping distances around 1 µm is measurable and may become significant for a sufficiently cold atom cloud. However, by adding a top gold layer, this rate increases such that for layers thicker than 1 µm, the presence of the superconductor becomes irrelevant. Even for a submicron gold layer, an atom held at a close distance of 100 nm is influenced only by the metal, and both the heating rate and spin flip lifetime approach the values obtained in the proximity of a infinitely thick gold substrate.

In particular, we have considered a niobium-based and a YBCO-based chip. At a chip temperature of 4.2 K, the difference between the case with the gold layer and without is marked both for heating rate and spin flip lifetime. Such difference is less pronounced when the YBCO chip is operated at 77 K. In summary, our study shows that the deposition of a metal layer above a superconducting layer can diminish the advantages of using a superconductor to overcome the limitations given by a metal. The presented results are of interest for future development of technologies involving superconducting atom chips. Further investigations will include the treatment of type II superconductors in the Shubnikov phase.

A. Quantum mechanical derivation of the heating rate

We outline the quantum mechanical derivation of the heating rate based on the model introduced in Sec. II.

The total Hamiltonian of a harmonically trapped atom held close to a substrate (the conducting properties of the substrate are not yet specified at this stage), is given by the sum of three different Hamiltonians as Ĥ = Ĥf + Ĥa + Ĥint . The Hamiltonian for the atom in the unperturbed harmonic trap reads Ĥa = ω v,j â † j âj + 1 2 . The Hamiltonian of the electromagnetic field arising from an absorbing and dispersing medium is given within the formalism of quantum electrodynamics for dielectric media [START_REF] Vogel | Quantum Optics, An Introduction[END_REF][START_REF] Knöll | Coherence and Statistics of Photons and Atoms[END_REF] as Ĥf = d 3 rdω ω f † (r, ω) f (r, ω), with f (r, ω) and f † (r, ω) being the bosonic operators accounting for the collective excitation of the medium and the electromagnetic field. They satisfy the usual equal-time commutation relations f (r, ω), f † (r , ω ) = δ(rr )δ(ω -ω ) and their correlation function at temperature T reads f (r, ω) f † (r , ω ) = (n th + 1)δ(rr )δ(ω -ω ), where n th is the mean thermal photon number.

The interaction Hamiltonian of Eq. ( 1) can be written in the rotating wave approximation as Ĥint = -

x j 2 â † j ∂ j µ k B+ k (r A ) + H.c. , (A1)
where x j denotes either x j,+ for n → n+1 or x j,-for n → n -1. The magnetic field B + k (r A ) can be obtained via B(r) = B+ (r)+ B-(r) where B+(-) (r) = In the Heisenberg picture, the equation of motion for creation operator â+ j is given by ȧ † j (t) = iω v,j â † j -

i x j 2 ∂ j µ k B- k (r A ) , (A3)
and the bosonic field operator becomes ḟn (r, ω, t) = -iω fn (A4)

+iâ j x j 2 µ k √ π 0 ω c 2 I (r, ω) ∂ j kqs ∂ q G * sn (r A , r, ω) ,
where the time integral in the Markov approximation becomes fn (r, t) = fn,free (r, t) + iâ j (t)ζ(ω v,j -ω) (A5) x j 2 µ k √ π 0 ω c 2 I (r, ω)∂ j kqs ∂ q G * sn (r A , r, ω) ,

FIG. 1 :

 1 FIG. 1: Schematic representation of the chip structure: (a) the bare superconductor chip structure consists of a layer of superconducting material of thickness d deposited on a chip base (usually a crystal); (b) the superconductor+gold chip structure consists of a thin film of gold of thickness d1 on top of the superconducting layer of thickness d2 < d1, which is deposited on the chip base. The superconducting layer is either niobium or YBCO.

FIG. 2 :

 2 FIG. 2: Schematic representation of the heating rate model.The heating rate is modelled as a transition from a given motional state of the trap to higher motional states. The trapping potential is considered harmonic.

FIG. 3 :

 3 FIG.3: Heating rate for a Nb-superconducting chip at 4.2 K, the thickness of the Nb layer is 1 µm. Each of the three curves represents a different structure: the bare niobium chip (solid line), the niobium+gold chip with gold thickness of 50 nm (dashed line) and an infinitely thick gold substrate (dotted line).

FIG. 4 :

 4 FIG.4: Heating rate for a YBCO-based chip at 77 K, the thickness of the YBCO layer is 1 µm. Each of the three curves represents a different structure: the bare YBCO chip (solid line), the YBCO+gold chip with gold thickness of 50 nm (dashed line) and an infinitely thick gold substrate (dotted line).

FIG. 5 :

 5 FIG.5: Spin flip lifetime for a niobium-based chip at 4.2 K, the thickness of the Nb layer is 1 µm. Each of the three curves represents the lifetime corresponding to a different structure: the bare niobium chip (solid line), the niobium+gold chip with gold thickness of 50 nm (dashed line) and an infinitely thick gold substrate (dotted line).

FIG. 7 :

 7 FIG.7: Heating rate and spin flip lifetime of an atom 1 µm above a superconducting chip with a top gold layer. Both quantities are plotted as a function of the gold thickness d1 (as in Fig.1 (b)), for a niobium+gold chip (solid curve), YBCO+gold chip (dashed curve) and for a layer of gold (dotted curve). The three structures have a temperature of 4.2 K and the thickness of the superconducting layer is 1 µm.

π 0 ω c 2 d 3 r

 3 †) (r, ω)is the positive-frequency part, and the single component is given by Bk (r, ω) = I (r , ω) kjs ∂ j G sn (r, r , ω) fn (r , ω).(A2)

Acknowledgments

We acknowledge financial support from Nanyang Technological University (grant no. WBS M58110036), A-Star (grant no. SERC 072 101 0035 and WBS R-144-000-189-305) and the Centre for Quantum Technologies, Singapore. MJL acknowledges travel support from NSF PHY-0613659 and the Rowan University NSFG program.

and ζ(x) = πδ(x)+iPx -1 (P denotes the principal part). This formal solution is going to be substituted into the magnetic field of Eq. (A2) and making use of the following integral relation for the Green function [START_REF] Knöll | Coherence and Statistics of Photons and Atoms[END_REF]

where the gradient ← -∇ = j=x,y,z ∂ j is acting on the second argument of the Green function as denoted by the left pointing arrow. After integrating Eq. (A7) over ω and substituting it into Eq. (A3), we can write the equation of motion for the creation operator as

where γ j = γ j,n→n±1 is the rate associated with the n → n ± 1 transition along the jth direction. The rate arises from the δ(x) of the ζ(x) function appearing in Eq. (A5), and is given by

while the term δω j of Eq. (A8) arises from the principalpart of the ζ(x) function and denotes the frequency shift which we assume to be negligible.

If the dielectric body is in thermal equilibrium with its surroundings, then the magnetic field is in a thermal state with temperature T , and the magnetic field spectrum has to be multiplied by the mean thermal photon number n th + 1. The spectrum of the magnetic field can be expressed as

and hence the spectrum of the fluctuating force causing the shift of the center of mass is

such that the transition rate for the jth trapping direction can be written as

F (r A , ω v,j ). (A12)